RegulatoryCeilingCert
A certificate structure records that the central binomial coefficient for an eight-element set equals 70 and is maximal among all entries in its row. It further confirms that this peak lies below twice the gap-45 threshold yet above the single gap, while the sum of all binomial coefficients recovers the full power-set size 256. Workers on state-counting bounds in recognition models cite the certificate to cap the number of coherent configurations at 70 for an eight-state regulatory module. The structure is assembled directly from five sibling 8
claim$C(8,4)=70$, $C(8,k)≤70$ for all $k≤8$, $70≤2·g_{45}$, $70>g_{45}$, and $∑_{k=0}^8 C(8,k)=256$, where $g_{45}$ is the gap-45 threshold drawn from the phi-ladder.
background
The CrossDomain module treats regulatory ceilings via the binomial row for eight recognition states, the size fixed by the eight-tick octave of the forcing chain. The gap-45 threshold supplies the numerical decoherence limit taken from the phi-ladder spacing in the mass formula. The module document states that this peak remains below the doubled gap-45 ceiling, yielding the concrete bound of at most 70 mutually consistent half-activated configurations before decoherence.
proof idea
The declaration is a bare structure definition. Each field is witnessed by a direct reference to a sibling lemma: choose_8_4 supplies the peak value, choose_8_4_is_max supplies maximality, peak_fits_double_gap supplies the double-gap fit, peak_exceeds_single_gap supplies the single-gap exceedance, and totalPowerSet supplies the sum. No additional tactics or reductions are performed.
why it matters in Recognition Science
The structure is instantiated by the downstream regulatoryCeilingCert definition. It realizes the C9 regulatory-ceiling claim that the maximum number of consistent states on the eight-element recognition lattice is 70. Within the forcing chain this corresponds to the T7 eight-tick octave and supplies a numerical cap any regulatory model must respect before crossing the Berry creation threshold. The result leaves open how the gap-45 value is itself obtained from the phi-ladder constants.
scope and limits
- Does not compute the numerical value of the gap-45 threshold.
- Does not derive the gene-regulation interpretation from the forcing chain.
- Does not extend the counting argument to state spaces larger than eight elements.
- Does not prove uniqueness of the maximum across all recognition lattices.
formal statement (Lean)
44structure RegulatoryCeilingCert where
45 peak : Nat.choose 8 4 = 70
46 peak_is_max : ∀ k, k ≤ 8 → Nat.choose 8 k ≤ Nat.choose 8 4
47 fits_double_gap : Nat.choose 8 4 ≤ 2 * gap45
48 exceeds_gap : Nat.choose 8 4 > gap45
49 total_256 : (Finset.range 9).sum (fun k => Nat.choose 8 k) = 256
50