# 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.