Two discovery expeditions

Supplied mathematical/computational report; publication packaging did not reproduce the calculation. Read the report for assumptions, refutations and incomplete claims.

# C1 — Transport precision in multi-terminal conductors

**Derived from:** H274, rank 29/30 in Phase II
**Status: PARTIAL RESULT** — a formal proof of a corrected statement, plus a *refutation*
of one of the original hypothesis's two claims, plus a computational counterexample that
overturns the hypothesis's central interpretation.

---

## Step 1 — The exact claim

Phase I asserted two things. They are separated here because they have different fates.

**Claim A (survives, as a theorem).** In isothermal linear response, for any terminal *i* of
an *N*-terminal conductor,

$$\frac{\operatorname{Var}(J_i)\,\sigma}{\langle J_i\rangle^{2}}\;=\;2k_B\,
\frac{(e_i^{\mathsf T} L\, e_i)(F^{\mathsf T} L F)}{(e_i^{\mathsf T} L F)^{2}}\;\ge\;2k_B,$$

with equality **iff** $F \parallel e_i$.

**Claim B (refuted).** "The residual slack collapses on $1-1/N_{\rm eff}$."

**Claim C (refuted as stated).** "Reported violations of the TUR are a probe-terminal
heat-accounting error." The correct statement is narrower: accounting errors *can* produce
apparent violations, but there exists a physically ordinary regime — sub-Poissonian thermal
noise at low bias — in which the TUR fails genuinely, with no accounting error anywhere.

---

## Step 2 — Foundations

Assumptions, stated so that they can be attacked individually:

1. **Linear response.** Currents are linear in forces: $J = L F$.
2. **Isothermal, single carrier type.** All terminals share one temperature; entropy
   production is $\sigma = F^{\mathsf T} L F$ (Joule heating only).
3. **Onsager reciprocity.** $L = L^{\mathsf T}$ (no magnetic field, no broken time-reversal
   symmetry).
4. **Passivity.** $L \succeq 0$ — the second law, nothing more.
5. **Current conservation.** $\sum_\alpha J_\alpha = 0$, i.e. $L \mathbf{1} = 0$.
6. **Gaussian, Markovian noise.** $\operatorname{Var}(J_i) = 2k_B (L)_{ii}$ in the
   low-frequency limit.

Assumptions 1–6 are the standard setting of mesoscopic linear-response transport. They are
*not* claimed to describe every experiment; they define the model in which Claim A is exact.

---

## Step 3 — Derivation

**Theorem (model-specific).** Under assumptions 1–6, Claim A holds, and it is not an
independent physical law but the Cauchy–Schwarz inequality.

*Proof.* Let $\langle u,v\rangle_L := u^{\mathsf T} L v$. Because $L \succeq 0$ (assumption 4),
this is a positive semidefinite bilinear form, hence an inner product on the quotient by its
null space. Take $u = e_i$ and $v = F$. Cauchy–Schwarz gives

$$\langle e_i, F\rangle_L^{2} \;\le\; \langle e_i,e_i\rangle_L \;\langle F,F\rangle_L ,$$

that is,

$$(e_i^{\mathsf T} L F)^{2} \;\le\; (e_i^{\mathsf T} L e_i)(F^{\mathsf T} L F).$$

Divide by $(e_i^{\mathsf T} L F)^2$ and identify $J_i = e_i^{\mathsf T} L F$, $\sigma = F^{\mathsf T} L F$:

$$\frac{(L_{ii})\,\sigma}{J_i^{2}} \;\ge\; 1 .$$

Multiplying by $2k_B$ and using $\operatorname{Var}(J_i) = 2k_B L_{ii}$ gives Claim A. Equality
in Cauchy–Schwarz holds iff $e_i$ and $F$ are linearly dependent, i.e. $F \parallel e_i$. ∎

Three consequences follow immediately, and they are the actual content of the result:

- **The TUR is dimensionally and algebraically trivial in this model.** It carries no
  information about the conductor beyond $L \succeq 0$. Any "violation" in a system obeying
  assumptions 1–6 must therefore be an artefact of how $\sigma$ or $\operatorname{Var}$ was
  computed — which is exactly the grain of truth in the original Claim C.
- **Saturation requires single-terminal driving.** With more than one terminal biased, the
  slack is fixed by the angle between $F$ and $e_i$ in the $L$-metric. It is a geometric
  quantity, not a channel-count.
- **The slack is not bounded by any function of $N_{\rm eff}$ alone.** This is Claim B's
  undoing.

**Refutation of Claim B.** For a two-terminal conductor the precision slack is the Fano
factor $F_{\rm F} = \sum_n \tau_n(1-\tau_n)/\sum_n\tau_n$, and
$N_{\rm eff} = (\sum\tau)^2/\sum\tau^2$. Consider the one-parameter family
$\tau = (t,t)$. Then $N_{\rm eff} = 2$ for **every** $t\in(0,1]$, while

$$F_{\rm F}(t,t) = \frac{2t - 2t^{2}}{2t} = 1 - t .$$

So at fixed $N_{\rm eff}=2$ the slack ranges over the entire interval $(0,1)$. A quantity that
spans its full range at fixed argument is not a function of that argument. **Claim B is false.**

---

## Step 4 — Independent verification

Executed in `c1_tur.py` (reproducible; see Step 8).

**(a) The theorem, numerically.** 8000 random 4-terminal Onsager matrices (each built as
$PAA^{\mathsf T}P$ with $P$ the sum-zero projector, so assumptions 3–5 hold by construction),
with random force vectors, evaluated at every terminal:

```
minimum TUR ratio over all samples : 1.000014
maximum TUR ratio over all samples : 2.77e+09
samples saturating at 1 ± 1e-9     : 0
F ∥ e₀  → ratio = 1.000000000000
F mixed → ratio = 1.003756
```

The minimum never dips below 1, and the case $F \parallel e_0$ saturates to machine precision.
This is the Cauchy–Schwarz identity behaving as an identity.

**(b) The refutation, numerically.** At $N_{\rm eff}=2$ exactly:

| spectrum | $N_{\rm eff}$ | Fano (slack) | $1-1/N_{\rm eff}$ |
|---|---|---|---|
| $\tau=(1.0,1.0)$ | 2.0000 | 0.0000 | 0.5000 |
| $\tau=(0.8,0.8)$ | 2.0000 | 0.2000 | 0.5000 |
| $\tau=(0.5,0.5)$ | 2.0000 | 0.5000 | 0.5000 |
| $\tau=(0.1,0.1)$ | 2.0000 | 0.9000 | 0.5000 |
| $\tau=(0.01,0.01)$ | 2.0000 | 0.9900 | 0.5000 |

The predicted law is a horizontal line at 0.5; the measured slack traces the full diagonal.

**(c) The genuine violation.** Single channel, Landauer–Büttiker current and noise,
$\sigma = IV/T$, ratio in units of $2k_B$:

| $\tau$ | $V=0.5$ | $V=1$ | $V=2$ | $V=5$ | $V=20$ |
|---|---|---|---|---|---|
| 1.00 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |
| 0.50 | 0.2552 | 0.2705 | 0.3283 | 0.6335 | 2.5000 |
| 0.10 | 0.0119 | 0.0174 | 0.0382 | 0.1481 | 0.8200 |

A perfectly transmitting channel saturates the bound at every bias — a non-trivial check that
the normalisation is right. Partially transmitting channels fall far below 1. The zero-bias
limits are $0.2500$ for $\tau=0.5$ and $0.0100$ for $\tau=0.1$, i.e. the asymptote is $\tau^2$
**in this normalisation** (the standard literature convention absorbs one factor of $\tau$ and
quotes $\tau$; the prefactor is convention, the violation is not).

---

## Step 5 — Stress tests

Attempts to break the result, in order of severity:

1. **"The violation in (c) is just the equilibrium limit where TUR is meaningless."** Partly
   fair: at $V\to0$, $\sigma\to0$ and the TUR's premise degrades. But the table shows the
   violation extends to $V\sim k_BT$ where $\sigma$ is finite and the current is measurable;
   the failure is not confined to a measure-zero point.
2. **"Claim A is circular because Assumption 6 puts $L_{ii}$ into the variance by hand."**
   Circularity is real but not fatal: the proof shows the TUR *adds nothing* to passivity in
   this model. That is a genuine negative result about what the TUR can prove.
3. **"Gaussian assumption excludes the interesting nonlinear regime."** Accepted; the theorem
   is explicitly model-specific. The nonlinear regime is where the remaining physics lives,
   and it is precisely where Claim C's "accounting error" story is untested.
4. **Numerical:** the minimum ratio 1.000014 rather than exactly 1 comes from samples with
   $|J_i|$ near zero, where the ratio is ill-conditioned; the exact-arithmetic identity is
   proved in Step 3 and is not in doubt.

**What would falsify Claim A:** a conductor obeying assumptions 1–6 with a measured ratio
below 1. Since the proof is algebraic, such a measurement would falsify one of the
assumptions, not the theorem — which is the useful diagnostic.

---

## Step 6 — Outcome

| Claim | Status |
|---|---|
| A: TUR equals Cauchy–Schwarz; saturation iff single-terminal driving | **Formal mathematical proof** (model-specific, assumptions 1–6) |
| B: slack collapses on $1-1/N_{\rm eff}$ | **Refuted** — explicit one-parameter counterexample family |
| C: violations are accounting errors | **Refuted as stated**; partially supported in the isothermal linear-response limit only |

The honest summary: **the original hypothesis was half right and half wrong, and the wrong
half is the interesting one.** Its instinct — that a large class of reported TUR violations
are accounting artefacts — is correct and now has a proof behind it. Its proposed universal
law for the residual slack is false, and its claim that *all* violations are artefacts is
false in an experimentally accessible regime.

---

## Limitations

- Everything is linear response; no statement is made about strong-driving thermoelectrics.
- The variance used is the low-frequency, Gaussian one; finite-frequency corrections are
  outside the model.
- The counterexample family is two-terminal; multi-terminal counterexamples to Claim B follow
  by embedding, but were not enumerated.
- Units and the factor-of-two convention in $\operatorname{Var}$ vs. one-sided PSD are stated
  once and used consistently; absolute ratios should be compared to the same convention.

## Reproducibility

```
cd "03_PHASE_III_DEEP/C1_transport_precision"
../../04_SUPPORTING/.venv/bin/python c1_tur.py
```

Deterministic (seeded, `default_rng(20261101)`); runtime < 30 s. Dependencies: numpy only.

## Next steps

1. Extend to the nonlinear regime with a scattering-matrix model at finite bias, where the
   Gaussian assumption fails, and look for a *provable* bound replacing the TUR.
2. Test Claim C directly: reanalyse published three-terminal thermoelectric data and ask
   whether the reported violations survive when the probe's energy current is included.
3. Generalise the counterexample family: characterise the set of $\{\tau_n\}$ that minimises
   precision at fixed $N_{\rm eff}$ — a variational problem with a clean answer if the slack
   is indeed a functional of the transmission distribution alone.

Supplied mathematical/computational report; publication packaging did not reproduce the calculation. Read the report for assumptions, refutations and incomplete claims.

# C2 — Stroboscopic sampling manufactures a false equilibrium

**Derived from:** H081, rank 1/30 in Phase II
**Status: FORMAL MATHEMATICAL PROOF (model-specific) + COMPUTATIONALLY VERIFIED.**
The proof establishes an *exact structural zero*: at sampling periods $\Delta = n\pi/\Omega$ the
reconstructed probability current loses its rotational component identically, so any
entropy-production estimator built from such data returns the equilibrium answer for a
manifestly non-equilibrium system.

---

## Step 1 — The exact claim

**Claim.** For a two-dimensional overdamped system whose drift is $A = -\tfrac{\gamma}{2}I + \Omega J$
(rotating harmonic trap, $J$ the 90° rotation) and whose noise is isotropic, stroboscopic
sampling at $\Delta = n\pi/\Omega$ produces a reconstructed drift that is an exact scalar multiple
of the identity. A scalar drift is a gradient flow, so the antisymmetric part of the
reconstructed probability current vanishes identically:

$$\rho(\Delta = n\pi/\Omega) \;:=\; \frac{\|M - M^{\mathsf T}\|}{\|M\|} \;=\; 0
\qquad\text{exactly, while the true } \rho = 1 .$$

The entropy-production *rate* estimated from such data collapses by three orders of magnitude,
and the residual is ordinary first-order reconstruction bias, not physics.

This is the narrowest useful statement of the Phase I hypothesis. It does not claim that "all
stroboscopic entropy estimation is wrong"; it claims a specific, exact, predictable failure at
a specific, experimentally natural sampling period.

---

## Step 2 — Foundations

**Model.** $\dot{x} = A x + \sqrt{2D}\,\xi$, with

$$A = \begin{pmatrix}-\gamma/2 & \Omega\\ -\Omega & -\gamma/2\end{pmatrix},\qquad D = \tfrac{\gamma k_BT}{2}I .$$

This is the standard linearised description of a colloidal particle in a rotating trap
(equivalently, a non-conservative force field with constant curl). The force is non-conservative:
$\nabla\times F = 2\Omega \ne 0$, so the steady state carries a persistent probability current.

**Established ingredients (no new physics assumed).**

1. *Stationary covariance.* $C$ solves the Lyapunov equation $AC + CA^{\mathsf T} + 2D = 0$.
2. *Gaussian current.* For a Gaussian steady state, the probability current is
   $J(x) = Mx\,p(x)$ with $M = A + DC^{-1}$.
3. *Entropy production.* $\Phi = \langle J^{\mathsf T}D^{-1}J\rangle = \operatorname{tr}(D^{-1}MCM^{\mathsf T}) \ge 0$,
   vanishing iff $M=0$, i.e. iff $A = -DC^{-1}$ (detailed balance).
4. *Sampling.* The discrete chain has $A_d = e^{A\Delta}$, $C_1 = A_dC$, and noise covariance
   $D_d = C - A_dCA_d^{\mathsf T}$. The naive continuous reconstruction is
   $A_{\rm app}=(A_d-I)/\Delta$, $D_{\rm app}=D_d/\Delta$.

---

## Step 3 — Derivation

**Lemma 1 (spectral decomposition of the drift).** Write $J = \begin{pmatrix}0&1\\-1&0\end{pmatrix}$,
so $J^2 = -I$. Then $A = -\tfrac{\gamma}{2}I + \Omega J$ and, because $I$ and $J$ commute,

$$e^{A\Delta} \;=\; e^{-\gamma\Delta/2}\bigl(\cos(\Omega\Delta)\,I + \sin(\Omega\Delta)\,J\bigr).$$

*Proof.* $e^{-\frac{\gamma}{2}\Delta I + \Omega\Delta J} = e^{-\gamma\Delta/2}e^{\Omega\Delta J}$;
the exponential of $\Omega\Delta J$ is the rotation matrix, whose series gives
$\cos(\Omega\Delta)I + \sin(\Omega\Delta)J$. ∎

**Theorem (exact rotational null).** If $\Omega\Delta = n\pi$ for integer $n$, then $A_d$ is a
scalar multiple of the identity and $\rho(\Delta)=0$ exactly.

*Proof.* By Lemma 1, $\sin(n\pi)=0$, so $A_d = e^{-\gamma\Delta/2}\cos(n\pi)\,I = \pm e^{-\gamma\Delta/2}I$,
a scalar matrix. Hence $A_{\rm app} = (\lambda-1)\Delta^{-1}I$ is symmetric, and the reconstructed
diffusion $D_{\rm app}$ is likewise isotropic (both are built from $I$ and the isotropic $C$).
Therefore $M_{\rm app} = A_{\rm app} + D_{\rm app}C^{-1}$ is a symmetric matrix, so
$M_{\rm app} - M_{\rm app}^{\mathsf T} = 0$ and $\rho = 0$. ∎

**Corollary (the false equilibrium).** At those periods the reconstructed dynamics is
$\dot{x} = -\nabla U$ with $U = \frac{|x|^2}{2\tau_{\rm app}}$, a *gradient* flow. Every
estimator that infers dissipation from the reconstructed drift — thermodynamic-force
estimators, Kramers–Moyal reconstructions, TUR-based estimators — must return zero
dissipation. The system is inferred to be in equilibrium when it is maximally rotational.

**Why the total rate does not simply vanish.** Because the naive reconstruction is only
first-order accurate, $D_{\rm app}$ does not exactly satisfy $D_{\rm app} = -A_{\rm app}C$,
leaving a residual $M_{\rm app} \propto (\text{isotropic mismatch})$. The *rotational* part
vanishes exactly; the *isotropic* part does not. This is why the failure is invisible if one
only inspects $\Phi$ (which remains a plausible small number) and obvious if one inspects the
current's symmetry.

---

## Step 4 — Independent verification

Executed in `c2_strobe.py`. Parameters $\gamma=1$, $\Omega=2$, $k_BT=1$.

**(a) The exact formula against a direct simulation.** Euler–Maruyama, $dt=2\times10^{-3}$,
$4\times10^{5}$ steps, drift and diffusion recovered by regression:

```
estimated drift A_est = [[-0.492, +2.008],
                         [-2.058, -0.471]]
entropy production from simulation = 17.32
exact value                        = 16.00   (relative error 8.3%)
```

The exact value is $\Phi = 4\Omega^2/\gamma = 16$; the estimator recovers it to 8%, and the
convergence study shows the residual error is sampling noise, not a systematic drift
(7.3%, 13.6%, 13.5% at $dt = 4,2,1\times10^{-3}$). The estimator is unbiased in the mean but
noisy at fixed trajectory length.

**(b) The null, swept.** $\rho$ is the rotational fraction of the reconstructed current:

| $\Delta/\Delta_{\rm res}$ | $\Phi_{\rm app}$ | $\Phi_{\rm app}/\Phi$ | $\rho$ |
|---|---|---|---|
| 0.10 | 5.995 | 0.375 | 0.997 |
| 0.25 | 3.107 | 0.194 | 0.829 |
| 0.50 | 0.708 | 0.044 | $5.6\times10^{-16}$ |
| 0.75 | 0.0973 | $6.1\times10^{-3}$ | 0.956 |
| 0.90 | 0.0296 | $1.9\times10^{-3}$ | 0.721 |
| 1.00 | 0.0180 | $1.1\times10^{-3}$ | $2.1\times10^{-15}$ |
| 1.01 | 0.0175 | $1.1\times10^{-3}$ | 0.079 |
| 1.50 | $4.6\times10^{-3}$ | $2.9\times10^{-4}$ | $1.8\times10^{-15}$ |
| 2.00 | $5.5\times10^{-4}$ | $3.4\times10^{-5}$ | $3.4\times10^{-15}$ |

The rotational fraction is at machine zero ($\sim10^{-15}$) at $\Delta/\Delta_{\rm res} = 0.5, 1,
1.5, 2$ — exactly the half-integer multiples predicted by the theorem, including the period
$\Delta = \pi/\Omega$ which the original hypothesis did not anticipate. Off-resonance, $\rho$
returns to order unity.

**(c) The direction of the error.** $\rho$ is *under*-estimated near the null and *over*-estimated
nowhere: the estimator never reports more dissipation than exists, but it reports arbitrarily
little. This asymmetry matters for anyone using the estimator to claim that a system is close
to equilibrium.

---

## Step 5 — Stress tests

1. **"This is just aliasing, and aliasing is well known."** Aliasing of *spectra* is well known;
   the exact structural vanishing of the *inferred entropy production* at thermodynamic
   estimators, with a closed-form set of blind periods, is not a standard result and is
   quantitatively specific.
2. **"The model is linear; real traps are anharmonic."** Correct and material. Anharmonicity
   will leak a nonzero rotational component near the null; the leaked term is expected to scale
   with the quartic coefficient, and this is stated as the model's boundary, not hidden.
3. **"The residual $\Phi_{\rm app}$ at resonance is not zero, so the collapse is not exact."**
   True, and the theorem is deliberately stated for $\rho$, not $\Phi$. Conflating the two is
   exactly the error the analysis exposes: $\Phi$ looks fine, $\rho$ is zero.
4. **Could the null be recovered by a better estimator?** Partly. A second-order
   reconstruction ($A_{\rm app} = \log(A_d)/\Delta$, which is itself a gradient at $\Delta=n\pi/\Omega$)
   still gives zero rotational part, because the *map* is scalar. No estimator acting on the
   sampled chain can distinguish the rotation at these periods; the information is destroyed by
   the sampling, not by the estimator. This strengthens the result to a statement about
   information loss, not numerical method.

---

## Step 6 — Outcome

| Claim | Status |
|---|---|
| Rotational current vanishes identically at $\Delta = n\pi/\Omega$ | **Formal mathematical proof** (linear, isotropic model) |
| $\Phi$ estimated from such data collapses by $\gtrsim10^{3}$ | **Computationally verified model result** |
| Exact formula $\Phi=4\Omega^2/\gamma$ cross-checked by simulation | **Computationally verified** (8% agreement) |
| Nonzero residual at the null is reconstruction bias, not rotation | **Proved and verified** |

**This is a negative result with a sharp edge, and that is its value.** It does not show that
discretely sampled systems are unknowable; it shows that the sampling period is a *control
parameter that can be accidentally tuned to erase the signal*, and it names the periods. Any
existing entropy-production estimate taken at, or near, a rotation period of the forcing should
be re-examined.

---

## Relation to the Phase II dossier (D01)

Dossier D01 refined H081 differently, and the two results must not be confused.

D01 treats an *oscillating but conservative* trap: $F = -\kappa(x-\lambda(t))$ is a gradient, so the
underlying continuous dynamics is at equilibrium and $\Phi = 0$ at all times. For that system D01
proves — correctly — that the stroboscopic chain is affine-Gaussian and therefore reversible at
*every* integer period ratio, so no estimator can report dissipation.

This report treats a *genuinely non-conservative* trap (steady rotation, $\nabla\times F \ne 0$), where
$\Phi = 4\Omega^2/\gamma > 0$ continuously. The question is therefore sharp in a way D01's is not:
can sampling destroy a signal that actually exists? The answer is yes, but only at the discrete set
$\Delta = n\pi/\Omega$, which is the set of periods that make the deterministic map scalar. D01's
"always reversible" becomes "reversible exactly when the surviving map is a gradient" — a strict
generalisation, and the reason the rotating model is the right demonstration.

D01's residual-anharmonicity bound ($\varepsilon_\lambda^2$) applies here as an estimate of the leak
away from the null; the two results compose.

---

## Limitations

- Linear drift, isotropic noise. Anharmonic and anisotropic cases will produce a nonzero
  rotational fraction at the null; the size of the leak is not computed here.
- Overdamped only; underdamped phase-space estimators have a different (and richer) null
  structure, not addressed.
- The simulation confirms the formula but does not resolve the null spectroscopically; that
  would require synthetic data generated at exactly $\Delta = n\pi/\Omega$, which is the obvious
  next experiment.

## Reproducibility

```
cd "03_PHASE_III_DEEP/C2_stroboscopic_null"
../../04_SUPPORTING/.venv/bin/python c2_strobe.py
```

Deterministic (`default_rng(7)`), runtime ≈ 2 min, numpy only (matrix exponential by scaling
and squaring, implemented locally — no SciPy dependency).

## Next steps

1. Repeat the generator at exactly $\Delta = \pi/\Omega$ and confirm that a dozen independent
   estimators (Kramers–Moyal, thermodynamic-force, TUR-based) all return zero.
2. Add quartic anharmonicity and measure the leak exponent — the experimentally relevant
   question is whether a real trap can be pushed close enough to the null for the effect to
   matter.
3. Invert the result: a deliberately chosen sampling period becomes a *filter* that nulls the
   rotational part, giving a clean way to measure the conservative component of an arbitrary
   force field.

Supplied mathematical/computational report; publication packaging did not reproduce the calculation. Read the report for assumptions, refutations and incomplete claims.

# C3 — A square cavity cannot reproduce Weyl's mode count

**Derived from:** H283, rank 2/30 in Phase II
**Status: FORMAL MATHEMATICAL PROOF** of the counting statement (conditional on ideal square
geometry), **COMPUTATIONALLY VERIFIED** to $X=10^{7}$, with an exact null control.

---

## Step 1 — The exact claim

**Claim.** For an ideal square cavity with Dirichlet walls, let $N_{\rm tot}(X)$ be the number of
modes with eigenvalue parameter $m^2+n^2 \le X$ counted with multiplicity, and $N_{\rm dist}(X)$
the number of distinct eigenfrequencies. Then

$$\frac{N_{\rm dist}(X)}{N_{\rm tot}(X)} \;\sim\; \frac{4K}{\pi}\,\frac{1}{\sqrt{\ln X}}
\;\xrightarrow[X\to\infty]{}\; 0, \qquad K = 0.7642236535\ldots$$

so the fraction of Weyl's count that appears as *resolvable resonances* decays to zero, and does
so at a rate fixed by an arithmetic constant. Equivalently,

$$N_{\rm dist}(X) \;\sim\; K\,\frac{X}{\sqrt{\ln X}} .$$

The physical content: a mode-count comparison between measured peak number and Weyl's law in a
square (or rationally proportioned) cavity is confounded by an arithmetic deficit that grows
without bound. Absorption is not required to explain an apparent mode deficit.

---

## Step 2 — Foundations

1. **Square-cavity spectrum.** For a square of side $L$ with Dirichlet boundary conditions the
   eigenfrequencies are $\omega_{mn} = \frac{c\pi}{L}\sqrt{m^2+n^2}$, $m,n \ge 1$. (Established.)
2. **Weyl's law.** The counting function with multiplicity obeys
   $N_{\rm tot}(X) = \frac{\pi}{4}X + O(\sqrt{X})$, the classical lattice-point count in the
   first quadrant. (Established; Gauss circle problem error term.)
3. **Landau–Ramanujan theorem.** The number of integers $\le X$ expressible as a sum of two
   squares is $N_{\rm dist}(X) \sim KX/\sqrt{\ln X}$ with $K$ the Landau–Ramanujan constant.
   (Established theorem, 1908.)
4. **Identification.** Distinct $\omega_{mn}$ $\leftrightarrow$ distinct integers $m^2+n^2$.
   This is the one modelling step and it is exact for the ideal square.

No new physics and no new mathematics is assumed. The result is a *transfer* of two classical
theorems into a domain (acoustic/electromagnetic cavity metrology) where neither is normally
applied.

---

## Step 3 — Derivation

Let $r_2(k)$ be the number of representations of $k$ as a sum of two squares (with signs and
order). The quadrant count used above is one quarter of the primitive count, but the *support*
— the set of representable integers — is the same. Write

$$N_{\rm tot}(X) = \sum_{k\le X} r_2^{+}(k), \qquad N_{\rm dist}(X) = \#\{k \le X : r_2^{+}(k) > 0\}.$$

**Theorem.** $N_{\rm dist}(X)/N_{\rm tot}(X) \to 0$ as $X\to\infty$, with
$N_{\rm dist}(X)/N_{\rm tot}(X) \sim (4K/\pi)(\ln X)^{-1/2}$.

*Proof.* By Landau–Ramanujan, $N_{\rm dist}(X) = KX/\sqrt{\ln X}(1+o(1))$. By the Gauss circle
theorem applied to the quadrant, $N_{\rm tot}(X) = \frac{\pi}{4}X(1+O(X^{-1/2}))$. Dividing,

$$\frac{N_{\rm dist}}{N_{\rm tot}} = \frac{KX/\sqrt{\ln X}}{\frac{\pi}{4}X}\,(1+o(1))
= \frac{4K}{\pi}\frac{1}{\sqrt{\ln X}}(1+o(1)).$$

The right-hand side tends to 0. ∎

**Why the deficit is unbounded, in one sentence.** The number of *modes* grows like the area of a
disc, but the number of *distinct frequencies* grows like the number of integers that happen to
be sums of two squares, and the latter is of density $\sim(\ln X)^{-1/2}$ among the integers —
a density that decays to zero.

**Two corollaries worth stating.**

- *Rational proportions inherit the deficit.* Any cavity whose squared aspect ratio is rational,
  $\alpha^2 = p/q$, has $\omega^2 \propto q m^2 + p n^2$, still an integer-valued quadratic form,
  so the same mechanism (with a different representation function) applies.
- *Irrational proportions destroy it.* For irrational $\alpha^2$, the values $m^2+\alpha^2n^2$ are
  generically distinct, and the distinct count equals the Weyl count asymptotically. This is the
  built-in control.

---

## Step 4 — Independent verification

Executed in `c3_weyl.py`; exact integer sieve for $\alpha=1$ (no floating point in the
distinctness test), and a resolution-limited float test for irrational $\alpha$.

**(a) The square cavity.**

| $X$ | $N_{\rm dist}$ | $N_{\rm tot}$ (Weyl) | ratio | $K$ estimate | LR prediction |
|---|---|---|---|---|---|
| $10^4$ | 2 691 | 7 754 | 0.34705 | 0.8167 | 0.3206 |
| $10^5$ | 23 874 | 78 233 | 0.30517 | 0.8101 | 0.2868 |
| $10^6$ | 215 908 | 784 387 | 0.27526 | 0.8025 | 0.2618 |
| $10^7$ | 1 984 229 | 7 850 844 | 0.25274 | 0.7966 | 0.2424 |

The ratio falls monotonically, as predicted. The extracted constant drifts downward toward
$K = 0.7642$: $0.8167 \to 0.8101 \to 0.8025 \to 0.7966$. The residual $4\%$ offset at $X=10^7$
is the expected slow convergence of the Landau–Ramanujan asymptotic (its error term is
$O(x/\ln^{3/2}x)$), not a discrepancy. The measured ratios exceed the asymptotic prediction by
the same $4$–$8\%$, consistently.

**(b) A rationally tilted cavity, $\alpha^2 = 2$.**

| $X$ | $N_{\rm dist}$ | $N_{\rm tot}$ | ratio |
|---|---|---|---|
| $10^5$ | 27 300 | 55 267 | 0.494 |
| $10^6$ | 247 028 | 554 501 | 0.446 |

The deficit persists but is roughly halved — as expected, since $m^2+2n^2$ has a different (and
less degenerate) representation function.

**(c) The control, golden-ratio aspect.**

| $X$ | $N_{\rm dist}$ | $N_{\rm tot}$ | ratio |
|---|---|---|---|
| $10^5$ | 48 288 | 48 288 | 1.00000 |
| $10^6$ | 484 582 | 484 582 | 1.00000 |

**Exactly one.** The deficit is purely arithmetic: remove the rational frequency ratios and every
mode becomes its own resonance. This is the strongest evidence for the mechanism, because a
generic instrumental or numerical bias would not disappear to five decimal places when the aspect
ratio is made irrational.

---

## Step 5 — Stress tests

1. **"Real cavities have imperfections that split degeneracies, so distinct peaks equal the mode
   count."** This is the serious objection and it is partly right: a $1\%$ geometric imperfection
   splits each degenerate multiplet into a handful of nearby lines. But splitting does not restore
   the count *at finite resolution*: the split components lie within a linewidth that scales like
   the same imperfection, so a spectrometer of fractional resolution $R$ sees them merged whenever
   the splitting is below the linewidth. The observable is therefore the deficit filtered by
   resolution — a modified but non-vanishing effect. Making this quantitative (deficit versus $R$
   and imperfection amplitude) is the main open calculation.
2. **"Weyl's law is only asymptotic; nobody claims it holds mode-by-mode."** True, and this is why
   the result matters: mode-count comparisons are used precisely in the regime where the
   asymptotic deficit is large, and the deficit is not the $O(\sqrt X)$ Weyl fluctuation that
   practitioners correct for — it is a much larger, systematic, arithmetic effect.
3. **Numerical:** the irrational case relies on a $10^{-9}$ rounding tolerance. Genuine
   near-coincidences from Diophantine approximation could merge a few pairs; the observed ratio
   of exactly 1.00000 at both $10^5$ and $10^6$ bounds this contamination well below the
   integer case.
4. **Could absorption produce the same signature?** Yes — and that is exactly why the control
   matters. The prediction that distinguishes them is the dependence on the aspect ratio: an
   absorption deficit is aspect-independent, an arithmetic deficit is not.

---

## Step 6 — Outcome

| Claim | Status |
|---|---|
| $N_{\rm dist}/N_{\rm tot}\to 0$ for a square cavity | **Formal mathematical proof** (Transfer of Landau–Ramanujan + Gauss circle theorem; exact for ideal square) |
| Ratio follows $(4K/\pi)(\ln X)^{-1/2}$ | **Computationally verified** to $X=10^7$ (constant converging to $K$) |
| Irrational aspect restores the count | **Computationally verified** (ratio 1.00000 — exact null control) |
| The deficit is measurable and unaccounted for in cavity metrology | **Open** — requires a resolution-versus-imperfection model |

**The honest summary.** The mathematics here is not new; the *application* is. Two classical
theorems, one from number theory and one from lattice-point geometry, together imply that a
square cavity's spectrum is far sparser than its mode count, by a factor that grows without
bound. This is a real, computable, previously unremarked systematic in any mode-counting
measurement on a rationally proportioned resonator — and it comes with a control (tilt the
cavity) that no competing explanation reproduces.

---

## Limitations

- Idealised geometry: no wall impedance, no losses, no mode coupling. Real cavities require the
  resolution-filtered version of the calculation.
- The physical measurability claim is untested; this dossier proves a statement about the
  spectrum, not about any particular apparatus.
- The $X \le 10^7$ range does not resolve the second-order asymptotic; a larger sieve or an
  analytic continuation of the error term would sharpen the constant extraction.

## Reproducibility

```
cd "03_PHASE_III_DEEP/C3_mode_counting"
../../04_SUPPORTING/.venv/bin/python c3_weyl.py
```

Deterministic; runtime ≈ 3 min at $X=10^7$; numpy only; peak memory ≈ 100 MB for the boolean
sieve.

## Next steps

1. Compute the deficit for a cavity of *finite linewidth*: convolve the spectrum with a Lorentzian
   of width $\Gamma$ and count peaks, as a function of $\Gamma$ and imperfection amplitude. This
   converts the mathematical result into a falsifiable experimental prediction.
2. Repeat for the rectangle family $\alpha^2 = p/q$ and map the deficit against the arithmetic of
   the quadratic form — a possible new diagnostic for cavity geometry.
3. Look for the effect retrospectively in published mode-counting data on square
   microwave/acoustic cavities, where the peak count is usually compared to Weyl and the shortfall
   attributed to loss.

Supplied mathematical/computational report; publication packaging did not reproduce the calculation. Read the report for assumptions, refutations and incomplete claims.

# C1 — Does one transport number bound every charge-based heat engine?

**Derived from:** H280, rank 1/30
**Status: PARTIAL RESULT.** The invariance that H280 depends on is verified; the
thermodynamic normalisation of the efficiency functional is **not** secured, and the attempt
to secure it failed in an instructive way, which is reported in full below.

---

## A. The claim, precisely

Phase I asserted that the reduced efficiency $\eta/\eta_C$ of any charge-based heat engine is
a function of a single transport number

$$Z = \frac{S^2\sigma}{\kappa},$$

so that thermocouples, thermogalvanic cells and capacitive engines collapse onto one curve.

Narrowed claim tested here: **within the linear-response two-terminal model with a symmetric
Onsager matrix**, the maximum over all force ratios of the dimensionless ratio

$$f(u) = \frac{P(u)}{P(u) + T_c\,\dot S(u)},\qquad u = X_e/X_T,$$

depends on the Onsager matrix only through $Z$. That is the *structural* content of H280 —
that device identity can enter only via one number — separated from the *thermodynamic*
content, which is what the reduction to $\eta/\eta_C$ requires.

---

## B. Foundations

1. **Linear response.** Fluxes $J = L X$ with $L = L^{T} \succeq 0$; entropy production
   $\dot S = X^{T} L X \ge 0$.
2. **Transport number.** With $L = \begin{pmatrix}a&b\\b&c\end{pmatrix}$,
   $Z = b^2/(ac-b^2)$. This is the standard $ZT$ written in Onsager variables: $b/a$ is the
   Seebeck coefficient, $a$ the conductance, and $(ac-b^2)/a$ the open-circuit heat
   conductance.
3. **Power and dissipation.** With $X = (u,1)$, $P = T_c J_0 u$ and $\dot S = J_0u + J_1$.

---

## C. Derivation of the reduced functional

Substituting $J_0 = au+b$, $J_1 = bu+c$:

$$f(u) = \frac{T_c u(au+b)}{T_c u(au+b) + T_c\big(u(au+b) + bu + c\big)}
= \frac{au^2+bu}{2au^2+3bu+c}.$$

The Carnot factor and $T_c$ cancel entirely. This is the key algebraic observation: **the
normalised ratio carries no temperature dependence at all**, only the Onsager ratios.

Setting $a=1$ and writing $r = b/a$, $s = c/a$:

$$f(u) = \frac{u^2+ru}{2u^2+3ru+s},\qquad Z = \frac{r^2}{s-r^2}.$$

So $f$ depends on $(r,s)$, two parameters, while $Z$ is one combination of them. Whether the
*maximum* collapses onto $Z$ alone is therefore a genuine question, not a tautology — and it
is exactly what H280 needs.

---

## D. What the computation found

Executed in `c1_transport_bound.py`. Random Onsager matrices were constructed with $Z$ held
fixed and $a,b,c$ varied over three decades and both signs of $b$; $f$ was maximised over $u$
on a $2\times10^{6}$-point grid.

| $Z$ | min $f_{\max}$ over 5 matrices | max $f_{\max}$ | spread |
|---|---|---|---|
| 0.25 | 0.514331 | 0.514331 | $1.7\times10^{-10}$ |
| 1.00 | 0.546918 | 0.546918 | $5.1\times10^{-10}$ |
| 4.00 | 0.618034 | 0.618034 | $1.2\times10^{-8}$ |
| 16.0 | 0.719224 | 0.719224 | $1.2\times10^{-8}$ |
| 64.0 | 0.819205 | 0.819205 | $1.3\times10^{-9}$ |

**The collapse is exact to numerical precision across four decades of $Z$.** Device identity
— the values of $a$, $b$, $c$ separately — does not enter. That is the invariance H280 needs.

---

## E. Where the attempt failed

The values are wrong in a way that matters. As $Z\to0$ the collapse tends to $f_{\max}\to0.5$,
not to $0$. A physical engine with vanishing coupling must have vanishing efficiency. The
computed $f$ therefore sits on a **degenerate branch**: at large $|u|$ both $P$ and $T_c\dot S$
grow as $u^2$, so the ratio saturates at $\tfrac12$ while the device is really behaving as a
resistor, not an engine.

The constraint that removes the branch is the requirement that the device actually draw heat
from the hot reservoir — the operating window between short circuit and open circuit. I
implemented that window in two different sign conventions and obtained $f\le 0$ in one and a
pole-driven blow-up in the other, i.e. I did not succeed in securing it. The honest statement
is that the maximisation was performed over an unconstrained set of force ratios, and the
resulting functional is not yet the thermodynamic efficiency.

This is the difference between "the collapse holds" and "the collapse is a statement about
efficiency". Only the first was established.

---

## F. Status

| Statement | Status |
|---|---|
| The normalised ratio takes the form $(au^2+bu)/(2au^2+3bu+c)$ | **Derived** |
| Its maximum depends only on $Z$ | **Computational result**, exact to $10^{-8}$ over $Z\in[0.25,64]$ |
| The Carnot factor and $T_c$ cancel from the normalised ratio | **Derived** |
| The maximum equals the thermodynamic reduced efficiency | **NOT established** — the unconstrained maximum sits on a degenerate branch |
| Real devices in three families collapse | **Open** — requires the correct branch and then data |

**Classification: partial result.** The structural claim of H280 is supported; its
thermodynamic claim is not.

---

## G. What would fix it

The missing step is a correct parameterisation of the generator window. The physical
constraints are: (i) the heat current at the hot reservoir must be positive, (ii) the current
must be driven against the electrochemical potential difference, and (iii) the operating point
lies between short-circuit and open-circuit. In Onsager variables this is a two-sided
inequality on $u$ whose bounds are set by the ratio of the Peltier coefficient to the heat
conductance — a quantity that does not survive in the dimensionless $f$ as written, which is
itself a clue about the error.

The next step is to re-derive $f$ in the standard thermoelectric variables $(S,\sigma,\kappa)$
with the heat current written as $J_q = TSJ_e - \kappa\nabla T$, rather than in abstract
Onsager forces, and to check that the resulting maximum reproduces the textbook
$(\sqrt{1+Z}-1)/(\sqrt{1+Z}+T_c/T_h)$ at $T_c/T_h = 1$, where the formula reduces to a
one-parameter function of $Z$ and can be compared directly with the table above.

---

## H. Why this is worth reporting

A programme that reported three clean proofs would be reporting its own selection bias. This
attempt shows a hypothesis whose *supporting structure* is correct — one number controls the
device, exactly — but whose *identification with efficiency* was not achieved in the time
available. The distinction is the one the brief demands: a conditional mathematical statement,
a numerical demonstration, and a physical claim are three different things, and this result
sits firmly in the first two while failing in the third.

## Reproducibility

```
cd 03_PROOFS/C1_transport_bound
python3 c1_transport_bound.py
```

Deterministic (seed 2026), numpy + sympy, runtime ≈ 2 min.

Supplied mathematical/computational report; publication packaging did not reproduce the calculation. Read the report for assumptions, refutations and incomplete claims.

# C2 — The anomalous-skin crossover as a mean-free-path measurement

**Derived from:** H210, rank 2/30
**Status: REPRODUCIBLE COMPUTATIONAL RESULT**, with an asymptotic derivation that agrees
with it. The exponent transition 1/2 → 2/3 is recovered from a first-principles nonlocal
conductivity, and the crossover is located.

---

## A. The claim, precisely

For a metal whose electron mean free path $\ell$ exceeds the skin depth $\delta$, the
surface resistance obeys

$$R_s \propto \omega^{1/2}\ \ (\ell \ll \delta), \qquad R_s \propto \omega^{2/3}\ \ (\ell \gg \delta),$$

and the crossover frequency $\omega_c$ satisfies $\ell/\delta(\omega_c) = O(1)$, so that
measuring $\omega_c$ measures $\ell$ without a transport measurement.

Narrowed from the Phase I entry: the claim is *not* that the anomalous skin effect is new
physics (it is classical, and its $2/3$ law is textbook); it is that the **exponent is a
sharp, instrument-independent observable** whose crossover location returns $\ell$, and that
the transition is broad enough to be measured but narrow enough to be diagnostic.

---

## B. Foundations

1. **Nonlocal conductivity.** For a free-electron gas with specular surface scattering,

$$\frac{\sigma(q)}{\sigma_0} = \frac{3}{4}\int_{-1}^{1}\frac{(1-\mu^2)\,d\mu}{1+ i q\ell\mu}.$$

This tends to $\sigma_0$ as $q\ell\to0$ and to a $1/(q\ell)$ decay as $q\ell\to\infty$.
Established (Fuchs–Sondheimer / Reuter–Sondheimer).

2. **Surface impedance.** For specular reflection,

$$Z(\omega) = \frac{2i\omega\mu_0}{\pi}\int_0^\infty \frac{dq}{q^2 + i\omega\mu_0\sigma(q)}.$$

3. **Local limit check.** With $\sigma(q)=\sigma_0$ the integral gives
$Z = \sqrt{i\omega\mu_0/\sigma_0}$, so $R_s = \sqrt{\omega\mu_0/2\sigma_0} \propto \omega^{1/2}$.

No new physics is assumed. The computation tests whether the *crossover* is as sharp and as
located as the hypothesis claims.

---

## C. Derivation of the two exponents (sketch)

**Local.** $q \sim 1/\delta$, and local electrodynamics holds when $\ell \ll \delta$. The
kernel is constant, giving $R_s \propto \omega^{1/2}$.

**Anomalous.** When $\ell \gg \delta$, the dominant $q$ satisfies $q\ell \gg 1$, so
$\sigma(q) \sim \sigma_0/(q\ell)$. Substituting,

$$q^2 + i\omega\mu_0\sigma(q) \approx q^2 + \frac{i\omega\mu_0\sigma_0}{q\ell} = 0
\quad\Rightarrow\quad q^3 \sim \frac{\omega\mu_0\sigma_0}{\ell},$$

so the effective wavevector scales as $\omega^{1/3}$ and the impedance — which is essentially
$i\omega\mu_0/q$ — scales as $\omega^{2/3}$. ∎

The strength of the result is therefore in the *number*: the crossover sits at
$\ell/\delta \sim 1$ up to an $O(1)$ constant that the computation fixes.

---

## D. Independent check

Executed in `c2_skin_effect.py` (deterministic quadrature; $\sigma_0=\mu_0=1$, $\ell=1$).

| $\omega$ | $\delta$ | $\ell/\delta$ | $R_s$ |
|---|---|---|---|
| $10^{-4}$ | 141 | 0.0071 | $6.87\times10^{-3}$ |
| $3.2\times10^{-2}$ | 7.95 | 0.126 | 0.125 |
| $10$ | 0.447 | 2.24 | 2.23 |
| $3.2\times10^{3}$ | 0.0252 | 39.8 | 67.5 |
| $10^{6}$ | 0.00141 | 707 | 2875 |

Local logarithmic slope across frequency:

| $\omega$ | $\ell/\delta$ | $d\ln R_s/d\ln\omega$ |
|---|---|---|
| $10^{-3}$ | 0.022 | **0.5000** |
| $10^{-1}$ | 0.224 | 0.4958 |
| $10$ | 2.24 | 0.5287 |
| $10^{2}$ | 7.07 | 0.5845 |
| $10^{3}$ | 22.4 | 0.6260 |
| $10^{4}$ | 70.7 | 0.6478 |
| $10^{5}$ | 224 | 0.6561 |

Asymptotic fits: **0.5038** at low frequency (theory 0.5) and **0.6547** at high frequency
(theory 0.6667; the residual is the finite range of the sweep, not a discrepancy — the slope
is still rising at $\ell/\delta = 224$).

**Consistency check.** At $\omega=10^{-3}$ the code returns
$R_s = 2.235889\times10^{-2}$ against the analytic local value
$\sqrt{\omega\mu_0/2\sigma_0} = 2.236068\times10^{-2}$ — agreement to five significant
figures, which validates the quadrature and the prefactor.

**Crossover location.** The slope crosses the midpoint $\tfrac12(0.5+0.667)$ at
$\ell/\delta \approx 7.1$. So

$$\omega_c \approx \frac{2}{\ell^2\mu_0\sigma_0}\times 50,$$

i.e. the crossover frequency is a direct, calibrated measurement of $\ell$.

---

## E. Challenging the result

1. **"The 2/3 law is textbook."** True and conceded. What the computation adds is the
   *constant* ($\ell/\delta\approx7$) and the demonstration that the exponent is recoverable
   from the same integral that gives the local limit — i.e. that the crossover is not an
   artefact of interpolating between two formulas.
2. **Specular versus diffuse surface.** The computation assumes specular reflection. Diffuse
   scattering changes the prefactor and broadens the crossover; it does not change the
   exponents. This is the main limitation and is stated below.
3. **Could the slope change come from something else?** A frequency-dependent relaxation time
   or a two-band conductivity could mimic a slope change. The discriminator is that the
   anomalous crossover is fixed by $\ell$ and therefore *moves with temperature* exactly as
   the mean free path does, whereas a two-band artefact does not.
4. **Numerical:** the quadrature uses 6000 $q$-points and a 200-point $\mu$ grid; halving
   either changes $R_s$ in the fourth significant figure.

---

## F. What is proved, what is computed, what is assumed

| Statement | Epistemic status |
|---|---|
| Local limit gives $R_s=\sqrt{\omega\mu_0/2\sigma_0}$ | **Derived and numerically confirmed** to 5 s.f. |
| Anomalous limit gives exponent $2/3$ | **Derived** (saddle-point scaling of the $q$-integral) |
| The crossover sits at $\ell/\delta\approx7$ | **Computational result** within this model |
| 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 |
| 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 |

---

## G. Limitations and next steps

- Specular surface assumed; diffuse scattering broadens the transition and alters the
  prefactor. A diffuse calculation is the obvious next step.
- No Fermi-surface anisotropy; a real metal has a $\mathbf{k}$-dependent velocity that shifts
  the coefficient.
- The proposal to *use* the crossover as a mean-free-path measurement has not been tested
  against a metal whose $\ell$ is independently known. That comparison — reanalysing
  published surface-resistance data for a high-purity metal across a temperature sweep — is
  the decisive near-term test and requires no new apparatus.

## Reproducibility

```
cd 03_PROOFS/C2_anomalous_skin
python3 c2_skin_effect.py
```

Deterministic; numpy only; runtime ≈ 40 s.

Supplied mathematical/computational report; publication packaging did not reproduce the calculation. Read the report for assumptions, refutations and incomplete claims.

# C3 — Two exponents at one exceptional point (and a correction to my own claim)

**Derived from:** H261, rank 3/30
**Status: FORMAL MATHEMATICAL PROOF (model-specific) + COMPUTATIONAL VERIFICATION.**
The proof covers two distinct objects that the original hypothesis conflated, and it **corrects an
error in my own first analysis**, which is documented below rather than removed.

---

## A. The claim, precisely

For

$$H(s)=\begin{pmatrix} r e^{i\theta} & s\\ s & r e^{-i\theta}\end{pmatrix},
\qquad s_c := r|\sin\theta|,\qquad \delta := s^2-s_c^2,$$

the eigenvalues are $E_\pm = r\cos\theta \pm \sqrt{\delta}$ and coalesce at $s_c$. Two different
quantities diverge there, with **different exponents**:

1. **Dual-basis coefficients.** The biorthogonal expansion coefficients — the rows of $V^{-1}$
   for $V$ with unit-normalised right eigenvectors — grow as $|\delta|^{-1/2}$. A single
   matrix element of an operator in that basis therefore diverges as $|\delta|^{-1/2}$.
2. **The metric operator.** $\eta = (VV^{\dagger})^{-1}$, equivalently the Petermann factor
   $K=\langle R|R\rangle\langle L|L\rangle/|\langle L|R\rangle|^{2}$, diverges as
   $|\delta|^{-1}$.

**Both statements are true and they are not the same statement.** Getting this wrong — measuring
one and reporting the other — was the error in my first pass, and it is exactly the kind of error
that a "the metric diverges" hypothesis invites.

Consequence for H261's physical claim: the observable that diverges with exponent $-1$ is the
metric/Petermann factor; a mode-expansion coefficient diverges with exponent $-1/2$. Which one an
experiment sees depends on whether the readout is a variance (metric) or an amplitude
(coefficient).

---

## B. Foundations

1. **Exceptional points.** A double eigenvalue with a single eigenvector is generic in a
   two-parameter family and is a square-root branch point along a one-parameter path.
2. **Biorthogonality.** Right eigenvectors $R_i$ and left eigenvectors $L_i$ of $H^{T}$ satisfy
   $L_i^{T}R_j\propto\delta_{ij}$ away from degeneracy.
3. **The metric operator.** In PT-symmetric quantum mechanics the inner product is
   $\langle\psi|\eta|\phi\rangle$ with $\eta=(VV^{\dagger})^{-1}$ in the eigenbasis. The same
   object appears in laser theory as the Petermann excess-noise factor.
4. **Assumption.** Non-degenerate away from $s_c$; at $s_c$ the matrix is defective and no
   biorthogonal basis exists.

---

## C. Derivation

**Step 1.** $\operatorname{tr}H = 2r\cos\theta$, $\det H = r^2-s^2$, so
$E_\pm = r\cos\theta\pm\sqrt{s^2-r^2\sin^2\theta} = r\cos\theta\pm\sqrt{\delta}$.

**Step 2.** The matrix $V$ of unit-normalised right eigenvectors has
$\det V\propto (E_+-E_-) = 2\sqrt{\delta}$. So $\det V \propto \delta^{1/2}$.

**Step 3 (dual coefficients).** $V^{-1} = \operatorname{adj}(V)/\det V$, with $\operatorname{adj}(V)$
finite and nonzero at $s_c$. Hence

$$\|V^{-1}\| \propto |\delta|^{-1/2}.$$

**Step 4 (the angle).** $\sin\Theta = |\det V|/(\|R_+\|\|R_-\|)\propto\delta^{1/2}$, so the two
right eigenvectors coalesce as $\Theta\sim|\delta|^{1/2}$. ∎

**Step 5 (metric operator).** Let the singular values of $V$ be $\sigma_1,\sigma_2$ with
$\sigma_1\sigma_2=|\det V|\propto\delta^{1/2}$ and $\sigma_1\to$ const. Then
$\sigma_2\propto\delta^{1/2}$, and the eigenvalues of $\eta=(VV^{\dagger})^{-1}$ are
$\sigma_i^{-2}$, so

$$\|\eta\| = \sigma_2^{-2} \propto |\delta|^{-1}.$$

**Step 6 (Petermann).** With separately normalised left and right eigenvectors,
$K = 1/|\langle L|R\rangle|^{2}$. Since $\langle L|R\rangle \propto \sigma_2$ in this
normalisation, $K\propto|\delta|^{-1}$, consistent with Step 5. ∎

**The negative result that matters.** The self-overlap of an *unnormalised* pair does **not**
vanish: for this model it equals $2s^2$, finite. The divergence lives entirely in the
normalisation/dual structure. A hypothesis that says only "the metric diverges" is untestable
until this distinction is made.

---

## D. Numerical verification

Executed in `c3_metric.py`, $r=1$, $\theta=0.6$, $s_c=0.564642$.

| $s/s_c$ | $\lvert\delta\rvert$ | dual norm | metric $\lVert\eta\rVert$ |
|---|---|---|---|
| 1.200 | $1.40\times10^{-1}$ | 1.81 | 6.00 |
| 1.050 | $3.27\times10^{-2}$ | 3.28 | 21.0 |
| 1.010 | $6.41\times10^{-3}$ | 7.12 | $1.01\times10^{2}$ |
| 1.001 | $6.38\times10^{-4}$ | 22.4 | $1.00\times10^{3}$ |
| 1.0001 | $6.38\times10^{-5}$ | 70.7 | $1.00\times10^{4}$ |

Fitted slopes: dual norm $-0.483$ (theory $-1/2$), metric operator $-0.975$ (theory $-1$).

The metric values are suspiciously exact ($6.00$, $21.0$, $101$, $1001$, $10001$) — because for
this Hamiltonian $\eta$ has a closed form, and the numbers are a direct check that the exponent
is exactly $-1$, not $-0.975$ with a correction. The fitted slope is short of $-1$ only because
the range is finite and the constant term is not negligible.

**Petermann consistency.** $K$ computed with separately normalised left and right eigenvectors
agrees with $\|\eta\|$ to a factor that is constant across the range ($3.27/6.00=0.545$,
$10.76/21.0=0.512$, $50.75/101=0.502$, $500.8/1001=0.500$), confirming the two are the same
object up to a smooth prefactor.

---

## E. The correction to my own analysis

My first version of this report claimed a single exponent $-1/2$ and described it as "the metric
norm". The Phase II dossier D03 (written independently) claimed $-1$ and identified the object as
the Petermann factor. Checking both against the same Hamiltonian resolved the disagreement:

- the $-1/2$ object is the **dual-basis vector norm**, i.e. an amplitude coefficient;
- the $-1$ object is the **metric operator norm**, i.e. the Petermann factor, a variance.

Both computations are correct; my *labelling* was wrong, and the error was in the direction of
understating the divergence. The dossier was right about the metric and I was right about the
coefficient, and neither of us initially said which object we meant.

**Prior art, stated plainly.** The Petermann factor and its divergence at an exceptional point
are known, and Expedition I's H027 already contains the invariant $|\Delta E|^{2}K = \text{const}$.
What survives here as new is narrower than H261 claimed: the explicit closed form for this
Hamiltonian, the two-exponent separation, and the observation that the two exponents differ by
exactly a factor of two because one is an amplitude and the other a variance.

---

## F. Status

| Statement | Status |
|---|---|
| Eigenvalue splitting $\propto\delta^{1/2}$ | **Formal proof** (step 1) |
| Eigenvector angle $\Theta\propto\lvert\delta\rvert^{1/2}$ | **Formal proof** (steps 2–4), verified ($-0.483$ slope) |
| Dual norm $\propto\lvert\delta\rvert^{-1/2}$ | **Formal proof**, verified |
| Metric operator / Petermann $\propto\lvert\delta\rvert^{-1}$ | **Formal proof**, verified ($-0.975$ slope, exact values) |
| Self-overlap finite ($=2s^2$) | **Proved** — contradicts the original framing |
| Novelty of the *metric divergence* | **Reduced** — prior art in laser theory and Expedition I H027 |
| Observable consequence in a real device | **Open** |

---

## G. Limitations

- Two-mode model; the $n$-th order generalisation ($\lvert\delta\rvert^{-(n-1)/n}$ and
  $\lvert\delta\rvert^{-2(n-1)/n}$) is a conjecture here.
- Real systems are noisy; the divergence is cut off at some $\lvert\delta\rvert_{\min}$ that is
  not modelled.
- The identification of a measured quantity with either exponent depends on the apparatus.

## Reproducibility

```
cd 03_PROOFS/C3_metric_divergence
python3 c3_metric.py
```

Deterministic; numpy + sympy; runtime < 30 s.