TickPhase
plain-language theorem explainer
The inductive definition enumerates an eight-element phase set with constructors t0 through t7. Cross-domain researchers cite it to anchor the uniform 8-count in attention networks, DFT decompositions, and Pauli structures for three spatial dimensions. The definition proceeds by direct constructor enumeration that automatically installs decidable equality, representation, boolean equality, and finite-type structure.
Claim. Let $P$ be the inductive type whose elements are the eight phases $t_0,t_1,t_2,t_3,t_4,t_5,t_6,t_7$, equipped with decidable equality, a representation, boolean equality, and finite-type cardinality.
background
The module establishes that the count 8 = 2^3 appears uniformly across Recognition Science domains as the maximal periodic structure for spatial dimension D = 3. TickPhase supplies the concrete 8-element set realizing the eight-tick fundamental period. It is imported from the sibling AttentionSpace module, where AttentionState is defined as the product of an attention network with this phase set.
proof idea
The declaration is an inductive definition consisting of eight nullary constructors. The deriving clause installs DecidableEq, Repr, BEq, and Fintype by standard enumeration.
why it matters
TickPhase supplies the cardinality statement used in AttentionSpaceCert to record tick_2cube : Fintype.card TickPhase = 2^3 and thereby close the attention-state gap relation. It realizes the T7 eight-tick octave landmark of the forcing chain and supplies the common count for the recognition cube in D = 3. The module collects parallel 8-element instances (DFT modes, Q3 vertices, Pauli elements) to establish the shared identity.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.