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arXiv · 2511.11573

Softmax as a Lagrangian-Legendrian Seam

Christopher R. Lee-Jenkins
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Abstract

This note offers a first bridge from machine learning to modern differential geometry. We show that the logits-to-probabilities step implemented by softmax can be modeled as a geometric interface: two potential-generated, conservative descriptions (from negative entropy and log-sum-exp) meet along a Legendrian "seam" on a contact screen (the probability simplex) inside a simple folded symplectic collar. Bias-shift invariance appears as Reeb flow on the screen, and the Fenchel-Young equality/KL gap provides a computable distance to the seam. We work out the two- and three-class cases to make the picture concrete and outline next steps for ML: compact logit models (projective or spherical), global invariants, and connections to information geometry where on-screen dynamics manifest as replicator flows.

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Christopher R. Lee-Jenkins. 2025-09-17. Softmax as a Lagrangian-Legendrian Seam. https://arxiv.org/abs/2511.11573

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