combinatoricsCert
The combinatorics certificate assembles the count of five families with the binomial identity binom(8,4) equals 70 and its comparison to 45 into one record. A researcher tracing combinatorial structure from the eight-tick octave in Recognition Science would cite this certificate to anchor the gap-45 relations. The definition is a direct record construction that substitutes the decided equalities and inequalities from the upstream binomial theorems.
claimThe combinatorics certificate is the structure asserting that the cardinality of the set of combinatorial families equals 5, that the binomial coefficient $binom(8,4)$ equals 70, and that $binom(8,4)$ exceeds 45.
background
CombinatoricsCert collects three facts required by the module: the number of combinatorial families is five, binom(8,4) equals 70, and binom(8,4) exceeds 45. The module derives these from the eight-tick octave where 8 equals 2 to the power D with D equal to 3 spatial dimensions, noting that binom(8,4) equals 70 equals gap45 plus 25 equals gap45 plus D to the power D-1. The upstream theorem combinatoricsFamilyCount establishes the cardinality equals 5 by direct decision, while choose84_eq_70 and choose84_gt_gap45 supply the binomial equality and the strict inequality over 45.
proof idea
The definition constructs the CombinatoricsCert record by assigning the five_families field from the cardinality theorem, the choose84 field from the equality theorem, and the choose84_gt field from the inequality theorem. Each source theorem is proved by the decide tactic, which reduces the statements to decidable equalities and inequalities over natural numbers.
why it matters in Recognition Science
This certificate supplies the combinatorial identities required for the Recognition Science derivation of five families from the eight-tick period. It fills the combinatorial step in the forcing chain leading to D equals 3 dimensions and confirms binom(8,4) equals 70 near the gap-45 threshold in the phi-ladder. The module document states that five canonical combinatorial families equal configDim D equals 5.
scope and limits
- Does not derive the binomial coefficients from the J-cost function or Recognition Composition Law.
- Does not prove the existence of the five families from the self-similar fixed point phi.
- Does not connect the combinatorial count to the Berry creation threshold or Z_cf.
- Does not address higher-dimensional extensions beyond D equals 3.
formal statement (Lean)
45def combinatoricsCert : CombinatoricsCert where
46 five_families := combinatoricsFamilyCount
proof body
Definition body.
47 choose84 := choose84_eq_70
48 choose84_gt := choose84_gt_gap45
49
50end IndisputableMonolith.Mathematics.CombinatoricsFromRS