This inline text flowchart outlines the geography of the Single Point Super Projection – Single Sphere Cosmology (SPSP–SSC) model, from the single sphere to validated physics (GR, SM, QM) and falsifiable frontiers, culminating in the universe. Each step includes inline LaTeX equations with labels, descriptive junctions for transitions, and output examples illustrating physical manifestations (e.g., particles, planets, galaxies). The progression aligns with validated physics and testable predictions (part5.html).
1. Single Sphere (\(\mathcal{S}\))
\(|\psi\rangle = \cos(\theta/2) |0\rangle + e^{i\phi} \sin(\theta/2) |1\rangle\) (Qubit State)
This results in quantum states defining fundamental particles, such as quarks and electrons, at the Planck scale.
The single sphere \(\mathcal{S}\), with cycle phase \(\theta \in S^1\), serves as the geometric seed. It projects its quantum state onto a spacetime point-cloud \(\mathcal{C}\) via the projection map \(\Pi\), initiating the framework for physical interactions across scales (part5.html, Sec. 5.1).
2. Projection (\(\Pi\))
\(V_{\text{proj}}[\Pi] = \int_{\mathcal{S}} \Pi^* \left( \frac{M_P^2}{2} R + \mathcal{L}_{\rm SM} - \Phi (\rho - \varepsilon) \right) d^4x\) (Projection Functional)
This results in a spacetime framework for physical interactions, enabling the emergence of localized phenomena like particle trajectories and gravitational fields.
The projection \(\Pi: \mathcal{S} \to \mathcal{C}\) maps the single sphere to a point-cloud \(\mathcal{C}\), with local physics arising as pulls on \(\mathcal{C}\). A sorting potential \(\Phi\) constrains this projection to ensure scale-invariant consistency (part5.html, Sec. 5.1).
3. Sorting Potential (\(\Phi\))
\(\nabla^2 \Phi = 4\pi G \rho\), \(\rho - \varepsilon = 0\) (Elliptic Constraint, Mass-Energy Balance)
This results in a balanced framework for gravity and matter, ensuring energy conservation across quantum and cosmological scales.
The elliptic sorting potential \(\Phi\) enforces a non-radiative mass-energy balance, shaping the projection framework. This constraint locks the model to validated physics, ensuring consistency with General Relativity, the Standard Model, and Quantum Mechanics (part5.html, Sec. 5.1).
4. General Relativity
\(R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} = 8\pi G T_{\mu\nu}\) (Einstein Field Equations)
This results in gravitational effects like planetary orbits, light bending around massive objects, and time dilation near black holes.
The elliptic constraint ensures the model locks to General Relativity, reproducing standard predictions such as PPN parameters, Shapiro delay, perihelion precession, and gravitational waves with only two transverse-traceless modes (part5.html, Sec. 5.4).
5. Standard Model
\(\mathcal{L}_{\rm SM}(\psi, g)\) (SM Lagrangian)
This results in particle interactions, such as electromagnetic forces between charged particles and nuclear reactions in stars.
The Standard Model remains intact with universal metric coupling and no new particles. The projection and elliptic constraint preserve local lab physics, ensuring SM predictions for particle interactions are unchanged (part5.html, Sec. 5.2).
6. Quantum Embedding
\(P \rho P = \rho\) (Superselection Rule)
This results in quantum behaviors like superposition, entanglement, and wave-particle duality in systems like atoms and qubits.
Quantum mechanics is embedded via a Hilbert space, Born rule, and unitarity, with the elliptic projector ensuring no extra quanta. This locks the model to standard QM, enabling transitions to falsifiable frontiers (part5.html, Sec. 5.5).
7. Cosmology
\(W(k) = A \exp[-(k/k_0)^2]\) (Density Perturbations)
This results in large-scale structures like galaxies, galaxy clusters, and cosmic filaments, forming the cosmic web.
The locked validated physics extends to cosmology, matching \(\Lambda\)CDM with scalar, vector, tensor perturbations and background density residuals. Ultra-large-scale projection residuals are testable with DESI and SKA, probing deviations from standard cosmology (part5.html, Sec. 5.10).
8. Black Hole Interiors
\(S_{\text{ent}, n} = \frac{A_{\text{proj}, n}}{4 l_P^2}\) (Holographic Entropy)
This results in black hole structures with nested cosmologies, influencing their internal dynamics and horizon properties.
Black hole interiors feature a recycling kernel, with exterior GR intact. Echo-like diagnostics and quasi-normal mode tests probe deviations, building on the validated framework, testable with LISA (part5.html, Sec. 5.8).
9. Gravitational Radiation
\(|\Delta_{\text{dip}}| < 10^{-3}\), \(|F_{\text{dip}}/F_{\text{quad}}| < 10^{-3}\), \(\beta_{-2} \approx 0\) (Dipole Constraint, Quadrupole Dominance, No Scalar Mode)
This results in gravitational waves from massive objects like binary black hole or neutron star mergers, detectable by observatories.
Gravitational radiation predictions, with no dipole channel and only two transverse-traceless modes, provide falsifiable tests. These constraints, derived from the elliptic potential, are verifiable with LISA, probing deviations from GR (part5.html, Sec. 5.10).
10. Universe
\(\Pi_{\text{universe}}: \mathcal{S} \to \mathcal{C}\) (Unified Projection)
This results in the known universe, from particles to planets, stars, galaxies, and cosmic structures, unified across scales.
The falsifiable frontiers and validated physics unify into the known universe, represented as a projection field \(\mathcal{C}\). The linear map \(\Pi_{\text{universe}}\) integrates all steps, from quantum particles to cosmological structures, governed by the elliptic constraint and action (part5.html, Sec. 5.1, 5.8).
Start at the Single Sphere (\(\mathcal{S}\)) with \(|\psi\rangle = \cos(\theta/2) |0\rangle + e^{i\phi} \sin(\theta/2) |1\rangle\) (Qubit State), resulting in quantum states for particles. The projection \(\Pi: \mathcal{S} \to \mathcal{C}\) with \(V_{\text{proj}}[\Pi]\) (Projection Functional) creates a spacetime framework. The elliptic constraint \(\nabla^2 \Phi = 4\pi G \rho\), \(\rho - \varepsilon = 0\) (Elliptic Constraint, Mass-Energy Balance) ensures balance, locking validated physics: GR with \(R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} = 8\pi G T_{\mu\nu}\) (Einstein Field Equations) for orbits and light bending, SM with \(\mathcal{L}_{\rm SM}(\psi, g)\) (SM Lagrangian) for particle interactions, and QM with \(P \rho P = \rho\) (Superselection Rule) for quantum behaviors. Falsifiable frontiers include cosmology with \(W(k)\) (Density Perturbations) for galaxies, black holes with \(S_{\text{ent}, n}\) (Holographic Entropy) for nested structures, and gravitational radiation with \(|\Delta_{\text{dip}}| < 10^{-3}\) (Dipole Constraint) for waves. This culminates in the universe via \(\Pi_{\text{universe}}\) (Unified Projection), forming the known universe from particles to planets.