27 Filling-pair coefficients and a weighted mixing converse

27.1 Two independent character variables retain the pair

Keep a closed marked hyperbolic surface $X$ as an auxiliary device, and let $\rho,\sigma$ be independent rank-two character variables. The length-valued product of Section 25 can be tensored with itself, retaining two separate length coordinates: $$\mathbf E_X(\rho)\boxtimes\mathbf E_X(\sigma) =\sum_{C,D} T_C(\rho)T_D(\sigma)\, (\nu_C^X\boxtimes\nu_D^X).$$ Here $\boxtimes$ is the external product of locally finite discrete measures, and both $C$ and $D$ range over integral weighted multicurves. The two variables are essential: multiplication in one character algebra would resolve crossings between $C$ and $D$, whereas their ordered pair is precisely the information required here.

After this extraction, the marked labels retain $i(C,D)$ and the topological filling condition. Thus one can restrict to filling pairs and apply weights to their associated flat differentials. The auxiliary lengths $\ell_X$ provide the triangular extraction; the area of the resulting differential will be $i(C,D)$, a different quantity. In particular, a bound on area is not a bound on both auxiliary hyperbolic lengths on the marked space. Passing to an area cutoff and a mapping-class quotient is an additional operation, whose quantitative cost is not supplied by fixed-coefficient convergence.

27.2 Filling pairs as actual coordinates for the flow

For definiteness let $S$ be closed, oriented, and of genus $g\geq2$. A pair $(A,B)$ of nonzero measured foliations is filling if $$i(A,C)+i(B,C)>0\qquad\text{for every nonzero measured foliation }C.$$ The condition is stronger than $i(A,B)>0$. The realization theorem of Gardiner–Masur, also proved by Wentworth using harmonic maps, identifies filling pairs with marked nonzero holomorphic quadratic differentials [1, Theorem 1.6 and Corollary 1.7]; see also [2, Theorem 1.6] for the equivariant homeomorphism and its integral interpretation. Use the convention that $A=\mathcal V(q)$ is the vertical foliation with transverse measure $|dx|$, and $B=\mathcal H(q)$ is the horizontal foliation with transverse measure $|dy|$, in coordinates $q=dz^2$. Write $q(A,B)$ for the resulting differential. Then $$\begin{aligned} \operatorname{Area}(q(A,B))&=i(A,B),\\ g_tq(A,B)&=q(e^tA,e^{-t}B), \qquad g_t=\operatorname{diag}(e^t,e^{-t}). \end{aligned}$$ In particular $$\widehat q(A,B):= q\left(\frac{A}{\sqrt{i(A,B)}}, \frac{B}{\sqrt{i(A,B)}}\right)$$ has unit area. This is a genuine parametrization of points of the flow space; the two foliations are the stable and unstable data of a single differential, rather than two unrelated curves.

For an integral pair $A=\sum_i a_i\alpha_i$, $B=\sum_j b_j\beta_j$ in minimal position, there is a concrete realization. Put a rectangle of horizontal width $a_i$ and vertical height $b_j$ at each intersection of $\alpha_i$ with $\beta_j$, and glue the sides in the cyclic orders supplied by the embedded curves. The filling condition makes the complementary regions disks. Minimal position excludes bigons. A complementary $2k$-gon yields total angle $k\pi$, hence a zero of order $k-2$; $k=2$ gives a regular point. Integral side lengths subdivide the rectangles into unit squares, with edge identifications by translations or half-turns. The total number of squares is $$\sum_{i,j}a_i b_j i(\alpha_i,\beta_j)=i(A,B).$$ Thus the discrete objects suggested by the pair coefficients are precisely square-tiled half-translation surfaces. Translation surfaces form the subcollection in which the half-translation structure has trivial linear holonomy. Formula (27.2) is then visible by stretching and contracting the rectangles.

27.3 The correct quotient and the two different counting scales

Let $\Gamma=\operatorname{Mod}(S)$. Equivariance of the pair parametrization identifies the stabilizer of the measured pair with the finite automorphism group of its differential: $$\operatorname{Stab}_\Gamma(A,B) =\operatorname{Stab}_\Gamma(A)\cap\operatorname{Stab}_\Gamma(B).$$ For integral pairs, finiteness also follows by acting on the finite filling graph $A\cup B$ and then on its complementary disks; for general measured foliations it follows from the realized conformal automorphism group. It matters here that one fixes the measured foliations, including their scales. Stabilizers of projective foliations can contain nontrivial flow translations.

One simultaneous $\Gamma$-orbit of a filling pair represents just one point in moduli space and has constant intersection number. To obtain a growing family, prescribe the two curve types separately and take the diagonal quotient: $$\bigl(\Gamma\alpha\times\Gamma\beta\bigr)_{\mathrm{fill}}/\Gamma.$$ If $\Gamma_\alpha=\operatorname{Stab}_\Gamma(\alpha)$ and $\Gamma_\beta=\operatorname{Stab}_\Gamma(\beta)$, its classes are the relevant double cosets in $\Gamma_\alpha\backslash\Gamma/\Gamma_\beta$, represented by $(\alpha,\phi\beta)$. For an orbifold volume comparison the natural weight of such a class is $$\frac{1}{|\Gamma_\alpha\cap \phi\Gamma_\beta\phi^{-1}|}.$$ This is the same stabilizer factor appearing in the square-tiled count of [2, Section 1 and Section 3].

Fix a connected component $\mathcal Q$ of a stratum and put $h=\dim_{\mathbb C}\mathcal Q$. The real period-coordinate dimension of its cone is $2h$, and the unit-area locus has real dimension $2h-1$. If $D_r$ multiplies every period by $r$, then area is multiplied by $r^2$. With the appropriate period lattice and its covolume fixed, Lebesgue measure $\nu$ scales by $$\nu(D_r E)=r^{2h}\nu(E).$$ Consequently an area cutoff $L$ has natural count exponent $h$, whereas a common linear scale $R=\sqrt L$ has exponent $2h$. These conventions must not be interchanged when converting powers into rates in flow time.

For the closed principal quadratic stratum $h=6g-6$. For a quadratic stratum with $n$ specified singularities, including marked regular points, $h=2g-2+n$ when the differential is not a square; the corresponding square locus has $h=2g-1+n$. Lower-dimensional strata require their own period-coordinate lattice and Masur–Veech measure. The ambient product of two full-dimensional Thurston measures cannot serve as their volume form. Likewise, counting all square-tiled surfaces in a stratum does not by itself approximate an arbitrary affine invariant measure on a smaller locus.

27.4 The existing square-tiled counting theorem and the new target

The passage from pairs to square tilings is established mathematics, not a new consequence of the coefficient construction. Arana-Herrera’s effective theorem counts square-tiled surfaces in a fixed connected component with prescribed horizontal and vertical multicurve types, with a power-saving error in the area cutoff [3, Theorem 1.1]. Its proof explicitly passes through Eskin–Mirzakhani–Mohammadi orbit counting and effective horosphere equidistribution, using exponential Teichmüller mixing [3, Theorem 3.2 and its proof]. Using that estimate to recover mixing would therefore not provide an independent proof of mixing.

There is a useful distinction already visible in that paper. When one foliation type is fixed and the other is unrestricted, standard lattice counting replaces the mapping-class-orbit estimate [3, Theorem 3.9]. This provides a tractable elementary benchmark. It does not control arbitrary initial and terminal observables at long times: their weights must still be tracked through the reciprocal scaling in (27.2).

27.5 What refining the coefficient mesh actually proves

There are two independent limiting operations: approximation of a phase-space integral by rational points, and factorization of that integral under a long flow. The first can be effective without implying the second. Here is a precise local calculation. Put $$A_t=\operatorname{diag}(e^t I_d,e^{-t}I_d),\qquad H_t(x)=F(x)G(A_tx),$$ where $F\in C_c^1(\mathbb R^{2d})$ has support in a fixed compact set $K$ and $G\in C_b^1(\mathbb R^{2d})$. For $R\geq1$ define $$Q_R(t)=R^{-2d}\sum_{k\in\mathbb Z^{2d}}H_t(k/R),\qquad I(t)=\int_{\mathbb R^{2d}}H_t(x)\,dx.$$ Then, for $t\geq0$, $$|Q_R(t)-I(t)|\leq C_K R^{-1}e^t \|F\|_{C^1}\|G\|_{C^1}.$$(27.3) Indeed, partition space into cubes of side $R^{-1}$, compare the value at one corner with the integral on its cube, and use $$\|\nabla H_t\|_\infty \leq \|\nabla F\|_\infty\|G\|_\infty +e^t\|F\|_\infty\|\nabla G\|_\infty.$$ Only $O_K(R^{2d})$ cubes contribute. A fixed smooth density can be absorbed in $F$. To model unit-area data by an ambient lattice, one can first use a smooth radial cutoff in a compact area shell bounded away from area zero; radial normalization is then smooth. This local argument does not supply uniform estimates across arbitrarily many period-coordinate charts, their boundaries, or the cusp.

At the diagonal scale $R=e^t$, the displayed error has no decay. Taking $R=e^{\sigma t}$, with $\sigma>1$, gives a quadrature error $O(e^{-(\sigma-1)t})$. This removes the resolution obstruction in this particular local model. It does not evaluate the remaining integral $I(t)$. For Schwartz data, Poisson summation makes the distinction exact: $$Q_R(t)=\sum_{n\in\mathbb Z^{2d}}\widehat H_t(Rn) =I(t)+\sum_{n\ne0}\widehat H_t(Rn).$$(27.4) The zero Poisson frequency is the two-time integral itself, not the product of its one-time means. Suppressing the nonzero Poisson frequencies therefore proves accurate quadrature, not mixing. On the quotient, one must still analyze the sum over changes of marking that returns $A_tK$ to a chosen collection of charts.

Even exact sampling identities do not force mixing. On $\mathbb R/\mathbb Z$, with $g_t(x)=x+t$ and $F(x)=G(x)=\cos(2\pi x)$, $$\int_0^1F(x)G(g_tx)\,dx=\tfrac12\cos(2\pi t).$$ For every integer $R\geq3$, the corresponding equal-mesh average is exactly this integral, by finite Fourier orthogonality. Both means vanish, while the correlation does not decay. This example isolates the logical issue; it is not a model of the full geometry of a stratum.

27.6 An exact weighted bisector kernel

There is a concrete discrete count whose remainder does imply mixing. The necessary weight can be specified exactly, and the passage from a cumulative count to a correlation has an elementary quantitative proof. This construction uses the full unit cotangent bundle, with its Masur–Veech measure supported on the principal stratum; it is not an automatic statement for an arbitrary lower-dimensional stratum or affine invariant measure.

Let $S$ be closed of genus $g\geq2$, put $h=6g-6$, and replace the mapping class group by a torsion-free finite-index subgroup $\Gamma$. This removes stabilizer factors. Write $$Y=\Gamma\backslash Q^1\mathcal T(S),\qquad B=\Gamma\backslash\mathcal T(S),$$ and normalize the invariant measure $\mu$ on $Y$ to have mass one. Its projection $m$ to $B$ also has mass one. Use the same notation for the lifted measures, and disintegrate $$\mathop{}\!\mathrm d\mu(q)=\mathop{}\!\mathrm ds_x(q)\,\mathop{}\!\mathrm dm(x),\qquad q\in S(x),$$ where $S(x)=\pi^{-1}(x)$ and $s_x$ is a probability measure. The Teichmüller polar-coordinate Jacobian $J_x(q,t)>0$ is defined almost everywhere by $$\mathop{}\!\mathrm dm\bigl(\pi(g_tq)\bigr) =J_x(q,t)\,\mathop{}\!\mathrm ds_x(q)\,\mathop{}\!\mathrm dt.$$(27.5) This is the polar-coordinate formula of [4, Section 2.4, equation (2.3)]. We use the conventional Teichmüller time $g_t=\mathop{\mathrm{diag}}(e^t,e^{-t})$. The equality is an equality of measures under the map $(q,t)\mapsto\pi(g_tq)$ on the marked universal space, with $t>0$; it is not a claim that this map is one-to-one on the quotient $B$.

Let $D\subset\mathcal T(S)$ be a measurable fundamental domain. For $x,y\in D$ and $\gamma\in\Gamma$, let $$\ell_\gamma=d_{\mathcal T}(x,\gamma y),\qquad q_\gamma=q_s(x,\gamma y)\in S(x)$$ be the length and initial unit quadratic differential of the unique Teichmüller segment. Its terminal differential, transported to $S(y)$, is $\gamma^{-1}g_{\ell_\gamma}q_\gamma$. For bounded observables $F,G$ on $Y$, understood through their $\Gamma$-invariant lifts, define $$\begin{split} K_T^{F,G}(x,y) =\sum_{\substack{\gamma\in\Gamma\\1\leq\ell_\gamma\leq T}} \frac{e^{h\ell_\gamma}}{J_x(q_\gamma,\ell_\gamma)}\, F(q_\gamma)\, G\bigl(\gamma^{-1}g_{\ell_\gamma}q_\gamma\bigr). \end{split}$$(27.6) The lower cutoff avoids the polar singularity at zero. The sum is finite for fixed $x,y,T$. Define it for $m\times m$-almost every $(x,y)$ where the polar data are regular for all contributing $\gamma$. Such a set has full measure, since $\Gamma$ is countable; values on its complement can be assigned arbitrarily. This convention makes no assertion about an exceptional prescribed pair of basepoints.

The count is genuinely discrete before the basepoint integrals are taken. It records both the initial and the terminal differential, and its geometric weight is explicit. The identity does not prove its asymptotic: that is the new counting theorem one would need.

27.7 Removing the time average with a controlled loss

The next lemma applies to any probability-preserving flow. Write $U_tG=G\circ g_t$ for its strongly continuous unitary action on $L^2(\mu)$, and let $V$ be the infinitesimal generator, so that $\partial_tU_tG=U_tVG$ on its domain. Put $$C_{F,G}(t)=\int_Y F\,(G\circ g_t)\,\mathop{}\!\mathrm d\mu, \qquad M_{F,G}=\mu(F)\mu(G).$$ There is no complex conjugation in this notation; the usual Hermitian correlation is obtained by replacing $F$ by $\overline F$.

Only derivatives along the flow enter this last step. In particular, it does not differentiate the spatially stretched observable $G\circ g_t$ in an expanding transverse direction. This removes the aliasing difficulty from the de-averaging step; the difficulty of proving the weighted count itself remains.

A convenient sufficient counting hypothesis is the almost-everywhere bisector estimate $$\begin{split} K_T^{F,G}(x,y) &=\frac{e^{hT}-e^h}{h}\, f_F(x)f_G(y)+\mathcal E_T^{F,G}(x,y),\\ f_F(x)&=\int_{S(x)}F\,\mathop{}\!\mathrm ds_x,\qquad f_G(y)=\int_{S(y)}G\,\mathop{}\!\mathrm ds_y,\\ |\mathcal E_T^{F,G}(x,y)| &\leq C(x,y)\,\mathcal N(F)\mathcal N(G) e^{(h-\delta)T},\qquad C\in L^1(D\times D,m\times m). \end{split}$$(27.11) Integrating gives (27.8) with $A(F,G)=\|C\|_1\mathcal N(F)\mathcal N(G)$. A direct estimate for the averaged kernel is sufficient and is weaker than this pointwise hypothesis. Uniformity over the observables, the two basepoints, and all large real values of $T$ is part of the assertion; one unweighted total count does not supply it.

A flow eigenfunction checks that the weighted hypothesis detects an actual spectral obstruction. If an admissible nonzero test $G$ satisfies $U_tG=e^{i\omega t}G$, $\mu(G)=0$, and $\|G\|_2=1$, take $F=\overline G$. Then exact unfolding gives $$A_{F,G}(T) =\frac{e^{(h+i\omega)T}-e^{h+i\omega}}{h+i\omega},$$ whose size is comparable to $e^{hT}$ for large $T$. It violates every positive-saving estimate in (27.8). This test is available only if the asserted counting estimate covers the relevant observables; an unweighted count does not test this eigenmode.

27.8 A derivative endpoint weight preserves the full exponent

For a counting estimate uniform over endpoint observables, there is a stronger converse than temporal smoothing. One can differentiate the endpoint test inside the discrete count and use an exact identity, without differentiating an asymptotic formula.

For example, if the count is uniform in a norm $\mathcal N$ with $$A(F,H)\leq C\mathcal N(F)\mathcal N(H),$$ the norm cost is explicit: $$\begin{split} |\operatorname{Corr}_{F,G}(T)| &\leq C\mathcal N(F) \bigl(h\mathcal N(G)+\mathcal N(VG)\bigr)e^{-\delta T}\\ &\quad+e^{-h(T-1)} \|F-\mu(F)\|_2\|G-\mu(G)\|_2. \end{split}$$(27.16) For a suitable smooth norm on compactly supported observables, this is one additional derivative along the flow in the terminal test. On a noncompact space, any growth of the coefficients of $V$ in the chosen norm must be retained in that estimate. The terminal weight $(h+V)G$ can have either sign, so the counting hypothesis must accommodate signed weights with a quantitative norm bound.

The distinction from Theorem 27.3 is precise. If only the primitive estimate for the particular pair $(F,G)$ is available, temporal smoothing gives $\delta k/(k+1)$ with $k$ flow derivatives. If the same weighted counting theorem also applies to $(F,(h+V)G)$, the exact identity above preserves $\delta$. Thus exponent loss is not intrinsic to the converse for a sufficiently rich family of endpoint weights.

In operator notation, the weighted time integral is governed by $h+V$; inserting $(h+V)G$ removes this integral by an exact boundary evaluation. No spectral gap is needed for this algebraic step. The unresolved input is the norm-controlled remainder for the corresponding discrete weighted count.

There is a corresponding improvement of the conditional cusp calculation. Suppose bounded observables can be cut off at height $H$ with correlation and mean errors $O(e^{-aH})$. Suppose the counting bound for the cutoff pair $$\bigl(F_H,(h+V)G_H\bigr)$$ has constant $O(e^{bH})$, including the derivative of the cutoff. Then (27.13) gives, after controlling the boundary term, $$|\operatorname{Corr}_{F,G}(t)| \ll e^{bH-\delta t}+e^{-aH}+e^{-ht}.$$ For $a>0$, $b\geq0$, and $0<\delta\leq h$, choose $H=\delta t/(a+b)$ to obtain $$\eta=\frac{\delta a}{a+b}.$$(27.17) As before, unbounded observables require a norm-controlled tail estimate in place of a bound on discarded measure alone.

27.9 Comparison with the usual bisector count

The exact weight $e^{ht}/J_x(q,t)$ cannot simply be omitted. To compare normalizations, let $\mu_0$ be the local-product normalization of [4, Section 2.2] and put $V_0=\mu_0(Y)$. Our probability normalization is $\mu=\mu_0/V_0$. The conditional probability measures $s_x$ are unchanged, whereas $m=m_0/V_0$ and $J_x=J_{0,x}/V_0$. Use the canonical Hubbard–Masur factors $\lambda^\pm$ belonging to $\mu_0$. For endpoints in a fixed compact subset of the principal stratum, and for segments spending at least half their time in a suitable larger compact set, [4, Proposition 2.5] then gives $$J_x(q,t) =V_0^{-1}e^{ht}\lambda^-(q)\lambda^+(g_tq) +O(e^{(h-\alpha)t}).$$(27.18) If the endpoint factors are bounded away from zero on the sets under consideration, then, uniformly there for sufficiently large $t$, the relative error in (27.18) is less than $1/2$. Taking the reciprocal yields $$\frac{e^{ht}}{J_x(q,t)} =\frac{V_0}{\lambda^-(q)\lambda^+(g_tq)} +O(e^{-\alpha t}).$$(27.19) Thus the natural approximation is an ordinary bisector count with the two endpoint observables divided by the corresponding Hubbard–Masur factors, with the factor $V_0$. Usual bisector counts often use the direction pointing back from the terminal point; that convention replaces the terminal differential here by its negative and requires the corresponding substitution in $G$. There are three distinct estimates to check: the weighted bisector remainder, the accumulated error in replacing the exact Jacobian, and the contribution of segments failing the recurrence or compactness conditions. Formula (27.18) is not a global uniform asymptotic.

This optional comparison also has a dependency worth making explicit: the proof of [4, Proposition 2.5 and Section 3.2] uses Forni’s contraction estimate for the Hodge norm of the Kontsevich–Zorich cocycle, quoted there as Theorem 3.3. The exact Jacobian unfolding and the elementary de-averaging theorem above do not use that estimate. Invoking the Hubbard–Masur asymptotic would, however, retain this part of the existing cocycle machinery.

Effective bisector counts are already part of [5, Section 10]; those results are obtained from exponential mixing. They locate the strength of the required counting statement, but do not give an independent proof of mixing when inserted into the converse above. An independent derivation from coefficient distributions or a different counting mechanism would have to establish the required weights and uniformity.

For completeness, the noncompact loss for the single-test smoothing argument of Theorem 27.3 can also be computed. For bounded observables, a discarded mass $O(e^{-aH})$ gives a correlation cutoff error of the same order by invariance. For general $L^2$ observables one must instead assume a corresponding norm-controlled tail estimate. Suppose cutoffs at height $H$ have this error, the counting constant for cutoff observables grows as $e^{bH}$, and their $k$th flow-derivative bounds are uniform. That smoothing proof gives an error of the form $$\varepsilon^k+ \varepsilon^{-1}e^{bH-\delta t}+e^{-aH}.$$ Balancing the three terms yields $$\eta_k=\frac{\delta}{1+b/a+1/k}.$$(27.20) This last formula is conditional on the stated cutoff estimates. If the product of the relevant $k$th derivative norm and source $L^2$ norm grows instead as $e^{dH}$, the same calculation gives $$\eta_k=\frac{\delta}{1+b/a+(1+d/a)/k}.$$ For fixed growth rates with $d=0$, increasing $k$ approaches $\delta a/(a+b)$. These formulae do not establish any of the required cusp or cutoff estimates.

27.10 A completely explicit arithmetic two-endpoint calculation

The modular case exhibits the additional cancellation that a two-endpoint count must detect. If $$\gamma=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in\mathrm{SL}_2(\mathbb Z), \qquad c>0,$$ then $$\gamma= \begin{pmatrix}1&a/c\\0&1\end{pmatrix} \begin{pmatrix}0&-1/c\\c&0\end{pmatrix} \begin{pmatrix}1&d/c\\0&1\end{pmatrix}.$$(27.21) Modulo left and right integral unipotents, the two coordinates are $a/c,d/c\pmod1$, with the coupled congruence $ad\equiv1\pmod c$. They are initial and terminal horocyclic coordinates; the corresponding cusp images are $\gamma\infty=a/c$ and $\gamma^{-1}\infty=-d/c$. They should not be identified, without an additional unfolding argument, with arbitrary phase-space flow boxes.

Let $e(x)=e^{2\pi ix}$ and, for periodic test functions $f,g$ with absolutely summable Fourier coefficients (in particular $C^2$ tests), set $$D_c(f,g)=\frac1{\varphi(c)} \sum_{\substack{a\bmod c\\(a,c)=1}} f(a/c)g(\overline a/c),\qquad a\overline a\equiv1\pmod c.$$ Writing $f(x)=\sum_m\widehat f(m)e(mx)$ and similarly for $g$ gives the exact identity $$D_c(f,g)-\widehat f(0)\widehat g(0) =\frac1{\varphi(c)} \sum_{(m,n)\ne(0,0)}\widehat f(m)\widehat g(n)S(m,n;c),$$(27.22) where $$S(m,n;c)=\sum_{a\bmod c}^{*}e\bigl((ma+n\overline a)/c\bigr).$$ The classical bound [6, Section 1.1 and equation (3.2)] $$|S(m,n;c)|\leq\tau(c)c^{1/2}(m,n,c)^{1/2}$$ therefore supplies an actual product main term and a nontrivial remainder. For example, defining $$\mathcal A(f)=\sum_{m\in\mathbb Z}(1+|m|)^{1/2}|\widehat f(m)|,$$ one obtains $$|D_c(f,g)-(\int f)(\int g)| \leq\frac{\tau(c)c^{1/2}}{\varphi(c)}\mathcal A(f)\mathcal A(g) \ll_\epsilon c^{-1/2+\epsilon}\|f\|_{C^2}\|g\|_{C^2}.$$(27.23) Here $(m,n)\ne(0,0)$ implies $(m,n,c)^{1/2}\leq(1+|m|)^{1/2}(1+|n|)^{1/2}$, and two integrations by parts bound $\mathcal A(f)$ by a constant times $\|f\|_{C^2}$. For trigonometric polynomials supported on $|m|,|n|\leq M$, the sharper explicit estimate is $$|D_c(f,g)-(\int f)(\int g)| \leq\frac{\tau(c)\sqrt{cM}}{\varphi(c)} \|\widehat f\|_{\ell^1}\|\widehat g\|_{\ell^1}, \qquad M\geq1.$$ For a prime $c=p$ and $M<p$, the numerator $\tau(c)\sqrt{cM}$ can be replaced by $2\sqrt p$; a frequency with exactly one zero coordinate gives the Ramanujan sum $-1$.

Uniformity cannot be omitted: for $f_c(x)=g_c(x)=e(cx)$, both continuous means vanish but $D_c(f_c,g_c)=1$. The growth of the test norms in (27.23) correctly prevents a false conclusion. This is the arithmetic counterpart of the expanding-frequency problem. The result is a rigorous benchmark for endpoint-residue decorrelation. A proof of mixing of the modular geodesic flow still requires the appropriate unfolding, radial/time weights, and cusp estimates; no higher-genus analogue of the displayed Kloosterman bound has been established here.

For comparison, fix two primitive oriented torus curve types. In the corresponding matrix model, with positive oriented intersection $c$, the remaining residue is $a\pmod c$, $(a,c)=1$, so there are exactly $\varphi(c)$ possibilities. Thus the unweighted count already has the power-saving asymptotic $$\sum_{c\leq L}\varphi(c) =\frac{3}{\pi^2}L^2+O(L\log L).$$ Indeed, substitute $\varphi(c)=c\sum_{d\mid c}\mu(d)/d$ and sum first over $c=dk$; the leading term is $\tfrac12L^2\sum_{d\leq L}\mu(d)/d^2$ and the remainder is $O(L\log L)$. These constants belong to this oriented matrix model; no unmentioned quotient by signs or orbifold automorphism weight is being included. The exact formula $\varphi(c)$ by itself says nothing about the values of $S(m,n;c)$ for $(m,n)\ne(0,0)$.