Elliptic Constraint Action — Short Reference

Core mechanism behind the GR/SM/QM lock-in of the Single Point Super Projection – Single Sphere Cosmology (SPSP–SSC).

Summary. The SPSP–SSC framework preserves all validated predictions of General Relativity (GR), the Standard Model (SM), and Quantum Mechanics (QM) by adding a single, non-radiative elliptic constraint to the action. This constraint enforces global consistency of the projection while introducing no new propagating degrees of freedom. As a result there is no scalar dipole radiation, cosmological linear observables match ΛCDM/GR, and standard quantum postulates remain intact. Novel structure is confined to bounded boundary terms (e.g., horizon diagnostics), yielding clear falsifiability criteria.

1. Action (validated exterior)

In regimes where precision tests exist (Solar System, binary pulsars, LIGO/Virgo/KAGRA, linear cosmology, laboratory quantum), the action is

\[ S \;=\; \int d^4x\,\sqrt{-g}\,\Big[\;\tfrac{M_P^2}{2}\,R \;+\; \mathcal{L}_{\mathrm{SM}} \;+\; \Phi\,(\rho-\varepsilon) \;\Big] \;+\; S_{\mathrm{bdy}}, \]

where \(R\) is the Ricci scalar, \(\mathcal{L}_{\mathrm{SM}}\) is the unmodified SM Lagrangian, and \(\Phi\) is the sorting potential that enforces the elliptic constraint. The boundary piece \(S_{\mathrm{bdy}}\) encodes projection consistency at horizons/outer boundaries and does not alter exterior equations of motion.

2. Variation & Constraint

Varying with respect to \(\Phi\) yields an elliptic (Poisson-type) constraint on each spatial slice:

\[ \nabla^2 \Phi \;=\; 4\pi G\,\rho \;-\; 4\pi G\,\varepsilon \;\;\xrightarrow{\;\text{validated exterior}\;}\;\; \nabla^2 \Phi \;=\; 4\pi G\,\rho , \]

which is non-propagating (no wave equation). Varying with respect to \(g_{\mu\nu}\) gives the Einstein equations with the usual SM stress–energy; the \(\Phi\)-sector contributes only constrained (instantaneous) balance, not radiative stress.

3. Immediate Consequences

No new radiative channel: \(\Phi\) does not propagate, hence no scalar dipole radiation. Gravitational radiation remains purely quadrupolar with two transverse–traceless modes at \(c\).
PPN lock-in: Post-Newtonian parameters match GR exactly (\(\gamma=\beta=1\), others zero) in screened/validated regimes.
Cosmology lock-in: Background expansion and linear perturbations reduce to ΛCDM/GR; no extra anisotropic stress at linear order.
Quantum exactness: Standard Hilbert space postulates, Born rule, entanglement and Bell/CHSH are preserved (the constraint acts as a fixed projector on the physical subspace).

4. Falsifiability (hard bounds)

Binary pulsars: fractional excess decay \(\Delta_{\rm dip} \equiv (\dot P_{\rm obs}-\dot P_{\rm GR})/|\dot P_{\rm GR}|\).
\[ \boxed{\,|\Delta_{\rm dip}| < 10^{-3}\,} \]
Any confirmed \(|\Delta_{\rm dip}|\ge 10^{-3}\) (after systematics) falsifies the model.
GW phasing: −1PN dipole term \(\beta_{-2}\) in \(\Psi(f)=\Psi_{\rm GR}(f)+\beta_{-2}f^{-7/3}+\cdots\).
\[ \boxed{\,|\beta_{-2}| < \beta_{\rm thr}\ \ \Longleftrightarrow\ \ |\mathcal F_{\rm dip}|/\mathcal F_{\rm quad} < 10^{-3}\,} \]
Detection at or above threshold falsifies.
Linear cosmology: Systematic departure from ΛCDM/GR bandpowers in precision windows falsifies.

5. Why the Constraint is Elliptic (and benign)

The \(\Phi\)-term enters the Lagrangian without a kinetic \( (\partial \Phi)^2 \) piece; its Euler–Lagrange equation is algebraic in time and second order in space (Poisson-like). In the Hamiltonian picture, the primary constraint \(\pi_\Phi\!\approx\!0\) is preserved by a secondary (elliptic) constraint, closing with the standard GR Dirac algebra. Hence no additional propagating degree of freedom is introduced, and radiative content remains that of GR.

6. How to Test

Pulsar timing: compute \(\dot P_{\rm GR}\) (quadrupole) and compare to \(\dot P_{\rm obs}\); check the bound above.
GW events: fit \(\beta_{-2}\) in frequency-domain phasing; compare to event-specific thresholds.
Cosmology: compare linear CMB/LSS residuals against ΛCDM/GR within precision windows.

Clarity statement. All exterior predictions use unmodified GR geodesics and SM/QM dynamics; SPSP–SSC effects appear only through bounded elliptic boundary diagnostics. This preserves every validated prediction while keeping novelty sharply testable.