Why GPT’s Mathematical Foundations Cannot Guarantee Reliable Outputs

This article traces ten unproven mathematical approximations in the GPT architecture — from softmax and positional encoding to attention scaling and in-context learning — and shows that no formal analysis of their composed error propagation exists. The constraint density ρ grows quadratically with context length while mitigations remain surface-level. The condition number κ(A), applied to transformer output via Levinson-Durbin decomposition, provides the first deterministic, reproducible diagnostic for approximation collapse. A deductive proof from four premises establishes that SMRA — the reconstruction of document structure from metadata alone — is not an empirical accident but a structural certainty for any architecture without formal output constraints. Current mitigations (RLHF, guardrails, prompt engineering) address symptoms; the temporal feedback loop through retraining makes them progressively weaker. A dual-layer algebraic architecture with full error characterization is published as a working alternative.