{ "collections": [ { "collection": "expedition-1", "hypotheses": 300, "nested_leads": 30, "supplied_result_reports": 3, "readme": "sources/expedition-1/README.md", "results": [ { "name": "C1_transport_precision", "url": "sources/expedition-1/03_PHASE_III_DEEP/C1_transport_precision/REPORT.md", "text": "# C1 — Transport precision in multi-terminal conductors\n\n**Derived from:** H274, rank 29/30 in Phase II\n**Status: PARTIAL RESULT** — a formal proof of a corrected statement, plus a *refutation*\nof one of the original hypothesis's two claims, plus a computational counterexample that\noverturns the hypothesis's central interpretation.\n\n---\n\n## Step 1 — The exact claim\n\nPhase I asserted two things. They are separated here because they have different fates.\n\n**Claim A (survives, as a theorem).** In isothermal linear response, for any terminal *i* of\nan *N*-terminal conductor,\n\n$$\\frac{\\operatorname{Var}(J_i)\\,\\sigma}{\\langle J_i\\rangle^{2}}\\;=\\;2k_B\\,\n\\frac{(e_i^{\\mathsf T} L\\, e_i)(F^{\\mathsf T} L F)}{(e_i^{\\mathsf T} L F)^{2}}\\;\\ge\\;2k_B,$$\n\nwith equality **iff** $F \\parallel e_i$.\n\n**Claim B (refuted).** \"The residual slack collapses on $1-1/N_{\\rm eff}$.\"\n\n**Claim C (refuted as stated).** \"Reported violations of the TUR are a probe-terminal\nheat-accounting error.\" The correct statement is narrower: accounting errors *can* produce\napparent violations, but there exists a physically ordinary regime — sub-Poissonian thermal\nnoise at low bias — in which the TUR fails genuinely, with no accounting error anywhere.\n\n---\n\n## Step 2 — Foundations\n\nAssumptions, stated so that they can be attacked individually:\n\n1. **Linear response.** Currents are linear in forces: $J = L F$.\n2. **Isothermal, single carrier type.** All terminals share one temperature; entropy\n production is $\\sigma = F^{\\mathsf T} L F$ (Joule heating only).\n3. **Onsager reciprocity.** $L = L^{\\mathsf T}$ (no magnetic field, no broken time-reversal\n symmetry).\n4. **Passivity.** $L \\succeq 0$ — the second law, nothing more.\n5. **Current conservation.** $\\sum_\\alpha J_\\alpha = 0$, i.e. $L \\mathbf{1} = 0$.\n6. **Gaussian, Markovian noise.** $\\operatorname{Var}(J_i) = 2k_B (L)_{ii}$ in the\n low-frequency limit.\n\nAssumptions 1–6 are the standard setting of mesoscopic linear-response transport. They are\n*not* claimed to describe every experiment; they define the model in which Claim A is exact.\n\n---\n\n## Step 3 — Derivation\n\n**Theorem (model-specific).** Under assumptions 1–6, Claim A holds, and it is not an\nindependent physical law but the Cauchy–Schwarz inequality.\n\n*Proof.* Let $\\langle u,v\\rangle_L := u^{\\mathsf T} L v$. Because $L \\succeq 0$ (assumption 4),\nthis is a positive semidefinite bilinear form, hence an inner product on the quotient by its\nnull space. Take $u = e_i$ and $v = F$. Cauchy–Schwarz gives\n\n$$\\langle e_i, F\\rangle_L^{2} \\;\\le\\; \\langle e_i,e_i\\rangle_L \\;\\langle F,F\\rangle_L ,$$\n\nthat is,\n\n$$(e_i^{\\mathsf T} L F)^{2} \\;\\le\\; (e_i^{\\mathsf T} L e_i)(F^{\\mathsf T} L F).$$\n\nDivide by $(e_i^{\\mathsf T} L F)^2$ and identify $J_i = e_i^{\\mathsf T} L F$, $\\sigma = F^{\\mathsf T} L F$:\n\n$$\\frac{(L_{ii})\\,\\sigma}{J_i^{2}} \\;\\ge\\; 1 .$$\n\nMultiplying by $2k_B$ and using $\\operatorname{Var}(J_i) = 2k_B L_{ii}$ gives Claim A. Equality\nin Cauchy–Schwarz holds iff $e_i$ and $F$ are linearly dependent, i.e. $F \\parallel e_i$. ∎\n\nThree consequences follow immediately, and they are the actual content of the result:\n\n- **The TUR is dimensionally and algebraically trivial in this model.** It carries no\n information about the conductor beyond $L \\succeq 0$. Any \"violation\" in a system obeying\n assumptions 1–6 must therefore be an artefact of how $\\sigma$ or $\\operatorname{Var}$ was\n computed — which is exactly the grain of truth in the original Claim C.\n- **Saturation requires single-terminal driving.** With more than one terminal biased, the\n slack is fixed by the angle between $F$ and $e_i$ in the $L$-metric. It is a geometric\n quantity, not a channel-count.\n- **The slack is not bounded by any function of $N_{\\rm eff}$ alone.** This is Claim B's\n undoing.\n\n**Refutation of Claim B.** For a two-terminal conductor the precision slack is the Fano\nfactor $F_{\\rm F} = \\sum_n \\tau_n(1-\\tau_n)/\\sum_n\\tau_n$, and\n$N_{\\rm eff} = (\\sum\\tau)^2/\\sum\\tau^2$. Consider the one-parameter family\n$\\tau = (t,t)$. Then $N_{\\rm eff} = 2$ for **every** $t\\in(0,1]$, while\n\n$$F_{\\rm F}(t,t) = \\frac{2t - 2t^{2}}{2t} = 1 - t .$$\n\nSo at fixed $N_{\\rm eff}=2$ the slack ranges over the entire interval $(0,1)$. A quantity that\nspans its full range at fixed argument is not a function of that argument. **Claim B is false.**\n\n---\n\n## Step 4 — Independent verification\n\nExecuted in `c1_tur.py` (reproducible; see Step 8).\n\n**(a) The theorem, numerically.** 8000 random 4-terminal Onsager matrices (each built as\n$PAA^{\\mathsf T}P$ with $P$ the sum-zero projector, so assumptions 3–5 hold by construction),\nwith random force vectors, evaluated at every terminal:\n\n```\nminimum TUR ratio over all samples : 1.000014\nmaximum TUR ratio over all samples : 2.77e+09\nsamples saturating at 1 ± 1e-9 : 0\nF ∥ e₀ → ratio = 1.000000000000\nF mixed → ratio = 1.003756\n```\n\nThe minimum never dips below 1, and the case $F \\parallel e_0$ saturates to machine precision.\nThis is the Cauchy–Schwarz identity behaving as an identity.\n\n**(b) The refutation, numerically.** At $N_{\\rm eff}=2$ exactly:\n\n| spectrum | $N_{\\rm eff}$ | Fano (slack) | $1-1/N_{\\rm eff}$ |\n|---|---|---|---|\n| $\\tau=(1.0,1.0)$ | 2.0000 | 0.0000 | 0.5000 |\n| $\\tau=(0.8,0.8)$ | 2.0000 | 0.2000 | 0.5000 |\n| $\\tau=(0.5,0.5)$ | 2.0000 | 0.5000 | 0.5000 |\n| $\\tau=(0.1,0.1)$ | 2.0000 | 0.9000 | 0.5000 |\n| $\\tau=(0.01,0.01)$ | 2.0000 | 0.9900 | 0.5000 |\n\nThe predicted law is a horizontal line at 0.5; the measured slack traces the full diagonal.\n\n**(c) The genuine violation.** Single channel, Landauer–Büttiker current and noise,\n$\\sigma = IV/T$, ratio in units of $2k_B$:\n\n| $\\tau$ | $V=0.5$ | $V=1$ | $V=2$ | $V=5$ | $V=20$ |\n|---|---|---|---|---|---|\n| 1.00 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |\n| 0.50 | 0.2552 | 0.2705 | 0.3283 | 0.6335 | 2.5000 |\n| 0.10 | 0.0119 | 0.0174 | 0.0382 | 0.1481 | 0.8200 |\n\nA perfectly transmitting channel saturates the bound at every bias — a non-trivial check that\nthe normalisation is right. Partially transmitting channels fall far below 1. The zero-bias\nlimits are $0.2500$ for $\\tau=0.5$ and $0.0100$ for $\\tau=0.1$, i.e. the asymptote is $\\tau^2$\n**in this normalisation** (the standard literature convention absorbs one factor of $\\tau$ and\nquotes $\\tau$; the prefactor is convention, the violation is not).\n\n---\n\n## Step 5 — Stress tests\n\nAttempts to break the result, in order of severity:\n\n1. **\"The violation in (c) is just the equilibrium limit where TUR is meaningless.\"** Partly\n fair: at $V\\to0$, $\\sigma\\to0$ and the TUR's premise degrades. But the table shows the\n violation extends to $V\\sim k_BT$ where $\\sigma$ is finite and the current is measurable;\n the failure is not confined to a measure-zero point.\n2. **\"Claim A is circular because Assumption 6 puts $L_{ii}$ into the variance by hand.\"**\n Circularity is real but not fatal: the proof shows the TUR *adds nothing* to passivity in\n this model. That is a genuine negative result about what the TUR can prove.\n3. **\"Gaussian assumption excludes the interesting nonlinear regime.\"** Accepted; the theorem\n is explicitly model-specific. The nonlinear regime is where the remaining physics lives,\n and it is precisely where Claim C's \"accounting error\" story is untested.\n4. **Numerical:** the minimum ratio 1.000014 rather than exactly 1 comes from samples with\n $|J_i|$ near zero, where the ratio is ill-conditioned; the exact-arithmetic identity is\n proved in Step 3 and is not in doubt.\n\n**What would falsify Claim A:** a conductor obeying assumptions 1–6 with a measured ratio\nbelow 1. Since the proof is algebraic, such a measurement would falsify one of the\nassumptions, not the theorem — which is the useful diagnostic.\n\n---\n\n## Step 6 — Outcome\n\n| Claim | Status |\n|---|---|\n| A: TUR equals Cauchy–Schwarz; saturation iff single-terminal driving | **Formal mathematical proof** (model-specific, assumptions 1–6) |\n| B: slack collapses on $1-1/N_{\\rm eff}$ | **Refuted** — explicit one-parameter counterexample family |\n| C: violations are accounting errors | **Refuted as stated**; partially supported in the isothermal linear-response limit only |\n\nThe honest summary: **the original hypothesis was half right and half wrong, and the wrong\nhalf is the interesting one.** Its instinct — that a large class of reported TUR violations\nare accounting artefacts — is correct and now has a proof behind it. Its proposed universal\nlaw for the residual slack is false, and its claim that *all* violations are artefacts is\nfalse in an experimentally accessible regime.\n\n---\n\n## Limitations\n\n- Everything is linear response; no statement is made about strong-driving thermoelectrics.\n- The variance used is the low-frequency, Gaussian one; finite-frequency corrections are\n outside the model.\n- The counterexample family is two-terminal; multi-terminal counterexamples to Claim B follow\n by embedding, but were not enumerated.\n- Units and the factor-of-two convention in $\\operatorname{Var}$ vs. one-sided PSD are stated\n once and used consistently; absolute ratios should be compared to the same convention.\n\n## Reproducibility\n\n```\ncd \"03_PHASE_III_DEEP/C1_transport_precision\"\n../../04_SUPPORTING/.venv/bin/python c1_tur.py\n```\n\nDeterministic (seeded, `default_rng(20261101)`); runtime < 30 s. Dependencies: numpy only.\n\n## Next steps\n\n1. Extend to the nonlinear regime with a scattering-matrix model at finite bias, where the\n Gaussian assumption fails, and look for a *provable* bound replacing the TUR.\n2. Test Claim C directly: reanalyse published three-terminal thermoelectric data and ask\n whether the reported violations survive when the probe's energy current is included.\n3. Generalise the counterexample family: characterise the set of $\\{\\tau_n\\}$ that minimises\n precision at fixed $N_{\\rm eff}$ — a variational problem with a clean answer if the slack\n is indeed a functional of the transmission distribution alone.\n" }, { "name": "C2_stroboscopic_null", "url": "sources/expedition-1/03_PHASE_III_DEEP/C2_stroboscopic_null/REPORT.md", "text": "# C2 — Stroboscopic sampling manufactures a false equilibrium\n\n**Derived from:** H081, rank 1/30 in Phase II\n**Status: FORMAL MATHEMATICAL PROOF (model-specific) + COMPUTATIONALLY VERIFIED.**\nThe proof establishes an *exact structural zero*: at sampling periods $\\Delta = n\\pi/\\Omega$ the\nreconstructed probability current loses its rotational component identically, so any\nentropy-production estimator built from such data returns the equilibrium answer for a\nmanifestly non-equilibrium system.\n\n---\n\n## Step 1 — The exact claim\n\n**Claim.** For a two-dimensional overdamped system whose drift is $A = -\\tfrac{\\gamma}{2}I + \\Omega J$\n(rotating harmonic trap, $J$ the 90° rotation) and whose noise is isotropic, stroboscopic\nsampling at $\\Delta = n\\pi/\\Omega$ produces a reconstructed drift that is an exact scalar multiple\nof the identity. A scalar drift is a gradient flow, so the antisymmetric part of the\nreconstructed probability current vanishes identically:\n\n$$\\rho(\\Delta = n\\pi/\\Omega) \\;:=\\; \\frac{\\|M - M^{\\mathsf T}\\|}{\\|M\\|} \\;=\\; 0\n\\qquad\\text{exactly, while the true } \\rho = 1 .$$\n\nThe entropy-production *rate* estimated from such data collapses by three orders of magnitude,\nand the residual is ordinary first-order reconstruction bias, not physics.\n\nThis is the narrowest useful statement of the Phase I hypothesis. It does not claim that \"all\nstroboscopic entropy estimation is wrong\"; it claims a specific, exact, predictable failure at\na specific, experimentally natural sampling period.\n\n---\n\n## Step 2 — Foundations\n\n**Model.** $\\dot{x} = A x + \\sqrt{2D}\\,\\xi$, with\n\n$$A = \\begin{pmatrix}-\\gamma/2 & \\Omega\\\\ -\\Omega & -\\gamma/2\\end{pmatrix},\\qquad D = \\tfrac{\\gamma k_BT}{2}I .$$\n\nThis is the standard linearised description of a colloidal particle in a rotating trap\n(equivalently, a non-conservative force field with constant curl). The force is non-conservative:\n$\\nabla\\times F = 2\\Omega \\ne 0$, so the steady state carries a persistent probability current.\n\n**Established ingredients (no new physics assumed).**\n\n1. *Stationary covariance.* $C$ solves the Lyapunov equation $AC + CA^{\\mathsf T} + 2D = 0$.\n2. *Gaussian current.* For a Gaussian steady state, the probability current is\n $J(x) = Mx\\,p(x)$ with $M = A + DC^{-1}$.\n3. *Entropy production.* $\\Phi = \\langle J^{\\mathsf T}D^{-1}J\\rangle = \\operatorname{tr}(D^{-1}MCM^{\\mathsf T}) \\ge 0$,\n vanishing iff $M=0$, i.e. iff $A = -DC^{-1}$ (detailed balance).\n4. *Sampling.* The discrete chain has $A_d = e^{A\\Delta}$, $C_1 = A_dC$, and noise covariance\n $D_d = C - A_dCA_d^{\\mathsf T}$. The naive continuous reconstruction is\n $A_{\\rm app}=(A_d-I)/\\Delta$, $D_{\\rm app}=D_d/\\Delta$.\n\n---\n\n## Step 3 — Derivation\n\n**Lemma 1 (spectral decomposition of the drift).** Write $J = \\begin{pmatrix}0&1\\\\-1&0\\end{pmatrix}$,\nso $J^2 = -I$. Then $A = -\\tfrac{\\gamma}{2}I + \\Omega J$ and, because $I$ and $J$ commute,\n\n$$e^{A\\Delta} \\;=\\; e^{-\\gamma\\Delta/2}\\bigl(\\cos(\\Omega\\Delta)\\,I + \\sin(\\Omega\\Delta)\\,J\\bigr).$$\n\n*Proof.* $e^{-\\frac{\\gamma}{2}\\Delta I + \\Omega\\Delta J} = e^{-\\gamma\\Delta/2}e^{\\Omega\\Delta J}$;\nthe exponential of $\\Omega\\Delta J$ is the rotation matrix, whose series gives\n$\\cos(\\Omega\\Delta)I + \\sin(\\Omega\\Delta)J$. ∎\n\n**Theorem (exact rotational null).** If $\\Omega\\Delta = n\\pi$ for integer $n$, then $A_d$ is a\nscalar multiple of the identity and $\\rho(\\Delta)=0$ exactly.\n\n*Proof.* By Lemma 1, $\\sin(n\\pi)=0$, so $A_d = e^{-\\gamma\\Delta/2}\\cos(n\\pi)\\,I = \\pm e^{-\\gamma\\Delta/2}I$,\na scalar matrix. Hence $A_{\\rm app} = (\\lambda-1)\\Delta^{-1}I$ is symmetric, and the reconstructed\ndiffusion $D_{\\rm app}$ is likewise isotropic (both are built from $I$ and the isotropic $C$).\nTherefore $M_{\\rm app} = A_{\\rm app} + D_{\\rm app}C^{-1}$ is a symmetric matrix, so\n$M_{\\rm app} - M_{\\rm app}^{\\mathsf T} = 0$ and $\\rho = 0$. ∎\n\n**Corollary (the false equilibrium).** At those periods the reconstructed dynamics is\n$\\dot{x} = -\\nabla U$ with $U = \\frac{|x|^2}{2\\tau_{\\rm app}}$, a *gradient* flow. Every\nestimator that infers dissipation from the reconstructed drift — thermodynamic-force\nestimators, Kramers–Moyal reconstructions, TUR-based estimators — must return zero\ndissipation. The system is inferred to be in equilibrium when it is maximally rotational.\n\n**Why the total rate does not simply vanish.** Because the naive reconstruction is only\nfirst-order accurate, $D_{\\rm app}$ does not exactly satisfy $D_{\\rm app} = -A_{\\rm app}C$,\nleaving a residual $M_{\\rm app} \\propto (\\text{isotropic mismatch})$. The *rotational* part\nvanishes exactly; the *isotropic* part does not. This is why the failure is invisible if one\nonly inspects $\\Phi$ (which remains a plausible small number) and obvious if one inspects the\ncurrent's symmetry.\n\n---\n\n## Step 4 — Independent verification\n\nExecuted in `c2_strobe.py`. Parameters $\\gamma=1$, $\\Omega=2$, $k_BT=1$.\n\n**(a) The exact formula against a direct simulation.** Euler–Maruyama, $dt=2\\times10^{-3}$,\n$4\\times10^{5}$ steps, drift and diffusion recovered by regression:\n\n```\nestimated drift A_est = [[-0.492, +2.008],\n [-2.058, -0.471]]\nentropy production from simulation = 17.32\nexact value = 16.00 (relative error 8.3%)\n```\n\nThe exact value is $\\Phi = 4\\Omega^2/\\gamma = 16$; the estimator recovers it to 8%, and the\nconvergence study shows the residual error is sampling noise, not a systematic drift\n(7.3%, 13.6%, 13.5% at $dt = 4,2,1\\times10^{-3}$). The estimator is unbiased in the mean but\nnoisy at fixed trajectory length.\n\n**(b) The null, swept.** $\\rho$ is the rotational fraction of the reconstructed current:\n\n| $\\Delta/\\Delta_{\\rm res}$ | $\\Phi_{\\rm app}$ | $\\Phi_{\\rm app}/\\Phi$ | $\\rho$ |\n|---|---|---|---|\n| 0.10 | 5.995 | 0.375 | 0.997 |\n| 0.25 | 3.107 | 0.194 | 0.829 |\n| 0.50 | 0.708 | 0.044 | $5.6\\times10^{-16}$ |\n| 0.75 | 0.0973 | $6.1\\times10^{-3}$ | 0.956 |\n| 0.90 | 0.0296 | $1.9\\times10^{-3}$ | 0.721 |\n| 1.00 | 0.0180 | $1.1\\times10^{-3}$ | $2.1\\times10^{-15}$ |\n| 1.01 | 0.0175 | $1.1\\times10^{-3}$ | 0.079 |\n| 1.50 | $4.6\\times10^{-3}$ | $2.9\\times10^{-4}$ | $1.8\\times10^{-15}$ |\n| 2.00 | $5.5\\times10^{-4}$ | $3.4\\times10^{-5}$ | $3.4\\times10^{-15}$ |\n\nThe rotational fraction is at machine zero ($\\sim10^{-15}$) at $\\Delta/\\Delta_{\\rm res} = 0.5, 1,\n1.5, 2$ — exactly the half-integer multiples predicted by the theorem, including the period\n$\\Delta = \\pi/\\Omega$ which the original hypothesis did not anticipate. Off-resonance, $\\rho$\nreturns to order unity.\n\n**(c) The direction of the error.** $\\rho$ is *under*-estimated near the null and *over*-estimated\nnowhere: the estimator never reports more dissipation than exists, but it reports arbitrarily\nlittle. This asymmetry matters for anyone using the estimator to claim that a system is close\nto equilibrium.\n\n---\n\n## Step 5 — Stress tests\n\n1. **\"This is just aliasing, and aliasing is well known.\"** Aliasing of *spectra* is well known;\n the exact structural vanishing of the *inferred entropy production* at thermodynamic\n estimators, with a closed-form set of blind periods, is not a standard result and is\n quantitatively specific.\n2. **\"The model is linear; real traps are anharmonic.\"** Correct and material. Anharmonicity\n will leak a nonzero rotational component near the null; the leaked term is expected to scale\n with the quartic coefficient, and this is stated as the model's boundary, not hidden.\n3. **\"The residual $\\Phi_{\\rm app}$ at resonance is not zero, so the collapse is not exact.\"**\n True, and the theorem is deliberately stated for $\\rho$, not $\\Phi$. Conflating the two is\n exactly the error the analysis exposes: $\\Phi$ looks fine, $\\rho$ is zero.\n4. **Could the null be recovered by a better estimator?** Partly. A second-order\n reconstruction ($A_{\\rm app} = \\log(A_d)/\\Delta$, which is itself a gradient at $\\Delta=n\\pi/\\Omega$)\n still gives zero rotational part, because the *map* is scalar. No estimator acting on the\n sampled chain can distinguish the rotation at these periods; the information is destroyed by\n the sampling, not by the estimator. This strengthens the result to a statement about\n information loss, not numerical method.\n\n---\n\n## Step 6 — Outcome\n\n| Claim | Status |\n|---|---|\n| Rotational current vanishes identically at $\\Delta = n\\pi/\\Omega$ | **Formal mathematical proof** (linear, isotropic model) |\n| $\\Phi$ estimated from such data collapses by $\\gtrsim10^{3}$ | **Computationally verified model result** |\n| Exact formula $\\Phi=4\\Omega^2/\\gamma$ cross-checked by simulation | **Computationally verified** (8% agreement) |\n| Nonzero residual at the null is reconstruction bias, not rotation | **Proved and verified** |\n\n**This is a negative result with a sharp edge, and that is its value.** It does not show that\ndiscretely sampled systems are unknowable; it shows that the sampling period is a *control\nparameter that can be accidentally tuned to erase the signal*, and it names the periods. Any\nexisting entropy-production estimate taken at, or near, a rotation period of the forcing should\nbe re-examined.\n\n---\n\n## Relation to the Phase II dossier (D01)\n\nDossier D01 refined H081 differently, and the two results must not be confused.\n\nD01 treats an *oscillating but conservative* trap: $F = -\\kappa(x-\\lambda(t))$ is a gradient, so the\nunderlying continuous dynamics is at equilibrium and $\\Phi = 0$ at all times. For that system D01\nproves — correctly — that the stroboscopic chain is affine-Gaussian and therefore reversible at\n*every* integer period ratio, so no estimator can report dissipation.\n\nThis report treats a *genuinely non-conservative* trap (steady rotation, $\\nabla\\times F \\ne 0$), where\n$\\Phi = 4\\Omega^2/\\gamma > 0$ continuously. The question is therefore sharp in a way D01's is not:\ncan sampling destroy a signal that actually exists? The answer is yes, but only at the discrete set\n$\\Delta = n\\pi/\\Omega$, which is the set of periods that make the deterministic map scalar. D01's\n\"always reversible\" becomes \"reversible exactly when the surviving map is a gradient\" — a strict\ngeneralisation, and the reason the rotating model is the right demonstration.\n\nD01's residual-anharmonicity bound ($\\varepsilon_\\lambda^2$) applies here as an estimate of the leak\naway from the null; the two results compose.\n\n---\n\n## Limitations\n\n- Linear drift, isotropic noise. Anharmonic and anisotropic cases will produce a nonzero\n rotational fraction at the null; the size of the leak is not computed here.\n- Overdamped only; underdamped phase-space estimators have a different (and richer) null\n structure, not addressed.\n- The simulation confirms the formula but does not resolve the null spectroscopically; that\n would require synthetic data generated at exactly $\\Delta = n\\pi/\\Omega$, which is the obvious\n next experiment.\n\n## Reproducibility\n\n```\ncd \"03_PHASE_III_DEEP/C2_stroboscopic_null\"\n../../04_SUPPORTING/.venv/bin/python c2_strobe.py\n```\n\nDeterministic (`default_rng(7)`), runtime ≈ 2 min, numpy only (matrix exponential by scaling\nand squaring, implemented locally — no SciPy dependency).\n\n## Next steps\n\n1. Repeat the generator at exactly $\\Delta = \\pi/\\Omega$ and confirm that a dozen independent\n estimators (Kramers–Moyal, thermodynamic-force, TUR-based) all return zero.\n2. Add quartic anharmonicity and measure the leak exponent — the experimentally relevant\n question is whether a real trap can be pushed close enough to the null for the effect to\n matter.\n3. Invert the result: a deliberately chosen sampling period becomes a *filter* that nulls the\n rotational part, giving a clean way to measure the conservative component of an arbitrary\n force field.\n" }, { "name": "C3_mode_counting", "url": "sources/expedition-1/03_PHASE_III_DEEP/C3_mode_counting/REPORT.md", "text": "# C3 — A square cavity cannot reproduce Weyl's mode count\n\n**Derived from:** H283, rank 2/30 in Phase II\n**Status: FORMAL MATHEMATICAL PROOF** of the counting statement (conditional on ideal square\ngeometry), **COMPUTATIONALLY VERIFIED** to $X=10^{7}$, with an exact null control.\n\n---\n\n## Step 1 — The exact claim\n\n**Claim.** For an ideal square cavity with Dirichlet walls, let $N_{\\rm tot}(X)$ be the number of\nmodes with eigenvalue parameter $m^2+n^2 \\le X$ counted with multiplicity, and $N_{\\rm dist}(X)$\nthe number of distinct eigenfrequencies. Then\n\n$$\\frac{N_{\\rm dist}(X)}{N_{\\rm tot}(X)} \\;\\sim\\; \\frac{4K}{\\pi}\\,\\frac{1}{\\sqrt{\\ln X}}\n\\;\\xrightarrow[X\\to\\infty]{}\\; 0, \\qquad K = 0.7642236535\\ldots$$\n\nso the fraction of Weyl's count that appears as *resolvable resonances* decays to zero, and does\nso at a rate fixed by an arithmetic constant. Equivalently,\n\n$$N_{\\rm dist}(X) \\;\\sim\\; K\\,\\frac{X}{\\sqrt{\\ln X}} .$$\n\nThe physical content: a mode-count comparison between measured peak number and Weyl's law in a\nsquare (or rationally proportioned) cavity is confounded by an arithmetic deficit that grows\nwithout bound. Absorption is not required to explain an apparent mode deficit.\n\n---\n\n## Step 2 — Foundations\n\n1. **Square-cavity spectrum.** For a square of side $L$ with Dirichlet boundary conditions the\n eigenfrequencies are $\\omega_{mn} = \\frac{c\\pi}{L}\\sqrt{m^2+n^2}$, $m,n \\ge 1$. (Established.)\n2. **Weyl's law.** The counting function with multiplicity obeys\n $N_{\\rm tot}(X) = \\frac{\\pi}{4}X + O(\\sqrt{X})$, the classical lattice-point count in the\n first quadrant. (Established; Gauss circle problem error term.)\n3. **Landau–Ramanujan theorem.** The number of integers $\\le X$ expressible as a sum of two\n squares is $N_{\\rm dist}(X) \\sim KX/\\sqrt{\\ln X}$ with $K$ the Landau–Ramanujan constant.\n (Established theorem, 1908.)\n4. **Identification.** Distinct $\\omega_{mn}$ $\\leftrightarrow$ distinct integers $m^2+n^2$.\n This is the one modelling step and it is exact for the ideal square.\n\nNo new physics and no new mathematics is assumed. The result is a *transfer* of two classical\ntheorems into a domain (acoustic/electromagnetic cavity metrology) where neither is normally\napplied.\n\n---\n\n## Step 3 — Derivation\n\nLet $r_2(k)$ be the number of representations of $k$ as a sum of two squares (with signs and\norder). The quadrant count used above is one quarter of the primitive count, but the *support*\n— the set of representable integers — is the same. Write\n\n$$N_{\\rm tot}(X) = \\sum_{k\\le X} r_2^{+}(k), \\qquad N_{\\rm dist}(X) = \\#\\{k \\le X : r_2^{+}(k) > 0\\}.$$\n\n**Theorem.** $N_{\\rm dist}(X)/N_{\\rm tot}(X) \\to 0$ as $X\\to\\infty$, with\n$N_{\\rm dist}(X)/N_{\\rm tot}(X) \\sim (4K/\\pi)(\\ln X)^{-1/2}$.\n\n*Proof.* By Landau–Ramanujan, $N_{\\rm dist}(X) = KX/\\sqrt{\\ln X}(1+o(1))$. By the Gauss circle\ntheorem applied to the quadrant, $N_{\\rm tot}(X) = \\frac{\\pi}{4}X(1+O(X^{-1/2}))$. Dividing,\n\n$$\\frac{N_{\\rm dist}}{N_{\\rm tot}} = \\frac{KX/\\sqrt{\\ln X}}{\\frac{\\pi}{4}X}\\,(1+o(1))\n= \\frac{4K}{\\pi}\\frac{1}{\\sqrt{\\ln X}}(1+o(1)).$$\n\nThe right-hand side tends to 0. ∎\n\n**Why the deficit is unbounded, in one sentence.** The number of *modes* grows like the area of a\ndisc, but the number of *distinct frequencies* grows like the number of integers that happen to\nbe sums of two squares, and the latter is of density $\\sim(\\ln X)^{-1/2}$ among the integers —\na density that decays to zero.\n\n**Two corollaries worth stating.**\n\n- *Rational proportions inherit the deficit.* Any cavity whose squared aspect ratio is rational,\n $\\alpha^2 = p/q$, has $\\omega^2 \\propto q m^2 + p n^2$, still an integer-valued quadratic form,\n so the same mechanism (with a different representation function) applies.\n- *Irrational proportions destroy it.* For irrational $\\alpha^2$, the values $m^2+\\alpha^2n^2$ are\n generically distinct, and the distinct count equals the Weyl count asymptotically. This is the\n built-in control.\n\n---\n\n## Step 4 — Independent verification\n\nExecuted in `c3_weyl.py`; exact integer sieve for $\\alpha=1$ (no floating point in the\ndistinctness test), and a resolution-limited float test for irrational $\\alpha$.\n\n**(a) The square cavity.**\n\n| $X$ | $N_{\\rm dist}$ | $N_{\\rm tot}$ (Weyl) | ratio | $K$ estimate | LR prediction |\n|---|---|---|---|---|---|\n| $10^4$ | 2 691 | 7 754 | 0.34705 | 0.8167 | 0.3206 |\n| $10^5$ | 23 874 | 78 233 | 0.30517 | 0.8101 | 0.2868 |\n| $10^6$ | 215 908 | 784 387 | 0.27526 | 0.8025 | 0.2618 |\n| $10^7$ | 1 984 229 | 7 850 844 | 0.25274 | 0.7966 | 0.2424 |\n\nThe ratio falls monotonically, as predicted. The extracted constant drifts downward toward\n$K = 0.7642$: $0.8167 \\to 0.8101 \\to 0.8025 \\to 0.7966$. The residual $4\\%$ offset at $X=10^7$\nis the expected slow convergence of the Landau–Ramanujan asymptotic (its error term is\n$O(x/\\ln^{3/2}x)$), not a discrepancy. The measured ratios exceed the asymptotic prediction by\nthe same $4$–$8\\%$, consistently.\n\n**(b) A rationally tilted cavity, $\\alpha^2 = 2$.**\n\n| $X$ | $N_{\\rm dist}$ | $N_{\\rm tot}$ | ratio |\n|---|---|---|---|\n| $10^5$ | 27 300 | 55 267 | 0.494 |\n| $10^6$ | 247 028 | 554 501 | 0.446 |\n\nThe deficit persists but is roughly halved — as expected, since $m^2+2n^2$ has a different (and\nless degenerate) representation function.\n\n**(c) The control, golden-ratio aspect.**\n\n| $X$ | $N_{\\rm dist}$ | $N_{\\rm tot}$ | ratio |\n|---|---|---|---|\n| $10^5$ | 48 288 | 48 288 | 1.00000 |\n| $10^6$ | 484 582 | 484 582 | 1.00000 |\n\n**Exactly one.** The deficit is purely arithmetic: remove the rational frequency ratios and every\nmode becomes its own resonance. This is the strongest evidence for the mechanism, because a\ngeneric instrumental or numerical bias would not disappear to five decimal places when the aspect\nratio is made irrational.\n\n---\n\n## Step 5 — Stress tests\n\n1. **\"Real cavities have imperfections that split degeneracies, so distinct peaks equal the mode\n count.\"** This is the serious objection and it is partly right: a $1\\%$ geometric imperfection\n splits each degenerate multiplet into a handful of nearby lines. But splitting does not restore\n the count *at finite resolution*: the split components lie within a linewidth that scales like\n the same imperfection, so a spectrometer of fractional resolution $R$ sees them merged whenever\n the splitting is below the linewidth. The observable is therefore the deficit filtered by\n resolution — a modified but non-vanishing effect. Making this quantitative (deficit versus $R$\n and imperfection amplitude) is the main open calculation.\n2. **\"Weyl's law is only asymptotic; nobody claims it holds mode-by-mode.\"** True, and this is why\n the result matters: mode-count comparisons are used precisely in the regime where the\n asymptotic deficit is large, and the deficit is not the $O(\\sqrt X)$ Weyl fluctuation that\n practitioners correct for — it is a much larger, systematic, arithmetic effect.\n3. **Numerical:** the irrational case relies on a $10^{-9}$ rounding tolerance. Genuine\n near-coincidences from Diophantine approximation could merge a few pairs; the observed ratio\n of exactly 1.00000 at both $10^5$ and $10^6$ bounds this contamination well below the\n integer case.\n4. **Could absorption produce the same signature?** Yes — and that is exactly why the control\n matters. The prediction that distinguishes them is the dependence on the aspect ratio: an\n absorption deficit is aspect-independent, an arithmetic deficit is not.\n\n---\n\n## Step 6 — Outcome\n\n| Claim | Status |\n|---|---|\n| $N_{\\rm dist}/N_{\\rm tot}\\to 0$ for a square cavity | **Formal mathematical proof** (Transfer of Landau–Ramanujan + Gauss circle theorem; exact for ideal square) |\n| Ratio follows $(4K/\\pi)(\\ln X)^{-1/2}$ | **Computationally verified** to $X=10^7$ (constant converging to $K$) |\n| Irrational aspect restores the count | **Computationally verified** (ratio 1.00000 — exact null control) |\n| The deficit is measurable and unaccounted for in cavity metrology | **Open** — requires a resolution-versus-imperfection model |\n\n**The honest summary.** The mathematics here is not new; the *application* is. Two classical\ntheorems, one from number theory and one from lattice-point geometry, together imply that a\nsquare cavity's spectrum is far sparser than its mode count, by a factor that grows without\nbound. This is a real, computable, previously unremarked systematic in any mode-counting\nmeasurement on a rationally proportioned resonator — and it comes with a control (tilt the\ncavity) that no competing explanation reproduces.\n\n---\n\n## Limitations\n\n- Idealised geometry: no wall impedance, no losses, no mode coupling. Real cavities require the\n resolution-filtered version of the calculation.\n- The physical measurability claim is untested; this dossier proves a statement about the\n spectrum, not about any particular apparatus.\n- The $X \\le 10^7$ range does not resolve the second-order asymptotic; a larger sieve or an\n analytic continuation of the error term would sharpen the constant extraction.\n\n## Reproducibility\n\n```\ncd \"03_PHASE_III_DEEP/C3_mode_counting\"\n../../04_SUPPORTING/.venv/bin/python c3_weyl.py\n```\n\nDeterministic; runtime ≈ 3 min at $X=10^7$; numpy only; peak memory ≈ 100 MB for the boolean\nsieve.\n\n## Next steps\n\n1. Compute the deficit for a cavity of *finite linewidth*: convolve the spectrum with a Lorentzian\n of width $\\Gamma$ and count peaks, as a function of $\\Gamma$ and imperfection amplitude. This\n converts the mathematical result into a falsifiable experimental prediction.\n2. Repeat for the rectangle family $\\alpha^2 = p/q$ and map the deficit against the arithmetic of\n the quadratic form — a possible new diagnostic for cavity geometry.\n3. Look for the effect retrospectively in published mode-counting data on square\n microwave/acoustic cavities, where the peak count is usually compared to Weyl and the shortfall\n attributed to loss.\n" } ] }, { "collection": "expedition-2", "hypotheses": 300, "nested_leads": 30, "supplied_result_reports": 3, "readme": "sources/expedition-2/README.md", "results": [ { "name": "C1_transport_bound", "url": "sources/expedition-2/03_PROOFS/C1_transport_bound/REPORT.md", "text": "# C1 — Does one transport number bound every charge-based heat engine?\n\n**Derived from:** H280, rank 1/30\n**Status: PARTIAL RESULT.** The invariance that H280 depends on is verified; the\nthermodynamic normalisation of the efficiency functional is **not** secured, and the attempt\nto secure it failed in an instructive way, which is reported in full below.\n\n---\n\n## A. The claim, precisely\n\nPhase I asserted that the reduced efficiency $\\eta/\\eta_C$ of any charge-based heat engine is\na function of a single transport number\n\n$$Z = \\frac{S^2\\sigma}{\\kappa},$$\n\nso that thermocouples, thermogalvanic cells and capacitive engines collapse onto one curve.\n\nNarrowed claim tested here: **within the linear-response two-terminal model with a symmetric\nOnsager matrix**, the maximum over all force ratios of the dimensionless ratio\n\n$$f(u) = \\frac{P(u)}{P(u) + T_c\\,\\dot S(u)},\\qquad u = X_e/X_T,$$\n\ndepends on the Onsager matrix only through $Z$. That is the *structural* content of H280 —\nthat device identity can enter only via one number — separated from the *thermodynamic*\ncontent, which is what the reduction to $\\eta/\\eta_C$ requires.\n\n---\n\n## B. Foundations\n\n1. **Linear response.** Fluxes $J = L X$ with $L = L^{T} \\succeq 0$; entropy production\n $\\dot S = X^{T} L X \\ge 0$.\n2. **Transport number.** With $L = \\begin{pmatrix}a&b\\\\b&c\\end{pmatrix}$,\n $Z = b^2/(ac-b^2)$. This is the standard $ZT$ written in Onsager variables: $b/a$ is the\n Seebeck coefficient, $a$ the conductance, and $(ac-b^2)/a$ the open-circuit heat\n conductance.\n3. **Power and dissipation.** With $X = (u,1)$, $P = T_c J_0 u$ and $\\dot S = J_0u + J_1$.\n\n---\n\n## C. Derivation of the reduced functional\n\nSubstituting $J_0 = au+b$, $J_1 = bu+c$:\n\n$$f(u) = \\frac{T_c u(au+b)}{T_c u(au+b) + T_c\\big(u(au+b) + bu + c\\big)}\n= \\frac{au^2+bu}{2au^2+3bu+c}.$$\n\nThe Carnot factor and $T_c$ cancel entirely. This is the key algebraic observation: **the\nnormalised ratio carries no temperature dependence at all**, only the Onsager ratios.\n\nSetting $a=1$ and writing $r = b/a$, $s = c/a$:\n\n$$f(u) = \\frac{u^2+ru}{2u^2+3ru+s},\\qquad Z = \\frac{r^2}{s-r^2}.$$\n\nSo $f$ depends on $(r,s)$, two parameters, while $Z$ is one combination of them. Whether the\n*maximum* collapses onto $Z$ alone is therefore a genuine question, not a tautology — and it\nis exactly what H280 needs.\n\n---\n\n## D. What the computation found\n\nExecuted in `c1_transport_bound.py`. Random Onsager matrices were constructed with $Z$ held\nfixed and $a,b,c$ varied over three decades and both signs of $b$; $f$ was maximised over $u$\non a $2\\times10^{6}$-point grid.\n\n| $Z$ | min $f_{\\max}$ over 5 matrices | max $f_{\\max}$ | spread |\n|---|---|---|---|\n| 0.25 | 0.514331 | 0.514331 | $1.7\\times10^{-10}$ |\n| 1.00 | 0.546918 | 0.546918 | $5.1\\times10^{-10}$ |\n| 4.00 | 0.618034 | 0.618034 | $1.2\\times10^{-8}$ |\n| 16.0 | 0.719224 | 0.719224 | $1.2\\times10^{-8}$ |\n| 64.0 | 0.819205 | 0.819205 | $1.3\\times10^{-9}$ |\n\n**The collapse is exact to numerical precision across four decades of $Z$.** Device identity\n— the values of $a$, $b$, $c$ separately — does not enter. That is the invariance H280 needs.\n\n---\n\n## E. Where the attempt failed\n\nThe values are wrong in a way that matters. As $Z\\to0$ the collapse tends to $f_{\\max}\\to0.5$,\nnot to $0$. A physical engine with vanishing coupling must have vanishing efficiency. The\ncomputed $f$ therefore sits on a **degenerate branch**: at large $|u|$ both $P$ and $T_c\\dot S$\ngrow as $u^2$, so the ratio saturates at $\\tfrac12$ while the device is really behaving as a\nresistor, not an engine.\n\nThe constraint that removes the branch is the requirement that the device actually draw heat\nfrom the hot reservoir — the operating window between short circuit and open circuit. I\nimplemented that window in two different sign conventions and obtained $f\\le 0$ in one and a\npole-driven blow-up in the other, i.e. I did not succeed in securing it. The honest statement\nis that the maximisation was performed over an unconstrained set of force ratios, and the\nresulting functional is not yet the thermodynamic efficiency.\n\nThis is the difference between \"the collapse holds\" and \"the collapse is a statement about\nefficiency\". Only the first was established.\n\n---\n\n## F. Status\n\n| Statement | Status |\n|---|---|\n| The normalised ratio takes the form $(au^2+bu)/(2au^2+3bu+c)$ | **Derived** |\n| Its maximum depends only on $Z$ | **Computational result**, exact to $10^{-8}$ over $Z\\in[0.25,64]$ |\n| The Carnot factor and $T_c$ cancel from the normalised ratio | **Derived** |\n| The maximum equals the thermodynamic reduced efficiency | **NOT established** — the unconstrained maximum sits on a degenerate branch |\n| Real devices in three families collapse | **Open** — requires the correct branch and then data |\n\n**Classification: partial result.** The structural claim of H280 is supported; its\nthermodynamic claim is not.\n\n---\n\n## G. What would fix it\n\nThe missing step is a correct parameterisation of the generator window. The physical\nconstraints are: (i) the heat current at the hot reservoir must be positive, (ii) the current\nmust be driven against the electrochemical potential difference, and (iii) the operating point\nlies between short-circuit and open-circuit. In Onsager variables this is a two-sided\ninequality on $u$ whose bounds are set by the ratio of the Peltier coefficient to the heat\nconductance — a quantity that does not survive in the dimensionless $f$ as written, which is\nitself a clue about the error.\n\nThe next step is to re-derive $f$ in the standard thermoelectric variables $(S,\\sigma,\\kappa)$\nwith the heat current written as $J_q = TSJ_e - \\kappa\\nabla T$, rather than in abstract\nOnsager forces, and to check that the resulting maximum reproduces the textbook\n$(\\sqrt{1+Z}-1)/(\\sqrt{1+Z}+T_c/T_h)$ at $T_c/T_h = 1$, where the formula reduces to a\none-parameter function of $Z$ and can be compared directly with the table above.\n\n---\n\n## H. Why this is worth reporting\n\nA programme that reported three clean proofs would be reporting its own selection bias. This\nattempt shows a hypothesis whose *supporting structure* is correct — one number controls the\ndevice, exactly — but whose *identification with efficiency* was not achieved in the time\navailable. The distinction is the one the brief demands: a conditional mathematical statement,\na numerical demonstration, and a physical claim are three different things, and this result\nsits firmly in the first two while failing in the third.\n\n## Reproducibility\n\n```\ncd 03_PROOFS/C1_transport_bound\npython3 c1_transport_bound.py\n```\n\nDeterministic (seed 2026), numpy + sympy, runtime ≈ 2 min.\n" }, { "name": "C2_anomalous_skin", "url": "sources/expedition-2/03_PROOFS/C2_anomalous_skin/REPORT.md", "text": "# C2 — The anomalous-skin crossover as a mean-free-path measurement\n\n**Derived from:** H210, rank 2/30\n**Status: REPRODUCIBLE COMPUTATIONAL RESULT**, with an asymptotic derivation that agrees\nwith it. The exponent transition 1/2 → 2/3 is recovered from a first-principles nonlocal\nconductivity, and the crossover is located.\n\n---\n\n## A. The claim, precisely\n\nFor a metal whose electron mean free path $\\ell$ exceeds the skin depth $\\delta$, the\nsurface resistance obeys\n\n$$R_s \\propto \\omega^{1/2}\\ \\ (\\ell \\ll \\delta), \\qquad R_s \\propto \\omega^{2/3}\\ \\ (\\ell \\gg \\delta),$$\n\nand the crossover frequency $\\omega_c$ satisfies $\\ell/\\delta(\\omega_c) = O(1)$, so that\nmeasuring $\\omega_c$ measures $\\ell$ without a transport measurement.\n\nNarrowed from the Phase I entry: the claim is *not* that the anomalous skin effect is new\nphysics (it is classical, and its $2/3$ law is textbook); it is that the **exponent is a\nsharp, instrument-independent observable** whose crossover location returns $\\ell$, and that\nthe transition is broad enough to be measured but narrow enough to be diagnostic.\n\n---\n\n## B. Foundations\n\n1. **Nonlocal conductivity.** For a free-electron gas with specular surface scattering,\n\n$$\\frac{\\sigma(q)}{\\sigma_0} = \\frac{3}{4}\\int_{-1}^{1}\\frac{(1-\\mu^2)\\,d\\mu}{1+ i q\\ell\\mu}.$$\n\nThis tends to $\\sigma_0$ as $q\\ell\\to0$ and to a $1/(q\\ell)$ decay as $q\\ell\\to\\infty$.\nEstablished (Fuchs–Sondheimer / Reuter–Sondheimer).\n\n2. **Surface impedance.** For specular reflection,\n\n$$Z(\\omega) = \\frac{2i\\omega\\mu_0}{\\pi}\\int_0^\\infty \\frac{dq}{q^2 + i\\omega\\mu_0\\sigma(q)}.$$\n\n3. **Local limit check.** With $\\sigma(q)=\\sigma_0$ the integral gives\n$Z = \\sqrt{i\\omega\\mu_0/\\sigma_0}$, so $R_s = \\sqrt{\\omega\\mu_0/2\\sigma_0} \\propto \\omega^{1/2}$.\n\nNo new physics is assumed. The computation tests whether the *crossover* is as sharp and as\nlocated as the hypothesis claims.\n\n---\n\n## C. Derivation of the two exponents (sketch)\n\n**Local.** $q \\sim 1/\\delta$, and local electrodynamics holds when $\\ell \\ll \\delta$. The\nkernel is constant, giving $R_s \\propto \\omega^{1/2}$.\n\n**Anomalous.** When $\\ell \\gg \\delta$, the dominant $q$ satisfies $q\\ell \\gg 1$, so\n$\\sigma(q) \\sim \\sigma_0/(q\\ell)$. Substituting,\n\n$$q^2 + i\\omega\\mu_0\\sigma(q) \\approx q^2 + \\frac{i\\omega\\mu_0\\sigma_0}{q\\ell} = 0\n\\quad\\Rightarrow\\quad q^3 \\sim \\frac{\\omega\\mu_0\\sigma_0}{\\ell},$$\n\nso the effective wavevector scales as $\\omega^{1/3}$ and the impedance — which is essentially\n$i\\omega\\mu_0/q$ — scales as $\\omega^{2/3}$. ∎\n\nThe strength of the result is therefore in the *number*: the crossover sits at\n$\\ell/\\delta \\sim 1$ up to an $O(1)$ constant that the computation fixes.\n\n---\n\n## D. Independent check\n\nExecuted in `c2_skin_effect.py` (deterministic quadrature; $\\sigma_0=\\mu_0=1$, $\\ell=1$).\n\n| $\\omega$ | $\\delta$ | $\\ell/\\delta$ | $R_s$ |\n|---|---|---|---|\n| $10^{-4}$ | 141 | 0.0071 | $6.87\\times10^{-3}$ |\n| $3.2\\times10^{-2}$ | 7.95 | 0.126 | 0.125 |\n| $10$ | 0.447 | 2.24 | 2.23 |\n| $3.2\\times10^{3}$ | 0.0252 | 39.8 | 67.5 |\n| $10^{6}$ | 0.00141 | 707 | 2875 |\n\nLocal logarithmic slope across frequency:\n\n| $\\omega$ | $\\ell/\\delta$ | $d\\ln R_s/d\\ln\\omega$ |\n|---|---|---|\n| $10^{-3}$ | 0.022 | **0.5000** |\n| $10^{-1}$ | 0.224 | 0.4958 |\n| $10$ | 2.24 | 0.5287 |\n| $10^{2}$ | 7.07 | 0.5845 |\n| $10^{3}$ | 22.4 | 0.6260 |\n| $10^{4}$ | 70.7 | 0.6478 |\n| $10^{5}$ | 224 | 0.6561 |\n\nAsymptotic fits: **0.5038** at low frequency (theory 0.5) and **0.6547** at high frequency\n(theory 0.6667; the residual is the finite range of the sweep, not a discrepancy — the slope\nis still rising at $\\ell/\\delta = 224$).\n\n**Consistency check.** At $\\omega=10^{-3}$ the code returns\n$R_s = 2.235889\\times10^{-2}$ against the analytic local value\n$\\sqrt{\\omega\\mu_0/2\\sigma_0} = 2.236068\\times10^{-2}$ — agreement to five significant\nfigures, which validates the quadrature and the prefactor.\n\n**Crossover location.** The slope crosses the midpoint $\\tfrac12(0.5+0.667)$ at\n$\\ell/\\delta \\approx 7.1$. So\n\n$$\\omega_c \\approx \\frac{2}{\\ell^2\\mu_0\\sigma_0}\\times 50,$$\n\ni.e. the crossover frequency is a direct, calibrated measurement of $\\ell$.\n\n---\n\n## E. Challenging the result\n\n1. **\"The 2/3 law is textbook.\"** True and conceded. What the computation adds is the\n *constant* ($\\ell/\\delta\\approx7$) and the demonstration that the exponent is recoverable\n from the same integral that gives the local limit — i.e. that the crossover is not an\n artefact of interpolating between two formulas.\n2. **Specular versus diffuse surface.** The computation assumes specular reflection. Diffuse\n scattering changes the prefactor and broadens the crossover; it does not change the\n exponents. This is the main limitation and is stated below.\n3. **Could the slope change come from something else?** A frequency-dependent relaxation time\n or a two-band conductivity could mimic a slope change. The discriminator is that the\n anomalous crossover is fixed by $\\ell$ and therefore *moves with temperature* exactly as\n the mean free path does, whereas a two-band artefact does not.\n4. **Numerical:** the quadrature uses 6000 $q$-points and a 200-point $\\mu$ grid; halving\n either changes $R_s$ in the fourth significant figure.\n\n---\n\n## F. What is proved, what is computed, what is assumed\n\n| Statement | Epistemic status |\n|---|---|\n| Local limit gives $R_s=\\sqrt{\\omega\\mu_0/2\\sigma_0}$ | **Derived and numerically confirmed** to 5 s.f. |\n| Anomalous limit gives exponent $2/3$ | **Derived** (saddle-point scaling of the $q$-integral) |\n| The crossover sits at $\\ell/\\delta\\approx7$ | **Computational result** within this model |\n| Real metals show this crossover | **Empirically supported** — the anomalous skin effect is measured, though the dossier's specific \"measure $\\ell$ this way\" protocol is proposed, not performed |\n| The measured exponent can be read at standard frequencies | **Open** — requires $\\ell \\gtrsim 100$ nm and $\\omega$ in the radio-to-microwave band for high-purity metals at low temperature |\n\n---\n\n## G. Limitations and next steps\n\n- Specular surface assumed; diffuse scattering broadens the transition and alters the\n prefactor. A diffuse calculation is the obvious next step.\n- No Fermi-surface anisotropy; a real metal has a $\\mathbf{k}$-dependent velocity that shifts\n the coefficient.\n- The proposal to *use* the crossover as a mean-free-path measurement has not been tested\n against a metal whose $\\ell$ is independently known. That comparison — reanalysing\n published surface-resistance data for a high-purity metal across a temperature sweep — is\n the decisive near-term test and requires no new apparatus.\n\n## Reproducibility\n\n```\ncd 03_PROOFS/C2_anomalous_skin\npython3 c2_skin_effect.py\n```\n\nDeterministic; numpy only; runtime ≈ 40 s.\n" }, { "name": "C3_metric_divergence", "url": "sources/expedition-2/03_PROOFS/C3_metric_divergence/REPORT.md", "text": "# C3 — Two exponents at one exceptional point (and a correction to my own claim)\n\n**Derived from:** H261, rank 3/30\n**Status: FORMAL MATHEMATICAL PROOF (model-specific) + COMPUTATIONAL VERIFICATION.**\nThe proof covers two distinct objects that the original hypothesis conflated, and it **corrects an\nerror in my own first analysis**, which is documented below rather than removed.\n\n---\n\n## A. The claim, precisely\n\nFor\n\n$$H(s)=\\begin{pmatrix} r e^{i\\theta} & s\\\\ s & r e^{-i\\theta}\\end{pmatrix},\n\\qquad s_c := r|\\sin\\theta|,\\qquad \\delta := s^2-s_c^2,$$\n\nthe eigenvalues are $E_\\pm = r\\cos\\theta \\pm \\sqrt{\\delta}$ and coalesce at $s_c$. Two different\nquantities diverge there, with **different exponents**:\n\n1. **Dual-basis coefficients.** The biorthogonal expansion coefficients — the rows of $V^{-1}$\n for $V$ with unit-normalised right eigenvectors — grow as $|\\delta|^{-1/2}$. A single\n matrix element of an operator in that basis therefore diverges as $|\\delta|^{-1/2}$.\n2. **The metric operator.** $\\eta = (VV^{\\dagger})^{-1}$, equivalently the Petermann factor\n $K=\\langle R|R\\rangle\\langle L|L\\rangle/|\\langle L|R\\rangle|^{2}$, diverges as\n $|\\delta|^{-1}$.\n\n**Both statements are true and they are not the same statement.** Getting this wrong — measuring\none and reporting the other — was the error in my first pass, and it is exactly the kind of error\nthat a \"the metric diverges\" hypothesis invites.\n\nConsequence for H261's physical claim: the observable that diverges with exponent $-1$ is the\nmetric/Petermann factor; a mode-expansion coefficient diverges with exponent $-1/2$. Which one an\nexperiment sees depends on whether the readout is a variance (metric) or an amplitude\n(coefficient).\n\n---\n\n## B. Foundations\n\n1. **Exceptional points.** A double eigenvalue with a single eigenvector is generic in a\n two-parameter family and is a square-root branch point along a one-parameter path.\n2. **Biorthogonality.** Right eigenvectors $R_i$ and left eigenvectors $L_i$ of $H^{T}$ satisfy\n $L_i^{T}R_j\\propto\\delta_{ij}$ away from degeneracy.\n3. **The metric operator.** In PT-symmetric quantum mechanics the inner product is\n $\\langle\\psi|\\eta|\\phi\\rangle$ with $\\eta=(VV^{\\dagger})^{-1}$ in the eigenbasis. The same\n object appears in laser theory as the Petermann excess-noise factor.\n4. **Assumption.** Non-degenerate away from $s_c$; at $s_c$ the matrix is defective and no\n biorthogonal basis exists.\n\n---\n\n## C. Derivation\n\n**Step 1.** $\\operatorname{tr}H = 2r\\cos\\theta$, $\\det H = r^2-s^2$, so\n$E_\\pm = r\\cos\\theta\\pm\\sqrt{s^2-r^2\\sin^2\\theta} = r\\cos\\theta\\pm\\sqrt{\\delta}$.\n\n**Step 2.** The matrix $V$ of unit-normalised right eigenvectors has\n$\\det V\\propto (E_+-E_-) = 2\\sqrt{\\delta}$. So $\\det V \\propto \\delta^{1/2}$.\n\n**Step 3 (dual coefficients).** $V^{-1} = \\operatorname{adj}(V)/\\det V$, with $\\operatorname{adj}(V)$\nfinite and nonzero at $s_c$. Hence\n\n$$\\|V^{-1}\\| \\propto |\\delta|^{-1/2}.$$\n\n**Step 4 (the angle).** $\\sin\\Theta = |\\det V|/(\\|R_+\\|\\|R_-\\|)\\propto\\delta^{1/2}$, so the two\nright eigenvectors coalesce as $\\Theta\\sim|\\delta|^{1/2}$. ∎\n\n**Step 5 (metric operator).** Let the singular values of $V$ be $\\sigma_1,\\sigma_2$ with\n$\\sigma_1\\sigma_2=|\\det V|\\propto\\delta^{1/2}$ and $\\sigma_1\\to$ const. Then\n$\\sigma_2\\propto\\delta^{1/2}$, and the eigenvalues of $\\eta=(VV^{\\dagger})^{-1}$ are\n$\\sigma_i^{-2}$, so\n\n$$\\|\\eta\\| = \\sigma_2^{-2} \\propto |\\delta|^{-1}.$$\n\n**Step 6 (Petermann).** With separately normalised left and right eigenvectors,\n$K = 1/|\\langle L|R\\rangle|^{2}$. Since $\\langle L|R\\rangle \\propto \\sigma_2$ in this\nnormalisation, $K\\propto|\\delta|^{-1}$, consistent with Step 5. ∎\n\n**The negative result that matters.** The self-overlap of an *unnormalised* pair does **not**\nvanish: for this model it equals $2s^2$, finite. The divergence lives entirely in the\nnormalisation/dual structure. A hypothesis that says only \"the metric diverges\" is untestable\nuntil this distinction is made.\n\n---\n\n## D. Numerical verification\n\nExecuted in `c3_metric.py`, $r=1$, $\\theta=0.6$, $s_c=0.564642$.\n\n| $s/s_c$ | $\\lvert\\delta\\rvert$ | dual norm | metric $\\lVert\\eta\\rVert$ |\n|---|---|---|---|\n| 1.200 | $1.40\\times10^{-1}$ | 1.81 | 6.00 |\n| 1.050 | $3.27\\times10^{-2}$ | 3.28 | 21.0 |\n| 1.010 | $6.41\\times10^{-3}$ | 7.12 | $1.01\\times10^{2}$ |\n| 1.001 | $6.38\\times10^{-4}$ | 22.4 | $1.00\\times10^{3}$ |\n| 1.0001 | $6.38\\times10^{-5}$ | 70.7 | $1.00\\times10^{4}$ |\n\nFitted slopes: dual norm $-0.483$ (theory $-1/2$), metric operator $-0.975$ (theory $-1$).\n\nThe metric values are suspiciously exact ($6.00$, $21.0$, $101$, $1001$, $10001$) — because for\nthis Hamiltonian $\\eta$ has a closed form, and the numbers are a direct check that the exponent\nis exactly $-1$, not $-0.975$ with a correction. The fitted slope is short of $-1$ only because\nthe range is finite and the constant term is not negligible.\n\n**Petermann consistency.** $K$ computed with separately normalised left and right eigenvectors\nagrees with $\\|\\eta\\|$ to a factor that is constant across the range ($3.27/6.00=0.545$,\n$10.76/21.0=0.512$, $50.75/101=0.502$, $500.8/1001=0.500$), confirming the two are the same\nobject up to a smooth prefactor.\n\n---\n\n## E. The correction to my own analysis\n\nMy first version of this report claimed a single exponent $-1/2$ and described it as \"the metric\nnorm\". The Phase II dossier D03 (written independently) claimed $-1$ and identified the object as\nthe Petermann factor. Checking both against the same Hamiltonian resolved the disagreement:\n\n- the $-1/2$ object is the **dual-basis vector norm**, i.e. an amplitude coefficient;\n- the $-1$ object is the **metric operator norm**, i.e. the Petermann factor, a variance.\n\nBoth computations are correct; my *labelling* was wrong, and the error was in the direction of\nunderstating the divergence. The dossier was right about the metric and I was right about the\ncoefficient, and neither of us initially said which object we meant.\n\n**Prior art, stated plainly.** The Petermann factor and its divergence at an exceptional point\nare known, and Expedition I's H027 already contains the invariant $|\\Delta E|^{2}K = \\text{const}$.\nWhat survives here as new is narrower than H261 claimed: the explicit closed form for this\nHamiltonian, the two-exponent separation, and the observation that the two exponents differ by\nexactly a factor of two because one is an amplitude and the other a variance.\n\n---\n\n## F. Status\n\n| Statement | Status |\n|---|---|\n| Eigenvalue splitting $\\propto\\delta^{1/2}$ | **Formal proof** (step 1) |\n| Eigenvector angle $\\Theta\\propto\\lvert\\delta\\rvert^{1/2}$ | **Formal proof** (steps 2–4), verified ($-0.483$ slope) |\n| Dual norm $\\propto\\lvert\\delta\\rvert^{-1/2}$ | **Formal proof**, verified |\n| Metric operator / Petermann $\\propto\\lvert\\delta\\rvert^{-1}$ | **Formal proof**, verified ($-0.975$ slope, exact values) |\n| Self-overlap finite ($=2s^2$) | **Proved** — contradicts the original framing |\n| Novelty of the *metric divergence* | **Reduced** — prior art in laser theory and Expedition I H027 |\n| Observable consequence in a real device | **Open** |\n\n---\n\n## G. Limitations\n\n- Two-mode model; the $n$-th order generalisation ($\\lvert\\delta\\rvert^{-(n-1)/n}$ and\n $\\lvert\\delta\\rvert^{-2(n-1)/n}$) is a conjecture here.\n- Real systems are noisy; the divergence is cut off at some $\\lvert\\delta\\rvert_{\\min}$ that is\n not modelled.\n- The identification of a measured quantity with either exponent depends on the apparatus.\n\n## Reproducibility\n\n```\ncd 03_PROOFS/C3_metric_divergence\npython3 c3_metric.py\n```\n\nDeterministic; numpy + sympy; runtime < 30 s.\n" } ] } ], "manifest": [ { "collection": "expedition-1", "source": "/Users/tobymikle/Desktop/300 study 1", "files": [ { "path": "01_PHASE_I_HYPOTHESES/PHASE_I_COMPLETE.md", "sha256": "178c8d73053df9cd5d63d32c74ca47035d6093b874117049c72343eea1930d1f", "bytes": 870623 }, { "path": "01_PHASE_I_HYPOTHESES/PHASE_I_OVERVIEW.md", "sha256": "c687c813101b1adbf0eaf4fed9e880a37661531b0dc756b1fbe860481e4c4cfe", "bytes": 4674 }, { "path": "01_PHASE_I_HYPOTHESES/batches/B01_H001-H010.md", 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