arXiv:2608.10288 · v2
Power law graph attention: exact generalization of scaled dot-product attention, empirical collapse at inference
摘要
The Large Language Model from Power Law Decoder Representations (PLDR-LLM) and its attention, Power Law Graph Attention (PLGA), replace the fixed bilinear form of scaled dot-product attention (SDPA) with a learned, input-generated bilinear operator $G_{\mathrm{LM}}$, built from a positive tensor $A_{\mathrm{LM}}$ by elementwise power laws. The architecture is fully specified, verified against pinned reference releases; claims are labeled theorem, conditional theorem, measurement, or conjecture. Unconditionally: PLGA contains SDPA exactly at $G_{\mathrm{LM}}=I$; $A_{\mathrm{LM}}$ and $A_{P}$ are strictly entrywise positive, with Perron-Frobenius structure on $A_{\mathrm{LM}}$; the DAG regularizer has the NOTEARS walk-counting form and positivity obstructs exact acyclicity; and, under nonresonance (satisfied by standard rotary frequencies), a commutant criterion identifies which operators preserve relative-position dependence. An inference-collapse theorem: exact input invariance of the head score operators $G_{\mathrm{LM}}$ collapses inference to generalized SDPA with a constant operator per head. Measured invariance: relative fluctuations of $10^{-6}$ and below; perturbation bounds quantify but do not certify cached inference; the assembled proxy misses the decoding margin. A conditional three-stage mechanism (rotary twirl, concentration, row-map contraction) is measured on a released checkpoint. Blockwise training and scoring under the global Gram are stated with explicit target exposure; on tested samples, block and sequential scoring select identical answers and agree on the published TruthfulQA probability-mass metric within $5\times10^{-5}$ per item. Self-organized criticality enters as a phenomenological framework with an intrinsic order parameter; open claims become falsifiable conjectures. Selected proof cores are machine-checked in Lean 4.
作者备注：66 pages, 1 figure, 8 tables. Revised with clarifications and improvements to content and presentation based on the published book "Power Law Graph Attention and PLDR-LLMs: Mathematical Foundations, Training Dynamics, and Predictive Inference" (2026), which incorporates this manuscript