From a Gravity-Calibrated Helix to a Matter–Spacetime Interface: Closure Obstruction and a Factorized Mass-Squared Ansatz

Xiaodan Wu

doi:10.5281/zenodo.22649267

Abstract

The Emergent Frame (TEF) investigates whether matter and open spacetime may be different global organizations of compatible local structure. In the working “track” picture developed here, the track is not a classical rail carrying a particle: it represents structural constraints on how energy may be organized, propagated, and rerouted. This paper studies a prerequisite for such a picture: whether the frozen native TEF helix can close directly into matter-like structures. For every oriented complete nonzero-pitch circular helix segment, the endpoint tangent and net displacement satisfy $t_{i} \cdot Δ r_{i} > 0$ . It follows that a finite sequence of such complete segments cannot form a closed oriented curve by direct sequential $C^{1}$ joining alone. Some additional completion mechanism—such as rerouting, reconnection, partial-turn structure, or more general geometry—is therefore required. When one frozen native turn is retained as part of a smooth closed completion, Fenchel’s theorem further gives the necessary bound $K_{int} \geq 2 π [1 - 1/ 1 + q^{2}]$ . For the previously frozen TEF shape parameter $q = 0.556324180045 \dots$ , the corresponding dimensionless Fenchel deficit is $0.1261280215 \dots$ . Motivated by this closure obstruction, the paper constructs a geometric interface map from prescribed post-rerouting closure data and studies the positive Gram operator $G_{N} = C_{N}^{†} C_{N}$ . For minimal Euclidean closures, its rank and kernel follow exactly from the specified geometry, including a single interface-null combination. The identification of this operator with a physical mass-squared contribution remains an ansatz. Particle masses, QCD dynamics, and a fundamental TEF energy law are not yet derived.

Research Context

This preprint asks whether the open-spacetime geometry inherited from Paper I can participate directly in the closed structures contemplated by Paper IV. It also follows Paper V by moving from a conditional closure-energy framework toward a more explicit matter–spacetime interface.

The broad TEF hypothesis is that matter and open spacetime may share compatible local framing primitives while differing in their global organization. For a tractable test, Paper VI imposes a stronger complete-native-segment ansatz: matter-like structures are assembled sequentially from complete, oriented, nonzero-pitch circular helix segments with direct $C^{1}$ tangent continuity.

The theorem applies to this restricted ansatz. It is not a theorem about arbitrary curves, branching networks, partial turns, variable local geometry, or all possible realizations of shared local framing.

Closure Obstruction

For each complete circular-helix segment, the endpoint tangent $t_{i}$ and net displacement $Δ r_{i}$ obey

t_{i} \cdot Δ r_{i} = \frac{2 π R _{i} q _{i}^{2}}{1 + q _{i}^{2}} > 0.

Direct sequential $C^{1}$ joining propagates a common endpoint tangent through the assembly. A closed curve would require $\sum_{i} Δ r_{i} = 0$ , but taking the dot product with that common tangent gives a sum of strictly positive terms. Therefore a finite sequence of the specified complete helical segments cannot close through direct sequential smooth joining alone.

If one frozen native turn is retained as a subcurve of a smooth closed completion, Fenchel’s theorem gives the separate necessary bound

K_{int} \geq 2 π (1 - \frac{1}{1 + q ^{2}}) .

For the frozen TEF value of $q$ , the dimensionless Fenchel deficit is $0.1261280215 \dots$ . This is an accumulated total-curvature lower bound, not a unique junction angle, a mass formula, or proof that the bound is attainable.

Interface Operator Ansatz

After the direct-gluing restriction is relaxed, the paper separately introduces a linear interface map $C_{N}$ from prescribed post-rerouting closure data and defines the positive Gram operator

G_{N} = C_{N}^{†} C_{N} .

Once the map and positive-definite inner products are specified, positivity, rank, and kernel statements follow exactly. For a minimal Euclidean $N$ -branch closure with nonzero weights and $2 \leq N \leq 4$ ,

rank G_{N} = N - 1, dim ker G_{N} = 1.

The closure obstruction does not uniquely derive $C_{N}$ , its weights, its normalization, or a completion geometry. Identifying $G_{N}$ with a relativistic mass-squared contribution is an additional physical ansatz. The algebraic resemblance of the two-branch case to a neutral electroweak mass matrix is structural compatibility only; gauge fields, symmetry protection, Higgs dynamics, couplings, and particle masses are not derived.

Publication Status

Version 3.2 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.22649267; the concept DOI for the complete version history is 10.5281/zenodo.22649266. 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-2026-006-v3.2. The verification script checks the helix geometry, closure-obstruction quantities, curvature bound, scale hierarchy, and symmetric Gram spectra. It validates reported algebra and arithmetic only; it does not validate the physical interface or mass-squared ansatz.

Relationship to the Earlier TEF Papers

Paper VI uses the frozen native helix from Paper I, addresses the matter-closure problem introduced in Paper IV, and supplies a geometric prerequisite for the interface and energy questions developed in Paper V. It does not replace those constructions with a complete microscopic model.

The paper’s staged logical chain is:

complete-native-segment ansatz ⟶ closure obstruction ⟶ completion or rerouting required ⟶ C_{N} ⟶ G_{N} = C_{N}^{†} C_{N} ⟶ candidate mass-squared sector .

The first implication is a theorem under the stated geometric assumptions. The interface map is a separately chosen operator ansatz, and the final mass-squared interpretation is a physical hypothesis.

Citation

Zenodo citation

Wu, X. (2026). From a Gravity-Calibrated Helix to a Matter–Spacetime Interface: Closure Obstruction and a Factorized Mass-Squared Ansatz (Version 3.2). Zenodo. https://doi.org/10.5281/zenodo.22649267

@misc{wu2026matterspacetime,
  author = {Wu, Xiaodan},
  title = {From a Gravity-Calibrated Helix to a Matter--Spacetime Interface: Closure Obstruction and a Factorized Mass-Squared Ansatz},
  year = {2026},
  publisher = {Zenodo},
  version = {3.2},
  doi = {10.5281/zenodo.22649267},
  url = {https://doi.org/10.5281/zenodo.22649267}
}

Version History

v3.2 - September 8, 2026

Public preprint release on Zenodo. Version DOI: 10.5281/zenodo.22649267.

v0.2 - September 7, 2026

Initial internal matter–spacetime interface manuscript.