Tribonacci Periodic Enumeration

Abstract

Exact computation supplies ten valid, disjoint Tribonacci cycles through period five.

Theorem 1.1 (Fixed-point equation counts through period five).

$$\mathrm{fixedPointEquationCountsThrough}\left(5\right)=\mathrm{list}\left(1,3,7,11,21\right)$$

Proof. Machine-checked in Lean as D5/S0/Tower/DBonacciGeneral/TribonacciPeriodicEnumeration.tribonacci_fixed_point_counts_through_five (✓ std3). ∎

Source. Repository-derived.

Commentary.

The closed-walk generator produces 1, 3, 7, 11, and 21 phase-marked fixed-point equations at periods one through five.

Theorem 1.2 (The ten orbit periods are explicit).

$$\mathrm{orbitPeriodList}\left(tribonacciPeriodicOrbitRepresentativesFive\right)=\mathrm{list}\left(1,2,3,3,4,4,5,5,5,5\right)$$

Proof. Machine-checked in Lean as D5/S0/Tower/DBonacciGeneral/TribonacciPeriodicEnumeration.tribonacci_periodic_orbit_period_distribution_five (✓ std3). ∎

Source. Repository-derived.

Commentary.

There is one primitive cycle of periods one and two, two cycles of periods three and four, and four cycles of period five.

Theorem 1.3 (Every representative uses valid branches).

$$\mathrm{Forall}\left(tribonacciPeriodicOrbitRepresentativesFive,tribonacciCodedOrbitValid\right)$$

Proof. Machine-checked in Lean as D5/S0/Tower/DBonacciGeneral/TribonacciPeriodicEnumeration.tribonacci_periodic_orbit_representatives_valid (✓ std3). ∎

Source. Repository-derived.

Commentary.

Exact cubic inequalities certify the source gap, branch side, target gap, and closure condition for every displayed representative.

Theorem 1.4 (Coded phase states are globally disjoint).

$$\mathrm{Nodup}\left(\mathrm{flatMapOrbitStates}\left(tribonacciPeriodicOrbitRepresentativesFive\right)\right)$$

Proof. Machine-checked in Lean as D5/S0/Tower/DBonacciGeneral/TribonacciPeriodicEnumeration.tribonacci_periodic_orbit_state_codes_nodup (✓ std3). ∎

Source. Repository-derived.

Commentary.

Flattening all ten cycles gives a list with no repeated exact cubic state code, including across different representatives.

References

• Truth anchor: D5/S0/Tower/DBonacciGeneral/TribonacciPeriodicEnumeration.tribonacci_fixed_point_counts_through_five
• Truth anchor: D5/S0/Tower/DBonacciGeneral/TribonacciPeriodicEnumeration.tribonacci_periodic_orbit_period_distribution_five
• Truth anchor: D5/S0/Tower/DBonacciGeneral/TribonacciPeriodicEnumeration.tribonacci_periodic_orbit_representatives_valid
• Truth anchor: D5/S0/Tower/DBonacciGeneral/TribonacciPeriodicEnumeration.tribonacci_periodic_orbit_state_codes_nodup
• Dependency: D5/S0/Tower/DBonacciGeneral/ChampionValue
• Dependency: D5/S0/Tower/DBonacciGeneral/TribonacciPeriodicGenerator