Complete Negative Base-Phi Tail Fibers

Abstract

Every nonempty complete negative base-phi tail of a positive natural has a singleton or three-consecutive fiber.

Theorem 1.1 (Complete negative tails have the singleton-trident dichotomy).

$$d_{-1}=1\Rightarrow F_N=\{N\},d_{-1}=0\Rightarrow \exists !q,F_N=\{q,q+1,q+2\}$$

Proof. Machine-checked in Lean as D5/S1/Words/Expansions/BasePhiTailFiber.negative_tail_fiber_shape (✓ std3). ∎

Source. Repository-derived.

Commentary.

For a positive natural whose canonical expansion reaches a negative exponent, the complete negative-position digit tail determines that natural uniquely when its first digit is one. When the first digit is zero, the same tail occurs at exactly three consecutive positive naturals, with a unique least member.

This is the singleton-trident consequence of Dekking’s recursive structure used by the frontier theorem. It is deliberately narrower than a formalization of the paper’s complete recursive word presentation.

References

• Truth anchor: D5/S1/Words/Expansions/BasePhiTailFiber.negative_tail_fiber_shape
• Dependency: D5/S1/Words/Expansions/BasePhiRecursiveStructure