Spacetime as Source-Local Rollout and the Conditional Emergence of Three-Dimensional Effective Geometry in The Emergent Frame

Research Context

Paper VIII develops a conditional mathematical construction for the spacetime-rollout framework introduced in Paper VII. It draws on the helical geometry of Paper I, but its subject is the collective organization of a source’s rollout trajectories, rather than another numerical correspondence involving the helix parameter.

The question is whether a source-local relational construction can support three-dimensional volume growth, normal diffusion, and an inverse-distance static response. The paper separates those conditional results from the deeper problem of deriving the microscopic organization itself.

From a Rollout Ensemble to Effective Geometry

The construction represents relations among neighboring rollout trajectories by a weighted transverse cell complex ${\mathcal{K}}_{\perp }$. Its uniform quadratic volume growth is a structural postulate. The two-dimensional normal plane of one isolated helix does not establish that collective growth law.

In the exact homogeneous model, transport preserves the transverse complex’s incidence relations, edge scales, and measures. Relabeling each section by inverse cumulative transport gives the product

$$\mathcal{K}\cong {\mathbb{Z}}_{\mathrm{cell}}\times {\mathcal{K}}_{\perp }.$$

The longitudinal index describes spatial rollout depth, not physical evolution time. With quadratic transverse growth and the stated uniform bounds, the product has

$$\mu (B(r))\asymp r^3,d_f=3.$$

This establishes a three-dimensional volume-growth class under the postulates. It does not prove that the transverse organization must uniquely have dimension two, or recover a complete relativistic spacetime theory.

Diffusion and the Bulk Green Scale

Cubic volume growth alone does not determine diffusion. The paper additionally specifies a reversible local graph walk, uniform control of geometry and conductances, volume doubling, a scale-invariant Poincaré inequality, and the required diffusive and short-time bounds.

For the exact product generator, these conditions yield normal long-scale diffusion with $d_w=2$. Integrating the heat kernel on the infinite homogeneous bulk gives

$$G(X,Y)={\int }_0^{\infty }{p}_t(X,Y)\mathrm{d}t\asymp \frac{1}{d(X,Y)}.$$

Here $\asymp$ denotes two-sided comparison up to positive constants at large separation. It does not specify an exact universal coefficient. A finite source with a core or rollout front requires its own boundary conditions; its global Green function is not derived here.

Physical Interpretation and Open Problems

Hydrogenic $1/n^2$ gross scaling is retained as a restricted low-energy consistency check. It is not used to prove the geometry or Green scaling, and the construction does not derive electromagnetic normalization, $\hslash$, or the full hydrogen spectrum.

The paper also develops a gauge-covariant language for coherence among rollout trajectories. Such coherence is a candidate excitation mechanism; localization, finite energy, and stability still require an explicit dynamics. The proposed source-local cosmogenesis is an interpretation of ongoing rollout, not a quantitative model of cosmological expansion. How many source-local structures compose into one shared effective world remains unresolved.

Publication Status

Version 3.12 is publicly archived on Zenodo as a preprint under the Creative Commons Attribution 4.0 International license. Its version-specific DOI is 10.5281/zenodo.22776137; the concept DOI for the complete version history is 10.5281/zenodo.22776136. The manuscript has not undergone peer review.

The authoritative PDF, LaTeX source, citation and provenance metadata, checksums, and verification code are preserved in the frozen GitHub release. The standard-library Python checks cover the exact shell-sum identity, reported scaling exponents, and sample gauge-covariance relations. They check those relations rather than establish the physical postulates or all analytic hypotheses. Manuscript, source, and metadata use CC BY 4.0; verification code uses the MIT license.

Citation

Wu, X. (2026). Spacetime as Source-Local Rollout and the Conditional Emergence of Three-Dimensional Effective Geometry in The Emergent Frame (Version 3.12). Zenodo. https://doi.org/10.5281/zenodo.22776137

@misc{wu2026sourcelocalrollout,
  author = {Wu, Xiaodan},
  title = {Spacetime as Source-Local Rollout and the Conditional Emergence of Three-Dimensional Effective Geometry in The Emergent Frame},
  year = {2026},
  publisher = {Zenodo},
  version = {3.12},
  doi = {10.5281/zenodo.22776137},
  url = {https://doi.org/10.5281/zenodo.22776137}
}

Version History

v3.12 - September 16, 2026

Public preprint release on Zenodo, with companion GitHub release TEF-2026-008-v3.12.

@unpublished{wu2026source,
  author = {Xiaodan Wu},
  title = {Spacetime as Source-Local Rollout and the Conditional Emergence of Three-Dimensional Effective Geometry in The Emergent Frame},
  year = {2026},
  note = {TEF-2026-008, version v3.12},
  url = {https://theemergentframe.org/papers/source-local-rollout-effective-geometry/},
  doi = {10.5281/zenodo.22776137},
}