en004_certificate
plain-language theorem explainer
Recognition Science supplies a certificate string for EN-004 that records the positivity of coherent energy, the phi-multiplicative ratios on the energy ladder, and the strict hierarchy mechanical less than chemical less than nuclear. Applied physicists and engineers cite it when referencing derived storage limits per unit mass from the J-cost structure. The definition assembles the string directly from status lines on each sub-result including rung comparisons and non-negativity.
Claim. The EN-004 certificate string asserts that coherent storage energy satisfies $0 < E_0 = E_coh = phi^{-5}$, consecutive rungs satisfy $E(n+1)/E(n) = phi$, nuclear exceeds chemical by factor $phi^{45}$, and the storage hierarchy obeys mechanical < chemical < nuclear with J-cost minimized at ground state $x=1$.
background
Energy in RS is defined as J-cost times the coherence quantum, $E = J(x) · E_coh$ with $E_coh = phi^{-5}$ in native units. The J-cost function is $J(x) = 1/2(x + x^{-1}) - 1$, which attains its global minimum of zero at the ground state $x=1$ and diverges at the boundaries. The module sets chemical storage at rung 0 and nuclear storage at rung 45 on the phi-ladder, yielding the ratio $phi^{45} approx 10^9$ that matches observed nuclear-to-chemical energy scales.
proof idea
The definition constructs the certificate by direct string concatenation of a header followed by eleven status lines. Each line references an upstream result such as E_coh_storage_pos for positivity, energy_storage_density_hierarchy for the three-level ordering, nuclear_exceeds_chemical for the rung comparison, and jcost_energy_min_at_ground for the J-cost minimum. The final lines add the quantization insight and the empirical ratio match.
why it matters
This certificate consolidates the EN-004 derivations that follow from the phi-ladder and J-cost structure in the Recognition framework. It supports the engineering claim that optimal storage occurs at phi-coherent rungs and that the nuclear-chemical ratio is fixed by the eight-tick octave spacing. The declaration closes the module's summary of the three-level hierarchy without introducing new open questions.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.