Sasaki Bridge Theorem: Spectral Gap Unifies Lyapunov Convergence and Commutator Norm Decay in Transformer Attention, with a Commutativity Scaling Law

We present the Sasaki Bridge Theorem, which establishes that under positive spectral gap λgap > 0, the commutator norm ‖Xₕ (t), X*‖F in Transformer attention matrices decays exponentially at rate μ/2, where μ is the Lyapunov decay rate of a structural divergence functional L (Lemma 1–5, proven). The theorem is conditioned on Assumption A4 (dL/dt ≤ −μL), empirically verified in 27. 3% of trials for GPT-2 Small; a weaker condition A4' is confirmed in all tested layers. We further report the Commutativity Scaling Law (CSL): C (W) = 0. 39·W^−1. 14 (R² = 0. 914) across seven Transformer models, where W is the attention window width. The scaling exponent b constitutes an empirical invariant (CV = 5. 0% across GPT-2 sizes) separating architecture classes: b ≈ 1. 05 (causal decoders), b ≈ 0. 65 (alternating attention), b ≈ 0. 35 (bidirectional encoders). Note: the lower bound constant c₁ = 2λgap² is near-vacuous for typical Transformer weights (c₁ ≈ 1. 4×10⁻⁶ for GPT-2 Small) ; the upper bound is the practically informative direction. All main results rely on the upper bound only. We additionally present — explicitly as a conjecture, not a theorem — a three-phase model in which intelligence emergence corresponds to a grokking-type phase transition (Phase 1: transient commutator defect spike, supported by Xu et al. arXiv: 2602. 16967) followed by re-commutatization (Phase 2: governed by the main theorem). This conjecture is not formally proven within the present framework. Scope: The Sasaki Bridge Theorem describes the post-grokking stabilization phase (Phase 2). The pre-grokking commutator defect spike (Phase 1) is outside the theorem's current scope and is identified as primary future work.