valueFunctional_at_optimum
The theorem establishes that the recognition value functional attains its maximum of 1 precisely when an agent's sigma charge equals the maximum healthy charge. Researchers constructing consent interfaces in Recognition Science cite this normalization to anchor the local consent criterion that an action preserves value to first order. The proof is a one-line wrapper that unfolds the definition, cancels the ratio via the nonzero hypothesis, invokes the J-cost unit lemma, and finishes by ring simplification.
claimLet $V(s, s) := 1 - J(s/s)$ denote the recognition value functional for nonzero real $s$. Then $V(s, s) = 1$.
background
The module defines the recognition value functional by $V_i = 1 - Jcost(sigma_i / sigma_max)$, where sigma_i is an agent's sigma charge and sigma_max is the maximum healthy charge. Consent for an action holds when the post-action J-cost on each affected sigma ratio does not increase. This rests on the J-cost lemma from the Cost module, which states $Jcost(1) = 0$ and expresses $J(x) = (x-1)^2/(2x)$.
proof idea
The proof is a one-line wrapper. It unfolds valueFunctional to obtain 1 - Jcost(s/s), rewrites the self-division using the hypothesis s ≠ 0, applies the Jcost_unit0 lemma to replace Jcost(1) by 0, and concludes by ring.
why it matters in Recognition Science
This supplies the value_at_opt field inside the ConsentInterfaceCert structure that upgrades the consent interface from scaffold to structural theorem. It normalizes the value functional at the healthy ratio, consistent with the J-uniqueness property and the Recognition Composition Law in the forcing chain. The remaining open question is the empirical bridge from this algebraic criterion to observable physical systems.
scope and limits
- Does not prove the sigma-preserving consensual property for arbitrary actions.
- Does not treat multi-agent interactions or global sigma preservation.
- Does not supply an empirical falsifier or observable test for the consent interface.
formal statement (Lean)
50theorem valueFunctional_at_optimum (s : ℝ) (h : s ≠ 0) :
51 valueFunctional s s = 1 := by
proof body
One-line wrapper that applies unfold.
52 unfold valueFunctional; rw [div_self h, Jcost_unit0]; ring
53