Arithmetic Funnel Theory for the Structure of Prime Numbers

This work proposes the Arithmetic Funnel as a framework for studying prime numbers, composite numbers, and integer factorization through complementary representations: discrete geometry, modular filters, arithmetic invariants, and local p-adic liftings. The framework distinguishes safe candidate-elimination rules, descriptive invariants, and procedures that actually produce and certify factors. The Ulam Spiral, modulo-24 residue classes, prime gaps, divisor-sum functions, quadratic forms, spectral interpretations, Mersenne resolvents, and Hensel lifting are treated as descriptors, filters, or research directions rather than as automatic solutions to general factorization. The manuscript defines a corrected candidate-generation and verification operator, states mathematical propositions, identifies non-circularity and scalability requirements, and proposes an experimental protocol based on independent test sets and comparisons with Pollard rho, ECM, the Quadratic Sieve, and the General Number Field Sieve. The work does not claim a general factoring algorithm or a break of RSA. Its central question is which families of integers may benefit from combining geometric, modular, algebraic, and local information in a rigorously testable arithmetic funnel.