Closure-First Riemann–Weil Transport Geometry in Mittermeier Attractor Theory (MAT): Entropy-Width Lock, Gamma-Refined Length Map, Prime-Transfer Law, and the Finite-Support Origin of the Spectral Length Scale

This paper treats the Riemann hypothesis—the central problem on the distribution of prime numbers—as a global positivity problem equipped with a fixed closure-first transport geometry. The framework fixes its scales, coordinates, and prime-transfer rules before any zeta-zero information is used. Its finite-support anchor is the MAT equality S_th^MAT = π R_th² = S_BH(m_cross) = log(T_ent^MAT/(2πe)). Here R_th is the finite-support radius of the smallest MAT support cell, S_th^MAT its support entropy, m_cross = R_th/2 the black-hole compactness crossing mass in Planck units, S_BH = 4πm² the Bekenstein–Hawking entropy at that crossing, and T_ent^MAT the Riemann height at which the logarithmic width L(T) = log(T/(2πe)) equals the same MAT entropy. The equality locks a black-hole entropy scale and the leading explicit-formula width to one finite-support normalization rather than treating them as independent inputs. From this anchor, the Mittermeier Flow Equation generates a logarithmic length coordinate, its gamma refinement produces the non-affine MAT length map Y_gamma, and the Schwarzian derivative of Y_gamma yields a positive Liouville counterterm on the audited domain. Prime powers are then transported by the MAT prime-transfer law x_(p,k) = k log p → Y_gamma(x_(p,k)), so the arithmetic lattice is tested in the same coordinate that carries the archimedean defect. The verifier establishes finite positive semidefiniteness, exact projective restriction, monotone cutoff increments, local Schur margins, completed-log-derivative residual identities, a Moore–Aronszajn RKHS corridor for the regularized kernel, and a zero-Poisson diagnostic profile. These components define a fixed scattering/canonical-system theorem target: the unregularized Guinand–Weil form must be identified with the MAT transport form, with zero-Poisson positivity and Schur-contractivity on the completed global test space.