The price of a read

At the monomial level, gate shadows are exactly the character kernels of the discrete-log torus (Z/n)^L — n = p − 1 the order of the channel's unit group, one coordinate per leaf — and a stable torsion orbit — the generators of the cyclic group C = ⟨(a, b, …)⟩ of order d — costs exactly m(C) + ω(d) monomial gates (ω the number of distinct prime factors), m(C) the minimal number of monomial equations cutting C exactly = the rank of the ambient modulo C (at two leaves: 1 if d = n, else 2) — coset gates plus order read. Upper: the canonical program — m(C) gates cutting C, then for each prime q | d one gate on a frame word — a word of full order d on the orbit. Lower: forced separators — per prime q | d, its own gate — plus covering rigidity (the trade-exclusion): any coset the membership gates fail to kill must be covered by separator slices — a separator gate's kernel cut down to that coset — and a slice is pinned — a coset of order coprime to d threads every slice through one point — so every rank the membership gates drop comes back, gate for gate, in separators — no covering system of t membership gates and s separators, repeated or composite moduli allowed, ever beats t + s ≥ m(C) + ω(d). The lower bound never uses the leaf structure, so the law holds in any finite abelian ambient group. At one leaf the law is the order-cost law (rule, proved field-free both directions): [ord x = d] costs exactly ω(d) + 1, dropping to ω(n) at d = n — primitivity reads one gate cheaper exactly at the top — and filters — [x = 1] and the gates gate_m(ab⁻¹) of the equality grading — cost exactly 1. On the orbits: the order-5 orbit of the pentagon conic reads at 3 = 2 + 1 gates against its cut cost of 5 collected monomials — deciding a locus and cutting it are different currencies, neither dominating; the Φ₆ stable orbit of the line a + b = 1 reads at 3 = 1 + 2 at p = 7, where d = n, and 4 = 2 + 2 at p = 13, so per-channel prices are not uniform across a rung. Character kernels make a readout's atoms — the smallest sets its gates can tell apart — obey Lagrange arithmetic: the subgroup-cell bound (rule, proved), the ambient form the law sharpens on cyclic targets.

Scope. Theorem proved both directions, any finite abelian ambient. Censuses stand as verification: all 337,054,176 structured ≤ 4-gate candidates at the three-leaf m(C) = 3 orbit, four fresh configurations sized so a trade would fit under the law; the one-leaf law exhaustive over lattice sweeps n = 6..360 plus boundary targets at n = 2310.

verifiers: explore_orbit_cost.py, explore_trade_exclusion.py, explore_gate_budget.py

The meadow level collapses gate count to 1 — gate₁ of a decision word — at a price moved wholly into word ops: only cum(L) distinct words — the running total of the formula-counting recurrence — exist at cost ≤ L, so the dearest residue's decision-word cost grows without bound with p. That cost is the constant's height h_p(c), the fewest word operations that build c from one leaf holding any residue other than 0 and 1, and it carries the whole price: [x = c] is decided by one gate and at most h_p(c) + 2 word ops. The deleted place returns as the price of omniscience: which residue you decide carries a height tax that grows without bound. The wall-breaker word is optimal — h_p(−1) = 6 exactly at p = 5, 7, the cheapest nontrivial constant (observation; heights are not monotone: h(4) < h(3) at p = 7) — and one ring gate reads all k channels in parallel, so every per-channel price above is already the ring's.

Scope. The counting bound and the one-gate-plus-h_p(c)+2 decision are proved; the optimality of the wall-breaker word is an observation at p = 5, 7 and the non-monotonicity of heights one at p = 7.

verifier: explore_gate_budget.py

For product-form f the ring-total factorizes per channel in any commutative semiring (the generalized distributive law + CRT), and f equals the normalized product of its own channel marginals iff f is product-form — a semiring-indexed rank-1 test, exact in each of the three semirings tested. On relations the verdicts are semiring-stable (a 0/1 product form can be taken with 0/1 factors — the lemma needs only that the semiring has no zero divisors): equality and divisibility pass, the order relation x ≤ y fails, at all three points tested. The walls do not depend on the temperature, the semiring the total is taken in: Bayesian sum, tropical max, and Boolean feasibility agree on what cannot factor.

Scope. Exhaustive at Z/30 (sum/max/Boolean) and Z/210 (Boolean + sum); the classical family is the generalized distributive law (Aji–McEliece) and valuation algebras (Shenoy–Shafer) — ours is the tower-side composition.

verifier: explore_evidence_calculus.py

An e-variable is a nonnegative statistic of mean at most 1 under the null — the capital of a bet against it. The support cylinder A_S = {x : x·e_S = x} has exact probability μ(e_S), so E_S = 1[A_S]/μ(e_S) is an exact e-variable, and the product of two is an e-variable iff the two events are CRT-independent (S ∪ T = all channels) — the multiplicative valuation law is the bookkeeping. The classicality e-variable reads the defect δ(x) = x² − x; composing it with the collapse x → x^λ — λ the ring's universal period, so the power map sends every unit to 1 — returns ∏_p∈S p/2 for every x (mean 105/8 at Z/210 with S = all channels): the measurement fabricates classicality, and the e-calculus is exactly the frame that rejects it — bets precede measurement, e-processes on the ring adapted to the pre-collapse filtration. Merging shows the same arrow: min and harmonic merges are e-valid on all pairs, the arithmetic merge is exactly tight (the Vovk–Wang canonical merge, exact on the ring), and past it merges die — quadratic with an exact witness of mean > 1.345, max-merge at worst mean exactly 2 − 1/N. The max-tropical end fabricates evidence, the min-tropical end only wastes it: the same one-directionality as the pair's destructive interference under OR — truth can be lost, never fabricated — now enforced by a formalism. The e-calculus is not a third semiring point but a second-order axis (bets over masses) anchored at the summing end, whose log-shadow is the tropical valuation.

Scope. The product rule exhaustive at Z/30 and Z/210 (all idempotent pairs; Z/510510 all 128² by count formula, spot-enumerated); the merge verdicts are the script's record. The e-value frame is Shafer–Vovk's, the merges Vovk–Wang's — the ring composition is the claim.

verifier: explore_evidence_calculus.py