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Holographic entropy inequalities without unscaled contraction proofs

Primarily AI-generated textHuman understanding: no partsmath-ph — Mathematical Physicsmath.CO — Combinatoricsquant-ph — Quantum Physics

Contributed by Antti Hippeläinen ↗

Version 1 / Nov 10, 2026 / CC BY 4.0

Abstract

We show that a valid holographic entropy inequality can lack a proof by contraction until all its coefficients are multiplied by a common integer, even when it is balanced, that is, every party has the same total coefficient on both sides, or superbalanced, that is, this remains true after any party is exchanged with the purifier. Every facet of the holographic entropy cone other than subadditivity is superbalanced. A proof by contraction is a map between Boolean cubes that respects the terms in which each party occurs and does not increase weighted Hamming distances. A balanced seven-party inequality and a superbalanced six-party inequality fail at a single point of the cube. Six of their right-hand side terms correspond to the edges of two disjoint triangles of parties, and at that point an unscaled map would have to choose three of these edges that together touch all six vertices, which is impossible. The seven-party one has a proof after doubling. Superbalanced eight- and seven-party inequalities have no unscaled proof even on their realizable domains, the subsets of the cube on which a map already proves validity, and they keep this property when instances of the monogamy of mutual information are subtracted in five successive steps. Splitting further into valid inequalities leads to a superbalanced facet F7F_7 of the seven-party cone that we did not find in the literature. It has unscaled maps on its realizable domain, all of which break its symmetry. For star inequalities, whose left-hand side contains only single-party entropies, the least scaling is the least kk for which an explicit polytope contains a point with coordinates in 1kZ\frac1k\mathbb{Z}. From the weighing matrices of Alon and V\~u we obtain valid NN-party star inequalities whose least scaling exceeds N(1/4−ε)NN^{(1/4-\varepsilon)N} for every ε>0\varepsilon>0 and large NN, while every finite least scaling of an NN-party star inequality is below NN/2N^{N/2}. Non-existence rests on DRAT proofs, validity on exact rational maps or identities, and the facet property on 126126 explicit graph models, except for the star inequalities of arbitrary size, whose results rest on written proofs.

Provenance statement

This paper was produced by large language models, instances of the Claude models of Anthropic running as autonomous agents, in October and November 2026. They found the inequalities, wrote the proofs and the programs that check the certificates, ran the computations and wrote the text. The arguments were checked only by other instances of the same models, acting as referees and recomputing the certificates with programs of their own. Such checks may share the blind spots of the models that wrote the proofs. The author has read the paper and checked that the references lead to the cited works, but the statements attributed to them were checked only by the models: every statement was matched to a verbatim passage of the cited work, and another instance of the model ruled on whether the passage supports the sentence. No human expert has checked the proofs. The steps regarded as least secure are the SAT encodings and the argued results on star inequalities. A DRAT proof certifies only that a formula is unsatisfiable, while the step from the map problem to the formula rests on the encoder. The programs that check the certificates, the certificates themselves, and a program that regenerates the Boolean formulas and the integer maps are available under the MIT license at https://github.com/ahippelainen/unscaled-contraction-public.

Tools used

Anthropic
Claude OpusVersion 5.5
PySAT
PySAT
CaDiCaL
CaDiCaL
drat-trim
drat-trim
Kissat
q
NetworkX
NetworkX

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Version history

  1. v1 · 37 pagesSubmitted by Antti HippeläinenInitial depositCurrentNov 09, 2026