\section[Factorial/Rees bridges and covariance] {Factorial clearing, joint rapid bridges, and handlebody covariance} \label{p4:chap:factorial-joint} A marked-spine move can introduce color-dependent denominators before a handlebody pairing is evaluated. To transport the state, we must clear those denominators and control the loss in the chosen completion. This section compares two proved ways to do so: a legacy all-rate Rees bridge and a factorially selected lattice. Their covariance statements concern these specified coefficient objects. Candidate~\ref{p4:thm:frame-no-go} formulates the proposed obstruction to the unrestricted product topology. It remains a candidate statement, and no argument below uses it as a theorem. The constructions instead use two proved replacements: a legacy all-rate Rees bridge in the recursively cleared gauge, and a new factorial selected lattice with a quadratic joint disk/cyclotomic topology. They are distinct completions; no equality theorem identifies them. The first theorem records the legacy all-rate Rees bridge and its quantifier order. The factorial Jones--Wenzl denominator-clearing factor then supplies a local denominator removal and its explicit loss estimate. These two repairs are combined in the joint coefficient object, after which the factorial--GM comparison and handlebody covariance transport the chosen lattice. The final stabilization result is stated only for the split zero-handle situation, so the section does not identify the two completions or promote their local covariance to a general boundary equivalence. \subsection{The legacy all-rate bridge} We first specify the regulator shared by the estimates. In this section \(A=s^2\) and \(q=A^2=s^4\). This \(q\) differs from the transported GM parameter \(-A\) of the preceding section. For \(N\ge1\), put \[ e_d(N)=\max\{0,\lfloor N/d\rfloor-1\},\qquad \overline K_N=\prod_{d=1}^N\Phi_d(A^2)^{e_d(N)},\qquad K_N=A^{4N-1}\overline K_N, \] with \(\overline K_0=K_0=1\), and set \(J_N=(K_N)\). The transition maps in both inverse systems are the natural quotient maps: \[ \cH_K=\varprojlim_N\mathbb Z[A]/(\overline K_N),\qquad \widehat{\cJ}=\varprojlim_N\mathbb Z[A]/J_N. \] The exponents increase with \(N\). Comparing each cyclotomic direction and the monomial orders gives \[ \overline K_D\overline K_N\mid\overline K_{D+N},\qquad K_DK_N\mid K_{D+N},\qquad \overline K_N\mid(A^2;A^2)_N\mid\overline K_{2N}. \] Indeed \(\max(0,a-1)+\max(0,b-1)\le \max(0,\lfloor(D+N)/d\rfloor-1)\) for \(a=\lfloor D/d\rfloor\), \(b=\lfloor N/d\rfloor\). The factorial has exponent \(\lfloor N/d\rfloor\), between the two displayed truncated exponents. If \(L_N=A^{2N}(A^2;A^2)_N\), then \(K_N\mid L_{2N}\) and \(L_N\mid K_{2N}\); thus the two joint filtrations are cofinal. These identities also give the loss rule: if a denominator divides \(K_D\), division carries \(J_{D+N}\widehat{\cJ}\) into \(J_N\widehat{\cJ}\) in the common fraction ring. The analogous rule uses \(\overline K_N\) on the cyclotomic factor. The legacy selected gauge uses the recursively squared coupon clearer \[ c_0=c_1=1,\qquad c_n=\prod_{k=2}^n u_k^{2^{n-k}},\qquad u_k=A^{1-k}\frac{1-q^k}{1-q}. \] Fix the source and target charts and a word of \(\ell\) Whitehead moves; \(E\) bounds their edge counts. In a standard chart use the proper height \(h_\Gamma=1+\sum_e n_e\), with \(n_e\) the doubled colors; in a normalized GM chart use one plus the sum of its nonnegative coordinates. For a product summand, \(H\) dominates the endpoint and intermediate heights. All these choices are fixed before a decay rate is requested. Here is the coefficient topology in the legacy theorem. Use the closed ideals \(I_N=\overline K_N\cH_K\) and \(v_K(x)=\sup\{N:x\in I_N\}\), with \(v_K(0)=\infty\). Each root coefficient is realized on its active line \(\epsilon_{\Gamma,\alpha,p}\), with quotient-first evaluation. The rapid sector consists of these realizations of coefficients \(x_\alpha\in\cH_K\) satisfying, for every integer \(s\ge1\), \[ v_K(x_\alpha)\ge\max\{0,2^{s h_\Gamma(\alpha)}-c_s\} \quad\text{for one }c_s\text{ and all }\alpha. \] Its zero-neighborhoods replace the right side by \(N+2^{s h_\Gamma(\alpha)}\), for \(N\ge0\). The algebraic core has finite label support and activated cofinite cyclotomic coefficients, together with finite exceptional-root corrections; it is given the discrete topology. The bridge is its sum with the rapid sector, with the quotient topology from the discrete core plus rapid sector modulo their exact diagonal intersection. One fixed exponential rate cannot replace the condition for every \(s\). \begin{theorem}[Uniform legacy Rees bridge] \label{p4:thm:legacy-rees} Fix a finite genus and a finite quo\-tient-first word. In the legacy recursively squared selected gauge, the finite localized core plus the all-exponential-rate K\"othe--Habiro tail is complete, Hausdorff, and has dense finite truncations. The exactly typed selected theta and normalized GM maps extend to mutually inverse bicontinuous maps. \end{theorem} \begin{proof} The legacy bridge is controlled by one quantifier order. The regulator and Whitehead estimate first determine the loss for a fixed state, so one chooses an input decay rate above that loss and fixes a finite core radius; only then is an arbitrary output seminorm requested. Closed regulator ideals give completeness, proper height gives dense truncations, and the finite-support inverse identities extend to the Hausdorff completions. The exact regulator multiplication law and the Whitehead factorial estimate bound the loss along a product summand by \(\rho_{E,\ell}(H)=(8E+80\ell+16)2^H\). For either typed orientation \(T\), affine support gives \(h_{\rm out}\le ah_{\rm in}+b\), finite columns, and an input denominator stage at most \(\lceil D_T2^{r h_{\rm in}}\rceil\), for fixed \(a,b,D_T,r\). Given a rapid state, choose once \(t_0>\max\{r,a\}\) and a cutoff \(H_0\) at which the rate-\(t_0\) bound absorbs this denominator and the output rate one, with the structural activation reserve. Its image \(c_T=T(x^{\le H_0})\) is one finite core, fixed before the output rate. For an arbitrary output rate \(s\), choose \(t>\max\{r,as\}\). Above a possibly larger cutoff, \[ 2^{t h}-c_t\ge D_T2^{r h}+2^{s(ah+b)}-c'_s. \] The loss rule puts this tail in the output rate-\(s\) ideals. The middle band above \(H_0\) is already cyclotomic-integral and has finite output support; enlarging \(c'_s\) absorbs it without changing \(c_T\). Thus \(T(x)-c_T\) satisfies every rate with that same core. Proper height and the remaining regulator order make every row converge; closed ideals preserve its estimates. A sufficiently deep input neighborhood satisfies the same inequality with any requested \(N\), giving continuity. Quotient-first activation avoids specializing an inactive pole. Closed regulator ideals give completeness, and the rate-\(s+1\) estimate makes finite truncations dense at rate \(s\). The finite-support inverse products are the activation identities, so continuity and Hausdorffness extend them to the completed bridges. \end{proof} This theorem is cyclotomic only and belongs to the legacy selected gauge. \subsection{The factorial Jones--Wenzl denominator-clearing factor} Use an auxiliary parameter \(s\) with \(A=s^2\), and put \[ d_n(A)=\prod_{k=1}^n \frac{s^{2k}-s^{-2k}}{s^2-s^{-2}}, \qquad N_n=\frac{n(n-1)}2. \] In the frozen Hecke convention \((T_i-s)(T_i+s^{-3})=0\), define \[ X_n=\sum_{w\in S_n}s^{3\ell(w)}T_w, \qquad f_n=\frac{X_n}{s^{2N_n}d_n(A)}. \] The symmetrizer enters for one local geometric reason: after a marked-spine move or handlebody pairing, the Jones--Wenzl coupon can carry a color-dependent denominator. Multiplying by $d_n(A)$ clears that local denominator before the transported coupons are assembled into a word; the subsequent factorial estimate measures the remaining loss in the joint topology. \begin{lemma}[All-rank Hecke symmetrizer] \label{p4:lem:hecke-symmetrizer} For every rank \(n\), \[ X_nT_i=T_iX_n=sX_n, \qquad X_n^2=s^{2N_n}d_n(A)X_n. \] Thus \(f_n\) is idempotent and is annihilated on both sides by every return generator \(e_i=s(T_i-s)\). \end{lemma} \begin{proof} Pair \(w\) with \(ws_i\) whenever \(\ell(ws_i)=\ell(w)+1\). If the coefficient of \(T_w\) is \(c\), the pair is \(cT_w+cs^3T_{ws_i}\). The quadratic Hecke relation sends its right product with \(T_i\) to \(s\) times the pair. Left descents give the left identity. Therefore \[ X_n^2=\left(\sum_{w\in S_n}s^{4\ell(w)}\right)X_n. \] The type-\(A\) Poincare product is \(s^{2N_n}d_n(A)\), proving the square and idempotence. The return-generator statement follows from the eigenvalue equation. No finite-rank extrapolation is used. \end{proof} \begin{theorem}[Factorial selected image] \label{p4:thm:factorial-clearer} On a fixed trivalent spine and frozen crossingless coupon/pants chart, multiplication of every color-\(n\) coupon by \(d_n(A)\) clears it exactly. If \(X_\alpha\) is the cleared graph vector, after equal crossingless coordinates have been combined, and \[ m_\alpha=-\min\operatorname{exp}_A(X_\alpha), \qquad t_\alpha=A^{m_\alpha}X_\alpha, \] then \(t_\alpha\) is \(A\)-primitive and \(|m_\alpha|=O_\Gamma(h(\alpha)^2)\). The rebuilt factorial selected image has the exact root specialization kernel and root faithfulness. The simple \(\Phi_{10}^{-1}\) born pole survives, but the old exponential \(\Phi_2\) loss becomes linear; a word of length \(\ell\) has cyclotomic stage at most \[ (2E+36\ell+16)h. \] \end{theorem} \begin{proof} The factorial repair acts locally on each color-\(n\) coupon, where the Hecke symmetrizer supplies the exact denominator-clearing factor. A fixed crossingless assembly contributes only the controlled word loss, so the primitive shift and cyclotomic valuation can be read off before passing to the global selected image. The surviving \(\Phi_{10}\) pole records the remaining local obstruction rather than being hidden by the estimate. Lemma~\ref{p4:lem:hecke-symmetrizer} gives exact clearing. Every cleared coupon has an explicit quadratic Laurent exponent interval; a fixed crossingless assembly adds only linearly many loop factors. The signed primitive shift therefore has two-sided quadratic size and records the exact raw disk order in this chart. Cyclotomic factorization of \(d_n\) gives the admissible-root unit range and exact valuations. Rebuilding the bad-prime selector then repeats the basiswise proof of Theorem~\ref{p4:thm:prime-selective}. The exact \(2222\) coefficient has one uncanceled \(\Phi_{10}^{-1}\). Summing the linear factorial losses over the fixed word gives the displayed stage. \end{proof} For clarity, the chart constants in the joint estimate are fixed as follows. If the crossingless assembly in chart \(\Gamma\) removes at most \(c_\Gamma S(\alpha)\) additional loops, put \[ \begin{gathered} S(\alpha)=\sum_e n_e,\quad N(\alpha)=\sum_e n_e(n_e-1)/2,\quad\\ M_\Gamma=|E(\Gamma)|(c_\Gamma+1),\quad C_\Gamma=M_\Gamma+|E(\Gamma)|+8. \end{gathered} \] A cleared color-\(n\) coupon has Laurent exponents in \([-n(n-1)/2,n(n-1)]\); assembly places surviving exponents in \([-N(\alpha)-c_\Gamma S(\alpha), 2N(\alpha)+c_\Gamma S(\alpha)]\). Negating the actual least exponent gives the signed shift \(m_{\Gamma,\alpha}\), possibly negative, and \(|m_{\Gamma,\alpha}|\le M_\Gamma h_\Gamma(\alpha)^2\). For the selected diagonal scalar \[ s_{\Gamma,\alpha} =A^{2b_\Gamma(\alpha)+m_{\Gamma,\alpha}} C_\alpha\prod_e d_{n_e}(A), \] \(b_\Gamma\) is the physical birth level and \(C_\alpha\) the fixed bad-prime selector, with \(C_\alpha(0)=1\). The whole primitive vector has disk order \(2b_\Gamma\), while its scalar has order \(2b_\Gamma+m_{\Gamma,\alpha}-N(\alpha)\), whose absolute value is at most \(C_\Gamma h_\Gamma(\alpha)^2\). In the fixed normalized GM product chart, polynomial unitriangularity gives the same bound with its declared chart constant. The symbols \(C_\Theta,C_L\) below mean these two endpoint constants; they are not universal constants over all charts. Throughout this argument, the Hecke convention and corrected Costantino coefficient are the ones displayed above; the incompatible symmetrizer convention printed in the cited source is not used. \subsection{The joint coefficient object} Use the regulator and its quotient transitions defined above. Since \(\overline K_N(0)=\pm1\), CRT and cofinality give \[ \widehat{\cJ}\cong\Z[[A]]\times\cH_K. \] Let \(S_{\rm ev}\) be generated by \(A\) and finite products of cyclotomic factors that are units at every sufficiently large admissible odd prime, excluding the persistent factor \(A+1\). Then \[ S_{\rm ev}^{-1}\widehat{\cJ} \cong\Z((A))\times S_{\rm ev}^{-1}\cH_K. \] This localization has a Laurent disk coordinate and only a cofinite root germ; exceptional root values are separate finite data. The reduced typed denominators belong to this fixed monoid. A disk-normalized quantum factorial contains only \(\Phi_d(q)\) with \(d\ge2\), since its \(\Phi_1(q)\) factor has canceled. Under \(q=A^2\) these split into nonpersistent cyclotomic factors in \(A\). The endpoint selectors contribute only finitely exceptional \(\Phi_{2p}(A)\) factors, the monic changes \(P,P^{-1}\) are polynomial, and the remaining denominators are \(A\)-monomials. Thus no persistent \(A+1\) denominator is hidden in a typed matrix. The CRT split follows because \(\overline K_N(0)=\pm1\), so the disk and cyclotomic ideals are comaximal at each stage. It supplies no map identifying the two inverse-limit factors. Define the discrete localized core \(\cC^{\rm loc}\) by finite support in stable labels. A coefficient is an actual pair: the Laurent disk coordinate of one localized joint fraction and its activated cofinite root tail, augmented by an arbitrary finite root correction. Different fraction presentations change only that finite correction. This paired core does not assign a disk coordinate to an unpaired eventual root sequence. Put \(v_J(x)=\sup\{N:x\in J_N\widehat{\cJ}\}\), with \(v_J(0)=\infty\). Define the all-slope quadratic rapid group \(\cS^{\rm quad}\) by the condition that for every \(s\ge1\) there is \(c_s\) with \[ v_J(x_\lambda)\ge \max\{0,sh(\lambda)^2-c_s\}. \] Here \(x_\lambda\in\widehat{\cJ}\) and the bound holds for all labels with one \(c_s\) for each integer \(s\ge1\). Its zero-neighborhoods \(V_{\Gamma;s,N}\) require \(v_J(x_\lambda)\ge N+s h_\Gamma(\lambda)^2\) for every label. Finally set \[ \cB^{\rm joint}= \frac{\cC^{\rm loc}_{\rm disc}\oplus\cS^{\rm quad}} {\{(i,-i):i\in\cC^{\rm loc}\cap\cS^{\rm quad}\}}. \] \begin{lemma}[Topology of the joint bridge] \label{p4:lem:joint-topology} The rapid group is complete and Hausdorff with dense finite truncations. The diagonal intersection is closed, so \(\cB^{\rm joint}\) is complete and Hausdorff. \end{lemma} \begin{proof} The joint object is formed by gluing a discrete finite-support core to the rapid sector along their common finite-support part. Coefficientwise limits preserve every slope estimate, finite truncations control the tail, and the closed diagonal intersection makes the quotient retain both completeness and separation. Every regulator ideal is closed. A coefficientwise Cauchy limit retains each uniform slope estimate, hence lies in their intersection. Proper height makes a truncation finite and raises its omitted tail into any prescribed neighborhood. Separation follows from \(\bigcap_NJ_N\widehat{\cJ}=0\). The rapid group is an open kernel in the bridge, whose quotient is the discrete core modulo its exact intersection. A Cauchy net is eventually in one open fiber, a translate of the complete rapid group, and converges there. Separation follows in that fiber or in the discrete quotient. Thus one discrete core is used for every slope and neighborhood. \end{proof} \subsection{The factorial theta--GM bridge} If \(U_{\rm std}\) is the standard chart transition, \(P\) the normalized-GM polynomial unitriangular change, and \(S_\Theta,S_L\) the selected endpoint diagonals just specified, the two orientations are \[ M_{\rm fac}=S_L^{-1}P^{-1}U_{\rm std}^{-1}S_\Theta,\qquad N_{\rm fac}=S_\Theta^{-1}U_{\rm std}PS_L. \] Their physical realizations are quotient-first, with the inverse selected scalar used only on an active coefficient line. For a word of length \(\ell\), put \[ R_{\rm joint}(H)= \max\left\{(2E+36\ell+16)H, \left\lceil\frac{(80\ell+C_\Theta+C_L)H^2+1}{4}\right\rceil \right\}. \] \begin{theorem}[Joint factorial rapid bridge] \label{p4:thm:joint-factorial} For fixed finite theta and normalized-GM charts and a fixed finite quotient-first word, the exactly typed factorial transition matrices have denominators dividing the regulator at \(R_{\rm joint}(H)\). They preserve the localized cores and extend to mutually inverse bicontinuous maps between the corresponding \(\cB^{\rm joint}\) spaces. The rapid and exact intersection sectors admit coherent common disk/cyclotomic polynomial approximants. \end{theorem} \begin{proof} For a fixed word, the factorial estimate controls the cyclotomic loss and the two endpoint gauges control the disk loss. Their maximum is the joint regulator. Finite columns are handled in the localized core; rapid states are handled by choosing an input slope above that regulator and then using a stronger slope for the requested output. Finite-stage CRT supplies the common polynomial approximants. The linear cyclotomic loss is Theorem~\ref{p4:thm:factorial-clearer}; the two signed endpoint gauges give the quadratic disk loss. Their maximum is the displayed regulator: the monomial order \(4R-1\) clears \((80\ell+C_\Theta+C_L)H^2\), while the cyclotomic factor clears the linear stage, with one structural activation factor retained. This also holds for \(\ell=0\). Affine support for either orientation gives \(h_{\rm out}\le a h_{\rm in}+b\) and an input denominator stage \(\lceil\delta_T h_{\rm in}^2\rceil\), with fixed \(a,b,\delta_T\). Fixed generic columns are finite and their denominators lie in \(S_{\rm ev}\), so the core is preserved; quotient-first disagreement is confined to finitely many root fibers. For a rapid state, choose once \(t_0>\delta_T\) and separate the fixed low-height core at which the denominator is absorbed. For a requested output slope \(s\), take \(t>\delta_T+s(a+b)^2\). The loss rule gives \[ v_J(T_{\mu\lambda}x_\lambda) \ge(t-\delta_T)h_{\rm in}^2-c_t \ge s h_{\rm out}^2-c_t. \] Finite middle terms enlarge the output constant, not the core. These orders tend to infinity along each row, giving convergence. For continuity choose \(t\ge\delta_T+s(a+b)^2+1\); then \(T(V_{\rm in;t,N})\subseteq V_{\rm out;s,N}\), with the activation reserve preventing a new finite-core correction. The core and rapid maps agree on their exact intersection. The inverse identities hold on dense finite support and extend to the completions. Modulo a fixed regulator only finitely many labels remain; CRT supplies one polynomial with the required disk and root residues, compatibly through the inverse system. \end{proof} The finite localized core has only finite-battery interpolation unless additional coherence is supplied. This is why finite representatives are stated on the rapid and intersection sectors only. At a rapid stage, choose \(u_N,v_N\in\mathbb Z[A]\) with \(u_NA^{4N-1}+v_N\overline K_N=1\). For disk residue \(d_N\) and cyclotomic residue \(c_N\), the single polynomial \[ f_N=c_Nu_NA^{4N-1}+d_Nv_N\overline K_N \] has both residues; compatibility comes from the given inverse-limit element. For a localized fraction, a prescribed disk order and finite root battery instead give a Laurent-polynomial interpolant after clearing its monomial denominator and inverting finitely many integer resultants. For this interpolation the active root coefficients are in the reduced scalar image \(\Z[A_p]=\Z[\zeta_p]\), with \(A_p=e^{\pi i/p}\), or in its specified finite integer localization. Finite root corrections have the same finite-battery meaning when their coefficients have this type. The precise localization is given in the proof of Theorem~\ref{p4:thm:handlebody-chambers}; it does not enlarge the joint carrier or identify a separate dimension line with scalar coefficients. These interpolants are not asserted to be \(J\)-Cauchy or coherent at all roots. \subsection{Arbitrary-genus handlebody covariance} The handlebody group acts by banded spine moves, not by a chosen spatial factorization \cite{Koda,GhigginiLisca,Marche}. The factorial fixed-word estimates and ribbon/framing translation bound every such action in the joint rapid topology. \begin{theorem}[Handlebody rapid covariance] \label{p4:thm:handlebody-covariance} For every fixed finite genus, the natural handlebody action extends to the localized joint rapid bridge, obeys the exact group law, and fixes the vacuum. The construction is independent of the chosen banded-spine factorization, and no hidden Maslov degree occurs on the handlebody side. \end{theorem} \begin{proof} A handlebody move is reduced to finitely many coupon, recoupling, and framing operations. The factorial bounds control the algebraic moves, the ribbon eigenvalue controls the framing height, and agreement on finite skeins passes the result to the joint completion. This is covariance of the selected handlebody action, not a new scalar stabilization theorem. The banded handlebody-spine theorem expresses every handlebody map as a finite sequence of coupon, recoupling, and framing operations. The factorial bound controls the first two and the exact ribbon eigenvalue gives a quadratic height bound for the third. The all-slope extension argument from Theorem~\ref{p4:thm:joint-factorial} gives a continuous action. Two factorizations represent isotopic handlebody cobordisms, so the generic skein maps agree on dense finite support and hence on the completion. The same reasoning proves the group law. The empty skein is fixed, and a handlebody mapping cylinder has zero relative Maslov contribution. \end{proof} \begin{theorem}[Handlebody chamber lattice and common orbit approximants] \label{p4:thm:handlebody-chambers} The paired orbit lattices of the positive and negative handlebody actions form an arbitrary-genus chamber lattice. For the approximation assertion, fix a finite genus, finite output charts, and a finite collection of transported state/cotest pairs. Suppose that both coefficient families of each pair, in those output charts, belong to the embedded \(\cS^{\rm quad}\), with coefficients in the unlocalized \(\widehat{\cJ}\). For a finite-core output this means membership of its full paired data in \(\cC^{\rm loc}\cap\cS^{\rm quad}\), not merely regularity of a scalar contraction. Then the transported vectors have simultaneous finite-support \(\Z[A]\)-vector approximants, compatible in their joint regulator residue classes and converging in every rapid seminorm. The transported vectors, their residues, and their defined pairings are word-representative independent; the polynomial lifts need not be unique. For finite localized-core outputs, a separate conclusion holds at a prescribed disk precision \(M\ge0\) and a finite set \(\mathcal P\) of admissible roots. Require one nonzero integer \(d\), fixed for this finite data set, to carry every prescribed active root coefficient into \(\Z[A_p]\). With the finite coefficient localization specified below, there are simultaneous Laurent-polynomial interpolants matching the disk data modulo \(A^M\) and the prescribed root values. No stage coherence, \(J\)-Cauchy property, or all-root convergence is claimed for these interpolants. \end{theorem} \begin{proof} Take the orbit of the declared joint carrier under the honest handlebody action and pair it with the contragredient cotest orbit, as before. Theorem~\ref{p4:thm:handlebody-covariance} supplies the exact group law on these full carriers. Two words representing the same map therefore give the same transported state and cotest. Their disk/root residues agree, and naturality of the declared pairing makes its value independent of the orbit chart. This construction imposes no rapid-sector restriction. Ordinary polynomial approximation uses the additional hypothesis on the outputs. The rapid subgroup embeds in \(\cB^{\rm joint}\): the diagonal relation identifies a finite core with a rapid element only on their exact intersection. Thus the stated hypothesis supplies actual coefficient families \(x_\lambda\in\widehat{\cJ}\), including both disk and cyclotomic data, for every transported state and cotest. It is not an invariance assertion about initially rapid states. Theorem~\ref{p4:thm:joint-factorial} preserves the enlarged joint bridge; its fixed-core argument alone does not put every output in the rapid subgroup. For the regulator used here, the kernel of projection to stage \(L\) is \(K_L\widehat{\cJ}\). Indeed, divide compatible kernel residues at stages \(T\ge L\) by \(K_L\). Their quotient moduli \(K_T/K_L\) are cofinal with the regulator moduli, since \(K_LK_m\mid K_{L+m}\); cancellation gives a compatible inverse-limit element. Moreover \(K_L\widehat{\cJ}\cap\Z[A]=K_L\Z[A]\), by projection to the polynomial quotient. Thus a coefficient lift with the right stage residues has the required \(J\)-valuation, and polynomial differences have the asserted regulator divisibility. We spell out the simultaneous all-slope approximation. For \(n\ge1\), use the slope \(n+1\) bounds of the finitely many output families to choose a common increasing height cutoff \(H_n\), so that outside that cutoff \[ v_J(x_\lambda)\ge n+n h(\lambda)^2. \] Proper height makes the retained support finite. Choose a common integer \(L_n\ge n+nH_n^2\). For each retained coefficient take its disk residue modulo \(A^{4L_n-1}\) and its cyclotomic residue modulo \(\overline K_{L_n}\). Since \(\overline K_{L_n}(0)=\pm1\), the integral CRT formula preceding this theorem gives one coefficient in \(\Z[A]\) with both residues. Put zero on the omitted labels and call the resulting polynomial vector \(F_n\). Then \[ x-F_n\in V_{\Gamma;n,n}. \] Hence \(F_n\) converges in every \(V_{\Gamma;s,N}\). At stage \(n\) its residue is the given residue of \(x\); these classes are compatible under the natural quotient transitions because \(x\) is one inverse-limit element. In particular the coefficients of \(F_{n+1}-F_n\) lie in \(K_n\Z[A]\). The same cutoff and stage work for the entire finite collection. At any root whose cyclotomic factor has entered that stage, quotient-first evaluation gives the prescribed active root residues. Different polynomial lifts can differ by the regulator ideal; the residue classes, rather than a canonical choice of lifts, are determined by the transported data. For the separate finite-battery assertion, let \(z_\lambda^{\rm disk}\) be the finitely many Laurent disk coordinates. Choose \(b\ge0\) with \(A^bz_\lambda^{\rm disk}\in\Z[[A]]\) for all of them. In the reduced scalar branch put \[ \pi_p(A)=\Phi_{2p}(A),\qquad \Delta=\left|d\prod_{\substack{p,r\in\mathcal P\\p