IndisputableMonolith.Gravity.WeakFieldSuperposition
The module shows J-cost additivity for combined small strains in the weak-field regime. Gravitational coherence and acoustic levitation work cites it to justify linear superposition. The argument expands the exact identity J(1+ε) = ε²/(2(1+ε)) to bound the cross term at O(ε1 ε2).
claim$J(1 + \epsilon_1 + \epsilon_2) = J(1 + \epsilon_1) + J(1 + \epsilon_2) + O(\epsilon_1 \epsilon_2)$ for small $|\epsilon_i|$, using the identity $J(1+\epsilon)= rac{\epsilon^2}{2(1+\epsilon)}$.
background
J-cost measures recognition defect via the Recognition Science function J(x)=(x + x^{-1})/2 - 1. The module works in the weak-field regime where strains ε are small deviations from the identity element. It imports coherence fall and energy processing bridge results to set up the additive splitting.
proof idea
The module structure applies the exact identity J(1+ε)=ε²/(2(1+ε)) to the combined argument 1+ε1+ε2, expands the quadratic terms, and isolates the bilinear cross term as higher order.
why it matters in Recognition Science
This supplies the weak-field superposition step required by AcousticPhaseLevitation. It closes the additive approximation needed for phase levitation calculations in the gravity domain.
scope and limits
- Does not treat strong-field regimes where |ε| is order one.
- Does not compute explicit numerical values for the cross term.
- Does not derive the J function or its uniqueness property.
- Does not address time-dependent or relativistic corrections.