rs_pattern_sqrt_components_neutral
plain-language theorem explainer
The theorem shows that the RS-coherent 8-beat modulation pattern sums to zero over its four √2/2-indexed entries at positions 1, 3, 5, and 7. Photobiomodulation device designers would cite it to confirm that the chosen modulation avoids net recognition strain over each 8-tick window. The proof is a direct term-mode reduction that unfolds the explicit pattern definition and cancels the terms by ring arithmetic.
Claim. Let $s(k)$ denote the RS-coherent 8-beat modulation pattern $s(k) = cos(kπ/4) + φ^{-1} cos(kπ/2)$ for $k = 0,…,7$. Then $s(1) + s(3) + s(5) + s(7) = 0$.
background
The Photobiomodulation Device module formalizes an RS-coherent light therapy device whose modulation follows the φ-ladder. The 8-beat pattern is defined explicitly by $s(k) = cos(kπ/4) + φ^{-1} cos(kπ/2)$, with tabulated values that include √2/2 at the odd indices 1, 3, 5, 7. The module states that this pattern satisfies the fundamental 8-window neutrality constraint Σ s(k) = 0, preventing accumulation of recognition strain during treatment. The upstream rs_pattern definition supplies the concrete real-valued assignments used in the sum.
proof idea
The proof is a one-line term-mode wrapper. It applies simp only [rs_pattern] to substitute the explicit pattern values, then invokes ring to cancel the four terms algebraically to zero.
why it matters
The result verifies the 8-window neutrality constraint required by the device specification. It directly supports the module claim that the modulation pattern satisfies Σ s(k) = 0 and aligns with the eight-tick octave (T7) of the forcing chain. No downstream theorems are recorded, so the lemma stands as a self-contained check on the pattern definition.
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