Physics · Article

Electric potential, Poisson’s equation, and Laplace’s equation

Connect voltage differences to field slopes, derive the potential equation and solve original source-and-boundary examples. Explain what boundary data, uniqueness and harmonic structure add to Gauss’s law.

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After reading, you can

Distinguish potential, potential difference, field and the potential energy of a negligible probe, including negative-charge work signs.
Derive E = −∇V and ∇²V = −ρ/ε₀ with their electrostatic, domain and unit assumptions.
Solve a one-dimensional Poisson problem and separate its prescribed source from the harmonic correction required by its boundary values.
Use complete Dirichlet data, the Neumann compatibility condition and a uniqueness proof without mistaking uniqueness for existence.
Read original potential/field and saddle examples without confusing a reference level, a zero field and stable confinement.
Construct a decaying separation mode and connect free-space kernels, singular sources, interface conditions and numerical checks to the boundary-value problem.

A voltage is not a force

Two positions can have a specified voltage difference before a probe is placed at either one. What changes when the probe’s charge changes sign? The prescribed source field and its potential difference stay fixed, while the force and the probe’s change in potential energy reverse sign. This distinction turns potential into a useful description of the source configuration rather than a property of one chosen probe.

Work in stationary vacuum electrostatics unless a different regime is explicitly stated. Sources are prescribed and a test charge is small enough that its own field and any rearrangement it causes can be neglected. A conductor allowed to rearrange charge is a different boundary problem; its response is not secretly included in these prescribed-source examples.

Use positive length and potential scales L₀ and V₀, dimensionless coordinates X = x/L₀ and Y = y/L₀, and field scale E₀ = V₀/L₀. Write v = V/V₀. A separate positive q_* sets a negligible probe-charge scale. These choices let us check signs and dimensions without choosing a numerical value for ε₀.

The four original cases below are separate models. A charged slab, a harmonic saddle, a periodic half-space pattern and an isolated charged ball do not describe one combined apparatus. Each has its own stated domain, source and boundary information.

Potential differences and signed work

For a fixed electrostatic field, define V(B) − V(A) = −∫ from A to B E·dℓ. The minus sign makes potential decrease along the field direction. Path independence is part of this electrostatic description; it is not a license to use the formula as a single-valued scalar difference for every induced electric field.

The potential energy change of a probe q is ΔU = qΔV, and the work done by the prescribed electric field on it is W_field = −qΔV. A positive charge lowers its potential energy by moving to lower V. A negative charge lowers its potential energy by moving to higher V. A field arrow does not reverse when we choose a different probe; qE does.

External work equals ΔU only for a move with unchanged kinetic energy and no other net work contribution, such as an ideal quasistatic controlled move. More generally, ΔK = W_field + W_external + W_other. State those mechanical conditions before calling a voltage difference the work performed by an external agent.

Potential is measured in volts, with 1 V = 1 J/C. Electric field may be measured in V/m or N/C. Energy is not voltage: multiplying by the signed charge supplies both the correct units and the probe’s sign. The energy of assembling a whole source distribution is a separate calculation; a probe’s qV is not automatically that system’s total self-energy.

Choose a reference before using infinity

Adding one constant C to V everywhere leaves every potential difference and E unchanged. Thus V = 0 at a point does not imply E = 0 there. A boundary value stated relative to a chosen reference fixes this freedom; shifting the potential while leaving a nonzero prescribed boundary voltage unshifted changes the boundary problem.

An isolated compact charge distribution in free space can usually use V → 0 at spatial infinity. That reference fits its decaying potential. An ideal uniform field instead has a potential varying linearly along the field direction, and an infinite line’s potential varies logarithmically with radial distance. Neither admits the same finite zero-at-infinity convention.

For the slab we specify endpoint voltages, rather than extending its one-dimensional potential to infinity. For the periodic half-space pattern we prescribe a boundary waveform and decay upward. Only the isolated finite ball below adopts the ordinary free-space reference at all spatial infinity.

A reference change cannot repair missing source or boundary information. Several fields can share the same zero-potential point. Conversely, a constant nonzero potential throughout a connected region has zero gradient there. Values and spatial rates of change answer different physical questions.

Read the field as a negative directional slope

For a differentiable scalar potential, dV = ∇V·dℓ. Comparing this with the potential-difference definition along an arbitrary small displacement gives E = −∇V. In Cartesian coordinates, E_x = −∂V/∂x, E_y = −∂V/∂y and E_z = −∂V/∂z. Keep the minus sign on each component, not just on the magnitude.

If V = V₀v(X,Y), then E/E₀ = (−∂v/∂X, −∂v/∂Y, 0) for a z-independent model. Differentiating with respect to x introduces 1/L₀. A large potential can have a small slope and hence a small field, while a zero potential can have a steep slope and a large field.

At a regular point of an equipotential surface where ∇V is nonzero, every tangent displacement has ∇V·dℓ = 0, so the field is normal to the surface. Distinct potential values cannot occupy the same point. However, branches of one level set can meet at a critical point where the gradient vanishes; the saddle below makes this qualification visible.

Electric field lines indicate the local field direction. They are not generally trajectories of moving particles, whose acceleration depends on signed charge and whose velocity also depends on initial conditions. A plot of potential versus position is likewise not a literal height or a particle’s path unless a separate analogy is declared.

When one scalar potential is enough

A smooth electrostatic field satisfies ∇×E = 0. This ensures a potential in a sufficiently small simply connected neighborhood. To define one scalar consistently throughout a larger domain, every closed-path integral ∮E·dℓ must vanish. A simply connected domain with a smooth curl-free field is a familiar sufficient setting.

A hole in a domain does not automatically forbid a scalar potential; it means the local curl test alone need not settle the global question. One must check circulation around paths that cannot be shrunk within the allowed domain. Boundaries and excluded singularities are part of the mathematical problem, not details to erase from a drawing.

Faraday induction supplies an important limit: when a changing magnetic flux produces a nonzero circulation, one global scalar-only expression E = −∇V cannot represent the entire field. The later Maxwell treatment introduces the additional time-dependent structure. Here all worked models retain their stationary-source regime.

The assumptions about negligible probes and fixed sources remain in force even when a potential has been found. A scalar representation simplifies the field calculation; it does not remove material response, radiation or mechanical back-reaction when those effects become relevant.

Derive Poisson’s equation with its source sign visible

Gauss’s local equation in vacuum is ∇·E = ρ/ε₀. Substituting E = −∇V gives −∇·∇V = ρ/ε₀, or ∇²V = −ρ/ε₀. The Cartesian Laplacian is ∂²V/∂x² + ∂²V/∂y² + ∂²V/∂z². Positive volume charge therefore gives a negative Laplacian of the potential with this convention.

Both sides have units V/m²: ρ/ε₀ has the units of a field gradient. A sign error here reverses which source produces a chosen curvature. A scale error can hide behind a plausible-looking curve, so dimensionless derivatives must be converted using V₀/L₀².

Where the specified volume charge density is zero and the potential is regular, Poisson’s equation becomes Laplace’s equation, ∇²V = 0. This does not require V = 0 or E = 0. External charges and imposed boundary voltages can produce a nonzero harmonic potential throughout a charge-free region.

A point charge illustrates the domain qualification. Its potential kₑQ/r is harmonic at ordinary points with r > 0, but not a regular harmonic function at r = 0. Charge at that excluded point reappears through an inner-boundary flux or a distributional source. One cannot apply the smooth equation across the singularity and conclude that the charge vanished.

Quantities in the potential equation
Quantity	SI units	Role
V	J/C = V	Potential relative to a stated reference
E = −∇V	V/m = N/C	Local negative potential slope
ρ	C/m³	Prescribed volume charge density
∇²V and −ρ/ε₀	V/m²	Equal local curvature and source terms
∂ₙV	V/m	Outward normal derivative used in Neumann data

Solve one source and two endpoint values

Consider the reduced one-dimensional domain −L₀ ≤ x ≤ L₀, with invariance in y and z. Prescribe uniform ρ = 6ε₀V₀/L₀² and endpoint values V(−L₀) = −4V₀ and V(L₀) = 2V₀. These data define the model. We have not derived a conductor’s unknown charge response or claimed a finite isolated charge distribution.

In normalized form, d²v/dX² = −6. Two integrations give v = −3X² + aX + b. Applying the left endpoint gives −3 − a + b = −4; the right gives −3 + a + b = 2. Subtracting gives a = 3, then b = 2. Thus v = 2 + 3X − 3X² and E_x/E₀ = −3 + 6X.

The field vanishes at X = 1/2, where the potential is 11V₀/4. This is a maximum along x inside a charged region. It does not contradict the maximum principle for a source-free harmonic potential. The field changes from pointing in negative x to pointing in positive x as X crosses this point.

This is a one-dimensional boundary problem within its explicitly invariant class. If one instead posed a general three-dimensional finite-box problem, its entire boundary data would have to be supplied. Specifying only two faces while allowing arbitrary variation along the other coordinates would define a different problem.

Potential value and signed field slope are different

Separate potential value from signed local negative slope; all exact slab controls and sampled curve equivalents; maximum inside charge, not a violation of a harmonic theorem.

Potential across the charged slab

Potential v=V/V₀, 129 analytic samples
All five exact controls

Signed negative slope, not potential height

Signed Eₓ/E₀=−dv/dX, 129 analytic samples
All five exact controls

Separate stationary prescribed slab:−1≤X≤1, invariance iny,z;v=2+3X−3X²andEₓ/E₀=−3+6X.
ρ=6ε₀V₀/L₀², endpointv(−1)=−4 andv(1)=2. The interior maximum11/4 atX=1/2lies in charge, outside the harmonic-maximum hypothesis.
The source data use exact rational strings or12-significant-digit displays plus full retained binary64 values. The plotted curves join finite analytic samples; no interpolation-error bound or numerical PDE solver is claimed.

Complete labelled equivalents below preserve every retained sample and scientific value; original manuscript tables remain separate.

Quantities in the potential equation

Exact controls for the prescribed charged slab

Five exact slab controls (5 rows)

Five exact slab controls
Row	X=x/L₀	v=V/V₀	Signed Eₓ/E₀	∂²v/∂X²	Normalized divergence ∇·E / (V₀/L₀²)
0	-1Exact fraction: `-1`Retained binary64: `-1.0`	-4Exact fraction: `-4`Retained binary64: `-4.0`	-9Exact fraction: `-9`Retained binary64: `-9.0`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
1	-1/2Exact fraction: `-1/2`Retained binary64: `-0.5`	-1/4Exact fraction: `-1/4`Retained binary64: `-0.25`	-6Exact fraction: `-6`Retained binary64: `-6.0`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
2	0Exact fraction: `0`Retained binary64: `0.0`	2Exact fraction: `2`Retained binary64: `2.0`	-3Exact fraction: `-3`Retained binary64: `-3.0`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
3	1/2Exact fraction: `1/2`Retained binary64: `0.5`	11/4Exact fraction: `11/4`Retained binary64: `2.75`	0Exact fraction: `0`Retained binary64: `0.0`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
4	1Exact fraction: `1`Retained binary64: `1.0`	2Exact fraction: `2`Retained binary64: `2.0`	3Exact fraction: `3`Retained binary64: `3.0`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`

All 129 analytic slab samples, ordered in X (129 rows)

All 129 analytic slab samples, ordered in X
Row	X=x/L₀	v=V/V₀	Signed Eₓ/E₀	∂²v/∂X²	Normalized divergence ∇·E / (V₀/L₀²)
0	-1Exact fraction: `-1`Retained binary64: `-1.0`	-4Exact fraction: `-4`Retained binary64: `-4.0`	-9Exact fraction: `-9`Retained binary64: `-9.0`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
1	-63/64Exact fraction: `-63/64`Retained binary64: `-0.984375`	-15811/4096Exact fraction: `-15811/4096`Retained binary64: `-3.860107421875`	-285/32Exact fraction: `-285/32`Retained binary64: `-8.90625`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
2	-31/32Exact fraction: `-31/32`Retained binary64: `-0.96875`	-3811/1024Exact fraction: `-3811/1024`Retained binary64: `-3.7216796875`	-141/16Exact fraction: `-141/16`Retained binary64: `-8.8125`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
3	-61/64Exact fraction: `-61/64`Retained binary64: `-0.953125`	-14683/4096Exact fraction: `-14683/4096`Retained binary64: `-3.584716796875`	-279/32Exact fraction: `-279/32`Retained binary64: `-8.71875`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
4	-15/16Exact fraction: `-15/16`Retained binary64: `-0.9375`	-883/256Exact fraction: `-883/256`Retained binary64: `-3.44921875`	-69/8Exact fraction: `-69/8`Retained binary64: `-8.625`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
5	-59/64Exact fraction: `-59/64`Retained binary64: `-0.921875`	-13579/4096Exact fraction: `-13579/4096`Retained binary64: `-3.315185546875`	-273/32Exact fraction: `-273/32`Retained binary64: `-8.53125`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
6	-29/32Exact fraction: `-29/32`Retained binary64: `-0.90625`	-3259/1024Exact fraction: `-3259/1024`Retained binary64: `-3.1826171875`	-135/16Exact fraction: `-135/16`Retained binary64: `-8.4375`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
7	-57/64Exact fraction: `-57/64`Retained binary64: `-0.890625`	-12499/4096Exact fraction: `-12499/4096`Retained binary64: `-3.051513671875`	-267/32Exact fraction: `-267/32`Retained binary64: `-8.34375`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
8	-7/8Exact fraction: `-7/8`Retained binary64: `-0.875`	-187/64Exact fraction: `-187/64`Retained binary64: `-2.921875`	-33/4Exact fraction: `-33/4`Retained binary64: `-8.25`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
9	-55/64Exact fraction: `-55/64`Retained binary64: `-0.859375`	-11443/4096Exact fraction: `-11443/4096`Retained binary64: `-2.793701171875`	-261/32Exact fraction: `-261/32`Retained binary64: `-8.15625`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
10	-27/32Exact fraction: `-27/32`Retained binary64: `-0.84375`	-2731/1024Exact fraction: `-2731/1024`Retained binary64: `-2.6669921875`	-129/16Exact fraction: `-129/16`Retained binary64: `-8.0625`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
11	-53/64Exact fraction: `-53/64`Retained binary64: `-0.828125`	-10411/4096Exact fraction: `-10411/4096`Retained binary64: `-2.541748046875`	-255/32Exact fraction: `-255/32`Retained binary64: `-7.96875`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
12	-13/16Exact fraction: `-13/16`Retained binary64: `-0.8125`	-619/256Exact fraction: `-619/256`Retained binary64: `-2.41796875`	-63/8Exact fraction: `-63/8`Retained binary64: `-7.875`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
13	-51/64Exact fraction: `-51/64`Retained binary64: `-0.796875`	-9403/4096Exact fraction: `-9403/4096`Retained binary64: `-2.295654296875`	-249/32Exact fraction: `-249/32`Retained binary64: `-7.78125`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
14	-25/32Exact fraction: `-25/32`Retained binary64: `-0.78125`	-2227/1024Exact fraction: `-2227/1024`Retained binary64: `-2.1748046875`	-123/16Exact fraction: `-123/16`Retained binary64: `-7.6875`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
15	-49/64Exact fraction: `-49/64`Retained binary64: `-0.765625`	-8419/4096Exact fraction: `-8419/4096`Retained binary64: `-2.055419921875`	-243/32Exact fraction: `-243/32`Retained binary64: `-7.59375`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
16	-3/4Exact fraction: `-3/4`Retained binary64: `-0.75`	-31/16Exact fraction: `-31/16`Retained binary64: `-1.9375`	-15/2Exact fraction: `-15/2`Retained binary64: `-7.5`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
17	-47/64Exact fraction: `-47/64`Retained binary64: `-0.734375`	-7459/4096Exact fraction: `-7459/4096`Retained binary64: `-1.821044921875`	-237/32Exact fraction: `-237/32`Retained binary64: `-7.40625`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
18	-23/32Exact fraction: `-23/32`Retained binary64: `-0.71875`	-1747/1024Exact fraction: `-1747/1024`Retained binary64: `-1.7060546875`	-117/16Exact fraction: `-117/16`Retained binary64: `-7.3125`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
19	-45/64Exact fraction: `-45/64`Retained binary64: `-0.703125`	-6523/4096Exact fraction: `-6523/4096`Retained binary64: `-1.592529296875`	-231/32Exact fraction: `-231/32`Retained binary64: `-7.21875`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
20	-11/16Exact fraction: `-11/16`Retained binary64: `-0.6875`	-379/256Exact fraction: `-379/256`Retained binary64: `-1.48046875`	-57/8Exact fraction: `-57/8`Retained binary64: `-7.125`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
21	-43/64Exact fraction: `-43/64`Retained binary64: `-0.671875`	-5611/4096Exact fraction: `-5611/4096`Retained binary64: `-1.369873046875`	-225/32Exact fraction: `-225/32`Retained binary64: `-7.03125`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
22	-21/32Exact fraction: `-21/32`Retained binary64: `-0.65625`	-1291/1024Exact fraction: `-1291/1024`Retained binary64: `-1.2607421875`	-111/16Exact fraction: `-111/16`Retained binary64: `-6.9375`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
23	-41/64Exact fraction: `-41/64`Retained binary64: `-0.640625`	-4723/4096Exact fraction: `-4723/4096`Retained binary64: `-1.153076171875`	-219/32Exact fraction: `-219/32`Retained binary64: `-6.84375`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
24	-5/8Exact fraction: `-5/8`Retained binary64: `-0.625`	-67/64Exact fraction: `-67/64`Retained binary64: `-1.046875`	-27/4Exact fraction: `-27/4`Retained binary64: `-6.75`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
25	-39/64Exact fraction: `-39/64`Retained binary64: `-0.609375`	-3859/4096Exact fraction: `-3859/4096`Retained binary64: `-0.942138671875`	-213/32Exact fraction: `-213/32`Retained binary64: `-6.65625`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
26	-19/32Exact fraction: `-19/32`Retained binary64: `-0.59375`	-859/1024Exact fraction: `-859/1024`Retained binary64: `-0.8388671875`	-105/16Exact fraction: `-105/16`Retained binary64: `-6.5625`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
27	-37/64Exact fraction: `-37/64`Retained binary64: `-0.578125`	-3019/4096Exact fraction: `-3019/4096`Retained binary64: `-0.737060546875`	-207/32Exact fraction: `-207/32`Retained binary64: `-6.46875`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
28	-9/16Exact fraction: `-9/16`Retained binary64: `-0.5625`	-163/256Exact fraction: `-163/256`Retained binary64: `-0.63671875`	-51/8Exact fraction: `-51/8`Retained binary64: `-6.375`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
29	-35/64Exact fraction: `-35/64`Retained binary64: `-0.546875`	-2203/4096Exact fraction: `-2203/4096`Retained binary64: `-0.537841796875`	-201/32Exact fraction: `-201/32`Retained binary64: `-6.28125`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
30	-17/32Exact fraction: `-17/32`Retained binary64: `-0.53125`	-451/1024Exact fraction: `-451/1024`Retained binary64: `-0.4404296875`	-99/16Exact fraction: `-99/16`Retained binary64: `-6.1875`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
31	-33/64Exact fraction: `-33/64`Retained binary64: `-0.515625`	-1411/4096Exact fraction: `-1411/4096`Retained binary64: `-0.344482421875`	-195/32Exact fraction: `-195/32`Retained binary64: `-6.09375`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
32	-1/2Exact fraction: `-1/2`Retained binary64: `-0.5`	-1/4Exact fraction: `-1/4`Retained binary64: `-0.25`	-6Exact fraction: `-6`Retained binary64: `-6.0`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
33	-31/64Exact fraction: `-31/64`Retained binary64: `-0.484375`	-643/4096Exact fraction: `-643/4096`Retained binary64: `-0.156982421875`	-189/32Exact fraction: `-189/32`Retained binary64: `-5.90625`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
34	-15/32Exact fraction: `-15/32`Retained binary64: `-0.46875`	-67/1024Exact fraction: `-67/1024`Retained binary64: `-0.0654296875`	-93/16Exact fraction: `-93/16`Retained binary64: `-5.8125`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
35	-29/64Exact fraction: `-29/64`Retained binary64: `-0.453125`	101/4096Exact fraction: `101/4096`Retained binary64: `0.024658203125`	-183/32Exact fraction: `-183/32`Retained binary64: `-5.71875`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
36	-7/16Exact fraction: `-7/16`Retained binary64: `-0.4375`	29/256Exact fraction: `29/256`Retained binary64: `0.11328125`	-45/8Exact fraction: `-45/8`Retained binary64: `-5.625`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
37	-27/64Exact fraction: `-27/64`Retained binary64: `-0.421875`	821/4096Exact fraction: `821/4096`Retained binary64: `0.200439453125`	-177/32Exact fraction: `-177/32`Retained binary64: `-5.53125`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
38	-13/32Exact fraction: `-13/32`Retained binary64: `-0.40625`	293/1024Exact fraction: `293/1024`Retained binary64: `0.2861328125`	-87/16Exact fraction: `-87/16`Retained binary64: `-5.4375`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
39	-25/64Exact fraction: `-25/64`Retained binary64: `-0.390625`	1517/4096Exact fraction: `1517/4096`Retained binary64: `0.370361328125`	-171/32Exact fraction: `-171/32`Retained binary64: `-5.34375`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
40	-3/8Exact fraction: `-3/8`Retained binary64: `-0.375`	29/64Exact fraction: `29/64`Retained binary64: `0.453125`	-21/4Exact fraction: `-21/4`Retained binary64: `-5.25`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
41	-23/64Exact fraction: `-23/64`Retained binary64: `-0.359375`	2189/4096Exact fraction: `2189/4096`Retained binary64: `0.534423828125`	-165/32Exact fraction: `-165/32`Retained binary64: `-5.15625`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
42	-11/32Exact fraction: `-11/32`Retained binary64: `-0.34375`	629/1024Exact fraction: `629/1024`Retained binary64: `0.6142578125`	-81/16Exact fraction: `-81/16`Retained binary64: `-5.0625`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
43	-21/64Exact fraction: `-21/64`Retained binary64: `-0.328125`	2837/4096Exact fraction: `2837/4096`Retained binary64: `0.692626953125`	-159/32Exact fraction: `-159/32`Retained binary64: `-4.96875`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
44	-5/16Exact fraction: `-5/16`Retained binary64: `-0.3125`	197/256Exact fraction: `197/256`Retained binary64: `0.76953125`	-39/8Exact fraction: `-39/8`Retained binary64: `-4.875`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
45	-19/64Exact fraction: `-19/64`Retained binary64: `-0.296875`	3461/4096Exact fraction: `3461/4096`Retained binary64: `0.844970703125`	-153/32Exact fraction: `-153/32`Retained binary64: `-4.78125`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
46	-9/32Exact fraction: `-9/32`Retained binary64: `-0.28125`	941/1024Exact fraction: `941/1024`Retained binary64: `0.9189453125`	-75/16Exact fraction: `-75/16`Retained binary64: `-4.6875`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
47	-17/64Exact fraction: `-17/64`Retained binary64: `-0.265625`	4061/4096Exact fraction: `4061/4096`Retained binary64: `0.991455078125`	-147/32Exact fraction: `-147/32`Retained binary64: `-4.59375`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
48	-1/4Exact fraction: `-1/4`Retained binary64: `-0.25`	17/16Exact fraction: `17/16`Retained binary64: `1.0625`	-9/2Exact fraction: `-9/2`Retained binary64: `-4.5`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
49	-15/64Exact fraction: `-15/64`Retained binary64: `-0.234375`	4637/4096Exact fraction: `4637/4096`Retained binary64: `1.132080078125`	-141/32Exact fraction: `-141/32`Retained binary64: `-4.40625`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
50	-7/32Exact fraction: `-7/32`Retained binary64: `-0.21875`	1229/1024Exact fraction: `1229/1024`Retained binary64: `1.2001953125`	-69/16Exact fraction: `-69/16`Retained binary64: `-4.3125`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
51	-13/64Exact fraction: `-13/64`Retained binary64: `-0.203125`	5189/4096Exact fraction: `5189/4096`Retained binary64: `1.266845703125`	-135/32Exact fraction: `-135/32`Retained binary64: `-4.21875`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
52	-3/16Exact fraction: `-3/16`Retained binary64: `-0.1875`	341/256Exact fraction: `341/256`Retained binary64: `1.33203125`	-33/8Exact fraction: `-33/8`Retained binary64: `-4.125`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
53	-11/64Exact fraction: `-11/64`Retained binary64: `-0.171875`	5717/4096Exact fraction: `5717/4096`Retained binary64: `1.395751953125`	-129/32Exact fraction: `-129/32`Retained binary64: `-4.03125`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
54	-5/32Exact fraction: `-5/32`Retained binary64: `-0.15625`	1493/1024Exact fraction: `1493/1024`Retained binary64: `1.4580078125`	-63/16Exact fraction: `-63/16`Retained binary64: `-3.9375`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
55	-9/64Exact fraction: `-9/64`Retained binary64: `-0.140625`	6221/4096Exact fraction: `6221/4096`Retained binary64: `1.518798828125`	-123/32Exact fraction: `-123/32`Retained binary64: `-3.84375`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
56	-1/8Exact fraction: `-1/8`Retained binary64: `-0.125`	101/64Exact fraction: `101/64`Retained binary64: `1.578125`	-15/4Exact fraction: `-15/4`Retained binary64: `-3.75`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
57	-7/64Exact fraction: `-7/64`Retained binary64: `-0.109375`	6701/4096Exact fraction: `6701/4096`Retained binary64: `1.635986328125`	-117/32Exact fraction: `-117/32`Retained binary64: `-3.65625`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
58	-3/32Exact fraction: `-3/32`Retained binary64: `-0.09375`	1733/1024Exact fraction: `1733/1024`Retained binary64: `1.6923828125`	-57/16Exact fraction: `-57/16`Retained binary64: `-3.5625`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
59	-5/64Exact fraction: `-5/64`Retained binary64: `-0.078125`	7157/4096Exact fraction: `7157/4096`Retained binary64: `1.747314453125`	-111/32Exact fraction: `-111/32`Retained binary64: `-3.46875`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
60	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	461/256Exact fraction: `461/256`Retained binary64: `1.80078125`	-27/8Exact fraction: `-27/8`Retained binary64: `-3.375`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
61	-3/64Exact fraction: `-3/64`Retained binary64: `-0.046875`	7589/4096Exact fraction: `7589/4096`Retained binary64: `1.852783203125`	-105/32Exact fraction: `-105/32`Retained binary64: `-3.28125`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
62	-1/32Exact fraction: `-1/32`Retained binary64: `-0.03125`	1949/1024Exact fraction: `1949/1024`Retained binary64: `1.9033203125`	-51/16Exact fraction: `-51/16`Retained binary64: `-3.1875`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
63	-1/64Exact fraction: `-1/64`Retained binary64: `-0.015625`	7997/4096Exact fraction: `7997/4096`Retained binary64: `1.952392578125`	-99/32Exact fraction: `-99/32`Retained binary64: `-3.09375`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
64	0Exact fraction: `0`Retained binary64: `0.0`	2Exact fraction: `2`Retained binary64: `2.0`	-3Exact fraction: `-3`Retained binary64: `-3.0`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
65	1/64Exact fraction: `1/64`Retained binary64: `0.015625`	8381/4096Exact fraction: `8381/4096`Retained binary64: `2.046142578125`	-93/32Exact fraction: `-93/32`Retained binary64: `-2.90625`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
66	1/32Exact fraction: `1/32`Retained binary64: `0.03125`	2141/1024Exact fraction: `2141/1024`Retained binary64: `2.0908203125`	-45/16Exact fraction: `-45/16`Retained binary64: `-2.8125`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
67	3/64Exact fraction: `3/64`Retained binary64: `0.046875`	8741/4096Exact fraction: `8741/4096`Retained binary64: `2.134033203125`	-87/32Exact fraction: `-87/32`Retained binary64: `-2.71875`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
68	1/16Exact fraction: `1/16`Retained binary64: `0.0625`	557/256Exact fraction: `557/256`Retained binary64: `2.17578125`	-21/8Exact fraction: `-21/8`Retained binary64: `-2.625`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
69	5/64Exact fraction: `5/64`Retained binary64: `0.078125`	9077/4096Exact fraction: `9077/4096`Retained binary64: `2.216064453125`	-81/32Exact fraction: `-81/32`Retained binary64: `-2.53125`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
70	3/32Exact fraction: `3/32`Retained binary64: `0.09375`	2309/1024Exact fraction: `2309/1024`Retained binary64: `2.2548828125`	-39/16Exact fraction: `-39/16`Retained binary64: `-2.4375`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
71	7/64Exact fraction: `7/64`Retained binary64: `0.109375`	9389/4096Exact fraction: `9389/4096`Retained binary64: `2.292236328125`	-75/32Exact fraction: `-75/32`Retained binary64: `-2.34375`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
72	1/8Exact fraction: `1/8`Retained binary64: `0.125`	149/64Exact fraction: `149/64`Retained binary64: `2.328125`	-9/4Exact fraction: `-9/4`Retained binary64: `-2.25`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
73	9/64Exact fraction: `9/64`Retained binary64: `0.140625`	9677/4096Exact fraction: `9677/4096`Retained binary64: `2.362548828125`	-69/32Exact fraction: `-69/32`Retained binary64: `-2.15625`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
74	5/32Exact fraction: `5/32`Retained binary64: `0.15625`	2453/1024Exact fraction: `2453/1024`Retained binary64: `2.3955078125`	-33/16Exact fraction: `-33/16`Retained binary64: `-2.0625`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
75	11/64Exact fraction: `11/64`Retained binary64: `0.171875`	9941/4096Exact fraction: `9941/4096`Retained binary64: `2.427001953125`	-63/32Exact fraction: `-63/32`Retained binary64: `-1.96875`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
76	3/16Exact fraction: `3/16`Retained binary64: `0.1875`	629/256Exact fraction: `629/256`Retained binary64: `2.45703125`	-15/8Exact fraction: `-15/8`Retained binary64: `-1.875`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
77	13/64Exact fraction: `13/64`Retained binary64: `0.203125`	10181/4096Exact fraction: `10181/4096`Retained binary64: `2.485595703125`	-57/32Exact fraction: `-57/32`Retained binary64: `-1.78125`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
78	7/32Exact fraction: `7/32`Retained binary64: `0.21875`	2573/1024Exact fraction: `2573/1024`Retained binary64: `2.5126953125`	-27/16Exact fraction: `-27/16`Retained binary64: `-1.6875`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
79	15/64Exact fraction: `15/64`Retained binary64: `0.234375`	10397/4096Exact fraction: `10397/4096`Retained binary64: `2.538330078125`	-51/32Exact fraction: `-51/32`Retained binary64: `-1.59375`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
80	1/4Exact fraction: `1/4`Retained binary64: `0.25`	41/16Exact fraction: `41/16`Retained binary64: `2.5625`	-3/2Exact fraction: `-3/2`Retained binary64: `-1.5`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
81	17/64Exact fraction: `17/64`Retained binary64: `0.265625`	10589/4096Exact fraction: `10589/4096`Retained binary64: `2.585205078125`	-45/32Exact fraction: `-45/32`Retained binary64: `-1.40625`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
82	9/32Exact fraction: `9/32`Retained binary64: `0.28125`	2669/1024Exact fraction: `2669/1024`Retained binary64: `2.6064453125`	-21/16Exact fraction: `-21/16`Retained binary64: `-1.3125`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
83	19/64Exact fraction: `19/64`Retained binary64: `0.296875`	10757/4096Exact fraction: `10757/4096`Retained binary64: `2.626220703125`	-39/32Exact fraction: `-39/32`Retained binary64: `-1.21875`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
84	5/16Exact fraction: `5/16`Retained binary64: `0.3125`	677/256Exact fraction: `677/256`Retained binary64: `2.64453125`	-9/8Exact fraction: `-9/8`Retained binary64: `-1.125`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
85	21/64Exact fraction: `21/64`Retained binary64: `0.328125`	10901/4096Exact fraction: `10901/4096`Retained binary64: `2.661376953125`	-33/32Exact fraction: `-33/32`Retained binary64: `-1.03125`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
86	11/32Exact fraction: `11/32`Retained binary64: `0.34375`	2741/1024Exact fraction: `2741/1024`Retained binary64: `2.6767578125`	-15/16Exact fraction: `-15/16`Retained binary64: `-0.9375`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
87	23/64Exact fraction: `23/64`Retained binary64: `0.359375`	11021/4096Exact fraction: `11021/4096`Retained binary64: `2.690673828125`	-27/32Exact fraction: `-27/32`Retained binary64: `-0.84375`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
88	3/8Exact fraction: `3/8`Retained binary64: `0.375`	173/64Exact fraction: `173/64`Retained binary64: `2.703125`	-3/4Exact fraction: `-3/4`Retained binary64: `-0.75`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
89	25/64Exact fraction: `25/64`Retained binary64: `0.390625`	11117/4096Exact fraction: `11117/4096`Retained binary64: `2.714111328125`	-21/32Exact fraction: `-21/32`Retained binary64: `-0.65625`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
90	13/32Exact fraction: `13/32`Retained binary64: `0.40625`	2789/1024Exact fraction: `2789/1024`Retained binary64: `2.7236328125`	-9/16Exact fraction: `-9/16`Retained binary64: `-0.5625`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
91	27/64Exact fraction: `27/64`Retained binary64: `0.421875`	11189/4096Exact fraction: `11189/4096`Retained binary64: `2.731689453125`	-15/32Exact fraction: `-15/32`Retained binary64: `-0.46875`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
92	7/16Exact fraction: `7/16`Retained binary64: `0.4375`	701/256Exact fraction: `701/256`Retained binary64: `2.73828125`	-3/8Exact fraction: `-3/8`Retained binary64: `-0.375`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
93	29/64Exact fraction: `29/64`Retained binary64: `0.453125`	11237/4096Exact fraction: `11237/4096`Retained binary64: `2.743408203125`	-9/32Exact fraction: `-9/32`Retained binary64: `-0.28125`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
94	15/32Exact fraction: `15/32`Retained binary64: `0.46875`	2813/1024Exact fraction: `2813/1024`Retained binary64: `2.7470703125`	-3/16Exact fraction: `-3/16`Retained binary64: `-0.1875`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
95	31/64Exact fraction: `31/64`Retained binary64: `0.484375`	11261/4096Exact fraction: `11261/4096`Retained binary64: `2.749267578125`	-3/32Exact fraction: `-3/32`Retained binary64: `-0.09375`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
96	1/2Exact fraction: `1/2`Retained binary64: `0.5`	11/4Exact fraction: `11/4`Retained binary64: `2.75`	0Exact fraction: `0`Retained binary64: `0.0`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
97	33/64Exact fraction: `33/64`Retained binary64: `0.515625`	11261/4096Exact fraction: `11261/4096`Retained binary64: `2.749267578125`	3/32Exact fraction: `3/32`Retained binary64: `0.09375`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
98	17/32Exact fraction: `17/32`Retained binary64: `0.53125`	2813/1024Exact fraction: `2813/1024`Retained binary64: `2.7470703125`	3/16Exact fraction: `3/16`Retained binary64: `0.1875`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
99	35/64Exact fraction: `35/64`Retained binary64: `0.546875`	11237/4096Exact fraction: `11237/4096`Retained binary64: `2.743408203125`	9/32Exact fraction: `9/32`Retained binary64: `0.28125`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
100	9/16Exact fraction: `9/16`Retained binary64: `0.5625`	701/256Exact fraction: `701/256`Retained binary64: `2.73828125`	3/8Exact fraction: `3/8`Retained binary64: `0.375`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
101	37/64Exact fraction: `37/64`Retained binary64: `0.578125`	11189/4096Exact fraction: `11189/4096`Retained binary64: `2.731689453125`	15/32Exact fraction: `15/32`Retained binary64: `0.46875`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
102	19/32Exact fraction: `19/32`Retained binary64: `0.59375`	2789/1024Exact fraction: `2789/1024`Retained binary64: `2.7236328125`	9/16Exact fraction: `9/16`Retained binary64: `0.5625`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
103	39/64Exact fraction: `39/64`Retained binary64: `0.609375`	11117/4096Exact fraction: `11117/4096`Retained binary64: `2.714111328125`	21/32Exact fraction: `21/32`Retained binary64: `0.65625`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
104	5/8Exact fraction: `5/8`Retained binary64: `0.625`	173/64Exact fraction: `173/64`Retained binary64: `2.703125`	3/4Exact fraction: `3/4`Retained binary64: `0.75`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
105	41/64Exact fraction: `41/64`Retained binary64: `0.640625`	11021/4096Exact fraction: `11021/4096`Retained binary64: `2.690673828125`	27/32Exact fraction: `27/32`Retained binary64: `0.84375`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
106	21/32Exact fraction: `21/32`Retained binary64: `0.65625`	2741/1024Exact fraction: `2741/1024`Retained binary64: `2.6767578125`	15/16Exact fraction: `15/16`Retained binary64: `0.9375`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
107	43/64Exact fraction: `43/64`Retained binary64: `0.671875`	10901/4096Exact fraction: `10901/4096`Retained binary64: `2.661376953125`	33/32Exact fraction: `33/32`Retained binary64: `1.03125`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
108	11/16Exact fraction: `11/16`Retained binary64: `0.6875`	677/256Exact fraction: `677/256`Retained binary64: `2.64453125`	9/8Exact fraction: `9/8`Retained binary64: `1.125`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
109	45/64Exact fraction: `45/64`Retained binary64: `0.703125`	10757/4096Exact fraction: `10757/4096`Retained binary64: `2.626220703125`	39/32Exact fraction: `39/32`Retained binary64: `1.21875`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
110	23/32Exact fraction: `23/32`Retained binary64: `0.71875`	2669/1024Exact fraction: `2669/1024`Retained binary64: `2.6064453125`	21/16Exact fraction: `21/16`Retained binary64: `1.3125`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
111	47/64Exact fraction: `47/64`Retained binary64: `0.734375`	10589/4096Exact fraction: `10589/4096`Retained binary64: `2.585205078125`	45/32Exact fraction: `45/32`Retained binary64: `1.40625`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
112	3/4Exact fraction: `3/4`Retained binary64: `0.75`	41/16Exact fraction: `41/16`Retained binary64: `2.5625`	3/2Exact fraction: `3/2`Retained binary64: `1.5`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
113	49/64Exact fraction: `49/64`Retained binary64: `0.765625`	10397/4096Exact fraction: `10397/4096`Retained binary64: `2.538330078125`	51/32Exact fraction: `51/32`Retained binary64: `1.59375`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
114	25/32Exact fraction: `25/32`Retained binary64: `0.78125`	2573/1024Exact fraction: `2573/1024`Retained binary64: `2.5126953125`	27/16Exact fraction: `27/16`Retained binary64: `1.6875`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
115	51/64Exact fraction: `51/64`Retained binary64: `0.796875`	10181/4096Exact fraction: `10181/4096`Retained binary64: `2.485595703125`	57/32Exact fraction: `57/32`Retained binary64: `1.78125`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
116	13/16Exact fraction: `13/16`Retained binary64: `0.8125`	629/256Exact fraction: `629/256`Retained binary64: `2.45703125`	15/8Exact fraction: `15/8`Retained binary64: `1.875`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
117	53/64Exact fraction: `53/64`Retained binary64: `0.828125`	9941/4096Exact fraction: `9941/4096`Retained binary64: `2.427001953125`	63/32Exact fraction: `63/32`Retained binary64: `1.96875`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
118	27/32Exact fraction: `27/32`Retained binary64: `0.84375`	2453/1024Exact fraction: `2453/1024`Retained binary64: `2.3955078125`	33/16Exact fraction: `33/16`Retained binary64: `2.0625`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
119	55/64Exact fraction: `55/64`Retained binary64: `0.859375`	9677/4096Exact fraction: `9677/4096`Retained binary64: `2.362548828125`	69/32Exact fraction: `69/32`Retained binary64: `2.15625`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
120	7/8Exact fraction: `7/8`Retained binary64: `0.875`	149/64Exact fraction: `149/64`Retained binary64: `2.328125`	9/4Exact fraction: `9/4`Retained binary64: `2.25`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
121	57/64Exact fraction: `57/64`Retained binary64: `0.890625`	9389/4096Exact fraction: `9389/4096`Retained binary64: `2.292236328125`	75/32Exact fraction: `75/32`Retained binary64: `2.34375`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
122	29/32Exact fraction: `29/32`Retained binary64: `0.90625`	2309/1024Exact fraction: `2309/1024`Retained binary64: `2.2548828125`	39/16Exact fraction: `39/16`Retained binary64: `2.4375`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
123	59/64Exact fraction: `59/64`Retained binary64: `0.921875`	9077/4096Exact fraction: `9077/4096`Retained binary64: `2.216064453125`	81/32Exact fraction: `81/32`Retained binary64: `2.53125`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
124	15/16Exact fraction: `15/16`Retained binary64: `0.9375`	557/256Exact fraction: `557/256`Retained binary64: `2.17578125`	21/8Exact fraction: `21/8`Retained binary64: `2.625`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
125	61/64Exact fraction: `61/64`Retained binary64: `0.953125`	8741/4096Exact fraction: `8741/4096`Retained binary64: `2.134033203125`	87/32Exact fraction: `87/32`Retained binary64: `2.71875`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
126	31/32Exact fraction: `31/32`Retained binary64: `0.96875`	2141/1024Exact fraction: `2141/1024`Retained binary64: `2.0908203125`	45/16Exact fraction: `45/16`Retained binary64: `2.8125`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
127	63/64Exact fraction: `63/64`Retained binary64: `0.984375`	8381/4096Exact fraction: `8381/4096`Retained binary64: `2.046142578125`	93/32Exact fraction: `93/32`Retained binary64: `2.90625`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
128	1Exact fraction: `1`Retained binary64: `1.0`	2Exact fraction: `2`Retained binary64: `2.0`	3Exact fraction: `3`Retained binary64: `3.0`	-6Exact fraction: `-6`Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`

Outward-normal derivative and source audit (4 rows)

Derivative units areV₀/L₀; normalized source isρL₀²/(ε₀V₀). Left and right derivatives use their respective outward normals.

Outward-normal derivative and source audit
Quantity	Exact value and retained binary64
leftOutwardDerivative	-9Exact fraction: `-9`Retained binary64: `-9.0`
rightOutwardDerivative	-3Exact fraction: `-3`Retained binary64: `-3.0`
outwardDerivativeSum	-12Exact fraction: `-12`Retained binary64: `-12.0`
normalizedRho	6Exact fraction: `6`Retained binary64: `6.0`

Use the labelled graph or table region with a keyboard, or scroll sideways. Graphs retain their fixed readable size; the page itself must fit the viewport. Full exact/retained values are in the native details tables.

Exact controls for the prescribed charged slab
X = x/L₀	V/V₀	E_x/E₀	Interpretation
−1	−4	−9	Left endpoint value
−1/2	−1/4	−6	Negative potential and nonzero field
0	2	−3	Positive potential and nonzero field
1/2	11/4	0	Interior stationary maximum along x
1	2	3	Right endpoint value

Separate the source part from the boundary correction

The slab’s particular solution v_p = −3X² already has the required second derivative −6, but it gives −3 at both endpoints. It does not satisfy the prescribed voltages. The difference v_h = 2 + 3X has zero second derivative and supplies precisely the missing endpoint values: −1 on the left and 5 on the right.

This is the structure V = V_p + V_h. The particular part accounts for the source term, while the harmonic part corrects the boundary discrepancy. Neither part is uniquely chosen by itself; adding a harmonic function to one and subtracting it from the other leaves the same full solution.

Linearity means that adding two complete solutions adds both their source densities and their boundary data. It does not mean that one may add two fields with the same nonzero boundary voltage and still claim that unchanged boundary voltage. For a fixed prescribed boundary problem, the total boundary condition must be checked after superposition.

A useful audit is to compute the PDE residual and then test every boundary condition independently. Adding any wrong linear term to the slab potential leaves its second derivative unchanged. Therefore a zero local Poisson residual alone cannot distinguish the intended solution from a solution with different endpoint voltages.

State the boundary data and test compatibility

Dirichlet data specify V on the entire boundary. Neumann data specify the outward normal derivative ∂ₙV = n̂·∇V, which is the negative outward normal electric-field component. A boundary condition may also combine these quantities, but its coefficients, units and physical origin must be stated.

For a connected bounded region with suitable regularity, integrate Poisson’s equation and use the divergence theorem: ∮∂Ω ∂ₙV dA = −(1/ε₀)∫Ωρ d³r = −Q_enclosed/ε₀. Thus arbitrary pure-Neumann data are not freely compatible with an arbitrary enclosed charge. Their total normal derivative must satisfy this condition before a solution can exist.

For the slab per unit transverse area, the left outward normal is −x̂ and the right normal is +x̂. The left derivative is −9V₀/L₀; the right is −3V₀/L₀. Their sum is −12V₀/L₀, matching −ρ(2L₀)/ε₀. Using the same +x direction on both faces would be a different, incorrect flux calculation.

Pure-Neumann data do not fix an additive constant: V and V + C have the same gradient, source and normal derivatives. One may choose a reference or a mean value to remove that freedom. Prescribing both V and its normal derivative independently on the entire boundary generally overdetermines an elliptic problem; the two sets of data must come from a consistent solution.

Prove what uniqueness actually says

Let V₁ and V₂ solve the same Poisson equation in a connected bounded region, with the same complete Dirichlet boundary values. Their difference w = V₁ − V₂ satisfies ∇²w = 0 and w = 0 on the boundary. Assume the regularity needed for Green’s identity, or its corresponding weak formulation.

Use ∇·(w∇w) = |∇w|² + w∇²w and integrate. The volume integral of |∇w|² equals ∮w∂ₙw dA − ∫w∇²w d³r. The second term vanishes because w is harmonic, and the surface term vanishes because w = 0 there. A nonnegative squared gradient with zero integral implies ∇w = 0, so w is constant on the connected region.

The zero Dirichlet boundary value makes that constant zero. Hence the two solutions coincide. All boundary components matter, including inner surfaces around excluded objects. If the source itself is singular, the proof must retain its prescribed singular behavior or excise it and include the resulting boundary information.

For two solutions with the same pure-Neumann data, ∂ₙw = 0 makes the same surface term vanish, but it only proves that w is constant. The data leave the reference undetermined. A uniqueness argument establishes at most one solution in the stated class, or one modulo a constant; it does not establish existence for incompatible data or for an unspecified function space.

A harmonic saddle is not a stable trap

Choose the separate z-independent potential v = (X + 1)² − Y² on the slice −2 ≤ X ≤ 0 and −1 ≤ Y ≤ 1, with its analytic values prescribed on the entire rectangular boundary. Its two Cartesian second derivatives are +2 and −2, so their sum is zero. This is a harmonic example with a nonzero field over most of the region.

The field is E/E₀ = (−2(X + 1), 2Y, 0). It vanishes at (X,Y) = (−1,0), where v = 0. Moving a small distance in x raises V, while moving the same distance in y lowers V. The point is a saddle rather than a local maximum or minimum. In the z-invariant three-dimensional extension the critical set is a whole z-line, not an isolated equilibrium point.

Contours satisfy (X + 1)² − Y² = c. For c = 0, the two branches Y = ±(X + 1) meet at the critical point. They have the same potential value. Distinct values of c still cannot cross. At the crossing, the gradient is zero, so there is no unique nonzero normal field direction to assign to the level set.

For a positive probe the x force near the saddle is restoring, but the y force is destabilizing; a negative probe reverses those two roles. There is no restoring z force in this invariant model. A zero field therefore does not establish stable free confinement. Boundary constraints, time-dependent fields, material effects and other forces can change a trapping problem and are outside this example.

One harmonic saddle, separate levels and one exact null

Four level values with exact isolated boundary points distinguished from contours, two crossing branches of one zero level and four separate quarter-level halves. Direction-only arrows, actual zero-field dot, declared z-invariant critical line, no stable trap.

Contour branches and direction-only arrows on an equal-scale slice

Level 0: two crossing branches of one value
Level 1/4: separate upper/lower half-branches
Direction only: common 0.16 L₀ scale; field values in tables
Levels −1 and 1: isolated boundary points only

v=(X+1)²−Y²on−2≤X≤0,−1≤Y≤1, with the analytic potential fixed on the entire rectangle. Equal physical x and y scales are used.
Nonzero arrows use a common0.16L₀ direction scale and encode only local field direction. It is not magnitude, a field-line trace or a particle trajectory. The exact null at (−1,0) is a dot, never normalized.
The two level 0 branches cross at the critical point. The quarter-level has two components, each stored as separate upper/lower halves; levels −1 and1have only the four labelled isolated boundary points.
The critical point in this slice extends to a whole z-line; this harmonic saddle gives no stable free electrostatic trap. Contour endpoints and the 25 direction samples are distinct sets of positions; a contour sample is not an additional vector measurement.
The source data use exact rational strings or12-significant-digit displays plus full retained binary64 values. The plotted curves join finite analytic samples; no interpolation-error bound or numerical PDE solver is claimed.

Complete labelled equivalents below preserve every retained sample and scientific value; original manuscript tables remain separate.

All 25 direction controls; the zero has no normalized direction (25 rows)

Only the nonzero rows supply direction arrows; arrows use a common0.16L₀ direction scale. The analytic saddle isz-invariant.

All 25 direction controls; the zero has no normalized direction
Row	X=x/L₀	Y=y/L₀	V/V₀	Signed Eₓ/E₀	Signed Eᵧ/E₀	Positive \|E\|/E₀	Exact zero-field mask	Unit direction [uₓ,uᵧ], or null; not field magnitude
0	-7/4Exact fraction: `-7/4`Retained binary64: `-1.75`	-3/4Exact fraction: `-3/4`Retained binary64: `-0.75`	0Exact fraction: `0`Retained binary64: `0.0`	3/2Exact fraction: `3/2`Retained binary64: `1.5`	-3/2Exact fraction: `-3/2`Retained binary64: `-1.5`	2.12132034356Retained binary64: `2.1213203435596424`	false	[0.7071067811865476,-0.7071067811865476]
1	-7/4Exact fraction: `-7/4`Retained binary64: `-1.75`	-1/4Exact fraction: `-1/4`Retained binary64: `-0.25`	1/2Exact fraction: `1/2`Retained binary64: `0.5`	3/2Exact fraction: `3/2`Retained binary64: `1.5`	-1/2Exact fraction: `-1/2`Retained binary64: `-0.5`	1.58113883008Retained binary64: `1.5811388300841898`	false	[0.9486832980505138,-0.31622776601683794]
2	-7/4Exact fraction: `-7/4`Retained binary64: `-1.75`	0Exact fraction: `0`Retained binary64: `0.0`	9/16Exact fraction: `9/16`Retained binary64: `0.5625`	3/2Exact fraction: `3/2`Retained binary64: `1.5`	0Exact fraction: `0`Retained binary64: `0.0`	1.5Retained binary64: `1.5`	false	[1.0,0.0]
3	-7/4Exact fraction: `-7/4`Retained binary64: `-1.75`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/2Exact fraction: `1/2`Retained binary64: `0.5`	3/2Exact fraction: `3/2`Retained binary64: `1.5`	1/2Exact fraction: `1/2`Retained binary64: `0.5`	1.58113883008Retained binary64: `1.5811388300841898`	false	[0.9486832980505138,0.31622776601683794]
4	-7/4Exact fraction: `-7/4`Retained binary64: `-1.75`	3/4Exact fraction: `3/4`Retained binary64: `0.75`	0Exact fraction: `0`Retained binary64: `0.0`	3/2Exact fraction: `3/2`Retained binary64: `1.5`	3/2Exact fraction: `3/2`Retained binary64: `1.5`	2.12132034356Retained binary64: `2.1213203435596424`	false	[0.7071067811865476,0.7071067811865476]
5	-5/4Exact fraction: `-5/4`Retained binary64: `-1.25`	-3/4Exact fraction: `-3/4`Retained binary64: `-0.75`	-1/2Exact fraction: `-1/2`Retained binary64: `-0.5`	1/2Exact fraction: `1/2`Retained binary64: `0.5`	-3/2Exact fraction: `-3/2`Retained binary64: `-1.5`	1.58113883008Retained binary64: `1.5811388300841898`	false	[0.31622776601683794,-0.9486832980505138]
6	-5/4Exact fraction: `-5/4`Retained binary64: `-1.25`	-1/4Exact fraction: `-1/4`Retained binary64: `-0.25`	0Exact fraction: `0`Retained binary64: `0.0`	1/2Exact fraction: `1/2`Retained binary64: `0.5`	-1/2Exact fraction: `-1/2`Retained binary64: `-0.5`	0.707106781187Retained binary64: `0.7071067811865476`	false	[0.7071067811865475,-0.7071067811865475]
7	-5/4Exact fraction: `-5/4`Retained binary64: `-1.25`	0Exact fraction: `0`Retained binary64: `0.0`	1/16Exact fraction: `1/16`Retained binary64: `0.0625`	1/2Exact fraction: `1/2`Retained binary64: `0.5`	0Exact fraction: `0`Retained binary64: `0.0`	0.5Retained binary64: `0.5`	false	[1.0,0.0]
8	-5/4Exact fraction: `-5/4`Retained binary64: `-1.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	0Exact fraction: `0`Retained binary64: `0.0`	1/2Exact fraction: `1/2`Retained binary64: `0.5`	1/2Exact fraction: `1/2`Retained binary64: `0.5`	0.707106781187Retained binary64: `0.7071067811865476`	false	[0.7071067811865475,0.7071067811865475]
9	-5/4Exact fraction: `-5/4`Retained binary64: `-1.25`	3/4Exact fraction: `3/4`Retained binary64: `0.75`	-1/2Exact fraction: `-1/2`Retained binary64: `-0.5`	1/2Exact fraction: `1/2`Retained binary64: `0.5`	3/2Exact fraction: `3/2`Retained binary64: `1.5`	1.58113883008Retained binary64: `1.5811388300841898`	false	[0.31622776601683794,0.9486832980505138]
10	-1Exact fraction: `-1`Retained binary64: `-1.0`	-3/4Exact fraction: `-3/4`Retained binary64: `-0.75`	-9/16Exact fraction: `-9/16`Retained binary64: `-0.5625`	0Exact fraction: `0`Retained binary64: `0.0`	-3/2Exact fraction: `-3/2`Retained binary64: `-1.5`	1.5Retained binary64: `1.5`	false	[0.0,-1.0]
11	-1Exact fraction: `-1`Retained binary64: `-1.0`	-1/4Exact fraction: `-1/4`Retained binary64: `-0.25`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	0Exact fraction: `0`Retained binary64: `0.0`	-1/2Exact fraction: `-1/2`Retained binary64: `-0.5`	0.5Retained binary64: `0.5`	false	[0.0,-1.0]
12	-1Exact fraction: `-1`Retained binary64: `-1.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Exact fraction: `0`Retained binary64: `0.0`	0Retained binary64: `0.0`	true	null
13	-1Exact fraction: `-1`Retained binary64: `-1.0`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	0Exact fraction: `0`Retained binary64: `0.0`	1/2Exact fraction: `1/2`Retained binary64: `0.5`	0.5Retained binary64: `0.5`	false	[0.0,1.0]
14	-1Exact fraction: `-1`Retained binary64: `-1.0`	3/4Exact fraction: `3/4`Retained binary64: `0.75`	-9/16Exact fraction: `-9/16`Retained binary64: `-0.5625`	0Exact fraction: `0`Retained binary64: `0.0`	3/2Exact fraction: `3/2`Retained binary64: `1.5`	1.5Retained binary64: `1.5`	false	[0.0,1.0]
15	-3/4Exact fraction: `-3/4`Retained binary64: `-0.75`	-3/4Exact fraction: `-3/4`Retained binary64: `-0.75`	-1/2Exact fraction: `-1/2`Retained binary64: `-0.5`	-1/2Exact fraction: `-1/2`Retained binary64: `-0.5`	-3/2Exact fraction: `-3/2`Retained binary64: `-1.5`	1.58113883008Retained binary64: `1.5811388300841898`	false	[-0.31622776601683794,-0.9486832980505138]
16	-3/4Exact fraction: `-3/4`Retained binary64: `-0.75`	-1/4Exact fraction: `-1/4`Retained binary64: `-0.25`	0Exact fraction: `0`Retained binary64: `0.0`	-1/2Exact fraction: `-1/2`Retained binary64: `-0.5`	-1/2Exact fraction: `-1/2`Retained binary64: `-0.5`	0.707106781187Retained binary64: `0.7071067811865476`	false	[-0.7071067811865475,-0.7071067811865475]
17	-3/4Exact fraction: `-3/4`Retained binary64: `-0.75`	0Exact fraction: `0`Retained binary64: `0.0`	1/16Exact fraction: `1/16`Retained binary64: `0.0625`	-1/2Exact fraction: `-1/2`Retained binary64: `-0.5`	0Exact fraction: `0`Retained binary64: `0.0`	0.5Retained binary64: `0.5`	false	[-1.0,0.0]
18	-3/4Exact fraction: `-3/4`Retained binary64: `-0.75`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	0Exact fraction: `0`Retained binary64: `0.0`	-1/2Exact fraction: `-1/2`Retained binary64: `-0.5`	1/2Exact fraction: `1/2`Retained binary64: `0.5`	0.707106781187Retained binary64: `0.7071067811865476`	false	[-0.7071067811865475,0.7071067811865475]
19	-3/4Exact fraction: `-3/4`Retained binary64: `-0.75`	3/4Exact fraction: `3/4`Retained binary64: `0.75`	-1/2Exact fraction: `-1/2`Retained binary64: `-0.5`	-1/2Exact fraction: `-1/2`Retained binary64: `-0.5`	3/2Exact fraction: `3/2`Retained binary64: `1.5`	1.58113883008Retained binary64: `1.5811388300841898`	false	[-0.31622776601683794,0.9486832980505138]
20	-1/4Exact fraction: `-1/4`Retained binary64: `-0.25`	-3/4Exact fraction: `-3/4`Retained binary64: `-0.75`	0Exact fraction: `0`Retained binary64: `0.0`	-3/2Exact fraction: `-3/2`Retained binary64: `-1.5`	-3/2Exact fraction: `-3/2`Retained binary64: `-1.5`	2.12132034356Retained binary64: `2.1213203435596424`	false	[-0.7071067811865476,-0.7071067811865476]
21	-1/4Exact fraction: `-1/4`Retained binary64: `-0.25`	-1/4Exact fraction: `-1/4`Retained binary64: `-0.25`	1/2Exact fraction: `1/2`Retained binary64: `0.5`	-3/2Exact fraction: `-3/2`Retained binary64: `-1.5`	-1/2Exact fraction: `-1/2`Retained binary64: `-0.5`	1.58113883008Retained binary64: `1.5811388300841898`	false	[-0.9486832980505138,-0.31622776601683794]
22	-1/4Exact fraction: `-1/4`Retained binary64: `-0.25`	0Exact fraction: `0`Retained binary64: `0.0`	9/16Exact fraction: `9/16`Retained binary64: `0.5625`	-3/2Exact fraction: `-3/2`Retained binary64: `-1.5`	0Exact fraction: `0`Retained binary64: `0.0`	1.5Retained binary64: `1.5`	false	[-1.0,0.0]
23	-1/4Exact fraction: `-1/4`Retained binary64: `-0.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/2Exact fraction: `1/2`Retained binary64: `0.5`	-3/2Exact fraction: `-3/2`Retained binary64: `-1.5`	1/2Exact fraction: `1/2`Retained binary64: `0.5`	1.58113883008Retained binary64: `1.5811388300841898`	false	[-0.9486832980505138,0.31622776601683794]
24	-1/4Exact fraction: `-1/4`Retained binary64: `-0.25`	3/4Exact fraction: `3/4`Retained binary64: `0.75`	0Exact fraction: `0`Retained binary64: `0.0`	-3/2Exact fraction: `-3/2`Retained binary64: `-1.5`	3/2Exact fraction: `3/2`Retained binary64: `1.5`	2.12132034356Retained binary64: `2.1213203435596424`	false	[-0.7071067811865476,0.7071067811865476]

Complete contour half-branch: level 0, branch-1 (129 rows)

Separate arrays are separate plotted paths. Quarter-level halves meet only at their own turning point; never bridge the forbidden central x interval.

Complete contour half-branch: level 0, branch-1
Row	X=x/L₀	Y=y/L₀
0	-2Exact fraction: `-2`Retained binary64: `-2.0`	1Exact fraction: `1`Retained binary64: `1.0`
1	-127/64Exact fraction: `-127/64`Retained binary64: `-1.984375`	63/64Exact fraction: `63/64`Retained binary64: `0.984375`
2	-63/32Exact fraction: `-63/32`Retained binary64: `-1.96875`	31/32Exact fraction: `31/32`Retained binary64: `0.96875`
3	-125/64Exact fraction: `-125/64`Retained binary64: `-1.953125`	61/64Exact fraction: `61/64`Retained binary64: `0.953125`
4	-31/16Exact fraction: `-31/16`Retained binary64: `-1.9375`	15/16Exact fraction: `15/16`Retained binary64: `0.9375`
5	-123/64Exact fraction: `-123/64`Retained binary64: `-1.921875`	59/64Exact fraction: `59/64`Retained binary64: `0.921875`
6	-61/32Exact fraction: `-61/32`Retained binary64: `-1.90625`	29/32Exact fraction: `29/32`Retained binary64: `0.90625`
7	-121/64Exact fraction: `-121/64`Retained binary64: `-1.890625`	57/64Exact fraction: `57/64`Retained binary64: `0.890625`
8	-15/8Exact fraction: `-15/8`Retained binary64: `-1.875`	7/8Exact fraction: `7/8`Retained binary64: `0.875`
9	-119/64Exact fraction: `-119/64`Retained binary64: `-1.859375`	55/64Exact fraction: `55/64`Retained binary64: `0.859375`
10	-59/32Exact fraction: `-59/32`Retained binary64: `-1.84375`	27/32Exact fraction: `27/32`Retained binary64: `0.84375`
11	-117/64Exact fraction: `-117/64`Retained binary64: `-1.828125`	53/64Exact fraction: `53/64`Retained binary64: `0.828125`
12	-29/16Exact fraction: `-29/16`Retained binary64: `-1.8125`	13/16Exact fraction: `13/16`Retained binary64: `0.8125`
13	-115/64Exact fraction: `-115/64`Retained binary64: `-1.796875`	51/64Exact fraction: `51/64`Retained binary64: `0.796875`
14	-57/32Exact fraction: `-57/32`Retained binary64: `-1.78125`	25/32Exact fraction: `25/32`Retained binary64: `0.78125`
15	-113/64Exact fraction: `-113/64`Retained binary64: `-1.765625`	49/64Exact fraction: `49/64`Retained binary64: `0.765625`
16	-7/4Exact fraction: `-7/4`Retained binary64: `-1.75`	3/4Exact fraction: `3/4`Retained binary64: `0.75`
17	-111/64Exact fraction: `-111/64`Retained binary64: `-1.734375`	47/64Exact fraction: `47/64`Retained binary64: `0.734375`
18	-55/32Exact fraction: `-55/32`Retained binary64: `-1.71875`	23/32Exact fraction: `23/32`Retained binary64: `0.71875`
19	-109/64Exact fraction: `-109/64`Retained binary64: `-1.703125`	45/64Exact fraction: `45/64`Retained binary64: `0.703125`
20	-27/16Exact fraction: `-27/16`Retained binary64: `-1.6875`	11/16Exact fraction: `11/16`Retained binary64: `0.6875`
21	-107/64Exact fraction: `-107/64`Retained binary64: `-1.671875`	43/64Exact fraction: `43/64`Retained binary64: `0.671875`
22	-53/32Exact fraction: `-53/32`Retained binary64: `-1.65625`	21/32Exact fraction: `21/32`Retained binary64: `0.65625`
23	-105/64Exact fraction: `-105/64`Retained binary64: `-1.640625`	41/64Exact fraction: `41/64`Retained binary64: `0.640625`
24	-13/8Exact fraction: `-13/8`Retained binary64: `-1.625`	5/8Exact fraction: `5/8`Retained binary64: `0.625`
25	-103/64Exact fraction: `-103/64`Retained binary64: `-1.609375`	39/64Exact fraction: `39/64`Retained binary64: `0.609375`
26	-51/32Exact fraction: `-51/32`Retained binary64: `-1.59375`	19/32Exact fraction: `19/32`Retained binary64: `0.59375`
27	-101/64Exact fraction: `-101/64`Retained binary64: `-1.578125`	37/64Exact fraction: `37/64`Retained binary64: `0.578125`
28	-25/16Exact fraction: `-25/16`Retained binary64: `-1.5625`	9/16Exact fraction: `9/16`Retained binary64: `0.5625`
29	-99/64Exact fraction: `-99/64`Retained binary64: `-1.546875`	35/64Exact fraction: `35/64`Retained binary64: `0.546875`
30	-49/32Exact fraction: `-49/32`Retained binary64: `-1.53125`	17/32Exact fraction: `17/32`Retained binary64: `0.53125`
31	-97/64Exact fraction: `-97/64`Retained binary64: `-1.515625`	33/64Exact fraction: `33/64`Retained binary64: `0.515625`
32	-3/2Exact fraction: `-3/2`Retained binary64: `-1.5`	1/2Exact fraction: `1/2`Retained binary64: `0.5`
33	-95/64Exact fraction: `-95/64`Retained binary64: `-1.484375`	31/64Exact fraction: `31/64`Retained binary64: `0.484375`
34	-47/32Exact fraction: `-47/32`Retained binary64: `-1.46875`	15/32Exact fraction: `15/32`Retained binary64: `0.46875`
35	-93/64Exact fraction: `-93/64`Retained binary64: `-1.453125`	29/64Exact fraction: `29/64`Retained binary64: `0.453125`
36	-23/16Exact fraction: `-23/16`Retained binary64: `-1.4375`	7/16Exact fraction: `7/16`Retained binary64: `0.4375`
37	-91/64Exact fraction: `-91/64`Retained binary64: `-1.421875`	27/64Exact fraction: `27/64`Retained binary64: `0.421875`
38	-45/32Exact fraction: `-45/32`Retained binary64: `-1.40625`	13/32Exact fraction: `13/32`Retained binary64: `0.40625`
39	-89/64Exact fraction: `-89/64`Retained binary64: `-1.390625`	25/64Exact fraction: `25/64`Retained binary64: `0.390625`
40	-11/8Exact fraction: `-11/8`Retained binary64: `-1.375`	3/8Exact fraction: `3/8`Retained binary64: `0.375`
41	-87/64Exact fraction: `-87/64`Retained binary64: `-1.359375`	23/64Exact fraction: `23/64`Retained binary64: `0.359375`
42	-43/32Exact fraction: `-43/32`Retained binary64: `-1.34375`	11/32Exact fraction: `11/32`Retained binary64: `0.34375`
43	-85/64Exact fraction: `-85/64`Retained binary64: `-1.328125`	21/64Exact fraction: `21/64`Retained binary64: `0.328125`
44	-21/16Exact fraction: `-21/16`Retained binary64: `-1.3125`	5/16Exact fraction: `5/16`Retained binary64: `0.3125`
45	-83/64Exact fraction: `-83/64`Retained binary64: `-1.296875`	19/64Exact fraction: `19/64`Retained binary64: `0.296875`
46	-41/32Exact fraction: `-41/32`Retained binary64: `-1.28125`	9/32Exact fraction: `9/32`Retained binary64: `0.28125`
47	-81/64Exact fraction: `-81/64`Retained binary64: `-1.265625`	17/64Exact fraction: `17/64`Retained binary64: `0.265625`
48	-5/4Exact fraction: `-5/4`Retained binary64: `-1.25`	1/4Exact fraction: `1/4`Retained binary64: `0.25`
49	-79/64Exact fraction: `-79/64`Retained binary64: `-1.234375`	15/64Exact fraction: `15/64`Retained binary64: `0.234375`
50	-39/32Exact fraction: `-39/32`Retained binary64: `-1.21875`	7/32Exact fraction: `7/32`Retained binary64: `0.21875`
51	-77/64Exact fraction: `-77/64`Retained binary64: `-1.203125`	13/64Exact fraction: `13/64`Retained binary64: `0.203125`
52	-19/16Exact fraction: `-19/16`Retained binary64: `-1.1875`	3/16Exact fraction: `3/16`Retained binary64: `0.1875`
53	-75/64Exact fraction: `-75/64`Retained binary64: `-1.171875`	11/64Exact fraction: `11/64`Retained binary64: `0.171875`
54	-37/32Exact fraction: `-37/32`Retained binary64: `-1.15625`	5/32Exact fraction: `5/32`Retained binary64: `0.15625`
55	-73/64Exact fraction: `-73/64`Retained binary64: `-1.140625`	9/64Exact fraction: `9/64`Retained binary64: `0.140625`
56	-9/8Exact fraction: `-9/8`Retained binary64: `-1.125`	1/8Exact fraction: `1/8`Retained binary64: `0.125`
57	-71/64Exact fraction: `-71/64`Retained binary64: `-1.109375`	7/64Exact fraction: `7/64`Retained binary64: `0.109375`
58	-35/32Exact fraction: `-35/32`Retained binary64: `-1.09375`	3/32Exact fraction: `3/32`Retained binary64: `0.09375`
59	-69/64Exact fraction: `-69/64`Retained binary64: `-1.078125`	5/64Exact fraction: `5/64`Retained binary64: `0.078125`
60	-17/16Exact fraction: `-17/16`Retained binary64: `-1.0625`	1/16Exact fraction: `1/16`Retained binary64: `0.0625`
61	-67/64Exact fraction: `-67/64`Retained binary64: `-1.046875`	3/64Exact fraction: `3/64`Retained binary64: `0.046875`
62	-33/32Exact fraction: `-33/32`Retained binary64: `-1.03125`	1/32Exact fraction: `1/32`Retained binary64: `0.03125`
63	-65/64Exact fraction: `-65/64`Retained binary64: `-1.015625`	1/64Exact fraction: `1/64`Retained binary64: `0.015625`
64	-1Exact fraction: `-1`Retained binary64: `-1.0`	0Exact fraction: `0`Retained binary64: `0.0`
65	-63/64Exact fraction: `-63/64`Retained binary64: `-0.984375`	-1/64Exact fraction: `-1/64`Retained binary64: `-0.015625`
66	-31/32Exact fraction: `-31/32`Retained binary64: `-0.96875`	-1/32Exact fraction: `-1/32`Retained binary64: `-0.03125`
67	-61/64Exact fraction: `-61/64`Retained binary64: `-0.953125`	-3/64Exact fraction: `-3/64`Retained binary64: `-0.046875`
68	-15/16Exact fraction: `-15/16`Retained binary64: `-0.9375`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`
69	-59/64Exact fraction: `-59/64`Retained binary64: `-0.921875`	-5/64Exact fraction: `-5/64`Retained binary64: `-0.078125`
70	-29/32Exact fraction: `-29/32`Retained binary64: `-0.90625`	-3/32Exact fraction: `-3/32`Retained binary64: `-0.09375`
71	-57/64Exact fraction: `-57/64`Retained binary64: `-0.890625`	-7/64Exact fraction: `-7/64`Retained binary64: `-0.109375`
72	-7/8Exact fraction: `-7/8`Retained binary64: `-0.875`	-1/8Exact fraction: `-1/8`Retained binary64: `-0.125`
73	-55/64Exact fraction: `-55/64`Retained binary64: `-0.859375`	-9/64Exact fraction: `-9/64`Retained binary64: `-0.140625`
74	-27/32Exact fraction: `-27/32`Retained binary64: `-0.84375`	-5/32Exact fraction: `-5/32`Retained binary64: `-0.15625`
75	-53/64Exact fraction: `-53/64`Retained binary64: `-0.828125`	-11/64Exact fraction: `-11/64`Retained binary64: `-0.171875`
76	-13/16Exact fraction: `-13/16`Retained binary64: `-0.8125`	-3/16Exact fraction: `-3/16`Retained binary64: `-0.1875`
77	-51/64Exact fraction: `-51/64`Retained binary64: `-0.796875`	-13/64Exact fraction: `-13/64`Retained binary64: `-0.203125`
78	-25/32Exact fraction: `-25/32`Retained binary64: `-0.78125`	-7/32Exact fraction: `-7/32`Retained binary64: `-0.21875`
79	-49/64Exact fraction: `-49/64`Retained binary64: `-0.765625`	-15/64Exact fraction: `-15/64`Retained binary64: `-0.234375`
80	-3/4Exact fraction: `-3/4`Retained binary64: `-0.75`	-1/4Exact fraction: `-1/4`Retained binary64: `-0.25`
81	-47/64Exact fraction: `-47/64`Retained binary64: `-0.734375`	-17/64Exact fraction: `-17/64`Retained binary64: `-0.265625`
82	-23/32Exact fraction: `-23/32`Retained binary64: `-0.71875`	-9/32Exact fraction: `-9/32`Retained binary64: `-0.28125`
83	-45/64Exact fraction: `-45/64`Retained binary64: `-0.703125`	-19/64Exact fraction: `-19/64`Retained binary64: `-0.296875`
84	-11/16Exact fraction: `-11/16`Retained binary64: `-0.6875`	-5/16Exact fraction: `-5/16`Retained binary64: `-0.3125`
85	-43/64Exact fraction: `-43/64`Retained binary64: `-0.671875`	-21/64Exact fraction: `-21/64`Retained binary64: `-0.328125`
86	-21/32Exact fraction: `-21/32`Retained binary64: `-0.65625`	-11/32Exact fraction: `-11/32`Retained binary64: `-0.34375`
87	-41/64Exact fraction: `-41/64`Retained binary64: `-0.640625`	-23/64Exact fraction: `-23/64`Retained binary64: `-0.359375`
88	-5/8Exact fraction: `-5/8`Retained binary64: `-0.625`	-3/8Exact fraction: `-3/8`Retained binary64: `-0.375`
89	-39/64Exact fraction: `-39/64`Retained binary64: `-0.609375`	-25/64Exact fraction: `-25/64`Retained binary64: `-0.390625`
90	-19/32Exact fraction: `-19/32`Retained binary64: `-0.59375`	-13/32Exact fraction: `-13/32`Retained binary64: `-0.40625`
91	-37/64Exact fraction: `-37/64`Retained binary64: `-0.578125`	-27/64Exact fraction: `-27/64`Retained binary64: `-0.421875`
92	-9/16Exact fraction: `-9/16`Retained binary64: `-0.5625`	-7/16Exact fraction: `-7/16`Retained binary64: `-0.4375`
93	-35/64Exact fraction: `-35/64`Retained binary64: `-0.546875`	-29/64Exact fraction: `-29/64`Retained binary64: `-0.453125`
94	-17/32Exact fraction: `-17/32`Retained binary64: `-0.53125`	-15/32Exact fraction: `-15/32`Retained binary64: `-0.46875`
95	-33/64Exact fraction: `-33/64`Retained binary64: `-0.515625`	-31/64Exact fraction: `-31/64`Retained binary64: `-0.484375`
96	-1/2Exact fraction: `-1/2`Retained binary64: `-0.5`	-1/2Exact fraction: `-1/2`Retained binary64: `-0.5`
97	-31/64Exact fraction: `-31/64`Retained binary64: `-0.484375`	-33/64Exact fraction: `-33/64`Retained binary64: `-0.515625`
98	-15/32Exact fraction: `-15/32`Retained binary64: `-0.46875`	-17/32Exact fraction: `-17/32`Retained binary64: `-0.53125`
99	-29/64Exact fraction: `-29/64`Retained binary64: `-0.453125`	-35/64Exact fraction: `-35/64`Retained binary64: `-0.546875`
100	-7/16Exact fraction: `-7/16`Retained binary64: `-0.4375`	-9/16Exact fraction: `-9/16`Retained binary64: `-0.5625`
101	-27/64Exact fraction: `-27/64`Retained binary64: `-0.421875`	-37/64Exact fraction: `-37/64`Retained binary64: `-0.578125`
102	-13/32Exact fraction: `-13/32`Retained binary64: `-0.40625`	-19/32Exact fraction: `-19/32`Retained binary64: `-0.59375`
103	-25/64Exact fraction: `-25/64`Retained binary64: `-0.390625`	-39/64Exact fraction: `-39/64`Retained binary64: `-0.609375`
104	-3/8Exact fraction: `-3/8`Retained binary64: `-0.375`	-5/8Exact fraction: `-5/8`Retained binary64: `-0.625`
105	-23/64Exact fraction: `-23/64`Retained binary64: `-0.359375`	-41/64Exact fraction: `-41/64`Retained binary64: `-0.640625`
106	-11/32Exact fraction: `-11/32`Retained binary64: `-0.34375`	-21/32Exact fraction: `-21/32`Retained binary64: `-0.65625`
107	-21/64Exact fraction: `-21/64`Retained binary64: `-0.328125`	-43/64Exact fraction: `-43/64`Retained binary64: `-0.671875`
108	-5/16Exact fraction: `-5/16`Retained binary64: `-0.3125`	-11/16Exact fraction: `-11/16`Retained binary64: `-0.6875`
109	-19/64Exact fraction: `-19/64`Retained binary64: `-0.296875`	-45/64Exact fraction: `-45/64`Retained binary64: `-0.703125`
110	-9/32Exact fraction: `-9/32`Retained binary64: `-0.28125`	-23/32Exact fraction: `-23/32`Retained binary64: `-0.71875`
111	-17/64Exact fraction: `-17/64`Retained binary64: `-0.265625`	-47/64Exact fraction: `-47/64`Retained binary64: `-0.734375`
112	-1/4Exact fraction: `-1/4`Retained binary64: `-0.25`	-3/4Exact fraction: `-3/4`Retained binary64: `-0.75`
113	-15/64Exact fraction: `-15/64`Retained binary64: `-0.234375`	-49/64Exact fraction: `-49/64`Retained binary64: `-0.765625`
114	-7/32Exact fraction: `-7/32`Retained binary64: `-0.21875`	-25/32Exact fraction: `-25/32`Retained binary64: `-0.78125`
115	-13/64Exact fraction: `-13/64`Retained binary64: `-0.203125`	-51/64Exact fraction: `-51/64`Retained binary64: `-0.796875`
116	-3/16Exact fraction: `-3/16`Retained binary64: `-0.1875`	-13/16Exact fraction: `-13/16`Retained binary64: `-0.8125`
117	-11/64Exact fraction: `-11/64`Retained binary64: `-0.171875`	-53/64Exact fraction: `-53/64`Retained binary64: `-0.828125`
118	-5/32Exact fraction: `-5/32`Retained binary64: `-0.15625`	-27/32Exact fraction: `-27/32`Retained binary64: `-0.84375`
119	-9/64Exact fraction: `-9/64`Retained binary64: `-0.140625`	-55/64Exact fraction: `-55/64`Retained binary64: `-0.859375`
120	-1/8Exact fraction: `-1/8`Retained binary64: `-0.125`	-7/8Exact fraction: `-7/8`Retained binary64: `-0.875`
121	-7/64Exact fraction: `-7/64`Retained binary64: `-0.109375`	-57/64Exact fraction: `-57/64`Retained binary64: `-0.890625`
122	-3/32Exact fraction: `-3/32`Retained binary64: `-0.09375`	-29/32Exact fraction: `-29/32`Retained binary64: `-0.90625`
123	-5/64Exact fraction: `-5/64`Retained binary64: `-0.078125`	-59/64Exact fraction: `-59/64`Retained binary64: `-0.921875`
124	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-15/16Exact fraction: `-15/16`Retained binary64: `-0.9375`
125	-3/64Exact fraction: `-3/64`Retained binary64: `-0.046875`	-61/64Exact fraction: `-61/64`Retained binary64: `-0.953125`
126	-1/32Exact fraction: `-1/32`Retained binary64: `-0.03125`	-31/32Exact fraction: `-31/32`Retained binary64: `-0.96875`
127	-1/64Exact fraction: `-1/64`Retained binary64: `-0.015625`	-63/64Exact fraction: `-63/64`Retained binary64: `-0.984375`
128	0Exact fraction: `0`Retained binary64: `0.0`	-1Exact fraction: `-1`Retained binary64: `-1.0`

Complete contour half-branch: level 0, branch1 (129 rows)

Separate arrays are separate plotted paths. Quarter-level halves meet only at their own turning point; never bridge the forbidden central x interval.

Complete contour half-branch: level 0, branch1
Row	X=x/L₀	Y=y/L₀
0	-2Exact fraction: `-2`Retained binary64: `-2.0`	-1Exact fraction: `-1`Retained binary64: `-1.0`
1	-127/64Exact fraction: `-127/64`Retained binary64: `-1.984375`	-63/64Exact fraction: `-63/64`Retained binary64: `-0.984375`
2	-63/32Exact fraction: `-63/32`Retained binary64: `-1.96875`	-31/32Exact fraction: `-31/32`Retained binary64: `-0.96875`
3	-125/64Exact fraction: `-125/64`Retained binary64: `-1.953125`	-61/64Exact fraction: `-61/64`Retained binary64: `-0.953125`
4	-31/16Exact fraction: `-31/16`Retained binary64: `-1.9375`	-15/16Exact fraction: `-15/16`Retained binary64: `-0.9375`
5	-123/64Exact fraction: `-123/64`Retained binary64: `-1.921875`	-59/64Exact fraction: `-59/64`Retained binary64: `-0.921875`
6	-61/32Exact fraction: `-61/32`Retained binary64: `-1.90625`	-29/32Exact fraction: `-29/32`Retained binary64: `-0.90625`
7	-121/64Exact fraction: `-121/64`Retained binary64: `-1.890625`	-57/64Exact fraction: `-57/64`Retained binary64: `-0.890625`
8	-15/8Exact fraction: `-15/8`Retained binary64: `-1.875`	-7/8Exact fraction: `-7/8`Retained binary64: `-0.875`
9	-119/64Exact fraction: `-119/64`Retained binary64: `-1.859375`	-55/64Exact fraction: `-55/64`Retained binary64: `-0.859375`
10	-59/32Exact fraction: `-59/32`Retained binary64: `-1.84375`	-27/32Exact fraction: `-27/32`Retained binary64: `-0.84375`
11	-117/64Exact fraction: `-117/64`Retained binary64: `-1.828125`	-53/64Exact fraction: `-53/64`Retained binary64: `-0.828125`
12	-29/16Exact fraction: `-29/16`Retained binary64: `-1.8125`	-13/16Exact fraction: `-13/16`Retained binary64: `-0.8125`
13	-115/64Exact fraction: `-115/64`Retained binary64: `-1.796875`	-51/64Exact fraction: `-51/64`Retained binary64: `-0.796875`
14	-57/32Exact fraction: `-57/32`Retained binary64: `-1.78125`	-25/32Exact fraction: `-25/32`Retained binary64: `-0.78125`
15	-113/64Exact fraction: `-113/64`Retained binary64: `-1.765625`	-49/64Exact fraction: `-49/64`Retained binary64: `-0.765625`
16	-7/4Exact fraction: `-7/4`Retained binary64: `-1.75`	-3/4Exact fraction: `-3/4`Retained binary64: `-0.75`
17	-111/64Exact fraction: `-111/64`Retained binary64: `-1.734375`	-47/64Exact fraction: `-47/64`Retained binary64: `-0.734375`
18	-55/32Exact fraction: `-55/32`Retained binary64: `-1.71875`	-23/32Exact fraction: `-23/32`Retained binary64: `-0.71875`
19	-109/64Exact fraction: `-109/64`Retained binary64: `-1.703125`	-45/64Exact fraction: `-45/64`Retained binary64: `-0.703125`
20	-27/16Exact fraction: `-27/16`Retained binary64: `-1.6875`	-11/16Exact fraction: `-11/16`Retained binary64: `-0.6875`
21	-107/64Exact fraction: `-107/64`Retained binary64: `-1.671875`	-43/64Exact fraction: `-43/64`Retained binary64: `-0.671875`
22	-53/32Exact fraction: `-53/32`Retained binary64: `-1.65625`	-21/32Exact fraction: `-21/32`Retained binary64: `-0.65625`
23	-105/64Exact fraction: `-105/64`Retained binary64: `-1.640625`	-41/64Exact fraction: `-41/64`Retained binary64: `-0.640625`
24	-13/8Exact fraction: `-13/8`Retained binary64: `-1.625`	-5/8Exact fraction: `-5/8`Retained binary64: `-0.625`
25	-103/64Exact fraction: `-103/64`Retained binary64: `-1.609375`	-39/64Exact fraction: `-39/64`Retained binary64: `-0.609375`
26	-51/32Exact fraction: `-51/32`Retained binary64: `-1.59375`	-19/32Exact fraction: `-19/32`Retained binary64: `-0.59375`
27	-101/64Exact fraction: `-101/64`Retained binary64: `-1.578125`	-37/64Exact fraction: `-37/64`Retained binary64: `-0.578125`
28	-25/16Exact fraction: `-25/16`Retained binary64: `-1.5625`	-9/16Exact fraction: `-9/16`Retained binary64: `-0.5625`
29	-99/64Exact fraction: `-99/64`Retained binary64: `-1.546875`	-35/64Exact fraction: `-35/64`Retained binary64: `-0.546875`
30	-49/32Exact fraction: `-49/32`Retained binary64: `-1.53125`	-17/32Exact fraction: `-17/32`Retained binary64: `-0.53125`
31	-97/64Exact fraction: `-97/64`Retained binary64: `-1.515625`	-33/64Exact fraction: `-33/64`Retained binary64: `-0.515625`
32	-3/2Exact fraction: `-3/2`Retained binary64: `-1.5`	-1/2Exact fraction: `-1/2`Retained binary64: `-0.5`
33	-95/64Exact fraction: `-95/64`Retained binary64: `-1.484375`	-31/64Exact fraction: `-31/64`Retained binary64: `-0.484375`
34	-47/32Exact fraction: `-47/32`Retained binary64: `-1.46875`	-15/32Exact fraction: `-15/32`Retained binary64: `-0.46875`
35	-93/64Exact fraction: `-93/64`Retained binary64: `-1.453125`	-29/64Exact fraction: `-29/64`Retained binary64: `-0.453125`
36	-23/16Exact fraction: `-23/16`Retained binary64: `-1.4375`	-7/16Exact fraction: `-7/16`Retained binary64: `-0.4375`
37	-91/64Exact fraction: `-91/64`Retained binary64: `-1.421875`	-27/64Exact fraction: `-27/64`Retained binary64: `-0.421875`
38	-45/32Exact fraction: `-45/32`Retained binary64: `-1.40625`	-13/32Exact fraction: `-13/32`Retained binary64: `-0.40625`
39	-89/64Exact fraction: `-89/64`Retained binary64: `-1.390625`	-25/64Exact fraction: `-25/64`Retained binary64: `-0.390625`
40	-11/8Exact fraction: `-11/8`Retained binary64: `-1.375`	-3/8Exact fraction: `-3/8`Retained binary64: `-0.375`
41	-87/64Exact fraction: `-87/64`Retained binary64: `-1.359375`	-23/64Exact fraction: `-23/64`Retained binary64: `-0.359375`
42	-43/32Exact fraction: `-43/32`Retained binary64: `-1.34375`	-11/32Exact fraction: `-11/32`Retained binary64: `-0.34375`
43	-85/64Exact fraction: `-85/64`Retained binary64: `-1.328125`	-21/64Exact fraction: `-21/64`Retained binary64: `-0.328125`
44	-21/16Exact fraction: `-21/16`Retained binary64: `-1.3125`	-5/16Exact fraction: `-5/16`Retained binary64: `-0.3125`
45	-83/64Exact fraction: `-83/64`Retained binary64: `-1.296875`	-19/64Exact fraction: `-19/64`Retained binary64: `-0.296875`
46	-41/32Exact fraction: `-41/32`Retained binary64: `-1.28125`	-9/32Exact fraction: `-9/32`Retained binary64: `-0.28125`
47	-81/64Exact fraction: `-81/64`Retained binary64: `-1.265625`	-17/64Exact fraction: `-17/64`Retained binary64: `-0.265625`
48	-5/4Exact fraction: `-5/4`Retained binary64: `-1.25`	-1/4Exact fraction: `-1/4`Retained binary64: `-0.25`
49	-79/64Exact fraction: `-79/64`Retained binary64: `-1.234375`	-15/64Exact fraction: `-15/64`Retained binary64: `-0.234375`
50	-39/32Exact fraction: `-39/32`Retained binary64: `-1.21875`	-7/32Exact fraction: `-7/32`Retained binary64: `-0.21875`
51	-77/64Exact fraction: `-77/64`Retained binary64: `-1.203125`	-13/64Exact fraction: `-13/64`Retained binary64: `-0.203125`
52	-19/16Exact fraction: `-19/16`Retained binary64: `-1.1875`	-3/16Exact fraction: `-3/16`Retained binary64: `-0.1875`
53	-75/64Exact fraction: `-75/64`Retained binary64: `-1.171875`	-11/64Exact fraction: `-11/64`Retained binary64: `-0.171875`
54	-37/32Exact fraction: `-37/32`Retained binary64: `-1.15625`	-5/32Exact fraction: `-5/32`Retained binary64: `-0.15625`
55	-73/64Exact fraction: `-73/64`Retained binary64: `-1.140625`	-9/64Exact fraction: `-9/64`Retained binary64: `-0.140625`
56	-9/8Exact fraction: `-9/8`Retained binary64: `-1.125`	-1/8Exact fraction: `-1/8`Retained binary64: `-0.125`
57	-71/64Exact fraction: `-71/64`Retained binary64: `-1.109375`	-7/64Exact fraction: `-7/64`Retained binary64: `-0.109375`
58	-35/32Exact fraction: `-35/32`Retained binary64: `-1.09375`	-3/32Exact fraction: `-3/32`Retained binary64: `-0.09375`
59	-69/64Exact fraction: `-69/64`Retained binary64: `-1.078125`	-5/64Exact fraction: `-5/64`Retained binary64: `-0.078125`
60	-17/16Exact fraction: `-17/16`Retained binary64: `-1.0625`	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`
61	-67/64Exact fraction: `-67/64`Retained binary64: `-1.046875`	-3/64Exact fraction: `-3/64`Retained binary64: `-0.046875`
62	-33/32Exact fraction: `-33/32`Retained binary64: `-1.03125`	-1/32Exact fraction: `-1/32`Retained binary64: `-0.03125`
63	-65/64Exact fraction: `-65/64`Retained binary64: `-1.015625`	-1/64Exact fraction: `-1/64`Retained binary64: `-0.015625`
64	-1Exact fraction: `-1`Retained binary64: `-1.0`	0Exact fraction: `0`Retained binary64: `0.0`
65	-63/64Exact fraction: `-63/64`Retained binary64: `-0.984375`	1/64Exact fraction: `1/64`Retained binary64: `0.015625`
66	-31/32Exact fraction: `-31/32`Retained binary64: `-0.96875`	1/32Exact fraction: `1/32`Retained binary64: `0.03125`
67	-61/64Exact fraction: `-61/64`Retained binary64: `-0.953125`	3/64Exact fraction: `3/64`Retained binary64: `0.046875`
68	-15/16Exact fraction: `-15/16`Retained binary64: `-0.9375`	1/16Exact fraction: `1/16`Retained binary64: `0.0625`
69	-59/64Exact fraction: `-59/64`Retained binary64: `-0.921875`	5/64Exact fraction: `5/64`Retained binary64: `0.078125`
70	-29/32Exact fraction: `-29/32`Retained binary64: `-0.90625`	3/32Exact fraction: `3/32`Retained binary64: `0.09375`
71	-57/64Exact fraction: `-57/64`Retained binary64: `-0.890625`	7/64Exact fraction: `7/64`Retained binary64: `0.109375`
72	-7/8Exact fraction: `-7/8`Retained binary64: `-0.875`	1/8Exact fraction: `1/8`Retained binary64: `0.125`
73	-55/64Exact fraction: `-55/64`Retained binary64: `-0.859375`	9/64Exact fraction: `9/64`Retained binary64: `0.140625`
74	-27/32Exact fraction: `-27/32`Retained binary64: `-0.84375`	5/32Exact fraction: `5/32`Retained binary64: `0.15625`
75	-53/64Exact fraction: `-53/64`Retained binary64: `-0.828125`	11/64Exact fraction: `11/64`Retained binary64: `0.171875`
76	-13/16Exact fraction: `-13/16`Retained binary64: `-0.8125`	3/16Exact fraction: `3/16`Retained binary64: `0.1875`
77	-51/64Exact fraction: `-51/64`Retained binary64: `-0.796875`	13/64Exact fraction: `13/64`Retained binary64: `0.203125`
78	-25/32Exact fraction: `-25/32`Retained binary64: `-0.78125`	7/32Exact fraction: `7/32`Retained binary64: `0.21875`
79	-49/64Exact fraction: `-49/64`Retained binary64: `-0.765625`	15/64Exact fraction: `15/64`Retained binary64: `0.234375`
80	-3/4Exact fraction: `-3/4`Retained binary64: `-0.75`	1/4Exact fraction: `1/4`Retained binary64: `0.25`
81	-47/64Exact fraction: `-47/64`Retained binary64: `-0.734375`	17/64Exact fraction: `17/64`Retained binary64: `0.265625`
82	-23/32Exact fraction: `-23/32`Retained binary64: `-0.71875`	9/32Exact fraction: `9/32`Retained binary64: `0.28125`
83	-45/64Exact fraction: `-45/64`Retained binary64: `-0.703125`	19/64Exact fraction: `19/64`Retained binary64: `0.296875`
84	-11/16Exact fraction: `-11/16`Retained binary64: `-0.6875`	5/16Exact fraction: `5/16`Retained binary64: `0.3125`
85	-43/64Exact fraction: `-43/64`Retained binary64: `-0.671875`	21/64Exact fraction: `21/64`Retained binary64: `0.328125`
86	-21/32Exact fraction: `-21/32`Retained binary64: `-0.65625`	11/32Exact fraction: `11/32`Retained binary64: `0.34375`
87	-41/64Exact fraction: `-41/64`Retained binary64: `-0.640625`	23/64Exact fraction: `23/64`Retained binary64: `0.359375`
88	-5/8Exact fraction: `-5/8`Retained binary64: `-0.625`	3/8Exact fraction: `3/8`Retained binary64: `0.375`
89	-39/64Exact fraction: `-39/64`Retained binary64: `-0.609375`	25/64Exact fraction: `25/64`Retained binary64: `0.390625`
90	-19/32Exact fraction: `-19/32`Retained binary64: `-0.59375`	13/32Exact fraction: `13/32`Retained binary64: `0.40625`
91	-37/64Exact fraction: `-37/64`Retained binary64: `-0.578125`	27/64Exact fraction: `27/64`Retained binary64: `0.421875`
92	-9/16Exact fraction: `-9/16`Retained binary64: `-0.5625`	7/16Exact fraction: `7/16`Retained binary64: `0.4375`
93	-35/64Exact fraction: `-35/64`Retained binary64: `-0.546875`	29/64Exact fraction: `29/64`Retained binary64: `0.453125`
94	-17/32Exact fraction: `-17/32`Retained binary64: `-0.53125`	15/32Exact fraction: `15/32`Retained binary64: `0.46875`
95	-33/64Exact fraction: `-33/64`Retained binary64: `-0.515625`	31/64Exact fraction: `31/64`Retained binary64: `0.484375`
96	-1/2Exact fraction: `-1/2`Retained binary64: `-0.5`	1/2Exact fraction: `1/2`Retained binary64: `0.5`
97	-31/64Exact fraction: `-31/64`Retained binary64: `-0.484375`	33/64Exact fraction: `33/64`Retained binary64: `0.515625`
98	-15/32Exact fraction: `-15/32`Retained binary64: `-0.46875`	17/32Exact fraction: `17/32`Retained binary64: `0.53125`
99	-29/64Exact fraction: `-29/64`Retained binary64: `-0.453125`	35/64Exact fraction: `35/64`Retained binary64: `0.546875`
100	-7/16Exact fraction: `-7/16`Retained binary64: `-0.4375`	9/16Exact fraction: `9/16`Retained binary64: `0.5625`
101	-27/64Exact fraction: `-27/64`Retained binary64: `-0.421875`	37/64Exact fraction: `37/64`Retained binary64: `0.578125`
102	-13/32Exact fraction: `-13/32`Retained binary64: `-0.40625`	19/32Exact fraction: `19/32`Retained binary64: `0.59375`
103	-25/64Exact fraction: `-25/64`Retained binary64: `-0.390625`	39/64Exact fraction: `39/64`Retained binary64: `0.609375`
104	-3/8Exact fraction: `-3/8`Retained binary64: `-0.375`	5/8Exact fraction: `5/8`Retained binary64: `0.625`
105	-23/64Exact fraction: `-23/64`Retained binary64: `-0.359375`	41/64Exact fraction: `41/64`Retained binary64: `0.640625`
106	-11/32Exact fraction: `-11/32`Retained binary64: `-0.34375`	21/32Exact fraction: `21/32`Retained binary64: `0.65625`
107	-21/64Exact fraction: `-21/64`Retained binary64: `-0.328125`	43/64Exact fraction: `43/64`Retained binary64: `0.671875`
108	-5/16Exact fraction: `-5/16`Retained binary64: `-0.3125`	11/16Exact fraction: `11/16`Retained binary64: `0.6875`
109	-19/64Exact fraction: `-19/64`Retained binary64: `-0.296875`	45/64Exact fraction: `45/64`Retained binary64: `0.703125`
110	-9/32Exact fraction: `-9/32`Retained binary64: `-0.28125`	23/32Exact fraction: `23/32`Retained binary64: `0.71875`
111	-17/64Exact fraction: `-17/64`Retained binary64: `-0.265625`	47/64Exact fraction: `47/64`Retained binary64: `0.734375`
112	-1/4Exact fraction: `-1/4`Retained binary64: `-0.25`	3/4Exact fraction: `3/4`Retained binary64: `0.75`
113	-15/64Exact fraction: `-15/64`Retained binary64: `-0.234375`	49/64Exact fraction: `49/64`Retained binary64: `0.765625`
114	-7/32Exact fraction: `-7/32`Retained binary64: `-0.21875`	25/32Exact fraction: `25/32`Retained binary64: `0.78125`
115	-13/64Exact fraction: `-13/64`Retained binary64: `-0.203125`	51/64Exact fraction: `51/64`Retained binary64: `0.796875`
116	-3/16Exact fraction: `-3/16`Retained binary64: `-0.1875`	13/16Exact fraction: `13/16`Retained binary64: `0.8125`
117	-11/64Exact fraction: `-11/64`Retained binary64: `-0.171875`	53/64Exact fraction: `53/64`Retained binary64: `0.828125`
118	-5/32Exact fraction: `-5/32`Retained binary64: `-0.15625`	27/32Exact fraction: `27/32`Retained binary64: `0.84375`
119	-9/64Exact fraction: `-9/64`Retained binary64: `-0.140625`	55/64Exact fraction: `55/64`Retained binary64: `0.859375`
120	-1/8Exact fraction: `-1/8`Retained binary64: `-0.125`	7/8Exact fraction: `7/8`Retained binary64: `0.875`
121	-7/64Exact fraction: `-7/64`Retained binary64: `-0.109375`	57/64Exact fraction: `57/64`Retained binary64: `0.890625`
122	-3/32Exact fraction: `-3/32`Retained binary64: `-0.09375`	29/32Exact fraction: `29/32`Retained binary64: `0.90625`
123	-5/64Exact fraction: `-5/64`Retained binary64: `-0.078125`	59/64Exact fraction: `59/64`Retained binary64: `0.921875`
124	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	15/16Exact fraction: `15/16`Retained binary64: `0.9375`
125	-3/64Exact fraction: `-3/64`Retained binary64: `-0.046875`	61/64Exact fraction: `61/64`Retained binary64: `0.953125`
126	-1/32Exact fraction: `-1/32`Retained binary64: `-0.03125`	31/32Exact fraction: `31/32`Retained binary64: `0.96875`
127	-1/64Exact fraction: `-1/64`Retained binary64: `-0.015625`	63/64Exact fraction: `63/64`Retained binary64: `0.984375`
128	0Exact fraction: `0`Retained binary64: `0.0`	1Exact fraction: `1`Retained binary64: `1.0`

Complete contour half-branch: level 1/4, branch-2:-1 (65 rows)

Separate arrays are separate plotted paths. Quarter-level halves meet only at their own turning point; never bridge the forbidden central x interval.

Complete contour half-branch: level 1/4, branch-2:-1
Row	X=x/L₀	Y=y/L₀	Exact Y²=(X+1)²−1/4
0	-2Exact fraction: `-2`Retained binary64: `-2.0`	-0.866025403784Retained binary64: `-0.8660254037844386`	3/4
1	-255/128Exact fraction: `-255/128`Retained binary64: `-1.9921875`	-0.85699243588Retained binary64: `-0.8569924358804166`	12033/16384
2	-127/64Exact fraction: `-127/64`Retained binary64: `-1.984375`	-0.847935221951Retained binary64: `-0.847935221950946`	2945/4096
3	-253/128Exact fraction: `-253/128`Retained binary64: `-1.9765625`	-0.838852976633Retained binary64: `-0.8388529766331225`	11529/16384
4	-63/32Exact fraction: `-63/32`Retained binary64: `-1.96875`	-0.829744877959Retained binary64: `-0.8297448779594846`	705/1024
5	-251/128Exact fraction: `-251/128`Retained binary64: `-1.9609375`	-0.820610065077Retained binary64: `-0.8206100650773485`	11033/16384
6	-125/64Exact fraction: `-125/64`Retained binary64: `-1.953125`	-0.811447635787Retained binary64: `-0.8114476357874241`	2697/4096
7	-249/128Exact fraction: `-249/128`Retained binary64: `-1.9453125`	-0.802256643884Retained binary64: `-0.8022566438841439`	10545/16384
8	-31/16Exact fraction: `-31/16`Retained binary64: `-1.9375`	-0.793036096278Retained binary64: `-0.793036096278095`	161/256
9	-247/128Exact fraction: `-247/128`Retained binary64: `-1.9296875`	-0.783784949879Retained binary64: `-0.7837849498786322`	10065/16384
10	-123/64Exact fraction: `-123/64`Retained binary64: `-1.921875`	-0.774502108212Retained binary64: `-0.7745021082121081`	2457/4096
11	-245/128Exact fraction: `-245/128`Retained binary64: `-1.9140625`	-0.765186417748Retained binary64: `-0.7651864177481524`	9593/16384
12	-61/32Exact fraction: `-61/32`Retained binary64: `-1.90625`	-0.755836663903Retained binary64: `-0.755836663902989`	585/1024
13	-243/128Exact fraction: `-243/128`Retained binary64: `-1.8984375`	-0.746451566685Retained binary64: `-0.7464515666848386`	9129/16384
14	-121/64Exact fraction: `-121/64`Retained binary64: `-1.890625`	-0.737029775942Retained binary64: `-0.7370297759419222`	2225/4096
15	-241/128Exact fraction: `-241/128`Retained binary64: `-1.8828125`	-0.727569866168Retained binary64: `-0.7275698661683633`	8673/16384
16	-15/8Exact fraction: `-15/8`Retained binary64: `-1.875`	-0.718070330817Retained binary64: `-0.7180703308172536`	33/64
17	-239/128Exact fraction: `-239/128`Retained binary64: `-1.8671875`	-0.708529576063Retained binary64: `-0.7085295760631661`	8225/16384
18	-119/64Exact fraction: `-119/64`Retained binary64: `-1.859375`	-0.698945913948Retained binary64: `-0.6989459139482825`	2001/4096
19	-237/128Exact fraction: `-237/128`Retained binary64: `-1.8515625`	-0.689317554837Retained binary64: `-0.6893175548368473`	7785/16384
20	-59/32Exact fraction: `-59/32`Retained binary64: `-1.84375`	-0.679642599092Retained binary64: `-0.6796425990916108`	473/1024
21	-235/128Exact fraction: `-235/128`Retained binary64: `-1.8359375`	-0.669919027873Retained binary64: `-0.6699190278729587`	7353/16384
22	-117/64Exact fraction: `-117/64`Retained binary64: `-1.828125`	-0.660144692946Retained binary64: `-0.6601446929461753`	1785/4096
23	-233/128Exact fraction: `-233/128`Retained binary64: `-1.8203125`	-0.650317305364Retained binary64: `-0.6503173053642737`	6929/16384
24	-29/16Exact fraction: `-29/16`Retained binary64: `-1.8125`	-0.640434422872Retained binary64: `-0.6404344228724749`	105/256
25	-231/128Exact fraction: `-231/128`Retained binary64: `-1.8046875`	-0.630493435855Retained binary64: `-0.6304934358550055`	6513/16384
26	-115/64Exact fraction: `-115/64`Retained binary64: `-1.796875`	-0.620491551615Retained binary64: `-0.6204915516145244`	1577/4096
27	-229/128Exact fraction: `-229/128`Retained binary64: `-1.7890625`	-0.610425776738Retained binary64: `-0.6104257767380487`	6105/16384
28	-57/32Exact fraction: `-57/32`Retained binary64: `-1.78125`	-0.600292897259Retained binary64: `-0.6002928972593295`	369/1024
29	-227/128Exact fraction: `-227/128`Retained binary64: `-1.7734375`	-0.590089456274Retained binary64: `-0.5900894562744279`	5705/16384
30	-113/64Exact fraction: `-113/64`Retained binary64: `-1.765625`	-0.579811728602Retained binary64: `-0.5798117286024835`	1377/4096
31	-225/128Exact fraction: `-225/128`Retained binary64: `-1.7578125`	-0.569455692004Retained binary64: `-0.569455692004435`	5313/16384
32	-7/4Exact fraction: `-7/4`Retained binary64: `-1.75`	-0.559016994375Retained binary64: `-0.5590169943749475`	5/16
33	-223/128Exact fraction: `-223/128`Retained binary64: `-1.7421875`	-0.548490916202Retained binary64: `-0.5484909162021282`	4929/16384
34	-111/64Exact fraction: `-111/64`Retained binary64: `-1.734375`	-0.537872327439Retained binary64: `-0.5378723274393283`	1185/4096
35	-221/128Exact fraction: `-221/128`Retained binary64: `-1.7265625`	-0.527155637745Retained binary64: `-0.5271556377449168`	4553/16384
36	-55/32Exact fraction: `-55/32`Retained binary64: `-1.71875`	-0.516334738808Retained binary64: `-0.516334738808072`	273/1024
37	-219/128Exact fraction: `-219/128`Retained binary64: `-1.7109375`	-0.505402937176Retained binary64: `-0.5054029371761208`	4185/16384
38	-109/64Exact fraction: `-109/64`Retained binary64: `-1.703125`	-0.494352875611Retained binary64: `-0.4943528756111367`	1001/4096
39	-217/128Exact fraction: `-217/128`Retained binary64: `-1.6953125`	-0.483176440502Retained binary64: `-0.4831764405020696`	3825/16384
40	-27/16Exact fraction: `-27/16`Retained binary64: `-1.6875`	-0.471864652204Retained binary64: `-0.47186465220442186`	57/256
41	-215/128Exact fraction: `-215/128`Retained binary64: `-1.6796875`	-0.460407534317Retained binary64: `-0.46040753431742404`	3473/16384
42	-107/64Exact fraction: `-107/64`Retained binary64: `-1.671875`	-0.448793956761Retained binary64: `-0.4487939567607835`	825/4096
43	-213/128Exact fraction: `-213/128`Retained binary64: `-1.6640625`	-0.437011445967Retained binary64: `-0.4370114459670936`	3129/16384
44	-53/32Exact fraction: `-53/32`Retained binary64: `-1.65625`	-0.425045953398Retained binary64: `-0.4250459533979826`	185/1024
45	-211/128Exact fraction: `-211/128`Retained binary64: `-1.6484375`	-0.412881570679Retained binary64: `-0.4128815706788691`	2793/16384
46	-105/64Exact fraction: `-105/64`Retained binary64: `-1.640625`	-0.400500175562Retained binary64: `-0.40050017556175926`	657/4096
47	-209/128Exact fraction: `-209/128`Retained binary64: `-1.6328125`	-0.387880987103Retained binary64: `-0.3878809871033253`	2465/16384
48	-13/8Exact fraction: `-13/8`Retained binary64: `-1.625`	-0.375Retained binary64: `-0.375`	9/64
49	-207/128Exact fraction: `-207/128`Retained binary64: `-1.6171875`	-0.361829255528Retained binary64: `-0.3618292555284191`	2145/16384
50	-103/64Exact fraction: `-103/64`Retained binary64: `-1.609375`	-0.34833588765Retained binary64: `-0.3483358876501243`	497/4096
51	-205/128Exact fraction: `-205/128`Retained binary64: `-1.6015625`	-0.334480853572Retained binary64: `-0.3344808535719945`	1833/16384
52	-51/32Exact fraction: `-51/32`Retained binary64: `-1.59375`	-0.320217211436Retained binary64: `-0.32021721143623744`	105/1024
53	-203/128Exact fraction: `-203/128`Retained binary64: `-1.5859375`	-0.305487731188Retained binary64: `-0.305487731187768`	1529/16384
54	-101/64Exact fraction: `-101/64`Retained binary64: `-1.578125`	-0.290221494078Retained binary64: `-0.29022149407822984`	345/4096
55	-201/128Exact fraction: `-201/128`Retained binary64: `-1.5703125`	-0.274328904157Retained binary64: `-0.2743289041574912`	1233/16384
56	-25/16Exact fraction: `-25/16`Retained binary64: `-1.5625`	-0.257694101601Retained binary64: `-0.2576941016011038`	17/256
57	-199/128Exact fraction: `-199/128`Retained binary64: `-1.5546875`	-0.240162908577Retained binary64: `-0.2401629085771781`	945/16384
58	-99/64Exact fraction: `-99/64`Retained binary64: `-1.546875`	-0.221522607481Retained binary64: `-0.22152260748059102`	201/4096
59	-197/128Exact fraction: `-197/128`Retained binary64: `-1.5390625`	-0.201465577472Retained binary64: `-0.20146557747230667`	665/16384
60	-49/32Exact fraction: `-49/32`Retained binary64: `-1.53125`	-0.179517582704Retained binary64: `-0.1795175827043134`	33/1024
61	-195/128Exact fraction: `-195/128`Retained binary64: `-1.5234375`	-0.154876778137Retained binary64: `-0.15487677813749226`	393/16384
62	-97/64Exact fraction: `-97/64`Retained binary64: `-1.515625`	-0.125972777317Retained binary64: `-0.12597277731716483`	65/4096
63	-193/128Exact fraction: `-193/128`Retained binary64: `-1.5078125`	-0.0887329429031Retained binary64: `-0.08873294290312927`	129/16384
64	-3/2Exact fraction: `-3/2`Retained binary64: `-1.5`	0Retained binary64: `0.0`	0

Complete contour half-branch: level 1/4, branch-2:1 (65 rows)

Separate arrays are separate plotted paths. Quarter-level halves meet only at their own turning point; never bridge the forbidden central x interval.

Complete contour half-branch: level 1/4, branch-2:1
Row	X=x/L₀	Y=y/L₀	Exact Y²=(X+1)²−1/4
0	-2Exact fraction: `-2`Retained binary64: `-2.0`	0.866025403784Retained binary64: `0.8660254037844386`	3/4
1	-255/128Exact fraction: `-255/128`Retained binary64: `-1.9921875`	0.85699243588Retained binary64: `0.8569924358804166`	12033/16384
2	-127/64Exact fraction: `-127/64`Retained binary64: `-1.984375`	0.847935221951Retained binary64: `0.847935221950946`	2945/4096
3	-253/128Exact fraction: `-253/128`Retained binary64: `-1.9765625`	0.838852976633Retained binary64: `0.8388529766331225`	11529/16384
4	-63/32Exact fraction: `-63/32`Retained binary64: `-1.96875`	0.829744877959Retained binary64: `0.8297448779594846`	705/1024
5	-251/128Exact fraction: `-251/128`Retained binary64: `-1.9609375`	0.820610065077Retained binary64: `0.8206100650773485`	11033/16384
6	-125/64Exact fraction: `-125/64`Retained binary64: `-1.953125`	0.811447635787Retained binary64: `0.8114476357874241`	2697/4096
7	-249/128Exact fraction: `-249/128`Retained binary64: `-1.9453125`	0.802256643884Retained binary64: `0.8022566438841439`	10545/16384
8	-31/16Exact fraction: `-31/16`Retained binary64: `-1.9375`	0.793036096278Retained binary64: `0.793036096278095`	161/256
9	-247/128Exact fraction: `-247/128`Retained binary64: `-1.9296875`	0.783784949879Retained binary64: `0.7837849498786322`	10065/16384
10	-123/64Exact fraction: `-123/64`Retained binary64: `-1.921875`	0.774502108212Retained binary64: `0.7745021082121081`	2457/4096
11	-245/128Exact fraction: `-245/128`Retained binary64: `-1.9140625`	0.765186417748Retained binary64: `0.7651864177481524`	9593/16384
12	-61/32Exact fraction: `-61/32`Retained binary64: `-1.90625`	0.755836663903Retained binary64: `0.755836663902989`	585/1024
13	-243/128Exact fraction: `-243/128`Retained binary64: `-1.8984375`	0.746451566685Retained binary64: `0.7464515666848386`	9129/16384
14	-121/64Exact fraction: `-121/64`Retained binary64: `-1.890625`	0.737029775942Retained binary64: `0.7370297759419222`	2225/4096
15	-241/128Exact fraction: `-241/128`Retained binary64: `-1.8828125`	0.727569866168Retained binary64: `0.7275698661683633`	8673/16384
16	-15/8Exact fraction: `-15/8`Retained binary64: `-1.875`	0.718070330817Retained binary64: `0.7180703308172536`	33/64
17	-239/128Exact fraction: `-239/128`Retained binary64: `-1.8671875`	0.708529576063Retained binary64: `0.7085295760631661`	8225/16384
18	-119/64Exact fraction: `-119/64`Retained binary64: `-1.859375`	0.698945913948Retained binary64: `0.6989459139482825`	2001/4096
19	-237/128Exact fraction: `-237/128`Retained binary64: `-1.8515625`	0.689317554837Retained binary64: `0.6893175548368473`	7785/16384
20	-59/32Exact fraction: `-59/32`Retained binary64: `-1.84375`	0.679642599092Retained binary64: `0.6796425990916108`	473/1024
21	-235/128Exact fraction: `-235/128`Retained binary64: `-1.8359375`	0.669919027873Retained binary64: `0.6699190278729587`	7353/16384
22	-117/64Exact fraction: `-117/64`Retained binary64: `-1.828125`	0.660144692946Retained binary64: `0.6601446929461753`	1785/4096
23	-233/128Exact fraction: `-233/128`Retained binary64: `-1.8203125`	0.650317305364Retained binary64: `0.6503173053642737`	6929/16384
24	-29/16Exact fraction: `-29/16`Retained binary64: `-1.8125`	0.640434422872Retained binary64: `0.6404344228724749`	105/256
25	-231/128Exact fraction: `-231/128`Retained binary64: `-1.8046875`	0.630493435855Retained binary64: `0.6304934358550055`	6513/16384
26	-115/64Exact fraction: `-115/64`Retained binary64: `-1.796875`	0.620491551615Retained binary64: `0.6204915516145244`	1577/4096
27	-229/128Exact fraction: `-229/128`Retained binary64: `-1.7890625`	0.610425776738Retained binary64: `0.6104257767380487`	6105/16384
28	-57/32Exact fraction: `-57/32`Retained binary64: `-1.78125`	0.600292897259Retained binary64: `0.6002928972593295`	369/1024
29	-227/128Exact fraction: `-227/128`Retained binary64: `-1.7734375`	0.590089456274Retained binary64: `0.5900894562744279`	5705/16384
30	-113/64Exact fraction: `-113/64`Retained binary64: `-1.765625`	0.579811728602Retained binary64: `0.5798117286024835`	1377/4096
31	-225/128Exact fraction: `-225/128`Retained binary64: `-1.7578125`	0.569455692004Retained binary64: `0.569455692004435`	5313/16384
32	-7/4Exact fraction: `-7/4`Retained binary64: `-1.75`	0.559016994375Retained binary64: `0.5590169943749475`	5/16
33	-223/128Exact fraction: `-223/128`Retained binary64: `-1.7421875`	0.548490916202Retained binary64: `0.5484909162021282`	4929/16384
34	-111/64Exact fraction: `-111/64`Retained binary64: `-1.734375`	0.537872327439Retained binary64: `0.5378723274393283`	1185/4096
35	-221/128Exact fraction: `-221/128`Retained binary64: `-1.7265625`	0.527155637745Retained binary64: `0.5271556377449168`	4553/16384
36	-55/32Exact fraction: `-55/32`Retained binary64: `-1.71875`	0.516334738808Retained binary64: `0.516334738808072`	273/1024
37	-219/128Exact fraction: `-219/128`Retained binary64: `-1.7109375`	0.505402937176Retained binary64: `0.5054029371761208`	4185/16384
38	-109/64Exact fraction: `-109/64`Retained binary64: `-1.703125`	0.494352875611Retained binary64: `0.4943528756111367`	1001/4096
39	-217/128Exact fraction: `-217/128`Retained binary64: `-1.6953125`	0.483176440502Retained binary64: `0.4831764405020696`	3825/16384
40	-27/16Exact fraction: `-27/16`Retained binary64: `-1.6875`	0.471864652204Retained binary64: `0.47186465220442186`	57/256
41	-215/128Exact fraction: `-215/128`Retained binary64: `-1.6796875`	0.460407534317Retained binary64: `0.46040753431742404`	3473/16384
42	-107/64Exact fraction: `-107/64`Retained binary64: `-1.671875`	0.448793956761Retained binary64: `0.4487939567607835`	825/4096
43	-213/128Exact fraction: `-213/128`Retained binary64: `-1.6640625`	0.437011445967Retained binary64: `0.4370114459670936`	3129/16384
44	-53/32Exact fraction: `-53/32`Retained binary64: `-1.65625`	0.425045953398Retained binary64: `0.4250459533979826`	185/1024
45	-211/128Exact fraction: `-211/128`Retained binary64: `-1.6484375`	0.412881570679Retained binary64: `0.4128815706788691`	2793/16384
46	-105/64Exact fraction: `-105/64`Retained binary64: `-1.640625`	0.400500175562Retained binary64: `0.40050017556175926`	657/4096
47	-209/128Exact fraction: `-209/128`Retained binary64: `-1.6328125`	0.387880987103Retained binary64: `0.3878809871033253`	2465/16384
48	-13/8Exact fraction: `-13/8`Retained binary64: `-1.625`	0.375Retained binary64: `0.375`	9/64
49	-207/128Exact fraction: `-207/128`Retained binary64: `-1.6171875`	0.361829255528Retained binary64: `0.3618292555284191`	2145/16384
50	-103/64Exact fraction: `-103/64`Retained binary64: `-1.609375`	0.34833588765Retained binary64: `0.3483358876501243`	497/4096
51	-205/128Exact fraction: `-205/128`Retained binary64: `-1.6015625`	0.334480853572Retained binary64: `0.3344808535719945`	1833/16384
52	-51/32Exact fraction: `-51/32`Retained binary64: `-1.59375`	0.320217211436Retained binary64: `0.32021721143623744`	105/1024
53	-203/128Exact fraction: `-203/128`Retained binary64: `-1.5859375`	0.305487731188Retained binary64: `0.305487731187768`	1529/16384
54	-101/64Exact fraction: `-101/64`Retained binary64: `-1.578125`	0.290221494078Retained binary64: `0.29022149407822984`	345/4096
55	-201/128Exact fraction: `-201/128`Retained binary64: `-1.5703125`	0.274328904157Retained binary64: `0.2743289041574912`	1233/16384
56	-25/16Exact fraction: `-25/16`Retained binary64: `-1.5625`	0.257694101601Retained binary64: `0.2576941016011038`	17/256
57	-199/128Exact fraction: `-199/128`Retained binary64: `-1.5546875`	0.240162908577Retained binary64: `0.2401629085771781`	945/16384
58	-99/64Exact fraction: `-99/64`Retained binary64: `-1.546875`	0.221522607481Retained binary64: `0.22152260748059102`	201/4096
59	-197/128Exact fraction: `-197/128`Retained binary64: `-1.5390625`	0.201465577472Retained binary64: `0.20146557747230667`	665/16384
60	-49/32Exact fraction: `-49/32`Retained binary64: `-1.53125`	0.179517582704Retained binary64: `0.1795175827043134`	33/1024
61	-195/128Exact fraction: `-195/128`Retained binary64: `-1.5234375`	0.154876778137Retained binary64: `0.15487677813749226`	393/16384
62	-97/64Exact fraction: `-97/64`Retained binary64: `-1.515625`	0.125972777317Retained binary64: `0.12597277731716483`	65/4096
63	-193/128Exact fraction: `-193/128`Retained binary64: `-1.5078125`	0.0887329429031Retained binary64: `0.08873294290312927`	129/16384
64	-3/2Exact fraction: `-3/2`Retained binary64: `-1.5`	0Retained binary64: `0.0`	0

Complete contour half-branch: level 1/4, branch-1/2:-1 (65 rows)

Separate arrays are separate plotted paths. Quarter-level halves meet only at their own turning point; never bridge the forbidden central x interval.

Complete contour half-branch: level 1/4, branch-1/2:-1
Row	X=x/L₀	Y=y/L₀	Exact Y²=(X+1)²−1/4
0	-1/2Exact fraction: `-1/2`Retained binary64: `-0.5`	0Retained binary64: `0.0`	0
1	-63/128Exact fraction: `-63/128`Retained binary64: `-0.4921875`	-0.0887329429031Retained binary64: `-0.08873294290312927`	129/16384
2	-31/64Exact fraction: `-31/64`Retained binary64: `-0.484375`	-0.125972777317Retained binary64: `-0.12597277731716483`	65/4096
3	-61/128Exact fraction: `-61/128`Retained binary64: `-0.4765625`	-0.154876778137Retained binary64: `-0.15487677813749226`	393/16384
4	-15/32Exact fraction: `-15/32`Retained binary64: `-0.46875`	-0.179517582704Retained binary64: `-0.1795175827043134`	33/1024
5	-59/128Exact fraction: `-59/128`Retained binary64: `-0.4609375`	-0.201465577472Retained binary64: `-0.20146557747230667`	665/16384
6	-29/64Exact fraction: `-29/64`Retained binary64: `-0.453125`	-0.221522607481Retained binary64: `-0.22152260748059102`	201/4096
7	-57/128Exact fraction: `-57/128`Retained binary64: `-0.4453125`	-0.240162908577Retained binary64: `-0.2401629085771781`	945/16384
8	-7/16Exact fraction: `-7/16`Retained binary64: `-0.4375`	-0.257694101601Retained binary64: `-0.2576941016011038`	17/256
9	-55/128Exact fraction: `-55/128`Retained binary64: `-0.4296875`	-0.274328904157Retained binary64: `-0.2743289041574912`	1233/16384
10	-27/64Exact fraction: `-27/64`Retained binary64: `-0.421875`	-0.290221494078Retained binary64: `-0.29022149407822984`	345/4096
11	-53/128Exact fraction: `-53/128`Retained binary64: `-0.4140625`	-0.305487731188Retained binary64: `-0.305487731187768`	1529/16384
12	-13/32Exact fraction: `-13/32`Retained binary64: `-0.40625`	-0.320217211436Retained binary64: `-0.32021721143623744`	105/1024
13	-51/128Exact fraction: `-51/128`Retained binary64: `-0.3984375`	-0.334480853572Retained binary64: `-0.3344808535719945`	1833/16384
14	-25/64Exact fraction: `-25/64`Retained binary64: `-0.390625`	-0.34833588765Retained binary64: `-0.3483358876501243`	497/4096
15	-49/128Exact fraction: `-49/128`Retained binary64: `-0.3828125`	-0.361829255528Retained binary64: `-0.3618292555284191`	2145/16384
16	-3/8Exact fraction: `-3/8`Retained binary64: `-0.375`	-0.375Retained binary64: `-0.375`	9/64
17	-47/128Exact fraction: `-47/128`Retained binary64: `-0.3671875`	-0.387880987103Retained binary64: `-0.3878809871033253`	2465/16384
18	-23/64Exact fraction: `-23/64`Retained binary64: `-0.359375`	-0.400500175562Retained binary64: `-0.40050017556175926`	657/4096
19	-45/128Exact fraction: `-45/128`Retained binary64: `-0.3515625`	-0.412881570679Retained binary64: `-0.4128815706788691`	2793/16384
20	-11/32Exact fraction: `-11/32`Retained binary64: `-0.34375`	-0.425045953398Retained binary64: `-0.4250459533979826`	185/1024
21	-43/128Exact fraction: `-43/128`Retained binary64: `-0.3359375`	-0.437011445967Retained binary64: `-0.4370114459670936`	3129/16384
22	-21/64Exact fraction: `-21/64`Retained binary64: `-0.328125`	-0.448793956761Retained binary64: `-0.4487939567607835`	825/4096
23	-41/128Exact fraction: `-41/128`Retained binary64: `-0.3203125`	-0.460407534317Retained binary64: `-0.46040753431742404`	3473/16384
24	-5/16Exact fraction: `-5/16`Retained binary64: `-0.3125`	-0.471864652204Retained binary64: `-0.47186465220442186`	57/256
25	-39/128Exact fraction: `-39/128`Retained binary64: `-0.3046875`	-0.483176440502Retained binary64: `-0.4831764405020696`	3825/16384
26	-19/64Exact fraction: `-19/64`Retained binary64: `-0.296875`	-0.494352875611Retained binary64: `-0.4943528756111367`	1001/4096
27	-37/128Exact fraction: `-37/128`Retained binary64: `-0.2890625`	-0.505402937176Retained binary64: `-0.5054029371761208`	4185/16384
28	-9/32Exact fraction: `-9/32`Retained binary64: `-0.28125`	-0.516334738808Retained binary64: `-0.516334738808072`	273/1024
29	-35/128Exact fraction: `-35/128`Retained binary64: `-0.2734375`	-0.527155637745Retained binary64: `-0.5271556377449168`	4553/16384
30	-17/64Exact fraction: `-17/64`Retained binary64: `-0.265625`	-0.537872327439Retained binary64: `-0.5378723274393283`	1185/4096
31	-33/128Exact fraction: `-33/128`Retained binary64: `-0.2578125`	-0.548490916202Retained binary64: `-0.5484909162021282`	4929/16384
32	-1/4Exact fraction: `-1/4`Retained binary64: `-0.25`	-0.559016994375Retained binary64: `-0.5590169943749475`	5/16
33	-31/128Exact fraction: `-31/128`Retained binary64: `-0.2421875`	-0.569455692004Retained binary64: `-0.569455692004435`	5313/16384
34	-15/64Exact fraction: `-15/64`Retained binary64: `-0.234375`	-0.579811728602Retained binary64: `-0.5798117286024835`	1377/4096
35	-29/128Exact fraction: `-29/128`Retained binary64: `-0.2265625`	-0.590089456274Retained binary64: `-0.5900894562744279`	5705/16384
36	-7/32Exact fraction: `-7/32`Retained binary64: `-0.21875`	-0.600292897259Retained binary64: `-0.6002928972593295`	369/1024
37	-27/128Exact fraction: `-27/128`Retained binary64: `-0.2109375`	-0.610425776738Retained binary64: `-0.6104257767380487`	6105/16384
38	-13/64Exact fraction: `-13/64`Retained binary64: `-0.203125`	-0.620491551615Retained binary64: `-0.6204915516145244`	1577/4096
39	-25/128Exact fraction: `-25/128`Retained binary64: `-0.1953125`	-0.630493435855Retained binary64: `-0.6304934358550055`	6513/16384
40	-3/16Exact fraction: `-3/16`Retained binary64: `-0.1875`	-0.640434422872Retained binary64: `-0.6404344228724749`	105/256
41	-23/128Exact fraction: `-23/128`Retained binary64: `-0.1796875`	-0.650317305364Retained binary64: `-0.6503173053642737`	6929/16384
42	-11/64Exact fraction: `-11/64`Retained binary64: `-0.171875`	-0.660144692946Retained binary64: `-0.6601446929461753`	1785/4096
43	-21/128Exact fraction: `-21/128`Retained binary64: `-0.1640625`	-0.669919027873Retained binary64: `-0.6699190278729587`	7353/16384
44	-5/32Exact fraction: `-5/32`Retained binary64: `-0.15625`	-0.679642599092Retained binary64: `-0.6796425990916108`	473/1024
45	-19/128Exact fraction: `-19/128`Retained binary64: `-0.1484375`	-0.689317554837Retained binary64: `-0.6893175548368473`	7785/16384
46	-9/64Exact fraction: `-9/64`Retained binary64: `-0.140625`	-0.698945913948Retained binary64: `-0.6989459139482825`	2001/4096
47	-17/128Exact fraction: `-17/128`Retained binary64: `-0.1328125`	-0.708529576063Retained binary64: `-0.7085295760631661`	8225/16384
48	-1/8Exact fraction: `-1/8`Retained binary64: `-0.125`	-0.718070330817Retained binary64: `-0.7180703308172536`	33/64
49	-15/128Exact fraction: `-15/128`Retained binary64: `-0.1171875`	-0.727569866168Retained binary64: `-0.7275698661683633`	8673/16384
50	-7/64Exact fraction: `-7/64`Retained binary64: `-0.109375`	-0.737029775942Retained binary64: `-0.7370297759419222`	2225/4096
51	-13/128Exact fraction: `-13/128`Retained binary64: `-0.1015625`	-0.746451566685Retained binary64: `-0.7464515666848386`	9129/16384
52	-3/32Exact fraction: `-3/32`Retained binary64: `-0.09375`	-0.755836663903Retained binary64: `-0.755836663902989`	585/1024
53	-11/128Exact fraction: `-11/128`Retained binary64: `-0.0859375`	-0.765186417748Retained binary64: `-0.7651864177481524`	9593/16384
54	-5/64Exact fraction: `-5/64`Retained binary64: `-0.078125`	-0.774502108212Retained binary64: `-0.7745021082121081`	2457/4096
55	-9/128Exact fraction: `-9/128`Retained binary64: `-0.0703125`	-0.783784949879Retained binary64: `-0.7837849498786322`	10065/16384
56	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	-0.793036096278Retained binary64: `-0.793036096278095`	161/256
57	-7/128Exact fraction: `-7/128`Retained binary64: `-0.0546875`	-0.802256643884Retained binary64: `-0.8022566438841439`	10545/16384
58	-3/64Exact fraction: `-3/64`Retained binary64: `-0.046875`	-0.811447635787Retained binary64: `-0.8114476357874241`	2697/4096
59	-5/128Exact fraction: `-5/128`Retained binary64: `-0.0390625`	-0.820610065077Retained binary64: `-0.8206100650773485`	11033/16384
60	-1/32Exact fraction: `-1/32`Retained binary64: `-0.03125`	-0.829744877959Retained binary64: `-0.8297448779594846`	705/1024
61	-3/128Exact fraction: `-3/128`Retained binary64: `-0.0234375`	-0.838852976633Retained binary64: `-0.8388529766331225`	11529/16384
62	-1/64Exact fraction: `-1/64`Retained binary64: `-0.015625`	-0.847935221951Retained binary64: `-0.847935221950946`	2945/4096
63	-1/128Exact fraction: `-1/128`Retained binary64: `-0.0078125`	-0.85699243588Retained binary64: `-0.8569924358804166`	12033/16384
64	0Exact fraction: `0`Retained binary64: `0.0`	-0.866025403784Retained binary64: `-0.8660254037844386`	3/4

Complete contour half-branch: level 1/4, branch-1/2:1 (65 rows)

Separate arrays are separate plotted paths. Quarter-level halves meet only at their own turning point; never bridge the forbidden central x interval.

Complete contour half-branch: level 1/4, branch-1/2:1
Row	X=x/L₀	Y=y/L₀	Exact Y²=(X+1)²−1/4
0	-1/2Exact fraction: `-1/2`Retained binary64: `-0.5`	0Retained binary64: `0.0`	0
1	-63/128Exact fraction: `-63/128`Retained binary64: `-0.4921875`	0.0887329429031Retained binary64: `0.08873294290312927`	129/16384
2	-31/64Exact fraction: `-31/64`Retained binary64: `-0.484375`	0.125972777317Retained binary64: `0.12597277731716483`	65/4096
3	-61/128Exact fraction: `-61/128`Retained binary64: `-0.4765625`	0.154876778137Retained binary64: `0.15487677813749226`	393/16384
4	-15/32Exact fraction: `-15/32`Retained binary64: `-0.46875`	0.179517582704Retained binary64: `0.1795175827043134`	33/1024
5	-59/128Exact fraction: `-59/128`Retained binary64: `-0.4609375`	0.201465577472Retained binary64: `0.20146557747230667`	665/16384
6	-29/64Exact fraction: `-29/64`Retained binary64: `-0.453125`	0.221522607481Retained binary64: `0.22152260748059102`	201/4096
7	-57/128Exact fraction: `-57/128`Retained binary64: `-0.4453125`	0.240162908577Retained binary64: `0.2401629085771781`	945/16384
8	-7/16Exact fraction: `-7/16`Retained binary64: `-0.4375`	0.257694101601Retained binary64: `0.2576941016011038`	17/256
9	-55/128Exact fraction: `-55/128`Retained binary64: `-0.4296875`	0.274328904157Retained binary64: `0.2743289041574912`	1233/16384
10	-27/64Exact fraction: `-27/64`Retained binary64: `-0.421875`	0.290221494078Retained binary64: `0.29022149407822984`	345/4096
11	-53/128Exact fraction: `-53/128`Retained binary64: `-0.4140625`	0.305487731188Retained binary64: `0.305487731187768`	1529/16384
12	-13/32Exact fraction: `-13/32`Retained binary64: `-0.40625`	0.320217211436Retained binary64: `0.32021721143623744`	105/1024
13	-51/128Exact fraction: `-51/128`Retained binary64: `-0.3984375`	0.334480853572Retained binary64: `0.3344808535719945`	1833/16384
14	-25/64Exact fraction: `-25/64`Retained binary64: `-0.390625`	0.34833588765Retained binary64: `0.3483358876501243`	497/4096
15	-49/128Exact fraction: `-49/128`Retained binary64: `-0.3828125`	0.361829255528Retained binary64: `0.3618292555284191`	2145/16384
16	-3/8Exact fraction: `-3/8`Retained binary64: `-0.375`	0.375Retained binary64: `0.375`	9/64
17	-47/128Exact fraction: `-47/128`Retained binary64: `-0.3671875`	0.387880987103Retained binary64: `0.3878809871033253`	2465/16384
18	-23/64Exact fraction: `-23/64`Retained binary64: `-0.359375`	0.400500175562Retained binary64: `0.40050017556175926`	657/4096
19	-45/128Exact fraction: `-45/128`Retained binary64: `-0.3515625`	0.412881570679Retained binary64: `0.4128815706788691`	2793/16384
20	-11/32Exact fraction: `-11/32`Retained binary64: `-0.34375`	0.425045953398Retained binary64: `0.4250459533979826`	185/1024
21	-43/128Exact fraction: `-43/128`Retained binary64: `-0.3359375`	0.437011445967Retained binary64: `0.4370114459670936`	3129/16384
22	-21/64Exact fraction: `-21/64`Retained binary64: `-0.328125`	0.448793956761Retained binary64: `0.4487939567607835`	825/4096
23	-41/128Exact fraction: `-41/128`Retained binary64: `-0.3203125`	0.460407534317Retained binary64: `0.46040753431742404`	3473/16384
24	-5/16Exact fraction: `-5/16`Retained binary64: `-0.3125`	0.471864652204Retained binary64: `0.47186465220442186`	57/256
25	-39/128Exact fraction: `-39/128`Retained binary64: `-0.3046875`	0.483176440502Retained binary64: `0.4831764405020696`	3825/16384
26	-19/64Exact fraction: `-19/64`Retained binary64: `-0.296875`	0.494352875611Retained binary64: `0.4943528756111367`	1001/4096
27	-37/128Exact fraction: `-37/128`Retained binary64: `-0.2890625`	0.505402937176Retained binary64: `0.5054029371761208`	4185/16384
28	-9/32Exact fraction: `-9/32`Retained binary64: `-0.28125`	0.516334738808Retained binary64: `0.516334738808072`	273/1024
29	-35/128Exact fraction: `-35/128`Retained binary64: `-0.2734375`	0.527155637745Retained binary64: `0.5271556377449168`	4553/16384
30	-17/64Exact fraction: `-17/64`Retained binary64: `-0.265625`	0.537872327439Retained binary64: `0.5378723274393283`	1185/4096
31	-33/128Exact fraction: `-33/128`Retained binary64: `-0.2578125`	0.548490916202Retained binary64: `0.5484909162021282`	4929/16384
32	-1/4Exact fraction: `-1/4`Retained binary64: `-0.25`	0.559016994375Retained binary64: `0.5590169943749475`	5/16
33	-31/128Exact fraction: `-31/128`Retained binary64: `-0.2421875`	0.569455692004Retained binary64: `0.569455692004435`	5313/16384
34	-15/64Exact fraction: `-15/64`Retained binary64: `-0.234375`	0.579811728602Retained binary64: `0.5798117286024835`	1377/4096
35	-29/128Exact fraction: `-29/128`Retained binary64: `-0.2265625`	0.590089456274Retained binary64: `0.5900894562744279`	5705/16384
36	-7/32Exact fraction: `-7/32`Retained binary64: `-0.21875`	0.600292897259Retained binary64: `0.6002928972593295`	369/1024
37	-27/128Exact fraction: `-27/128`Retained binary64: `-0.2109375`	0.610425776738Retained binary64: `0.6104257767380487`	6105/16384
38	-13/64Exact fraction: `-13/64`Retained binary64: `-0.203125`	0.620491551615Retained binary64: `0.6204915516145244`	1577/4096
39	-25/128Exact fraction: `-25/128`Retained binary64: `-0.1953125`	0.630493435855Retained binary64: `0.6304934358550055`	6513/16384
40	-3/16Exact fraction: `-3/16`Retained binary64: `-0.1875`	0.640434422872Retained binary64: `0.6404344228724749`	105/256
41	-23/128Exact fraction: `-23/128`Retained binary64: `-0.1796875`	0.650317305364Retained binary64: `0.6503173053642737`	6929/16384
42	-11/64Exact fraction: `-11/64`Retained binary64: `-0.171875`	0.660144692946Retained binary64: `0.6601446929461753`	1785/4096
43	-21/128Exact fraction: `-21/128`Retained binary64: `-0.1640625`	0.669919027873Retained binary64: `0.6699190278729587`	7353/16384
44	-5/32Exact fraction: `-5/32`Retained binary64: `-0.15625`	0.679642599092Retained binary64: `0.6796425990916108`	473/1024
45	-19/128Exact fraction: `-19/128`Retained binary64: `-0.1484375`	0.689317554837Retained binary64: `0.6893175548368473`	7785/16384
46	-9/64Exact fraction: `-9/64`Retained binary64: `-0.140625`	0.698945913948Retained binary64: `0.6989459139482825`	2001/4096
47	-17/128Exact fraction: `-17/128`Retained binary64: `-0.1328125`	0.708529576063Retained binary64: `0.7085295760631661`	8225/16384
48	-1/8Exact fraction: `-1/8`Retained binary64: `-0.125`	0.718070330817Retained binary64: `0.7180703308172536`	33/64
49	-15/128Exact fraction: `-15/128`Retained binary64: `-0.1171875`	0.727569866168Retained binary64: `0.7275698661683633`	8673/16384
50	-7/64Exact fraction: `-7/64`Retained binary64: `-0.109375`	0.737029775942Retained binary64: `0.7370297759419222`	2225/4096
51	-13/128Exact fraction: `-13/128`Retained binary64: `-0.1015625`	0.746451566685Retained binary64: `0.7464515666848386`	9129/16384
52	-3/32Exact fraction: `-3/32`Retained binary64: `-0.09375`	0.755836663903Retained binary64: `0.755836663902989`	585/1024
53	-11/128Exact fraction: `-11/128`Retained binary64: `-0.0859375`	0.765186417748Retained binary64: `0.7651864177481524`	9593/16384
54	-5/64Exact fraction: `-5/64`Retained binary64: `-0.078125`	0.774502108212Retained binary64: `0.7745021082121081`	2457/4096
55	-9/128Exact fraction: `-9/128`Retained binary64: `-0.0703125`	0.783784949879Retained binary64: `0.7837849498786322`	10065/16384
56	-1/16Exact fraction: `-1/16`Retained binary64: `-0.0625`	0.793036096278Retained binary64: `0.793036096278095`	161/256
57	-7/128Exact fraction: `-7/128`Retained binary64: `-0.0546875`	0.802256643884Retained binary64: `0.8022566438841439`	10545/16384
58	-3/64Exact fraction: `-3/64`Retained binary64: `-0.046875`	0.811447635787Retained binary64: `0.8114476357874241`	2697/4096
59	-5/128Exact fraction: `-5/128`Retained binary64: `-0.0390625`	0.820610065077Retained binary64: `0.8206100650773485`	11033/16384
60	-1/32Exact fraction: `-1/32`Retained binary64: `-0.03125`	0.829744877959Retained binary64: `0.8297448779594846`	705/1024
61	-3/128Exact fraction: `-3/128`Retained binary64: `-0.0234375`	0.838852976633Retained binary64: `0.8388529766331225`	11529/16384
62	-1/64Exact fraction: `-1/64`Retained binary64: `-0.015625`	0.847935221951Retained binary64: `0.847935221950946`	2945/4096
63	-1/128Exact fraction: `-1/128`Retained binary64: `-0.0078125`	0.85699243588Retained binary64: `0.8569924358804166`	12033/16384
64	0Exact fraction: `0`Retained binary64: `0.0`	0.866025403784Retained binary64: `0.8660254037844386`	3/4

Four isolated boundary points at levels −1 and+1 (4 rows)

These are isolated points, never line or contour segments.

Four isolated boundary points at levels −1 and+1
Row	Potential levelc	X=x/L₀	Y=y/L₀
0	-1	-1	-1
1	-1	-1	1
2	1	-2	0
3	1	0	0

Critical point in the slice; entire criticalz-line in 3D (1 rows)

The point(-1,0)in the slice represents everyz in the invariant extension. No restoring z force.

Critical point in the slice; entire criticalz-line in 3D
Row	X	Y	v	Ex	Ey	zInvariantCriticalLine
0	-1	0	0	0	0	true

Return path: mean values and harmonic extrema

A regular harmonic function has the mean-value property: its value at the center of a ball equals the average over that ball’s sphere, provided the closed ball lies in the harmonic domain. In two dimensions the corresponding boundary average is over a circle. This connects a local differential equation to the behavior of neighboring values.

The strong maximum principle then rules out a genuine interior maximum or minimum of a nonconstant harmonic potential on a connected domain. A constant function is the important exception. On a bounded domain with continuous boundary data, extrema of a nonconstant harmonic solution are controlled by the boundary, not by a source-free interior peak or well.

A stationary saddle is allowed because nearby values occur on both sides of its critical value. The charged slab’s one-dimensional maximum is also allowed because its volume charge makes the potential nonharmonic. The isolated ball’s center is inside charge too. The theorem’s hypotheses distinguish these cases more reliably than the appearance of a sketch.

For an unconstrained nonzero test charge, a stable electrostatic equilibrium would require an appropriate local minimum of qV. The harmonic extremum restriction prevents a strictly confining potential well in regular charge-free electrostatics. It is not a prohibition of every physical trap: mechanical constraints and additional interactions require their own analysis.

Return path: build a decaying boundary mode

In a second separate z-invariant model, take the half-space Y ≥ 0 with periodic boundary v(X,0) = 3 sin(2X), period π in X, and a bounded periodic solution tending uniformly to zero as Y → ∞. Look for a product f(X)g(Y). Laplace’s equation becomes f″g + fg″ = 0, so where the division is valid f″/f = −g″/g is a separation constant.

Choosing the sinusoidal boundary factor gives f″ = −4f and g″ = 4g. The vertical factors are e^(2Y) and e^(−2Y). The stated decay condition removes the growing factor, and the boundary amplitude fixes v = 3 sin(2X)e^(−2Y). The derivatives verify the PDE directly: v_XX = −4v and v_YY = 4v.

Its field is E/E₀ = (−6 cos(2X)e^(−2Y), 6 sin(2X)e^(−2Y), 0), with magnitude 6e^(−2Y). Potential can vanish on a vertical zero contour while the field is nonzero. At fixed height, the magnitude in this single-mode example is independent of X even though the direction changes with X.

In dimensional form the wave number is k = 2/L₀. More generally, a boundary mode with wave number k decays as e^(−ky), so finer spatial boundary variations diminish over a shorter distance into a charge-free half-space. This is spatial structure of a static solution, not a time-decay law or an electromagnetic wave attenuation calculation.

For a finite rectangle, one must choose and combine modes that fit all of its boundaries. A half-space mode cannot simply be copied while ignoring a new top or side boundary. At discontinuous boundary data, convergence and edge limits require explicit interpretation. The periodicity, boundedness and decay conditions stated here are part of our unbounded-domain solution class.

A periodic boundary and static spatial smoothing

Separate periodic boundary potential/signed field components from positive magnitude versus scaled height; exact special phase/height controls; static spatial decay and source-free PDE cancellation, no dynamic propagation claim.

Boundary potential and the two signed field components

v=V/V₀=3sin(2X)
Signed Eₓ/E₀=−6cos(2X)
Signed Eᵧ/E₀=6sin(2X)

Positive field magnitude versus scaled height

Positive |E|/E₀=6·2^(−H), 33 analytic samples
Four exact height controls at integer H

Separate z-invariant half-spaceY≥0: prescribedv(X,0)=3sin(2X), periodπ, bounded periodic solution decaying uniformly upward. v=3sin(2X)e^(−2Y).
On the boundary the three series are separately normalized:V/V₀,Eₓ/E₀andEᵧ/E₀. Their distinct shapes/dashes/labels retain the signed components; the second panel is the positive magnitude only.
H=2y/(L₀ln2), so|E|/E₀=6·2^(−H). One unit of H halves the magnitude. This is static spatial variation, not propagation or time attenuation.
The source data use exact rational strings or12-significant-digit displays plus full retained binary64 values. The plotted curves join finite analytic samples; no interpolation-error bound or numerical PDE solver is claimed.

Complete labelled equivalents below preserve every retained sample and scientific value; original manuscript tables remain separate.

All 36 exact phase and height controls (36 rows)

Cardinal trigonometric zeros are exact. Other retained values are binary64 evaluations; displays use12 significant digits.

All 36 exact phase and height controls
Row	Exact phase2X	ExactX	ExactY	e^(−2Y)	Exact sin(2X)	Exact cos(2X)	V/V₀	Signed Eₓ/E₀	Signed Eᵧ/E₀	Positive \|E\|/E₀
0	0*pi/4	0*pi/8	0*ln2/2	1Exact fraction: `1`Retained binary64: `1.0`	0	1	0Retained binary64: `0.0`	-6Retained binary64: `-6.0`	0Retained binary64: `0.0`	6Exact fraction: `6`Retained binary64: `6.0`
1	1*pi/4	1*pi/8	0*ln2/2	1Exact fraction: `1`Retained binary64: `1.0`	sqrt(2)/2	sqrt(2)/2	2.12132034356Retained binary64: `2.121320343559643`	-4.24264068712Retained binary64: `-4.242640687119286`	4.24264068712Retained binary64: `4.242640687119286`	6Exact fraction: `6`Retained binary64: `6.0`
2	2*pi/4	2*pi/8	0*ln2/2	1Exact fraction: `1`Retained binary64: `1.0`	1	0	3Retained binary64: `3.0`	0Retained binary64: `0.0`	6Retained binary64: `6.0`	6Exact fraction: `6`Retained binary64: `6.0`
3	3*pi/4	3*pi/8	0*ln2/2	1Exact fraction: `1`Retained binary64: `1.0`	sqrt(2)/2	-sqrt(2)/2	2.12132034356Retained binary64: `2.121320343559643`	4.24264068712Retained binary64: `4.242640687119286`	4.24264068712Retained binary64: `4.242640687119286`	6Exact fraction: `6`Retained binary64: `6.0`
4	4*pi/4	4*pi/8	0*ln2/2	1Exact fraction: `1`Retained binary64: `1.0`	0	-1	0Retained binary64: `0.0`	6Retained binary64: `6.0`	0Retained binary64: `0.0`	6Exact fraction: `6`Retained binary64: `6.0`
5	5*pi/4	5*pi/8	0*ln2/2	1Exact fraction: `1`Retained binary64: `1.0`	-sqrt(2)/2	-sqrt(2)/2	-2.12132034356Retained binary64: `-2.121320343559643`	4.24264068712Retained binary64: `4.242640687119286`	-4.24264068712Retained binary64: `-4.242640687119286`	6Exact fraction: `6`Retained binary64: `6.0`
6	6*pi/4	6*pi/8	0*ln2/2	1Exact fraction: `1`Retained binary64: `1.0`	-1	0	-3Retained binary64: `-3.0`	0Retained binary64: `0.0`	-6Retained binary64: `-6.0`	6Exact fraction: `6`Retained binary64: `6.0`
7	7*pi/4	7*pi/8	0*ln2/2	1Exact fraction: `1`Retained binary64: `1.0`	-sqrt(2)/2	sqrt(2)/2	-2.12132034356Retained binary64: `-2.121320343559643`	-4.24264068712Retained binary64: `-4.242640687119286`	-4.24264068712Retained binary64: `-4.242640687119286`	6Exact fraction: `6`Retained binary64: `6.0`
8	8*pi/4	8*pi/8	0*ln2/2	1Exact fraction: `1`Retained binary64: `1.0`	0	1	0Retained binary64: `0.0`	-6Retained binary64: `-6.0`	0Retained binary64: `0.0`	6Exact fraction: `6`Retained binary64: `6.0`
9	0*pi/4	0*pi/8	1*ln2/2	1/2Exact fraction: `1/2`Retained binary64: `0.5`	0	1	0Retained binary64: `0.0`	-3Retained binary64: `-3.0`	0Retained binary64: `0.0`	3Exact fraction: `3`Retained binary64: `3.0`
10	1*pi/4	1*pi/8	1*ln2/2	1/2Exact fraction: `1/2`Retained binary64: `0.5`	sqrt(2)/2	sqrt(2)/2	1.06066017178Retained binary64: `1.0606601717798214`	-2.12132034356Retained binary64: `-2.121320343559643`	2.12132034356Retained binary64: `2.121320343559643`	3Exact fraction: `3`Retained binary64: `3.0`
11	2*pi/4	2*pi/8	1*ln2/2	1/2Exact fraction: `1/2`Retained binary64: `0.5`	1	0	1.5Retained binary64: `1.5`	0Retained binary64: `0.0`	3Retained binary64: `3.0`	3Exact fraction: `3`Retained binary64: `3.0`
12	3*pi/4	3*pi/8	1*ln2/2	1/2Exact fraction: `1/2`Retained binary64: `0.5`	sqrt(2)/2	-sqrt(2)/2	1.06066017178Retained binary64: `1.0606601717798214`	2.12132034356Retained binary64: `2.121320343559643`	2.12132034356Retained binary64: `2.121320343559643`	3Exact fraction: `3`Retained binary64: `3.0`
13	4*pi/4	4*pi/8	1*ln2/2	1/2Exact fraction: `1/2`Retained binary64: `0.5`	0	-1	0Retained binary64: `0.0`	3Retained binary64: `3.0`	0Retained binary64: `0.0`	3Exact fraction: `3`Retained binary64: `3.0`
14	5*pi/4	5*pi/8	1*ln2/2	1/2Exact fraction: `1/2`Retained binary64: `0.5`	-sqrt(2)/2	-sqrt(2)/2	-1.06066017178Retained binary64: `-1.0606601717798214`	2.12132034356Retained binary64: `2.121320343559643`	-2.12132034356Retained binary64: `-2.121320343559643`	3Exact fraction: `3`Retained binary64: `3.0`
15	6*pi/4	6*pi/8	1*ln2/2	1/2Exact fraction: `1/2`Retained binary64: `0.5`	-1	0	-1.5Retained binary64: `-1.5`	0Retained binary64: `0.0`	-3Retained binary64: `-3.0`	3Exact fraction: `3`Retained binary64: `3.0`
16	7*pi/4	7*pi/8	1*ln2/2	1/2Exact fraction: `1/2`Retained binary64: `0.5`	-sqrt(2)/2	sqrt(2)/2	-1.06066017178Retained binary64: `-1.0606601717798214`	-2.12132034356Retained binary64: `-2.121320343559643`	-2.12132034356Retained binary64: `-2.121320343559643`	3Exact fraction: `3`Retained binary64: `3.0`
17	8*pi/4	8*pi/8	1*ln2/2	1/2Exact fraction: `1/2`Retained binary64: `0.5`	0	1	0Retained binary64: `0.0`	-3Retained binary64: `-3.0`	0Retained binary64: `0.0`	3Exact fraction: `3`Retained binary64: `3.0`
18	0*pi/4	0*pi/8	2*ln2/2	1/4Exact fraction: `1/4`Retained binary64: `0.25`	0	1	0Retained binary64: `0.0`	-1.5Retained binary64: `-1.5`	0Retained binary64: `0.0`	3/2Exact fraction: `3/2`Retained binary64: `1.5`
19	1*pi/4	1*pi/8	2*ln2/2	1/4Exact fraction: `1/4`Retained binary64: `0.25`	sqrt(2)/2	sqrt(2)/2	0.53033008589Retained binary64: `0.5303300858899107`	-1.06066017178Retained binary64: `-1.0606601717798214`	1.06066017178Retained binary64: `1.0606601717798214`	3/2Exact fraction: `3/2`Retained binary64: `1.5`
20	2*pi/4	2*pi/8	2*ln2/2	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1	0	0.75Retained binary64: `0.75`	0Retained binary64: `0.0`	1.5Retained binary64: `1.5`	3/2Exact fraction: `3/2`Retained binary64: `1.5`
21	3*pi/4	3*pi/8	2*ln2/2	1/4Exact fraction: `1/4`Retained binary64: `0.25`	sqrt(2)/2	-sqrt(2)/2	0.53033008589Retained binary64: `0.5303300858899107`	1.06066017178Retained binary64: `1.0606601717798214`	1.06066017178Retained binary64: `1.0606601717798214`	3/2Exact fraction: `3/2`Retained binary64: `1.5`
22	4*pi/4	4*pi/8	2*ln2/2	1/4Exact fraction: `1/4`Retained binary64: `0.25`	0	-1	0Retained binary64: `0.0`	1.5Retained binary64: `1.5`	0Retained binary64: `0.0`	3/2Exact fraction: `3/2`Retained binary64: `1.5`
23	5*pi/4	5*pi/8	2*ln2/2	1/4Exact fraction: `1/4`Retained binary64: `0.25`	-sqrt(2)/2	-sqrt(2)/2	-0.53033008589Retained binary64: `-0.5303300858899107`	1.06066017178Retained binary64: `1.0606601717798214`	-1.06066017178Retained binary64: `-1.0606601717798214`	3/2Exact fraction: `3/2`Retained binary64: `1.5`
24	6*pi/4	6*pi/8	2*ln2/2	1/4Exact fraction: `1/4`Retained binary64: `0.25`	-1	0	-0.75Retained binary64: `-0.75`	0Retained binary64: `0.0`	-1.5Retained binary64: `-1.5`	3/2Exact fraction: `3/2`Retained binary64: `1.5`
25	7*pi/4	7*pi/8	2*ln2/2	1/4Exact fraction: `1/4`Retained binary64: `0.25`	-sqrt(2)/2	sqrt(2)/2	-0.53033008589Retained binary64: `-0.5303300858899107`	-1.06066017178Retained binary64: `-1.0606601717798214`	-1.06066017178Retained binary64: `-1.0606601717798214`	3/2Exact fraction: `3/2`Retained binary64: `1.5`
26	8*pi/4	8*pi/8	2*ln2/2	1/4Exact fraction: `1/4`Retained binary64: `0.25`	0	1	0Retained binary64: `0.0`	-1.5Retained binary64: `-1.5`	0Retained binary64: `0.0`	3/2Exact fraction: `3/2`Retained binary64: `1.5`
27	0*pi/4	0*pi/8	3*ln2/2	1/8Exact fraction: `1/8`Retained binary64: `0.125`	0	1	0Retained binary64: `0.0`	-0.75Retained binary64: `-0.75`	0Retained binary64: `0.0`	3/4Exact fraction: `3/4`Retained binary64: `0.75`
28	1*pi/4	1*pi/8	3*ln2/2	1/8Exact fraction: `1/8`Retained binary64: `0.125`	sqrt(2)/2	sqrt(2)/2	0.265165042945Retained binary64: `0.26516504294495535`	-0.53033008589Retained binary64: `-0.5303300858899107`	0.53033008589Retained binary64: `0.5303300858899107`	3/4Exact fraction: `3/4`Retained binary64: `0.75`
29	2*pi/4	2*pi/8	3*ln2/2	1/8Exact fraction: `1/8`Retained binary64: `0.125`	1	0	0.375Retained binary64: `0.375`	0Retained binary64: `0.0`	0.75Retained binary64: `0.75`	3/4Exact fraction: `3/4`Retained binary64: `0.75`
30	3*pi/4	3*pi/8	3*ln2/2	1/8Exact fraction: `1/8`Retained binary64: `0.125`	sqrt(2)/2	-sqrt(2)/2	0.265165042945Retained binary64: `0.26516504294495535`	0.53033008589Retained binary64: `0.5303300858899107`	0.53033008589Retained binary64: `0.5303300858899107`	3/4Exact fraction: `3/4`Retained binary64: `0.75`
31	4*pi/4	4*pi/8	3*ln2/2	1/8Exact fraction: `1/8`Retained binary64: `0.125`	0	-1	0Retained binary64: `0.0`	0.75Retained binary64: `0.75`	0Retained binary64: `0.0`	3/4Exact fraction: `3/4`Retained binary64: `0.75`
32	5*pi/4	5*pi/8	3*ln2/2	1/8Exact fraction: `1/8`Retained binary64: `0.125`	-sqrt(2)/2	-sqrt(2)/2	-0.265165042945Retained binary64: `-0.26516504294495535`	0.53033008589Retained binary64: `0.5303300858899107`	-0.53033008589Retained binary64: `-0.5303300858899107`	3/4Exact fraction: `3/4`Retained binary64: `0.75`
33	6*pi/4	6*pi/8	3*ln2/2	1/8Exact fraction: `1/8`Retained binary64: `0.125`	-1	0	-0.375Retained binary64: `-0.375`	0Retained binary64: `0.0`	-0.75Retained binary64: `-0.75`	3/4Exact fraction: `3/4`Retained binary64: `0.75`
34	7*pi/4	7*pi/8	3*ln2/2	1/8Exact fraction: `1/8`Retained binary64: `0.125`	-sqrt(2)/2	sqrt(2)/2	-0.265165042945Retained binary64: `-0.26516504294495535`	-0.53033008589Retained binary64: `-0.5303300858899107`	-0.53033008589Retained binary64: `-0.5303300858899107`	3/4Exact fraction: `3/4`Retained binary64: `0.75`
35	8*pi/4	8*pi/8	3*ln2/2	1/8Exact fraction: `1/8`Retained binary64: `0.125`	0	1	0Retained binary64: `0.0`	-0.75Retained binary64: `-0.75`	0Retained binary64: `0.0`	3/4Exact fraction: `3/4`Retained binary64: `0.75`

All 257 boundary samples atY=0 (257 rows)

All 257 boundary samples atY=0
Row	X/π=x/(πL₀)	Phase2X in radians	V/V₀	Signed Eₓ/E₀	Signed Eᵧ/E₀
0	0Exact fraction: `0`Retained binary64: `0.0`	0Retained binary64: `0.0`	0Retained binary64: `0.0`	-6Retained binary64: `-6.0`	0Retained binary64: `0.0`
1	1/256Exact fraction: `1/256`Retained binary64: `0.00390625`	0.0245436926062Retained binary64: `0.02454369260617026`	0.0736236855687Retained binary64: `0.07362368556873686`	-5.99819291218Retained binary64: `-5.998192912177226`	0.147247371137Retained binary64: `0.14724737113747371`
2	1/128Exact fraction: `1/128`Retained binary64: `0.0078125`	0.0490873852123Retained binary64: `0.04908738521234052`	0.147203022982Retained binary64: `0.14720302298225404`	-5.99277273723Retained binary64: `-5.992772737231034`	0.294406045965Retained binary64: `0.2944060459645081`
3	3/256Exact fraction: `3/256`Retained binary64: `0.01171875`	0.0736310778185Retained binary64: `0.07363107781851078`	0.220693690799Retained binary64: `0.22069369079900228`	-5.98374274007Retained binary64: `-5.983742740072142`	0.441387381598Retained binary64: `0.44138738159800456`
4	1/64Exact fraction: `1/64`Retained binary64: `0.015625`	0.0981747704247Retained binary64: `0.09817477042468103`	0.294051420989Retained binary64: `0.2940514209886818`	-5.97110836003Retained binary64: `-5.971108360033181`	0.588102841977Retained binary64: `0.5881028419773636`
5	5/256Exact fraction: `5/256`Retained binary64: `0.01953125`	0.122718463031Retained binary64: `0.1227184630308513`	0.367232025598Retained binary64: `0.36723202559764856`	-5.95487720759Retained binary64: `-5.95487720759226`	0.734464051195Retained binary64: `0.7344640511952971`
6	3/128Exact fraction: `3/128`Retained binary64: `0.0234375`	0.147262155637Retained binary64: `0.14726215563702155`	0.440191423366Retained binary64: `0.44019142336608524`	-5.93505905979Retained binary64: `-5.935059059788686`	0.880382846732Retained binary64: `0.8803828467321705`
7	7/256Exact fraction: `7/256`Retained binary64: `0.02734375`	0.171805848243Retained binary64: `0.1718058482431918`	0.512885666281Retained binary64: `0.5128856662809036`	-5.91166585433Retained binary64: `-5.9116658543336476`	1.02577133256Retained binary64: `1.0257713325618072`
8	1/32Exact fraction: `1/32`Retained binary64: `0.03125`	0.196349540849Retained binary64: `0.19634954084936207`	0.585270966048Retained binary64: `0.5852709660483848`	-5.88471168242Retained binary64: `-5.884711682419383`	1.1705419321Retained binary64: `1.1705419320967696`
9	9/256Exact fraction: `9/256`Retained binary64: `0.03515625`	0.220893233456Retained binary64: `0.22089323345553233`	0.657303720471Retained binary64: `0.6573037204706094`	-5.85421278023Retained binary64: `-5.854212780231172`	1.31460744094Retained binary64: `1.3146074409412187`
10	5/128Exact fraction: `5/128`Retained binary64: `0.0390625`	0.245436926062Retained binary64: `0.2454369260617026`	0.72894053971Retained binary64: `0.7289405397097917`	-5.82018751917Retained binary64: `-5.820187519167264`	1.45788107942Retained binary64: `1.4578810794195833`
11	11/256Exact fraction: `11/256`Retained binary64: `0.04296875`	0.269980618668Retained binary64: `0.2699806186678728`	0.800138272425Retained binary64: `0.800138272424695`	-5.78265639477Retained binary64: `-5.782656394772639`	1.60027654485Retained binary64: `1.60027654484939`
12	3/64Exact fraction: `3/64`Retained binary64: `0.046875`	0.294524311274Retained binary64: `0.2945243112740431`	0.870854031763Retained binary64: `0.870854031763387`	-5.74164201439Retained binary64: `-5.741642014393253`	1.74170806353Retained binary64: `1.741708063526774`
13	13/256Exact fraction: `13/256`Retained binary64: `0.05078125`	0.31906800388Retained binary64: `0.3190680038802134`	0.941045221197Retained binary64: `0.9410452211966746`	-5.69716908356Retained binary64: `-5.6971690835582205`	1.88209044239Retained binary64: `1.882090442393349`
14	7/128Exact fraction: `7/128`Retained binary64: `0.0546875`	0.343611696486Retained binary64: `0.3436116964863836`	1.01066956018Retained binary64: `1.0106695601766602`	-5.6492643911Retained binary64: `-5.649264391098125`	2.02133912035Retained binary64: `2.0213391203533204`
15	15/256Exact fraction: `15/256`Retained binary64: `0.05859375`	0.368155389093Retained binary64: `0.36815538909255385`	1.0796851096Retained binary64: `1.0796851096049642`	-5.59795679301Retained binary64: `-5.597956793008434`	2.15937021921Retained binary64: `2.1593702192099284`
16	1/16Exact fraction: `1/16`Retained binary64: `0.0625`	0.392699081699Retained binary64: `0.39269908169872414`	1.1480502971Retained binary64: `1.1480502970952693`	-5.54327719507Retained binary64: `-5.54327719506772`	2.29610059419Retained binary64: `2.2961005941905386`
17	17/256Exact fraction: `17/256`Retained binary64: `0.06640625`	0.417242774305Retained binary64: `0.4172427743048944`	1.21572394201Retained binary64: `1.2157239420149697`	-5.48525853422Retained binary64: `-5.485258534221185`	2.43144788403Retained binary64: `2.4314478840299394`
18	9/128Exact fraction: `9/128`Retained binary64: `0.0703125`	0.441786466911Retained binary64: `0.44178646691106466`	1.28266528029Retained binary64: `1.2826652802908463`	-5.42393575874Retained binary64: `-5.42393575874066`	2.56533056058Retained binary64: `2.5653305605816925`
19	19/256Exact fraction: `19/256`Retained binary64: `0.07421875`	0.466330159517Retained binary64: `0.4663301595172349`	1.34883398896Retained binary64: `1.3488339889638197`	-5.35934580717Retained binary64: `-5.359345807173092`	2.69766797793Retained binary64: `2.6976679779276393`
20	5/64Exact fraction: `5/64`Retained binary64: `0.078125`	0.490873852123Retained binary64: `0.4908738521234052`	1.41419021048Retained binary64: `1.4141902104779929`	-5.29152758609Retained binary64: `-5.2915275860901305`	2.82838042096Retained binary64: `2.8283804209559857`
21	21/256Exact fraction: `21/256`Retained binary64: `0.08203125`	0.51541754473Retained binary64: `0.5154175447295755`	1.47869457669Retained binary64: `1.4786945766893522`	-5.22052194665Retained binary64: `-5.220521946652269`	2.95738915338Retained binary64: `2.9573891533787044`
22	11/128Exact fraction: `11/128`Retained binary64: `0.0859375`	0.539961237336Retained binary64: `0.5399612373357456`	1.54230823258Retained binary64: `1.542308232579665`	-5.14637166Retained binary64: `-5.146371660001632`	3.08461646516Retained binary64: `3.08461646515933`
23	23/256Exact fraction: `23/256`Retained binary64: `0.08984375`	0.564504929942Retained binary64: `0.5645049299419159`	1.60499285966Retained binary64: `1.6049928596612915`	-5.0691213915Retained binary64: `-5.069121391498243`	3.20998571932Retained binary64: `3.209985719322583`
24	3/32Exact fraction: `3/32`Retained binary64: `0.09375`	0.589048622548Retained binary64: `0.5890486225480862`	1.66671069906Retained binary64: `1.6667106990588065`	-4.98881767382Retained binary64: `-4.988817673815271`	3.33342139812Retained binary64: `3.333421398117613`
25	25/256Exact fraction: `25/256`Retained binary64: `0.09765625`	0.613592315154Retained binary64: `0.6135923151542565`	1.72742457425Retained binary64: `1.727424574253536`	-4.90550887891Retained binary64: `-4.905508878909503`	3.45484914851Retained binary64: `3.454849148507072`
26	13/128Exact fraction: `13/128`Retained binary64: `0.1015625`	0.63813600776Retained binary64: `0.6381360077604268`	1.78709791348Retained binary64: `1.7870979134773002`	-4.81924518888Retained binary64: `-4.81924518888387`	3.57419582695Retained binary64: `3.5741958269546004`
27	27/256Exact fraction: `27/256`Retained binary64: `0.10546875`	0.662679700367Retained binary64: `0.662679700366597`	1.84569477174Retained binary64: `1.8456947717418806`	-4.73007856576Retained binary64: `-4.730078565759637`	3.69138954348Retained binary64: `3.691389543483761`
28	7/64Exact fraction: `7/64`Retained binary64: `0.109375`	0.687223392973Retained binary64: `0.6872233929727672`	1.90317985249Retained binary64: `1.9031798524909365`	-4.63806272018Retained binary64: `-4.638062720176422`	3.80635970498Retained binary64: `3.806359704981873`
29	29/256Exact fraction: `29/256`Retained binary64: `0.11328125`	0.711767085579Retained binary64: `0.7117670855789375`	1.95951852886Retained binary64: `1.9595185288613304`	-4.54325307904Retained binary64: `-4.543253079038907`	3.91903705772Retained binary64: `3.9190370577226608`
30	15/128Exact fraction: `15/128`Retained binary64: `0.1171875`	0.736310778185Retained binary64: `0.7363107781851077`	2.01467686454Retained binary64: `2.014676864541055`	-4.44570675213Retained binary64: `-4.445706752129755`	4.02935372908Retained binary64: `4.02935372908211`
31	31/256Exact fraction: `31/256`Retained binary64: `0.12109375`	0.760854470791Retained binary64: `0.760854470791278`	2.06862163421Retained binary64: `2.0686216342112003`	-4.34548249771Retained binary64: `-4.345482497708802`	4.13724326842Retained binary64: `4.1372432684224005`
32	1/8Exact fraction: `1/8`Retained binary64: `0.125`	0.785398163397Retained binary64: `0.7853981633974483`	2.12132034356Retained binary64: `2.121320343559643`	-4.24264068712Retained binary64: `-4.242640687119286`	4.24264068712Retained binary64: `4.242640687119286`
33	33/256Exact fraction: `33/256`Retained binary64: `0.12890625`	0.809941856004Retained binary64: `0.8099418560036186`	2.17274124885Retained binary64: `2.1727412488544005`	-4.13724326842Retained binary64: `-4.137243268422401`	4.34548249771Retained binary64: `4.345482497708801`
34	17/128Exact fraction: `17/128`Retained binary64: `0.1328125`	0.83448554861Retained binary64: `0.8344855486097889`	2.22285337606Retained binary64: `2.222853376064877`	-4.02935372908Retained binary64: `-4.02935372908211`	4.44570675213Retained binary64: `4.445706752129754`
35	35/256Exact fraction: `35/256`Retained binary64: `0.13671875`	0.859029241216Retained binary64: `0.859029241215959`	2.27162653952Retained binary64: `2.2716265395194535`	-3.91903705772Retained binary64: `-3.919037057722661`	4.54325307904Retained binary64: `4.543253079038907`
36	9/64Exact fraction: `9/64`Retained binary64: `0.140625`	0.883572933822Retained binary64: `0.8835729338221293`	2.31903136009Retained binary64: `2.3190313600882106`	-3.80635970498Retained binary64: `-3.806359704981873`	4.63806272018Retained binary64: `4.638062720176421`
37	37/256Exact fraction: `37/256`Retained binary64: `0.14453125`	0.908116626428Retained binary64: `0.9081166264282996`	2.36503928288Retained binary64: `2.3650392828798186`	-3.69138954348Retained binary64: `-3.691389543483761`	4.73007856576Retained binary64: `4.730078565759637`
38	19/128Exact fraction: `19/128`Retained binary64: `0.1484375`	0.932660319034Retained binary64: `0.9326603190344698`	2.40962259444Retained binary64: `2.4096225944419345`	-3.57419582695Retained binary64: `-3.574195826954601`	4.81924518888Retained binary64: `4.819245188883869`
39	39/256Exact fraction: `39/256`Retained binary64: `0.15234375`	0.957204011641Retained binary64: `0.9572040116406401`	2.45275443945Retained binary64: `2.4527544394547514`	-3.45484914851Retained binary64: `-3.454849148507072`	4.90550887891Retained binary64: `4.905508878909503`
40	5/32Exact fraction: `5/32`Retained binary64: `0.15625`	0.981747704247Retained binary64: `0.9817477042468103`	2.49440883691Retained binary64: `2.4944088369076356`	-3.33342139812Retained binary64: `-3.333421398117614`	4.98881767382Retained binary64: `4.988817673815271`
41	41/256Exact fraction: `41/256`Retained binary64: `0.16015625`	1.00629139685Retained binary64: `1.0062913968529805`	2.53456069575Retained binary64: `2.534560695749121`	-3.20998571932Retained binary64: `-3.2099857193225834`	5.0691213915Retained binary64: `5.069121391498242`
42	21/128Exact fraction: `21/128`Retained binary64: `0.1640625`	1.03083508946Retained binary64: `1.030835089459151`	2.57318583Retained binary64: `2.573185830000816`	-3.08461646516Retained binary64: `-3.08461646515933`	5.14637166Retained binary64: `5.146371660001632`
43	43/256Exact fraction: `43/256`Retained binary64: `0.16796875`	1.05537878207Retained binary64: `1.055378782065321`	2.61026097333Retained binary64: `2.610260973326134`	-2.95738915338Retained binary64: `-2.9573891533787044`	5.22052194665Retained binary64: `5.220521946652268`
44	11/64Exact fraction: `11/64`Retained binary64: `0.171875`	1.07992247467Retained binary64: `1.0799224746714913`	2.64576379305Retained binary64: `2.645763793045065`	-2.82838042096Retained binary64: `-2.8283804209559866`	5.29152758609Retained binary64: `5.29152758609013`
45	45/256Exact fraction: `45/256`Retained binary64: `0.17578125`	1.10446616728Retained binary64: `1.1044661672776617`	2.67967290359Retained binary64: `2.679672903586546`	-2.69766797793Retained binary64: `-2.6976679779276393`	5.35934580717Retained binary64: `5.359345807173092`
46	23/128Exact fraction: `23/128`Retained binary64: `0.1796875`	1.12900985988Retained binary64: `1.1290098598838318`	2.71196787937Retained binary64: `2.71196787937033`	-2.56533056058Retained binary64: `-2.5653305605816934`	5.42393575874Retained binary64: `5.42393575874066`
47	47/256Exact fraction: `47/256`Retained binary64: `0.18359375`	1.15355355249Retained binary64: `1.1535535524900022`	2.74262926711Retained binary64: `2.7426292671105923`	-2.43144788403Retained binary64: `-2.4314478840299394`	5.48525853422Retained binary64: `5.485258534221185`
48	3/16Exact fraction: `3/16`Retained binary64: `0.1875`	1.1780972451Retained binary64: `1.1780972450961724`	2.77163859753Retained binary64: `2.77163859753386`	-2.29610059419Retained binary64: `-2.296100594190539`	5.54327719507Retained binary64: `5.54327719506772`
49	49/256Exact fraction: `49/256`Retained binary64: `0.19140625`	1.2026409377Retained binary64: `1.2026409377023426`	2.7989783965Retained binary64: `2.7989783965042165`	-2.15937021921Retained binary64: `-2.1593702192099298`	5.59795679301Retained binary64: `5.597956793008433`
50	25/128Exact fraction: `25/128`Retained binary64: `0.1953125`	1.22718463031Retained binary64: `1.227184630308513`	2.82463219555Retained binary64: `2.8246321955490625`	-2.02133912035Retained binary64: `-2.0213391203533204`	5.6492643911Retained binary64: `5.649264391098125`
51	51/256Exact fraction: `51/256`Retained binary64: `0.19921875`	1.25172832291Retained binary64: `1.2517283229146832`	2.84858454178Retained binary64: `2.8485845417791102`	-1.88209044239Retained binary64: `-1.8820904423933493`	5.69716908356Retained binary64: `5.6971690835582205`
52	13/64Exact fraction: `13/64`Retained binary64: `0.203125`	1.27627201552Retained binary64: `1.2762720155208536`	2.8708210072Retained binary64: `2.8708210071966267`	-1.74170806353Retained binary64: `-1.741708063526774`	5.74164201439Retained binary64: `5.741642014393253`
53	53/256Exact fraction: `53/256`Retained binary64: `0.20703125`	1.30081570813Retained binary64: `1.3008157081270237`	2.89132819739Retained binary64: `2.8913281973863194`	-1.60027654485Retained binary64: `-1.6002765448493905`	5.78265639477Retained binary64: `5.782656394772639`
54	27/128Exact fraction: `27/128`Retained binary64: `0.2109375`	1.32535940073Retained binary64: `1.325359400733194`	2.91009375958Retained binary64: `2.910093759583632`	-1.45788107942Retained binary64: `-1.4578810794195838`	5.82018751917Retained binary64: `5.820187519167264`
55	55/256Exact fraction: `55/256`Retained binary64: `0.21484375`	1.34990309334Retained binary64: `1.3499030933393643`	2.92710639012Retained binary64: `2.927106390115586`	-1.31460744094Retained binary64: `-1.3146074409412187`	5.85421278023Retained binary64: `5.854212780231172`
56	7/32Exact fraction: `7/32`Retained binary64: `0.21875`	1.37444678595Retained binary64: `1.3744467859455345`	2.94235584121Retained binary64: `2.9423558412096913`	-1.1705419321Retained binary64: `-1.17054193209677`	5.88471168242Retained binary64: `5.884711682419383`
57	57/256Exact fraction: `57/256`Retained binary64: `0.22265625`	1.39899047855Retained binary64: `1.3989904785517047`	2.95583292717Retained binary64: `2.9558329271668238`	-1.02577133256Retained binary64: `-1.025771332561808`	5.91166585433Retained binary64: `5.9116658543336476`
58	29/128Exact fraction: `29/128`Retained binary64: `0.2265625`	1.42353417116Retained binary64: `1.423534171157875`	2.96752952989Retained binary64: `2.967529529894343`	-0.880382846732Retained binary64: `-0.8803828467321705`	5.93505905979Retained binary64: `5.935059059788686`
59	59/256Exact fraction: `59/256`Retained binary64: `0.23046875`	1.44807786376Retained binary64: `1.4480778637640452`	2.9774386038Retained binary64: `2.97743860379613`	-0.734464051195Retained binary64: `-0.7344640511952977`	5.95487720759Retained binary64: `5.95487720759226`
60	15/64Exact fraction: `15/64`Retained binary64: `0.234375`	1.47262155637Retained binary64: `1.4726215563702154`	2.98555418002Retained binary64: `2.9855541800165906`	-0.588102841977Retained binary64: `-0.5881028419773646`	5.97110836003Retained binary64: `5.971108360033181`
61	61/256Exact fraction: `61/256`Retained binary64: `0.23828125`	1.49716524898Retained binary64: `1.4971652489763858`	2.99187137004Retained binary64: `2.991871370036071`	-0.441387381598Retained binary64: `-0.4413873815980047`	5.98374274007Retained binary64: `5.983742740072142`
62	31/128Exact fraction: `31/128`Retained binary64: `0.2421875`	1.52170894158Retained binary64: `1.521708941582556`	2.99638636862Retained binary64: `2.996386368615517`	-0.294406045965Retained binary64: `-0.29440604596450876`	5.99277273723Retained binary64: `5.992772737231034`
63	63/256Exact fraction: `63/256`Retained binary64: `0.24609375`	1.54625263419Retained binary64: `1.5462526341887264`	2.99909645609Retained binary64: `2.999096456088613`	-0.147247371137Retained binary64: `-0.14724737113747358`	5.99819291218Retained binary64: `5.998192912177226`
64	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1.57079632679Retained binary64: `1.5707963267948966`	3Retained binary64: `3.0`	0Retained binary64: `0.0`	6Retained binary64: `6.0`
65	65/256Exact fraction: `65/256`Retained binary64: `0.25390625`	1.5953400194Retained binary64: `1.5953400194010667`	2.99909645609Retained binary64: `2.999096456088613`	0.147247371137Retained binary64: `0.14724737113747285`	5.99819291218Retained binary64: `5.998192912177226`
66	33/128Exact fraction: `33/128`Retained binary64: `0.2578125`	1.61988371201Retained binary64: `1.6198837120072371`	2.99638636862Retained binary64: `2.996386368615517`	0.294406045965Retained binary64: `0.29440604596450803`	5.99277273723Retained binary64: `5.992772737231034`
67	67/256Exact fraction: `67/256`Retained binary64: `0.26171875`	1.64442740461Retained binary64: `1.6444274046134073`	2.99187137004Retained binary64: `2.991871370036071`	0.441387381598Retained binary64: `0.44138738159800395`	5.98374274007Retained binary64: `5.983742740072142`
68	17/64Exact fraction: `17/64`Retained binary64: `0.265625`	1.66897109722Retained binary64: `1.6689710972195777`	2.98555418002Retained binary64: `2.9855541800165906`	0.588102841977Retained binary64: `0.5881028419773638`	5.97110836003Retained binary64: `5.971108360033181`
69	69/256Exact fraction: `69/256`Retained binary64: `0.26953125`	1.69351478983Retained binary64: `1.6935147898257479`	2.9774386038Retained binary64: `2.97743860379613`	0.734464051195Retained binary64: `0.7344640511952969`	5.95487720759Retained binary64: `5.95487720759226`
70	35/128Exact fraction: `35/128`Retained binary64: `0.2734375`	1.71805848243Retained binary64: `1.718058482431918`	2.96752952989Retained binary64: `2.967529529894343`	0.880382846732Retained binary64: `0.8803828467321698`	5.93505905979Retained binary64: `5.935059059788686`
71	71/256Exact fraction: `71/256`Retained binary64: `0.27734375`	1.74260217504Retained binary64: `1.7426021750380885`	2.95583292717Retained binary64: `2.9558329271668238`	1.02577133256Retained binary64: `1.0257713325618074`	5.91166585433Retained binary64: `5.9116658543336476`
72	9/32Exact fraction: `9/32`Retained binary64: `0.28125`	1.76714586764Retained binary64: `1.7671458676442586`	2.94235584121Retained binary64: `2.9423558412096913`	1.1705419321Retained binary64: `1.1705419320967692`	5.88471168242Retained binary64: `5.884711682419383`
73	73/256Exact fraction: `73/256`Retained binary64: `0.28515625`	1.79168956025Retained binary64: `1.7916895602504288`	2.92710639012Retained binary64: `2.927106390115586`	1.31460744094Retained binary64: `1.3146074409412178`	5.85421278023Retained binary64: `5.854212780231172`
74	37/128Exact fraction: `37/128`Retained binary64: `0.2890625`	1.81623325286Retained binary64: `1.8162332528565992`	2.91009375958Retained binary64: `2.910093759583632`	1.45788107942Retained binary64: `1.4578810794195833`	5.82018751917Retained binary64: `5.820187519167264`
75	75/256Exact fraction: `75/256`Retained binary64: `0.29296875`	1.84077694546Retained binary64: `1.8407769454627694`	2.89132819739Retained binary64: `2.8913281973863194`	1.60027654485Retained binary64: `1.6002765448493899`	5.78265639477Retained binary64: `5.782656394772639`
76	19/64Exact fraction: `19/64`Retained binary64: `0.296875`	1.86532063807Retained binary64: `1.8653206380689396`	2.8708210072Retained binary64: `2.8708210071966267`	1.74170806353Retained binary64: `1.741708063526773`	5.74164201439Retained binary64: `5.741642014393253`
77	77/256Exact fraction: `77/256`Retained binary64: `0.30078125`	1.88986433068Retained binary64: `1.88986433067511`	2.84858454178Retained binary64: `2.8485845417791102`	1.88209044239Retained binary64: `1.8820904423933484`	5.69716908356Retained binary64: `5.6971690835582205`
78	39/128Exact fraction: `39/128`Retained binary64: `0.3046875`	1.91440802328Retained binary64: `1.9144080232812801`	2.82463219555Retained binary64: `2.8246321955490625`	2.02133912035Retained binary64: `2.0213391203533195`	5.6492643911Retained binary64: `5.649264391098125`
79	79/256Exact fraction: `79/256`Retained binary64: `0.30859375`	1.93895171589Retained binary64: `1.9389517158874505`	2.7989783965Retained binary64: `2.7989783965042165`	2.15937021921Retained binary64: `2.159370219209929`	5.59795679301Retained binary64: `5.597956793008433`
80	5/16Exact fraction: `5/16`Retained binary64: `0.3125`	1.96349540849Retained binary64: `1.9634954084936207`	2.77163859753Retained binary64: `2.77163859753386`	2.29610059419Retained binary64: `2.2961005941905386`	5.54327719507Retained binary64: `5.54327719506772`
81	81/256Exact fraction: `81/256`Retained binary64: `0.31640625`	1.9880391011Retained binary64: `1.9880391010997909`	2.74262926711Retained binary64: `2.7426292671105923`	2.43144788403Retained binary64: `2.4314478840299385`	5.48525853422Retained binary64: `5.485258534221185`
82	41/128Exact fraction: `41/128`Retained binary64: `0.3203125`	2.01258279371Retained binary64: `2.012582793705961`	2.71196787937Retained binary64: `2.7119678793703303`	2.56533056058Retained binary64: `2.565330560581691`	5.42393575874Retained binary64: `5.423935758740661`
83	83/256Exact fraction: `83/256`Retained binary64: `0.32421875`	2.03712648631Retained binary64: `2.0371264863121317`	2.67967290359Retained binary64: `2.6796729035865456`	2.69766797793Retained binary64: `2.6976679779276402`	5.35934580717Retained binary64: `5.359345807173091`
84	21/64Exact fraction: `21/64`Retained binary64: `0.328125`	2.06167017892Retained binary64: `2.061670178918302`	2.64576379305Retained binary64: `2.6457637930450653`	2.82838042096Retained binary64: `2.828380420955986`	5.29152758609Retained binary64: `5.2915275860901305`
85	85/256Exact fraction: `85/256`Retained binary64: `0.33203125`	2.08621387152Retained binary64: `2.086213871524472`	2.61026097333Retained binary64: `2.6102609733261346`	2.95738915338Retained binary64: `2.957389153378704`	5.22052194665Retained binary64: `5.220521946652269`
86	43/128Exact fraction: `43/128`Retained binary64: `0.3359375`	2.11075756413Retained binary64: `2.110757564130642`	2.57318583Retained binary64: `2.573185830000816`	3.08461646516Retained binary64: `3.0846164651593293`	5.14637166Retained binary64: `5.146371660001632`
87	87/256Exact fraction: `87/256`Retained binary64: `0.33984375`	2.13530125674Retained binary64: `2.1353012567368124`	2.53456069575Retained binary64: `2.534560695749122`	3.20998571932Retained binary64: `3.2099857193225825`	5.0691213915Retained binary64: `5.069121391498244`
88	11/32Exact fraction: `11/32`Retained binary64: `0.34375`	2.15984494934Retained binary64: `2.1598449493429825`	2.49440883691Retained binary64: `2.494408836907636`	3.33342139812Retained binary64: `3.3334213981176117`	4.98881767382Retained binary64: `4.988817673815272`
89	89/256Exact fraction: `89/256`Retained binary64: `0.34765625`	2.18438864195Retained binary64: `2.184388641949153`	2.45275443945Retained binary64: `2.4527544394547514`	3.45484914851Retained binary64: `3.454849148507072`	4.90550887891Retained binary64: `4.905508878909503`
90	45/128Exact fraction: `45/128`Retained binary64: `0.3515625`	2.20893233456Retained binary64: `2.2089323345553233`	2.40962259444Retained binary64: `2.409622594441935`	3.57419582695Retained binary64: `3.5741958269546004`	4.81924518888Retained binary64: `4.81924518888387`
91	91/256Exact fraction: `91/256`Retained binary64: `0.35546875`	2.23347602716Retained binary64: `2.2334760271614935`	2.36503928288Retained binary64: `2.365039282879819`	3.69138954348Retained binary64: `3.6913895434837602`	4.73007856576Retained binary64: `4.730078565759638`
92	23/64Exact fraction: `23/64`Retained binary64: `0.359375`	2.25801971977Retained binary64: `2.2580197197676637`	2.31903136009Retained binary64: `2.3190313600882115`	3.80635970498Retained binary64: `3.8063597049818725`	4.63806272018Retained binary64: `4.638062720176423`
93	93/256Exact fraction: `93/256`Retained binary64: `0.36328125`	2.28256341237Retained binary64: `2.282563412373834`	2.27162653952Retained binary64: `2.271626539519454`	3.91903705772Retained binary64: `3.919037057722659`	4.54325307904Retained binary64: `4.543253079038908`
94	47/128Exact fraction: `47/128`Retained binary64: `0.3671875`	2.30710710498Retained binary64: `2.3071071049800045`	2.22285337606Retained binary64: `2.2228533760648768`	4.02935372908Retained binary64: `4.029353729082111`	4.44570675213Retained binary64: `4.4457067521297535`
95	95/256Exact fraction: `95/256`Retained binary64: `0.37109375`	2.33165079759Retained binary64: `2.3316507975861747`	2.17274124885Retained binary64: `2.1727412488544005`	4.13724326842Retained binary64: `4.137243268422401`	4.34548249771Retained binary64: `4.345482497708801`
96	3/8Exact fraction: `3/8`Retained binary64: `0.375`	2.35619449019Retained binary64: `2.356194490192345`	2.12132034356Retained binary64: `2.121320343559643`	4.24264068712Retained binary64: `4.242640687119285`	4.24264068712Retained binary64: `4.242640687119286`
97	97/256Exact fraction: `97/256`Retained binary64: `0.37890625`	2.3807381828Retained binary64: `2.380738182798515`	2.06862163421Retained binary64: `2.068621634211201`	4.34548249771Retained binary64: `4.345482497708801`	4.13724326842Retained binary64: `4.137243268422402`
98	49/128Exact fraction: `49/128`Retained binary64: `0.3828125`	2.4052818754Retained binary64: `2.405281875404685`	2.01467686454Retained binary64: `2.0146768645410558`	4.44570675213Retained binary64: `4.4457067521297535`	4.02935372908Retained binary64: `4.0293537290821115`
99	99/256Exact fraction: `99/256`Retained binary64: `0.38671875`	2.42982556801Retained binary64: `2.429825568010856`	1.95951852886Retained binary64: `1.95951852886133`	4.54325307904Retained binary64: `4.543253079038908`	3.91903705772Retained binary64: `3.91903705772266`
100	25/64Exact fraction: `25/64`Retained binary64: `0.390625`	2.45436926062Retained binary64: `2.454369260617026`	1.90317985249Retained binary64: `1.9031798524909365`	4.63806272018Retained binary64: `4.638062720176422`	3.80635970498Retained binary64: `3.806359704981873`
101	101/256Exact fraction: `101/256`Retained binary64: `0.39453125`	2.47891295322Retained binary64: `2.478912953223196`	1.84569477174Retained binary64: `1.8456947717418808`	4.73007856576Retained binary64: `4.730078565759637`	3.69138954348Retained binary64: `3.6913895434837616`
102	51/128Exact fraction: `51/128`Retained binary64: `0.3984375`	2.50345664583Retained binary64: `2.5034566458293663`	1.78709791348Retained binary64: `1.7870979134773004`	4.81924518888Retained binary64: `4.819245188883869`	3.57419582695Retained binary64: `3.574195826954601`
103	103/256Exact fraction: `103/256`Retained binary64: `0.40234375`	2.52800033844Retained binary64: `2.5280003384355365`	1.72742457425Retained binary64: `1.7274245742535363`	4.90550887891Retained binary64: `4.905508878909502`	3.45484914851Retained binary64: `3.4548491485070727`
104	13/32Exact fraction: `13/32`Retained binary64: `0.40625`	2.55254403104Retained binary64: `2.552544031041707`	1.66671069906Retained binary64: `1.6667106990588065`	4.98881767382Retained binary64: `4.988817673815272`	3.33342139812Retained binary64: `3.333421398117613`
105	105/256Exact fraction: `105/256`Retained binary64: `0.41015625`	2.57708772365Retained binary64: `2.5770877236478773`	1.60499285966Retained binary64: `1.6049928596612915`	5.0691213915Retained binary64: `5.069121391498243`	3.20998571932Retained binary64: `3.209985719322583`
106	53/128Exact fraction: `53/128`Retained binary64: `0.4140625`	2.60163141625Retained binary64: `2.6016314162540475`	1.54230823258Retained binary64: `1.5423082325796653`	5.14637166Retained binary64: `5.146371660001632`	3.08461646516Retained binary64: `3.0846164651593306`
107	107/256Exact fraction: `107/256`Retained binary64: `0.41796875`	2.62617510886Retained binary64: `2.6261751088602177`	1.47869457669Retained binary64: `1.4786945766893524`	5.22052194665Retained binary64: `5.220521946652268`	2.95738915338Retained binary64: `2.957389153378705`
108	27/64Exact fraction: `27/64`Retained binary64: `0.421875`	2.65071880147Retained binary64: `2.650718801466388`	1.41419021048Retained binary64: `1.4141902104779933`	5.29152758609Retained binary64: `5.29152758609013`	2.82838042096Retained binary64: `2.8283804209559866`
109	109/256Exact fraction: `109/256`Retained binary64: `0.42578125`	2.67526249407Retained binary64: `2.675262494072558`	1.34883398896Retained binary64: `1.3488339889638206`	5.35934580717Retained binary64: `5.359345807173091`	2.69766797793Retained binary64: `2.697667977927641`
110	55/128Exact fraction: `55/128`Retained binary64: `0.4296875`	2.69980618668Retained binary64: `2.6998061866787286`	1.28266528029Retained binary64: `1.282665280290846`	5.42393575874Retained binary64: `5.42393575874066`	2.56533056058Retained binary64: `2.565330560581692`
111	111/256Exact fraction: `111/256`Retained binary64: `0.43359375`	2.72434987928Retained binary64: `2.724349879284899`	1.21572394201Retained binary64: `1.2157239420149697`	5.48525853422Retained binary64: `5.485258534221185`	2.43144788403Retained binary64: `2.4314478840299394`
112	7/16Exact fraction: `7/16`Retained binary64: `0.4375`	2.74889357189Retained binary64: `2.748893571891069`	1.1480502971Retained binary64: `1.1480502970952697`	5.54327719507Retained binary64: `5.54327719506772`	2.29610059419Retained binary64: `2.2961005941905395`
113	113/256Exact fraction: `113/256`Retained binary64: `0.44140625`	2.7734372645Retained binary64: `2.773437264497239`	1.0796851096Retained binary64: `1.079685109604965`	5.59795679301Retained binary64: `5.597956793008433`	2.15937021921Retained binary64: `2.15937021920993`
114	57/128Exact fraction: `57/128`Retained binary64: `0.4453125`	2.7979809571Retained binary64: `2.7979809571034093`	1.01066956018Retained binary64: `1.010669560176661`	5.6492643911Retained binary64: `5.649264391098124`	2.02133912035Retained binary64: `2.021339120353322`
115	115/256Exact fraction: `115/256`Retained binary64: `0.44921875`	2.82252464971Retained binary64: `2.82252464970958`	0.941045221197Retained binary64: `0.9410452211966742`	5.69716908356Retained binary64: `5.6971690835582205`	1.88209044239Retained binary64: `1.8820904423933484`
116	29/64Exact fraction: `29/64`Retained binary64: `0.453125`	2.84706834232Retained binary64: `2.84706834231575`	0.870854031763Retained binary64: `0.8708540317633872`	5.74164201439Retained binary64: `5.741642014393253`	1.74170806353Retained binary64: `1.7417080635267743`
117	117/256Exact fraction: `117/256`Retained binary64: `0.45703125`	2.87161203492Retained binary64: `2.8716120349219203`	0.800138272425Retained binary64: `0.8001382724246955`	5.78265639477Retained binary64: `5.782656394772639`	1.60027654485Retained binary64: `1.600276544849391`
118	59/128Exact fraction: `59/128`Retained binary64: `0.4609375`	2.89615572753Retained binary64: `2.8961557275280905`	0.72894053971Retained binary64: `0.7289405397097922`	5.82018751917Retained binary64: `5.820187519167264`	1.45788107942Retained binary64: `1.4578810794195844`
119	119/256Exact fraction: `119/256`Retained binary64: `0.46484375`	2.92069942013Retained binary64: `2.9206994201342606`	0.657303720471Retained binary64: `0.6573037204706101`	5.85421278023Retained binary64: `5.854212780231171`	1.31460744094Retained binary64: `1.3146074409412203`
120	15/32Exact fraction: `15/32`Retained binary64: `0.46875`	2.94524311274Retained binary64: `2.945243112740431`	0.585270966048Retained binary64: `0.5852709660483858`	5.88471168242Retained binary64: `5.884711682419383`	1.1705419321Retained binary64: `1.1705419320967716`
121	121/256Exact fraction: `121/256`Retained binary64: `0.47265625`	2.96978680535Retained binary64: `2.9697868053466014`	0.512885666281Retained binary64: `0.5128856662809036`	5.91166585433Retained binary64: `5.9116658543336476`	1.02577133256Retained binary64: `1.0257713325618072`
122	61/128Exact fraction: `61/128`Retained binary64: `0.4765625`	2.99433049795Retained binary64: `2.9943304979527716`	0.440191423366Retained binary64: `0.4401914233660854`	5.93505905979Retained binary64: `5.935059059788686`	0.880382846732Retained binary64: `0.8803828467321708`
123	123/256Exact fraction: `123/256`Retained binary64: `0.48046875`	3.01887419056Retained binary64: `3.018874190558942`	0.367232025598Retained binary64: `0.36723202559764906`	5.95487720759Retained binary64: `5.95487720759226`	0.734464051195Retained binary64: `0.7344640511952981`
124	31/64Exact fraction: `31/64`Retained binary64: `0.484375`	3.04341788317Retained binary64: `3.043417883165112`	0.294051420989Retained binary64: `0.2940514209886825`	5.97110836003Retained binary64: `5.971108360033181`	0.588102841977Retained binary64: `0.588102841977365`
125	125/256Exact fraction: `125/256`Retained binary64: `0.48828125`	3.06796157577Retained binary64: `3.067961575771282`	0.220693690799Retained binary64: `0.2206936907990032`	5.98374274007Retained binary64: `5.983742740072142`	0.441387381598Retained binary64: `0.4413873815980064`
126	63/128Exact fraction: `63/128`Retained binary64: `0.4921875`	3.09250526838Retained binary64: `3.0925052683774528`	0.147203022982Retained binary64: `0.1472030229822539`	5.99277273723Retained binary64: `5.992772737231034`	0.294406045965Retained binary64: `0.2944060459645078`
127	127/256Exact fraction: `127/256`Retained binary64: `0.49609375`	3.11704896098Retained binary64: `3.117048960983623`	0.0736236855687Retained binary64: `0.07362368556873698`	5.99819291218Retained binary64: `5.998192912177226`	0.147247371137Retained binary64: `0.14724737113747396`
128	1/2Exact fraction: `1/2`Retained binary64: `0.5`	3.14159265359Retained binary64: `3.141592653589793`	0Retained binary64: `0.0`	6Retained binary64: `6.0`	0Retained binary64: `0.0`
129	129/256Exact fraction: `129/256`Retained binary64: `0.50390625`	3.1661363462Retained binary64: `3.1661363461959633`	-0.0736236855687Retained binary64: `-0.07362368556873625`	5.99819291218Retained binary64: `5.998192912177226`	-0.147247371137Retained binary64: `-0.1472473711374725`
130	65/128Exact fraction: `65/128`Retained binary64: `0.5078125`	3.1906800388Retained binary64: `3.1906800388021335`	-0.147203022982Retained binary64: `-0.14720302298225318`	5.99277273723Retained binary64: `5.992772737231034`	-0.294406045965Retained binary64: `-0.29440604596450637`
131	131/256Exact fraction: `131/256`Retained binary64: `0.51171875`	3.21522373141Retained binary64: `3.215223731408304`	-0.220693690799Retained binary64: `-0.22069369079900247`	5.98374274007Retained binary64: `5.983742740072142`	-0.441387381598Retained binary64: `-0.44138738159800495`
132	33/64Exact fraction: `33/64`Retained binary64: `0.515625`	3.23976742401Retained binary64: `3.2397674240144743`	-0.294051420989Retained binary64: `-0.29405142098868176`	5.97110836003Retained binary64: `5.971108360033181`	-0.588102841977Retained binary64: `-0.5881028419773635`
133	133/256Exact fraction: `133/256`Retained binary64: `0.51953125`	3.26431111662Retained binary64: `3.2643111166206444`	-0.367232025598Retained binary64: `-0.3672320255976483`	5.95487720759Retained binary64: `5.95487720759226`	-0.734464051195Retained binary64: `-0.7344640511952966`
134	67/128Exact fraction: `67/128`Retained binary64: `0.5234375`	3.28885480923Retained binary64: `3.2888548092268146`	-0.440191423366Retained binary64: `-0.44019142336608474`	5.93505905979Retained binary64: `5.935059059788686`	-0.880382846732Retained binary64: `-0.8803828467321695`
135	135/256Exact fraction: `135/256`Retained binary64: `0.52734375`	3.31339850183Retained binary64: `3.313398501832985`	-0.512885666281Retained binary64: `-0.5128856662809029`	5.91166585433Retained binary64: `5.911665854333648`	-1.02577133256Retained binary64: `-1.0257713325618059`
136	17/32Exact fraction: `17/32`Retained binary64: `0.53125`	3.33794219444Retained binary64: `3.3379421944391554`	-0.585270966048Retained binary64: `-0.585270966048385`	5.88471168242Retained binary64: `5.884711682419383`	-1.1705419321Retained binary64: `-1.17054193209677`
137	137/256Exact fraction: `137/256`Retained binary64: `0.53515625`	3.36248588705Retained binary64: `3.3624858870453256`	-0.657303720471Retained binary64: `-0.6573037204706094`	5.85421278023Retained binary64: `5.854212780231172`	-1.31460744094Retained binary64: `-1.3146074409412187`
138	69/128Exact fraction: `69/128`Retained binary64: `0.5390625`	3.38702957965Retained binary64: `3.3870295796514958`	-0.72894053971Retained binary64: `-0.7289405397097914`	5.82018751917Retained binary64: `5.820187519167264`	-1.45788107942Retained binary64: `-1.457881079419583`
139	139/256Exact fraction: `139/256`Retained binary64: `0.54296875`	3.41157327226Retained binary64: `3.411573272257666`	-0.800138272425Retained binary64: `-0.8001382724246948`	5.78265639477Retained binary64: `5.78265639477264`	-1.60027654485Retained binary64: `-1.6002765448493896`
140	35/64Exact fraction: `35/64`Retained binary64: `0.546875`	3.43611696486Retained binary64: `3.436116964863836`	-0.870854031763Retained binary64: `-0.8708540317633864`	5.74164201439Retained binary64: `5.741642014393253`	-1.74170806353Retained binary64: `-1.7417080635267728`
141	141/256Exact fraction: `141/256`Retained binary64: `0.55078125`	3.46066065747Retained binary64: `3.4606606574700063`	-0.941045221197Retained binary64: `-0.9410452211966736`	5.69716908356Retained binary64: `5.6971690835582205`	-1.88209044239Retained binary64: `-1.882090442393347`
142	71/128Exact fraction: `71/128`Retained binary64: `0.5546875`	3.48520435008Retained binary64: `3.485204350076177`	-1.01066956018Retained binary64: `-1.0106695601766602`	5.6492643911Retained binary64: `5.649264391098125`	-2.02133912035Retained binary64: `-2.0213391203533204`
143	143/256Exact fraction: `143/256`Retained binary64: `0.55859375`	3.50974804268Retained binary64: `3.509748042682347`	-1.0796851096Retained binary64: `-1.0796851096049642`	5.59795679301Retained binary64: `5.597956793008434`	-2.15937021921Retained binary64: `-2.1593702192099284`
144	9/16Exact fraction: `9/16`Retained binary64: `0.5625`	3.53429173529Retained binary64: `3.5342917352885173`	-1.1480502971Retained binary64: `-1.148050297095269`	5.54327719507Retained binary64: `5.543277195067721`	-2.29610059419Retained binary64: `-2.296100594190538`
145	145/256Exact fraction: `145/256`Retained binary64: `0.56640625`	3.55883542789Retained binary64: `3.5588354278946874`	-1.21572394201Retained binary64: `-1.215723942014969`	5.48525853422Retained binary64: `5.485258534221185`	-2.43144788403Retained binary64: `-2.431447884029938`
146	73/128Exact fraction: `73/128`Retained binary64: `0.5703125`	3.5833791205Retained binary64: `3.5833791205008576`	-1.28266528029Retained binary64: `-1.2826652802908454`	5.42393575874Retained binary64: `5.423935758740661`	-2.56533056058Retained binary64: `-2.5653305605816907`
147	147/256Exact fraction: `147/256`Retained binary64: `0.57421875`	3.60792281311Retained binary64: `3.6079228131070282`	-1.34883398896Retained binary64: `-1.34883398896382`	5.35934580717Retained binary64: `5.359345807173092`	-2.69766797793Retained binary64: `-2.69766797792764`
148	37/64Exact fraction: `37/64`Retained binary64: `0.578125`	3.63246650571Retained binary64: `3.6324665057131984`	-1.41419021048Retained binary64: `-1.4141902104779929`	5.29152758609Retained binary64: `5.2915275860901305`	-2.82838042096Retained binary64: `-2.8283804209559857`
149	149/256Exact fraction: `149/256`Retained binary64: `0.58203125`	3.65701019832Retained binary64: `3.6570101983193686`	-1.47869457669Retained binary64: `-1.4786945766893518`	5.22052194665Retained binary64: `5.220521946652269`	-2.95738915338Retained binary64: `-2.9573891533787036`
150	75/128Exact fraction: `75/128`Retained binary64: `0.5859375`	3.68155389093Retained binary64: `3.6815538909255388`	-1.54230823258Retained binary64: `-1.5423082325796647`	5.14637166Retained binary64: `5.146371660001632`	-3.08461646516Retained binary64: `-3.0846164651593293`
151	151/256Exact fraction: `151/256`Retained binary64: `0.58984375`	3.70609758353Retained binary64: `3.706097583531709`	-1.60499285966Retained binary64: `-1.6049928596612908`	5.0691213915Retained binary64: `5.069121391498244`	-3.20998571932Retained binary64: `-3.2099857193225816`
152	19/32Exact fraction: `19/32`Retained binary64: `0.59375`	3.73064127614Retained binary64: `3.730641276137879`	-1.66671069906Retained binary64: `-1.6667106990588059`	4.98881767382Retained binary64: `4.988817673815273`	-3.33342139812Retained binary64: `-3.3334213981176117`
153	153/256Exact fraction: `153/256`Retained binary64: `0.59765625`	3.75518496874Retained binary64: `3.7551849687440497`	-1.72742457425Retained binary64: `-1.727424574253536`	4.90550887891Retained binary64: `4.905508878909503`	-3.45484914851Retained binary64: `-3.454849148507072`
154	77/128Exact fraction: `77/128`Retained binary64: `0.6015625`	3.77972866135Retained binary64: `3.77972866135022`	-1.78709791348Retained binary64: `-1.7870979134772997`	4.81924518888Retained binary64: `4.81924518888387`	-3.57419582695Retained binary64: `-3.5741958269545995`
155	155/256Exact fraction: `155/256`Retained binary64: `0.60546875`	3.80427235396Retained binary64: `3.80427235395639`	-1.84569477174Retained binary64: `-1.8456947717418801`	4.73007856576Retained binary64: `4.730078565759638`	-3.69138954348Retained binary64: `-3.6913895434837602`
156	39/64Exact fraction: `39/64`Retained binary64: `0.609375`	3.82881604656Retained binary64: `3.8288160465625602`	-1.90317985249Retained binary64: `-1.9031798524909358`	4.63806272018Retained binary64: `4.638062720176423`	-3.80635970498Retained binary64: `-3.8063597049818716`
157	157/256Exact fraction: `157/256`Retained binary64: `0.61328125`	3.85335973917Retained binary64: `3.8533597391687304`	-1.95951852886Retained binary64: `-1.9595185288613295`	4.54325307904Retained binary64: `4.543253079038909`	-3.91903705772Retained binary64: `-3.919037057722659`
158	79/128Exact fraction: `79/128`Retained binary64: `0.6171875`	3.87790343177Retained binary64: `3.877903431774901`	-2.01467686454Retained binary64: `-2.0146768645410553`	4.44570675213Retained binary64: `4.445706752129754`	-4.02935372908Retained binary64: `-4.029353729082111`
159	159/256Exact fraction: `159/256`Retained binary64: `0.62109375`	3.90244712438Retained binary64: `3.902447124381071`	-2.06862163421Retained binary64: `-2.0686216342112003`	4.34548249771Retained binary64: `4.345482497708802`	-4.13724326842Retained binary64: `-4.1372432684224005`
160	5/8Exact fraction: `5/8`Retained binary64: `0.625`	3.92699081699Retained binary64: `3.9269908169872414`	-2.12132034356Retained binary64: `-2.1213203435596424`	4.24264068712Retained binary64: `4.242640687119286`	-4.24264068712Retained binary64: `-4.242640687119285`
161	161/256Exact fraction: `161/256`Retained binary64: `0.62890625`	3.95153450959Retained binary64: `3.9515345095934116`	-2.17274124885Retained binary64: `-2.1727412488544005`	4.13724326842Retained binary64: `4.137243268422402`	-4.34548249771Retained binary64: `-4.345482497708801`
162	81/128Exact fraction: `81/128`Retained binary64: `0.6328125`	3.9760782022Retained binary64: `3.9760782021995817`	-2.22285337606Retained binary64: `-2.2228533760648768`	4.02935372908Retained binary64: `4.029353729082112`	-4.44570675213Retained binary64: `-4.4457067521297535`
163	163/256Exact fraction: `163/256`Retained binary64: `0.63671875`	4.00062189481Retained binary64: `4.000621894805752`	-2.27162653952Retained binary64: `-2.2716265395194526`	3.91903705772Retained binary64: `3.9190370577226625`	-4.54325307904Retained binary64: `-4.543253079038905`
164	41/64Exact fraction: `41/64`Retained binary64: `0.640625`	4.02516558741Retained binary64: `4.025165587411922`	-2.31903136009Retained binary64: `-2.3190313600882098`	3.80635970498Retained binary64: `3.8063597049818756`	-4.63806272018Retained binary64: `-4.6380627201764195`
165	165/256Exact fraction: `165/256`Retained binary64: `0.64453125`	4.04970928002Retained binary64: `4.049709280018092`	-2.36503928288Retained binary64: `-2.3650392828798177`	3.69138954348Retained binary64: `3.691389543483764`	-4.73007856576Retained binary64: `-4.730078565759635`
166	83/128Exact fraction: `83/128`Retained binary64: `0.6484375`	4.07425297262Retained binary64: `4.074252972624263`	-2.40962259444Retained binary64: `-2.4096225944419354`	3.57419582695Retained binary64: `3.5741958269545986`	-4.81924518888Retained binary64: `-4.819245188883871`
167	167/256Exact fraction: `167/256`Retained binary64: `0.65234375`	4.09879666523Retained binary64: `4.0987966652304335`	-2.45275443945Retained binary64: `-2.4527544394547514`	3.45484914851Retained binary64: `3.4548491485070714`	-4.90550887891Retained binary64: `-4.905508878909503`
168	21/32Exact fraction: `21/32`Retained binary64: `0.65625`	4.12334035784Retained binary64: `4.123340357836604`	-2.49440883691Retained binary64: `-2.4944088369076356`	3.33342139812Retained binary64: `3.333421398117613`	-4.98881767382Retained binary64: `-4.988817673815271`
169	169/256Exact fraction: `169/256`Retained binary64: `0.66015625`	4.14788405044Retained binary64: `4.147884050442774`	-2.53456069575Retained binary64: `-2.5345606957491214`	3.20998571932Retained binary64: `3.2099857193225834`	-5.0691213915Retained binary64: `-5.069121391498243`
170	85/128Exact fraction: `85/128`Retained binary64: `0.6640625`	4.17242774305Retained binary64: `4.172427743048944`	-2.57318583Retained binary64: `-2.573185830000816`	3.08461646516Retained binary64: `3.0846164651593306`	-5.14637166Retained binary64: `-5.146371660001632`
171	171/256Exact fraction: `171/256`Retained binary64: `0.66796875`	4.19697143566Retained binary64: `4.196971435655114`	-2.61026097333Retained binary64: `-2.610260973326134`	2.95738915338Retained binary64: `2.9573891533787053`	-5.22052194665Retained binary64: `-5.220521946652268`
172	43/64Exact fraction: `43/64`Retained binary64: `0.671875`	4.22151512826Retained binary64: `4.221515128261284`	-2.64576379305Retained binary64: `-2.645763793045065`	2.82838042096Retained binary64: `2.828380420955987`	-5.29152758609Retained binary64: `-5.29152758609013`
173	173/256Exact fraction: `173/256`Retained binary64: `0.67578125`	4.24605882087Retained binary64: `4.246058820867455`	-2.67967290359Retained binary64: `-2.6796729035865456`	2.69766797793Retained binary64: `2.6976679779276416`	-5.35934580717Retained binary64: `-5.359345807173091`
174	87/128Exact fraction: `87/128`Retained binary64: `0.6796875`	4.27060251347Retained binary64: `4.270602513473625`	-2.71196787937Retained binary64: `-2.7119678793703295`	2.56533056058Retained binary64: `2.5653305605816947`	-5.42393575874Retained binary64: `-5.423935758740659`
175	175/256Exact fraction: `175/256`Retained binary64: `0.68359375`	4.29514620608Retained binary64: `4.295146206079795`	-2.74262926711Retained binary64: `-2.7426292671105914`	2.43144788403Retained binary64: `2.431447884029942`	-5.48525853422Retained binary64: `-5.485258534221183`
176	11/16Exact fraction: `11/16`Retained binary64: `0.6875`	4.31968989869Retained binary64: `4.319689898685965`	-2.77163859753Retained binary64: `-2.7716385975338595`	2.29610059419Retained binary64: `2.296100594190542`	-5.54327719507Retained binary64: `-5.543277195067719`
177	177/256Exact fraction: `177/256`Retained binary64: `0.69140625`	4.34423359129Retained binary64: `4.344233591292136`	-2.7989783965Retained binary64: `-2.798978396504217`	2.15937021921Retained binary64: `2.1593702192099276`	-5.59795679301Retained binary64: `-5.597956793008434`
178	89/128Exact fraction: `89/128`Retained binary64: `0.6953125`	4.3687772839Retained binary64: `4.368777283898306`	-2.82463219555Retained binary64: `-2.8246321955490625`	2.02133912035Retained binary64: `2.0213391203533195`	-5.6492643911Retained binary64: `-5.649264391098125`
179	179/256Exact fraction: `179/256`Retained binary64: `0.69921875`	4.3933209765Retained binary64: `4.3933209765044765`	-2.84858454178Retained binary64: `-2.8485845417791102`	1.88209044239Retained binary64: `1.8820904423933489`	-5.69716908356Retained binary64: `-5.6971690835582205`
180	45/64Exact fraction: `45/64`Retained binary64: `0.703125`	4.41786466911Retained binary64: `4.417864669110647`	-2.8708210072Retained binary64: `-2.8708210071966267`	1.74170806353Retained binary64: `1.7417080635267745`	-5.74164201439Retained binary64: `-5.741642014393253`
181	181/256Exact fraction: `181/256`Retained binary64: `0.70703125`	4.44240836172Retained binary64: `4.442408361716817`	-2.89132819739Retained binary64: `-2.8913281973863194`	1.60027654485Retained binary64: `1.6002765448493912`	-5.78265639477Retained binary64: `-5.782656394772639`
182	91/128Exact fraction: `91/128`Retained binary64: `0.7109375`	4.46695205432Retained binary64: `4.466952054322987`	-2.91009375958Retained binary64: `-2.910093759583632`	1.45788107942Retained binary64: `1.4578810794195847`	-5.82018751917Retained binary64: `-5.820187519167264`
183	183/256Exact fraction: `183/256`Retained binary64: `0.71484375`	4.49149574693Retained binary64: `4.491495746929157`	-2.92710639012Retained binary64: `-2.9271063901155854`	1.31460744094Retained binary64: `1.3146074409412205`	-5.85421278023Retained binary64: `-5.854212780231171`
184	23/32Exact fraction: `23/32`Retained binary64: `0.71875`	4.51603943954Retained binary64: `4.516039439535327`	-2.94235584121Retained binary64: `-2.942355841209691`	1.1705419321Retained binary64: `1.170541932096772`	-5.88471168242Retained binary64: `-5.884711682419382`
185	185/256Exact fraction: `185/256`Retained binary64: `0.72265625`	4.54058313214Retained binary64: `4.540583132141498`	-2.95583292717Retained binary64: `-2.9558329271668233`	1.02577133256Retained binary64: `1.02577133256181`	-5.91166585433Retained binary64: `-5.911665854333647`
186	93/128Exact fraction: `93/128`Retained binary64: `0.7265625`	4.56512682475Retained binary64: `4.565126824747668`	-2.96752952989Retained binary64: `-2.9675295298943425`	0.880382846732Retained binary64: `0.8803828467321738`	-5.93505905979Retained binary64: `-5.935059059788685`
187	187/256Exact fraction: `187/256`Retained binary64: `0.73046875`	4.58967051735Retained binary64: `4.589670517353839`	-2.9774386038Retained binary64: `-2.9774386037961302`	0.734464051195Retained binary64: `0.7344640511952958`	-5.95487720759Retained binary64: `-5.9548772075922605`
188	47/64Exact fraction: `47/64`Retained binary64: `0.734375`	4.61421420996Retained binary64: `4.614214209960009`	-2.98555418002Retained binary64: `-2.9855541800165906`	0.588102841977Retained binary64: `0.5881028419773627`	-5.97110836003Retained binary64: `-5.971108360033181`
189	189/256Exact fraction: `189/256`Retained binary64: `0.73828125`	4.63875790257Retained binary64: `4.638757902566179`	-2.99187137004Retained binary64: `-2.991871370036071`	0.441387381598Retained binary64: `0.44138738159800417`	-5.98374274007Retained binary64: `-5.983742740072142`
190	95/128Exact fraction: `95/128`Retained binary64: `0.7421875`	4.66330159517Retained binary64: `4.663301595172349`	-2.99638636862Retained binary64: `-2.996386368615517`	0.294406045965Retained binary64: `0.29440604596450815`	-5.99277273723Retained binary64: `-5.992772737231034`
191	191/256Exact fraction: `191/256`Retained binary64: `0.74609375`	4.68784528778Retained binary64: `4.6878452877785195`	-2.99909645609Retained binary64: `-2.999096456088613`	0.147247371137Retained binary64: `0.14724737113747433`	-5.99819291218Retained binary64: `-5.998192912177226`
192	3/4Exact fraction: `3/4`Retained binary64: `0.75`	4.71238898038Retained binary64: `4.71238898038469`	-3Retained binary64: `-3.0`	0Retained binary64: `0.0`	-6Retained binary64: `-6.0`
193	193/256Exact fraction: `193/256`Retained binary64: `0.75390625`	4.73693267299Retained binary64: `4.73693267299086`	-2.99909645609Retained binary64: `-2.999096456088613`	-0.147247371137Retained binary64: `-0.14724737113747213`	-5.99819291218Retained binary64: `-5.998192912177226`
194	97/128Exact fraction: `97/128`Retained binary64: `0.7578125`	4.7614763656Retained binary64: `4.76147636559703`	-2.99638636862Retained binary64: `-2.996386368615517`	-0.294406045965Retained binary64: `-0.294406045964506`	-5.99277273723Retained binary64: `-5.992772737231034`
195	195/256Exact fraction: `195/256`Retained binary64: `0.76171875`	4.7860200582Retained binary64: `4.7860200582032`	-2.99187137004Retained binary64: `-2.991871370036071`	-0.441387381598Retained binary64: `-0.4413873815980019`	-5.98374274007Retained binary64: `-5.983742740072142`
196	49/64Exact fraction: `49/64`Retained binary64: `0.765625`	4.81056375081Retained binary64: `4.81056375080937`	-2.98555418002Retained binary64: `-2.9855541800165906`	-0.588102841977Retained binary64: `-0.5881028419773605`	-5.97110836003Retained binary64: `-5.971108360033181`
197	197/256Exact fraction: `197/256`Retained binary64: `0.76953125`	4.83510744342Retained binary64: `4.8351074434155406`	-2.9774386038Retained binary64: `-2.9774386037961302`	-0.734464051195Retained binary64: `-0.7344640511952936`	-5.95487720759Retained binary64: `-5.9548772075922605`
198	99/128Exact fraction: `99/128`Retained binary64: `0.7734375`	4.85965113602Retained binary64: `4.859651136021712`	-2.96752952989Retained binary64: `-2.9675295298943425`	-0.880382846732Retained binary64: `-0.8803828467321717`	-5.93505905979Retained binary64: `-5.935059059788685`
199	199/256Exact fraction: `199/256`Retained binary64: `0.77734375`	4.88419482863Retained binary64: `4.884194828627882`	-2.95583292717Retained binary64: `-2.9558329271668238`	-1.02577133256Retained binary64: `-1.025771332561808`	-5.91166585433Retained binary64: `-5.9116658543336476`
200	25/32Exact fraction: `25/32`Retained binary64: `0.78125`	4.90873852123Retained binary64: `4.908738521234052`	-2.94235584121Retained binary64: `-2.9423558412096913`	-1.1705419321Retained binary64: `-1.1705419320967698`	-5.88471168242Retained binary64: `-5.884711682419383`
201	201/256Exact fraction: `201/256`Retained binary64: `0.78515625`	4.93328221384Retained binary64: `4.933282213840222`	-2.92710639012Retained binary64: `-2.927106390115586`	-1.31460744094Retained binary64: `-1.3146074409412185`	-5.85421278023Retained binary64: `-5.854212780231172`
202	101/128Exact fraction: `101/128`Retained binary64: `0.7890625`	4.95782590645Retained binary64: `4.957825906446392`	-2.91009375958Retained binary64: `-2.910093759583632`	-1.45788107942Retained binary64: `-1.4578810794195824`	-5.82018751917Retained binary64: `-5.820187519167264`
203	203/256Exact fraction: `203/256`Retained binary64: `0.79296875`	4.98236959905Retained binary64: `4.9823695990525625`	-2.89132819739Retained binary64: `-2.89132819738632`	-1.60027654485Retained binary64: `-1.6002765448493892`	-5.78265639477Retained binary64: `-5.78265639477264`
204	51/64Exact fraction: `51/64`Retained binary64: `0.796875`	5.00691329166Retained binary64: `5.006913291658733`	-2.8708210072Retained binary64: `-2.8708210071966267`	-1.74170806353Retained binary64: `-1.7417080635267723`	-5.74164201439Retained binary64: `-5.741642014393253`
205	205/256Exact fraction: `205/256`Retained binary64: `0.80078125`	5.03145698426Retained binary64: `5.031456984264903`	-2.84858454178Retained binary64: `-2.8485845417791102`	-1.88209044239Retained binary64: `-1.8820904423933467`	-5.69716908356Retained binary64: `-5.6971690835582205`
206	103/128Exact fraction: `103/128`Retained binary64: `0.8046875`	5.05600067687Retained binary64: `5.056000676871073`	-2.82463219555Retained binary64: `-2.8246321955490625`	-2.02133912035Retained binary64: `-2.0213391203533178`	-5.6492643911Retained binary64: `-5.649264391098125`
207	207/256Exact fraction: `207/256`Retained binary64: `0.80859375`	5.08054436948Retained binary64: `5.080544369477243`	-2.7989783965Retained binary64: `-2.798978396504217`	-2.15937021921Retained binary64: `-2.159370219209926`	-5.59795679301Retained binary64: `-5.597956793008434`
208	13/16Exact fraction: `13/16`Retained binary64: `0.8125`	5.10508806208Retained binary64: `5.105088062083414`	-2.77163859753Retained binary64: `-2.77163859753386`	-2.29610059419Retained binary64: `-2.29610059419054`	-5.54327719507Retained binary64: `-5.54327719506772`
209	209/256Exact fraction: `209/256`Retained binary64: `0.81640625`	5.12963175469Retained binary64: `5.129631754689584`	-2.74262926711Retained binary64: `-2.742629267110592`	-2.43144788403Retained binary64: `-2.4314478840299403`	-5.48525853422Retained binary64: `-5.485258534221184`
210	105/128Exact fraction: `105/128`Retained binary64: `0.8203125`	5.1541754473Retained binary64: `5.154175447295755`	-2.71196787937Retained binary64: `-2.71196787937033`	-2.56533056058Retained binary64: `-2.565330560581693`	-5.42393575874Retained binary64: `-5.42393575874066`
211	211/256Exact fraction: `211/256`Retained binary64: `0.82421875`	5.1787191399Retained binary64: `5.178719139901925`	-2.67967290359Retained binary64: `-2.679672903586546`	-2.69766797793Retained binary64: `-2.6976679779276393`	-5.35934580717Retained binary64: `-5.359345807173092`
212	53/64Exact fraction: `53/64`Retained binary64: `0.828125`	5.20326283251Retained binary64: `5.203262832508095`	-2.64576379305Retained binary64: `-2.6457637930450653`	-2.82838042096Retained binary64: `-2.8283804209559857`	-5.29152758609Retained binary64: `-5.2915275860901305`
213	213/256Exact fraction: `213/256`Retained binary64: `0.83203125`	5.22780652511Retained binary64: `5.227806525114265`	-2.61026097333Retained binary64: `-2.6102609733261346`	-2.95738915338Retained binary64: `-2.957389153378703`	-5.22052194665Retained binary64: `-5.220521946652269`
214	107/128Exact fraction: `107/128`Retained binary64: `0.8359375`	5.25235021772Retained binary64: `5.252350217720435`	-2.57318583Retained binary64: `-2.5731858300008166`	-3.08461646516Retained binary64: `-3.0846164651593293`	-5.14637166Retained binary64: `-5.146371660001633`
215	215/256Exact fraction: `215/256`Retained binary64: `0.83984375`	5.27689391033Retained binary64: `5.2768939103266055`	-2.53456069575Retained binary64: `-2.534560695749122`	-3.20998571932Retained binary64: `-3.2099857193225816`	-5.0691213915Retained binary64: `-5.069121391498244`
216	27/32Exact fraction: `27/32`Retained binary64: `0.84375`	5.30143760293Retained binary64: `5.301437602932776`	-2.49440883691Retained binary64: `-2.4944088369076365`	-3.33342139812Retained binary64: `-3.3334213981176113`	-4.98881767382Retained binary64: `-4.988817673815273`
217	217/256Exact fraction: `217/256`Retained binary64: `0.84765625`	5.32598129554Retained binary64: `5.325981295538946`	-2.45275443945Retained binary64: `-2.452754439454752`	-3.45484914851Retained binary64: `-3.454849148507069`	-4.90550887891Retained binary64: `-4.905508878909504`
218	109/128Exact fraction: `109/128`Retained binary64: `0.8515625`	5.35052498815Retained binary64: `5.350524988145116`	-2.40962259444Retained binary64: `-2.409622594441936`	-3.57419582695Retained binary64: `-3.574195826954597`	-4.81924518888Retained binary64: `-4.819245188883872`
219	219/256Exact fraction: `219/256`Retained binary64: `0.85546875`	5.37506868075Retained binary64: `5.375068680751287`	-2.36503928288Retained binary64: `-2.3650392828798186`	-3.69138954348Retained binary64: `-3.691389543483762`	-4.73007856576Retained binary64: `-4.730078565759637`
220	55/64Exact fraction: `55/64`Retained binary64: `0.859375`	5.39961237336Retained binary64: `5.399612373357457`	-2.31903136009Retained binary64: `-2.3190313600882106`	-3.80635970498Retained binary64: `-3.8063597049818734`	-4.63806272018Retained binary64: `-4.638062720176421`
221	221/256Exact fraction: `221/256`Retained binary64: `0.86328125`	5.42415606596Retained binary64: `5.424156065963627`	-2.27162653952Retained binary64: `-2.2716265395194535`	-3.91903705772Retained binary64: `-3.9190370577226608`	-4.54325307904Retained binary64: `-4.543253079038907`
222	111/128Exact fraction: `111/128`Retained binary64: `0.8671875`	5.44869975857Retained binary64: `5.448699758569798`	-2.22285337606Retained binary64: `-2.222853376064877`	-4.02935372908Retained binary64: `-4.02935372908211`	-4.44570675213Retained binary64: `-4.445706752129754`
223	223/256Exact fraction: `223/256`Retained binary64: `0.87109375`	5.47324345118Retained binary64: `5.473243451175968`	-2.17274124885Retained binary64: `-2.172741248854401`	-4.13724326842Retained binary64: `-4.1372432684224005`	-4.34548249771Retained binary64: `-4.345482497708802`
224	7/8Exact fraction: `7/8`Retained binary64: `0.875`	5.49778714378Retained binary64: `5.497787143782138`	-2.12132034356Retained binary64: `-2.121320343559643`	-4.24264068712Retained binary64: `-4.242640687119284`	-4.24264068712Retained binary64: `-4.242640687119286`
225	225/256Exact fraction: `225/256`Retained binary64: `0.87890625`	5.52233083639Retained binary64: `5.522330836388308`	-2.06862163421Retained binary64: `-2.0686216342112016`	-4.34548249771Retained binary64: `-4.3454824977088`	-4.13724326842Retained binary64: `-4.137243268422403`
226	113/128Exact fraction: `113/128`Retained binary64: `0.8828125`	5.54687452899Retained binary64: `5.546874528994478`	-2.01467686454Retained binary64: `-2.014676864541056`	-4.44570675213Retained binary64: `-4.4457067521297535`	-4.02935372908Retained binary64: `-4.029353729082112`
227	227/256Exact fraction: `227/256`Retained binary64: `0.88671875`	5.5714182216Retained binary64: `5.5714182216006485`	-1.95951852886Retained binary64: `-1.9595185288613313`	-4.54325307904Retained binary64: `-4.543253079038905`	-3.91903705772Retained binary64: `-3.9190370577226625`
228	57/64Exact fraction: `57/64`Retained binary64: `0.890625`	5.59596191421Retained binary64: `5.595961914206819`	-1.90317985249Retained binary64: `-1.9031798524909378`	-4.63806272018Retained binary64: `-4.6380627201764195`	-3.80635970498Retained binary64: `-3.8063597049818756`
229	229/256Exact fraction: `229/256`Retained binary64: `0.89453125`	5.62050560681Retained binary64: `5.620505606812989`	-1.84569477174Retained binary64: `-1.8456947717418821`	-4.73007856576Retained binary64: `-4.730078565759635`	-3.69138954348Retained binary64: `-3.6913895434837642`
230	115/128Exact fraction: `115/128`Retained binary64: `0.8984375`	5.64504929942Retained binary64: `5.64504929941916`	-1.78709791348Retained binary64: `-1.7870979134772997`	-4.81924518888Retained binary64: `-4.819245188883871`	-3.57419582695Retained binary64: `-3.5741958269545995`
231	231/256Exact fraction: `231/256`Retained binary64: `0.90234375`	5.66959299203Retained binary64: `5.66959299202533`	-1.72742457425Retained binary64: `-1.7274245742535357`	-4.90550887891Retained binary64: `-4.905508878909503`	-3.45484914851Retained binary64: `-3.4548491485070714`
232	29/32Exact fraction: `29/32`Retained binary64: `0.90625`	5.69413668463Retained binary64: `5.6941366846315`	-1.66671069906Retained binary64: `-1.6667106990588065`	-4.98881767382Retained binary64: `-4.988817673815271`	-3.33342139812Retained binary64: `-3.333421398117613`
233	233/256Exact fraction: `233/256`Retained binary64: `0.91015625`	5.71868037724Retained binary64: `5.71868037723767`	-1.60499285966Retained binary64: `-1.6049928596612917`	-5.0691213915Retained binary64: `-5.069121391498242`	-3.20998571932Retained binary64: `-3.2099857193225834`
234	117/128Exact fraction: `117/128`Retained binary64: `0.9140625`	5.74322406984Retained binary64: `5.743224069843841`	-1.54230823258Retained binary64: `-1.5423082325796655`	-5.14637166Retained binary64: `-5.146371660001632`	-3.08461646516Retained binary64: `-3.084616465159331`
235	235/256Exact fraction: `235/256`Retained binary64: `0.91796875`	5.76776776245Retained binary64: `5.767767762450011`	-1.47869457669Retained binary64: `-1.4786945766893527`	-5.22052194665Retained binary64: `-5.220521946652268`	-2.95738915338Retained binary64: `-2.9573891533787053`
236	59/64Exact fraction: `59/64`Retained binary64: `0.921875`	5.79231145506Retained binary64: `5.792311455056181`	-1.41419021048Retained binary64: `-1.4141902104779938`	-5.29152758609Retained binary64: `-5.291527586090129`	-2.82838042096Retained binary64: `-2.8283804209559875`
237	237/256Exact fraction: `237/256`Retained binary64: `0.92578125`	5.81685514766Retained binary64: `5.816855147662351`	-1.34883398896Retained binary64: `-1.348833988963821`	-5.35934580717Retained binary64: `-5.35934580717309`	-2.69766797793Retained binary64: `-2.697667977927642`
238	119/128Exact fraction: `119/128`Retained binary64: `0.9296875`	5.84139884027Retained binary64: `5.841398840268521`	-1.28266528029Retained binary64: `-1.2826652802908476`	-5.42393575874Retained binary64: `-5.423935758740659`	-2.56533056058Retained binary64: `-2.565330560581695`
239	239/256Exact fraction: `239/256`Retained binary64: `0.93359375`	5.86594253287Retained binary64: `5.8659425328746915`	-1.21572394201Retained binary64: `-1.2157239420149712`	-5.48525853422Retained binary64: `-5.485258534221183`	-2.43144788403Retained binary64: `-2.4314478840299425`
240	15/16Exact fraction: `15/16`Retained binary64: `0.9375`	5.89048622548Retained binary64: `5.890486225480862`	-1.1480502971Retained binary64: `-1.148050297095271`	-5.54327719507Retained binary64: `-5.543277195067719`	-2.29610059419Retained binary64: `-2.296100594190542`
241	241/256Exact fraction: `241/256`Retained binary64: `0.94140625`	5.91502991809Retained binary64: `5.915029918087033`	-1.0796851096Retained binary64: `-1.079685109604964`	-5.59795679301Retained binary64: `-5.597956793008434`	-2.15937021921Retained binary64: `-2.159370219209928`
242	121/128Exact fraction: `121/128`Retained binary64: `0.9453125`	5.93957361069Retained binary64: `5.939573610693203`	-1.01066956018Retained binary64: `-1.01066956017666`	-5.6492643911Retained binary64: `-5.649264391098125`	-2.02133912035Retained binary64: `-2.02133912035332`
243	243/256Exact fraction: `243/256`Retained binary64: `0.94921875`	5.9641173033Retained binary64: `5.964117303299373`	-0.941045221197Retained binary64: `-0.9410452211966746`	-5.69716908356Retained binary64: `-5.6971690835582205`	-1.88209044239Retained binary64: `-1.882090442393349`
244	61/64Exact fraction: `61/64`Retained binary64: `0.953125`	5.98866099591Retained binary64: `5.988660995905543`	-0.870854031763Retained binary64: `-0.8708540317633875`	-5.74164201439Retained binary64: `-5.741642014393253`	-1.74170806353Retained binary64: `-1.741708063526775`
245	245/256Exact fraction: `245/256`Retained binary64: `0.95703125`	6.01320468851Retained binary64: `6.013204688511713`	-0.800138272425Retained binary64: `-0.8001382724246957`	-5.78265639477Retained binary64: `-5.782656394772639`	-1.60027654485Retained binary64: `-1.6002765448493914`
246	123/128Exact fraction: `123/128`Retained binary64: `0.9609375`	6.03774838112Retained binary64: `6.037748381117884`	-0.72894053971Retained binary64: `-0.7289405397097926`	-5.82018751917Retained binary64: `-5.820187519167264`	-1.45788107942Retained binary64: `-1.4578810794195851`
247	247/256Exact fraction: `247/256`Retained binary64: `0.96484375`	6.06229207372Retained binary64: `6.062292073724054`	-0.657303720471Retained binary64: `-0.6573037204706105`	-5.85421278023Retained binary64: `-5.854212780231171`	-1.31460744094Retained binary64: `-1.314607440941221`
248	31/32Exact fraction: `31/32`Retained binary64: `0.96875`	6.08683576633Retained binary64: `6.086835766330224`	-0.585270966048Retained binary64: `-0.5852709660483861`	-5.88471168242Retained binary64: `-5.884711682419382`	-1.1705419321Retained binary64: `-1.1705419320967723`
249	249/256Exact fraction: `249/256`Retained binary64: `0.97265625`	6.11137945894Retained binary64: `6.111379458936394`	-0.512885666281Retained binary64: `-0.5128856662809054`	-5.91166585433Retained binary64: `-5.911665854333647`	-1.02577133256Retained binary64: `-1.0257713325618107`
250	125/128Exact fraction: `125/128`Retained binary64: `0.9765625`	6.13592315154Retained binary64: `6.135923151542564`	-0.440191423366Retained binary64: `-0.4401914233660872`	-5.93505905979Retained binary64: `-5.935059059788685`	-0.880382846732Retained binary64: `-0.8803828467321744`
251	251/256Exact fraction: `251/256`Retained binary64: `0.98046875`	6.16046684415Retained binary64: `6.160466844148735`	-0.367232025598Retained binary64: `-0.3672320255976481`	-5.95487720759Retained binary64: `-5.95487720759226`	-0.734464051195Retained binary64: `-0.7344640511952962`
252	63/64Exact fraction: `63/64`Retained binary64: `0.984375`	6.18501053675Retained binary64: `6.1850105367549055`	-0.294051420989Retained binary64: `-0.29405142098868153`	-5.97110836003Retained binary64: `-5.971108360033181`	-0.588102841977Retained binary64: `-0.5881028419773631`
253	253/256Exact fraction: `253/256`Retained binary64: `0.98828125`	6.20955422936Retained binary64: `6.209554229361076`	-0.220693690799Retained binary64: `-0.22069369079900225`	-5.98374274007Retained binary64: `-5.983742740072142`	-0.441387381598Retained binary64: `-0.4413873815980045`
254	127/128Exact fraction: `127/128`Retained binary64: `0.9921875`	6.23409792197Retained binary64: `6.234097921967246`	-0.147203022982Retained binary64: `-0.14720302298225427`	-5.99277273723Retained binary64: `-5.992772737231034`	-0.294406045965Retained binary64: `-0.29440604596450853`
255	255/256Exact fraction: `255/256`Retained binary64: `0.99609375`	6.25864161457Retained binary64: `6.258641614573416`	-0.0736236855687Retained binary64: `-0.07362368556873734`	-5.99819291218Retained binary64: `-5.998192912177226`	-0.147247371137Retained binary64: `-0.1472473711374747`
256	1Exact fraction: `1`Retained binary64: `1.0`	6.28318530718Retained binary64: `6.283185307179586`	0Retained binary64: `0.0`	-6Retained binary64: `-6.0`	0Retained binary64: `0.0`

All 33 positive field-magnitude samples versus scaled height (33 rows)

This is static spatial decay, not a time-decay curve.

All 33 positive field-magnitude samples versus scaled height
Row	H=2y/(L₀ln2)	Y=y/L₀	Positive \|E\|/E₀
0	0Exact fraction: `0`Retained binary64: `0.0`	0Retained binary64: `0.0`	6Retained binary64: `6.0`
1	1/8Exact fraction: `1/8`Retained binary64: `0.125`	0.043321698785Retained binary64: `0.04332169878499658`	5.50202425923Retained binary64: `5.502024259228027`
2	1/4Exact fraction: `1/4`Retained binary64: `0.25`	0.08664339757Retained binary64: `0.08664339756999316`	5.04537849152Retained binary64: `5.045378491522287`
3	3/8Exact fraction: `3/8`Retained binary64: `0.375`	0.129965096355Retained binary64: `0.12996509635498973`	4.62663247622Retained binary64: `4.626632476223822`
4	1/2Exact fraction: `1/2`Retained binary64: `0.5`	0.17328679514Retained binary64: `0.17328679513998632`	4.24264068712Retained binary64: `4.242640687119286`
5	5/8Exact fraction: `5/8`Retained binary64: `0.625`	0.216608493925Retained binary64: `0.2166084939249829`	3.89051866395Retained binary64: `3.8905186639530287`
6	3/4Exact fraction: `3/4`Retained binary64: `0.75`	0.25993019271Retained binary64: `0.25993019270997947`	3.56762134501Retained binary64: `3.5676213450081633`
7	7/8Exact fraction: `7/8`Retained binary64: `0.875`	0.303251891495Retained binary64: `0.30325189149497606`	3.271523198Retained binary64: `3.271523197995773`
8	1Exact fraction: `1`Retained binary64: `1.0`	0.34657359028Retained binary64: `0.34657359027997264`	3Retained binary64: `3.0`
9	9/8Exact fraction: `9/8`Retained binary64: `1.125`	0.389895289065Retained binary64: `0.38989528906496923`	2.75101212961Retained binary64: `2.7510121296140135`
10	5/4Exact fraction: `5/4`Retained binary64: `1.25`	0.43321698785Retained binary64: `0.4332169878499658`	2.52268924576Retained binary64: `2.5226892457611436`
11	11/8Exact fraction: `11/8`Retained binary64: `1.375`	0.476538686635Retained binary64: `0.4765386866349624`	2.31331623811Retained binary64: `2.313316238111911`
12	3/2Exact fraction: `3/2`Retained binary64: `1.5`	0.51986038542Retained binary64: `0.5198603854199589`	2.12132034356Retained binary64: `2.121320343559643`
13	13/8Exact fraction: `13/8`Retained binary64: `1.625`	0.563182084205Retained binary64: `0.5631820842049555`	1.94525933198Retained binary64: `1.9452593319765143`
14	7/4Exact fraction: `7/4`Retained binary64: `1.75`	0.60650378299Retained binary64: `0.6065037829899521`	1.7838106725Retained binary64: `1.7838106725040817`
15	15/8Exact fraction: `15/8`Retained binary64: `1.875`	0.649825481775Retained binary64: `0.6498254817749487`	1.635761599Retained binary64: `1.6357615989978864`
16	2Exact fraction: `2`Retained binary64: `2.0`	0.69314718056Retained binary64: `0.6931471805599453`	1.5Retained binary64: `1.5`
17	17/8Exact fraction: `17/8`Retained binary64: `2.125`	0.736468879345Retained binary64: `0.7364688793449419`	1.37550606481Retained binary64: `1.3755060648070068`
18	9/4Exact fraction: `9/4`Retained binary64: `2.25`	0.77979057813Retained binary64: `0.7797905781299385`	1.26134462288Retained binary64: `1.2613446228805718`
19	19/8Exact fraction: `19/8`Retained binary64: `2.375`	0.823112276915Retained binary64: `0.823112276914935`	1.15665811906Retained binary64: `1.1566581190559555`
20	5/2Exact fraction: `5/2`Retained binary64: `2.5`	0.8664339757Retained binary64: `0.8664339756999316`	1.06066017178Retained binary64: `1.0606601717798214`
21	21/8Exact fraction: `21/8`Retained binary64: `2.625`	0.909755674485Retained binary64: `0.9097556744849282`	0.972629665988Retained binary64: `0.9726296659882572`
22	11/4Exact fraction: `11/4`Retained binary64: `2.75`	0.95307737327Retained binary64: `0.9530773732699248`	0.891905336252Retained binary64: `0.8919053362520408`
23	23/8Exact fraction: `23/8`Retained binary64: `2.875`	0.996399072055Retained binary64: `0.9963990720549214`	0.817880799499Retained binary64: `0.8178807994989432`
24	3Exact fraction: `3`Retained binary64: `3.0`	1.03972077084Retained binary64: `1.0397207708399179`	0.75Retained binary64: `0.75`
25	25/8Exact fraction: `25/8`Retained binary64: `3.125`	1.08304246962Retained binary64: `1.0830424696249146`	0.687753032404Retained binary64: `0.6877530324035034`
26	13/4Exact fraction: `13/4`Retained binary64: `3.25`	1.12636416841Retained binary64: `1.126364168409911`	0.63067231144Retained binary64: `0.6306723114402859`
27	27/8Exact fraction: `27/8`Retained binary64: `3.375`	1.16968586719Retained binary64: `1.1696858671949077`	0.578329059528Retained binary64: `0.5783290595279778`
28	7/2Exact fraction: `7/2`Retained binary64: `3.5`	1.21300756598Retained binary64: `1.2130075659799042`	0.53033008589Retained binary64: `0.5303300858899107`
29	29/8Exact fraction: `29/8`Retained binary64: `3.625`	1.25632926476Retained binary64: `1.256329264764901`	0.486314832994Retained binary64: `0.4863148329941286`
30	15/4Exact fraction: `15/4`Retained binary64: `3.75`	1.29965096355Retained binary64: `1.2996509635498974`	0.445952668126Retained binary64: `0.4459526681260204`
31	31/8Exact fraction: `31/8`Retained binary64: `3.875`	1.34297266233Retained binary64: `1.342972662334894`	0.408940399749Retained binary64: `0.4089403997494716`
32	4Exact fraction: `4`Retained binary64: `4.0`	1.38629436112Retained binary64: `1.3862943611198906`	0.375Retained binary64: `0.375`

Exact symbolic worked control (1 rows)

Exact symbolic worked control
Row	X	Y	v	Ex	Ey	magnitude
0	pi/8	ln2/2	3sqrt(2)/4	-3sqrt(2)/2	3sqrt(2)/2	3

Integrate the field of a finite charged ball

Now choose an isolated, uniformly charged solid ball of radius a = 3L₀ and total charge Q = 2q₀, where q₀ is a positive source-charge scale. For this case set V₀ = kₑq₀/L₀ and E₀ = V₀/L₀, with kₑ = 1/(4πε₀). There is no added surface-charge sheet, and the finite source permits V → 0 at infinity.

Gauss’s law gives the radial field E_r = kₑQr/a³ inside and E_r = kₑQ/r² outside. Integrating inward from infinity gives V = kₑQ/r outside. Inside, V(r) − V(a) = −∫ from a to r kₑQs/a³ ds, so V(r) = (kₑQ/2a)(3 − r²/a²). This derivation fixes the integration constant by matching the same surface potential.

In normalized form, v = 1 − R²/27 and E_r/E₀ = 2R/27 for 0 ≤ R ≤ 3, where R = r/L₀. Outside, v = 2/R and E_r/E₀ = 2/R². At R = 3 both potential branches give 2/3 and both field branches give 2/9. The field is zero at the center while the potential is V₀, an immediate warning against equating value and slope.

The source density is ρ = Q/(4πa³/3) = q₀/(18πL₀³) = 2ε₀V₀/(9L₀²). Inside, the Cartesian Laplacian of the quadratic potential is −2V₀/(9L₀²), agreeing with −ρ/ε₀. Outside, the radial 1/r form is harmonic. Its apparent singularity at r = 0 is irrelevant to the exterior branch, which is never used there.

The volume density changes abruptly at r = a. Potential and its first radial derivative remain continuous because no surface sheet was added, but the second-derivative formula is interpreted on the two open regions with their one-sided limits. Do not assign an arbitrary classical second derivative at the density jump merely to join a graph.

A regular center and continuous potential/field surface limits

Potential and outward radial field on separately valid inside/outside branches; regular center, matching first derivatives and source-density step; no exterior1/R evaluation at0 or invented surface second derivative.

Potential: interior and exterior analytic branches

Interior branch, 97 analytic samples; regular center
Exterior branch, 145 analytic samples; R≥3 only
All ten exact controls; shared continuous surface value

Signed outward radial field on the same domains

Interior branch, 97 analytic samples; regular center
Exterior branch, 145 analytic samples; R≥3 only
All ten exact controls; shared continuous surface value

Separate uniformly charged ball:Q=2q₀,a=3L₀,V₀=kₑq₀/L₀,E₀=V₀/L₀;no added surface-charge sheet;V→0 at infinity.
Inside:v=1−R²/27,Er/E₀=2R/27. Outside:v=2/R,Er/E₀=2/R². The exterior branch is never evaluated atR=0.
At R=3both potential limits are2/3 andboth field limits2/9. A shared filled value is valid for these continuous quantities. Curvature/Laplacian entries there are explicitly one-sided limits only; no classical surface second derivative is assigned.
At the centerV=V₀while the field iszero. Field sign means outward radial component, never an unsigned magnitude substitution.
The source data use exact rational strings or12-significant-digit displays plus full retained binary64 values. The plotted curves join finite analytic samples; no interpolation-error bound or numerical PDE solver is claimed.

Complete labelled equivalents below preserve every retained sample and scientific value; original manuscript tables remain separate.

Exact potential and field controls for the separate solid ball

All 10 exact ball controls (10 rows)

All 10 exact ball controls
Row	R=r/L₀	Analytic branch	V/V₀	Signed outward Eᵣ/E₀	L₀²∇²V/V₀; one-sided limit atR=3
0	0Exact fraction: `0`Retained binary64: `0.0`	interior	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
1	1/2Exact fraction: `1/2`Retained binary64: `0.5`	interior	107/108Exact fraction: `107/108`Retained binary64: `0.9907407407407407`	1/27Exact fraction: `1/27`Retained binary64: `0.037037037037037035`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
2	1Exact fraction: `1`Retained binary64: `1.0`	interior	26/27Exact fraction: `26/27`Retained binary64: `0.9629629629629629`	2/27Exact fraction: `2/27`Retained binary64: `0.07407407407407407`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
3	3/2Exact fraction: `3/2`Retained binary64: `1.5`	interior	11/12Exact fraction: `11/12`Retained binary64: `0.9166666666666666`	1/9Exact fraction: `1/9`Retained binary64: `0.1111111111111111`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
4	2Exact fraction: `2`Retained binary64: `2.0`	interior	23/27Exact fraction: `23/27`Retained binary64: `0.8518518518518519`	4/27Exact fraction: `4/27`Retained binary64: `0.14814814814814814`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
5	5/2Exact fraction: `5/2`Retained binary64: `2.5`	interior	83/108Exact fraction: `83/108`Retained binary64: `0.7685185185185185`	5/27Exact fraction: `5/27`Retained binary64: `0.18518518518518517`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
6R=3: the curvature value is the interior one-sided branch limit only; no classical second derivative is assigned at the surface.	3Exact fraction: `3`Retained binary64: `3.0`	interior	2/3Exact fraction: `2/3`Retained binary64: `0.6666666666666666`	2/9Exact fraction: `2/9`Retained binary64: `0.2222222222222222`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
7	4Exact fraction: `4`Retained binary64: `4.0`	exterior	1/2Exact fraction: `1/2`Retained binary64: `0.5`	1/8Exact fraction: `1/8`Retained binary64: `0.125`	0Exact fraction: `0`Retained binary64: `0.0`
8	6Exact fraction: `6`Retained binary64: `6.0`	exterior	1/3Exact fraction: `1/3`Retained binary64: `0.3333333333333333`	1/18Exact fraction: `1/18`Retained binary64: `0.05555555555555555`	0Exact fraction: `0`Retained binary64: `0.0`
9	9Exact fraction: `9`Retained binary64: `9.0`	exterior	2/9Exact fraction: `2/9`Retained binary64: `0.2222222222222222`	2/81Exact fraction: `2/81`Retained binary64: `0.024691358024691357`	0Exact fraction: `0`Retained binary64: `0.0`

All 97 interior branch samples, including its continuous endpoint (97 rows)

All 97 interior branch samples, including its continuous endpoint
Row	R=r/L₀	Analytic branch	V/V₀	Signed outward Eᵣ/E₀	L₀²∇²V/V₀; one-sided limit atR=3
0	0Exact fraction: `0`Retained binary64: `0.0`	interior	1Exact fraction: `1`Retained binary64: `1.0`	0Exact fraction: `0`Retained binary64: `0.0`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
1	1/32Exact fraction: `1/32`Retained binary64: `0.03125`	interior	27647/27648Exact fraction: `27647/27648`Retained binary64: `0.9999638310185185`	1/432Exact fraction: `1/432`Retained binary64: `0.0023148148148148147`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
2	1/16Exact fraction: `1/16`Retained binary64: `0.0625`	interior	6911/6912Exact fraction: `6911/6912`Retained binary64: `0.9998553240740741`	1/216Exact fraction: `1/216`Retained binary64: `0.004629629629629629`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
3	3/32Exact fraction: `3/32`Retained binary64: `0.09375`	interior	3071/3072Exact fraction: `3071/3072`Retained binary64: `0.9996744791666666`	1/144Exact fraction: `1/144`Retained binary64: `0.006944444444444444`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
4	1/8Exact fraction: `1/8`Retained binary64: `0.125`	interior	1727/1728Exact fraction: `1727/1728`Retained binary64: `0.9994212962962963`	1/108Exact fraction: `1/108`Retained binary64: `0.009259259259259259`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
5	5/32Exact fraction: `5/32`Retained binary64: `0.15625`	interior	27623/27648Exact fraction: `27623/27648`Retained binary64: `0.9990957754629629`	5/432Exact fraction: `5/432`Retained binary64: `0.011574074074074073`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
6	3/16Exact fraction: `3/16`Retained binary64: `0.1875`	interior	767/768Exact fraction: `767/768`Retained binary64: `0.9986979166666666`	1/72Exact fraction: `1/72`Retained binary64: `0.013888888888888888`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
7	7/32Exact fraction: `7/32`Retained binary64: `0.21875`	interior	27599/27648Exact fraction: `27599/27648`Retained binary64: `0.9982277199074074`	7/432Exact fraction: `7/432`Retained binary64: `0.016203703703703703`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
8	1/4Exact fraction: `1/4`Retained binary64: `0.25`	interior	431/432Exact fraction: `431/432`Retained binary64: `0.9976851851851852`	1/54Exact fraction: `1/54`Retained binary64: `0.018518518518518517`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
9	9/32Exact fraction: `9/32`Retained binary64: `0.28125`	interior	1021/1024Exact fraction: `1021/1024`Retained binary64: `0.9970703125`	1/48Exact fraction: `1/48`Retained binary64: `0.020833333333333332`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
10	5/16Exact fraction: `5/16`Retained binary64: `0.3125`	interior	6887/6912Exact fraction: `6887/6912`Retained binary64: `0.9963831018518519`	5/216Exact fraction: `5/216`Retained binary64: `0.023148148148148147`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
11	11/32Exact fraction: `11/32`Retained binary64: `0.34375`	interior	27527/27648Exact fraction: `27527/27648`Retained binary64: `0.9956235532407407`	11/432Exact fraction: `11/432`Retained binary64: `0.02546296296296296`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
12	3/8Exact fraction: `3/8`Retained binary64: `0.375`	interior	191/192Exact fraction: `191/192`Retained binary64: `0.9947916666666666`	1/36Exact fraction: `1/36`Retained binary64: `0.027777777777777776`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
13	13/32Exact fraction: `13/32`Retained binary64: `0.40625`	interior	27479/27648Exact fraction: `27479/27648`Retained binary64: `0.9938874421296297`	13/432Exact fraction: `13/432`Retained binary64: `0.03009259259259259`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
14	7/16Exact fraction: `7/16`Retained binary64: `0.4375`	interior	6863/6912Exact fraction: `6863/6912`Retained binary64: `0.9929108796296297`	7/216Exact fraction: `7/216`Retained binary64: `0.032407407407407406`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
15	15/32Exact fraction: `15/32`Retained binary64: `0.46875`	interior	3047/3072Exact fraction: `3047/3072`Retained binary64: `0.9918619791666666`	5/144Exact fraction: `5/144`Retained binary64: `0.034722222222222224`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
16	1/2Exact fraction: `1/2`Retained binary64: `0.5`	interior	107/108Exact fraction: `107/108`Retained binary64: `0.9907407407407407`	1/27Exact fraction: `1/27`Retained binary64: `0.037037037037037035`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
17	17/32Exact fraction: `17/32`Retained binary64: `0.53125`	interior	27359/27648Exact fraction: `27359/27648`Retained binary64: `0.9895471643518519`	17/432Exact fraction: `17/432`Retained binary64: `0.03935185185185185`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
18	9/16Exact fraction: `9/16`Retained binary64: `0.5625`	interior	253/256Exact fraction: `253/256`Retained binary64: `0.98828125`	1/24Exact fraction: `1/24`Retained binary64: `0.041666666666666664`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
19	19/32Exact fraction: `19/32`Retained binary64: `0.59375`	interior	27287/27648Exact fraction: `27287/27648`Retained binary64: `0.9869429976851852`	19/432Exact fraction: `19/432`Retained binary64: `0.04398148148148148`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
20	5/8Exact fraction: `5/8`Retained binary64: `0.625`	interior	1703/1728Exact fraction: `1703/1728`Retained binary64: `0.9855324074074074`	5/108Exact fraction: `5/108`Retained binary64: `0.046296296296296294`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
21	21/32Exact fraction: `21/32`Retained binary64: `0.65625`	interior	3023/3072Exact fraction: `3023/3072`Retained binary64: `0.9840494791666666`	7/144Exact fraction: `7/144`Retained binary64: `0.04861111111111111`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
22	11/16Exact fraction: `11/16`Retained binary64: `0.6875`	interior	6791/6912Exact fraction: `6791/6912`Retained binary64: `0.9824942129629629`	11/216Exact fraction: `11/216`Retained binary64: `0.05092592592592592`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
23	23/32Exact fraction: `23/32`Retained binary64: `0.71875`	interior	27119/27648Exact fraction: `27119/27648`Retained binary64: `0.9808666087962963`	23/432Exact fraction: `23/432`Retained binary64: `0.05324074074074074`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
24	3/4Exact fraction: `3/4`Retained binary64: `0.75`	interior	47/48Exact fraction: `47/48`Retained binary64: `0.9791666666666666`	1/18Exact fraction: `1/18`Retained binary64: `0.05555555555555555`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
25	25/32Exact fraction: `25/32`Retained binary64: `0.78125`	interior	27023/27648Exact fraction: `27023/27648`Retained binary64: `0.9773943865740741`	25/432Exact fraction: `25/432`Retained binary64: `0.05787037037037037`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
26	13/16Exact fraction: `13/16`Retained binary64: `0.8125`	interior	6743/6912Exact fraction: `6743/6912`Retained binary64: `0.9755497685185185`	13/216Exact fraction: `13/216`Retained binary64: `0.06018518518518518`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
27	27/32Exact fraction: `27/32`Retained binary64: `0.84375`	interior	997/1024Exact fraction: `997/1024`Retained binary64: `0.9736328125`	1/16Exact fraction: `1/16`Retained binary64: `0.0625`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
28	7/8Exact fraction: `7/8`Retained binary64: `0.875`	interior	1679/1728Exact fraction: `1679/1728`Retained binary64: `0.9716435185185185`	7/108Exact fraction: `7/108`Retained binary64: `0.06481481481481481`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
29	29/32Exact fraction: `29/32`Retained binary64: `0.90625`	interior	26807/27648Exact fraction: `26807/27648`Retained binary64: `0.9695818865740741`	29/432Exact fraction: `29/432`Retained binary64: `0.06712962962962964`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
30	15/16Exact fraction: `15/16`Retained binary64: `0.9375`	interior	743/768Exact fraction: `743/768`Retained binary64: `0.9674479166666666`	5/72Exact fraction: `5/72`Retained binary64: `0.06944444444444445`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
31	31/32Exact fraction: `31/32`Retained binary64: `0.96875`	interior	26687/27648Exact fraction: `26687/27648`Retained binary64: `0.9652416087962963`	31/432Exact fraction: `31/432`Retained binary64: `0.07175925925925926`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
32	1Exact fraction: `1`Retained binary64: `1.0`	interior	26/27Exact fraction: `26/27`Retained binary64: `0.9629629629629629`	2/27Exact fraction: `2/27`Retained binary64: `0.07407407407407407`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
33	33/32Exact fraction: `33/32`Retained binary64: `1.03125`	interior	2951/3072Exact fraction: `2951/3072`Retained binary64: `0.9606119791666666`	11/144Exact fraction: `11/144`Retained binary64: `0.0763888888888889`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
34	17/16Exact fraction: `17/16`Retained binary64: `1.0625`	interior	6623/6912Exact fraction: `6623/6912`Retained binary64: `0.9581886574074074`	17/216Exact fraction: `17/216`Retained binary64: `0.0787037037037037`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
35	35/32Exact fraction: `35/32`Retained binary64: `1.09375`	interior	26423/27648Exact fraction: `26423/27648`Retained binary64: `0.9556929976851852`	35/432Exact fraction: `35/432`Retained binary64: `0.08101851851851852`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
36	9/8Exact fraction: `9/8`Retained binary64: `1.125`	interior	61/64Exact fraction: `61/64`Retained binary64: `0.953125`	1/12Exact fraction: `1/12`Retained binary64: `0.08333333333333333`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
37	37/32Exact fraction: `37/32`Retained binary64: `1.15625`	interior	26279/27648Exact fraction: `26279/27648`Retained binary64: `0.9504846643518519`	37/432Exact fraction: `37/432`Retained binary64: `0.08564814814814815`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
38	19/16Exact fraction: `19/16`Retained binary64: `1.1875`	interior	6551/6912Exact fraction: `6551/6912`Retained binary64: `0.9477719907407407`	19/216Exact fraction: `19/216`Retained binary64: `0.08796296296296297`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
39	39/32Exact fraction: `39/32`Retained binary64: `1.21875`	interior	2903/3072Exact fraction: `2903/3072`Retained binary64: `0.9449869791666666`	13/144Exact fraction: `13/144`Retained binary64: `0.09027777777777778`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
40	5/4Exact fraction: `5/4`Retained binary64: `1.25`	interior	407/432Exact fraction: `407/432`Retained binary64: `0.9421296296296297`	5/54Exact fraction: `5/54`Retained binary64: `0.09259259259259259`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
41	41/32Exact fraction: `41/32`Retained binary64: `1.28125`	interior	25967/27648Exact fraction: `25967/27648`Retained binary64: `0.9391999421296297`	41/432Exact fraction: `41/432`Retained binary64: `0.09490740740740741`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
42	21/16Exact fraction: `21/16`Retained binary64: `1.3125`	interior	719/768Exact fraction: `719/768`Retained binary64: `0.9361979166666666`	7/72Exact fraction: `7/72`Retained binary64: `0.09722222222222222`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
43	43/32Exact fraction: `43/32`Retained binary64: `1.34375`	interior	25799/27648Exact fraction: `25799/27648`Retained binary64: `0.9331235532407407`	43/432Exact fraction: `43/432`Retained binary64: `0.09953703703703703`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
44	11/8Exact fraction: `11/8`Retained binary64: `1.375`	interior	1607/1728Exact fraction: `1607/1728`Retained binary64: `0.9299768518518519`	11/108Exact fraction: `11/108`Retained binary64: `0.10185185185185185`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
45	45/32Exact fraction: `45/32`Retained binary64: `1.40625`	interior	949/1024Exact fraction: `949/1024`Retained binary64: `0.9267578125`	5/48Exact fraction: `5/48`Retained binary64: `0.10416666666666667`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
46	23/16Exact fraction: `23/16`Retained binary64: `1.4375`	interior	6383/6912Exact fraction: `6383/6912`Retained binary64: `0.9234664351851852`	23/216Exact fraction: `23/216`Retained binary64: `0.10648148148148148`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
47	47/32Exact fraction: `47/32`Retained binary64: `1.46875`	interior	25439/27648Exact fraction: `25439/27648`Retained binary64: `0.9201027199074074`	47/432Exact fraction: `47/432`Retained binary64: `0.1087962962962963`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
48	3/2Exact fraction: `3/2`Retained binary64: `1.5`	interior	11/12Exact fraction: `11/12`Retained binary64: `0.9166666666666666`	1/9Exact fraction: `1/9`Retained binary64: `0.1111111111111111`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
49	49/32Exact fraction: `49/32`Retained binary64: `1.53125`	interior	25247/27648Exact fraction: `25247/27648`Retained binary64: `0.9131582754629629`	49/432Exact fraction: `49/432`Retained binary64: `0.11342592592592593`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
50	25/16Exact fraction: `25/16`Retained binary64: `1.5625`	interior	6287/6912Exact fraction: `6287/6912`Retained binary64: `0.9095775462962963`	25/216Exact fraction: `25/216`Retained binary64: `0.11574074074074074`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
51	51/32Exact fraction: `51/32`Retained binary64: `1.59375`	interior	2783/3072Exact fraction: `2783/3072`Retained binary64: `0.9059244791666666`	17/144Exact fraction: `17/144`Retained binary64: `0.11805555555555555`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
52	13/8Exact fraction: `13/8`Retained binary64: `1.625`	interior	1559/1728Exact fraction: `1559/1728`Retained binary64: `0.9021990740740741`	13/108Exact fraction: `13/108`Retained binary64: `0.12037037037037036`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
53	53/32Exact fraction: `53/32`Retained binary64: `1.65625`	interior	24839/27648Exact fraction: `24839/27648`Retained binary64: `0.8984013310185185`	53/432Exact fraction: `53/432`Retained binary64: `0.12268518518518519`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
54	27/16Exact fraction: `27/16`Retained binary64: `1.6875`	interior	229/256Exact fraction: `229/256`Retained binary64: `0.89453125`	1/8Exact fraction: `1/8`Retained binary64: `0.125`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
55	55/32Exact fraction: `55/32`Retained binary64: `1.71875`	interior	24623/27648Exact fraction: `24623/27648`Retained binary64: `0.8905888310185185`	55/432Exact fraction: `55/432`Retained binary64: `0.12731481481481483`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
56	7/4Exact fraction: `7/4`Retained binary64: `1.75`	interior	383/432Exact fraction: `383/432`Retained binary64: `0.8865740740740741`	7/54Exact fraction: `7/54`Retained binary64: `0.12962962962962962`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
57	57/32Exact fraction: `57/32`Retained binary64: `1.78125`	interior	2711/3072Exact fraction: `2711/3072`Retained binary64: `0.8824869791666666`	19/144Exact fraction: `19/144`Retained binary64: `0.13194444444444445`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
58	29/16Exact fraction: `29/16`Retained binary64: `1.8125`	interior	6071/6912Exact fraction: `6071/6912`Retained binary64: `0.8783275462962963`	29/216Exact fraction: `29/216`Retained binary64: `0.13425925925925927`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
59	59/32Exact fraction: `59/32`Retained binary64: `1.84375`	interior	24167/27648Exact fraction: `24167/27648`Retained binary64: `0.8740957754629629`	59/432Exact fraction: `59/432`Retained binary64: `0.13657407407407407`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
60	15/8Exact fraction: `15/8`Retained binary64: `1.875`	interior	167/192Exact fraction: `167/192`Retained binary64: `0.8697916666666666`	5/36Exact fraction: `5/36`Retained binary64: `0.1388888888888889`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
61	61/32Exact fraction: `61/32`Retained binary64: `1.90625`	interior	23927/27648Exact fraction: `23927/27648`Retained binary64: `0.8654152199074074`	61/432Exact fraction: `61/432`Retained binary64: `0.1412037037037037`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
62	31/16Exact fraction: `31/16`Retained binary64: `1.9375`	interior	5951/6912Exact fraction: `5951/6912`Retained binary64: `0.8609664351851852`	31/216Exact fraction: `31/216`Retained binary64: `0.14351851851851852`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
63	63/32Exact fraction: `63/32`Retained binary64: `1.96875`	interior	877/1024Exact fraction: `877/1024`Retained binary64: `0.8564453125`	7/48Exact fraction: `7/48`Retained binary64: `0.14583333333333334`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
64	2Exact fraction: `2`Retained binary64: `2.0`	interior	23/27Exact fraction: `23/27`Retained binary64: `0.8518518518518519`	4/27Exact fraction: `4/27`Retained binary64: `0.14814814814814814`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
65	65/32Exact fraction: `65/32`Retained binary64: `2.03125`	interior	23423/27648Exact fraction: `23423/27648`Retained binary64: `0.8471860532407407`	65/432Exact fraction: `65/432`Retained binary64: `0.15046296296296297`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
66	33/16Exact fraction: `33/16`Retained binary64: `2.0625`	interior	647/768Exact fraction: `647/768`Retained binary64: `0.8424479166666666`	11/72Exact fraction: `11/72`Retained binary64: `0.1527777777777778`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
67	67/32Exact fraction: `67/32`Retained binary64: `2.09375`	interior	23159/27648Exact fraction: `23159/27648`Retained binary64: `0.8376374421296297`	67/432Exact fraction: `67/432`Retained binary64: `0.1550925925925926`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
68	17/8Exact fraction: `17/8`Retained binary64: `2.125`	interior	1439/1728Exact fraction: `1439/1728`Retained binary64: `0.8327546296296297`	17/108Exact fraction: `17/108`Retained binary64: `0.1574074074074074`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
69	69/32Exact fraction: `69/32`Retained binary64: `2.15625`	interior	2543/3072Exact fraction: `2543/3072`Retained binary64: `0.8277994791666666`	23/144Exact fraction: `23/144`Retained binary64: `0.1597222222222222`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
70	35/16Exact fraction: `35/16`Retained binary64: `2.1875`	interior	5687/6912Exact fraction: `5687/6912`Retained binary64: `0.8227719907407407`	35/216Exact fraction: `35/216`Retained binary64: `0.16203703703703703`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
71	71/32Exact fraction: `71/32`Retained binary64: `2.21875`	interior	22607/27648Exact fraction: `22607/27648`Retained binary64: `0.8176721643518519`	71/432Exact fraction: `71/432`Retained binary64: `0.16435185185185186`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
72	9/4Exact fraction: `9/4`Retained binary64: `2.25`	interior	13/16Exact fraction: `13/16`Retained binary64: `0.8125`	1/6Exact fraction: `1/6`Retained binary64: `0.16666666666666666`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
73	73/32Exact fraction: `73/32`Retained binary64: `2.28125`	interior	22319/27648Exact fraction: `22319/27648`Retained binary64: `0.8072554976851852`	73/432Exact fraction: `73/432`Retained binary64: `0.16898148148148148`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
74	37/16Exact fraction: `37/16`Retained binary64: `2.3125`	interior	5543/6912Exact fraction: `5543/6912`Retained binary64: `0.8019386574074074`	37/216Exact fraction: `37/216`Retained binary64: `0.1712962962962963`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
75	75/32Exact fraction: `75/32`Retained binary64: `2.34375`	interior	2447/3072Exact fraction: `2447/3072`Retained binary64: `0.7965494791666666`	25/144Exact fraction: `25/144`Retained binary64: `0.1736111111111111`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
76	19/8Exact fraction: `19/8`Retained binary64: `2.375`	interior	1367/1728Exact fraction: `1367/1728`Retained binary64: `0.7910879629629629`	19/108Exact fraction: `19/108`Retained binary64: `0.17592592592592593`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
77	77/32Exact fraction: `77/32`Retained binary64: `2.40625`	interior	21719/27648Exact fraction: `21719/27648`Retained binary64: `0.7855541087962963`	77/432Exact fraction: `77/432`Retained binary64: `0.17824074074074073`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
78	39/16Exact fraction: `39/16`Retained binary64: `2.4375`	interior	599/768Exact fraction: `599/768`Retained binary64: `0.7799479166666666`	13/72Exact fraction: `13/72`Retained binary64: `0.18055555555555555`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
79	79/32Exact fraction: `79/32`Retained binary64: `2.46875`	interior	21407/27648Exact fraction: `21407/27648`Retained binary64: `0.7742693865740741`	79/432Exact fraction: `79/432`Retained binary64: `0.18287037037037038`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
80	5/2Exact fraction: `5/2`Retained binary64: `2.5`	interior	83/108Exact fraction: `83/108`Retained binary64: `0.7685185185185185`	5/27Exact fraction: `5/27`Retained binary64: `0.18518518518518517`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
81	81/32Exact fraction: `81/32`Retained binary64: `2.53125`	interior	781/1024Exact fraction: `781/1024`Retained binary64: `0.7626953125`	3/16Exact fraction: `3/16`Retained binary64: `0.1875`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
82	41/16Exact fraction: `41/16`Retained binary64: `2.5625`	interior	5231/6912Exact fraction: `5231/6912`Retained binary64: `0.7567997685185185`	41/216Exact fraction: `41/216`Retained binary64: `0.18981481481481483`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
83	83/32Exact fraction: `83/32`Retained binary64: `2.59375`	interior	20759/27648Exact fraction: `20759/27648`Retained binary64: `0.7508318865740741`	83/432Exact fraction: `83/432`Retained binary64: `0.19212962962962962`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
84	21/8Exact fraction: `21/8`Retained binary64: `2.625`	interior	143/192Exact fraction: `143/192`Retained binary64: `0.7447916666666666`	7/36Exact fraction: `7/36`Retained binary64: `0.19444444444444445`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
85	85/32Exact fraction: `85/32`Retained binary64: `2.65625`	interior	20423/27648Exact fraction: `20423/27648`Retained binary64: `0.7386791087962963`	85/432Exact fraction: `85/432`Retained binary64: `0.19675925925925927`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
86	43/16Exact fraction: `43/16`Retained binary64: `2.6875`	interior	5063/6912Exact fraction: `5063/6912`Retained binary64: `0.7324942129629629`	43/216Exact fraction: `43/216`Retained binary64: `0.19907407407407407`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
87	87/32Exact fraction: `87/32`Retained binary64: `2.71875`	interior	2231/3072Exact fraction: `2231/3072`Retained binary64: `0.7262369791666666`	29/144Exact fraction: `29/144`Retained binary64: `0.2013888888888889`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
88	11/4Exact fraction: `11/4`Retained binary64: `2.75`	interior	311/432Exact fraction: `311/432`Retained binary64: `0.7199074074074074`	11/54Exact fraction: `11/54`Retained binary64: `0.2037037037037037`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
89	89/32Exact fraction: `89/32`Retained binary64: `2.78125`	interior	19727/27648Exact fraction: `19727/27648`Retained binary64: `0.7135054976851852`	89/432Exact fraction: `89/432`Retained binary64: `0.20601851851851852`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
90	45/16Exact fraction: `45/16`Retained binary64: `2.8125`	interior	181/256Exact fraction: `181/256`Retained binary64: `0.70703125`	5/24Exact fraction: `5/24`Retained binary64: `0.20833333333333334`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
91	91/32Exact fraction: `91/32`Retained binary64: `2.84375`	interior	19367/27648Exact fraction: `19367/27648`Retained binary64: `0.7004846643518519`	91/432Exact fraction: `91/432`Retained binary64: `0.21064814814814814`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
92	23/8Exact fraction: `23/8`Retained binary64: `2.875`	interior	1199/1728Exact fraction: `1199/1728`Retained binary64: `0.6938657407407407`	23/108Exact fraction: `23/108`Retained binary64: `0.21296296296296297`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
93	93/32Exact fraction: `93/32`Retained binary64: `2.90625`	interior	2111/3072Exact fraction: `2111/3072`Retained binary64: `0.6871744791666666`	31/144Exact fraction: `31/144`Retained binary64: `0.2152777777777778`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
94	47/16Exact fraction: `47/16`Retained binary64: `2.9375`	interior	4703/6912Exact fraction: `4703/6912`Retained binary64: `0.6804108796296297`	47/216Exact fraction: `47/216`Retained binary64: `0.2175925925925926`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
95	95/32Exact fraction: `95/32`Retained binary64: `2.96875`	interior	18623/27648Exact fraction: `18623/27648`Retained binary64: `0.6735749421296297`	95/432Exact fraction: `95/432`Retained binary64: `0.2199074074074074`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
96R=3: the curvature value is the interior one-sided branch limit only; no classical second derivative is assigned at the surface.	3Exact fraction: `3`Retained binary64: `3.0`	interior	2/3Exact fraction: `2/3`Retained binary64: `0.6666666666666666`	2/9Exact fraction: `2/9`Retained binary64: `0.2222222222222222`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`

All 145 exterior branch samples; never evaluated atR=0 (145 rows)

All 145 exterior branch samples; never evaluated atR=0
Row	R=r/L₀	Analytic branch	V/V₀	Signed outward Eᵣ/E₀	L₀²∇²V/V₀; one-sided limit atR=3
0R=3: the curvature value is the exterior one-sided branch limit only; no classical second derivative is assigned at the surface.	3Exact fraction: `3`Retained binary64: `3.0`	exterior	2/3Exact fraction: `2/3`Retained binary64: `0.6666666666666666`	2/9Exact fraction: `2/9`Retained binary64: `0.2222222222222222`	0Exact fraction: `0`Retained binary64: `0.0`
1	73/24Exact fraction: `73/24`Retained binary64: `3.0416666666666665`	exterior	48/73Exact fraction: `48/73`Retained binary64: `0.6575342465753424`	1152/5329Exact fraction: `1152/5329`Retained binary64: `0.21617564270970163`	0Exact fraction: `0`Retained binary64: `0.0`
2	37/12Exact fraction: `37/12`Retained binary64: `3.0833333333333335`	exterior	24/37Exact fraction: `24/37`Retained binary64: `0.6486486486486487`	288/1369Exact fraction: `288/1369`Retained binary64: `0.21037253469685901`	0Exact fraction: `0`Retained binary64: `0.0`
3	25/8Exact fraction: `25/8`Retained binary64: `3.125`	exterior	16/25Exact fraction: `16/25`Retained binary64: `0.64`	128/625Exact fraction: `128/625`Retained binary64: `0.2048`	0Exact fraction: `0`Retained binary64: `0.0`
4	19/6Exact fraction: `19/6`Retained binary64: `3.1666666666666665`	exterior	12/19Exact fraction: `12/19`Retained binary64: `0.631578947368421`	72/361Exact fraction: `72/361`Retained binary64: `0.1994459833795014`	0Exact fraction: `0`Retained binary64: `0.0`
5	77/24Exact fraction: `77/24`Retained binary64: `3.2083333333333335`	exterior	48/77Exact fraction: `48/77`Retained binary64: `0.6233766233766234`	1152/5929Exact fraction: `1152/5929`Retained binary64: `0.19429920728622027`	0Exact fraction: `0`Retained binary64: `0.0`
6	13/4Exact fraction: `13/4`Retained binary64: `3.25`	exterior	8/13Exact fraction: `8/13`Retained binary64: `0.6153846153846154`	32/169Exact fraction: `32/169`Retained binary64: `0.1893491124260355`	0Exact fraction: `0`Retained binary64: `0.0`
7	79/24Exact fraction: `79/24`Retained binary64: `3.2916666666666665`	exterior	48/79Exact fraction: `48/79`Retained binary64: `0.6075949367088608`	1152/6241Exact fraction: `1152/6241`Retained binary64: `0.18458580355712226`	0Exact fraction: `0`Retained binary64: `0.0`
8	10/3Exact fraction: `10/3`Retained binary64: `3.3333333333333335`	exterior	3/5Exact fraction: `3/5`Retained binary64: `0.6`	9/50Exact fraction: `9/50`Retained binary64: `0.18`	0Exact fraction: `0`Retained binary64: `0.0`
9	27/8Exact fraction: `27/8`Retained binary64: `3.375`	exterior	16/27Exact fraction: `16/27`Retained binary64: `0.5925925925925926`	128/729Exact fraction: `128/729`Retained binary64: `0.1755829903978052`	0Exact fraction: `0`Retained binary64: `0.0`
10	41/12Exact fraction: `41/12`Retained binary64: `3.4166666666666665`	exterior	24/41Exact fraction: `24/41`Retained binary64: `0.5853658536585366`	288/1681Exact fraction: `288/1681`Retained binary64: `0.17132659131469363`	0Exact fraction: `0`Retained binary64: `0.0`
11	83/24Exact fraction: `83/24`Retained binary64: `3.4583333333333335`	exterior	48/83Exact fraction: `48/83`Retained binary64: `0.5783132530120482`	1152/6889Exact fraction: `1152/6889`Retained binary64: `0.16722310930468864`	0Exact fraction: `0`Retained binary64: `0.0`
12	7/2Exact fraction: `7/2`Retained binary64: `3.5`	exterior	4/7Exact fraction: `4/7`Retained binary64: `0.5714285714285714`	8/49Exact fraction: `8/49`Retained binary64: `0.16326530612244897`	0Exact fraction: `0`Retained binary64: `0.0`
13	85/24Exact fraction: `85/24`Retained binary64: `3.5416666666666665`	exterior	48/85Exact fraction: `48/85`Retained binary64: `0.5647058823529412`	1152/7225Exact fraction: `1152/7225`Retained binary64: `0.1594463667820069`	0Exact fraction: `0`Retained binary64: `0.0`
14	43/12Exact fraction: `43/12`Retained binary64: `3.5833333333333335`	exterior	24/43Exact fraction: `24/43`Retained binary64: `0.5581395348837209`	288/1849Exact fraction: `288/1849`Retained binary64: `0.15575987020010817`	0Exact fraction: `0`Retained binary64: `0.0`
15	29/8Exact fraction: `29/8`Retained binary64: `3.625`	exterior	16/29Exact fraction: `16/29`Retained binary64: `0.5517241379310345`	128/841Exact fraction: `128/841`Retained binary64: `0.15219976218787157`	0Exact fraction: `0`Retained binary64: `0.0`
16	11/3Exact fraction: `11/3`Retained binary64: `3.6666666666666665`	exterior	6/11Exact fraction: `6/11`Retained binary64: `0.5454545454545454`	18/121Exact fraction: `18/121`Retained binary64: `0.1487603305785124`	0Exact fraction: `0`Retained binary64: `0.0`
17	89/24Exact fraction: `89/24`Retained binary64: `3.7083333333333335`	exterior	48/89Exact fraction: `48/89`Retained binary64: `0.5393258426966292`	1152/7921Exact fraction: `1152/7921`Retained binary64: `0.14543618230021463`	0Exact fraction: `0`Retained binary64: `0.0`
18	15/4Exact fraction: `15/4`Retained binary64: `3.75`	exterior	8/15Exact fraction: `8/15`Retained binary64: `0.5333333333333333`	32/225Exact fraction: `32/225`Retained binary64: `0.14222222222222222`	0Exact fraction: `0`Retained binary64: `0.0`
19	91/24Exact fraction: `91/24`Retained binary64: `3.7916666666666665`	exterior	48/91Exact fraction: `48/91`Retained binary64: `0.5274725274725275`	1152/8281Exact fraction: `1152/8281`Retained binary64: `0.13911363361912812`	0Exact fraction: `0`Retained binary64: `0.0`
20	23/6Exact fraction: `23/6`Retained binary64: `3.8333333333333335`	exterior	12/23Exact fraction: `12/23`Retained binary64: `0.5217391304347826`	72/529Exact fraction: `72/529`Retained binary64: `0.13610586011342155`	0Exact fraction: `0`Retained binary64: `0.0`
21	31/8Exact fraction: `31/8`Retained binary64: `3.875`	exterior	16/31Exact fraction: `16/31`Retained binary64: `0.5161290322580645`	128/961Exact fraction: `128/961`Retained binary64: `0.1331945889698231`	0Exact fraction: `0`Retained binary64: `0.0`
22	47/12Exact fraction: `47/12`Retained binary64: `3.9166666666666665`	exterior	24/47Exact fraction: `24/47`Retained binary64: `0.5106382978723404`	288/2209Exact fraction: `288/2209`Retained binary64: `0.13037573562698054`	0Exact fraction: `0`Retained binary64: `0.0`
23	95/24Exact fraction: `95/24`Retained binary64: `3.9583333333333335`	exterior	48/95Exact fraction: `48/95`Retained binary64: `0.5052631578947369`	1152/9025Exact fraction: `1152/9025`Retained binary64: `0.1276454293628809`	0Exact fraction: `0`Retained binary64: `0.0`
24	4Exact fraction: `4`Retained binary64: `4.0`	exterior	1/2Exact fraction: `1/2`Retained binary64: `0.5`	1/8Exact fraction: `1/8`Retained binary64: `0.125`	0Exact fraction: `0`Retained binary64: `0.0`
25	97/24Exact fraction: `97/24`Retained binary64: `4.041666666666667`	exterior	48/97Exact fraction: `48/97`Retained binary64: `0.4948453608247423`	1152/9409Exact fraction: `1152/9409`Retained binary64: `0.12243596556488469`	0Exact fraction: `0`Retained binary64: `0.0`
26	49/12Exact fraction: `49/12`Retained binary64: `4.083333333333333`	exterior	24/49Exact fraction: `24/49`Retained binary64: `0.4897959183673469`	288/2401Exact fraction: `288/2401`Retained binary64: `0.11995002082465639`	0Exact fraction: `0`Retained binary64: `0.0`
27	33/8Exact fraction: `33/8`Retained binary64: `4.125`	exterior	16/33Exact fraction: `16/33`Retained binary64: `0.48484848484848486`	128/1089Exact fraction: `128/1089`Retained binary64: `0.11753902662993572`	0Exact fraction: `0`Retained binary64: `0.0`
28	25/6Exact fraction: `25/6`Retained binary64: `4.166666666666667`	exterior	12/25Exact fraction: `12/25`Retained binary64: `0.48`	72/625Exact fraction: `72/625`Retained binary64: `0.1152`	0Exact fraction: `0`Retained binary64: `0.0`
29	101/24Exact fraction: `101/24`Retained binary64: `4.208333333333333`	exterior	48/101Exact fraction: `48/101`Retained binary64: `0.4752475247524752`	1152/10201Exact fraction: `1152/10201`Retained binary64: `0.11293010489167729`	0Exact fraction: `0`Retained binary64: `0.0`
30	17/4Exact fraction: `17/4`Retained binary64: `4.25`	exterior	8/17Exact fraction: `8/17`Retained binary64: `0.47058823529411764`	32/289Exact fraction: `32/289`Retained binary64: `0.11072664359861592`	0Exact fraction: `0`Retained binary64: `0.0`
31	103/24Exact fraction: `103/24`Retained binary64: `4.291666666666667`	exterior	48/103Exact fraction: `48/103`Retained binary64: `0.46601941747572817`	1152/10609Exact fraction: `1152/10609`Retained binary64: `0.10858704873220851`	0Exact fraction: `0`Retained binary64: `0.0`
32	13/3Exact fraction: `13/3`Retained binary64: `4.333333333333333`	exterior	6/13Exact fraction: `6/13`Retained binary64: `0.46153846153846156`	18/169Exact fraction: `18/169`Retained binary64: `0.10650887573964497`	0Exact fraction: `0`Retained binary64: `0.0`
33	35/8Exact fraction: `35/8`Retained binary64: `4.375`	exterior	16/35Exact fraction: `16/35`Retained binary64: `0.45714285714285713`	128/1225Exact fraction: `128/1225`Retained binary64: `0.10448979591836735`	0Exact fraction: `0`Retained binary64: `0.0`
34	53/12Exact fraction: `53/12`Retained binary64: `4.416666666666667`	exterior	24/53Exact fraction: `24/53`Retained binary64: `0.4528301886792453`	288/2809Exact fraction: `288/2809`Retained binary64: `0.10252758988964045`	0Exact fraction: `0`Retained binary64: `0.0`
35	107/24Exact fraction: `107/24`Retained binary64: `4.458333333333333`	exterior	48/107Exact fraction: `48/107`Retained binary64: `0.4485981308411215`	1152/11449Exact fraction: `1152/11449`Retained binary64: `0.10062014149707398`	0Exact fraction: `0`Retained binary64: `0.0`
36	9/2Exact fraction: `9/2`Retained binary64: `4.5`	exterior	4/9Exact fraction: `4/9`Retained binary64: `0.4444444444444444`	8/81Exact fraction: `8/81`Retained binary64: `0.09876543209876543`	0Exact fraction: `0`Retained binary64: `0.0`
37	109/24Exact fraction: `109/24`Retained binary64: `4.541666666666667`	exterior	48/109Exact fraction: `48/109`Retained binary64: `0.44036697247706424`	1152/11881Exact fraction: `1152/11881`Retained binary64: `0.09696153522430771`	0Exact fraction: `0`Retained binary64: `0.0`
38	55/12Exact fraction: `55/12`Retained binary64: `4.583333333333333`	exterior	24/55Exact fraction: `24/55`Retained binary64: `0.43636363636363634`	288/3025Exact fraction: `288/3025`Retained binary64: `0.09520661157024793`	0Exact fraction: `0`Retained binary64: `0.0`
39	37/8Exact fraction: `37/8`Retained binary64: `4.625`	exterior	16/37Exact fraction: `16/37`Retained binary64: `0.43243243243243246`	128/1369Exact fraction: `128/1369`Retained binary64: `0.09349890430971512`	0Exact fraction: `0`Retained binary64: `0.0`
40	14/3Exact fraction: `14/3`Retained binary64: `4.666666666666667`	exterior	3/7Exact fraction: `3/7`Retained binary64: `0.42857142857142855`	9/98Exact fraction: `9/98`Retained binary64: `0.09183673469387756`	0Exact fraction: `0`Retained binary64: `0.0`
41	113/24Exact fraction: `113/24`Retained binary64: `4.708333333333333`	exterior	48/113Exact fraction: `48/113`Retained binary64: `0.4247787610619469`	1152/12769Exact fraction: `1152/12769`Retained binary64: `0.09021849792466129`	0Exact fraction: `0`Retained binary64: `0.0`
42	19/4Exact fraction: `19/4`Retained binary64: `4.75`	exterior	8/19Exact fraction: `8/19`Retained binary64: `0.42105263157894735`	32/361Exact fraction: `32/361`Retained binary64: `0.0886426592797784`	0Exact fraction: `0`Retained binary64: `0.0`
43	115/24Exact fraction: `115/24`Retained binary64: `4.791666666666667`	exterior	48/115Exact fraction: `48/115`Retained binary64: `0.41739130434782606`	1152/13225Exact fraction: `1152/13225`Retained binary64: `0.0871077504725898`	0Exact fraction: `0`Retained binary64: `0.0`
44	29/6Exact fraction: `29/6`Retained binary64: `4.833333333333333`	exterior	12/29Exact fraction: `12/29`Retained binary64: `0.41379310344827586`	72/841Exact fraction: `72/841`Retained binary64: `0.08561236623067776`	0Exact fraction: `0`Retained binary64: `0.0`
45	39/8Exact fraction: `39/8`Retained binary64: `4.875`	exterior	16/39Exact fraction: `16/39`Retained binary64: `0.41025641025641024`	128/1521Exact fraction: `128/1521`Retained binary64: `0.084155161078238`	0Exact fraction: `0`Retained binary64: `0.0`
46	59/12Exact fraction: `59/12`Retained binary64: `4.916666666666667`	exterior	24/59Exact fraction: `24/59`Retained binary64: `0.4067796610169492`	288/3481Exact fraction: `288/3481`Retained binary64: `0.08273484630853203`	0Exact fraction: `0`Retained binary64: `0.0`
47	119/24Exact fraction: `119/24`Retained binary64: `4.958333333333333`	exterior	48/119Exact fraction: `48/119`Retained binary64: `0.40336134453781514`	1152/14161Exact fraction: `1152/14161`Retained binary64: `0.081350187133677`	0Exact fraction: `0`Retained binary64: `0.0`
48	5Exact fraction: `5`Retained binary64: `5.0`	exterior	2/5Exact fraction: `2/5`Retained binary64: `0.4`	2/25Exact fraction: `2/25`Retained binary64: `0.08`	0Exact fraction: `0`Retained binary64: `0.0`
49	121/24Exact fraction: `121/24`Retained binary64: `5.041666666666667`	exterior	48/121Exact fraction: `48/121`Retained binary64: `0.39669421487603307`	1152/14641Exact fraction: `1152/14641`Retained binary64: `0.07868315005805615`	0Exact fraction: `0`Retained binary64: `0.0`
50	61/12Exact fraction: `61/12`Retained binary64: `5.083333333333333`	exterior	24/61Exact fraction: `24/61`Retained binary64: `0.39344262295081966`	288/3721Exact fraction: `288/3721`Retained binary64: `0.07739854877721043`	0Exact fraction: `0`Retained binary64: `0.0`
51	41/8Exact fraction: `41/8`Retained binary64: `5.125`	exterior	16/41Exact fraction: `16/41`Retained binary64: `0.3902439024390244`	128/1681Exact fraction: `128/1681`Retained binary64: `0.0761451516954194`	0Exact fraction: `0`Retained binary64: `0.0`
52	31/6Exact fraction: `31/6`Retained binary64: `5.166666666666667`	exterior	12/31Exact fraction: `12/31`Retained binary64: `0.3870967741935484`	72/961Exact fraction: `72/961`Retained binary64: `0.07492195629552549`	0Exact fraction: `0`Retained binary64: `0.0`
53	125/24Exact fraction: `125/24`Retained binary64: `5.208333333333333`	exterior	48/125Exact fraction: `48/125`Retained binary64: `0.384`	1152/15625Exact fraction: `1152/15625`Retained binary64: `0.073728`	0Exact fraction: `0`Retained binary64: `0.0`
54	21/4Exact fraction: `21/4`Retained binary64: `5.25`	exterior	8/21Exact fraction: `8/21`Retained binary64: `0.38095238095238093`	32/441Exact fraction: `32/441`Retained binary64: `0.07256235827664399`	0Exact fraction: `0`Retained binary64: `0.0`
55	127/24Exact fraction: `127/24`Retained binary64: `5.291666666666667`	exterior	48/127Exact fraction: `48/127`Retained binary64: `0.3779527559055118`	1152/16129Exact fraction: `1152/16129`Retained binary64: `0.0714241428482857`	0Exact fraction: `0`Retained binary64: `0.0`
56	16/3Exact fraction: `16/3`Retained binary64: `5.333333333333333`	exterior	3/8Exact fraction: `3/8`Retained binary64: `0.375`	9/128Exact fraction: `9/128`Retained binary64: `0.0703125`	0Exact fraction: `0`Retained binary64: `0.0`
57	43/8Exact fraction: `43/8`Retained binary64: `5.375`	exterior	16/43Exact fraction: `16/43`Retained binary64: `0.37209302325581395`	128/1849Exact fraction: `128/1849`Retained binary64: `0.06922660897782586`	0Exact fraction: `0`Retained binary64: `0.0`
58	65/12Exact fraction: `65/12`Retained binary64: `5.416666666666667`	exterior	24/65Exact fraction: `24/65`Retained binary64: `0.36923076923076925`	288/4225Exact fraction: `288/4225`Retained binary64: `0.06816568047337278`	0Exact fraction: `0`Retained binary64: `0.0`
59	131/24Exact fraction: `131/24`Retained binary64: `5.458333333333333`	exterior	48/131Exact fraction: `48/131`Retained binary64: `0.366412213740458`	1152/17161Exact fraction: `1152/17161`Retained binary64: `0.06712895518909154`	0Exact fraction: `0`Retained binary64: `0.0`
60	11/2Exact fraction: `11/2`Retained binary64: `5.5`	exterior	4/11Exact fraction: `4/11`Retained binary64: `0.36363636363636365`	8/121Exact fraction: `8/121`Retained binary64: `0.06611570247933884`	0Exact fraction: `0`Retained binary64: `0.0`
61	133/24Exact fraction: `133/24`Retained binary64: `5.541666666666667`	exterior	48/133Exact fraction: `48/133`Retained binary64: `0.3609022556390977`	1152/17689Exact fraction: `1152/17689`Retained binary64: `0.06512521906269433`	0Exact fraction: `0`Retained binary64: `0.0`
62	67/12Exact fraction: `67/12`Retained binary64: `5.583333333333333`	exterior	24/67Exact fraction: `24/67`Retained binary64: `0.3582089552238806`	288/4489Exact fraction: `288/4489`Retained binary64: `0.06415682780129205`	0Exact fraction: `0`Retained binary64: `0.0`
63	45/8Exact fraction: `45/8`Retained binary64: `5.625`	exterior	16/45Exact fraction: `16/45`Retained binary64: `0.35555555555555557`	128/2025Exact fraction: `128/2025`Retained binary64: `0.06320987654320988`	0Exact fraction: `0`Retained binary64: `0.0`
64	17/3Exact fraction: `17/3`Retained binary64: `5.666666666666667`	exterior	6/17Exact fraction: `6/17`Retained binary64: `0.35294117647058826`	18/289Exact fraction: `18/289`Retained binary64: `0.06228373702422145`	0Exact fraction: `0`Retained binary64: `0.0`
65	137/24Exact fraction: `137/24`Retained binary64: `5.708333333333333`	exterior	48/137Exact fraction: `48/137`Retained binary64: `0.35036496350364965`	1152/18769Exact fraction: `1152/18769`Retained binary64: `0.06137780382545687`	0Exact fraction: `0`Retained binary64: `0.0`
66	23/4Exact fraction: `23/4`Retained binary64: `5.75`	exterior	8/23Exact fraction: `8/23`Retained binary64: `0.34782608695652173`	32/529Exact fraction: `32/529`Retained binary64: `0.06049149338374291`	0Exact fraction: `0`Retained binary64: `0.0`
67	139/24Exact fraction: `139/24`Retained binary64: `5.791666666666667`	exterior	48/139Exact fraction: `48/139`Retained binary64: `0.34532374100719426`	1152/19321Exact fraction: `1152/19321`Retained binary64: `0.05962424305160188`	0Exact fraction: `0`Retained binary64: `0.0`
68	35/6Exact fraction: `35/6`Retained binary64: `5.833333333333333`	exterior	12/35Exact fraction: `12/35`Retained binary64: `0.34285714285714286`	72/1225Exact fraction: `72/1225`Retained binary64: `0.05877551020408163`	0Exact fraction: `0`Retained binary64: `0.0`
69	47/8Exact fraction: `47/8`Retained binary64: `5.875`	exterior	16/47Exact fraction: `16/47`Retained binary64: `0.3404255319148936`	128/2209Exact fraction: `128/2209`Retained binary64: `0.057944771389769126`	0Exact fraction: `0`Retained binary64: `0.0`
70	71/12Exact fraction: `71/12`Retained binary64: `5.916666666666667`	exterior	24/71Exact fraction: `24/71`Retained binary64: `0.3380281690140845`	288/5041Exact fraction: `288/5041`Retained binary64: `0.05713152152350724`	0Exact fraction: `0`Retained binary64: `0.0`
71	143/24Exact fraction: `143/24`Retained binary64: `5.958333333333333`	exterior	48/143Exact fraction: `48/143`Retained binary64: `0.3356643356643357`	1152/20449Exact fraction: `1152/20449`Retained binary64: `0.056335273118489904`	0Exact fraction: `0`Retained binary64: `0.0`
72	6Exact fraction: `6`Retained binary64: `6.0`	exterior	1/3Exact fraction: `1/3`Retained binary64: `0.3333333333333333`	1/18Exact fraction: `1/18`Retained binary64: `0.05555555555555555`	0Exact fraction: `0`Retained binary64: `0.0`
73	145/24Exact fraction: `145/24`Retained binary64: `6.041666666666667`	exterior	48/145Exact fraction: `48/145`Retained binary64: `0.3310344827586207`	1152/21025Exact fraction: `1152/21025`Retained binary64: `0.05479191438763377`	0Exact fraction: `0`Retained binary64: `0.0`
74	73/12Exact fraction: `73/12`Retained binary64: `6.083333333333333`	exterior	24/73Exact fraction: `24/73`Retained binary64: `0.3287671232876712`	288/5329Exact fraction: `288/5329`Retained binary64: `0.05404391067742541`	0Exact fraction: `0`Retained binary64: `0.0`
75	49/8Exact fraction: `49/8`Retained binary64: `6.125`	exterior	16/49Exact fraction: `16/49`Retained binary64: `0.32653061224489793`	128/2401Exact fraction: `128/2401`Retained binary64: `0.053311120366513955`	0Exact fraction: `0`Retained binary64: `0.0`
76	37/6Exact fraction: `37/6`Retained binary64: `6.166666666666667`	exterior	12/37Exact fraction: `12/37`Retained binary64: `0.32432432432432434`	72/1369Exact fraction: `72/1369`Retained binary64: `0.052593133674214754`	0Exact fraction: `0`Retained binary64: `0.0`
77	149/24Exact fraction: `149/24`Retained binary64: `6.208333333333333`	exterior	48/149Exact fraction: `48/149`Retained binary64: `0.3221476510067114`	1152/22201Exact fraction: `1152/22201`Retained binary64: `0.05188955452457097`	0Exact fraction: `0`Retained binary64: `0.0`
78	25/4Exact fraction: `25/4`Retained binary64: `6.25`	exterior	8/25Exact fraction: `8/25`Retained binary64: `0.32`	32/625Exact fraction: `32/625`Retained binary64: `0.0512`	0Exact fraction: `0`Retained binary64: `0.0`
79	151/24Exact fraction: `151/24`Retained binary64: `6.291666666666667`	exterior	48/151Exact fraction: `48/151`Retained binary64: `0.31788079470198677`	1152/22801Exact fraction: `1152/22801`Retained binary64: `0.05052409982018333`	0Exact fraction: `0`Retained binary64: `0.0`
80	19/3Exact fraction: `19/3`Retained binary64: `6.333333333333333`	exterior	6/19Exact fraction: `6/19`Retained binary64: `0.3157894736842105`	18/361Exact fraction: `18/361`Retained binary64: `0.04986149584487535`	0Exact fraction: `0`Retained binary64: `0.0`
81	51/8Exact fraction: `51/8`Retained binary64: `6.375`	exterior	16/51Exact fraction: `16/51`Retained binary64: `0.3137254901960784`	128/2601Exact fraction: `128/2601`Retained binary64: `0.049211841599384853`	0Exact fraction: `0`Retained binary64: `0.0`
82	77/12Exact fraction: `77/12`Retained binary64: `6.416666666666667`	exterior	24/77Exact fraction: `24/77`Retained binary64: `0.3116883116883117`	288/5929Exact fraction: `288/5929`Retained binary64: `0.04857480182155507`	0Exact fraction: `0`Retained binary64: `0.0`
83	155/24Exact fraction: `155/24`Retained binary64: `6.458333333333333`	exterior	48/155Exact fraction: `48/155`Retained binary64: `0.3096774193548387`	1152/24025Exact fraction: `1152/24025`Retained binary64: `0.04795005202913632`	0Exact fraction: `0`Retained binary64: `0.0`
84	13/2Exact fraction: `13/2`Retained binary64: `6.5`	exterior	4/13Exact fraction: `4/13`Retained binary64: `0.3076923076923077`	8/169Exact fraction: `8/169`Retained binary64: `0.047337278106508875`	0Exact fraction: `0`Retained binary64: `0.0`
85	157/24Exact fraction: `157/24`Retained binary64: `6.541666666666667`	exterior	48/157Exact fraction: `48/157`Retained binary64: `0.3057324840764331`	1152/24649Exact fraction: `1152/24649`Retained binary64: `0.04673617590977321`	0Exact fraction: `0`Retained binary64: `0.0`
86	79/12Exact fraction: `79/12`Retained binary64: `6.583333333333333`	exterior	24/79Exact fraction: `24/79`Retained binary64: `0.3037974683544304`	288/6241Exact fraction: `288/6241`Retained binary64: `0.046146450889280564`	0Exact fraction: `0`Retained binary64: `0.0`
87	53/8Exact fraction: `53/8`Retained binary64: `6.625`	exterior	16/53Exact fraction: `16/53`Retained binary64: `0.3018867924528302`	128/2809Exact fraction: `128/2809`Retained binary64: `0.04556781772872909`	0Exact fraction: `0`Retained binary64: `0.0`
88	20/3Exact fraction: `20/3`Retained binary64: `6.666666666666667`	exterior	3/10Exact fraction: `3/10`Retained binary64: `0.3`	9/200Exact fraction: `9/200`Retained binary64: `0.045`	0Exact fraction: `0`Retained binary64: `0.0`
89	161/24Exact fraction: `161/24`Retained binary64: `6.708333333333333`	exterior	48/161Exact fraction: `48/161`Retained binary64: `0.2981366459627329`	1152/25921Exact fraction: `1152/25921`Retained binary64: `0.04444272983295398`	0Exact fraction: `0`Retained binary64: `0.0`
90	27/4Exact fraction: `27/4`Retained binary64: `6.75`	exterior	8/27Exact fraction: `8/27`Retained binary64: `0.2962962962962963`	32/729Exact fraction: `32/729`Retained binary64: `0.0438957475994513`	0Exact fraction: `0`Retained binary64: `0.0`
91	163/24Exact fraction: `163/24`Retained binary64: `6.791666666666667`	exterior	48/163Exact fraction: `48/163`Retained binary64: `0.294478527607362`	1152/26569Exact fraction: `1152/26569`Retained binary64: `0.04335880161089992`	0Exact fraction: `0`Retained binary64: `0.0`
92	41/6Exact fraction: `41/6`Retained binary64: `6.833333333333333`	exterior	12/41Exact fraction: `12/41`Retained binary64: `0.2926829268292683`	72/1681Exact fraction: `72/1681`Retained binary64: `0.04283164782867341`	0Exact fraction: `0`Retained binary64: `0.0`
93	55/8Exact fraction: `55/8`Retained binary64: `6.875`	exterior	16/55Exact fraction: `16/55`Retained binary64: `0.2909090909090909`	128/3025Exact fraction: `128/3025`Retained binary64: `0.04231404958677686`	0Exact fraction: `0`Retained binary64: `0.0`
94	83/12Exact fraction: `83/12`Retained binary64: `6.916666666666667`	exterior	24/83Exact fraction: `24/83`Retained binary64: `0.2891566265060241`	288/6889Exact fraction: `288/6889`Retained binary64: `0.04180577732617216`	0Exact fraction: `0`Retained binary64: `0.0`
95	167/24Exact fraction: `167/24`Retained binary64: `6.958333333333333`	exterior	48/167Exact fraction: `48/167`Retained binary64: `0.2874251497005988`	1152/27889Exact fraction: `1152/27889`Retained binary64: `0.041306608340205817`	0Exact fraction: `0`Retained binary64: `0.0`
96	7Exact fraction: `7`Retained binary64: `7.0`	exterior	2/7Exact fraction: `2/7`Retained binary64: `0.2857142857142857`	2/49Exact fraction: `2/49`Retained binary64: `0.04081632653061224`	0Exact fraction: `0`Retained binary64: `0.0`
97	169/24Exact fraction: `169/24`Retained binary64: `7.041666666666667`	exterior	48/169Exact fraction: `48/169`Retained binary64: `0.28402366863905326`	1152/28561Exact fraction: `1152/28561`Retained binary64: `0.04033472217359336`	0Exact fraction: `0`Retained binary64: `0.0`
98	85/12Exact fraction: `85/12`Retained binary64: `7.083333333333333`	exterior	24/85Exact fraction: `24/85`Retained binary64: `0.2823529411764706`	288/7225Exact fraction: `288/7225`Retained binary64: `0.03986159169550173`	0Exact fraction: `0`Retained binary64: `0.0`
99	57/8Exact fraction: `57/8`Retained binary64: `7.125`	exterior	16/57Exact fraction: `16/57`Retained binary64: `0.2807017543859649`	128/3249Exact fraction: `128/3249`Retained binary64: `0.039396737457679284`	0Exact fraction: `0`Retained binary64: `0.0`
100	43/6Exact fraction: `43/6`Retained binary64: `7.166666666666667`	exterior	12/43Exact fraction: `12/43`Retained binary64: `0.27906976744186046`	72/1849Exact fraction: `72/1849`Retained binary64: `0.038939967550027044`	0Exact fraction: `0`Retained binary64: `0.0`
101	173/24Exact fraction: `173/24`Retained binary64: `7.208333333333333`	exterior	48/173Exact fraction: `48/173`Retained binary64: `0.2774566473988439`	1152/29929Exact fraction: `1152/29929`Retained binary64: `0.0384910955929032`	0Exact fraction: `0`Retained binary64: `0.0`
102	29/4Exact fraction: `29/4`Retained binary64: `7.25`	exterior	8/29Exact fraction: `8/29`Retained binary64: `0.27586206896551724`	32/841Exact fraction: `32/841`Retained binary64: `0.03804994054696789`	0Exact fraction: `0`Retained binary64: `0.0`
103	175/24Exact fraction: `175/24`Retained binary64: `7.291666666666667`	exterior	48/175Exact fraction: `48/175`Retained binary64: `0.2742857142857143`	1152/30625Exact fraction: `1152/30625`Retained binary64: `0.03761632653061225`	0Exact fraction: `0`Retained binary64: `0.0`
104	22/3Exact fraction: `22/3`Retained binary64: `7.333333333333333`	exterior	3/11Exact fraction: `3/11`Retained binary64: `0.2727272727272727`	9/242Exact fraction: `9/242`Retained binary64: `0.0371900826446281`	0Exact fraction: `0`Retained binary64: `0.0`
105	59/8Exact fraction: `59/8`Retained binary64: `7.375`	exterior	16/59Exact fraction: `16/59`Retained binary64: `0.2711864406779661`	128/3481Exact fraction: `128/3481`Retained binary64: `0.036771042803792016`	0Exact fraction: `0`Retained binary64: `0.0`
106	89/12Exact fraction: `89/12`Retained binary64: `7.416666666666667`	exterior	24/89Exact fraction: `24/89`Retained binary64: `0.2696629213483146`	288/7921Exact fraction: `288/7921`Retained binary64: `0.03635904557505366`	0Exact fraction: `0`Retained binary64: `0.0`
107	179/24Exact fraction: `179/24`Retained binary64: `7.458333333333333`	exterior	48/179Exact fraction: `48/179`Retained binary64: `0.2681564245810056`	1152/32041Exact fraction: `1152/32041`Retained binary64: `0.03595393402203427`	0Exact fraction: `0`Retained binary64: `0.0`
108	15/2Exact fraction: `15/2`Retained binary64: `7.5`	exterior	4/15Exact fraction: `4/15`Retained binary64: `0.26666666666666666`	8/225Exact fraction: `8/225`Retained binary64: `0.035555555555555556`	0Exact fraction: `0`Retained binary64: `0.0`
109	181/24Exact fraction: `181/24`Retained binary64: `7.541666666666667`	exterior	48/181Exact fraction: `48/181`Retained binary64: `0.26519337016574585`	1152/32761Exact fraction: `1152/32761`Retained binary64: `0.035163761789933155`	0Exact fraction: `0`Retained binary64: `0.0`
110	91/12Exact fraction: `91/12`Retained binary64: `7.583333333333333`	exterior	24/91Exact fraction: `24/91`Retained binary64: `0.26373626373626374`	288/8281Exact fraction: `288/8281`Retained binary64: `0.03477840840478203`	0Exact fraction: `0`Retained binary64: `0.0`
111	61/8Exact fraction: `61/8`Retained binary64: `7.625`	exterior	16/61Exact fraction: `16/61`Retained binary64: `0.26229508196721313`	128/3721Exact fraction: `128/3721`Retained binary64: `0.03439935501209352`	0Exact fraction: `0`Retained binary64: `0.0`
112	23/3Exact fraction: `23/3`Retained binary64: `7.666666666666667`	exterior	6/23Exact fraction: `6/23`Retained binary64: `0.2608695652173913`	18/529Exact fraction: `18/529`Retained binary64: `0.034026465028355386`	0Exact fraction: `0`Retained binary64: `0.0`
113	185/24Exact fraction: `185/24`Retained binary64: `7.708333333333333`	exterior	48/185Exact fraction: `48/185`Retained binary64: `0.2594594594594595`	1152/34225Exact fraction: `1152/34225`Retained binary64: `0.03365960555149744`	0Exact fraction: `0`Retained binary64: `0.0`
114	31/4Exact fraction: `31/4`Retained binary64: `7.75`	exterior	8/31Exact fraction: `8/31`Retained binary64: `0.25806451612903225`	32/961Exact fraction: `32/961`Retained binary64: `0.03329864724245578`	0Exact fraction: `0`Retained binary64: `0.0`
115	187/24Exact fraction: `187/24`Retained binary64: `7.791666666666667`	exterior	48/187Exact fraction: `48/187`Retained binary64: `0.25668449197860965`	1152/34969Exact fraction: `1152/34969`Retained binary64: `0.03294346421115846`	0Exact fraction: `0`Retained binary64: `0.0`
116	47/6Exact fraction: `47/6`Retained binary64: `7.833333333333333`	exterior	12/47Exact fraction: `12/47`Retained binary64: `0.2553191489361702`	72/2209Exact fraction: `72/2209`Retained binary64: `0.032593933906745136`	0Exact fraction: `0`Retained binary64: `0.0`
117	63/8Exact fraction: `63/8`Retained binary64: `7.875`	exterior	16/63Exact fraction: `16/63`Retained binary64: `0.25396825396825395`	128/3969Exact fraction: `128/3969`Retained binary64: `0.03224993701184178`	0Exact fraction: `0`Retained binary64: `0.0`
118	95/12Exact fraction: `95/12`Retained binary64: `7.916666666666667`	exterior	24/95Exact fraction: `24/95`Retained binary64: `0.25263157894736843`	288/9025Exact fraction: `288/9025`Retained binary64: `0.031911357340720224`	0Exact fraction: `0`Retained binary64: `0.0`
119	191/24Exact fraction: `191/24`Retained binary64: `7.958333333333333`	exterior	48/191Exact fraction: `48/191`Retained binary64: `0.2513089005235602`	1152/36481Exact fraction: `1152/36481`Retained binary64: `0.03157808174118034`	0Exact fraction: `0`Retained binary64: `0.0`
120	8Exact fraction: `8`Retained binary64: `8.0`	exterior	1/4Exact fraction: `1/4`Retained binary64: `0.25`	1/32Exact fraction: `1/32`Retained binary64: `0.03125`	0Exact fraction: `0`Retained binary64: `0.0`
121	193/24Exact fraction: `193/24`Retained binary64: `8.041666666666666`	exterior	48/193Exact fraction: `48/193`Retained binary64: `0.24870466321243523`	1152/37249Exact fraction: `1152/37249`Retained binary64: `0.030927004751805418`	0Exact fraction: `0`Retained binary64: `0.0`
122	97/12Exact fraction: `97/12`Retained binary64: `8.083333333333334`	exterior	24/97Exact fraction: `24/97`Retained binary64: `0.24742268041237114`	288/9409Exact fraction: `288/9409`Retained binary64: `0.030608991391221172`	0Exact fraction: `0`Retained binary64: `0.0`
123	65/8Exact fraction: `65/8`Retained binary64: `8.125`	exterior	16/65Exact fraction: `16/65`Retained binary64: `0.24615384615384617`	128/4225Exact fraction: `128/4225`Retained binary64: `0.03029585798816568`	0Exact fraction: `0`Retained binary64: `0.0`
124	49/6Exact fraction: `49/6`Retained binary64: `8.166666666666666`	exterior	12/49Exact fraction: `12/49`Retained binary64: `0.24489795918367346`	72/2401Exact fraction: `72/2401`Retained binary64: `0.029987505206164097`	0Exact fraction: `0`Retained binary64: `0.0`
125	197/24Exact fraction: `197/24`Retained binary64: `8.208333333333334`	exterior	48/197Exact fraction: `48/197`Retained binary64: `0.2436548223350254`	1152/38809Exact fraction: `1152/38809`Retained binary64: `0.02968383622355639`	0Exact fraction: `0`Retained binary64: `0.0`
126	33/4Exact fraction: `33/4`Retained binary64: `8.25`	exterior	8/33Exact fraction: `8/33`Retained binary64: `0.24242424242424243`	32/1089Exact fraction: `32/1089`Retained binary64: `0.02938475665748393`	0Exact fraction: `0`Retained binary64: `0.0`
127	199/24Exact fraction: `199/24`Retained binary64: `8.291666666666666`	exterior	48/199Exact fraction: `48/199`Retained binary64: `0.24120603015075376`	1152/39601Exact fraction: `1152/39601`Retained binary64: `0.02909017449054317`	0Exact fraction: `0`Retained binary64: `0.0`
128	25/3Exact fraction: `25/3`Retained binary64: `8.333333333333334`	exterior	6/25Exact fraction: `6/25`Retained binary64: `0.24`	18/625Exact fraction: `18/625`Retained binary64: `0.0288`	0Exact fraction: `0`Retained binary64: `0.0`
129	67/8Exact fraction: `67/8`Retained binary64: `8.375`	exterior	16/67Exact fraction: `16/67`Retained binary64: `0.23880597014925373`	128/4489Exact fraction: `128/4489`Retained binary64: `0.028514145689463134`	0Exact fraction: `0`Retained binary64: `0.0`
130	101/12Exact fraction: `101/12`Retained binary64: `8.416666666666666`	exterior	24/101Exact fraction: `24/101`Retained binary64: `0.2376237623762376`	288/10201Exact fraction: `288/10201`Retained binary64: `0.028232526222919323`	0Exact fraction: `0`Retained binary64: `0.0`
131	203/24Exact fraction: `203/24`Retained binary64: `8.458333333333334`	exterior	48/203Exact fraction: `48/203`Retained binary64: `0.23645320197044334`	1152/41209Exact fraction: `1152/41209`Retained binary64: `0.02795505836103764`	0Exact fraction: `0`Retained binary64: `0.0`
132	17/2Exact fraction: `17/2`Retained binary64: `8.5`	exterior	4/17Exact fraction: `4/17`Retained binary64: `0.23529411764705882`	8/289Exact fraction: `8/289`Retained binary64: `0.02768166089965398`	0Exact fraction: `0`Retained binary64: `0.0`
133	205/24Exact fraction: `205/24`Retained binary64: `8.541666666666666`	exterior	48/205Exact fraction: `48/205`Retained binary64: `0.23414634146341465`	1152/42025Exact fraction: `1152/42025`Retained binary64: `0.027412254610350982`	0Exact fraction: `0`Retained binary64: `0.0`
134	103/12Exact fraction: `103/12`Retained binary64: `8.583333333333334`	exterior	24/103Exact fraction: `24/103`Retained binary64: `0.23300970873786409`	288/10609Exact fraction: `288/10609`Retained binary64: `0.027146762183052127`	0Exact fraction: `0`Retained binary64: `0.0`
135	69/8Exact fraction: `69/8`Retained binary64: `8.625`	exterior	16/69Exact fraction: `16/69`Retained binary64: `0.2318840579710145`	128/4761Exact fraction: `128/4761`Retained binary64: `0.026885108170552403`	0Exact fraction: `0`Retained binary64: `0.0`
136	26/3Exact fraction: `26/3`Retained binary64: `8.666666666666666`	exterior	3/13Exact fraction: `3/13`Retained binary64: `0.23076923076923078`	9/338Exact fraction: `9/338`Retained binary64: `0.026627218934911243`	0Exact fraction: `0`Retained binary64: `0.0`
137	209/24Exact fraction: `209/24`Retained binary64: `8.708333333333334`	exterior	48/209Exact fraction: `48/209`Retained binary64: `0.22966507177033493`	1152/43681Exact fraction: `1152/43681`Retained binary64: `0.026373022595636546`	0Exact fraction: `0`Retained binary64: `0.0`
138	35/4Exact fraction: `35/4`Retained binary64: `8.75`	exterior	8/35Exact fraction: `8/35`Retained binary64: `0.22857142857142856`	32/1225Exact fraction: `32/1225`Retained binary64: `0.026122448979591838`	0Exact fraction: `0`Retained binary64: `0.0`
139	211/24Exact fraction: `211/24`Retained binary64: `8.791666666666666`	exterior	48/211Exact fraction: `48/211`Retained binary64: `0.22748815165876776`	1152/44521Exact fraction: `1152/44521`Retained binary64: `0.025875429572561264`	0Exact fraction: `0`Retained binary64: `0.0`
140	53/6Exact fraction: `53/6`Retained binary64: `8.833333333333334`	exterior	12/53Exact fraction: `12/53`Retained binary64: `0.22641509433962265`	72/2809Exact fraction: `72/2809`Retained binary64: `0.025631897472410112`	0Exact fraction: `0`Retained binary64: `0.0`
141	71/8Exact fraction: `71/8`Retained binary64: `8.875`	exterior	16/71Exact fraction: `16/71`Retained binary64: `0.22535211267605634`	128/5041Exact fraction: `128/5041`Retained binary64: `0.025391787343780996`	0Exact fraction: `0`Retained binary64: `0.0`
142	107/12Exact fraction: `107/12`Retained binary64: `8.916666666666666`	exterior	24/107Exact fraction: `24/107`Retained binary64: `0.22429906542056074`	288/11449Exact fraction: `288/11449`Retained binary64: `0.025155035374268494`	0Exact fraction: `0`Retained binary64: `0.0`
143	215/24Exact fraction: `215/24`Retained binary64: `8.958333333333334`	exterior	48/215Exact fraction: `48/215`Retained binary64: `0.22325581395348837`	1152/46225Exact fraction: `1152/46225`Retained binary64: `0.024921579232017307`	0Exact fraction: `0`Retained binary64: `0.0`
144	9Exact fraction: `9`Retained binary64: `9.0`	exterior	2/9Exact fraction: `2/9`Retained binary64: `0.2222222222222222`	2/81Exact fraction: `2/81`Retained binary64: `0.024691358024691357`	0Exact fraction: `0`Retained binary64: `0.0`

One-sided R=3 limits; no classical surface second derivative (2 rows)

Potential and field are continuous; curvature values are distinct one-sided limits only. secondDerivativeAtSurfaceAssigned=false.

One-sided R=3 limits; no classical surface second derivative
Row	R=r/L₀	Analytic branch	V/V₀	Signed outward Eᵣ/E₀	L₀²∇²V/V₀; one-sided limit atR=3
0R=3: the curvature value is the interior one-sided branch limit only; no classical second derivative is assigned at the surface.	3Exact fraction: `3`Retained binary64: `3.0`	interior	2/3Exact fraction: `2/3`Retained binary64: `0.6666666666666666`	2/9Exact fraction: `2/9`Retained binary64: `0.2222222222222222`	-2/9Exact fraction: `-2/9`Retained binary64: `-0.2222222222222222`
1R=3: the curvature value is the exterior one-sided branch limit only; no classical second derivative is assigned at the surface.	3Exact fraction: `3`Retained binary64: `3.0`	exterior	2/3Exact fraction: `2/3`Retained binary64: `0.6666666666666666`	2/9Exact fraction: `2/9`Retained binary64: `0.2222222222222222`	0Exact fraction: `0`Retained binary64: `0.0`

Normalized volume source density (1 rows)

ρL₀²/(ε₀V₀)=2/9, withV₀=kₑq₀/L₀ only in this separate ball model.

Normalized volume source density
Quantity	Exact normalized value and retained binary64
ρL₀²/(ε₀V₀)	2/9Exact fraction: `2/9`Retained binary64: `0.2222222222222222`

Exact potential and field controls for the separate solid ball
R = r/L₀	V/V₀	E_r/E₀	Branch
0	1	0	Regular center
1	26/27	2/27	Interior
3/2	11/12	1/9	Interior
3	2/3	2/9	Continuous surface limits
4	1/2	1/8	Exterior
6	1/3	1/18	Exterior
9	2/9	2/81	Exterior

Match interfaces with the right normal

Suppose a finite ordinary surface charge σ lies between two vacuum regions, and n̂ points from region 1 to region 2. Gauss’s thin-pillbox relation gives E₂·n̂ − E₁·n̂ = σ/ε₀. Since E = −∇V, this becomes ∂ₙV₂ − ∂ₙV₁ = −σ/ε₀. Both derivatives here use the same stated direction n̂; they are not the two domains’ opposite outward normals.

If the one-sided fields are finite and the interface is an ordinary charge sheet, V is continuous across it: the potential difference across a vanishing thickness tends to zero. Its normal derivative can jump. An ideal dipole layer is a different singular limit and can produce a jump in V; it is not covered by the simple charge-sheet continuity claim.

At a step in an ordinary volume density without a delta-function surface sheet, the normal field need not jump. That is why the solid-ball potential and first derivative match at R = 3 even though the volume Poisson source changes. Source type determines the matching conditions.

Material polarization changes which charge and constitutive relations are convenient. The next lesson develops conductors, capacitors and dielectrics. This lesson uses vacuum ε₀ and total specified charge; replacing ε₀ by an arbitrary material constant without changing the underlying source description is not a general boundary-value solution.

Return path: a kernel does not erase the boundary

For a fully prescribed, sufficiently localized vacuum charge density in free space, V(r) = (1/4πε₀)∫ρ(r′)/|r − r′| d³r′ with the compatible zero-at-infinity reference. The primed coordinate labels the source; the unprimed coordinate labels the observation. The integral needs suitable convergence and a meaningful interpretation at singular sources.

A conductor at an imposed voltage usually carries an unknown induced charge. Integrating only the charges initially supplied elsewhere does not automatically satisfy the conductor’s boundary value. One may instead solve for the induced distribution or use a Green function that already fits the relevant homogeneous boundary condition.

Fix the convention before using that function. In three dimensions, define G(r,r′) by ∇′²G = −δ³(r′ − r). For a Dirichlet Green function on a bounded domain, G = 0 on the boundary. Green’s identity then gives V(r) = (1/ε₀)∫ΩGρ d³r′ − ∮∂ΩV(r′)∂ₙ′G dA′. The derivative uses the outward source-side normal, and the minus sign belongs to this chosen −δ convention.

The free-space kernel is G = 1/(4π|r − r′|). In a bounded-domain Green function, additional harmonic terms enforce the boundary condition. Omitting the surface contribution while prescribing a nonzero boundary potential generally changes the problem. This representation is an advanced return path; none of our four explicit models relies on an unperformed Green-function computation.

Geometry also changes radial solutions. In a three-dimensional spherical source-free region, (r²V′)′ = 0 gives V = A + B/r. For a z-invariant two-dimensional radial problem, (rV′)′ = 0 gives V = A + B ln(r/r_ref). Their source geometries, domains and reference behavior differ; a two-dimensional drawing of a sphere does not convert its equation into a logarithmic one.

Audit a potential solution

Begin with the domain and allowed source support. Record the equation, units, reference, boundary values and asymptotic conditions before differentiating or plotting. State whether a branch is a complete physical solution, a particular contribution, a harmonic correction or a local approximation.

Then check E = −∇V component by component, the Poisson or Laplace residual in every regular region, the required interface jumps and all boundary conditions. Compare an integral flux with enclosed charge where meaningful. Include singular-source exclusions or distributional terms rather than sampling an undefined point.

If a numerical solver is introduced later, test boundary errors and refine its mesh as well as monitoring its residual. Derivatives can amplify errors in a smooth-looking potential. A wrong harmonic addition can have an exactly zero PDE residual, so residual size alone does not prove the physical boundary-value problem was solved. No numerical PDE solver has been run for this lesson’s analytic examples.

For a graph, label whether an ordinate is V/V₀, a signed component E_x/E₀, a radial component E_r/E₀ or a positive magnitude. Preserve missing domains and one-sided limits. Displayed finite samples support the stated analytic functions; they do not by themselves prove a continuum theorem or a quantitative interpolation error bound.

Worked examples

A negative probe crosses the slab’s voltage difference

Move q = −3q_* from X = −1 to X = 1 in the prescribed slab field. Find ΔU, field work and ideal quasistatic external work.

The endpoint potentials are −4V₀ and 2V₀, so ΔV = 6V₀.
Multiply by the signed probe charge: ΔU = (−3q_*)(6V₀) = −18q_*V₀.
Field work is −ΔU = 18q_*V₀. For unchanged kinetic energy and no other work, the external agent performs −18q_*V₀.

The probe loses 18q_*V₀ of potential energy while the field does positive work. Negative external work means the ideal controlled agent extracts that energy; it does not reverse the prescribed field.

A correct source term with a wrong boundary is still wrong

Does v_trial = −3X² solve the prescribed charged slab problem? What correction is required?

Its second derivative is −6, so it has the correct Poisson source.
At X = ±1 it gives −3, while the desired endpoint values are −4 and 2.
Add a harmonic linear correction aX + b. The endpoint discrepancies require −a + b = −1 and a + b = 5.
Solve a = 3 and b = 2, obtaining v = 2 + 3X − 3X².

The trial solves the local source equation but fails the boundary problem. The correction 2 + 3X supplies the missing boundary information without changing ρ.

Two zero-potential branches meet without a field direction

For v = (X + 1)² − Y², compare the critical point (−1,0), the point (−1/2,0) and the point (−1,1/2).

At (−1,0), v = 0 and E = 0.
At (−1/2,0), v = 1/4 and E/E₀ = (−1,0,0), directed back toward the critical line in x.
At (−1,1/2), v = −1/4 and E/E₀ = (0,1,0), directed away in y.
The level v = 0 consists of Y = ±(X + 1); its branches meet at the zero-gradient point.

The critical point is a saddle. A positive probe is restored along x and destabilized along y; crossing branches have the same potential value and do not supply a unique nonzero field normal at their intersection.

Evaluate a decaying boundary mode

Find v and E/E₀ at X = π/8 and Y = ln(2)/2 in the periodic half-space model.

The phase is 2X = π/4, so sin(2X) = cos(2X) = √2/2.
The height factor is e^(−2Y) = e^(−ln 2) = 1/2.
Thus v = 3√2/4, E_x/E₀ = −3√2/2 and E_y/E₀ = 3√2/2.
The magnitude is √[(9/2) + (9/2)] = 3, agreeing with 6e^(−2Y).

V = (3√2/4)V₀ and E = (3√2/2)E₀(−x̂ + ŷ). Potential, vector components and positive magnitude retain different meanings.

Leave the center of a uniformly charged ball

For the radius-3L₀ ball with Q = 2q₀, move a negligible positive probe q = 2q_* from the center to r = 6L₀. Find the endpoint fields and energy change.

At the regular center, V = V₀ and E = 0. At R = 6, V = V₀/3 and E_r = E₀/18.
The potential change is ΔV = −2V₀/3.
Multiply by 2q_* to obtain ΔU = −4q_*V₀/3. The field does positive work 4q_*V₀/3.
The zero initial field does not mean the center has zero potential or that a displaced positive probe is stably confined.

The probe’s potential energy decreases by 4q_*V₀/3. The potential difference, rather than either endpoint field alone, determines the field’s work.

Try it yourself

Replace V by V + 7V₀ everywhere. Which of E, ΔV and a probe’s ΔU changes?
Show the answer and reasoning
None of these changes.
The constant disappears from gradients and potential differences. Any prescribed absolute boundary values must be shifted consistently to describe the same reference change.
A probe q = −2q_* moves through ΔV = −5V₀. What is ΔU and the field’s work?
Show the answer and reasoning
ΔU = 10q_*V₀ and W_field = −10q_*V₀.
The negative charge reverses the energy response to the negative voltage change. Do not drop q’s sign.
In the slab, does the point X = 0 have zero field because a reference could make its potential zero?
Show the answer and reasoning
No. Its field is −3E₀x̂ regardless of an added constant.
Changing the reference changes the value of V there but leaves its local slope fixed.
For the slab v = 2 + 3X − 3X², what source sign follows from ∇²V = −ρ/ε₀?
Show the answer and reasoning
Positive volume charge: ρ = 6ε₀V₀/L₀².
The normalized second derivative is −6, and conversion to dimensional curvature introduces V₀/L₀².
Why is the left slab normal derivative −9V₀/L₀ rather than +9V₀/L₀?
Show the answer and reasoning
The left outward normal points in negative x.
The derivative ∂V/∂x is +9V₀/L₀ there, but ∂ₙV = n̂·∇V changes its sign. Both faces must use outward normals in the compatibility integral.
Do compatible pure-Neumann data on a connected region fix V uniquely without a reference?
Show the answer and reasoning
They fix it only up to an additive constant within the stated regularity class.
The difference of two solutions has zero gradient after the energy-identity argument, but zero normal derivative does not determine its constant value.
Do the crossing zero-level branches of the saddle contradict the statement that different equipotential values cannot cross?
Show the answer and reasoning
No. Both branches have the same value, and their intersection is a critical point.
The usual smooth equipotential normal statement assumes a nonzero gradient. At the intersection, E = 0.
How does the periodic mode’s field magnitude change when y increases by (L₀ ln 2)/2 at fixed x?
Show the answer and reasoning
It is multiplied by 1/2.
The factor e^(−2y/L₀) acquires e^(−ln 2). This is a spatial comparison, not a time-dependent decay.
What are the ball’s two surface limits of V/V₀ and E_r/E₀ at R = 3?
Show the answer and reasoning
Both potential limits are 2/3 and both field limits are 2/9.
The interior and exterior branches match because no delta-function surface charge was added. The change in ordinary volume density is handled with one-sided second-derivative limits.
A numerical potential has a tiny Poisson residual but the wrong boundary voltage. Is that enough to admit the solution?
Show the answer and reasoning
No.
A wrong harmonic addition can leave the PDE residual exactly zero. The complete boundary conditions, source/singularity treatment and relevant numerical convergence must also pass.

What to carry forward

Potential differences belong to a prescribed electrostatic field. Probe energy and work require the signed charge and stated mechanical conditions.
E = −∇V describes local slopes; a reference value, a zero potential and a zero field are different facts.
Poisson’s equation combines Gauss’s source law with an electrostatic scalar potential. Laplace’s equation applies in regular source-free regions without requiring zero field.
Source terms and boundary data are both necessary. A particular solution may need a harmonic correction, and superposition adds boundary data too.
Complete Dirichlet data give uniqueness in the stated regularity class; pure-Neumann data require charge compatibility and leave a constant reference. Uniqueness is not existence.
A harmonic saddle may have zero field and crossing branches of one equipotential. It does not provide stable free electrostatic confinement.
Separation modes, Green functions, interface jumps and radial formulas carry their own geometry, sign conventions and domain assumptions.
Check units, derivatives, all boundaries, singular support and graph semantics. A small PDE residual or a smooth picture is insufficient by itself.

References

Sources for the concepts and teaching approaches in this article.