1 The volume problem and its normalization

Let $\mathcal M_{g,n}(L_1,\ldots,L_n)$ denote the moduli space of oriented hyperbolic surfaces of genus $g$ with $n$ labeled, unparametrized geodesic boundary components of prescribed lengths. A zero length denotes a cusp. Throughout, the curvature is $-1$, the parameters are fixed before taking a volume, and $2g-2+n>0$. Put $$d=3g-3+n,\qquad V_{g,n}(\mathbf L)=\int_{\mathcal M_{g,n}(\mathbf L)} \frac{\omega_{\mathrm{WP}}^{d}}{d!}.$$ We use the Fenchel–Nielsen convention $$\omega_{\mathrm{WP}}=\sum_{j=1}^{d}\mathop{}\!\mathrm d\ell_j\wedge\mathop{}\!\mathrm d\tau_j, \qquad \text{one full Dehn twist: }\tau_j\longmapsto\tau_j+\ell_j.$$(1.1) This fixes the scale of the form independently of the convention for orbifold integration. For comparison with complex coordinates $t_j$, let $\mu_j$ be the harmonic Beltrami representative of $\partial/\partial t_j$ and let $H_{j\bar k}=\int_X\mu_j\overline{\mu_k}\,\mathop{}\!\mathrm dA$ use the curvature $-1$ area element. Our positive complex convention is $$\omega_{\mathrm{WP}}=i\sum_{j,k}H_{j\bar k}\,\mathop{}\!\mathrm dt_j\wedge\mathop{}\!\mathrm d\bar t_k.$$ The factor is $i$, rather than $i/2$ with the same raw coefficients. For closed surfaces the conversion is explicit in [64, Section 6.1, equation (38)]; that paper writes $\mathop{}\!\mathrm d\tau\wedge\mathop{}\!\mathrm d\ell$, so its twist direction is opposite to ours. The conical extension used in Section 8 retains this scale; its geometric hypotheses are stated there.

Volumes are orbifold, or stack, volumes: generic ineffective stabilizers are included. In particular, $$V_{0,3}=1,\qquad V_{0,4}(\mathbf L)=2\pi^2+\frac12\sum_{i=1}^4L_i^2, \qquad V_{1,1}(L)=\frac{L^2+4\pi^2}{48}.$$(1.2) The effective torus volume is twice the last expression. This convention matters both in unfolding and in the torus–sphere correspondence; see Do–Norbury [55] and Section 6. More generally, a generic ineffective stabilizer of order $h$ divides the effective volume by $h$ without rescaling the local two-form. Rescaling $\omega_{\mathrm{WP}}$ by a constant $c$ instead multiplies a $2d$-dimensional volume by $c^d$. These are separate normalization operations.

The original geometric proposal has three separate ingredients:

1. realize an intrinsic surface by a canonical convex polyhedron;

2. describe the allowed polyhedra by cells, ideally linear polytopes;

3. identify the pullback of $\omega_{\mathrm{WP}}^{d}/d!$ on each cell.

The realization and the cell decomposition identify the integration domain. They do not, by themselves, identify its measure. In the ideal genus-zero case the measure comparison is particularly favorable; in the hyperideal case a useful local density is still available, but it is generally no longer constant in the natural angle variables.

There are also two different cross-ratios in the four-point problem. Four ideal vertices on $\partial\mathbb H^3$ have an ambient cross-ratio. The intrinsic metric on the polyhedral boundary has a conformal uniformization, and its four punctures have a conformal cross-ratio. The realization theorem relates the two parameter spaces, but these two numerical coordinates are not generally equal. Keeping them distinct is essential to the first explicit integral below.

The computational companion in experiments/normalizations/README.md records exact finite checks of the determinants, factors, and low-dimensional intersection formulas used in Sections 1–8 and Appendix A; its machine-readable output is artifacts/normalizations/result.json. These fixtures check algebraic consequences and conventions. They do not prove the cited realization, measure-comparison, or convergence theorems.