The Foundation of Projection Relativity

We present Projection Relativity (PR), a spectral--projection framework in which gravitational stiffness, displacement-generated inertia, compact electromagnetic phase, and projection-response effects arise as observable sectors of a single master projection field \(\Psi(x,\xi)\) over an internal spectral manifold. The framework is built from a minimal radial--compact internal geometry,$\mathcal M_{\rm int}=\mathbb R_w\times S^1_\theta,$ whose radial spectral gap supplies a finite coercive scale and whose compact phase coordinate supplies the electromagnetic projection sector. The projected metric recovers the Einstein field equations in the low-energy limit and reduces to the Kerr exterior outside the finite projection core. The same nonzero radial spectral gap replaces the classical \(r=0\) singular endpoint with a bounded finite-core geometry, while preserving standard exterior black-hole phenomenology. In the dynamical ringdown sector, PR inherits the Kerr quasi-normal-mode hierarchy in the exterior limit. Projection-sector corrections are gap-suppressed, so current gravitational-wave observations primarily test the Kerr-recovery limit rather than providing a positive residual detection.

We find a luminosity-linked negative velocity residual in the quasar sample. Across the primary warning-clean estimators, the predicted sign appears in \(140/144\), \(134/144\), \(136/144\), and \(133/144\) matched bins for DR16Q, PIPE, PCA, and MgII, respectively. We also map the foreground-cleaned LoTSS residual, $|{\rm RRM}_{2022}|=1.254538\,{\rm rad\,m^{-2}}$, through the compact-phase magnetic area law, turning the observed Faraday-rotation residual into a projected compact-domain constraint rather than a fixed primordial field amplitude. A first Hubble--DESI pressure test maps the local--CMB Hubble split to a positive projected phase-density offset, \(\Delta\rho_\theta\simeq1.6\times10^{-27}\,{\rm kg\,m^{-3}}\), and finds that a high-\(H_0\) mixed phase-response diagnostic can remain close to compressed DESI BAO distances while approximately preserving the CMB acoustic calibration.

The resulting architecture provides a compact, testable projection framework for finite-core black holes, Kerr-consistent ringdown phenomenology, magnetically constrained compact phase structure, and luminosity-linked quasar velocity residuals. The complete derivation ledger, symbolic audits, numerical validation scripts, and figure-generation workflows are provided in the companion supplementary material and public repository. The scope of this manuscript is the minimal bosonic projection chain: gravitational stiffness, displacement-generated inertia, compact electromagnetic phase, projection response, and their first observational consequences.

GitHub Repository