modified_total_potential
plain-language theorem explainer
This definition assembles the first-order total potential for an extended object under simultaneous gravitational processing, external phase, and frame acceleration. Modelers of acoustic levitation or coherence-adjusted gravity cite it when deriving potential differences across finite objects. The construction is a direct linearization: Taylor expansion of each potential to first order around the center of mass plus the explicit acceleration term.
Claim. Let $h_{cm}$ be the center-of-mass position of an extended object, $z$ a small displacement, $a$ the frame acceleration, $Φ_{grav}(h)$ the gravitational processing potential, and $Φ_{ext}(h)$ the external phase potential. The total potential is $Φ_{total}(z) = Φ_{grav}(h_{cm}+z) + Φ_{ext}(h_{cm}+z) + a z$, obtained by linearizing each field to first order around $h_{cm}$.
background
The AcousticPhaseLevitation module examines how external phase mechanisms alter gravitational coherence for extended objects. ProcessingField supplies the gravitational potential function $Φ_{grav} : Position → ℝ$. ExternalPhaseField supplies an independent potential $Φ_{ext}$ arising from acoustic standing waves or phase-locked fields. ExtendedObject encodes an object with center-of-mass height $h_{cm}$ and positive spatial extent. Upstream results include the defect functional (equal to the J-cost) from LawOfExistence, the eight-tick phase values from EightTick, and the instantaneous present snapshot from TimeEmergence. These supply the periodic and defect structure underlying all coherence calculations in the module.
proof idea
The definition is a direct algebraic construction. It forms three terms: the first-order Taylor expansion of the gravitational potential from ProcessingField, the matching expansion of the external potential from ExternalPhaseField, and the linear acceleration contribution $a z$. No external lemmas are applied; the result is the explicit sum of these linearized contributions.
why it matters
The definition feeds directly into modified_coherence_defect, which computes the head-to-feet potential difference used to derive equilibrium acceleration and effective gravitational coupling under external fields. It extends the Recognition Science coherence-fall analysis by incorporating phase modifications, consistent with eight-tick resonance and coherence-gain mechanisms. The construction supports modeling of acoustic levitation thresholds on the phi-ladder while remaining within the linearized, frame-dependent setting.
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