Position in the Foundations Landscape

Constraint-Surface Dynamics (CSD) is a realist, deterministic, single-world and contextual framework for quantum mechanics. It belongs to the foundations landscape alongside Copenhagen, Many-Worlds, Bohmian mechanics, GRW and QBism, but occupies a distinct position through its combination of objective ontic dynamics, an epistemic wavefunction, geometric Born weights and machine-checked Bell-violating correlations.

CSD is deterministic

A single ontic microstate evolves continuously under Hamiltonian flow on a compact geometric surface. There is no fundamental randomness in the dynamics. Apparent stochasticity arises from ignorance of the microstate's precise location within this surface, not from indeterminacy in the laws of motion. In this respect, CSD aligns with deterministic programmes such as Bohmian mechanics and contrasts with Copenhagen and GRW, which treat randomness as primitive.

CSD is single-world

There is one geometric surface, one trajectory, and one outcome per measurement. Branching in the Many-Worlds sense does not occur. What appears as branching is instead the dynamical partitioning of the surface into disjoint outcome regions during measurement interactions. The underlying surface remains unified at all times. This places CSD against Many-Worlds and alongside single-world realist approaches.

CSD treats the wavefunction as epistemic in role rather than ontic in substance

The wavefunction does not represent a physical field or object. Instead, it encodes an observer's coarse-grained information about which region of the geometric surface the microstate occupies. Measurement updates this information by revealing the realised outcome region. While this informational role parallels aspects of QBism, CSD sharply diverges from QBism's subjectivism: the underlying geometric surface and its dynamics are fully objective and mind-independent.

CSD fixes the Born rule through symmetry and geometry rather than postulating it

The SU(n) symmetry of complex projective space selects a unique invariant measure, the Fubini–Study measure. That fixes the reference measure and nothing more. The toric moment map then pushes it forward to the uniform measure on the probability simplex, so the state-dependent barycentric regions have Fubini-Study volume exactly the squared amplitudes. Gleason-class operational constraints characterise the same trace rule independently, but they are not a premise of the volume theorem. Combined with the deterministic volume-typicality result of Paper A, this yields outcome frequencies matching |ψ|² as a consequence of geometric structure. By contrast, Bohmian mechanics requires an additional quantum equilibrium hypothesis, Many-Worlds relies on decision-theoretic arguments, and Copenhagen and GRW postulate the Born rule directly.

CSD introduces no collapse mechanism, no stochastic dynamics, and no dual ontology

Unlike Bohmian mechanics, CSD does not posit a guiding wave. Its ontic microstate determines the realised outcome only within the measurement context that actually occurs; it does not assign pre-existing values to every incompatible observable. Unlike GRW, it does not modify Schrödinger evolution. Unlike Copenhagen, it does not appeal to an undefined measurement cut. The framework operates within geometric quantum mechanics, treating the underlying geometric structure as physically real.

Where CSD stands against the no-go theorems

A ψ-epistemic realist framework has to say where it sits with respect to the theorems that constrain ontological models, and the Pusey-Barrett-Rudolph theorem in particular.

PBR does not rule out ψ-epistemic models as such. It rules them out given a further assumption, preparation independence, which requires that independently prepared systems have a joint ontic state factorising as a product of the individual ones.

CSD satisfies PBR’s disjointness conclusion rather than evading it. On the canonical exact preparation interface the projection is deterministic, so distinct pure-state preparations are mutually singular, which makes CSD psi-ontic in the Harrigan-Spekkens sense. That has been a machine-checked theorem since 25 August 2026. The classification tracks whether the ontic state determines psi, not whether psi is physically real, so it is consistent with the wavefunction being epistemic in CSD’s own sense: the projection is many-to-one and the fibre carries structure the base does not. Finite-resolution preparations over positive-volume regions do overlap, but that is a theorem about a different preparation class and neither establishes nor softens the classification of the exact interface. Operational independence and marginal stability still hold, so this is not a conspiratorial correlation. Paper D sets out CSD's compliance with the Bell, Kochen-Specker, Fine and PBR constraints; a dedicated companion paper gives the full argument.

Kochen-Specker and the contextuality results are addressed differently. CSD is contextual by construction, since measurement outcomes are context-dependent regions rather than pre-assigned values.

CSD occupies a distinctive position in the foundations landscape

Its defining combination is a deterministic single-world ontology, an epistemic wavefunction, Born probabilities fixed by geometry, contextual measurement outcomes, Bell-violating singlet correlations and operational no-signalling. The central finite-dimensional results are machine-checked in Lean 4, making this a formal reconstruction rather than only an interpretive proposal.

CSD is contextual, Bell-violating and operationally no-signalling

In CSD, determinism applies to the outcome realised in the measurement context that actually occurs. It does not require a single context-independent assignment of values to all possible incompatible measurements.

C1 machine-checks an explicit model that reproduces the complete singlet probability table, the correlation E(a,b) = −a · b, and the maximal CHSH value 2√2. It proves that no compatible global assignment across the four CHSH settings can reproduce these correlations and verifies uniform local marginals and operational no-signalling.

CSD is therefore not Bell-local in the factorisation sense ruled out by Bell’s theorem. Its locality result concerns the factorisation of measurement dynamics, not the factorisation of the joint outcome statistics. Deterministic realised outcomes, Bell violation and operational no-signalling coexist within the same contextual framework.