jcost_energy
jcost_energy defines stored energy for a recognition event of ratio x as the product of the fixed coherence quantum E_coh and the J-cost of that ratio. Storage engineers and RS theorists cite it when bounding energy densities on the phi-ladder. The definition is a direct multiplication that inherits non-negativity and ground-state minimum from the upstream Jcost function.
claimThe energy stored in a recognition event with positive ratio $x$ is defined by $E(x) := E_0 J(x)$, where $E_0 = phi^{-5}$ and $J(x) = (x + x^{-1})/2 - 1$.
background
In the Recognition Science framework energy is the product of a coherence quantum and the J-cost of the recognition ratio. Module EN-004 derives storage density limits from this structure, with E_coh_storage defined as phi to the power of negative five. Upstream cost functions arise from multiplicative recognizers and observer forcing events, both of which reduce to the same J-cost expression.
proof idea
One-line definition that multiplies the coherence energy E_coh_storage by the Jcost function applied to x.
why it matters in Recognition Science
This definition supplies the energy quantity used throughout the EN-004 theorems on storage hierarchy. It feeds jcost_energy_nonneg, jcost_energy_min_at_ground, phi_coherent_minimizes_jcost_per_energy and phi_rung_jcost_energy, which establish non-negativity, ground-state minimum and phi-rung scaling. Within the framework it realizes the claim that energy equals J-cost times E_coh, linking directly to the phi-ladder and the eight-tick octave.
scope and limits
- Does not compute numerical values for specific chemical or nuclear storage media.
- Does not prove positivity of E_coh_storage.
- Does not address continuous versus discrete tuning of storage ratios.
- Does not derive mass-energy equivalence from this expression.
Lean usage
example (x : ℝ) (hx : 0 < x) : jcost_energy x hx = E_coh_storage * Jcost x := rflformal statement (Lean)
110def jcost_energy (x : ℝ) (hx : 0 < x) : ℝ := E_coh_storage * Jcost x
proof body
Definition body.
111
112/-- **THEOREM EN-004.7**: J-cost energy is non-negative. -/