bandRatio
plain-language theorem explainer
The ratio of successive frequencies on the phi-ladder equals the golden ratio phi for every natural number index. Physicists constructing the electromagnetic spectrum from Recognition Science scaling would cite this to verify uniform phi-decade spacing across the five canonical bands. The proof unfolds the bandFrequency definition, invokes positivity of phi to the power k, rewrites via successor power and division equivalence, then normalizes algebraically.
Claim. For every natural number $k$, the ratio of the frequency of spectral band $k+1$ to the frequency of spectral band $k$ equals the golden ratio $phi$, where frequencies follow the self-similar ladder $nu_k = nu_0 times phi^k$.
background
The module derives the electromagnetic spectrum from the phi-ladder under Recognition Science, with each band spanning one phi-decade so that $nu_k = nu_0 times phi^k$. Five canonical bands (radio through UV+X+gamma) correspond to configDim $D=5$. The cortical carrier is defined upstream as $5 times phi$ Hz in both the neuromodulation device and phantom antenna modules, supplying the biological reference frequency that anchors the ladder. Band frequency itself is the upstream definition that expresses the ladder scaling via exponentiation.
proof idea
Term-mode proof that unfolds bandFrequency, obtains positivity of $phi^k$ from pow_pos, rewrites the successor power and applies div_eq_iff using the positivity hypothesis, then closes with ring normalization.
why it matters
Supplies the phi_ratio field inside the emSpectrumCert structure that certifies the full spectrum construction. It directly instantiates the self-similar fixed point of the forcing chain (T6) and the eight-tick octave scaling, confirming that the Recognition Composition Law produces uniform phi ratios across the electromagnetic bands. The result closes the algebraic verification step for the phi-ladder in the electromagnetic domain.
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