A Formal Uniqueness Proof for the Recognition Ledger

We derive, in five strictly formal steps, the uniqueness of the recognition ledger—a commutative group equipped with a self-dual cost functional. Step 1 fixes the functional J(x) = 1/2 (x+ 1/x) by syntactic completeness of a terminating, confluent rewrite system. Step 2 proves categorical equivalence between the class of cost models and a single commutative group, eliminating all alternative ledgers. Step 3 shows that every physical constant (α^(-1),G,ℓ₁,ℓ₂) is a categorical invariant; the Poisson and Dirac equations appear as the sole natural transformations of the cost groupoid. Step 4 embeds Peano Arithmetic into the ledger calculus, transferring ω-consistency and closing Gödel loopholes. Step 5 enumerates four minimal empirical counter-models—axial pseudo-boson, neutron electric dipole moment, photon-bath Gdrift, and CMB likelihood Δχ²—whose single failure would falsify the framework. No external assumptions, dials, or supplementary codes are invoked.