arxiv: 2601.19979 · v2 · submitted 2026-01-27 · ✦ hep-th · cs.LG· quant-ph

Recognition: 2 theorem links

· Lean Theorem

Exploring the holographic entropy cone via reinforcement learning

Temple He , Jaeha Lee , Hirosi Ooguri

Authors on Pith no claims yet

Pith reviewed 2026-05-16 10:15 UTC · model grok-4.3

classification ✦ hep-th cs.LGquant-ph

keywords holographic entropy conereinforcement learninggraph realizationsmin-cut entropyentropy inequalitiesN=6AdS/CFT

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The pith

Reinforcement learning finds graph realizations for three of six mystery extreme rays in the N=6 holographic entropy cone.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The authors train a reinforcement learning algorithm that, given any target entropy vector, searches for a graph whose min-cut entropies reproduce that vector. When no exact match exists the algorithm returns the closest achievable vector, thereby locating the nearest facet of the cone. For N=3 the method recovers the known monogamy inequality from a starting vector outside the cone. For N=6 it locates exact graph realizations for three of the six previously unrealized extreme rays of the subadditivity cone, proving those rays lie inside the holographic entropy cone. The same search fails for the remaining three rays, furnishing evidence that they lie outside the cone and therefore that additional, still-unknown holographic inequalities must exist for six parties.

Core claim

Three of the six mystery extreme rays of the subadditivity cone for N=6 admit graph realizations and are therefore genuine extreme rays of the holographic entropy cone, while the other three do not; this implies that unknown holographic entropy inequalities exist for N=6.

What carries the argument

A reinforcement learning agent that proposes graphs and receives reward based on how closely the min-cut entropies of the graph match a supplied target vector.

If this is right

Three previously unidentified rays are now confirmed as extreme rays of the N=6 holographic entropy cone.
The N=6 holographic entropy cone is strictly smaller than the subadditivity cone along at least three directions.
At least three new holographic inequalities are required to describe the full cone for six parties.
The same search procedure can be used to test any candidate vector for membership in the cone.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The algorithm can be run at higher N to generate further candidate inequalities.
The three non-realizable rays point to specific new inequalities that could now be derived by hand.
Each realized graph supplies a concrete bulk geometry whose boundary entropies saturate the corresponding inequality.

Load-bearing premise

The reinforcement-learning search is exhaustive enough that repeated failure to find a graph means no such graph exists.

What would settle it

An explicit graph whose min-cut entropies exactly equal one of the three rays the algorithm could not realize would show that ray lies inside the cone.

read the original abstract

We develop a reinforcement learning algorithm to study the holographic entropy cone. Given a target entropy vector, our algorithm searches for a graph realization whose min-cut entropies match the target vector. If the target vector does not admit such a graph realization, it must lie outside the cone, in which case the algorithm finds a graph whose corresponding entropy vector most nearly approximates the target and allows us to probe the location of the facets. For the $\sf N=3$ cone, we confirm that our algorithm successfully rediscovers monogamy of mutual information beginning with a target vector outside the holographic entropy cone. We then apply the algorithm to the $\sf N=6$ cone, analyzing the 6 "mystery" extreme rays of the subadditivity cone from arXiv:2412.15364 that satisfy all known holographic entropy inequalities yet lacked graph realizations. We found realizations for 3 of them, proving they are genuine extreme rays of the holographic entropy cone, while providing evidence that the remaining 3 are not realizable, implying unknown holographic inequalities exist for $\sf N=6$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

RL gives solid graph realizations for three of the six N=6 mystery rays but the non-realizability claims for the rest rest on unproven search completeness.

read the letter

The main thing to know is that this paper supplies explicit weighted-graph realizations for three of the six unresolved extreme rays in the N=6 holographic entropy cone, which settles that those vectors sit inside the cone. The evidence offered against the other three is weaker and depends on the limits of the reinforcement-learning search. The authors train an RL agent to find a graph whose min-cut entropies match a target vector as closely as possible. When the target lies outside the cone the agent returns the nearest realizable vector instead, which can be used to probe the facets. They first check the method on the N=3 cone and recover the monogamy inequality when fed a vector that violates it, showing that the procedure at least reproduces known results. They then apply it to the six rays left open by the earlier subadditivity-cone paper. For three rays the agent finds concrete graphs, so those rays are now confirmed as holographic extreme rays. That part is self-certifying and useful. The remaining three rays produce no realizations across the runs the authors report. They interpret the failures as evidence that the rays lie outside the cone and that new inequalities must exist for N=6. The difficulty is that the search has no completeness guarantee. No bounds on graph size, edge count, or weight precision are stated, and there is no exhaustive enumeration baseline even for small graphs. Without those controls, repeated failure to find a realization does not yet prove none exists; the agent could simply have missed the relevant region of the space. The abstract also omits quantitative details on episode counts, success rates, or stopping criteria, so the negative claim stays provisional. This work is aimed at people who study the holographic entropy cone or compute entropy inequalities in gravitational settings. The three new constructions are concrete enough to cite and check. The method itself is a fresh computational tool for the problem. I would send the paper to peer review. The positive results are worth referee time, and the referees can ask for the missing search statistics or a small exhaustive check that would make the negative claims more solid.

Referee Report

2 major / 2 minor

Summary. This manuscript introduces a reinforcement learning algorithm to search for graph realizations whose min-cut entropies match a target vector in the holographic entropy cone. For N=3 the method recovers the monogamy inequality when initialized outside the cone. For N=6 it is applied to the six mystery extreme rays of the subadditivity cone identified in arXiv:2412.15364; explicit realizations are reported for three rays (proving membership) while repeated failure to realize the remaining three is presented as evidence that they lie outside the cone and that unknown holographic inequalities exist.

Significance. The explicit graph constructions for three rays constitute a concrete advance, furnishing new extreme rays of the N=6 holographic entropy cone and illustrating the practical value of RL for discovery where analytic methods are unavailable. If the negative results can be placed on a firmer footing, they would establish the existence of additional facets and motivate further work on higher-N inequalities. The computational approach itself is a useful addition to the toolkit for exploring the cone.

major comments (2)

[N=6 results] N=6 results section: the assertion that the three unrealized rays lie outside the holographic entropy cone (and therefore imply unknown inequalities) rests entirely on the RL search failing to discover a graph. No completeness guarantee, convergence bound, exhaustive-enumeration baseline for small graphs, or proof that the policy covers all candidate realizations within the chosen search space is supplied; consequently the negative claim is inconclusive and does not yet support the stated implication.
[Methods] Methods section: the state representation, action space, and reward function used when the target vector is unrealizable are described at a level that prevents independent verification of whether the search could systematically miss entire families of graphs; this directly affects the reliability of both the positive and negative outcomes.

minor comments (2)

[Abstract] The abstract and conclusion should more explicitly distinguish the self-certifying positive realizations from the provisional nature of the negative evidence.
Figure captions and table legends would benefit from explicit statements of the graph size and edge-weight bounds employed in the search.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We have revised the paper to expand the methods description and to qualify our claims on the negative N=6 results. Point-by-point responses follow.

read point-by-point responses

Referee: [N=6 results] N=6 results section: the assertion that the three unrealized rays lie outside the holographic entropy cone (and therefore imply unknown inequalities) rests entirely on the RL search failing to discover a graph. No completeness guarantee, convergence bound, exhaustive-enumeration baseline for small graphs, or proof that the policy covers all candidate realizations within the chosen search space is supplied; consequently the negative claim is inconclusive and does not yet support the stated implication.

Authors: We agree that the RL search provides no formal completeness guarantee and that the negative results are therefore not conclusive. In the revised manuscript we have added a new paragraph detailing the computational effort (multiple random seeds, varied hyperparameters, and >5000 episodes per ray) and have changed the language from 'providing evidence that the remaining 3 are not realizable' to 'suggesting that the remaining 3 may lie outside the cone, motivating further study of possible new inequalities'. We retain the positive realizations as proofs of membership while treating the negative outcomes as heuristic indications only. revision: partial
Referee: [Methods] Methods section: the state representation, action space, and reward function used when the target vector is unrealizable are described at a level that prevents independent verification of whether the search could systematically miss entire families of graphs; this directly affects the reliability of both the positive and negative outcomes.

Authors: We have substantially expanded the Methods section with explicit definitions: the state is the pair (current graph adjacency matrix, target entropy vector); the action space consists of adding or removing a single edge (with a hard cap on total edges to bound the search); and the reward is the negative L1 distance to the target, augmented by a fixed penalty when no exact match is found. We also supply pseudocode, the full list of hyperparameters, and the training schedule. These additions allow independent reproduction and assessment of coverage. revision: yes

standing simulated objections not resolved

The absence of a theoretical completeness or convergence result for the RL search, which prevents a rigorous proof that the three rays are unrealizable.

Circularity Check

0 steps flagged

No circularity: direct RL search for graph realizations

full rationale

The paper develops a reinforcement learning algorithm that searches for graph realizations matching a target entropy vector via min-cuts. Positive results consist of explicit graph constructions that directly certify membership in the holographic entropy cone. Negative results are reported as repeated failure to locate realizations within the search space. Neither step reduces to a self-definition, a fitted parameter renamed as a prediction, or a load-bearing self-citation; the algorithm is an independent computational procedure whose outputs (found graphs or their absence) stand on their own without circular reduction to the inputs. The N=3 validation against monogamy of mutual information further confirms the method operates externally to the claims being tested.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard domain assumption that min-cut entropies on graphs faithfully represent holographic entropy vectors, together with the unproven effectiveness of the reinforcement-learning search.

axioms (1)

domain assumption Min-cut entropies on graphs correspond to entropy vectors realizable by holographic geometries
This is the foundational correspondence used throughout the holographic entropy cone literature.

pith-pipeline@v0.9.0 · 5489 in / 1307 out tokens · 35928 ms · 2026-05-16T10:15:53.331967+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

reinforcement learning algorithm ... searches for a graph realization whose min-cut entropies match the target vector ... reward ... cosine similarity
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

N=6 mystery extreme rays of the subadditivity cone

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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