HECATE — Holographic Entropy Cone Analysis & Theorem Engine

Named for the goddess of boundaries and crossroads — fitting for an engine that maps the boundary of which entanglement patterns can be geometry.

The holographic entropy cone reduces to graph combinatorics (BNOSSW, arXiv:1505.07839): an entropy vector is holographic iff some weighted graph realizes it via min-cuts. This repo builds the engine, the verifiers, and the search loops to attack the open n=6 frontier. Full plan: SPEC.md. The Python package keeps the short import name hec.

Status

Milestone	State
M0 prereqs + env	done — engine hand-verified on star/cycle graphs
M1 engine + cone tools	done — 10⁶ graphs zero violations (40s); 5-party cone (P2) reproduced exactly incl. Normaliz facet↔ray duality (2267 rays)
M2 prover + canonicalization	done — MMI re-derived; CP-SAT proves 5/6 nontrivial C5 orbits (Q3: map requires the unit-expanded cube — pending, via arXiv:2403.13283)
P3 reproduction	done — all 208 six-party orbits re-verified (extreme rank-62, SSA-compatible); 52-violator index set == the paper's published list
C5 ground truth	done — all 19 published graph models transcribed & exactly verified (hec/c5_graphs.py); 19/19 ray realizations certified in reports/realizations/: 16 fully independent (13 LP-cold + 3 annealed) and 3 second-tier (rays 10, 12, 19) — second-tier means the topology comes from the published figure while the weights are independently re-derived by LP, with the certificate verified exactly like any other; derived split-tree oracles for 8/10 non-chordal rays
M3 the hunt	reframed by the field — {146, 180, 181} were realized via RL search in arXiv:2601.19979 (our independent verification: done — all three exact, scales 12/12/9, all simple-but-cyclic as chordality requires; hec/hlo_data.py); {110, 145, 168} remain open with non-realizability conjectured. All six fail the chordality gate: no simple-forest model exists for any of them (theorem, arXiv:2512.24490). The campaign on the suspects is two-track: (A) realization search over non-simple trees and cyclic graphs (slack-guided topology annealing), and (B) a separating-inequality hunt; absence-of-realization from annealing is never treated as evidence — exclusion claims come only from exclusion-grade machinery (bounded exhaustion of chordal fine-grainings)

Layout

• hec/entropy.py — graph → S-vector via min-cut (networkx exact path; igraph fast path, exact for integer weights)
• hec/inequalities.py — SA / AL / SSA / WM / MMI checks on entropy vectors
• hec/graphs.py — random graph generators (stars, trees, bulk cycles)
• hec/subsets.py — bitmask ↔ (size, lex) paper-order bookkeeping
• hec/cone.py — Sym(n+1) purified symmetry action, orbits, SA/SSA instance families, exact extremality
• hec/prover.py — contraction maps: checker, brute force, CP-SAT search
• hec/canon.py — dedupe keys: Sym(n+1) vector canon + pynauty weighted-graph canon
• hec/realize.py — target ray → graph search (v1 hill-climb)
• hec/lp_realize.py — LP cut-assignment engine: slack-LP descent → hard-pin constraint generation → exact integer certification
• hec/anneal.py — topology annealing with LP slack as fitness (the Sparse Oracle Principle: search sparse structure space, not dense supergraphs)
• hec/chordality.py — the arXiv:2512.24490 iff-gate for simple-forest realizability (correlation hypergraph → line graph → chordal); calibrated on C5: 9 must-pass (known simple trees) + 10 derived facts (non-chordality is a new, criterion-dependent claim)
• hec/tree_builder.py — Algorithm 1 of arXiv:2512.18702: constructive simple-tree builder, 9/9 exact on the chordal C5 rays
• hec/c5_graphs.py — the 19 published C5 graph models, transcribed and exactly verified
• hec/splits.py + entropy_vector_labeled — non-simple models (multi-vertex parties); boundary-splitting transform with exact re-verification
• hec/n6er_data.py — the two public tree-resistant n=6 models (verified; not SAC-extreme, ranks 58/61)
• hec/c5_data.py, hec/hei_data.py — arXiv:1903.09148 tables; arXiv:2309.06296 loader
• scripts/ — day1_mmi, verify_engine (10⁶ run), verify_c5 (P2 + --duality), verify_hei6, verify_er6 (P3 + target extraction), prove_inequalities, prove_q3_expanded, realize_c5 / attempt_mystery (--engine lp|hill), fetch_papers.sh
• data/targets/mystery_orbits.json — the 6 open T1 targets, validated, with provenance
• reports/ — machine-checkable certificates (contraction maps, realizations)
• NOTES.md — running lab notebook

Pending: party-splitting moves for the annealer (non-simple tree search), the Bao–Naskar deterministic prover (arXiv:2403.13283; the Q3 unit-expanded proof waits on it), chordal fine-graining exclusion machinery (campaign track B), falsify.py, search.py evolutionary loop, db.py store.

Setup & run

uv venv --python 3.12 .venv
uv pip install -p .venv/bin/python -e '.[dev]' sympy   # core
uv pip install -p .venv/bin/python '.[search]'         # igraph, ortools, numpy
./scripts/fetch_papers.sh                               # papers + ancillary data

.venv/bin/python -m pytest                              # ground-truth tests
.venv/bin/python scripts/day1_mmi.py --count 10000      # the Day-1 assertion

Data & references

Results in this repo are computed against, and verified to reproduce, the following sources:

• arXiv:1505.07839 (Bao–Nezami–Ooguri–Stoica–Sully–Walter) — the graph-model theorem and contraction-map proof method this entire toolkit is built on.
• arXiv:1903.09148 (Hernández Cuenca) — the 5-party cone; Tables 1–2 transcribed in hec/c5_data.py and re-verified here (372 facets ↔ 2,267 extreme rays, including an independent Normaliz double-description run).
• arXiv:2309.06296 (Czech–Dong–Hernández Cuenca et al.) — the 1,877 known 6-party holographic entropy inequalities; loaded from the paper's arXiv ancillary files (HEIvectors.txt).
• arXiv:2412.15364 (He–Hubeny–Rota) — the 208 SSA-compatible extreme-ray orbits of the 6-party subadditivity cone; full classification re-verified here. Ray data from the authors' repository Max-Rota/SSA-compatible-Extreme-Rays-of-the-Subadditivity-Cone (DOI: 10.5281/zenodo.14983856), pinned at commit e973df6e0aa6 — scripts/fetch_papers.sh clones exactly that snapshot.
• Methods arc for the prover: arXiv:2403.13283 → 2409.17317 → 2506.18086 (contraction-map algorithmics and completeness).
• arXiv:2204.00075 (Hernández Cuenca–Hubeny–Rota) — the tree conjecture, the fig. N5trees non-simple trees, and the entropy-preserving graph operations; source of the two tree-resistant n=6 models verified in hec/n6er_data.py.
• arXiv:2412.18018 (Hubeny–Rota) — the correlation hypergraph and the original chordality necessary condition.
• arXiv:2512.18702 (Hubeny–Rota) — the constructive simple-tree algorithm (our hec/tree_builder.py) and the fine-graining toolkit.
• arXiv:2512.24490 (Hubeny–Rota) — chordality is necessary AND sufficient (the iff-gate in hec/chordality.py).
• arXiv:2601.19979 (He–Lee–Ooguri) — RL realization of mystery orbits {146, 180, 181}; code at Jaeha0526/EntropyCone_RL, pinned at commit 193d994c8b3a. Their three published edge lists are transcribed in hec/hlo_data.py and verified exactly here (tests/test_hlo.py); certificates and reduced models in reports/realizations/.

The distilled T1 targets (data/targets/mystery_orbits.json) ship in-repo with provenance and validation steps recorded in the file itself.

Conventions

Parties are integer vertices 0..n-1, the purifier is vertex "O", subsets of parties are bitmasks (bit i ⇔ party i), entropy vectors are {mask: S(mask)} for mask = 1..2ⁿ−1. Edge weights live in the capacity attribute and are integers unless a claim needs Fractions — floats explore, they never prove (SPEC.md §5, rule 1).