neuromod_one_statement
plain-language theorem explainer
The declaration asserts that the cortical carrier frequency 5φ lies in (8.05, 8.10) Hz, the pulse spacing in (0.123, 0.125) s, and that entrainment confidence 1/φ^k decreases strictly with rung index k. Device engineers working on transcranial neuromodulation would cite this to fix the phi-rational operating schedule. The proof is a direct term that assembles the pre-established band theorems and the strict anti-monotonicity lemma.
Claim. Let $f_c = 5φ$ be the cortical-column carrier frequency in Hz and $τ = 1/f_c$ the pulse spacing in seconds. The statement asserts $8.05 < f_c < 8.10$, $0.123 < τ < 0.125$, and that the entrainment confidence $c(k) = φ^{-k}$ satisfies $c(k_2) < c(k_1)$ for all natural numbers $k_1 < k_2$.
background
The module specifies a transcranial neuromodulation device whose carrier frequency is defined as five times the golden-ratio fixed point phi. Pulse spacing is the reciprocal of this carrier. Entrainment confidence at rung k is defined as the reciprocal of phi to the power k. Upstream results establish the numerical band for the carrier via bounds on phi, the corresponding band for pulse spacing, and the strict decrease of confidence with increasing k using properties of positive powers and reciprocals.
proof idea
The proof is a one-line term that packages the left and right bounds from the carrier_band theorem, the left and right bounds from the pulseSpacing_band theorem, and the entrainmentConfidence_strict_anti lemma. The anti-monotonicity lemma unfolds the definition and applies the one_div_lt_one_div_of_lt rule together with the fact that phi exceeds one to obtain the strict inequality on the powers.
why it matters
This theorem supplies the single consolidated statement for the cortical neuromodulation device specification. It directly implements the phi-ladder entrainment model, where the self-similar fixed point phi determines the operating frequency and the confidence decay. The module falsifier is any EEG study placing optimal entrainment outside [7.5, 8.1] Hz. No downstream uses are recorded.
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