Primitive Golden Period-Nine Certificates

Abstract

Eight exact primitive period-nine orbit certificates extend the golden data table.

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

Theorem 1.1 (Eight primitive period-nine orbits).

$$\mathrm{length}\left(goldenPeriodicOrbitRepresentativesExactlyNine\right)=8$$

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

Source. Repository-derived.

Commentary.

The exact table contains eight representatives, each carrying a nine-step closed itinerary.

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

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

Proof. Machine-checked in Lean as D5/S0/Tower/GoldenPeriodic/EnumerationNineData.golden_new_periodic_orbit_representatives_valid_nine (✓ 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 seventy-two new state codes are distinct).

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

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

Source. Repository-derived.

Commentary.

Flattening the eight nine-cycles produces no repeated exact state code.

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

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

Proof. Machine-checked in Lean as D5/S0/Tower/GoldenPeriodic/EnumerationNineData.golden_new_periodic_orbit_low_arms_bounded_nine (✓ 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/EnumerationNineData.golden_new_periodic_orbit_count_nine
• Truth anchor: D5/S0/Tower/GoldenPeriodic/EnumerationNineData.golden_new_periodic_orbit_low_arms_bounded_nine
• Truth anchor: D5/S0/Tower/GoldenPeriodic/EnumerationNineData.golden_new_periodic_orbit_representatives_valid_nine
• Truth anchor: D5/S0/Tower/GoldenPeriodic/EnumerationNineData.golden_new_periodic_orbit_state_codes_nodup_nine
• Dependency: D5/S0/Tower/GoldenPeriodic/EnumerationEight