classS-schur-db

classS-schur-db — exact Schur indices of four-dimensional N=2 class S theories (release 5)

A database of exact finite expansions of the Schur index of four-dimensional N=2 superconformal theories of class S. Release 5 holds 118,867 records, one per construction or Argyres–Douglas label: untwisted theories of type A₁–A₄, 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 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 thirteen types, not within class S.

Release 5 is the revision tagged v5 of this dataset repository, https://huggingface.co/datasets/cms1308/classS-schur-db. A search site that reads these files, with a page per record, is at https://huggingface.co/spaces/cms1308/classS-schur-db. Releases 1 to 4 stay at the tags v1, v2, v3 and v4 (section "Versions").

Contents

In all 118,867 records: 97,237 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 (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. 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, an irreducible singularity 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 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 5

Release 5 contains every candidate of the thirteen 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.

A1, 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 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 5: class S theories of type A_{N−1} with N > 5 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, k < 0 and a puncture other than the full one and [N−2, 1, 1], and with reducible singularities.

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 with gcd(k, h) > 1 except the four Lagrangian quivers of subset (D), which have no deep series. For the 454 Argyres–Douglas records in the finite window, the dependence on the irregular singularity is in these fields only.

Index formulas

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 every stored unflavored series was recomputed by the independent reviews (below); 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 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 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

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); 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; 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 (11,793 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, 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 no source calls the identification a conjecture (179 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). 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). 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) only; the two stored series agree through x⁴⁰. 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 (13,490 rows, 39,485 members) lists the groups of records with identical stored series through x²⁰. 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.
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.

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 1,297 admitted records and missing on 95,940, 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 152 caveats: 148 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 such a caveat 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) and three ℤ₃-twisted D₄ records (d4z3-printed-conflicts);
• 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).

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 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 (through x²⁰, through x⁴⁰ where stored, and through K_deep for the Argyres–Douglas records) 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. 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 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 or ad-gcd-frames-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 (this release): 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.
• 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 5},
  year         = {2026},
  version      = {v5},
  howpublished = {Dataset},
  url          = {https://huggingface.co/datasets/cms1308/classS-schur-db}
}