A Coherence Theorem for Conserved Comparison Ledgers: Structural Axioms Pin Down the Scale, the Cost, and the Ratio

A conserved comparison ledger is an abstract system in which each state carries a positive ratio r(s), every measurement factors through that ratio, and pairwise comparisons are scored by an admissible cost J. A separate classification fixes the admissible costs with polynomial combiner and, under a units normalization, selects the representative Jcost(x)=12(x+x−1)−1. This paper installs that classified cost in a ledger and determines what the ledger assumptions force. The argument has three structural steps and one neutrality axiom: H1 forces the scale ratio to σ=φ; H2, using that classification, forces J=Jcost; H3 reads r(s) from a constrained cost minimizer of an internal positive vector, and A3 imposes zero total log-charge; together they force r≡1, make the measurement map constant, and collapse the observational quotient to a singleton. The assumptions are also sharp in the following sense: removing H1, H2, H3, or A3 admits an explicit ledger satisfying the remaining meaningful assumptions while losing the corresponding conclusion. The examples verify joint satisfiability, include non-uniform internal data whose variational minimum gives r=1, and show nontrivial sector dynamics when A3 is not imposed. A final example realizes the full axiom package on the patch space of the Fibonacci quasicrystal, with the scale and the charge-sector structure supplied intrinsically by the tiling and the remaining installations stated explicitly. We frame this result as a coherence theorem for a deliberately chosen axiom system: each hypothesis is individually natural and, by design, controls a single output, so their joint forcing of (σ,J,r) exhibits the internal consistency of the axiom package rather than asserting that any independently given system must realize it.