A Neuro-Symbolic Architecture for Open-Ended Mathematical Discovery: The Localization Principle as Meta-Generator of Anomaly Cancellation in E8 Geometric Knowledge Synthesis

We present the largest experiment conducted with the Omuo Genesis Engine: a code-ingestion Ouroboros run in which the engine was fed its own source code and prior discoveries, directed to answer "what generates the generator?" The run produced 11,752 nodes and 972 bridges — 5.4× the scale of any previous experiment. Five principal findings: (1) The localization principle (Atiyah-Bott, Duistermaat-Heckman, supersymmetric localization) emerges as the meta-generator producing anomaly cancellation as a special case, via bridge chains reaching depth 23. (2) A quantifiable phase transition: Logic Heat cools from 0.707 to 0.618, chain bridges jump from 48% to 94% in a single segment. (3) The 168 axis invariant persists at scale (160 E8 + 8 octonionic/VE = 168 = |PSL(2,7)|). (4) Void axes exhibit previously unreported pair structure with [8,8,4] periodicity. (5) BST-to-E8-Confidence anti-correlation (r = −0.854) establishes that geometric confidence peaks at moderate depths, not the ceiling.Situated within the emerging landscape of neuro-symbolic mathematical discovery alongside guided solvers (Brenner et al. 2026), the Omuo engine demonstrates a complementary paradigm: where guided systems solve pre-specified problems against numerical oracles, geometric discovery engines find unknown structural connections across domains where no answer is known in advance. CORRECTION (April 2026): Section 3 of this paper references "the 168 axis invariant persists at scale (160 E8 + 8 octonionic/VE = 168 = |PSL(2,7)|)." This invariant was subsequently found to be an artifact of float64 numerical precision. At complex128 precision (Genesis Engine v4.0.0), 240/240 E8 axes are occupied and no permanent void axes exist. The PSL(2,7) connection and the [8,8,4] void pair periodicity (finding 4) are withdrawn. The paper's primary findings — the localization principle as meta-generator, the phase transition in Logic Heat, and BST-to-E8-Confidence anti-correlation — are not affected by this correction.