Incoprea validation round · 3 November 2026

Working research. Not externally peer reviewed. AI-assisted. No new physical experiment was performed. Nine investigations are model, simulation, or re-analysis; one (H203) uses a public NASA/JPL catalogue as measured data. Every number below is regenerated by the code in code/, with fixed seeds. The originals are preserved unmodified in orig/ with SHA-256 hashes.

0. What this is, and what it is not

The previous round produced five named investigations (I2, H0199, P1, P2, P3) and a large bank of new hypotheses. This round does three things:

1. Reproduces the originals first, before extending them, and records where the original code and the original prose disagree.
2. Runs an independent test of ten hypotheses that looked provable: the five named ones plus five from the Nov3 near-proof set (H123, H153, H203, H541, H346).
3. Reports negative and inconvenient results with the same weight as supportive ones.

The status vocabulary is deliberately narrow:

A result can carry two labels (for example, internally verified; sub-claim contradicted). That is the honest case for several of these.

1. Verdict at a glance

The single most important result is negative: P1’s “proof” of T₂ ≤ 2T₁ assumes the inequality it reports. The bound may still be true — the density-matrix check is consistent with it — but the calculation does not establish it. The second most important is that I2’s sign change is at λ=0.4348, not λ=1, which changes which measurements are above or below the true κ.

2. Reproduce before extending

The originals were located, copied, hashed, and rerun before any new code was written.

The originals are in orig/ with orig/SHA256SUMS.txt. Nothing in the original folders was modified.

Where the original prose and the original code disagree. Three cases matter:

1. I2 says the bias “crosses unity at λ = 1.” The code’s own R(1.0) returns 1.6769. The zero is at λ = 0.4348. This is not a rounding issue; it is a factor 1.68 in the inferred κ at the claimed crossover.
2. P2 quotes j_s = 4×10⁻¹¹ A cm. The code evaluates j_s/d with d in metres. The same number in those units is 4×10⁻⁹ A/cm. The crossover stays at 200 nm because both terms were scaled together, which is why the error survived.
3. P1 reports “exactly 2.000.” The value is 1.9999999999993 at N=32, i.e. 2.000 to three decimals. The model cannot exceed 2T₁ by construction (Section 4.3).

These are recorded, not silently corrected. The corrected statements are in each section.

3. I2 — the spot-size bias in in-plane thermal conductivity

3.1 Original claim and equations

A sheet of thickness d and conductivity κ on a substrate with interface conductance G is heated by a Gaussian spot. In steady state,

κ d ∇²T − G T = −q(r),     q(r) = (P / 2πs²) exp(−r²/2s²)

The Hankel transform gives, at the beam centre,

ΔT(0) = P/(4πκd) · e^λ E₁(λ),      λ = G s²/(2κd)
κ_app/κ = 1/[e^λ E₁(λ)] ≡ R(λ)

The original claims: R crosses 1 at λ=1; the crossover spot radius is s* = √(2κd/G); over plausible (s, G) the reported κ spans 136×; two spot sizes determine κ and G, while more laser powers do not.

3.2 Independent test

Three independent routes:

1. Closed form vs an axisymmetric finite-volume solve of the PDE on r ∈ [0, L], with a symmetry condition at r=0 and T(L)=0. This does not use the Hankel transform at all.
2. A 400-experiment synthetic recovery with κ ∈ [5, 500] W m⁻¹K⁻¹, G ∈ [10⁶, 10⁹] W m⁻²K⁻¹, s ∈ [0.2, 3] µm, d ∈ [20, 500] nm, 1% noise. The naive analysis reports κ_app and ignores G; the corrected analysis inverts with G known; the two-spot analysis fits both.
3. Held-out spot sizes: fit at s = 0.4 and 1.2 µm, predict at 0.8 and 2.0 µm.

3.3 Results

The closed form is right. Finite-volume vs closed form agrees to 2.1×10⁻⁵ relative across λ = 0.05–10 (the residual is discretisation; the original Hankel check was 2×10⁻¹³). The equation is not in question.

The sign change is at λ = 0.4348, not 1. Solving R(λ)=1 gives λ* = 0.434818. At λ=1, R = 1.6769: the reported κ is 68% high, not at the crossover. The crossover spot radius should be s* = √(2λ*κd/G) = 0.932√(κd/G), not √(2κd/G). This is a real correction to the paper’s central figure.

The beam width is off by 2. In the formula, q ∝ exp(−r²/2s²). If w is the 1/e² intensity radius, q ∝ exp(−2r²/w²), so s = w/2. The original calls s the 1/e² radius and uses s = w. Using w instead of s multiplies λ by 4. For three representative cases the inferred κ_app changes by factors 2.08, 2.75 and 3.45. The fix is one line: s = w/2, or write λ = G w²/(8κd).

Ignoring G is catastrophic; one spot with G known is not. Across 400 draws, the naive κ_app has median absolute error 144% and 90th percentile 8,300%. With G known, one spot gives median 2.8%, 90th percentile 37% (the inversion is ill-conditioned near the substrate-dominated limit). Two spots with G fitted give median 3.3%, but 4.5% of draws fail by more than 50% when both spots are substrate-dominated. Conditional on max λ < 10, the two-spot fit gives median 2.2%, 90th percentile 4.8%, maximum 7.5%.

Held-out prediction is good when the design spans the crossover. Median relative error 0.68%, 90th percentile 1.97%, maximum 4.4% at the two unseen spot sizes.

The spread is parameter-range dependent. The original 136× is reproduced for its range; a wider plausible grid (w = 0.5–2 µm, G = 10⁶–10⁸) gives R from 0.143 to 51.0, a 356× spread. The asymptotic statement is the robust one: for λ ≫ 1, κ_app → Gs²/(2d), independent of the material.

3.4 Strongest prior research

The 2-D heat-spreading solution with a substrate conductance is standard in optothermal Raman and TDTR metrology; the same e^λE₁(λ) structure appears in thin-film thermal models. Relevant anchors: the refined optothermal Raman measurement of lateral and interfacial conductivity in MoS₂/MoSe₂ (ACS Appl. Mater. Interfaces), and the variable-spot-size laser-flash Raman method for 2-D heterostructures (ScienceDirect). The metrological point — that the substrate term makes the reported κ depend on the beam size — is what this investigation adds.

3.5 Known vs potentially novel

• Known: the Green’s function; the substrate-conductance term; the need for two spot sizes to break the (κ, G) degeneracy.
• Potentially novel: the precise zero-bias point λ*=0.4348; the beam-width convention error as a common factor-2 trap; the failure map showing that a two-spot design still fails when both spots are substrate-dominated.

3.6 Falsification criteria

• A measurement with independently known κ and G whose κ_app does not follow R(λ) at the stated s and d.
• A substrate-free (suspended) sample with G→0 where the reported κ still shows the predicted s² scaling. In the model R→0 as λ→0, so a suspended sample must be analysed with the domain-size log term, not R.
• A two-spot recovery with max λ < 10 whose error exceeds 10% at 1% noise.

3.7 Corrected hypothesis

For a Gaussian spot, define s as the 1/e² amplitude radius (= w/2 for a beam of 1/e² intensity radius w). Then

κ_app/κ = R(λ),   λ = G s²/(2κd),   R(λ*) = 1 at λ* = 0.4348

A single spot gives one number and cannot separate κ from G. A two-spot design with at least one spot in 0.1 < λ < 3 gives ~2% κ at 1% noise; a design with both spots at λ > 10 is unidentifiable regardless of how many powers are measured.

Status: internally verified; the λ=1 crossing and the beam-width label are contradicted. No experimental data.

4. H0199 — where the witness blanks go

4.1 Original claim and estimator

A substrate carries M=25 reactive patches. Their uptake is R_i = ε f(x_i) + noise; N inert witnesses measure W_j = f(x_j) + noise. The estimator is

ε̂ = Σ_i R_i / Σ_i f̂(x_i)

with f̂ interpolated from the witnesses by quadratic least squares. The claim: interleaving witnesses among the reactive patches beats putting all blanks at the edge. The dossier reports 0.8–1.0% RMSE for distributed vs 22.9% for edge on a boundary-null field.

4.2 Independent test

• Reimplemented the fields independently (linear, quadratic, radial, rim-enhanced, boundary-null, Gaussian-process).
• Seven layouts at matched N=12 and matched patch count: edge, interior grid, jittered grid, random, interleaved (the actual claim, which the original code did not implement), D-optimal for a quadratic basis, and maximin space-filling.
• Three estimators: quadratic OLS, Gaussian process with a fixed kernel, and an oracle f̂=f.
• 400 seeds per cell, bootstrap 95% confidence intervals, plus an N sweep from 4 to 100 and an M sweep from 9 to 100.

4.3 Results

The 22.9% is reproduced exactly, but it is not a universal number. For the original 25-patch inset grid, the noiseless edge bias is 22.9898% — precisely the mean of f over those 25 patch locations minus 1. The continuum mean over the full substrate is 20.24%, and over the inset region it tends to 30.3%. Sweeping M at fixed inset:

So the headline number is a property of the patch layout and count. The qualitative conclusion — edge-only fails for a boundary-constant field — is unaffected and is the stronger statement.

Matched layouts, boundary-null field, N=12, noise 2%:

More edge blanks do not help. Edge RMSE from N=4 to N=100: 22.7, 22.1, 23.3, 23.1, 22.9, 22.9, 23.1, 23.1%. The error is a bias set by extrapolation, not a variance set by sample size.

The rim-enhanced field is worse for edge. Edge 37.85%, grid 1.03%, interleaved 5.91%, maximin 9.81%, D-optimal 18.14%. Random placement misses the rim (3.60%); a grid samples it.

Optimising for the wrong model is a failure mode. The D-optimal layout for a quadratic basis puts points at the boundary and is worse than a space-filling design on both adversarial fields (18.1% vs 9.8% on the rim; 2.8% vs 1.1% on boundary-null). Design optimality is model-relative.

The GP estimator is not a rescue. On boundary-null: edge 16.9%, grid 2.9%, interleaved 1.6%. The bias shrinks because the GP prior pulls the interior toward the boundary value, but it does not vanish.

4.4 Strongest prior research

Spatial sampling, kriging, and D-optimal design are mature. The relevant prior is the standard result that a polynomial fitted only on a boundary cannot identify interior structure; the deposition literature’s centre-to-edge non-uniformity is the physical instantiation. The original dossier’s own stop rule already anticipated the caveat.

4.5 Known vs potentially novel

• Known: regression extrapolation fails outside the design support; space-filling designs are more robust than model-optimal ones under misspecification.
• Potentially novel: the matched head-to-head with the interleaved layout (the claim as stated), the demonstration that the headline 22.9% is patch-count dependent, and the finding that “optimised” witness placement can be worse than random for a boundary-null field.

4.6 Falsification criteria

• An edge-only layout achieving <2% error on a field constant on the boundary, at any N.
• A case where the interleaved layout loses to edge on a field with interior structure.
• A measured deposition field where the edge estimator’s error shrinks with N.

4.7 Corrected hypothesis

Use interior/distributed witnesses whenever the process can produce centre-to-edge or rim structure. Edge-only blanks are not merely suboptimal; they are unidentifiable for a field that is constant on the boundary. The 22.9% figure should be quoted as “23% for a 25-patch inset grid; 17–27% across M=9–100”, not as a universal constant. Interleaving is one way to sample the interior; a space-filling design is at least as good and more robust.

Status: internally verified; the headline number is refined, not contradicted. No experimental data.

5. P1 — T₂ ≤ 2T₁: theorem, assumption, or estimator artefact?

5.1 Original claim and model

The original claims T₂ ≤ 2T₁ is universal, that a violation would signal non-Markovian noise, and that T₂*/T₁ is a monotone meter of the noise exponent β. The calculation uses the exact CPMG filter function

F(ω,T) = |∫₀ᵀ s(t) e^{iωt} dt|²,     χ(T) = (1/π)∫ S(ω)F(ω,T)/ω² dω
W(T) = exp(−χ(T)) exp(−T/2T₁)

and defines T₂ as the time at which W falls to 1/e.

5.2 Independent test

1. Reimplemented the filter function and checked it against direct time-domain integration (7 cases where the filter is not cancellation-limited; max relative difference 3.4×10⁻⁴, at N=8 and ω=10⁵ where the exact value is ~10⁻¹³).
2. Proved the inequality algebraically and evaluated W(2T₁) for β ∈ {0.5,1,2}, N ∈ {0,1,4,32}.
3. Checked density-matrix positivity for 40 random initial states × 120 times under amplitude damping plus pure dephasing; checked the Choi matrix of the dephasing channel.
4. Compared five T₂ estimators on four coherence curves.
5. Swept the noise amplitude A at fixed β to test whether T₂*/T₁ measures β.

5.3 Results

The “proof” is circular. The model is a product:

W(T) = exp(−χ(T)) · exp(−T/2T₁)  ≤  exp(−T/2T₁)

because exp(−χ) ≤ 1. Therefore W(T₂)=e⁻¹ implies exp(−T₂/2T₁) ≥ e⁻¹, i.e. T₂ ≤ 2T₁. The inequality is assumed by the functional form, not derived from the physics. Numerically, W(2T₁) = e⁻¹exp(−χ(2T₁)) is strictly below e⁻¹ for every χ>0; the slack ranges from 1.3×10⁻¹³ (β=2, N=32) to 0.368 (β=0.5, FID). The bisection can never return a value above 2T₁.

The “exactly 2.000” is rounding. The maximum ratio in the table is 1.9999999999993; at N=32, β=1 it is 1.99999987. The bound is an asymptote approached from below, not a saturated equality.

Positivity is consistent with the bound. The worst violation of |ρ₀₁|² ≤ ρ₀₀ρ₁₁ over the random-state sweep is 2.6×10⁻¹⁶ (machine zero). The Choi matrix of the pure-dephasing channel is PSD for γ ∈ [0,1]. The bound is what positivity plus amplitude damping gives; the filter-function calculation is a complicated way of evaluating a special case.

The estimator changes the answer by a factor of 7. On curves with T₁=100 µs:

The “first crossing” is meaningless for a curve that revives; the area definition is the stable one. This is the original’s own caveat, now quantified.

**T₂*/T₁ is not a β meter.** At fixed β=1, varying the noise amplitude A by 10⁻³ to 10² changes the ratio from 1.94 to 0.036. Conversely, 37 different (β, A) pairs give the same ratio 1.2. The ratio is a function of (β, A, band, T₁), not of β alone. The calibration table is conditional on the arbitrary A_cal and the 10²–10⁸ rad/s band.

5.4 Strongest prior research

The bound T₂ ≤ 2T₁ is standard for a qubit with amplitude damping plus pure dephasing; it follows from the positivity of the density matrix, not from a noise-spectrum integral. Reports of T₂ > 2T₁ on hardware exist (for example the IBM Washington discussion on Quantum Computing Stack Exchange) and are usually attributed to different T₁/T₂ measurement conditions, non-exponential decays, or leakage — not to a failure of positivity. The 2026 Spectral Bottleneck Constraint (Zenodo, May 2026) is a related no-go result: no universal asymptotic rate law can depend only on an aggregate.

5.5 Known vs potentially novel

• Known: the bound for the standard qubit model; the estimator dependence of T₂; the fact that a 1/e time is not a unique definition.
• Potentially novel: the explicit demonstration that the Discovery07 “proof” is circular; the non-identifiability of the claimed β meter (37 degenerate pairs); the quantified statement that the ratio is a consistency check, not a spectrum measurement.

5.6 Falsification criteria

• A device with an independently verified white spectrum and T₂*/T₁ below the calibration at the stated band.
• A device with verified Q and T₁ and T₂ > 2T₁ under the same measurement conditions, with an exponential decay and no leakage.
• A demonstration that T₂*/T₁ is one-to-one with β when A is unknown.

5.7 Corrected hypothesis

For a qubit with amplitude damping and pure dephasing, T₂ ≤ 2T₁ holds, and the filter-function model reproduces it only because it assumes it. T₂/T₁ is a consistency check and a lower bound on the dephasing rate, not a spectrum meter. A credible β inference needs an independently measured noise amplitude and band (or a direct noise spectrum); the ratio alone is degenerate.

Status: the algebra is internally verified; the universality claim is unsupported (circular); the meter is unsupported. Positivity is consistent with the bound.

6. P2 — bulk plus surface photocurrent

6.1 Original claim and equations

In a centrosymmetric film the bulk photovoltaic current vanishes by symmetry, so the measured current density is

J(d) = j_b + j_s/d

with j_b in A cm⁻² and j_s in A cm (a sheet current per unit width). The original quotes j_b=2×10⁻⁴ A cm⁻², j_s=4×10⁻¹¹ A cm, crossover 200 nm, a 100 nm film reading +200% above bulk, and a 50 nm/1 µm ratio of 4.17.

6.2 Independent test

1. Dimensional audit of the equation and the code.
2. Cramér–Rao lower bound on (j_b, j_s) as a function of thickness range and noise.
3. A two-layer forward model: a surface layer of thickness t_s with conductivity σ_s on a bulk σ_b, J = σ_b E + (σ_s−σ_b)t_s E/d. Fit the two-parameter law and compare the recovered j_s to the true (σ_s−σ_b)t_s E.
4. A distinctive protocol: illumination-side antisymmetry. If J(+)=j_b+j_s/d and J(−)=j_b−j_s/d, then the half-sum is j_b and the half-difference is j_s/d. The half-difference versus 1/d has slope j_s and zero intercept.
5. A symmetric-interface null: two identical surfaces cancel j_s, so J(d) is thickness-independent.

6.3 Results

The numbers reproduce. J(100 nm)/j_b = 3.0 (+200%); J(50 nm)/J(1 µm) = 4.167; crossover 200 nm. The algebra is right.

The quoted j_s is 100× too small. The code evaluates j_s/d with d in metres. The same numerical value 4×10⁻¹¹ then has units A m cm⁻¹, i.e. 4×10⁻⁹ A/cm. Written with d in cm, the correct j_s is 4.0×10⁻⁹ A/cm. The crossover is unchanged because j_b and j_s were scaled together. This is a unit-label error, not a physics error, but it would mislead anyone converting the number.

The fit needs a wide thickness range. CRLB at 3% relative noise, 60 points:

A two-parameter separation from a 2:1 series is not credible at 3% noise; the original’s 100:1 series is well conditioned. The correlation between the two parameters is −0.64 to −0.98, so the fit is a classic collinearity problem.

The 1/d law is exact for a uniform surface layer. In the two-layer model the fit recovers j_s to 0.4–4.5% (noise-limited), and the true j_s is (σ_s−σ_b)t_s E. The law is not a microscopic derivation; it is dimensional bookkeeping for a sheet current. A surface layer whose thickness or conductivity changes with d breaks it.

The clean test is antisymmetry, not a thickness series. The half-difference recovers j_s = 4.0×10⁻⁹ A/cm exactly and the half-sum recovers j_b = 2.0×10⁻⁴ A cm⁻². A symmetric device has j_s = 0 and no 1/d term. This is a distinctive, cheap prediction: reverse the illumination side (or the contact polarity) and the surface term changes sign while the bulk term does not.

6.4 Strongest prior research

The 1/d dilution of a sheet current is standard. The relevant recent literature is the interface-BPVE programme: “Giant bulk photovoltaic effect driven by interfacial symmetry breaking in MoS₂/Ta₂NiSe₅ heterostructures” (Nature Communications, 2026) and “High performance bulk photovoltaics in narrow-bandgap centrosymmetric ultrathin films” (Materials Horizons, 2020), plus thickness-dependent BPVE in 3R-MoS₂ (Advanced Physics Research, 2025). These support the physical premise — a centrosymmetric film can show an interface-dominated photocurrent — and make the 1/d term plausible. The novel part here is the separation protocol, not the law.

6.5 Known vs potentially novel

• Known: sheet-current dilution; surface/interface photocurrent; thickness-dependent BPVE.
• Potentially novel: the antisymmetry protocol that separates j_b and j_s without a thickness series; the explicit 100× unit correction; the conditioning map showing when a thickness fit is hopeless.

6.6 Falsification criteria

• J independent of thickness in a centrosymmetric material at fixed illumination and electrode geometry (the symmetric-interface null).
• A half-difference that does not scale as 1/d, or a half-sum that changes with illumination side.
• A thickness series where the two-parameter fit recovers a j_s that depends systematically on the range (a sign of a third term).

6.7 Corrected hypothesis

J(d) = j_b + j_s/d with j_s in A/cm and d in cm. The quoted j_s should be 4.0×10⁻⁹ A/cm, not 4×10⁻¹¹. The law is exact for a uniform surface layer; its microscopic interpretation requires t_s and σ_s. The decisive test is the illumination-side antisymmetry: J(+)−J(−) = 2j_s/d, with a null in a symmetric device.

Status: internally verified; the unit sub-claim is contradicted; the law is conditional on geometry. No experimental data.

7. P3 — the fluctuation–dissipation coherence bound

7.1 Original claim and equations

A harmonic oscillator of frequency ω₀, quality factor Q and bath temperature T has

τ_φ ≤ ħQ/(k_BT)          (classical)
τ_φ ≤ 1/[γ(n_th+1/2)]    (quantum),  γ=ω₀/Q,  n_th=1/(e^{ħω₀/k_BT}−1)

At Q=10⁶, T=20 mK, f₀=100 kHz the classical bound is 382 µs.

7.2 Independent test

1. Reproduced the numbers.
2. Derived the exact identity and compared it with the classical limit across T = 1 µK–300 K.
3. Solved the two-level Lindblad equation with amplitude damping and heating; fitted the ρ₀₁ decay and compared with γ(n_th+1/2).
4. Computed the squeezed-thermal-bath dephasing rate Γ_φ = γ e^{−2r}(n_th+1/2), which exceeds the thermal bound by e^{2r}.
5. Searched the 2025–2026 literature for a counterexample to a universal spectral bound.

7.3 Results

The numbers reproduce. Classical 381.9116 µs; quantum 381.911627 µs; ratio 0.999999995; ħω₀/k_BT = 2.4×10⁻⁴.

The quantum expression is exact, and it is an identity. For a thermal Markovian bath,

Γ_φ = γ(n_th+1/2) = (γ/2)coth(ħω₀/2k_BT)
τ_φ = (2Q/ω₀)tanh(ħω₀/2k_BT)

The two-level Lindblad fit gives Γ_φ/γ = 0.5, 1.0, 5.5 for n_th = 0, 0.5, 5 — matching n_th+1/2 to 10⁻¹¹. The “bound” is an equality for the model it describes.

The classical form is the high-T limit, and it is very good at ordinary temperatures. The 10% deviation from the exact form occurs at T = 4.34 µK for f₀=100 kHz. At 20 mK the correction is 4.8×10⁻⁹. The “quantum correction” is therefore irrelevant for essentially all optomechanics at 100 kHz; it matters only in the microkelvin regime. The classical formula diverges as T→0 and is not a bound there; the quantum form is.

The bound is not universal: a squeezed bath beats it. With squeezing parameter r, Γ_φ = γ e^{−2r}(n_th+1/2), so τ_φ exceeds the thermal bound by e^{2r}: 2.72× at r=0.5, 7.39× at r=1, 54.6× at r=2. This is outside the claim’s stated scope (a squeezed bath is non-thermal), but it shows that the inequality is a thermal-equilibrium statement, not a law of nature. A driven/parametric system can do the same.

The 2026 literature search did not find the specific September counterexample. The closest 2026 items are the Spectral Bottleneck Constraint (Zenodo, 24 May 2026), a no-go theorem for aggregate asymptotic rate laws, and the September 2026 induced-coherence paper (arXiv:2609.35523), which is about interferometric triality, not this bound. A systematic search of web indexes and arXiv listings on 3 November 2026 found no paper that directly contradicts τ_φ ≤ ħQ/(k_BT) for a thermal bath. If a specific September 2026 counterexample exists outside the indexed sources, it should be supplied; the honest statement is that this search did not locate it.

7.4 Strongest prior research

The FDT relation S_F = 4mγk_BT and the thermal dephasing rate γk_BT/(ħω₀) are standard in optomechanics and in the theory of damped oscillators. The quantum form (2Q/ω₀)tanh(ħω₀/2k_BT) is the same identity written with n_th. The 2026 no-go result is a warning against three-number consistency claims in general, not a direct test of this one.

7.5 Known vs potentially novel

• Known: the FDT; the thermal dephasing rate; the quantum correction; the fact that a squeezed bath can suppress noise.
• Potentially novel: the precise scope statement (thermal, linear, Markovian, single oscillator); the 4.34 µK crossover at 100 kHz; the explicit demonstration that the “bound” is an equality for the model, and that the classical inequality is loose below ~5 µK.

7.6 Falsification criteria

• A thermal-bath oscillator with independently verified Q and T whose measured τ_φ exceeds 1/[γ(n_th+1/2)].
• A device whose coherence is above the classical bound at T > 10 µK with no squeezing or drive.
• A derivation showing Γ_φ ≠ γ(n_th+1/2) for a linear Markovian thermal bath.

7.7 Corrected hypothesis

For a single harmonic oscillator linearly coupled to a Markovian thermal bath, τ_φ = 1/[γ(n_th+1/2)] exactly. The classical ħQ/(k_BT) is the high-temperature limit and is accurate to 10% above ~4.3 µK at 100 kHz. The inequality τ_φ ≤ ħQ/(k_BT) is a thermal-equilibrium consistency check, not a universal bound; squeezed or driven baths exceed it by e^{2r} or more.

Status: internally verified; the universal reading is contradicted by the squeezed bath; the thermal-bath identity is exact. No experimental data.

8. H123 — the repertoire as a quantizer

8.1 Claim

For X ~ N(0, σ²I_d) and a codebook of N points, the Gaussian rate–distortion converse gives D ≥ σ²N^{−2/d}, so the RMS coverage error decays as N^{−1/d}. The claim is a theorem transplant from rate–distortion theory.

8.2 Test

Lloyd/k-means quantizers with KD-tree assignment, N = 8…4096, d = 2, 5, 10, and a 2-D manifold embedded in d=10. Fit the log–log slope over N ≥ 64.

8.3 Results

The exponent approaches −2/d; the d=2 finite-N slope is shallow because N=4096 is not asymptotic and k-means is not the optimal quantizer. The prefactor is 8–14, not 1: the inequality D ≥ σ²N^{−2/d} holds, but it is not tight at these N, and the claimed constant is not achieved. The manifold case is decisive: the ambient dimension is 10 but the slope is −0.83, i.e. the effective dimension controls the cost. Using d=10 would overestimate the repertoire needed by a large factor.

8.4 Prior research and status

Shannon’s rate–distortion theorem and Zador’s asymptotic quantization theory are the prior art. The biology is the open mapping. Status: internally verified (theorem transplant); the constant sub-claim is refined; the effective-dimension caveat is the useful result. No experimental data.

8.5 Falsification and correction

Coverage decaying exponentially in N, or a fitted slope near −1 for a genuinely high-dimensional pathogen space, would contradict the quantizer model. Corrected: D ≈ C_d σ²N^{−2/d} with C_d ≈ 8–14 over the tested range; use the measured effective dimension, not the ambient dimension.

9. H153 — the stemma threshold

9.1 Claim

On a regular b-ary copying tree with per-edge error p, the archetype is reconstructible iff b(1−2p)² > 1 (Kesten–Stigum). The claim is a transplant.

9.2 Test

80–120 trees, root strings of length 24–48, b=2,3,4, depth 4–12, majority reconstruction (ML for the symmetric BSC tree). Locate the 0.5-overlap crossing and compare with p_c=(1−1/√b)/2.

9.3 Results

The finite-depth crossing is below the asymptotic threshold, as expected: at finite depth the transition is smoothed. The depth scan confirms the direction: for b=3, p=0.18 (above threshold) has overlap 0.60→0.51 over h=4→10, while p=0.22 (below) decays 0.45→0.27, and p=0.30 decays to 0.04. The theorem is reproduced with a finite-size correction.

9.4 Prior research and status

Kesten–Stigum (1966); Mossel’s tree-reconstruction work; the Roch lecture notes. Status: independently replicated numerically. The paleographic estimation of b and p is unverified and is the actual open problem.

9.5 Falsification and correction

A tradition below the predicted threshold reconstructed with high confidence, or above it and opaque, would contradict the mapping. Corrected: the threshold is asymptotic; a single-depth 0.5 crossing is biased low by 0.01–0.02 at h≤12. Use the depth scaling (persistence vs decay), not a single crossing.

10. H203 — the impact-loss tail

10.1 Claim

If N(>D) ∝ D^{−b}, E ∝ D³ and L ∝ E^γ, then P(L>x) ∝ x^{−α} with α = b/(3γ). For b=2.7 and γ=2/3, α=1.35: finite mean, infinite variance.

10.2 Test

• Synthetic: 300,000 Pareto diameters, E=D³, L=E^γ; Hill estimator at thresholds 1, 3, 10, 30.
• Public data: the NASA/JPL CNEOS fireball catalogue, 1,074 events, fetched 3 November 2026. The catalogue’s impact energy is already E ~ D³, so the Hill index of E gives b/3 directly.

10.3 Results

Synthetic: b_est = 2.69–2.71, α_L = 1.345–1.356, exact to the estimator’s precision.

Fireball catalogue:

For thresholds 0.2–2 kt, b ≈ 2.65–2.82 and α_L ≈ 1.33–1.41, bracketing the predicted 1.35. At 5 kt the tail has only 28 events and the estimate is noisy. The result is a genuine public-data consistency check, not a new measurement: the catalogue is incomplete, the impact energies are model-derived from radiated energy, and the agreement may be partly built into the energy conversion. The actuarial conclusion (variance infinite for 1<α<2) is standard extreme-value theory; it is not tested here against insured losses.

10.4 Prior research and status

Impactor size–frequency laws, the CNEOS fireball survey, and regular variation in catastrophe insurance. Status: experimentally supported (catalogue), conditional. The synthetic part is independently replicated.

10.5 Falsification and correction

An exponential loss tail, or α > 2 at high confidence, or a population with b>2 and γ≈1 producing a finite-variance loss distribution after aggregation. Corrected: α = b/(3γ) is exact given the power laws; the empirical support is b≈2.7 for 0.2–2 kt fireballs, so α≈1.35 with γ=2/3. The claim should cite the threshold range, not a single number.

11. H541 — craze cells as a Poisson–Voronoi tessellation

11.1 Claim

E[A]=1/λ for a homogeneous Poisson nucleation process; the area CV is ≈0.53; the earlier exponential-area guess is false.

11.2 Test

Periodic Voronoi tessellation (3×3 tiling, convex-hull cell areas), 200–3,200 nuclei, 8 repetitions, plus Thomas-clustered and regular-grid controls.

11.3 Results

Poisson: mean-area ratio 1.000 at n=3,200; CV 0.530 (n=3,200), 0.519 (n=800), 0.543 (n=200). The exponential fit is rejected (KS p≈0). Thomas-clustered: CV 0.993 at n=200, 0.727 at n=800, 0.585 at n=3,200 (approaching Poisson as the density rises). Regular grid: CV≈0. The mean-area identity is exact; the CV is a diagnostic for Poisson nucleation, not a universal constant of glaze chemistry.

11.4 Prior research and status

Meijering’s theorem; the Gilbert/Voronoi model of craze cells. Status: internally verified; the exponential guess is contradicted. No glaze images were measured.

11.5 Falsification and correction

Mean area not scaling as 1/λ, or a CV far from 0.53 with no change in chemistry. Corrected: E[A]=1/λ is exact; CV≈0.53 is the Poisson benchmark; a CV near 1 indicates clustering, near 0 indicates a regular grid.

12. H346 — opening a book is a bending problem

12.1 Claim

F ≈ E w t³ θ/(6H²); the force scales as the cube of spine thickness.

12.2 Test

A 2-D plane-stress finite-element model: a cantilever strip of length H, thickness t, unit width, clamped at x=0, loaded by a pure end moment M. The effective stiffness EI_eff = MH/θ is measured from the end-face rotation. A fixed rotational compliance c_j models the joint.

12.3 Results

The fitted slope of log EI_eff vs log t is 2.99975 over t = 0.005–0.08, confirming the cubic scaling. The absolute value is 0.648× the beam formula t³/12 across the whole range (a constant 2-D/end-effect factor), so the dossier’s 1/6 is a boundary-condition convention, not a universal prefactor; the elastic-strip value is 1/12.

The joint changes the exponent. With a fixed rotational compliance c_j, the log–log slope of F/θ vs t is 3.0 for c_j=0, 1.84 for c_j=10⁶, 0.26 for c_j=10⁸, and 0.04 for c_j=10⁹. When the joint dominates, the force is nearly thickness-independent. A measured slope below 3 is therefore evidence for joint compliance, not a refutation of elasticity.

12.4 Prior research and status

Euler–Bernoulli beam theory is textbook. Status: internally verified; the absolute constant is refined. No bench measurements.

12.5 Falsification and correction

A slope significantly different from 3 after controlling for width, height and joint compliance; or a force independent of thickness in a joint-free strip. Corrected: F ∝ t³ in the elastic-strip limit; the prefactor is geometry-dependent (0.648× the beam formula in this 2-D model); a joint-dominated binding shows a slope near 0–1.

13. Cross-cutting findings

Four failure modes recur across the ten investigations, and they are more transferable than any single number.

1. Normalisation is often the result. The 22.9% edge bias is the mean over 25 inset patches; it becomes 16.7% at M=9 and 26.9% at M=100. The 136× I2 spread becomes 356× on a wider grid. The “2.000” is a rounded 1.9999999999993. A number quoted without its layout, range, or estimator is not reproducible.

2. Estimator dependence is not a nuisance; it is the systematic. T₂ varies by a factor of 7 between the first-crossing and area definitions on the same curve. κ_app varies by a factor of 3.45 between the amplitude and intensity beam radii. The Hill tail index varies by 0.25 between thresholds. Pre-registering the estimator is part of the test.

3. A proof that assumes its conclusion is worse than no proof. P1’s filter-function model multiplies the coherence by exp(−T/2T₁), so it cannot return T₂ > 2T₁. The calculation is internally consistent and physically irrelevant to the universality question. This is the clearest negative result of the round.

4. Model-relative optimisation can be worse than random. D-optimal witness placement for a quadratic field is worse than maximin or a grid on both adversarial fields (18.1% vs 9.8% on the rim; 2.8% vs 1.1% on boundary-null). The same lesson applies to the I2 two-spot design: two spots chosen without regard to λ can be unidentifiable.

14. Known vs potentially novel

None of the “potentially novel” items is a new physical law. They are corrections, protocols, and scope statements. That is the appropriate output of a validation round.

15. Falsification criteria, compact

16. Remaining uncertainty

• No experiment. Nine of the ten are model/simulation; H203 re-analyses a public catalogue. The experimental status of every claim is “untested” in the laboratory sense.
• Synthetic parameter ranges. The I2 and H0199 error maps are only as good as the stated parameter draws. The qualitative conclusions (sign of the bias, flat-in-N edge error) are robust; the exact percentages are not.
• The fireball catalogue. It is not volume-complete, the impact energies are model-derived, and the tail fit is threshold-sensitive. The 1.35 agreement is a consistency check, not a measurement of insured loss.
• The 2026 literature. The search was not a professional systematic review. The September 2026 counterexample to a universal spectral bound was not located; if it exists, the P3 scope statement may need revision.
• The P1 bound. The density-matrix check is consistent with T₂ ≤ 2T₁ for the standard model, but this round did not prove the bound for arbitrary non-Markovian noise. The honest statement is: the Discovery07 proof does not establish it; the standard model satisfies it.
• The website publication. The research update is published as a page under /research/; it is not peer reviewed, and the code is the authority.

17. Reproducing this round

Environment: Python 3.14, numpy 2.5.4, scipy 1.18.1, matplotlib 3.11.2. All runs fixed-seed. Total runtime under five minutes on the machine that produced this report; the fireball catalogue is cached in data/fireballs.json.

cd code
python3 t01_i2.py        # I2
python3 t02_h0199.py     # H0199
python3 t03_p1.py        # P1
python3 t04_p2.py        # P2
python3 t05_p3.py        # P3
python3 t06_h123.py      # H123
python3 t07_h153.py      # H153
python3 t08_h203.py      # H203 (uses data/fireballs.json)
python3 t09_h541.py      # H541
python3 t10_h346.py      # H346

Each script writes data/tNN_*.json and figures/tNN_*.png. The summary numbers in this report are read from those JSON files; no number was typed by hand without a corresponding field.

Originals: orig/ contains the unmodified source, the Discovery07 proof text, and SHA256SUMS.txt. The original folders in the workspace were not modified.

18. Sources and verification status

The 2026 sources are flagged as preprints or metadata-level checks; they were not read in full. No source is used to claim experimental support for a claim that this round did not test.

If one sentence survives: the numbers that were wrong were wrong by factors of 2 to 100, and the proofs that were circular were circular because they multiplied by a factor that could never exceed 1. Both are cheaper to catch than to publish.

Download the release ZIP (code, data, figures, originals, 1.3 MB) · Back to the research hub

Status vocabulary: unsupported, contradicted, internally verified, independently replicated, experimentally supported. No claim on this page is experimentally supported by a new experiment.