Thermodynamic Routing of Stale Gradients via Non-Associative Information Geometry: Bypassing the Ring All-Reduce Bottleneck via the $G_2$ Fano-Fisher Metric

Modern large language model training is constrained by the Ring All-Reduce bottleneck: synchronous gradient aggregation forces thousands of GPUs to idle while waiting for the slowest node (the straggler problem). Asynchronous alternatives such as Hogwild! relax this constraint but fail in deep networks due to gradient staleness and divergence at the Edge of Stability.

This paper replaces Euclidean staleness heuristics with a fundamental law of topological physics. We introduce Non-Associative Information Geometry (NAIG) Routing: incoming gradient drifts are projected into the 14-dimensional Lie algebra $\mathfrak{g}_2$ and evaluated not by Euclidean magnitude but by topological contradiction against a ground-truth reference, via the Fano-Fisher metric characterised in (doi:10.5281/zenodo.20076498).

The central result (Theorem 4.1 of Paper 221) is that the Hessian $\Psi(\theta_\mathrm{ref}) = 2V^\top V$ of the associator energy functional has rank exactly 4, with all four non-zero eigenvalues pinned at $8/3$. This rank-4 friction subspace acts as a geometric lock-out: drifts aligned with the Fano-compatible null space ($E_k \to 0$) are smoothly integrated; drifts entering the 4D Information Ridge ($E_k \to E_\mathrm{max} = \tfrac{8}{3}|\Delta g|^2$) are thermodynamically frozen out by the Maslov-Gibbs Einsum (MGE).

NAIG operates as a pure topological control layer over standard Euclidean SGD. The $G_2$ geometry evaluates compatibility (Control Plane); thermodynamic weights $\pi_k = \exp(-\beta E_k)/\sum_j \exp(-\beta E_j)$ are applied to native gradients; standard Async-SGD executes unchanged (Data Plane). No modification to the optimizer, hardware stack, or training hyperparameters is required.

Two experiments are reported. Experiment A (Gram-Schmidt Topological Audit): a controlled falsification audit constructs workers with identical cosine similarity by design (Gram-Schmidt orthogonalisation against the oracle). NAIG correctly assigns zero weight to topologically contradictory workers ($E_k = 9.23$ and $57.68$) while executing Topological Rescue on a highly stale but Fano-compatible worker (lag 35, $E_k = 0.00$) — a distinction entirely invisible to cosine similarity or Hogwild!. Experiment B (GPT-2 cluster): NAIG routing on a 124M-parameter GPT-2 with four asynchronous workers (lags 0, 10, 20, 35) achieves superior stabilisation over Hogwild! ($-31%$ final loss) with a $35{,}000\times$ dimensional compression of the routing signal (effective routing dimensionality: 4, not $5 \times 10^6$). The system auto-anneals without any schedule: $G_2$ geometry spontaneously freezes out contradictions during chaotic exploration and thaws to uniform weighting at the convergence basin.

This paper is part of the Adelic Simplicial Architecture (ASA), Portfolio C (AI & Deep Learning). It depends on the papers: Fano-Fisher Decomposition Theorem (doi:10.5281/zenodo.20076498) and Maslov-Gibbs Einsum (doi:10.5281/zenodo.17981393).