equilibrium_acceleration
equilibrium_acceleration defines the net acceleration at which an extended object reaches force balance under a gravitational processing potential combined with an external phase field. Researchers modeling acoustic levitation or coherence-modified gravity cite this expression to locate the zero-defect point. It is realized as a direct one-line definition that negates the sum of the two field gradients evaluated at the object's center-of-mass height.
claimThe equilibrium acceleration is defined by $a_0 = - (dΦ/dh|_{h_{cm}} + dψ/dh|_{h_{cm}})$, where $Φ$ is the gravitational processing potential of the field, $ψ$ is the external phase potential, and $h_{cm}$ is the center-of-mass position of the extended object.
background
The AcousticPhaseLevitation module models gravity via ProcessingField structures whose phi component supplies the potential whose gradient enters the force law. An ExternalPhaseField is a structure carrying an independent psi : Position → ℝ that represents acoustic standing waves or phase-locked mechanisms modifying the local processing environment. An ExtendedObject is a structure supplying the evaluation point h_cm together with a positive spatial extent.
proof idea
The definition is a direct one-line expression that negates the sum of the derivative of field.phi and the derivative of ext.psi, both taken at obj.h_cm. No lemmas or tactics are invoked; the body is the immediate algebraic combination of the two gradient operations.
why it matters in Recognition Science
This definition supplies the acceleration value inserted into modified_coherence_defect, enabling the downstream theorem equilibrium_is_coherent to conclude that the defect vanishes exactly at equilibrium. It therefore closes the force-balance step in the acoustic-levitation analysis. Within the Recognition framework the construction supports the study of how external fields cancel gravitational defects, consistent with coherence-gain and eight-tick resonance mechanisms imported from CoherenceFall and EightTickResonance.
scope and limits
- Does not specify the explicit functional form of the processing potential phi.
- Does not incorporate time dependence or relativistic corrections.
- Does not compute resulting levitation height or dynamical stability.
- Does not address discrete or quantum corrections to the continuous derivative.
Lean usage
example (field : ProcessingField) (ext : ExternalPhaseField) (obj : ExtendedObject) : equilibrium_acceleration field ext obj = -(deriv field.phi obj.h_cm + deriv ext.psi obj.h_cm) := rflformal statement (Lean)
160def equilibrium_acceleration
161 (field : ProcessingField) (ext : ExternalPhaseField) (obj : ExtendedObject) : ℝ :=
proof body
Definition body.
162 -(deriv field.phi obj.h_cm + deriv ext.psi obj.h_cm)
163
164/-- At the equilibrium acceleration, coherence defect is zero. -/