IndisputableMonolith.Chemistry.PhaseSeparationFromJCost
Defines phase separation in RS chemistry from the J-cost: a domain cost, a positive canonical threshold, and a PhaseSepCert witnessing separation when cost exceeds threshold. Chemists and RS auditors tracking immiscibility from the cost law would cite it. Mostly definitional, with nonnegativity and positivity lemmas over Cost and Constants.
claimThe module introduces a domain cost built from the J-cost $J(x)=\frac{x+x^{-1}}{2}-1$, a canonical threshold $\tau>0$, and a phase-separation certificate asserting that domains separate whenever the domain cost meets or exceeds $\tau$. Supporting facts: domain cost is nonnegative and vanishes at equilibrium equality; the threshold is strictly positive; the certificate type is inhabited.
background
Recognition Science forces the unique cost $J(x)=\cosh(\log x)-1$ (equivalently $(x+x^{-1})/2-1$) via the Recognition Composition Law. The Cost import supplies $J$ and its algebraic identities; Constants supplies RS-native units (including the tick $\tau_0$).
In the chemistry layer, mixing versus pure-phase configurations is scored by excess $J$-cost. This module packages that score as domainCost, fixes a canonicalThreshold for the onset of separation, and wraps the comparison in the structure PhaseSepCert. Sibling lemmas record nonnegativity, the on-equilibrium value, and positivity of the threshold.
proof idea
Definition-and-certificate module, not a deep proof development. domainCost is defined from $J$; domainCost_nonneg and domainCost_at_eq are elementary consequences of Cost facts. canonicalThreshold is a fixed positive scale with canonicalThreshold_pos. PhaseSepCert is a Prop/structure bundling the cost-versus-threshold inequality; cert and cert_inhabited supply a canonical inhabitant. No multi-step tactic argument beyond unfolding and Cost lemmas.
why it matters in Recognition Science
Gives Lean infrastructure for reading chemical phase separation off the same $J$ that the forcing chain uniquely determines (T5 J-uniqueness, RCL). No downstream used_by edges are recorded yet; the module is a chemistry-side landing pad for later theorems that connect domain cost excess to immiscibility, binodals, or multi-phase equilibria. It keeps the chemistry story pinned to the cost law rather than to phenomenological free-energy fits.
scope and limits
- Does not derive experimental phase diagrams or critical exponents.
- Does not treat multi-component or continuum field models beyond the certificate interface.
- Does not prove the numerical value of the threshold from T0–T8 alone.
- Does not link domain cost to the mass ladder, alpha band, or eight-tick octave.
- Does not claim thermodynamic stability proofs outside the PhaseSepCert inequality.