An Electron-Like Charged-Textured Excitation with Spinorial Configuration Topology in The Emergent Frame

Research Context

Paper IX developed an effective rollout connection with a low-energy sector conditionally compatible with classical Maxwell–Lorentz electrodynamics. It left the charged matter sector unresolved. Paper X asks whether one rollout-based effective state can jointly support localization, electric charge, finite energy, internal texture, and configuration-space topology compatible with a spinorial sector.

The candidate is not a point moving on one rollout helix. It is a localized coherent charged resonance distributed across neighboring helices on an already-realized rollout substrate, coupled to an independent degree-one rollout-frame texture. The distinction between rollout sources and excitations follows the framework established in Paper VII.

The result is an effective compatibility construction, not a first-principles derivation of the observed electron.

Unified Effective Model

The dynamical fields are a complex collective field $z$, a canonically normalized electromagnetic connection $A_{\mu }$, and an internal rollout-frame order parameter $\mathcal{R}\in SO(3)$:

$${\mathcal{E}}_{\mathrm{dyn}}=\{z,A_{\mu },\mathcal{R}\}.$$

The charged collective mode uses

$$D_{\mu }z=({\partial }_{\mu }+in\zeta A_{\mu })z,$$

while the frame field is described by

$$L_{\mu }={\mathcal{R}}^{-1}{\partial }_{\mu }\mathcal{R}\in \mathfrak{so}(3).$$

These ingredients enter one effective action,

$$\begin{array}{rl}{S}_{\mathrm{eff}}=\int d^{4}x\sqrt{-g_{\mathrm{roll}}}[& -\frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu }+\frac{1}{2}(D_{\mu }z{)}^{\ast }{D}^{\mu }z\\ & -U(|z|)+{\mathcal{L}}_R].\end{array}$$

The scalar potential and bounded scalar-dependent softening of the frame stiffness provide a test mechanism by which a localized charged core can energetically favor overlap with a nontrivial frame texture. They are effective model ingredients and have not been derived from primitive multi-helix dynamics.

Finite-Box Stationary Solution

On a prescribed flat effective background, the calculation uses the radial ansatz

$$z(r,t)=u(r)e^{-i\omega t},A_0=\varphi (r),A_i=0,$$

together with a degree-one lift of the rollout-frame texture. The coupled radial Euler–Lagrange equations admit a nontrivial numerical solution in which the charged core, electrostatic field, and frame texture coexist with mutual backreaction.

For the primary finite domain with $R=40$, the reported values are

$$\mathcal{N}\simeq 82.382379,E\simeq 174.450853,$$

with texture degree $0.999999998383$ and relative Gauss-law mismatch $1.55\times 10^{-10}$. At matched normalized phase/Noether charge, the joint solution lies below the specified finite-box reference sum of separately solved isolated-core and isolated-texture energies by

$$\Delta E_{\mathrm{ref}}^{(40)}\simeq \mathrm{7.887096.}$$

This is a reference-channel comparison on the stated finite domain. It does not prove nonlinear stability, global energetic preference, or a general binding theorem.

Spinorial Configuration Topology

The finite numerical domain is not treated as a smooth field on all of ${\mathbb{R}}^3$. Instead, the paper separately constructs a smooth finite-energy full-space representative whose scalar amplitude and degree-one texture agree with the numerical profiles only on a prescribed bulk region.

The spatial-rotation orbit of this full-space representative projects to the standard odd mapping-space class in the independent texture sector. Consequently,

$$2\pi \text{rotation loop: noncontractible},$$ $$4\pi \text{twice-traversed loop: contractible}.$$

This is configuration-space topology compatible with a spinorial quantum sector. It does not itself choose the nontrivial Finkelstein–Rubinstein representation, derive a rotational spectrum, or establish $j=1/2$.

The finite-box stationary solution and the full-space topology construction are therefore associated but distinct. No numerical value of the full-space representative’s charge or energy is identified with the finite-box values.

Scope and Open Problems

Paper X establishes one normalized effective action, one coupled finite-box stationary solution, and one related full-space representative with the required $2\pi /4\pi$ configuration-space topology. It does not yet identify the candidate with the physical electron.

That identification would require, among other steps:

• nonlinear and spectral stability;
• selection of a Finkelstein–Rubinstein representation;
• dynamical isolation of $j=1/2$;
• derivation of the observed unit charge and mass;
• construction of exchange statistics and a two-object sector;
• relativistic propagation and Dirac-like closure; and
• a consistent interacting quantum theory comparable with QED.

The effective action also remains to be derived from underlying rollout microphysics rather than adopted as a test structure.

Publication Status

Version 4.0 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.22923781; the concept DOI for the complete version history is 10.5281/zenodo.22923780. The manuscript has not undergone peer review.

The authoritative PDF, LaTeX source, required source figure, citation and provenance metadata, complete supplementary package, checksums, and standard-library audit code are preserved in the frozen GitHub release. The repository verification checks the supplementary manifest, preserved execution record, finite-box observables, reference-channel energy gap, convergence diagnostics, and degree-parity table. It does not rerun the nonlinear solver or validate the physical assumptions. Manuscript, source, supplementary materials, and metadata use CC BY 4.0; repository verification code uses the MIT license.

Citation

Wu, X. (2026). An Electron-Like Charged-Textured Excitation with Spinorial Configuration Topology in The Emergent Frame (Version 4.0). Zenodo. https://doi.org/10.5281/zenodo.22923781

@misc{wu2026electronlike,
  author = {Wu, Xiaodan},
  title = {An Electron-Like Charged-Textured Excitation with Spinorial Configuration Topology in The Emergent Frame},
  year = {2026},
  publisher = {Zenodo},
  version = {4.0},
  doi = {10.5281/zenodo.22923781},
  url = {https://doi.org/10.5281/zenodo.22923781}
}

Version History

v4.0 - September 24, 2026

Public preprint release on Zenodo, with companion GitHub release TEF-2026-010-v4.0.

@unpublished{wu2026electron,
  author = {Xiaodan Wu},
  title = {An Electron-Like Charged-Textured Excitation with Spinorial Configuration Topology in The Emergent Frame},
  year = {2026},
  note = {TEF-2026-010, version v4.0},
  url = {https://theemergentframe.org/papers/electron-like-charged-textured-excitation/},
  doi = {10.5281/zenodo.22923781},
}