neutralityScore_shift1_of_periodic8
plain-language theorem explainer
The lemma establishes that any 8-periodic real sequence on the naturals yields identical neutrality scores for 8-tick windows starting at t0 and at t0+1. Flight schedule designers and 8-phase control theorists cite it when confirming shift-stability of gate diagnostics. The tactic proof expands both scores as Finset.range 8 sums, invokes periodicity to match the wrap-around term, then chains sum_range_succ, sum_range_succ', and sum_congr reindexing to equate the expressions.
Claim. Let $w : ℕ → ℝ$ satisfy $w(t + 8) = w(t)$ for every natural number $t$. Then the 8-gate neutrality score of $w$ at $t_0 + 1$ equals the 8-gate neutrality score of $w$ at $t_0$, for any natural number $t_0$.
background
The Flight.Schedule module models control schedules as functions $w : ℕ → ℝ$ equipped with 8-periodicity to enforce phase-locked behavior over the eight-tick octave. neutralityScore is the direct re-export of the 8-gate sum diagnostic from SpiralField, used to test neutral-gate conditions. Periodic8 is the predicate ∀ t, w(t + 8) = w(t). Upstream arithmetic results supply add_assoc and add_comm for the reindexing steps inside the finite sums.
proof idea
The proof unfolds neutralityScore into an 8-term Finset sum. It derives w(t0 + 8) = w(t0) directly from the periodicity hypothesis. Two have statements apply sum_range_succ and sum_range_succ'.symm to peel the endpoint terms, while sum_congr aligns the seven-term partial sums by the shift k ↦ k + 1. The pieces are combined in a calc chain that equates the shifted window sum to the original.
why it matters
The result is invoked inside eightGateNeutral_shift_invariance to obtain the forward direction of the full shift equivalence for the eightGateNeutral predicate. It supplies the elementary invariance step required by the 8-tick control discipline in the Flight module and aligns with the eight-tick octave (T7) of the Recognition Science forcing chain. No scaffolding or open questions remain; the lemma closes a basic stability property for periodic schedules.
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