Signed Zeckendorf Orbit Code

Abstract

The three zero symmetries act on a signed W code by flipping its two sign coordinates.

Theorem 1.1 (Klein actions are the two independent sign flips).

$$\begin{array}{r}\forall delta,gamma:\mathbb{R},N,multiplicity:\mathbb{N},\\ letrho:\mathbb{C}=\frac{1}{2}+delta+\mathrm{ComplexI}\left(\right)\times gamma;\\ letunsignedThread:\mathbb{R}\to WDigitString=(x\mapsto \mathrm{wEncoding}\left(\lfloor {\phi }^N\cdot \Vert x\Vert \rfloor \right));\\ letcode:\mathbb{C}\to SignType\times WDigitString\times SignType\times WDigitString\times WDigitString=(z\mapsto (\mathrm{sign}\left(\mathrm{re}\left(z\right)-\frac{1}{2}\right),unsignedThread\left(\mathrm{re}\left(z\right)-\frac{1}{2}\right),\mathrm{sign}\left(\mathrm{im}\left(z\right)\right),unsignedThread\left(\mathrm{im}\left(z\right)\right),\mathrm{wEncoding}\left(multiplicity\right)));\\ letstates:\mathrm{List}\left(SignType\times WDigitString\times SignType\times WDigitString\times WDigitString\right)=[(\mathrm{sign}\left(delta\right),unsignedThread\left(delta\right),\mathrm{sign}\left(gamma\right),unsignedThread\left(gamma\right),\mathrm{wEncoding}\left(multiplicity\right)),(\mathrm{sign}\left(delta\right),unsignedThread\left(delta\right),-\mathrm{sign}\left(gamma\right),unsignedThread\left(gamma\right),\mathrm{wEncoding}\left(multiplicity\right)),(-\mathrm{sign}\left(delta\right),unsignedThread\left(delta\right),\mathrm{sign}\left(gamma\right),unsignedThread\left(gamma\right),\mathrm{wEncoding}\left(multiplicity\right)),(-\mathrm{sign}\left(delta\right),unsignedThread\left(delta\right),-\mathrm{sign}\left(gamma\right),unsignedThread\left(gamma\right),\mathrm{wEncoding}\left(multiplicity\right))];\\ letorbitCodes:\mathrm{List}\left(SignType\times WDigitString\times SignType\times WDigitString\times WDigitString\right)=[code\left(rho\right),code\left(\mathrm{conj}\left(rho\right)\right),code\left(1-\mathrm{conj}\left(rho\right)\right),code\left(1-rho\right)];\\ code\left(\mathrm{conj}\left(rho\right)\right)=(\mathrm{sign}\left(delta\right),unsignedThread\left(delta\right),-\mathrm{sign}\left(gamma\right),unsignedThread\left(gamma\right),\mathrm{wEncoding}\left(multiplicity\right))\wedge \\ code\left(1-\mathrm{conj}\left(rho\right)\right)=(-\mathrm{sign}\left(delta\right),unsignedThread\left(delta\right),\mathrm{sign}\left(gamma\right),unsignedThread\left(gamma\right),\mathrm{wEncoding}\left(multiplicity\right))\wedge \\ code\left(1-rho\right)=(-\mathrm{sign}\left(delta\right),unsignedThread\left(delta\right),-\mathrm{sign}\left(gamma\right),unsignedThread\left(gamma\right),\mathrm{wEncoding}\left(multiplicity\right))\wedge \\ orbitCodes=states\wedge \\ delta\ne 0\Rightarrow \left(gamma\ne 0\Rightarrow \mathrm{Nodup}\left(orbitCodes\right)\right).\end{array}$$

Proof. Machine-checked in Lean as D5/S3/Zeros/Symmetry/SignedZeckendorfOrbitCode.klein_actions_two_sign_bits (✓ std3). ∎

Source. Repository-derived.

Commentary.

The public code is constructed from the centered real coordinate, the height, their golden-scale W encodings, and the multiplicity W word. Conjugation, conjugate reflection, and reflection induce the three displayed sign transformations while preserving every unsigned word.

The orbit-code list is equal to the explicitly listed sign-state list. When both centered coordinates are nonzero, the sign values are nonzero and each differs from its negative, so the four entries are pairwise distinct.

References

• Truth anchor: D5/S3/Zeros/Symmetry/SignedZeckendorfOrbitCode.klein_actions_two_sign_bits
• Dependency: D5/S0/Conventions/WDigits