attentionSpaceCert
plain-language theorem explainer
The attention space certificate definition assembles the verified cardinality and surjectivity properties of the attention states into a single record. Cross-domain researchers in Recognition Science cite it to confirm that the product of the five-element network and eight-element tick phase yields exactly 40 states under the gap45 ceiling with five overflow slots. It is assembled as a direct record instantiation using the pre-established count and projection lemmas.
Claim. The attention space certificate is the structure satisfying $|$AttentionState$| = 40$, gap45 $- |$AttentionState$| = 5$, $|$AttentionState$| + 5 = $gap45$, $|$TickPhase$| = 2^3$, and the projections from attention states onto the network and tick components are surjective.
background
In the Attention Space module the attention state space is the cartesian product of the attention network (cardinality 5) and tick phase (cardinality 8). The complexity ceiling gap45 equals 45, so the product leaves exactly five overflow slots. This factorization matches the eight-tick octave structure from the forcing chain. The AttentionSpaceCert structure collects five required properties: state cardinality 40, overflow size 5, their sum recovering gap45, tick cardinality 8, and surjectivity of both projections. Upstream results establish the cardinalities by rewriting to the product definition and deciding the numerical values, while surjectivity is shown by explicit preimage construction.
proof idea
The definition is a one-line wrapper that populates the attention space certificate record by assigning the state cardinality theorem, the overflow size theorem, the sum identity theorem, the tick cardinality theorem, and the two surjectivity theorems to the respective fields.
why it matters
This definition completes the C5 structural claim by certifying that the 40-state attention space fits under gap45 with five overflow slots, aligning with the eight-tick octave from T7. It supports the module prediction of 40 stable plateaus plus five transients in attentional-blink experiments. No downstream theorems depend on it yet, leaving open its role in larger cross-domain compositions.
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