Frontier

The Linear Communication Bottleneck Theorem

Spectral Frustration of Procrustes Cocycles on Random Encoder Graphs

DemonstratedTrichotomy proved at paper level (Theorem 5): aligned frustration O(θ²), independent (misaligned) encoders matching the Haar-random O(B) spectral-gap baseline to within O(B/d), and shared-subspace interpolation. Rests on a Haar-universality lemma (the Procrustes polar factor of random encoder overlaps is Haar on O(B), adjacent edges exactly pairwise independent) plus Grassmannian concentration and triangle-holonomy bounds; the paper reports Monte Carlo confirmation of the predictions. Analytical, not mechanized. Harvested from the SHEAF program. Draft, 2026.
Key result

Below a critical subspace-sharing threshold, the communication channel completely randomizes gauge structure regardless of representation dimension d — reducing coordination frustration requires shared representation subspaces, not more bandwidth

Falsification

Independent Haar-random encoders whose Procrustes cocycle deviates from the Haar O(B) spectral-gap baseline by more than O(B/d); or a bandwidth increase (larger B) that reduces frustration below the subspace-sharing threshold

Abstract

When n agents encode d-dimensional representations into B-dimensional messages (B < d) and align them via the orthogonal Procrustes map, the resulting cocycle on the communication graph K_n determines a connection Laplacian whose spectral gap measures the limit of bandwidth-limited coordination. The paper proves a trichotomy: in the aligned regime frustration scales as O(θ²); in the misaligned regime (independent Haar-random encoders) each Procrustes rotation is Haar on O(B) with adjacent edges pairwise independent, and the expected spectral gap matches the Haar baseline to within O(B/d), the correction localized to triangle holonomy; an intermediate shared-subspace regime interpolates. It identifies a sharp phase transition: below a critical subspace-sharing threshold the channel randomizes gauge structure regardless of d.

Resources