Modern Mathematical Physics

We introduce a toroidal representation of Bekenstein capacity for finite near-horizon quantum information. The energy E of a bounded region defines an associated Schwarzschild length , an energy-associated length that need not equal a horizon radius. Together with the operational radius R, it yields the two-cycle capacity area . As a modeling choice, we promote these two physical scales into the noncontractible cycles of a minimal genus-one code support. The result is a finite-capacity toric-code model in which the intersection form fixes the logical algebra, the capacity conditionally bounds admissible microscopic support and code distance, and an explicit two-logical-qubit Clifford model illustrates sequential purifier transfer without double-counting the interior purifier. The same structure gives a laboratory-testable quantum information analogue: anisotropic logical protection and code-deformation transfer in finite surface-code memories.

We investigate the non-linear dynamics of de Broglie gravitational waves within a post-Minkowskian framework. Starting from the perturbative solution proposed by Feoli and Scarpetta, we derive the second-order gravitational field generated by the wave’s self-interaction. Owing to the extremely high oscillation frequency of de Broglie gravitational waves, we employ the Isaacson/Misner–Thorne–Wheeler averaging procedure to obtain an effective coarse-grained geometry. The resulting averaged theory takes the form of an effective linear perturbation theory at second order in the wave amplitude. Near the symmetry axis, the resulting spacetime exhibits a non-vanishing static curvature generated entirely by non-linear effects. We show that geodesics in this effective geometry undergo bounded transverse oscillations governed by a harmonic equation. Possible implications for de Broglie’s wave–particle framework are also discussed.

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