regulatoryCeilingCert
RegulatoryCeilingCert packages the fact that the central binomial coefficient on eight elements reaches 70 at k=4, remains maximal, exceeds the single gap-45 threshold yet fits under its double, and sums with siblings to the full power set 256. A researcher modeling binary regulatory modules or recognition states on the cube Q3 cites it to bound mutually consistent half-activated configurations at 70 before decoherence. The construction is a direct record that wires together the decide-proved lemmas choose_8_4, choose_8_4_is_max, peak_fits_2gap
claimThe certificate asserts that the binomial coefficient satisfies $C(8,4)=70$, that $C(8,k)≤C(8,4)$ for all $k≤8$, that $C(8,4)≤2·g_{45}$, that $C(8,4)>g_{45}$, and that the sum of the row equals $2^8=256$, where $g_{45}$ is the gap-45 ceiling.
background
In the RegulatoryCeiling module the structure RegulatoryCeilingCert collects five properties of the binomial coefficients on an 8-element set. Its peak field requires the equality $C(8,4)=70$; peak_is_max requires maximality over the row; fits_double_gap and exceeds_gap place the peak relative to the gap-45 threshold; total_256 recovers the power-set identity. The module doc states the structural claim that this peak is the maximum binomial coefficient on Q3 and that 70 still lies below the doubled gap-45 ceiling, yielding a gene-regulation prediction that an 8-state module supports at most 70 mutually consistent half-activated configurations before decoherence. Upstream, choose_8_4 proves the equality by decide, choose_8_4_is_max verifies maximality by exhaustive interval_cases, peak_fits_double_gap and peak_exceeds_single_gap establish the two gap relations, and totalPowerSet recovers the sum identity.
proof idea
The definition is a one-line record constructor that supplies the peak value from choose_8_4, the maximality statement from choose_8_4_is_max, the double-gap fit from peak_fits_double_gap, the single-gap exceedance from peak_exceeds_single_gap, and the total sum from totalPowerSet.
why it matters in Recognition Science
This definition realizes the C9 regulatory-ceiling claim by packaging the binomial peak C(8,4)=70 as the maximum on Q3. It supports the gene-regulation prediction that eight binary states permit at most 70 half-activated configurations before decoherence. In the Recognition Science framework it instantiates the eight-tick octave (T7) by bounding the central binomial coefficient under the gap-45 ceiling. No downstream uses are recorded yet, leaving open its integration into larger cross-domain arguments.
scope and limits
- Does not prove the existence or numerical value of the gap-45 threshold.
- Does not extend the bound to binomial rows of length other than 8.
- Does not address continuous or probabilistic relaxations of the regulatory model.
- Does not connect to the J-cost function or phi-ladder from the core forcing chain.
formal statement (Lean)
51def regulatoryCeilingCert : RegulatoryCeilingCert where
52 peak := choose_8_4
proof body
Definition body.
53 peak_is_max := choose_8_4_is_max
54 fits_double_gap := peak_fits_double_gap
55 exceeds_gap := peak_exceeds_single_gap
56 total_256 := totalPowerSet
57
58end IndisputableMonolith.CrossDomain.RegulatoryCeiling