energy_pos
The energy positivity theorem asserts that every energy level E_n in the Recognition Science harmonic oscillator model is strictly positive. Cosmologists deriving constant energy contributions and ethicists constructing lattice states cite this result to guarantee positive densities. The proof is a term-mode reduction that unfolds the energy level definition and applies multiplication positivity using the known positivity of omega_RS.
claimFor every natural number $n$, the energy level $E_n = phi^{-5/2} (n + 1/2)$ satisfies $E_n > 0$.
background
The Oscillatory Dynamics module derives the simple harmonic oscillator from the leading-order expansion of J-cost near equilibrium, where J(1 + ε) ≈ ε²/2, yielding spring constant k = 1. In RS-native units the angular frequency is omega_RS = phi^{5/2}, so the energy level is defined by energy_level n := omega_RS / phi^5 * (n + 1/2), which equals phi^{-5/2} (n + 1/2). This rests on the upstream theorem omega_pos establishing 0 < omega_RS via square-root and power positivity, together with the abbrev Energy := ℝ.
proof idea
The term proof unfolds energy_level, applies mul_pos to the product, invokes div_pos on omega_pos and pow_pos phi_pos 5 to obtain positivity of the first factor, and closes with linarith on Nat.cast_nonneg n for the second factor.
why it matters in Recognition Science
This supplies the energy_pos field required by ConstantEnergyContribution, which in turn enables w_eq_neg_one proving the dark-energy equation of state equals -1. It likewise feeds vacuumUniformityCert and LatticeState constructions. In the Recognition framework it confirms positivity of the phi-ladder levels arising from T5 J-uniqueness and the eight-tick octave, completing a required step in the oscillatory dynamics master certificate.
scope and limits
- Does not compute explicit numerical values for any E_n.
- Does not address relativistic or quantum-field corrections to the SHO.
- Does not treat interactions among multiple oscillators.
- Does not derive the underlying J-cost function itself.
formal statement (Lean)
51theorem energy_pos (n : ℕ) : 0 < energy_level n := by
proof body
Term-mode proof.
52 unfold energy_level
53 apply mul_pos
54 · exact div_pos omega_pos (pow_pos phi_pos 5)
55 · linarith [Nat.cast_nonneg n]
56
57/-- Energy levels are strictly increasing. -/