Langlands duality with no Galois side at all — the centre of an affine vertex algebra at the critical level is the W-algebra of the dual. WP-82 corrected Volume XV's seed to this and measured one file using the word. Here is the floor under it, and an honest measure of the distance left.
WP-82 §3 first gave Volume XV — rung 32, Motivic / Langlands — the seed “Monstrous Moonshine — 25 files, and Vol VIII entire — is the road from the Monster to modular functions, which is half the distance already walked.” Its own correction of 2026-09-11 withdrew that, and the withdrawal is the better piece of mathematics:
The replacement is Feigin–Frenkel. At the critical level the centre of the affine vertex algebra is the classical W-algebra of the Langlands dual — a duality living entirely inside vertex algebra theory, with no Galois side needed, and reachable from material Volume VIII already holds. Measured at HEAD, entity-aware, this page classified out:
| pattern | files | chapters |
|---|---|---|
| E8 | 158 | 141 |
| Moonshine | 28 | 22 |
| vertex operator | 9 | 8 |
| central charge | 8 | 7 |
| Virasoro | 7 | 7 |
| Kac–Moody / affine Lie | 3 | 2 |
| Langlands | 3 | 2 |
| critical level | 1 | 1 |
| W-algebra | 1 | 1 |
| Feigin | 1 | 1 |
| Sugawara | 0 | 0 |
| dual Coxeter number | 0 | 0 |
| Zamolodchikov | 0 | 0 |
The single chapter using “W-algebra” is WP-82 itself — the paper that named the gap. Everything the theorem is stated in terms of reads zero: no Sugawara construction, no dual Coxeter number, no critical level anywhere but in that one correction. Those are what this page builds.
Ln = −zn+1d/dz on Laurent monomials gives Lnzj = −j zn+j, and the bracket follows in one line, checked on 2197 integer triples with no floating point. The interesting question is the central extension, and block [2] computes it rather than quoting it:
Two columns of that table are the methodological part. Truncating the algebra to |n| ≤ N drops every cocycle condition that reaches outside the window, so the raw truncated cocycle space is inflated at the edge and the answer comes out wrong. The fix is to solve on [−N, N] and then restrict to pairs inside a smaller [−M, M] before quotienting — the same lesson ch-gelfand met from the other end, where an algebra generated to a fixed word length reported dimension 33 where the commutant forces 36. H²(Witt) is one-dimensional, which is why Virasoro is the central extension and not a central extension.
In the degree-zero part, ω(Lm, L−m) = c(m), the cocycle condition is a linear system. Solved exactly: the solution space is two-dimensional, spanned by m and m³ — and m² and m⁵ are not cocycles at all, which the block checks so that the reading cannot be an accident of the ansatz. The coboundaries in degree zero are c(m) = 2m f(L0), exactly the multiples of m. So every representative is m³ + t m, and
t = −1 is the unique value with c(1) = 0, and c(1) = 0 is the statement that L−1, L0, L1 span an uncentred sl2 — the Möbius subalgebra. The 12 is a normalisation; the −m is forced. With it, c(2) = 1/2, which is [L2, L−2] = 4L0 + c/2.
The Sugawara construction puts a Virasoro action on a level-k module with central charge c(k) = k dim g / (k + h∨), and a normalisation 1/(2(k + h∨)) that is undefined at k = −h∨. That is the critical level, and the fact that the Virasoro description breaks exactly there is the reason something else has to take over. Block [4] computes h∨, the root count and dim g for fifteen simple types from the Cartan matrix alone — building the root system by reflection closure, finding the highest root, and converting to coroot coefficients through the symmetriser — and every value agrees with the known one:
The check that makes this more than bookkeeping is in block [6], and it is deliberately not circular: at level 1, simply laced, c must equal the rank. Nothing in the computation was arranged to make that so — it tests h∨ and dim g together — and it holds for every simply-laced type tested. For E8: 248/31 = 8.
E8 is the object this corpus names in 141 chapters. From its Cartan matrix: h∨ = 30, 240 roots, dim 248. The Langlands dual is the transpose of the Cartan matrix, and E8 is self-dual. So the critical level is k = −30, the central charge is c(k) = 248k/(k+30), and the centre of the affine vertex algebra there is W(e8) — the dual is not some other algebra to be hunted for.
Block [5] computes the duals by transposition: A, D and E self-dual, G2 and F4 self-dual up to relabelling the nodes, and Bn ↔ Cn. The last pair is the whole content of the duality on this side, and it is a transpose.
No W-algebra is constructed on this page. Feigin–Frenkel is quoted; what is computed is the data the theorem is stated in terms of — which dual, which critical level, which central charge. The Sugawara operator is not built, the centre at the critical level is not exhibited, and the isomorphism is not checked in any case, not even sl2.
That gap has a reason worth printing, because it is exactly why the corpus's existing Virasoro vocabulary does not extend for free. A W-algebra is not a Lie algebra. Zamolodchikov's W3 adds a weight-3 field whose bracket closes only on the composite field :TT: − (3/10)∂²T, so the structure “constants” depend on the central charge and nothing in Part II carries over. Getting there needs operator product expansions and normal ordering, and neither is in this corpus. By WP-82's own admissibility bar — a machine-checked core, “or it is a reading list with a DOI on it” — Volume XV is not opened by this page. What the page supplies is the floor under its seed and an honest measure of what is left.
| Feigin–Frenkel | B. Feigin and E. Frenkel, on the centre of the affine Kac–Moody algebra at the critical level and W-algebras, from 1991 onward. The statement used here is as given in E. Frenkel, Langlands Correspondence for Loop Groups, Cambridge University Press, 2007. |
| Zamolodchikov 1985 | A. B. Zamolodchikov, on additional symmetries in two-dimensional conformal field theory — the W3 algebra and its composite field. |
| Kac 1990 | V. G. Kac, Infinite Dimensional Lie Algebras, 3rd ed., Cambridge University Press — Cartan matrices, dual Coxeter numbers, the Sugawara construction. |
| in-corpus | WP-82 §3, the XV row and its 2026-09-11 correction · ch-gelfand (the same truncation lesson, from the other end) · ch-conley |
| verification | book7/ch-feigin-verify.py — eight blocks, standard library only, exact over ℚ throughout. Every number on this page is printed by it. |
No priority is claimed for anything on this page. The Witt algebra, its second cohomology, the Virasoro cocycle, Cartan matrices, dual Coxeter numbers, the Sugawara central charge and Langlands duality by transposition are all classical. They are derived here rather than looked up, which is this corpus's practice, and that is a statement about method and not about priority.