Quantitative Refinements of Arithmetic Holonomy Bounds: Mixed Source Metrics and Joint Moment Determinants

This preprint develops quantitative refinements of the arithmetic holonomy framework of Calegari–Dimitrov–Tang. Using unequal row degrees, mixed source metrics, and joint moment determinants, it establishes the following upper bounds for the irrationality exponent μ: μ(π) ≤ 6.70μ(arctan(1/2)) ≤ 5.97μ(π/√3) ≤ 4.226μ(L(2, χ₋₃)) ≤ 240μ(log 2) ≤ 3.6555768 Here χ₋₃ is the nonprincipal Dirichlet character modulo 3. The three real-period bounds follow from a common three-function construction. The Dirichlet L-value bound uses a mixture of thirteen source radii applied to the fourteen-function input of Calegari–Dimitrov–Tang. The logarithm argument uses a direct joint-moment determinant and includes a nearly matching obstruction at 3.65557675 for the specified determinant model. The logarithm bound is weaker than the previously established bound of 3.57455391; its contribution is the explicit construction and the quantitative analysis of that model’s limitations. The deposit contains the main manuscript, two companion proof documents, LaTeX sources, numerical certificates, verification programs, and documentation of the verification scope. The proofs use fixed-input interval computations. All irrationality-exponent bounds are asymptotic; numerical exceptional-height thresholds are not supplied. GPT and Claude were used for mathematical exploration, programming, verification work, and drafting. The accompanying audit records document AI-assisted reviews. This is an unrefereed, computer-assisted proof manuscript; no external human peer review or formal proof verification is claimed.