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"Numerical fixed trigonometric identities; no tetrahedron realization or rank certification" } ] } ], "coverage_limits": [ "Finite exact algebra checks use the printed geometric identities as inputs; they do not prove their geometric hypotheses.", "Euler quadrature and special-function evaluations are independent formulas using the same mpmath arithmetic library.", "Numerical tolerances, two-precision agreement, and finite-series differences are not interval certificates or rigorous tail bounds.", "Balanced periods use a separate logarithmic basis, never substitution into a singular nonresonant connection matrix.", "No full double-sine or DOZZ evaluation, spectral integral, quantum tail estimate, global Airy claim, or arbitrary rank/intersection engine.", "Formal filling and fixed Hessian tensors do not compute a filled manifold, a verified filling volume, or a torsion invariant.", "The two-cusp example is conditional matrix algebra, not the construction of a manifold or deformation family.", "Regular fold polynomial/Taylor checks come directly from Section14; no code or numerical outputs from the previous Section15 companion are reused." ], "elapsed_seconds": 4.025452999980189 }