The dimension test for balanced holomorphic vertex operator algebras at central charges 32 and 40

The value of m = dim V₂ − dim L_𝔤̂(k,0)₂ at the three cells treated here, and its check against the two realised lattice theories E_(8,1)⁴ and D_(32,1), where m = 0

Erratum: the displayed closed form m = dim V₁ + 139504 − dim Sym²𝔤 in arXiv:2602.12312v3 §5.1 is wrong; the coefficient of dim V₁ should be 247

The dimension test excludes the root system A_(5,64) at c = 32 (arXiv:2602.12312v3 Appendix A row 26), one of the two cells the source states it could not run

The dimension test does not exclude the root system A_(1,64)³ A_(4,160) at c = 32 (arXiv:2602.12312v3 Appendix A row 2), the other cell the source states it could not run

Control: the dimension test excludes A_(6,14) (arXiv:2602.12312v3 Appendix A row 72), reproducing the source's own X (dim) label through the same machinery and by the source's own gcd(D) ∤ m criterion

Audit of arXiv:2602.12312v3 Appendix A: 443 of the 447 decidable cells reproduced; the printed table carries 122 X (dim) rows against the prose's '121'; four cells named where the label is not reproduced

The smallest conformal-weight-2 module of A₅¹ at level 64 has top-level dimension exactly 3 845 961, at λ = ω₁ + 4ω₄ + 11ω₅ and its dual; there are 80 such highest weights and every one has level at most 19

Negative controls: the bound of BR7 is sharp, the exclusion of BR3 is about the target and not about D (3 845 961 is representable), 23 ∤ 147094, 23p + 247q = 100 is unsolvable, the A₁ weights 21 and 23 do not have conformal weight 2, and casA/dimA reproduce the adjoint and fundamental dimensions

The census of semisimple balanced root systems is kernel-bound: # BRS(40,0) = 1277 (and 449 at c = 32, 3294 at c = 48, and every value the source prints for c ≤ 9, 16, 24), together with the finiteness theorem dim 𝔤 ≤ c(1+h^∨) that makes the search finite

Erratum: arXiv:2602.12312v3 Table tab:numBRS prints # BRS(10,0) = 42; the correct value is 22

The dimension test excludes A_(6,35) at c = 40 (row 137 of the ancillary Tablesc40all.pdf, which records no result for it); the smallest conformal-weight-2 module of A₆¹ at level 35 has top-level dimension exactly 194 480, and there are 56 such weights, all of level ≤ 13

The dimension test excludes A_(3,1)⁴ G_(2,1)¹⁰ at c = 40 (row 1221, no result recorded) by a genuine coin-problem obstruction: neither of the two criteria the source states decides it

The dimension test excludes A_(40,1) at c = 40 (row 1276, the largest single-factor row of the table, d₁ = 1680, no result recorded): A₄₀¹ at level 1 has no module of conformal weight 2 at all

The c = 40 dimension test run on all 1 277 rows: 94 of the 1 123 rows the source's ancillary table leaves blank are excluded (62 by gcd(D) ∤ m, 10 by a size obstruction, 22 by a coin-problem failure); and an audit of the ancillary's own labels — 94 of its 118 X (d) rows reproduced, the 24 exceptions named, 36 of 36 X (J) consistent

Negative controls for c = 40: the bound of BR11 is sharp and attained twice, 194 480 is representable, the A₆ weight condition and the level both bite, A₄₀ has 41 level-one modules but none of conformal weight 2, neither part of the row-1221 certificate alone divides m, 90 515 is representable, and gcd(D) = 1 with m below the Frobenius number

The c = 40 value of m, checked against the two realised lattice rows E_(8,1)⁵ (row 1274) and D_(40,1) (row 1277), both of which give m = 0; and the level-one correction shown to be mandatory — without it D_(40,1) gets m = −3 409 560

The dimension test excludes 6 balanced root systems of central charge 40 with d₁ = 41 — ancillary rows 2, 3, 4, 5, 7, 8 of Tablesc40all.pdf, for none of which the source records any result (3 by gcd(D ∩ [1,m]) ∤ m, 2 by a size obstruction, 1 by a coin-problem failure)

The dimension test excludes 9 balanced root systems of central charge 40 with d₁ = 42 — ancillary rows 26, 27, 29, 30, 31, 32, 33, 40, 45 of Tablesc40all.pdf, for none of which the source records any result (0 by gcd(D ∩ [1,m]) ∤ m, 5 by a size obstruction, 4 by a coin-problem failure)

The dimension test excludes 4 balanced root systems of central charge 40 with d₁ = 43 and 44 — ancillary rows 57, 62, 68, 71 of Tablesc40all.pdf, for none of which the source records any result (2 by gcd(D ∩ [1,m]) ∤ m, 0 by a size obstruction, 2 by a coin-problem failure)

The dimension test excludes 8 balanced root systems of central charge 40 with d₁ = 48 — ancillary rows 138, 139, 140, 148, 155, 158, 163, 164 of Tablesc40all.pdf, for none of which the source records any result (7 by gcd(D ∩ [1,m]) ∤ m, 0 by a size obstruction, 1 by a coin-problem failure)

The dimension test excludes 3 balanced root systems of central charge 40 with d₁ = 60 and 105 — ancillary rows 274, 280, 924 of Tablesc40all.pdf, for none of which the source records any result (0 by gcd(D ∩ [1,m]) ∤ m, 1 by a size obstruction, 2 by a coin-problem failure)

The dimension test excludes 10 balanced root systems of central charge 40 with d₁ = 120 (1 of 4) — ancillary rows 950, 951, 954, 955, 956, 958, 962, 963, 970, 971 of Tablesc40all.pdf, for none of which the source records any result (7 by gcd(D ∩ [1,m]) ∤ m, 0 by a size obstruction, 3 by a coin-problem failure)

The dimension test excludes 10 balanced root systems of central charge 40 with d₁ = 120 (2 of 4) — ancillary rows 975, 976, 979, 981, 982, 984, 985, 988, 989, 990 of Tablesc40all.pdf, for none of which the source records any result (9 by gcd(D ∩ [1,m]) ∤ m, 0 by a size obstruction, 1 by a coin-problem failure)

The dimension test excludes 10 balanced root systems of central charge 40 with d₁ = 120 (3 of 4) — ancillary rows 1003, 1006, 1018, 1019, 1025, 1026, 1028, 1039, 1040, 1066 of Tablesc40all.pdf, for none of which the source records any result (6 by gcd(D ∩ [1,m]) ∤ m, 0 by a size obstruction, 4 by a coin-problem failure)

The dimension test excludes 12 balanced root systems of central charge 40 with d₁ = 120 (4 of 4) — ancillary rows 1067, 1101, 1104, 1109, 1110, 1113, 1114, 1115, 1117, 1118, 1119, 1120 of Tablesc40all.pdf, for none of which the source records any result (12 by gcd(D ∩ [1,m]) ∤ m, 0 by a size obstruction, 0 by a coin-problem failure)

The dimension test excludes 7 balanced root systems of central charge 40 with d₁ = 160 and 180 — ancillary rows 1138, 1142, 1143, 1144, 1184, 1191, 1192 of Tablesc40all.pdf, for none of which the source records any result (5 by gcd(D ∩ [1,m]) ∤ m, 0 by a size obstruction, 2 by a coin-problem failure)

The dimension test excludes 11 balanced root systems of central charge 40 with d₁ = 200 and 240 — ancillary rows 1207, 1208, 1209, 1210, 1211, 1214, 1215, 1218, 1222, 1223, 1241 of Tablesc40all.pdf, for none of which the source records any result (10 by gcd(D ∩ [1,m]) ∤ m, 0 by a size obstruction, 1 by a coin-problem failure)

Correction: the ancillary Tablesc40all.pdf of arXiv:2602.12312v3 records X (d) for row 93, A_(5,48) B_(2,24) (c = 40, d₁ = 45), but the dimension test does not exclude it — m = 41 860 = 88·385 + 4·1995, and the two conformal-weight-2 modules of dimensions 385 and 1995 are exhibited

Negative controls for the 90 c = 40 exclusions: the size obstruction is about size and not emptiness (B_(7,8) has exactly three conformal-weight-2 modules, of dimensions 230 945, 271 700, 388 960, all above m = 67 030; B_(3,100)² has one of dimension 419 881 above m = 40 507), a too-small claim is refuted, the coin obstruction is one unit wide at row 274 (m − 1 = 48 489 is representable, m = 48 490 is not), the level-one correction moves m by 1127 at row 1223, and the conformal-weight condition bites

The c = 40 dimension test as a kernel statement about modules: RepByModules and the completeness theorem mem_wt2DimsSmall, with a dimension prune, two interchangeable complete per-factor enumerations and deduplication — the engine that turns each of the 94 exclusions into three data (factor list, common denominator, m)