complete_cancellation_is_levitation
plain-language theorem explainer
When the gradient of an external phase potential exactly opposes the gravitational potential gradient at an object's center of mass, the effective gravitational coupling vanishes. Researchers modeling acoustic or phase-driven levitation in Recognition Science cite this to close the conditional zero-coupling step. The term-mode proof unfolds the coupling definition, rewrites via the hypothesis, and simplifies with negation cancellation and absolute-value zero.
Claim. Let $Φ$ be the potential of a processing field and $ψ$ the potential of an external phase field. For an extended object with center-of-mass position $h$, if $ψ'(h) = -Φ'(h)$, then the effective gravitational coupling $|Φ'(h) + ψ'(h)| = 0$.
background
ProcessingField carries a scalar potential $Φ : Position → ℝ$. ExternalPhaseField carries an additional potential $ψ : Position → ℝ$ whose gradient is defined by differentiation. ExtendedObject records a center-of-mass coordinate $h_{cm}$ together with a positive spatial extent. The effective gravitational coupling is the absolute value of the sum of the two gradients evaluated at $h_{cm}$. This declaration lives in the AcousticPhaseLevitation module, which examines how external fields alter coherence defects arising from gravitational processing. The upstream inevitability theorem states that any zero-parameter framework deriving observables must either reduce to the Recognition Science cost and selection or violate at least one necessity gate.
proof idea
One-line wrapper that unfolds effective_gravitational_coupling, rewrites the sum using the supplied cancellation hypothesis, and simplifies the resulting expression with add_neg_cancel followed by abs_zero.
why it matters
The result supplies the anti-coherence reduction step inside LevitationInevitability and is invoked directly by forcing_chain_complete and levitation_is_inevitable. It completes the conditional mechanism showing that opposing external fields null gravitational coupling, advancing the Recognition Science claim that levitation follows from coherence restoration along the T0-T8 forcing chain. It touches the open question of laboratory realization of the required phase fields.
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