Projection Bridge — a rigid lock between GR, SM, and QM

About: a compact, three-layer presentation of the projection bridge that locks General Relativity (GR), the Standard Model (SM), and standard Quantum Mechanics (QM) without adding propagating fields.

1) Rigorous statement (variational lock and consequences)

Bridge action (validated exterior regimes):

\[ S=\!\int d^4x\,\sqrt{-g}\Big[ \tfrac{M_P^2}{2}R + \mathcal L_{\rm SM}(\psi,g)\;-\;\Phi\,(\rho-\varepsilon) \Big] \quad (+\ \text{screened/boundary terms}). \]

Elliptic constraint (non-radiative)

Variation with respect to \(\Phi\) gives an elliptic (instantaneous) balance relation:

\[ \delta_\Phi S = 0 \ \Longrightarrow\ \rho-\varepsilon = 0 \ \Longrightarrow\ \nabla^2\Phi = 4\pi G\,\rho \quad (\text{validated Poisson-type balance}). \]

This constraint introduces no additional propagating degree of freedom and no extra polarization.

Metric equations (GR with minimal coupling)

Variation with respect to \(g_{\mu\nu}\), then taking the validated limit (screening + no anisotropic stress from \(\Phi\)), yields Einstein’s equations with the usual SM stress–energy:

\[ \delta_g S = 0 \ \Longrightarrow\ \tfrac{M_P^2}{2}G_{\mu\nu} = T^{\rm SM}_{\mu\nu} + T^{(\Phi)}_{\mu\nu} \ \Longrightarrow_{\text{validated}}\ G_{\mu\nu}=8\pi G\,T^{\rm SM}_{\mu\nu}. \]

Propagating content remains the two transverse–traceless tensor modes. No scalar dipole channel; geodesics and redshifts follow GR exactly in validated regimes.

Quantum embedding (no change to QM postulates)

On the quantum side, the bridge acts as a fixed projector \(P\) determined by exterior boundary data. Physical states satisfy \(P\rho P=\rho\). The Born rule, unitary evolution on \(\mathrm{Im}\,P\), composition, and Bell/CHSH statistics are unchanged. No new dissipative or signalling channels are introduced by the constraint.

Bottom line (rigorous): in validated domains, the bridge imposes an elliptic constraint that locks the theory to GR+SM and standard QM without tuning any coefficients or adding propagating degrees of freedom.

2) Layman’s picture (the escrow conduit)

Think of two ledgers that must always balance: curvature on the GR side and energy–momentum on the SM side. The projection bridge is a rigid escrow conduit that enforces the balance:

Whenever matter or energy appears on the SM ledger, the bridge demands exactly the matching curvature on the GR ledger.
Whenever curvature appears on the GR ledger, the bridge demands the SM ledger can support it (no “free” extra gravity channels).
In quantum experiments, the bridge behaves like a fixed gate: it never alters the probabilities or allows new signals—just ensures consistency.

The conduit is bidirectional (GR ↔ SM) but non-translating: it does not convert fields; it enforces compatibility. That is why the familiar GR/QM formulas reappear unchanged where they have already been validated.

3) Flow diagram (one-glance schematic)

Flow: SM observables and GR geometry are coupled only through the non-radiative elliptic constraint. QM lives on the same physical subspace via a fixed projector. In validated regimes, this reproduces all standard GR/QM predictions exactly; novelty is confined to bounded diagnostics (e.g., interiors) with falsifiable signatures.