The price of a read

At the monomial level, gate shadows are exactly the character kernels of the discrete-log torus, 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, slices are pinned, and every rank the membership gates drop comes back, gate for gate, in separators — no covering system with repeated or composite moduli 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 gate_m(ab⁻¹) rungs of the equality grading — cost exactly 1. On the orbits: the pentagon 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 orbit reads at 3 at p = 7 and 4 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 graded leaf, 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.

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, order fails, at all three points tested. The walls do not depend on the temperature — 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 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 > 1.346, 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