corticalResonance_fifth_mode_band
plain-language theorem explainer
The result shows that five times the golden ratio lies strictly inside the open interval from 7.5 to 8.1. Researchers calibrating cortical resonance frequencies from the RS time unit τ₀ cite this interval to match the Fifth Mode paper's predicted band. The proof is a one-line wrapper that unfolds the resonance definition and applies linear arithmetic to the established bounds on the golden ratio.
Claim. $7.5 < 5φ < 8.1$ where $φ = (1 + √5)/2$ is the golden ratio.
background
The Tau-Zero Calibrator module treats τ₀ as the single positive calibration scalar that fixes the RS time unit, with the predicted scale 7.3 × 10^{-15} s. Under this choice the fifth-mode cortical resonance appears as the quantity 5φ when time is expressed in units of τ₀. Upstream lemmas supply the concrete bounds φ > 1.5 (from √5 > 2) and φ < 1.62 (from √5 < 2.24). The module states that every Hz-scale prediction, including this resonance, follows uniquely once τ₀ is fixed.
proof idea
The tactic proof first unfolds the definition of the resonance quantity, exposing the product 5φ. It then splits the conjunction into two inequalities and invokes linarith on each, supplying the upstream lemmas that bound φ from below by 1.5 and from above by 1.62.
why it matters
This bound supplies the fifth_mode_band component inside the tauZeroCert definition that assembles the complete τ₀ certification. It closes the numerical check for the Fifth Mode paper's prediction band. In the Recognition Science chain it anchors the forced value of φ (step T6) to an observable frequency interval, supporting the eight-tick octave without additional hypotheses.
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