Published April 16, 2026 | Version v13_r22
Preprint Open

Yang–Mills Mass Gap in Four Dimensions: Reduction and Closure for the BT–2 Route

Authors/Creators

Description

 

This upload presents a single-manuscript reduction-and-closure framework for the four-dimensional SU(2) Yang-Mills mass-gap problem along the BT-2 route.

Main idea

The manuscript is organized in two parts:

  • Part I (Reduction) develops the Osterwalder–Schrader (OS) window, the analytic and geometric infrastructure needed for the mass-gap transfer, and the reduction of the remaining model-specific target-side burden to one canonical shell-side specialization theorem on a canonical $SU(2)$ family.

  • Part II (Closure) proves that remaining shell-side theorem on one fixed canonical good packet chart and one fixed packet depth, and then follows the already prepared downstream continuation to the final mass-gap conclusion.

Global theorem-facing chain

The overall route is established through the following chain:

$$\delta_{\phi}(t_0) > 0 \implies \lambda_{\min}(-\Delta_M) \ge \kappa_{\mathrm{geo}}\,\delta_{\phi}(t_0)^2$$
$$\implies m_0 \ge c_{\mathrm{dom}}\,\lambda_{\min}(-\Delta_M) \implies m_0 \ge c_{\mathrm{gap}}\,\delta_{\phi}^2 > 0,$$

with the gap constant defined as

$$c_{\mathrm{gap}} := c_{\mathrm{dom}}\kappa_{\mathrm{geo}}.$$

Thus, the essential bottleneck is the target-side production of residual-density positivity.

What Part I establishes

Part I establishes the foundational analytic and geometric modules, including:

  • The parameter-box feasibility theorem and the residual topology density $\delta_\phi$.

  • The topological-coercivity-to-gap route.

  • The FRD / KP / CM.1 / entropy-trace / EVI analytic modules.

  • The strict interchange-of-limits package and OS reconstruction.

  • The global operator-gap transfer.

What Part II closes

Part II closes the remaining shell-side theorem on one fixed canonical chart. It specifically proves:

  1. A shell-center principal separation theorem.

  2. A fixed-chart one-point shell-margin theorem via the affine donor chain.

  3. A near-good branch entry theorem and a fixed-chart probability-floor transfer theorem.

  4. The canonical shell-side activation theorem.

The representative local closure score on the canonical target $SU(2)$ family is summarized by:

$$\Upsilon_{SU(2)} < 1.$$

Downstream continuation after shell-side closure

Once the shell-side theorem is in force, the remaining continuation is internal to the framework:

$$\text{shell-side closure} \implies \text{codimension-one continuation}$$
$$\implies \text{target-law two-tail floor} \implies \text{block-variance floor}$$
$$\implies \delta_{\phi}(t_0) > 0 \implies \delta_{\phi} > 0 \implies m_0 \ge c_{\mathrm{gap}}\delta_{\phi}^2 > 0.$$

Reading guide

This manuscript is intended to be read as a single-manuscript reduction-and-closure theorem package for the canonical target $SU(2)$ BT-2 route. The appendices contain the supporting analytic, geometric, and target-specialization proofs underlying the main theorem chain.

 

Files

Yang_Mills_Mass_Gap_single_manuscript_v13r22.pdf

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