Finite Description Codes in PZG Tables

Abstract

Shifted prime-sequence description codes embed into canonical PZG tables.

Theorem 1.1 (Finite description codes have canonical PZG tables).

$$\forall D\in \mathrm{List}(\mathbb{N}),\mathrm{primeAxisEncoding}(\mathrm{finiteDescriptionPZGCode}(D))=\mathrm{positivePrimeSequenceCode}(D)\wedge \mathrm{decodePrimeAxisTable}(\mathrm{finiteDescriptionPZGCode}(D))=\mathrm{primeSequenceCode}(D).$$

Proof. Machine-checked in Lean as D5/S1/Digit/PrimeAxis/FiniteDescriptionPZGCode.finite_description_pzg_code_spec (✓ std3). ∎

Source. Repository-derived.

Commentary.

A finite description is represented here by a finite sequence of natural numbers. Its shifted prime-power code is always positive, including for the empty description, and therefore lies in the positive-natural codomain of the established primeAxisEncoding equivalence.

Applying the inverse equivalence produces a canonical PrimeAxisTable. The forward equivalence returns exactly the original shifted prime-sequence code, and decodePrimeAxisTable recovers its underlying natural number.

This theorem supplies only the generic PZG membership bridge. It does not assert a kernel self-code fixed point: the repository does not yet define a particular kernel, its finite syntax description, or a kernel self-code operator.

References

• Truth anchor: D5/S1/Digit/PrimeAxis/FiniteDescriptionPZGCode.finite_description_pzg_code_spec
• Dependency: D5/S0/History/PrimeSequenceCode
• Dependency: D5/S1/Digit/PrimeAxisEncoding