Order, Event Propagation, and the Conditional Relativistic Spacetime Bridge: From Unitary Order to Spectral Dimension, Lorentz/Dirac Structure, Localized Relativistic Mass, Universal Coupling, and the Spin‑2 Gravity Gate

This paper formulates the cleaned spacetime sector of the Canvas programme at full mathematical depth. Finite internal recurrence is separated from physical spacetime: unbounded Order supplies dynamical evolution, while physical space is assigned to propagation relations among activated events.

The Logical Architecture

The cleaned architecture is:

· Finite recurrence → internal/configurational geometry
· Unbounded Order → dynamical evolution
· Activated-event propagation → candidate physical space

Throughout, "derived" means mathematical consequence of stated premises; "conditional" means exact after an unresolved physical premise; "constitutive" means presently supplied; and "no-go" means a stronger candidate claim fails.

Order and Unitary Evolution

Strongly continuous homogeneous norm-preserving composition gives, by Stone's theorem:

U(t)=e^{-itH}, \qquad i\partial_t\psi = H\psi.

The first-order law is therefore derived once continuous homogeneous norm-preserving Order evolution is supplied. The existence of unbounded Order itself is not derived from finite recurrence.

Spectral Dimension and the Dimension Criterion

For a homogeneous event network, infrared dimension is controlled by the Hessian rank of the propagation dispersion:

\lambda(k)=\lambda_0+\frac12 k_i A_{ij}k_j+O(|k|^4), \qquad d_{\rm IR}=\rank A.

Independently, the heat-kernel return probability gives the spectral dimension:

d_s=-2\frac{d\log P(s)}{d\log s}.

Three-space therefore requires a derived kernel with three gapless quadratic directions and P(s)\sim s^{-3/2}. This criterion is derived, while d=3 itself remains upstream. Conditional on rank three, isotropy, hyperbolicity, and a universal limiting speed, the continuum has Lorentzian principal symbol.

Dirac Factorization and the Clifford Algebra

The first-order factorization of the Lorentzian quadratic form forces:

\{\gamma^\mu,\gamma^\nu\}=2\eta^{\mu\nu}I,

giving the Dirac/Spin(1,3) structure required by the matter sector:

(i\hbar\gamma^\mu\partial_\mu-mc)\psi=0.

Localized Relativistic Mass and the Equivalence Principle

For any finite-energy localized solution of a Lorentz-invariant continuum theory, Lorentz covariance fixes the leading collective-coordinate action to:

S_{\rm eff}=-E_0\int dt\sqrt{1-v^2/c^2},

hence M_i=E_0/c^2. If the emergent metric couples universally to the same conserved stress tensor, weak-field gravitational mass satisfies M_g=E_0/c^2, so M_g=M_i. This is exact within its premises; it does not derive those premises from the voxel kernel.

The Gravity Gate

Universal coupling is distinguished from a derivation of general relativity. Pure massless spin-2 mode content, gauge redundancy, nonlinear constraints, and the continuum realization of diffeomorphism invariance remain separate gates. A microscopic theory claiming emergent GR must recover the appropriate continuum gauge-equivalence and constraint structure. Graph relabeling is not diffeomorphism invariance: vertex relabeling removes dependence on names assigned to discrete events, but continuum diffeomorphism invariance is implemented dynamically by first-class constraints with a local algebra.

No-Go Results and Obstructions

· Finite voxel support does not imply spin geometry: a discrete graph can provide adjacency and a positive Laplacian, but not automatically a Lorentzian signature, vierbein, spin structure, spin connection, continuum Clifford bundle, or exact Lorentz covariance.
· The naive lattice Dirac operator has fermion doubling; Nielsen–Ninomiya obstructs a broad class of local translationally invariant chiral lattice discretizations from producing one isolated chiral species under the standard assumptions. Overlap/Ginsparg–Wilson constructions are not forced by finite polynomial shifts.
· Finite voxels do not imply finite information capacity: a spatially discrete site can carry an infinite-dimensional oscillator Hilbert space. An area-law entropy bound cannot be derived merely by counting voxels unless local capacity and constraints are independently established.
· Universal metric coupling does not imply general relativity: a scalar–tensor theory can still couple matter universally to the metric while propagating an additional scalar.

The Complete Derivation Ladder

The spacetime programme can now be written:

\text{Order}\to U(t)\to H,

\text{event network}\to K\to\lambda(k)\to\rank(\nabla\nabla\lambda),

d_{\rm IR}=3+\text{isotropy}+\text{hyperbolicity}\to\eta_{\mu\nu}\to Cl(1,3)\to Spin(1,3),

\text{deformation kernel}\to P^{(2)}\text{ pole}\to\text{universal }T_{\mu\nu}\text{ coupling}\to\text{nonlinear constraints}.

Only some arrows are presently derived. The microscopic propagation/deformation kernel is the principal missing object.

Status Ledger

The paper provides a comprehensive status ledger with 24 items classified as retained, conditional, derived, open, constitutive, or no-go. Key entries include:

· Three spatial dimensions: open/constitutive (requires rank-three Hessian and d_s=3)
· Dirac factorization: derived conditionally (gives Cl(1,3))
· Voxel \Rightarrow spin geometry: no-go (discreteness alone is insufficient)
· Universal coupling \Rightarrow GR: no-go (extra modes can remain)
· Graph relabeling \Rightarrow diffeomorphisms: no-go (constraint algebra not implied)

Why This Matters

This paper does not claim that finite recurrence already is spacetime. Unbounded Order supports unitary evolution; physical space must emerge from event propagation. A derived kernel with exactly three gapless quadratic directions would explain three-dimensional infrared propagation and must independently satisfy P(s)\sim s^{-3/2}. Isotropy and hyperbolicity then provide the conditional Lorentzian bridge, whose first-order factorization supplies the spacetime Clifford/Dirac algebra.

The defensible endpoint is a mathematically explicit conditional spacetime bridge, not a completed microscopic derivation of spacetime. The propagation/deformation kernel K is the central object on which closure depends.

Keywords: canvas model, spacetime emergence, Stone's theorem, spectral dimension, Dirac equation, Clifford algebra, equivalence principle, universal coupling, massless spin-2, general relativity, no-go theorems, fermion doubling, diffeomorphism invariance, conditional bridge