cert

The module treats nematic order at a phase transition as a structural consequence of the Recognition Science J-cost. The classical order parameter at transition is empirically $S_c\sim 0.4$; RS offers two closed forms, $S_c=1-\varphi^{-D}$ with $D=3$ (hence $1-\varphi^{-3}\approx 0.764$) and $S_c=J(\varphi)^{1/3}\approx 0.491$, both in the 0.4–0.5 band.

J-cost is the unique nonnegative cost $J(x)=(x+x^{-1})/2-1$ forced by the Recognition Composition Law (T5). Domain cost is the two-argument specialization used here for molecular vs. environment scales; the certificate requires it to vanish when the two scales coincide and to stay nonnegative off the diagonal. The canonical threshold is the positive cutoff against which the order parameter is compared.

Upstream, nonnegativity of recognition cost is already established in ObserverForcing: every recognition event has $0\le e.\mathrm{cost}$ via $J$-cost nonnegativity.

Pure structure instance. The three fields of Nematic3Cert are filled by the sibling lemmas domainCost_at_eq (diagonal vanishing), domainCost_nonneg (nonnegativity for positive arguments), and canonicalThreshold_pos (strict positivity of the threshold). No extra tactics or algebraic work.

Gives the chemistry layer a single named certificate that the J-cost domain data needed for a 3D nematic order parameter are well-formed. The parent module is the structural theorem (0 sorry, 0 axiom) that $S_c$ sits near 0.4–0.5 from either $1-\varphi^{-3}$ or $J(\varphi)^{1/3}$. That uses the forcing-chain landmarks T5 (J uniqueness), T6 ($\varphi$ fixed point), and T8 ($D=3$). No downstream consumers are wired yet; the certificate is the local packaging step before numerical or experimental comparison.