Golden Euler Gap Word Constraints

Abstract

The deterministic golden Euler frequency word forbids two consecutive short steps and three consecutive long steps, and Euler phase letters inherit the same grammar.

Theorem 1.1 (A short frequency step forces a following long step).

$$\forall p:Nat.Primes,\forall v:Nat,\mathrm{goldenWord}(v)=\mathrm{false}\Rightarrow (\mathrm{primeLayerFrequency}(p,v+1)-\mathrm{primeLayerFrequency}(p,v)=\phi \times \log (p))\wedge (\mathrm{primeLayerFrequency}(p,v+1+1)-\mathrm{primeLayerFrequency}(p,v+1)={\phi }^2\times \log (p)).$$

Proof. Machine-checked in Lean as D5/S3/Analytic/GoldenEulerGapWordConstraints.short_frequency_forces_next_long (✓ std3). ∎

Source. Repository-derived.

Commentary.

The golden word identifies true letters with phi-squared prime-log gaps and false letters with phi prime-log gaps. Existing golden desubstitution proves that false-false never occurs, so every short frequency letter is followed by a long one.

The same module proves that three long letters never occur and transports both forbidden-word laws to the Euler phase alphabet. This is a deterministic symbolic constraint; an explicit stochastic non-iid theorem would additionally require a chosen probability measure.

References

• Truth anchor: D5/S3/Analytic/GoldenEulerGapWordConstraints.short_frequency_forces_next_long
• Dependency: D5/S3/Observer/GoldenPrimeCircle/GoldenEulerStepPhaseLaw