halfRowSum_eq
The equality asserts that the partial binomial sum from k=0 to 3 for n=8 equals 93. Researchers bounding consistent configurations in 8-state recognition systems cite this to confirm the half-power-set lies below the gap-45 threshold. The proof is a direct evaluation by decision procedure on the finite natural-number expression.
claimLet $S = C(8,0) + C(8,1) + C(8,2) + C(8,3)$. Then $S = 93$.
background
The RegulatoryCeiling module treats 8-element recognition states whose power set is partitioned into rows of binomial coefficients. The half-row sum is the explicit definition summing the lower four coefficients, which counts configurations with at most three active bits. This local setting is the structural claim that the central coefficient C(8,4) = 70 remains the unique maximum and satisfies 70 < 2 * gap45. The upstream definition supplies the exact expression whose numerical value is asserted here.
proof idea
The proof is a one-line wrapper that applies the decide tactic to evaluate the closed-form sum of four binomial coefficients.
why it matters in Recognition Science
This equality supplies the concrete numerical anchor for the regulatory-ceiling certification that follows in the same module. It directly supports the C9 claim that an 8-state system admits at most 70 mutually consistent half-activated configurations before decoherence. Within the Recognition framework it closes the arithmetic step needed to compare the central binomial peak against the doubled gap-45 bound.
scope and limits
- Does not compute or prove the central coefficient C(8,4).
- Does not derive the gap-45 threshold or the regulatory ceiling itself.
- Does not address gene-regulation predictions or cosmological spectra.
formal statement (Lean)
38theorem halfRowSum_eq : halfRowSum = 93 := by decide
proof body
Term-mode proof.
39
40/-- Total Q₃ power set: Σ_k C(8,k) = 2^8 = 256. -/