Golden Realization Certificate

Abstract

One certificate packages the quadratic, Fibonacci, rotation-trace, Mobius-fixed, and projective-attraction realizations of the golden structure while exhibiting a repelling countermodel.

Theorem 1.1 (Canonical Golden Cross Representation Certificate).

$(G o l d e n C ross R e p rese n t a t i o n C er t i f i c a t e) .$

Proof. Machine-checked in Lean as D5/S3/Observer/WorldModel/GoldenRealizationCertificate.canonical_golden_cross_representation_certificate (✓ std3). ∎

Source. Repository-derived.

Commentary.

The canonical golden structure satisfies the full cross-representation certificate.

The declaration keeps its parameters and hypotheses explicit; the result makes no converse or broader existence claim beyond that scope.

Theorem 1.2 (Golden Repelling Affine Fixed).

$(F u n c t i o n . I s F i x e d Pt g o l d e n R e p e ll in g A ff in e R e a l . g o l d e n R a t i o) .$

Proof. Machine-checked in Lean as D5/S3/Observer/WorldModel/GoldenRealizationCertificate.golden_repelling_affine_fixed (✓ std3). ∎

Source. Repository-derived.

Commentary.

The same golden point can be fixed in a different dynamical system.

The declaration keeps its parameters and hypotheses explicit; the result makes no converse or broader existence claim beyond that scope.

Theorem 1.3 (Golden Repelling Affine Has Deriv At).

$(H a sDer i v A t g o l d e n R e p e ll in g A ff in e (R e a l . g o l d e n R a t i o^{2}) R e a l . g o l d e n R a t i o) .$

Proof. Machine-checked in Lean as D5/S3/Observer/WorldModel/GoldenRealizationCertificate.golden_repelling_affine_hasDerivAt (✓ std3). ∎

Source. Repository-derived.

Commentary.

The affine countermodel has derivative φ² at the fixed point.

The declaration keeps its parameters and hypotheses explicit; the result makes no converse or broader existence claim beyond that scope.

Theorem 1.4 (Golden Repelling Affine Multiplier Gt One).

$(1 < ∣ R e a l . g o l d e n R a t i o^{2} ∣) .$

Proof. Machine-checked in Lean as D5/S3/Observer/WorldModel/GoldenRealizationCertificate.golden_repelling_affine_multiplier_gt_one (✓ std3). ∎

Source. Repository-derived.

Commentary.

The affine countermodel is strictly repelling.

The declaration keeps its parameters and hypotheses explicit; the result makes no converse or broader existence claim beyond that scope.

Theorem 1.5 (Golden Fixed Does Not Force Attraction).

$(F u n c t i o n . I s F i x e d Pt g o l d e n R e p e ll in g A ff in e R e a l . g o l d e n R a t i o \land 1 < ∣ R e a l . g o l d e n R a t i o^{2} ∣) .$

Proof. Machine-checked in Lean as D5/S3/Observer/WorldModel/GoldenRealizationCertificate.golden_fixed_does_not_force_attraction (✓ std3). ∎

Source. Repository-derived.

Commentary.

Hence fixedness of the golden point alone does not imply attraction.

The declaration keeps its parameters and hypotheses explicit; the result makes no converse or broader existence claim beyond that scope.

References

Truth anchor: D5/S3/Observer/WorldModel/GoldenRealizationCertificate.canonical_golden_cross_representation_certificate
Truth anchor: D5/S3/Observer/WorldModel/GoldenRealizationCertificate.golden_fixed_does_not_force_attraction
Truth anchor: D5/S3/Observer/WorldModel/GoldenRealizationCertificate.golden_repelling_affine_fixed
Truth anchor: D5/S3/Observer/WorldModel/GoldenRealizationCertificate.golden_repelling_affine_hasDerivAt
Truth anchor: D5/S3/Observer/WorldModel/GoldenRealizationCertificate.golden_repelling_affine_multiplier_gt_one
Dependency: D5/S3/Observer/GoldenCoding/GoldenAngleTraceBridge
Dependency: D5/S3/Observer/WorldModel/FixedPointStabilityProfile

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Golden Realization Certificate

Abstract

References