rs_pattern_peak
plain-language theorem explainer
The theorem states that the RS-coherent 8-beat modulation pattern reaches its maximum value of φ at index k=0. Photobiomodulation device designers would cite this result to fix the starting point of the φ-ladder energy alignment. The proof is a direct reflexivity reduction that matches the explicit case in the pattern definition.
Claim. Let $s: Fin 8 → ℝ$ be the modulation pattern defined by $s(k) = cos(kπ/4) + φ^{-1} cos(kπ/2)$. Then $s(0) = φ$.
background
The module formalizes an RS-coherent photobiomodulation device whose energy rungs follow E(n) = E_base · φ^n with E_coh = φ^{-5} eV. Rung 6 yields the therapeutic wavelength near 766 nm. The 8-beat modulation pattern is introduced to enforce 8-window neutrality, preventing recognition strain accumulation during treatment. The upstream definition rs_pattern supplies the explicit values: s(0) = φ, s(1) = √2/2, s(2) = 1 - φ, and so on, derived from the cosine sum formula.
proof idea
The proof is a one-line reflexivity that directly matches the explicit case ⟨0, _⟩ => phi in the definition of rs_pattern.
why it matters
This anchors the initial value of the modulation pattern at φ, which is required for the 8-window neutrality constraint Σ s(k) = 0 stated in the module documentation. It supplies the starting condition for the φ-energy ladder calculations that produce the red/near-IR therapeutic band. The result sits inside the applied layer that connects the core T7 eight-tick octave to device-level coherence.
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