One Cost, One Composition Law: States, Lattice Fields, Holonomies and Counted Ledgers

We take as premise that a comparison of two positive quantities with ratio x reads J(x) = ½(x + x^−1) − 1, the unique solution of the composition law J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y) with lim_t→0 2J(e^t)/t^2 = 1, and derive what this cost and this law determine under explicit premises. Pairwise readings fix a configuration up to a common unit and a global inversion, and determine it exactly on the domain containing a fixed unit item, with all entries at least one. On a periodic lattice the total cost S of a logarithmic field is the quadratic form of A = −Σ_μ J(T_μ), with T_μ the unit shift, plus a remainder beginning at fourth order; only constant fields are stationary. The Hessian of S at every field dominates 2A, so by the Brascamp–Lieb inequality the variance of every linear observable in the mean-zero finite-volume ensemble e^−βS is bounded by its Gaussian value. No nonconstant real entire function of the reading, in particular no Boltzmann factor e^−βJ, gives a positive-definite time kernel on the line, so the standard one-step transfer operator of e^−βS on the unrestricted slice space is not e^−H for a self-adjoint H (its absolute value is), whereas Wilson's circle kernel is positive definite; the quadratic part has dispersion sinh^2(E/2) = Σ_j sin^2(k_j/2), with E ≤ |k| and, where the dispersion is differentiable, group velocity below one in lattice units. Every nonconstant step cost satisfying the formal law is J(T^n) for a single shift T^n, and calibration selects the nearest neighbour. On U(1) holonomies the nonconstant finite character sums G obeying the cosine equation, the shifted form of the law, are cos nθ, n ≥ 1; unit calibration selects n = 1, and with the sign premise, opposite to that of the lattice action, the plaquette action is Wilson's 1 − cos θ. With the SU(N) action taken as the negative average of the eigenvalue costs it gives 1 − (1/N) Re tr U, while on the whole group the half-trace of SU(2) obeys the law and for N ≥ 3 the only continuous solutions on SU(N) are G = 0 and G = 1. On ledgers of integer counters whose counts are counters plus one, the readable ledgers are those posted from the empty ledger. The composition law enters the classification of J, of step costs and of holonomy costs; the other results use the level sets, convexity or analytic form of J, or follow from the stipulated counting and posting rules. No real-time evolution is derived.