plateVelocityRatio
plain-language theorem explainer
The theorem establishes that plate velocities assigned to consecutive rungs on the phi-ladder differ exactly by the golden ratio factor phi. Geologists working within Recognition Science would cite it to justify the observed spread between slow continental and fast oceanic plates. The proof is a direct algebraic reduction that unfolds the exponential definition and simplifies via ring after invoking positivity.
Claim. For every natural number $k$, if the velocity at rung $k$ is defined as $v(k) = phi^k$, then $v(k+1)/v(k) = phi$.
background
The module treats plate velocities as lying on the phi-ladder, with rung index $k$ giving velocity $phi^k$ in RS-native units. This follows the self-similar fixed point phi forced in the unified chain and the eight-tick octave structure. The local setting records that observed velocities range from roughly 1 cm/year to 17 cm/year, with the ratio approximating $phi^5$, and assigns five canonical plate types to match configDim $D=5$. Upstream the rung velocity is introduced as the pure exponential $phi^k$.
proof idea
Unfold the definition of velocity at rung to expose the powers of phi. Invoke positivity of $phi^k$ to justify the division step. Rewrite the successor exponent and apply the equivalence that converts the ratio into an equality, then finish with ring simplification.
why it matters
The result supplies the phi-ratio field required by the plate motion certificate, which records both the count of five plate types and this exact ratio as the core invariants of the model. It directly realizes the module's RS prediction that adjacent plates differ by factor phi, linking the geology tier to the phi-ladder construction in the forcing chain. No open scaffolding remains in this declaration.
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