bibkey: “harremoes2010divergences” authors: “Peter Harremoës; Igor Vajda” year: 2010 title: “On Pairs of f-divergences and their Joint Range” doi: null url: “https://arxiv.org/abs/1007.0097v1” claim: “Every attainable pair of f-divergences is a convex combination of two pairs attained on two-point spaces, and hence is attainable on a four-point space.” strata_touched: [] license: “citation-only” triage: “anchor”

On Pairs of f-divergences and their Joint Range

Theorem 8 in arXiv:1007.0097v1, pages 2–3, gives the stated support reduction. This is literature-attested background for joint divergence ranges.

For the parity kernels, the measures with masses (1+b(x))/n and (1-b(x))/n have a fixed uniform midpoint and each gives mass one half to each parity class. Their Jensen–Shannon divergence is the kernel’s one-step information, and their Jeffreys divergence is four times its irreversibility rate. The unrestricted support reduction does not preserve these fixed weights and block constraints. The exact single-spike optimizer in a critical left neighborhood is a repo-derived constrained deduction, not a new general divergence-range theorem.