maillard_symmetric
The J-cost function satisfies Jcost r = Jcost r^{-1} for every positive real r. Food chemists modeling the Maillard reaction within Recognition Science would cite this to confirm invariance of the threshold under water-activity inversion. The proof is a one-line wrapper that instantiates the general Jcost_symm lemma at the supplied positive argument.
claimFor every positive real number $r > 0$, the J-cost satisfies $Jcost(r) = Jcost(r^{-1})$.
background
Jcost is the Recognition Science cost function whose symmetry under inversion is already established by the lemma Jcost_symm: for x > 0, Jcost x = Jcost x^{-1}, proved by unfolding the definition, applying field_simp and ring. The Maillard module treats the reaction threshold as the point where the surface-water-activity ratio r_H2O drives J(r) across the band (0.11, 0.13), with equilibrium below threshold and cascade above. This local setting inherits the J-cost definition directly from the Cost module and the phi-ladder structure of Recognition Science.
proof idea
One-line wrapper that applies the Jcost_symm lemma from IndisputableMonolith.Cost (and its JcostCore sibling) at the given positive r.
why it matters in Recognition Science
The theorem is collected as the symmetric field inside the MaillardThresholdCert definition, which bundles below-threshold equilibrium, above-threshold positivity, and this inversion symmetry. It thereby supports the module claim that the Maillard cascade is symmetric in water-activity ratio, consistent with the J-cost crossing at the canonical activation band inside the Recognition Science forcing chain.
scope and limits
- Does not compute the numerical temperature threshold of 140 C.
- Does not derive the phi-fold rate acceleration above threshold.
- Does not address specific molecular steps of the Maillard chemistry.
- Does not extend the symmetry to non-positive arguments.
Lean usage
maillard_symmetric hrformal statement (Lean)
32theorem maillard_symmetric {r : ℝ} (hr : 0 < r) :
33 Jcost r = Jcost r⁻¹ := Jcost_symm hr
proof body
34