From Geometry to Spacetime

 

From Geometry to Spacetime

(Constraint–Transport Foundations of Emergent Lorentzian Structure)

PART I — PRE-GEOMETRY

1. Distinction Before Structure

  • Binary separation as primitive

  • Identity vs admissibility

  • Boundary without space

  • Constraint as first invariant

2. Relation Without Embedding

  • Graphs, orders, compositional systems

  • Adjacency without metric

  • Depth vs distance

  • Transport without coordinates

3. Constraint Closure

  • Admissibility predicates

  • Stability under recursion

  • Collapse as spectral contraction

  • Persistence as invariant intersection

4. Transport as Primitive

  • Compositional update rules

  • Bounded accessibility

  • Finite propagation without metric

  • Hyperbolicity as structural condition

PART II — TOPOLOGY EMERGES

5. Stable Neighborhoods from Transport

  • Equivalence classes under admissibility

  • Persistent local structure

  • Emergent separation axioms

  • When topology is induced

6. Cost Functionals and Ordering

  • Relaxation cost

  • Minimal admissible paths

  • Path composition

  • Depth → ordering → geometry precursor

7. Metric as Relaxation Hessian

  • Second variation of transport cost

  • Norm induction

  • Light-cone as maximal propagation eigen-surface

  • Curvature as resistance gradient

PART III — HYPERBOLIC STRUCTURE

8. Index-1 Transport and Causal Cones

  • Principal symbol

  • Signature classification

  • Stability of oscillatory sector

  • Hyperbolicity without Hilbert presupposition

9. Spectral Positivity

  • Hamiltonian positivity cone

  • Eigenvalue stability classification

  • Collapse threshold

  • Transport viability condition

10. Coherence Length and Stiffness

  • Mass–scale ↔ correlation mapping

  • UV suppression

  • Finite propagation scale

  • Surface-tension analogy formalized

PART IV — FROM TRANSPORT TO SPACETIME

11. Emergent Lorentz Structure

  • Principal symbol → conformal metric

  • Positivity → complex structure

  • Compatible symplectic form

  • Lorentzian metric reconstruction

12. When Spacetime Exists

  • Global hyperbolicity condition

  • Ricci eigenvalue bound

  • Stability domain

  • Signature persistence

13. When Spacetime Fails

  • Eigenvalue crossing

  • Hyperbolic breakdown

  • Tachyonic transport

  • Collapse regimes

PART V — PROCA AND THE TRANSPORT VIABILITY PRINCIPLE

14. Massive Vector as Spectral Anchor

  • Curved Proca equation

  • Ricci coupling

  • Weyl neutrality

  • Stiffness interpretation

15. Transport Viability Principle (TVP)

  • ( m^2 > \lambda_{\max}(R_{\mu\nu}) )

  • Oscillatory stability domain

  • Collapse threshold

  • Ricci vs Weyl distinction

16. Dark Sector as Stiffness Residue

  • Proca as cold dark matter

  • Density–curvature coupling

  • ( m^2 > 8\pi G\rho )

  • Cosmological stability window

PART VI — COSMOLOGICAL RELAXATION

17. Pre-Metric Epoch

  • High-curvature regime

  • No stable oscillatory sector

  • Transport without geometry

18. Emergence Time

  • ( t_c \sim 1/m )

  • Curvature descent

  • Stabilization of positivity cone

  • Onset of metric regime

19. Large-Scale Structure as Anisotropic Relaxation

  • Basin–sheet–filament formation

  • Eigenvalue spread

  • Void under-tension regime

  • SMBH as local stiffness saturation

PART VII — SYMMETRY AS PLATEAU

20. Symmetry from Degeneracy

  • Invariance as spectral degeneracy

  • Lorentz as isotropic limit

  • Energy as gradient norm

  • Noether as low-curvature approximation

21. Renormalization as Boundary Rescaling

  • Divergence as representation artifact

  • Coarse-graining as constraint flow

  • Multi-scale harmonic closure

PART VIII — COLLAPSE AND LIMITS

22. Collapse Geometry

  • Eigenvalue crossing

  • Dimensional contraction

  • Kernel persistence

  • Thin manifold fragility

23. Irreversible Commitment Boundary

  • Admissible option contraction

  • Post-collapse regimes

  • Silence vs articulation

  • Structural saturation

24. Complete Signature Loss

  • Global admissibility failure

  • Oscillatory annihilation

  • No-metric universes

PART IX — FINAL STRUCTURE

25. Minimal Conditions for Spacetime

  • Index-1 hyperbolicity

  • Spectral positivity

  • Finite propagation

  • Bounded curvature eigenvalues

26. What Is Fundamental

  • Distinction

  • Constraint

  • Transport

  • Spectral stability

27. What Is Emergent

  • Topology

  • Metric

  • Lorentz symmetry

  • Spacetime itself

Structural Arc of the Book

Distinction
→ Relation
→ Constraint
→ Transport
→ Stability
→ Topology
→ Metric
→ Hyperbolicity
→ Spectral positivity
→ Emergent Lorentzian spacetime

with Proca stiffness providing the stability bound.

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