below_threshold_equilibrium
plain-language theorem explainer
The theorem states that the J-cost at ratio 1 is exactly zero, confirming recognition equilibrium for normal hydration below the Maillard threshold. Chemists modeling activation energies in the Recognition Science framework cite it as the base case for J-cost crossing. The proof is a direct one-line application of the unit lemma for the squared-ratio cost function.
Claim. The recognition cost function satisfies $J(1) = 0$, where $J(x) = (x-1)^2/(2x)$ expresses the squared deviation from unity.
background
The J-cost function is defined by $J(x) = (x-1)^2/(2x)$, the squared ratio that quantifies deviation from multiplicative equilibrium in recognition events. In this module the same function measures the recognition cost of surface-water-activity ratios during dehydration. The local setting is the Maillard reaction threshold (F7), where normal hydration keeps $J(r_{H_2O}) = 0$ and equilibrium holds; the upstream lemma Jcost_unit0 establishes the unit case by direct simplification of the definition.
proof idea
One-line wrapper that applies the Jcost_unit0 lemma from the Cost module.
why it matters
This supplies the equilibrium_below field of the MaillardThresholdCert structure, which bundles the full threshold certificate together with the positive-cost and symmetry statements. It anchors the J-cost description of the Maillard cascade to the Recognition Composition Law and the canonical activation band $J(r) ∈ (0.11, 0.13)$.
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