grayCodeCycleLength
The definition assigns Gray code cycle length in D dimensions to equal 2^D. Ledger uniqueness proofs cite it to quantify the period needed for complete hypercube traversal in the RS ledger. It is realized as a direct power-of-two expression that specializes immediately to the value 8 when D equals 3.
claimThe length of a Gray code cycle on a $D$-dimensional hypercube equals $2^D$.
background
Module Meta.LedgerUniqueness addresses Gap 9 by showing that any discrete conservative ledger must adopt the specific triple of phi, the 3-cube, and an 8-tick cycle. The Gray code cycle length supplies the combinatorial period that forces the third component: cycles shorter than 2^D cannot visit every vertex of the D-cube without repetition. Upstream results on completeness predicates and simplicial edge lengths supply the requirement that every state must be reached exactly once in a closed traversal.
proof idea
The declaration is a direct definition that sets the return value to the power of two in the supplied dimension. No lemmas are invoked; the expression is the mathematical content itself.
why it matters in Recognition Science
It supplies the cycle-length constraint used by complete_ledger_uniqueness and rs_ledger_is_unique to close the uniqueness argument for the RS ledger. The definition thereby realizes the eight-tick octave (T7) of the forcing chain by making the period for D=3 equal to 8, the smallest integer compatible with Gray-code coverage of the 3-cube. It is cited in the module's resolution of the objection that other discrete ledgers might exist.
scope and limits
- Does not construct or verify an explicit Gray code sequence.
- Does not prove that 2^D is the minimal period for any traversal.
- Does not connect the length to the golden-ratio cost function or to physical constants.
- Does not address non-hypercube state spaces or non-conservative dynamics.
Lean usage
theorem eight_tick_minimal : grayCodeCycleLength 3 = 8 := by unfold grayCodeCycleLength; norm_numformal statement (Lean)
153def grayCodeCycleLength (D : ℕ) : ℕ := 2^D
proof body
Definition body.
154
155/-- For D=3, the minimal complete cycle is 8 = 2³. -/