Classify negative base-phi prefix occurrence sequences

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.”

Proposed formal target: 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. Do not weaken this 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.

• 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.

• Frozen digits are nonnegative Fibonacci-index coordinates; beta^-(N) uses negative powers of phi and is not represented.
• No theorem currently converts a canonical Zeckendorf expansion to the two-sided base-phi expansion.
• The paper's morphisms f, g, h, the parameterized sequence families V_F, V_G, V_H, and the union-of-three data are absent.
• Existing return-word theorems concern factors of the frozen golden word; it remains to prove that negative-prefix cylinders land in those exact subshifts.

1. Port the two-sided base-phi expansion and prove value/uniqueness by clearing negative powers with a suitable phi power and invoking GoldenInt/WDigits normalization.
2. Construct a finite carry transducer from a Zeckendorf word to the first m digits of beta^-; its state should be a bounded conjugate/deficit residue because the negative tail is contractive.
3. Identify the output cylinder's return itinerary with x_F, x_G, x_H, or a three-state interleaving. Use frozen return-word and occurrence-gap results after this identification, not before it.
4. Prove Lucas parameters by induction/desubstitution on w; use the frozen Beatty displacement reading to close the affine occurrence formula.
5. Start with a declaration-ready restricted theorem for prefixes ending in a state whose transducer is a single V_F component, then generalize to the trident.

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.

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.

theorem. The missing two-sided conversion is substantial, but the repository already owns precisely the normalization, mechanical-word, Beatty, and return-gap ingredients suggested by the conjecture's shape.

• The paper's phrase "union of three" has a unique intended formal parameterization and does not require extra overlap/multiplicity conventions.
• A bounded-state transducer from WDigits to every fixed negative prefix exists in a form compatible with current definitions.
• The frozen golden return-word theorems apply after a finite shift/intercept change; this is the main bridge to prove.
• Whether the conjecture was resolved after arXiv v1 is unverified, and any novelty of intermediate bridge theorems is unassessed.