Quadratic windows

At four such windows — the golden 1/φ = [0; 1, 1, …]; the silver √2 − 1 = [0; 2, 2, …] (Pell weights 1, 2, 5, 12, 29, …); the bronze [0; 3, 3, …]; and √3 − 1 = [0; 1, 2, 1, 2, …] (period 2) — the completion is an odometer at every one: the successor and the unit's shift action read at bounded lookahead (≤ period + 1), while 2n, 3n and ⌊n/2⌋ are gated with delay growing with range — the deepest agreeing pair whose images differ grows in step with the deepest agreement the scan realizes at all, at every range tried, no plateau — and the count of realized depth-t cells grows by ε itself. The same scan at the period-3 windows [0; 1, 1, a], a = 2, 3, 4 returns the same verdict. So the reading is not decided by membership in Z[α], which every integer has, but by the unit group: the unit shift reads and the tested non-units fail.

Scope. Rule at scanned scope: exhaustive to 3·10⁵ per window, the failure-depth reading taken at three ranges, and at the same bound for the three period-3 windows. Beyond the tested windows the unit-gate statement is a conjecture; the blocks below carry the failing half at named families — by proof where a comb telescopes, at scanned scope where the boundary family does the work instead.

verifiers: explore_ostrowski_window.py, explore_period3_borrow.py

At every constant-a window with a ≥ 2 the ×a half is proved, no range cap: the even comb Y_K = q₂ + q₄ + ⋯ + q_K maps under ×a, by the telescoping borrow, to q_K+1 − a with bottom digit 0, while Y_K + q_K — the same string with its top digit raised — maps to q_K+1 + a·q_K − a with bottom digit 1: pairs agreeing to unbounded depth K whose images differ at the lowest digit, unconditionally in both K and a — one comb with no parity condition where the Zeckendorf family needed a parity stripe. Both inputs converge in the completion to the infinite even comb, their images to two distinct points — ×a has no continuous extension, and no constant-a completion is a ring; silver's ×2 and bronze's ×3 are the a = 2 and a = 3 instances. The one degeneracy is a = 1, where legality caps the bottom digit at 0 and squeezes the witness string out — exactly where Zeckendorf's parity-striped comb takes over.

An alternating alphabet relocates the flip, not the shape: at every period-2 window [0; 1, a, 1, a, …] with a ≥ 2 — √3 − 1 the tested a = 2 instance — the same down-borrow holds at every odd position, the positions whose digit cap is a, so the odd comb q₁ + q₃ + ⋯ + q_K telescopes under ×a to q_K+1 − 1 with b₁ = a, while the comb plus q_K maps to q_K+1 + a·q_K − 1, whose legal string has b₁ = 0 at odd K ≥ 3 (a − 2 at the K = 1 edge — distinct either way): pairs agreeing to unbounded depth whose images differ at b₁ — here a₁ = 1 pins b₀ at 0 exactly as Zeckendorf's alphabet does, yet no parity stripe appears; the degeneracy moves the flip one position up, ×a again has no continuous extension, and no period-2 completion is a ring either.

And when the borrow's span mismatches the comb's spacing, the discontinuity survives with a new carrier: at the period-3 window [0; 1, 1, 2, …] the large quotient's positions sit three apart while its borrow still spans two, so no single identity telescopes — the intermediate quotients' own rewrites chain, and every extension of the comb u_M = q₃ + q₆ + ⋯ + q_3M (teeth at the positions ≡ 0 mod 3, whose digit cap is 1) cascades to the bottom, landing the image 2·u_M on one of two legal strings by the parity of M: Zeckendorf's parity stripe, returned one level up, the flip still at b₁ — the lowest position whose cap admits a nonzero digit — so ×2 has no continuous extension and that completion is not a ring either.

Scope. The ×a discontinuity is a theorem at every constant-a window and at every period-2 window [0; 1, a, 1, a, …], both for every a ≥ 2, the comb families checked by greedy extraction at a = 2..7 (constant-a to depth 40, period-2 at odd K ≤ 41); ×2 is a theorem at the period-3 window [0; 1, 1, 2, …], its comb checked at M ≤ 26. The period-3 gate scans are scoped with the gate above.

verifiers: explore_silver_discontinuity.py, explore_constant_a_borrow.py, explore_period2_borrow.py, explore_period3_borrow.py

Across the period-3 family [0; 1, 1, a] the parity of a splits the comb's fate, and the split is a theorem at every a ≥ 3 (a = 2 is the cascade above): at each position K holding a large quotient, the two unit quotients above it give q_K+2 = 2q_K + q_K−1, so a top coefficient c on q_K−1 can absorb the next comb extension's cost a·q_K in place — everything below untouched — precisely when 2c = a − 1. Odd a carries that conserved half-coefficient (a − 1)/2 upward forever and freezes the image bottom at (0, 1, (a + 1)/2), the comb read with no flip; even a's best integer coefficient a/2 − 1 sheds one unit per extension, and b₂ stripes between a/2 and a/2 + 1 by the parity of M, both closed forms proved. But the split is a fact about the comb, never about the gate: every window a ≤ 7 carries the boundary family of the next section instead, whose witness never consults the parity of a at all.

Scope. Theorem at every a ≥ 3, its closed forms checked at a = 3..9, M ≤ 26.

verifier: explore_odd_a_freeze.py

The engine is the maximal string — the legal string of q_K − 1 — and that string is a theorem at arbitrary tail, periodicity never used: greedy on q_K − 1 = a_K q_K−1 + (q_K−2 − 1) recurses in steps of two, so the largest integer supported below position K writes as the alternating cap-filling — digit a_K−2i at position K − 1 − 2i, plus b₀ = a₁ − 1 at odd K — supported on one position parity and flipping whole with the parity of K, the flip at the lowest position whose cap admits a nonzero digit: the flip address derived for this family at every window at once.

Some reals carry two legal codings — two distinct points of the completion over one point of the line, the coding boundary. Dividing near-denominator combinations q_K + u q_K−r − t (u, r, t small and fixed, K running) by a inside one residue class of the convergent pair (q_K, p_K) mod a — the class chosen so the resulting inputs' limit misses the coding boundary while the image limit sits on it; a | t with p-residue 0 is exactly the boundary case and is excluded — yields one convergent input family whose images land beside the boundary on the side the convergent's sign (−1)^K dictates, alternating between the two codings of one point forever: the gate witnessed at every window a = 2..7, the flip at b₁ except a = 7's b₂, the parity of a never consulted.

And a level further out, at arbitrary period, by the same engine with no comb at all: at four fresh windows of periods 4 and 5, alphabets mixed and a₁ up to 3, the family witnesses ×m gated at all 24 (window, m) cells, m = 2..7, the flip at b₀ or b₁, the successor reading at lookahead 1 beside it. Where each comb family above needed a telescope built for it, this engine needs only the max-string theorem plus per-cell class existence — and that leg is settled: the class condition is periodic in the convergent pair mod m, so one member of each parity certifies it forever, and every cell here is certified. Nor is the certificate a formality — existence is provably impossible at some windows: at the golden window no residue class holds both parities for m = 3, 5, or 7, so where the certificate cannot exist the gate needs different machinery, which is exactly what the comb telescopes are. The excluded class is its own specimen: there the input family itself lands on the boundary and never converges — the max-string mechanism one level down.

Scope. The max-string identity — the legal string of q_K − 1 itself — is a theorem at arbitrary tail, verified K = 1..40 at the four period-4/5 windows; the boundary family's class membership is certified outright — one member of each parity plus the period of the convergent pair mod m — at every cell of both families, while its input convergence and wider image strings are a rule at scanned scope (at [0; 1, 1, a]: classes to K ≤ 400, agreement depths growing to 140, strings by greedy extraction at a = 2..7; at the four period-4/5 windows: all 24 (window, m) cells, m = 2..7, classes to K ≤ 400, depths to 225, one cell needing r = 3).

verifiers: explore_max_string_witness.py, explore_general_max_string.py, explore_class_criterion.py

How common the certificate is turns out to be a 2-adic question. At a constant-a window, run the denominator recurrence one step lower — U₀ = 0, U₁ = 1, U_n+1 = a·U_n + U_n−1 — and for an odd prime m not dividing a² + 4 let z(m), the rank of apparition, be the first index at which m divides a term. The boundary family's certificate exists iff z(m) ≡ 2 mod 4 (at m = 2 it always does), and that criterion reads off the unit: z(m) ≡ 2 mod 4 exactly when the multiplicative order of ε mod m has 2-adic valuation v₂ ≤ 1. So a prime inert at the window (a² + 4 not a square mod m) never qualifies — inert m ≡ 1 mod 4 has z odd, inert m ≡ 3 mod 4 has v₂(z) = v₂(m + 1) ≥ 2 — while a split m ≡ 3 mod 4 always does. Among all odd primes the qualifying density lands at 1/3 for a generic window, and at 7/24 exactly when a² + 4 is 8·(square) — the Pell family a = 2, 14, 82, …, where splitting ties to m mod 8. Windows whose ε are odd powers of one unit share v₂(z) at every prime, so the golden pair a = 1, 4 (ε = φ, φ³) and the Pell rows print identical columns. The certificate-less windows are thus the two-thirds majority — why the comb telescopes were needed at all.

Scope. The criterion, the unit law and the inert/split laws are theorems, asserted at every scanned prime, as is the odd-power sharing; the densities are a pattern at 12 windows over odd primes to 10⁶, matching the derivation within 2·10⁻³ — the one unproved ingredient a Kummer independence at split m ≡ 1 mod 4.

verifiers: explore_class_criterion.py, explore_apparition_density.py