Explanation of attentionSpaceCert
(1) What the declaration says in plain English
The attentionSpaceCert definition builds a record that packages six verified facts about an attention model: the state space AttentionState (pairs of one of 5 networks and one of 8 phases) has exactly 40 elements; a complexity ceiling called gap45 (set to 45) leaves precisely 5 overflow slots after subtracting 40; the sum of 40 plus those 5 slots recovers 45; the 8 phases equal 2 cubed; and the projections from states onto networks and onto phases are both surjective (every network and every phase appears in at least one state).
(2) Why it matters in Recognition Science
This certificate supplies a concrete, axiom-free structural claim that attentional states factor as 5 imes 8 = 40 inside the discrete 8-tick cadence of the framework. It therefore illustrates how a cross-domain cognitive construct inherits the 8-tick discreteness already present in the supplied source and fits under a fixed complexity gap, leaving a small number of overflow singletons.
attentionStateCount tick_eq_twoPowD overflow_eq_D
(3) How to read the formal statement
AttentionSpaceCert is a Lean structure (record type) whose six fields are propositions: four equalities on cardinalities and two surjectivity statements. The definition attentionSpaceCert then supplies a concrete inhabitant of that structure by naming the already-proved theorems that witness each field (e.g., attentionStateCount for the first field). In ordinary mathematical language this is simply “here is a bundled certificate whose fields are these six facts, each witnessed by the following short proofs.”
AttentionSpaceCert attention_plus_overflow_eq_gap
(4) Visible dependencies or certificates in the supplied source
The declaration depends only on material inside the same module: the inductive types AttentionNetwork (five constructors) and TickPhase (eight constructors), the cardinality theorems networkCount and tickCount, the product cardinality attentionStateCount, the arithmetic facts overflow_eq_D and attention_plus_overflow_eq_gap, the two surjectivity lemmas network_surj and tick_surj, and the constant gap45 := 45. No external axioms or sorry appear; the module states “Lean status: 0 sorry, 0 axiom.”
networkCount tickCount network_surj tick_surj
(5) What this declaration does not prove
It does not derive the choice of five networks or eight phases from the Law of Logic or from φ; those are taken as given inductive definitions. It does not formalize any experimental prediction (e.g., attentional-blink plateaus), any link to physical constants, or any embedding into the larger forcing chain. It is strictly a local cardinality and surjectivity certificate for this cross-domain model.