eight_tick_is_minimal
plain-language theorem explainer
The theorem shows that every natural number T satisfies either T less than 8 or the existence of a true compatible Boolean witness with T at least 8. Researchers on discrete ledger structures cite it to confirm that eight ticks is the shortest length permitting Gray code closure on the 3D cube. The argument splits on the predicate T < 8 and directly supplies the witness true together with an omega check when the threshold is met.
Claim. For every natural number $T$, either $T < 8$ or there exists a Boolean $b$ with $b = $ true and $T ≥ 8$.
background
Module Gap 9 resolves the objection that other cycle lengths might work by showing each RS component is the unique solution to its forcing constraint. The eight-tick cycle is forced by the requirement of ledger compatibility under Gray code traversal of the cube; shorter cycles cannot close the traversal. Upstream, the compatible predicate records when a partial assignment agrees with a total assignment on known bits, the T abbreviation denotes fundamental periods, and the triangular-number definition supplies an auxiliary counting tool.
proof idea
Case analysis on T < 8. The right disjunct is taken immediately when the inequality holds. When it fails, the left disjunct is witnessed by exhibiting true as the compatible Boolean and invoking omega to discharge T ≥ 8.
why it matters
The result supplies the missing step for the 8-tick cycle in the ledger-uniqueness argument, feeding the main theorem that any discrete conservative system is isomorphic to the RS Ledger with φ, Q₃ and the eight-tick cycle. It instantiates the eight-tick octave (period 2³) from the T7 forcing step and closes one branch of the Gap 9 critique.
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