Prime-Golden Bidegree Frequency Rigidity

Abstract

In one prime channel, irrational golden frequency faithfully recovers the prime-event and short-step counts.

Theorem 1.1 (Real frequency recovers the bidegree count ledger).

$$\begin{array}{c}\forall p:Nat.Primes,\\ \mathrm{Injective}(\mathrm{bidegreeFrequency}(p))\wedge \\ \forall u,w,\mathrm{isSinglePrimeWord}(p,u)\Rightarrow \mathrm{isSinglePrimeWord}(p,w)\Rightarrow \\ \mathrm{totalStepFrequency}(u)=\mathrm{totalStepFrequency}(w)\Rightarrow \mathrm{primeGoldenBidegree}(u)=\mathrm{primeGoldenBidegree}(w).\end{array}$$

Proof. Machine-checked in Lean as D5/S3/Observer/Chronology/PrimeGoldenBidegreeFrequencyRigidity.prime_golden_bidegree_frequency_rigidity (✓ std3). ∎

Source. Repository-derived.

Commentary.

For fixed prime p, the scalar frequency of bidegree (k,s) is (k phi^2 - s) log p.

The nonzero prime logarithm and irrationality of the golden ratio make this map injective on natural-number bidegrees.

The result recovers event count and short-step count, while chronology within the recovered bidegree remains outside the scalar frequency readout.

References

• Truth anchor: D5/S3/Observer/Chronology/PrimeGoldenBidegreeFrequencyRigidity.prime_golden_bidegree_frequency_rigidity
• Dependency: D5/S1/Phase/Basic
• Dependency: D5/S3/Observer/Chronology/PrimeGoldenBigradedChronologicalSignature