The Freeze–Melt Interface: Fluctuation Symmetry, Ledger Quantization, and Membrane Percolation

Entropy production holds a theorem‑level status in statistical mechanics; the production of information — the crystallization of difference, the inverse process — is usually treated as bookkeeping. Starting from two exact postulates — the nonexistence of a deletion operator among the allowed dynamics, and the structural discreteness of memory (transitions book integer face units, with the Gibbs potential itself the grading) — we derive their thermodynamic consequences for a minimal memory system (a bit, or a membrane of bits, coupled to a single bath). (i) No‑deletion is equivalent to unitality of the allowed channels — for qubits, to random‑unitary structure. (ii) A bidirectional Landauer‑style bound forces a unique, direction‑independent bath temperature (Lemma 2). (iii) For face‑graded chains, reversibility is equivalent to the affinity being a pure gradient of the integer ledger; entropy production along a trajectory is the ledger gradient, quantized in units of the spacing \(\lambda=\beta\varepsilon\), and specializes to \(k\ln 2\) per bit at the Landauer point \(\beta\varepsilon=\ln 2\). (iv) The freeze‑melt fluctuation relation — the \(I=0\) slice of information thermodynamics — governs both directions of the interface, exactly at finite bias,\(\ln \big[P_{F}(W)/P_{R}(-W)\big]=\beta\big(W-\Delta F(\varepsilon_{\mathrm{max}})\big)\)with \(\Delta F(\varepsilon_{\mathrm{max}})=\beta^{-1}\big[\ln 2-\ln(1+e^{-\beta\varepsilon_{\mathrm{max}}})\big]\), reducing in the ideal memory limit to a fiberwise joint form\(\ln \big[P_{F}(W,\Delta\iota)/P_{R}(-W,-\Delta\iota)\big]=\beta W-\ln 2,\)identically on every populated fiber — the information current labeling the exact fibers of the symmetry, the \(\ln 2\) carried once by the conversion free energy. The symmetry is distributional, not trajectory‑wise; two failed strengthenings are reported with autopsies, and a third candidate was buried after numerical verification showed it reduces to detailed balance. The same postulates, imposed on spatially extended dynamics, exclude internal energetics, exclude absorbing‑state criticality, and confine the membrane transition to two families — protocol‑field percolation or kinetically constrained dynamics; a finite‑size scaling pilot (\(L=32{-}128\), up to \(4\times 10^{3}\) configurations per point, parametric‑bootstrap errors) locates the threshold at \(p_{c}=0.592\pm 0.001\) and gives \(\nu=1.34\,[1.26,1.43]\), consistent with the \(q=1\) class, for the minimal rule; a correlated symmetric‑rule scan finds near‑adiabatic thresholds for all rules at scanned rates, with the constrained rule retaining clearly larger residual hysteresis than the independent rule at the slowest ramp. A quench‑loop identity converts the directional hysteresis‑area difference into the entropy‑production difference of the two directions. A backwards‑run feedback‑demon experiment is specified with frozen observables and a 5 % exclusion criterion.