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Metadata 
slug: base-phi-negative-prefix-trident
bibkey: dekking2023structure
doi: 10.48550/arXiv.2305.08349
triage: theorem
motivation_gids:
  - D5/S0/Conventions/WDigits
  - D5/S1/Words/ZeckendorfOrder
  - D5/S1/Words/ZeckendorfBeattyBridge
  - D5/S1/Words/GoldenMechanicalWord
  - D5/S1/Words/GoldenSubstFixed
  - D5/S1/Words/Complexity/MechanicalSubshiftIntercept
  - D5/S1/Words/ReturnWords/GoldenReturnWords
  - D5/S1/Words/ReturnWords/GoldenReturnWordsExact
  - D5/S1/Words/ReturnWords/GoldenOccurrenceGaps
  - D5/S1/Words/ReturnWords/GoldenReturnItinerary
  - D5/S1/Deficit/ZeckendorfDisplacementReading

Classify negative base-phi prefix occurrence sequences

Problem

Write the canonical base-phi expansion as beta(N) = beta^+(N) . beta^-(N), and, for a word w of length m, let R_{.w} be the increasing sequence of natural numbers N whose first m negative-position digits are w, that is d_{-1}...d_{-m}(N) = w. The paper defines three families V_F, V_G, V_H through first-difference words x_F, x_G, x_H arising from three Sturmian morphisms.

The conjecture, quoted from arXiv:2305.08349v1:

“Let (\beta(N)=\beta^+(N)\cdot\beta^-(N)) be the base phi expansion of the number (N). Let (w) be a word of length (m). Let (R_{\cdot w}) be the sequence of occurrences of numbers (N) such that the first (m) digits of (\beta^-(N)) are equal to (w), i.e., (d_{-1}\ldots d_{-m}(N)=w). Then there exist two Lucas numbers (a) and (b) such that either (R_{\cdot w}=V_F), or (R_{\cdot w}=V_G), or (R_{\cdot w}=V_H). A second possibility is that (R_{\cdot w}) is a union of three of such sequences.”

Formal target, in two parts. Port the paper’s parameterized definitions faithfully, then prove that every admissible negative prefix cylinder has an occurrence set represented by one trident component V_F, V_G, or V_H, or a union of three such components with Lucas parameters. For the repository’s depth reading that is done and frozen; see Gap. For the paper’s reading, which is a different predicate on the same words, it is open, and that is what this entry tracks. Neither part may be weakened to mere eventual periodicity or occurrence.

The paper states the obstruction:

“However, this does not work. The reason is that the (\beta^-(N)) words do not occur in lexicographical order, in contrast with the (\beta^+(N)) words.”

It adds that some occurrence sequences are Lucas-Wythoff and some are not, although they remain close to that form. It exhibits the first V_G, the first V_H, and a three-component case, but not the general classification.

Motivation

  • The positive Zeckendorf side already has numerical lexicographic order and an exact Beatty/mechanical least-digit bridge.
  • The return-word layer is closer than a generic Sturmian fact: it already turns golden factor cylinders into exact return itineraries and finite adjacent-gap spectra.
  • The conjectural V_F/V_G/V_H alternatives are classifications by first-difference words. A plausible bridge is therefore: negative prefix cylinder, then a finite-state transducer over canonical Zeckendorf digits, then a factor/return itinerary in one of three shifted golden subshifts, then an occurrence-gap sequence.
  • ZeckendorfDisplacementReading supplies an exact digit-upshift/Beatty identity that may convert transducer states into Lucas-affine occurrence formulas.

Gap

  • The two-sided base-phi expansion is represented: BasePhiNegative carries basePhiValue, BasePhiNegativeExpansion, negativeDigit, reachesNegativeDepth, NegativePrefixOccurs and occurrenceSet, and BasePhiCanonicalExpansion proves basePhiExpansion_existsUnique and canonical_two_sided_digits_unique.
  • A partial Zeckendorf bridge exists, and it is conditional and one-sided: BasePhiCarryTransducer proves carrySkipRun_zeckendorf (the carry-skip run’s positive component, unconditional), mem_rawToZeckendorf_iff (under CanonicalRaw) and nonnegative_digit_iff_mem_zeckendorf_of_realizes, whose hypothesis is realizes : forall N, CarrySkipRealizes expansion N and whose conclusion covers only nonnegative exponents. The obligation stands: no theorem yet converts a canonical Zeckendorf expansion to the two-sided base-phi expansion without that hypothesis.
  • The three families are present as first-difference words: GapFamily, fibonacciGapLetter, familyLetter (F, G = bF, H = aF), gapSequence, vForFamily, LucasPair, prefixMultiplicity, CoreLucasWitness. The paper’s substitutions f, g, h themselves are not formalized; the families are characterized by their first-difference words instead, which is the paper’s own equivalent description.
  • A six-state prefix machine exists: FrontierPhase (F0o, F1o, F0e, G1e, G0o, H0e), its ten transitions FrontierPhaseTransition, the base cases PrefixPhaseMachineFor ([0] to <F0o,4,3>, [1] to <F1o,7,4>), and FrontierReturnWord.
  • The reading question is settled by BasePhiNegativePrefixPaddedReading: NegativePrefixOccurs carries the extra conjunct reachesNegativeDepth expansion N w.length, which is not the paper’s reading. Under that conjunct the paper’s own Proposition 7.8 d) identity R_{.01} = R_{.010} fails, with symmetric difference exactly {2, 3, 4}; under the zero-padded reading it holds and is a corollary of canonicality.
  • The classification under the repository reading is proved and frozen. D5/S1/Words/NegativeExpansions/BasePhiNegativePrefixTridentClassification carries negative_prefix_trident_classification: for every BasePhiNegativeExpansion and every admissible w, occurrenceSet expansion w equals sequenceRange (vF a b r), vG, or vH with LucasPair a b, or a union of three pairwise disjoint arms vForFamily family a b (r + i), i : Fin 3. AdmissibleNegativePrefix expansion w unfolds to nonemptiness of occurrenceSet expansion w (admissible_negative_prefix_iff_occurrence_set_nonempty, proved by rfl), so the statement carries no hypothesis beyond the word occurring. A source scan of the classification module and its 41 transitively imported D5 files, 42 files in total, finds no sorry, no admit, no native_decide and no axiom, and its five load-bearing _proved lemmas are theorems, not axiom debt. That scan is over D5 source only; it is not a fresh kernel run or a transitive axiom-closure check.
  • What remains open is therefore the classification under the paper’s reading, not the classification as such. Three readings of “the first m digits of beta^-(N) are w” must be separated, and the paper prints two facts that any faithful reading has to satisfy at once: Proposition 7.8 d) R_{.01} = R_{.010}, and the trident classification. Proved in the tree. The depth reading satisfies the classification and fails the identity: BasePhiNegativePrefixPaddedReading gives occurrenceSet 01 \ occurrenceSet 010 = {2, 3, 4}. The zero-padded reading satisfies the identity in that same module, and a set equality survives restriction, so it holds under N >= 2 as well. How the three domains relate. The two zero-padded readings differ only at N = 1, which is what restricting to N >= 2 removes. The depth reading differs from both of them at points other than N = 1: N = 2, 3, 4 for the word 010, all of which survive the N >= 2 restriction. So the divergence is a single point only between the two padded domains, not among the three readings. Not proved here. Whether either zero-padded reading satisfies the classification is open. The finite measurement recorded under Evidence exhibits failures at the all-zero words when N = 1 is included, and no failure under N >= 2; the window is finite, so neither the failure at every all-zero word nor the success under N >= 2 is established. Closing that is the remaining obligation.

Route

This route was written before the depth-reading classification was proved. It is kept as the record of what was anticipated, with each step marked against what actually happened.

  1. Done. The two-sided base-phi expansion is ported with value and uniqueness.
  2. Open, and independent of the reading question. Convert a canonical Zeckendorf expansion to the two-sided base-phi expansion without the CarrySkipRealizes hypothesis and for negative exponents as well; see Gap. A finite carry transducer whose state is a bounded conjugate/deficit residue remains the proposed shape.
  3. Done for the depth reading, by a different route than anticipated. The occurrence set is identified with a single family member or a disjoint union of three. The transitive import closure of the frozen classification is 42 D5 files and contains no ReturnWords module, so the return-word and occurrence-gap results were not the bridge.
  4. Done for the depth reading. The parameters are established as a LucasPair, two consecutive Lucas numbers. The frozen Beatty displacement reading is not in that import closure either.
  5. Superseded for the depth reading: the frozen theorem already proves the general classification, so no restricted first step is outstanding.

The paper-reading classification remains open. The frozen depth-reading proof does not determine which construction or bridge steps a proof for that reading will require; the import readings above record what that proof did not use, not what a future one cannot use. One thing is settled about the relation: it is not a same-word set equality with the frozen depth-reading result, which already fails at N = 2, 3, 4 for the word 010; see Gap.

Falsifier

An admissible word w for which the exact occurrence sequence has a first-difference factor impossible in all three of x_F, x_G, x_H, even after testing every Lucas parameter and every allowed three-component interleaving, falsifies the conjecture.

A finite prefix alone cannot refute equality of infinite sequences unless it contradicts a necessary invariant. Use invariants such as allowed gap alphabet, factor complexity, return-word count, and Lucas congruence classes; report the first violating index and exact base-phi expansion.

Evidence

For all admissible w of length at most 14:

  1. compute exact two-sided base-phi expansions for 1 <= N <= 2,000,000 using integer pairs in Z[phi], not floating point;
  2. extract R_{.w} and its first differences;
  3. infer a candidate F/G/H state and Lucas pair from a training prefix;
  4. verify on a disjoint tail and check necessary return-word/factor invariants;
  5. emit the smallest unresolved or contradictory w with a reproducible integer-coordinate trace.

The first Evidence goal is to validate the finite transducer and discover its states, not to certify the infinite conjecture from samples.

ASSUMED-UNVERIFIED – reading trichotomy, measured outside the repository with programs that are not committed here. Window 1 <= N <= 1,000,000, exact integer-pair arithmetic in Z[phi], the 374 admissible words of length at most ten whose depth-reading occurrence set has at least 60 elements in the window. Counts of words whose occurrence set is neither a single family member nor a union of three: depth 0, zero-padded 10, zero-padded with N >= 2 0. The 10 are exactly the all-zero words 0^1 to 0^10, and for each of them the zero-padded set exceeds the depth set by exactly {1}. Under the depth reading each 0^m is a union of three V_F arms with (a, b) = (L_{m+2}, L_{m+1}), measured as (4,3), (7,4), (11,7), (18,11), (29,18), (47,29), (76,47), (123,76), (199,123), (322,199). The window is finite and this settles no infinite statement.

Steps 1 and 2 of the protocol above have been run, and step 3 only in the degenerate form of classifying every word directly rather than inferring from a training prefix. Step 4 – verification on a disjoint tail and the return-word/factor invariant checks – and step 5 have NOT been run.

The reproducible part of this is already in the repository: Evidence/D5/S1/Words/BasePhiNegativePrefixTrident.result.json records a scan with limit 2000000, max_prefix 14, min_exponent_guard -96, and method: exact integer-pair arithmetic in Z[phi], classifying single_family 986, three_family 1595, unresolved_count 0. That artifact also records 01 with 633438 occurrences against 010 with 633435 – a difference of exactly 3, which is the {2, 3, 4} that occurrenceSet_prefix01_symmDiff_prefix010 now proves, and which that artifact reports as matched: true for both words.

ASSUMED-UNVERIFIED – the following were run outside the repository and their programs are not committed here, so a reader cannot recompute them: an independent enumeration to 1 <= N <= 4,000,000 over the same 2,581 admissible words of length at most 14, by two implementations agreeing on 20,000 x 10 digits; and the selector counts below. The counts that the committed artifact does cover – 986 single, 1595 trident, 0 unresolved, and 987 admissible words at length 14 – agree with that independent run. What was additionally measured: window 1 <= N <= 4,000,000, all 2,581 admissible w of length at most 14, exact integer arithmetic in Z[phi] (two independent implementations agreeing on 20,000 x 10 digits). Every occurrence set is a single family member or a union of three, and every step pair is a pair of consecutive Lucas numbers; zero violations. Leading-component families: F 1232, G 979, H 370. Under the zero-padded reading the frozen frontierFamily selector has zero counterexamples and dataFrontierFamily has 276; under the reachesNegativeDepth reading the two are exactly exchanged. This does not settle the conjecture; the window is finite.

Triage

theorem. The depth-reading classification is proved and frozen; the remaining classification target concerns the paper’s reading, whose truth remains open as described in Gap. The two-sided conversion named in Gap is a separate outstanding obligation that does not depend on the reading question.

ASSUMED-UNVERIFIED

  • The paper’s phrase “union of three” has a unique intended formal parameterization and does not require extra overlap/multiplicity conventions.
  • The frozen return-word theorems apply after a finite shift/intercept change. This was recorded as the main bridge to prove. The frozen depth-reading classification did not use it: no ReturnWords module appears in its 42-file transitive import closure. The statement is therefore unused here, not established.
  • Novelty of the intermediate bridge theorems is unassessed.
  • Post-v1 literature status: arXiv:2305.08349 has one version only (15 May 2023) and was published as Communications in Mathematics 33 (2025) no. 2; a search for a later proof of the conjecture returned nothing. That search was the orchestrator’s own and is not machine-checkable, so the question of whether the conjecture was resolved elsewhere stays ASSUMED-UNVERIFIED.
  • A bounded-state transducer from WDigits to every fixed negative prefix exists in a form compatible with current definitions. This obligation stands: the frozen FrontierPhase machine is not it. That machine consumes the prefix word and emits a phase with Lucas parameters (PrefixPhaseMachineFor, FrontierPhaseTransition); it does not convert Zeckendorf digits into negative-position digits.
  • Separately, and not discharging the line above: the FrontierPhase machine reproduces the measured Lucas step pair (a, b) for every admissible w of length at most 14 over 1 <= N <= 4,000,000, with zero counterexamples under both readings. Whether it does so for every w is unverified here, and that correspondence is a repository-internal obligation, separate from the classification already proved for the depth reading.
  • prefixMultiplicity w is if w.head? = some true then 1 else 3. Under the depth reading that split is proved, not assumed; it is recorded in Gap and is not an item of this section. What belongs here is the paper’s reading, for which the corresponding assertion is open.