beat_diff
The arithmetic identity establishing the beat frequency between the 45-fold phase structure and the 8-tick cycle equals 37 is recorded here. Researchers examining the Gap-45 Recognition Barrier cite this to confirm that the numerator of the beat frequency 37/360 is prime and therefore irreducible. The proof is a one-line numerical normalization of the subtraction.
claimThe beat frequency between the 45-fold phase structure and the 8-tick cycle satisfies $45 - 8 = 37$.
background
The Gap-45 Recognition Barrier module formalizes the mathematical core arising from coprimality gcd(8,45)=1. No proper divisor of 360 simultaneously divides both 8 and 45, so any finite window shorter than 360 ticks encounters incompatible demands from the two periodicities. The beat frequency 37/360 is highlighted because 37 is prime, rendering the frequency irreducible in the discrete setting.
proof idea
The proof is a one-line wrapper that applies numerical normalization to evaluate the subtraction directly.
why it matters in Recognition Science
This supplies the explicit beat frequency used to demonstrate window insufficiency in the Gap-45 barrier. It supports the claim that finite-horizon decision procedures cannot resolve the 8-tick and 45-fold constraints simultaneously, feeding the parent BarrierCert structure. The result sits inside the eight-tick octave of the forcing chain and leaves the physical interpretation of consciousness emergence as an open hypothesis.
scope and limits
- Does not prove any physical or phenomenological consequence of the barrier.
- Does not address continuous-time or infinite-horizon alignment.
- Does not establish irrationality beyond the primality of the numerator.
formal statement (Lean)
40theorem beat_diff : 45 - 8 = 37 := by norm_num
proof body
Term-mode proof.
41
42/-- 37 is prime: the beat frequency is irreducible. -/