cost_algebra_certificate
plain-language theorem explainer
The cost algebra certificate bundles five algebraic properties of the J-cost function and its shift H: satisfaction of the Recognition Composition Law, normalization at unity, explicit control of the associator defect under raw composition, left cancellation on the nonnegative domain, and the d'Alembert equation for H. A physicist or mathematician verifying the algebraic backbone of Recognition Science before proceeding to forcing chains or mass ladders would reference this result. The proof is a direct term that packages five upstream theorems
Claim. Let $J(x) = ½(x + x^{-1}) - 1$. Then $J$ satisfies the Recognition Composition Law $J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y)$ for $x,y > 0$, $J(1) = 0$, the operation $a ★ b := a + b + 2ab$ satisfies $(a ★ b) ★ c = a ★ (b ★ c) + 2(a - c)$ for all real $a,b,c$, left cancellation holds ($a ≥ 0$ and $a ★ b_1 = a ★ b_2$ imply $b_1 = b_2$), and $H(x) := J(x) + 1$ satisfies $H(xy) + H(x/y) = 2H(x)H(y)$ for $x,y > 0$.
background
The Recognition Composition Law is the primitive of Recognition Science: for a function F, F(xy) + F(x/y) = 2F(x)F(y) + 2F(x) + 2F(y) when x,y > 0. The J-cost is the specific solution J(x) = ½(x + x^{-1}) - 1, whose satisfaction of the law is recorded in RCL_holds. Its shift H(x) = J(x) + 1 converts the law into the standard d'Alembert equation H(xy) + H(x/y) = 2H(x)H(y). The raw composition a ★ b is the unnormalized form whose associator defect equals 2(a - c), as stated in costCompose_assoc_defect. Left cancellation on the nonnegative range is given by costCompose_left_cancel. The local setting is the CostAlgebra module, which packages these properties as the algebraic certificate for the forcing chain.
proof idea
The proof is a term-mode construction that directly returns the five-tuple consisting of RCL_holds, J_at_one, costCompose_assoc_defect, the lambda fun _ _ _ ha h => costCompose_left_cancel ha h, and H_dAlembert. No tactics are applied beyond the direct assembly of these previously established results.
why it matters
This declaration collects the core algebraic properties required for the Recognition Science framework, as listed in its documentation: RCL satisfaction, normalization, reciprocal symmetry, non-negativity, controlled defect, cancellation, monoid structure on the shifted operation, d'Alembert for H, and uniqueness modulo regularity. It supports the transition from the functional equation to the forcing chain steps T5 through T8, including J-uniqueness and the derivation of three spatial dimensions. Since used_by is empty in the current graph, it serves as a summary interface rather than an intermediate lemma. It touches the open question of regularity assumptions needed for full uniqueness of J.
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