The Geometry of Semantic Space: A Continuous Geometric Framework for the Transformer Architecture

Hugging Face Daily Papers Papers
transformer-architecture geometric-framework differential-geometry attention language-models

Summary

Presents a continuous geometric framework modeling Transformer operations as integro-differential equations on a semantic fiber bundle, validated across multiple architectures.

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 calE = calM times R^d. 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\nobreakdash-3.1, Gemma\nobreakdash-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 (R^2 = 1.000), the Lie--Trotter operator-splitting torsion, the symmetric ablation instability confirming the Dual-Law of Topological Stability, the calO(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.
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Paper page - The Geometry of Semantic Space: A Continuous Geometric Framework for the Transformer Architecture

Source: https://huggingface.co/papers/2607.17146

Abstract

WepresentacontinuousgeometricframeworkthatmodelsthediscretealgebraicoperationsoftheTransformerarchitectureasanintegro-differentialequation(IDE)onasemanticfiberbundlecalE=calMtimesR^d.Beginningfromasinglegeometricaxiom--thatthetokensequenceformsadiscrete1-manifoldequippedwithacanonicalmeasurelattice--wetranslateeverycorecomponentofthemodernTransformer(RMSNorm,RoPE,SoftmaxAttention,FFN,ResidualStream,SGD,WeightDecay)intoacohesivevocabularyofdifferentialgeometry,measuretheory,andstochasticcalculus.Theresultingframeworkyieldsquantitativepredictionsspanningentropicoptimaltransport(AttentionasaSchrödingerbridge)andnon-equilibriumthermodynamics(SGDasItôdiffusionviolatingdetailedbalance).Weconductasix-partexperimentalcampaignacrossfivearchitectures(Qwen3,LLaMA\nobreakdash-3.1,Gemma\nobreakdash-3,GPT-2,Mistral)spanning124Mto8Bparameters.Theempiricalobservablesarequantitativelyconsistentwiththegeometricpredictions:theε^{-1/2}Lipschitzscalingcalibrationatmachineprecision(R^2=1.000),theLie--Trotteroperator-splittingtorsion,thesymmetricablationinstabilityconfirmingtheDual-LawofTopologicalStability,thecalO(1/k)thermodynamicsuppressionofPoincarérecurrenceontheRoPEtorus,thethermodynamiccontext-limitphasetransition,andtheNon-EquilibriumSteadyStateparametervortex--verifiedacrosstwooptimizers(AdamWandPureSGD)toexcludemomentumartifacts.TheresultsdemonstratethatanalyzingTransformersthroughthelensofcontinuousstochasticdifferentialgeometryprovidesapredictivedescriptivevocabularyforthestabilitylimits,contextbounds,andoptimizationdynamicsofLargeLanguageModels.

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