Measure

The ring polynomials AND = ab, OR = a+b−ab, NOT = 1−a, IMP = 1−a+ab restrict to exact Boolean logic on the 2^k idempotents — intersection, union, complement — and the residuation law (x AND a) ≤ b iff x ≤ (a IMP b) holds over all 128³ idempotent triples of Z/510510. On the rest of the ring the same polynomials give a graded logic whose every failing Boolean law fails inside the defect ideal generated by δ(x) = x² − x: the idempotents of any commutative ring form a Boolean algebra under these connectives (standard), so a law's LHS − RHS vanishes in Z[x]/(x_i² − x_i) — every failure is a combination Σ δ(x_i)·g_i. The eleven charted laws each deform in a single variable — each holds whenever that one variable is classical, the others arbitrary: excluded middle by +δ, contradiction by −δ, absorption by δ(1−b), the two distributivities by δ·bc and −δ(1−b)(1−c), modus ponens by −δ(1−b); De Morgan, double negation, and contraposition survive identically. Truth is per-channel: δ vanishes on channel p iff the residue reads 0 or 1, so supp δ(a) is exactly the non-classical channels of a. The graded region has no internal order (a ≤ a already needs idempotence): degrees are not ordered as in fuzzy logic, they are located.

Scope. The defect-ideal law proved; the deformation identities verified exhaustively at Z/30 and on 20,000 triples at Z/510510.

verifier: explore_idempotent_logic.py

The defect composes exactly: δ(NOT a) = δ(a), and δ(a AND b) = δ(a)δ(b) + δ(a)b + aδ(b) — a twisted Leibniz rule (twist a → a²) whose NOT-conjugates are the OR and IMP laws. Additively δ(a+b) = δ(a) + δ(b) + 2ab, the polarization that dies in characteristic 2 — channel 2 is wholly classical (F₂ = {0, 1}), so the graded logic lives on the odd channels. Nesting in the four connectives therefore never creates non-classicality: supp δ(F) lies inside the union of the inputs' defect supports for every AND/OR/NOT/IMP formula F. Bare ring + escapes the scope, its polarization 2ab creating defect from two classical truths (δ(1+1) = 2). A non-classical channel is classicalized in exactly two ways per binary connective — absorption (partner 0 for AND, 1 for OR) or its own inverse (partner x⁻¹ for AND, the NOT-mirror for OR) — and NOT never cancels.

Scope. Proved by substitution; verified exhaustively at Z/30 and on 20,000 pairs at Z/510510; the classicalization census over all p ≤ 23, all non-classical residues.

verifier: explore_idempotent_logic.py

◇a = a^λ reads “true to some degree” (channel 1 iff residue ≠ 0); its De Morgan conjugate □a = 1 − (1−a)^λ reads “fully true” (1 iff residue = 1); and ◇NOT = NOT□ exactly — the measurement's NOT-failure is NOT exchanging the pair, not NOT breaking. ◇ commutes with AND exactly (fields), □ with OR exactly; each leaks on the dual connective at the same per-channel count: two true channels disjoin to false under ◇ iff (1−x)(1−y) = 1 — exactly p − 2 of the (p−1)² true-true pairs per channel, so under ◇ truth can be lost, never fabricated — exactly as □ gains truth over AND by inversion. Over uniform random pairs the per-channel mismatch (p−2)/p² predicts a tuple-level rate of 0.4237 at Z/510510; measured 0.4234. Mixed-form T, 4/5, K, necessitation, and the Halmos closure law hold exactly: an S5-shaped pair. The pair's gap is the defect measurement: ◇a − □a = ◇δ(a) — “true to some degree and false to some degree” is exactly the non-classical region.

Scope. Leak laws proved by substitution, census p ≤ 23; the gap identity exhaustive over all channels.

verifiers: explore_idempotent_logic.py, explore_quantifier_envelope.py

The pair turns measurement into a two-sided certificate for every connective formula (AND/OR/NOT/IMP; bare + stays outside the scope): push NOT to the leaves — the input variables; De Morgan survives off the lattice — and keep per leaf only the two bits (◇a, □a). The Boolean evaluations roof U (◇ at positive leaves, NOT□ at negated — exact there, by the conjugacy) and floor L (□ / NOT◇) bracket the true shadow channel-wise, L ≤ □F ≤ ◇F ≤ U, by an induction whose two inequality slots are exactly the one-directional leak laws. U = 0 certifies the shadow false, L = 1 fully true, U = L the verdict outright — a Kleene-style two-sided evaluation computed in-ring; ◇ alone certifies only falsity, □ alone only truth, so the two-sided decision needs the pair. Slack is located — the uncertainty region lies inside the union of the inputs' defect supports, so classical-input channels are always certified — and counted: the same p − 2 mismatch sets, now as envelope slack at the dual-connective nodes. Read-once formulas achieve both ends of the bracket at p = 5, 7, 11 (rule for the swept family — exhaustive, trees to 4 leaves — observation beyond); at p = 3 the envelope can overshoot — the graded region of F₃ is the single self-inverse point −1, forcing (−1) OR (−1) = 0 and (−1) AND (−1) = 1, the p = 3 rigidity — and dependent shadows sit inside the bracket at p ≡ 1 mod 3: a AND NOT a is fully true wherever δ(x) = −1 is solvable, the floor stuck at 0. The worked decision procedure — certified verdicts under a classical-leaning mixture, zero wrong certificates over censuses and ring-level streams, the ◇-only mask erring at an exactly enumerated nonzero rate — is the script's record.

Scope. The bracket proved; sharpness as tiered above.

verifier: explore_quantifier_envelope.py

The pair is the two ends of one family: for m | λ the partial collapse ◇_m a = a^m filters — m | m′ makes a^m′ a function of a^m — and the bit-valued gate gate_m(a) = □(a^m) reads the order: [ord a | m] per channel. gate₁ = □, gate_λ = ◇, monotone along the divisor lattice, meet-exact (gate_gcd = gate_m AND gate_m′), gcd-complete (any integer exponent's gate is a divisor rung's), outputs on the idempotent lattice — d(λ) genuine quantifiers between “fully true” and “true to some degree” (at Z/510510: λ = 240 and 20 of them, all distinct). Repeated leaves are the ladder: □ of the m-fold AND of one leaf is gate_m — what a repeated leaf computes beyond the pair is exactly order information. The ladder bar (a gate must decide what the pair cannot) is met by the envelope's own dependent shadow: □(a AND NOT a) = [Φ₆(a) = 0] identically, and 6 — the smallest non-prime-power order — is the ladder's first join defect (gate joins are exact below it and first leak at lcm(2, 3)), so the shadow is gate₆ AND NOT(gate₂ OR gate₃). Pair-decidability of this shadow is the splitting law of Q(ζ₆): split channels p ≡ 1 mod 3 blind the pair, inert channels are vacuously false, and the ramified p = 3 is pinned by the rigidity — which is Φ₆'s ramification (disc −3), the ladder folding there (gate₆ = gate₂). The ladder decides the shadow on every channel with zero errors, on a formula where the pair-only floor is identically 0 (at Z/510510 the ladder earns exactly channels 7 and 13). Its own wall: the leaf ladder measures exactly the joint order profile (ord a, ord(1−a)) — shadows with non-cyclotomic root conditions escape ([a²(1−a) = 1] first collides at p = 11), and [ab = 1] escapes every single-leaf measurement: relative order is not a leaf quantity. Beneath the gates, the power maps that are honest projections are exactly the idempotent exponents e² ≡ e mod λ — 2^ω(λ) Hall projections splitting truth-degree by λ's primes (at Z/510510: 8): the logic's projection lattice is the idempotent lattice of the exponent ring Z/λ, one level down (index coordinates).

Scope. Proved: the gate identities, the lattice laws, and the Φ₆ splitting reading. The shadow decision is exhaustive plus ring-level streams; the counts at Z/510510 and the p = 11 collision are censuses.

verifier: explore_quantifier_ladder.py

The whole hierarchy is one chart. A measurement program is a word — a formula over the leaves in MUL/INV/NOT, the meadow alphabet, INV being the meadow inverse that inverts each unit channel and fixes 0, so division is total; dropping NOT leaves the monomial alphabet, whose words are the aⁱb^j with exponents in Z. A word is read by gates gate_m(w) = [w^m = 1] under a Boolean readout, and its three budgets — alphabet level, word ops, gate count — bind one per level, the gate-budget grading. The measurement walls are the table's ∞ rows: monomial gates are constant on conjugacy orbits — the sets the power automorphisms x → x^u sweep — so an orbit-splitting shadow has no finite monomial-gate price at any count — and every alphabet wall falls at a deeper row until the meadow closure breaks the last, the wall returning as a price that grows without bound (the height floor).

The symbols the chart carries: dlog is the discrete log of a unit in its channel; ω(d) is the number of distinct prime factors of d and q_min(d) its least prime factor; m(C) is the minimal number of monomial equations cutting an orbit's group C exactly; and h_p(c) is the height of the residue c, the fewest word operations that build it from one graded leaf. A torsion order is of the codeword species for a shadow when the shadow's fiber over it is a nonzero codeword of a cyclotomic cyclic code, and of the whole-group species when that fiber vanishes identically. The word and cut rows are charted on Words and curves, the price column on The price of a read.

Scope. Every row is one of this section's claims at its stated tier; the ∞-row mechanism (orbit-constancy of monomial gates) is proved.

verifier: explore_gate_budget.py