Kevin L. Brown
August 2026
This paper develops an adversarial preservation-and-recovery stress test for Unified Informational Mathematics (UIM), a formal framework derived from the Unified Informational Physics Ontology (UIPO), by applying it to ten landmark problems spanning analysis, partial differential equations, complexity theory, quantum field theory, arithmetic geometry, topology, and number theory.
The central purpose is not to claim solutions to open problems, but to determine exactly what mathematical conclusions can be supported, which representations preserve the structure required by each problem, and where the smallest unresolved proof obligation remains. The ten dossiers address the Riemann hypothesis, three-dimensional Navier-Stokes regularity, P versus NP, Yang-Mills existence and mass gap, the Birch-Swinnerton-Dyer conjecture, the Hodge conjecture, the Poincare conjecture, even Goldbach, Collatz, and the twin-prime conjecture.
A common UIM architecture is imposed across all ten cases. Each problem is expressed through a source domain, an informational representation, a recovery map, and a preservation requirement. Exact recovery is treated as sufficient for transferring isomorphism-invariant properties, while lossy representations are tested through fiber collisions: if two mathematically distinct source objects map to the same representation but differ on the target property, that representation cannot decide the target.
The paper proves several cross-cutting obstruction results. These include the finite-agreement obstruction, showing why finite data cannot establish an unbounded universal claim; a finite-runtime extension theorem showing why benchmarked runtimes cannot by themselves determine asymptotic complexity class; a finite-gap obstruction for continuum spectral claims; scale non-identifiability results; exact finite-certificate soundness; quantifier-order and average-versus-worst-case obstructions; and a theorem showing that downstream computation cannot restore distinctions already discarded by a lossy representation.
Each landmark problem is then given a proof-status ledger. The manuscript establishes valid local results including zeta-zero symmetry, the smooth periodic Navier-Stokes energy identity, P contained in NP, analytic order factorization near the Birch-Swinnerton-Dyer critical point, the Hodge type of smooth cycle classes, a restricted spherical Poincare theorem, bounded Goldbach-certificate soundness, a Collatz descent reduction, and admissibility of the twin-prime pattern. The full Poincare conjecture is treated as an externally solved calibration case through Hamilton-Perelman theory. No new proof of any unresolved headline conjecture is claimed.
The paper also introduces an internal Archion proof-audit protocol that separates document-level completeness from target-level proof completion. A claim remains unproved whenever any essential inference, recovery, invariance, limit, or falsification gate remains open. The result is a reusable research methodology for distinguishing computation from proof, finite evidence from universal theorems, and suggestive representations from mathematically valid recovery. UIM is presented not as a shortcut around established mathematics, but as a disciplined framework for exposing exactly what must still be proved.
A further contribution is a smallest-missing-lemma catalogue that converts each open dossier into a concrete next theorem rather than a vague research direction. The manuscript also identifies a formal-verification kernel suitable for proof-assistant implementation, while requiring imported theorems to remain distinct from locally derived results. This creates a reproducible path from exploratory representation, through adversarial testing, to formal mathematical verification and independent external review.