closure_number_eq_9
plain-language theorem explainer
The declaration fixes the closure number at 9 for an 8-tick cycle. Researchers deriving the 45-tick synchronization from cumulative phase accumulation in the Recognition Science ledger model would cite it. The proof is a direct reflexivity step on the definition of closure number.
Claim. The number of steps required to close an 8-tick cycle equals 9.
background
In the 8-tick framework the fundamental period is 8 ticks, arising from the octave with period 2^3 that forces three spatial dimensions. The closure principle adds one extra step to return to the initial phase state after traversing the cycle, following the fence-post counting rule. Cumulative phase then sums linearly over these 9 steps to give the triangular number T(9) = 45. Upstream constants define the tick as the RS-native time quantum and supply the phase values kπ/4 for k = 0 to 7.
proof idea
The proof is a one-line term that applies reflexivity to the definition of the closure number.
why it matters
This equality supplies the missing physical count that converts the 8-tick cycle into a closed 9-step process whose cumulative phase is exactly 45. It directly addresses the paper gap on the physical motivation for 45-tick synchronization and feeds the triangular-number identities used to equate ledger neutrality with phase closure. The result sits inside the forcing chain that derives D = 3 from the eight-tick octave.
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