Mathematical Appendix

This appendix states and proves three results the rest of the reference relies on informally. Each is elementary and standard; collecting them here, with explicit proofs, makes the framework’s load-bearing claims checkable rather than asserted. They correspond to the three mechanisms developed narratively in the deep dives: coherent enhancement, the equilibrium no-go, and rectification.

Lemma 1 — Coherent enhancement is bounded by N

Statement. For $N$ identical scattering centres, each contributing amplitude $a$, the effective cross-section satisfies $\sigma_{\text{eff}} = N^2\,\sigma_1$ in the fully coherent (forward) direction and $\sigma_{\text{eff}} = N\,\sigma_1$ when the contributions add incoherently, where $\sigma_1 \propto |a|^2$. The coherent enhancement factor over the incoherent case is therefore at most $N$.

Proof. The total scattering amplitude is the phased sum $A(\mathbf{q}) = \sum_{j=1}^{N} a\,e^{i\mathbf{q}\cdot\mathbf{r}_j}$, where $\mathbf{q}$ is the momentum transfer and $\mathbf{r}_j$ the centre positions. The cross-section is proportional to $|A|^2$:

$$|A(\mathbf{q})|^2 = |a|^2 \sum_{j,k} e^{i\mathbf{q}\cdot(\mathbf{r}_j - \mathbf{r}_k)}. \tag{1}$$

In the forward limit $\mathbf{q}\to 0$ every phase is unity, the double sum has $N^2$ equal terms, and $|A|^2 = N^2|a|^2$, giving $\sigma_{\text{eff}} = N^2\sigma_1$. If instead the phases are random and uncorrelated, the off-diagonal terms average to zero, leaving the $N$ diagonal terms, so $\langle|A|^2\rangle = N|a|^2$ and $\sigma_{\text{eff}} = N\sigma_1$. The ratio of the two is $N$, attained only where the phases align; at finite $\mathbf{q}$ a form factor $F(\mathbf{q}) = N^{-2}\sum_{j,k} e^{i\mathbf{q}\cdot(\mathbf{r}_j-\mathbf{r}_k)} \le 1$ interpolates between them. $\blacksquare$

The enhancement is thus real but bounded: it can lift a per-particle cross-section by up to a factor $N$, and only within the coherence window set by $F(\mathbf{q})$. This is the quantitative content behind the $N^2$ law.

Lemma 2 — No net current at equilibrium

Statement. A system in thermal equilibrium produces no directed steady current: $\langle J\rangle_{\text{eq}} = 0$. Consequently the net conversion efficiency $\eta_{\text{conv}}$ vanishes in equilibrium, and any nonzero output requires a non-equilibrium drive.

Proof. Let $P(x\to x')$ be the transition rate between microstates. Detailed balance at temperature $T$ requires

$$P(x\to x')\,\rho_{\text{eq}}(x) = P(x'\to x)\,\rho_{\text{eq}}(x'), \tag{2}$$

with $\rho_{\text{eq}}(x)\propto e^{-\beta E(x)}$. The steady current associated with any antisymmetric observable $\phi(x,x') = -\phi(x',x)$ is $J = \tfrac12\sum_{x,x'}\phi(x,x')\big[P(x\to x')\rho_{\text{eq}}(x) - P(x'\to x)\rho_{\text{eq}}(x')\big]$. By (2) the bracket is zero term by term, so $J=0$. Equivalently, the fluctuation–dissipation theorem fixes the equilibrium fluctuation spectrum to the dissipative response at the same $T$, leaving no surplus to rectify. $\blacksquare$

This is the precise sense in which the framework does not describe a perpetual-motion machine: at equilibrium it returns exactly zero, and it is defined only for open systems held away from equilibrium (open thermodynamic systems).

Lemma 3 — Directed output requires broken symmetry

Statement (Curie’s principle). If the dynamics and the drive of a system are invariant under spatial inversion $\mathbf{r}\to-\mathbf{r}$, then the steady directed current vanishes, $\langle\mathbf{J}\rangle = 0$. A nonzero directed current therefore requires broken inversion symmetry.

Proof. Let $\mathcal{P}$ denote spatial inversion. The current is a polar vector, $\mathcal{P}\mathbf{J} = -\mathbf{J}$. If the generator of the dynamics (Hamiltonian plus drive, and the stationary distribution it induces) commutes with $\mathcal{P}$, then the expectation is inversion-invariant, $\langle\mathbf{J}\rangle = \langle\mathcal{P}\mathbf{J}\rangle = -\langle\mathbf{J}\rangle$, whence $\langle\mathbf{J}\rangle = 0$. A nonvanishing $\langle\mathbf{J}\rangle$ is possible only if $\mathcal{P}$ is not a symmetry — the system must carry a structural asymmetry (a ratchet). $\blacksquare$

Together with Lemma 2 this gives the two necessary conditions for directed output: a broken spatial symmetry (Lemma 3) and a non-equilibrium drive (Lemma 2). Neither alone suffices — a ratchet at equilibrium still yields nothing — and this pair is exactly what the rectification factor in $P(t)$ presupposes (stochastic rectification).

What the appendix secures

The three lemmas are the mathematical backbone of the framework’s physical story: enhancement is real but capped at $N$ (Lemma 1); equilibrium yields nothing, so no thermodynamic law is violated (Lemma 2); and any net directed output demands both a structural asymmetry and a non-equilibrium drive (Lemma 3). None of these is novel physics — they are standard results from scattering theory, statistical mechanics and symmetry analysis — and that is the point: the load-bearing steps of $P(t)$ rest on settled mathematics, with the empirical magnitudes left to measurement (measurement and falsifiability).