Recognition Algebra: The Cost Layer Forced by a Single Composition Law, and the Canonical Structures Built on It

A single functional equation on ℝ₍>0₎, J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y), has, among continuous functions, exactly four solution classes: the degenerate constants F ≡ 0 and F ≡ −1, the hyperbolic family cosh(κ ln x) − 1, and the circular family cos(κ ln x) − 1. The normalization J(1) = 0 and the calibration J″_log(0) = 1 select J(x) = ½(x + x⁻¹) − 1, and we show precisely how: the calibration performs two logically distinct acts, a signature choice (hyperbolic over circular and trivial) and a gauge choice (the unit κ = 1 within the hyperbolic class). This is the exact content of the "zero free parameters" claim, and we prove it categorically: the category of continuous solutions splits into three connected components, each nontrivial component is an indiscrete groupoid (so uniqueness up to unique isomorphism holds fiberwise and initiality does no selective work), and the category as a whole has no initial object. An axiom ledger separates what the equation assumes: existence of a combiner (a congruence condition, not free), its polynomial form, coupling, continuity (load-bearing: a discontinuous solution of the exact equation exists on a Hamel basis), and calibration (which must act on the cost, since the equation itself is blind to κ). We then correct the algebraic profile of the induced cost composition a ⋆ b = 2ab + 2a + 2b: it is commutative, flexible, and third-power associative, but not power-associative (the fourth powers of 1 differ, 106 ≠ 96); its associator 2(a − c) is the coboundary of the potential a → 2a, and is therefore orthogonal to every conserved flow, which is the honest bridge between the cost layer and the ledger layer. The golden-ratio ring ℤ[φ], the ledger of integer circulations (a free abelian group of rank |E| − |V| + c), and the period-8 phase structure are canonical constructions on named additional postulates, not consequences of the equation; we tag each one and prove exactly the part that is theorem.