What question did this study set out to answer?

This work aims to improve the understanding of symmetric diagonal affine systems through new mathematical approaches.

June 8, 2026Open Access

Exact Finite Packet Sieves and Survivor Convolutions for Symmetric Diagonal Affine Systems

Key Points

This work aims to improve the understanding of symmetric diagonal affine systems through new mathematical approaches.
Consolidated revision of finite local sieve theory and associated mathematical concepts.
Merged previous research to enhance clarity and coherence in presenting findings.
Introduced new techniques that validate finite results independently from classical theories.
Outlined significant relationships between residue-cloud obstructions and finite sieve counts.

Abstract

This version substantially revises and consolidates the previous finite local sieve draft. It presents the work as the merged Paper II of the current program, combining local residue-cloud obstruction geometry with finite packet-sieve counts, survivor convolutions, and fiber-moment identities. The exposition has been recalibrated so that classical analytic theories are used only as external comparison and validation points, not as inputs to the finite results. Previous version of this preprint:https://zenodo.org/records/20447546 Companion preprint:https://zenodo.org/records/20259817

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