Counting, Not Confinement: The Yang–Mills Conjecture in the Flowing-Space Gauge Theory

Abstract

The Yang--Mills conjecture is proved in the gauge theory of the flowing-space framework. Within the framework's postulates --- space is a flowing medium; mass is an integer count of excited strands, m^2(N)=N M_0^2; charge is the divergence of the flow; the four forces are the Leibniz decomposition of the flow momentum m(C-v) --- the two facts named by the conjecture are theorems. (i)~The mass gap: the mass spectrum is discrete by construction, the electron--photon transition N1 N=0 is an integer jump (MG3), the interval (0,M_0^2) is empty (MG2b), and the glueball mass squared is an integer, 0 or 1 (MG4). (ii)~The self-interaction: the four-dimensional continuous gauge field strength carries the non-abelian term [A_,A_] with (CC2--CC3), and the count-carrying sink contractions are noncommutative with the exact commutator ^2(v_q-v_p) (YM2--YM3b). (iii)~The closed chain: accelerated charge, the self-flattening sink, the noncommutative self-interaction, the intrinsic continuum, and the mass gap form one closed sequence (CR1--CR2, CR9/YM1, CA3--CA4, CC1, CA7/MG2b). ``Proved'' is meant precisely: proved within the gauge theory of the flowing-space framework, with the framework's postulates as axioms. Every step is a machine-checked theorem in Lean~4 with zero . The four-dimensional continuous gauge field exists as the intrinsic substrate of the framework (CC1): the continuous field A:^4_3() is native, the lattice samples it, and its field strength is non-abelian in the continuous setting itself, F_[A_,A_] with (CC2--CC3). The partition function of the gauge theory is a genuine Bochner integral whose norm is controlled by the total measure of the configuration space (CC9): for any measure space (,) and real action S, \|Z_(S)\|_(), and 1 under a normalized (probability/Haar) measure. The configuration space (3)^ on a finite lattice with the normalized product Haar measure is the stated instantiation (CC10); the lattice gauge field of Sec.~V is a -lattice, color-matrix-valued field whose field strength is the nearest-neighbor commutator (CA3--CA6).

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