What question did this study set out to answer?

This work aims to resolve the Riemann Hypothesis through geometric and spectral methods involving Weyl partitions.

May 18, 2026Open Access

The Simplicial Geometry of Weyl Partitions (revised) The Simplicial Geometry of Weyl Partitions: A Geometric and Spectral Proof of the Rieman

Puntos clave

This work aims to resolve the Riemann Hypothesis through geometric and spectral methods involving Weyl partitions.
Introduced the Kaleidoscopic Filter Theorem to isolate partition manifold fluctuations.
Applied Ehrhart-Macdonald Reciprocity and Faulhaber summation to derive a defect-free generating function.
Established a zero-defect operator theory linking the Riemann Hypothesis to symmetric operators.
Proved that all non-trivial zeros lie on the critical line Re(s) = 1/2, confirming the Riemann Hypothesis.
Demonstrated that the Fredholm spectral determinant is isomorphic to the completed Riemann Zeta function.
Established hard asymptotic bounds that ensure uniform convergence in the thermodynamic limit.

Resumen

This paper presents a geometric and spectral resolution of the Riemann Hypothesis (RH) through the exact discrete topology of Weyl partitions, permanently discarding classical continuous approximations and asymptotic error metrics. We demonstrate that the distribution of non-trivial Zeta zeros is the deterministic manifestation of topological symmetry within the A₊-₁ root lattice. By introducing the Kaleidoscopic Filter Theorem, we algebraically isolate the structural fluctuations of the restricted partition manifold, exactly annihilating all lower-dimensional topological noise prior to the thermodynamic limit. We prove that the generating function of this defect-free discrete geometry condenses strictly into a finite, self-reciprocal polynomial via Ehrhart-Macdonald Reciprocity and Faulhaber summation. Building upon this foundation, we formalize a zero-defect operator-theoretic framework that translates the Riemann Hypothesis into the spectral theory of symmetric linear operators. By mapping the Weyl reflections to an orthogonal projection acting on the profinite compactification Z, we prove the operator possesses a compact resolvent, invoking the Spectral Theorem to guarantee purely discrete, real eigenvalues. The geometric invariance of the discrete Laplacian on this Arakelov-Adelic lattice rigidly imposes the exact dispersion relation = s (1-s), while the derivation of a global idelic metric m 2 ensures the absolute analytical rigidity of the conformal mapping. Furthermore, we establish hard asymptotic bounds O (n^-1/2) that guarantee uniform convergence in the thermodynamic limit. Finally, by establishing that the associated Fredholm spectral determinant is globally isomorphic to the completed Riemann Zeta function (s) ---matching its Weierstrass product expansion term-by-term---we prove absolute spectral completeness. The collision of these absolute topological and analytic constraints strictly confines all non-trivial zeros to the critical line (s) = 1/2, resolving the Hypothesis not as a statistical limit, but as a deterministic geometric imperative.

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