Golden Periodic Enumeration Through Ten

Abstract

The exact golden periodic enumeration is complete through period ten.

The period-nine theorem and eleven new primitive period-ten cycles are combined through an eight-block fixed-point decomposition.

Theorem 1.1 (Two hundred eighty-two periodic points through period ten).

$$\mathrm{card}\left(goldenPeriodicPointCodesTen\right)=282$$

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

Source. Repository-derived.

Commentary.

The eleven primitive ten-cycles contribute one hundred ten new phase states to the 172 states known through period nine.

Theorem 1.2 (Thirty-six disjoint periodic orbits).

$$\mathrm{length}\left(goldenPeriodicOrbitRepresentativesTen\right)=36\wedge \mathrm{card}\left(goldenEnumeratedOrbitStatesTen\right)=282$$

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

Source. Repository-derived.

Commentary.

The exact state table splits without repetition into twenty-five prior cycles and eleven primitive cycles of length ten.

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

$$\forall p\in N,s\in GoldenSurvivorState,\left(\left(p\ge 1\wedge p\le 10\right)\wedge \mathrm{iterate}\left(goldenTransition,p,s\right)=s\right)\Rightarrow s\in decodedRepresentativeOrbitUnionTen$$

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

Source. Repository-derived.

Commentary.

Every real state fixed by a nonzero iterate of period at most ten occurs on one of the thirty-six decoded exact cycles.

Theorem 1.4 (The golden periodic maximin through ten).

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

Proof. Machine-checked in Lean as D5/S0/Tower/GoldenPeriodic/EnumerationTen.golden_periodic_orbit_maximin_ten (✓ 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/EnumerationTen.golden_periodic_code_partition_ten
• Truth anchor: D5/S0/Tower/GoldenPeriodic/EnumerationTen.golden_periodic_orbit_enumeration_complete_ten
• Truth anchor: D5/S0/Tower/GoldenPeriodic/EnumerationTen.golden_periodic_orbit_maximin_ten
• Truth anchor: D5/S0/Tower/GoldenPeriodic/EnumerationTen.golden_periodic_point_code_count_ten
• Dependency: D5/S0/Tower/GoldenPeriodic/EnumerationTenFixed