# 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.