Completions

The units read and every tested non-unit fails. The base's own multiplication and division by φ are the digit shifts, carry-free on strings — readable at lookahead 0 and 1 — and the successor n + 1 reads at exactly 1. Against them, 3n, 4n and ⌊n/2⌋ are unreadable at every lookahead ≤ 12 at every depth ≤ 11, and ×2 is unreadable at every depth and every lookahead, with no range cap — the theorem in the next block. The window's roof is exactly two-valued at scanned depths: every depth-t cell holds one of two counts of integers below the scan bound, by its last-digit class — two consecutive Fibonacci numbers, ratio → φ. The mechanism is not the continued fraction's: a near-constant roof makes rate forcing unavailable, and what fails instead is carry alignment. Zeckendorf doubling has a down carry — 2F_k = F_k+1 + F_k−2 — that descends, through a receptive pattern of low digits, to the lowest digit, where base-b carries go only up.

Scope. Rule at scanned scope, exhaustive below F₂₆ = 121393: the unit delays uniform over depths 1–14, the non-unit failures at every lookahead ≤ 12 and depth ≤ 11, the roof at depths 6, 8, 10. The ×2 half is a theorem — next block.

verifier: explore_zeckendorf_window.py

The window forces the gate's native form. At trailing base b, multiplication by any integer reads at lookahead 0 and unit-ness gates only division; at Zeckendorf the gate binds multiplication itself, though every integer is an element of Z[φ]. The statement uniform across both: λ is readable at bounded lookahead iff λ extends to a Lipschitz self-map of the window's completion — the limit object the nested cells define. That is an identity rather than a conjecture, because lookahead c says exactly that agreement to depth t + c forces image agreement to depth t, which is the Lipschitz bound 2^c in the cells' own ultrametric. Continuity alone is strictly weaker and does not suffice: between Lipschitz and discontinuous sits a middle class — maps continuous but readable only at a lookahead growing with depth — and even-position digit extraction, reading every other digit of a trailing base-2 string, inhabits it: its least lookahead at depth t is exactly t − 1 there, so it is readable at no bounded lookahead at all. What the gate rests on is that the arithmetic maps — integer multiplication, floor division, the successor — never land in that class, and that is proved at the ring and odometer shapes — the completion a positional base reaches, a ring, and the one an irrational's digit window reaches, whose cells are cut by a circle rotation's coding: at every trailing base-b window and at the trailing digit window of every irrational — the quadratic windows with their quadratic restriction lifted — ×m and ⌊n/m⌋ are continuous iff Lipschitz. On the ring completion multiplication is Lipschitz outright, and floor division is the residue's visibility: rad v | rad b makes n mod v locally constant, and otherwise some prime of v has a residue no trailing cell settles, so the division tears at every point. On an odometer completion — Zeckendorf's three-distance partition above is the golden instance — the cells are intervals, so a continuous ×m would descend to multiplication on that circle, which tears where a point that is no cell endpoint lands on one, while ⌊n/m⌋ reads a residue the cells leave dense — torn at every point. The scans stand as the theorem's independent instruments: the base-10 divisions split at rad v | rad b with no third behaviour, and at four continued-fraction windows and four quadratic windows, every tested integer map beyond the successor — ×2, ×3, ×4 and both floor divisions — is discontinuous outright: the deepest agreement its pairs realize stays pinned to the range's own ceiling as that ceiling moves, where a middle map's would fall away from it. The extraction map itself does not cross families: carried to Zeckendorf — read every other digit, write them as consecutive digits — it falls to the same down-carry that kills doubling (re-legalizing the packed string resolves duplicates by 2F_m = F_m+1 + F_m−2, rewriting a lower position). The middle class itself survives on the odometer completions, but only by design: the map constant on each “lowest nonzero digit at position k” cell, sent to an integer whose lowest nonzero digit sits at position ⌈k/2⌉, is continuous and Lipschitz at no constant — a stretcher, where the extraction compresses. So the extraction enters the middle only where carries go up, the stretcher enters everywhere, and arithmetic enters nowhere — proved at the ring and odometer shapes, scanned at the fourth shape's degree-3 member below. What stays conjectural is the fourth shape and past it, never the ring or the odometer, where the verdict is proved at every positional base and every irrational α and not only the scanned ones. The b-adic completion is a ring, so all integer multiplication is continuous and only inversion is gated. The Zeckendorf completion is an odometer and not a ring: along the step-3 comb family C_K = F₄ + F₇ + ⋯ + F_K, the double is 2C_K = F_K+2 − 2, whose lowest digit reads the parity of K (1 exactly when K is even) — so the pairs (C_K, C_K+3) agree to depth K + 1 while their doubles differ at the lowest digit, and the images 2C_K converge, along the two parity classes of K, to two distinct points of the completion: ×2 has no continuous extension, and no continuous addition on the completion extends the integers' — x ↦ x + x is one. The golden positional control isolates the axis. Base-φ digits — n a finite sum of powers φ^k over k ∈ Z, digits {0, 1}, no two adjacent 1s — have no bottom position, so a trailing cell is agreement at every position ≤ a cutoff c; that window is rigid: the conjugation φ ↦ −1/φ, which fixes every integer, bounds a cell's diameter by φ^1−c, so cells are singletons from cutoff 1 up, consecutive pairs only at cutoff 0, every cell finite — the completion adds no new points, discrete, and no gate question exists. Infinite trailing cells are bought by a non-unit base's index (b-adic cosets) or by non-positional cells (the Fibonacci roof); the golden positional base has neither. The trichotomy on the completion: ring (base b — all integer multiplication continuous, division gated), odometer (Zeckendorf — units only), discrete (golden positional — vacuous). The action on the completion, not membership in the structure, is the trailing gate in its native form — at the Lipschitz strength above — and unit-ness is what the inverse pair costs in both. And the shapes are not three. A census over the window families here closes every remaining row: a chain of moduli — mixed radix, factorial — completes to a profinite ring by construction, its division gate generalizing the base-b criterion (⌊n/v⌋ is Lipschitz iff the chain absorbs v cofinally — v divides a ratio of place values from every depth on, which at a fixed base b is exactly rad v | rad b; the factorial chain absorbs every v, so its gate stands fully open, the opposite extreme from a fixed base); a redundant digit system — one whose digit set is larger than its base, so its cells overlap instead of partitioning — has a quotient, representations identified when they name the same integer, that is the base ring itself, the redundancy invisible; and the continued-fraction window poses no completion question, an integer's expansion being the single quotient [n]. But the trailing Tribonacci window — q_k = q_k−1 + q_k−2 + q_k−3, digits {0, 1}, no three consecutive 1s — realizes a fourth shape: not discrete (q_t, q_t+1, q_t+2 share the depth-t zero cell), classically completing almost one-to-one over a translation of the two-torus rather than a circle, and not a ring. And the refusal generalizes: at every degree d ≥ 2 the trailing d-bonacci window — q_k = q_k−1 + ⋯ + q_k−d, digits {0, 1}, no d consecutive 1s — refuses the ring, by one proof uniform in d. The down-carry 2q_k = q_k+1 + q_k−d cascades tooth by tooth down the step-(d + 1) comb T_K = q_d + q_2d+1 + ⋯ + q_K, so the double is a step-d comb whose lowest digits cycle through d phases with period d(d + 1) in K: the pairs (T_K, T_K+d+1) agree to depth K + d + 1 while their doubles differ at a digit ≤ d − 2, and the images converge to d distinct points of the completion over the single input limit — ×2 has no continuous extension at any degree. Zeckendorf's two phases and Tribonacci's three are the d = 2, 3 rows of this one theorem; the four-phase degree-4 row is proved in full, the generic table checked at degrees 5 and 6. The completions' positive side is classical at every member — each d-bonacci window completes almost one-to-one over a translation of the (d − 1)-torus — so the odometer shape above is the degree-2 member of a tower of torus translations, one per degree, the circle its ground floor: every floor above it is neither a ring, nor cut by a circle rotation's coding as the odometers above are, nor discrete. And the arithmetic maps still keep clear of the middle at the scanned member: the same two-class verdict, successor Lipschitz and multiplication and division torn, prints at degree 3; the ×2 tear is the comb theorem's, at every degree, and past ×2 the floors above degree 3 are unscanned.

Scope. The ×2 discontinuity and the odometer-not-a-ring statement proved, no range cap; the positional rigidity proved by the conjugation bound, its cutoff-0 collision census — consecutive pairs only — a rule below 5000; the b-adic completion classical; the odometer shape's units-only reading is the gate rule above. The middle class's emptiness on the arithmetic maps is a theorem at the ring and odometer shapes — every base and every irrational's window, the discrete shape holding it trivially — with the scans as its independent instruments: the base-10 division row, four continued-fraction windows, and four quadratic windows read against ranges whose realized depth moves by construction; the extraction's t − 1 lookahead exact at every scanned range; the Zeckendorf fate of the extraction map an observation at its scanned frames; the designed stretcher a property, following from its construction. Each shape is held by its own window; the odometer shape is populated by a whole family — Quadratic windows. The census rows are properties, each shape decided by the window's own construction (the factorial gate's delays verified by sampling; the redundant-quotient construction executed at signed-digit base 2); the ×2 discontinuity on the d-bonacci completions is a theorem at every degree d ≥ 2, no range cap — one proof uniform in d, its phase table verified by greedy extraction at degrees 4, 5 and 6; the torus factors are classical (Rauzy at degree 3; in general the d-bonacci substitution is the substitutive Arnoux–Rauzy sequence, and those have pure discrete spectrum — Berthé–Steiner–Thuswaldner); and the no-middle verdict beyond ×2 is a rule at scanned scope at degree 3 (agreement depth 17 below 200000), unscanned above. What stays conjectural is the classification itself: which shape a completion lands in, and how many shapes there are — the shapes do not stop at four (the fifth shape).

verifiers: explore_zeckendorf_discontinuity.py, explore_goldenbase_control.py, explore_continuity_converse.py, explore_quadratic_middle.py, explore_arithmetic_gate.py, explore_completion_atlas.py, explore_tribonacci_discontinuity.py, explore_tetranacci_window.py

The d-bonacci tower is one line — the spine — through a wider family. A Pisot base — an algebraic integer above 1 whose other conjugates all lie inside the unit circle — whose own expansion of 1 terminates carries a trailing window whose admissible strings are fixed by that finite expansion — a word, which on the spine is the run-length condition “no d consecutive ones” and off it reads as something else. Over the legal binary words to length 6 — 21 of them, five all-ones and so the spine — the comb witness of the trichotomy returns a verdict at only eight — the five spine words among them, refusing by the theorem there — and the thirteen silences are the instrument's. Reading the phase count — how many image limit points sit over the one input limit — needs the doubled comb to be a single-step comb above its bottom, which the spine's theorem hands over for free; the refusal needs none of it, since a family converging to one input limit whose images take two values at a bounded digit is the whole of the argument, and a digit set that is no comb separates as well as one that is. Drop that demand and all thirteen speak: each carries admissible growing combs — all 38 steps in 3..40 at twelve of them and 35 at the thirteenth — and at each, some step's image takes between 2 and 13 values at depth 4, which pins the separating digit at 3 or less with no further argument. Read off the prefixes it is 0 at eleven of the thirteen and 1 at the other two, while the two comb members whose images differ agree to depth 813 or beyond. So ×2 has no continuous extension at any legal binary window to length 6: the refusal is the whole binary family's rather than the eight windows an instrument happened to reach.

And the count parts from the separation — which is what a comb image can never show, the two readings being one there. Ten of the thirteen have a measured least period in K, between 6 and 51. Three do not: at 100001, 101001 and 110101 every depth-80 prefix sampled is distinct at each of the tails 120, 240 and 480, where the controls saturate over those same three — Tribonacci, whose count of 3 is the theorem, prints 3, 3, 3, and 10101 prints 5, 5, 5 — so the count is reading phases and not samples. Two prefixes differing below depth 80 are two limit points over the one input limit, so those three windows carry unboundedly many: neither a ring, nor finitely many image limit points over one input limit. That is a fifth shape, and the fourth is the finite case of something wider. The single-step demand is exactly what hid it — a comb image has finitely many phases by construction, so an instrument that reads only comb images cannot return this answer at all. Their low digits settle even so, the depth-4 counts holding at 5, 8 and 13 across tails from 60 to 480: the separation is settled where the phase count is not. And the mechanism at the shortest silent word is a carry with three low teeth, 2q_k = q_k+1 + q_k−2 + q_k−9 + q_k−11 at 10101, against the spine's single lag — which is why no one cascade closes there, and why nothing forces the image to be a comb. Which windows take the fifth shape is read off one hypothesis (the hypothesis law).

Scope. Observation at scanned scope throughout, and the scope is the binary alphabet: legal words to length 6, comb steps 3..40, the separations recurring over 120 samples and the periods searched to 100 steps over 200. Off that alphabet the question is untouched: over the 172 legal words carrying a digit 2 or more the witness's count is 1 at 169, so it reports nothing there. The unboundedness is a distinguishing sweep and not a demonstration: no cascade theorem is derived for any silent window, and the counts are finite runs. Proved here: the 10101 carry, a rule derived algebraically and canonical at every k from 11 to 60. The comb's bottom offset is free and was swept — it moves where a separation shows and never whether, the depth-4 reading dropping to a count of 1 at one window and two offsets while the separation still stands, at digits 4 and 5.

verifiers: explore_silent_window.py, explore_pisot_confound.py

Which shape a binary window takes is decided by one hypothesis, and past it not decided at all. A window's Parry polynomial is the characteristic polynomial of the recurrence its word seeds, with the base β as its largest root; the base is Pisot when, any factor whose roots are roots of unity set aside, every other root lies inside the unit circle, and the polynomial is then exactly β's minimal polynomial when it is irreducible. Read against the count above — a window settles when its depth-80 prefix count saturates across the tails 120, 240 and 480, and tracks its tail when every sample is distinct at every tail — the 125 legal binary words to length 9 sort in two moves. Where the Parry polynomial is exactly a Pisot minimal polynomial, the count settles: 39 of 39 such windows. That is the hypothesis of Frougny's theorem read to the letter — for a linear recurrence whose characteristic polynomial is exactly the minimal polynomial of a Pisot number, rewriting any digit string into the window's admissible form is computable by a finite automaton — and the doubled comb's digits are that rewriting of the digit 2 placed on the comb's positions. That the automaton makes the low digits eventually periodic in K, which is what a settled count reads, is argued from the automaton's decomposition into a left- and a right-sequential pass and not proved here at this depth. The fifth shape therefore needs the hypothesis to fail — a root of the Parry polynomial other than β outside the unit circle, or a root-of-unity factor beside a Pisot minimal polynomial — and every window that tracks its tail is one of those, 125 of 125; at length 6 and under, the three fifth-shape windows are exactly the three non-Pisot ones.

And where the hypothesis fails, nothing read so far decides. Of the 62 non-Pisot windows 38 track their tail, 23 settle and one, 100000001, repeats one or two prefixes in every tail; of the 24 windows carrying a root-of-unity factor beside a Pisot minimal polynomial, 2 track it — 1001001 and 10100101, every sample distinct to tail 1920, and again on a second reader at depth 120 over 300 samples with no period to 120 — and 22 settle. The size of the root outside the circle does not sort them (every window with it below 1.020 settles, every one above 1.048 tracks, and between the two they are mixed), and neither does whether 2 itself has a finite expansion in the base (it does at 2 of the 62, one of each shape). The converse theorem — no finite automaton at a base that is not Pisot — is a statement about every digit string, and the 23 settling non-Pisot windows measure the gap between that and one string family.

Scope. The settled count at the hypothesis is a rule at scanned scope, 39 of 39 windows, with the periodicity behind it argued and not proved; the fifth shape occurring only outside the hypothesis is a rule in range, 125 of 125 legal binary words to length 9; everything off the hypothesis is observation at that scope, tails to 480 and to 1920 at the two exceptions, and the question there is open — what the doubled comb's rewriting needs beyond the base is derived for no window. Frougny's theorem and its converse are classical and keep their names. The instrument, its dials and its controls are the fifth shape's, carried unchanged.

verifier: explore_conjugate_spectrum.py