Critical L¹-Morrey Control of Covariant Dissipation in Four-Dimensional Yang–Mills Flow
- 1Independent Researcher; DIVITIA LLC, Milton, Georgia, USA
Abstract
We formulate a depletion–absorption–persistence architecture for dissipative systems whose cubic production is controlled, with the solution itself as weight, by a codimension-two critical L¹ Morrey density of the state variable, and we develop its Yang–Mills realization in full. The central chain is: M^{1,2} depletion ⇒ full covariant dissipation budget ⇒ finite caloric size. For a smooth connection A on a trivial G-bundle over ℝ⁴, G compact, with θ(A) = sup_{x,r} r⁻² ∫_{B_r(x)} |F_A|, the analytic input is the critical Morrey–cubic inequality |∫⟨F_A, Q(F_A,F_A)⟩| ≤ C_Q C_M θ(A) ‖∇_A F_A‖²_{L²} (Q the Weitzenböck bracket), proved at the q = 1 Morrey endpoint — where the bilinear Fefferman–Phong inequality fails — by combining a heat-kernel Morrey characterization with the refined Sobolev inequality of Gérard–Meyer–Oru; the scalar core assembles known tools and is not claimed as new. The consequences are: under a uniform margin C_Q C_M θ(A(t)) ≤ 1 − η along the Yang–Mills gradient flow, the dissipation upgrades from the tension ‖D_A*F_A‖²_{L²} to the full covariant budget ∫₀ᵀ ‖∇_A F_A‖²_{L²} dt ≤ Y(0)/η; every non-flat critical connection with F_A, ∇_A F_A ∈ L² obeys the explicit M^{1,2} gap C_Q C_M θ(A) ≥ 1; the Oh–Tataru caloric size obeys Q³_{[0,T]} ≤ S₄² (2Y(0))^{1/2} Y(0)/η uniformly in T, so that (under an explicit class-compatibility hypothesis, in the topologically trivial Ḣ¹ sector) a globally persistent margin forces convergence to a flat connection, excluding the infinite-time bubbling of Sire–Wei–Zheng(–Zhou); and the density persists by a Grönwall estimate. A compressed, self-contained three-dimensional Navier–Stokes instantiation of the same triad — vorticity density θ*(ω) = sup_{x,r} r⁻¹ ∫_{B_r(x)} |ω|, enstrophy absorption, and a heat-kernel persistence inequality — is included to fix the cross-equation formulation, with one illustrative consequence and explicit non-claims. No new Yang–Mills continuation theorem and no unconditional Navier–Stokes continuation theorem is claimed: finite-time regularity of four-dimensional Yang–Mills flow on compact manifolds is a theorem of Waldron, and no unconditional Navier–Stokes regularity theorem, improvement of the classical Leray exponent, or resolution of the Caffarelli–Kohn–Nirenberg obstruction is claimed. The record also includes the LaTeX source and a one-page visual abstract intended as an accessible guide to the paper's depletion–absorption–persistence architecture. The manuscript is authoritative for all hypotheses, constants, proofs, comparisons, and limitations.
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BibTeX
@misc{das2026critical,
title = {{Critical L¹-Morrey Control of Covariant Dissipation in Four-Dimensional Yang–Mills Flow}},
author = {Suvajit Das},
howpublished = {Zenodo (CERN European Organization for Nuclear Research)},
year = {2026},
month = aug,
doi = {10.5281/zenodo.21840810},
url = {https://doi.org/10.5281/zenodo.21840810},
note = {CC BY},
}
RIS
TY - UNPB
TI - Critical L¹-Morrey Control of Covariant Dissipation in Four-Dimensional Yang–Mills Flow
AU - Suvajit Das
PY - 2026
DA - 2026/08/08
PB - Zenodo (CERN European Organization for Nuclear Research)
DO - 10.5281/zenodo.21840810
UR - https://doi.org/10.5281/zenodo.21840810
AB - We formulate a depletion–absorption–persistence architecture for dissipative systems whose cubic production is controlled, with the solution itself as weight, by a codimension-two critical L¹ Morrey density of the state variable, and we develop its Yang–Mills realization in full. The central chain is: M^{1,2} depletion ⇒ full covariant dissipation budget ⇒ finite caloric size. For a smooth connection A on a trivial G-bundle over ℝ⁴, G compact, with θ(A) = sup_{x,r} r⁻² ∫_{B_r(x)} |F_A|, the analytic input is the critical Morrey–cubic inequality |∫⟨F_A, Q(F_A,F_A)⟩| ≤ C_Q C_M θ(A) ‖∇_A F_A‖²_{L²} (Q the Weitzenböck bracket), proved at the q = 1 Morrey endpoint — where the bilinear Fefferman–Phong inequality fails — by combining a heat-kernel Morrey characterization with the refined Sobolev inequality of Gérard–Meyer–Oru; the scalar core assembles known tools and is not claimed as new. The consequences are: under a uniform margin C_Q C_M θ(A(t)) ≤ 1 − η along the Yang–Mills gradient flow, the dissipation upgrades from the tension ‖D_A*F_A‖²_{L²} to the full covariant budget ∫₀ᵀ ‖∇_A F_A‖²_{L²} dt ≤ Y(0)/η; every non-flat critical connection with F_A, ∇_A F_A ∈ L² obeys the explicit M^{1,2} gap C_Q C_M θ(A) ≥ 1; the Oh–Tataru caloric size obeys Q³_{[0,T]} ≤ S₄² (2Y(0))^{1/2} Y(0)/η uniformly in T, so that (under an explicit class-compatibility hypothesis, in the topologically trivial Ḣ¹ sector) a globally persistent margin forces convergence to a flat connection, excluding the infinite-time bubbling of Sire–Wei–Zheng(–Zhou); and the density persists by a Grönwall estimate. A compressed, self-contained three-dimensional Navier–Stokes instantiation of the same triad — vorticity density θ*(ω) = sup_{x,r} r⁻¹ ∫_{B_r(x)} |ω|, enstrophy absorption, and a heat-kernel persistence inequality — is included to fix the cross-equation formulation, with one illustrative consequence and explicit non-claims. No new Yang–Mills continuation theorem and no unconditional Navier–Stokes continuation theorem is claimed: finite-time regularity of four-dimensional Yang–Mills flow on compact manifolds is a theorem of Waldron, and no unconditional Navier–Stokes regularity theorem, improvement of the classical Leray exponent, or resolution of the Caffarelli–Kohn–Nirenberg obstruction is claimed. The record also includes the LaTeX source and a one-page visual abstract intended as an accessible guide to the paper's depletion–absorption–persistence architecture. The manuscript is authoritative for all hypotheses, constants, proofs, comparisons, and limitations.
ER -