overflow_eq_D
plain-language theorem explainer
The equality gap45 minus the cardinality of attentional states equals 5 follows from the product structure of the state space. Cross-domain recognition researchers cite this when confirming the five overflow slots after the 5 by 8 factorization. The proof is a one-line wrapper that rewrites via the precomputed cardinality and applies a decision tactic.
Claim. $45 - |S| = 5$, where $S$ is the attentional state space given by the Cartesian product of the attention network and tick phase spaces.
background
The module defines AttentionState as the product AttentionNetwork × TickPhase. The upstream theorem attentionStateCount states that the cardinality of this product equals 40, obtained via networkCount = 5 and tickCount = 8. The local setting is the structural claim for wave 62 cross-domain attention: the complexity ceiling gap45 equals 45, leaving exactly 5 overflow slots after the 40 states. The module documentation notes that this predicts 40 stable plateaus plus 5 transient ones in attentional-blink experiments, with zero sorry or axiom.
proof idea
The proof is a one-line wrapper. It rewrites the left-hand side using the attentionStateCount theorem to insert the value 40, then applies the decide tactic to verify the numerical equality 45 - 40 = 5.
why it matters
This supplies the overflow_D component inside the attentionSpaceCert definition, which also collects the state count, the sum equaling the gap, the tick power-of-two relation, and network surjectivity. It realizes the module claim that gap45 leaves exactly 5 overflow slots after the 5 × 8 factorization. In the Recognition framework it extends the eight-tick octave structure to cross-domain attention, with the overflow slots corresponding to the five singletons under saturation and linking to the phi-ladder predictions.
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