Part 4 — Quantum Embedding of SPSP–SSC

In this section we demonstrate that the Single Point Super Projection – Single Sphere Cosmology (SPSP–SSC) admits a rigorous quantum formulation. The aim is to show that, when treated as a quantum system, SPSP–SSC reproduces the exact postulates and predictive structure of standard quantum mechanics (QM), with novelty confined to an elliptic constraint projector derived from the projection formalism. This ensures consistency with all known laboratory quantum tests while providing a precise interface between quantum information, gravity, and cosmology.

Within this embedding, the single-sphere projection also functions as a fundamental qubit: a solitary component that, when split across projection states, naturally yields entanglement and encodes quantum amplitudes within the same geometry that locks gravity to GR. This places SPSP–SSC as a unified framework where quantum mechanics, general relativity, and cosmology are structurally reconciled.

SPSP–SSC should therefore be regarded not as an alternative to quantum mechanics, but as a minimal extension: it preserves the full Hilbert-space structure, unitary evolution, and Born-rule probabilities of standard QM, while enforcing a single additional condition. This elliptic constraint removes one otherwise-allowed radiative channel, locking the quantum sector to the same two tensorial degrees of freedom as in GR. In this way, SPSP–SSC functions as a rigorous extension-by-subtraction of QM, fully consistent with laboratory tests yet predictive at the projection and horizon scale.

4.1 State Space and Postulates

Axiom Q1 (State space). Each SPS is associated with a complex Hilbert space \(\mathcal H_{\rm SPS}\cong\mathbb C^2\). Pure states are rays \(|\psi\rangle=\alpha|0\rangle+\beta|1\rangle\) with \(|\alpha|^2+|\beta|^2=1\). Mixed states are density operators \(\rho\ge0\) with \(\mathrm{tr}\,\rho=1\).

Axiom Q2 (Born rule). Measurement outcomes are described by a POVM \(\{\Pi_i\}\), and outcome probabilities are \(p(i)=\mathrm{tr}(\rho\,\Pi_i)\).

Axiom Q3 (Composition). For two SPS units, the joint space is \(\mathcal H_{12}=\mathcal H_1\otimes\mathcal H_2\). Entanglement is defined in the usual way: \(\rho_{12}\) is entangled iff it cannot be written as a convex combination of product states.

These axioms establish SPS as a qubit-like system, with no deviation from standard quantum postulates.

4.2 Projection Constraint as Elliptic Superselection

SPSP–SSC differs from ordinary QM only by the presence of an elliptic projection constraint. This is formalized as a fixed projector \(P\) determined by SSC boundary data, such that physical states satisfy \(P\rho P=\rho\). Thus allowed superpositions live entirely in the subspace \(\mathrm{Im}\,P\). This acts like a superselection rule: it does not radiate, it introduces no new degrees of freedom, and it preserves the structure of probability calculus.

4.3 Dynamics and Evolution

4.3.1 Constrained Unitary Dynamics

In validated regimes (outside horizons), evolution is unitary with effective Hamiltonian \(H_{\rm eff}\) from GR+SM:

\[ \rho(t+\Delta t)= \frac{P\,U(\Delta t)\,\rho(t)\,U^\dagger(\Delta t)\,P}{\mathrm{tr}\!\left(P\,U(\Delta t)\,\rho(t)\,U^\dagger(\Delta t)\,P\right)}, \quad U(\Delta t)=e^{-\,\tfrac{i}{\hbar}H_{\rm eff}\Delta t}. \]

Since \(P\) is time-independent in validated domains, this reduces to ordinary unitary evolution on the physical subspace. No Lindblad dissipator or exotic dynamics are introduced.

4.3.2 Measurement and Collapse

Measurement is described by a POVM \(\{\Pi_i\}\) on \(\mathrm{Im}\,P\). The Born rule holds exactly: \(p(i)=\mathrm{tr}(\rho\,\Pi_i)\). State update is given by the Lüders rule \(\rho\mapsto \Pi_i^{1/2}\rho\Pi_i^{1/2}/p(i)\). Thus SPSP–SSC retains the standard collapse postulate.

4.4 Entanglement and No-Signalling

Consider two SPS units with joint state \(\rho_{AB}\). If \((P_A\otimes P_B)\rho_{AB}(P_A\otimes P_B)=\rho_{AB}\), then for any local POVM on \(A\), the reduced state on \(B\) remains unchanged. Formally:

\[ \rho'_B=\sum_a \mathrm{tr}_A\!\left((M_a\otimes\mathbb I)\rho_{AB}\right) =\mathrm{tr}_A(\rho_{AB})=\rho_B. \]

This guarantees that entanglement persists, but superluminal signalling does not. The elliptic constraint supports Bell-nonlocal correlations while respecting relativistic causality.

4.4.1 Bell/CHSH Compliance

The constrained subspace supports standard CHSH violations: for a maximally entangled state \(|\Phi^+\rangle\in\mathrm{Im}\,P\otimes\mathrm{Im}\,P\), the CHSH parameter reaches \(S=2\sqrt{2}\). Thus SPSP–SSC reproduces quantum nonlocality exactly, with no hidden-variable loopholes.

4.5 Gravity–Quantum Interface

In SPSP–SSC, gravity does not require quantizing the metric. Instead, the constraint projector \(P\) is fixed by SSC boundary data sourced by GR+SM. This ensures: (i) all exterior evolution is standard QM; (ii) projection constraints implement the lock to GR; (iii) no additional propagating degrees of freedom appear.

4.6 Falsifiable Quantum Tests

Laboratory QM: SPSP–SSC must reproduce all quantum experiments — interference, Bell tests, tomography. Any mismatch falsifies the model.
Horizon diagnostics: Echo-phase structures bounded by \(\Delta B_{\rm norm}\approx0.02\) must be reproducible; instability or drift would falsify.
Curved-space interferometry: Phase shifts must match GR+QM predictions. Any additional dispersive channel beyond environment decoherence falsifies the elliptic constraint.

Summary. SPSP–SSC as a quantum system is a qubit with a fixed elliptic projection constraint. It inherits the exact structure of standard QM — Hilbert space, unitary dynamics, Born rule, entanglement, Bell nonlocality, and no-signalling — while introducing no new radiative channels. This provides a rigorous, falsifiable bridge between quantum mechanics and the SSC gravitational framework.

4.x Conclusion

The quantum embedding shows that SPSP–SSC is not merely a reformulation of gravity, but a framework that unifies general relativity, the Standard Model, mathematics, and quantum mechanics under a single projection-based formalism. Gravity is locked to GR in all validated regimes, particle physics remains that of the SM, quantum postulates are exactly reproduced, and the elliptic constraint provides the structural bridge to mathematics. Thus SPSP–SSC offers a rigorously defined, testable synthesis in which known physics is preserved and novel structure is confined to precisely falsifiable frontiers.

In summary, SPSP–SSC extends QM only by enforcing a universal elliptic constraint: a subtraction rather than an addition. This preserves every validated quantum prediction, while providing a decisive lock to GR and a pathway for projection-scale falsification.

Read Part V