Single Point Super Projection — A Single Sphere Cosmology (SPSP–SSC): Rigorous Gravity Lock-In

Introduction

Modern cosmology and gravitational physics are dominated by the ΛCDM + GR paradigm, which has passed stringent observational tests from laboratory scales to cosmological surveys. At the same time, deep conceptual puzzles remain: the nature of dark matter and dark energy, the singularity problem inside black holes, and the apparent nonlocality of quantum entanglement. These motivate the search for frameworks that preserve all validated predictions of GR while opening conceptual pathways to unify mass–energy duality and holographic or projected descriptions of the universe.

The Single Point Super Projection – Single Sphere Cosmology (SPSP–SSC) model is proposed as such a framework. Its central idea is that the universe can be described as a single spherical instance projected outward into a cloud of points, with every observed coordinate corresponding to a projection of the same underlying core. Matter and energy are balanced by a sorting potential \(\Phi\), while projection consistency is maintained by auxiliary fields \(\Pi\). General relativity, the Standard Model, and the Higgs mechanism remain intact, but are reinterpreted in this projected geometry.

A key feature of SPSP–SSC is that the sorting potential \(\Phi\) is not a new wave-like field but a constraint that enforces balance across the projection. Locally, its equations may resemble those of a wave, but globally they are elliptic — like a static balance condition rather than a propagating signal. This ensures that no new radiation channel is introduced: there is no scalar dipole emission, and only the two familiar tensorial gravitational waves of general relativity remain. This structural choice both preserves consistency with pulsar timing and gravitational-wave observations, and provides a clear falsifiability criterion for the model.

Structure of the paper. Section 2 introduces the geometric and projectional structure. Section 3 defines the validated-regime principle. Section 4 gives the full action with its sectors. Section 5 derives the PPN limit. Section 6 analyzes linearized theory and degrees of freedom. Section 7 covers binary radiation. Section 8 discusses cosmology and perturbations. Section 9–12 address black holes, compliance with canonical tests, and predictions. Appendices A–L provide technical details and proofs.

1. Core Statement & Axioms

Core Statement. Every single point in the universe is an instance of SPSP–SSC: a unique projection from the same core sphere. The core sphere is generated by dual poles \(c\) (energy/light) and \(m\) (mass/density). A projection map \(\Pi:\mathcal{S}\!\to\!\mathcal{C}\) produces a connected yet unbounded point-cloud \(\mathcal{C}\). Each atom, field excitation, and spacetime coordinate is one such projection-point—independent in expression, connected in origin.

1. Single Sphere \(\mathcal{S}\): Core \(c\); shell \(m\).
2. Projection: \(\Pi:\mathcal{S}\to\mathcal{C}\); our observable universe is a subset of \(\mathcal{C}\).
3. Universality: Every local degree of freedom is a projection-point of \(\mathcal{S}\).
4. Cycle: Endless \(c\to m\to c\) encoded by phase \(\theta\in S^1\); the present sits near \(\chi\) where \(c\) and \(m\) balance.
5. Gravity: Vacuum sorting potential \(\Phi\) produces observed gravitational phenomenology.
6. Recycling: Boundary collapse (BH) recycles \(m\to c\) internally; exterior equals GR.
7. Intact Limits: GR, SM, and \(\Lambda\)CDM are recovered when SSC corrections vanish in tested regimes.

2. Geometry & Diagrams

Infinity loop (c/m) → 3D single sphere (core \(c\), shell \(m\)) → projection into a point cloud where every observed point is an instance of the same sphere.

3. Formalism & Projection

• Objects: \(\mathcal{S}\) (single sphere), \(\theta\in S^1\) (cycle phase), \(\mathcal{C}\) (point cloud), \(\Pi:\mathcal{S}\to\mathcal{C}\), \(\chi\) (balance region).
• Fields: \(\rho\) (mass density), \(\varepsilon\) (energy/flux), \(\theta\) (cycle order parameter), \(\Phi\) (sorting potential), \(g_{\mu\nu}\) (effective metric on the inner surface).
• Observables: pullbacks \(F_p=i_p^* F\) at \(p\in\mathcal{C}\); intact-physics limit coincides with GR+SM.

All exterior predictions use unmodified GR geodesics; SPSP–SSC effects appear only through bounded elliptic boundary terms and diagnostic overlays. This ensures that standard propagation results remain intact, with novel structure confined to validated boundary sectors.

4. Action (Lagrangian) & Sectors

Elliptic constraint (no kinetic term for \(\Phi\)). The sorting potential \(\Phi\) acts as a Lagrange multiplier. Varying the action gives \(\delta S/\delta\Phi=0 \Rightarrow \mathcal{C}[g_{\mu\nu},T_{\alpha\beta}]=0\). In the non-relativistic validated limit we take \(\mathcal{C}\xrightarrow{\rm NR}\nabla^2\Phi - 4\pi G\,\rho = 0\), so \(\Phi\) is elliptic and non-radiative (no wave operator, no extra GW polarization).

Notes: (i) In earlier drafts a wave-like kinetic term \(-\tfrac{Z(\theta)}{2}\,\partial_\mu\Phi\,\partial^\mu\Phi\) appeared. In the validated SPSP–SSC action this term is absent; \(\Phi\) acts as a Lagrange multiplier (elliptic constraint). (ii) Any explicit source product \(\Phi(\rho-\varepsilon)\) is not retained as a standalone term in the covariant action; it represents only the nonrelativistic limit of the constraint, \(\mathcal C \xrightarrow{\rm NR} \nabla^2\Phi - 4\pi G\,\rho = 0\). (iii) \(S_{\text{recycle}}^{(\text{phen})}\) is an exterior-safe phenomenological statement about BH interiors and is kept out of the variational dynamics.

5. Post-Newtonian (PPN) Derivation

5.1 Setup: action, matter, bookkeeping

We work in the validated regime with matter minimally coupled and the sorting potential elliptic (constraint) as in Section 4:

Thus \(\Phi\) is non-radiative (no wave operator), enforces the Poisson constraint, and introduces no new propagating DOF. PN bookkeeping follows GR with \(U\) from \(\nabla^2 U=-4\pi G\rho\).

5.2 Variation and field equations

Varying \(g_{\mu\nu}\) yields Einstein equations with matter sources; \(\Phi\) enforces \(\nabla^2\Phi=4\pi G\rho\) and does not propagate. There is no additional kinetic stress from \(\Phi\) in the validated (elliptic) formulation; thus the PN sector reduces to the GR system, recovering \(\gamma=1\), \(\beta=1\) and vanishing preferred-frame/non-conservative parameters.

5.3 PN gauge and metric ansatz

Adopt the standard PPN-isotropic ansatz (indices \(i,j=1..3\)):

where \(U(\mathbf x)=G\!\int d^3x'\,\rho(\mathbf x')/|\mathbf x-\mathbf x'|\) and \(V_i,W_i,\Xi\) denote the standard PN vector and composite scalar potentials built from \(\rho,\mathbf v,p,\Pi\) (internal energy). We will derive the values of the PPN parameters \((\gamma,\beta,\xi,\alpha_{1,2,3},\zeta_{1..4})\).

5.4 Solve order-by-order

(i) Newtonian order \(\mathcal O(\epsilon)\)

The \(00\) Einstein equation reduces to Poisson for \(U\):

The \(ij\) trace at this order fixes spatial curvature: absence of anisotropic stress at \(\mathcal O(\epsilon)\) implies

Reason: the scalar contributes only via gradients of \(\Phi\) that source \(U\) itself; its stress is isotropic at this order (no shear), so the two Newtonian metric potentials coincide.

(ii) Gravito-magnetic order \(\mathcal O(\epsilon^{3/2})\)

The \(0i\) equations determine \(g_{0i}\) in terms of the mass-current. Universal metric coupling (no extra long-range vectors) yields the GR coefficients; hence all **preferred-frame** parameters vanish:

Moreover, no non-conservative terms appear (screening removes local \(\delta\mathcal P\)), so **\(\zeta_{1,2,3,4}=0\)** and **\(\xi=0\)**.

(iii) Post-Newtonian nonlinearity \(\mathcal O(\epsilon^{2})\)

At \(\mathcal O(v^4/c^4)\) the nonlinearity in \(g_{00}\) arises from the Einstein–Hilbert sector; the scalar sector does not introduce a new \(U^2\) term because its energy density is already accounted for via the Newtonian field and screening suppresses further local corrections. Therefore the coefficient of \(U^2\) is the GR value:

5.5 Assemble the metric and extract parameters

Collecting results:

Thus, to first post-Newtonian order, SPSP–SSC is observationally **degenerate with GR** in validated regimes.

5.6 Observables (explicit)

• Light deflection by a mass \(M\) with impact parameter \(b\): \(\Delta\phi = \dfrac{1+\gamma}{2}\,\dfrac{4GM}{bc^2} \Rightarrow \Delta\phi=\dfrac{4GM}{bc^2}\).
• Shapiro delay: \(\Delta t \propto (1+\gamma)\Rightarrow\) GR value.
• Perihelion advance (Schwarzschild): \(\Delta\varpi = \dfrac{6\pi GM}{a(1-e^2)c^2}\).
• Redshift & time dilation, LLR, geodetic & frame-dragging: GR values.

5.7 Consistency checks internal to SPSP–SSC

• No extra long-range vectors: guaranteed by the action; hence no preferred-frame parameters.
• Scalar non-radiative locally: \(\Phi\) obeys Poisson; screening \(\delta\mathcal P\!\to\!0\) prevents local fifth forces at measured sensitivity.
• Energy-momentum conservation: follows from diffeomorphism invariance and the \(\Phi\) EOM (shown in Sec. 7); required for vanishing \(\zeta_i\).

5.8 What could shift PPN away from GR (and how it’s avoided)

Deviations would occur if (i) \(Z(\theta)\) varied significantly locally (introducing anisotropic stress), (ii) screening failed so that \(\delta\mathcal P\) was non-negligible, or (iii) matter coupled non-universally. The validated-regime postulates of SPSP–SSC exclude these cases; any dynamics that relaxes them is confined to untested domains (ultra-large scales or BH interiors) and does not leak into PPN.

6. Linearized Theory: Constraints, DOF, Stability

6.1 Quadratic action & gauge choice

Expand around Minkowski: \(g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}\), keep terms to \(\mathcal O(h^2,\varphi^2)\), with \(\Phi=\Phi_0+\varphi\) and \(\partial\theta_0=0\) in validated regimes. The relevant (validated) action pieces are

Here \(\varphi\) is the linearized sorting potential acting as a constraint (no wave kinetic term); its variation yields the Poisson relation in §6.2.

6.2 Linearized field equations & constraints

Varying the quadratic action yields

The graviton sector is hyperbolic (two TT modes at \(c\)); \(\Phi\) is elliptic (constraint), so it does not radiate and cannot produce dipole emission.

6.3 Propagators & dispersion relations

In Fourier space, the graviton propagator in de Donder gauge has the GR form; the TT sector satisfies

i.e. two tensor polarizations at luminal speed. The scalar \(\varphi\) is instantaneous (elliptic) in the validated limit and does not appear as a radiative pole. No extra vector modes are present.

6.4 Well-posedness in the validated regime

Lemma (Cauchy well-posedness). Consider the SPSP–SSC action restricted to the validated regime, where: (i) the projection functional is screened (\(\eta,\epsilon \to 0\)), (ii) the cycle field \(\theta\) is frozen to a constant background, and (iii) the sorting potential \(\Phi\) enters only as an elliptic constraint \(\nabla^2\Phi = 4\pi G\,\delta\rho\). Then the Cauchy problem is well posed on globally hyperbolic spacetimes: there exist unique solutions depending continuously on initial data.

Proof sketch. In this regime, the Einstein–Hilbert sector reduces to the standard GR initial-value problem, for which Choquet-Bruhat’s theorem guarantees well-posedness. The elliptic constraint for \(\Phi\) is solved instantaneously on each spatial slice and introduces no new propagating degrees of freedom, so it does not alter the hyperbolic character of the Einstein equations. Matter fields evolve under standard covariant conservation, with their initial data constrained by the usual Hamiltonian and momentum constraints. Thus the combined system admits a consistent initial-value formulation with only the two transverse–traceless tensorial degrees of freedom propagating.

6.5 Stability: ghosts, gradients, tachyons

• No ghosts: The graviton kinetic term has coefficient \(+M_P^2/8\); the sorting sector is a constraint (no propagating scalar) and adds no ghostly mode.
• No gradient instabilities: The TT sector is hyperbolic with real sound speed \(c\); the scalar constraint is elliptic and non-radiative locally.
• No tachyons: No mass term for the graviton; any cycle-sector mass \(U''(\theta_0)\ge 0\) is frozen in validated regimes.

6.6 Hamiltonian (ADM) Constraint Proof

Perform a 3+1 split with lapse \(N\), shift \(N^i\), spatial metric \(h_{ij}\):

The canonical momenta are

Elliptic \(\Phi\) (validated regime). Treat \(\Phi\) as a Lagrange multiplier: the primary constraint is \(\pi_\Phi \approx 0\), and variation enforces \(\mathcal C=\nabla^2\Phi-4\pi G\,\rho=0\) on each spatial slice. The Hamiltonian density becomes

Preservation of \(\pi_\Phi\approx 0\) gives the (elliptic) secondary constraint \(\mathcal C=0\). The GR Dirac algebra of \((\mathcal H_0,\mathcal H_i)\) closes as usual; \(\Phi\) adds no propagating DOF.

6.6.1 Secondary constraints & algebra

Preserving \(\pi_N,\pi_i\) in time yields the secondary constraints \(\mathcal H_0\approx0,\ \mathcal H_i\approx0\). The smeared constraints close first-class (Dirac/DeWitt algebra), and the minimally coupled scalar does not spoil closure:

6.6.2 Degrees of freedom counting

The metric sector: 6 components \(h_{ij}\) \(\Rightarrow\) 12 canonical vars \((h_{ij},\pi^{ij})\); 4 first-class constraints + 4 gauge generators.

The scalar pair \((\Phi,\pi_\Phi)\) is constrained to be non-radiative (elliptic Poisson equation in validated regimes), so it does not contribute a propagating GW DOF locally.

6.6.3 Positivity & absence of ghosts

The kinetic terms for physical modes are positive for \(M_P^2>0\) and \(Z_0>0\). There is no Ostrogradsky instability since the action is second order and constraints remove non-physical components.

6.6.4 Conclusion of ADM proof

The canonical analysis matches the Lagrangian result: exactly two propagating tensor modes, luminal propagation, and no extra radiative scalars/vectors in validated regimes.

6.7 Energy flux & conservation (linear)

The Isaacson effective stress-energy tensor for high-frequency TT waves is

leading to the standard energy flux and ensuring consistency with binary-radiation results (see Section 7).

6.8 Radiative content & phenomenology

• Propagating modes: two TT polarizations; no extra polarizations observed.
• Speed/dispersion: luminal and non-dispersive at leading order.
• Scalar sector: non-radiative locally; any cosmological dynamics is screened in validated domains.

6.9 Summary

Linearized SPSP–SSC is equivalent to GR in validated regimes: two healthy tensor DOF with \(\omega^2=c^2 k^2\); scalar \(\varphi\) is elliptic and non-radiative locally; no ghosts or gradient instabilities. This underpins the PPN lock-in (Section 5) and the GR-identical GW phenomenology (Section 7).

7. Binary Radiation & Gravitational Waves

7.1 Linear TT modes & Isaacson tensor

In the weak-field, far-zone regime, SPSP–SSC reduces to linearized GR for the transverse–traceless (TT) metric perturbation. The wave equation and effective GW stress-energy tensor are:

7.2 Quadrupole flux & orbital decay

The leading energy flux is quadrupolar, as in GR:

For a quasi-circular binary with component masses \(m_1,m_2\) and separation \(a\):

These results reproduce the GR orbital decay law verified in pulsar timing.

7.3 Absence of dipole radiation (elliptic \(\Phi\))

Because the sorting potential \(\Phi\) is an elliptic constraint field, it obeys \(\nabla^2 \varphi = 4\pi G\,\delta\rho\) with no time derivatives. This means \(\Phi\) has no propagator, no radiative Green's function, and therefore cannot carry energy away from a binary system through a dipole channel.

The only propagating gravitational degrees of freedom in SPSP–SSC are the two transverse–traceless tensor polarizations of the metric. Consequently:

• The leading gravitational-wave energy loss is quadrupolar, identical to GR.
• No scalar dipole contribution exists, consistent with pulsar timing bounds (e.g., Hulse–Taylor and double pulsar systems).
• Gravitational-wave detections (LIGO–Virgo–KAGRA) observe the same waveform structure as GR, with no additional polarizations.

This explicit absence of dipole radiation locks the model to GR predictions for orbital decay, satisfying observational tests at the \(10^{-3}\)–\(10^{-4}\) level.

For explicit falsifiability criteria, see Sec. 12. Strict falsifiability bounds where hard inequalities are given for pulsar timing (\(\Delta_{\rm dip}\)), flux ratios (\(\mathcal F_{\rm dip}/\mathcal F_{\rm quad}\)), and GW phasing coefficients (\(\beta_{-2}\)).

7.4 GW speed & dispersion

The tensor sector propagates luminally:

No extra polarizations or dispersion arise in validated regimes, consistent with LIGO–Virgo–KAGRA constraints and the multi-messenger bound from GW170817/GRB170817A.

7.5 GW bounds summary

Combining the above results:

• Energy loss: Matches GR quadrupole formula within current experimental precision.
• Polarizations: Only the two TT modes are present; no scalar or vector modes propagate.
• Speed: Exactly luminal (\(|v_{\rm GW}-c|/c \lesssim 10^{-15}\)).
• Dipole channel: Absent due to universal coupling and screening.

Hence SPSP–SSC passes all current gravitational-wave observational tests.

8. Cosmology: SVT Derivation & Observables

8.1 Background & validated limit

In conformal time \(\tau\), the FRW metric is \(ds^2 = a^2(\tau)\left[-d\tau^2 + d\mathbf{x}^2\right]\). The effective expansion history is chosen to satisfy \(H_{\rm eff}(\tau) \approx H_{\Lambda{\rm CDM}}(\tau)\). Screening suppresses SSC corrections at high density/early epochs, so linear perturbations reduce to GR+ΛCDM in validated regimes.

8.2 Scalar–Vector–Tensor decomposition

In Newtonian gauge, set \(B=H_T=0\). Screening enforces \(\Phi_g=\Psi\) at linear order (no gravitational slip).

8.3 Linearized Einstein constraints

Here primes denote derivatives with respect to \(\tau\), and \(\mathcal H=a'/a\). In validated regimes, \(\Phi_g=\Psi\) and sources match GR.

8.4 Fluid equations

For each species \(X\) (photons, neutrinos, baryons, CDM):

These are the same as in GR when SSC corrections are screened.

8.5 Tensor sector

giving two TT polarizations at luminal speed, as in GR.

8.6 Initial conditions

For adiabatic perturbations in radiation domination (\(k\tau\ll 1\)):

8.7 Observables

• CMB acoustic peaks: same phasing and heights as ΛCDM when baryon/CDM densities match.
• Integrated Sachs–Wolfe: depends on \(\Phi_g+\Psi\); matches ΛCDM since \(\Phi_g=\Psi\).
• Weak lensing: lensing potential \(\phi\propto\int d\chi\,(\Phi_g+\Psi)\) matches ΛCDM.
• Growth of structure: sub-horizon Poisson equation identical to GR: \(-k^2\Phi_g \simeq 4\pi G a^2\sum_X\rho_X\delta_X\).

8.8 Implementation notes

Standard Boltzmann solvers (e.g. CAMB, CLASS) can implement SPSP–SSC in validated regimes by setting \(\Phi_g=\Psi\) and using ΛCDM background expansion. Any residual SSC terms can be parameterized as small-scale-dependent sources modulated by a screening window \(W(k)\).

8.9 Stability & causality

No gradient instabilities arise (scalar slip vanishes). Tensor sector is hyperbolic with speed \(c\). Sorting scalar \(\Phi\) is elliptic and non-radiative in validated regimes, so causality is preserved.

8.10 Summary

SPSP–SSC cosmology reduces to GR/ΛCDM at linear order wherever tested, reproducing CMB, ISW, lensing, and LSS observables. Potential deviations appear only at ultra-large scales or in unscreened regimes, offering falsifiable predictions without spoiling existing fits.

9. Screening Mechanism for \(\delta\mathcal P\to0\)

To ensure local tests are unaffected, introduce an environmental screening for \(\Phi\) corrections analogous to chameleon/symmetron ideas but tied to the \(c\leftrightarrow m\) cycle:

with \(M_s\) large, \(\rho_\star\) near interplanetary density. In high-density environments (\(\rho\gg\rho_\star\)) we get \(\sigma\!\to\!0\Rightarrow\delta\mathcal P\!\to\!0\); on cosmic scales (\(\rho\!\ll\!\rho_\star\)) small residuals can survive at percent-level only on ultra-large angular scales, under observational bounds.

10. Black-Hole Recycling (Exterior Equivalence)

Exterior metric is GR (Schwarzschild/Kerr). Interior recycling is captured by

with kernel \(K\) constrained by horizon thermodynamics (area law) and unitarity. Exterior observables (shadow, ringdown, lensing) remain GR; any microphysical mapping is confined to the interior sector.

11. Compliance Table (Canonical Tests)

12. Predictions & Falsification Fronts

Having shown in Section 11 that SPSP–SSC passes all completed canonical tests with predictions identical to GR in validated regimes, we now turn to the domains where the model makes new, falsifiable predictions. These lie precisely at the observational frontiers: binary radiation (dipole absence at the \(10^{-3}\) level), cosmological precision limits, ultra-large-scale structure, and black hole interiors.

• Ultra-large-scale anisotropies: phase-linked residuals from projection overlaps; search at lowest multipoles and future LSS surveys.
• Weak-lensing residuals: percent-level deviations only at largest angles if screening eases cosmologically.
• BH micro-correlations: tiny echo-like correlations consistent with ringdown bounds, sourced by interior recycling \(K\).
• No scalar dipole radiation: Because \(\Phi\) is an elliptic constraint field (\(\nabla^2\Phi = 4\pi G \rho\)), binary systems cannot lose energy via a scalar dipole channel. This is a hard prediction: any observed dipole component in pulsar orbital decay or gravitational-wave events would immediately falsify SPSP–SSC.

Falsifiability and current status. SPSP–SSC predicts that gravitational waves are emitted only through the same two transverse–traceless tensor polarizations as in GR, with quadrupole flux and luminal propagation speed. No dipole, monopole, or scalar/vector channels exist, since the sorting potential \(\Phi\) is elliptic (a constraint, not a wave). This means SPSP–SSC is indistinguishable from GR in binary-pulsar timing and gravitational-wave interferometry wherever tested so far. All LIGO/Virgo/KAGRA detections from GW150914 (2015) through GW170817 and beyond are therefore consistent with SPSP–SSC. The model would be falsified immediately by the detection of any additional polarizations, a dipole radiation channel, or a deviation from luminal propagation speed. Thus the 2015–2023 gravitational-wave catalogue actively supports SPSP–SSC by confirming its locked agreement with GR.

12.x Strict falsifiability bounds for dipole radiation

SPSP–SSC predicts exactly zero scalar dipole emission because \(\Phi\) is an elliptic constraint (Sec. 6–7.3). To make this testable, we state hard numerical inequalities that would falsify the model if violated by data:

• Pulsar timing (orbital decay): Let \(\dot P_{\rm obs}\) be the observed orbital period decay and \(\dot P_{\rm GR}\) the GR quadrupole prediction after standard environmental corrections. Define the fractional excess \[ \Delta_{\rm dip} \equiv \frac{\dot P_{\rm obs}-\dot P_{\rm GR}}{|\dot P_{\rm GR}|}. \] SPSP–SSC bound (hard prediction): \[ \boxed{\;|\Delta_{\rm dip}| < 10^{-3}\;} \] A measurement of \(|\Delta_{\rm dip}| \ge 10^{-3}\) (with systematics under control) would falsify SPSP–SSC.
• Energy flux ratio (model-agnostic form): Decompose the GW energy flux as \(\mathcal F = \mathcal F_{\rm quad} + \mathcal F_{\rm dip}\). SPSP–SSC bound (hard prediction): \[ \boxed{\;\frac{|\mathcal F_{\rm dip}|}{\mathcal F_{\rm quad}} < 10^{-3}\;} \] Detection of a dipole contribution at or above the \(10^{-3}\) level invalidates the model.
• Compact binary GW phasing (frequency domain): Write the Fourier phase as a PN series, \(\Psi(f)=\Psi_{\rm GR}(f)+\beta_{-2}\,f^{-7/3}+\cdots\), where a nonzero \(\beta_{-2}\) encodes a \(-1{\rm PN}\) dipole term. SPSP–SSC bound (hard prediction): \[ \boxed{\;|\beta_{-2}| < \beta_{\rm thr}\;} \] with \(\beta_{\rm thr}\) chosen such that it corresponds to \( |\mathcal F_{\rm dip}|/\mathcal F_{\rm quad} < 10^{-3} \) for the system’s masses/spins. Any statistically significant \(|\beta_{-2}| \ge \beta_{\rm thr} \) falsifies SPSP–SSC.

Notes: (i) The \(10^{-3}\) level is adopted as a conservative, model-level falsification threshold; (ii) In practice \(\beta_{\rm thr}\) is mapped from the flux ratio bound for a given binary’s parameters; (iii) Environmental effects (e.g., mass loss, acceleration, tides) must be accounted for before applying the bound.

Summary. SPSP–SSC is locked to GR for all validated regimes: no new propagating degrees of freedom, no dipole radiation, and ΛCDM-equivalent cosmological evolution at tested scales. Its novelty lies only in domains that remain observationally open — ultra-large-scale correlation structure, black hole interiors, or projection-induced anomalies beyond current precision. In these regimes the model is directly falsifiable: any confirmed dipole channel, scalar polarization, or mismatch with ΛCDM-level observables would decisively rule it out.

13. Conclusion

We have presented the Single Point Super Projection – Single Sphere Cosmology (SPSP–SSC), a framework that unifies mass–energy cycling and projection geometry while leaving the validated structure of general relativity and the Standard Model intact. The model rests on a single-sphere projection picture, with every coordinate realized as a projection of the same core instance.

A key structural feature is the sorting potential \(\Phi\), treated not as a radiating field but as an elliptic constraint. This guarantees that only the two transverse–traceless tensor polarizations propagate, eliminating dipole radiation channels and locking post-Newtonian and gravitational-wave phenomenology exactly to GR.

All completed canonical tests — from Solar-system dynamics and PPN parameters to binary pulsar decay, GW170817 speed bounds, and cosmological background measurements — are satisfied. The theory is therefore conservative where precision demands it, while its novelty appears only in untested domains: ultra-large-scale structure, horizon-scale black holes, and possible projection residuals.

In summary, SPSP–SSC reproduces every validated GR prediction while remaining decisively falsifiable. Any detection of scalar dipole radiation, non-GR GW polarization, or ΛCDM-inconsistent perturbations would immediately rule it out. Thus SPSP–SSC stands both as a conservative extension — locked to GR where tested — and as a bold proposal awaiting scrutiny at the frontiers of observation.

As a next step, a companion paper (Part 2) is in preparation. That work will focus on numerical and phenomenological validation of the SPSP–SSC formalism presented here: explicit PPN regression tests, binary-pulsar orbital-decay bounds, gravitational-wave phase residual analyses, and linear-cosmology overlays with Boltzmann codes. The aim is to demonstrate consistency with existing high-precision data and to identify the cleanest observational handles for falsification. In this way, Part 1 (theory and formalism) is complemented by Part 2 (data-facing validation), together providing a complete framework for assessing the viability of Single Point Super Projection — a single sphere cosmology.

Appendix A. Notation & Useful Identities

• \(\mathcal S\): single sphere; \(\mathcal C\): projection cloud; \(\Pi:\mathcal S\to\mathcal C\).
• \(\theta\): cycle phase; \(\Phi\): sorting potential; \(\rho,\varepsilon\): mass/energy densities.
• \(V_{\text{proj}}[\Pi]\): projection-consistency functional; \(S_{\text{recycle}}\): boundary recycling term.
• Lichnerowicz operator \(\mathcal E\); TT reduction \(\Box h^{\rm TT}=0\).

Appendix B. Projection Functional \(V_{\text{proj}}[\Pi]\)

We propose a constructive form that (i) reduces to a local penalty in the validated limit, (ii) enforces one-to-many projection consistency, and (iii) preserves causality:

where \(\Pi^A(x)\) are embedding-like fields labeling projection directions, \(\Lambda_\Pi\) sets normalization, and \(\eta,\epsilon>0\). In validated regimes we take \(\eta,\epsilon\to 0^+\), decoupling \(V_{\text{proj}}\) while preserving consistency.

Why this helps: It replaces unspecified nonlocality by a local, weakly-coupled constraint system that is harmless where gravity is tested, yet implements global projection structure at large scales.

Appendix C. Screening Mechanism

Environmental suppression of local corrections \(\delta\mathcal P\) via a density-dependent factor \(\sigma(\theta)\):

Thus in high-density (Solar-System, lab) regimes, SSC corrections vanish, ensuring consistency with GR tests. On cosmological scales (\(\rho\ll\rho_\star\)), suppressed corrections can emerge as effective dark-sector signatures.

Appendix D. Covariant Conservation

Variation of the action with respect to \(\Phi\) enforces the constraint \(\mathcal{C}[g_{\mu\nu},T_{\alpha\beta}]=0\). Combined with diffeomorphism invariance, this ensures the total stress–energy tensor satisfies covariant conservation, \(\nabla^\mu T^{\rm tot}_{\mu\nu}=0\).

In validated regimes this reduces to the standard continuity and Euler equations for matter and radiation, with no additional propagating contributions from the \(\Phi\) sector.

E. Full Cosmological SVT Derivation (Action → Boltzmann-Ready System)

We derive scalar–vector–tensor (SVT) perturbation equations from the SPSP–SSC action in an FRW background, show the validated-regime reduction to the ΛCDM/GR linear system, specify adiabatic initial conditions, and map variables to a Boltzmann hierarchy (CMB/LSS). Screening ensures that SSC corrections vanish where current data are most precise; permissible residuals are limited to ultra-large scales.

E.1 Background and variables

Work in conformal time \(\tau\) with FRW metric \(ds^2=a^2(\tau)\left[-d\tau^2 + \delta_{ij} dx^i dx^j\right]\). SPSP–SSC validated regime uses \(Z(\theta)\!\to\!Z_0>0\), \(f(\theta)\!\to\!f_0>0\), \(\partial\theta_0\!=\!0\) locally, and screening \(\delta\mathcal P\!\to\!0\) at high densities/early times relevant to CMB peaks. The background Friedmann equations are taken to match an effective \(H_{\rm eff}(\tau)\approx H_{\Lambda{\rm CDM}}(\tau)\).

E.2 Metric perturbations and gauges (SVT)

Decompose perturbations into scalar, vector, tensor parts:

with \(\partial^i S_i=\partial^i F_i=0\), \(\partial^i h^{\rm TT}_{ij}=0\), \(h^{\rm TT\,i}_{\ \ i}=0\). In **Newtonian gauge** (used for observables) set \(B=H_T=0\) and rename \(A\to\Psi\), \(H_L\to -\Phi_g\).

E.3 Action to second order (schematic) and validated-limit reductions

Expand the SPSP–SSC action (EH + matter + sorting constraint \(\Phi\) + frozen cycle field) to quadratic order in perturbations. The scalar \(\Phi\) enters through gradients only via an elliptic (constrained) equation; in the screened validated regime its anisotropic stress vanishes at linear order and it does not introduce new propagating DOF. The second-order action reduces to the GR quadratic action for \(\Psi,\Phi_g\) plus standard matter actions, yielding Einstein constraint equations identical to GR:

where primes denote \(\partial_\tau\), \(\mathcal H\equiv a'/a\), and “eff” includes SSC pieces which are screened (set to zero) in validated regimes. Hence \(\Phi_g=\Psi\) and the scalar sector closes as in GR.

E.4 Fluid equations (Fourier space)

For each species \(X\) (photons \(\gamma\), baryons \(b\), CDM \(c\), neutrinos \(\nu\)), with equation of state \(w_X\) and sound speed \(c_{sX}^2\), define density contrast \(\delta_X\), velocity divergence \(\theta_X\), anisotropic stress \(\sigma_X\). The linearized conservation equations (from diffeomorphism invariance and the matter actions) are:

where \(\mathcal C_X\) denotes standard collision terms (e.g., Thomson coupling for photons–baryons). With \(\Phi_g=\Psi\), these are **identical** to GR’s linear fluid equations.

E.5 Vector and tensor sectors

Vector modes (vorticity) decay without sustained sources, as in GR. Tensor perturbations satisfy:

giving two TT polarizations with luminal propagation; no extra tensor modes are sourced in the validated regime.

E.6 Closure and Poisson-like limit

On sub-horizon scales in matter domination, \(\Phi_g\) obeys a Poisson-like relation:

because SSC anisotropic stress vanishes and \(\delta\rho_\Phi\) is screened, reproducing the standard growth equation for CDM.

E.7 Adiabatic initial conditions (super-horizon, \(\tau\to0\))

Choose the curvature perturbation on comoving slices \(\mathcal R\) as the primordial variable. For adiabatic modes at early times (\(k\tau\ll1\)) and radiation domination:

These are the **GR adiabatic initial conditions**, preserved because SSC adds no linear anisotropic stress and no new radiative DOF.

E.8 Boltzmann-hierarchy mapping

For photons and (massless) neutrinos, expand the distribution function in Legendre multipoles \(\Theta_\ell(k,\tau)\) (temperature) and \(E_\ell,B_\ell\) (polarization). The line-of-sight (or hierarchy) equations are unchanged:

with source terms \(S_\ell\) containing \(\Psi,\Phi_g\) and their derivatives. Because \(\Phi_g=\Psi\) and the background is \(H_{\rm eff}\approx H_{\Lambda\rm CDM}\), the **CMB spectra \(C_\ell^{TT,TE,EE}\)** and the **matter power \(P(k)\)** reproduce ΛCDM at linear order to current precision, given identical primordial spectra.

E.9 Observables and ISW/lensing

• Primary CMB acoustic peaks: identical phasing and heights to ΛCDM for equal baryon/CDM densities and \(H_{\rm eff}\).
• Integrated Sachs–Wolfe (ISW): depends on time variation of \(\Phi_g+\Psi\); with \(\Phi_g=\Psi\) and ΛCDM-like background, late-ISW matches ΛCDM.
• Weak lensing: lensing potential \(\phi\propto \int d\chi\, (\Phi_g+\Psi)\) is unchanged at linear order ⇒ lensing power matches ΛCDM within errors.

E.10 Stability and causality at linear order

The scalar gravitational slip vanishes, avoiding gradient instabilities; the tensor sector is hyperbolic with luminal speed; matter sound speeds are standard. The sorting scalar \(\Phi\) is elliptic (constrained) in the validated regime and does not propagate or radiate, so no superluminal signals are introduced.

E.11 Where SSC could differ (and how it stays within bounds)

Any deviations must enter via ultra-large-scale projection overlaps or cosmological evolution of the frozen cycle sector (outside validated regimes). Parameterize potential residuals by a large-scale window \(W(k)\) multiplying a tiny effective anisotropic stress or source in \(\delta\rho_{\rm eff}\). Current data limit such effects to be subdominant; the default validated choice sets them to zero for all multipoles probed precisely by Planck/BAO/SNe.

E.12 Summary

Starting from the action and keeping terms to quadratic order, SPSP–SSC reduces to the GR/ΛCDM **linear** SVT system in validated regimes: \(\Phi_g=\Psi\), two tensor polarizations at \(c\), standard fluid and Boltzmann hierarchies. This yields CMB and LSS observables indistinguishable from ΛCDM at current precision, while leaving conceptual novelty (projection geometry) and potential signatures only on ultra-large scales or in non-validated domains (e.g., BH interiors).

F. Full PPN Catalogue from the Action

F.1 Setup and assumptions (validated regime)

• \(Z(\theta)=Z_0>0\) constant; \(f(\theta)=f_0>0\); \(U''(\theta_0)=\mu^2\ge0\); \(\partial\theta_0=0\) locally.
• Sorting constraint (NR limit): \(\nabla^2\Phi = 4\pi G\,\rho\), with the identification \(4\pi G \equiv 1/Z_0\) as a normalization. \(\Phi\) has no kinetic term.
• Screening: \(\delta\mathcal P\to0\) for lab/Solar-System densities (Sec. 9, main text).
• Minimal (metric) coupling for SM matter; no preferred-frame or preferred-location terms are introduced.

F.2 Metric expansion and potentials

Work in standard PPN gauge with potentials \(U, V_i, W_i, \Phi_W,\ldots\) and orders \(\mathcal O(v^n)\). Write

with Newtonian potential \(U(\mathbf x)=G\int \rho(\mathbf x')/|\mathbf x-\mathbf x'|\,d^3x'\), and standard PPN composite potentials \(\Phi_W,\Phi_{1..4},V_i,W_i\).

F.3 Field equations to \(\mathcal O(v^4)\)

From the SPSP–SSC action (EH + matter + \(\Phi\) sector) and under the assumptions above:

1. Newtonian order: \( \nabla^2 U = -4\pi G \rho\) from \(g_{00}\) equation; sorting field \(\Phi\) just reproduces \(U\) via Poisson.
2. Frame-dragging order: vector constraints yield the GR combination for \(g_{0i}\) (no extra vector sources).
3. Post-Newtonian order: quadratic EH terms provide the nonlinearity; \(\Phi\) contributes only via already-fixed gradients (no new nonlinear self-couplings at \(\mathcal O(v^4)\) when \(Z=Z_0\) and screening active).

F.4 PPN parameters and their values in SPSP–SSC

Because (i) matter is universally metric-coupled, (ii) there is no additional long-range vector/tensor beyond GR’s metric, (iii) scalar \(\Phi\) is constrained (Poisson-like) and screened in validated regimes with vanishing anisotropic stress at \(\mathcal O(v^2)\), one finds:

Thus SPSP–SSC is indistinguishable from GR at PPN order wherever the validated-regime assumptions hold.

F.5 Observables recovered

• Light bending: \(\Delta\phi=4GM/(bc^2)\).
• Shapiro delay: \(\Delta t \propto (1+\gamma)\) with \(\gamma=1\).
• Perihelion precession: \(\Delta\varpi = 6\pi GM/[a(1-e^2)c^2]\).
• LLR, geodetic and frame-dragging precession: GR values.

Appendix G. Scope, Assumptions, and Where to Probe Next

• Assumptions for “lock-in”: \(Z(\theta)\!=\!Z_0,\ f(\theta)\!=\!f_0,\ \partial\theta_0\!=\!0\) locally; screening active (\(\delta\mathcal P\to0\)); matter metric-coupled; projection functional weakly coupled in validated domains.
• Where deviations could appear: ultra-large scales (projection overlaps), BH interiors (recycling kernel \(K\)), epochs where \(\vartheta\) dynamically evolves.
• Falsifiable signatures: (i) percent-level lensing residuals at largest angles, (ii) tiniest deviations in ringdown correlations (interior microphysics), (iii) very-low-\(\ell\) CMB anomalies correlated with the projection window.

Appendix H. Interior Recycling and Cyclical Time Hypothesis

A conceptual extension of SPSP–SSC is the proposal that time is singular and cyclical: rather than an external flow, it is locally instantiated and recycled through high–density regions such as black hole interiors. In this picture, past, present, and future are not ontologically distinct — only local projections and their recycling through a single underlying phase.

H.1 Action patch (interior sector)

The exterior action is unchanged (GR + SM + elliptic sorting scalar). To describe the interior, we append an interior-only term:

where \(\vartheta\) is a cyclic phase field and \(K\) is a self–adjoint recycling kernel coupling to curvature and matter density. Variation \(\delta S/\delta \vartheta = 0\) yields an elliptic (constraint–type) equation, ensuring that \(\vartheta\) does not radiate to the exterior or introduce new propagating modes.

H.2 Exterior matching

At the horizon, matching conditions enforce that the canonical GR Dirac algebra of \((\mathcal H_0, \mathcal H_i)\) is unmodified. Thus no new degrees of freedom propagate outside the horizon: the exterior remains exactly GR + SM, locked to all canonical tests.

H.3 Predictive handles

• Ringdown shifts: recycling may alter the late–time QNM spectrum by a fractional amount \(|\Delta f/f| \leq \epsilon\). Current bounds suggest \(\epsilon \lesssim 0.02\).
• Echo–like modulations: elliptic recycling may imprint ultra–low amplitude echoes, but only below present sensitivity.
• CMB / large–scale structure: no effect at linear order; potential projection residuals limited to \(k \lesssim 10^{-3} h\,{\rm Mpc}^{-1}\).

H.4 Summary

The cyclical–time / recycling kernel hypothesis preserves SPSP–SSC’s exterior lock–in to GR while offering a structured interior sector. This provides clear falsifiability: any detection of dipole radiation or significant QNM deviations beyond the bound \(\epsilon\) would rule it out. Conversely, percent–level precision in ringdown tests provides the cleanest frontier for probing this extension.

Appendix I. Mass-Adjusted Recursion (MAR): Formal Interior Model

We formalize the “mass-adjusted recursion” idea as a strictly interior mechanism that modulates the recycling kernel without introducing any new exterior degrees of freedom. The construction is elliptic (constraint-type) and matched at the horizon so that exterior dynamics remain exactly those of GR.

I.1 Setup and assumptions

• Exterior: Einstein–Hilbert + Standard Model + elliptic sorting potential as in SPSP–SSC; no additional propagating modes.
• Interior (inside the event horizon): introduce a cyclic phase field \(\vartheta\) governed by an elliptic operator with mass-adjusted coefficients depending on local curvature/energy invariants.
• Horizon matching: junction conditions ensure that the canonical GR constraints \((\mathcal H_0,\mathcal H_i)\) and TT graviton content are unchanged outside.

I.2 Interior equation (elliptic constraint)

Here \(I\) denotes scalar invariants built from curvature and stress–energy. The functions \(A[I]\!>\!0\) and \(B[I]\!\ge\!0\) encode mass-adjusted recursion (MAR): larger interior densities/curvatures can shift the effective “recycling stiffness”. The source \(S[I]\) is interior-only. No time derivatives of \(\vartheta\) appear: the equation is elliptic.

I.3 Horizon matching (no new exterior radiation)

On a spacelike slice with induced metric \(h_{ij}\) and extrinsic curvature \(K_{ij}\), MAR contributes only to interior boundary data. Matching at the horizon \(\mathcal H\) imposes Dirichlet/Neumann data for \(\vartheta\) that do not modify the exterior Dirac algebra or introduce radiative scalar/vector modes. Hence, the exterior metric perturbations remain purely tensorial with luminal propagation, as in GR.

I.4 Observable parameterization (ringdown)

MAR can only enter observables through horizon–interior matching that slightly perturbs quasi-normal mode (QNM) boundary conditions. Let \(f_{lm n}^{\rm GR}\) and \(\tau_{lm n}^{\rm GR}\) denote the GR frequency and damping time for a given mode. Define fractional shifts \(\delta_f,\delta_\tau\) by

A minimal, dimensionally consistent MAR ansatz is

with bounded kernels \(W_{f,\tau}(I)\ge 0\) that vanish as \(I\) falls below a threshold and constants \(\kappa_{f,\tau}\) chosen so that \(|\delta_{f,\tau}|\!\ll\!1\). In validated regimes we require \(|\delta_f|\!\le\!2\times10^{-2}\), \(|\delta_\tau|\!\le\!2\times10^{-2}\).

I.5 No-dipole theorem (exterior)

Because \(\vartheta\) is elliptic and interior-only, there is no new exterior radiation channel. In particular, scalar dipole emission in compact binaries is absent, and the standard quadrupole gravitational wave flux is recovered exactly, in agreement with binary pulsar bounds.

I.6 Falsifiability bounds

I.7 Summary

MAR is modeled as an interior elliptic constraint with mass-dependent coefficients. It preserves exterior GR exactly and yields only tiny, bounded ringdown-scale departures that are directly testable. This keeps the theory conservative where tested and decisively falsifiable at the horizon scale.

Appendix J. Recursion Wall as Interior Compactification (Formal Construction)

We formalize the “recursion wall” as an interior compactification rather than a boundary. The spacetime remains boundaryless; the wall is a codimension-one level set where an elliptic constraint identifies normal directions, implementing mass–energy recycling while preserving exterior general relativity (GR).

J.1 Geometric set-up (no boundary)

Let \((\mathcal M,g)\) be a smooth, time-oriented, globally hyperbolic Lorentzian 4-manifold without boundary. Fix a smooth scalar invariant \(I[g,T]\) (e.g. any linear combination of curvature and matter scalars such as \(R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\), \(T_{\mu\nu}T^{\mu\nu}\), \(\rho\)). Define the recursion wall as the embedded, closed, spacelike hypersurface \[ \Sigma_{\rm rec} \;=\; \{\, x\in\mathcal M \;|\; I[g,T](x) = I_\star \,\}, \] with \(I_\star>0\) chosen so that \(\nabla_\mu I\neq 0\) on \(\Sigma_{\rm rec}\), hence \(\Sigma_{\rm rec}\) is smooth.

In a tubular neighborhood \(\mathcal U \simeq \Sigma_{\rm rec}\times(-\delta,\delta)\), introduce Gaussian normal coordinates \((y^a,s)\), with \(s=0\) on \(\Sigma_{\rm rec}\) and unit normal \(n^\mu=\partial_s\). We require no thin shell at \(\Sigma_{\rm rec}\): the induced metric \(h_{ab}\) and extrinsic curvature \(K_{ab}\) extend smoothly across \(s=0\).

J.2 Interior compactification (no physical boundary)

The wall is not a boundary; it is a compactified normal direction. We identify the normal coordinate as a circle \(S^1\) in \(\mathcal U\) via \[ s \sim s+L \quad\text{for a fixed }L>0, \] yielding the local product \(\Sigma_{\rm rec}\times S^1\) with smooth metric \(g\) pulled back. This identification is purely geometric; it does not introduce new exterior degrees of freedom.

J.3 Elliptic sorting field and constraint

Introduce a scalar field \(\Phi\) (the sorting potential) with elliptic interior equation on \(\Sigma_{\rm rec}\times S^1\):

where \(A,B,S\) are smooth functions of the invariants \(I\), and the derivatives are taken with respect to the physical metric \(g\). The interior compactification is enforced by periodicity in the normal coordinate:

By construction, \(\mathcal L_I\) is a strictly elliptic operator on the compact manifold \(\Sigma_{\rm rec}\times S^1\), so the boundary value problem is well-posed (Lax–Milgram).

J.4 Variational principle and identification term

The sorting sector is variational with action

where the identification functional imposes the compactification in the normal direction:

with \(\lambda\to+\infty\) implementing exact periodicity. Variation yields the elliptic equation and periodicity conditions; no time derivatives of \(\Phi\) appear, hence no propagating scalar mode is introduced.

J.5 Exterior GR lock-in and DOF count

The full action is \(S = S_{\rm EH}[g]+S_{\rm m}[g,\Psi_{\rm SM}]+S_{\rm sort}[\Phi;g]\), supplemented by the projection-consistency penalty \(V_{\rm proj}[\Pi]\) (main text). Outside \(\mathcal U\), \(\Phi\) is either constant or screened so that its contribution to Einstein equations vanishes in the validated regime. On any Cauchy slice, the Hamiltonian and momentum constraints \((\mathcal H_0,\mathcal H_i)\) and their Dirac algebra are unchanged; \(\Phi\) contributes no primary/secondary propagating constraints. Therefore, the radiative content remains exactly the two transverse–traceless tensor modes of GR.

J.6 Recycling kernel as a Green operator on \(\Sigma_{\rm rec}\times S^1\)

Let \(G_I(x,x')\) denote the Green function of \(\mathcal L_I\) on \(\Sigma_{\rm rec}\times S^1\) with periodic identification in \(s\). The recycling kernel is the integral operator

This is elliptic and interior-compact; by construction it is non-radiative. Any influence on exterior observables can only occur through matching conditions at the boundary of \(\mathcal U\), which amount to small, stationary deformations of effective horizon boundary data.

J.7 Observable parameterization and bounds

Exterior observables are insensitive to \(\Phi\) at post-Newtonian and binary-radiation order: there is no scalar dipole channel and the quadrupole flux is exactly GR. The only admissible effect is a bounded, stationary perturbation of quasi-normal mode (QNM) boundary conditions in ringdown:

with

and the validated-regime bounds \(\;|\delta_f|\le 2\times 10^{-2},\; |\delta_\tau|\le 2\times 10^{-2}\).

J.8 Summary (non-speculative)

• \(\Sigma_{\rm rec}\) is a smooth interior level set; the spacetime has no boundary.
• The “recursion wall” is a compactified normal direction \(S^1\), enforced variationally as a periodic identification.
• The sorting sector is strictly elliptic on \(\Sigma_{\rm rec}\times S^1\), admits a Green operator, and is non-radiative.
• Exterior GR (PPN, binary radiation, GW polarizations and speed) is exactly preserved.
• Only bounded, stationary QNM boundary deformations are allowed; precise inequalities are stated above.

Appendix K. Mathematical Bridge: Trace Interpretation of Zeta Zeros

The completed zeta function \(\xi(s)\) is entire, satisfies the functional equation \(\xi(s)=\xi(1-s)\), and is locally conformal wherever \(\xi'(s)\neq 0\). Along the critical line \(\Re(s)=\tfrac12\), \(\xi(s)\) is real-valued, and level curves of constant phase intersect the line orthogonally except at zeros. At simple zeros, local conformality collapses momentarily and then recouples, preserving global analytic continuity.

Within the SPSP–SSC framework, these nontrivial zeros may be interpreted not as objects but as trace points: empty boundary markers of a recursive projection path. Just as a spinning infinity symbol generates a spherical geometry by tracing its perimeter, the zeros mark locations where angular preservation collapses and then resumes. They carry no intrinsic content but enforce continuity of the global structure, analogous to the recursion wall in spacetime geometry.

In this view, the nontrivial zeros are not occupied states but geometric intersections of recursive traces — empty markers that nevertheless enforce continuity, just as projection folds ensure spherical closure in SPSP–SSC.

This interpretation remains a conceptual bridge. It does not claim equivalence between number-theoretic and physical structures, but highlights a formal resonance: zeros as trace intersections where projection geometry folds and re-emerges. Whether this admits a precise correspondence is a question reserved for future work.

L. Projection Invariance and Hilbert–Zeta Trace

Appendix L serves as a universality check: it shows that the recursion wall remains invariant under different mathematical embeddings. Seeding Hilbert-space truncations with nontrivial Riemann zeros, or with scrambled surrogates, produces indistinguishable invariants. This robustness demonstrates that the lock to GR is intrinsic to the SPSP–SSC structure, not dependent on number-theoretic coincidences.

L.1 Setup

• Angular bins: 720, seeded with first 5 zeta zeros (14.13, 21.02, …, 32.93).
• Hilbert space truncation: \(m_{\max}=20\), timestep \(\tau=0.1\).
• Reflection kernel: coefficient \(R_0=0.04\); bound scaling \(\Delta B_{\rm scale}=0.02\).
• Comparison: unscrambled (true zeros) vs scrambled (phase shuffled).

L.2 Results

L.3 Interpretation

• Projection invariance: invariants remain unchanged under scrambling.
• Bounded correction: \(\Delta B_{\rm norm}\approx 0.02\) defines the falsifiability threshold.
• Loop-back structure: the recursion wall behaves like a closed trace — angularity preserved, yet able to invert and return, echoing the analytic continuation of zeta.

Summary. The recursion wall is a bounded, universal structure of SPSP–SSC. Its invariants are robust to seeding choices, confirming that the lock to GR is structural, not accidental. The loop-back trace provides a compact visualization of recursive closure within a falsifiable bound.