Proof. The group \(\pi_1(M)\) of the manifold in Theorem 1 is finitely presented because compact smooth manifolds have finite CW homotopy type. It is not good: otherwise (Powell et al. 2025, Theorem A), applied to the discs and spheres in (2), would supply the forbidden embedded discs.

Choose a finite generating list for \(\pi_1(M)\) of length \(r\geq2\), padding if necessary. There is then a surjection \(F_r\twoheadrightarrow\pi_1(M)\). The elements \(a^jba^{-j}\) for \(0\leq j<r\) freely generate a rank-\(r\) subgroup of \(F_2=\langle a,b\rangle\), as follows from reduced-word normal form. Subgroups and quotients of good groups are good (Powell et al. 2025, sec. 1.1, p. 4). If \(F_2\) were good, its subgroup \(F_r\) and then the quotient \(\pi_1(M)\) would be good, a contradiction. Finally, a good group containing a subgroup isomorphic to \(F_2\) would make \(F_2\) good by subgroup closure, which proves the last assertion. ◻

Proof. Together these assertions imply the unrestricted disc embedding conjecture (Kim, Powell, et al. 2021, sec. 3). Its algebraic hypotheses apply to the finite family of discs and framed spheres in the compact manifold of Theorem 1, so it would give the excluded boundary filling. ◻

Proof. In an end summand, deleting \(S^1\times\operatorname{int}D^2\) leaves \(S^1\times D^2\) with a ball removed. The old meridian bounds the complementary \(D^2\), whereas the old longitude generates its fundamental group. Van Kampen across the connected-sum spheres gives the displayed free product and proves connectedness.

All attachments are collared cofibrations, so the ordinary gluing computes the homotopy pushout. Replace \(V\) by its bouquet of handle circles, each \(N_a\) by its generator circle, and each \(P_e\) by a point. For an edge \(e\) the resulting pushout cones off the two independent circles \(z_{e,0},z_{e,1}\) to the same new point. Each of the two cone discs retracts onto an arc from the old bouquet point to this cone point, keeping both endpoints fixed. The two arcs together give one circle \(q_e\). The other circles are the three \(h\) circles. The only point of each boundary circle shared with the other summands is the old bouquet point, so these retractions are compatible. This proves the asserted graph homotopy type.

On \(V\) take the handle map retaining the \(h\) circles and constant on each \(N_a\). On \(P_e\) prescribe a function to \([0,1]\) equal to \(0\) and \(1\) on its two attaching tori and to the product coordinate on \(C_e\). The compatible boundary function extends over the ball: extend it as a real function and compose with the retraction of \(\mathbb R\) onto \([0,1]\). Compose with the edge \(q_e\), whose endpoints are the common vertex. The resulting map induces the graph identification just described, and is therefore a homotopy equivalence. All boundary values may be taken productwise in collars. ◻

Proof. Represent the path by an embedded arc disjoint from the knots except at its endpoints. Along a narrow finger following this arc move a short strand of the second knot to the first, stopping when the two strands meet transversely in a small two-dimensional patch. Before the final contact the move is an isotopy. Thicken the resulting crossed graph to an oriented ribbon with alternating cyclic order of the four half-edges at the crossing. Its regular neighborhood is a punctured torus. Choose the normal resolution determined by the pre-contact strands. Reversing that specific finger isotopy recovers the original two-component link. The boundary traverses the circuits in the order \(uvu^{-1}v^{-1}\), after a choice of orientation and basing. The two sides of the finger insert precisely the chosen change-of-basing path and its inverse.

The ribbon away from the crossing consists of separate bands. Insert full twists in either band, supported in a short segment away from the crossing and the other band. Take its supporting tube disjoint from the other resolved knot as well. Shrinking the displacement in this tube gives an ambient isotopy of the resolved link that fixes its other component. Thus a twist preserves the chosen two-component resolution, including its linking. The lateral ribbon vector rotates once per full twist, giving independent integer changes to the two knot framings.

Here is the normal-bundle calculation needed to transfer that integer to a cap attachment. Let \(t\) be tangent to a basis curve, \(b\) its lateral ribbon vector, \(n\) the normal to the ribbon in the level three-manifold, and \(v\) the positive collar direction. A rising annulus initially has tangent plane spanned by \(t\) and \(r=a n+c v\), with \(a^2+c^2=1\) and \(c>0\). Set \(w=c n-a v\). Its oriented normal quotient has frame \(([b],[w])\). Subtracting the \(v\) component divided by \(c\) times \(r\) identifies this quotient with the normal plane of the movie knot in the level slice, and gives \[ [b]\longmapsto[b],\qquad [w]\longmapsto[n/c]. \tag{6}\] Thus it carries exactly the ribbon framing, with a positive rescaling of the second vector. During one band twist the pair changes to \[b_\theta=\cos\theta\,b+\sin\theta\,n, \qquad n_\theta=-\sin\theta\,b+\cos\theta\,n,\] where \(\theta\) makes a full turn. Equation (6) has degree one in the oriented rank-two frame group, up to the chosen sign convention. This is an integer calculation in \(\pi_1(\operatorname{GL}^+(2,\mathbb R))\), rather than a calculation of the parity of an ambient three-frame.

The separation of the attaching annuli can be checked in local collar coordinates \((x,y,n,v)\), with the body in \(n=v=0\) and its basis curves the \(x\)- and \(y\)-axes. Choose the signs of these coordinates to match the selected normal resolution, and take the initial annuli to be \[(x,0,s,s),\qquad (0,y,-s,s),\qquad s\geq0.\] They meet on the body at \(s=0\) and are disjoint for \(s>0\). At a small positive level they give the chosen normal resolution; continue them by the inverse finger isotopy of the whole link. The band-twist isotopies just described are supported away from the other component, so their traces also preserve this disjointness. Figure 2 separates the body coordinates from the normal coordinates in this local model. Along a rising movie the same quotient identification transports the framing to the given knot. Consequently a full band twist changes by one the obstruction to extending a prescribed ordered normal frame over an upper cap disc. Killing that integer extends both vectors: an extended nonzero first vector and its oriented quarter-turn give a frame, whose second vector can be adjusted at the boundary through positive independent vectors. This also explains why the statement controls a prescribed boundary framing, not just abstract triviality of a disc bundle. ◻

Proof. Give each leaf occurrence its own parallel longitude knot on the \(Q\) side of its end torus, even when several occurrences have the same label. Reserve a separate return lane inside the corresponding solid torus. Each lane leads to its own framed core-disc parallel in \(P_e\). There are finitely many occurrences, so all lanes may be chosen disjoint within each end family.

Choose narrow, separated ranges of collar levels centered at \[0<t_B<t_S<t_T<\epsilon.\] The base-body operations occupy the range about \(t_B\), the inner-body operations the range about \(t_S\), and the transfers into the end solid tori the range about \(t_T\). Each body-operation range includes room above its body level for the rising resolution and inverse finger isotopy of Lemma 5. Between ranges, continuing knots run in narrow disjoint pipes; at an active range all inactive continuations remain in their pipes.

To trace one relation through these heights, its base boundary rises from level zero to \(B_r\) near \(t_B\). The shallow attachment at \(\alpha_r\) continues to the transfer range. The other attachment rises from \(\beta_r\) to the boundary of \(S_r\) near \(t_S\); the two attachments at \(x_r,y_r\) then rise through their own pipes to the transfer range. There every leaf moves from the \(Q\) side into its end solid torus and returns down its reserved lane in the torus interior to its cap in \(P_e\). These downward lanes lie outside the \(Q\)-side body neighborhoods. The construction order is upper stage first: after making \(S_r\), we carry its boundary down to a handoff knot at the upper side of the base-operation range. With this knot and the slot-\(1\) leaf fixed above the lower body level, we choose their connecting arc. The lower ribbon’s rising movie then joins the body to those two inputs. Thus the later base operation is below the earlier upper body and avoids the inactive leaf pipes.

For either body-operation range, once its input knots have been fixed, choose the connecting arcs with separate endpoints and the prescribed relative path classes in \(Q\). General position in this three-manifold makes the finitely many arcs embedded, mutually disjoint and disjoint from the other knots. No prescribed path class in a fixed link complement is needed. Small neighborhoods of these graphs accommodate the ribbons, isotopies and local twists.

Choose the connecting arcs for the slot-\(2,3\) pairs as disjoint arcs in \(Q\), and first apply Lemma 5 to every pair of leaf occurrences in slots \(2,3\), to make \(S_r\), avoiding all slot-\(1\) leaf knots. Attach the rising annuli to the leaf knots, and match the \(x_r\) band framing to the framed \(x_r\) cap. Cutting out an annulus along \(x_r\) leaves a pair of pants. Capping its two new boundary circles by nearby copies of the \(x_r\) cap gives a disc \(D_r\). The extended lateral normal field supplied by Lemma 5 places the two copies at the cut band sides. This disc is embedded: the retained body and its collars are disjoint, and the only cap pieces retained have the same end type. In particular, crossings with the unused \(y_r\) cap or with another leaf are not self-crossings of \(D_r\).

Return the boundary knot of \(S_r\) along an attaching collar to a lower level. Its contracting disc supplies a normal-framing class at this handoff. With the returned boundary knots now fixed, choose the lower connecting arcs and pair each knot with its slot-\(1\) leaf, again by Lemma 5, to make \(B_r\). Choose the \(\beta_r\) band twist so that the transported ordered frame extends over this already constructed contracting disc. At the chosen normal phase of \(\beta_r\times S^1\), the vector \(w\) in (6) is its angular tangent. Extension of the ordered frame therefore extends this particular angular vector, as required for compression of the product torus.

These choices are sequential. If \(n_x,n_\beta\) are the two twists, the two relative framing obstructions have the form \[o_x(n_x,n_\beta)=o_x(0,0)+\varepsilon_x n_x, \qquad o_\beta(n_x,n_\beta)=o_\beta(n_x,0) +\varepsilon_\beta n_\beta, \qquad \varepsilon_x,\varepsilon_\beta\in\{1,-1\}.\] The lower \(\beta_r\) twist does not change the fixed upper disc or its \(x_r\) framing. One first sets \(o_x=0\), records the resulting handoff framing, and then sets \(o_\beta=0\). There is no simultaneous parity assumption hidden in these choices.

Chosen changes of basing make the leaf circuits the conjugates in (5). Applying the commutator rule first to \(S_r\) and then to \(B_r\) gives the asserted words. Paths recording basings need not themselves be physical attachments from a single basepoint.

Finally extend the base boundaries down to level zero in \(Q\). This makes the \(B_r\) proper in \(V\). Choose the remaining small parallel displacements and radii after all finitely many bodies and collars have been fixed. Outside the attachment charts and the cap-crossing charts their compact disjoint pieces have positive separation; inside the charts use the product choices (4). This gives simultaneous smooth choices with exactly the asserted intersections. ◻

Proof. The chosen angular normal field extends over the embedded disc \(D_r\), so a narrow interval thickening of that disc replaces the annulus on \(T_r\) by its two disjoint parallel boundary discs. Away from the attaching curve the thickening misses \(T_r\) if it is sufficiently small. All cap sheets in this one sphere come from its own \(x_r\) cap and hence have the same end type. The disjoint-body construction and the local surgery model give embeddedness. The phase containing the shallow-cap intersection is disjoint from the annulus being replaced, so that point remains. The separation choices in Lemma 6 account for all other possible intersections.

The interval thickening is an oriented three-chain in \(M\) whose boundary is the difference between the original annulus and the two replacement discs. Consequently \([g_r]=[T_r]\) in \(H_2(M;\mathbb Z)\), up to a harmless orientation sign. The torus has an explicit trivial normal bundle, with one vector lateral to \(\beta_r\) in the base and one vector radial in the normal disc to the base. Thus its ordinary self-intersection is zero. Homology invariance of that pairing gives \(g_r\cdot g_r=0\). For an embedded oriented sphere this is the Euler number of its oriented normal plane bundle. Such a bundle on \(S^2\) is classified by its Euler number, so \(g_r\) has a disjoint normal pushoff. The pushoff computes zero in the full group-ring diagonal pairing. Embeddedness gives no double points at all in the Wall invariant, and in particular gives the stated reduced invariant. ◻

Proof. Use the product attachment collars to make the carrier and its comparison homotopy explicit in the exterior. Choose a common small radius \(\varepsilon\) for the deleted base tubes, smaller than all the finitely many collar lengths. Take \(T_s\) at radius \(2\varepsilon\). Shorten the initial collar of \(S_s\) at radius \(3\varepsilon\) and join it back to \(T_s\) by the radial annulus between these two radii; call this retained upper stage \(S_s\) as well. This collar shortening changes neither its two basis curves nor its fundamental group. All caps are retained. Choose subsequent parallel displacements smaller than \(\varepsilon/10\) and than the positive clearances from the other deleted tubes. The cap-parallel collapse and the band comparison below can be fixed on this radial collar, so their images stay outside every deleted tube.

For this retained upper stage of entry \(s\) form the abstract complexes \[K_x=S_s\cup_{x_s}D_x,\qquad K=S_s\cup_{x_s}D_x\cup_{y_s}D_y.\] Attaching annuli are included in the cap notation. The actual configuration gives a map \(j:K\longrightarrow M\) with the collar just specified. The caps may cross other cap images; none of those images has been deleted from \(M\). The presentations \[ \pi_1(K_x)=\langle x_s,y_s\mid x_s\rangle =\langle y_s\rangle, \qquad \pi_1(K)=\langle x_s,y_s\mid x_s,y_s\rangle=1 \tag{8}\] are presentations of these abstract complexes.

Collapse the small parallel displacement in the compressed disc \(D_s\) and reglue its cut \(x_s\) band. This gives a map \(c:D_s\longrightarrow K_x\) identifying corresponding points of its two cap copies with the same point of \(D_x\). The actual map of \(D_s\) into \(M\) is homotopic to \(j\circ c\) by the normal intervals of its cap parallels and the matching band collars. The homotopy stays in a neighborhood of the upper stage, away from every deleted base tube. It is a homotopy of maps and is allowed to cross the other cap images.

Fix one of the two outer copies of \(D_s\) in \(g_s\). From a point on its retained upper body choose paths \(\gamma_+,\gamma_-\) to corresponding crossings on its two inner cap sheets. In a crossing chart these sheets and the fixed other sheet are \[D^2\times\{a_+\},\quad D^2\times\{a_-\}, \quad \{b\}\times D^2.\] The short comparison path \(\tau(u)=(b,(1-u)a_++u a_-)\) is simultaneously a path on the fixed other sheet and the normal interval between the paired cap points. Choose the paths on the fixed sheet using this segment. Up to conjugation by the common path from the sphere’s whisker, the quotient of the two Wall labels is represented by \[\gamma_+\,\tau\,\gamma_-^{-1}.\] Under the collapse comparison this becomes \(c(\gamma_+)c(\gamma_-)^{-1}\) in \(K_x\): the segment \(\tau\) collapses to the single point of \(D_x\). The loop might represent a nonzero power of \(y_s\) in \(K_x\); Equation (8) kills it in \(K\), using the actual unused \(y_s\) cap. Composing its nullhomotopy with \(j\) proves equality of the labels in \(M\). Arbitrary conjugators are already part of the attaching paths; they do not alter the simple connectivity of \(K\).

The two inner cap copies cap the two boundary components of the annulus cut from \(S_s\), so have opposite induced orientations. Their signs against any fixed sheet are therefore opposite. Reversing the orientation of an outer contracting-disc copy reverses both signs and preserves this conclusion. Apply the pairing separately within each outer copy, and separately for each fixed cap sheet on the other surface. This accounts for every cap crossing, including repeated end labels and opposite end labels within one entry. In particular no comparison of paths between the two outer phases, and no contraction of a base meridian in \(M\), has been used. ◻

Proof. Lemmas 7 and 8 leave exactly one signed group element in \(\lambda(f_r,g_r)\) and no other contributions. Orient \(g_r\) and change its whisker so that this coefficient is \(+1\). Whisker changes multiply or conjugate zero entries by units and hence preserve the other vanishings. Triviality of the normal bundles proves the Stiefel–Whitney assertion. The manifold is oriented, so its orientation character is trivial and the involution on \(R\) is \(h\mapsto h^{-1}\). The reduced Wall invariant vanishes even before imposing its additive relations \(h-h^{-1}\) and the identity coefficient. The discs have disjoint embedded boundary circles and the proper collar model. There is no requirement to make their mutual intersections or self-intersection invariants vanish. ◻

Proof. Join the boundary of each actual embedded disc back to \(\alpha_r\) by the radial annulus in the deleted base tube. The bases and these return annuli lie in \(V\) and hence already map into \(\Gamma_h\). The boundary map of each disc lies in \(\Gamma_h\) and is nullhomotopic in \(\Gamma\) because the actual disc gives a filling. The inclusion \(\pi_1(\Gamma_h)\longrightarrow\pi_1(\Gamma)\) is injective, so it has a filling in \(\Gamma_h\). The original filling and this new filling are homotopic relative boundary since \(\pi_2(\Gamma)=0\). Transfer this homotopy to the disc image using the embedding parametrization. This step uses no homotopy between an output disc and its input cap.

Let \(B\) be the union of the bases, return annuli, and disc images, and put \(A=B\cup\partial G\). Prescribe the just constructed homotopies on the discs and keep the other parts fixed. These prescriptions agree on intersections: disc boundaries are fixed, disc interiors are mutually disjoint and proper in \(M\), and the only intersections with the returned pieces are the stated attaching circles. Pasting finitely many compact closed sets gives a continuous homotopy on \(A\). Only \(B\) is to map into \(\Gamma_h\); the \(q\)-collar portions of \(\partial G\) keep their prescribed product values.

Here the extension to \(G\) need not assume that the topological disc images are smooth or that their inclusion is a cofibration. A finite graph is an absolute neighborhood retract for metric spaces. Apply its neighborhood-extension property to the prescribed map on the closed subset \[(G\times\{0\})\cup(A\times[0,1])\ \subset G\times[0,1].\] It extends to an open neighborhood \(O\) of that subset. Compactness of \(A\times[0,1]\) gives a neighborhood \(U\) of \(A\) with \(U\times[0,1]\subset O\). Choose a continuous function \(\chi:G\longrightarrow[0,1]\) equal to one on \(A\) and supported in \(U\). If \(\widetilde H\) is the neighborhood extension, then \[H(x,t)=\widetilde H(x,\chi(x)t)\] is defined on all of \(G\times[0,1]\): when \(\chi(x)>0\) it uses \(U\times[0,1]\), and otherwise it uses the time-zero slice. It is the required extension, fixed on \(\partial G\). Its final map sends \(B\) into \(\Gamma_h\).

Choose mutually disjoint small open intervals about the midpoints of the \(q_e\) edges. Their closed smaller intervals miss \(\Gamma_h\), so their inverse images miss the protected compact set \(B\). On \(\partial G\) the map is already the fixed coordinate on the single collar \(C_e=T^2\times[0,1]\). In a small boundary collar straighten its real interval coordinate to be constant in the inward normal direction, using an interpolation fixed on the boundary. Use a slightly larger target interval to localize this operation away from \(B\). Relative smooth approximation of real functions, followed by relative transversality in the remaining interior, makes the map smooth and transverse near a smaller middle level, preserving the boundary product and staying off \(B\). The intervals for different edges have disjoint supports. These changes are homotopies within intervals, so the resulting \(p\) is still homotopic to \(p_0\) relative boundary. The regular-value and boundary regular-fiber statements are those of (Milnor 1965, sec. 2, pp. 10–13); the fixed product collar supplies regularity on the boundary.

Each level \(J_e\) is therefore a compact smooth hypersurface, cooriented by the interval and oriented by \(G\). Its boundary is exactly the boundary level \(T^2\times\{\mathrm{point}\}\) in \(C_e\); \(Q\) contributes no boundary point at that level. There is one component with this boundary torus, and every other component is closed. All the topological disc images and returned pieces remain disjoint from the cuts. ◻

Proof. The input assertions are Corollary 9. Use Lemma 10 to obtain \(p\) and \(J_e\). Since \(p\) is homotopic to \(p_0\), Lemma 4 makes it a homotopy equivalence. Cutting along the disjoint level hypersurfaces gives the stated collared manifold \(X\), whose corners can be rounded. Connectedness of \(X\) was not used and is not asserted. The connected region \(Q\) singles out one component \(X_Q\).

For each \(r\), the union of \(B_r\), its return annulus and its new disc misses every cut and meets \(Q\). It therefore lies in \(X_Q\). In that union the curve \(\alpha_r\) bounds the returned disc, so the base boundary, which represents \([\alpha_r,\beta_r]\), is nullhomotopic. Its word in \(Q\) is exactly \(R_r\), up to the already fixed orientation and basing conventions. Any overall conjugate or inverse has the same nullity, so all based relations in (5) hold in \(\pi_1(X_Q)\). This group argument does not assert that an entire base has been compressed to an embedded disc. The map to the cut graph and all the peripheral assertions follow directly from the fixed product boundary values. Pullback from the graph gives the last local-system assertion. ◻

Proof. Take the two-sheeted cover of \(G\) classified by the homomorphism sending every \(q_i\) to the nontrivial element of \(\mathbb Z/2\) and every \(h\)-loop to zero. The lift of \(p\) is a homotopy equivalence with the corresponding graph cover. Thus the cover has zero cohomology in degrees at least two, with the lifted local system as well as with constant coefficients. After adjoining small product collars, its decomposition has two vertex spaces, both copies of \(X\), and two edge spaces per label, both copies of \(J_i\). The pairings go from one vertex copy to the opposite side of the other, giving exactly (11). The relevant part of Mayer–Vietoris is \[H^1(\widetilde G)\longrightarrow H^1(X)^{\oplus2} \xrightarrow{\Delta}\bigoplus_i H^1(J_i)^{\oplus2} \longrightarrow H^2(\widetilde G)=0\] and \[0=H^2(\widetilde G)\longrightarrow H^2(X)^{\oplus2} \xrightarrow{\Delta}\bigoplus_i H^2(J_i)^{\oplus2} \longrightarrow H^3(\widetilde G)=0 .\] The omitted coefficients in these two displays are the specified local systems.

The image in the middle degree-one group is precisely \(W_h\oplus W_h\). In one direction, a class restricted from the graph cover factors on each vertex through its cut-open graph, and hence through the associated \(h\)-rose. Conversely, choose cellular one-cochains representing any two classes on the two \(h\)-roses. Extend them to the graph cover by arbitrary values, for example zero, on the remaining edges. There are no two-cells, so the extensions are cocycles. Their restrictions give the chosen pair of classes on \(X\). This argument uses direct sums over all components of \(X\), and imposes no connectedness assumption on it. ◻

Proof. Poincare–Lefschetz duality (Hatcher 2002, Theorem 3.43) and the exact sequence of the pair imply that the restriction image in the cohomology of the boundary of an oriented compact three-manifold is its own annihilator for the intersection form. For completeness, the connecting map \(H^1(\partial J_i)\to H^2(J_i,\partial J_i)\) is dual, up to sign, to the restriction map. Exactness therefore identifies the kernel of that connecting map, which is the restriction image, with the annihilator of that image. On a single torus a subspace equal to its annihilator is a line. Closed components of \(J_i\) make no contribution to the boundary map. Denote this line by \(L_i\).

Tensoring with \(\mathfrak l\) gives the corresponding assertion for the trivial wall system. A restriction from the positive or negative copy of \(X\) has zero evaluation on the slope which bounds in \(Q\), because the local system is trivial on that torus. These two constraints are the distinct coordinate axes interchanged by the Hopf gluing. If \(L_i\) were neither axis, both restrictions would have zero torus value. This contradicts surjectivity in Lemma 12, since \(H^1(J_i;\mathfrak l)\) has a nonzero boundary restriction. Thus \(L_i\) is one axis; designate the contributing end positive and call this the first slope.

The map \(H^0(J_i;\mathfrak l)\to H^0(\partial J_i;\mathfrak l)\) is onto: constants on the unique component with boundary suffice. The exact sequence consequently identifies \[ H^1(J_i,\partial J_i;\mathfrak l) \ \cong\ \ker\bigl(H^1(J_i;\mathfrak l) \longrightarrow H^1(\partial J_i;\mathfrak l)\bigr). \tag{12}\] These are the \(n_i\) internal directions, including all degree-one cohomology of every closed component. They pair perfectly with \(H^2(J_i;\mathfrak l)\) by integration and the trace form. In particular \(\dim H^1(J_i;\mathfrak l)=3+n_i\).

The isomorphism in Lemma 12 now gives \(\dim U=3d+D\). Its first-slope coordinates are the first-slope values of \(u\), and the negatives of those of \(v\), because all opposite-side values are zero. Surjectivity implies independent surjectivity onto the \(3d\) first-slope coordinates. On their kernel the same isomorphism becomes \[U_0\oplus U_0\longrightarrow\mathcal Y,\qquad (u,v)\longmapsto R(u)-\sigma R(v),\] where \(R\) is paired restriction and \(\sigma\) interchanges the two slots for each \(i\). This proves the last assertions. ◻

Proof. Use a finite good cover, with contractible connected nonempty intersections, obtained from a finite smooth triangulation and boundary collars. Components are covered independently. Fix reference parallel frames. For each oriented edge of the nerve, vary a matrix \(g_{\alpha\beta}\) in an algebraic group coordinate patch, impose \(g_{\beta\alpha}=g_{\alpha\beta}^{-1}\), and set \(g_{\alpha\alpha}=1\). For an oriented triangle use the traceless part of \[g_{\alpha\beta}g_{\beta\gamma}g_{\gamma\alpha}-1\] as its three equation coordinates. These coordinates generate the full matrix defect ideal locally. Indeed, write the triangle product as \((1+x)1+T\), where \(T\) is traceless and \(x(0)=0\). Its determinant equation reads \(x(2+x)=-\det T\). The factor \(2+x\) is a unit, so \(x\) belongs to the ideal generated by the entries of \(T\) (in fact to its square).

In reference frames, the linearized defect map is the cochain differential \[d_\rho:C^1(Z;\mathfrak l_{\rho_Z}) \longrightarrow C^2(Z;\mathfrak l_{\rho_Z})\] on the good-cover nerve. Choose decompositions \[C^1=\operatorname{im}d_\rho^0\oplus H\oplus L,\qquad \ker d_\rho^1=\operatorname{im}d_\rho^0\oplus H,\] and choose a projection \(\pi:C^2\to\operatorname{im}d_\rho^1\) which is the identity on that image. Set the first summand of the coordinate variables equal to zero. Solve \(\pi(\text{defect})=0\) for the \(L\)-variables. Its derivative in those variables is the isomorphism \(d_\rho^1:L\to\operatorname{im}d_\rho^1\). The algebraic implicit function theorem gives the solution in an etale neighborhood. Its remaining coordinates are \(H\), and its defect has no linear term. Finally restrict the \(H\)-parameters to the chosen linear subspace \(T\). The identity \(\pi(\text{defect})=0\) survives this restriction. Good-cover cohomology agrees with local-system cohomology: the cover and every intersection are acyclic for these locally constant coefficients. Thus these are the asserted tangent coordinates.

Write \(B=\mathbb C[[T]]\), \(\mathfrak r=\mathfrak r_Z\), and suppose \(\lambda\ne0\). Its inverse-image kernel is the ideal \[\mathfrak s =\{b\in\mathfrak r:\lambda(b\bmod\mathfrak m\mathfrak r)=0\}.\] It is an ideal because multiplication acts on \(\mathfrak r/\mathfrak m\mathfrak r\) by its constant term. There is an exact sequence \[ 0\longrightarrow I=\mathfrak r/\mathfrak s \longrightarrow B/\mathfrak s \longrightarrow B/\mathfrak r\longrightarrow0, \qquad \dim_{\mathbb C}I=1,\quad \mathfrak m I=I^2=0. \tag{13}\] Here \(\mathfrak r^2\subset\mathfrak m\mathfrak r\) proves the last equality. Evaluating the triangle defects in \(I\) gives a reference two-cochain \(c\) with \(\pi c=0\). It is nonzero, since the entries of the defects generate \(\mathfrak r\) and hence span its quotient by \(\mathfrak m\mathfrak r\).

A flat lift changes edge matrices by \(I\)-valued corrections. Since \(\mathfrak mI=0\), their effect on triangle products is the reference differential, so flatness would give \(c+d_\rho u=0\). Applying \(\pi\) gives \(d_\rho u=0\), and then \(c=0\), a contradiction. If the corrections are general-linear, project them to their traceless parts. Conjugation preserves the splitting into scalar and traceless matrices, so this projection commutes with \(d_\rho\); the same contradiction follows. If the putative flat lift is specified only up to an isomorphism of its reduction, choose parallel frames on the contractible patches with initial values lifting the given reduced frames. Their transition matrices then lift exactly the original matrices, which is the case just proved. ◻

Proof. Apply Lemma 15 to each \(J_i\) at its trivial reference representation, taking the full \(H^1\), in the tree frames of Remark 16. A fixed edge word for the first torus generator gives an algebraic matrix function even when the transition cochain has nonzero defects. By Lemma 13 its three group coordinates have independent linear terms. Replace three of the \(H^1\)-coordinates by these coordinates, preserving complementary \(y_i\)-coordinates. The algebraic inverse function theorem permits this change.

Apply Lemma 15 to \(X\), retaining \(U\). Restriction of a flat cochain to a wall, followed by tree normalization, is computed by finitely many based path words. One can arrange compatible refinements of the covers, or subdivide each of these finitely many paths into patches; in either description all resulting matrix expressions are finite products of transitions and their inverses. Taking the free coordinates of the wall chart therefore gives algebraic lifts of the restriction parameters. Over the quotient by \(\mathfrak r\) the solved coordinates reproduce the same cochain by the uniqueness just proved.

The joint positive first-slope values on \(U\) have surjective derivative. Use their actual group coordinates as \(a\), retaining complementary coordinates \(t\). On the negative side the first slope bounds in \(Q\), so its holonomy is exactly \(1\) modulo \(\mathfrak r\); its coordinate lift may thus be set equal to zero. This gives (14). Basing comparisons can also be lifted algebraically: they are products along finitely many paths and changes between chosen local frames. At reference all wall holonomies are \(1\), so a varying basing frame makes no first-order change to their conjugated holonomies. The tangent maps are consequently the ordinary restriction maps. Lemma 13 gives their asserted complementarity. ◻

Proof. Use the ordered basis \(E=\left(\begin{smallmatrix}0&1\\0&0\end{smallmatrix}\right)\), \(H=\left(\begin{smallmatrix}1&0\\0&-1\end{smallmatrix}\right)\), \(F=\left(\begin{smallmatrix}0&0\\1&0\end{smallmatrix}\right)\). The operator \(A=\operatorname{Ad}\rho(h_0)\) has distinct eigenvalues \(4,1,1/4\). Its three eigenprojections \(P_E,P_H,P_F\) are polynomials of degree at most two in \(A\). If \(B_\pm=\operatorname{Ad}\rho(h_\pm)\), then \[\begin{array}{lll} B_+E=E,&B_+H=H-2E,&B_+F=F+H-E,\\ B_-E=E-H-F,&B_-H=H+2F,&B_-F=F. \end{array}\] For every two distinct basis lines, one of \(P_iB_+P_j,P_iB_-P_j\) is a nonzero multiple of the corresponding matrix unit. These operators are linear combinations of \(A^aB_\pm A^b\) with \(0\leq a,b\leq2\), hence of adjoints of words of length at most five. The diagonal matrix units are the \(P_i\) themselves. Thus the adjoints indexed by \(\mathcal H\) span all nine matrix units. ◻

Proof. All computations initially take place in the Noetherian local ring \(B/\mathfrak r\). Let \(Z=z-1\) for an end matrix. The determinant equation gives \(\operatorname{tr}Z=-\det Z\in\mathfrak n^2\). Thus one may replace \(Z\) by its traceless part when computing triple products modulo \(\mathfrak m\mathfrak n^3\). Indeed the discarded products lie in \(\mathfrak n^4\subset\mathfrak m\mathfrak n^3\).

The formal group-commutator expansion is \[[1+A,[1+B,1+C]] =1+[A,[B,C]]+\text{terms of total degree at least four}.\] It follows by expanding each inverse as \((1+A)^{-1}=1-A+A^2-\cdots\); the inner commutator starts with \(1+[B,C]\), and its commutator with \(1+A\) first contributes \([A,[B,C]]\). In (16), replacing a conjugating matrix by its reference value changes the displayed cubic term only by \(\mathfrak m\mathfrak n^3\). Every relation therefore implies, in \(\mathfrak n^3/\mathfrak m\mathfrak n^3\), \[[\operatorname{Ad}\rho(c_1)Z_1, [\operatorname{Ad}\rho(c_2)Z_2, \operatorname{Ad}\rho(c_3)Z_3]]=0,\] with traceless \(Z_j\). The independent choice of all three conjugators, followed by linear combination and Lemma 18, replaces the three adjoint maps by arbitrary endomorphisms of \(\mathfrak l\).

For any three scalar linear functionals \(\ell_1,\ell_2,\ell_3\) on \(\mathfrak l\), choose these endomorphisms as \[x\longmapsto\ell_1(x)E,\qquad x\longmapsto\ell_2(x)H,\qquad x\longmapsto\ell_3(x)F,\] where \[E=\begin{pmatrix}0&1\\0&0\end{pmatrix},\quad H=\begin{pmatrix}1&0\\0&-1\end{pmatrix},\quad F=\begin{pmatrix}0&0\\1&0\end{pmatrix}.\] Since \([E,[H,F]]=-2H\ne0\), every product \(\ell_1(Z_1)\ell_2(Z_2)\ell_3(Z_3)\) vanishes in this quotient. The relation list permits repeated ends, so this includes all generators of \(\mathfrak n^3/\mathfrak m\mathfrak n^3\). Hence \(\mathfrak n^3=\mathfrak m\mathfrak n^3\); Nakayama’s lemma, applied to the finitely generated ideal \(\mathfrak n^3\), proves (17).

Under either restriction, the wall defects vanish modulo \(\mathfrak r\). Each slope holonomy is either \(1\) or the holonomy of its end longitude, with a possible inverse and a change of basing. Entries of its deviation therefore belong to \(\mathfrak n\). This remains true for inverses, since \(z^{-1}-1=-z^{-1}(z-1)\), and for conjugation. The images of all the \(\mathfrak k_i\) are consequently contained in \(\mathfrak n\), proving the final statements. ◻

Proof. Choose smooth functions \(\phi_\alpha\) subordinate to the good cover, with support contained in their patches and \(\sum_\alpha\phi_\alpha^2=1\). In a sufficiently large trivial bundle form the block matrix \[e_{\alpha\beta}=\phi_\alpha\phi_\beta g_{\alpha\beta}.\] Terms with disjoint supports are zero; support containment makes the extensions by zero smooth. The triangle identities imply \(e^2-e\in\mathfrak d\). Put \(\Delta=e^2-e\) and define \[ E=\frac{1+(2e-1)(1+4\Delta)^{-1/2}}2 . \tag{19}\] The binomial series is defined formally because \(\Delta\in\mathfrak m\); indeed our charts have \(\mathfrak d\subset\mathfrak m^2\). All factors commute with \(e\), and \((2e-1)^2=1+4\Delta\), so \(E^2=E\). Moreover \(E=e\) modulo \(\mathfrak d\). Its image has rank two, as its reference image does.

On a patch indexed by \(\alpha\), set \[C_\alpha=(\phi_\beta g_{\beta\alpha})_\beta,\qquad R_\alpha=(\phi_\beta g_{\alpha\beta})_\beta,\qquad S_\alpha=EC_\alpha .\] The first is a block column, the second a block row. One has the exact identity \[ R_\alpha C_\alpha =\sum_\beta\phi_\beta^2g_{\alpha\beta}g_{\beta\alpha} =1. \tag{20}\] Also \(eC_\alpha=C_\alpha\) modulo \(\mathfrak d\), so \[T_\alpha:=R_\alpha S_\alpha-1\in\mathfrak d,\qquad L_\alpha=(1+T_\alpha)^{-1}R_\alpha\big|_{\operatorname{im}E}.\] These give a frame and its inverse on the rank-two image bundle. For example \(L_\alpha S_\alpha=1\), and a rank-two frame with this left inverse is an isomorphism onto the rank-two image. In the reduction, the frames satisfy \(S_\beta=S_\alpha g_{\alpha\beta}\) on intersections.

Choose a subordinate partition \((\chi_\alpha)\) and blend the connections which are trivial in these frames: \[ \nabla s=\sum_\alpha\chi_\alpha S_\alpha\,d_{\mathrm{tot}}(L_\alpha s). \tag{21}\] This is a connection since each summand satisfies the Leibniz rule and \(\sum\chi_\alpha=1\). In the reduction modulo \(\mathfrak d\) its fiber part is exactly the flat connection defined by the constant-in-fiber transitions. Its fiber curvature therefore has coefficients in \(\mathfrak d\). Using the full differential in (21) also supplies a total connection, regardless of the parameter dependence of the reduced transition matrices.

If a reference global frame is given, view it as a matrix of sections of \(\operatorname{im}E(0)\), extend its coefficients constantly in the parameters, and apply \(E\). The resulting sections reduce to that frame, and hence are a formal global frame. Inverses exist by the maximal-ideal Neumann expansion. ◻

Proof. Temporarily write \(U,V,\mathfrak d\) for the wall data. Modulo \(\mathfrak d\) the two torus holonomies commute. Their formal logarithms commute as well: they are power series in the two commuting matrices. On the simply connected cover of the collar, take the flat parallel frame \(P\) issuing from the prescribed based frame. In the chosen convention the frame \[P\exp(\theta U+\eta V)\] is periodic, since the exponential cancels the deck holonomies \(\exp(-U)\), \(\exp(-V)\). Its fiber connection form is \(U\,d\theta+V\,d\eta\).

To lift it, use finitely many small collar patches with lifted torus coordinates. On each patch a lift of \(P\) is expressed using an image frame \(S_\alpha\) and a fixed finite transition word. Multiply by the displayed exponential. These local lifted frames have the same reduction modulo \(\mathfrak d\); their partition-of-unity average is therefore a formal frame with that reduction. In this frame prescribe (22) near the boundary, including its zero parameter components, and use a collar cutoff to interpolate with the previous total connection. Their fiber reductions are the same flat connection. Thus the interpolation preserves the reduction and keeps the fiber curvature in \(\mathfrak d\). This reasoning does not require the off-equation matrices \(U,V\) to commute.

At the origin the representation on each wall component is trivial, and the based choices make the collar frame the restriction of a global parallel frame. Lift that frame globally as in Lemma 20. On a collar its discrepancy from the prescribed frame is a matrix \(G=1+O(\mathfrak m)\). Multiplication of the global frame by \(1+\chi(G-1)\), extending coefficients by zero using a collar cutoff \(\chi\), makes it agree with the prescribed frame on a shorter collar. This matrix is formally invertible. The resulting global frame has the asserted properties. Closed wall components require only the initial global parallel frame and its formal lift. ◻

Proof. Differentiate the integrand in (23). Cyclic trace, with the signs from degrees of forms, gives \[\delta\operatorname{tr} \left(A\wedge dA+\frac23 A^3\right) =2\operatorname{tr}(\delta A\wedge F_A) -d\operatorname{tr}(A\wedge\delta A).\] On the torus the integrated boundary term is \[-\epsilon_i\operatorname{tr} (U_i\,\delta V_i-V_i\,\delta U_i).\] Adding the variation of \(\epsilon_i\operatorname{tr}(U_iV_i)\) gives (24). The fiber curvature is in \(\mathfrak d_i\subset\mathfrak m_i^2\), and \(V_i\in\mathfrak m_i^2\). Thus each first derivative of \(P_i\) has order at least two. Since \(A_i(0)=0\), also \(P_i(0)=0\), which proves \(P_i\in\mathfrak m_i^3\). Finally \(\mathfrak d_i\) and \(V_i\) are contained in the completed ideal \(\mathfrak k_i\); integration preserves that ideal by Definition 14. ◻

Proof. Apply Lemma 20. The reference bundle comes from a graph; a complex rank-two bundle with connected structure group on a graph is trivial, by choosing a frame on its vertices and extending over its edges. Its pullback to each component of \(X\) is therefore trivial. A continuous reference frame may be smoothly approximated in the smooth bundle, preserving invertibility, and then lifted formally.

Modulo \(\mathfrak r\), the special-linear transitions give a parallel determinant section. Lift it to a nonvanishing formal section of the determinant bundle; local coefficient lifts and a partition suffice, and the reference section is nonzero. If its induced determinant connection has one-form \(\alpha\), then \(\alpha\in\mathfrak r\), because the reduced section is parallel. Subtract \(\frac12\alpha\,1\) from the rank-two connection. This preserves its reduction and makes the determinant section parallel, so the resulting curvature has trace zero. ◻

Proof. In a \(y_i\)-variation, \(U_i\) is fixed. Thus the boundary term of (24) is zero. After either substitution \(\iota_i^\pm\), the wall curvature belongs to \(\mathfrak r\), since the restriction is exactly flat modulo \(\mathfrak r\). All the derivatives in (26) consequently belong to \(\mathfrak r\).

It remains to prove that their images span \(\mathfrak r/\mathfrak m\mathfrak r\). Suppose a nonzero functional \(\lambda\) on this space annihilates all those images. Form its small extension (13), so that \(I=\mathfrak r/\mathfrak s\) is one-dimensional and \(\mathfrak mI=I^2=0\). Reduce the connection from Lemma 24 to \(B/\mathfrak s\). Its curvature is an \(I\)-valued traceless two-form. Since multiplying \(I\) by a positive-order parameter coefficient gives zero, Bianchi becomes the reference equation \(d_{A_X(0)}F_{A_X}=0\). The curvature thus defines \[\kappa\in H^2(X;\mathfrak l_\rho)\otimes_{\mathbb C} I.\]

The wall connections and the restrictions of \(A_X\) are isomorphic modulo \(\mathfrak r\), by the based restriction construction. Lift a smooth formal identification of their bundles over \(B/\mathfrak s\). In that identification their difference is an \(I\)-valued one-form \(\eta\). Curvature changes by exactly \(d_{A(0)}\eta\): its quadratic term is zero since \(I^2=0\), and the positive-order part of \(A\) acts trivially since \(\mathfrak mI=0\). Thus the wall curvature classes are the restrictions of \(\kappa\), in the common reference frames.

At the wall origin a \(y_i\)-derivative \(\partial_{y_{i,j}}A_i(0)\) is a closed one-form, because wall curvature has order at least two. Its tangential boundary value is zero: \(U_i\) does not vary with \(y_i\), and \(V_i\) has zero linear term. Its absolute cohomology class is the corresponding chart tangent, as is seen by first-order holonomy. Hence these forms represent all of \(H^1(J_i,\partial J_i;\mathfrak l)\), by (12). If a general-linear frame adds a scalar exact term, traceless projection has the same \(\mathfrak l\)-class and the same pairing with the traceless curvature. The assertion includes every closed wall component.

Now apply \(\lambda\) to the internal derivatives. The polarized variation formula reduces to the nonzero scalar \(2\) times the trace integration pairing of these relative one-forms with the wall restrictions of \(\kappa\), with an irrelevant overall minus sign on negative copies. Poincare–Lefschetz duality (Hatcher 2002, Theorem 3.43) gives a perfect pairing of these relative classes with \(H^2(J_i;\mathfrak l)\): the manifolds are compact oriented three-manifolds and the trace form identifies the coefficient system with its dual. Thus every wall restriction of \(\kappa\) vanishes. Apply the degree-two isomorphism of Lemma 12 to the pair \((\kappa,0)\). Its image is zero, so \(\kappa=0\) on every component of \(X\).

De Rham cohomology with a flat smooth coefficient bundle computes this local-system cohomology. Consequently there is a traceless reference one-form \(\eta_X\), with values in \(I\), satisfying \[d_{A_X(0)}\eta_X=-F_{A_X}.\] Then \(A_X+\eta_X\) is exactly flat over \(B/\mathfrak s\); all omitted nonlinear terms are zero by \(\mathfrak mI=I^2=0\). It has the prescribed reduction over \(B/\mathfrak r\).

On every contractible patch choose the original parallel frame of that reduction, namely the one realizing its chart transitions. Lift its initial value at a point and parallel-transport with the corrected connection. Uniqueness of parallel transport makes the reduced frame equal to the original specified frame. The lifted constant transition matrices therefore give a flat lift of the actual vertex cochain, not merely of an unbased isomorphism class. This contradicts Lemma 15. The contradiction does not require the corrected connection to remain in the chosen \(U\)-slice; the equation-chart lemma excludes arbitrary cochain corrections. No nonzero annihilating functional exists.

The quotient \(\mathfrak r/\mathfrak m\mathfrak r\) is finite-dimensional. Its derivative images therefore span, and Nakayama’s lemma applied to the quotient of \(\mathfrak r\) by the derivative ideal proves (26). ◻

Proof. For the first assertion put \(A_u=A+u\eta\), \(0\leq u\leq1\). Its curvature \(F_A+u\,d_A\eta+u^2\eta\wedge\eta\) lies in \(I\). The variation formula on the closed manifold gives \[\operatorname{CS}_N(A+\eta)-\operatorname{CS}_N(A) =2\int_0^1\int_N\operatorname{tr}(\eta\wedge F_{A_u})\,du \in I^2.\] For the second use \(A^g=g^{-1}Ag+g^{-1}d_Ng\). For a parameter variation set \(\xi=g^{-1}\delta g\). Then \[\delta A^g=g^{-1}(\delta A)g+d_{A^g}\xi,\qquad F_{A^g}=g^{-1}F_Ag.\] Bianchi and the closedness of \(N\) give \[\delta\bigl(\operatorname{CS}_N(A^g) -\operatorname{CS}_N(A)\bigr) =2\int_N d_N\operatorname{tr}(\xi F_{A^g})=0.\] In characteristic zero a formal series with zero first derivatives is constant. This argument permits any reference homotopy class of \(g\); its possible degree affects only that constant. ◻

Proof. All wall connections in this proof are first pulled back by their maps \(\iota_i^\pm\) to \(B=\mathbb C[[a,t]]\). Their curvature lies in \(\mathfrak r\). On a positive wall the second torus slope bounds in \(Q\), so \(V_i(a_i,Y_i^+)\in\mathfrak r\). Choose a fiber cutoff \(\chi\) supported on its collar, equal to one near the boundary, and put \[A_{i,u}=A_i-u\chi V_i\,d\eta_i,\qquad 0\leq u\leq1 .\] The connection change lies in \(\mathfrak r\), and every interpolated curvature also lies in \(\mathfrak r\). This is immediate from the curvature change formula and does not assume commutation away from the defect quotient. Use the polarized functional along this path: \[P_{i,u}=\operatorname{CS}_{J_i}(A_{i,u}) +\epsilon_i\operatorname{tr}(U_i(1-u)V_i).\] Here \(U_i\) is held fixed. Formula (24) has zero boundary term for the \(u\)-variation; its bulk term is a product of two \(\mathfrak r\)-valued factors. Thus \(P_{i,1}-P_{i,0}\in\mathfrak r^2\). At \(u=1\) the polarization correction is zero and the collar form is the pure form \(U_i\,d\theta_i\). On a negative wall \(U_i(0)=0\) already, so its functional is raw Chern–Simons and its collar form is the pure form \(V_i(0,Y_i^-)\,d\eta_i\). It follows that, modulo \(\mathfrak r^2\), \(\Psi(a,Y)\) is the oriented sum of the raw wall Chern–Simons integrals of these modified connections.

We next construct an exactly flat formal connection on \(Q\) with those pure collar forms. Its reduction must be the actual restriction from the flat vertex family, including based transports. Fix the standard paths from the main basepoint of \(Q\) to the seam basepoints. Lift their comparison transport matrices from \(B/\mathfrak r\) to \(B\). Assign each free end generator the required pure collar holonomy, conjugated by its lifted comparison transport and adjusted for its orientation. The other slope at that seam bounds in \(Q\). Lift the matrices of the \(h\)-generators from the vertex restriction arbitrarily in \(\operatorname{SL}_2(B)\). Such lifts exist because the group is smooth at every reference matrix. Since the end generators and \(h\)-generators freely generate \(\pi_1(Q)\), these choices impose no relations and define a representation over \(B\) lifting the specified one over \(B/\mathfrak r\). Its associated flat bundle has an exactly flat connection. Choose its frame at each seam basepoint using the lifted comparison transport; parallel transport followed by a one-slope exponential gives the prescribed pure connection form on that collar.

These collar frames extend to a global frame on \(Q\). At the origin this is a relative section problem for a special-linear bundle on a compact three-manifold, with prescribed section on its boundary collars and at its basepoint. The group \(\operatorname{SL}_2(\mathbb C)\) retracts onto \(\operatorname{SU}_2\cong S^3\), so its \(\pi_0,\pi_1,\pi_2\) vanish. The obstruction to extending a section across a relative cell of dimension \(j\leq3\) lies in \(\pi_{j-1}\) of this group; all such obstructions vanish. A smooth reference extension follows by relative smooth approximation. At successive formal orders the discrepancy is identity modulo the maximal ideal and extends coefficientwise with a cutoff, just as in Lemma 21. This yields a global formal frame agreeing with every prescribed collar frame.

In this frame \(\operatorname{CS}_Q\) is constant in the parameters. Its variation has zero bulk term by exact flatness. On each torus both its connection form and its variation lie along the same single one-form \(d\theta_i\) or \(d\eta_i\), so \(\operatorname{tr}(A_Q\wedge\delta A_Q)\) is zero. The boundary term of the raw Chern–Simons variation therefore also vanishes.

Glue these framed pieces to form a connection \(A_{\partial}\) in a global frame on the closed three-manifold \(\partial X\). The forms agree on the product collars, so their integrals add. We must compare the assembled flat reduction with the actual restriction from \(X\), including its transports. The component \(X_0=X_Q\) containing \(Q\) contains all the torus-bearing wall copies in its boundary. Each such copy attaches to \(Q\) along exactly one connected torus. The incidence graph has one \(Q\)-vertex and one leaf for each such wall copy. It has no cycles. Fix the flat comparison on \(Q\), including its seam-basepoint values. On a wall its based representation is the prescribed restriction, so choose the flat comparison with that same value at its single seam basepoint. Parallel transport makes the two comparisons agree on the entire connected torus. There is no second attachment imposing an additional transport condition. Every closed wall component is an isolated boundary piece for this purpose and is compared separately, whether it belongs to \(X_0\) or to another component of \(X\). Thus the assembled reduction is isomorphic to the full boundary restriction of the vertex family.

Lift this boundary bundle isomorphism formally. In global frames this amounts to lifting smooth matrix coefficients; the reference determinant is nonzero, so the lifted matrix has a formal inverse. No extension of this identification over \(X\) is required. Denote by \(g\) the resulting boundary gauge comparison with the already chosen global frame of \(A_X\). Then in a common boundary frame \[A_{\partial}=A_X^g|_{\partial X}+\eta, \qquad \eta\in\mathfrak r .\] Both compared curvatures are in \(\mathfrak r\). The first part of Lemma 26 shows that this connection change contributes only an element of \(\mathfrak r^2\). The second part shows that the separate frame change contributes a parameter-constant term, which disappears after subtracting reference values. The gauge \(g(0)\) may have nonzero degree; it is not asserted to extend over \(X\).

Finally apply Stokes in the global frame already chosen on \(X\). The identity \[d\,\operatorname{tr} \left(A_X\wedge dA_X+\frac23 A_X^3\right) =\operatorname{tr}(F_{A_X}\wedge F_{A_X})\] follows by the graded cyclic trace rule. Hence \[\operatorname{CS}_{\partial X}(A_X|_{\partial X}) =\int_X\operatorname{tr}(F_{A_X}\wedge F_{A_X}) \in\mathfrak r^2.\] Stokes applies componentwise to the compact oriented smooth four-manifold with its rounded collared corners, including any components with empty boundary. The \(Q\)-contribution is constant, all \(P_i(0)\) are zero, and the preceding connection and gauge comparisons hold after subtracting the origin values. These facts prove (27). ◻

Proof. Fix a wall \(Z=J_i\), put \(v=(a_i,y_i)\), and write \(B=\mathbb C[[v]]\), \(\mathfrak d=\mathfrak d_i\), and \(\mathfrak k=\mathfrak k_i\). Choose the common etale chart ring \(A^*\) once, containing the finite transition cochain, its defects, and the two holonomy matrices \(H_{i,1},H_{i,2}\).

The global frame defining \(P_i\) has only formal control in the maximal ideal; this does not give algebraic coefficients modulo powers of \(\mathfrak k\). We first express the gradient without derivatives of that frame, then control the local expressions that remain.

Suppress the wall index. In the global frame defining \(P\), write the total connection as \[\nabla=d_Z+d_{\mathrm{param}}+A+\Gamma .\] Here \(A\) and \(\Gamma\) are its fiber and parameter components. For a parameter vector \(\delta\), define the mixed curvature with the parameter argument first: \[ K_\delta=F_{\mathrm{tot}}(\delta,\,\cdot\,) =\delta A-d_A\Gamma_\delta . \tag{28}\] Bianchi’s identity \(d_AF_A=0\) and trace invariance give \[\operatorname{tr}(d_A\Gamma_\delta\wedge F_A) =d_Z\operatorname{tr}(\Gamma_\delta F_A).\] Consequently the exact polarized variation is \[\begin{align*} \delta P &=2\int_Z\operatorname{tr}(K_\delta\wedge F_A) +2\int_{\partial Z}\operatorname{tr}(\Gamma_\delta F_A) +2\epsilon\operatorname{tr}(V\,\delta U) \\ &=2\int_Z\operatorname{tr}(K_\delta\wedge F_A) +2\epsilon\operatorname{tr}(V\,\delta U). \tag{29}\end{align*}\] The second equality holds because the global frame agrees with the prescribed collar frame for every parameter, and the total connection has \(\Gamma_\delta=0\) there. Both curvature components in the bulk trace transform by conjugation under every parameter-dependent frame change. We can therefore compute that trace in the local image and collar frames of Lemmas 20 and 21, without differentiating the global gauge.

We now check that the coefficients needed in (29) can be computed in \(A^*\) to any prescribed \(\mathfrak k\)-adic order. On each smooth coordinate patch we maintain the following property: modulo \(\mathfrak k^b\), each required coefficient is a finite sum \[ \sum_{\nu=1}^{N_b} f_\nu(x)\,a_\nu(v), \qquad f_\nu\text{ smooth},\quad a_\nu\in A^* . \tag{30}\] The functions, constants, and number of summands may depend on \(b\); the complex algebra \(A^*\) does not. We verify this for every operation actually used.

Recall the image-frame construction: \(S_\alpha=EC_\alpha\), \(T_\alpha=R_\alpha S_\alpha-1\in\mathfrak d\), and \(L_\alpha=(1+T_\alpha)^{-1}R_\alpha|_{\operatorname{im}E}\). The block matrix \(e_{\alpha\beta}=\phi_\alpha\phi_\beta g_{\alpha\beta}\) has the finite-sums property exactly. Since \(\Delta=e^2-e\in\mathfrak d\subset\mathfrak k\), the binomial series in (19) truncates after finitely many terms modulo \(\mathfrak k^b\). The explicit inverse (20) is equally important. In the notation of Lemma 20, \[(1+T_\alpha)^{-1} \equiv\sum_{j=0}^{b-1}(-T_\alpha)^j \pmod{\mathfrak k^b}.\] Thus image frames and their inverses satisfy (30); no smooth partition function has been inverted. In frame \(S_\beta\), the total connection (21) has matrix \[\sum_\alpha\chi_\alpha (L_\beta S_\alpha)\, d_{\mathrm{tot}}(L_\alpha S_\beta).\] This expression introduces only the already specified inverse matrices, multiplication, and differentiation.

Next consider the averaged collar frame. Its local lifts have the form \[T_\gamma=S_{\alpha(\gamma)}W_\gamma \exp(\theta_\gamma U+\eta_\gamma V),\] where \(W_\gamma\) is a fixed transition word. Both \(U,V\) belong to \(\mathfrak kB\), since the corresponding matrix deviations generate part of \(\mathfrak k\). The logarithms and this exponential consequently truncate to finite expressions modulo each \(\mathfrak k^b\). Write \(T=\sum_\gamma\chi_\gamma T_\gamma\) for the average. On a fixed small patch choose one participating lift \(T_0\). All the lifts agree modulo \(\mathfrak d\); hence \[T-T_0=\sum_\gamma\chi_\gamma(T_\gamma-T_0) \in\mathfrak d.\] Modulo \(\mathfrak k\), the exponential is \(1\). An explicit leading inverse on that patch is therefore \[V_0=W_0^{-1}R_{\alpha(0)} \big|_{\operatorname{im}E}.\] It satisfies \(V_0T=1+N\) with \(N\in\mathfrak k\), and the inverse of the average is \[ T^{-1}\equiv \left(\sum_{j=0}^{b-1}(-N)^j\right)V_0 \pmod{\mathfrak k^b}. \tag{31}\] This proves the finite-sums property for the average and its inverse. Agreement of its reductions, not merely invertibility of its summands, is what permits this calculation. The collar interpolation uses only fixed smooth cutoffs and these expressions, so it preserves the same property.

The coordinate derivations of the affine parameter space extend uniquely to its etale algebra and preserve its local ring \(A^*\). In an implicit presentation \(\mathcal F(v,w)=0\), differentiation gives \[\partial_{v_j}w =-\bigl(\partial_w\mathcal F(v,w)\bigr)^{-1} \partial_{v_j}\mathcal F(v,w).\] The variable Jacobian determinant is a unit germ in \(A^*\), because its value at the origin is nonzero. Its inverse therefore already belongs to this local etale ring. The product rule gives \[ \partial_{v_j}(\mathfrak k^{b+1}B) \subset\mathfrak k^bB. \tag{32}\] Fiber derivatives preserve scalar ideal order. Accordingly, compute primitive frame and logarithm expressions one ideal order higher when taking a parameter derivative.

The fiber connection and fiber curvature use only fiber derivatives of primitive frame expressions. The parameter connection component uses one parameter derivative, and \(K_\delta\) uses at most one: its two terms are a parameter derivative of \(A\) and a fiber covariant derivative of \(\Gamma_\delta\). Thus calculations modulo \(\mathfrak k^{b+1}\), followed by (32), suffice to compute every term of (29) modulo \(\mathfrak k^b\). The torus term is likewise controlled by finite logarithm expansions one order higher.

Finally take a finite partition to integrate the local expressions. The integral of (30) is \(\sum_\nu(\int f_\nu)a_\nu\), which belongs to \(A^*\) because every integration constant is a complex scalar. For the remainder, work modulo \(\mathfrak m^N\). It has values in the finite-dimensional scalar subspace which is the image of \(\mathfrak k^b\), and integration preserves that subspace. The integrated remainder therefore lies in \(\mathfrak k^b+\mathfrak m^N\) for every \(N\). Ideals of a complete Noetherian local ring are adically closed, so it lies in \(\mathfrak k^b\). This justifies the assertion with the completed smooth coefficient modules actually in use. ◻

Proof. Only the last item remains to be checked. Work modulo \(\mathfrak r\), where the end-deviation ideal \(\mathfrak n\) satisfies \(\mathfrak n^3=0\). Each \(a_{i,1}\) is an algebraic coordinate of an end holonomy at the identity, and hence belongs to \(\mathfrak n\). Evaluate (24) with the \(a_{i,1}\)-variation on the negative copy, at \((0,Y_i^-(a,t))\). Its curvature term vanishes modulo \(\mathfrak r\), while its torus term is \[2\epsilon_i\operatorname{tr} \left(V_i(0,Y_i^-)\, \partial_{a_{i,1}}U_i(0)\right).\] The remaining second-slope matrix on that copy is the end holonomy, with the prescribed orientation and basing. Its negative logarithm consequently has entries in \(\mathfrak n\), so \(p_i(Y_i^-)\in\mathfrak n\). Both kinds of triple products vanish modulo \(\mathfrak r\). Proposition 25 identifies that ideal with the derivative ideal, which gives (34). ◻

Proof. Choose a minimal prime of \(S\) not containing \(b\) and pass to the corresponding complete local domain \(S_0\). Its dimension \(r\) is positive. Choose \(r-1\) elements \(x_2,\ldots,x_r\) such that \((b,x_2,\ldots,x_r)\) is primary to the maximal ideal. Such a choice is possible because \(b\ne0\) in a domain and \(\dim S_0/(b)\leq r-1\). A minimal prime \(\mathfrak q\) over \((x_2,\ldots,x_r)\) has height at most \(r-1\), so it is not maximal. It cannot contain \(b\), since otherwise it would contain the displayed maximal-primary ideal. In \(S_1=S_0/\mathfrak q\) the ideal \((b)\) is maximal-primary. The principal ideal theorem and nonmaximality of \(\mathfrak q\) give \(\dim S_1=1\). These dimension and height steps use the Noetherian local dimension theorems (The Stacks Project Authors 2026, Tags 00KQ, 0BBZ, and 00KW).

The continuous map \(\mathbb C[[b]]\longrightarrow S_1\) is injective: a nonzero one-variable series is a power of \(b\) times a unit. It is finite. Indeed \(S_1/(b)\) is a finite-dimensional complex vector space; lift a basis, and successively expand remainders in powers of \(b\). Completeness, together with equivalence of the \((b)\)-adic and maximal ideal topologies, expresses every element as a \(\mathbb C[[b]]\)-linear combination of those finitely many lifts. The normalization of the complete discrete valuation ring \(\mathbb C[[b]]\) in the finite fraction-field extension \(\operatorname{Frac}(S_1)\) is a finite complete discrete valuation ring. Its residue field is finite over \(\mathbb C\), hence equals \(\mathbb C\). These conclusions use the complete-DVR extension theorem, applicable because the fraction-field extension is finite and separable, and completeness of finite algebras (The Stacks Project Authors 2026, Tags 09E8 and 0325). Choosing a uniformizer identifies the normalization with \(\mathbb C[[u]]\) as a \(\mathbb C\)-algebra: subtract successive complex residue coefficients and use completeness (The Stacks Project Authors 2026, Tag 0C0S(2)). Since \(S_1\) is integral over \(\mathbb C[[b]]\), it embeds in this normalization, locally and continuously. The resulting arc detects \(b\).

For the consequence, suppose the class of \(P-P(0)\) in \(B/I\) is nonzero. An arc as above detects it. If \(v_j\) maps to \(v_j(u)\in(u)\), the formal chain rule gives \[\frac{d}{du}P(v(u)) =\sum_j(\partial_{v_j}P)(v(u))\,v_j'(u)=0.\] In characteristic zero a one-variable series with zero derivative is constant. Its constant is \(P(0)\), contradicting detection. ◻

Proof. We first descend a finite Taylor tuple of \(P\) along the reduced support of \(K\) to the Taylor tuple of a single element of \(A\). We then choose the length of that tuple so that a function with zero tuple belongs to \(K^m\), including at embedded primary components.

If \(K=A\) there is nothing to prove. Otherwise put \(I=\sqrt K\). The reduced ring \(A/I\) is essentially of finite type over \(\mathbb C\), hence excellent. Its completion is reduced: excellence makes the completion map regular, and reducedness ascends along a regular map. Also completion of the quotient identifies this completion with \(B/IB\). These results apply to a Noetherian local ring here (The Stacks Project Authors 2026, Tags 07QW, 07QU, 07GH, 07QK, and 00MA). Thus \(IB\) is radical, and Lemma 32 gives \[ P\equiv P(0)\pmod{IB}. \tag{40}\]

The coordinate derivations extend uniquely to \(A\), because the chart is étale, and they commute. For a positive integer \(q\), let \[T_q=(A/I)[s_1,\ldots,s_n]/(s_1,\ldots,s_n)^q,\qquad \tau_q(f)=\sum_{|\alpha|<q} \frac{\partial^\alpha f}{\alpha!}s^\alpha\pmod I.\] The product rule says that \(\tau_q\) is a ring homomorphism; on the coordinate ring it is the shift \(v\mapsto v+s\). Give \(T_q\) its \(A\)-module structure through \(\tau_q\). Its filtration by \(S=(s_1,\ldots,s_n)\) has successive quotients that are finite sums of \(A/I\) with their ordinary coefficient action. The shifted module \(T_q\) is therefore finite over \(A\).

For clarity, its completion is the usual finite Taylor target with a shifted scalar action. If \(\mathfrak m_A\) is the maximal ideal of \(A\), put \(J_q=\tau_q(\mathfrak m_A)T_q\) and \(H_q=\mathfrak m_AT_q\), where the latter uses the ordinary coefficient map. Since \(J_q+S=H_q+S\) and \(S^q=0\), expansion of ideal powers gives \[ J_q^{\,N+q-1}\subset H_q^N,\qquad H_q^{\,N+q-1}\subset J_q^N \quad(N\geq1). \tag{41}\] The identity of the underlying ring thus identifies the two completions, and yields \[ T_q\otimes_{A,\tau_q}B \cong (B/IB)[s_1,\ldots,s_n]/(s_1,\ldots,s_n)^q. \tag{42}\] The \(B\)-action on the right is still the Taylor-shifted action. Under this identification the completed map \(\widehat\tau_q\) is the formal Taylor map on \(B\). Indeed \(A\) is dense in \(B\) and each of the finitely many derivatives in the formula is continuous, losing at most \(|\alpha|\) powers of the maximal ideal; (41) identifies that coefficientwise limit with the shifted-module limit.

Every coefficient of \(\widehat\tau_q(P)\) has a representative in \(A/I\). The constant coefficient is supplied by (40). For \(1\leq|\alpha|<q\), choose \(j\) with \(\alpha_j>0\), put \(\beta=\alpha-e_j\), and use (39) to choose \(g_j\in A\) with \(\partial_{v_j}P-g_j\in K^qB\). The product rule gives the explicit loss estimate \[ \partial^\beta(K^qB)\subset K^{q-|\beta|}B\subset IB. \tag{43}\] Hence \(\partial^\alpha P/\alpha!\) is represented modulo \(IB\) by \(\partial^\beta g_j/\alpha!\). These finitely many coefficients define one element \(x\in T_q\) whose image in (42) is \(\widehat\tau_q(P)\).

It is essential to descend the image condition for this whole tuple. Set \(L_q=\ker\tau_q\), \(U_q=\operatorname{im}\tau_q\), and \(Q_q=T_q/U_q\), using the shifted module structure throughout. There are exact sequences of finite \(A\)-modules \[ \begin{split} 0&\longrightarrow L_q\longrightarrow A\longrightarrow U_q \longrightarrow0,\\ 0&\longrightarrow U_q\longrightarrow T_q\longrightarrow Q_q \longrightarrow0. \end{split} \tag{44}\] Completion of finite modules is exact and agrees with tensoring by \(B\); \(A\to B\) is faithfully flat (The Stacks Project Authors 2026, Tags 00MA–00MC). The class of \(x\) in \(Q_q\) becomes zero after this tensor product, so it was zero already. Thus \(x=\tau_q(P_q)\) for a single element \(P_q\in A\). Exactness of the first sequence additionally gives \[ \ker\widehat\tau_q=L_qB,\qquad P-P_q\in L_qB. \tag{45}\]

The whole Taylor tuple has now descended to one algebraic primitive. It remains to ensure that equality of that tuple measures the required order along every component of the original ideal, including embedded components. We now choose \(q\) so that \(L_q\subset K^m\). Take a finite primary decomposition \[K^m=\bigcap_{\lambda=1}^h Q_\lambda,\qquad \mathfrak p_\lambda=\sqrt{Q_\lambda},\] including all embedded components. For each \(\lambda\), choose \(N_\lambda\) with \(\mathfrak p_\lambda^{N_\lambda}\subset Q_\lambda\); this follows by taking powers of finitely many generators of the radical. For \(f\in L_q\) we have \[\partial^\alpha f\in I\subset\mathfrak p_\lambda \qquad(|\alpha|<q).\] The Zariski–Nagata differential-power theorem gives \[ f\in\mathfrak p_\lambda^{(q)} =\mathfrak p_\lambda^qA_{\mathfrak p_\lambda}\cap A. \tag{46}\] Its hypotheses are satisfied at every \(\mathfrak p_\lambda\): \(A\) is essentially smooth over the characteristic-zero field \(\mathbb C\), and \(\kappa(\mathfrak p_\lambda)/\mathbb C\) is separable. The coordinate operators \(\partial^\alpha/\alpha!\) of order less than \(q\) form an \(A\)-basis of the differential operators of that order, by the étale coordinate description. These are precisely the operators tested above; see (De Stefani et al. 2020, Theorem 3.6 and Lemma 2.3).

Choose \(q\geq\max_\lambda N_\lambda\). Equation (46) then puts \(f\) in \(Q_\lambda A_{\mathfrak p_\lambda}\) for every \(\lambda\). A primary ideal contracts from localization at its radical: \[Q_\lambda A_{\mathfrak p_\lambda}\cap A=Q_\lambda,\] since \(sf\in Q_\lambda\) with \(s\notin\mathfrak p_\lambda\) forces \(f\in Q_\lambda\). Intersecting gives \(L_q\subset K^m\). Together with (45) this proves the lemma. ◻

Proof. Apply Lemma 33 with \(K=\mathfrak k_i\) and \(m=9\). Since \(\mathfrak k_i\subset\mathfrak m_i\), the replacement changes no terms of degree less than nine, in particular preserving cubic order. For \(E_i=P_i-P_i^{\mathrm{o}}\), differentiate first in the wall variables and then substitute the original maps \(\phi_i^\pm\). The product rule gives \(\partial E_i\in\mathfrak k_i^8B_i\), so \[ \begin{split} \phi_i^{\pm *}E_i&\in(L_i^\pm)^9\subset\mathfrak r^3,\\ \phi_i^{\pm *}(\partial E_i)&\in(L_i^\pm)^8 \subset\mathfrak r^2\mathfrak m^2\subset\mathfrak m\mathfrak r. \end{split} \tag{48}\] The middle containment uses \((L_i^\pm)^3\subset\mathfrak r\) and \(L_i^\pm\subset\mathfrak m\), not an estimate for a derivative of a pulled-back function. The value estimate preserves \(g\in\mathfrak r^2\). If \(v^{\mathrm{o}}=(\partial_C\Psi^{\mathrm{o}})(a,Y)\), then \(v-v^{\mathrm{o}}\in(\mathfrak m\mathfrak r)^{2D}\). Thus \((v)\subset\mathfrak r\) and \(\mathfrak r=(v)+\mathfrak m\mathfrak r\). Nakayama’s lemma applied to the finite \(B_X\)-module \(\mathfrak r/(v)\) proves \((v)=\mathfrak r\).

The positive triple products have not changed. For each negative factor, \(\nu_i-p_i^{\mathrm{o}}(Y_i^-)\in\mathfrak r\) by the same derivative estimate. Telescoping a product of three factors therefore preserves its membership in \(\mathfrak r=(v)\), including repeated labels.

All the new germs \(P_i\), the original maps \(Y_i^\pm\), their indicated derivatives, and their compositions are algebraic. Place the finitely many germs in \(z\) in a common local étale ring \(E\): pull back the wall presentations along the maps \((a_i,Y_i^+)\) and \((0,Y_i^-)\), take finite fiber products, and localize at the specified point. Then \(\widehat E=B_X\). Faithful flatness contracts the ideals \((v)^2B_X\) and \((v)B_X\) to \((v)^2E\) and \((v)E\). Consequently the three membership assertions admit the coefficients in (47) in \(E\). We use these algebraic coefficients, rather than any previously chosen formal witnesses. ◻

Proof. Use finite étale presentations, near the chosen points, of the form \[\mathcal F(x,w)=0,\qquad w(0)=c,\qquad J_0=\partial_w\mathcal F(0,c),\quad\det J_0\ne0.\] Regular-function denominators can be included as additional variables by equations \(r f(x,w)=1\). Adjoin to \(\mathbb Z\) the finitely many polynomial coefficients, the entries of \(c\), the inverses of all the initial Jacobian determinants and denominator values, and the specified inverses. This defines a finitely generated subring \(R\).

Solve the equations by total degree in \(x\). At each positive degree the unknown homogeneous coefficient vector is multiplied by the same matrix \(J_0\); the other terms are expressions in previously determined coefficients. The inverse \(J_0^{-1}=\operatorname{adj}(J_0)/\det J_0\) already has entries in \(R\). Induction therefore puts every coefficient in \(R\). This procedure introduces no division by the degree or by a growing family of factorials. Apply it to all the finite presentations at once. ◻

Proof. Expand a monomial \(c_{\alpha\beta}a^\alpha C^\beta\) by the binomial formula. After subtracting its constant and linear terms in \(\delta\), the remaining terms are \[c_{\alpha\beta}\binom\beta\gamma a^\alpha C^{\beta-\gamma}\delta^\gamma, \qquad |\gamma|\geq2.\] Choose two occurrences among the \(\delta\) factors by a fixed rule, allowing the same index twice, and assign the coefficient after removing those factors to the corresponding entry of \(H\). The remaining total degree is \(|\alpha|+|\beta|-2\geq1\). Summing defines \(H\) coefficientwise; each coefficient receives only finitely many contributions. The other assertions follow by applying the one-factor version of the same construction. For a first partial of \(\Phi\), of order at least two, its one-factor remainder has order at least one, giving the claim for \(G\). ◻

Proof. Start with Proposition 35. Apply Lemma 36 to the unspecialized wall germs \(P_i\), all entries of \(Y\), and all algebraic witnesses \(b,h_+^{ijk},h_-^{ijk}\) in (47). Include \(\Delta^{-1}\) among the specified inverses. Partial differentiation multiplies coefficients by integers and centered substitution uses finite sums at each degree, so \(\Psi,p_i,v,g,\nu_i\) also have coefficients in this one ring \(R\). All identities in (47) hold in \(R[[z]]\), since \(R\subset\mathbb C\). Because \(\Psi\) has order at least three and \(Y\) is centered, \(v\in(z)^2R[[z]]^{2D}\).

Apply Lemma 37 to \(\Psi\), with \(N=2D\), and look for \[\widetilde Y=Y+Bv,\] where \(B\) is an \(N\)-by-\(N\) formal matrix. For \(\delta=Bv\), (49) and \(g=v^{\mathsf T}bv\) give \[\Psi(a,Y+Bv) =v^{\mathsf T}\bigl(b+B+B^{\mathsf T}H(a,Y,Bv)B\bigr)v.\] It suffices to solve the matrix equation \[ B=-b-B^{\mathsf T}H(a,Y,Bv)B. \tag{52}\]

Here is the coefficient construction over \(R\). Set \(\mathcal T(B)=-b-B^{\mathsf T}H(a,Y,Bv)B\). The entries of \(H(a,Y,Bv)\) lie in \((z)\) for every \(B\), since \(a,Y,Bv\) are centered and \(H\) has positive order. If \(B-B'\) has entries in \((z)^n\), changing either outer matrix factor changes \(B^{\mathsf T}H(a,Y,Bv)B\) only in \((z)^{n+1}\). Changing the inner argument \(Bv\) changes it in \((z)^{n+2}\) because \(v\in(z)^2\). Thus \[ B-B'\in(z)^n\quad\Longrightarrow\quad \mathcal T(B)-\mathcal T(B')\in(z)^{n+1}. \tag{53}\] The constant coefficient is forced to be \(B(0)=-b(0)\). For every higher degree the coefficient on the right of (52) depends only on previously determined coefficients of \(B\) and on the already known coefficients of \(b,H,Y,v\). This determines a unique solution in \(\operatorname{Mat}_N(R[[z]])\). Equivalently, iterate \(\mathcal T\) and use (53). The recursion uses only ring operations; there are no new numerical inverses. It proves the first identity in (51) exactly.

We next transport the two triple laws to this corrected graph. Equation (50) gives \[ \widetilde v=(1+M)v,\qquad M=G(a,Y,Bv)B\in(z)\operatorname{Mat}_N(R[[z]]). \tag{54}\] Hence \(A_1=1+M\) is invertible over the same ring, with \(A_1^{-1}=\sum_{n\geq0}(-M)^n\). Let \(\widetilde\nu_i=p_i(\widetilde Y_i^-)\). The one-factor difference formula of Lemma 37, followed by projection of \(Bv\) to its \(i\)th negative block, gives a row \(q_i\in R[[z]]^{1\times N}\) satisfying \[ \widetilde\nu_i-\nu_i=q_i v. \tag{55}\] For all ordered triples the exact telescoping formula reads \[\widetilde\nu_i\widetilde\nu_j\widetilde\nu_k-\nu_i\nu_j\nu_k =\bigl(\widetilde\nu_j\widetilde\nu_kq_i +\nu_i\widetilde\nu_kq_j+\nu_i\nu_jq_k\bigr)v.\] Consequently the required witnesses can be chosen as \[ \begin{split} \widetilde h_+^{ijk}&=h_+^{ijk}A_1^{-1},\\ \widetilde h_-^{ijk}&= \bigl(h_-^{ijk}+\widetilde\nu_j\widetilde\nu_kq_i +\nu_i\widetilde\nu_kq_j+\nu_i\nu_jq_k\bigr)A_1^{-1}. \end{split} \tag{56}\] These identities also hold when labels coincide. All coefficients stay in \(R\): the difference factorizations use integer coefficients, and centered substitution and the geometric inverse series involve only finitely many terms at each degree. This proves both remaining identities in (51).

Finally \(Bv\in(z)^2\), so the first-order part of \(Y\) and its paired tangent determinant are unchanged. This determinant is a unit of \(R\) by its choice. The map \[(t,u)\longmapsto\widetilde Y(0,t)+\sigma J u\] has invertible linear term \([J\ \sigma J]\), and therefore has a formal inverse over \(R\). To see coefficient control directly, solve for each homogeneous degree using this same invertible constant matrix. This identifies the first graph with the coordinate subspace \(u=0\); the swapped graph is treated identically. They have complementary tangent spaces. The value identity at \(a=0\) makes \(F\) vanish on the first; \(F(\sigma C)=-F(C)\) makes it vanish on the second. ◻

Proof. Because \(R\subset\mathbb C\), the finite type \(\mathbb Q\)-algebra \(R_{\mathbb Q}=R\otimes_{\mathbb Z}\mathbb Q\) is nonzero. Choose a maximal ideal. The weak Nullstellensatz, or Zariski’s lemma, says that its residue field \(L\) is a finite extension of \(\mathbb Q\). We obtain a homomorphism \(R\to L\). The images of the finitely many ring generators, including all designated inverses, lie in \(\mathcal O_L[1/N]\) for some positive integer \(N\): multiplying each algebraic number by a suitable nonzero integer makes it integral. Thus the homomorphism factors as \[R\longrightarrow\mathcal O_L[1/N]\longrightarrow \mathcal O_L/\mathfrak l=\Bbbk\] for any prime ideal \(\mathfrak l\) above a rational prime \(\ell\nmid N\). The residue field is finite. Every designated determinant stays a unit, because its inverse is already in \(R\).

Coefficientwise reduction commutes with multiplication, centered substitution, and formal differentiation; each coefficient involves only a finite sum and differentiation multiplies by an integer. It therefore preserves every identity in (51), including the membership coefficients, and the unit tangent determinant. It also preserves the order-at-least-three condition and the separated formula for \(\Psi\). The formal inverse construction in the theorem reduces over the same ring, giving the claimed smooth zero graphs. Only the algebraic replacements and their subsequent formal recursions are reduced. The original complex Chern–Simons series, and all integration used to obtain them, are absent from this specialization step. ◻

Proof. A centered formal coordinate change and its inverse preserve the coordinate Frobenius ideals. For a centered series \(f(z)=\sum_{e\ne0}c_ez^e\) in characteristic \(\ell\), the coefficientwise Frobenius identity gives \[f(z)^\ell=\sum_{e\ne0}c_e^\ell z^{\ell e} \in(z_1^\ell,\ldots,z_{2D}^\ell).\] Apply this to every component of each inverse coordinate map. Hence we may calculate in \(\Bbbk[[t,u]]/(t^\ell,u^\ell)\), where the notation means the separate \(\ell\)th powers of all coordinates. Multiplication by \(u_b\) annihilates (63). Its ordinary differential is zero because every term differentiating its coefficient repeats one of its normal differentials. Write \(F(t,u)\in(u)\). The tangential coefficients of \(\mathrm dF\) belong to \((u)\), whereas each normal differential wedges to zero with the full normal volume. Therefore \(\delta\gamma=0\).

The difference \(g-\sum_\nu\widetilde h_\nu\partial_{C_\nu}F\) lies in \((u)\) before truncation and continues to do so after truncation. The contraction identity in the original coordinates is the coefficient-linear extension of the exterior and interior product identity in (Koszul 1950, sec. 2, p. 72, (2.9)): \[ \delta\iota_\nu+\iota_\nu\delta =(\partial_{C_\nu}F)\operatorname{id}. \tag{66}\] Since \(\delta\) is linear over \(T\), annihilation of \((u)\) and \(\delta\gamma=0\) prove (65). There are no derivatives of the coefficients \(\widetilde h_\nu\) in this calculation.

For the last assertion let \(u\) and \(v\) be the respective normal coordinates. Complementarity of the tangent spaces says exactly that \((u,v)\) has invertible linear part, so it too is a full coordinate system. The socle of \(T\), namely the annihilator of its maximal ideal, is the one-dimensional space spanned by \(C_1^{\ell-1}\cdots C_{2D}^{\ell-1}\). This follows directly by multiplying a monomial expansion by each \(C_\nu\). The coordinate automorphism carries its nonzero socle generator to \[q=\prod_b u_b^{\ell-1}\prod_b v_b^{\ell-1}\ne0.\] The wedge of the two normal forms is \(qJ(C)\) times the original coordinate volume, where \(J(C)\) is the unit Jacobian determinant of \((u,v)\). Since \(q\) is in the socle, \(qJ(C)=J(0)q\ne0\). Its residue is therefore nonzero. This proves nonvanishing in the fixed original coordinates without a coordinate-invariance formula for the residue. ◻

Proof. The subspace \(K\) is a subcomplex: if \(x\in K\), then \(\mathcal Dx\in N\) and \(\mathcal D^2x=0\in N\). Choose a graded complement \(N=K\oplus E\). If \(e\in E\) and \(\mathcal De\in N\), then \(e\in K\), hence \(e=0\). Thus \(\mathcal D\) is injective on \(E\), and \(\mathcal DE\cap N=0\). In particular \(K\oplus E\oplus\mathcal DE\) is a graded direct sum. Also \(E\oplus\mathcal DE\) injects into \(V/K\): if \(e+\mathcal De'\in K\), then \(\mathcal De'\in N\), so \(e'=e=0\).

Take any graded vector-space complement \(C'\) to that direct sum, and define \[h(\mathcal De)=e\quad(e\in E),\qquad h|_{K\oplus E\oplus C'}=0.\] Set \(Q=\mathcal Dh+h\mathcal D\). The image of \(\mathcal Dh\) lies in \(\mathcal DE\), and the image of \(h\mathcal D\) lies in \(E\), even when \(\mathcal DC'\) has components in the other summands. Moreover \(Q\) is the identity on \(E\) and on \(\mathcal DE\), and is zero on \(K\). Consequently \(Q^2=Q\). The equation \(\mathcal D^2=0\) gives \[\mathcal DQ=\mathcal Dh\mathcal D=Q\mathcal D.\] Now \(r=1-Q\) has all the asserted properties. This calculation is why an arbitrary complement is allowed; an arbitrary vector-space projection by itself would not give a chain map. ◻

Proof. Choose complements \(V_i=N_i\oplus M_i\). The total tensor product is the direct sum of subspaces \(V_A\), indexed by subsets \(A\subset\{1,\ldots,d\}\), using \(M_i\) when \(i\in A\) and \(N_i\) when \(i\notin A\), together with the passive factor. Fix a triple \(I\). The operator \(S_I=\prod_{i\in I}S_i\) kills \(\bigoplus_{I\not\subset A}V_A\). Its restriction to the complementary subspace, which has \(M_i\) in every slot \(i\in I\) and unrestricted factors elsewhere, is injective: it is a tensor product of the injections \(S_i|_{M_i}\) and identity maps over a field. Thus \[\ker S_I=\bigoplus_{I\not\subset A}V_A.\] Intersecting these coordinate direct sums over all triples leaves exactly \(|A|\leq2\), which is the right-hand side of (68). The images \(S_i(M_i)\) may intersect \(N_i\); this does not affect injectivity on the specified complementary domain subspace. Passive factors have no effect on the argument. ◻

Proof of Theorem 41. Suppose the stated data exist. If \(D=0\), the ideal \(J_Y\) is zero, so the first membership in (59) already contradicts the nonzero monomial \(a_{1,1}a_{2,1}a_{3,1}\). We may assume \(D>0\). Set \[Q_i(y_i)=P_i(0,y_i),\qquad F(C)=\Psi(0,C)=\sum_i\bigl(Q_i(C_i^+)-Q_i(C_i^-)\bigr),\] and use the complexes and residue in (60). The two graphs \[\Gamma_+=Y(0,t),\qquad \Gamma_-=\sigma Y(0,t)\] have complementary tangent spaces. They are smooth formal graphs because their tangent maps are injective and can be completed to invertible coordinate maps. The function \(F\) vanishes on both: this follows from (58) and \(F(\sigma C)=-F(C)\).

Choose once and for all a constant matrix \(L\) whose columns complement the image of \(A\), and use the inverse coordinates of \[ C=Y(0,t)+Lu \tag{71}\] to define the normal cycle \(\alpha\) for \(\Gamma_+\). Choose any fixed ordered normal coordinates for \(\Gamma_-\) and define its normal cycle \(\beta\). Lemma 42 gives \[ \delta\alpha=\delta\beta=\mathrm d\alpha=\mathrm d\beta=0, \qquad \langle[\alpha],[\beta]\rangle\ne0. \tag{72}\]

The second membership in (59), at \(a=0\), transfers to \(\Gamma_-\) as follows. Regard \(\sigma\) also as the permutation of scalar-coordinate indices. Differentiating \(F(\sigma C)=-F(C)\) gives \[(\partial_{C_\nu}F)(Y(0,t)) =-(\partial_{C_{\sigma(\nu)}}F)(\sigma Y(0,t)).\] On \(\Gamma_-\), the positive block \(C_i^+\) equals \(Y_i^-(0,t)\). Thus permuting the derivative entries and negating their witnesses turns that membership into \[\bigl(p_i(C_i^+)p_j(C_j^+)p_k(C_k^+)\bigr)|_{\Gamma_-} \in\bigl((\partial_{C_\nu}F)|_{\Gamma_-}\bigr)_\nu.\] The explicit primitive in Lemma 42 therefore proves \[ p_i(C_i^+)p_j(C_j^+)p_k(C_k^+)[\beta]=0 \quad\hbox{in }H \tag{73}\] for every distinct triple. This is an ambient Koszul boundary statement, not just an identity after restriction to the graph.

The separated expression for \(F\) identifies \((\Omega,\delta)\) with the graded tensor product of the \(2d\) block complexes. A positive block has differential \(\delta_i^+=\mathrm dQ_i\wedge{}\) and a negative block has differential \(\delta_i^-=-\mathrm dQ_i\wedge{}\). Under the identification a tensor is sent to the wedge of its factors in the fixed block order; both the Koszul differential and ordinary differentiation carry the sign \((-1)^{\sum_{c<b}|x_c|}\) in block \(b\). Consequently there is a natural identification \[ H=\bigotimes_{i=1}^d(H_i^+\otimes H_i^-),\qquad \mathcal D=\hbox{the signed tensor differential induced by ordinary } \mathrm d_i^\pm. \tag{74}\] For completeness, the cohomology assertion uses only field linear algebra: split each finite complex into representatives for its cohomology with zero differential and pairs \(e,\delta e\). Each pair is contractible, and its tensor product with any complex is contractible by the signed tensor homotopy. The remaining tensor of the representative spaces is precisely the displayed cohomology. Naturality of the tensor cycle map gives the asserted ordinary differential on it.

On \(H_i^+\) let \(S_i\) be multiplication by \(p_i\), and let \(\mathcal D_i\) be the differential induced by ordinary differentiation. Define \[ N_i=\ker S_i,\qquad K_i=N_i\cap\mathcal D_i^{-1}N_i. \tag{75}\] Multiplication by \(p_i\) commutes with the block Koszul differential. The degree-one operator \(\mathrm dp_i\wedge{}\) anticommutes with that differential and induces on \(H_i^+\) the commutator \[ [\mathcal D_i,S_i]=\mathcal D_iS_i-S_i\mathcal D_i =[\mathrm dp_i\wedge{}]. \tag{76}\] This operator vanishes on \(K_i\), since both \(S_i x\) and \(S_i\mathcal D_i x\) vanish there.

Let \(z=[\beta]\). By (73) and Lemma 44, \(z\) lies in the sum of tensor subspaces having at least \(d-2\) positive slots in \(N_i\). Apply the retractions of Lemma 43 in all positive slots, taking \(r_b=1\) and \(h_b=0\) in negative slots. Their tensor \(R\) sends that sum into the analogous sum with at least \(d-2\) slots in \(K_i\). Since \(\mathcal Dz=0\), (70) gives \[Rz-z=-\mathcal D(\mathcal H z).\] Pairing with \([\alpha]\) kills this ordinary-differential boundary. Indeed if \(\omega\) is any Koszul-cycle representative of a class, then \(\mathrm d\alpha=0\) and (61) imply \[ \operatorname{Res}_C(\alpha\wedge\mathrm d\omega) =(-1)^D\operatorname{Res}_C\bigl(\mathrm d(\alpha\wedge\omega)\bigr) =0. \tag{77}\] Changing representatives adds a Koszul boundary, which pairs to zero by (62); ordinary differentiation of a changed representative adds another Koszul boundary by anticommutation. Thus (77) is genuinely an identity on \(H\). It follows from (72) that \(\langle[\alpha],Rz\rangle\ne0\).

All subspaces and complements used above are graded, and \(Rz\) has form degree \(D\). Express it as a finite sum of homogeneous simple tensors in the indicated sum of subspaces, keeping only total degree \(D\). One such tensor, denoted \(k\), satisfies \[ \langle[\alpha],k\rangle\ne0 \tag{78}\] and has at least \(d-2\geq3\) positive factors in their \(K_i\). Fix three of these labels \(i_1,i_2,i_3\). The selected tensor need not be \(\mathcal D\)-closed. What will be used is that each of its homogeneous factors has a homogeneous cycle representative for its block Koszul differential.