Automated Theory Formation via Structural Functor Search, Adversarial Inconsistency Auditing, and Complexity Taxation

## Abstract Contemporary artificial intelligence architectures—including large autoregressive language models (LLMs), gradient-descent optimizers, and dense vector retrieval systems (RAG)—suffer from two fundamental structural pathologies when applied to automated scientific discovery:1. **The In-System Epicycle Trap:** When confronting foundational anomalies that violate an initial model's ontology, continuous optimizers construct *Ptolemaic epicycles* (adding polynomial parameters to fit anomalous observations) because model complexity is metabolically unpenalized.2. **The Semantic Cosine Trap:** Dense vector embedding models (`bge-small-en-v1.5`, OpenAI text-embeddings) retrieve knowledge based on lexical and contextual proximity (cos θ). When the solution to a problem resides in a semantically distant domain that shares exact mathematical structure but zero vocabulary overlap, vector similarity collapses (cos θ ≈ 0), rendering cross-disciplinary breakthroughs invisible. Furthermore, when generative systems *do* propose analogies, they lack internal verification mechanisms, frequently hallucinating mathematically invalid isomorphisms. We address these challenges in the **Epistemic Life Engine**, a neuro-symbolic artificial life framework that couples:* **Structural Functor Search:** Extracting domain-invariant relational category skeletons S(C) and spectral graph invariants to retrieve structural analogies across semantically disjoint corpora.* **Adversarial Inconsistency Auditing:** An active falsification subsystem (composed of "Killer Cells" and automated SMT/relational model checkers) that tests candidate analogies against domain axioms, identifies concrete counterexamples, and drives autonomous repair loops.* **Algorithmic Information Economics:** A metabolic energy budget penalizing tunable parameter complexity (M(T) = α |V| + β k1.5), systematically starving over-parameterized epicycles in favor of minimal-parameter symmetry groups. We demonstrate the framework across seven foundational impasses, culminating in two closed-form, machine-verified discoveries: (1) the Universal Grassmannian Quorum Theorem in $PG(5, 4)$, which resolves the 25-year symmetry-efficiency dilemma by achieving O(√N) scaling with 100% $PGL$ transitivity; and (2) non-distillative fault-tolerant quantum universality, circumventing the Eastin–Knill barrier via 3D/2D gauge color code switching ([[15, 1, 3]] ↔ [[7, 1, 3]]), evaluated across all 16,384 trajectory branches to exact machine fidelity ($F = 1.000000000000$) in $0.33$ seconds using an $O(1)$ even-weight feed-forward decoder. We report on eleven reproducible computational evaluations across four core foundations and seven advanced grand frontiers:1. **Topological Impasse Resolution (K3,3 Kuratowski 3D Lift):** Dissolving non-planar crossing frustration (Φ → 114.59) via orthogonal dimensional lift ($cr = 0, E = 2.88$).2. **Universal Property Verification (Colimit Commutativity):** Verifying categorical pushouts to exact machine precision (\|u ∘ j - h\| < 10⁻¹⁶, generating 170 emergent cycles).3. **Occam–Gödel Pareto Optimization (Epicycle Starvation):** Total metabolic extinction of 11 polynomial epicycle models ($k=1$ to $55$) in favor of single-parameter Lorentz invariance (SO(1,1), BIC = -1107.85).4. **Information Retrieval Benchmark (11 Real Disciplines, 38 Corpora):** Structural Functor Search achieves **$100.0%$ Precision@1 and $1.000$ MRR**, whereas dense vector retrieval collapses to **$18.2%$ Precision@1** due to lexical blindness.5. **Grand Frontier 1 (Unsupervised Discovery & Stratification in T₃₂):** Auditing all 465 primitive binomials from raw multiplication tables, discovering all **294 zero divisors** partitioned into three exact topological strata (84 of N4, 84 of N8, 126 of N12) governed by the exact trace invariant L(eᵢ+eⱼ) = (32 + (Q²))/4 (100.0% verification).6. **Grand Frontier 2 (Dynamical Superselection & The Auditor Centrality Filter):** Demonstrating how the Adversarial Auditor catches that naive Fano commutator sums produce a central scalar matrix (HFano = 48 I₃₂), rejecting trivial commutation ($[cI, P]=0$), and evaluating genuine non-scalar cross-sector tunneling where Nullity 4 zero divisors achieve a **2.24× mean (4.61× peak) topological protection advantage** over Nullity 12.7. **Grand Frontier 3 (Scaling to 64 Dimensions — T₆₄ Chingons):** Auditing all 1,953 binomials in 0.08 seconds, discovering **1,518 zero divisors** across 7 nullity strata with an exact symmetric 6-strata core $[252, 168, 168, 168, 168, 252]$ and terminal stratum 342, verifying the generalized trace theorem = (64 + (Q²))/4 to 100.0% accuracy.8. **Grand Frontier 4 (The Projective Dimensional Lift — PG(2,2) → PG(3,2) BFT Masking):** When the Adversarial Auditor catches that the 2D Fano plane fails $f=1$ BFT because |Lᵢ ∩ Lⱼ| = 1 < 3, the engine executes a dimensional lift into 3D projective space $PG(3, 2)$, where 15 hyperplanes of size 7 with pairwise intersection strictly 3 **provably achieve full unconditional $f=1$ Byzantine Masking**.9. **Grand Frontier 5 (The 4-Term Quadrinomial Barrier & 42-Orbit Universality):** Auditing all 31,465 primitive 4-term elements in T₃₂, proving that all 12,264 zero divisors partition into exact multiples of 42 (191 × 42, 77 × 42, 24 × 42), and autonomously synthesizing 4th-degree irrational spectral projectors (λ = 4 ± 2√2) to 10⁻¹⁵ machine precision.10. **Grand Frontier 6 (The Frontier Limit-Breaker — Parity Wall, 128D/256D Algebras, & The Tower Law):** Proving the Odd-Parity Regularity Theorem across 174,406 multi-term sums ($k=3, 5$), auditing 8,001 pairs in T₁₂₈ and 32,385 pairs in T₂₅₆, and establishing the closed-form Cayley–Dickson Tower Law (V(n) = 42 × 4ⁿ⁻⁴) and asymptotic invertibility decay (limD → ∞ ρ(D) = 1.0).11. **Grand Frontier 7 (Non-Distillative Quantum Fault Tolerance — 3D/2D Gauge Color Code Switching):** Circumventing the Eastin–Knill barrier without magic-state distillation by executing dynamic dimensional gauge switching between the 3D tetrahedral [[15, 1, 3]] Reed–Muller code (where the non-Clifford T-gate is transversal) and the 2D [[7, 1, 3]] Steane code (where Clifford gates are transversal). Proving the Even-Weight Commutation Invariant [CZ, XL⁽⁷⁾] = 0 via the punctured $RM(1,3)$ syndrome matrix M, achieving **exact $1.000000000000$ logical fidelity across all 16,384 trajectory branches in 0.33 seconds** on full 32,768-dimensional Hilbert space state vectors. All algorithmic hyperparameters, scope boundaries, and verification harnesses are disclosed in full.