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arXiv · 2610.38538

Curvature-Dimension Conditions Along Discrete Refinement Sequences

Chen Tao
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Abstract

We introduce the sequence curvature--dimension condition $SCD(K,N)$ for complete refinement systems of finite metric probability spaces. The condition couples a transport action bound with entropy comparisons along a single probability section through the prescribed refinement layers. Within a bounded comparison range, we prove that midpoint comparisons on three consecutive layers characterize the condition on every fixed finite segment. If, in addition, the refinement sequence converges in the measured sense in its intrinsic bridge realization, then $SCD(K,N)$ is equivalent both to the vanishing of single-layer midpoint defects and to the entropic curvature--dimension condition $CD^{e}(K,N)$ of the limit. In the reverse direction, a $CD^{e}(K,N)$ bound on the limit yields quantitative recovery estimates that identify shrinking transport scales on which finite layers retain its second-order curvature comparison. These estimates are uniform over endpoint measures and include Dirac masses. Examples illustrate the effects of the refinement rule, the layer metrics and the reference measures.

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Chen Tao. 2026-10-28. Curvature-Dimension Conditions Along Discrete Refinement Sequences. https://arxiv.org/abs/2610.38538

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