Period-Twelve Fixed-Point Decomposition

Abstract

The 322 period-twelve fixed equations decompose into 21 prefix blocks.

Every block contains only eight, thirteen, or twenty-one exact affine fixed equations.

Theorem 1.1 (There are exactly 322 period-twelve fixed codes).

$$\mathrm{length}\left(\mathrm{goldenFixedPointCodes}\left(12\right)\right)=322$$

Proof. Machine-checked in Lean as D5/S0/Tower/GoldenPeriodicTwelve/EnumerationTwelveFixed.golden_fixed_point_code_count_exactly_twelve (✓ std3). ∎

Source. Repository-derived.

Commentary.

The twenty-one prefix-block counts sum exactly to 322.

Theorem 1.2 (Every period-twelve fixed point is an enumerated phase).

$$\mathrm{toFinset}\left(\mathrm{goldenFixedPointCodes}\left(12\right)\right)=goldenExpectedPointCodesTwelve$$

Proof. Machine-checked in Lean as D5/S0/Tower/GoldenPeriodicTwelve/EnumerationTwelveFixed.golden_fixed_point_codes_twelve_decompose (✓ std3). ∎

Source. Repository-derived.

Commentary.

The exact fixed-point codes equal all inherited phases and all 300 phases on the primitive twelve-cycles.

References

• Truth anchor: D5/S0/Tower/GoldenPeriodicTwelve/EnumerationTwelveFixed.golden_fixed_point_code_count_exactly_twelve
• Truth anchor: D5/S0/Tower/GoldenPeriodicTwelve/EnumerationTwelveFixed.golden_fixed_point_codes_twelve_decompose
• Dependency: D5/S0/Tower/GoldenPeriodicTwelve/EnumerationTwelveSeparation