% Provenance: corrected synthesis of the generic lollipop and common-approximant results developed in stage 17. \section{Regulated Lattices from the Generic Lollipop Basis} \label{p4:retained:ch:lollipop-lattices} To make the two completion problems calculable, we first need a lattice whose basis records color growth and whose pairing is controlled. This section starts from the established generic lollipop basis and builds the particular Newton-corrected, regulated polynomial lattice used by the later chamber and cap constructions. Thus ``lollipop lattice'' below names the present lattice construction, not a claim that every lattice associated with a lollipop basis has the same definition. The construction has three separate ingredients: a generic orthogonal lollipop basis, a proper height on its color set, and a regulator which is simultaneously small in the disk and cyclotomic topologies. Proper height makes each cutoff finite; the regulator places the pairing increments in both filtration ideals, so the finite pairings define one joint inverse-limit class. No mapping-class or filling assertion is needed for this first layer. The relative and higher-genus lollipop normalizations are those of integral TQFT \cite{GilmerMasbaum}; the cyclotomic regulator follows the Habiro filtration \cite{Habiro}. The intended topological test is to cap a marked handlebody state with its complementary handlebody. The non-Laurent lattice records the states used in this pairing, and the joint inverse-limit class supplies controlled finite approximants for both specializations. The choice of cap remains separate from the basis and regulator. \subsection{The coefficient system and the joint ideals} Let \[ \Lambda=\mathbb Z[A],\qquad q=A^2,\qquad \FA=\mathbb Q(A), \] and, for $N\geq 0$, put \[ C_N=(q;q)_N=\prod_{j=1}^N(1-q^j),\qquad J_N=q^NC_N=A^{2N}(A^2;A^2)_N. \] The ideals are nested because \[ \frac{J_h}{J_N} =J_{h-N}\genfrac{[}{]}{0pt}{}{h}{N}_q\in\Lambda \qquad (h\geq N). \] We write \[ \widehat\Lambda_J=\varprojlim_N\Lambda/(J_N). \] The factor $A^{2N}$ makes a $J_N$-small term small at $A=0$, while $(q;q)_N$ kills it at every $q$-root whose order is at most $N$. A single sequence with increments in $(J_N)$ is therefore a literal common-origin sequence for the two topologies. \subsection{Relative Newton lollipops} Fix a one-holed torus whose boundary has color $2c$. Let $L_{c,n}$ denote the usual generic lollipop basis and let $z$ be the core curve. Set \[ \lambda_i=-q^{i+1}-q^{-i-1},\qquad Q_n^{(c)}=L_{c,0}\prod_{i=c}^{c+n-1}(z-\lambda_i). \] These are Newton interpolation products in the core variable $z$, at the successive nodes $\lambda_c,\lambda_{c+1},\ldots$, multiplied by $L_{c,0}$. The basis assertion below is over $\FA$; polynomial clearing and regulation are separate steps. The relative Hopf pairing will be denoted by $B_c$. \begin{theorem}[Generic relative Newton basis] \label{p4:retained:thm:c11-newton} For every $c\geq0$, the vectors $Q_n^{(c)}$, $n\geq0$, form a monic triangular basis over $\FA$ and are pairwise orthogonal for $B_c$. \end{theorem} \begin{proof} Multiplication by $z$ is monic raising in the lollipop filtration, so the change from $L_{c,n}$ to $Q_n^{(c)}$ is unitriangular. Encircling $L_{c,m}$ by a color-one core has eigenvalue $\lambda_{c+m}$. If $k>0$, the first argument of $B_c(Q_{n+k}^{(c)},Q_n^{(c)})$ contains $z-\lambda_{c+m}$ for every $0\leq m\leq n$ appearing in the second argument, hence annihilates each term. Symmetry gives orthogonality in the reverse order. \end{proof} Write \[ \eta_{c,n}=B_c(Q_n^{(c)},Q_n^{(c)}),\quad L=c+n+1,\quad M=2c+n+1, \] and \[ D(c,n)=(c+n)^2+3c+2n. \] The generic norm obtained from the lollipop evaluation is \[ \eta_{c,n}=q^{-c(c+1)/2}\{n\}!\{2c+2n+1\}!! \{2c+n+1\}^{+}! \frac{(\{c\}_q!)^2}{\{1\}_q\{2c\}_q!}. \] \begin{theorem}[Exact norm factorization] \label{p4:retained:thm:c11-norm} One has \[ \eta_{c,n}=(-1)^cq^{-D(c,n)} \frac{(q;q)_n(q;q^2)_L(-q;q)_M} {(1-q^2)\genfrac{[}{]}{0pt}{}{2c}{c}_{q^2}}. \] The quotient after the Laurent monomial is removed lies in $\mathbb Z[q]$ and is divisible by $(q;q)_n$. Consequently \[ \mathcal D_{c,n}:=q^{D(c,n)+n}\eta_{c,n} \in q^n(q;q)_n\mathbb Z[q], \qquad \operatorname{ord}_q\mathcal D_{c,n}=n. \] \end{theorem} \begin{proof} The norm factorization is obtained by substituting the three quantum-integer identities, collecting the Laurent powers into the declared exponent, and then comparing cyclotomic exponents by residue class. The residual polynomial has constant coefficient one, so the exponent comparison gives the stated divisibility and sharp disk order; the extra monomial is what supplies the nonzero disk order. Use \[ \{j\}=-q^{-j/2}(1-q^j),\quad \{j\}^{+}=q^{-j/2}(1+q^j),\quad \{j\}_q=-q^{-j}(1-q^{2j}). \] Collecting Laurent powers gives $-D(c,n)$. For a cyclotomic polynomial $\Phi_d(q)$, the exponent in the numerator is \[ e_d=\left\lfloor\frac nd\right\rfloor +\mathbf1_{d\,\mathrm{odd}} \left\lfloor\frac{L+(d-1)/2}{d}\right\rfloor +\mathbf1_{d\,\mathrm{even}} \left\lfloor\frac{M+d/2}{d}\right\rfloor, \] whereas the denominator exponent is \[ \delta_d=\mathbf1_{d=1\text{ or }2} +\left\lfloor\frac{2c\gcd(d,2)}d\right\rfloor -2\left\lfloor\frac{c\gcd(d,2)}d\right\rfloor. \] A residue-class check, separately for odd $d$, even $d>2$, and $d=1,2$, gives \[ e_d-\delta_d\geq\left\lfloor\frac nd\right\rfloor. \] These are precisely the cyclotomic exponents of $(q;q)_n$. The residual polynomial has constant coefficient one, proving both divisibility and the sharp disk order. \end{proof} The extra factor $q^n$ is essential: multiplying only by $q^{D(c,n)}$ clears the Laurent pole but leaves disk order zero. \paragraph{Worked checkpoint.} For the first nonvacuum relative lollipop, $c=0$ and $n=1$, the normalized norm is \[ \mathcal D_{0,1}=q(1-q)^2(1+q^2)(1+q+q^2) =q(1-q)(1-q^3)(1+q^2). \] Thus \[ \frac{\mathcal D_{0,1}}{J_1}=(1-q^3)(1+q^2)\in\mathbb Z[q], \qquad \ord_q\mathcal D_{0,1}=1. \] This is the smallest nontrivial instance in which the disk and cyclotomic factors occur simultaneously. \subsection{Orthogonal lollipops in arbitrary genus} Let $G$ be a lollipop spine in a genus-$g$ handlebody. Write a generic admissible coloring as $\lambda=(a,b,c_{\rm tree})$: the stick colors are $2a_i$, the loop colors are $a_i+b_i$, and $c_{\rm tree}$ records the remaining tree colors. Denote the countable color set by $X_G$. The generic divided lollipop vectors $\widetilde b_\lambda$ are obtained by taking the Newton factor in each handle and the ordinary clasped graph on the remaining tree. \begin{theorem}[Generic arbitrary-genus orthogonality] \label{p4:retained:thm:c11-global-orthogonality} The family $\{\widetilde b_\lambda\}_{\lambda\in X_G}$ is a $\FA$-basis of the generic graph state space $V_G$. For the higher-genus Hopf pairing, \[ B_G(\widetilde b_\lambda,\widetilde b_\mu) =\delta_{\lambda\mu}\eta_\lambda, \qquad \eta_\lambda\in \FA^\times. \] \end{theorem} \begin{proof} The arbitrary-genus statement is assembled handle by handle. Each Newton change is triangular and invertible over the generic field, while the fixed-tree factors supply the remaining basis. Orthogonality is first known at sufficiently many admissible roots; because each fixed pair gives a rational function of the parameter, those infinitely many equalities promote to generic equality. In each handle the Newton change of basis is triangular and invertible over $\FA$, so their tensor product, together with the fixed tree basis, is a basis. Orthogonality and the factorized norm hold at every sufficiently large admissible odd level. For any fixed pair of generic colorings both sides are rational functions of $A$; equality at infinitely many roots therefore gives equality in $\FA$. The diagonal evaluations are nonzero generically. \end{proof} This does not identify one generic $\mathbb Z[q]$-lattice whose specializations are all prime-level integral lattices. Jones--Wenzl recursions have color-dependent denominators, and unbounded colors would require infinitely many inversions. The next definition instead chooses a lattice directly inside $V_G$. \subsection{Proper-height primal lattices and lattices of test functionals} Here a \emph{cotest} is a test functional in the algebraic dual. \begin{definition} A height $H:X_G\to\mathbb Z_{\geq0}$ is \emph{proper} when $H^{-1}([0,N))$ is finite for every $N$. We require $H(0)=0$ in the closed vacuum sector and may take the maximum of all dynamic edge colors and derived vertex half-sums. Put \[ \rho_\lambda=J_{H(\lambda)},\qquad P_\lambda=\rho_\lambda\widetilde b_\lambda,\qquad S_\lambda=\eta_\lambda^{-1}\widetilde b_\lambda, \] and define \[ L_G^P=\bigoplus_{\lambda}\Lambda P_\lambda,\qquad L_G^S=\bigoplus_{\lambda}\Lambda S_\lambda. \] If $\widetilde b^\lambda(\widetilde b_\mu)=\delta_{\lambda\mu}$, let \[ L_G^{S,*}=\bigoplus_\lambda\Lambda\widetilde b^\lambda. \] \end{definition} The generators may be rational vectors; integrality refers to their declared coefficient lattices. \begin{theorem}[Regulated Hopf pairing] \label{p4:retained:thm:c11-regulated} For all $\lambda,\mu$, \[ B_G(P_\lambda,S_\mu) =\widetilde b^\mu(P_\lambda) =\delta_{\lambda\mu}J_{H(\lambda)}. \] For arbitrary coefficient families $x_\lambda,y_\lambda\in\Lambda$, the sum \[ \sum_{\lambda\in X_G} x_\lambda y_\lambda J_{H(\lambda)} \] is well defined in $\widehat\Lambda_J$. \end{theorem} \begin{proof} The first identity is immediate from orthogonality. Modulo $J_N$, only the finitely many labels with $H(\lambda)