Quadratic Character Quantum Postprocessing

Abstract

Three-ring fibers remain indistinguishable under character processing and quantum preparation.

Theorem 1.1 (Character processing cannot split a three-ring fiber).

$$\begin{gathered} \forall u,v:\mathrm{Units}\left(\mathrm{ZMod}\left(60\right)\right),triRingImage\left(u\right)=triRingImage\left(v\right)\Rightarrow \\ gaussianCharacter\left(u\right)=gaussianCharacter\left(v\right)\wedge \\ eisensteinCharacter\left(u\right)=eisensteinCharacter\left(v\right)\wedge \\ goldenCharacter\left(u\right)=goldenCharacter\left(v\right)\wedge \\ (\forall chi:\mathrm{QuadraticObserver}\left(\mathrm{Units}\left(\mathrm{ZMod}\left(60\right)\right)\right),chi\left(u\right)=chi\left(v\right))\wedge \\ \forall C,Q,O:Type,f:\mathrm{Function}\left(ThreeRingProfile,C\right), \\ P:\mathrm{Function}\left(C,Q\right),R:\mathrm{Function}\left(Q,O\right), \\ R\left(P\left(f\left(triRingImage\left(u\right)\right)\right)\right)=R\left(P\left(f\left(triRingImage\left(v\right)\right)\right)\right). \end{gathered}$$

Proof. Machine-checked in Lean as D5/S3/PrimeForms/Splitting/QuadraticCharacterQuantumPostprocessing.quadratic_character_quantum_postprocessing_no_go (✓ std3). ∎

Source. Repository-derived.

Commentary.

The public input is the canonical three-ring image of units modulo sixty. Equal images force equal Gaussian, Eisenstein, and golden splitting characters, as well as every quadratic character.

The classical postprocessor is an arbitrary function of the entire three-coordinate output, so it includes addition, multiplication, and functional calculus wherever those operations are available.

Quantum preparation is constructed only from that processed classical value. Applying any final observation therefore gives the same result on both members of the fiber.

References

• Truth anchor: D5/S3/PrimeForms/Splitting/QuadraticCharacterQuantumPostprocessing.quadratic_character_quantum_postprocessing_no_go
• Dependency: D5/S3/PrimeForms/Splitting/QuadraticCharacterProfileRedundancy