TickIndex
plain-language theorem explainer
TickIndex indexes the 8 phases of the fundamental octave in Recognition Science. Physicists deriving the three fermion generations from 8-tick by 3D structure cite it when mapping discrete time to spatial bits. The declaration is a direct abbreviation with no lemmas or reduction steps.
Claim. Let $TickIndex$ denote the finite set $Fin 8$, which indexes the phases of the 8-tick cycle and equals the set of integers modulo 8.
background
Recognition Science obtains the 8-tick octave from the forcing chain at step T7, where the period is $2^3$. The upstream constant tick supplies the fundamental time quantum equal to 1 in RS-native units, and one octave is defined as exactly 8 ticks. The module setting treats the 8 phases as distributing across three orthogonal spatial dimensions (T8), with each phase labeled by a 3-bit tuple whose parity combinations label the generations.
proof idea
The declaration is a direct abbreviation with no lemmas applied and no tactic steps.
why it matters
TickIndex supplies the domain for the bit-decomposition functions that feed the bijection theorem bits_bijection and the maps bitsToTick and tickToBits. These realize the module hypothesis that exactly three generations arise from the parity patterns of the 8-tick cycle across three dimensions. It therefore closes the link between T7 (eight-tick octave) and the open question of the generation number in the Standard Model.
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