classS-schur-db — exact Schur indices of four-dimensional N=2 class S theories (release 7)

A database of exact finite expansions of the Schur index of four-dimensional N=2 superconformal theories of class S. Release 7 holds 140,340 records, one per construction or Argyres–Douglas label: untwisted theories of type A₁–A₁₀, D₄, E₆, E₇ and E₈, ℤ₂-twisted theories of type A₃, D₄ and E₆, ℤ₃-twisted theories of type D₄, and Argyres–Douglas theories with one irregular puncture. Every series is computed from a published index formula with exact integer arithmetic and stored through x²⁰ (x = q^(1/2)), and every record names its sources: the index formula, the catalogue location or the generating rule of every piece of its construction, the sources of its annotations, and earlier computations of the same index in the literature.

The release covers a stated finite range (below) and is complete within it: every candidate inside the range is present as a record, admitted, excluded or unresolved. It is complete within its nineteen types, not within class S.

Release 7 is the revision tagged v7 of this dataset repository, https://huggingface.co/datasets/cms1308/classS-schur-db. Access to the files is gated: a request made on this page is approved by the maintainer. The search site that read earlier releases is private and on hold. Releases 1 to 6 stay at the tags v1, v2, v3, v4, v5 and v6 (section "Versions"), under the same gate.

Contents

type	category	what	records	admitted	excluded	unresolved	finite window	series through x^40
A1	A-untwisted	untwisted A₁, genus g with s punctures	526	526	0	0	386	0
A2	A-untwisted	untwisted A₂	456	453	3	0	27	164
A3	A-untwisted	untwisted A₃	2,620	2,593	27	0	231	2,077
A4	A-untwisted	untwisted A₄	22,673	22,571	102	0	7,897	19,976
A5	A-untwisted	untwisted A₅ (N = 6), genus 0 and 1	2,774	2,774	0	0	0	0
A6	A-untwisted	untwisted A₆ (N = 7), genus 0 and 1	3,380	3,380	0	0	0	0
A7	A-untwisted	untwisted A₇ (N = 8), genus 0 and 1	5,158	5,158	0	0	0	0
A8	A-untwisted	untwisted A₈ (N = 9), genus 0 and 1	3,986	3,986	0	0	0	0
A9	A-untwisted	untwisted A₉ (N = 10), genus 0 and 1	3,622	3,622	0	0	0	0
A10	A-untwisted	untwisted A₁₀ (N = 11), genus 0 and 1	1,853	1,853	0	0	0	0
D4	D-untwisted	untwisted D₄	22,704	22,524	180	0	5,888	8,314
E6	E-untwisted	untwisted E₆, genus 0	2,133	954	1,179	0	0	0
E7	E-untwisted	untwisted E₇, genus 0	8,518	1,134	7,384	0	0	0
E8	E-untwisted	untwisted E₈, genus 0	10,299	68	10,231	0	0	0
A3tw	A-twisted-Z2	ℤ₂-twisted A₃, nontrivial twist sector	18,678	18,607	71	0	233	5,864
D4tw	D-twisted-Z2	ℤ₂-twisted D₄, nontrivial twist sector	22,972	21,991	981	0	0	0
E6tw	E-twisted-Z2	ℤ₂-twisted E₆, genus 0, nontrivial twist sector	2,922	1,550	1,372	0	0	0
D4z3	D-twisted-Z3	ℤ₃-twisted D₄, genus 0 and 1, nontrivial twist sector	3,010	2,927	83	0	0	0
AD	AD	Argyres–Douglas theories (J^h[k], Y)	2,056	2,039	0	17	454	0

In all 140,340 records: 118,710 admitted with a series, 21,613 excluded without one, 17 unresolved without one. The type is the prefix of the identifier id: <type>-g<genus>-s<punctures>-<sorted puncture labels> for class S constructions, AD-<J>-b<b>-k<k>-<Y> for the Argyres–Douglas labels. The last column counts the records with series_K40; an Argyres–Douglas record with a series stores instead, except for four Lagrangian labels of type D and 26 labels of type A with gcd(k, N) > 1 (below), its series through x^K_deep with K_deep ≥ 40 (series_deep).

A construction is a genus g, a multiset of s regular punctures with 2g − 2 + s ≥ 1 and, for the twisted types, the twist sector, together with a witness: a sewing of fixtures and cylinders taken from the published catalogues (or, for a three-punctured sphere, one catalogue fixture). It is admitted when its graded Coulomb-branch dimensions are nonnegative, its cutoff weights are positive and a sourced witness exists, and excluded otherwise. For the types A₅–A₁₀, added in release 7, no table of fixtures and cylinders is printed: the witness is a tree of the fixtures, irregular punctures and cylinders that the rules of 1008.5203 Sections 2.2–2.4 generate at the given N (witness kind generated, construction key A-generated; the rules reproduce the printed tables at N = 3, 4, 5, and reading them as valid for every N is this database's reading, caveat an-catalogue), and a construction with a negative graded dimension is not a record of these types (below), so that all their records are admitted. Admission does not certify that a construction is a superconformal field theory, and distinct constructions are not proven to be inequivalent theories. An Argyres–Douglas record is a label (J^h[k], Y) of arXiv:1706.01607 with b = h and c ≤ 40: J of type A with a regular puncture Y or none, or J of type D or E with the full puncture or none. The labels of type A with gcd(k, N) > 1, added in release 6 (700 labels, every regular puncture for k ≥ 1 and the full puncture for k ≤ 0, all admitted), have a reducible irregular singularity with gcd(k, N) − 1 mass parameters; the other labels have an irreducible one. The labels of type D and E with gcd(k, h) > 1, added in release 4, have no single formula: 34 of them are admitted with a series from the formula of their subset (below), two of them, D₂(E₆) and (E₇, A₈), since release 5, and 17 are unresolved. A record is unresolved when it lies inside the range but no series is stored, because no sourced Schur-index formula was found for it. It carries its annotations, its construction and the reason in status_reasons; it has no series, is in no equal window and does not count in the literature level. Most E-type candidates are excluded: a sphere with three punctures of an E type is a fixture only when every graded Coulomb dimension is nonnegative, and most multisets of the range fail that.

Range of release 7

Release 7 contains every candidate of the nineteen types above that satisfies at least one of two conditions:

1. it lies in the (genus, s) box of its type, where s is the number of punctures (n_punctures);
2. the formula value of its central charge, virtual_annotations.c, satisfies c ≤ 40.

For the types A₅–A₁₀ (untwisted A_{N−1}, N = 6–11), added in release 7, the range is every genus and multiset of regular punctures with c ≤ 40 and every graded dimension d_k ≥ 0. Unlike the other types, a candidate with c ≤ 40 and a negative graded dimension is not a record (209,788 such multisets at N = 6–11, which the manifest counts by type in range.types), so c_at_most_40 of these types counts their records (caveat an-range).

types	genus 0	genus 1	genus 2	genus 3
A2, A3, A4	s = 3–12	s = 1–8	s = 0–6	s = 0–4
D4, A3tw	s = 3–8	s = 1–6	s = 0–4	s = 0–2

A1, A5–A10, D4tw, AD, E6, E7, E8, E6tw and D4z3 have no box and enter by the bound on c alone; every E-type record has genus 0 and every A5–A10 and D4z3 record genus 0 or 1, since a surface of higher genus with these punctures has c > 40. For a class S construction c = (2n_v + n_h)/12, with n_v and n_h computed from the puncture data; it is evaluated for every candidate, excluded ones included. For an Argyres–Douglas label c is taken from the sources named in its provenance_annotations.

Not in release 7: class S theories of type A_{N−1} with N ≥ 12 (N = 12 was removed by the cost rule declared before the computation, about 114 core-hours projected for its 1,886 candidates, and N = 13–29 are deferred for the cost of the sum), the candidates of A₅–A₁₀ with a negative graded dimension, and D_N with N > 4; twists other than the ℤ₂ twists of A₃, D₄ and E₆ and the ℤ₃ twist of D₄ (in particular the full S₃ twist of D₄); Argyres–Douglas theories with b ≠ h, with J of type D or E and a general puncture, with J of type A, gcd(k, N) = 1, k < 0 and a puncture other than the full one and [N−2, 1, 1], with J of type A, gcd(k, N) > 1, k ≤ 0 and a puncture other than the full one, and with reducible singularities of type D or E.

The stored series

The index is I(q) = Tr (−1)^F q^(E−R) with R = 1/2 on the hypermultiplet scalars, unrefined (every flavor fugacity set to one), with the vacuum-energy factor of the source stripped. A record stores I(x²) = Σ b_n x^n for n = 0, …, K with K = 20 inclusive and b₀ = 1, as exact signed integers written as decimal strings. Zeros are explicit; the tail beyond x^K is uncomputed, not zero.

With each series the record stores:

• pl: the plethystic logarithm PL[I] through x^K;
• quotient: S = I/P, where P = 𝒜^(2g−2+s) Π_p K_p is the prefactor of the index formula of the type, 𝒜 independent of the punctures and K_p the factor of puncture p (for an Argyres–Douglas record, P is the factor K_Y of its regular puncture); with quotient_pl = PL[S] and prefactor_pl = PL[P]. S belongs to the construction, not to the theory: two constructions of one theory with different prefactors have different quotients.
• finite_window: true when the smallest cutoff weight exceeds K, so that the stored series equals the prefactor P alone through x²⁰;
• series_K40, quotient_K40, quotient_pl_K40: the same through x⁴⁰ where stored (36,395 records);
• series_deep, pl_deep (Argyres–Douglas only): the series through x^K_deep, with K_deep = max(40, 2(h+k+j_max)) for the labels with gcd(k, h) = 1 (at most x⁶⁴⁶) and K_deep = 40 for the admitted labels of type D and E with gcd(k, h) > 1 except the four Lagrangian quivers of subset (D), which have no deep series, and K_deep = 40 for 674 of the 700 labels of type A with gcd(k, N) > 1 (the other 26 have none, caveat ad-gcd-A-deep). For the 454 Argyres–Douglas records in the finite window, the dependence on the irregular singularity is in these fields only.

Index formulas

formula	types	records with a series	status	sources	restatements
A	A2, A3, A4, A5, A6, A7, A8, A9, A10	46,390	conjectural	1104.3850, 1110.3740	1408.6522
A1	A1	526	proven	1104.3850, 1110.3740, 0910.2225	1707.07679, 2112.10715
D4	D4	22,524	conjectural	1212.1271, 1212.0545	—
E6	E6	954	conjectural	1212.0545	—
E7	E7	1,134	conjectural	1212.0545	—
E8	E8	68	conjectural	1212.0545	—
A3tw	A3tw	18,607	conjectural	1212.1271, 1212.0545, 1501.00357	1711.04727
D4tw	D4tw	21,991	conjectural	1212.1271, 1212.0545, 1501.00357	1711.04727
E6tw	E6tw	1,550	conjectural	1501.00357, 1212.0545	1711.04727
D4z3	D4z3	2,927	conjectural	1601.02077, 2112.10227, 1212.0545	1711.04727
AD	AD	1,305	conjectural	1706.01607	—
ADgcd-A	AD	20	conjectural	2609.02437	—
ADgcd-B	AD	2	conjectural	2012.12827, 2106.12579	—
ADgcd-C	AD	4	conjectural	1006.3435, 1706.01607	—
ADgcd-D	AD	4	conjectural	1303.3149	1110.3740
ADgcd-E	AD	2	conjectural	1303.3149, 1910.02281	1110.3740
ADgcd-F	AD	1	conjectural	2012.12827, 1910.02281, 1303.3149	1110.3740
ADgcd-G	AD	1	conjectural	1303.3149, 0906.0359, 1110.3740, 1212.0545	—
ADgcdA-A	AD	317	conjectural	1705.07173	—
ADgcdA-BC	AD	383	conjectural	2311.12123, 1706.01607	1110.3740

The status is that of the formula in its sources. The sources column names the papers that are the original source of some part of the formula; the restatements column names papers that print the formula in the form used here but credit it to earlier work (SOURCES.json gives the originality of every entry). The class S formulas are the index TQFT sums over representations with the puncture factors of the sources: for E₆, E₇ and E₈ the general simply-laced sum of 1212.0545 Section 4.1 at t = q; for twisted E₆ the fixture formula of 1501.00357 Section 4.2 with the K-factor of 1711.04727 Section 3, sewn to spheres with any number of punctures as 2201.13435 Section 2.3 prints it for a general ℤ₂ twist (an additional formula entry of these records). For ℤ₃-twisted D₄ the series are the sums of 1601.02077 Section 4 over G₂ representations, mapped to triality-invariant D₄ representations at the untwisted punctures, with characters and the Schur K-factors: for the sector with one untwisted and two twisted punctures the Schur form of 1711.04727 Section 3, with the K-factor conventions of 2112.10227 Section 3, and for the sector of three twisted punctures this database's reading of the Hall–Littlewood sum of 1601.02077 Section 4.2. They are sewn to spheres with pairs of ω and ω² punctures, as 2201.13435 Section 4 states for the vertex algebras, and to genus 1; the genus-1 surfaces without twisted punctures follow from the untwisted D₄ sum with the ℤ₃-twisted Spin(8) cylinder (1311.2945 Section 6.1), and the other sewings with fixtures of three twisted punctures are this database's (caveat d4z3-formula-status). The Argyres–Douglas formula is the closed plethystic form of 1706.01607, which rests on a conjectured irregular wave function. For A₁ the formula is an identity for the Lagrangian descriptions, whose proof its source outlines (1110.3740, Appendix E). A stored series is compared with a second implementation or route where one exists, through the order that the caveats of its archive state, and the independent reviews (below) recomputed every stored unflavored series of the archives of releases 1 to 6 and 2,174 of the 20,773 of untwisted A₅–A₁₀; nothing is claimed beyond the stored order. For 495 Argyres–Douglas records no second route applies and the series rests on the formula alone (caveat ad-formula-alone, attached to them: its field records lists 628 records, of which its scope excepts the 133 that a review compared with a second route). The Argyres–Douglas formula of 1706.01607 has earlier sources for part of its labels: the type-A irregular wave function was conjectured in 1509.06730, the Schur index with the full puncture or without puncture was proposed as a vacuum character in 1604.02155; and the Schur indices of the (A₁, G) theories, among them (A₁, D_{2n+1}) with the full puncture and the type-A labels without puncture, were computed earlier in 1506.00265 and 1505.05884. SOURCES.json records the originality of each (the entries of 1706.01607, 1506.00265 and 1505.05884), and a record lists the earlier computations of its theory in provenance_literature.

The 20,773 records of untwisted A₅–A₁₀, added in release 7, take the formula A of A₂–A₄ at N = 6–11: the sum over the representations of SU(N) with 𝒜 = Π_{d=2}^{N}(x^{2d}; x²)∞ and the K-factor of each puncture, the characters at the torus points of the punctures computed by branching over the groups of equal eigenvalues (caveat an-formula-status). A second implementation, the alternant form of the A₂–A₄ engine, agreed at generation with every record of N = 6 through x²⁰, N = 7 through x¹² and N = 8 through x⁶; for N = 9–11 the characters were compared with semistandard tableaux on a sample, and the independent review (below) recomputed 175 records of N = 9–11 among 2,174 of N = 6–11 through x²⁰. Five candidates of N = 10 and 11 with more than four million representations were evaluated as sums of parts of the unchanged sum. Routes that do not rest on the class S formula agree where they apply: the closed sums for N=4 SU(N) (1507.08659 (21) with its footnote 3, the original, restated as 1510.07041 (4.8)) on the tori with one minimal puncture at N = 6–11, the two-node circular quiver of 1510.07041 (4.16) on the tori with two at N = 6–8, the series of the rank-three d₄ instanton SCFT that 1907.08629 (6.1) computes from its Lagrangian, and the review's Haar integrals of SU(N) with 2N flavors and other Lagrangian records; further, resting on other conjectures, the rank-two E₇ series that 1905.00864 predicts beyond q² from elliptic genera of instanton strings, the column sums of its Table tb:E6red, the series of 2110.14897 from giant gravitons below their stated error orders, and the series of 1912.09348 from an N=1 Lagrangian that flows to R{2,5}, shown on the A₅ fixture of fig133 with its six free hypermultiplets. The other series of 1905.00864 (rank-two E₆, rank-three d₄, rank-three E₆ through q², printed through q^{7/2}) were computed with the class S formula itself (caveats an-second-implementation, an-literature, an-review).

The closed form of 1706.01607 is established for gcd(k, h) = 1 and has the wrong high-temperature growth at every label of type D or E with gcd(k, h) > 1, so these labels (added in release 4; subsets (F) and (G) admitted in release 5) take the formula of their subset, each conjectural as a Schur index:

• (A) the twenty (D_{3m+1}, A₂): the closed form (B47) of 2609.02437, which that paper proves equal to the SU(2) Haar integral of its frame (for m = 1 the elliptic E₆ model of 1103.5832 Section 7.1, SU(2) coupled to three (A₁, A₃) theories); 2609.04099 proves it independently;
• (B) (E₆, A₃) and (E₈, A₅): the SU(3) and SU(5) Haar integrals of 2106.12579 over coprime D_p(SU(N)) blocks, the frames that 2012.12827 names;
• (C) (E₆, A₁), (E₈, A₁), (E₈, A₂), (E₈, A₃): the formula of 1706.01607 at the coprime label with the same 3-fold, by the statement (G, G′) ∼ (G′, G) of 1006.3435;
• (D) D₃(SO(8)), D₅(SO(12)), D₃(SO(14)), D₃(SO(20)): the Schur index of the Lagrangian quiver of 1303.3149 (5.27), exact for the quiver;
• (E) D₉(SO(8)) and D₁₅(SO(8)): the rule of 1910.02281 Section 6.2 and (7.17), PE[28(q − q^u)/((1 − q)(1 − q^u))] times the index of the u = 1 quiver D₃(SO(8)) with q replaced by q^u (u = 3, 5), resting on that paper alone (caveat ad-gcd-single-source);
• (F) (E₇, A₈): the Spin(8) gauging of D₉(SO(8)), D⁴₃(SO(8)) and SU(2) with four flavors that 2012.12827 prints (restated in 2203.16550), with the indices of the two non-Lagrangian blocks by the rule of 1910.02281 Section 6.2: the index of the u = 1 theory with q replaced by q³ and its SO(8) fugacities kept, times PE[(q − q³)χ_adj/((1 − q)(1 − q³))], the u = 1 theory being the quiver D₃(SO(8)) of 1303.3149 (5.27) for D₉(SO(8)) and SU(2) with four flavors for D⁴₃(SO(8)), with half-hypermultiplets in the bifundamentals where 1910.02281 (6.10) prints full ones (caveat ad-gcd-frames-formula-status);
• (G) D₂(E₆): the SU(3) gauging of the E₈ Minahan–Nemeschansky theory of 1303.3149 Section 7.3.3, the E₈ theory being the A₅ fixture [1⁶], [2³], [3²] of 0906.0359 Section 4.4 with the Schur index of 1110.3740. Its series differs from the closed form of 1706.01607 and 2106.12579 (3.7) from x⁴ on and exceeds the vacuum character of the affine algebra (e₆)₋₆ that 1910.02281 and 1604.02155 assign to the theory, a conflict that this release records and does not decide (caveat ad-gcd-frames-conflicts).

The subset of a record is the formula ADgcd-A to ADgcd-G whose sources are those of its provenance_formula.

The 700 labels of type A with gcd(k, N) > 1, added in release 6, carry m − 1 = gcd(k, N) − 1 mass parameters, which the closed form of 1706.01607 does not allow; it is not their index (caveat ad-gcd-A-closed-form). Each label takes the formula of its subset, each conjectural as a Schur index:

• (A) N | k, 317 labels (ADgcdA-A): the sum over representations of SU(N) of the irregular wave function of 1705.07173 ((1.3) without puncture, (1.6)–(1.7) with a regular puncture), which the source checks for the theory without puncture and, by its high-temperature behavior, with the full puncture; for every other puncture the agreement with the frame below at all 317 labels, through x⁴⁰ at 291, is this database's check;
• (B) k ≥ 1, N ∤ k, 356 labels, and (C) k ≤ 0 with the full puncture, 27 labels (ADgcdA-BC): the Haar integral of the frame of 2311.12123 (3.85), a chain of su(n) gaugings of coprime type I theories with fundamental hypermultiplets, with the index of each coprime block by 1706.01607 (in the form of 2509.20439 (3.4)); for (C) with p | N the frame is the Lagrangian quiver of 1303.3149 (5.12)–(5.13). 23 labels have no second route (caveat ad-gcd-A-routes). The records of D₂(E₆) and (E₇, A₈) come from the archive ad-gcd-frames, and link the unresolved lines of the archive ad-gcd that release 4 published second.

Files

file	rows	content
records.parquet	140,340	the records, one row per construction or label; nested fields as JSON strings
records.jsonl.gz	140,340	the same records as gzip JSON Lines, nested fields as objects
equal_windows.parquet	14,614	groups of records with identical stored series
equal_windows.jsonl.gz	14,614	the same, JSON Lines
known_equivalences.parquet	12,010	sets of records that a source states to be one theory
known_equivalences.jsonl.gz	12,010	the same, JSON Lines
caveats.json	169	the caveats that records refer to by caveat_ids, with their scope and release notes
schema.json	51	the record fields with their types and descriptions
SOURCES.json	233	the registry of cited locations: one entry per location in a paper, with its verification
RELEASE-CAVEATS.json	4	the release layer over the archive caveats: the scopes and release notes written into caveats.json, and the 4 caveats of the release
RELEASE-STATEMENTS.json	79	the release layer of printed statements: statements about fixtures, read in the sources at the reviews of release 2, that the records carry besides those of their archive files
BUILD.json	—	the build record: input hashes, counts and the provenance report
manifest.json	—	the manifest of the database: release files, range, formulas, relations, and the archive files the release is built from (for each archive the formulas of the records that link it in any position)
README.md	—	this card
SHA256SUMS	—	SHA-256 of every other file

Records

The 51 fields are described in schema.json. The main ones:

• identity: id, category, algebra, twist, twist_sector, genus, n_punctures, punctures (label, partition as SU(2)-block dimensions, twisted flag, color of a D₄ puncture, the label of the source), irregular (Argyres–Douglas: J, b, k);
• status: status (admitted, excluded, unresolved) and status_reasons;
• series: series_K, series, pl, quotient, quotient_pl, prefactor_pl, finite_window, cutoff_weights, the x⁴⁰ and deep fields above;
• annotations: a, c (exact rationals), d_k, n_h, n_v, coulomb_rank, coulomb_dimensions, manifest_flavor_dimension, manifest_flavor_rank, known_free_hypers, named_theory, enhanced_flavor, source_discrepancy, and virtual_annotations (the same quantities computed formally for every candidate);
• construction: witnesses;
• provenance: provenance_formula, provenance_construction, provenance_annotations, provenance_literature, provenance_missing;
• caveat_ids and archive_records.

Puncture labels. A label is the partition joined by dashes, with t in front for a twisted puncture. Untwisted D₄ punctures of the two triality families carry a color in the label and in color: green, red, blue for [3,1⁵], [2⁴], [2⁴] (3-1-1-1-1-1g, 2-2-2-2r, 2-2-2-2b) and for [5,1³], [4²], [4²] (5-1-1-1g, 4-4r, 4-4b). Triality permutes the three colors on all punctures at once and leaves the index unchanged, and a construction is stored once, under the image whose identifier is lexicographically smallest; a search by partition must take every image (caveat release-d4-triality). Twisted A₃ punctures are stored in the D₃ frame of the engine: untwisted ones as D₃ partitions of 6, twisted ones as C₂ partitions of 4, with the label of 1212.3952 in source_label ([3,1²]_t is t2-2, [2²,1]_t is t2-1-1; caveat release-a3tw-puncture-labels, which also says where the stored dictionary departs from the one printed in 1212.3952). Punctures of E₆, E₇ and E₈ are nilpotent orbits with their Bala–Carter labels: the label is the ASCII spelling (p for + and, for E₇, for a prime: A3pA1p is (A₃+A₁)′, E6a1 is E₆(a₁)), partition is null and source_label holds the label printed in the source. The twisted punctures of twisted E₆ are F₄ orbits, labelled t and the F₄ Bala–Carter label, with s after the rank of a short-root component (tA2s is Ã₂, tF4a1 is F₄(a₁)). The untwisted punctures of ℤ₃-twisted D₄ are the seven triality classes of 1601.02077, labelled by a stand-in partition (2-2-2-2 stands for [3,1⁵], [2⁴]_r, [2⁴]_b and 4-4 for [5,1³], [4²]_r, [4²]_b), and its twisted punctures are G₂ orbits, labelled w (monodromy ω) or ww (monodromy ω²) and the G₂ Bala–Carter label (wA1s is Ã₁ with monodromy ω, wwG2a1 is G₂(a₁) with monodromy ω²; a reader tests the prefix ww before w), with partition and color null. The assignment of ω and ω² is part of the construction, defined up to their simultaneous exchange, and the identifier is the smaller of the two; constructions that differ only by the assignment have equal stored series (caveat d4z3-assignments). A genus-1 record of this type without twisted punctures carries the twisted handle.

A null annotation means unrecorded, not zero or absent. known_free_hypers, named_theory and enhanced_flavor are filled only where a source prints the statement (a catalogue row, a labelled figure or a sentence), and not for every such statement: the tori with one minimal puncture, which the entries of their literature describe as N=4 SU(N) with one free hypermultiplet, and the A_{kn−1} spheres that 1907.08629 (2.28) states to be rank-n instanton SCFTs with one free hypermultiplet, crediting the proposal of 1203.5517 (Sections 3.2–3.3 for the A₃ sphere and the E₆ spheres, Appendix A, Figure 10, for the E₇ sphere; the d₄ spheres of rank three to five do not appear there) (the A₃ sphere of release 1 and six spheres of A₅–A₉), have known_free_hypers null and no name; enhanced_flavor of an E-type three-punctured sphere lists every such statement with its location in statements, where the source's own data contradict it the values of the series in formula_values. A record is named where a printed row or a statement that names its sector as an SCFT is read by its archive or by the release layer, by its printed flavor symmetry: for E₇, E₈, twisted E₆ and ℤ₃-twisted D₄ the caveats of the archive (e7-statement-reading, e8-section3-reading, e6tw-statement-reading, d4z3-statement-reading) list these statements; for E₆ they are the rows of 1403.4604 that print an SCFT, the preamble of its Section 3.5 and its Section 5, 1501.00357 Section 7, 2203.05040 Section 3.1 and 1704.07890 Sections 4.3.2–4.3.3 and 4.4.1, each cited in the record's provenance, the earliest print first; the names of the records of v1 are those of their catalogue rows; for A₅–A₁₀ the names are those of the series of 1008.5203 Section 7 (T₆ also named in 0904.2715), given to the three-punctured spheres that carry the printed graded dimensions with a and c (with n_v and n_h for V_N and W_N), and the E₈ theory as the A₅ fixture [1⁶], [2³], [3²] of 0906.0359 Section 4.4, 31 records, with the printed global symmetry of each in enhanced_flavor (for S₆ the printed group has dimension 72 and b₂ of the series is 78, a declared source issue); the release adds R_{2,5}, R_{2,7} and R_{2,9} with 6, 8 and 10 free hypermultiplets for the fixtures [1^N], [N/2, N/2−1, 1]² of N = 6, 8, 10, by the sentence of that section on the fixture of its fig133 (caveat an-names); a product of sectors is written with x, and a sector whose name has several factors in brackets, for example E_7 x [(E_7)_16 x SU(2)_9]. In Parquet the coefficients are lists of decimal strings and the fields irregular, refined_probes, virtual_annotations, enhanced_flavor, source_discrepancy, the provenance objects of the formula and the annotations, and the data of a witness are JSON strings; the JSON Lines file keeps them as objects.

Relations between records

• Two records are never merged; the identifier is the construction.
• known_equivalences (12,010 rows) is the only relation that groups records as one theory. Each row gives its statement and the registry entries of its sources, and every row has one stored series. A row is declared when a source states the identification (for the Argyres–Douglas labels of type A with gcd(k, N) > 1 also as part of a conjecture of the source: the frames of 2311.12123 (3.85), and D₂(SU(4)) with SU(2) with four flavors, which 1204.2270 and 1210.2886 conjecture and 1303.3149 states; the registry gives the status of each source), or when two catalogue rows print the same global symmetry with levels, graded dimensions and (n_h, n_v) and a source uses that symmetry as the name of an SCFT, provided the stored series agree and, outside those labels, no source calls the identification a conjecture (396 rows, each with a name; 166 of them are the k ↔ N pairs of Argyres–Douglas labels, and 4 group a label of type D or E with gcd(k, h) > 1 with the coprime labels of the same 3-fold, by 1006.3435, each named after the coprime label; 217, with 550 members, group the labels of type A with gcd(k, N) > 1 with each other, with records of ad and with the A₁ record of SU(2) with four flavors: the pairs (A_{N−1}, A_{k−1}) ∼ (A_{k−1}, A_{N−1}) of 1006.3435, the N = 2 identifications of 1706.01607 Section 3 (restated in 2311.12123 (3.20); its family (A₁, A_{2N−1}) stated earlier in 1301.0210, and its two smallest cases, (A₁, A₃) = (A₁, D₃) and D₂(SU(3)) = (A₁, D₄), in 1011.1911 and 1210.2886), the frames of 2311.12123 (3.85) that are one coprime block or, after one step, one theory with mass parameters (after one step stated earlier in 1301.0210 and, with distinct indices, derived as the case n = 1 of 2012.12852 (3.26); for the member (A₂, A₂) of the row of (A₁, D₄) earlier still in 1006.3435 table2 and 1210.2886 Section 4.1.2), and D₂(SU(4)) and D₃(SU(3)) with SU(2) with four flavors (1204.2270 Section 6.3.1, which states both with the four-punctured A₁ sphere, D₂(SU(4)) as a conjecture, restated in 1210.2886); their names label the rows only). An identification that follows only by combining two sources is not declared: the label D₃(SO(8)) and the ℤ₃-twisted D₄ fixture D4z3-g0-s3-w0_wG2a1_wG2, which 1303.3149 and 1601.02077 print as the same gauge theory, share a row of equal_windows only (with the four-punctured sphere that the rule of the atypical punctures identifies with that fixture, and two ℤ₂-twisted A₃ records). Twenty Lagrangian labels of type A with gcd(k, N) > 1, such as D₂(SU(6)) (SU(3) with six flavors) and D₂(SU(12)) (SU(6) with twelve flavors), share rows of equal_windows with the class S spheres of their gauge theories, seven with spheres of A₂–A₄ and thirteen with spheres of A₅–A₁₀; these identifications follow by combining two sources (1210.2886 or 1303.3149 with 0904.2715) and are not declared. Likewise the untwisted D₄ sphere D4-g0-s4-1-1-1-1-1-1-1-1_3-3-1-1_5-3_5-3, whose witness is the SU(3) gauging of the E₈ theory (1106.5410), the frame of D₂(E₆) (1303.3149), shares a row of equal_windows with D₂(E₆), and since release 7 with the A₅ sphere A5-g0-s4-2-2-2_2-2-2_3-3_5-1, and with no other record; the stored series of the D₄ sphere and of D₂(E₆) agree through x⁴⁰. The A₅ fixture [1⁶], [2³], [3²] (A5-g0-s3-1-1-1-1-1-1_2-2-2_3-3), which 0906.0359 Section 4.4 names the E₈ theory, and the untwisted D₄ record D4-g0-s3-1-1-1-1-1-1-1-1_1-1-1-1-1-1-1-1_5-3, which the D₄ catalogue row D4fig93 of 1106.5410 names the E₈ SCFT, both carry the name E_8 and share a row of equal_windows only: their identification follows by combining the two sources, the A₅ side being a statement and not a catalogue row (caveat an-relations). The name of a declared row is also the named_theory of its members that have none, except for the row of the rank-two d₄ instanton SCFT, which 2007.00647 Section 5 realizes as a twisted A₃ fixture and as a ℤ₃-twisted D₄ fixture: its name labels the row only. A record can lie in a declared row and in a row of a rule (below): the twisted A₃ record of that theory does. A record can also lie in two declared rows: the four coprime Argyres–Douglas labels AD-A2-b3-k4-3, AD-A3-b4-k3-4, AD-A2-b3-k5-3 and AD-A4-b5-k3-5 lie in a k ↔ N row and in the row of the label with gcd(k, h) > 1 that has their 3-fold. The pairs of E₆ fixtures that 1403.4604 Section 6 conjectures to be isomorphic are not declared, nor the D₄ fixture 120 of 1106.5410, which prints the data of the first pair. The other 11,614 rows, without a name, come from a rule: a record with an atypical twisted puncture (t4-2, t3-3 of twisted D₄; t2-2 of twisted A₃; tF4a1 of twisted E₆; wG2a1, wwG2a1 of ℤ₃-twisted D₄) and the records obtained by resolving such punctures into their pairs are one theory, by 1309.2299 Section 2.5, 1212.3952 Section 4.1 (stated there for the fixture and the four-punctured sphere; the formula of 1309.2299 at N = 3 gives the same resolution for twisted A₃), 1501.00357 Section 3.6 with 1711.04727 Section 5, and 1601.02077 Section 3.5 (5,857 rows with 16,801 twisted D₄ records, 4,761 rows with 13,681 twisted A₃ records, 291 rows with 621 twisted E₆ records, 705 rows with 1,724 ℤ₃-twisted D₄ records, whose classes are taken up to the simultaneous exchange of ω and ω²; caveat release-atypical-punctures).
• Fixtures that a source conjectures to be isomorphic, or whose interacting parts a source identifies while their free hypermultiplets differ, are not known equivalences; the caveats of their archives name them, and for E₇ and E₈ records enhanced_flavor lists the partner records (e6_sector_records, e7_sector_records).
• equal_windows (14,614 rows, 42,164 members) lists the groups of records with identical stored series through x²⁰. Of the records of A₅–A₁₀, 103 have the stored series of a record of an earlier type (a record of v6), and 2,027 fall into 962 groups among themselves (58 of them cross_type; caveat an-relations). It serves a reader who wants one record per distinct series. Equal stored series are not an identification. compared_members lists the members whose series were compared beyond x²⁰, through x^compared_through; persists_through says that they agree through that order, first_differing_order where they first differ (null when no member was compared beyond x²⁰). Imported records were compared through x⁴⁰; a group of Argyres–Douglas records only is compared through the smallest stored order of its members (K_deep, or x²⁰ for a member without a deep series), and a group that mixes them with records of another type is compared as the other groups are. An unresolved record is in no group. relation is the first class that fits: cross_type (members of more than one type), e12 (twisted A₃ records that agree after every t2-2 is resolved into 3-3 and t4, the atypical puncture above; the label is that of the imported relation file), color_only (untwisted D₄ records that differ only in the colors of their punctures), prefactor_only (every member in the finite window), other.

Provenance

Each record carries its sources at four levels. Every source identifier is an entry of SOURCES.json, which gives the arXiv identifier, the location in the paper (section, equation label, table or figure), an excerpt of at most 25 words, the status of the statement, whether the location supports the use made of it, and whether the paper is the original source of the statement or restates an earlier one.

1. Formula (provenance_formula): the source identifiers and status of the index formula, and the files of the code that computed the series, by path and SHA-256.
2. Construction (provenance_construction): the catalogue location of every fixture and cylinder of the witness; a fixture that its catalogue counts without printing it is cited by the count statement with the rule that recovers it; for a generated witness (A₅–A₁₀) every fixture, irregular puncture and cylinder by its pole structures with the rule of 1008.5203 Sections 2.2–2.4 that it satisfies (key A-generated).
3. Annotations (provenance_annotations): the sources of every annotation, and whether a value is quoted from a source or computed from its formulas.
4. Literature (provenance_literature): earlier computations of the same index; same_index is false for a related quantity, such as a Macdonald or full index whose Schur limit the source does not write out, or a series printed for the interacting part or a factor of the record, which the release check compares after removing the free hypermultiplets or dividing by the release records of the other factors. One convention, kept since release 1, is the exception: the series of N=4 SU(N) on the tori with one minimal puncture, each entry naming the free hypermultiplet that the record has besides that theory, are same_index true and are compared with that free hypermultiplet, while the series of the rank-n instanton SCFTs on their spheres with one free hypermultiplet are same_index false.

provenance_missing lists the levels not filled. The formula, construction and annotation levels are filled on every record to which they apply; the literature level is filled on 1,583 admitted records and missing on 117,127, for which no earlier computation of the same Schur index was found (computations of a related index are listed with same_index false and do not fill the level). Some entries rest on the full text of a paper as read on alphaXiv, not on a copy of its source; their registry entry says so (read_via).

archive_records links every record to the lines of the archive files it was converted from, with their SHA-256. The archive files and the code that computed the series are not part of this release; they are named here by path and hash. Paths under imported/ are archive files imported byte for byte from an earlier project of the author.

Caveats

caveats.json holds 169 caveats: 165 quoted from the documentation of the archive files and 4 of the release itself (archives: release). A record lists in caveat_ids the caveats of the archive files whose content it carries, each restricted by its scope: the field applies of a caveat states the records it concerns (null: every record of its archives), for example only the records in the finite window, or only the records listed in its field records less those that its scope excepts. The quotes are verbatim; part of them name results, open questions or files of the earlier project in which an archive was first produced, or files of the code repository, none of which is part of this release, and the field release_note of most such caveats says what the reference means for a reader of the release. The main caveats:

• the index formulas are conjectural in their sources except for A₁, and a series is certified through its stored order only;
• admission does not certify a superconformal field theory, and distinct constructions are not proven to be inequivalent theories;
• in a finite-window record the stored series is the prefactor alone through x²⁰;
• one witness is stored per construction, and the witness search has no second implementation; the E-type fixtures that a catalogue counts without printing are recovered by the rule that every graded Coulomb dimension be nonnegative;
• free-hypermultiplet counts, theory names and enhanced flavor symmetries are recorded only where an archive or the release reads them from a printed statement of a source (for A₁, only on the three-punctured sphere), and null means unrecorded;
• printed rows that the stored series contradict are kept as printed and name the conflict: one twisted A₃ fixture (1212.3952, A3twistedinteracting7; caveats oq30-open, oq30-constraints on every twisted A₃ record), six E₆ rows (e6-flavor-conflicts), nine E₇ rows (e7-printed-conflicts), one E₈ statement (e8-printed-conflict), three twisted E₆ records (e6tw-printed-conflicts), three ℤ₃-twisted D₄ records (d4z3-printed-conflicts) and one A₅ fixture (S₆, caveat an-names);
• the punctures of twisted A₃, the colors of untwisted D₄ and the monodromies of ℤ₃-twisted D₄ follow conventions stated in release-a3tw-puncture-labels, release-d4-triality and d4z3-assignments;
• for the Argyres–Douglas labels of type D and E with gcd(k, h) > 1: the 17 unresolved labels with their reasons (ad-gcd-unresolved); the two series that rest on one source (ad-gcd-single-source); the four Lagrangian quivers, whose series the author's second evaluation checks through x⁴ to x²⁰ and the independent review through x²⁰ (ad-gcd-quivers); for D₂(E₆) and (E₇, A₈), whose records come from the archive ad-gcd-frames: the formulas of their frames (ad-gcd-frames-formula-status), the conflict of the series of D₂(E₆) with the closed form of 1706.01607 and with the vertex operator algebra (e₆)₋₆ (ad-gcd-frames-conflicts), and the second routes, among them the D₄ sphere above (ad-gcd-frames-routes);
• for the Argyres–Douglas labels of type A with gcd(k, N) > 1 (archive ad-gcd-A): the formulas and their checks (ad-gcd-A-formula-status), the 23 labels without a second route (ad-gcd-A-routes), the 26 without a deep series (ad-gcd-A-deep), the eight records whose x² coefficient exceeds the manifest flavor dimension (ad-gcd-A-flavor), the closed form of 1706.01607 that is not their index (ad-gcd-A-closed-form), the relations and the identifications not declared (ad-gcd-A-relations), and three printed statements not reproduced (ad-gcd-A-source-issues);
• for untwisted A₅–A₁₀ (archive an): the formula and the character routine (an-formula-status), the catalogue generated by the rules of 1008.5203 (an-catalogue), the second implementation on N = 6–8 only and what checks N = 9–11 (an-second-implementation), the names and the printed data declared, with the names of the release layer (an-names), three printed statements of 1008.5203 not used as printed (an-source-issues), the range with N ≥ 12 left out (an-range), the relations and the E₈ pair not declared (an-relations) and the literature (an-literature).

Independent review

Each archive was reviewed independently before it was published, by a Claude subagent (Claude Opus 5.5) in a fresh context that wrote no file of the database: the eight archives of release 1 on 2026-10-02, the four added in release 2 (e6, e7, e8, e6tw) on 2026-10-05, the one added in release 3 (d4z3) on 2026-10-07, the one added in release 4 (ad-gcd) on 2026-10-13, the one added in release 5 (ad-gcd-frames) on 2026-10-16, the one added in release 6 (ad-gcd-A) on 2026-10-20, the one added in release 7 (an) on 2026-10-27. The reviewers are the same model family as the author and read the author's notes, so the reviews are not blind comparisons. With code of their own, which shares no code with the engines, the reviewers recomputed every stored unflavored series of every archive of releases 1 to 6 (through x²⁰, through x⁴⁰ where stored, and through K_deep for the Argyres–Douglas records, except six deep series of ad-gcd-A, checked through x²⁰), 2,174 of the 20,773 series of an (below) and 600 of the 9,994 stored flavored probe series (refined_probes), enumerated the candidate sets of the range, checked the witnesses and compared the release with the archive files; they found no differing coefficient, status, witness or annotation value. For E₆ and twisted E₆ the reviewers used character methods other than those of the author's engines (the Weyl character formula at a regular torus element, a Brauer–Klimyk recursion); for E₇ and E₈ they used the method of the engines, the Weyl character formula restricted to a Levi subalgebra, in code of their own, which differs from the method of the second implementation. For twisted E₆ they also used routes without class S: the E-string series of 1609.00310, which give the rank-one factors, and a modular differential equation of 1907.08629, which gives the rank-two E₆ Minahan–Nemeschansky series of the records through x¹⁸. For ℤ₃-twisted D₄ the reviewer used the Weyl character formula in a limit at the torus element of each puncture, checked against a Brauer–Klimyk recursion, with K-factors from the grading of the adjoint, and compared three Lagrangian fixtures with Haar integrals of their printed matter through x¹² to x¹⁶ and the record of the rank-two d₄ instanton SCFT with the series printed in 1905.00864 (5.40) through x²⁰. For the Argyres–Douglas labels of type D and E with gcd(k, h) > 1 the reviewer evaluated every Haar integral by a finite-field cubature over Weyl-orbit representatives, a method of neither of the author's kinds, built the SU(2) frame of subset (A) itself, used the Frenkel–Kac–Wakimoto vacuum characters at the coprime labels of subset (C) and the Nahm sums of 2609.04099 (3.6) and 2607.19021 (2.6), and continued the four Lagrangian quivers to x⁴⁰, x²⁶, x³⁰ and x²⁴; D₃(SO(8)) through x⁴⁰ equals the stored series of two ℤ₂-twisted A₃ records. For D₂(E₆) and (E₇, A₈) the reviewer recomputed both series through x⁴⁰ by routes other than the author's: for (E₇, A₈) the block SU(2) with four flavors by the A₁ class S sum, the block D₉(SO(8)) by Haar integrals of its u = 1 quiver and the Spin(8) integral by projection onto characters; for D₂(E₆) the A₅ fixture by a class S sum of its own with the SU(3) integral by Kostant's multiplicity formula, and the D₄ sphere above by a D₄ class S sum of its own, which is the same frame in another realization. For the Argyres–Douglas labels of type A with gcd(k, N) > 1 the reviewer recomputed the 317 series of (A) through x⁴⁰ by a sum of 1705.07173 of its own; the 677 labels that are not Lagrangian through x²⁰, and 342 of the 348 deep series of (B) and (C) among them through x⁴⁰, by a builder of the frames of 2311.12123 (3.85) of its own integrated by discrete Fourier sums over Weyl orbits modulo four primes; the 22 Lagrangian labels by a class S sum of type A of its own; and D₂(SU(2)) as two free hypermultiplets. For untwisted A₅–A₁₀ the reviewer recomputed 2,174 records through x²⁰ by Jacobi–Trudi characters evaluated at roots of unity modulo three or four primes (1,187, 647, 165, 84, 43 and 48 of N = 6 to 11, covering every puncture type of every N, the 57 records of genus 1 and the 31 named records), the five candidates evaluated in parts at their lowest orders (each whole candidate through x¹⁰), eight Lagrangian records (seven spheres and the torus with one minimal puncture of N = 6) by Haar integrals (SU(N) with 2N flavors for N = 6, 7, 8 through x²⁰ and N = 9 through x¹⁴, N=4 SU(6) with one free hypermultiplet, usp(2n) with four fundamentals and one antisymmetric hypermultiplet for n = 3, 4, 5), and the views and annotations of all records; it also generated the catalogue from the text of 1008.5203 and checked every witness. The series of the other 18,599 records rest on the generation, with its second implementation through x²⁰, x¹² and x⁶ at N = 6, 7, 8; of those of N = 9–11, seven have another evaluation (the five candidates in parts, the rank-five d₄ sphere by its Haar integral, the rank-three E₆ sphere by printed series), and 9,279 have no second evaluation of their series, 36 of them having the stored series of a record of v6 of another type (caveat an-review). Lagrangian routes (Haar integrals) and character formulas that differ from the index formula cover part of the records; elsewhere a recomputation of the same formula supports the coefficients, not the formula.

The corrections the reviews of release 2 required concerned printed statements that the archive transcriptions lacked (79 statements on 64 records, from tables, labelled figures and sentences of 1704.07890, 1802.09626, 1501.00357, 1403.4604, 2203.05040, 2212.11983, 1803.02425, 1503.06217 and 1504.08348, now in RELEASE-STATEMENTS.json; a further review of the corrections found the statements of 1403.4604 Section 5 and 1803.02425, which name 24 more records), names, literature entries, originality verdicts and the text of caveats; no stored coefficient, status or witness changed. The corrections the review of release 3 required concerned an earlier print of a series (1905.00864, attached to the records of the rank-two d₄ instanton SCFT), the identification of 2007.00647 Section 5 declared as a known equivalence, a name from 2102.02443 Section 4.8, the printed number of free hypermultiplets of one statement and the text of the caveats; again no stored coefficient, status or witness changed. The corrections the review of release 4 required concerned the erratum on the stored reason of (E₇, A₈) (its frame and the indices of its blocks are sourced, so it has a formula that this release does not evaluate), the meaning of the status unresolved, the attribution of the frame of (A₂, D₄) to 1103.5832, literature entries (2603.29829, 2607.19021), three source discrepancies recorded in two registry entries of 1910.02281, and the text of caveats and release notes; no stored coefficient, status or witness changed. The corrections the review of release 5 required concerned the D₄ sphere with the series of D₂(E₆) (a second realization of its frame, not an independent check), the citations of 1604.02155 and 2106.12579 in the account of the conflict of D₂(E₆), the sources of the blocks of (E₇, A₈), registry entries and this card; no stored coefficient, status or witness changed. The corrections the review of release 6 required concerned the original source of the identification of D₂(SU(4)) and D₃(SU(3)) with SU(2) with four flavors (1210.2886, and, as a re-review found, the earlier 1204.2270), 42 identifications of one step of 2311.12123 (3.85) declared (41 of them not implied by other edges), earlier originals of some identifications (1011.1911, 1210.2886, 1006.3435, 1301.0210), the reason why seven cross-type identifications are not declared, the citations in the verification texts, registry entries and this card; no stored coefficient, status or witness changed. The corrections the review of release 7 required concerned three names with their free hypermultiplets that 1008.5203 Section 7 states (the fixtures of its fig133 at N = 6, 8, 10, named in the release), the stated reason why the E₈ pair above is not declared, the statement of the range of A₅–A₁₀ in the manifest, the verification text of S₆, the level of the E₈ fixture, the schema text of enhanced_flavor, a release note, registry entries, the caveats of the archive and this card; no stored coefficient, status or witness changed. The outcome for each archive is its caveat release-review-imported, c40-review, a1-review, twisted-d4-review, ad-review, e6-review, e7-review, e8-review, e6tw-review, d4z3-review, ad-gcd-review, ad-gcd-frames-review, ad-gcd-A-review or an-review.

Versions

• v1 (2026-10-03): release 1, 91,934 records of the types A₁–A₄, D₄, ℤ₂-twisted A₃ and D₄ and the Argyres–Douglas labels, at commit 22dc4f0 of this repository. It is unchanged.
• v2 (2026-10-06): release 2, release 1 with the 23,872 records of untwisted E₆, E₇, E₈ and ℤ₂-twisted E₆ added, 115,806 records, at commit 84feb10. It is unchanged.
• v3 (2026-10-12): release 3, release 2 with the 3,010 records of ℤ₃-twisted D₄ added, 118,816 records, at commit 713adb9. It is unchanged.
• v4 (2026-10-15): release 4, release 3 with the 51 records of the Argyres–Douglas labels of type D and E with gcd(k, h) > 1 added (32 admitted, 19 unresolved), 118,867 records, at commit ec9925c. It is unchanged. Every record of v1, v2 and v3 is a record of v4 with the same identity, status, series and views, cutoff data, annotations and witnesses. Of the records of v3 only 121 differ, all Argyres–Douglas labels with k < 0, in the text of the first element of their construction level, which reads "no 3-fold (k < 0)" where it read "3-fold None"; the records of v1 and v2 differ in addition as they did in v3 (below).
• v5 (2026-10-17): release 5, release 4 with D₂(E₆) and (E₇, A₈) admitted from the archive ad-gcd-frames, with series through x⁴⁰ from their sourced frames; the 17 other unresolved labels stay unresolved, and the count of records is unchanged. Every other record of v4 is a record of v5 unchanged in every field. The two records change from unresolved to admitted in their status and reasons, series and views, stratum, cutoff data, formula and construction provenance, missing provenance, caveats, archive links and the archive named by their witness; their identity, label and annotations are those of v4. The records of v1, v2 and v3 differ from those of v5 as from those of v4. One row of equal_windows is new (D₂(E₆) with the D₄ sphere above), and the 7,167 rows after it have bucket numbers one higher than in v4: bucket numbers are not stable across releases. It is at commit 1f056e9 and unchanged.
• v6 (2026-10-22): release 6, release 5 with the 700 records of the Argyres–Douglas labels of type A with gcd(k, N) > 1 added (archive ad-gcd-A, all admitted), 119,567 records, at commit c3b72e0. It is unchanged. Every record of v5 is a record of v6 unchanged in every field except the A₁ record of SU(2) with four flavors (A1-g0-s4-1-1_1-1_1-1_1-1), whose literature level gains the series of 1606.08429 through its known equivalence with D₂(SU(4)) and D₃(SU(3)); the records of v1 to v4 differ from those of v6 as from those of v5, with this change in addition. The 11,793 rows of known_equivalences of v5 are unchanged and 217 are new. Of the rows of equal_windows, 90 are new and 7 rows of v5 gain members: 4 rows of type cross_type gain the type AD, and 3 rows of Argyres–Douglas records also change their compared members, compared order and cutoff weights; bucket numbers are not stable across releases.
• v7 (this release): release 6 with the 20,773 records of untwisted A₅–A₁₀ added (archive an, all admitted), 140,340 records. Every record of v6 is a record of v7 with the same identity, status, series and views, cutoff data, annotations and witnesses; seven differ in their literature level, which gains the closed sums evaluated for this release: the tori with one minimal puncture of A₁–A₄ the entry of 1507.08659 (21) with its footnote 3, and the tori with two minimal punctures of A₂–A₄ that of 1510.07041 (4.16), whose level, missing in v6 for A₃ and A₄, is now filled. The records of v1 to v5 differ from those of v7 as from those of v6, with these changes in addition. The 12,010 rows of known_equivalences are unchanged. Of the rows of equal_windows, 1,034 are new (962 of records of A₅–A₁₀ only, 72 with records of earlier types) and 31 rows of v6 gain members of A₅–A₁₀, among them the row of D₂(E₆): all 31 change their types, 19 their relation to cross_type, 5 their cutoff weights and 28 their bucket number; bucket numbers are not stable across releases.
• In v3, compared with v2 and v1: of the records of v2 three differed, in their literature level (the A₃ sphere and the two twisted A₃ records of the rank-two d₄ instanton SCFT gained the series of 1907.08629 and 1905.00864, and the first twisted A₃ record shares its row of known_equivalences with the ℤ₃-twisted D₄ record of that theory); of the records of v1 five differed in that level, these three and the two D₄ records that gained literature in v2.
• The relations, caveats and registry grow with the new records of each release.

Reading the data

The Parquet files can be read with pyarrow, pandas or the Hugging Face datasets library; the coefficients are decimal strings, to be converted to arbitrary-precision integers.

import json
import pyarrow.parquet as pq

columns = ['id', 'category', 'genus', 'n_punctures', 'status', 'c', 'named_theory', 'series', 'provenance_formula']
records = pq.read_table('records.parquet', columns=columns).to_pylist()
admitted = [r for r in records if r['status'] == 'admitted']
print(len(records), 'records,', len(admitted), 'admitted')

r = next(r for r in records if r['named_theory'] == 'E_6')
print(r['id'], 'c =', r['c'])
print([int(b) for b in r['series'][:9]])                 # b_0, ..., b_8 of I(x^2)
print(json.loads(r['provenance_formula'])['source_ids'])  # entries of SOURCES.json

The JSON Lines files are read one line at a time with gzip.open(path, 'rt') and json.loads; loading every record at once with its nested fields takes several GiB.

Verifying the files

SHA256SUMS gives the SHA-256 of every file. manifest.json lists the release files with their hashes, row counts and the hash of each gzip payload (compressed bytes can differ between zlib builds; the payload hash does not), states the range and the counts by type, and lists the formulas with their source identifiers.

License

The database (records, relations, caveats, schema, registry, release layers, manifest, build record) is released under the Creative Commons Attribution 4.0 International license (CC BY 4.0). The excerpts in SOURCES.json and the quotes in RELEASE-STATEMENTS.json are short quotations from the cited papers, each with its arXiv identifier and location; they remain under the terms of those papers.

Citation

If you use the database, please cite it and the sources of the formulas for the records you use (the sources column of the formula table above, which names the original sources, and for Argyres–Douglas records the earlier sources named below the table that apply to them; each record lists every entry, restatements included, in provenance_formula).

@misc{classS-schur-db,
  author       = {Cho, Minseok},
  title        = {{classS-schur-db}: exact Schur indices of four-dimensional {$\mathcal N=2$} class {S} theories, release 7},
  year         = {2026},
  version      = {v7},
  howpublished = {Dataset},
  url          = {https://huggingface.co/datasets/cms1308/classS-schur-db}
}