bibkey: dekking2023structure authors: F. Michel Dekking year: 2023 title: The structure of base phi expansions doi: 10.48550/arXiv.2305.08349 claim: Occurrence sequences of negative-position digit prefixes in base-phi expansions are conjecturally Lucas-parameterized sequences drawn from three Sturmian families. strata_touched:

• D5/S1/Words/NegativeExpansions/BasePhiNegativePrefixTridentClassification
• D5/S1/Words/ZeckendorfOrder
• D5/S1/Words/ZeckendorfBeattyBridge
• D5/S1/Words/ReturnWords/GoldenReturnWords
• D5/S1/Deficit/ZeckendorfDisplacementReading license: citation-only triage: anchor

The structure of base phi expansions

Dekking studies the two-sided base-phi expansion beta(N) = beta^+(N) . beta^-(N) and the sequences R_{.w} of natural numbers whose first m negative-position digits equal a fixed word w. The paper introduces three families V_F, V_G, V_H built from first-difference words of three Sturmian morphisms, and conjectures that every such occurrence sequence is one of those families, or a union of three of them, with Lucas-number parameters. It records that the obstruction is the failure of beta^-(N) words to appear in lexicographic order, unlike the positive side.

This note is the literature anchor for the problem candidate Problems/base-phi-negative-prefix-trident.md.

The paper locates the phenomenon the formalization needs: Section 7.1 defines the singleton/trident dichotomy of equal complete negative tails, and Theorem 7.5 states the recursion the paper uses to prove it. The repository formalization does not derive Theorem 7.5’s recursion; it proves the singleton/trident fiber shape directly with repo-native Beatty floor coordinate arguments. The paper reference is provenance for the statement, not for the proof route. On PDF page 16, Theorem 7.5 gives the gamma^- recursion: its odd branches append 10, 0010, and 00 in Equations (15a-c), while its even branches append 00, 01, and 01 in Equations (16a-c). Section 7.1 defines equal complete negative tails as either singletons or three consecutive inputs (“tridents”); Lemma 7.1 proves the boundary trident splitting by induction from Theorem 3.3. Section 7.2 then starts the conjectural classification of arbitrary finite negative prefixes, which is strictly stronger and is not claimed by the present formalization.

The Lean proof takes only the cropped algebraic consequence needed by the frontier: split a canonical expansion at exponent zero, identify its nonnegative GoldenInt coordinate through the pinned Zeckendorf/Beatty bridge, bound the nonempty negative tail on the two sides of phi^-1, classify the corresponding floor fiber, and use the canonical seam digit to reduce the upper-side two-coordinate fiber to a singleton. On the lower side, canonical tail gluing realizes all three consecutive coordinates. This is equivalent to the singleton/trident consequence of the recursive append structure; it does not formalize the paper’s full interval recursion or prefix-family panorama.

Search log

• 2026-08-18: Queried the arXiv Atom API for id_list=2305.08349. HTTP 200 with totalResults=1; the entry resolved to http://arxiv.org/abs/2305.08349v1, title The structure of base phi expansions, sole author F. Michel Dekking, published 2023-05-15, primary category math.NT. The API reported no arxiv:doi and no arxiv:journal_ref, so the arXiv-assigned DOI is used.
• 2026-08-18: Issued HEAD https://doi.org/10.48550/arXiv.2305.08349, which returned HTTP 302 redirecting to https://arxiv.org/abs/2305.08349.
• 2026-08-22: Searched arXiv for 2305.08349, fetched https://arxiv.org/pdf/2305.08349v1, and extracted the theorem text. The PDF fetch and arXiv API query both returned HTTP 200. Located Theorem 7.5 on PDF page 16, Lemma 7.1 on page 15, and the conjectural finite-prefix program at the start of Section 7.2 on page 20.

No literature search for a later resolution of the conjecture was performed; the open status recorded in the problem candidate is the status stated in this arXiv version, not an assessment of the subsequent literature.

Verified locator

• arXiv: https://arxiv.org/abs/2305.08349
• DOI: https://doi.org/10.48550/arXiv.2305.08349