The Architecture of State-Dependence: Nonlinearity, Backreaction, and Background Independence
Why state-dependent geometry is natural in sPNP. sPNP is formulated on the relational 3N−6 configuration space obtained by quotienting out global translations and rotations, leaving only the intrinsic shape-and-scale degrees of freedom. On this space there is no background metric — no geometric structure imposed from outside the theory against which the wavefunctional is measured. This is not an oversight but a feature: the theory is background-independent at the level of configuration space in the same sense that general relativity is background-independent at the level of spacetime. Because no external metric exists, the only way to define distinguishability between nearby configurations, to construct geodesics, and to perform coarse-graining is to induce the metric from the wavefunctional itself. The Fisher metric Gᴵᴶ[R], constructed from ρ = |Ψ|², is the unique monotone Riemannian metric on the space of probability distributions under admissible coarse-grainings, as guaranteed by Čencov's theorem. State-dependent geometry is therefore not a dynamical pathology imported into an otherwise clean theory — it is the minimal background-independent way for the theory to have a geometry at all. A fixed background metric on 3N−6 would be an additional unrelational structure with no justification; the Fisher metric induced by Ψ is what background independence demands. This makes the answer to the question of why sPNP has a state-dependent metric the same as the answer to why general relativity has a dynamical metric: because there is nothing else for the geometry to depend on.
This point sets the context for the nonlinearity discussion that follows. Because sPNP is formulated on relational 3N−6 shape space rather than on a fixed background geometry, state-dependent geometry is not an extra dynamical assumption but the minimal background-independent way to define distinguishability, geodesics, and coarse-graining. This makes the Fisher metric kinematic at the level of relational geometry: it is constructed from the state, but it does not by itself imply nonlinear Hilbert-space evolution. The question of whether it does so is the subject of the following analysis.
Summary. sPNP treats the nonlinearity objection in two regimes. In the kinematic regime, the Fisher metric Gᴵᴶ[R] is a derived geometric re-description of linear quantum motion: the wavefunctional evolves under the standard unmodified Hamiltonian, the phase S fixes the topological path through configuration space, and the metric affects only the rate at which that path is traversed. There is no nonlinear term in the fundamental generator, so the standard objection has no target. In the reflexive regime, where the geometry becomes genuinely state-dependent, sPNP structurally excludes the Gisin signaling mechanism by restricting backreaction to the amplitude sector Gᴵᴶ[R], constructed from ρ = |Ψ|² alone, so phase-distinct states with identical ρ induce identical geometry. The Pusey–Barrett–Rudolph theorem supports the ontic status of Ψ, removing the ψ-epistemic mixture loophole. However, full no-signaling and norm conservation in the reflexive regime are not automatic consequences of this architecture: they require the specific reflexive Hamiltonian to preserve the continuity equation and the relevant symplectic structure, and those are verification obligations rather than established results. The kinematic case is safe outright; the reflexive case is a precisely constrained open problem, not an unsupported claim.
Full treatment.
The nonlinearity objection splits cleanly into two regimes, and the two regimes require categorically different responses.
The kinematic regime. In the kinematic limit, the fundamental Hamiltonian Ĥ is the standard linear operator and the wavefunctional evolves as iħ ∂_τΨ = ĤΨ without modification. The Fisher metric Gᴵᴶ[R] is computed from the solution R = |Ψ|¹ᐟ² after the evolution, not inserted into the generator — it is a functional of the output, not an input to the dynamics. The Bohmian trajectories are governed by Ẋᴵ = Gᴵᴶ[R] ∂ⱼS, where the phase S is determined entirely by the linearly-evolved wavefunctional. The level surfaces of S fix which configurations are visited and in what sequence — the topological path through configuration space. The metric Gᴵᴶ[R] raises the index on ∂ⱼS, affecting the rate at which that path is traversed, not the path itself. This is the quantum analog of the classical Jacobi reparametrization, and it maps directly onto a structure already present and accepted in standard Bohmian mechanics: the quantum potential Q = −(ħ²/2m)(∇²R / R) already depends on R directly, yet is not regarded as a dangerous nonlinearity because it is computed from the linearly-evolved Ψ rather than fed back into its generator. sPNP replaces the flat Laplacian with the curved Laplace–Beltrami Δ_G, yielding
Qₛᴾᴺᴾ = −(ħ² / 2m) (Δ_G R / R)
which is a geometric upgrade of an existing state-dependent structure, not a novel nonlinear coupling. In the kinematic regime the objection has no target, because there is no nonlinear term anywhere in the dynamics.
The reflexive regime. When the Hamiltonian becomes a functional ℋ[Gᴵᴶ[ρ]] of the state, the defense must be stated with precision rather than confidence, because not all of its components have the same logical status.
The first and only unconditional protection follows from an architectural choice: the reflexive coupling is restricted to Gᴵᴶ[R], the amplitude sector of the metric, with phase contributions structurally excluded from the feedback loop. Because Gᴵᴶ depends only on ρ = |Ψ|² and not on the full complex amplitude, two pure states with identical ρ but different phase structures generate identical metrics. The Gisin algebraic mechanism requires distinguishing pure states that share ρ but differ in phase; since Gᴵᴶ[ρ] is phase-blind by construction, this condition is never satisfied. This protection is unconditional because it follows from the definition of the coupling rather than from any dynamical theorem.
The second protection is PBR-supported but logically narrower than it might appear. The Pusey–Barrett–Rudolph theorem establishes under preparation independence that Ψ is ontic, removing the ψ-epistemic escape hatch and ensuring the nonlinear dynamics acts on a single definite wavefunctional rather than on a fundamental mixture. This eliminates the claim that hidden mixtures supply the Gisin mechanism with its target. It does not, however, by itself prove no-signaling. Even in a ψ-ontic theory, if Alice's different preparation choices produce different observable statistics for Bob through the nonlinear dynamics, signaling occurs. The full no-signaling argument requires equivariance.
Equivariance — the preservation of Pₜ = |Ψₜ|² under the nonlinear evolution — holds if and only if the continuity equation
∂_τρ + ∇ᴵ(ρ Gᴵᴶ[R] ∂ⱼS) = 0
continues to hold in the reflexive regime. This is a non-trivial condition on the specific form of the reflexive Hamiltonian, not a consequence of general principles, and it must be verified rather than assumed. If equivariance holds, Bob's marginal statistics are always determined by the linearly-evolved quantum mechanics and no-signaling follows. If it fails, the theory faces a deeper problem than signaling, because Born-rule statistics would no longer match quantum predictions.
Norm conservation carries an analogous obligation. U(1) invariance of ℋ[Gᴵᴶ[ρ]] is immediate: ρ is invariant under Ψ → eⁱᵅΨ, which makes Gᴵᴶ[ρ] and therefore ℋ invariant. In a linear theory this would immediately give norm conservation by Noether's theorem. In a nonlinear theory it does not, because Noether's theorem in the relevant sense requires that the nonlinear flow preserve the symplectic structure on the space of wavefunctionals, so that U(1) generates a genuine symplectic symmetry with a conserved charge. Whether the sPNP reflexive flow preserves that symplectic structure must be established for the specific ℋ, not assumed from the U(1) property alone.
The reflexive defense is therefore a precisely constrained set of verification obligations rather than an established result. The phase-blind amplitude restriction defeats the Gisin algebraic mechanism unconditionally. PBR-backed ψ-ontology removes the mixture loophole. But whether the theory is fully safe in the reflexive regime depends on whether the specific reflexive Hamiltonian preserves the continuity equation and the symplectic structure — obligations that must be discharged explicitly in a complete treatment, not read off from the architecture alone.
One further point limits the practical stakes. Backreaction enters at order Q₀² times the local Fisher curvature, so at physical scales where Q₀ ∼ ℓₚ the correction is of order ℓₚ² / L² relative to the leading classical geometry — undetectable on all accessible scales and vanishing exactly in the limit Q₀ → 0. The kinematic regime is therefore not only the logically safer case but the physically dominant one across the entire accessible energy range. The reflexive corrections matter in principle and must be proved safe, but they are not where the observable physics of sPNP lives.
