Proof. We first estimate the vertical faces and corners, then prove uniform regularity on the remaining outer strata.

At \(s=0\) the vertical equation is \(yz=w^k\). Its link on \(\partial Y_t\) is the cyclic quotient of a 3-sphere, hence a lens space. The cone is smooth away from its vertex. The \(y\)- and \(z\)-faces cannot meet: if both were active, then \(|yz|=M^2t^k>|w|^k\). On the \(y\)-face, \[ \left|\frac{z^b}{y^a}\right| \le M^{-(a+b)}\left(\frac{|w|}{t}\right)^{kb}<1. \tag{6}\] After eliminating \(z=w^k/y\), the free derivative is \(P_w=y^a+(kb+1)z^b\), which is nonzero by the stronger choice of \(M\). The \(z\)-face is symmetric, with \(P_w=z^b+(ka+1)y^a\). These estimates persist for small positive \(s\) by uniform convergence of \(R_s\) and \(\partial_wR_s\) on the fixed outer face. At a \(y/w\) corner the bound \[ |P|\ge t^{1+a\alpha}(M^a-M^{-b})>0 \tag{7}\] holds; similarly at a \(z/w\) corner. Choose \(\varepsilon+\delta\) below both lower bounds at \(t=\rho\). On the \(w\)-face, near a zero of \(P\) with \(w\ne0\), use \(y\) as the free coordinate and \(z=w^k/y\). Then \(dP(\partial_{\log y})=w(ay^a-bz^b) =w(a+b)y^a\ne0\) at \(P=0\). On the compact outer \(w\)-face, this nonvanishing persists in one uniform small-value neighborhood; choose \(\varepsilon+\delta\) within it. The quotient expression \(\widetilde P=UV(U^{ka}+V^{kb})\) also shows submersivity of \(P\) in the nonvertex cone interior: if \(U,V\ne0\), its two partials cannot vanish simultaneously because that would imply \((ka+1)(kb+1)=1\); on each axis the transverse partial is nonzero. These calculations cover the vertical faces and corners near the small values used in (5).

To finish the regularity claim on the genuine outer strata, consider a sequence approaching the cone vertex while \(s\to0\), and let \(\varrho=\max(|y|^{1/\alpha},|z|^{1/\beta},|w|)\). If \(\varrho/s\to\infty\), weighted rescaling by \(\varrho\) gives smooth convergence at a nonvertex cone point; all lower Weierstrass terms vanish because \(A_j\) is bounded and \((s/\varrho)^{k-j}\to0\). If \(\varrho/s\) is bounded, rescale by \(s\), with \((Y,Z,u)=(y/s^\alpha,z/s^\beta,w/s)\); the limit is the smooth model \(YZ=R(p,u)\). For \((Y,Z)\ne(0,0)\) the base projection is a submersion; at \(Y=Z=0\) its differential contains the tangent image of the smooth root graph. The specified \(h,d\) Milnor cuts have the needed transversality on \(\partial\mathcal B\), including corners with \(|h|=\varepsilon\). For the actual function \(f_s\), these arguments give the stated outer lateral regularity over the whole value disk. Indeed \(\lvert P\rvert\le\varepsilon+\delta\) whenever \(\lvert f_s\rvert\le\delta\), so all the outer vertical-face estimates apply throughout that disk, also at their intersections with base faces because the free vertical directions keep the base fixed. On an outer stratum with no vertical face, a hypothetical failure with unbounded \(\varrho/s\) would contradict submersivity of \(P\) on the nonvertex cone after rescaling. If \(\varrho/s\) is bounded, then \(P=s^{1+a\alpha}u(Y^a+Z^b)=o(1)\) in the smooth rescaled model. The face \(\lvert h\rvert=\varepsilon\) is then absent over \(\lvert f_s\rvert\le\delta<\varepsilon/4\). On the remaining base-ball face, the boundary fibrations of \(h\) and \(d\) over their full small value disks give rank two, respectively where \((Y,Z)\ne(0,0)\) and on the smooth root graph. Compactness and the preceding smooth convergence preserve these ranks for small \(s\). This assertion concerns the outer cuts of \(X_s\); it does not assert disk-wide submersivity on the auxiliary \(J\). The remaining interior critical-point possibility is excluded in Lemma 9. All statements here are open in compact families of smooth strata, so one uniform upper bound on \(s\) works for all strata. ◻

Proof. At the inner interface, (4) and \(|h+\theta P|=\delta\) imply \[ 3\delta/4\le |h|\le5\delta/4<\varepsilon \quad(0\le\theta\le1). \tag{8}\] Thus \(h\) is regular there, including on its base-ball cut, and the \(|h|=\varepsilon\) cut is absent. On the inner \(w\)-face, (8) makes \(h\) regular, and \(\lambda\) can be decreased until \(P\) is small in \(C^1\) there. The zero-parameter cone is smooth along this interface, so the same assertion holds throughout the \((s,\theta)\) rectangle for small \(s\). We use regularity of the horizontal boundary to isotope the sublevel totals as manifolds with corners; we do not assert that \(h\) is submersive through every value in the disk at \(\theta=0\).

We first construct the prescribed circle family. For fixed \((p_0,\phi,\theta,s)\), solve \[q+\theta P\left(y,\frac{R_s(\Phi(p_0,q),w_0)}y,w_0\right)=c\] for \(q\) near \(c\), then set \(p=\Phi(p_0,q)\) and \(z=R_s(p,w_0)/y\). The root bound gives \(\lvert R_s(p,w_0)\rvert\le(3\lambda/4)^k\), so these circles lie on the \(y\)-face and strictly away from the \(z\)- and \(w\)-faces. Thus the small bound for \(P\) applies there. The solution is unique and smooth by the real implicit-function theorem: \(P\) is uniformly small there, and its real derivative \(D_qP\to0\) as \(s\to0\), since every \(p\)-dependent term of \(R_s(p,w_0)\) contains a positive power of \(s\). Decrease \(s\) until \(\lVert D_qP\rVert<1/2\) uniformly on this neighborhood. This lift keeps the \(h\)-page label \(p_0\), \(w_0\), and the exact oriented \(y\)-angle \(\phi\) fixed; the physical \(p\) may move. At \(\theta=0\), the \(s\)-motion keeps \(p_0,y,w_0\) fixed and only changes \(z=R_s(p_0,w_0)/y\). Hence the family of anchored circles is \(\mathcal H_c\times S^1\), with no base-to-circle shear. To make this family the restriction of the boundary isotopy, use the same coordinates near its compact trace and put \[\mathcal P_s(p_0,q,y,w) =P\left(y,\frac{R_s(\Phi(p_0,q),w)}{y},w\right).\] The real operator \(I+\theta D_q\mathcal P_s\) is uniformly invertible there. The local lifts that fix \(p_0,y,w\) have \[\dot q_\theta=-(I+\theta D_q\mathcal P_s)^{-1}\mathcal P_s, \qquad \dot q_s=-(I+\theta D_q\mathcal P_s)^{-1} \theta\,\partial_s\mathcal P_s.\] Differentiating \(q+\theta\mathcal P_s\) verifies that both lifts preserve its complex value. They are tangent to the \(y\)-face and to the base-ball face, because \(y\) and the pair-trivialization label \(p_0\) are fixed. Their restrictions to the circles are exactly the derivatives of the displayed implicit solution.

We extend these local lifts to the sublevel continuations. Near \(J\), vary \(\lambda\) slightly to obtain bicollars. For the \(\theta\)-motion the ambient \(C_s\) is fixed. On a collar of \(|h+\theta P|=\delta\), the map consisting of \(h+\theta P\) and the active lateral defining functions is a submersion. Choose a parameter field \(V\) tangent to every active lateral face and satisfying \(d(h+\theta P)(V)+P=0\) there. Near the compact anchor trace, patch it with the local \(\theta\)-lift above by a cutoff equal to one on that trace. The lift equation and all active-face tangency conditions are affine, so this patch remains a valid lift. Extend only its spatial correction by a cutoff supported in that collar and zero near the critical region, retaining the parameter direction, and integrate. This isotopes the smooth sublevel totals and preserves the complex value on their horizontal boundary. For the \(s\)-motion on the exterior, identify the smooth ambient family away from the cone vertex and make the same horizontal-collar correction, patched with the displayed local \(s\)-lift near the anchors. Uniqueness of the flows makes their restrictions exactly the anchored family, fixing \(p_0,w_0,\phi\). On the interface rectangle (8) keeps \(h\) regular, so its collar lifts exist along every edge and path homotopy gives isotopic boundary identifications relative to these anchors. Locally this is explicit in the coordinates \((p_0,q+\theta\mathcal P_s,y,w,\theta,s)\), in which the two anchored lifts are parameter coordinate fields. At \(s=0\), the cone is independent of \(p\), so on the entire page interface this normalization can be chosen explicitly as \[(p_0,v)\longmapsto\bigl(\Phi(p_0,c-\theta P(v)),v\bigr), \qquad v=(y,z,w)\in L=\{yz=w^k\}\cap\partial Y_\lambda.\] The bound \(|P(v)|<\delta/4\) keeps this formula in the chosen trivialization. It agrees with the anchored circles and identifies the interface with \(\mathcal H_c\times L\) while retaining the same base label. No value-preserving lift is used through the critical inner total. ◻

Proof. The gradient ideal of the isolated plane singularity \(h\) contains a power of the maximal ideal. For this \(h\), use the same \(L_0=4\) supplied by (1): on each nonconstant analytic arc \(p\to0\), \(\nu(h_p)\le L_0\nu(p)\). For fixed \(s\), at a constrained critical point in the ambient coordinates, write the spatial differential equation \[d(h+P)=\chi_{\rm orig}\,d(yz-R_s).\] After the change \(y=s^\alpha Y\), \(z=s^\beta Z\), \(w=su\), put \(\xi=s^{1+a\alpha}\). Then \(h+P=h+\xi u(Y^a+Z^b)\), while the constraint is \(s^k(YZ-R)=0\). We write \(\chi=s^k\chi_{\rm orig}\) for the multiplier of the rescaled constraint \(YZ-R=0\); the two symbols will be kept distinct. The rescaled spatial critical equations are \[ \begin{aligned} h_p&=-\chi R_p,& \xi(Y^a+Z^b)&=-\chi R_u,\\ \xi a uY^{a-1}&=\chi Z,& \xi b uZ^{b-1}&=\chi Y. \end{aligned} \tag{9}\]

We first dispose of points over \(p=0\), then of points with \(y=z=w=0\) and \(p\ne0\). Any remaining hypothetical critical sequence as \(s\to0\) has, after a subsequence, either \(p\to p_0\ne0\) or \(p\to0\). We exclude the first alternative by the cone and root-graph calculations and the second by valuations.

At \(p=0\), \(R_s(0,w)=w^k\) exactly, so the vertical constrained critical equations are those of \(P\) on \(yz=w^k\). If \(w\ne0\), then \(y,z\ne0\). Multiplying the \(y\)- and \(z\)-equations by \(y\) and \(z\), respectively, and eliminating \(\chi_{\rm orig}w^{k-1}\) from the \(w\)-equation gives \[(ka+1)y^a+z^b=0,\qquad y^a+(kb+1)z^b=0.\] Their determinant is \(k(kab+a+b)\ne0\), which would force \(y^a=z^b=0\), a contradiction. If \(w=0\), then \(yz=0\); at a nonzero point on either axis \(P_w=y^a+z^b\ne0\), whereas the \(w\)-derivative of \(yz-w^k\) is zero. Thus the vertical equations exclude every point over \(p=0\) except the origin, without a condition on \(w/s\). At \(y=z=w=0\) with \(p\ne0\), constrained criticality is exactly criticality of \(d\) on its smooth root graph and is excluded near the origin.

First exclude critical points whose base coordinate tends to \(p_0\ne0\) while \(s\to0\). Let \(\varrho=\max(|y|^{1/\alpha},|z|^{1/\beta},|w|)\). If \(\varrho/s\to\infty\), weighted rescaling by \(\varrho\) leaves a nonvertex cone point: every lower term in \(R_s/\varrho^k\) tends to zero. The normalized vertical critical equations would make \(P\) critical on the smooth nonvertex cone, which is impossible. If \(\varrho/s\) is bounded, use the rescaled variables and multiplier in (9). The variables \(Y,Z,u\) are bounded and \(\xi\to0\). On the bounded \((p,u)\)-set, \(R_p\) is bounded, while \(h_p(p)\) stays bounded away from zero because \(h_p(p_0)\ne0\). The base equation \(h_p=-\chi R_p\) therefore bounds \(\lvert\chi\rvert\) away from zero. The \(Y\)- and \(Z\)-equations in (9), with bounded \(u,Y,Z\) and \(\xi\to0\), now force \(Y,Z\to0\). After passing to a subsequence \(u\to u_0\), the constraint gives \(R(p_0,u_0)=0\). All roots over the selected base cut lie in the chosen local graph representative; only at this point do we use its unit component of \(R_p\). The base equation then bounds \(\chi\) above, so a further subsequence has a nonzero finite limit \(\chi_{\rm lim}\). The \(u\)-equation in (9) gives \(R_u(p_0,u_0)=0\). Together with \(R(p_0,u_0)=0\) and \(h_p(p_0)=-\chi_{\rm lim}R_p(p_0,u_0)\), this says \(h\) is critical on the smooth root graph \(R=0\) away from its origin, contrary to isolation of \(d\).

Thus a remaining critical sequence has \(p\to0\). Real-analytic curve selection gives an arc; write \(\nu\) for its valuation, and set \[m_0=\nu(p)>0,\qquad \nu_{\rm v}=\min\bigl(\nu(y)/\alpha,\nu(z)/\beta,\nu(w)\bigr).\] The case \(\nu_{\rm v}=\infty\) was already covered by the root graph, so assume it is finite. Write \(\varrho\) for an arc scale with \(\nu(\varrho)=\nu_{\rm v}\). If the lower Weierstrass terms all vanish under rescaling by \(\varrho\), the vertical equation limits to \(yz=w^k\) at a nonvertex point. Normalize the original critical multiplier to \(\chi_{\rm cone}=\chi_{\rm orig}/\varrho^{d_P-k}\), where \(d_P=1+a\alpha\). A nonzero component of the cone gradient bounds this multiplier through the corresponding vertical critical equation; a convergent subsequence of all the equations then gives \(d(P|_{yz=w^k})=0\), contradicting the submersivity just proved. Thus, for some \(j<k\), \[ (k-j)(\nu s-\nu_{\rm v})+\nu A_j(p)\le0. \tag{10}\] Since \(A_j(0)=0\), \(\nu A_j(p)\ge m_0\), and therefore \[ \nu_{\rm v}-\nu s\ge m_0/k. \tag{11}\] Return to the rescaled variables and multiplier in (9). By (11), \(p,u,Y,Z\to0\) and \(\nu Y\ge\alpha m_0/k\), \(\nu Z\ge\beta m_0/k\). The unit component of \(R_p\) and the gradient bound give \(\nu\chi\le L_0m_0=4m_0\). If \(Y,Z,u\ne0\), the \(Y\)- and \(Z\)-equations in (9) give \[\begin{align*} \nu Z-\nu Y &=\nu\xi+\nu u+(a-2)\nu Y-\nu\chi\\ &\ge \nu\xi+\nu u+ \left(\frac{(a-2)\alpha}{k}-L_0\right)m_0>0,\\ \nu Y-\nu Z &=\nu\xi+\nu u+(b-2)\nu Z-\nu\chi\\ &\ge \nu\xi+\nu u+ \left(\frac{(b-2)\beta}{k}-L_0\right)m_0>0. \end{align*}\] Here \(\nu\xi,\nu u>0\), and the strict signs use the two inequalities on \(a,b\) in Theorem 6, together with \(m_0>0\). The two conclusions are incompatible. If exactly one of \(Y,Z\) vanishes, or if \(u=0\) while both are nonzero, (9) forces \(\chi=0\), hence \(h_p=0\), contrary to \(p\ne0\). If both vanish, then \(R=0\) and the remaining critical equations make \(h\) critical on the root graph, contrary to isolation of \(d\). This exhausts the arc. ◻

Proof. Replace \(h+P\) by \(h\) on the inner piece using Lemma 8. The base-with-\(w\) page is \[G=(\mathcal B\cap\{h=c\})\times\{|w|\le\lambda\},\] and its root subpage \(\mathcal D\) is a page of \(d\). For \(R_s\ne0\), we retract the vertical hyperbola to a circle; over the root page, that circle collapses. We realize this model by balancing the two moduli inside the inner block. For fixed \((p,w)\), write \(A=|y|^2/\sigma^2\), \(B=\sigma^2|z|^2\), so \(AB=|R_s|^2\), and choose \(\sigma\) near \(\lambda^{(\alpha-\beta)/2}\). For \(0\le t\le1\), replace \(A-B\) by \((1-t)(A-B)\) while retaining \(AB\) and the circle angle. The positive solution for \(A,B\) is continuous at \(R_s=0\); on an axis it shrinks the active coordinate to zero. Both moduli stay between their starting and final values. Since \(\max|R_s|\le(1+o(1))\lambda^k\), the core \(|y|=\sigma\sqrt{|R_s|}\), \(|z|=\sigma^{-1}\sqrt{|R_s|}\) stays strictly within the \(M\)-bounds. This is a deformation retraction, relative to the constant-angle section, onto \[ X(G,\mathcal D)=(G\times S^1)/ (\{g\}\times S^1\text{ collapsed for each }g\in\mathcal D). \tag{13}\] The constant-angle section splits off the homology of \(G\). Quotienting that section gives \((G/\mathcal D)\wedge S^1\), so the remaining homology is \(H_{*-1}(G,\mathcal D)\), naturally for moving roots. The spaces \(G\) and \(\mathcal D\) are connected and have only first reduced homology. Their relative long exact sequence therefore gives \[0\longrightarrow H_2(G,\mathcal D)\longrightarrow D \xrightarrow{\pi_*}H\longrightarrow H_1(G,\mathcal D)\longrightarrow0.\] Thus the underlying groups are \(H_3(W)\cong K\) and \(H_2(W)\cong H/\pi_*D\). We next fix the signs of these identifications and of the mod-\(k\) injection; these signs determine the residue condition in Theorem 6.

Orient this \(S^1\) by increasing \(\arg y\). For an arc \(I\) from \(w_0\) to a root, use the circle-first orientation \([S^1_y]\times I\) on its sweep; the collapsed root end leaves boundary \(+\gamma\) at \(w_0\). Write \(\mathfrak S(C)=[S^1_y]\times C\) for circle-first suspension. The cross-product boundary rule gives \[\partial\mathfrak S(C)=-\mathfrak S(\partial C),\qquad \mathfrak S(a_0\times I)=-a_0\times([S^1_y]\times I)\] in a product chart over an oriented base interval \(a_0\), with its endpoint faces taken relatively. The root part of the boundary of \(a_0\times I\) is \(-[a_{0,\rm root}]\); write \(\partial_{\mathcal D}\) for this root-endpoint boundary, and for the resulting relative connecting map on relative cycles. Define \(\Psi\) as the negative of the relative root-endpoint connecting map under this circle-first suspension, namely \(-\partial_{\mathcal D}\mathfrak S^{-1}\). Thus the local chain \(-a_0\times(S^1_y\times I)\) has signed root contribution \(+[a_{0,\rm root}]\). The interval endpoint faces and the remaining \(w_0\)-boundary must cancel before a homology class is defined. The next section supplies this cancellation by a lens chain with a cap at the seam, whose local root vector is diagonal. Then \(\Psi\) applies to the resulting mod-\(k\) class with the sign just computed; no individual root is assumed to close around the whole base loop. This fixes the sign of \(\Psi\) throughout. We use the same suspension convention for \(H_3(W)=K\). In degree two, identify \(H_2(W)\) with \(H/\pi_*D\) by the negative of circle-first unsuspension. Künneth products on \(J\simeq\mathcal H\times L\) are ordered with the base factor first: for a closed oriented base loop \(a\), \([a\times\gamma]=-\mathfrak S(a)\). This normalization therefore sends the interface torus \([a\times\gamma]\) to \(+a\bmod\pi_*D\). The quotient is defined on \(H/kH\) because \(kH\subseteq\pi_*D\), by \(\pi_*\pi^!=k\). With \(\mathbb Z/k\) coefficients, the same relative sequence injects \(H_2(G,\mathcal D;\mathbb Z/k)\) into \(H_1(\mathcal D;\mathbb Z/k)=D/kD\). Under suspension, the signed injection from \(H_3(W;\mathbb Z/k)\) is the negative relative boundary just fixed, namely \(\Psi\) in (12).

It remains to prove acyclicity of the large total. The same collapse argument for total disks first shows that \(\widehat W\) is acyclic, because the total tubes of \(h,d\) are acyclic. We control the exterior attachment using the explicit contraction of the cone model.

At \(s=0\), put \(d_P=1+a\alpha=1+b\beta\). The explicit contraction \[ (X,T;y,z,w)\longmapsto \bigl(t^{2/5}X,t^{1/5}T;\ t^{\alpha/d_P}y,t^{\beta/d_P}z,t^{1/d_P}w\bigr), \qquad 0\le t\le1, \tag{14}\] scales both \(h\) and \(P\) by \(t\), and scales both \(yz\) and \(w^k\) by \(t^{k/d_P}\). It preserves the base ball, its \(|h|\)-cut, the vertical block cuts, and the \(|h+P|\)-cut. Hence \(\widehat\mathcal T_0\) and its inner part are contractible. Mayer–Vietoris therefore makes \(\widehat J_0\to\widehat M_{{\rm ext},0}\) a homology equivalence. Transport by Lemma 8 gives the same statement at positive \(s\). Attaching the acyclic \(\widehat W\) along \(\widehat J\) leaves \(\widehat\mathcal T_s\) acyclic. ◻

Proof. Fix \(s>0\) after Lemmas 7 and 9. In a holomorphic chart on the smooth germ \(C_s\), choose a closed ordinary Milnor ball \(B_{\rm m}\) inside the interior of every outer cut of \(X_s\). Choose \(0<\delta'\le\delta\) so that its sphere is transverse to all fibers over \(\{|v|\le\delta'\}\). The corresponding ordinary analytic total \[\widehat N_s=B_{\rm m}\cap f_s^{-1}(\{|v|\le\delta'\})\] is contractible; this is the ordinary tube consequence of Milnor’s proof recorded in (Massey 2015, Theorem 1.8 and Remark 1.9). It does not apply by itself to the custom outer cuts.

Let \(\widehat\mathcal T_s'\) be the large total with value radius \(\delta'\), and set \[\widehat A_s=(X_s\setminus\operatorname{int}B_{\rm m}) \cap f_s^{-1}(\{|v|\le\delta'\}),\qquad \widehat S_s=\partial B_{\rm m} \cap f_s^{-1}(\{|v|\le\delta'\}).\] Write \(N_s,A_s,S_s,\mathcal T_s'\) for the corresponding pages at a fixed \(\lvert c'\rvert=\delta'\). Stratify \(\widehat A_s\) by the genuine outer faces and the new Milnor-sphere face. The auxiliary cut \(J\) remains ordinary interior in this comparison. Lemma 7 gives rank two on the outer strata over the whole disk, the Milnor choice gives it on the sphere, and Lemma 9 gives it in the intervening interior. The cuts are compact, so \(f_s\) is a proper stratified submersion on \(\widehat A_s\) and on \(\widehat S_s\) over this disk, stratified by its interior and boundary circle. The face-preserving Ehresmann argument over the contractible disk therefore makes \(A_s\hookrightarrow\widehat A_s\) and \(S_s\hookrightarrow\widehat S_s\) homotopy equivalences. Their relative homology groups vanish. Relative Mayer–Vietoris for \[(\widehat\mathcal T_s',\mathcal T_s') =(\widehat N_s,N_s)\ \cup\ (\widehat A_s,A_s), \qquad (\widehat N_s,N_s)\cap(\widehat A_s,A_s) =(\widehat S_s,S_s)\] now identifies the large total/page relative homology with the ordinary local one.

The disk-wide outer regularity and isolation away from value zero also permit the large radius to shrink from \(\delta\) to \(\delta'\) through all positive intermediate radii: horizontal collar fields are tangent to every outer face and vanish near the critical region. Lemma 10 proves that \(\widehat\mathcal T_s\) is acyclic at the original radius; this isotopy therefore makes \(\widehat\mathcal T_s'\) acyclic. The ordinary total \(\widehat N_s\) is contractible as above. The connecting maps in the relative long exact sequences of these two acyclic totals identify their relative \(H_4\)-groups with the page \(H_3\)-groups, proving the homology assertion. A local filling can be used unchanged in the large total. Transporting the two page cycles from the smaller to the original value circle along two disjoint angular rays adds collars at disjoint complex values, hence no new intersections. Thus the filling pairing is identical. ◻

Proof. Use Lemma 8 to compute the exterior at \(s=0\). Normalize \(c>0\), and put \(q=h\). Cut at \(\operatorname{Re}q=c/2\), as in Figure 1, calling the left and right pieces \(A\) and \(B\), respectively. On \(A\), \(|P|=|c-q|\ge c/2\), so the removed inner block is absent. The allowed \(P\)-values form a convex region avoiding zero, and the boundary estimates of Lemma 7 trivialize its \(P\)-pages with their outer cuts. Contracting the base by the positive weights of \(h\) and carrying these pages through the trivialization retracts \(A\) to \(Q\).

On \(B\), choose one boundary-respecting trivialization of the \(h\)-pages over the convex regular-value region \(\{|q|\le\varepsilon,\ \operatorname{Re}q\ge c/2\}\). It can agree near \(c\) with the page labels used in Lemma 8. For \(v=(y,z,w)\), define the radius of its block by \[r(v)=\max\left\{ (|y|/M)^{1/\alpha},\ (|z|/M)^{1/\beta},\ |w| \right\}.\] The exterior condition is \(\lambda\le r(v)\le\rho\). Scale \(v\) by the cone weights, from \(t=1\) down to \(t=\lambda/r(v)\), while retaining its \(h\)-page label and setting \[v_t=(t^\alpha y,t^\beta z,tw),\qquad q_t=c-t^{d_P}P(v)=(1-t^{d_P})c+t^{d_P}q, \qquad d_P=1+a\alpha.\] This keeps \(\lambda\le r(v_t)\le\rho\), and \(q_t\) stays in the same convex \(h\)-region. The base-ball cut is preserved by the trivialization. The homotopy therefore stays in \(B\) and stops at the actual interface \(J\), fixing \(J\) pointwise. At its endpoint \(v_L\in L\), the value is \(q=c-P(v_L)\), generally different from \(c\). The \(s=0\) interface continuation of Lemma 8 retains \(v_L\) and the base label while changing this value to \(c\). It identifies the retract with \(\mathcal H\times L\).

The seam values form the contractible interval \(I=\{q=c/2+iu:|q|\le\varepsilon\}\). Trivializing both page factors along \(I\) identifies the overlap \(C=A\cap B\) with \(\mathcal H\times Q\times I\). After contracting \(I\), its map to \(J\simeq\mathcal H\times L\) is homotopic to \(\mathrm{id}_{\mathcal H}\times(Q\to L)\), where \(Q\to L\) is weighted radial projection. Both \(\mathcal H,Q\) have homotopy dimension one, so overlap \(H_3=0\), while overlap \(H_2=H\otimes E\). Moreover \(H_2(J)=H\otimes\mathbb Z/k=H/kH\) and \(H_3(J)=\mathbb Z\). The latter generator maps to zero in \(H_3(W)=K\): it is the full inner fiber over a regular base point, whose \(H_3\) vanishes in the collapse model. We record the orientation of the remaining interface map. For a full-page 3-cycle write its exterior chain as \(e=e_A+e_B\), with the inner chain \(w\) satisfying \(\partial w=b_J,\ \partial e=-b_J\). Define \(t_C=-\partial e_A\in Z_2(C)\). Then \[ \partial e_B=t_C-b_J,\qquad z:=[t_C]\in H_2(C)=H\otimes E. \tag{16}\] This fixes the sign of the Mayer–Vietoris coordinate. Let \(r_t:C\to B\), \(0\le t\le1\), be the stopped radial homotopy just constructed, reparametrized to end in \(J\). Its endpoint is the actual interface, with the same \(h\)-page label under the compatible parameter rectangle. For the cycle \(t_C\), the interval-first prism chain \(S=r_*([0,1]\times t_C)\) has \(\partial S=r_{1*}t_C-t_C\). Consequently \[ \partial(e_B+S)=r_{1*}t_C-b_J,\qquad [b_J]=r_{1*}[t_C] =(1\otimes\tau)z\quad\hbox{in }H_2(J)=H/kH. \tag{17}\] Under the degree-two normalization fixed after (13), this is the positive sign of the last arrow in (15). The Mayer–Vietoris sequence now gives (15), with final map the quotient of \(1\otimes\tau\). Before \(W\) is attached, exterior \(H_3\) differs from its \(H\otimes\ker\tau\) contribution by precisely the image of \(H_3(J)\); that image dies after attachment. This proves the last statement. ◻

Proof. The weighted rotation preserves each axis, hence its boundary homology class. Transfer of either axis boundary is a single boundary circle of the upstairs plane-curve page. In a positive plane-curve Seifert convention, the pairing of two distinct complex branch boundaries is their positive local intersection multiplicity. The sum of all oriented boundary classes is zero, so the self-pairing of the \(V=0\) boundary is \(-(1+4a)=-n_0^2\), and that of \(U=0\) is \(-(1+4b)=-n_1^2\). The two axes meet once. Reversing the second orientation gives the mixed value \(-1\) upstairs. Division by the covering degree proves the first two formulas.

The integer-valued functional \(x\mapsto4e(b_j,x)\) reduces modulo \(4\) through \(\tau\). Indeed a loop whose \(\tau\)-class is zero lifts closed; its full transfer consists of four deck translates, and the transferred axis class is deck invariant. The four Seifert pairings are equal, so their sum is divisible by \(4\). A class in \(\ker\tau\) can be represented by such loop systems, and \(\tau(b_j)=1\) generates \(\mathbb Z/4\). Evaluation at \(b_j\) gives \(4e(b_j,b_j)=-n_j^2\equiv-1\pmod4\), proving the last congruence. ◻

Proof. The coefficient of \(x\in E_{\mathbb Z_\ell}\) on \(v_j\) is \(c_j(x)=-4e(v_j,x)\). It is integral because \(4e(b_j,x)\) is integral and \(n_j\) is a unit in \(\mathbb Z_\ell\), and it sends \(v_j\) to \(1\) by (27). Its kernel \(U_{j,\ell}\) is therefore an integral orthogonal summand. The local congruence above gives \(\tau_{\mathbb Z_\ell}(U_{j,\ell})=0\). At \(\ell=2\), \(c_j(x)\equiv\tau_{\mathbb Z_2}(x)\pmod{4\mathbb Z_2}\) because \(n_j\equiv1\pmod4\); at odd primes the target of \(\tau_{\mathbb Z_\ell}\) is zero. Thus the projection \(x\mapsto c_j(x)v_j\) is integral also at \(\ell=2\): all divisions are by the unit \(n_j\).

The image of (28) satisfies the lattice congruence (24). Conversely, write the second coordinate of an arbitrary lattice element as \(z=x_H\otimes v_j+u\). Its congruence says \(x_H-\pi_{\nu*}l\in4H_{\mathbb Z_\ell}\). Replacing \(l\) by \[l+\pi_\nu^!\left((x_H-\pi_{\nu*}l)/4\right)\] is legitimate even at \(\ell=2\), because the numerator belongs to \(4H_{\mathbb Z_\ell}\). It does not alter \(\operatorname{pr}_\nu l\) and makes its image under \(\pi_{\nu*}\) equal to \(x_H\). This proves surjectivity; injectivity follows from the rational direct sum.

The \(H\otimes U_{j,\ell}\) summand is orthogonal to the image of \(D_\nu\). On two vectors from \(D_\nu\), transfer adjunction gives \[s_{d_\nu}(\operatorname{pr}_\nu l, \operatorname{pr}_\nu l') =s_{d_\nu}(l,l')- \frac14s_h(\pi_{\nu*}l,\pi_{\nu*}l').\] The product-form contribution on \(v_j\) is the negative of the last fraction, by \(e(v_j,v_j)=-1/4\). Combined with the leading minus sign on \(K_\nu\) in (24), the result is \(-s_{d_\nu}(l,l')\), as claimed. ◻

Proof. The two \(v_j\) have the same norm \(-1/4\), while \(e(v_0,v_1)=-1/(4n_0n_1)\). Therefore \[w=v_0-v_1,\qquad e(w,w)=-\frac12\left(1-\frac1{n_0n_1}\right)\ne0.\] The vector \(w\) is monodromy fixed and hence pairs symmetrically for \(e\). Reflection \[R_w(x)=x-\frac{2e(x,w)}{e(w,w)}w\] is consequently an \(e\)-isometry commuting with monodromy, and it exchanges \(v_0,v_1\). Define \(\rho_\nu\) on \(F_\nu\) as identity on \((K_\nu)_{\mathbb Q}\) and \(1_H\otimes R_w\) on \((H\otimes E)_{\mathbb Q}\). This is a rational Seifert involution. Let \(\mathcal R_\nu\) be identity on \(D_\nu\) and \(1_H\otimes R_w\) from \(H\otimes U_{0,\mathbb Q}\) to \(H\otimes U_{1,\mathbb Q}\). Equation (28) gives \(\rho_\nu\alpha_{\nu,0}=\alpha_{\nu,1}\mathcal R_\nu\). The same \(R_w\) occurs for both coverings, so the square formed by \(\mathcal R_0,\mathcal R_1\) and \(\mathcal I\oplus1\) commutes. Consequently \[ J_1=\rho_1J_0\rho_0^{-1},\qquad \Delta=(J_0^{-1}\rho_1J_0)\rho_0^{-1}. \tag{30}\] The minus space of \(\rho_\nu\) is \(W_\nu=H_{\mathbb Q}\otimes\mathbb Qw\). Its restricted Seifert form is \(e(w,w)s_h\), so its Seifert operator is \(A_h\). In particular it lies entirely in \(F_{\nu,1}\oplus F_{\nu,5}\). Both factors of (30), and therefore \(\Delta\), act identically on every other rational primary summand.

The minus spaces within the two primary sectors are \[W_{\nu,1}=H^{(1)}\otimes\mathbb Qw,\qquad W_{\nu,5}=H^{(5)}\otimes\mathbb Qw.\] The canonical map \(x\otimes w\mapsto x\otimes w\) from \(W_0\) to \(W_1\) preserves the restricted Seifert and symmetric forms and intertwines \(A_h\). It therefore restricts to isometries on the individual sectors displayed above. These restrictions are nondegenerate: \(e(w,w)\ne0\), and \(s_h+s_h^{\mathsf t}\) is nondegenerate on both \(H^{(1)}\) and \(H^{(5)}\). Since the Seifert isometry \(J_0\) intertwines \(A_0\) and \(A_1\), composition with \(J_0^{-1}\) identifies, separately on each sector, the minus spaces of \(\rho_0\) and \(J_0^{-1}\rho_1J_0\). On \(F_{0,5}\) the isometry \(\Delta\) commutes with the \(L\)-action and preserves the Hermitian form recovered from \(B_0\). The two involutions in (30) have the same \(L\)-dimension of their minus spaces, so their \(L\)-determinants cancel: \(\det_L\Delta=1\). It is therefore an \(L^+\)-rational special-unitary element.

On \(F_{0,1}\) the same comparison gives determinant \(1\). The spinor norm of an involution with nondegenerate minus space is the product, modulo rational squares, of the quadratic values of an orthogonal basis of that space. The minus spaces just identified are isometric, so their spinor norms are equal and cancel in the product. Thus \(\Delta\) has trivial spinor norm. The standard exact sequence for \(\operatorname{Spin}\to\operatorname{SO}\) now supplies a rational spin lift (Conrad 2014, sec. C.5, paragraph before Example C.5.5 and Lemma C.5.8). ◻

Proof. By Lemma 2, the rational primary parts \(H^{(1)}\) and \(H^{(5)}\) have dimensions two and four, respectively, and together equal \(H_{\mathbb Q}\). Their eigenvalue sets are stable under inversion, so they are nondegenerate orthogonal summands for \(s_h\). The restriction to \(H^{(1)}\) is symmetric, and each complex primitive fifth eigenspace is one-dimensional and pairs nondegenerately with its inverse eigenspace.

To obtain both Hodge signs in the primitive fifth eigenspaces of \(E_{\mathbb C}\), we construct holomorphic forms with inverse primitive fifth characters. Their conjugate antiholomorphic classes will give opposite cup-product signs. Set \(m=4ab+a+b\), and write \(a=ga'\), \(b=gb'\) with \(\gcd(a',b')=1\). The weights of \(U,V\) in (25) are \(b/m,a/m\). For \(q=1\) and \(q=4\) there are positive integers \(i,j\), congruent modulo \(4\), with \[ \frac{bi+aj}{m}=\frac q5. \tag{31}\] The existence and positivity of these two index pairs were proved in Lemma 17.

On the smooth affine curve \(\widetilde P=1\), the differential \[ \omega_{i,j}=U^{i-1}V^{j-1} \frac{dU}{\partial_V\widetilde P} =-U^{i-1}V^{j-1} \frac{dV}{\partial_U\widetilde P} \tag{32}\] is regular and nonzero. The second expression is used where \(\partial_V\widetilde P=0\). Its deck character is \(\omega^{i-j}=1\), so it descends to the quotient, and its weighted rotation character is \(e^{2\pi iq/5}\). To obtain the needed holomorphic classes, we check that these differentials extend across every end of the normalized compactification. At an end put \(u=\operatorname{ord}U\), \(v=\operatorname{ord}V\). If \(4au<4bv\), then \(u<0\) and \(v=-(4a+1)u\); the order of (32) is \((i-(4a+1)j)u-1\ge0\). Here \(i<4a+1\), because (31), \(q\le4\) and \(j\ge1\) give \(bi\le4m/5-a<b(4a+1)\). The case \(4bv<4au\) is symmetric, using the second expression and \(j<4b+1\). In the remaining case \(4au=4bv<0\), leading terms in \(U^{4a}+V^{4b}\) cancel and \(\operatorname{ord}(\partial_V\widetilde P)=u+4bv\). There is a positive integer \(c\) with \((u,v)=-c(b',a')\); the order is \[(i-1)u+(j-1-4b)v-1 =c\left(1-\frac q5\right)\frac mg-1\ge0.\] These are all possible ends, since one of \(U,V\) has a pole.

The two differentials for \(q=1,4\) thus give nonzero holomorphic forms on the smooth compactification with characters \(\zeta\) and \(\bar\zeta\). Complex conjugation gives antiholomorphic forms of the opposite characters. The cup-product Hermitian form has opposite signs on holomorphic and antiholomorphic forms. Poincaré duality exchanges reciprocal characters and gives opposite signs for the Hermitian intersection form on the corresponding homology classes. Since both characters have been constructed, each of the two homology eigenspaces has both signs. With complex coefficients, the map from affine-page homology to compactification homology is surjective and preserves intersections. Average a linear section of this map over the finite group generated by the commuting deck transformations and weighted rotation. The resulting equivariant section lifts the classes with their characters and intersection numbers unchanged. Transfer over \(\mathbb C\) identifies the deck-invariant part with \(E_{\mathbb C}\) and preserves the monodromy characters. Since \(e-e^{\mathsf t}\) is, up to our common Seifert convention, the page intersection form, the Hermitian form \(i(e-e^{\mathsf t})(x,\bar x)\) has both signs in each of \(E_\zeta\) and \(E_{\bar\zeta}\).

We transfer these signs to the fixed quadratic sector by tensoring a primitive base eigenspace with the inverse eigenspace of \(E\). After complexifying, choose \(\lambda\in\{\zeta,\bar\zeta\}\) and a nonzero \(x\in\ker(A_h-\lambda)\subset H_{\mathbb C}\). This eigenspace is one-dimensional and pairs nondegenerately with its inverse eigenspace. The preceding statement that both signs occur applies to \(\ker(A_e-\lambda^{-1})\subset E_{\mathbb C}\) under either inverse monodromy convention. For \(y\) in this eigenspace, put \[I_H=i(s_h-s_h^{\mathsf t})(x,\bar x),\qquad I_E=i(e-e^{\mathsf t})(y,\bar y).\] These are real, and \(I_H\ne0\). The relation \(s^{\mathsf t}=sA\) in each factor, together with the vanishing of \(s_h(x,x)\) and \(e(y,y)\) when \(\lambda^2\ne1\), gives \[B_\nu(x\otimes y+\bar x\otimes\bar y,\, x\otimes y+\bar x\otimes\bar y) =-\frac{4I_HI_E}{|1-\lambda|^2}.\] The vectors in this display lie in \(F_{\nu,1}\otimes_{\mathbb Q}\mathbb R\), and the two signs of \(I_E\) prove that \(B_\nu|F_{\nu,1}\) is indefinite. These tensors and their conjugates already contribute at least four real dimensions. The invariant part of \(H\) has dimension two, and \(b_0\) is a nonzero invariant vector of \(E\) by (26); its tensor contributes at least two more dimensions.

For the Hermitian sector, we tensor the primitive eigenspace of \(E\) with a fixed base vector. Choose \(x\in H\cap H^{(1)}\) with \(s_h(x,x)\ne0\). Such an integral vector exists: the nondegenerate symmetric restriction to \(H^{(1)}\) has a rational vector of nonzero norm, and clearing denominators puts it in \(H\). Tensor \(x\) with \(E_\zeta\). This lies in the \(\zeta\) component of \(F_{\nu,5}\) for one of the two inverse monodromy conventions. If the eigenvalue of \(A_e\) on \(E_\zeta\) is \(\lambda\in\{\zeta,\zeta^{-1}\}\), then on this eigenspace \[(e+e^{\mathsf t})(y,\bar y) =\frac{1+\lambda^{-1}}{i(1-\lambda^{-1})} i(e-e^{\mathsf t})(y,\bar y),\] and the displayed ratio is a nonzero real number. The two signs found above therefore give both signs for the Hermitian form of \(B_\nu\) at this real embedding. Its dimension is at least two. ◻

Proof. By Lemma 21, \(G_1=\operatorname{Spin}(F_{0,1},B_0)\) is a connected, simply connected, absolutely almost simple group over \(\mathbb Q\) of quadratic dimension at least five (Conrad 2014, Proposition C.3.10, Lemma C.4.1, Proposition C.4.10, and Example C.6.6). The proved indefiniteness makes it noncompact over \(\mathbb R\). The second correction group \(G_5=\operatorname{SU}(F_{0,5},\mathfrak h_{0,5})\) is connected, simply connected, and absolutely almost simple over \(L^+\): after algebraic closure it is \(\operatorname{SL}_d\), with \(d\ge2\) (Milne 2022, Proposition 24.40(b), equation (158), and Summary 21.96). It is noncompact at the real place found in Lemma 21. We use \(G=G_1\times\operatorname{Res}_{L^+/\mathbb Q}G_5\), acting on the indicated primary spaces and as identity on their \(s_0\)-orthogonal primary complement. The Spin action preserves \(B_0\) and commutes with \(A_0=1\) on \(F_{0,1}\). The special-unitary action preserves the Hermitian form, hence \(B_0\), and commutes with \(A_0\) on \(F_{0,5}\) by \(L\)-linearity. On both sectors \(1+A_0\) is invertible and \[s_0=B_0(1+A_0)^{-1}.\] Consequently this action of \(G\) preserves the Seifert form \(s_0\). We obtain the density assertion for \(G\) by treating its two factors separately.

The strong-approximation theorem of Kneser and Platonov, in the precise form of (Rapinchuk 2014, Theorem 2.3), says that for a connected, simply connected, absolutely almost simple group over a number field, noncompactness at a place in the omitted finite set implies density of rational points in the adeles away from that set. Apply it to \(G_1\) over \(\mathbb Q\) and to \(G_5\) over \(L^+\), omitting all archimedean places. Restriction of scalars and a finite product give density of \(G(\mathbb Q)\) in \(G(\mathbb A_f)\).

Write \(\rho\colon G\to\operatorname{GL}(F_0)\) for this rational action. Lemma 20 gives a rational lift \(\widetilde\Delta\in G(\mathbb Q)\) with \(\rho(\widetilde\Delta)=\Delta\). For each rational prime \(\ell\), put \(a_\ell=\widetilde\Delta\) if \(\ell\mid n_0\), and \(a_\ell=1\) otherwise. The tuple \(\mathbf a=(a_\ell)_\ell\) has finite support and lies in \(G(\mathbb A_f)\). We seek one rational point \(g_{\mathrm{rat}}\in G(\mathbb Q)\) whose localizations \(g_{\mathrm{rat},\ell}\) differ from these selected corrections by automorphisms of the full source lattice: \[\rho(a_\ell^{-1}g_{\mathrm{rat},\ell}) (\Lambda_0\otimes\mathbb Z_\ell) =\Lambda_0\otimes\mathbb Z_\ell \quad\text{for every }\ell.\] To construct a suitable open neighborhood of \(\mathbf a\), define the exact source-lattice stabilizer \[C_\ell=\left\{\gamma\in G(\mathbb Q_\ell): \rho(\gamma)(\Lambda_0\otimes\mathbb Z_\ell) =\Lambda_0\otimes\mathbb Z_\ell\right\}.\] Each \(C_\ell\) is compact open. Here the spin action has finite central kernel, so it need not be faithful; compactness follows from finiteness of the covering onto its algebraic image. For the adelic topology we only need compact opens inside these stabilizers. Spread out \(G\) and its rational action on \(F_0\) to a smooth affine group scheme \(\mathcal G\) over \(\mathbb Z[1/N]\), enlarging \(N\) so that the action preserves \(\Lambda_0\otimes\mathbb Z[1/N]\). At every \(\ell\nmid N\) its standard compact open \(\mathcal G(\mathbb Z_\ell)\) lies in \(C_\ell\). For each of the finitely many primes \(\ell\mid N\), choose a compact open subgroup \(K_\ell\subset C_\ell\); at the other primes put \(K_\ell=\mathcal G(\mathbb Z_\ell)\). Then \(\mathcal K=\prod_\ell K_\ell\) is compact open in \(G(\mathbb A_f)\) and \(\mathbf a\mathcal K\) is an open coset contained in the desired set of local integral corrections.

At \(\ell\nmid n_0\), Lemma 19 makes \(J_0\) an integral lattice isometry. At \(\ell\mid n_0\) we have \(\ell\nmid n_1\), so \(J_0\rho(a_\ell)=J_1\) is integral. Consequently \[J_0\rho(a_\ell)(\Lambda_0\otimes\mathbb Z_\ell) =\Lambda_1\otimes\mathbb Z_\ell \quad\text{for every }\ell.\] Density supplies \(g_{\mathrm{rat}}\in G(\mathbb Q)\cap\mathbf a\mathcal K\). Writing \(g_{\mathrm{rat},\ell}=a_\ell k_\ell\) with \(k_\ell\in K_\ell\subset C_\ell\) shows that \(J_0\rho(g_{\mathrm{rat}})\) induces the same local lattice equality at every prime. A full \(\mathbb Z\)-lattice in a rational vector space is the intersection of its localizations at all primes; applying this to both sides yields \(J_0\rho(g_{\mathrm{rat}})\Lambda_0=\Lambda_1\). Since \(J_0\) and \(\rho(g_{\mathrm{rat}})\) preserve their respective Seifert forms, this is the asserted integral congruence. ◻

Proof of Theorem 1. Proposition 5 and Lemma 9 give smooth ambient fourfold germs and reduced isolated functions of orders five and four. Theorem 6 identifies their actual integral Milnor Seifert forms, and Proposition 22 gives an integral congruence. The Kato argument above identifies their embedded links. The analytic pair-cone construction above extends that equivalence to an ambient homeomorphism germ, proving the assertion. ◻