phi_cubed_in_theta_band
plain-language theorem explainer
The theorem places the frequency phi cubed hertz strictly inside the open interval from 4 to 8 hertz, locating it in the theta EEG band. Photobiomodulation device designers following the Recognition Science protocol cite this to justify the theta modulation component of the rs_device specification. The proof is a direct one-line wrapper that invokes the tighter phi_cubed_bounds lemma and discharges both sides of the conjunction with linear arithmetic.
Claim. The cube of the golden ratio satisfies the bounds $4 < phi^3 land phi^3 < 8$.
background
The Photobiomodulation Device module formalizes an RS-coherent light therapy device whose parameters are derived from the phi-energy ladder E(n) = E_base * phi^n. Brainwave entrainment frequencies are assigned as phi^3 hertz for theta, phi^5 hertz for alpha, and phi^8 hertz for gamma, with the modulation pattern required to satisfy 8-window neutrality. The local setting is the phi-ladder applied to therapeutic wavelengths (rung 6 yields approximately 766 nm) and EEG-band frequencies for coherence with healing states.
proof idea
The proof is a one-line wrapper that applies the phi_cubed_bounds lemma from the Constants module. It uses constructor to split the conjunction, then linarith on each component of the tighter interval 4.0 < phi^3 < 4.25 to obtain the looser theta-band bounds.
why it matters
This result supplies the theta-band justification required by the downstream rs_device definition, which hard-codes theta_freq_Hz := phi^3. It completes the brainwave entrainment section of the module specification, aligning the phi-ladder frequencies with documented EEG bands and the eight-tick octave structure of the Recognition Science framework. The parent object is the canonical PBM device specification in the same module.
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