\section{The affine spectral filtration}\label{sec:spectral}

Write \(e_i\) for the \(i\)th standard coordinate vector in \(\mathbb R^n\) and \(W_0\) for the finite group of signed permutations of these coordinates. For the affine Coxeter systems here the translation lattice \(X\) is \(\mathbb Z^n\) in type \(\widetilde C_n\), with finite root system \(B_n\) in the Bernstein presentation, and is \(\{x\in\mathbb Z^n:\sum_i x_i\in 2\mathbb Z\}\) in type \(\widetilde B_n\), with finite root system \(C_n\) in that presentation. In both cases \(W=X\rtimes W_0\). The Bernstein lattice elements \(\theta_x\) for \(x\in X\) generate a Laurent group algebra, with \(\theta_x\theta_y=\theta_{x+y}\). The \(W_0\)-invariants are central. The group ring in these elements and the finite Hecke algebra give the integral Bernstein presentation \cite[Theorem~2.1 and \S2.0.4]{DelormeOpdam}.

\subsection{Tempered families and their exponents}

Specialize \(v\) to a real number greater than one. We use the Plancherel decomposition of the trace \cite[Theorem~4.43 and Corollary~4.45]{Opdam}. Its data are parabolically induced representations \(\pi_{\alpha,v,z}\) indexed by finitely many association classes \(\alpha\), with \(z\) in a compact torus \(T_{\alpha,u}\). More precisely, choose one finite Levi and one of its discrete-series representations for each class and vary its twist by unitary unramified characters. The torus is the compact part of a complex algebraic torus \(T_\alpha\). The finite group \(G_\alpha\) of association maps of this family acts by affine monomial automorphisms of \(T_\alpha\), using conjugation by normalized intertwiners on its compact-model matrices. The Plancherel measure on the chosen torus includes the reciprocal finite-group factors. Write \(d\mu_{\alpha,v}\) for that measure and \({\rm Tr}\) for the unnormalized matrix trace. Thus
\[
 \tau(h)=\sum_\alpha\int_{T_{\alpha,u}}
              {\rm Tr}\bigl(\pi_{\alpha,v,z}(h)\bigr)\,d\mu_{\alpha,v}(z).
\]
The associated Fourier isometry on the Hilbert completion in the standard basis has image the square-integrable equivariant matrix sections. At each unitary point the commutant of the intertwiner action of its stabilizer is the algebra of blocks corresponding to the irreducible tempered constituents \cite[Theorem~5.5]{DelormeOpdam}.

The central character on a family is the \(W_0\)-orbit of a character of \(X\) with values \(\zeta(x)v^{2\langle x,R\rangle}\), where \(\zeta\) is unitary, \(X^\vee=\operatorname{Hom}(X,\mathbb Z)\), \(R\in X^\vee\otimes_{\mathbb Z}\mathbb Q\), and \(\langle x,R\rangle=R(x)\). The coordinates \(R_i\) are those of \(R\) in the dual of the standard coordinate basis of \(X\otimes_{\mathbb Z}\mathbb R\). Its real exponent \(R\) is constant on the torus up to \(W_0\). On a fixed integral parameter ray there are finitely many such orbits. Indeed the discrete-series residual equations are root-character equations with signed monomial right sides and the semisimple Levi residual points are isolated. Choosing a linearly independent set of equations expresses their coordinates as roots of unity times fixed rational powers of \(v\). Unramified twists only change the unitary part. We take \(v\) sufficiently large in what follows.

For a real number \(r\geq0\) put \(K_r(t)=\max(0,r-|t|)\). The Plancherel exponent of a family is the following \(W_0\)-invariant function of its real exponent:
\begin{equation}\label{eq:root-degree}
 A_R=\sum_{i<j}\bigl(K_a(R_i-R_j)+K_a(R_i+R_j)\bigr)
       +\sum_i\bigl(K_{(b+c)/2}(R_i)+K_{|b-c|/2}(R_i)\bigr).
\end{equation}
In the \(\widetilde B_n\) case put \(c=0\) in this expression. Specifically, with Haar probability measure \(dz\) on the compact torus,
\begin{equation}\label{eq:measure-limit}
 d\mu_{\alpha,v}(z)=v^{-2A_R}m_{\alpha,v}(z)\,dz,\qquad
 m_{\alpha,v}\longrightarrow m_{\alpha,0}
\end{equation}
uniformly, where \(m_{\alpha,0}\) is Laurent polynomial, nonnegative on the compact torus, and positive on a nonempty open set.

For the degree calculation use the product of reciprocal \(c\)-functions with the inverse finite longest-element parameter as prefactor, as in \cite[(3.57)]{Opdam}. The prime product at a residual torus omits identically zero numerator and denominator factors \cite[Theorem~3.25]{Opdam}. Inverting the monomial inside any such factor to change conventions has no effect on the degree of the prime product, since that monomial is identically one on the residual torus. The labels in \cite[Convention~2.1]{Opdam} allow exponent zero. A factor \(1-\epsilon v^j z^\nu\) with \(|\epsilon|=1\), a torus character \(z^\nu\), and \(j\ne0\) has degree \(\max(j,0)\). The same degree rule applies generically on the torus for \(j=0\), and deleted factors have exponent zero. For a root \(e_i\pm e_j\) let \(t=R_i\pm R_j\). The numerator for the root and its negative contributes \(2|t|\). The denominator contributes \(2\max(|t|-a,0)\), and the prefactor contributes \(-2a\). This gives \(-2K_a(t)\). For an endpoint root in the \(\widetilde C_n\) datum the numerator contributes \(4|R_i|\). The denominator contains the two pairs with parameter exponents \(-(b+c)\) and \(-(b-c)\), respectively. If \(b\geq c\) their degree is \(2\max(|R_i|-(b+c)/2,0)+2\max(|R_i|-(b-c)/2,0)\). If \(b<c\), replace \(b-c\) by \(c-b\) and add \(2(c-b)\) to that degree. Combining with the prefactor \(-2b\) gives the endpoint term in \(-2A_R\). In the reduced \(\widetilde B_n\) datum the endpoint root \(2e_i\) gives \(-2K_b(2R_i)\), which is \(-4K_{b/2}(R_i)\). These computations include the formal-degree factors of the inducing discrete series, using the same prime product on the Levi roots. The residual-degree constants are positive and independent of both \(v\) on the fixed ray and the torus variable \cite[Corollary~3.32 and Theorem~5.6]{Opdam}. Each nonstationary factor, after its leading monomial is removed, has uniform limit \(1\) in absolute square. Stationary binomial denominators cancel on the unitary torus, by real analyticity of the exact density. Each irreducible component of a stationary binomial divisor has a Zariski-dense unitary locus, so cancellation is algebraic. This proves \eqref{eq:measure-limit}.

\subsection{Leading compact matrices}

The compact induction model is the tensor product of the inducing space and the span of the finite minimal coset representatives, with positive form the tensor product of the inducing form and the standard orthonormal coset form \cite[Proposition~4.19]{Opdam}. In particular its Gram matrix is independent of \(z\). For each fixed algebra element with Laurent coefficients, its compact matrix is Laurent polynomial in \(z\), with finite support independent of \(v\). This follows by expressing multiplication in the right \(H^P\)-coset basis for the finite Levi with full Bernstein lattice. The quotient and twist to its semisimple inducing algebra send a Bernstein element to the product of its projected Bernstein element and a torus character \cite[Proposition~4.16]{Opdam}.

The coefficients of the inducing matrices can be taken algebraic over \(\mathbb C(v)\). For completeness, the algebra at a prescribed algebraic residual central character is finite dimensional over the function field since the affine algebra is finite over its Bernstein center. Over a finite extension of the function field its semisimple quotient is a direct sum of split matrix algebras. After localizing the finite algebra on the parameter curve, its radical and its split quotient specialize as such: the former is nilpotent and the latter a sum of matrix algebras. Thus for all sufficiently large \(v\) each branch of simple modules is given by an algebraic matrix representation. The strict weight inequalities for a discrete series distinguish its branches and are constant on the ray. The invariant Hermitian form on a unitary simple branch can also be chosen algebraic on the real ray, by solving the linear equations for the adjoint relations and choosing the eventual positive sign. Applying Hermitian elimination and rescaling the positive diagonal pivots gives a torus-independent algebraic basis change with orthonormal form. The matrices have convergent Puiseux expansions at infinity. These arguments can be carried out after choosing a single finite extension for the finitely many branches and taking its real positive determinations where needed.

Write
\[
 c_{\alpha,w}(z)=\lim_{v\longrightarrow\infty}v^{-A_R}\pi_{\alpha,v,z}(N_w).
\]
To justify this limit, an entry of degree \(d\) in its Puiseux expansion is bounded from below in absolute value by a positive constant times \(v^d\) on a compact patch with nonempty interior where its leading coefficient is nonzero and \(m_{\alpha,0}\) is positive. The Plancherel squared norm of \(N_w\) is \(1\), so \eqref{eq:measure-limit} forces \(d\leq A_R\). The limit is uniform since the torus support is finite. It is unchanged for \(C_w\), by its finite strictly negative triangular transition from \(N_w\).

The finite torus twisting groups depend only on the root datum. To test association of two branches, use the twisting and Weyl isomorphisms of the Levi algebras \cite[\S3.2]{DelormeOpdam} and solve the equations for an intertwiner on a finite generating set. The coefficient matrix of this linear system is algebraic in \(v\), so its rank is constant for all sufficiently large \(v\). For two simple branches a nonzero solution is an isomorphism. Thus the families and their finite association groups can be fixed on that interval. A nonzero algebraic solution can be chosen by elimination, and in the orthonormal Levi models its adjoint times itself is a positive scalar by Schur's lemma. Dividing by the positive square root of this scalar gives a unitary identification. These identifications are independent of the induction variable.

The normalized intertwiners and their inverses have uniform limits \(U_g(z)\) in the orthonormal compact model, including at stabilizer points. Indeed the formula \cite[Theorem~4.33, (4.74)]{Opdam} uses the inverse of a product of Bernstein factors on the indicated Levi subspace. Its torus denominators are products of binomials \(1-\epsilon v^j z^\nu\) on that subspace. Nilpotent parts can increase powers of these denominators, since the inverse on a generalized weight is a finite geometric expression. Each stationary divisor has a Zariski-dense unitary locus. Unitarity at every sufficiently large \(v\) therefore cancels the corresponding pole at each such \(v\). This cancellation holds over the parameter field as well: division by a power of a fixed primitive binomial leaves a remainder with algebraic coefficients in \(v\), which vanish identically if they vanish at all sufficiently large \(v\). The remaining denominators, with their leading monomials removed, have uniform nonzero limits. Boundedness removes the positive powers of \(v\) in the resulting expansion. The same argument for the inverse shows that \(U_g\) is an invertible Laurent matrix. These limiting operators are projective intertwiners, so conjugation gives an ordinary group action. Its invariant polynomial sections form
\[
 \mathcal B_\alpha=\bigl(\operatorname{Mat}_{d_\alpha}(\mathbb C[T_\alpha])\bigr)^{G_\alpha},
\]
where \(d_\alpha\) is the compact model dimension and \(\mathbb C[T_\alpha]\) its Laurent coordinate ring.

\subsection{The polynomial Fourier image}\label{sec:polynomial-image}

The Plancherel identity for fixed pairs of standard basis vectors, after taking the uniform limits above, gives an isometry
\begin{equation}\label{eq:f0}
 F_0:\ell^2(W)\longrightarrow
 \bigoplus_\alpha L^2(T_{\alpha,u},m_{\alpha,0}dz;\operatorname{Mat}_{d_\alpha})^{G_\alpha},
 \qquad N_w\longmapsto (c_{\alpha,w})_\alpha .
\end{equation}
Here \(\ell^2(W)\) is the Hilbert space with orthonormal basis \((N_w)_{w\in W}\), and a matrix \(M\) in a fiber has squared norm \({\rm Tr}(MM^*)\). We prove that \(F_0\) identifies the finite-support complex vector space on \(W\) with \(\bigoplus_\alpha\mathcal B_\alpha\).

Write an element of \(W\) as \(u t_x u'\), where \(u,u'\in W_0\), \(x\in X\) is dominant, and \(t_x\) is translation by \(x\). Opdam's formula \cite[(2.25)]{Opdam} expresses \(N_{u t_x u'}\) as a finite sum of fixed finite factors on both sides of \(\theta_x\), with coefficients constant as \(x\) varies in the relative interior of one face of the dominant cone. Subdivide the faces into relative interiors of rational simplicial cones. Choose lattice generators on the rays of each cone. These generate a finite-index sublattice of the lattice in its span. Subtracting integral multiples of the ray generators leaves finitely many lattice offsets in a bounded parallelepiped. For a relative interior, choose the fractional coefficients of the offsets in \((0,1]\), so that the remaining integral coefficients are nonnegative. In the resulting parametrization the coordinates \(m_i\) are nonnegative integers, \(x\) is a fixed offset plus the sum of \(m_i\) times fixed generators, and each entry is a finite sum of terms of the form
\begin{equation}\label{eq:affine-exponential}
 R_j(v,z)P_j(m)\epsilon_j(m)v^{\kappa_j(m)}z^{\nu_j(m)}.
\end{equation}
In this formula \(P_j\) is polynomial, \(\epsilon_j\) is a unitary lattice character, \(\kappa_j\) is a real linear form nonpositive on the cone, and \(\nu_j\) is a linear map to the torus character lattice. The finite factors and the offsets are absorbed into \(R_j\). To obtain the formula, use the simultaneous generalized translation weights. On one generalized weight space the unipotent parts of the commuting translations have commuting nilpotent logarithms. Their powers give a polynomial in \(m\) of bounded degree. The projection onto a generalized weight space is a product of univariate Hermite spectral projectors for the members of a lattice basis. Its denominators are powers of differences of distinct monomial eigenvalues. After taking Puiseux coefficients, all possible torus poles of \(R_j\) are therefore stationary unitary binomial divisors.

For each term the only point of the nonnegative real cone with \(\kappa_j(m)=0\) and \(\nu_j(m)=0\) is \(m=0\). In the induced model, the Bernstein relations give a triangular action of \(\theta_x\) on the finite coset basis ordered by Bruhat order. Indeed commuting \(\theta_x\) past a reduced word for a representative \(u\) gives the top term \(N_u\theta_{u^{-1}x}\), and each correction omits a simple factor and has lower Bruhat order. Replacing the lower terms by their minimal right coset representatives preserves the triangular order. Thus every joint weight comes from evaluating \(\theta_{u^{-1}x}\) on a Levi weight times the unramified character. Since \(u\) is minimal in its right coset, it sends positive Levi roots to positive roots, so \(u^{-1}x\) is Levi-dominant for dominant \(x\). Its projection to the unramified quotient is measured by \(\nu_j\). If that projection vanishes, the pullback lies in the Levi span, where the strict discrete-series Casselman inequality is negative on every nonzero dominant translation. The criterion applies first to lattice points \cite[Lemmas~2.20 and~2.22]{Opdam}. The equations \(\kappa_j=\nu_j=0\) and the cone are rational, so a nonzero real solution would give a nonzero rational solution and hence a lattice solution after scaling. Consequently, for \(\nu_j(m)\) in a fixed finite set, there are constants \(C,c>0\) with
\begin{equation}\label{eq:frequency-decay}
 \kappa_j(m)\leq C-c\sum_i m_i .
\end{equation}
Indeed a sequence with bounded frequencies, unbounded \(\sum_i m_i\), and \(\kappa_j(m)/\sum_i m_i\to0\) would yield a common zero on the compact simplex \(\sum_i m_i=1\). Choose \(c\) smaller than the resulting uniform lower bound for \(-\kappa_j(m)/\sum_i m_i\) at large \(\sum_i m_i\), and enlarge \(C\) for the bounded part.

First consider a polynomial matrix \(B(z)\) on one family. Let \(B_v(z)\) be its average under the exact conjugation action of \(G_\alpha\) and \(B_0(z)\) its limiting average. These are equivariant sections. Their convergence is uniform on the compact torus. Choose a scalar invariant Laurent polynomial \(f(v,z)\) of degree zero in \(v\), with nonzero limit \(f_0(z)\), clearing the torus denominators of all products used in the Fourier coefficients of \(B_v\). Concretely take the product of the finitely many denominators of \(B_v\), the exact density, and the \(R_j^*\), and their \(G_\alpha\)-translates, then divide by the leading power of \(v\). The \(R_j^*\) here are adjoints for real \(v\) and unitary \(z\), which extend rationally to the torus variables. An additional invariant Laurent factor does not affect the argument. The exact packet \(v^{A_R} f B_v\) on this family, zero on the other families, has inverse Fourier coefficients given by integrating its matrix pairings with \(\pi_{\alpha,v,z}(N_w)\) against \(d\mu_{\alpha,v}\). They are finite constant terms in \(z\). In \eqref{eq:affine-exponential} only a fixed finite set of \(\nu_j(m)\) can contribute. By \eqref{eq:frequency-decay} their degrees tend linearly to \(-\infty\) with \(\sum m_i\). Moreover all their coefficient functions in \(v\) come from a fixed finite list, up to polynomial factors in \(m\) and \(v^{\kappa_j(m)}\). Hence each coefficient is bounded by \(C'(1+\sum_i m_i)^k v^{D-c\sum_i m_i}\) for some constants \(C',D,k\) independent of \(m\) and sufficiently large \(v\). Once \(c\sum_i m_i>D\), the bound tends to zero and is dominated by its value at any fixed sufficiently large \(v\). The number of nonnegative integer tuples with fixed sum grows at most polynomially in that sum. The squared bounds are therefore summable. The fixed-coordinate limits, which exist by uniform convergence in the Plancherel pairing, give a finite vector \(x\), and dominated convergence gives convergence of the inverse vectors in \(\ell^2(W)\).

The norms of the exact packets tend to the norm of \(f_0B_0\) in \eqref{eq:f0}. Also the limiting coordinates are exactly the pairings of \(f_0B_0\) with \(F_0(N_w)\), by the fixed-vector limit. Since \(F_0\) is isometric, the equality of norms shows that \(f_0B_0=F_0(x)\). The same holds for \(g f_0B_0\) for any invariant Laurent polynomial \(g\). The continuous invariant functions \(\overline{f_0}/(|f_0|^2+\varepsilon)\) can be uniformly approximated on the compact torus by invariant Laurent polynomials, by trigonometric approximation followed by averaging. Multiplying by \(f_0 B_0\) and taking \(\varepsilon\downarrow0\) shows \(B_0\) is in the closed range of \(F_0\). Here the zero set of the nonzero Laurent polynomial \(f_0\) has Haar measure zero. Laurent matrices are dense, and averaging for the limiting unitary conjugation action is a bounded projection. Thus \(F_0\) is surjective on the Hilbert spaces.

We next check that the inverse image of a Laurent matrix in the invariant range has finite support. On each lattice parametrization, the top Puiseux degree of each \(R_j\) is fixed. If \(\kappa_j\) is strictly negative on a coordinate direction, choose a cutoff so that the top degree plus \(\kappa_j(m)\) is less than \(A_R\) once that coordinate is above the cutoff. The nonpositivity of \(\kappa_j\) on the other directions makes the cutoff independent of their values. Subdivide into finitely many strata where each coordinate is either fixed below the cutoff or free above it. After shifting the free coordinates, on a stratum with \(r\) free coordinates the leading matrix has the expression
\[
 c_m(z)=\sum_j Q_j(m,z)\chi_j(m,z),\qquad m\in\mathbb N^r.
\]
Here \(\chi_j\) are distinct unitary lattice characters with torus frequencies \(\nu_j(m)\), and the \(Q_j\) are polynomials in \(m\) with rational torus coefficients. Terms with identical characters have been combined. For every surviving term \(\kappa_j\) vanishes in the free directions, so \(\|\nu_j(m)\|\geq c_j\sum_i m_i\) for some \(c_j>0\). Distinct characters have distinct phase vectors for almost all \(z\). If the maximum degree in \(m\) of the \(Q_j\) is \(d\), the average of \(\|c_m(z)\|^2\) over \(0\leq m_i<M\), divided by \(M^{2d}\), tends pointwise almost everywhere to
\[
 \sum_j\int_{[0,1]^r}\|Q_{j,d}(x,z)\|^2\,dx .
\]
The diagonal terms are Riemann sums. For a fixed phase \(\omega\ne1\), the partial sums of \(\omega^\ell\) are bounded, so summation by parts gives \(\sum_{\ell=0}^{M-1}\ell^h\omega^\ell=O_\omega(M^h)\) for every integer \(h\geq0\). Applying this in a coordinate with distinct phases makes each off-diagonal term \(O(M^{-1})\) after normalization. Since each \(c_m\) has squared \(L^2\)-norm at most one, Fatou's lemma forces the degree-\(d\) terms to vanish if \(d>0\). Thus each amplitude is independent of \(m\), say \(Q_j(z)\). Taking the same average with \(d=0\) shows that each \(Q_j\) is square-integrable for \(m_{\alpha,0}dz\).

A pole of order \(k\) of an entry of \(Q_j\) along a primitive stationary binomial divisor is detected near a generic unitary point of that divisor. With \(u\) a real transverse coordinate, a squared entry times the density is bounded below by a positive multiple of \(|u|^{s-2k}\) near a generic point, where \(s\) is the vanishing order of the density. Integrability gives \(s-2k>-1\), hence \(s\geq2k\). As these are the only possible poles and the Laurent ring is factorial, \(m_{\alpha,0}Q_j^*\) is Laurent polynomial. Pairing \(c_m\) with a fixed Laurent matrix therefore gives only a finite set of possible frequencies \(\nu_j(m)\). The lower bound for \(\|\nu_j(m)\|\) restricts \(m\) to a finite set. Strata with no free coordinate are already finite. This proves finite support and the asserted polynomial Fourier identification.

\subsection{Integral level projections and multiplication}

The following construction identifies the original index sets of the spectral levels. For a remaining real exponent orbit choose a representative \(R\) of largest norm for a rational \(W_0\)-invariant inner product, and take a positive integral multiple \(\lambda\in X\) of its metric dual. Write \(D=2\langle\lambda,R\rangle\). The central element
\[
 z_\lambda=\sum_{\mu\in W_0\lambda}\theta_\mu
\]
sums over distinct orbit elements. On the chosen orbit \(v^{-D}z_\lambda\) has one leading monomial of modulus one. On the other remaining orbits its value tends uniformly to zero. Indeed Cauchy--Schwarz gives \(2\langle\mu,R'\rangle\leq D\), with equality only for \(R'\) in the orbit of \(R\) and \(\mu\) its chosen multiple of the metric dual. If the last remaining orbit is zero, use \(z_\lambda=1\).

Identify the regular Hilbert spaces at different \(v\) by the orthonormal standard basis, and choose \(t=v^{-1/e}\) for a positive integer \(e\) clearing the finitely many exponents. The left regular operator of a fixed Laurent Hecke element is a Laurent polynomial in \(t\) with bounded operator coefficients, integer entries, and finite row and column supports. This follows from the quadratic multiplication formula for the finitely many simple factors in each term. If such a family, or an operator-meromorphic family with a finite-order pole, is bounded as \(t\downarrow0\), its negative coefficients vanish, as can be checked in each matrix entry.

Suppose the projections already constructed sum to \(P(t)\) and commute with the regular actions. Begin with \(P=0\). Apply the preceding observation to
\[
 B(t)=v^{-D}(1-P(t))L_{z_\lambda},
\]
where \(L_h\) denotes ordinary left multiplication by \(h\). Its norm is bounded by the supremum of its central spectral values. Put \(Y=B^\sharp B\), where \(\sharp\) is adjunction of Taylor coefficients, agreeing with Hilbert adjunction for real \(t\). The constant coefficient \(Y(0)\) is an orthogonal projection because on the positive real axis \(\|Y^2-Y\|\to0\). It has integer matrix entries. An integral orthogonal projection is coordinate in an orthonormal basis: its diagonal entries are in \(\{0,1\}\), and the squared column norm equals that diagonal entry.

Set \(f(Y)=3Y^2-2Y^3\). The identities
\[
 f(Y)-Y=-(2Y-1)(Y^2-Y),\qquad
 f(Y)^2-f(Y)=(Y^2-Y)^2(4Y^2-4Y-3)
\]
give uniform convergence on a small closed disk about \(t=0\). In fact shrink the disk so that the initial norm is at most \(5/4\) and the initial defect norm at most \(1/100\). While the norm is at most \(2\), a defect of norm \(\delta\) gives an increment of norm at most \(5\delta\) and a new defect of norm at most \(27\delta^2\). The sum of all increment bounds is at most \(5/(100-27)<3/4\), so the iterates stay in that norm bound. On the real axis the spectrum of \(Y\) is uniformly close to \(0\) or \(1\). The scalar iterations stay in the respective neighborhoods and converge to \(0\) or \(1\), by the same bounds. Functional calculus therefore identifies the limit with the exact projection \(E_R(t)\) onto the cluster near \(1\). The order of vanishing of the defect at \(t=0\) at least doubles at each step. Each Taylor coefficient therefore stabilizes and has integral entries and finite row and column supports. The projection is zero on \(P(t)\), and recursion gives mutually orthogonal projections summing to the identity. By the fixed-pair Plancherel limit their constant coefficients correspond under \(F_0\) to the real-exponent-orbit cutoffs. Summing over equal values of \(A_R\) therefore gives a partition of \(W\) into coordinate sets \(S_A\), where \(A\) runs over the finitely many spectral degrees. The involution \(w\mapsto w^{-1}\) preserves these sets, since the corresponding matrix cutoffs commute with the adjoint on sections.

\begin{proposition}\label{prop:leading}
The leading matrices give a unital associative \(\mathbb Z\)-algebra \(J\) on the basis \((t_w)_{w\in W}\), with an involution \(t_w^*=t_{w^{-1}}\). Its complexification is \(\bigoplus_\alpha\mathcal B_\alpha\). The real-exponent cutoffs decompose \(J\) into its spectral levels. The symmetric trace on \(J\), given by integrating matrix traces against the measures \(m_{\alpha,0}dz\), makes the original basis orthonormal for \(\langle x,y\rangle=\tau_J(xy^*)\).
\end{proposition}

\begin{proof}
On the regular Hilbert space define the central operators, summing over real exponent orbits
\[
 P_-(v)=\sum_R v^{-A_R}E_R,\qquad P_+(v)=\sum_R v^{A_R}E_R .
\]
Their series at \(t=0\) are meromorphic with finite-order poles, with integral coefficient matrices of finite row and column support. For fixed \(x\), the operator \(P_-(v)L_{N_x}\) is bounded because its Fourier matrix is \(v^{-A_R}\pi_{\alpha,v,z}(N_x)\). Its constant column at \(N_y\) is therefore a finite integral vector. Its coefficient at \(N_w\) is the limit of
\[
 \begin{aligned}
 \sum_\alpha\int_{T_{\alpha,u}} {\rm Tr}\bigl(
    &v^{-A_R}\pi_{\alpha,v,z}(N_x)\,
     v^{-A_R}\pi_{\alpha,v,z}(N_y)\\
    &\qquad\cdot(v^{-A_R}\pi_{\alpha,v,z}(N_w))^*\bigr)
     m_{\alpha,v}(z)\,dz .
 \end{aligned}
\]
Uniform convergence identifies these limits with the pairings of the pointwise product \((c_{\alpha,x}c_{\alpha,y})_\alpha\) against \(F_0(N_w)\). The polynomial image result shows that this pointwise product lies in the image. Thus the constant columns give the stated multiplication, and associativity and the involution follow in the Fourier model.

Likewise \(\|P_+(v)N_1\|^2=\sum_\alpha\int d_\alpha m_{\alpha,v}\,dz\) is bounded. The constant coefficient is a finite integral vector. Pairing with each \(N_w\) and taking the limit identifies its Fourier image with the identity sections. This gives the unit. The trace and orthonormality follow from \eqref{eq:f0}. The direct sum by spectral levels is a decomposition into ideals because it is pointwise on the families.
\end{proof}