network_surj
plain-language theorem explainer
Surjectivity of the first-component projection from AttentionState to AttentionNetwork. Cross-domain attention researchers cite it to confirm every network is realized across the eight tick phases. The proof is a one-line wrapper constructing the pair with the base phase t0.
Claim. The projection map $pi : AttentionNetwork times TickPhase to AttentionNetwork$ given by the first component is surjective.
background
AttentionState is the product type AttentionNetwork times TickPhase. TickPhase is the inductive type with eight constructors t0 through t7. The module establishes that the attentional state space factors as 5 networks times 8 phases equaling 40, leaving 5 overflow slots under the complexity ceiling gap45. The TickPhase definition is shared with the TwoCubeUniversality module, enforcing the 2^3 cardinality.
proof idea
The proof is a one-line wrapper. It introduces an arbitrary network x and constructs the pair (x, TickPhase.t0), whose first component equals x by reflexivity.
why it matters
This theorem supplies the network_surj field of the AttentionSpaceCert structure, which attentionSpaceCert assembles into the full certificate. It verifies network coverage in the 5 times 8 factorization and connects to the eight-tick octave from the forcing chain T7. No open questions are addressed.
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