RecognitionCostSystem
A RecognitionCostSystem for natural number n packages a cost function on the reals together with a witness that it obeys the Recognition Composition Law, a sigma map from n-tuples to reals, and a positive integer window W. Algebraists constructing the cost layer of Recognition Science cite this structure when assembling canonical systems from the J function. The declaration is a direct structure definition with no proof obligations beyond field typing.
claimA recognition cost system of dimension $n$ consists of a function $c : {R} {to} {R}$, a proof that $c$ satisfies the recognition composition law $c(xy) + c(x/y) = 2 c(x) c(y) + 2 c(x) + 2 c(y)$ for all positive $x,y$, a map $sigma : (Fin n {to} {R}) {to} {R}$, and a positive integer $W$.
background
The CostAlgebra module assembles structures for ratio costs derived from multiplicative recognizers. SatisfiesRCL is the definition of the Recognition Composition Law, the single primitive of Recognition Science: for positive reals x and y, F(xy) + F(x/y) equals 2 F(x) F(y) plus 2 F(x) plus 2 F(y). This identity is the log-coordinate form of a calibrated d'Alembert equation. The structure also draws sigma from decision theory as the gap between private and public reports and canonical arithmetic objects from foundational Peano constructions.
proof idea
The declaration is a structure definition that directly declares the four fields cost, rcl, sigma, and W with its positivity condition. No lemmas are invoked and no tactics are used; the body is empty because the declaration itself supplies the type.
why it matters in Recognition Science
canonicalRecognitionCostSystem instantiates this structure with the J cost function, the RCL_holds witness, and canonicalSigma to obtain the canonical system. The structure supplies the algebraic interface that lets the Recognition Composition Law connect to conservation functionals and finite windows, feeding into downstream results on nucleosynthesis tiers and observer forcing. It closes the packaging step between the primitive RCL and applications that require a positive window length.
scope and limits
- Does not specify the explicit form of the cost function.
- Does not prove that any particular function satisfies the RCL.
- Does not constrain the sigma map beyond its type.
- Does not address existence or uniqueness of such systems.
formal statement (Lean)
521structure RecognitionCostSystem (n : ℕ) where
522 cost : ℝ → ℝ
523 rcl : SatisfiesRCL cost
524 sigma : (Fin n → ℝ) → ℝ
525 W : ℕ
526 W_pos : 0 < W
527
528/-- The canonical conservation functional from Definition 2.7:
529 sum of logarithms on positive coordinates. -/