lambda_PBM_pos
The lemma proves that the photobiomodulation wavelength lambda_PBM, recovered as h c over E_PBM, is strictly positive. Modelers assembling RS-coherent light therapy devices cite it when populating the rs_device record with derived parameters. The proof is a term-mode one-liner that unfolds the definition and chains the library lemmas div_pos and mul_pos over the three positivity facts for planck_h, speed_of_light, and E_PBM.
claimLet $E_{PBM} > 0$ be the photon energy at rung 6 of the phi-ladder, $h > 0$ Planck's constant in RS units, and $c > 0$ the speed of light. Define the therapeutic wavelength by $lambda_{PBM} := hc / E_{PBM}$. Then $lambda_{PBM} > 0$.
background
The PhotobiomodulationDevice module builds an RS-coherent light therapy device whose parameters are fixed by the phi-ladder with no free constants. Energy levels follow $E(n) = E_{base} cdot phi^n$, and rung 6 supplies the therapeutic photon energy $E_{PBM} in (1.61, 1.62)$ eV. The wavelength is recovered from the Planck relation as the quotient $hc / E_{PBM}$ (meters).
proof idea
The proof is a term-mode one-liner. It unfolds lambda_PBM to expose the quotient (planck_h * speed_of_light) / E_PBM, then applies div_pos to the product of planck_h_pos and speed_of_light_pos together with E_PBM_pos.
why it matters in Recognition Science
The result is required to construct the rs_device record, which packages lambda_PBM with the rs_neutral_pattern and the frequencies phi^3, phi^5, phi^8. It completes the chain from the phi-energy ladder (T6) to a concrete device whose wavelength sits inside the 750-780 nm window noted in the module documentation. The downstream definition rs_device therefore inherits positivity directly from this lemma.
scope and limits
- Does not compute the numerical value of lambda_PBM or confirm it lies in 750-780 nm.
- Does not model tissue absorption, penetration depth, or biological response.
- Does not address coherence maintenance or phase stability beyond the 8-beat neutrality constraint.
- Does not extend to energies outside the phi-ladder or to non-RS modulation patterns.
formal statement (Lean)
144lemma lambda_PBM_pos : 0 < lambda_PBM := by
proof body
Term-mode proof.
145 unfold lambda_PBM
146 exact div_pos (mul_pos planck_h_pos speed_of_light_pos) E_PBM_pos
147
148/-- λ_PBM falls in the therapeutic photobiomodulation window (750–780 nm). -/