International Journal of Research in Science and Technology (IJRST)

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The Critical Line from First Principles: A Complete Unconditional Liouville-Collar Closure of the Riemann Hypothesis

Deep Bhattacharjee

Electro-Gravitational Space Propulsion Laboratory (EGSPL), Bhubaneswar, Odisha, India

9-54

Vol: 16, Issue: 2, 2026

Receiving Date: 2026-02-15 Acceptance Date:

2026-03-28

Publication Date:

2026-04-11

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http://doi.org/10.37648/ijrst.v16i02.002

Abstract

Let X be a smooth complex projective variety and let Hdg?(X) = H²?(X, ?) ? H?,?(X, ?) denote the space of rational Hodge classes of codimension p. This paper formulates the zero-defect support-descent mechanism as a Chow-level realization procedure for rational Hodge classes. André’s motivated-cycle theorem places every class ? ? Hdg?(X) inside a finite support presentation whose edges consist of algebraic correspondences, pull-backs, push-forwards, cup-products with divisor classes, and inverse Lefschetz operators. The obstruction to algebraicity is measured by a defect filtration: a defect is contributed exactly by an inverse Lefschetz operator carried by a non-abelian support.

The closure mechanism is the abelian lowering move. Each non-abelian inverse-Lefschetz edge is factored through an auxiliary abelian variety A? by algebraic correspondences ??,? = (P?)* ? ?_A?,?? ? (Q?)*. Because Lieberman’s theorem makes ?_A?,?? algebraic on abelian varieties, every lowered edge becomes a Chow-level correspondence. Finite lowering reduces the defect to zero, and upward induction on the resulting graph produces a cycle Z ? CH?(X)? with cl(Z) = ?. The final closure identity is therefore cl(CH?(X)?) = Hdg?(X).

The strengthened form is expressed as universal algebraic integrability of Hodge classes. The comparison morphism from Chow correspondences to Hodge correspondences is identified on every product X × Y, giving the functorial equality Hom_Mrat(h(X), h(Y)) ? CH??? X(X × Y)? ? Hdg??? X(X × Y) ? Hom_MHdg(h(X), h(Y)). Thus the zero-defect calculus identifies the Chow-motive and Hodge-motive realization on the generated support category. The admissible class CHC, the abelian-motive cases, and the universal zero-defect descent are synthesized into a single unconditional closure formalism for converting rational Hodge classes into rational algebraic cycle classes.

Keywords: Riemann Hypothesis; Liouville function; spectral large sieve; Kuznetsov formula; collar dispersion.

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The Critical Line from First Principles: A Complete Unconditional Liouville-Collar Closure of the Riemann Hypothesis

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