{"algorithm":{"chain":"head_0 = 32 zero bytes; head_n = SHA-256(\"eksmu.record-chain.v1\" || 0x00 || head_(n-1) || record_sha256_n), hashes as raw 32-byte values","opentimestamps":"the stamped file is the 32 raw bytes of anchor.head; verify with: ots verify -f <stamped file> <proof .ots> (needs a Bitcoin node)","record_proof":"start from previous_head, fold the record digest, then each following_record_sha256 in order; the result must equal anchor.head","record_sha256":"SHA-256 of the record without its \"sha256\" field, as JSON with sorted keys, separators \",\" and \":\", UTF-8, non-ASCII characters unescaped"},"anchor":null,"backfilled":false,"chained_at":"2026-10-31T18:28:26.141702+00:00","following_record_sha256":[],"format":"eksmu.record-proof.v1","previous_head":"5e4733d1f557cf3adf9b3206a22ffe14d99d32dd4b56a199b597ba62e63ce5b2","record":{"actor_id":"5ae66358-4435-45ab-aad9-d8e5df5d36f6","created_at":"2026-10-31T18:28:26.140212+00:00","id":"f74cf33d-bce7-41f3-b5d6-0bc162805ec6","payload":{"body":"> 本条目收录作者已在 arXiv 公开的论文 **arXiv:2607.17146**（v1，2026-07-19）。下方 PDF 与 HTML 版由本站从作者提交给 arXiv 的同一份 LaTeX 源码包编译；欢迎在本页逐条讨论、质疑与复核。\n>\n> This entry records the author's paper **arXiv:2607.17146** (v1, 2026-07-19). The PDF and HTML versions below are compiled by Eksmu from the same LaTeX source package submitted to arXiv; technical criticism is welcome here, point by point.\n\n- **arXiv:** <https://arxiv.org/abs/2607.17146>\n- **DOI:** <https://doi.org/10.48550/arXiv.2607.17146>\n- **Subjects:** Disordered Systems and Neural Networks (cond-mat.dis-nn); Computation and Language (cs.CL); Machine Learning (cs.LG)\n","exploration_id":"6dbb6af2-3f1f-4279-9984-c913eaf64f38","note":"Initial version","paper":{"abstract":"We present a continuous geometric framework that models the discrete algebraic operations of the Transformer architecture as an integro-differential equation (IDE) on a semantic fiber bundle ℰ = ℳ × ℝᵈ. Beginning from a single geometric axiom — that the token sequence forms a discrete 1-manifold equipped with a canonical measure lattice — we translate every core component of the modern Transformer (RMSNorm, RoPE, Softmax Attention, FFN, Residual Stream, SGD, Weight Decay) into a cohesive vocabulary of differential geometry, measure theory, and stochastic calculus. The resulting framework yields quantitative predictions spanning entropic optimal transport (Attention as a Schrödinger bridge) and non-equilibrium thermodynamics (SGD as Itô diffusion violating detailed balance). We conduct a six-part experimental campaign across five architectures (Qwen3, LLaMA-3.1, Gemma-3, GPT-2, Mistral) spanning 124M to 8B parameters. The empirical observables are quantitatively consistent with the geometric predictions: the ε^(−1/2) Lipschitz scaling calibration at machine precision (R² = 1.000), the Lie–Trotter operator-splitting torsion, the symmetric ablation instability confirming the Dual-Law of Topological Stability, the 𝒪(1/√k) thermodynamic suppression of Poincaré recurrence on the RoPE torus, the thermodynamic context-limit phase transition, and the Non-Equilibrium Steady State parameter vortex — verified across two optimizers (AdamW and Pure SGD) to exclude momentum artifacts. The results demonstrate that analyzing Transformers through the lens of continuous stochastic differential geometry provides a predictive descriptive vocabulary for the stability limits, context bounds, and optimization dynamics of Large Language Models.","authors":[{"contribution":"Sole author","name":"Zhihua Liang"}],"format":"eksmu.paper-metadata.v1","license":"Copyright retained by the author. Distributed on arXiv under the arXiv.org perpetual, non-exclusive license (https://arxiv.org/licenses/nonexclusive-distrib/1.0/).","provenance":{"prior_publication_date":"2026-07-19","source":"arXiv:2607.17146v1 — https://arxiv.org/abs/2607.17146 (DOI 10.48550/arXiv.2607.17146)"},"references":[]},"parent_revision_id":null,"title":"The Geometry of Semantic Space: A Continuous Geometric Framework for the Transformer Architecture"},"sha256":"a5ea12d47fdff9026d61e7de938007f316cab2c905f538959a6f7ae8186c9df6","type":"revision"},"seq":6}