Canonical Negative Base-Phi Tail Bounds

Abstract

A nonempty canonical negative base-phi tail lies in the unit interval, with its first digit selecting the side of the inverse-golden cut.

Theorem 1.1 (The first negative digit selects the inverse-golden interval).

$$0<T_N<1,d_{-1}=1\Rightarrow {\phi }^{-1}\le T_N,d_{-1}=0\Rightarrow T_N<{\phi }^{-1}$$

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

Source. Repository-derived.

Commentary.

Reindexing the finite negative support turns it into a binary nonadjacent word. Its inverse-golden evaluation is positive and below one; a leading one gives the closed upper side of the inverse-golden cut, while a leading zero gives the open lower side.

References

• Truth anchor: D5/S1/Words/Expansions/BasePhiTailBounds.negative_tail_real_bounds
• Dependency: D5/S1/Words/Expansions/BasePhiCanonicalExpansion