equilibrium_is_coherent
plain-language theorem explainer
Equilibrium acceleration under combined gravitational and external phase fields yields zero modified coherence defect. Researchers modeling acoustic levitation or phase-modified gravity cite this identity to anchor effective coupling calculations. The tactic proof simplifies the defect, substitutes the equilibrium definition, cancels the derivative sum by ring, and concludes zero via mul_zero and abs_zero.
Claim. Let $a_{eq} = - (D_h Φ + D_h Ψ)$ be the equilibrium acceleration for gravitational potential $Φ$ and external phase potential $Ψ$. Then the modified coherence defect evaluated at $a_{eq}$ equals zero: $D_{mod}(a_{eq}) = 0$.
background
The AcousticPhaseLevitation module treats gravity via processing fields whose gradients determine accelerations and coherence defects. ExternalPhaseField is a structure supplying an auxiliary potential psi whose gradient modifies the local environment, as in acoustic standing waves or phase-locked fields. equilibrium_acceleration is defined as the negative sum of the gravitational and external gradients at the object's center of mass. modified_coherence_defect quantifies the absolute deviation from coherence under the total field. The theorem follows the definition of equilibrium_acceleration and precedes the anti-coherence coupling discussion.
proof idea
Apply the simplification lemma modified_coherence_defect_simplify. Unfold equilibrium_acceleration to expose the negative gradient sum. Construct the ring identity showing the derivative sum minus itself equals zero. Rewrite with that identity, then apply mul_zero and abs_zero to reach the goal.
why it matters
The result shows the equilibrium point is coherent by algebraic construction, supplying the base case for effective_gravitational_coupling and anti_coherence_reduces_coupling in the same module. It occupies the coherence-gain step in the gravity sector of Recognition Science, consistent with the forcing chain where defect zero corresponds to balanced J-cost. No open scaffolding is closed here.
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