29 From simple-curve coefficients to controlled observables

29.1 The norm-ball obstruction and the correct error scale

The genus-one length norm gives a necessary benchmark for any passage from smooth return estimates to sharp simple-curve counting. Let $X$ be a fixed complete hyperbolic once-punctured torus, and let $B_X$ be the unit ball of the McShane–Rivin length norm on $H_1(X;\mathbb R)\simeq\mathbb R^2$. Put $$A_X(L)=\#(LB_X\cap\mathbb Z^2),\qquad P_X(L)=\#(LB_X\cap\mathbb Z^2_{\mathrm{prim}}).$$ The first count includes the origin and nonprimitive classes. The second counts oriented primitive classes; the count of unoriented simple closed geodesics is $N_X(L)=\tfrac12P_X(L)$. Elementary planar lattice counting and Möbius inversion give $$\begin{aligned} A_X(L)&=\operatorname{Area}(B_X)L^2+O_X(L),\\ P_X(L)&=\frac{\operatorname{Area}(B_X)}{\zeta(2)}L^2 +O_X(L\log L). \end{aligned}$$ Indeed, $P_X(L)=\sum_d\mu(d)(A_X(L/d)-1)$, with $d\ll_X L$; the boundary errors sum as $L\sum_{d\ll_X L}d^{-1}$. These are the upper estimates behind [16, Theorem 4.1], with the orientation factor made explicit here.

The next theorem in that paper concerns a dense set of hyperbolic structures. Its mechanism is an infinitely flat boundary point whose supporting normal is rational with respect to the homology lattice. The subsequent Conjecture 4.3 proposes an $O(L^{1/2+\varepsilon})$ error for the modular torus; thus the obstruction is not asserted there for every individual surface. The phrase “at least $O(L)$” in Theorem 4.2 should be separated from the literal modern assertion $\limsup|E_X(L)|/L>0$. The elementary argument below proves the precise consequence needed here: no fixed improvement of the exponent below one is possible under the stated flat-point hypothesis. We do not use it to claim a literal $\Omega(L)$ bound.

The proof works directly with primitive classes, so no cancellation in a Möbius sum needs to be controlled. The short interval $[L_n,L_n+L_n^{-1}]$ acquires many new primitive points along a nearly flat supporting row, while its predicted main-term increment stays bounded. Infinite flatness alone does not force the row length to be a positive fraction of $L_n$; that is why the proof gives an exponent-one barrier, not a literal linear lower bound. The distinction between a fixed-surface estimate, an estimate uniform over moduli, and a smoothed estimate must be retained.

29.2 The Fourier and spectral meaning of the obstruction

The flatness obstruction has a direct oscillatory-integral interpretation. For illustration, suppose a localized smooth boundary arc has graph $$y=h-f(x),\qquad f(x)=a x^k+O(x^{k+1}),\quad a>0,$$ where $k\geq2$ is even. Apply smooth cutoffs supported near this arc and equal to one at the supporting point. Integrating the localized indicator in the normal coordinate first gives, up to rapidly decreasing terms, $$-\frac{e^{-2\pi i r h}}{2\pi i r} \int \chi(x)e^{2\pi i r f(x)}\,\mathop{}\!\mathrm dx.$$ The substitution $x=r^{-1/k}u$ in the stationary-phase calculation gives $$\widehat{\chi_0\mathbf1_B}(r\nu) =c_{a,k}\,e^{-2\pi i r h}r^{-1-1/k} +o(r^{-1-1/k}),\qquad c_{a,k}\ne0.$$(29.4) where $\nu$ is the chosen normal direction and $\chi_0$ is the localized cutoff. The familiar nondegenerate exponent is $r^{-3/2}$ when $k=2$; larger contact order weakens it toward $r^{-1}$. Localization matters: the contributions of different supporting points can interfere in the Fourier transform of the whole ball. No global lower bound follows from discarding those phases.

Under dilation, $$\widehat{\mathbf1_{LB}}(m)=L^2\widehat{\mathbf1_B}(Lm).$$ Thus a finite-type flat point with a lattice-rational normal contributes on the scale $L^{1-1/k}$ to the corresponding localized Poisson terms. For an infinitely flat point there is no fixed finite type to insert in this formula. Proposition 29.1 supplies a direct counting argument in that case, without assuming smooth boundary regularity at all nearby points or invoking a formal stationary-phase expansion there.

There is also an exact elementary spectral model. On an auxiliary flat torus $\mathbb R^2/\mathbb Z^2$, define the positive self-adjoint Fourier multiplier $$\mathsf P_X e^{2\pi i\langle m,x\rangle} =\|m\|_X e^{2\pi i\langle m,x\rangle}.$$ Norm equivalence gives a discrete spectrum tending to infinity. Its counting function is $A_X(L)$, and restriction to the closed span of primitive Fourier modes has counting function $P_X(L)$. Restricting further to the even part identifies $m$ and $-m$ and gives $N_X(L)$. This multiplier is defined spectrally; the nonsmooth length norm is not being asserted to be a classical smooth pseudodifferential symbol. At a differentiable point $p$ of its unit ball, a supporting covector $\nu$ satisfies $$\nabla\|\cdot\|_X(p)=\frac{\nu}{\langle\nu,p\rangle}.$$ Thus a lattice-rational normal is exactly a rational direction for the translation velocity of the associated Hamiltonian. This explains the periodic-phase interpretation, without assuming that a global smooth trace formula applies to the length norm.

The norm-ball discrepancy is therefore a spectral-projector remainder for this particular length multiplier. Its flat normal directions produce degenerate phases in a regularized Poisson or trace calculation. This is distinct from complementary-series spectrum for the $SL_2(\mathbb R)$ action on the marked-torus moduli space. In particular, it does not contradict the temperedness statement or the smooth correlation estimate in Section 28. The rotation-Sobolev issue depends on what is held fixed. A sharp indicator of a nonradial region in fixed physical vector coordinates generally has rotational jumps, so smoothing it introduces growing derivative norms. By contrast, hyperbolic length is unchanged when the flat charts are rotated while the underlying conformal surface stays fixed. A sharp cutoff in that invariant length can therefore lie in the rotation generator’s domain, and a radial flat cutoff can be rotation invariant as well. Sharpness alone is not an obstruction to the stated mixing bound. For those invariant observables, the remaining boundary and regularity issues enter when taking flow derivatives or passing from an averaged correlation to a fixed-surface counting remainder.

29.3 A coefficient observable from primitive pole-to-pole arcs

The marked-torus construction supplies an explicit family of simple curves for which the coefficient data become observables with controlled norms. This family carries a flat-length cutoff as well as a hyperbolic-length weight. The two length parameters remain distinct.

Work on the finite labeled model $Y_2$ of Section 28.5. Write $E=\mathbb R^2/g\mathbb Z^2$, with its origin one of the four ramification points of the pillowcase map. Fix a nonzero parity vector $\delta\in(\mathbb Z/2\mathbb Z)^2$. For a nonzero primitive integer vector $m\equiv\delta\pmod2$, project the straight segment $$[0,gm/2]\subset\mathbb R^2$$ first to $E$ and then to $E/\{\pm1\}$. It joins the pole at the origin to the pole corresponding to $\delta/2$.

Write $\ell_m(q_X)=\ell_X(\alpha_m)$ for the hyperbolic length of the same topological curve. Proposition 23.3 identifies it as the position of the first atom of $\sigma^X_{\alpha_m}$. In particular, if a local cutoff $\chi_m$ is one near that atom and vanishes on the remaining support, then for a bounded length weight $H$, $$H(\ell_m)= \frac{\langle\sigma^X_{\alpha_m},H(s)\chi_m(s)\rangle} {\langle\sigma^X_{\alpha_m},\chi_m(s)\rangle}.$$(29.6) For arbitrary bounded Borel $H$, the same statement is interpreted using the already recovered first-atom location. No assertion that first-atom extraction is a bounded linear functional is being made.

Choose an even $f\in C_c^2(\mathbb R^2)$ supported in the unit disk, a bounded Borel function $h:(0,\infty)\to\mathbb C$, and parameters $R\geq1$, $L>0$. Define $$\mathcal F_{\delta;f,h}^{R,L}(q_X) =\frac1{2R^2} \sum_{\substack{m\in\mathbb Z^2\ \mathrm{primitive}\\m\equiv\delta\ (2)}} f(gm/R)\,h\bigl(\ell_m(q_X)/L\bigr).$$(29.7) The factor $1/2$ identifies opposite directions. The function is well defined on $Y_2$: changes of lattice basis preserve the parity class and relabel the sum. More precisely, define the geometric formula on the invariant conull admissible set of Lemma 29.2 and set it to zero on its complement; this fixes a measurable representative of the same $L^2$ class. Averaging over the finite labeling choices gives observables on the unlabelled stratum, with the same norm bounds. Such an average is an average of selected families; it need not count their union without multiplicities.

29.4 Uniform rotation norms, including a sharp hyperbolic cutoff

Put $$J=\begin{pmatrix}0&-1\\1&0\end{pmatrix},\qquad A=\begin{pmatrix}1&0\\0&-1\end{pmatrix},\qquad D_Bf(x)=(Bx)\cdot\nabla f(x).$$ The notation $D_B$ here is a linear vector field on $\mathbb R^2$.

Rotation of the flat charts fixes the underlying conformal surface and the intrinsic pole-to-pole arcs. Consequently $$\Theta\ell_m=0,$$ and differentiation of (29.7) under rotation only replaces $f$ by $D_Jf$. Set $$C_*=1+\pi,\qquad N_{f,h}=C_*\|h\|_\infty \bigl(\|f\|_\infty^2+\|D_Jf\|_\infty^2\bigr)^{1/2}.$$ The primitive-vector bound proves, uniformly in $R\geq1$ and $L>0$, $$\mathcal S_1(\mathcal F_{\delta;f,h}^{R,L})\leq N_{f,h}.$$(29.9) This is global, including the modular cusp. Although the curve labels can jump as a regular mark crosses a selected arc, the exceptional walls are $SL_2$-invariant. Along almost every group orbit the labels are fixed under the natural marking, and the preceding derivatives are legitimate. Uniform bounds on the difference quotients give the corresponding strong $L^2$ generator statements.

In particular, Theorem 28.1 now applies to two observables of the form (29.7), with $$|\operatorname{Corr}_t(\mathcal F,\mathcal G)| \leq4\coth\pi\,(1+t)e^{-t}\,N_{f,h}N_{\widetilde f,\widetilde h},$$(29.10) uniformly in their two flat cutoffs and two hyperbolic-length scales. The hyperbolic weight may be the sharp cutoff $h(s)=\mathbf1_{s\leq1}$: its discontinuity costs no rotation derivative. If the flat weight is radial as well, the whole observable is $K$-invariant. In that case a sharp disk cutoff is also allowed directly, and the single-$K$-type matrix-coefficient bound gives the sharper estimate $$|\operatorname{Corr}_t(F,G)| \leq\Xi(a_t)\|F^\circ\|_2\|G^\circ\|_2 \leq\frac4\pi(1+t)e^{-t}\|F^\circ\|_2\|G^\circ\|_2.$$(29.11) No rotational smoothing is needed for these radial tests.

29.5 Control of the terminal derivative

For the derivative counting converse, require $h\in C^1$ and $$H_0=\|h\|_\infty<\infty,\qquad H_1=\sup_{s>0}s|h'(s)|<\infty.$$ Wolpert’s quasiconformal length inequality gives $$|V\ell_m|\leq2\ell_m, \qquad |\Theta V\ell_m|\leq4\ell_m.$$(29.12) For completeness, the first inequality follows by lifting a $K$-quasiconformal map to the cyclic annular covers associated with the curve. Their conformal moduli are $\pi/\ell_X$ and $\pi/\ell_Y$; modulus distortion gives $K^{-1}\leq\ell_Y/\ell_X\leq K$. The map associated with $a_t$ has dilatation $e^{2|t|}$. For the second inequality use $\Theta\ell_m=0$ and $[\Theta,V]=\pm2W$, where $W$ is the generator of a rotated diagonal subgroup. The same first inequality applies to $W$. See also [110, Equation (1.1)] for the quasiconformal inequality and its attribution to Wolpert.

Let $F=\mathcal F_{\delta;f,h}^{R,L}$, and abbreviate $$f_0=\|f\|_\infty,\quad f_K=\|D_Jf\|_\infty,\quad f_A=\|D_Af\|_\infty,\quad f_{KA}=\|D_JD_Af\|_\infty.$$ Direct differentiation and (29.12) give $$\begin{aligned} \|VF\|_\infty &\leq C_*\bigl(f_AH_0+2f_0H_1\bigr),\\ \|\Theta VF\|_\infty &\leq C_*\bigl(f_{KA}H_0+2f_KH_1+4f_0H_1\bigr). \end{aligned}$$ These constants are again independent of $R\geq1$ and $L>0$. No second derivative of $h$ occurs, because rotation fixes $\ell_m$. Thus $\mathcal S_1(VF)$, and hence $\mathcal S_1((4+V)F)$, have explicit uniform bounds. In particular the terminal identity extends to these observables: $$\int_1^T e^{4t}\operatorname{Corr}_t(F,(4+V)G)\,\mathop{}\!\mathrm dt =e^{4T}\operatorname{Corr}_T(F,G) -e^4\operatorname{Corr}_1(F,G).$$(29.14) Here $G$ has a smooth length weight as above; invariance of probability measure gives $\mu(VG)=0$. This proves applicability of the integrated counting converse to these endpoints. It does not identify an individual curve sum with the finite affine return sum of the preceding section. Sharp cutoffs require a separate smoothing or boundary argument when the terminal derivative is used.

29.6 What the bridge retains

For $h\equiv1$, the observable reduces to a primitive parity Siegel transform on the modular base and has no nonzero marked-point Fourier mode. For a disk cutoff and fixed $g$, elementary lattice counting and Möbius inversion give $$\frac12\#\{m\ \mathrm{primitive}:m\equiv\delta\ (2),\ |gm|\leq R\} =\frac{\pi}{6\zeta(2)}R^2+O_g(R\log R) =\frac1\pi R^2+O_g(R\log R).$$ Indeed, only odd common divisors occur in a nonzero parity class, and $\sum_{d\ \mathrm{odd}}\mu(d)/d^2=4/(3\zeta(2))$. Adding the hyperbolic weight $h(\ell_m/L)$ can retain dependence on the two regular marks and uses the simple-curve first-atom data. For $h\equiv1$ the exact mean is $$\mu(\mathcal F_{\delta;f,1}^{R,L}) =\frac1{6\zeta(2)}\int_{\mathbb R^2}f(x)\,\mathop{}\!\mathrm dx,$$ by the primitive Siegel formula and the equal densities of the three nonzero parity classes.

There is a useful fixed-surface sparsity bound. On a fixed compact $(X,q_X)$ put $$C(X,q_X)=\sup_X\frac{|q_X|^{1/2}}{\mathop{}\!\mathrm ds_{\mathrm{hyp}}}<\infty.$$ The selected flat geodesic minimizes flat length in its free homotopy class, so comparison with the hyperbolic geodesic gives $$|gm|=\ell_{q_X}(\alpha_m) \leq C(X,q_X)\ell_X(\alpha_m).$$ Consequently the whole selected family with hyperbolic length at most $L$ lies in the flat window $R=C(X,q_X)L$ and has size $O_{X,q_X}(L^2)$. It is a sparse family relative to the full genus-two $L^6$ simple-curve count. The bound is for a fixed surface and differential; no uniform comparison constant over moduli is asserted.

Equations (29.6)–(29.13) are a norm-controlled bridge for this selected family. They do not identify the full connected-simple-curve count with an affine return sum. Nor do they prove uniform stability of extracting first atoms from perturbed spectral or coefficient data: the shrinking-gap issue of Section 24.4 remains. The output observables have controlled geometric norms, while the quantitative conditioning of the nonlinear coefficient extraction is a different question.

29.7 Affine displacement sums: an exact isometric model

The homogeneous calculation can be tested on explicit geometric sums. These are the relative-period Siegel–Veech transforms discussed in [108, Section 6.1]. On the marked-torus model write a point as $(g,v_1,v_2)$, with the physical marked points understood modulo $g\mathbb Z^2$. For $f\in\mathscr S(\mathbb R^2)$ define $$S_{f,j}(g,v_1,v_2)=\sum_{m\in\mathbb Z^2}f(gm+v_j),\qquad F_{f,j}=S_{f,j}-\int_{\mathbb R^2}f(x)\,\mathop{}\!\mathrm dx.$$ The summands test the displacement vectors of straight segments from the torus origin to its $j$th marked point. These are flat geometric observables; a hyperbolic simple-curve interpretation requires a separate choice of a lift and of curve weights.

Use the Fourier transform $\widehat f(\xi)=\int f(x)e^{-2\pi i x\cdot\xi}\mathop{}\!\mathrm dx$. With $v_j=gu_j$, Poisson summation gives $$F_{f,j}(g,u_1,u_2)= \sum_{n\in\mathbb Z^2\setminus\{0\}} \widehat f(g^{-T}n)e^{2\pi i n\cdot u_j}.$$ Fiber Parseval followed by the usual Siegel mean formula for all nonzero lattice vectors gives the exact identity $$\|F_{f,j}\|_{L^2(Y_2)}^2=\|f\|_{L^2(\mathbb R^2)}^2.$$(29.15) This is also the one-relative-mark case of [108, Proposition 6.11]. The level-two probability cover changes neither side: the integrand descends to the full modular quotient. By polarization and equivariance, $$\boxed{ \int_{Y_2}\overline{F_{f,j}(x)}F_{h,j}(a_tx)\,\mathop{}\!\mathrm d\mu(x) =\int_{\mathbb R^2}\overline{f(x)}h(a_tx)\,\mathop{}\!\mathrm dx. }$$(29.16) Indeed the Fourier-side integral is $\int\overline{\widehat f(\xi)}\widehat h(a_{-t}\xi)\mathop{}\!\mathrm d\xi$. These identities extend by density to $L^2$ tests, using the isometric definition of the centered transform when the original sum is unavailable. For indicator tests the variance identity is also [109, Lemma 4]; the mean is Lemma 3 there. For different relative marks, the analogous centered cross-correlation vanishes by integration in the two independent torus fibers.

For the Gaussian $f(x)=e^{-\pi|x|^2}$, the formula becomes $$\int_{Y_2}\overline{F_{f,j}}F_{f,j}\circ a_t\,\mathop{}\!\mathrm d\mu =\frac1{\sqrt{(1+e^{2t})(1+e^{-2t})}} =\frac1{2\cosh t}.$$(29.17) Thus an $e^{-t}$ correlation rate is attained by a natural smooth observable. More generally, substitution $x_1=e^{-t}u$ in (29.16) gives, for Schwartz tests, $$e^t\int\overline f(x)h(a_tx)\,\mathop{}\!\mathrm dx \longrightarrow \left(\int_\mathbb R\overline{f(0,y)}\,\mathop{}\!\mathrm dy\right) \left(\int_\mathbb Rh(u,0)\,\mathop{}\!\mathrm du\right).$$ Schwartz domination justifies the limit.

29.8 Two marks: the rank-one divergence and its removal

The raw product $F_{f_1,1}F_{f_2,2}$ need not be in $L^2$, even for smooth rapidly decreasing tests. Its conditional squared norm is $$A_1(g)A_2(g),\qquad A_j(g)=\sum_{n\ne0}|\widehat f_j(g^{-T}n)|^2.$$ For Gaussians $f_1=f_2=e^{-\pi|x|^2}$ the divergence can be seen exactly. Write the collinear frequency pairs as $(an,bn)$, where $n$ is primitive modulo sign and $a,b\in\mathbb Z\setminus\{0\}$. Primitive Siegel unfolding gives the rank-one contribution $$\frac1{2\zeta(2)}\sum_{a,b\ne0} \int_{\mathbb R^2}e^{-2\pi(a^2+b^2)|x|^2}\,\mathop{}\!\mathrm dx =\frac1{4\zeta(2)}\sum_{a,b\ne0}\frac1{a^2+b^2} =\infty.$$ The divergence is logarithmic. More generally, if both $\int f_j$ are nonzero, Fourier continuity gives $A_j(g)\gg\varepsilon^{-1}$ when the dual lattice has a primitive vector of length $\varepsilon\ll1$; the cusp measure is comparable to $\varepsilon\mathop{}\!\mathrm d\varepsilon$. Centering has removed zero frequencies but leaves precisely this collinear-frequency obstruction.

There is an explicit $L^2$ replacement. Retain only the full-rank frequency matrices: $$H_{f_1,f_2}^{(2)}(g,u_1,u_2) =\sum_{\substack{n_1,n_2\in\mathbb Z^2\\\det(n_1,n_2)\ne0}} \widehat f_1(g^{-T}n_1)\widehat f_2(g^{-T}n_2) e^{2\pi i(n_1\cdot u_1+n_2\cdot u_2)}.$$(29.18) The series is absolutely convergent on each fixed torus fiber. The rank restriction is invariant under the modular transition and under the $SL_2$ action. It is a fiber Fourier projection, not a positivity-preserving deletion of physical segments. It is defined fiberwise before global integrability has been proved; no orthogonal projection on the global $L^2$ space is being applied to the raw product.

Here is a norm computation which proves global integrability. Put $a_j=|\widehat f_j|^2$ and let $\sigma_1(k)=\sum_{d\mid k}d$. Hermite normal form gives $\sigma_1(|k|)$ full-modular orbits of integer matrices of determinant $k\ne0$, each with trivial stabilizer. Thus Parseval and unfolding give $$\|H_{f_1,f_2}^{(2)}\|_2^2 =\sum_{k\ne0}\sigma_1(|k|)I_k(a_1,a_2),$$(29.19) where, with probability-covolume Haar normalization, $$I_k(a_1,a_2) =\frac1{|k|\zeta(2)} \int_{\mathbb R^2\setminus\{0\}}\int_\mathbb R a_1(x)a_2\left(\frac{kJx}{|x|^2}+sx\right) \,\mathop{}\!\mathrm ds\,\mathop{}\!\mathrm dx, \qquad J(x_1,x_2)=(-x_2,x_1).$$ For completeness the Haar-coordinate formula behind this expression is $$g=\left[x,\frac{Jx}{|x|^2}+sx\right],\qquad \mathop{}\!\mathrm dg=\frac1{\zeta(2)}\,\mathop{}\!\mathrm dx\,\mathop{}\!\mathrm ds.$$ The normalization follows by unfolding the primitive first-column Siegel transform: the stabilizer shifts $s$ by integers, and its mean is $\zeta(2)^{-1}\int f$. At level two each full-rank modular orbit splits into six orbits while Haar measure is divided by six, leaving (29.19) unchanged.

An explicit Schwartz bound is convenient. Define $$A(f)=\sup_{\xi\in\mathbb R^2} (1+|\xi|^2)^{3/2}|\widehat f(\xi)|.$$ For $k>0$ the orbit integral is bounded by $$I_{\pm k}\leq A(f_1)^2A(f_2)^2 \int_G(1+k\operatorname{tr}(g^Tg))^{-3}\,\mathop{}\!\mathrm dg =\frac{\pi^2 A(f_1)^2A(f_2)^2} {2\zeta(2)k(1+2k)^2}.$$ One way to verify the last integral is the Gaussian identity $$\int_G e^{-\alpha k\operatorname{tr}(g^Tg)}\,\mathop{}\!\mathrm dg =\frac{\pi^2}{\alpha k\zeta(2)}e^{-2\alpha k}, \qquad \alpha,k>0,$$ followed by the Laplace representation of $(1+u)^{-3}$. Since $\sum_{k\geq1}\sigma_1(k)k^{-3}=\zeta(3)\zeta(2)$, $$\boxed{ \|H_{f_1,f_2}^{(2)}\|_2 \leq\frac\pi2\sqrt{\zeta(3)}\,A(f_1)A(f_2). }$$(29.20)

Let $Df(x)=(Jx)\cdot\nabla f(x)$ and set $B(f)=A(f)+A(Df)$. Rotation commutes with the rank projection, so $$\Theta H_{f_1,f_2}^{(2)} =H_{Df_1,f_2}^{(2)}+H_{f_1,Df_2}^{(2)}.$$ Therefore $$\mathcal S_1(H_{f_1,f_2}^{(2)}) \leq\frac\pi2\sqrt{\zeta(3)}\,B(f_1)B(f_2).$$(29.21) The preceding mixing theorem now applies, with constants controlled explicitly by the four test functions. In particular, $$\left|\left\langle H_{f_1,f_2}^{(2)}, H_{h_1,h_2}^{(2)}\circ a_t\right\rangle\right| \leq\pi^2\zeta(3)\coth\pi\,(1+t)e^{-t} B(f_1)B(f_2)B(h_1)B(h_2).$$ The mean is zero because every retained fiber frequency is nonzero.

29.9 An exact Gaussian two-mark correlation

Take $f_1=f_2=f=e^{-\pi|x|^2}$ and write $H^{(2)}=H_{f,f}^{(2)}$. The determinant-orbit calculation can be done at every time: $$\boxed{ \left\langle H^{(2)},H^{(2)}\circ a_t\right\rangle =\frac{\pi}{\zeta(2)\cosh t} \sum_{k=1}^\infty\frac{\sigma_1(k)}k e^{-4\pi k\cosh t}. }$$(29.22) To see this, the product of the Fourier coefficients is $$\exp\bigl[-\pi\operatorname{tr}(Z^TC_tZ)\bigr], \qquad C_t=I+a_{-2t},\qquad \det Z=k.$$ Since $\sqrt{\det C_t}=2\cosh t$, an $SL_2$ change of variables reduces this to the radial Gaussian integral with $\alpha=2\pi\cosh t$. Summing the positive and negative determinant orbits gives (29.22). At $t=0$ this gives the exact squared norm. The identity $$\sum_{k\geq1}\frac{\sigma_1(k)}k q^k =-\sum_{n\geq1}\log(1-q^n),\qquad 0<q<1,$$ also expresses the answer by the ordinary partition Euler product. The resulting correlation decays faster than exponentially in $t$. This strong decay is a feature of these particular Gaussian tests; it is consistent with the $e^{-t}$ one-mark example and the uniform tempered bound. For a finite evaluation, put $q=e^{-4\pi\cosh t}$. The elementary bound $\sigma_1(k)/k\leq k$ gives the explicit truncation estimate $$0\leq\sum_{k>N}\frac{\sigma_1(k)}kq^k \leq\frac{q^{N+1}((N+1)-Nq)}{(1-q)^2}.$$ This controls truncation of the analytic series. Floating-point evaluation of this bound is not an interval enclosure of the total computational error, and checking finitely many times is not a proof of the general mixing theorem.

29.10 Sharp boundaries and the rotation norm

The exact isometry also displays a regularity cost of sharp cutoffs. For a bounded convex body $B\subset\mathbb R^2$, $$\|F_{\mathbf1_B,j}\circ k_\theta-F_{\mathbf1_B,j}\|_2^2 =\operatorname{area}(B\mathbin\triangle k_\theta B).$$ The first variation of area under rotation, interpreted using the almost-everywhere outer normal of a convex boundary, gives $$\operatorname{area}(B\mathbin\triangle k_\theta B) =|\theta|\int_{\partial B}|Jx\cdot n(x)|\,\mathop{}\!\mathrm ds+o(|\theta|).$$ Unless $B$ is rotationally invariant, this rules out a finite $L^2$ rotation derivative. Equivalently, the sharp transform has angular Sobolev regularity of order $s<1/2$, but not $s\geq1/2$. The direct summation of tempered $K$-type estimates requires an order greater than $1/2$; the earlier theorem uses order one.

This is a generic boundary-regularity phenomenon. It is not by itself the more specific arithmetic obstruction from the unusually flat McShane–Rivin norm ball. It explains why a smooth correlation bound cannot be applied to a sharp norm-ball count while leaving the observable norm unchanged. Here the body $B$ is held fixed in physical Euclidean coordinates. A cutoff depending only on the hyperbolic length $\ell_X(\alpha)$ is unchanged when the quadratic differential is rotated over the same conformal surface $X$, and does not incur this particular rotation loss. These two kinds of cutoffs must be distinguished. The fixed-surface primitive-lattice count, its Möbius inversion, and its unsmoothing error remain distinct from the affine mean-square identities proved here.