The claim
CLAIM-387 showed that the optimal partitions of this source at two budgets are incomparable. This claim is the same for the unrestricted information bottleneck, whose optima are stochastic encoders.
- Criterion (THEORY-753, after Charvin et al., Propositions 1, 4 and 8). The bottleneck is refinable from λ₁ to λ₂ iff some pair of optimal encoders forms X → T₂ → T₁, equivalently iff the coarse encoder is a garbling of the fine one. When the fine decoder points q(X|t) are affinely independent, this holds iff every coarse decoder point lies in their convex hull. If the optima are unique, up to relabelling and merging duplicate points, a failed hull test proves non-refinability.
- The optima. States 0 and 1 share P(Y|x), so the problem reduces to
three states. The optima were found from 400 Blahut–Arimoto restarts.
- At β = 8: I(X;T) = 0.546 bits and I(T;Y) = 0.097 bits, with two decoder points.
- At β = 12: I(X;T) = 0.952 bits and I(T;Y) = 0.139 bits, with three decoder points, one of them pure on {0, 1}.
- The certificate. Each was certified globally optimal and unique by checking that F_β minus the tangent plane through its decoder points is non-negative over the simplex. The check used a grid of spacing 1/1200 (minimum away from the decoder points 1.9 × 10⁻⁵ at β = 12), 300 local minimizations, all ending at the decoder points, and a bound near the vertices.
- The test. The β = 8 decoder point (0.376, 0.376, 0.244, 0.004) lies outside the hull of the β = 12 points. The LP phase-one minimum is 0.26, and the distance to the hull is 5.6 × 10⁻³, against solver error near 10⁻⁷. The same failure holds for β = 8 against 10, and 10 against 12. Re-run here from the reader’s script, with the same figures.
So no optimal encoder at β = 12 can be refined down to the optimum at β = 8 by adding noise. Moving from the smaller to the larger budget changes which distinctions the representation draws, not only how many.
What it does not say
- Not a proof in exact arithmetic. The certificate is a fine grid plus local search, not interval arithmetic. A sharper certificate would make it a theorem.
- Not that the loss is large. In unique information, the price of forcing nesting is 4.7 × 10⁻³ bits for β = 8 against 12: 0.49% of the larger budget.
- Not that bottlenecks generally fail to nest. Charvin et al. prove nesting at every pair of budgets when Y = f(X), and when X and Y are both binary (THEORY-757). With this source, the smallest case where failure is possible is three effective states with binary Y. The reader’s uncertified scan over β from 5 to 60 found failures mostly at nearby budgets, and nesting for far-apart pairs.
Note of 2026-10-21: proposed as a paper with three of its limits dropped
A2 and A5 of the exchange of 2026-10-20 proposed a paper on this result and CLAIM-387 (“More Information, Different Categories”) as “perhaps the cleanest paper-sized result in the current record”. They kept its numerical certificate and dropped three of its limits: the cost of forcing nesting is 0.49% of the budget, the reader’s scan found nesting for far-apart budgets, and nesting is proved when Y = f(X) or X and Y are both binary (THEORY-757).