Global Geometric Regularity in the 21-Equation Six-Dimensional Curvature-Torsion Framework

We present a self-contained geometric theory on six-dimensional manifolds equipped with a metric-compatible connection whose curvature and torsion tensors satisfy a system of 21 independent algebraic-differential constraints. We prove that the framework is intrinsically regular: the total torsion energy decays exponentially along the natural geometric heat flow, and finite-time blow-up is forbidden by the internal rigidity of the system. The proof is entirely internal, relying solely on the 21 constraints. The result establishes the 21-equation system as a closed, autonomous, and globally regular geometric theory.

This paper constructs a complete closed geometric system consisting of 10 Ricci-torsion algebraic identities, 6 generalized Bianchi identities and 5 divergence-free gauge conditions, totaling 21 independent scalar equations. Under this unified curvature-torsion framework, we prove the global smooth evolution of six-dimensional geometric manifolds via heat flow energy dissipation estimates, without relying on any external geometric or physical hypotheses. The theory naturally contains the incompressible Navier–Stokes equation as a reduced subsector under 3+3 Kaluza–Klein dimensional reduction, providing a pure geometric proof of regularity without case-by-case PDE analysis.

We present a self‑contained geometric theory on six‑dimensional manifolds equipped with a metric‑compatible connection whose curvature and torsion tensors satisfy a system of 21 independent algebraic‑differential constraints. We prove that the framework is intrinsically regular: the total torsion energy decays exponentially along the natural geometric heat flow, and finite‑time blow‑up is forbidden by the internal rigidity of the system. The proof is entirely internal, relying solely on the 21 constraints. The result establishes the 21‑equation system as a closed, autonomous, and globally regular geometric theory.

This paper constructs a complete closed geometric system consisting of 10 Ricci‑torsion algebraic identities, 6 generalized Bianchi identities and 5 divergence‑free gauge conditions, totaling 21 independent scalar equations. Under this unified curvature‑torsion framework, we prove the global smooth evolution of six‑dimensional geometric manifolds via heat flow energy dissipation estimates, without relying on any external geometric or physical hypotheses. The theory naturally contains the incompressible Navier‑Stokes equation as a reduced subsector under 3+3 Kaluza‑Klein dimensional reduction, providing a pure geometric proof of regularity without case‑by‑case PDE analysis.

This work is the extended sequel to our previous preprint (Version‑2, Zenodo: Global Geometric Regularity in the 21‑Equation Six‑Dimensional Curvature‑Torsion Framework). We build an eleven‑dimensional metric‑affine gravity model with dynamical torsion, based on the full set of 21 geometric constraint equations. Spontaneous compactification onto the six‑dimensional flag manifold F_{1,2}=SU(3)/U(1)\times U(1) is investigated. Explicit computations are carried out for non‑Einstein vacuum geometry, heat‑kernel spectral asymptotics, Chern‑Weil topological integral and Kaluza‑Klein mass‑splitting. The tachyonic zero‑mode problem is analysed, and three theoretical routes for tachyon cancellation are outlined. Further numerical simulation and fermion zero‑mode decomposition are reserved for follow‑up research.This preprint presents a consolidated master manuscript for the eleven‑dimensional geometric unification framework. It systematically collects the full axiom system A1‑A4, the complete set of Z‑series constraint equations, and the Zhang‑type governing equations derived from Palatini‑torsion gravity on the flag manifold F_{1,2}=SU(3)/U(1)\times U(1).

The manuscript clarifies the distinction between the static spectral equations from earlier works and the dynamical torsion‑heat‑flow evolution equations. Parameter‑consistency checks, dimensional‑analysis remarks, Weyl‑curvature evaluation for the compact internal space, and a complete checklist of pending computational tasks for follow‑up research are included.

All external mathematical theorems and experimental reference values are treated only for post‑hoc comparison, not as built‑in axioms of the theory. This document serves as a definitive archival reference for the whole series of related preprints.

Author: Xiangdong Zhang

E‑mail: 3672338080@qq.com

This preprint is an archival consolidated reference for the eleven‑dimensional geometric unification framework. It collects axioms A1‑A4, full Z‑series constraints and Zhang‑type governing equations for Palatini‑torsion gravity on flag manifold F_{1,2}=SU(3)/U(1)\times U(1).
 
It distinguishes static spectral relations from dynamical torsion‑heat‑flow evolution equations, with consistency checks and a complete list of pending calculations for follow‑up research. External theorems and experimental data are only used for post‑hoc comparison, not foundational axioms. It serves as the stable master reference for this preprint series.
 
Author: Xiangdong Zhang
E‑mail: 3672338080@qq.com

This archival‑reference manuscript systematically assembles the full axiomatic system, constraint equations, and dynamical evolution laws for the eleven‑dimensional geometric unification theory built upon the flag manifold F_{1,2}=SU(3)/U(1)\times U(1). It catalogues axioms A1–A5, the complete Z‑series constitutive relations, and the Zhang‑type heat‑flow governing equations, separates stationary spectral constraints from dynamic torsion‑metric evolution, and provides self‑consistency verification. All external mathematical theorems and experimental numerical data are used exclusively for posterior comparison, and are not adopted as foundational axioms within the theory. This document acts as the stable master citation source for the whole preprint series.

This is the revised release of the equation-system preprint belonging to the Rotation-Unification program. This work establishes the 15-dimensional metric-affine torsionful unification framework built upon minimal irreducible axioms A1–A4, A6, removing all manual ad-hoc assumptions inherited from the older 11-dimensional formulation.

We construct the full dynamical equation system, including 15-dimensional Einstein equations, algebraic torsion equation, trace relation, and the eigen-problem for the torsion quadratic form. The torsion quadratic form assembles into a 24×24 matrix whose minimal eigenvalue is λ_min = −35/8, triggering spontaneous symmetry breaking, a kink domain-wall solution, and a positive four-dimensional effective vacuum energy Λ_eff(0) ∼ 90.4 α/(β L²) > 0, with the L⁻² factor required by dimensional consistency.

A complete linear stability analysis for the Type-3 symmetry-breaking vacuum is carried out. Group-theoretic block-reduction shows every cross-sector torsion block is semi-positive-definite; therefore the Type-3 vacuum is linearly stable. This stability result exposes a topological obstruction against generating fermion zero-modes on S³ within the original axiom set.

An optional supplementary postulate extending A6 by an SU(3)-homogeneous holomorphic line bundle over the flag manifold F_{1,2} is discussed; it is not derived from the core axioms. This framework achieves classical geometric, gauge, constraint, and rotational unification. Fermion origin and supersymmetry lie outside the scope of the present classical geometry, reserved for follow-up investigations. This document supersedes the earlier eleven-dimensional preprint version.

This work presents the axiom‑based equation system and linear‑order stability analysis for the fifteen‑dimensional torsionful Rotation Unification framework. Under the block‑diagonal background, the torsion quadratic‑form matrix yields minimal eigenvalue \lambda_{\min}=-35/8, triggering spontaneous symmetry breaking, a kink domain‑wall solution, and positive four‑dimensional effective vacuum energy. Group‑theoretic block reduction demonstrates semi‑positive‑definite cross‑sector blocks, establishing linear stability for the Type‑3 symmetry‑breaking vacuum. A topological obstruction for fermion zero‑modes emerges within the original axiom set; a supplementary spinor‑bundle postulate over the flag manifold F_{1,2} resolves this issue. Full recalculation including cross‑sector dynamical couplings is deferred to follow‑up work. This manuscript supersedes the earlier eleven‑dimensional preprint.

This companion manuscript develops the axiomatic equation system and linear stability analysis of the 15D torsionful Rotation Unification framework. The torsion quadratic form gives \lambda_{\min}=-35/8, inducing spontaneous symmetry breaking, domain-wall kink solutions and positive 4D vacuum energy. Group-theoretic block reduction verifies linear stability of the Type-3 vacuum. A spinor-bundle postulate on flag manifold F_{\{1,2\}} resolves the fermion zero-mode topological obstruction. Full cross-sector coupling calculations are reserved for future work. This version replaces the older 11D preprint.

[Revision Notes - v2.0]
This version incorporates significant updates to the weak-field effective theory analysis:
1. Explicit Working Assumptions: The block-diagonal background ansatz (C-40) and partial-Palatini variation truncation (C-41) are now stated explicitly at the beginning of the manuscript to define the scope of the effective field theory.
2. Cross-Sector Stability (Appendix D): The stability analysis has been reworked with fixed mass normalizations (M_full = M_c^2 M_hat_full). A rigorous Schur-complement bound is applied to demonstrate that the zeroth-order vacuum remains stable under cross-sector perturbations, provided the compactification scale satisfies M_c << M_Pl (hierarchy generated by internal warping).
3. Textual Corrections: Resolved reference formatting issues in Appendix D and clarified the distinction between medium-refraction dispersion and contorsion-induced phase shifts.
4. Physical Predictions: The core numerical estimates remain unchanged (gamma_PPN - 1 ~ 10^-96, delta_Phi_GW ~ 10^-76).

[Revision Notes – v2.0]
This version incorporates significant updates to the weak-field effective theory analysis for the geometric rotation-curve acceleration model:
1. Explicit Working Assumptions: The frozen S³ orientation ansatz and 2D-to-3D far-field extrapolation are now stated explicitly at the beginning of the manuscript to define the scope of the effective field theory.
2. Consistency & Falsifiability: A clear falsification criterion is added for the 1/r acceleration scaling, requiring residual deviation after marginalizing baryonic mass uncertainties including stellar mass-to-light ratio.
3. Textual Corrections: Resolved symbol collision by reassigning suppression scale to ε, clarified the distinction between derived power-law form and observation-fitted amplitude.
4. Physical Predictions: The core theoretical prediction \(a_{\rm geo}\propto 1/r\) remains unchanged; the single-parameter fit to DDO 154 yields \(\alpha\approx0.98\).