carrier_pos
The cortical-column carrier frequency equals 5φ Hz and is strictly positive. Neuromodulation device designers cite this to validate pulse spacing and sensitivity calculations. The proof is a one-line wrapper that unfolds the carrier definition and applies the multiplication-positivity lemma together with the positivity of phi.
claim$0 < 5 phi$, where the left-hand side is the cortical-column carrier frequency in hertz and $phi$ is the golden ratio.
background
The Cortical Neuromodulation Device module derives parameters for transcranial neuromodulation (RS_PAT_025) tuned to cortical-column resonance at 5φ Hz, with optimal pulse spacing τ = 1/(5φ) s. Carrier is defined as carrier := 5 * phi. This definition is shared verbatim with the PhantomCoupledGWAntennaSensitivity module. The module doc states the device operates with a φ-rational pulse schedule and an entrainment-confidence ladder.
proof idea
This is a one-line wrapper proof. It unfolds carrier to expose the product 5 * phi, then invokes mul_pos with a norm_num tactic for the positive constant 5 and the phi_pos lemma for the positivity of the golden ratio.
why it matters in Recognition Science
The result feeds pulseSpacing_pos in the same module via div_pos and multiple sensitivity theorems in PhantomCoupledGWAntennaSensitivity, including sensitivity_at_carrier, sensitivity_pos, and sensitivity_strict_anti. It supports the engineering derivation in the J10 track of Plan v5. The positivity ensures derived quantities remain physically meaningful, consistent with the Recognition Science constants where phi is the self-similar fixed point.
scope and limits
- Does not compute the numerical value of carrier in hertz.
- Does not prove the resonance frequency lies inside the EEG band.
- Does not address device safety or clinical validation.
- Does not extend to complex frequencies or analytic continuation.
Lean usage
theorem pulseSpacing_pos : 0 < pulseSpacing := div_pos one_pos carrier_posformal statement (Lean)
34theorem carrier_pos : 0 < carrier := by
proof body
One-line wrapper that applies unfold.
35 unfold carrier; exact mul_pos (by norm_num) phi_pos
36