oscillatoryDynamicsCert_inhabited
plain-language theorem explainer
The theorem establishes existence of the OscillatoryDynamicsCert structure that bundles positivity of the RS angular frequency with positivity and strict increase of the energy levels. A physicist working on the harmonic-oscillator limit of J-cost near equilibrium would cite it to guarantee a well-defined spectrum in native units. The proof is a one-line term that directly supplies the pre-built instance from the module's three field lemmas.
Claim. The type of certificates for oscillatory dynamics is inhabited: there exists a structure containing proofs that the RS angular frequency satisfies $0 < omega_RS$, that the energy level $E_n > 0$ for every natural number $n$, and that $E_n < E_{n+1}$ for every $n$.
background
In Recognition Science the J-cost function admits the local expansion $J(1 + epsilon) = epsilon^2/2 + O(epsilon^3)$ near its minimum, reproducing the quadratic potential of a simple harmonic oscillator with Hessian calibration $k = 1$. In RS units the coherent mass is $phi^{-5}$, so the angular frequency becomes $omega_RS = phi^{5/2}$ and the energy levels read $E_n = phi^{-5/2}(n + 1/2)$. The module packages these facts into the structure OscillatoryDynamicsCert whose three fields are exactly the positivity of $omega_RS$, positivity of each $E_n$, and monotonic increase of the sequence.
proof idea
The proof is a term-mode one-liner that returns the inhabitant oscillatoryDynamicsCert. That definition in turn assembles the three required fields by direct reference to the sibling lemmas omega_pos, energy_pos and energy_increasing.
why it matters
The declaration closes the existence question for the master certificate inside the oscillatory-dynamics module, confirming that the J-cost-derived harmonic oscillator yields a positive, strictly increasing energy spectrum. It thereby supports the structural claims that link the Recognition Composition Law and the phi-ladder to quantized levels, forming part of the chain from T5 J-uniqueness through the eight-tick octave. No downstream uses are recorded yet.
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