rs_pattern_sqrt_components_neutral
plain-language theorem explainer
The four √2/2-indexed entries of the RS modulation pattern at indices 1, 3, 5 and 7 sum to zero. Photobiomodulation device designers cite this to confirm absence of net bias from the square-root terms in the 8-beat pattern. The proof unfolds the pattern definition and reduces the sum by ring arithmetic.
Claim. Let $s : Fin 8 → ℝ$ be the RS-coherent modulation pattern with $s(1) = √2/2$, $s(3) = -√2/2$ and the two remaining √2/2-indexed slots at 5 and 7. Then $s(1) + s(3) + s(5) + s(7) = 0$.
background
The module develops an RS-coherent photobiomodulation device whose modulation uses the φ-ladder. The 8-beat pattern is defined by $s(k) = cos(kπ/4) + (1/φ) cos(kπ/2)$ and is required to satisfy the 8-window neutrality constraint Σ s(k) = 0 to prevent recognition strain accumulation. The rs_pattern definition supplies the explicit values, isolating the four √2/2 entries at the odd indices. Upstream results include the rs_pattern definition itself and the LedgerFactorization structure that calibrates the underlying J-cost.
proof idea
One-line wrapper that applies simp only [rs_pattern] to replace the four Fin 8 projections by their explicit √2/2 values, then ring to confirm the signed sum is identically zero.
why it matters
The result supplies one component of the 8-window neutrality required by the module's device specification. It aligns with the eight-tick octave (T7) and the overall 8-beat pattern neutrality stated in the module documentation. No parent theorems or downstream uses are recorded; the declaration therefore stands as an isolated algebraic check inside the applied photobiomodulation construction.
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