IndisputableMonolith.Foundation.OscillatoryDynamicsDeep
The module introduces oscillatory dynamics in the Recognition Science foundation by defining RS-native frequency omega_RS and quantized energy levels on the phi-ladder. It establishes that the Hessian of J-cost at equilibrium equals 1. Researchers modeling equilibrium stability in RS would cite the jcost_hessian result. The module consists of supporting definitions and a single algebraic identity drawn from the Cost import.
claim$H(J_ {cost})(x_{eq}) = 1$, where $J_{cost}$ is the J-cost function and $x_{eq}$ its equilibrium point; auxiliary objects include the RS frequency $omega_{RS}$ and energy levels $E_n$.
background
The module sits in the Foundation domain and imports the RS time quantum tau_0 = 1 tick from Constants together with J-cost definitions from Cost. It introduces omega_RS as the native angular frequency, energy_level as the quantized energies, and energy_pos together with energy_increasing as monotonicity statements. The local setting assumes an equilibrium configuration of the J-cost potential whose second derivative supplies the curvature for small oscillations.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module supplies the equilibrium Hessian that later oscillatory certificates rely on for stability. It directly encodes the J-uniqueness property (T5) into a concrete curvature value and prepares the ground for energy quantization on the phi-ladder. No downstream theorems are recorded yet, but the structure aligns with the eight-tick octave periodicity.
scope and limits
- Does not derive explicit time-dependent solutions.
- Does not incorporate spatial dimensions D=3.
- Does not compute numerical values for omega_RS.
- Does not address non-equilibrium trajectories.