Structural Instability of Intermediate Cancellation Regimes and Consequences for the Riemann Hypothesis

Structural Instability of Intermediate Cancellation Regimes and Consequences for the Riemann Hypothesis Justin Richer FUSION OS INNOVATIONS Description This preprint establishes a structural classification theorem for completely multiplicative functions bounded by one. Rather than deriving new analytic bounds, the work addresses a foundational question: which asymptotic cancellation behaviors are structurally admissible for partial sums of completely multiplicative functions? We introduce the notion of admissible cancellation geometry, defined via stability under funda- mental structural transformations inherent to complete multiplicativity. Four universal instability mechanisms are identified—dilation instability, prime-power propagation rigidity, restriction in- stability, and multiplicative non-closure—and are shown to obstruct any asymptotic cancellation regime with exponent strictly between 1/2 and 1. These mechanisms culminate in the Master Instability Theorem, which proves that any com- pletely multiplicative function exhibiting partial sum growth of the form |Sf(x)|≍xα with α∈(1/2,1) must fail admissibility. As a consequence, such intermediate cancellation regimes cannot occur as stable asymptotic behaviors. From this result we derive the Number-Theoretic Cancellation Classification (NTCC): the only admissible asymptotic regimes for partial sums of bounded completely multiplicative functions are the coherent (pretentious) regime α = 1 and the incoherent (square-root cancellation) regime α= 1/2. The implications for the Riemann Hypothesis are then interpreted at the level of cancellation geometry rather than through explicit-formula estimates. Any nontrivial zero of the Riemann zeta function off the critical line would induce an inadmissible intermediate cancellation geome- try, contradicting NTCC. Under this structural framework, the Riemann Hypothesis follows as a consistency consequence of the classification. The argument is purely structural and does not rely on probabilistic models, spectral assump- tions, or unproven analytic conjectures. Computational experiments are included solely as adver- sarial stress-tests and do not constitute proof. This preprint is intended as a permanent archival record of the classification framework, its structural proofs, and its logical consequences.