Golden Periodic Enumeration Through Eight

Abstract

One incremental exact certificate extends the golden periodic enumeration through period eight.

The frozen period-at-most-seven certificate is reused without expansion. Only period-eight branch words are solved over Q(phi), then split by their first transition so each finite check remains bounded.

Theorem 1.1 (One hundred periodic points through period eight).

$$\mathrm{card}\left(goldenPeriodicPointCodesEight\right)=100$$

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

Source. Repository-derived.

Commentary.

The five new primitive period-eight cycles contribute forty new phase states; together with the frozen sixty states this gives one hundred.

Theorem 1.2 (Seventeen disjoint periodic orbits).

$$\mathrm{length}\left(goldenPeriodicOrbitRepresentativesEight\right)=17\wedge \mathrm{card}\left(goldenEnumeratedOrbitStatesEight\right)=100$$

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

Source. Repository-derived.

Commentary.

The one hundred exact state codes split without repetition into the twelve frozen cycles and five primitive cycles of length eight.

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

$$\forall p\in N,s\in GoldenSurvivorState,\left(\left(p\ge 1\wedge p\le 8\right)\wedge \mathrm{iterate}\left(goldenTransition,p,s\right)=s\right)\Rightarrow s\in decodedRepresentativeOrbitUnionEight$$

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

Source. Repository-derived.

Commentary.

Every real state fixed by a nonzero iterate of period at most eight occurs on one of the seventeen decoded exact cycles.

Theorem 1.4 (The golden periodic maximin through eight).

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

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

Source. Repository-derived.

Commentary.

Each new cycle has a certified low state whose arm is at most phi inverse squared over two, while the frozen period-three cycle continues to attain equality.

References

• Truth anchor: D5/S0/Tower/GoldenPeriodic/EnumerationEight.golden_periodic_code_partition_eight
• Truth anchor: D5/S0/Tower/GoldenPeriodic/EnumerationEight.golden_periodic_orbit_enumeration_complete_eight
• Truth anchor: D5/S0/Tower/GoldenPeriodic/EnumerationEight.golden_periodic_orbit_maximin_eight
• Truth anchor: D5/S0/Tower/GoldenPeriodic/EnumerationEight.golden_periodic_point_code_count_eight
• Dependency: D5/S0/Tower/Champions/GoldenPeriodicEnumeration