Measured 2026-09-11, before this paper existed, and recomputed on every run by block [5] of the script:
On its first run after this paper was written, block [5] failed:
“conformal geometry” had gone from 0 files to 2, and the two were this paper and the
index row that lists it. Reporting the finding put the phrase into the corpus the finding was about.
The check now reads the way it was always meant — outside the pages that exist because
of the finding, the phrase appears nowhere — and the baseline is kept beside the live number so
the drift stays visible. The first attempt at that exclusion was also wrong: excluding every file
that merely names WP-109 dropped ch8-0-monster and ch7-holographic,
two genuine CFT chapters that had only been annotated with a cross-link, and the CFT count fell from
20 to 17. A page belongs to a finding if it would not exist without it, not if it cites it. Both
errors are in the script’s comments where the next reader will hit them.
This is the same shape as the corpus’s other overloads and should be read beside them: composition appears in 138 files, every one categorical, while the n-bonacci numbers count integer compositions and nothing says so; eigenvalue appears in 124 files and the spectral theorem is stated in two. A word in heavy circulation is not evidence that its subject has been written about.
One page already polices this boundary properly and is worth the citation: WP-79 writes “Theirs is a ratio of conformal descendant levels fixed by a selection rule; ours is √(c/κnoise), a stochastic Lyapunov radius. No shared mechanism.” That is the register this paper generalises.
For n ≥ 3, every conformal map of a domain in ℝn is a Möbius transformation: a composition of similarities and inversions. The conformal group is finite-dimensional, of dimension (n+1)(n+2)/2 — 10 in three dimensions, 15 in four, 28 in six.
In n = 2 that rigidity fails. Every holomorphic map with non-vanishing derivative is conformal, so the local conformal algebra is infinite-dimensional — the Witt algebra, and its central extension is Virasoro. The global Möbius group of the sphere is still finite, (2+1)(2+2)/2 = 6, which is dim PSL(2,ℂ) as a real group; the exception is local, not global, and the distinction is the whole of it.
Block [4] takes the inversion x ↦ x/|x|² in ℝ³ and finds conformal defect 4.1×10−11: it is a Möbius generator and it survives. It then takes the naive analogue of z ↦ z² — double the azimuth, square the radius — and finds defect 0.44. In the plane that same construction is conformal; one dimension up it is not.
That is a demonstration of the contrast, not a proof of the theorem. Liouville’s statement is that the conformal maps of ℝn≥3 are exactly the Möbius transformations, and no finite computation establishes an “exactly”. The script says so in its own honesty block.
Licenses. Escher’s Droste map z ↦ zα is conformal — block [6] measures defect 1.4×10−10 — and it is conformal because it is holomorphic. Its existence is an instance of the two-dimensional freedom. Likewise Poincaré’s disk model is conformal and Klein’s is not, and both live in the plane where the question can even be interesting. Virasoro exists for the same reason, which is why Book 8 has the subject it has.
Forbids. Conformal intuition from a two-dimensional field theory does not transport to the corpus’s higher-dimensional objects. In six dimensions the conformal group has 28 parameters and no more; there is no infinite tower of local conformal transformations to inherit, and any argument that reaches for one has crossed a theorem. This is a guard of the same kind as the threshold-versus-scale rule, and it should be applied before any six-dimensional claim borrows a two-dimensional mechanism.
book8/ch8-0-monster.html states that the Monster’s vertex operator algebra is
“the chiral algebra of a conformal field theory whose contact structure on the unit tangent
bundle T¹ℍ is exactly the dm³ contact form with SL(2,ℤ)…”
T¹ℍ is the unit tangent bundle of the hyperbolic plane. That is Beltrami’s
object. Book 7 · Beltrami states the general
theorem — the geodesic flow of a Riemannian manifold is the Reeb flow of the canonical contact
form on its unit cotangent bundle — and cross-links to Ch Fy and Mirzakhani without
finding that the corpus already held the most famous instance, T¹ℍ/SL(2,ℤ), the
modular surface. The chapter was written from the general statement and did not check the
population first. That is the same error as the 1 255 broken links and the stale
origin/main, and it is recorded here rather than quietly fixed.
That T¹M carries a contact form whose Reeb flow is the geodesic flow
is standard. That the form on T¹ℍ is exactly the dm³ contact form is this
corpus’s own identification, and nothing in ch8-0-monster or in this paper
establishes it. It is the kind of sentence that should carry its own check before anything is built
on it, and no one has written that check.
Searching the corpus for Liouville returns the Liouville 1-form of contact and symplectic
geometry — vol2-contact, book7/ch-hamilton, and the Beltrami chapter
— not Liouville’s conformal rigidity theorem. At least four distinct results carry the
name in mathematics: the 1-form, the rigidity theorem used above, the theorem that a bounded entire
function is constant, and the Liouville measure. Two of them are now in this repository under one
word. Cite the statement, not the name.
Liouville’s theorem is quoted, not proved, and so is the infinite-dimensionality of the two-dimensional local conformal algebra. The counts in block [5] are file counts and a file mentioning a word once counts the same as a file about it. The reading of Book 8’s usage as field-theoretic rests on sampling four chapters, not on classifying all 35 files. Nothing here touches whether the dm³ identification on T¹ℍ is correct; it is flagged as unchecked and left that way.
| Liouville 1850 | Note VI in the third edition of Monge, Application de l'analyse à la géométrie — conformal rigidity in dimension three and above. |
| Internal · geometry | ch-beltrami · ch-felix-klein · ch-escher §6 and §12 |
| Internal · field theory | book8/ch8-0-monster · ch8-6-voa · ch7-holographic · ch13-holology |
| Internal · the guard | WP-79, which draws the same boundary for one number |
| Verification | book6/wp109-verify.py — 6 blocks, standard library only. |