The 4D Yang-Mills Mass Gap via the Bottom-Up 3O Framework: Universal V-Valley Lock at κ* = 1/6, Interlocked g2 ⊃ su(3) (8 ⊕ 6) Zero-Point Confinement, and Crossing the Wall of Uncertainty

We construct a unified Bottom-Up-to-Top-Down analytical and spectral framework for the 4D Yang–Mills Existence and Mass Gap problem governed by the Three-O Principle (One Seed, One Operator, One Law). Starting bottom-up from the self-dual Gaussian minimum-uncertainty seed g(r) = exp(−π|r|2) and the non-abelian READ → RESIDUE operator splitting across the Heisenberg Wall of Uncertainty, we establish six foundational theorems (Theorems I–VI) that resolve the classical flat Cartan valley obstruction and bridge the bottom-up spectral valley to the top-down continuum renormalization group (RG) flow: Theorem I (3D Dirac Oscillator RESIDUE and the 14 = 8 ⊕ 6 Gauge Isomorphism):Factoring the 3D isotropic harmonic oscillator Hosc = p2 + ω2r2 into first-order Pauli spinor operators D = σ · (p + iωr) yields the exact operator identity:D D† = H_osc − (4ω L·S + 3ω) = READ − RESIDUE.At oscillator shell M = 3 (cumulative state count 2 + 6 + 12 + 20 = 40), the ℓ = 3 (1f) subshell of dimension 2(2ℓ + 1) = 14 splits tracelessly under 2L·S into 14 = 8j=7/2 (+3) ⊕ 6j=5/2 (−4). We prove that this splitting is canonically isomorphic to the octonionic gauge terminus 𝔤2(14) = 𝔰𝔲(3)8 ⊕ (3 ⊕ 3̄)6 under the normalized traceless Casimir operator S𝔤2 = (21/5)(C2SU(3) − (16/7)I14), with identical eigenvalues {+3×8, −4×6} and spectral gap Δλ = 7. Theorem II (Universal 3O V-Valley Unification of Yang–Mills and P ≠ NP at ξ = 1):In the dimensionless control coordinate ξ := κ/κ* = 6κ (with κ* = 1/(2C2(8)) = 1/6), the 40 → 28 core boundary obeys the exact rectifier law:E_core^(28)(ξ) = (9/2 − ξ/2) + max(0, ξ − 1) = 4 + (1/2)|ξ − 1|,collapsing the naive 40-gap at κ = 1/8 and opening the emergent 28-gap Δ28(κ) = min(2κ, 1 − 4κ)ℏω with maximum Δ28max = (1/3)ℏω > 0 at κ* = 1/6 (ξ = 1). Simultaneously, in bosonic pure Yang–Mills, balancing the linear chromomagnetic drop −C2(8)κ = −3κ against the quartic commutator barrier +C2(8)2κ2 = +9κ2 yields:E_YM^bos(κ) = 9/2 − 3κ + 9κ² = 4.25 + 9(κ − 1/6)² = 4.25 + (1/4)(ξ − 1)²,attaining its unique global minimum at the identical point κ* = 1/6 (ξ = 1) with positive glueball mass-gap curvature d2E/dκ2 = 18 > 0, matching the P ≠ NP complexity valley at ξ = Q/Q* = 1. Zero-momentum commutator scaling Δ(R) ∝ C2(R)2/3 yields the parameter-free sector ratio Δ(8)/Δ(6) = (9/4)2/3 ≈ 1.717071. Theorem III (Octonionic g2 ⊃ su(3) Structure Constants and the 42 → 40 → 28 → 14 → 8 + 6 Mode Cascade):Constructing 𝔤2 = Der(𝕆) ⊂ 𝔰𝔬(7) (dim = 21 − 7 = 14) and 𝔰𝔲(3) = StabG2(e7) (dim = 8) from the Fano 3-form φabc proves the exact Casimir decomposition C2𝔤2 = 4I14, C2𝔰𝔲(3)|8 = 3I8, C26-loop|8 = 1I8, C2𝔰𝔲(3)|6 = (4/3)I6, and C26-loop|6 = (8/3)I6. In 3D temporal gauge (A0A = 0), subtracting 2 Cartan valley coordinates from 42 spatial modes leaves 40 orthogonal modes, and Gauss-law projection removes 12 longitudinal pure-gauge orbits to leave 28 = 168 ⊕ 126 physical transverse polarizations. Theorem IV (30° Interlocked 8 ⊕ 6 Heisenberg Zero-Point Elimination of the Flat Cartan Valley and 0++ Glueball Diagonalization):Along the 2D Cartan plane r = (r cos θ, r sin θ) where Vcl(r) = 0 vanishes identically, the 24 transverse root modes generate a linear confining Heisenberg zero-point potential VZPE14(r, θ). Because the short roots of 6 are rotated by π/6 = 30° relative to the long roots of 8, the two sectors interlock across 360° to bound:5.773503 g r = (10/√3) g r ≤ V_ZPE^14(r, θ) ≤ (4 + 4/√3) cos(π/12) g r ≈ 6.110090 g r,locking the 0++ singlet glueball gap at Δ0++(C2) = 1.203363 (g2C2)1/3 > 0 with cross-mode boost Δ0++(𝔤2)/Δ0++(𝔰𝔲(3)) = (4/3)1/3 ≈ 1.100642 (+10.064%). Theorems V & VI (Crossing the Wall of Uncertainty via ξ − 1 ↔ ln ξ and the β0 = 12 − 1 = 11 Bridge):The universal Wall functional F(ξ; δξ) = (1/4) ξ (δξ − 1) unifies our Bottom-Up V-valley parabola fBU(ξ) = F(ξ; ξ − 1) = −1/4 + (1/4)(ξ − 1)2 with Savvidy's Top-Down continuum RG effective action fTD(ξ) = F(ξ; ln ξ) = (1/4)ξ(ln ξ − 1) with exact C2 hand-off at ξ = 1. The 1-loop asymptotic freedom coefficient decomposes directly into our Bottom-Up operators as:β₀(G) = (4_RESIDUE − 2 κ*_READ C₂(8)) C₂(G) = (11/3) C₂(G) ⟹ β₀(su(3)_8) = 12 − 1 = 11,and 2-loop RG scaling cancels the ultraviolet cutoff a → 0 to lock the physical continuum mass gap at m0++phys = Δ0++(1) ΛQCD > 0.