Step 8a of 8. Isolates and classifies the sole remaining discrepancy after Schrödinger recovery.
The closure residual is exactly zero at equilibrium, perturbatively small nearby, phase-invariant, local, probability-conserving, and not removable by gauge transformation.
The Equilibrium-Centred Normal Form
Paper XVIII derived the full nonlinear wave equation. This paper rewrites it in equilibrium-centred normal form — separating the linear Witten Hamiltonian from the nonlinear remainder:
The Residual
The nonlinear residual has an explicit form. It measures how far the wavefunction amplitude deviates from the equilibrium curvature profile:
Structural Properties
Five properties of the residual are proved, establishing its mathematical character:
- (a) Exactly zero at equilibrium — when |Ψ| = Rst, the two curvature terms cancel identically
- (b) Phase-invariant — δUrel depends only on |Ψ|, not on the phase of Ψ
- (c) Local — depends only on |Ψ| and its spatial derivatives at each point
- (d) O(ε) near equilibrium — for perturbations |Ψ| = Rst + εη, the residual is first-order in ε
- (e) Not removable by gauge transformation — no phase redefinition Ψ → eiχΨ can eliminate the residual
The residual is a local, phase-invariant, non-gauge-removable functional of the wavefunction amplitude. It vanishes exactly at equilibrium and is perturbatively small nearby. It is a genuine nonlinear correction, not an artefact of the construction.
Norm Conservation
Despite the nonlinearity, the full equation preserves the L² norm exactly. The residual term, being real-valued and multiplicative, does not break unitarity in the sense of probability conservation:
∂t∫|Ψ|² dx = 0 for all solutions. Probability is conserved even in the nonlinear regime. This is not a perturbative statement — it holds exactly for the full nonlinear evolution.
The Regime Classifier
A dimensionless parameter Ξ quantifies the relative importance of the nonlinear residual:
- Ξ ≪ 1 — linear Schrödinger regime. The Witten Hamiltonian dominates; standard quantum mechanics applies to leading order.
- Ξ ~ 1 — mixed regime. Both the linear Hamiltonian and the nonlinear residual contribute comparably.
- Ξ ≫ 1 — nonlinear closure-dominated regime. The residual dominates; the dynamics is genuinely nonlinear and departs significantly from Schrödinger.
Stated Limitations
- No physical interpretation of the residual — classified mathematically only
- No prediction of when Ξ ~ 1 occurs in nature — the regime boundary is defined but not located
- No experimental signature derived — the residual is characterised, not tested
- Norm conservation proved for smooth solutions — regularity assumptions required
- The non-gauge-removability proof assumes standard gauge transformations — exotic redefinitions not excluded
The residual is classified. The dynamics is approximately Schrödinger near equilibrium and genuinely nonlinear away from it.