rs_pattern_window_neutral
plain-language theorem explainer
The sum of the RS-coherent 8-beat modulation pattern over one complete cycle equals zero. Photobiomodulation device designers cite this to confirm the pattern produces no net recognition strain. The term proof expands the finite sum into explicit cases for indices 0 through 7 and reduces the resulting algebraic identity with ring normalization.
Claim. $s(0) + s(1) + s(2) + s(3) + s(4) + s(5) + s(6) + s(7) = 0$, where $s :$ Fin $8$ $→$ ℝ is the modulation pattern with values $s(0) = φ$, $s(1) = √2/2$, $s(2) = 1 - φ$, $s(3) = -√2/2$, $s(4) = φ - 2$ and symmetric continuation for the remaining indices.
background
This theorem belongs to the PhotobiomodulationDevice module, which formalizes RS foundations for a light-therapy device operating on the φ-ladder. The module states that E(n) = E_base · φⁿ yields rung 6 at φ eV (≈1.618 eV, λ≈766 nm) inside the red/near-IR therapeutic window, and that the 8-beat pattern s(k) = cos(kπ/4) + φ⁻¹ cos(kπ/2) must satisfy 8-window neutrality to avoid strain accumulation. Upstream, rs_pattern supplies the explicit Fin 8 → ℝ map derived from that trigonometric form; tick is the fundamental time quantum τ₀ = 1; and A = 1 records active edges per tick. The module doc-comment records: 'The RS-coherent 8-beat modulation pattern values. Derived from s(k) = cos(kπ/4) + (1/φ)·cos(kπ/2).'
proof idea
The term proof rewrites Finset.univ.sum to sum_range, then uses simp to unfold rs_pattern at each index while supplying dif_pos witnesses that 0 < 8 through 7 < 8. The final ring tactic normalizes the eight-term polynomial to zero.
why it matters
The result is consumed by the downstream definition rs_neutral_pattern, which packages the pattern and its neutrality proof into a WindowNeutralPattern instance. It realizes the eight-tick octave neutrality (T7) inside the applied photobiomodulation setting, satisfying the recognition-ledger constraint that the pattern produces no net strain. The module doc notes this supports brainwave entrainment at φⁿ Hz frequencies while referencing the device specification RS_Biophase_Light_Device_Spec.tex.
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