Golden Periodic Enumeration Through Nine

Abstract

The exact golden periodic enumeration is complete through period nine.

The period-eight theorem and eight new primitive period-nine cycles are combined without expanding one monolithic arithmetic proof. A first- and second-step partition keeps the finite comparisons bounded.

Theorem 1.1 (One hundred seventy-two periodic points through period nine).

$$\mathrm{card}\left(goldenPeriodicPointCodesNine\right)=172$$

Proof. Machine-checked in Lean as D5/S0/Tower/GoldenPeriodic/EnumerationNine.golden_periodic_point_code_count_nine (✓ std3). ∎

Source. Repository-derived.

Commentary.

The eight primitive nine-cycles contribute seventy-two new phase states to the one hundred states known through period eight.

Theorem 1.2 (Twenty-five disjoint periodic orbits).

$$\mathrm{length}\left(goldenPeriodicOrbitRepresentativesNine\right)=25\wedge \mathrm{card}\left(goldenEnumeratedOrbitStatesNine\right)=172$$

Proof. Machine-checked in Lean as D5/S0/Tower/GoldenPeriodic/EnumerationNine.golden_periodic_code_partition_nine (✓ std3). ∎

Source. Repository-derived.

Commentary.

The exact state table splits without repetition into seventeen prior cycles and eight primitive cycles of length nine.

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

$$\forall p\in N,s\in GoldenSurvivorState,\left(\left(p\ge 1\wedge p\le 9\right)\wedge \mathrm{iterate}\left(goldenTransition,p,s\right)=s\right)\Rightarrow s\in decodedRepresentativeOrbitUnionNine$$

Proof. Machine-checked in Lean as D5/S0/Tower/GoldenPeriodic/EnumerationNine.golden_periodic_orbit_enumeration_complete_nine (✓ std3). ∎

Source. Repository-derived.

Commentary.

Every real state fixed by a nonzero iterate of period at most nine occurs on one of the twenty-five decoded exact cycles.

Theorem 1.4 (The golden periodic maximin through nine).

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

Proof. Machine-checked in Lean as D5/S0/Tower/GoldenPeriodic/EnumerationNine.golden_periodic_orbit_maximin_nine (✓ std3). ∎

Source. Repository-derived.

Commentary.

All new cycles stay below the exact threshold, while the inherited period-three champion continues to attain equality.

References

• Truth anchor: D5/S0/Tower/GoldenPeriodic/EnumerationNine.golden_periodic_code_partition_nine
• Truth anchor: D5/S0/Tower/GoldenPeriodic/EnumerationNine.golden_periodic_orbit_enumeration_complete_nine
• Truth anchor: D5/S0/Tower/GoldenPeriodic/EnumerationNine.golden_periodic_orbit_maximin_nine
• Truth anchor: D5/S0/Tower/GoldenPeriodic/EnumerationNine.golden_periodic_point_code_count_nine
• Dependency: D5/S0/Tower/GoldenPeriodic/EnumerationNineData