From a Gravity-Calibrated Helix to a Strong-Interaction Confinement Correspondence

Xiaodan Wu

doi:10.5281/zenodo.22412464

Abstract

We formulate a conditional closure-energy framework for a restricted connected static-source channel in The Emergent Frame (TEF). The construction joins the inherited open-space helix of Paper I to the matter-closure question of Paper IV without specifying a microscopic quark topology. Closure-compatible states are organized into integer rollout sectors, and a closure-preserving map advances a discrete branch by a fixed separation increment $P$ while increasing the sector count by one. Assuming that this connected branch admits no lower-count shortcut, that the static quadratic form reduces across number sectors, and that the residual energy remains subextensive after optimization over all higher-count sectors, we show that the averaged discrete static-energy slope tends to $σ = Δ E / P$ . A separate finite-window bound states the accuracy required before string breaking, while any identification with a continuous- $r$ constrained energy requires additional within-cell control. The TEF specialization sets $P = 2 π n q R_{G}$ ; neither $n$ nor the positive increment $Δ E$ is derived here. A future gauge completion could relate the linear term to rectangular Wilson-loop behavior only under additional spectral and limit assumptions. The result is therefore a conditional statement about connected-channel closure energetics, not a microscopic derivation of QCD confinement. String breaking, flux-tube fluctuations, $SU (3)_{c}$ dynamics, the numerical origin of $Δ E$ , and ultraviolet running remain unresolved.

Research Context

This preprint connects two previously separate branches of TEF. Paper I supplies the gravity-calibrated circular-helix geometry and its frozen parameters. Paper IV introduces the complementary distinction between closed matter and open space, and identifies the energy required to maintain composite closure under separation as an unresolved strong-sector question.

Paper V develops that question into a matter-shape-agnostic mathematical interface. Closure-compatible states are arranged into integer rollout sectors. Along a restricted connected static-source branch, a closure-preserving step increases the sector count by one and advances the separation by a fixed amount,

P = 2 π n q R_{G} .

Under the paper’s explicit assumptions, the optimized residual energy is subextensive and the averaged discrete static-energy slope approaches

σ = \frac{Δ E}{P} .

The construction also gives a finite-window error bound for the regime before string breaking. Extending the discrete result to a continuously constrained separation requires additional within-cell control.

Assumptions and Boundaries

The asymptotic result is conditional on a fixed-increment closure extension, the absence of a lower-count shortcut on the selected connected branch, reduction of the static quadratic form across number sectors, and a subextensive optimized residual-energy envelope.

The TEF specialization does not derive the turn multiplicity $n$ or the positive energy increment $Δ E$ . It does not derive microscopic QCD confinement, the QCD string tension, $SU (3)_{c}$ gauge dynamics, Wilson operators, string breaking, flux-tube fluctuations, or ultraviolet running. Any future connection to rectangular Wilson-loop behavior would require additional spectral and limiting assumptions.

The result should therefore be read as a conditional correspondence between closure energetics and an asymptotically linear averaged slope on a restricted connected channel, not as a parameter-free derivation of strong-interaction dynamics.

Publication Status

Version 4.1 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.22412464; the concept DOI for the complete version history is 10.5281/zenodo.22412463. The manuscript has not undergone peer review.

The authoritative Zenodo PDF, author-supplied LaTeX source, numerical checks, release metadata, and checksums are maintained in the public TEF research repository, with the frozen release tagged as TEF-2026-005-v4.1. The verification script checks the reported geometric scales, rollout lengths, optional string-tension diagnostic, and partition-invariance identity. These checks validate arithmetic only; they do not validate the physical assumptions or establish confinement.

Relationship to the Earlier TEF Papers

Paper V inherits $R_{G}$ and $q$ from Paper I without refitting them to strong-interaction data. It takes up the confinement-energetics problem posed by Paper IV while deliberately leaving the microscopic topology of quark-like matter unspecified.

Unlike Papers II and III, this work is not another comparison between a power of the frozen shape parameter and a measured dimensionless observable. It formulates a conditional structural result about how fixed-increment closure extension can produce an asymptotically linear averaged energy slope.

Citation

Zenodo citation

Wu, X. (2026). From a Gravity-Calibrated Helix to a Strong-Interaction Confinement Correspondence (Version 4.1). Zenodo. https://doi.org/10.5281/zenodo.22412464

@misc{wu2026confinement,
  author = {Wu, Xiaodan},
  title = {From a Gravity-Calibrated Helix to a Strong-Interaction Confinement Correspondence},
  year = {2026},
  publisher = {Zenodo},
  version = {4.1},
  doi = {10.5281/zenodo.22412464},
  url = {https://doi.org/10.5281/zenodo.22412464}
}

Version History

v4.1 - September 6, 2026

Public preprint release on Zenodo. Version DOI: 10.5281/zenodo.22412464.

v0.1 - August 30, 2026

Initial internal manuscript.