energy_processing_bridge
plain-language theorem explainer
The energy-processing bridge asserts that the J-cost vanishes at unit equilibrium, is strictly positive for deviations from it, recovers the quadratic weak-field energy expression, and that any energy distribution sources a processing field. Researchers closing the energy-to-gravity gap in acoustic levitation models would cite this result to justify the first link in the forcing chain. The proof is a direct structure instantiation that unfolds the J-cost definition for the zero-cost case and applies non-negativity plus zero-iff-one lemmas for a
Claim. The cost functional satisfies $J(1)=0$, $J(x)>0$ for all $x>0$ with $x≠1$, and $J(1+ε)=ε²/(2(1+ε))$ whenever $ε>-1$. Every energy distribution $e$ together with a constant $G$ sources a processing field via the constructor that maps $e$ to the associated processing gradient.
background
J-cost is the function $J(x)=(x+x^{-1})/2-1$ for $x>0$, the unique cost functional forced by the Recognition Composition Law. EnergyProcessingEquivalence packages four properties: zero cost at balance, positive cost for deviations, the quadratic energy match, and the sourcing of a processing field from any energy distribution. The module sits inside the gravity sector and imports the CPM2D bridge together with the RS-native constants; it relies on the upstream lemma that $J(x)≥0$ for $x>0$ (AM-GM inequality) and the companion result that $J(x)=0$ forces $x=1$.
proof idea
The proof instantiates the four fields of EnergyProcessingEquivalence. balance_zero_cost unfolds Jcost and simplifies directly. deviation_positive_cost invokes Jcost_nonneg to obtain non-negativity, then cases on equality versus strict inequality and rules out the zero case with Jcost_zero_iff_one. quadratic_energy_bridge is supplied by the sibling lemma Jcost_one_plus_exact. energy_sources_processing applies the energy_to_processing_field constructor with a reflexivity proof.
why it matters
This theorem supplies the gap1_energy_is_processing field required by both full_levitation_cert and levitation_unconditional in the AcousticPhaseLevitation module, thereby completing the energy-to-processing step that precedes coherence gain and resonance. It realizes the framework claim that energy creates a processing gradient capable of opposing gravity, directly supporting the T5 J-uniqueness and the eight-tick octave chain to D=3 levitation. The result closes one of the explicit gaps listed in the forcing-chain-complete certificate.
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