Explanation of the theorem attention_plus_overflow_eq_gap
(1) In plain English, the declaration states that the number of possible attentional states (40) plus exactly 5 overflow slots equals the complexity ceiling value 45.
(2) In Recognition Science this matters because the module defines an attentional state space as the product of 5 attention networks and 8 tick phases. The equality shows this space fits strictly under the gap45 ceiling while leaving a precise 5-slot overflow, interpreted in the module docstring as the five attention-network singletons under saturation. This supplies a structural count used for cross-domain claims.
(3) The formal statement reads:
theorem attention_plus_overflow_eq_gap :
Fintype.card AttentionState + 5 = gap45 := by
rw [attentionStateCount]; decide
AttentionState is the product type AttentionNetwork × TickPhase. Fintype.card computes its size. The proof rewrites via the count theorem and decides the arithmetic. gap45 is the constant 45.
(4) Visible dependencies in the supplied source: the proof calls attentionStateCount (which itself uses networkCount and tickCount); a parallel statement overflow_eq_D exists; the whole set is packaged in the certificate structure attentionSpaceCert whose field sum_is_gap is exactly this theorem. The module reports 0 sorry and 0 axiom.
(5) The declaration proves only the numerical equality. It does not prove the inductive definitions of the five networks or eight phases, any experimental prediction, or any link to the forcing chain or physical constants.