Algebraic Resolution of the Strong CP Problem and Preclusion of Dynamical Axion Fields in SU(3)_c Gauge Theory: Mass–Wave Orthogonality, Mittag-Leffler Imaginary-Pole Residue Locking, and Non-Singular U(1)_A Throat Impedance on the 3-Point Harmonic Lattice

### Overview & Archival Release (Version 1 Master Edition) This self-contained monograph establishes a rigorous, zero-parameter operator-algebraic resolution of the **Strong CP Problem** (`θ_QCD ≡ θ_YM + arg det M_q ≡ 0`) and the **Axial U(1)_A Problem** in `SU(3)_c` Quantum Chromodynamics (QCD), proving that hypothetical Peccei–Quinn `U(1)_PQ` symmetries and dynamical axion fields `a(x)` are mathematically superfluous. Grounded in the 3-point harmonic discrete lattice algebra on the Hilbert sequence space `ℓ²(N≥2)` established in the foundational Tri-Core monographs (Moon, 2026) and the spectral generating hierarchy of odd Riemann zeta values (v5.0), all gauge-theoretic and spectral identities are derived from first principles without empirical fitting parameters: 1. **Mass–Wave Transcendental Orthogonality & `arg det M_q ≡ 0` (Section 2):** On `ℓ²(N≥2)` equipped with the Gauss continued-fraction cylinder curvature weight `w(k) = 2Δ²(1/k) = 4/[k(k²-1)]` (`∑ w(k) ≡ 1`), opposite-sign step-2 quadratic telescoping identically expels Euler's circular phase constant `ζ(2) = π²/6` from the diagonal rest-mass operator `[W_w]_kk = 4/[k³(k²-1)²]` (`Z_diag = 4ζ(3) - 19/4 ≈ 0.058228 > 0`), while constructively preserving `ζ(2)` exclusively in the off-diagonal nearest-neighbor flavor-hopping channel `J_total² = ∑ w(k)w(k+1) = 238/9 - 16ζ(2) ≈ 0.125499`. Taking the unique positive operator square root `[W_w^(1/2)]_kk = 2/[k^(3/2)(k²-1)]` partitions the pre-valve (`k < 5`) fermion mass spectrum via Besicovitch's theorem into three algebraically orthogonal, strictly positive radical generation sectors `Q(√2)`, `Q(√3)`, and `Q(1)` (`95.0329%` trace capture). Consequently, `det M_q > 0 ⟹ arg det M_q ≡ 0` holds identically, while weak Cabibbo–Kobayashi–Maskawa (CKM) CP phases are isolated strictly within the unimodular off-diagonal hopping sector (`det V_CKM ≡ 1`). 2. **Mittag-Leffler Imaginary-Pole Residue Locking & `θ_YM ≡ 0` (Section 3):** Decomposing the non-perturbative vacuum generating function `Φ̃(u) = ∑ (ζ(2m+1)-1)u^m` (`u = t²`) into `Z_4` Hodge-dual parity sub-towers `Φ̃_odd(u)` and `Φ̃_even(u)` proves that at every Euclidean imaginary-axis pole `u = -n²` (`t = ±in, n ≥ 2`), the residues are locked to exact opposite rational half-integers `∓n/2 ∈ Q`. Any non-zero topological phase twist `θ_YM` induces an uncompensated imaginary residue `Resu=-n² Φ̃_θ(u) = -i n sin(θ_YM / 2)`, violating Euclidean holomorphy and Schwinger–Bianchi vacuum persistence (`Im W_eff ≡ 0`, `∇^μ G_μν ≡ 0`) unless `θ_YM ≡ 0 (mod 2π)`. 3. **Kato–Ponce Commutator Quenching & Centrifugal Instanton Preclusion (Section 4):** Under logarithmic radial magnification `u = -ln(r/l₀) ∈ [0, ∞)` inside the sub-Planckian throat `r ∈ (0, l₀]`, the disformal hyperbolic funnel (`R = -2/l₀²`) is geodesically complete as `u → ∞` (`r → 0⁺`), and double-exponential WKB centrifugal shielding `T_WKB(l ≥ 1) ≤ exp(-2√(l(l+1)) e^U) → 0` reflects all anisotropic modes. In the continuum boundary limit `k = 1 + δ → 1⁺` of the isotropic `l = 0` channel, the apparent simple pole `w(1+δ) = 2/δ - 3 + (7/2)δ - (15/4)δ² + O(δ³)` couples identically to zero under transversal gauge fixing (`∇ · A^a = 0`) in a Kato–Ponce commutator formulation (`(2/δ)(η · Â^b) ≡ 0`), precluding singular instanton pinching. 4. **Non-Singular Resolution of the Axial `U(1)_A` Mass Gap & BESIII `X(2370)` Concordance (Section 5):** The parity-odd pseudoscalar (`0⁻⁺`) mass gap arises without singular instantons from the unrelaxed throat boundary reflection impedance `κ₀ = V₂ = 2w(2) + Δ_⊥²(2) = 4/3 + 1/8 = 35/24` acting on the 5-body `0⁺⁺` scalar glueball mass gap `Δ_IR = 5Λ_QCD = 1650.0 MeV`. This yields `M_bare(0⁻⁺) = κ₀ Δ_IR = 2406.25 MeV` and doublet-screened mass `M_screened(0⁻⁺) = η2,3 κ₀ Δ_IR = 2391.49 MeV`, matching the BESIII measurement of `X(2370)` (`2395 ± 14 MeV`) to `0.25σ`, while 50-mode bulk Jacobi hopping relaxes the continued fraction back to `K_50(0) Δ_IR = 1649.32 MeV`. --- ### Interconnected Foundational Archives (Permanent DOIs) * **[Foundational Mathematical Suite (Tri-Core Monographs I, II, III & Companion Guides I–IV, v6, 82 pp)]** * *Geometric Emergence of Spacetime from Prime Spectral Scattering: Complex Phase Rotations, Sakharov Induced Gravity, and Critical Line Vacuum Stability* * **DOI:** [https://doi.org/10.5281/zenodo.23117791](https://doi.org/10.5281/zenodo.23117791)* **[Definitive Master Monograph on the Odd Zeta Hierarchy (v5.0, 27 pp)]** * *On ζ(3), ζ(5), and Odd ζ_n: Geometric Deficits, the Center of Mass on a Closed Ladder, Dual Obstructions, and Spectral Coherence Rigidity* * **DOI:** [https://doi.org/10.5281/zenodo.23355922](https://doi.org/10.5281/zenodo.23355922)* **[Higher-Order Harmonic Difference Stencils & Basel Annihilation]** * *Higher-Order Harmonic Difference Stencils, Algebraic Basel Annihilation, and the Universal Operator Structure of Odd Zeta Values* * **DOI:** [https://doi.org/10.5281/zenodo.23359104](https://doi.org/10.5281/zenodo.23359104)