peak_exceeds_single_gap
The theorem asserts that the central binomial coefficient for an 8-element set exceeds the gap-45 threshold. Cross-domain regulatory models cite it to bound consistent configurations in Q3 recognition states before decoherence sets in. The proof is a direct computational verification that evaluates the inequality outright.
claimThe central binomial coefficient satisfies $C(8,4) >$ gap-45.
background
The CrossDomain.RegulatoryCeiling module treats the regulatory ceiling for Q3, the 8-element recognition state space whose structure is supplied by the spectral emergence result. That upstream structure forces the SU(3) x SU(2) x U(1) gauge content, exactly three particle generations, and 24 chiral fermion flavors. gap45 is the single-gap threshold calibrated from the J-cost function (strictly convex with minimum at unity) and the ledger factorization of positive reals. The local setting is the eight-tick dynamics in which each recognition step updates at most eight neighbors.
proof idea
The proof is a term-mode application of the decide tactic that reduces the concrete numerical comparison between the binomial coefficient and the gap threshold to a decidable proposition.
why it matters in Recognition Science
The result supplies the exceeds_gap field inside the regulatoryCeilingCert definition, completing the C9 structural claim that the peak of 70 still lies below the doubled gap-45 ceiling of 90. It aligns with the eight-tick octave (T7) and the emergence of D=3 spatial dimensions while supporting the gene-regulation prediction for at most 70 mutually consistent half-activated states in an 8-binary-state module.
scope and limits
- Does not compute or derive the explicit numerical value of gap45.
- Does not establish that C(8,4) is maximal among binomial coefficients on eight elements.
- Does not extend the comparison to binomial rows of length other than eight.
- Does not incorporate the double-gap fitting or total power-set cardinality.
formal statement (Lean)
33theorem peak_exceeds_single_gap : Nat.choose 8 4 > gap45 := by decide
proof body
Term-mode proof.
34
35/-- Sum of Q₃ half-rows: C(8,0)+C(8,1)+C(8,2)+C(8,3) = 93. -/