phase_45
The declaration defines phase_45 as the cumulative phase accumulated over a closed 8-tick cycle, realized as the 9th triangular number T(9). Researchers addressing the physical motivation for 45-tick synchronization in the dimension-forcing chain would cite it to ground the number 45 in the ledger model. It is realized as a direct abbreviation of cumulative_phase applied to the closure step of 9.
claimLet $P_{45}$ denote the cumulative phase over a closed 8-tick cycle. Then $P_{45} := T(9) = 45$, where $T(n) = n(n+1)/2$ is the $n$th triangular number and the closure step augments the 8-tick octave by one tick to restore the initial phase state.
background
In Recognition Science the eight-tick octave (period $2^3$) is forced at T7 of the unified forcing chain and corresponds to $D=3$ spatial dimensions. The module supplies the physical motivation for the 45-tick synchronization by interpreting 45 as the cumulative phase sum over a closed cycle: each tick $k$ contributes phase proportional to $k$, so the total is the triangular number $T(9)$. Upstream results include the fundamental tick $τ_0 = 1$ (Constants.tick) and the abbreviation $T$ for fundamental periods (Breath1024.T).
proof idea
This is a one-line definition that directly assigns phase_45 to the value of cumulative_phase applied to the closure number 9. It inherits the triangular-number identity and the 8-tick closure principle already established in the module derivation and the imported eight_tick constant.
why it matters in Recognition Science
The definition grounds the number 45 in the Gap45.PhysicalMotivation module, directly feeding the theorem gap_45_from_phase (phase_45 = 45) and the synchronization period sync_period = lcm(8,45) = 360. It closes the paper-noted gap on physical motivation for the 45-tick argument, linking the triangular accumulation to the eight-tick octave and the forcing of $D=3$.
scope and limits
- Does not derive the closure number 9 from first principles.
- Does not compute J-cost or defectDist quantities.
- Does not address synchronization outside the 8-tick ledger model.
- Does not prove numerical stability under perturbations of the tick quantum.
Lean usage
def sync_period : ℕ := Nat.lcm eight_tick phase_45formal statement (Lean)
132def phase_45 : ℕ := cumulative_phase closure_number
proof body
Definition body.
133
134/-- **MAIN THEOREM**: The 45-tick period emerges from cumulative phase
135 accumulation over a closed 8-tick cycle.
136
137 45 = T(9) = T(8+1) = cumulative phase of closed 8-tick cycle. -/