Predictions, targets and levels of evidence

A target says what would be useful

An ambitious design may require a strength, conductivity or stability beyond established products. That requirement is a target. A calculation can propose a structure that meets it under assumptions; an experiment can investigate whether that structure and response occur. These are distinct claims.

A useful evidence level is attached to a particular claim. A verified numerical solution can support a conditional property prediction without showing that its structure can be synthesized. A synthesis signal can support formation of something without uniquely identifying its phase. A measured specimen can show a response without establishing a bulk product. These claims depend on different observations; they do not form one score that rises whenever another technique is added.

Consider a fictional candidate P for a heat-spreading material. A design target requires bulk, freestanding material with axial thermal conductivity κ ≥ 100 W/(m·K) at 450 K. A supplied calculation record predicts 120 W/(m·K) for a specified, defect-free periodic structure at that temperature. The record is a hypothetical input to this lesson, not an atomistic calculation or material discovery performed here. Numerical convergence, interaction approximations and the structural state would still need examination before trusting that prediction.

The same dossier describes a supported coating with the proposed composition. Its structural signal is consistent with P but also with another arrangement. That is incomplete identity evidence. The heat-flow readings below are also invented, with deliberately supplied uncertainties. Our actual numerical work solves the stated device models; it does not create experimental evidence for P.

Predict the observable before inverting a property

Let the coating have width w = 1.00 mm, thickness h and reservoir separation L. At a mean temperature of 450 K, assume steady one-dimensional diffusive conduction, constant axial κ over the small reservoir temperature difference ΔT, a uniform conducting section wh and known temperatures at the coating ends. Fourier conduction then gives heat rate Q̇_coating = κwhΔT/L. Heat rate is measured in watts; κ is a material coefficient in watts per metre kelvin.

Define total conductance G = Q̇_total/ΔT, in W/K. Model A includes the coating and a separate parallel background conductance G₀: G_A = κ_Awh/L + G₀. The background is stipulated to be independent of L and h over these comparisons. Parallel paths share ΔT, so their heat rates add. A series contact resistance would enter a different balance; it is not absorbed into this constant additive conductance.

The assumed end temperatures exclude an unknown temperature drop at the coating contacts; side losses and other paths are omitted unless explicitly included. Thermal conductance is therefore the more direct device result. Turning it into a coating conductivity needs this geometry and heat-path model. Heat capacity does not enter this steady balance and cannot be inferred from these records.

Actual suspended-device measurements of individual nanotubes inferred conductance from heater and sensor temperatures through a heat-transfer model. The conversion to conductivity also required geometric and contact assumptions. This illustrates the distinction between an observed thermal response and an inferred material coefficient; none of that study’s material values or device dimensions is used in our fictional case. Background heat transfer can persist even without a specimen, while contact resistance can change a series thermal path.

For the baseline thickness h₁ = 10.0 µm, wh₁ = 1.00 × 10⁻⁸ m². Denote the four readings by G₅, G₁₀, G₂₀ and G_new as in the table. Every supplied reading has a complete standard measurement uncertainty u_G = 2.00 µW/K, and the four input estimates are mutually independent. Geometry and temperature are exact in this teaching budget. This does not include uncertainty from a wrong heat-path model or an uncontrolled material change.

A fitted agreement and an independent comparison answer different questions

At h₁, the 5 mm specimen has twice the coating conductance of the 10 mm specimen in model A. Subtracting their equations gives κ_A = (G₅ − G₁₀)L₁₀/(wh₁) = 120 W/(m·K), with L₁₀ = 0.0100 m. The remaining background is G₀ = 2G₁₀ − G₅ = 20 µW/K. This matches the supplied ideal prediction numerically, but the fitted coefficient and the calculated ideal structure need not describe the same material state.

There are two adjustable parameters and two fitted central readings. Exact agreement at those points leaves no residual degrees of freedom for checking lack of fit. Parameter calibration means using observations to choose model parameters; here it does not independently test the fitted curve. Numerical verification asks whether the stated equations have been solved correctly. Validation comparisons ask whether the model represents the relevant physical observations with adequate accuracy in specified conditions. Success in the first task cannot supply the missing evidence for the others.

Before examining G₂₀, model A predicts G_pred,20 = −½G₅ + 3⁄2G₁₀ = 80 µW/K. The independently withheld reading is also 80 µW/K. This is useful evidence that the fitted total conductance describes a changed length at the original thickness, within the supplied measurement uncertainties. It does not prove the coating’s identity, uniqueness of its inferred κ, or accuracy at a different thickness, temperature or physical scale.

The plot shows why numerical agreement alone cannot assign a heat path. Against inverse length, the total conductance follows a straight line. Its slope can come entirely from the coating in model A, or from a coating plus another length-dependent path in model B below. Withholding a length point is a real test of the predicted observable; it is simply not a test that distinguishes those two explanations.

Independent readings produce correlated fitted parameters

Use the measured inputs, rather than treating fitted parameters as independent. For a linear combination y = Σa_iG_i of these uncorrelated readings, u²(y) = Σa_i²u_G². This follows from uncertainty propagation; for correlated inputs the additional terms are 2Σ_{i<j}a_i a_j Cov(G_i,G_j). Standard uncertainty is an estimated standard deviation of the specified measurement result, not an automatic confidence interval or a bound on unknown model error.

When G₅ and G₁₀ are entered in µW/K, their difference has the same numerical value as κ_A in W/(m·K), because L₁₀/(wh₁) = 10⁶ m⁻¹. Thus u(κ_A) = √8 = 2.828 W/(m·K), while u(G₀) = √20 = 4.472 µW/K. Their covariance is −12 [W/(m·K)]·[µW/K], and their correlation is about −0.94868. Both estimates reuse the same two readings with different signs.

In the displayed units, the predicted 20 mm conductance is 0.5κ_A + G₀. Its variance is 0.25(8) + 20 + 2(0.5)(−12) = 10 (µW/K)². Omitting the covariance would give 22 instead. The direct expression −½G₅ + 3⁄2G₁₀ gives the correct √10 = 3.162 µW/K without any fitted-parameter independence assumption.

For residual d = G₂₀ − G_pred,20, independence of the withheld reading gives u(d) = √(4 + 10) = √14 = 3.742 µW/K. Declare in advance a numerical consistency screen |d| ≤ 3u(d). Here d = 0 passes. This chosen screen has no automatic tail probability: no sampling distribution, effective degrees of freedom or model-discrepancy distribution has been supplied. Passing it describes this comparison, not the whole candidate.

An unmeasured path can reproduce every length point

Introduce a second positive parallel heat path with conductance H/L. Its coefficient H has units W·m/K. Model B is G_B = (κ_Bwh + H)/L + G₀. H could represent a separate bridge or host channel that shares the same end temperatures and the same 1/L scaling. Its existence and invariance are hypotheses, not observations established by the readings.

At h₁, choose κ_B = 60 W/(m·K), H = 6.00 × 10⁻⁷ W·m/K and G₀ = 20 µW/K. Then κ_Bwh₁ + H = 1.20 × 10⁻⁶ W·m/K, exactly the numerator κ_Awh₁ of model A. The models predict identical conductance for every length at h₁, not just the three tabulated lengths. More precise length data cannot separate these parameters at that thickness.

This is a structural identifiability issue: the observable determines the combination κ_Bwh₁ + H, rather than κ_B and H separately. A good fit does not resolve it. It is also why a fitted value equal to an ideal prediction does not, by itself, demonstrate the proposed microscopic mechanism. Another forward model can explain the same measured response with a different coating coefficient.

Change a condition that separates the models

Now compare h₂ = 20.0 µm at L₁₀. Assume for this calculation that changing thickness leaves composition, phase, defects, texture, contact conditions, G₀ and any H unchanged. These are consequential controls to check, not consequences of depositing a thicker coating. If they fail, a discrepancy can indicate a changed material or boundary state as well as an omitted path.

Model A predicts 260 µW/K for this thicker specimen: 2κ_Awh₁/L₁₀ + G₀ = G₅. The later reading is 200 µW/K. Because the prediction equals the original G₅ algebraically, its standard uncertainty is 2.00 µW/K. The independent residual is −60 µW/K with u(d) = √8 = 2.828 µW/K; |d|/u(d) ≈ 21.21, so it fails the declared 3u screen. This challenges model A together with its stated controls. It does not uniquely prove model B.

The proposed model B predicts 200 µW/K at h₂. Using the new thickness reading to determine its parameters gives κ_B = (G_new − G₁₀)L₁₀/[w(h₂ − h₁)] = 60 W/(m·K), and H = L₁₀(G₅ − G_new) = 6.00 × 10⁻⁷ W·m/K. Those parameters now use the changed-thickness datum: agreement there is a fit, not independent validation of the added path. The earlier withheld length cannot resolve H either, because both models already coincide at h₁.

A genuinely discriminating future comparison is a matched blank with h = 0 at L₁₀, provided removing the coating leaves the hypothesized background and bridge paths unchanged. A predicts 20 µW/K; B predicts 80 µW/K. These are prospective predictions, not measured blank readings. Their separation directs attention to a specific alternative. An unmatched substrate whose heat paths change during processing would not provide this clean test.

Structural identity still requires its own discrimination. Select additional structural/compositional observations whose predicted outcomes differ between P and the competing arrangement, for the same region and material state. Several signals derived from the same ambiguous structural assumption are not independent identification. A thermal blank does not identify the coating phase, and an indexed phase does not by itself measure its conductivity.

A narrower target can be supported without a bulk-product claim

A different fictional dossier R supplies independent structural evidence that resolves the coating phase within stated detection limits. At w = 1.00 mm, h = 10.0 µm and L = 10.0 mm, its coated device has conductance 126 µW/K; a matched blank has 20 µW/K. Each mean has complete standard measurement uncertainty 2.00 µW/K, and they are uncorrelated. Assume the blank subtraction isolates one uniform coating path, with negligible series contact drops and no change in background. These controls are stipulated for the exercise.

The conditional coating coefficient is κ_R = (126 − 20) × 10⁻⁶ × L/(wh) = 106 W/(m·K), with u(κ_R) = √8 ≈ 2.828 W/(m·K). For a declared local screen κ_R − 2u(κ_R) ≥ 100 W/(m·K), the lower numerical screen is 100.343 W/(m·K), so this record passes. The factor two is a decision convention here, not a supplied 95% confidence guarantee. A central estimate above a threshold alone would not necessarily pass this screen.

R therefore supports a useful, bounded achievement: a phase-identified supported coating meets the stated numerical conductivity screen within its measurement model and conditions. That claim is stronger than an ideal calculation or an ambiguous synthesis signal. It still does not establish the original bulk material target, a freestanding article, long-term retention, batch reproducibility or component performance. No persistence record or bulk test is supplied; those extensions remain separate questions.

Progress can be described precisely

Record what was made or represented, at what scale and conditions, and which function was measured. A result may be valuable even if synthesis, persistence or integration remains unresolved. Avoid converting a conditional prediction into an existing usable material.

For P, report the total device conductances and their conditions, the agreement at a withheld length, the changed-thickness discrepancy, and the nonunique phase evidence. The 60 W/(m·K) coating coefficient is conditional on model B; it is not an identified intrinsic property of P. Its difference from the ideal 120 does not by itself refute a calculation for a different, defect-free structure.

A concise defensible conclusion preserves both progress and uncertainty: a proposed structure has a conditional predicted property; a coated device exhibits specified conductance; one device model fails a changed-condition screen; another remains consistent but needs a discriminating comparison and identity evidence. In R, a local target screen passes. Neither conclusion supplies a service-life prediction. Atomistic models, assumptions and validation develops the model assumptions further; Synthesis, scale and testing extraordinary claims develops the subsequent synthesis and scale tests.

Worked examples

Try the reasoning

1. Does failure to manufacture a candidate refute every model property?

   Show the answer and reasoning

   Not necessarily.

   Manufacturability and predicted response are different questions.

2. Why label scale?

   Show the answer and reasoning

   A microscopic state may not become a bulk product.

   Assembly adds new requirements.

3. At h₁ = 10.0 µm, a further hypothetical L = 40.0 mm reading is 50.0 µW/K. Use A and B. Does perfect agreement at this fourth length establish κ_coating = 120 W/(m·K)?

   Show the answer and reasoning

   Both give 50.0 µW/K, so the additional length still cannot separate coating conduction from the proposed H/L path.

   At this thickness both numerators are 1.20 × 10⁻⁶ W·m/K. Division by 0.0400 m gives 30 µW/K, plus the 20 µW/K background. Changing length repeats the same parameter combination; a different condition is needed for identification.

4. For the fitted A and B models at L₁₀, predict a future h = 30.0 µm specimen under unchanged controls. Express the predictions in the original mean readings and find each prediction’s standard uncertainty.

   Show the answer and reasoning

   A predicts 380 µW/K = 2G₅ − G₁₀; B predicts 260 µW/K = 2G_new − G₁₀. Each prediction has u = √20 ≈ 4.472 µW/K.

   The A coating term triples from h₁ while G₀ stays fixed; substituting κ_A and G₀ gives 2G₅ − G₁₀. B increases only its coating term, giving 2G_new − G₁₀. In each case the independent input coefficients 2 and −1 give variance (4 + 1) × 4 = 20. No future measurement has been supplied, and the two predictions are themselves correlated through G₁₀.

5. A blank made by another process gives 80 µW/K, but microscopy shows a changed bridge geometry and no relation between its background and the coated device is supplied. Does the number confirm B’s H/L term?

   Show the answer and reasoning

   No. It equals B’s matched-blank prediction numerically, but the matching assumptions are not established.

   The proposed test requires the same H and G₀ paths after removing the coating. A process-dependent change can alter those quantities. The next evidence must address that physical control, not just repeat the blank more precisely.

6. Change only R’s coated central reading to 120 µW/K, retaining its 20 µW/K blank, independent u = 2.00 µW/K for each and exact geometry. Does its estimate pass the same κ − 2u ≥ 100 screen?

   Show the answer and reasoning

   κ = 100 W/(m·K), u = 2.828 W/(m·K), and κ − 2u ≈ 94.343 W/(m·K). It does not pass.

   Equality of the central estimate with the target is different from passing the declared lower-screen rule. This failure does not show a known probability that the coating is inadequate; the rule and standard uncertainties alone do not supply such a probability.

7. In a changed R budget, both coated and blank readings share exactly the same additive offset b. Each also has independent read-noise standard uncertainty 2.00 µW/K. The shared offset has u(b) = 5.00 µW/K. What is u(κ) for their difference if geometry is exact?

   Show the answer and reasoning

   The identical additive offset cancels, so u(κ) remains √8 ≈ 2.828 W/(m·K).

   Write G_coated = C + b and G_blank = B + b. Their difference is C − B. Alternatively, each total variance is 4 + 25 and their covariance is 25; the difference variance is 29 + 29 − 2(25) = 8. Cancellation requires an identical additive offset, not an unknown multiplicative gain or a drift that differs between readings.

8. The ideal calculation refers to a defect-free P crystal; a thermal specimen is later shown to be a porous mixture of P and another phase. Its conditional effective conductivity is 60 W/(m·K). Which computational claim does this directly refute?

   Show the answer and reasoning

   It does not directly refute the stated ideal-crystal value, because the measured material state is different. It can challenge a separate claim that this porous mixture has the ideal value.

   Compare the same structural state, temperature, direction and property definition, or supply a justified model connecting them. The mismatch is not a reason to ignore the measurement; it identifies the next modelling and synthesis questions.

9. A press release combines the fictional ideal value 120, the ambiguous structural signal and a 140 µW/K device reading into “bulk P with proven conductivity and long life.” Write the strongest supported replacement before the thickness test.

   Show the answer and reasoning

   A proposed ideal structure has a conditional prediction of 120 W/(m·K); a coated device with unresolved coating-phase identity has total conductance 140 µW/K at the stated geometry and temperature. The coating’s bulk conductivity and persistence have not been established.

   The device reading is meaningful. Assigning it to phase P requires identity and path evidence; extending it to bulk material or lifetime requires additional records. Repeating the same signal or quoting the ideal prediction does not supply those missing links.

What to carry forward

• Name the evidence stage.
• Describe achievements without erasing remaining questions.
• Separate solving a model, fitting its parameters and comparing a fixed prediction with independently withheld observations.
• An independent observation can test an observable without identifying a unique material coefficient or mechanism.
• Use covariance when predictions reuse fitted inputs; more precise data do not resolve an unidentifiable parameter combination.
• Choose a changed condition with different competing predictions, and keep identity, local properties, persistence and bulk use as distinct claims.