Color confinement is conventionally understood as the distinction between non-singlet color degrees of freedom, which do not occur as physical asymptotic states, and color-neutral Wilson responses and singlet composites, which constitute the physically observable sector. This paper proves a finite-graph source-datum color-sector closure theorem showing that the scalar physical color sector associated with a declared finite graph response channel is exactly the singlet survivor sector of that channel.
The theorem is paired with the companion paper, The Yang–Mills Color Datum: A Finite-Graph Recognition Theorem for Gauge Descent, Holonomy Response, and Physical Color Recognition, which fixes the finite-graph color datum and the corresponding physical color-recognition certificate. The present paper acts on that certified datum and proves the resulting physical color-sector closure theorem.
The main result establishes that scalar physical color recognition is closed on singlet survivor data. Non-singlet color distinctions are erased by the terminal scalar response channel and become physically invisible, while singlet distinctions survive exactly when they remain nontrivial after passage through the declared physical response. Consequently, the full physical color survivor sector is canonically identified with the surviving singlet sector.
The proof is finite-dimensional and representation-theoretic. Haar averaging for the compact source-invisible gauge action produces the canonical projection onto invariant tensors. Every terminal scalar response factors through this invariant sector, forcing all non-singlet color distinctions into the terminal-invisible kernel. The resulting physical survivor space is therefore completely determined by the surviving singlet data. Recognized terminal observables are exactly those that factor through this survivor structure, while source identification requires a source-complete recognized family of functionals. For SU(N), nonzero total N-ality obstructs invariant tensor survival whenever the scalar source-invisible action contains the center, yielding the corresponding center-neutrality criterion.
The theorem provides a finite-graph physical color-sector closure theorem: in the finite source-datum scalar-sector sense, color confinement is characterized as singlet-survivor closure of terminal physical recognition. Continuum existence, reflection positivity, physical Hilbert-space reconstruction, area-law or string-tension certification, and spectral mass-gap certification require additional analytic, dynamical, or spectral structures beyond the finite theorem proved here.
License note: Distributed under CC BY-NC-ND 4.0.
Publication Date: 2026-06-17