Words and curves

The word family — gates on aⁱb^j, exponents in Z via the meadow inverse — measures exactly the leaf discrete logs up to one shared unit: profiles coincide iff (a′, b′) = (a^u, b^u) for a single automorphism x → x^u (a cyclic-group fact, any letter count; the two pencils ord(aⁱb), ord(ab^j) already generate, the three words ord a, ord b, ord ab do not). A single leaf is the one-letter case — an order is a discrete log up to its own unit — so the genuinely two-leaf content is the relative position: where a is a primitive root the profile pins dlog_a b exactly, and [ab = 1], which walls the one-leaf ladder, is one gate. A two-variable shadow is therefore word-decidable iff constant on conjugacy orbits: every multiplicative coset [aⁱb^j = 1] is decided, and the additive world is blind except its crystallographic skeleton (below). The two-leaf read gate_m(ab⁻¹) is the equality grading on the units: [dlog a ≡ dlog b mod (p−1)/gcd(m, p−1)] — reflexive, symmetric, AND-transitive, unit-translation-invariant — a divisor-indexed congruence ultrametric on the diagonal; it is not an inner product: every word's discrete log is a linear form in the leaf discrete logs, and the bilinear candidate factors through the two single-leaf orders.

Scope. Proved (the theorem and the grading).

verifier: explore_joint_measurement.py

A point (a, b) of a locus is stable under the automorphism x → x^u when its conjugate (a^u, b^u) lies on the locus too. On a line a + b = c with c ≠ 0, the inversion u = −1 forces stable points onto ab = 1 and stability under every u forces every order-d element to one trace, φ(d) ≤ 2 — only torsion of order 1, 2, 3, 6 survives (order 4 lands on c = 0 = [ab⁻¹ = −1], wholly decided): c = ±2 keep (±1, ±1), c = 1 keeps the Φ₆ pair (iff p ≡ 1 mod 6 — the word-visible part of the dependent shadow a AND NOT a), c = −1 the Φ₃ pair (iff p ≡ 1 mod 3), and every other line is word-invisible. Deciding channels are exactly p ≤ 3 — the trivial Galois group, and the same arithmetic one level up is the p = 3 rigidity.

Scope. Proved, plus census over all p ≤ 100.

verifier: explore_joint_measurement.py

Reach is alphabet-graded. Mixed words on one leaf's letters {a, 1−a} already refine the order profile — first at p = 11, where the residues 9 and 3 share the profile (ord a, ord(1−a)) = (5, 5) and the word a²(1−a) separates them: the wall witness was itself a word. The four letters {a, b, 1−a, 1−b} add the six ±1-ratio affine lines ([a + b = 3] stays blind, witnessed at the Φ₆ points of p = 7). And the full meadow closure is complete: NOT(a⁻¹)·(NOT a)⁻¹·a = −1 on units a ≠ 1, so every constant is a one-leaf word and [x = c] = gate₁(xc⁻¹) pins every residue (0 and 1 need no word: [x = 1] is □x and [x = 0] is NOT ◇x, the two bits of the pair). The meadow inverse is the wall-breaker: every measurement wall in the hierarchy, the single-leaf wall included, is a fact about inverse-free alphabets.

Scope. Completeness proved; the alphabet grading a rule for the swept ranges.

verifier: explore_joint_measurement.py

A unit pair of torsion order d = lcm(ord a, ord b) is orbit-stable for a shadow [f = 0] iff its collected monomial fiber Σ c_mw_m^t (w_m = a^i_mb^j_m, a character sum on Z/d) either vanishes identically — f contains the whole cyclic group ⟨(a, b)⟩, the finite shadow of Lang's torsion-coset theorem — or is a codeword of the cyclotomic cyclic code (Φ_d) ⊆ F_p[T]/(T^d − 1). Those two cases are the two species: the whole-group species, and the codeword species.

Scope. Proved, both directions.

verifier: explore_curve_skeleton.py

That code's minimum distance is exactly q_min(d), the least prime factor of d: the longest run of consecutive units mod d is q_min − 1 (CRT), no nonzero M-term character sum vanishes at M consecutive points (Vandermonde), and the q-gon (T^d − 1)/(T^d/q − 1) realizes weight q. So a shadow with M monomials admits codeword-species torsion of order d only if q_min(d) ≤ M — the crystallographic restriction generalized (the line family is the M = 3 row, its unit coefficients and constant term cutting that menu down to φ(d) ≤ 2) — and a new weight class arrives at every prime. First at M = 5, inside the conic budget: the pentagon conic y² + xy + x + y + 1 = 0 holds exactly one stable order-5 orbit, (θ, θ²) under conjugacy, at every p ≡ 1 mod 5 — degree-2 shadows see torsion no line can. Weight 2 is the antipodal class (even torsion unbounded); weight 1 is empty, and odd d admits no weight 2. The run bound is field-free, so characteristic p adds no sporadic minima (in characteristic 0 the same minimum follows from Mann's vanishing-sums theorem).

Scope. Proved field-free (the menu law); the pentagon a rule.

verifier: explore_curve_skeleton.py

Bounded degree changes realizability, not the menu. Degree D reaches order d through the codeword species iff some exponent pair A, B with gcd(A, B, d) = 1 puts a nonzero (Φ_d)-codeword inside the triangle exponent set {Ai + Bj mod d : i + j ≤ D} — the exponent reduction (criterion, proved). The minimum degree is 1 for every even d (the antipodal word); for odd prime d = q the code has dimension one, so the triangle must cover all of Z/q — a modular postage-stamp problem, the triangle cover number c(q) = Θ(√q), bracketed between √(2q) − 3/2 and 2⌈√q⌉ − 1 and censused exactly for the 34 odd primes ≤ 149. c(q) is exactly the minimal diameter of a double-loop network on Z/q — the least D such that some step pair reaches every residue as Ai + Bj mod q with i + j ≤ D — whose literature (Wong–Coppersmith 1974; Fiol et al.) proves the sharper floor ⌈√(3q)⌉ − 2, and the census collapses onto it: 28 of the 34 values sit on the floor exactly (c(5) = 2, the pentagon's tightness, among them), the other six one above. The minimum degree is q_min(d) − 1 for every other odd d. The composite tax prices primality itself: at equal least prime, prime orders cost √-scale degree where composite orders pay their full least prime linearly (5 vs 25 and 35: degree 2 vs 4; 13 vs 143: 5 vs 10) — degree reads compositeness where monomial count cannot.

Scope. Criterion and the even/other-odd prices proved; the census a rule as stated; the double-loop identification proved, the ⌈√(3q)⌉ − 2 floor cited from the literature, the collapse arithmetic on both. Coverage is whole: prime powers at any exponent ≥ 2 by the line pigeonhole (a mod-q line meets the degree triangle in at most D + 1 points); composites with q_min ≤ 7, with q_min² | d, or with second prime > 3(q_min − 1)/2 by the slice forcing (a codeword's mod-q slices are pairwise congruent mod the inner code, and the menu law with line counts forces full q-gons); two close primes by the two-fat-slice exclusion; everything past two prime factors, counted with multiplicity, by the gon theorem (both below) — so the q_min(d) − 1 price is proved at every other odd d.

verifier: explore_realizability.py

For every two-prime d = q·q′ — two close primes, past the slice-forcing boundary — the question turns combinatorial and closes. The rook reduction (criterion, proved): with q′ prime the slices differ by constants, an evader — a nonzero codeword inside the triangle exponent set — is a tensor-sum, a row term plus a column term on the CRT grid, and a codeword lives in the triangle iff the triangle's complement in the q × q′ CRT grid is rook-disconnected — matched row/column partitions with every off-diagonal block inside the triangle; counting the blocks alone re-proves the forcing boundary q′ > (3q − 4)/2, and no prime lives between it and 3(q − 1)/2. The arithmetic is closed by the staircase lemma: the weak form (≤ 2s + 1 lines carry ≥ D + 1 − s lattice points, proved by pure counting for every s < (D + 1)/2) closes 46 of the 98 close pairs to q ≤ 97 on its own, and the strong form (at most s + 1 lines at depth D + 1 − s) is proved at every modulus and depth and every unit B. Consequence: no covering system with two fat slices evades at any two-prime d, any D ≤ q − 2, and the sweeps stand as verification — complement connected at every swept close pair and unit B, no witness.

Scope. The rook reduction and both forms of the staircase lemma are proved; the sweeps stand as verification. Composites of three or more prime factors, with multiplicity, close by the gon theorem.

verifiers: explore_cover_exclusion.py, explore_staircase_reduction.py

Everything past two prime factors closes by codeword geometry alone. The gon theorem: for every odd d, a nonzero (Φ_d)-codeword of weight below 2q_min(d) is a scalar gon — one constant on one coset of a prime-order subgroup of Z/d — by induction over the largest prime's slices, a codeword's restrictions to the residue classes mod one prime q | d, each a vector of length d/q; a congruence among the slices within one coset carries the non-squarefree case. The coset collapse: an evader — a nonzero codeword inside the triangle exponent set — has q_min-slices differing by multiples of one common gon (two different gons would overrun the 2q_min − 2 weight budget), so its support collapses into a coset of a two-prime-order subgroup holding at most D + 1 < q_min triangle points, and the menu law's own minimum distance finishes it. The collapse covers every odd d with three or more prime factors counted with multiplicity — 11·13² is inside — and q_min² ∤ d, field-free, at every D ≤ q_min − 2; at two primes the coset is everything — exactly where the rook machinery was needed. The composite row is closed: D_min(d) = q_min(d) − 1 is a proved rule at every odd composite non-prime-power d, the census empty at every exponent-pair class of the six smallest moduli with a close second prime, a 300-class four-prime sample beside them — 7,828 checks in all, no evader — and the q-gon entering at D = q_min − 1 exactly.

Scope. Both proved — the gon theorem at every odd modulus, the collapse field-free at three or more prime factors with multiplicity and q_min unsquared; the census stands as verification.

verifier: explore_multiprime_exclusion.py