Independent thermal-sheet validation

The stated infinite-plane PDE has a correct Gaussian center-temperature identity. It does not define a universal apparent-conductivity bias or a sign-changing experimental bias. The source’s Fisher calculation uses the reciprocal of the physical temperature response, and its inversion contradicts its own declared ratio. This assessment establishes mathematical/model consistency only; no physical observations, novelty search or new hypothesis test were performed.

Independent derivation and convention

Let K=kappa*d, a=G*sigma^2/(2*K) and F(a)=exp(a)*E1(a). The Fourier solution gives

• Center: T(0)=P*F(a)/(4*pi*K).
• Gaussian probe with width sigma_p: T_probe=P*F(G*(sigma^2+sigma_p^2)/(2*K))/(4*pi*K).
• Same pump/probe width: T_probe=P*F(2*a)/(4*pi*K).
• Intensity exp(-2*r^2/w^2) has 1/e² radius w=2*sigma; center argument is G*w^2/(8*K), matched-probe argument G*w^2/(4*K).

Here P is absorbed W, K is W/K, G is W/(m² K), sigma and w are m, and temperature rise is K. This is an intensity-weighted Gaussian temperature; a different detection law needs its own weighting. The thin-sheet model assumes constant, isotropic material properties, an infinite plane, a linear local areal sink and steady state.

The derivation and checks were frozen before source reading in FROZEN-DERIVATION-AND-PLAN.md at 2026-11-02T15:26:51.641909+00:00, SHA-256 d61172c5456ab742dd8feaf38476968c6462c632e591fbecccdbcbf9835f4955 (receipt FREEZE.json).

Bounded independent numerical checks

Independent standard-library code evaluated Fourier quadrature, an independently implemented series/continued-fraction special function, and a conservative radial finite-volume boundary-value solution. The supplied source code was read statically only and never imported or executed. Five predeclared a values and three grid refinements each were used; no fourth refinement was required. Outer radius was max(12*sigma,16*sqrt(K/G)).

All available predeclared checks passed, including spatial convergence, normalization, energy conservation, positive probe ordering, dimensional scaling, Gaussian-probe reduction and limiting approximations. Maximum finest-grid relative error was 1.94e-5 at center and 2.06e-5 for the probe (predeclared tolerance 1e-3). Maximum energy residual over all 15 finite-volume runs was 5.18e-12 (tolerance 1e-9). Gaussian source normalization used exact annular integrals and the omitted analytic tail. The G=0 finite-disk negative control showed the predicted increase P*ln(2)/(2*pi*K) when radius doubled, within 7.40e-6 relative error.

The independent special-function algorithm and Fourier quadrature agreed within 2.56e-15 at the tested center arguments. A third-party scipy/mpmath special-function comparison remains unavailable: those packages are absent and an official PyPI mpmath installation attempt failed DNS. This missing library check is explicitly recorded separately, not counted as passed. The finite-volume/quadrature cross-check does not depend on that package.

Detailed results: results.json, finite_volume_results.csv, independent rerunnable code independent_validation.py.

Static comparison against supplied claims

Line references are for /Volumes/Storage/DeepSeek/exports/2026-11-01-blind-spots/investigations/I2_boundary_bias.py.

1. Lines 7–10: the stated center expression is correct when s means Gaussian sigma. Line 166 instead labels s a 1/e² radius. If s is that optical radius, the dimensionless argument is four times too large. The matched Gaussian probe is also distinct from the center result.
2. Lines 11–13 and 52–65: defining R=1/F is a legitimate mathematical normalization, but calling it the standard or universal kappa_app/kappa_true needs an explicit estimator and reference geometry. The G→0 infinite 2-D sheet has no bounded decaying steady state for positive net heating, so its divergent temperature cannot supply the claimed default zero-sink calibration. A finite-boundary or differential-temperature estimator gives a different prefactor and potentially a different bias narrative. This invalidates universal experimental interpretation, not the center identity.
3. Independently R(1)=1.6768750281786993; the mathematical crossing R=1 is at a=0.4348182043849037. A plotted vertical line at a=1 (line 143) is not the R=1 crossing. The script’s print at lines 74–75 does compute the nonunit value. These numbers describe this normalized function, not an established conductivity-bias sign change.
4. Lines 16 and 77–81 solve kappa=kappa_app*R(a(kappa)), whereas their declared relation is kappa_app=kappa*R(a(kappa)). Its consistent implicit inversion is kappa=kappa_app/R(a(kappa)). Under this chosen normalization, the response also has a positive limiting floor kappa_app→G*sigma^2/(2*d) as kappa→0; an inversion needs an existence/conditioning check. At large a physical sensitivity to kappa becomes weak.
5. Lines 85–95 use log(T)=log(P/(4*pi*kappa*d))+log(R). The PDE requires minus log(R). With H=1/F(a)-a, correct log sensitivities are (S_kappa,S_G)=(-1+H,-H); source response gives (-1-H,+H). Thus positive G falsely increases temperature in that Fisher model. At a=0.45, physical sensitivities are (-0.43033549,-0.56966451), while the source response gives (-1.56966451,+0.56966451).
6. Line 102 computes abs(Jk*JG)/(Jk²+JG²), whose maximum is 1/2. This is not the cosine between parameter-sensitivity columns. A single scalar observation has rank-one information; repeated powers at fixed spot width scale the same response vector and do not create a second independent parameter direction.
7. Lines 118–124: the quoted kappa CRBs coincide algebraically with the corrected center-temperature model under the exact two-parameter, known-power, independent 1% log-noise assumptions, despite the wrong response. They must not all be labeled numerically false. The G CRBs differ substantially (below). With J_source=(-1-H,H) and J_physical=(-1+H,-H), the corresponding local coordinate relation is delta ln(kappa)_physical=delta ln(kappa)_source, delta ln(G)_physical=2*delta ln(kappa)_source-delta ln(G)_source, explaining the kappa coincidence.

These percentages are local Fisher bounds for log parameters (approximately fractional standard errors), not measured errors or guaranteed practical performance. They assume center detection, known P and widths, known d, exactly this PDE and independent 1% Gaussian log-temperature errors. Treating the listed widths instead as physical 1/e² radii changes a by a factor of four and changes these numerical bounds.

All 400 supplied investigations/i2_R_of_lambda.csv rows were compared statically to independent evaluation of 1/F. Largest relative difference was 2.244e-5 at the rounded a=30.9905, consistent with the source’s truncated large-a approximation plus CSV rounding. This is numerical-function agreement only. Details and source hash are in source_csv_comparison.json and source_csv_comparison.csv.

The README line 35 and lines 48–50 promote this normalized large-spot relation into a general reported-conductivity statement. The finite local-sink limit itself is correct for the model temperature, but a claimed reported conductivity needs the actual estimator and optics. No observations were checked here.

Limits, inference and disposition

F(a)~−gamma−ln(a) at small a, so G→0 or a point-source limit gives a logarithmically divergent center temperature at fixed absorbed P; exactly zero G has no infinite-plane decaying steady state. At large a, F(a)~1/a−1/a²+2/a³, giving T_center≈P/(2*pi*G*sigma²) and matched T_probe≈P/(4*pi*G*sigma²). Broad spots cool as sigma^-2 at fixed P. For K→infinity the temperature tends to zero approximately as ln(K)/K. These limiting statements concern the specified mathematical model only.

One spot’s power sweep cannot separately identify K and G from one slope. Multiple known widths can identify two parameters locally over a suitable range, but do not guarantee good conditioning, especially when all widths lie in a strong-sink limit. Unknown common absorption creates an exact scaling ambiguity even across widths: if P=eta*P_incident, simultaneous (eta,K,G)→(c*eta,c*K,c*G) preserves every predicted temperature. Bulk kappa also requires known thickness. Detection weighting, boundaries, anisotropy, finite thickness and temperature-dependent parameters can change experimental interpretation.

Disposition: preserve the original source and negative results. Retain the center identity and model-based width/identifiability discussion with precise assumptions. Hold the universal apparent-conductivity ratio/sign-flip interpretation, reversed inversion, radius label and physical Fisher-response interpretation for correction. This package establishes no experiment, novelty or breakthrough.

I2: primary-literature and measurement-readiness audit

Review date: 2 November 2026. This is a bounded literature audit, not exhaustive novelty clearance or external peer review. No numerical observations were extracted, digitized or fitted. Equations and method descriptions were examined; search/abstract pages incidentally displayed aggregate measurements, as recorded below.

Assessment

The broad physical setting is established: supported thin sheets heated by a Gaussian beam, substrate heat leakage, laser-weighted Raman temperature and multiple beam sizes for separating lateral conductivity from interface conductance all appear in primary literature. I2 should not be promoted as the invention of those principles. Its claimed universal apparent-conductivity bias still needs an explicitly defined measurement estimator; the heat equation alone does not specify what a laboratory reports as conductivity.

A useful contribution from this project could be a transparent correction/reproduction case with precisely matched beam conventions, observable, boundary assumptions and inference procedure. A new physical effect or broadly new measurement method has not been established by this review.

Closest primary work actually examined

Judek et al. (2015)

High accuracy determination of the thermal properties of supported 2D materials, Scientific Reports 5,12422. DOI 10.1038/srep12422; primary full text.

Read depth: introduction and Theory section through the beam-characterization discussion; equation images1–5 inspected directly. Experimental result figures/values were not analyzed. Equations1–5 treat Gaussian heating, sheet conduction and substrate leakage, then average the temperature with the optical weighting. The method explicitly varies beam size to infer conductivity and interface conductance jointly. This is close prior art for I2’s general mechanism and experimental lever. The article also explains why a constant-temperature substrate is an approximation requiring care.

Notation caution: displayed Eq.1 uses κh in the diffusion term, whereas the Eq.3 image shows sqrt(g/κ) in the kernel argument. Taken literally with Eq.1’s units, an h factor appears missing. This may be a notation or typographical issue; do not silently adopt that kernel in place of an independent dimensional derivation. The conceptual overlap does not require accepting every displayed expression uncritically.

License: the primary full text states Creative Commons Attribution4.0. Five downloaded equation images in this directory are from this article and are retained only as theory-reading evidence with this attribution, not as a measured dataset.

Zhang et al. (2015)

Measurement of Lateral and Interfacial Thermal Conductivity of Single- and Bilayer MoS2 and MoSe2 Using Refined Optothermal Raman Technique, ACS Applied Materials & Interfaces. DOI 10.1021/acsami.5b08580; author-submitted abstract; author-hosted article.

Read depth: arXiv abstract and primary/author-hosted bibliographic search extracts only; not full experimental tables or supplementary data. The abstract describes direct absorption characterization and supported/suspended responses at different laser sizes to determine lateral and interfacial transport. This independently establishes that spot variation and treatment of substrate coupling predate I2. It does not verify I2’s exact reciprocal-exponential-integral bias relation. The arXiv license link is the nonexclusive distribution license, not a general open-data/reuse license.

Yalon et al. (2017; arXiv revision2020)

Temperature Dependent Thermal Boundary Conductance of Monolayer MoS2 by Raman Thermometry, ACS Applied Materials & Interfaces. DOI 10.1021/acsami.7b11641; primary author record; author-hosted paper/support.

Read depth: arXiv abstract/metadata plus author-hosted method-related search excerpts. No supplemental numerical data or plots were digitized or fitted. The paper identifies temperature-dependent absorption and substrate heating as relevant to Raman thermal inference. These are confounders for applying a fixed absorbed-power, isothermal-sink model across studies. This paper is not an untouched test dataset in this audit: search results incidentally exposed figure descriptions and aggregate results. The arXiv record has a nonexclusive distribution license; no separate raw-data license was verified.

Jaramillo-Fernandez, Chavez-Angel and Sotomayor-Torres (2018)

Raman thermometry analysis: Modelling assumptions revisited, Applied Thermal Engineering130,1175–1181. DOI 10.1016/j.applthermaleng.2017.11.033; institutional accepted manuscript.

Read depth: publisher bibliographic/abstract search extract and institutional manuscript method-section2.2 search extract. Direct publisher access returned403; the full paper was not read. The work examines finite geometry/light penetration and uses a spatially weighted effective Raman temperature. It is relevant background on measurement-model assumptions, not evidence that the specific I2 estimator is standard. The method here concerns bulk/mesa geometries, so it is not a direct thin-sheet replication candidate.

The supplied arXiv2602.02418 reference

Unbounded Systematic Error in Thin Film Conductivity Measurements, Gao et al.2026; primary abstract/version record, DOI 10.48550/arXiv.2602.02418.

Read depth: primary arXiv abstract and version history. The supplied identifier and title are genuine. Its subject is electrical four-bar conductivity measurement and finite-conductivity metal electrodes. It is a possible analogy for measurement artifacts, not direct support for Gaussian Raman thermal-conductivity equations or their claimed universal bias. The source CSV associates it with P09/R05; this audit does not allege that the collection explicitly cited it as a direct proof of I2.

Exact contribution versus known work

Public measurement candidates: metadata readiness, not a data analysis

No standalone machine-readable raw dataset with all required metadata and independently constrained κ/G was identified in this bounded search. This is not a claim that such data do not exist. Three published measurement sources are worth a targeted data-availability request or later protocol assessment:

1. Judek2015 supported-flake beam-size sweeps — closest geometry/experimental lever; primary full text is CC BY4.0. Publication text/figures are accessible, but a raw numerical table, covariance and separate data license were not located. κ and G are extracted jointly from the Raman procedure, so treating the published fitted pair as independent ground truth for the same observations would be circular. Substrate heating is explicitly part of their enhanced analysis.
2. Zhang2015 supported/suspended variable-spot measurements — absorption and geometry are explicitly considered in the abstract; author-hosted paper and supporting-information availability are metadata leads. No raw dataset or data license was verified; numerical outcomes were not extracted. Multiple geometries and jointly inferred parameters require sample-level mapping, not a pooled cross-paper collapse.
3. Yalon2017 Raman/absorption measurements — useful for checking absorption/substrate-heating assumptions, but not selected as a clean variable-spot validation dataset. Raw machine-readable series, spot-by-spot independence, complete uncertainty and a raw-data license were not verified. Do not retrofit the simple fixed-sink model to these measurements without compatibility checks.

Required fields before accepting any comparison: sample and substrate identities; thickness/geometry; absorbed rather than merely incident power; temperature calibration/readout weighting; calibrated beam-width definition and uncertainty; same-sample spot series; temperature-rise uncertainty; boundary conditions and substrate temperature; which κ/G constraints are independent versus fitted from the same data; excluded observations and licensing/provenance. No missing values should be invented or silently inferred from a fitted conclusion.

Observation-test gate and exposure record

Do not announce an observation test yet. First settle whether I2 survives the independent derivation and define the exact observable. Then freeze the sample eligibility rules, prediction, error model, tolerances and refutation condition before retrieving numerical comparison series. If those conditions cannot be met, publish the mathematical audit plus a data request, not a purported experimental validation.

This review is not wholly blind to published outcomes: search results/abstracts incidentally displayed aggregate values from Zhang2015 and Yalon2017, including figure descriptions in a Yalon search result. Those values were not transcribed into a dataset, digitized, fitted or used to choose thresholds. An eventual analysis must record that prior exposure and should not call these sources unseen holdouts. No new contact with authors or laboratories was made.

Search scope: focused queries for supported2D Raman/Gaussian heating, variable spot size, the supplied arXiv identifier, and metadata for the three named candidate measurement papers. Patent search, exhaustive forward/backward citation review and full-text assessment of every cited paper were not performed.

Assessment against the frozen observation plan

The project has now frozen observations/FROZEN-OBSERVATION-PLAN.md. Its target is the corrected Gaussian-weighted temperature-rise per absorbed power, not a universal apparent-conductivity ratio. It requires at least four calibrated radii on one unchanged specimen, fits only the extreme radii, and holds out at least two intermediate radii.

None of these candidates passed the complete eligibility gate in this bounded metadata review. No numerical observations were retrieved for fitting, digitized or analyzed after the freeze; no thresholds were changed. Distinguish unknown metadata from evidence that a dataset is physically incompatible. The next legitimate action is a targeted request for raw same-sample measurements and calibrations, or a separately planned new measurement—not an invented validation or retrospective relaxation of the rules.

I2: prospective observation comparison plan

2 November 2026. This plan is frozen before acquiring, digitizing, fitting or inspecting the numerical comparison observations for this task. Literature methods and some abstract-level aggregate results have already been visible to the literature reviewer; this is not a wholly blinded literature search. A local timestamp and hash are not an external preregistration.

Claim under test

Test the corrected, conditional forward model for a supported isotropic thin sheet, not a universal bias law for reported thermal conductivities. For sheet conductivity K = kappa*d, absorbed Gaussian power P, calibrated 1/e^2 radius w, and a matched Gaussian-weighted thermometer, predict:

Y(w) = DeltaT_weighted/P = F(Gw^2/(4K))/(4piK), F(a) = integral_0^infinity exp(-u)/(u+a) du.

For a center-point thermometer use F(Gw^2/(8K))/(4piK), explicitly identified as a different observable. A probe-weighted observation must not be compared to the center-point formula. At large dimensionless argument, predict Y proportional to w^-2 and independent of K to leading order. This is a conditional consequence of the model, not a novelty claim. Do not test the original universal kappa_app or radius-of-crossover claim without an independently specified conductivity estimator.

Eligibility before reading numerical comparison observations

• Publicly accessible original measurements or author-supplied raw data with a clear reuse basis; no invented or simulated substitute labelled as observations.
• One unchanged supported specimen with known thickness and geometry; at least four independently calibrated beam radii, with the heating and thermometer weighting specified.
• Absorbed-power calibration and its uncertainty, temperature/power response and uncertainty, beam-radius convention and uncertainty, substrate boundary conditions and specimen dimensions.
• Documented small-signal linear power regime; no pooling different specimens or treating repeated power levels as independent spot sizes.
• Independent geometric/parameter bounds must justify the infinite-sheet and local areal-sink approximation. A finite-substrate or anisotropic model requires a separate plan, not a post-hoc adjustment to this one.
• An abstract, a table of final fitted conductivity values, or a figure without sufficient calibration is not automatically an eligible validation dataset.

Locked analysis and controls

Sort eligible radii by size. Fit positive K and G using only the smallest and largest radii and declared measurement uncertainties; hold out all intermediate radii (at least two). Fix absorption, thickness and radius calibration distributions from independent metadata, not by tuning them against held-out temperatures. If those quantities cannot be independently constrained, mark identifiability insufficient and do not report a successful two-parameter validation.

Use the independently implemented forward calculation, not the supplied DeepSeek program. Propagate declared calibration and response uncertainties to prediction intervals. Report covariance/conditioning and sensitivity to independently justified calibration bounds. Repeat powers at one radius are a linearity control, not a new shape constraint. Center-point versus weighted readout is a normalization control, not an interchangeable fit choice.

Outcomes fixed before data access

• Contradicted under the stated assumptions: at least one held-out normalized response lies outside its two-sided prediction interval after Bonferroni control of family-wise alpha=0.05 across the held-out radii, provided the calibration and model-eligibility requirements hold.
• Consistent within resolution: no held-out response breaches that threshold and all required inputs are independently constrained. Report residuals and interval widths; this does not prove the physical model, novelty or generality.
• Inconclusive / not testable: no eligible dataset, unresolved calibration, insufficient radii, unstable parameter identification, missing uncertainties, or incompatible geometry/readout. Do not silently relax the eligibility rules or substitute a synthetic result.

The high-argument w^-2 asymptote is secondary and will only be assessed when an independently justified lower bound places every point in that regime. No retrospective selection of a visually convenient power-law window.

Access log rule

Record candidate metadata and reasons for inclusion/exclusion. Keep the observation access log empty until a dataset passes eligibility. Hash the chosen inputs before fitting. Any change to this plan gets a separately dated version with its reason and disclosure of any data already examined.

H0199: reference placement — proposed physical-map test

Prepared 2 November 2026. Protocol proposal; no experiment, laboratory commitment or registration is claimed. This tests a narrow reconstruction question. It does not establish a universally best layout or validate reactive capture chemistry.

Original question: can interleaving reference measurements improve deposition-bias estimates relative to edge-only references? Read the separate correction and retained report first. Saved rankings depend on the estimator, and the supplied random-field normalization needs repair. The proposed test uses one common, physically measured field; it does not reuse that simulation.

Smallest useful first step

A collaborator supplies access to an existing calibrated scanner or an openly reusable, sufficiently documented deposition/thickness/fluorescence map. First determine whether a reliable reference map and separate measurement passes are possible. A measured-map comparison can precede fabrication of actual witness patches. If no adequate reference map is available, stop at feasibility; do not turn an uncalibrated image into ground truth.

Stage A — physical map, virtual layouts

Measure a benign nonreactive coating or other spatial field with a linear calibrated readout. Compare equal-count virtual reference layouts against a separate dense reference scan. This isolates reconstruction and location choice. It does not test whether physical witness patches shadow the deposition or whether reactive and inert chemistries respond equivalently.

Use three deliberately stated field families: (i) uniform/approximately affine field as the edge-friendly control, (ii) a positive field with an interior bump and nearly unchanged boundary, and (iii) a rim-enhanced field as a stress case. Classify the intended families before scanning and report the actual maps even if imperfect. Do not discard a map because a favored layout loses.

Stage B — physical patches, if Stage A is informative

Only after a useful Stage A result, fabricate equal-area reference patches at the two prescribed layouts. Compare their readings with independent maps and unpatched companion specimens. Estimate patch perturbation, calibration differences and any position-dependent capture. This is a separate experiment; Stage A cannot stand in for it.

Apparatus and required access

• A stable spatial field and specimen positioning fiducials; an established coating/deposition process or a benign optical analogue, selected by the laboratory.
• A calibrated full-field fluorescence/thickness scanner, profilometer, or equivalent with a documented linear range and spatial resolution finer than the reference-patch footprint. One instrument used in separate repeat passes is acceptable for the pilot, with shared calibration uncertainty stated; an independent mapping modality is stronger.
• Dark/zero reference, uniform reference, spatial scale standard and repeat-positioning check. Record exposure, gain, illumination and temperature.
• An analyst able to register coordinates, retain raw maps and uncertainty, and fit a predeclared polynomial without tuning its degree after seeing which layout wins.

Freeze before confirmatory data

After an explicitly labeled feasibility pilot, freeze specimen area, patch footprint, coordinate registration, calibration, eligibility rules, layout coordinates, estimator and analysis version. Hash the protocol and code. Pilot maps may not become confirmatory maps.

Equal-count comparison: 12 readings per layout, equal total sampled area. On a normalized unit square use the interior 3-by-4 grid at x=(j+0.5)/4, j=0..3; y=(k+0.5)/3, k=0..2. Edge samples are at the three quarter positions on each of the four edges. Confirm that patch footprints fit the specimen: if a finite edge footprint requires an inset, predeclare its exact geometry and do not apply an exact boundary-only theorem to it. Verify the six-column quadratic design matrix has full column rank and record its condition number before measurements. Do not use four- or eight-point underidentified quadratic controls.

Primary estimator: unregularized degree-2 least squares fitted only to the 12 reference readings; predict the mean at a fixed 5-by-5 interior target grid. The reference mean is the area-matched dense-map mean over those 25 target footprints. Keep one coordinate-wide normalization/calibration for all sampled locations; never renormalize the witness and target subsets separately. An affine estimator is a prespecified secondary analysis, reported even if it reverses the ranking.

For map k and layout L, report e(L,k)=(estimated target mean/reference target mean)-1. The primary outcome for each field family is RMSE(L)=sqrt(mean(e(L,k)^2)); the paired comparison is Q=RMSE(grid)/RMSE(edge). Also report signed bias, standard deviation, individual map errors and dense-map uncertainty. This avoids pretending a chemical capture efficiency was measured.

Controls, sample plan and decision criteria

• Use one scan pass for reference-location readings and a separate, registered pass for the dense reference mean. Repeat both passes in a random order to quantify drift. Include the uniform/affine control and unpatched controls where patches exist.
• Feasibility pilot: two independently prepared maps per family, six total; useful for instrument qualification only. Confirmatory planning target: ten independent preparations per family, 30 total. This is an initial design allowance, not a power calculation. A smaller number may be reported as a pilot but cannot be silently relabeled confirmatory.
• Avoid pseudoreplication: pixels and repeat scans are not independently prepared specimens. Bootstrap paired whole specimens, within family, for a 95% interval for Q. State the bootstrap seed and algorithm before data collection.
• Proposed practical target: at least 20% RMSE reduction, Q≤0.80, for the interior-bump family. Support for that narrowly specified target requires the upper 95% interval ≤0.80. An interval wholly above 0.80 contradicts that practical target in the tested setting; an interval crossing it is inconclusive. Q>1 indicates a point-estimate loss, not automatically a statistically established loss.
• Report the affine and rim cases separately. No success in the interior-bump case licenses “distributed always wins.”
• Feasibility gates: full-map uncertainty and registration error must be quantified; signals must lie within calibrated range; reference mean must remain sufficiently above its uncertainty to make relative error stable. Freeze numerical gates after pilot calibration and before confirmatory maps. Stop if combined reference/registration uncertainty exceeds one-third of the target error difference; redesign rather than infer a layout advantage from unresolved ground truth.

Uncertainty and retained data

Retain raw scans, exposure/calibration files, dark and uniform frames, registration transforms, target/reference footprint definitions, layout coordinates, specimen IDs, excluded-map reasons and analysis code. Propagate shared calibration and scan-repeat uncertainty; do not treat a dense image as exact. Record temperature/drift, spatial blur, saturation, edge clipping, specimen preparation and patch interference separately.

Planning costs and effort

These are illustrative budget allowances, not vendor quotes or measured costs. Confirm them with the host facility before selecting a route; no purchase is authorized by this protocol.

Recommendation: prioritize Stage A when a collaborator can provide independently calibrated maps. Otherwise N0751 may be the clearer first bench project for a lab with humidity control and displacement metrology. Neither has a completed physical result in this package.

N0751: differential swelling — proposed controlled-response test

Prepared 2 November 2026. Protocol proposal; no device measurement or laboratory engagement is claimed. Aim: test whether measured individual swelling responses predict their differential signal on unseen input waveforms. A successful digital-difference test would support the response mechanism, not yet demonstrate a standalone mechanical instrument or its novelty.

Source: retained N0751 report. The supplied first-order model is H_i(s)=g_i/(1+τ_i s), with gain-matched differential H_D(s)=g(τ₂−τ₁)s/[(1+τ₁s)(1+τ₂s)]. Its millihertz bandwidth, assumed diffusion coefficient, noise floor, matching tolerance and budget are model/design claims, not measured specifications. Use measured parameters for the chosen material.

Smallest decisive apparatus

• Two swelling films or elements exposed side by side to the same controlled humidity input, with different measured response times. Start with one material and two thicknesses; characterize the actual thickness and mechanical boundary conditions.
• A temperature-controlled humidity enclosure with programmable input. A reference hygrometer near each arm must have characterized timing, calibration and uncertainty; measure chamber transport and spatial gradients. A controller set point is not the stimulus record.
• Displacement readout for each path, synchronized with humidity and temperature. A calibrated microscope/camera, interferometer or displacement gauge can be considered if its measured noise and drift pass the feasibility gate. Do not assume the source’s 0.05 µm RMS noise is achieved.
• A same-material, approximately matched-response pair as the null control, and rigid/nonresponsive references for readout drift. “Same nominal thickness” does not prove equal gains or time constants.

First record the two paths separately and form a digital difference. This separates chemistry/transport from lever mechanics. Only after this passes should a mechanical lever/head be built and its own friction, creep, loading and gain transfer measured.

Characterization, then a frozen prediction

1. On a pilot pair, map each arm’s equilibrium displacement over the intended humidity interval and test reversibility. Select a small-signal range with documented hysteresis and no saturation. Include temperature monitoring and fixed preconditioning.
2. Apply repeated steps in both directions. Estimate g_i, τ_i and their uncertainty against the measured local humidity input. Inspect residuals for multi-time-constant behavior, creep and chamber lag. A first-order fit is a hypothesis, not a required successful answer.
3. Characterize the matched-time pair and rigid reference. Measure baseline/readout covariance; the paths share disturbances and cannot be assumed independent.
4. Freeze the calibration, gain ratio r=g₁/g₂, fitted response parameters, input alignment, time window, exclusions and analysis code before any held-out waveform. Use D(t)=y₁(t)−r y₂(t), with no per-waveform gain, delay, baseline polynomial or time-constant retuning. If the chamber/reference transfer function is corrected, its parameters also belong to the freeze.
5. In a separate held-out sequence, predict steps, positive and negative ramps, and sinusoids. Set sinusoidal frequencies at 0.3, 1 and 3 times f*=1/(2π√(τ₁τ₂)) using calibration-only parameters. Use small, preregistered amplitudes within the qualified range, randomize sequence order, and retain washout/re-equilibration periods. Record at least five cycles per sinusoid; allow at least five measured slow time constants after steps. At source-like time constants this can take hours, so do not promise a one-afternoon result.

A ramp rate is not sufficient to define an experiment: record its amplitude, duration, starting humidity and whether it remains within the qualified operating range. Stimulus units must be explicit (fraction RH versus percentage points RH).

Primary outcome and controls

For each waveform and pair, compare D_measured with the frozen prediction evaluated on the measured input. Report normalized RMS residual E=RMS(D_measured−D_predicted)/RMS(D_predicted), signed offset, amplitude ratio and phase error. Do not use correlation as the primary endpoint. Where predicted RMS is near zero, report absolute residual in displacement units and units of the measured baseline noise; do not divide by a vanishing denominator.

The matched-time control is exposed to the same measured stimulus, after its own calibration-based gain matching. Its expected residual signal is computed from its measured mismatch and uncertainty, not asserted to be exactly zero. Rigid references and swapping arm positions help distinguish optical drift, temperature gradients and chamber asymmetry from a swelling-rate channel. Apply arm swaps only in prespecified blocks with identical calibration rules.

Proposed gates and failure conditions

These numerical engineering thresholds are proposed acceptance choices, not consequences of the original model. Freeze them after the feasibility stage and before confirmatory held-out trials.

• Feasibility: unequal-path predicted RMS must exceed ten times the combined standard uncertainty over a representative waveform; local humidity differences and reference lag must be small enough that their propagated error is below 10% of predicted differential RMS. Otherwise improve metrology or choose a slower stimulus before continuing.
• A useful initial model target is E≤0.20 for each prespecified waveform, with amplitude error ≤20% and phase error ≤15° for sinusoids after propagating parameter/input uncertainty. Report uncertainty intervals, not just whether a point estimate passes.
• The first-order prediction fails for the tested material/range if residuals are reproducibly outside the frozen prediction envelope or require waveform-specific retuning. This does not refute every possible swelling mechanism.
• The proposed control target is residual RMS ≤20% of the unequal-time predicted RMS after accounting for the control’s measured mismatch, with uncertainty propagated. A control residual ≥50% of that signal in two prespecified repeat blocks is a strong apparatus-confounding flag, rather than evidence for the intended mechanism. Intervals spanning the control target are inconclusive. These are engineering decision thresholds, not universal physical constants.
• A predicted signal below the metrology floor, or an interval spanning the acceptance boundary, is inconclusive. A detection floor must be estimated from measured false alarms and misses across repeated trials, not inherited from the supplied Monte Carlo table.

One pilot pair can qualify the apparatus. For a separately labeled repeatability study, plan at least three independently fabricated unequal-time pairs plus matched-time controls and repeated held-out sequences on separate days. This is a planning minimum, not a powered population claim or manufacturing-tolerance estimate. Choose any confirmatory sample-size calculation using pilot variance before confirmatory measurements.

Uncertainty and records

Retain local RH, temperature, two displacement streams and rigid-reference stream with timestamps; calibration certificates or uncertainty estimates; sample thickness/geometry; preconditioning; loading history; characterization fits/residuals; the frozen parameter file and code hash; full stimulus order; excluded trials; paired raw controls; and day/pair-level results. Propagate gains, response times, reference lag, synchronization, drift, hysteresis and between-pair variability. Separate instrument noise from physical sample variation.

Planning costs, access and effort

Illustrative allowances only; not a priced bill of materials. Confirm capability, quotes and shared-facility charges before any commitment. The supplied “under €3,000” figure is not treated as a verified quote.

Next milestone: a frozen-parameter response comparison with measured controls, not a new assumed sensitivity budget. If promising, develop and test the mechanical head as a separately versioned follow-up.