What question did this study set out to answer?

This research aims to explore the Riemann hypothesis through a fixed transport geometry framework, establishing new relationships between entropy and prime distribution.

June 6, 2026Open Access

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

Puntos clave

This research aims to explore the Riemann hypothesis through a fixed transport geometry framework, establishing new relationships between entropy and prime distribution.
Utilized a closure-first transport geometry to analyze prime number distribution.
Developed the MAT equality framework linking various entropy scales and MAP parameters.
Examined prime powers using the MAT prime-transfer law within a logarithmic length coordinate.
Demonstrated a fixed entropy scale tied to black-hole entropy, confirming its dependence on MAT framework.
Establish significant connections between transport geometry and the Liouville counterterm's positivity.
Validated the unregularized Guinand-Weil form's identification with the MAT transport form, establishing positivity in testing.

Resumen

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 SₜhMAT = π Rₜh² = SBH (mcross) = log (TₑntMAT/ (2πe) ). Here Rₜh is the finite-support radius of the smallest MAT support cell, SₜhMAT its support entropy, mcross = Rₜh/2 the black-hole compactness crossing mass in Planck units, SBH = 4πm² the Bekenstein–Hawking entropy at that crossing, and TₑntMAT 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 Ygamma, and the Schwarzian derivative of Ygamma 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 → Ygamma (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.

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