Golden Permanent Survivors
Abstract
Strict golden survival has no permanent state; the closed threshold has a larger proved preperiodic carrier.
The source conflated two different constructions. Intersecting the closures of the four strict finite-depth tubes gives four limiting points, but permanent survival for the closed threshold also retains boundary preimages. The strict threshold is the usable replacement for the upper-bound argument.
Theorem 1.1 (The strict permanent survivor set is empty).
Proof. Machine-checked in Lean as D5/S0/Tower/Champions/GoldenPermanentSurvivors.golden_strict_permanent_set_eq_empty (✓ std3). ∎
Source. Repository-derived.
Commentary.
The pointwise classification places any hypothetical strict permanent state in the four-point closed-tube limit. Each of those four points is an excluded endpoint of the open depth-two tubes, so no state survives every strict backward depth.
Theorem 1.2 (The known closed preperiodic carrier survives).
Proof. Machine-checked in Lean as D5/S0/Tower/Champions/GoldenPermanentSurvivors.golden_known_closed_preperiodic_carrier_subset (✓ std3). ∎
Source. Repository-derived.
Commentary.
The proved carrier has eight states: the large-gap threshold point plus the frozen seven-state carrier. The threshold point maps directly to the tail. The counterexample used by the frozen theorem is the large-gap state with coordinate (9 minus 5 phi) over 2; its orbit passes through the large coordinate (4 phi minus 5) over 2, the small and large tail coordinates phi inverse over 2, and then the three-state champion cycle. The inclusion is deliberately not stated as equality and does not claim a complete closed-set classification.
References
- Truth anchor:
D5/S0/Tower/Champions/GoldenPermanentSurvivors.golden_known_closed_preperiodic_carrier_subset - Truth anchor:
D5/S0/Tower/Champions/GoldenPermanentSurvivors.golden_strict_permanent_set_eq_empty - Dependency: D5/S0/Tower/Champions/GoldenSurvivorClassification