rs_pattern_sqrt_components_neutral
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.
claimLet $s(k)$ 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 in Recognition Science
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.
scope and limits
- Does not evaluate the pattern at non-integer or non-Fin-8 indices.
- Does not prove the full eight-term sum equals zero.
- Does not derive the functional form of the pattern from the Recognition Composition Law.
- Does not address numerical stability or floating-point evaluation of the sum.
formal statement (Lean)
250theorem rs_pattern_sqrt_components_neutral :
251 rs_pattern ⟨1, by omega⟩ + rs_pattern ⟨3, by omega⟩ +
252 rs_pattern ⟨5, by omega⟩ + rs_pattern ⟨7, by omega⟩ = 0 := by
proof body
Term-mode proof.
253 simp only [rs_pattern]
254 ring
255
256/-! ## Section 4: Brainwave Entrainment Frequencies
257
258The device modulation frequencies are powers of φ in Hz, which
259fall naturally into known EEG bands:
260- φ³ ≈ 4.24 Hz → theta (4–8 Hz)
261- φ⁵ ≈ 11.09 Hz → alpha (8–13 Hz)
262- φ⁸ ≈ 46.98 Hz → gamma (30–100 Hz)
263
264This is not tuned — it falls out of the φ-ladder structure. -/
265
266/-- φ³ ≈ 4.24 Hz falls in the theta EEG band (4–8 Hz).
267 Theta waves: deep meditation, memory consolidation, healing. -/