Transformers as In-Context Mesa-Optimizers: Canonical Invariants, Tripartite Regime Decomposition, and The Lie Commutator of Causal Order-Dependence

The emergent capability of autoregressive transformers to perform In-Context Learning (ICL) without physical parameter adaptation is among the most profound open problems in machine learning. While prior constructive proofs (von Oswald et al., 2023; Akyürek et al., 2023) established that linear attention can emulate gradient updates, pre-trained language models diverge from classical gradient descent due to severe demonstration order-dependence and output distribution discrepancies. In this work, we present an exhaustive architectural and algebraic audit of the Attention-Gradient Descent equivalence across three rigorous epistemic tiers: Tier 1 (Constructive Realizability): We prove that canonical Transformer attention with an identity residual connection requires a Tri-Register Hidden Dimension (d ≥ d_x + 2d_y) to prevent catastrophic label memory corruption across multi-layer unrolling, enabling time-invariant frozen projection weights to compute exact L-step gradient descent trajectories with machine precision (||y_attn - y_gd||_inf < 2.66 × 10-15). Tier 2 (Architectural Invariant Classification): We formalize the unmasked query self-attention bias, proving that evaluating the diagonal induces an implicit directional Tikhonov (L2) shrinkage penalty. Resolving the asymptotic transition from Softmax Attention as β 0, we derive the first-order Taylor expansion proving that high-temperature attention computes Joint Slope-Intercept Gradient Descent with Automatic Feature Centering. Tier 3 (Mechanistic Reality & Order-Dependence): We formulate the Tripartite Algorithmic Decomposition Theorem and prove that the order-dependence of causal attention does not exist at depth L = 1 without positional encodings, but emerges strictly at depth L ≥ 2, where the permutation trajectory discrepancy is governed by the Lie bracket of sample Hessians [H_a, H_b] = x_a x_aT x_b x_bT - x_b x_bT x_a x_aT. Backed by exact symbolic derivations and an automated mathematical test suite.