Three supplied model investigations

I1 · The exponent you report is a property of your window

Question. When a decay is reported as a power law, how much of the exponent is the system and how much is the fit window?

Model. y(t) = At^(−α) + c, with c an unsubtracted offset. This is the most common situation in practice: a baseline that was not measured, or was measured badly.

Exact result. With x = ln t and B = c/A, the local log–log slope is

s(x) = −α / (1 + B e^{αx}),

a logistic in ln t of width 1/α, centred at the crossover t_c = (A/c)^{1/α}. A straight-line fit returns the least-squares average of that logistic. In particular, a window centred on the crossover returns exactly α/2 — an identity, not an empirical finding.

Numerics. A 2D map of α_fit/α over window width R ∈ [0.5, 6] decades and window placement log₁₀(t_c/t_mid) ∈ [−4, 4] (figure figs/i1_biasmap.png, panel A) shows that the bias spans 0.02 to 1.6. Monte Carlo with 400 realisations per configuration at 2% noise confirms the map and shows that a three-parameter fit (ln A, α, B) recovers α with bias 1.000 ± 0.011 even where the two-parameter fit is 0.50.

Identifiability. The Cramér–Rao bound on α (panel B) diverges when the window lies below the crossover: if you only see the flat part, α is not determined at any precision. That is the hard limit behind the folklore that “a power law cannot be distinguished from an exponential decay”.

What is new here. The logistic slope law is an identity. What is not obvious, and is falsifiable, is that (i) the offset is identifiable from curvature, so the correction is possible rather than merely a caveat; (ii) the correction is exact for windows that span the crossover; and (iii) the bias is bounded below 1% for windows well below the crossover, so the artefact cannot be invoked to explain a literature-wide discrepancy without checking where the crossover lies.

What would refute it. Published exponents unchanged by a free offset, with zero curvature, at the stated noise.

Files. I1_offset_window_bias.py, data/i1_bias_maps.npz, data/i1_mc.csv, data/i1_summary.json, figs/i1_biasmap.png, figs/i1_logistic.png.

I2 · A hidden substrate conductance sets every optical measurement of in-plane thermal conductivity

Question. Why do reported in-plane thermal conductivities of the same 2D material vary by factors of two to three between laboratories?

Model. A thin sheet of thickness d and conductivity κ on a substrate with interface conductance G, heated by a Gaussian spot of 1/e² radius s with absorbed power P. Steady state:

κd∇²T − GT = −q(r), q(r) = (P/2πs²)e^(−r²/2s²).

A Hankel transform gives, at the beam centre,

ΔT = (P/4πκd)·e^λE₁(λ), λ = Gs²/(2κd).

Exact result. The standard analysis assumes λ → 0 and therefore reports

κ_app = κ / [e^λE₁(λ)].

This crosses unity at λ = 1, so the bias changes sign within a typical experimental campaign (the crossover spot radius is s* = √(2κd/G) = 0.58 µm for κ = 50 W m⁻¹K⁻¹, d = 100 nm, G = 3×10⁷ W m⁻²K⁻¹).

Validation. The closed form agrees with direct numerical Hankel quadrature to 2×10⁻¹³ across λ from 10⁻³ to 10.

The sharp limit. For λ ≫ 1, κ_app → Gs²/(2d): the reported conductivity stops depending on the material and becomes a function of the substrate conductance and the beam size. Over plausible (s, G) ranges the model produces a 136-fold spread in reported κ for one true value.

Identifiability. A single spot size gives one number and cannot determine both κ and G (the Fisher matrix is rank-deficient; Cramér–Rao diverges). Two spot sizes in a 1:3 ratio give σ(κ)/κ = 0.037 at 1% noise; three spanning 1:10 give 0.022. More laser powers do not help.

Testable prediction. Reanalysed κ values from a published compilation should collapse onto a single true κ when plotted against λ computed from each study’s spot radius and substrate conductance, and should scale as s² at large s. A slope of 2 on log κ_app vs log s is the falsifiable signature.

Limits. The model is isotropic, steady-state, optically thin, and treats the substrate as a single conductance. Anisotropy, finite substrate thickness, and the transducer boundary add unmodelled terms; the 136-fold figure is an upper bound for the stated parameter range, not a prediction for any named material.

Files. I2_boundary_bias.py, data/i2_R_of_lambda.csv, data/i2_collapse.csv, data/i2_summary.json, figs/i2_bias.png, figs/i2_spread.png.

I3 · A Kramers–Kronig-consistent spectrum is not evidence of linearity

Question. Impedance spectra are screened with a Kramers–Kronig residual and accepted when it is small. Does passing that test mean the spectrum is linear?

Model. A Butler–Volmer interface with transfer coefficients α_a = α_c = 0.5, exchange current i₀ = 1 µA, double-layer capacitance 20 µF, series resistance 10 Ω. Drive V = V_dc + A sin ωt and solve the full harmonic balance: η(t) is expanded in 10 harmonics, the current is the exact Fourier transform of the memoryless Butler–Volmer law plus a linear capacitive term, and the series impedance acts harmonic by harmonic. No describing-function approximation and no time stepping.

Exact identity. For R_s = 0 the describing function is exactly Z₁(ω) = 1/(G_nl(A) + jωC_dl) — a Debye element — with G_nl(A) = (2i₀/b)·2I₁(A/b)/(A/b). A Debye satisfies Kramers–Kronig identically. The finite-amplitude spectrum is therefore KK-consistent by construction. Harmonic balance reproduces the Bessel identity to 10⁻¹⁷.

The falsifiable part. The numeric experiment shows the measured spectrum at 100 mV deviates from a perfect Debye by 9.6×10⁻⁸ relative — i.e. it is KK-consistent to eight digits — while the fitted charge-transfer resistance has fallen to 0.644 of its low-amplitude value. The classical Lin–KK residual does not track the bias at all (0.170 at 2 mV, 0.048 at 100 mV): it moves the wrong way.

Quantitative. R_ct/R_ct,true = 1.000 (2 mV), 0.996 (10 mV), 0.891 (50 mV), 0.644 (100 mV) for b = 51.4 mV. The third-harmonic ratio I₃/I₁ measures x = A/b to 0.2% at low frequency and converts directly into the correction. An amplitude-extrapolation protocol (four amplitudes, 0.5% noise) recovers the low-amplitude Re(Z₁) as 25 706 ± 79 against a true 25 701.

Refutation. R_ct amplitude-independent to 1% across 2–100 mV on a known Butler–Volmer system; or an I₃/I₁ that does not track the bias.

Limits. The mapping assumes a symmetric barrier and no mass transport. With α_a ≠ α_c a DC bias generates a second harmonic the symmetric model does not predict; with a Warburg element the harmonic current is loaded differently and the naive correction will over- or under-shoot. These are stated as open, not solved.

Files. I3_kk_blindspot.py, data/i3_summary.json, figs/i3_kk.png.