Counting, Not Confinement

Abstract

The Yang–Mills mass gap — the assertion that the lightest excitation of a non-abelian gauge theory is strictly massive — has stood for half a century as one of the deepest open questions at the boundary of mathematics and physics, underpinning the standard model yet resisting proof in its standard formulation. Its two structural facts, the discreteness of the confining spectrum and the nonlinear self-interaction of the gauge field, are postulates or measurements in the standard framework; neither is derived. Here both are proved, within the gauge theory of the flowing-space framework, in which space is a flowing medium and mass is an integer count of excited strands, m^2(N)=N M_0^2 (MG1). The mass gap becomes a theorem of integer arithmetic: the electron–photon transition is a jump across the empty interval (0,M_0^2) (MG2b, MG3), and every glueball mass squared is an integer (MG4). The self-interaction becomes a theorem of geometry: contractions of the count-carrying sinks at distinct points fail to commute, with the exact commutator ^2(v_q-v_p) (YM2–YM3b). The identification of this noncommutativity with the non-abelian term [A_,A_] of a four-dimensional continuous gauge field — native to the flowing space rather than manufactured by a lattice limit (CC1, CC8) — is structural and partial: the sink commutator is linear in the field difference, whereas [A_,A_] is quadratic in the field (Sec.~IV). The partition function of the gauge theory is a genuine Bochner integral bounded by the total measure of its configuration space (CC9), with the non-abelian group instance stated as the open type-level item (CC10 instantiates the abelian T^4). The counting law is no longer an input: a closed configuration of the field has integer winding (LT1–LT6), so the count is an integer because the field closes, and the gap is quantified over every closed configuration (Sec.~:closure). What remains an input is the price M_0 — and, for ratios between sectors, the second price: a shared price is excluded by four measured ratios at 10^4–10^5 (MB4), while a freely varying one fits anything and predicts nothing (MB5). The colour-resolved form of the chain carries one premise more — that the links are diagonal in the frame the flow supplies — and the premise is not innocent: diagonal links commute, so the field strength vanishes on the patch where colour windings exist. Unconditionally the gap survives in the coarse form N=k^2; the ladder, and the spectrum compared with data, live on the aligned sector. Every theorem stated is machine-checked in Lean~4 with zero sorry; the identifications that link theorems to the problem's two facts are stated explicitly as proposals of the framework, and the proof is complete within the framework's axioms. Translation of the intrinsic continuum into the standard Wightman axioms is posed explicitly as the open program.

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