Non-Relativistic Reduction of the Existence Equation and Observable Deviations from the Gross-Pitaevskii Limit

Evidence Paper VII of the Existence Equation series The Gross–Pitaevskii equation has governed the theory of macroscopically coherent matter fields for over sixty years. Its nonlinear term g|φ|²φ is introduced phenomenologically — as the mean-field limit of contact scattering in dilute gases, as the BCS condensate amplitude in superconductors, as a self-coupling postulated by the physics of each application domain. The Mexican-hat vacuum structure implicit in GP is not derived within GP itself. It is imported. This paper inverts the hierarchy. The Existence Equation 1 contains the condensate structure as a primitive: the Mexican-hat potential V(A) = −(λ/2)A² + (α/4)A is inherent to the theory's action, not added to account for observed coherence. GP is shown to be precisely the slow-manifold projection of ED, obtained by freezing two degrees of freedom. The reduction proceeds in three structural steps: Step Operation Result 1. Linearization Expand around vacuum A0 = √(λ/α) Goldstone branch (gapless, phase) + Higgs branch (gap μED = c√(2λ)) 2. Phase separation Ψ = e−iμtφ iφ̇ = ĤGPφ + φ̈/(2μ) ← residual 3. Slow manifold Drop φ̈/(2μ) → GP ε = k²/(4λ) controls validity The reduction eliminates two real degrees of freedom per spatial point (ED: 4, GP: 2). The two lost sectors are: (i) the antiparticle channel (e+iμtχ*, frozen by the rotating-frame ansatz), and (ii) the amplitude inertial mode (Ä, Φ̈, slaved to first-order GP flow). These are what GP structurally cannot represent. A single dimensionless parameter controls everything: ε = k² / (4λ) ε range Regime Example systems ε ≲ 10−3 GP exact Low-k BEC Bragg, sub-gap Josephson, ULDM ε ~ 0.05–0.2 ED correction (5–20%) High-k BEC Bragg ε ~ O(1) ED required Gap-scale Josephson, neutron-star superfluid cores The numerical verification uses a 1D Josephson junction simulated from ED alone (512 grid, split-operator leapfrog): Test ED result GP prediction DC Josephson J = Jc sin(Δφ), R² = 0.998 J = Jc sin(Δφ) AC Josephson dΔφ/dt ≈ −0.184 Δμ dΔφ/dt = +Δμ The DC result is a clean verification: ED reproduces the standard current–phase relation to R² = 0.998. The AC result is the structurally novel observation. The sign is inverted and the magnitude reduced — not as an ε-correction (which vanishes by reflection symmetry for the symmetric voltage assignment used), but because ED’s second-order temporal structure imposes an initial-condition over-constraint absent from GP. The system cannot simultaneously satisfy rotating-frame stationarity and lab-frame rest. A transient ensues that GP cannot describe. Four falsifiable predictions are identified at ε ~ 0.1: 1. High-momentum Bogoliubov dispersion: systematic downward shift of 5–20% at k ~ 2π/ξheal.2. Asymmetric AC Josephson: GP predicts equal slopes for symmetric and asymmetric voltage; ED predicts a specific functional difference with no free parameters.3. Deep Shapiro delay: ED phase accumulation deviates from GP in deep-well configurations (>50% amplitude suppression).4. Condensate fragmentation near 2μED: Higgs branch excitation produces fragmentation absent from GP’s parametric-instability analysis — a new-spectrum prediction. Existing empirical GP corrections — Beliaev damping, Lee–Huang–Yang energy, relativistic GP — are reinterpreted as leading terms of a single ε-expansion anchored in ED. What each phenomenological derivation lacked, ED supplies: one organizing parameter controlling all corrections simultaneously. The structural hierarchy is therefore: ED is the mother theory. GP is its non-relativistic slow-manifold projection. Linear quantum mechanics (Schrödinger without interaction) is not a limit of either — it is the degenerate case α → 0, λ → 0 in which the Mexican-hat vacuum disappears and all condensate phenomena are structurally absent. The common narrative that GP is obtained by “adding” nonlinearity to LQM inverts the structural hierarchy. References 1 J.-A. Shin, "The Existence Equation: The Grammar of Persistence," Zenodo (2026). doi: 10.5281/zenodo.18639316 All simulation code and raw data are publicly available at https://github.com/Galileo-leo/existence-equation.