Yang--Mills Existence and Mass Gap via Mock Modular K₈.: Paper XVII in MEF Series

We address the Yang--Mills Existence and Mass Gap Millennium Prize Problem for gauge group \ (G = SU (3) \). The proof proceeds in two pillars. Pillar I (Existence). We construct the 4D Euclidean theory as the unique Kaluza--Klein reduction of a 12-dimensional geometry₁₂ = (S¹ₒ₎₋ S³) K₈ the internal 8-manifold₈ =[ (CP^2 S^2) ₖ (T^2/Z₂) ₒ₏₈₍^₂, \]whose partition function is a mock Jacobi form of weight \ (0\), level \ (2\). We verify all four Osterwalder--Schrader axioms \ (OS-0\) through \ (OS-3\) for the 12-dimensional theory using the arithmetic of the odd divisor function \ (d₎₃₃ (n) \), the Brünier--Funke exact sequence, and the Białynicki--Birula cellular decomposition of instanton moduli spaces on \ (CP^2\). The physical Hilbert space of the internal sector is isolated via an intrinsic BRST cohomology — the Mock Modular BRST Theorem — which identifies the Brünier--Funke \ (₊\) operator as the nilpotent BRST charge and projects out the indefinite Zwegers shadow. Standard Kaluza--Klein reduction over a compact internal manifold with finite volume transmits these OS properties to the 4D effective theory, yielding a reflection-positive Euclidean QFT on \ (R^4\). Pillar II (Mass Gap). The topological rigidity of the \ (q^0\) sector uniquely determines the 4D theory as pure \ (SU (3) \) Yang--Mills with\ₐ₂₃ = 0 zero tunable parameters. Two independent derivations of\ > 0 provided. Route A (\ (\) -theorem chain). The Komargodski--Schwimmer \ (\) -theorem, combined with systematic topological exclusion of all known gapless phases, forces any hypothetical IR fixed point to be a ``Dead-End CFT'' with scalar gap\_ 4. Dead-End CFT Bootstrap Lemma, Lemma 6. 1, excludes this remaining possibility. The RG flow is thereby forced to end at the trivial fixed point\₈ₑ = 0, by the Hofman--Maldacena bounds rigorously precludes massless states, establishing\ > 0. 6. 1 is classified Motivated in this paper; Route A's closure of Theorem B is accordingly established only up to this lemma. Route B establishes the same conclusion independently of the Bootstrap Lemma. Route B (corner-geodesic theorem). Paper XXI 24 §7 establishes \ (\) as the minimum geodesic distance on \ (K₈\) between distinct pillowcase corners of \ (T^2/Z₂\) under (2) Sp (1) ; Kloosterman alternation₂ (n, -1;2) = (-1) ^n derived algebraically as the orbit signature of₀ₔₓ Z the quaternion group \ (Q₈\) over \ (n\) topological wraps of the \ (T^2\) base. Positivity of \ (\) follows from the distinctness of the four corners and the compactness of \ (K₈\). Route B is summarised in §7. 3. The mass gap is therefore Derived without residual gap. The construction uses zero supersymmetry, zero continuous free parameters, and zero string-theoretic dualities. The geometric foundation is established in Paper XV, the Twisted Torus Theorem. Every claim carries the four-tier classification: Rigorous, Derived, Motivated, or Gap.