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 ($\cos \theta$). When the solution to a problem resides in a semantically distant domain that shares exact mathematical structure but zero vocabulary overlap, vector similarity collapses ($\cos \theta \approx 0$), 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 $\mathcal{S}(\mathcal{C})$ 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 ($\mathcal{M}(T) = \alpha |V| + \beta k^{1.5}$), 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 $O(\sqrt{N})$ 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 ($[[15, 1, 3]] \leftrightarrow [[7, 1, 3]]$), 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 ($K_{3,3}$ Kuratowski 3D Lift):** Dissolving non-planar crossing frustration ($\Phi \to 114.59$) via orthogonal dimensional lift ($cr = 0, E = 2.88$).2. **Universal Property Verification (Colimit Commutativity):** Verifying categorical pushouts to exact machine precision ($\|u \circ j - h\| < 10^{-16}$, 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 ($SO(1,1), \mathrm{BIC} = -1107.85$).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 $\mathbb{T}_{32}$):** 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 $\dim\ker L(e_i+e_j) = (32 + \tr(Q^2))/4$ (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 ($H_{\mathrm{Fano}} = 48 I_{32}$), rejecting trivial commutation ($[cI, P]=0$), and evaluating genuine non-scalar cross-sector tunneling where Nullity 4 zero divisors achieve a **$2.24\times$ mean ($4.61\times$ peak) topological protection advantage** over Nullity 12.7. **Grand Frontier 3 (Scaling to 64 Dimensions — $\mathbb{T}_{64}$ 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 $\dim\ker = (64 + \tr(Q^2))/4$ to 100.0% accuracy.8. **Grand Frontier 4 (The Projective Dimensional Lift — $PG(2,2) \to PG(3,2)$ BFT Masking):** When the Adversarial Auditor catches that the 2D Fano plane fails $f=1$ BFT because $|L_i \cap L_j| = 1 < 3$, 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 $\mathbb{T}_{32}$, proving that all 12,264 zero divisors partition into exact multiples of 42 ($191 \times 42$, $77 \times 42$, $24 \times 42$), and autonomously synthesizing 4th-degree irrational spectral projectors ($\lambda = 4 \pm 2\sqrt{2}$) 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 $\mathbb{T}_{128}$ and 32,385 pairs in $\mathbb{T}_{256}$, and establishing the closed-form Cayley–Dickson Tower Law ($V(n) = 42 \times 4^{n-4}$) and asymptotic invertibility decay ($\lim_{D \to \infty} \rho(D) = 1.0$).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 $[C_Z, X_L^{(7)}] = 0$ 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.