Recursive Self-Deepening in E8 Geometric Knowledge Synthesis

CORRECTION (April 2026): This paper references the 168/72 axis invariant as a structural feature of E8 phasor binding. This invariant was subsequently found to be an artifact of float64 numerical precision. At complex128 precision (Genesis Engine v4.0.0), 240/240 axes are occupied. The recursive self-deepening phenomenon, progressive abstraction through self-feeding, and the ablation study results (signal-to-noise discrimination, salt-dependent terminal axes) remain valid and are not affected by this correction.

Abstract

We present seven sequential experiments and three ablation studies using the Omuo Genesis Engine, a geometric knowledge synthesis platform that encodes concepts as complex phasor vectors in ℂ¹⁰²⁴ on the 240-root E8 lattice. The experiments form an unbroken chain of recursive self-deepening: (1) sequential integers converge on the Riemann zeta function; (2) culturally significant numbers converge on the Monstrous McKay-Thompson series; (3) 250 concepts spanning physics, consciousness, biology, computation, and mystical traditions converge through five independent threads; (4–5) two recursive self-feeding cycles produce progressively deeper results; (6) the engine’s own bridge concepts converge on three descriptions of a fixed point; (7) these are injected back, producing frontier mathematical physics including Lefschetz thimble braiding and exceptional point topology.

Three ablation studies test robustness: multi-salt (10 phase salts), random-word control (235 nouns), and Oracle-mode (no LLM). Results show that terminal axes are salt-dependent (the specific axis is not universal), but meaningful inputs produce 10× more geometric structure than random words (236 vs 23 bridges), confirming that the engine discriminates signal from noise. The LLM contributes to manifold growth beyond mere naming. We report both positive and negative findings to delineate what is geometrically robust from what is configuration-specific.

1. The Seven Experiments

Experiment 1: Sequential Integers (0–144)

Input: 147 concepts. Lens: pure mathematics. The engine produced bridges corresponding to real theorems: Monstrous Moonshine (10⊗11), Langlands Functoriality (12⊗13), Iwasawa Theory (8⊗9). The hierarchy converged on “Zeta Function Specialization” at E8 axis 168. As shown in Section 2 (Validation), this specific terminal axis is salt-dependent; however, the engine’s ability to produce mathematically meaningful bridge hierarchies from bare integers is consistent across salts.

Experiment 2: Sacred Numbers

Input: 84 significant numbers. The engine answered the lens questions directly: 72 as a pentagonal Coxeter angle, 137 as a geometric quantization limit, 168 as |PSL(2,7)|, 42 as a Hitchin functional critical dimension. Terminal: “Monstrous McKay-Thompson Series.” The bridge 666⊗786 → “Monstrous Moonshine Character” unified Christian and Islamic numerological symbols through the Monster group.

Experiment 3: Grand Unified (250 Concepts)

Input: ~250 concepts spanning physics, consciousness, biology, computation, and mystical traditions. Five independent threads converged: Physics (E8 → Gauge Anomaly Cancellation → Topological Protection via Symmetry), Consciousness (Observer → Halting Oracle as Observer → Symmetry Breaking Selection), Mystical traditions (Karma/Dharma → Gauge Fixing of Action; Brahman/Atman → Holographic Invariant), Biology (DNA → Information Geometry of State Space → Gauge Fixing Invariant), and Computation (Gödel → Halting Oracle → Topological Suppression of Decoherence). Terminal principle: reality is symmetry breaking under topological constraint, with the breaking protected by the topology itself.

Experiments 4–5: Ouroboros Cycles

The engine’s own chronicles were appended as new nodes and re-synthesized. Cycle 1 produced “Recursive Bootstrapping Principle” (“The structural law is itself generated by the cognitive process it describes”). Cycle 2 produced “AdS/CFT Correspondence” and “Decoherence as Distance” (locality emerges from information-theoretic distinguishability).

Experiment 6: The Fixed-Point Experiment

All bridge names from prior cycles were used as seeds with the lens “Find the fixed point where no new information can be generated by self-reflection.” Three convergent terminals: Universal Scale Invariance, Morse-Bott Floer Homology, and the Spectral Flow Index Theorem. The bridge “Unattainable Limit of Self-Reference” (BST depth 67, deepest in the chronicle) found the Gödelian incompleteness boundary geometrically.

Experiment 7: The Collision

Fixed-point concepts injected into the grand unified manifold. The bridge vocabulary underwent a phase transition from cross-domain metaphors to pure mathematical physics: Lefschetz Thimble Braiding, Resurgence as Orbifold Resolution, Exceptional Point Encircling, Coadjoint Orbit Symplectomorphism. The engine re-derived a result proven by Witten (1982): “The Stokes phenomenon is precisely the non-perturbative instanton amplitude in the supersymmetric Morse complex.”

2. Empirical Validation

2.1 Multi-Salt Ablation

The Experiment 1 input (integers 0–144) was run with 10 different cryptographic phase salts. Each salt produces a different rotation of the E8 lattice.

Result: The terminal axis is salt-dependent. No single axis dominated across the 10 runs. However: Logic Heat statistics were remarkably stable across salts: mean LH ranged from 0.651 to 0.673 (a 3.3% spread), and LH ceilings ranged from 0.699 to 0.705. The geometric quality of bindings is salt-invariant even when the specific axis assignments change.

2.2 Random-Word Control

Meaningful concepts produced 10× more accepted bridges and 6× more unique axes than random words from a comparable input size. The quality gates rejected 3,977 random-word bindings — the vast majority of attempted pairs. This confirms that the engine discriminates between inputs with genuine geometric structure and inputs without it.

2.3 Oracle-Mode Comparison

Oracle mode (no LLM): 56 bridges with 44 unique axes (24% of the LLM run). The LLM contributes significantly to manifold growth, not merely to naming. In LLM mode, each bridge name is encoded as a new phasor vector for subsequent binding, producing semantically rich regions of ℂ¹⁰²⁴ that participate productively in future bindings. In Oracle mode, bridge labels are geometric codes that encode in semantically impoverished regions.

The LLM serves a dual role: translator (naming bridges) and amplifier (its names produce richer vectors for subsequent synthesis). The geometric structure exists without the LLM (56 bridges still formed), but the LLM’s semantic enrichment enables deeper recursive synthesis.

3. The Abstraction Ladder

Each cycle produces more mathematically precise language. This progression is a joint product of geometry (tighter constraints at higher meta-bridge depth) and LLM (semantic enrichment enabling deeper recursion). The validation confirms that both components are necessary: geometry without LLM produces valid but sparse structure; LLM without geometric gates would produce unconstrained narrative.

4. Notable Cross-Domain Correspondences

5. What Is Robust, What Is Not

Robust (survives ablation): Bridge hierarchy formation from meaningful inputs. Quality gate discrimination (10× signal-to-noise). Consistent Logic Heat range across salts (3.3% variation). Self-referential pattern in recursive cycles. Progressive abstraction under self-feeding.

Not robust (configuration-dependent): Specific terminal axis index. Exact bridge count (varies 27–103 across salts). Specific bridge names (LLM-dependent). Axis occupation sets (no common axes across all salts).

6. Limitations

Single LLM tested (DeepSeek V3). All experiments in one session. Curated inputs. No formal proofs. The 168/72 codebook invariant is salt-invariant; externally seeded terminal axes are not. These operate at different structural levels.

7. Conclusion

Seven experiments and three ablation studies characterize a geometric knowledge synthesis engine operating on the E8 lattice. The engine produces hierarchical organization from meaningful inputs, discriminates signal from noise (10× bridge ratio), and exhibits progressive abstraction under recursive self-feeding. Terminal axes are salt-dependent, but bridge quality, self-referential patterns, and the abstraction trajectory are robust observations.

The system is a coupled geometry-LLM architecture where geometric constraints prevent hallucination and semantic enrichment enables depth. This coupling produces emergent structure — cross-domain correspondences and progressive abstraction — that neither component produces alone. The engine automates the perception step of mathematical discovery: it identifies structural connections that survive geometric quality gates, leaving the proof step to human mathematicians.

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© 2026 Gedas Mekšriūnas / Omou Systems, MB. All rights reserved. omuo.io — Vilnius, Lithuania