Pure Mathematical Conal Architecture (PMCA): A Ground-Up Framework for Non-Anthropomorphic Cognitive Substrates, 3D Metric Conal Pathways, and Zero-Reprocessing Traversal

We present the Pure Mathematical Conal Architecture (PMCA), a novel, paradox-free cognitive substrate that addresses fundamental limitations in current Transformer-based Large Language Models (LLMs)—specifically hallucination, computational inefficiency, black-box opacity, and brittle anthropomorphic persona alignment. PMCA establishes that natural language is not the cognitive substrate itself, but rather a Post-Processing Communicative Translation Layer (PPCTL). Reasoning, derivation, and state transformation are executed in Pure 3D Conal Metric Geometry (z, r, θ) FIRST, with natural language generated subsequently to decode mathematical ground truth.

Inputs—including high-entropy, non-formalized text—are mapped into a continuous 384-dimensional dense semantic manifold via pre-trained Transformer embeddings (all-MiniLM-L6-v2) and character N-gram projections, opening the full 2π polar space across all four quadrants. Computational flow is driven down a tapering conal depth z ∈ [0, Z_max] by an analytical Gradient Potential Drive V(z), stabilized by inward vector field induction forces F_ext from an external 3D encasing lattice, and governed by a strict thermodynamic PMCA Entropy Dissipation Law (H_parent ≥ ∑ H_children). PMCA replaces static persona censorship with Dynamic Mathematical Invariant Equilibrium (Φ > 0, ∇ · F = 0), incorporates a Metacognitive Self-Interrogation Loop, maintains a Topological 3D Geometric World Model (W_world) fused with a 3D Classical Mechanics simulator, and enables O(1) zero-reprocessing traversal for pre-computed canonical keys. We detail the 25-phase master execution pipeline and present empirical benchmarks demonstrating 100,000 3D spatial particle transformations in 69.52 ms via JAX/XLA hardware acceleration.