\section[Rootwise graph and theta endpoint] {The rootwise graph and the canonical theta endpoint} \label{p4:chap:root-rees-theta} The strict rootwise/Rees module records a formal Rees coordinate together with separately typed reduced-root coordinates. This section constructs their compatible finite-stage representatives and the theta endpoint, then identifies the boundary of the construction. The formal Rees reading here is distinct from the disk coordinate of the marked boundary-module joint completion. Keep a fixed marked handlebody and its finite boundary states in view. A finite-stage representative is one finite state before the reduced-root and formal Rees readings are taken. Rootwise saturation supplies the integral fibers, the generic fixed-label trunk supplies the formal coordinate, and strict closure selects pairs with these common finite-stage representatives. Their compatibility records a common origin; it does not identify the completed targets or evaluate the entire Rees completion. The section first builds the finite-prime theta lattice and the generic fixed-label trunk, then identifies the continuity obstruction that prevents the full Rees completion from being root-valued. It next characterizes the strict graph of compatible finite-stage representatives and adjoins finite-support root data, before transporting the result to the canonical theta endpoint. The final Newton pullback records the maximal two-sided domain on which the rootwise and Rees readings can be combined; it does not enlarge either completion beyond that typed domain. \subsection{Rootwise saturation in the theta chart} Put \[ R=\Z[A,A^{-1}],\qquad \lambda=1+A,\qquad q=-A=1-\lambda, \qquad R^{\mathrm{ev}}=S_{\mathrm{ev}}^{-1}R, \] where \(S_{\mathrm{ev}}\) contains precisely the previously declared eventually invertible cyclotomic factors and does not contain \(\lambda\). Let \[ \mathcal R_{\lambda}=\varprojlim_N R^{\mathrm{ev}}/(\lambda^N). \] The factor \(\Phi_4(A)=A^2+1\) is inverted and is congruent to \(2\) modulo \(\lambda\); hence \(2\) is a unit in \(\mathcal R_\lambda\). Fix a finite genus \(g\), a lollipop spine \(L_g\), and a theta/ladder spine \(\Gamma_g\) in the same handlebody. At an odd prime \(p\ge5\), put \[ d_p=\frac{p-1}{2},\qquad \mathcal O_p=\Z[\zeta_p],\qquad h_p=1+A_p. \] For a stable small-lollipop label \(\alpha=(a,b,c)\), with trunk half-color \(e\) and \(s_\alpha=\sum_i a_i\), define \[ \kappa_\alpha=\sum_i b_i+ \left\lfloor\frac{s_\alpha-e}{2}\right\rfloor, \qquad \kappa_\alpha^\sharp=\sum_i b_i+ \left\lceil\frac{s_\alpha+e}{2}\right\rceil, \qquad \nu_\alpha=\kappa_\alpha^\sharp-\kappa_\alpha. \] If \(w^L_{\alpha,p}\) is the saturated lollipop frame and \(U_{\Gamma L,p}\in\operatorname{GL}(\mathcal O_p)\) is the actual quotient-first endpoint matrix, put \(w^\Gamma_{\alpha,p}=U_{\Gamma L,p}w^L_{\alpha,p}\). The indices $\kappa_\alpha$ and $\kappa_\alpha^\sharp$ measure the primal and dual denominator clearings, while $\nu_\alpha$ is the remaining gap. The matrix $U_{\Gamma L,p}$ transports those clearings from the lollipop frame to the theta chart. \begin{theorem}[All-prime theta saturation] \label{p4:thm:theta-saturation-rootwise} For every fixed finite genus and every odd prime \(p\ge5\), the transported numerator frame \(Q^\Gamma_{\alpha,p}=h_p^{\kappa_\alpha} w^\Gamma_{\alpha,p}\) is an \(\mathcal O_p\)-basis of the native graph lattice \(\mathcal G_{\Gamma,p}\), while \(\{w^\Gamma_{\alpha,p}\}\) is an orthogonal basis of the invariant lattice \(\mathcal S^+_{p,g}\). There is an exact typed chain \[ \mathcal G_{\Gamma,p}\subset\mathcal S^+_{p,g} \subset(\mathcal S^+_{p,g})^\sharp \subset\mathcal G_{\Gamma,p}^\sharp, \] whose three successive diagonal profiles are respectively \(\kappa_\alpha,\nu_\alpha,\kappa_\alpha\). The full graph-pairing profile is \[ 2\kappa_\alpha+\nu_\alpha =\kappa_\alpha+\kappa_\alpha^\sharp =2\sum_i b_i+s_\alpha. \] For the closed uncolored genus-two theta spine, \(\nu_\alpha=0\) and \(\kappa_\alpha=\kappa_\alpha^\sharp=b_1+b_2+a\); hence the invariant lattice is the canonical self-dual midpoint. \end{theorem} \begin{proof} The argument first moves the rootwise lollipop frame into the theta frame by a quotient-first unit matrix. It then tracks the powers of \(h_p\) in the numerator frame and in its pairing norm. The resulting diagonal calculation is what produces the three inclusions and the self-dual midpoint; no prime-independent frame is being constructed. The small-to-even reflection in the reduced root-of-unity quotient is a monomial unit matrix. A quotient-first Whitehead move between all-even frames has only quantum integers of index below \(p\) in its retained denominators, so it and its inverse lie in \(\operatorname{GL}(\mathcal O_p)\). The lollipop numerators are unitriangular in the graph frame and satisfy \(Q_{\alpha,p}=h_p^{\kappa_\alpha}w_{\alpha,p}\). Their orthogonal norms are units times \(h_p^{\nu_\alpha}\). Transport preserves the intrinsic states and pairing, so all four lattices become diagonal in the displayed frame. Taking diagonal quotients gives the three profiles. The identity \(\kappa_\alpha+\kappa_\alpha^\sharp=2\sum_i b_i+s_\alpha\) gives the full profile. In closed genus two, \(e=0\), \(a_1=a_2=a\), and the middle profile vanishes. \end{proof} The theorem is rootwise. The active label set and the small/even reflection depend on \(p\), so it does not manufacture one prime-independent generic theta matrix. \subsection{The cofinite generic trunk} Write \(z_i=h v_i-2\), where \(h=1-q=\lambda\), and put \[ \beta_m(q)=\frac{2-q^m-q^{-m}}{1-q}. \] \begin{theorem}[Fixed-label generic saturated trunk] \label{p4:thm:generic-saturated-trunk} For every fixed stable label \(\alpha=(a,b,c)\), the frozen planar lollipop banding has a vector in the unspecialized coefficient parameter \[ T_\alpha(A)=h^{-\lfloor(s_\alpha-e)/2\rfloor}g(a,0,c) \quad\text{over }R^{\mathrm{ev}}, \] and a finite common column \[ W^L_\alpha(A)=T_\alpha(A) \prod_{i=1}^g\prod_{j=a_i}^{a_i+b_i-1} \bigl(v_i-\beta_{j+1}(q)\bigr). \] At every sufficiently large active prime this column specializes exactly to \(w^L_{\alpha,p}\). Exceptional active primes retain their separately typed integral reductions. \end{theorem} \begin{proof} For one fixed label, the generic calculation is local: braid the bands, expand the eyeglass and tripod projectors, and count the factors of \(h=1-q\). The finite set of denominators is harmless on the eventual prime tail, and the last displayed factor records the remaining color dependence. This explains both the common column and the exceptional-prime qualification. The pairing-change braid on \(2a\) strands has permutation \([1,2a,2,2a-1,\ldots,a,a+1]\), whose inversion number \(a(a-1)\) is even. Jones--Wenzl absorption therefore contributes an even Laurent monomial in the bracket square root, hence a Laurent monomial in \(q\). Expanding the color-two eyeglass and tripod projectors gives literal generic formulas divisible by \(h\); the tripod identity is \[ z_{123}+[2]^{-1}(z_1z_{23}+z_2z_{13}+z_3z_{12}) +2[2]^{-2}z_1z_2z_3. \] After \(z_I=hv_I-2\), it is divisible by \(h\). The three-ball reduction produces at least \(\lfloor(s_\alpha-e)/2\rfloor\) such pieces, proving the declared trunk divisibility. Only finitely many quantum integers and the factor \(\Phi_{12}(q)\) occur for fixed \(\alpha\); they are units at every sufficiently large active prime. Finally, \((z_i+q^{j+1}+q^{-j-1})/h=v_i-\beta_{j+1}(q)\), which gives the column. \end{proof} \subsection{Why the Rees completion is not a root-valued object} \begin{theorem}[Obstruction to root specialization on the completed Rees source] \label{p4:thm:rees-no-evaluation} For each odd prime \(p\ge5\), no continuous homomorphism from all of \(\mathcal R_\lambda\) to the reduced \(p\)-th coefficient field extends \(A\mapsto A_p\). This is false even with the \(h_p\)-adic topology on the target. Consequently neither the uniform Rees product nor its \(c_0\) submodule has root specialization on every element. \end{theorem} \begin{proof} The proposed evaluation would have to send an element that is invertible in the localized Rees ring to zero at the chosen admissible reduced root. The explicit series gives the independent topological obstruction: after substitution, its terms fail to tend to zero in the target adic topology. The failure is therefore continuity of the ambient completion, not a missing formula. The factor \(\Phi_{2p}(A)\) is in \(S_{\mathrm{ev}}\), because it fails at only the \(p\)-th reduced root-of-unity fiber. It is therefore a unit in \(\mathcal R_\lambda\), but any extension of \(A\mapsto A_p\) would send it to zero. There is also a topological obstruction. Since \(\Phi_{2p}(-1)=p\), the element \(p\) is a unit in the localized completion, so \[ \sum_{m\ge0}p^{-m^2}\lambda^m\in\mathcal R_\lambda. \] Under the formal substitution \(\lambda=h_p\), its \(m\)-th term has \(h_p\)-valuation \(m-(p-1)m^2\to-\infty\). The terms do not tend to zero. \end{proof} This theorem forces a graph of compatible finite-stage representatives rather than an evaluation map. \subsection{The strict common rootwise/Rees module} Let \(\Lambda_g^{\mathrm{st}}\) be the stable label set and let \(\mathfrak P(\alpha)\) be the cofinite set of primes at which \(\alpha\) is active. Let \(R_\alpha^{\mathrm{act}}\subset R^{\mathrm{ev}}\) consist of fractions having a reduced denominator that is a unit at every incidence \((\alpha,p)\) with \(p\in\mathfrak P(\alpha)\), and put \[ \mathcal A_g^{\mathrm{fin}} =\bigoplus_{\alpha\in\Lambda_g^{\mathrm{st}}} R_\alpha^{\mathrm{act}}e_\alpha. \] Define \[ \mathscr P^0_{\lambda,g}=c_0(\Lambda_g^{\mathrm{st}},\mathcal R_\lambda), \qquad \mathscr R_g^+=\prod_{p\ge5}\mathcal S^+_{p,g}, \] where the first factor has the uniform \(\lambda\)-adic topology and the second is a product of discrete finite modules. For finite \(x=(x_\alpha)\), set \[ j_L(x)=\left( (\operatorname{Taylor}_\lambda x_\alpha)_\alpha, \left(\sum_{\alpha\in\Lambda_{g,p}} \operatorname{Ev}_p(x_\alpha)w^L_{\alpha,p}\right)_{p\ge5} \right). \] \begin{definition}[Strict saturated rootwise/Rees module] \label{p4:def:strict-root-rees-carrier} The strict lollipop module is \[ \mathscr C^{\mathrm{sat}}_{L,g} =\overline{j_L(\mathcal A_g^{\mathrm{fin}})} \subset\mathscr P^0_{\lambda,g}\times\mathscr R_g^+. \] \end{definition} \begin{theorem}[Characterization by compatible finite-stage representatives] \label{p4:thm:common-approximant-carrier} The module is complete and Hausdorff, and its finite common states are dense. Enumerate the odd primes \(p_1,p_2,\ldots\). A pair \((x,(X_p)_p)\) belongs to the module exactly when, for every \(N\), one finite \(a_N\in\mathcal A_g^{\mathrm{fin}}\) satisfies \[ \operatorname{Taylor}_\lambda(a_N)-x \in\lambda^N\mathscr P^0_{\lambda,g}, \qquad (j_L(a_N))_{p_j}=X_{p_j}\quad(1\le j\le N). \] The Rees projection \[ \operatorname{pr}_\lambda: \mathscr C^{\mathrm{sat}}_{L,g}\longrightarrow \mathscr P^0_{\lambda,g},\qquad (x,(X_p)_p)\longmapsto x, \] is injective. Thus the module is the graph of a uniquely determined root family over the proper submodule \(\operatorname{pr}_\lambda(\mathscr C^{\mathrm{sat}}_{L,g})\); it is a proper submodule of the ambient product and contains no independent root-product summand. \end{theorem} \begin{proof} The module is defined as the closure of finite states, so membership is tested by a finite-stage representative that matches a prescribed Rees precision and a finite list of root coordinates. If the Rees coordinate vanishes, the activation-safe denominator condition forces every root coordinate into arbitrarily high \(h_p\)-power. Krull intersection then removes the apparently independent root component. Both ambient factors are complete and Hausdorff, and the \(c_0\) condition is closed. Their basic neighborhoods are precisely a uniform Rees congruence and finitely many exact discrete-root conditions, proving the approximant criterion. For injectivity, suppose the Rees coordinate is zero. Fix \(p\) and \(N\). A compatible finite-stage representative has coefficients in \(\lambda^N\mathcal R_\lambda\cap R^{\mathrm{ev}}\). Activation-safe denominators are prime to \(\lambda\), so Gauss's lemma gives \[ \lambda^N\mathcal R_\lambda\cap R^{\mathrm{ev}} =\lambda^N R^{\mathrm{ev}}. \] Its root-fiber coefficient is therefore divisible by \(h_p^N\). Since the \(w^L_{\alpha,p}\) are an \(\mathcal O_p\)-basis, \(X_p\in h_p^N\mathcal S^+_{p,g}\) for every \(N\). Krull intersection in the finite free module forces \(X_p=0\), for every \(p\). Hence \(\operatorname{pr}_\lambda\) is injective. The ambient root product is nonzero, so a pair \((0,(Y_p)_p)\) with some \(Y_p\ne0\) belongs to the ambient product but not to the module. This proves properness and the absence of an independent root-product summand. \end{proof} \begin{theorem}[Finite-prime-support data must be adjoined after closure] \label{p4:thm:finite-defects-after-closure} Let \(\mathscr D_g^+=\bigoplus_{p\ge5}\mathcal S^+_{p,g}\) with the discrete topology. The correct extension is the external direct sum \[ \mathscr B^{\mathrm{sat}}_{L,g} =\mathscr C^{\mathrm{sat}}_{L,g}\oplus(\mathscr D_g^+)^{\mathrm{disc}}. \] It is complete and Hausdorff and has an injective algebraic realization in the Rees/\allowbreak root ambient object, \[ \rho\bigl((x,(X_p)_p),(d_p)_p\bigr) =\bigl(x,(X_p+d_p)_p\bigr). \] This realization is algebraic, not a topological embedding: the defect summand is declared discrete, while its induced root-product topology is nondiscrete. If the finite-root corrections are inserted before taking closure, their closure contains the whole independent root product and destroys the common-origin condition. \end{theorem} \begin{proof} The closure module already controls the common-origin condition. The finite-root corrections are therefore adjoined as a separately declared discrete direct sum, and the injective realization simply adds them to the root coordinates. Taking that correction before closure would erase the distinction by filling the whole independent product. If \(\rho((x,X),d)=0\), then \(x=0\). Injectivity of the strict Rees projection gives \(X=0\), after which \(d=0\); hence \(\rho\) is injective. A discrete module and the strict rootwise/Rees module are complete Hausdorff, so their external topological direct sum is complete Hausdorff. The algebraic direct sum of the discrete reduced root-of-unity fibers is dense in their product. Placing it in the dense core before closure therefore produces \(\{0\}\times\prod_p\mathcal S^+_{p,g}\). \end{proof} \subsection{Canonical theta transport} Define \(j_\Gamma\) by applying \(U_{\Gamma L,p}\) only in the \(p\)-th reduced-root coordinate, while retaining the same saturated-frame coefficient chart in the Rees coordinate, and put \(\mathscr C^{\mathrm{sat}}_{\Gamma,g} =\overline{j_\Gamma(\mathcal A_g^{\mathrm{fin}})}\). \begin{theorem}[Arbitrary-genus canonical theta endpoint] \label{p4:thm:theta-root-rees-endpoint} The map \[ \left(\id,\prod_{p\ge5}U_{\Gamma L,p}\right): \mathscr C^{\mathrm{sat}}_{L,g}\xrightarrow{\sim} \mathscr C^{\mathrm{sat}}_{\Gamma,g} \] is inverse bicontinuous, extends to the discrete defect summands, and is independent of every quotient-first Whitehead factorization. Sharp cotests transport contragrediently. In closed genus two the primal and sharp theta endpoints have the self-dual midpoint identification. \end{theorem} \begin{proof} The canonical endpoint is obtained by transporting the saturated lollipop frame through the quotient-first theta matrix and then taking the prescribed closed uncolored specialization. The rootwise diagonal profiles collapse at the theta endpoint, while the earlier evaluation obstruction prevents the same conclusion from being read as a global Rees evaluation. Every root-specialized endpoint matrix is an isomorphism of a finite discrete module, so their product is a homeomorphism. It sends the finite lollipop core exactly to the finite theta core and therefore sends one closure onto the other. Rootwise bigon, square, and pentagon identities give path independence. The inverse transpose gives the sharp statement; the genus-two assertion is the case \(\nu_\alpha=0\) of Theorem~\ref{p4:thm:theta-saturation-rootwise}. \end{proof} \begin{scopefirewall}[Theta is canonical but not prime-independent] \label{p4:firewall:no-generic-theta-matrix} The reduced-root state graph is the theta/ladder graph, with theta canonical in genus two. The Rees coordinate remains the transported saturated-frame coefficient chart. The theorem does not assert one generic theta transition matrix specializing at every prime. \end{scopefirewall} \subsection{The maximal genus-one Newton pullback} For genus one define \[ r_i(A)=\left(\frac{A^{2i+1}-A^{-2i-1}}{\lambda}\right)^2, \qquad W_n(A,v)=\prod_{i=0}^{n-1} (\lambda v^2-4v-\lambda r_i) =\sum_k t_{kn}(A)v^k. \] \begin{theorem}[Exact Newton matrix] \label{p4:thm:exact-newton-matrix} One has \(t_{kn}\ne0\) exactly for \(0\le k\le2n\), and throughout that support \[ \operatorname{ord}_\lambda(t_{kn})=|k-n|, \qquad t_{nn}\equiv(-4)^n\pmod{\lambda^2}. \] After normalizing the diagonal by \((-4)^n\), the matrix is \(I+\mathsf N\) with \(\mathsf N\) raising uniform \(\lambda\)-order. It therefore has a Neumann inverse on the localized uniform product and on its \(\lambda\)-adic \(c_0\) submodule. The older all-slope rapid bridge is not surjective: the unique inverse Newton expansion of \(v\) has coefficients \[ u_n=\lambda^{-n}g[r_0,\ldots,r_n],\qquad \operatorname{ord}_\lambda(u_n)=n-1, \] where \[ g(y)=2\lambda^{-1}\!\left(1-\sqrt{1+\lambda^2y/4}\right), \] with nonintegral leading constants at infinitely many labels. \end{theorem} \begin{proof} The product expansion is organized by how far a coefficient lies from the middle power of \(v\): each off-middle choice contributes one visible \(\lambda\)-order, while the middle coefficient has no order-one correction. After diagonal normalization this makes the Newton operator a uniformly topologically nilpotent perturbation of the identity, so the inverse is a Neumann series; interpolation of the displayed branch then identifies the older rapid-tail obstruction. In expanding \(W_n\), choose constant, linear, and quadratic terms. The unique term of smallest \(\lambda\)-order gives \(|k-n|\), and the diagonal has no order-one correction. Thus the normalized off-diagonal entries are in \(\lambda\mathcal R_\lambda\), while diagonal corrections are in \(\lambda^2\mathcal R_\lambda\), so the Neumann series converges uniformly. Newton interpolation of the displayed branch \(g(y)\) gives the inverse. Its leading coefficient is \((-1)^n(2n-3)!!/(2^{3n-1}n!)\), which excludes the integral old rapid tail; its disk order also has only one fixed quadratic slope rather than every slope required there. \end{proof} At an admissible odd prime define the quotient-first monomial module \[ V_p=\mathcal O_p[v]/(Q_{d_p,p}(v)),\qquad Q_{d_p,p}(v)=\prod_{i=0}^{d_p-1}(v-\rho_{i,p}), \quad \rho_{i,p}=\frac{2-q_p^{i+1}-q_p^{-i-1}}{h_p}. \] Let \(\mathscr C_v^{\mathrm{base}}\) be its strict graph of compatible finite-stage representatives, let \(\mathscr C_W=\mathscr C^{\mathrm{sat}}_{L,1}\), and let \(T_{\mathrm{pair}}\) denote the Rees Newton transform together with the finite quotient-first matrices. \begin{proposition}[The full strict rootwise/Rees module is not Newton invariant] \label{p4:prop:newton-full-invariance-no-go} \[ T_{\mathrm{pair}}(\mathscr C_W) \not\subset\mathscr C_v^{\mathrm{base}}. \] The obstruction occurs at \(p=5\), source label \(2\): the activation-safe coefficient \(\Phi_{10}(A)^{-1}\) produces the target vacuum coefficient \[ \frac{t_{0,2}}{\Phi_{10}(A)} =\frac{144}{5}\lambda^2+O(\lambda^3), \] which cannot be the Taylor germ of an all-incidence-safe target coefficient. \end{proposition} \begin{proof} The no-go is detected at one inactive prime rather than by comparing whole completed modules. An activation-safe source denominator is allowed to produce the indicated \(5\)-adic pole at the inactive label, whereas a target coefficient safe at every active incidence must be \(5\)-integral. The first nonintegral Taylor coefficient therefore blocks the criterion for compatible finite-stage representatives. Label \(2\) is inactive at \(p=5\), so \(\Phi_{10}^{-1}e_2\) is allowed in the source core. Every target denominator must be a unit at \(p=5\), and therefore every Taylor coefficient of an all-safe target fraction is \(5\)-integral. The coefficient \(144/5\) violates the target criterion for compatible finite-stage representatives modulo \(\lambda^3\). \end{proof} \begin{theorem}[Maximal genus-one two-sided transform] \label{p4:thm:maximal-newton-pullback} Put \[ \mathscr C_W^T=\mathscr C_W\cap T_{\mathrm{pair}}^{-1}(\mathscr C_v^{\mathrm{base}}). \] Then \(\mathscr C_W^T\) is closed, complete, and Hausdorff, and \[ T_{\mathrm{pair}}:\mathscr C_W^T\xrightarrow{\sim} \mathscr C_v^{\mathrm{base}} \] is inverse bicontinuous. This is maximal among submodules of \(\mathscr C_W\) whose paired image lands in the monomial module. Inverse images of the finite target core are dense; finite-support source vectors need not be dense in the pullback. \end{theorem} \begin{proof} The admissible two-sided object is cut out before inversion by the pullback condition. On the dense finite target core, divided-difference formulas give finite Newton rows whose active-incidence denominators are units; truncations then converge in the Rees \(c_0\) topology and stabilize in each finite quotient. Density and closedness extend this finite-stage inverse to the maximal module. Closedness and maximality follow from the defining pullback. For a target monomial \(c v^k\), the inverse row-\(n\) coefficient is \(\lambda^{-n}(g^k)[r_0,\ldots,r_n]\). Divided-difference denominators are products of node differences that are units at every active incidence, and the numerator contains the required \(\lambda^n\) before reduction. Finite truncations therefore belong to the source finite core, converge uniformly in Rees \(c_0\), and become exact in the \(p\)-th quotient once all its active rows have appeared. The compatible finite-stage representative criterion puts the full inverse of the dense finite target core in \(\mathscr C_W\). Continuity and closedness extend this to the whole target module. \end{proof} \begin{proposition}[Completed padding no-go] \label{p4:prop:completed-padding-no-go} For a finite monomial vector, quotient-first reduction differs from active-zero padding at only finitely many primes and is repaired by the discrete defect summand. There exists, however, a \(c_0\) common vector whose discrepancy is nonzero at every prime in an infinite sequence. Hence finite-support root data do not produce a completed padded isomorphism. \end{proposition} \begin{proof} Finite monomials have only finitely many primes at which their quotient degree reaches the root-admissibility cutoff, so their padding discrepancy has finite prime support. The completed witness chooses a new cutoff at each prime and uses earlier cyclotomic factors to freeze earlier coordinates; this turns the same finite-support root discrepancy into an infinite-support sequence, beyond the discrete finite-support root repair. For \(v^k\), a discrepancy can occur only when \(k\ge d_p\), hence at finitely many primes. For the completed witness enumerate \(p_j\), set \(k_j=d_{p_j}\), and take coefficients \(\lambda^j\prod_{i