Zeckendorf Real Thread Reconstruction

Abstract

The complete Zeckendorf thread reconstructs a nonnegative real number.

Theorem 1.1 (The complete Zeckendorf thread is injective).

$$\begin{array}{c}letq:\mathbb{N}\to {\mathbb{R}}_{\ge 0}\to \mathbb{N}:=(N\mapsto (x\mapsto \lfloor {\phi }^N\cdot x\rfloor ));\\ letZ:{\mathbb{R}}_{\ge 0}\to \mathbb{N}\to \mathrm{WDigitString}:=(x\mapsto (N\mapsto \mathrm{wEncoding}\left(q\left(N,x\right)\right)));\\ \mathrm{Injective}\left(Z\right).\end{array}$$

Proof. Machine-checked in Lean as D5/S3/Observer/GoldenCoding/ZeckendorfRealThread.zeckendorf_real_thread_injective (✓ std3). ∎

Source. Repository-derived.

Commentary.

At level N, the source quantization is the natural floor of phi to the N times x. Its thread coordinate is the repository’s canonical W encoding of that natural number.

Equal threads have equal quantizations because the W encoding is an equivalence. Distinct nonnegative reals have a positive gap, and some golden power expands that gap beyond the width of a single natural-floor interval.

References

• Truth anchor: D5/S3/Observer/GoldenCoding/ZeckendorfRealThread.zeckendorf_real_thread_injective
• Dependency: D5/S0/Conventions/WDigits