eight_tick_period
The definition sets the eight-tick period to the constant 8, encoding T7 of the Recognition Science forcing chain as the cycle length 2^D for D=3 spatial dimensions. Researchers deriving the full synchronization period or closure factors in Gap45.Derivation cite this value when forming the lcm with the gap. The assignment is a direct constant that matches the eight-tick octave requirement from the T0-T8 chain.
claimThe eight-tick period is the natural number $8 = 2^3$, where the exponent 3 equals the number of spatial dimensions forced by the framework.
background
The StillnessGenerative module shows that the zero-defect ground state x=1 is unstable and must generate non-trivial structure. From the T0-T8 chain, T7 requires an eight-tick cycle that visits eight distinct states on the three-dimensional configuration space rather than collapsing to a degenerate period-1 case. This definition supplies the period length 8 used to build the full synchronization period via lcm with the gap of 45.
proof idea
The declaration is a direct definition that assigns the constant 8. It encodes the T7 period requirement with no lemmas or tactics applied.
why it matters in Recognition Science
This supplies the period length required by T7 that feeds into downstream results in Gap45.Derivation, including closure_factor defined as eight_tick_period plus one, cycles_of_eight, and D_3_forced_from_structure which asserts eight_tick_period equals 2 cubed together with gap equals 45 and full_period equals 360. It anchors the eight-tick octave landmark, enabling the 360-unit synchronization period.
scope and limits
- Does not derive the value 8 from the functional equation or J-cost.
- Does not specify the states visited or prove non-degeneracy of the cycle.
- Does not address the phi-ladder or recognition costs within the cycle.
- Does not compute the full period or gap; those appear in downstream declarations.
Lean usage
def closure_factor : ℕ := eight_tick_period + 1formal statement (Lean)
212def eight_tick_period : ℕ := 8
proof body
Definition body.
213
214/-- A cycle of length k through a configuration is **non-degenerate** if
215 it visits at least two distinct values. A degenerate cycle has
216 effective period 1, not 8. -/