Golden Periodic Enumeration Through Twelve

Abstract

The exact golden periodic enumeration is complete through period twelve.

The period-eleven certificate and twenty-five primitive twelve-cycles are combined through a 322-equation exact fixed-point census.

Theorem 1.1 (Seven hundred eighty periodic points through twelve).

$$\mathrm{card}\left(goldenPeriodicPointCodesTwelve\right)=780$$

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

Source. Repository-derived.

Commentary.

The twenty-five primitive twelve-cycles add 300 phases to the 480 phases enumerated through period eleven.

Theorem 1.2 (Seventy-nine disjoint periodic orbits).

$$\mathrm{length}\left(goldenPeriodicOrbitRepresentativesTwelve\right)=79\wedge \mathrm{card}\left(goldenEnumeratedOrbitStatesTwelve\right)=780$$

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

Source. Repository-derived.

Commentary.

The 54 prior cycles and twenty-five primitive twelve-cycles have no repeated exact phase code.

Theorem 1.3 (The enumeration through period twelve is complete).

$$\forall p\in N,s\in GoldenSurvivorState,\left(\left(p\ge 1\wedge p\le 12\right)\wedge \mathrm{iterate}\left(goldenTransition,p,s\right)=s\right)\Rightarrow s\in decodedRepresentativeOrbitUnionTwelve$$

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

Source. Repository-derived.

Commentary.

Every real state fixed by a nonzero iterate of period at most twelve lies on one of the 79 decoded exact cycles.

Theorem 1.4 (The golden periodic maximin through twelve).

$$\mathrm{IsGreatest}\left(goldenPeriodicOrbitMinimaTwelve,goldenThreshold\right)$$

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

Source. Repository-derived.

Commentary.

Every new low-arm witness is bounded by the threshold, and the period-three champion continues to attain equality.

References

• Truth anchor: D5/S0/Tower/GoldenPeriodicTwelve/EnumerationTwelve.golden_periodic_code_partition_twelve
• Truth anchor: D5/S0/Tower/GoldenPeriodicTwelve/EnumerationTwelve.golden_periodic_orbit_enumeration_complete_twelve
• Truth anchor: D5/S0/Tower/GoldenPeriodicTwelve/EnumerationTwelve.golden_periodic_orbit_maximin_twelve
• Truth anchor: D5/S0/Tower/GoldenPeriodicTwelve/EnumerationTwelve.golden_periodic_point_code_count_twelve
• Dependency: D5/S0/Tower/GoldenPeriodicTwelve/EnumerationTwelveFixed