32 Questions suggested by the calculations

Question 32.1 (Direct hyperideal integration). For four labeled geodesic boundaries of fixed positive lengths, can the domains and local density (4.5) be integrated directly, with all cell and symmetry factors, to recover the already known volume $2\pi^2+\frac12\sum_iL_i^2$ in (1.2)? Which change of variables makes the polynomiality visible in these hyperideal coordinates? The positive-boundary extension described in Section 4 uses a selected pair $L_1<L_2$ to give a full-measure decomposition into tree and double-tree cells [99, Theorem 3]. Equal-length volume values follow by continuity. These spine/tree cells are not identified with the hyperideal angle cells.

Question 32.2 (A usable measure on general polyhedral cells). The inverse-Poisson formula (4.7) is already a local density on a symplectic leaf. Can it be expressed in independent dihedral-angle variables in a way that makes higher-dimensional cell integrals useful? A calculation must specify the coordinate Jacobian, the fixed hyperideal boundary lengths or compact cone openings, and the geometric chamber and quotient. In the ideal case the disintegrated measure (2.4) supplies a simple model; a determinant in redundant angle coordinates does not by itself supply the replacement.

Question 32.3 (Gluing and topology). Can a polyhedral description be combined with Fenchel–Nielsen gluing to give a practical calculation for arbitrary $(g,n)$? A proposed formula must specify both the cutting multiplicity and the full mapping-class quotient, including stabilizer weights. The twist periods in (1.1) remove the Dehn-twist ambiguity along a chosen cut; they do not make all pants-length coordinates a fundamental domain for moduli space.

Question 32.4 (Curvature corrections in period coordinates). Can the correction class $\Xi_\mu$ in (8.8) be represented by an explicit first curvature correction to the WP form in the actual finite-area conformal coordinates? Equation (12.19) answers the local four-point calculation in a blown-up character chart, whose conversion to conformal coordinates is still required. How does the local density behave in a joint limit toward a balanced collision wall, where the final-chamber expansion is not uniform in the weights?

Question 32.5 (Beyond the regular quantum degeneration). Can Proposition 15.1, for the pole-cleared equal-label family and bounded real transition parameter, be extended uniformly over compact sets of nondegenerate Euclidean tetrahedral shapes, with a geometric description of the required scalar normalization? Can a geometric state integral in that regime produce the Thurston–Deligne–Mostow measure on the boundary-shape space? The model (11.8) specifies a classical comparison, not an identification with an existing quantum invariant. The exact Liouville identity (12.6) and the torsion limit (12.14) identify two precise boundary constructions; the auxiliary-tetrahedron obstruction and the different regimes in (12.24) explain what remains to connect them to that joint asymptotic. The finite one-screening fusion calculation in Section 13 computes the finite period and residue-sector comparison, with a logarithmic limit at balanced weights. Its direct infinite-spectral derivation uses the continuation and tail-control input (13.3); Section 14 fixes the quantum-group normalization, proves the rank-two obstruction, and computes the classical rank-three fold in the regular compact model. Section 15 proves the regular-family formula with its first correction using the exact integration-coordinate dictionary, pole clearing, contour and tail estimates. Its finite exact-integrand evaluations test that normalization on specified inputs. They give no uniform theorem for arbitrary shapes, unbounded transition parameter, or global gluing. The fixed-shape root-of-unity theorems in Section 11 have their own admissible label sequences; the refined Chen–Murakami prefactor has stricter sequence hypotheses than the leading volume limit. Neither supplies this joint degeneration or identifies its measure.

Question 32.6 (Uniform quantum Dehn filling). For a knot complement admitting the positive geometric optimization of Section 16, can an exact quantum surgery representation be controlled uniformly in a complex neighborhood of complete peripheral holonomy, with a nonzero geometric contribution and bounds on all competing terms? Proposition 16.1 already fixes every classical long-slope coefficient, and (16.26) specifies the torsion check. The further issue is the analytic input of Proposition 16.2, especially in a joint limit where the slope varies with the quantum level.

Question 32.7 (Certified realization and quantitative angle bounds). For the planar problem of Section 9, can the finite-centering certificate and short-step iteration estimate (9.14) be combined with positive-feasibility search, controlled inexact solves, and geometric error certification into a complete complexity theorem with explicit input and precision dependence? The existing iteration estimate starts from a specified centered feasible point; the dense finite examples do not bound the cost of finding that point or certify their floating-point errors. For a three-manifold angle polytope whose maximum is interior, which additional geometric hypotheses bound its smallest angle in terms of the combinatorial input and a quantitative feasible margin? The planar Laplacian sign structure does not extend automatically to the full three-dimensional Hessian.

Question 32.8 (A continuum comparison of second variations). Can a family of refined polyhedral functionals, with specified geometric convergence and normalization, have a second-variation limit related to a WP potential? The exact cotangent identity, Schur complement, and Gaussian determinant in Section 9 provide discrete objects for such a comparison, with the stated boundary grounding or quotient by constants. Control of their refinement limits and an explicit identification of the target metric are still required; finite determinant identities supply neither.

Question 32.9 (Continuation of a coefficient Laplace transform). For a fixed nonempty multicurve on the closed surface of Section 23, does $D_C(s)$ admit a meromorphic continuation beyond the proved half-plane $\Re s>1$? The kernel in (23.13) is $e^{-sL}$. A proof must control coefficient extraction in a suitable topology on the continued character-valued expression: extraction is not bounded for the uniform norm on unitary characters. Pointwise continuation of twisted Selberg functions does not justify that step.

Question 32.10 (Control of the globally filtered remainder). Proposition 25.1 gives a bounded global cancellation of every one-crossing primitive and all its iterates. Can the signed residual $\mathcal R_X$ in (25.6) be estimated with constants controlled by the geometry of $X$? The exact obstruction (25.7) rules out a perfect selector in the specified class of fixed linear functionals. Length-dependent cancellation or the nonlinear diagonal extraction in (25.10) retains information that such a selector discards. The latter is an exact length-valued construction, whose quantitative stability under approximation remains an additional task.

Question 32.11 (Effective selection of connected supports). Can the signed coefficient relations produce the smoothed estimate (24.22) with specified $\eta,m$ and geometric constants? The elementary lattice remainder, the stable finite support projection (24.19), and the constructive isolation estimate are available. The global one-crossing filter and the length-valued Euler product now provide concrete aggregate objects. An estimate for the higher-intersection residual or for the nonlinear first-atom extraction, uniform over the growing family of supports and strong enough to yield a counting saving, remains to be proved. Fixed-coefficient isolation does not supply this aggregate estimate.

Question 32.12 (From weighted curve counts to mixing). Can the filling-pair coefficient construction of Section 27 produce a norm-controlled estimate for the weighted orbit kernel (27.6) without assuming mixing? Proposition 27.2 gives the exact unfolding with the reciprocal polar Jacobian. Under the assumed estimate for terminal tests $H_G=(h+V)G$, with $F\in L^2(\mu)$, $G\in\operatorname{Dom}(V)$ and $0<\delta\le h$, Theorem 27.4 preserves the counting exponent $\delta$. A bound for one endpoint test alone has the weaker de-averaging conclusion of Theorem 27.3. The norm-controlled two-endpoint estimate, its Jacobian weights, and the noncompact tail remain to be established by this coefficient route for the principal-stratum kernel; a scalar area or length count is insufficient. Section 28 settles the separate affine return estimate on the nonsquare $(2,2)$ stratum, and Section 29 gives selected primitive-arc coefficient observables with uniform geometric norms. Can their class be enlarged to control the full two-endpoint kernel, with quantitative stability of the nonlinear coefficient extraction and passage from auxiliary hyperbolic lengths to area on the simultaneous mapping-class quotient? Can the calculation be extended beyond the marked-torus locus with its distinct invariant measure?

Question 32.13 (The sharp norm-ball remainder). Can the qualitative $o_X(L\log L)$ remainder of Proposition 30.1 be made effective through a decay modulus satisfying (30.2), or through cancellation in (30.1)? The inverse-flatness formula (30.7) quantifies a local lower obstruction, while (30.3) gives a test independent of the leading coefficient. Determining whether the primitive remainder is $O_X(L)$, or has an unbounded slowly varying factor, remains open here. Section 31 makes the two inputs explicit: $\Delta_B(u)=O(u/\log u)$ together with any fixed Mertens power saving gives $O_X(L)$, whereas an integrable geometric modulus and the prime number theorem already give $o_X(L)$. The all-point target is itself conditional on compatibility with the local geometry: under the supporting-row hypotheses of Proposition 30.2, it requires $W(h/L^2)=O(1/\log L)$ at the contact scales $L=n/h$. The double-exponential model in Section 30.5 violates that necessary condition. No such supporting configuration or compatible modulus is established for the two computed tori. The maintained length-$8192$ campaign evaluates the finite diagnostic (31.9) in hyperbolic length units; the unit-shortest-vector gauge requires rescaling by the systole. It reproduces the displayed counts and block values, but the original precision and sensitivity archive remains unrecovered. Its new sampled precision checks and conditional sensitivity envelopes must retain their own provenance. Neither a finite range nor an assumed arithmetic saving determines the missing geometric modulus.

Question 32.14 (Beyond closure operations). Which positive constant series have a geometric partition or a useful functional equation behind them? The figure-eight kernel (18.7) and the twist kernel (20.12) provide two concrete models. Their different index sets suggest separating global curve identities from identities for a distinguished twist orbit.

Question 32.15 (A boundary phase-space formulation). Can the endpoint-and-direction matching relations in Section 21 be organized into an integration scheme for general filling curves? Orthogeodesic lengths give the canonical local geometry, while the endpoint phase variables retain the missing gluing information. A useful result would specify the invariant measure, matching multiplicities and convergence of both the itinerary sum and the phase-variable integrals. A cusp version must also retain the truncation convention of Section 22; its almost-everywhere cusp-normal-ray discussion assumes no geodesic boundary. Surfaces with geodesic boundary require the additional exiting-ray contributions.