25 Global coefficient filters and a length-valued Euler product

25.1 An explicit filter for the entire one-crossing sector

Write $n(C)=\#C$, counting parallel copies, and $n(\varnothing)=0$. This differs from the number $r(C)$ of distinct supports in Section 24.6. We seek a functional $$\Lambda(T_C)=w_{n(C)}$$ which preserves simple primitive curves and annihilates the complete iterate family of every primitive figure-eight.

For $0\leq\theta\leq\pi$, put $v=2\cos\theta$ and let $u_+,u_-$ be the two roots of $$u^2-u=v.$$ The unordered pair of roots is unambiguous, including where the discriminant vanishes. Define $$w_n=\frac1\pi\int_0^\pi \bigl(u_+(\theta)^n+u_-(\theta)^n\bigr)\,\mathop{}\!\mathrm d\theta, \qquad n\geq0.$$(25.1) In particular $w_0=2$ and $w_1=1$. If $v\geq-1/4$, the roots are real and lie in $[-1,2]$. If $v<-1/4$, they are complex conjugates of modulus $\sqrt{-v}\leq\sqrt2$. Thus $$|w_n|\leq2^{n+1},\qquad |\Lambda(f)|\leq2\|f\|_2.$$(25.2) Here $\|\cdot\|_2$ is the weighted coefficient norm from Section 23, not a norm obtained by evaluating characters.

The weights have a closed generating function. The root sums $S_n(v)=u_+^n+u_-^n$ satisfy $S_0=2$, $S_1=1$, and $S_n=S_{n-1}+vS_{n-2}$, whence $$\boxed{\quad \sum_{n\geq0}w_n q^n =\frac{2-q}{\sqrt{(1-q)^2-4q^4}}, \qquad |q|<\frac12. \quad}$$(25.3) The square root has value one at zero. To derive the formula, average $(2-q)/(1-q-vq^2)$ over $v=2\cos\theta$ and use the elementary integral of $(a-b\cos\theta)^{-1}$. The first weights are $$2,\ 1,\ 1,\ 1,\ 5,\ 11,\ 19,\ 29,\ 53,\ldots.$$ They are nonnegative integers: the coefficients of $S_n(v)$ are nonnegative integers and the averages of $v^{2j}$ and $v^{2j+1}$ are $\binom{2j}{j}$ and zero, respectively.

This is a single global functional, so no choice of a pants decomposition and no sum of separately normalized pants projections is required. It is not evaluation at a flat representation having the same trace on every simple curve; no such assertion is used.

Equation (23.15) and (25.2) justify the convergent aggregate $$\mathcal H_X^\Lambda(t)=\sum_C w_{n(C)}H_C^X(t), \qquad \Re t>0.$$(25.5) It is holomorphic in that half-plane. Since $\Lambda(1)=2$, the constant identity contribution must also be multiplied by two if one applies the functional before subtracting that contribution. Absolute convergence of the orbital weighted-norm estimate permits applying $\Lambda$ before the orbital sum. Every one-crossing primitive and all its powers therefore disappear. Simple iterates remain with the explicit coefficients $\Lambda(P_k(T_\alpha))$; their first values are $1,-3,-2,5,11,-6,\ldots$. Primitives with at least two self-intersections also remain.

If the higher symmetric-power twisting spectra used for primitive extraction in Section 23 are supplied, let $$\Pi_\rho^X=\sum_{\delta\in\mathcal P_{\mathrm{un}}} T_\delta(\rho)\delta_{\ell_\delta}.$$ This is the primitive measure from that section divided by its positive length variable. Then the filtered primitive measure satisfies $$\Lambda(\Pi_\rho^X) =\sum_{\alpha\ {\mathrm{simple}}}\delta_{\ell_\alpha} +\mathcal R_X, \qquad \mathcal R_X= \sum_{\substack{\delta\in\mathcal P_{\mathrm{un}}\\i(\delta,\delta)\geq2}} \Lambda(T_\delta)\delta_{\ell_\delta}.$$(25.6) Here $i(\delta,\delta)$ is the minimal self-intersection number. The remainder is signed. Its Gaussian transforms converge by the same norm estimate; no smaller-order counting bound for it has been established.

25.2 Why a fixed linear filter cannot finish the job

There is an exact obstruction already in one pair of pants. Choose embedded geodesic pants with three distinct essential cuffs, let $a,b$ generate its fundamental group, and put $$\gamma_1=a^2ba^{-1}b^{-1},\qquad \gamma_2=a^3b^2,\qquad \gamma_3=a^2bab.$$ Then, as functions on the $\mathrm{SL}_2$ character variety, $$\boxed{T_a=T_{\gamma_1}+T_{\gamma_2}-T_{\gamma_3}.}$$(25.7) For a direct verification set $x=T_a$, $y=T_b$, $z=T_{ab}$ and $k=x^2+y^2+z^2-xyz-2$. Cayley–Hamilton gives $$\begin{aligned} T_{\gamma_1}&=x(k-1),\\ T_{\gamma_2}&=(x^2-1)yz-xy^2-x^3+3x,\\ T_{\gamma_3}&=xz^2-yz-x. \end{aligned}$$ The indicated combination is $x$.

All three words are primitive in the geodesic sense of not being proper powers: their homology vectors are $(1,0)$, $(3,2)$, and $(3,2)$. They are nonperipheral cyclically reduced words in the pants group, so their geodesics are nonsimple. Embedded geodesic pants are convex; these intersections persist in the ambient surface. An ambient proper root would traverse the same geodesic image inside the pants and give a root in the pants group, which the homology vectors exclude. Here the word primitive does not assert membership in a free basis of the pants group.

Consequently, any linear functional $L$ satisfying $L(T_a)=1$ must satisfy $$\max_{1\leq j\leq3}|L(T_{\gamma_j})|\geq\frac13.$$(25.8) In particular, no linear functional on this character algebra preserves every primitive simple trace and annihilates every primitive nonsimple trace. The assertion applies also to cutoff-dependent functionals once the same functional is applied to all four labels. It does not exclude functionals depending on each atom’s length, nonlinear operations, or richer representation data. Our explicit filter has values $-5,5,-1$ on these three words, in agreement with the identity.

25.3 Retaining lengths: an Euler product with a unit diagonal

To preserve the information discarded by a fixed character functional, work in the convolution algebra of locally finite length measures supported on $[0,\infty)$, with $\delta_u*\delta_v=\delta_{u+v}$. Its coefficients lie in the character algebra and its unit is $\mathbf1=T_\varnothing\delta_0$. All products are first defined by finite length cutoffs, so each retained coefficient is a finite character polynomial. Define $$\mathbf E_X(\rho)= \prod_{\delta\in\mathcal P_{\mathrm{un}}} \bigl(\mathbf1-T_\delta(\rho)\delta_{\ell_\delta}\bigr)^{*-1}.$$(25.9) This is a product of scalar trace geometric series, not the determinant Selberg product. The positive systole makes every length cutoff finite. Equivalently, $$\mathbf E_X(\rho)= \sum_{\mathfrak m} \left(\prod_\delta T_\delta(\rho)^{m_\delta}\right) \delta_{\sum_\delta m_\delta\ell_\delta} =\sum_C T_C(\rho)\nu_C^X,$$ where $\mathfrak m$ runs over finite multisets of primitive unoriented geodesics, including the empty multiset.

25.4 Absolute convergence and spectral input

The fixed-coefficient bracelet estimate extends to multicomponent diagrams. With $t(u)=2\cosh(u/2)$ and the same $K_{C,X}$ as before, $$\left|[T_C]\prod_\delta T_\delta^{m_\delta}\right| \leq K_{C,X}\prod_\delta t(\ell_\delta)^{m_\delta}.$$ Indeed, sign coherence and positive Fuchsian evaluation apply to the entire diagram, whose absolute evaluation is that product. It follows that $$\int e^{-sL}\,\mathop{}\!\mathrm d|\nu_C^X|(L) \leq K_{C,X} \prod_{\delta\in\mathcal P_{\mathrm{un}}} \bigl(1-t(\ell_\delta)e^{-s\ell_\delta}\bigr)^{-1} <\infty$$(25.11) for real $$s>s_X^*= \max\left\{\frac32, \frac{\log(2\cosh(\varepsilon/2))}{\varepsilon}\right\}, \qquad \varepsilon=\operatorname{sys}(X).$$(25.12) The second condition makes every geometric-series ratio less than one: $\log t(u)/u$ decreases in $u$. The first makes the sum of those ratios converge using the elementary $O_X(e^R)$ primitive-orbit bound. This also proves coefficient holomorphy for $\Re s>s_X^*$ and convergence of every coefficient Gaussian transform for $\Re t>0$. For the latter claim fix any real $s_0>s_X^*$ in (25.11); on a compact subset of $\Re t>0$, the factor $\exp(-L^2\Re(1/t)/4+s_0L)$ is uniformly bounded. This supplies an integrable majorant against $e^{-s_0L}\,\mathop{}\!\mathrm d|\nu_C^X|$. No norm convergence of the whole character-valued product is needed for this fixed-coefficient assertion.

Recovering this product spectrally requires its higher twisting data to be kept straight. For a representation $V$, write $\Pi_V=\sum_\delta\chi_V(\delta)\delta_{\ell_\delta}$ and let $D_m\delta_L=\delta_{mL}$. The precise reconstruction formula is $$\boxed{ \log_*\mathbf E_X(\rho) =\sum_{m\geq1}\frac1m D_m\Pi_{\rho^{\otimes m}}. }$$(25.13) The tensor-power character is $T_\delta^m$. It must not be replaced by $P_m(T_\delta)=\operatorname{tr}\rho(\delta^m)$. Tensor powers decompose into finite sums of $\mathrm{SU}(2)$ symmetric powers, and the earlier Adams inversion recovers their primitive measures from the corresponding higher-twist spectra. Thus all those twisting spectra suffice for (25.13). A single rank-two primitive trace measure does not directly provide the required power sums at coincident lengths.

25.5 What these constructions settle, and what remains

The global filter gives an actual convergent aggregate with the entire one-crossing sector removed. Equation (25.7) proves that eliminating all nonsimple primitives by a fixed linear character functional is impossible. The length-valued product avoids that obstruction: its first-atom extraction is nonlinear and retains the source lengths.

In particular, the positive support partition function of Section 24.6 has the exact reconstruction $$\mathcal P_X(s,z) =\sum_C z^{r(C)} \exp\!\left[-s\,\inf\operatorname{supp}\nu_C^X\right], \qquad \Re s>0.$$ The unit first atoms make this identity unambiguous. It is a reconstruction statement, not a remainder estimate. Quantitative counting now requires either control of the residual $\mathcal R_X$ in (25.6), using cancellations among higher-intersection curves, or uniform control of the nonlinear diagonal extraction from $\mathbf E_X$. Neither follows from fixed-coefficient convergence alone.