Selected-Branch Energetic Confinement

Status

Valid conditional synthesis.

Open finite certificate — DP2/P06. The criterion is proved, but a completed numerical pure confinement certificate still requires

and a rigorous positive lower endpoint for the least generalized eigenvalue of every normalized passage block, together with the named imported completion certificates.

Statement

Persistent adjoint-unscreened center charge, a coherent selected carrier, a uniform positive passage floor, oriented Stokes propagation, linear separator packing, Morita noncollapse, and controlled refinement imply a linear lower bound for static-source separation energy on the selected branch.

Assume the seven packages stated in Theorem 10.1: persistent unscreened center classes; coherent selected fusion and cut lifts; uniform positive passage pencils; positive seam reserve and noncollapse; linear separator packing; controlled refinement and Mosco completion; and positive cofinal lower floors for every factor in the finite slope.

Then there are and such that

The conjugate nontrivial center sectors have equal lower tension.

Why it works

The topological class keeps the same nontrivial sector from disappearing. The finite Dirichlet pencil assigns a positive price to one ordinary passage. The seam estimate retains a fixed fraction of that price. Morita noncollapse and selected covariance keep the repeated separator views from losing norm. Linear packing supplies order disjoint collars. Renormalization-curvature density and Mosco lower convergence carry the resulting finite-scale slope to the completed branch.

Example (illustrative)

If a corridor contains at least certified disjoint passages, and every passage retains one uniform positive lower contribution after all frame, allocation, seam, and local-to-global factors, the total lower bound is linear in . The theorem supplies the typed version of that argument and states which certificate owns each factor.

Scope and limits

This theorem does not identify confinement with the mass gap. It does not construct a microscopic flux string or a Nambu–Goto worldsheet. A selected Wilson-tube area upper bound requires an additional reflection-positive transfer identity. Most importantly, the theorem remains conditional until DP2/P06 and the named imported branch certificates are closed.

Source

Phenomenological Implications I, §§8–10, especially equations (32)–(39) and Theorem 10.1; status map in §12.2.

Read the source manuscript.