B Conventions at a glance

These conventions locate the normalizations used in the chapter formulas. Their geometric and analytic hypotheses remain those of the referenced statements; a convention does not assert an existence or convergence theorem.

Quantity	Convention in this monograph
Surface/ambient hyperbolic curvature	$-1$ ; the area-one rescaling $\widehat g=A^{-1}g$ has curvature $-A$ ; (8.3).
Complex hyperbolic curvature	Holomorphic sectional curvature $-1$ , with $\omega_{\mathrm{CH}}=-2i\partial\bar\partial\log\mathcal A$ ; (7.6).
Flat weights	Deficit $2\pi\mu_i$ , opening $2\pi(1-\mu_i)$ , $0<\mu_i<1$ , $\sum_i\mu_i=2$ ; (7.1).
Three volume scales	$A$ is boundary area, $W$ is enclosed three-volume, and $V_{\mathrm{WP}}$ is moduli volume; Section 11.
Flat WP limit	$A^{-1}\omega_{\mathrm{WP}}\to\omega_{\mathrm{CH}}/2$ , locally on open moduli at fixed weights; (8.5).
WP form	$\sum_j\mathop{}\!\mathrm d\ell_j\wedge\mathop{}\!\mathrm d\tau_j$ , with full twist period $\ell_j$ ; (1.1). For the raw Beltrami pairing $H$ , the complex form is $i\sum_{j,k}H_{j\bar k}\mathop{}\!\mathrm dt_j\wedge\mathop{}\!\mathrm d\bar t_k$ ; Section 1.
Boundary components	Labeled, unparametrized, with lengths fixed before integration; a zero length denotes a cusp; Section 1.
Volumes	Orbifold/stack volumes include ineffective stabilizers; $V=\int\omega_{\mathrm{WP}}^{d}/d!$ ; Section 1.
Punctured-torus volume	$V_{1,1}(0)=\pi^2/12$ ; effective volume $\pi^2/6$ ; (1.2).
Four-punctured-sphere volume	$V_{0,4}(0)=2\pi^2$ ; the direct integral includes both labeled orientation sheets; (3.5).
Cone parameter	Opening $\theta$ , formal length $L=i\theta$ , deficit $2\pi-\theta$ . Geometric volumes require positive area and the specified chamber corrections; (5.1), (5.6).
Ideal tetrahedral $z$	Cross-ratio of ambient ideal vertices; it need not equal the intrinsic conformal cross-ratio; Section 3.
Exterior and interior angles	Exterior $\theta_e$ is a dual metric edge length, with sum $2\pi$ around an ideal vertex; (2.2). Schläfli uses interior $\alpha_e$ : $\mathop{}\!\mathrm dW=-\frac12\sum_e\ell_e\mathop{}\!\mathrm d\alpha_e$ ; (11.4).
Lambda length	$e^{\delta/2}$ for signed decorated distance $\delta$ , with no additional $\sqrt2$ ; (3.2), (22.3).
Planar Hessian and gauge	Leading–trailing coordinates have $u\|_B=0$ and $-D_u^2\mathcal V=2L_{II}$ ; (9.3), (9.4). The boundary response is grounded or taken modulo constants; (9.7).
Compact quantum integers	$[n]_{q_{\mathrm{alg}}}=(q_{\mathrm{alg}}^n-q_{\mathrm{alg}}^{-n})/(q_{\mathrm{alg}}-q_{\mathrm{alg}}^{-1})$ ; the compact root is $e^{\pi i/r}$ ; (10.2).
Colored Jones and volume roots	Reduced $J_N(\text{unknot};q_J)=1$ ; form the polynomial before $q_J=e^{2\pi i/N}$ . Chen–Yang RT/TV use their separate odd- $r$ root convention; (10.4), (10.5).
Chen–Murakami $6j$	$\xi_r=e^{2\pi i/r}$ , $r$ odd, source $q=\xi_r^2$ and $q_{\mathrm{alg}}=\xi_r$ . The refined prefactor in (11.6) requires the exact label sequence stated there, beyond the leading-limit hypotheses.
Continuous semiclassical scale	$h=2\pi b^2$ , volume-action $\hbar_{\mathrm{vol}}=\pi b^2=h/2$ ; (14.24). These differ from $\hbar_r=4\pi i/r$ in (10.20).
Regular Barnes normalization	Upward contour with measure $\mathop{}\!\mathrm dw/i$ ; the Airy expansion is for the pole-cleared $\widehat{\mathcal R}_b$ and bounded real $\zeta$ ; (15.1), (15.3), (15.4).
Liouville charges and marks	Physics $\alpha_i=\mu_i/(2b)$ , probabilistic $\beta_i=2\alpha_i$ , $Q=b+b^{-1}$ ; marks $0,1,z,\infty$ carry labels $1,2,3,4$ ; (10.21), (13.1).
Relative torsion normalization	The $B=-\mathop{\mathrm{tr}}$ peripheral factor in (12.13) is specific to that bilinear form; torsion, Goldman form and peripheral factors must be rescaled together.
Closed-curve sums	Unoriented free-homotopy classes unless stated otherwise; primitivity and iteration are specified separately; Sections 18, 23.
Positive geometric trace	$t_\gamma=2\cosh(\ell_\gamma/2)=\|\mathop{\mathrm{tr}}\widetilde\rho(\gamma)\|$ for a hyperbolic element; compatible lift signs are retained in trace identities; (18.1).
Coefficient basis	$T_C=\prod_{\alpha\subset C}\mathop{\mathrm{tr}}\rho(\alpha)$ , $T_\varnothing=1$ , with unnegated traces and parallel multiplicities; (23.2).
Twisted Selberg sum	Closed genus- $g\ge2$ surface; unoriented primitive $\delta$ and iterate $k$ have weight $\ell_\delta/\sinh(k\ell_\delta/2)$ ; (23.4).
Coefficient heat transform	$H_C(t)=e^{-t/4}(4\pi t)^{-1/2}\int e^{-L^2/(4t)}\,\mathop{}\!\mathrm d\sigma_C(L)$ ; (23.12).
Coefficient Laplace parameter	Kernel $e^{-sL}$ ; product parameter $v=s+1/2$ ; (23.13), (23.31).
Total multicurve count	All nonzero integral laminations, including disconnected and parallel supports; (24.2).
Primitive lamination	Gcd of distinct-support weights is one; connectedness is an additional condition; (24.5).
Support variable $z$	Exponent counts distinct geodesic supports, not parallel copies; (24.16).
One-crossing torus curves	Primitive curves with one transverse double point; each simple class gives two N-type and one H-type labels, including equal-length multiplicity; (18.8).
Basmajian sum	Coefficient $2$ for oriented orthogeodesics, $4$ for unoriented arcs counted once; (22.6).
Cusp orthospectrum	Simple cusp-to-cusp arcs, signed truncation $d$ at horocycle length $H$ : $H\mapsto kH$ gives $d\mapsto d-2\log k$ ; $e^{-d}/H^2$ is invariant; (22.4).
Cusp-normal rays	The almost-everywhere intersection/gap discussion is for complete finite-area surfaces without geodesic boundary; Section 22.
Cuff phase matching	Use one common cuff coordinate after reversing one induced orientation: $(s,p)\mapsto(s+\tau,p)$ , with $p$ the signed tangential unit velocity; (21.1), (21.2).
Peripheral logarithms	PSL $_2(\mathbb C)$ multiplier logs $u,v$ ; eigenvalues $m=e^{u/2}$ , $l=e^{v/2}$ with fixed lift signs; smooth filling $pu+qv=2\pi i$ ; (16.1), (16.2).
Peripheral polarization	$\Omega=\frac12\mathop{}\!\mathrm dv\wedge\mathop{}\!\mathrm du$ , $P=v/2$ , $[\widehat P,\widehat u]=\hbar$ ; $\widehat l$ shifts $u$ by $\hbar$ ; (10.15), (10.18).
Real filling source	Adding $jLx$ fixes the source conjugate to angular holonomy $\theta=Lx$ ; on a geometric cone branch, $j=\ell_{\mathrm{core}}/2$ and $\theta$ is the cone angle; (16.4), (16.5).
Holomorphic filling action	$F^{\prime}=v$ and $\Delta V=\frac12\Im\mathcal S^*$ , with $\mathcal S$ as in (16.10); (16.11).
Norm-ball gauge and counts	$A_B$ counts all nonzero signed lattice points, $P_B$ the primitive ones, boundary included. The shortest gauge value is one; $\lambda_\gamma=\ell_\gamma/s_X$ for systole $s_X$ ; Sections 30.1, 31. The earlier $A_X$ in Section 29.1 includes the origin and uses hyperbolic length units.
Flow and area scales	Teichmüller time uses $g_t=\operatorname{diag}(e^t,e^{-t})$ ; for $h=\dim_{\mathbb C}\mathcal Q$ , area cutoff $L$ has exponent $h$ and period scale $\sqrt L$ has exponent $2h$ ; (27.2), Section 27.3. The unit-speed geodesic flow in Section 26.1 uses $\operatorname{diag}(e^{t/2},e^{-t/2})$ .
Weighted correlations	$C_{F,G}(t)=\int F(G\circ g_t)\,\mathop{}\!\mathrm d\mu$ is bilinear; use $\overline F$ for the Hermitian version. The full-rate converse assumes the count for $(h+V)G$ ; Theorem 27.4.

Source conventions. The local form scale is fixed by (1.1) and the raw Beltrami pairing in Section 1. Ineffective stabilizer weights are a separate operation. Penner’s logarithmic triangle form is used with the ordered-endpoint conversion from Mondello in Section 3; the direct tetrahedral integral (3.5) and trace density (A.2) give independent normalization checks within the monograph. The Delaunay-form constants of [50, Sections 3.2.3–3.2.4] are therefore not imported unchanged into (3.1). The metric-map formula (2.5) uses Budd’s disintegrated measure and label-and-orientation-preserving automorphism weights [48, Section 1.1 and Corollary 2]. The ledger book/claims/section32-appendixb.json records inherited source-statement audits, current convention dependencies, and finite computational support. It does not claim a new complete review of the external proofs.