Recognition Cost from the Symplectic Trace Identity The Reciprocal Composition Law and the Reciprocal Symmetry are the Conservation Identities of the Area-Preserving Ledger

The canonical recognition cost J(x) = ½(x + x⁻¹) − 1 is the unique function that is reciprocal, normalized, calibrated, continuous, and obeys a reciprocal composition law (RCL). In that account both the RCL and the reciprocal symmetry are inputs. This paper shows that, once the cost is taken to be the affine trace character of an area-preserving ledger, both relations are forced as algebraic identities rather than assumed. A double-entry ledger is a two-dimensional phase space, a recognition event is a linear map, and the principle that an event creates no net imbalance is exactly preservation of the oriented area form. For two-dimensional maps that is the determinant condition det M = 1, membership in SL(2, ℝ), which in this dimension is the symplectic group Sp(2, ℝ). We choose as the scalar cost observable the calibrated trace character C(M) = ½ tr M − 1; area preservation does not by itself exclude nonlinear class functions of the trace, so this choice is a stated hypothesis, fixed by normalization and unit curvature. The trace functional sorts the conserving events into three regimes: it equals the canonical cost J on the positive-eigenvalue hyperbolic branch, it vanishes on the parabolic shears, and it is nonpositive on the elliptic rotations. For the trace character the two-dimensional Cayley-Hamilton identity yields the trace identity tr(AB) + tr(AB⁻¹) = tr(A) tr(B), which read on the split torus is exactly the RCL; the same identity gives tr(M⁻¹) = tr(M), which is the reciprocal symmetry. So conservation, with the trace-character cost choice, supplies as theorems the two relations the uniqueness account assumed. The cost is a function of the hyperbolic translation length ℓ(M) of the conserving motion, C(M) = cosh(ℓ(M)/2) − 1 with ℓ(M) = 2 arcosh(½ tr M) on the hyperbolic branch. One further input is named: the convex branch, which makes the balanced ledger a stable minimum. The trichotomy locates a clean division of labor for the wider program: the hyperbolic sector carries the nonnegative cost, while the compact elliptic sector SO(2) ≅ U(1) carries no positive cost and is the natural geometric home of the quantum phase developed separately.