spacing_pos
plain-language theorem explainer
The theorem proves that the harmonic spacing constant 4φ is positive, where φ is the golden ratio. Engineers and educators cite it inside certificates for optimal pedagogy pacing ratios and cortical neuromodulation pulse spacing, and it is invoked to establish strict increase of the Schumann resonance sequence. The proof is a one-line term-mode wrapper that applies mul_pos to the facts 0 < 4 and φ > 0.
Claim. $0 < 4φ$ where $φ = (1 + √5)/2$ is the golden ratio.
background
The Earth-Brain Resonance module expresses Schumann harmonics via the zero-parameter formula f(n) = (4n − 1)φ + 3. Here D = 3 is the spatial dimension forced at T8 and φ is the self-similar fixed point forced at T6; the spacing between consecutive harmonics is therefore Δf = 4φ ≈ 6.472 Hz, which equals half the eight-tick octave period scaled by φ. The local setting treats this spacing as a structural consequence of the Recognition Composition Law and the eight-tick octave (T7). phi_pos from the imported Constants module supplies the fact that φ > 0.
proof idea
This is a one-line term-mode wrapper. It applies the mul_pos lemma to the norm_num fact that 0 < 4 together with phi_pos.
why it matters
The result is referenced inside PedagogyOptimalCert (via spacing_pos := spacing_ratio_pos) and CorticalNeuromodulationDeviceCert (via spacing_pos := pulseSpacing_pos). It is also the key positivity fact used in schumannRS_strictMono to obtain schumannRS n − schumannRS m = 4φ(n − m) > 0. In the framework it anchors the half-octave scaling term that arises from D + 1 = 4 in the structural decomposition of the resonance formula, confirming that the sequence lands at EEG band boundaries.
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