rs_pattern
rs_pattern supplies the explicit 8-value sequence for the RS-coherent modulation used in photobiomodulation devices. Device physicists and RS theorists cite the sequence when specifying the beat pattern that enforces zero net strain over each cycle. The definition proceeds by direct case analysis on the Fin 8 index, assigning phi, sqrt(2)/2 and their signed variants in alternation.
claim$s : Fin 8 → ℝ$ with $s(0) = φ$, $s(1) = √2/2$, $s(2) = 1-φ$, $s(3) = -√2/2$, $s(4) = φ-2$, $s(5) = -√2/2$, $s(6) = 1-φ$, $s(7) = √2/2$, where $φ$ is the golden ratio.
background
The PhotobiomodulationDevice module applies Recognition Science to light therapy by defining discrete energy rungs on the φ-ladder via E(n) = E_base · φ^n, with rung 6 producing the therapeutic wavelength near 766 nm in the red/near-IR window. The 8-beat modulation pattern is introduced to satisfy the fundamental 8-window neutrality constraint that prevents net strain accumulation during treatment cycles. The module documentation states that the pattern is derived from the expression s(k) = cos(kπ/4) + (1/φ) cos(kπ/2).
proof idea
The declaration is a direct definition by pattern matching on the eight constructors of Fin 8. No lemmas are applied; the eight real values are assigned explicitly to realize the cosine-derived formula at successive integer points.
why it matters in Recognition Science
rs_pattern supplies the concrete sequence that is packaged into the WindowNeutralPattern structure via rs_neutral_pattern. It realizes the 8-window neutrality required for the photobiomodulation device, consistent with the eight-tick octave (T7) of the forcing chain. Downstream theorems rs_pattern_phi_components_neutral and rs_pattern_sqrt_components_neutral separately verify the cancellations of the φ-terms and the √2/2-terms that together yield the full sum zero.
scope and limits
- Does not derive the pattern from the cosine formula (the derivation is stated only in the module documentation).
- Does not prove the sum over one 8-window cycle equals zero (that is rs_pattern_window_neutral).
- Does not connect the pattern values to specific EEG bands or clinical efficacy data.
- Does not address hardware implementation or wavelength selection.
formal statement (Lean)
200noncomputable def rs_pattern : Fin 8 → ℝ
201 | ⟨0, _⟩ => phi
202 | ⟨1, _⟩ => Real.sqrt 2 / 2
203 | ⟨2, _⟩ => 1 - phi
204 | ⟨3, _⟩ => -(Real.sqrt 2 / 2)
205 | ⟨4, _⟩ => phi - 2
206 | ⟨5, _⟩ => -(Real.sqrt 2 / 2)
207 | ⟨6, _⟩ => 1 - phi
208 | ⟨7, _⟩ => Real.sqrt 2 / 2
209
210/-- **8-WINDOW NEUTRALITY**: The RS modulation pattern sums to zero over
211 one complete 8-beat cycle. This is the fundamental recognition ledger
212 constraint — the pattern produces no net strain accumulation.
213
214 Algebraically: the φ terms cancel (φ + (1-φ) + (φ-2) + (1-φ) = 0)
215 and the √2/2 terms cancel (alternating signs). -/