POLYDIM & EinsofOS: Breaking the 1D Token Bottleneck via Continuous Stiefel Manifold Geometry

Community
1

Summary: Breaking the 1D Token Serialization Bottleneck

Modern Large Language Models are universally constrained by 1D discrete tokenization: continuous, high-dimensional latent representations inside deep neural networks must collapse into scalar integer token IDs to cross process, network, or agent boundaries—only to be re-embedded at the next node. This incurs severe Shannon Data Processing Inequality (DPI) entropy loss, latency serialization tax, and quadratic API overhead.

POLYDIM / EinsofOS introduces a continuous geometric paradigm: inter-agent communication directly over the Stiefel manifold St(k, n) = {X in R^(n x k) : X^T X = I_k}. By combining isometric Lie algebra generators, compensated Neumaier summation (numerical drift <= 10^-15 in FP64), and native Shared Memory slabs (PmtpShmBus), latent activations travel vector-to-vector with zero internal tokens serialized.

Key Theoretical & Empirical Highlights

  1. Zero-Token Latent Telepathy: Verified in native silicon across an 8-hop multi-agent vector swarm:
    • Mini SLM Nexus: Qwen 2.5 Coder (896 → 1024) → Qwen 3.8 / 27B (5120 → 2048) → DeepSeek-v4 Distill (5120 → 2048).
    • Hardware Contract: Full C-ABI interop in local execution (llama_batch with token = NULL and embd != NULL), enabling continuous prefix tuning and zero token billing between sovereign nodes.
  2. Formal Foundation: 1,740 formally certified theorems in PostgreSQL 16 proving isometric volume conservation, Riemannian QR polar retractions, and spectral stability under adversarial noise (10^-310).
  3. Open Weight Adapters: LoRA adapters trained on the Stiefel manifold and published openly on Hugging Face:
    • Frontier Models: Qwen 2.5 72B Instruct, LLaMA 3.3 70B Instruct, DeepSeek-R1 (14B/32B/70B), Nous Hermes 3 (8B/70B), Phi-4 (14B), and Gemma 2 (9B/27B).
    • Vector Bridges: SLM adapters (Qwen 2.5 0.5B/1.5B/7B, LLaMA 3.2 1B/3B) acting as real-time continuous routers between cloud orchestration planes and sovereign local hardware.

Primary Canonical Links (Open Access)

(Additional canonical citations: HAL CNRS hal-05847496v1 | JST Jxiv 7040 | OSF osf.io/za3s7 | GitHub github.com/AGT1973/POLYDIM_EinSof | Dissertation DOI 10.5281/zenodo.21136742)

We invite the Google AI community, mathematicians, and systems researchers to audit the mathematical proofs, test the Colab benchmarks, and evaluate the continuous vector transfer paradigm.

(Author: Ariel García Traba — Independent Research / EinsofOS)