Primitive Golden Period-Ten Certificates

Abstract

Eleven exact primitive period-ten orbit certificates extend the golden data table.

The period-ten branch words are solved exactly over Q(phi). Their closure, validity, separation from earlier periods, and low-arm witnesses are checked in bounded pairs and a final singleton.

Theorem 1.1 (Eleven primitive period-ten orbits).

$$\mathrm{length}\left(goldenPeriodicOrbitRepresentativesExactlyTen\right)=11$$

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

Source. Repository-derived.

Commentary.

The exact table contains eleven representatives, each carrying a ten-step closed itinerary.

Theorem 1.2 (The period-ten representatives are valid).

$$\forall O\in goldenPeriodicOrbitRepresentativesExactlyTen,\mathrm{goldenCodedOrbitValid}\left(O\right)$$

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

Source. Repository-derived.

Commentary.

Every displayed code remains in the unit interval and follows the source, target, and affine rules of its branch word.

Theorem 1.3 (The one hundred ten new state codes are distinct).

$$\mathrm{Nodup}\left(\mathrm{flatMap}\left(goldenOrbitStates,goldenPeriodicOrbitRepresentativesExactlyTen\right)\right)$$

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

Source. Repository-derived.

Commentary.

Flattening the eleven ten-cycles produces no repeated exact state code and no collision with the earlier enumeration.

Theorem 1.4 (Period-ten low arms obey the golden bound).

$$\forall O\in goldenPeriodicOrbitRepresentativesExactlyTen,\mathrm{goldenStateArm}\left(\mathrm{decodeGoldenState}\left(\mathrm{lowState}\left(O\right)\right)\right)\le goldenThreshold$$

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

Source. Repository-derived.

Commentary.

Each new cycle has an explicit member whose arm is at most the exact golden threshold.

References

• Truth anchor: D5/S0/Tower/GoldenPeriodic/EnumerationTenData.golden_new_periodic_orbit_count_ten
• Truth anchor: D5/S0/Tower/GoldenPeriodic/EnumerationTenData.golden_new_periodic_orbit_low_arms_bounded_ten
• Truth anchor: D5/S0/Tower/GoldenPeriodic/EnumerationTenData.golden_new_periodic_orbit_representatives_valid_ten
• Truth anchor: D5/S0/Tower/GoldenPeriodic/EnumerationTenData.golden_new_periodic_orbit_state_codes_nodup_ten
• Dependency: D5/S0/Tower/GoldenPeriodic/EnumerationNine