Idea of the proof

We first construct a fixed immersed disk problem in the exterior of a finite collection of capped gropes. A capped grope here is a surface tree with disks attached at its terminal curves; its boundary records an iterated commutator in the terminal curves. The construction installs finitely many such commutator relations, with independently chosen based conjugations, in a smooth plumbing model. It also supplies the algebraic dual spheres required in Theorem 1. These data are fixed before any hypothetical embedded replacements are chosen.

Suppose this particular disk problem has replacements with the required boundary maps and framings. They allow the plumbing model to be cut into two vertex pieces \(A,B\), joined by compact three-manifolds \(Y_1,\ldots,Y_{2m}\) with torus boundary, for an integer \(m\geq5\) chosen in the construction. The cuts are smooth and miss the locally flat disks. The installed boundary words are now relations in the vertex groups, and restriction in cohomology gives complementary linear data on the two sides. Proposition 3 records this marked geometry; Lemma 5 verifies the equivariant disk-and-sphere input.

For each edge \(Y_i\), we study based flat \(SL_2\)-connections with one torus holonomy fixed. Basing records holonomy as a matrix, with no quotient by simultaneous conjugation. A potential is a function in finitely many formal coordinates whose gradient equations describe these flat connections. The central potential fixes this torus holonomy to the identity. Proposition 20 constructs it from the Chern–Simons functional, including a first-order variation of the fixed boundary holonomy. Theorem 21 then replaces the central potential by an algebraic function through a formal coordinate change. The comparison preserves based holonomy on the full critical scheme: its coordinate ring is the power-series ring modulo the actual gradient ideal, with its nilpotent elements retained. The geometric reduction selects \(m\) edges for the first-order boundary variations; we call these edges active.

The two vertex pieces determine formal graphs in the product of the edge coordinate spaces. Their central tangent spaces are complementary, and suitable sums of the edge potentials vanish identically on the graphs. The boundary relations imply more than vanishing at common critical points: products associated to every three distinct active edges belong to the restricted gradient ideals. On one side these products are \(t_it_jt_k\), where each \(t_i^2=0\) and mixed products remain in the parameter ring. On the other they are products of three algebraic holonomy functions. Proposition 25 establishes the exact vanishing and ideal memberships. The graph parametrizations may mix all edge coordinates, while each fixed potential depends only on its own edge block.

Theorem 8 rules out precisely these algebraic data when at least five edges are active. Artin approximation and finite specialization first reduce them to large positive characteristic. Truncating each internal coordinate by its characteristic power makes the associated Koszul complexes finite. The two transverse graphs determine classes with nonzero pairing. The target triple relations, together with ordinary exterior differentiation, decompose one class into tensors admitting three independent first-order parameter lifts. The source triple relations force each resulting pairing to vanish, giving the contradiction.

Three features are useful beyond the final contradiction. First, the geometry imposes a finite family of independently conjugated laws before the cuts are chosen. Second, the algebraicity argument compares full relative closed shifted forms and preserves the marked holonomies; an algebraic model of the reduced representation locus would be insufficient. Third, the tensor argument uses actual gradient ideals over nonreduced parameter rings and makes no isolated-critical-point or regular-sequence assumption.

The geometric and algebraic antecedents

Capped-grope contraction and transverse grope constructions provide the local geometric operations (Krushkal and Quinn 2000, sec. 1.2 and 1.4). Their use to relate surface intersections and commutator words also appears in the work of Freedman and Krushkal. Using the repeated-variable 2-Engel relation, they constructed homotopy solutions to the A–B slice problem and new universal surgery models (Freedman and Krushkal 2016, Theorems 1 and 2); their later work established universality for half-\(\pi_1\)-null surgery models, with a capped grope on one side and a \(\pi_1\)-null sphere on the other (Freedman and Krushkal 2020, Theorem 1 and Section 3). These results concern particular surgery or slicing formulations, with their own embedding and complement conditions. They explain the value of commutator calculus in this setting without supplying the marked geometric reduction used here.

Our relations use three distinct group inputs and become trivial when any one is the identity. Their logarithmic expansion begins with the trilinear bracket \([Z_1,[Z_2,Z_3]]\), where \(Z_a\) is the logarithm of the \(a\)-th input. The deletion property is the Brunnian property defined in Section 2. We install a finite family of such words with independently chosen conjugations, before knowing the cut pieces. The later argument uses those independent choices to recover every triple coefficient. It neither imposes a universal Engel relation nor requires the vertex groups to be nilpotent. The simultaneous placement, framing, and algebraic-dual calculations needed for this finite family are given in Section 2.

The deformation-theoretic part has a different origin. Goldman and Millson describe Artin-algebra families of flat connections by differential graded Lie algebras and identify their based holonomies with representation deformations (Goldman and Millson 1988, Corollary 6.4 and Proposition 6.6). Their quadraticity results use additional formality hypotheses in the compact Kähler setting. The edge manifolds here are arbitrary compact three-manifolds, so higher deformation terms remain. Chern and Simons constructed the classical transgression form whose exterior derivative is the corresponding curvature characteristic form (Chern and Simons 1974, sec. 3, especially Proposition 3.2). Section 4 uses this form to construct the potential, computes its boundary adjustment, and proves equality of its full curvature and gradient ideals.

The algebraic model comes from shifted symplectic geometry. The Betti transgression theorem of Pantev, Toën, Vaquié, and Vezzosi produces shifted symplectic forms on local-system stacks; their Lagrangian intersection theorem lowers the shift by one (Pantev et al. 2013, Theorems 2.5 and 2.9). Calaque’s relative orientation construction supplies the Lagrangian restriction map associated to an oriented manifold with boundary (Calaque 2015, Theorem 2.9 and §3.1). After a punctured filling of the edge, these results put us in the setting of Brav, Bussi, and Joyce’s algebraic Darboux theorem in shift \(-1\), which supplies an algebraic critical chart (Brav et al. 2019, Example 5.15 and Theorem 5.18(i)).

To use that chart, we must identify its full relative closed form with the form underlying the particular Chern–Simons potential already constructed. Proposition 22, proved in Appendix 8, carries out this comparison on derived coefficients and source cochain models. Its inputs include Getzler’s simplicial Maurer–Cartan comparison for nilpotent Lie algebras (Getzler 2009, Theorem 5.4 and Corollary 5.11) and the simplicial de Rham contraction of Dupont (Dupont 1976). Section 5 then turns the closed-form comparison into a coordinate change fixing the critical scheme and its based holonomies. Only after this step, and the polynomial replacement of the holonomy logarithms, do we apply ordinary Artin approximation (Artin 1969, Theorem 1.10). The approximation is applied to the final graph equations with their fixed, separately algebraic edge functions.

Conventions and organization

We assume familiarity with basic four-manifold topology, cohomology with local coefficients, and formal deformation theory. The algebraicity argument additionally uses derived mapping stacks and shifted symplectic forms; its required comparison is stated in the body and proved in full in Appendix 8.

All manifolds and maps in the initial model are smooth, with corners rounded when necessary. Hypothetical output disks are only locally flat. The subsequent cuts are smooth hypersurfaces in the original smooth manifold and miss those disks. Thus the vertex and edge calculations use ordinary smooth forms and Stokes’ theorem.

Unless a field is specified, cohomology has characteristic-zero coefficients. We use \(\mathfrak{g}=\mathfrak{sl}_2(\mathbb C)\) and the pairing \(\kappa(X,Y)=\mathop{\mathrm{tr}}(XY)\). Connections and coordinate changes are formal power series, coefficient by coefficient. A statement about an ideal is a statement in the complete coordinate ring, including its nilpotents. In Sections 4–6, \(d\) denotes the differential on source forms and \(\delta\) differentiation in formal coordinates. In the tensor argument of Section 3, \(d\) is the coordinate de Rham differential and \(\delta\) its induced operator on Koszul homology. We specify total signs when using both differentials.

Section 2 proves the geometric reduction. Section 3 states and proves the finite obstruction before we construct its inputs, making the target algebraic interface explicit. Sections 4 and 5 construct the edge germs and their algebraic models. Section 6 constructs the vertex graphs and checks every hypothesis of the tensor obstruction. Section 7 completes the proof of Theorem 1.

A finite list of relations

Our commutator convention is \([a,b]=aba^{-1}b^{-1}\). A word \(R\) in three free generators \(X_1,X_2,X_3\) will be called a three-slot grope word if it is the boundary word of the rooted surface tree consisting of a punctured torus with a single tip on its first branch and a second punctured torus on its other branch. Paths to its three tips are part of its marking. Changing those paths inside the body can conjugate a tip by words in the three based tips, and can conjugate the inner commutator by such a word. Overall conjugation of the boundary word is irrelevant to its being a relation. In particular, \[ \left.R\right|_{X_a=1}=1\quad(a=1,2,3), \qquad R\equiv[X_1,[X_2,X_3]] \pmod{\Gamma_4F(X_1,X_2,X_3)}. \tag{1}\] Here \(\Gamma_rF\) denotes the lower central series of the free group \(F\). The first property is the Brunnian property in the three slots: setting any one input equal to the identity trivializes the word. The orientations of the two surfaces can be fixed to give the displayed sign. Indeed a tip conjugation changes its image modulo \(\Gamma_2\) by zero, and hence changes this weight-three commutator only modulo \(\Gamma_4\). The first assertion follows directly by deleting a branch of the rooted tree. Equivalently, in the completed free associative algebra over \(\mathbb Q\), \[ \log R(e^{Z_1},e^{Z_2},e^{Z_3}) =[Z_1,[Z_2,Z_3]]+\text{terms of total degree at least four}; \tag{2}\] each term contains a letter from every slot. These two properties, rather than a claim that an entire vertex group is nilpotent, will be used.

Fix \(m\geq5\) and \(p\geq9\). Each plumbing block will have \(2m\) solid-torus boundary regions, called ports, through which it joins other blocks. Removing these regions leaves \(2m\) boundary tori. Write \(x_1,\ldots,x_{2m}\) for their meridians and \(l_1,\ldots,l_p\) for the extra one-handle generators, and put \[\mathcal C=\{1,l_1,\ldots,l_p\}.\] Before making any cut, install one marked grope body for every ordered triple of distinct port indices, every independent choice \(c_1,c_2,c_3\in\mathcal C\), and every independent choice \(\delta_1,\delta_2,\delta_3\in\{1,-1\}\). Its three based tips represent \[ X_a=c_a x_{r_a}^{\delta_a}c_a^{-1}, \qquad a=1,2,3. \tag{3}\] There are at most \[8(2m)(2m-1)(2m-2)(p+1)^3\] such bodies. Any additional fixed finite list of words in the extra letters could be included in \(\mathcal C\) by the same construction. In particular, neither the list of bodies nor their markings depends on the slopes selected by the cuts below.

The proof constructs a particular exterior \(E\) and a finite disk family in it before making any cut. A simultaneous framed replacement of that family alone implies all the cut conclusions of the proposition; the universal Assumption 2 is used only to provide that replacement. Lemma 5 specifies the fixed family and verifies its algebraic duals. This local implication will give the fixed counterexample in Theorem 1.

The plumbing blocks

Start with a standard four-dimensional one-handlebody with \(2m+p\) one-handles. Attach a cancelling two-handle to each of the first \(2m\) one-handles, using the product framing and disjoint standard attaching regions. Before plumbing, the resulting block \(B\) is a one-handlebody on the remaining \(p\) handles. The belt circles of the cancelling two-handles are a standard unlink in a ball in \(\partial B\). One can see each component separately from the cancelling pair: the complement of its attaching solid torus in \(S^1\times S^2\) is a solid torus, which becomes the complement of the belt unknot after cancellation. The framed attaching pushoff is its belt meridian. The cancellation regions for different pairs are disjoint.

Pair the \(2m\) two-handles. In each handle choose a small product patch \(D^2_{\mathrm{base}}\times D^2_{\mathrm{fibre}}\), and plumb the patches of its pair by interchanging the two factors. This is mutual plumbing; there are no self-plumbings. To specify the local manifold structure, the two branches at a plumbing can be represented by \[\bigl(D^2_R\times D^2_r\bigr) \ \cup\ \bigl(D^2_r\times D^2_R\bigr) \subset\mathbb R^2\times\mathbb R^2,\qquad 0<r<R.\] Their overlap is the product ball \(D^2_r\times D^2_r\). Rounding its corners gives the usual smooth plumbing chart. Distinct patches are disjoint. The factor interchange preserves the orientation in dimension four, and the product framings fix the signs of the resulting transverse core intersections.

Equivalently, take one copy \(B_v\) for each vertex of the Cayley tree of \(F_m=\langle a_1,\ldots,a_m\rangle\), and identify the designated patches of \(B_v\) and \(B_{va_j}\) by this chart. Neighbouring blocks intersect in one product ball, and there are no other or triple intersections. Left translation acts freely. The quotient is the finite plumbing \(X\), and the union is its regular cover \(\widetilde X\). Since all attaching intersections are contractible cofibrations, the homotopy type of the union is obtained by joining the block homotopy types along the tree. Thus \[ \widetilde X\simeq \text{the Cayley tree with \(p\) circles attached at each vertex}. \tag{5}\] This is not an assertion that \(\widetilde X\) is the universal cover: the extra circles remain.

At a product patch the removed part of the unplumbed block boundary is a solid-torus neighbourhood of the belt circle. Its boundary is \(S^1_{\mathrm{base}}\times S^1_{\mathrm{fibre}}\). Consequently the exposed boundary is exactly the piece \(K_v\) in Proposition 3, with a torus collar at each join. The factor interchange exchanges the two peripheral slopes. An unlink longitude bounds a disk in \(K_v\), whereas its meridian is one of its free generators. Adding the \(p\) extra one-handles gives the remaining free generators \(l_s\).

Fix boundary whiskers for these generators. In the cover, fix the boundary map to the tree which is constant on each \(K_v\) core and crosses the corresponding tree edge on every torus collar. The middle of each collar maps to the midpoint of that edge. The construction is equivariant. It descends both to the one-vertex graph and to the parity quotient.

Marked grope bodies and their framings

The plumbing fixes the port meridians and peripheral slopes. We next place the finite list of surface bodies with their prescribed based tip curves, and match their framings to the handle cores. These choices must be made before the exterior disk problem is defined.

In particular the finite family is constructed without knowing the future cut pieces. Each cap is individually embedded: a core has no self-plumbing. Caps of different labels may intersect, and no disjointness of their spanning surfaces or of all their interiors is asserted.

The disk problem in the exterior

The marked bodies have now been placed with framed caps. Removing only their bottom surfaces leaves an exterior in which one cap per body is the input disk. The remaining upper stage will supply its algebraic dual sphere; the unused upper cap will identify the group-ring labels needed for cancellation.

Denote the disjoint bottom punctured tori by \(S_1,\ldots,S_k\). Choose their open product tubes, including their proper boundary collars, and put \[E=X\setminus\bigcup_{j=1}^k\operatorname{int}\nu(S_j).\] The tubes are chosen after the finite placements of Lemma 4. They can be sufficiently thin that they avoid all the upper caps and meet the upper bodies only in their short attaching collars. The exterior is connected: a path can be moved off finitely many proper codimension-two surfaces and then off their sufficiently small tubes. It is a compact oriented smooth four-manifold after rounding.

Assumption 2, applied to \(E\) and Lemma 5, now gives pairwise disjoint locally flat embedded replacement caps with the specified boundary framings. Compress each bottom \(S_j\) with two parallels of its replacement cap, using the returning annuli in the removed tube. The result is a collection of pairwise disjoint properly embedded disks in \(X\), bounded by the initial grope boundaries. The framing is what makes this compression locally flat with the required product collar. No relative homotopy of the replacement cap to \(f_j\) is used. The output dual-sphere part of a disk embedding theorem is not used either.

Equivariant cuts with connected pieces

The preceding compression uses the assumed replacement disks to make the initial grope boundaries bound disjoint disks. We now keep those disks in their prescribed vertex regions while cutting the ambient plumbing into smooth pieces. Connectedness of these pieces will be needed for the restriction maps in the final part of this section.

Each initial boundary is therefore null-homotopic in its vertex piece. Its actual boundary word has the three-slot properties (1) with inputs (3). This proves precisely the relation statement in Proposition 3.

The parity quotient and its peripheral homology

Send every plumbing generator of \(F_m\) to \(1\in\mathbb Z/2\). The quotient of \(\widetilde X\) by the kernel has two principal vertex pieces \(A,B\) and two lifts of each of the \(m\) plumbing edges. Thus there are \(2m\) edges. The free translation interchanging parity gives a deck involution interchanging \(A\) and \(B\), and pairing these edge lifts. Equation (5), or the same contractible-overlap argument in this quotient, gives \[ X_2\simeq\Gamma\ \text{with \(p\) extra circles at each vertex}, \qquad |V\Gamma|=2,\quad |E\Gamma|=2m. \tag{8}\] In particular \(H^q(X_2;k)=0\) for \(q\geq2\), and \(\dim_kH^1(X_2;k)=2m-1+2p\). The extra loops at the two vertices can be assigned independently in the free ambient fundamental group. All port meridians are trivial there, since they bound the corresponding handle-core disks in \(X_2\).

Algebraic approximation and finite specialization

Truncated forms and transverse graph classes

The approximation step preserves all graph identities and actual ideal memberships. We now turn their positive-characteristic specializations into identities in finite-dimensional complexes, where transverse graphs can be tested by a nonzero pairing.

Fix a specialization supplied by Lemma 9 in characteristic \(p>2\), and denote its field by \(\Bbbk\). Continue to use the same letters for the specialized germs. Write \(z_1,\ldots,z_N\) for all internal coordinates and set \[A_p=\Bbbk[z_1,\ldots,z_N]/(z_1^p,\ldots,z_N^p),\qquad \Omega_p=A_p\otimes_\Bbbk\bigwedge\nolimits_\Bbbk \langle \mathrm{d}z_1,\ldots,\mathrm{d}z_N\rangle .\] This is the ordinary algebra of differential forms of \(A_p\). Its differential is well defined because \(\mathrm{d}(z_\nu^p)=0\). With external parameters, use \[T_p=\Bbbk[t_i:i\in I]/(t_i^2:i\in I), \qquad \Omega_{p,T}=\Omega_p\otimes_\Bbbk T_p,\] and differentiate only in the internal coordinates. All these spaces are finite over the indicated coefficient rings.

Define a \(\Bbbk\)-linear functional on forms, zero outside exterior degree \(N\), by \[\int f(z)\,\mathrm{d}z_1\wedge\cdots\wedge\mathrm{d}z_N =[z_1^{p-1}\cdots z_N^{p-1}]f .\] Extend it \(T_p\)-linearly in the presence of parameters. It vanishes on exact forms: a derivative contributing exponent \(p-1\) in its differentiated variable would have to come from exponent \(p\), whose derivative is zero. Consequently, for homogeneous forms, \[ \int\mathrm{d}\alpha\wedge\beta =-(-1)^{|\alpha|}\int\alpha\wedge\mathrm{d}\beta . \tag{12}\]

Let \(c_1,\ldots,c_r\) be graph equations for a smooth formal graph over the parameter ring. Their joint center is zero modulo \((t)\). Its graph form is \[ U_c=\bigl(c_1^{p-1}\mathrm{d}c_1\bigr)\wedge\cdots\wedge \bigl(c_r^{p-1}\mathrm{d}c_r\bigr) \in\Omega_{p,T}^{r}. \tag{13}\] This expression involves only a finite jet after truncation.

Let \(U_t\) be the graph form for \(L\), and let \(V\) be the graph form for \(R\). Define the degree-one operators \[D_t=\mathrm{d}S_t\wedge(-),\qquad D=\mathrm{d}S_0\wedge(-).\] Both square to zero. The graph identities and Lemma 12 give \[ D_tU_t=0,\qquad DV=0,\qquad \mathrm{d}U_t=0,\qquad\mathrm{d}V=0 . \tag{15}\] In particular \(DU_0=0\) and the nonzero pairing (14) is defined.

We will use two elementary facts about this pairing. If \(\theta\) is a one-form and \(D_\theta=\theta\wedge(-)\), then, for homogeneous \(\alpha\), \[ (D_\theta\alpha)\wedge\beta =(-1)^{|\alpha|}\alpha\wedge D_\theta\beta. \tag{17}\] Thus a \(D_\theta\)-boundary wedges to zero with a \(D_\theta\)-cycle, even before integration. Also \(\mathrm{d}D+D\mathrm{d}=0\), since \(\mathrm{d}^2S_0=0\). Ordinary differentiation therefore induces a differential on the homology of \(D\); Equation (12) continues to test its boundaries against actual closed representatives.

Two homologies and a compatible projection

We have graph cycles with a nonzero pairing and actual boundary identities for every triple. The separate edge blocks now let us analyze the target graph class one factor at a time. We first take Koszul homology, then use the differential induced on it by ordinary exterior differentiation.

Let \(\Omega_i\) be the truncated form algebra in the block \(h_i\), and write \[D_i=\mathrm{d}F_i\wedge(-),\qquad H_i=H(\Omega_i,D_i).\] The tensor products of graded spaces and operators in what follows use the usual Koszul signs. The block separation gives \[(\Omega_p,D)=\bigotimes_{i=1}^n(\Omega_i,D_i),\qquad H(\Omega_p,D)=\bigotimes_{i=1}^n H_i .\] The second equality is the finite-dimensional Künneth isomorphism over the field \(\Bbbk\). Let \(\delta_i\) denote the differential induced on \(H_i\) by ordinary \(\mathrm{d}\), and let \(\delta\) be their tensor differential. For \(i\in I\), multiplication by \(b_i\) induces a degree-zero endomorphism \(B_i\) of \(H_i\), since it commutes with \(D_i\). Set \[K_i=\ker B_i .\] There is no assertion that \(K_i\) is a \(\delta_i\)-subcomplex.

Apply Lemma 15 to each active factor \((H_i,\delta_i,K_i)\), and choose an ordinary homology projection in each inactive factor. Denote these maps by \(P_i\), and put \(\Pi=\bigotimes_iP_i\). This tensor projection is chain-homotopic to the identity. Indeed, the telescoping identity for \(\mathop{\mathrm{id}}-\bigotimes_iP_i\), together with the individual homotopies and the graded tensor convention, supplies a homotopy for the total differential \(\delta\).

Let \(v=[V]\in H(\Omega_p,D)\). Equations (16) imply the hypotheses of Lemma 14, and \(\delta v=0\) by (15). Thus \(\Pi v\) is a sum of homogeneous pure tensors \[ v_1\otimes\cdots\otimes v_n \tag{18}\] such that every \(v_i\) is a \(\delta_i\)-cycle and at least \(m-2\) active factors lie in \(K_i\cap\ker\delta_i\). This conclusion follows by applying \(\Pi\) to the tensor subspaces in Lemma 14; the projection of a kernel factor is still a kernel factor. Since \(m\geq5\), every nonzero pure tensor in this sum has at least three such active factors.

Choose \(D_i\)-cycle representatives for all the factors in (18), and let \(V'\) be the resulting sum of actual forms. The chain homotopy for \(\Pi\) shows that \([V']-[V]=\delta[Z]\) for some class in \(H(\Omega_p,D)\). Choose a \(D\)-cycle representative \(Z\). There is then an actual form \(E\) such that \[ V'-V=\mathrm{d}Z+D E . \tag{19}\] Pairing with \(U_0\), equations (12), (15), and (17) give \[ \int U_0\wedge V'=\int U_0\wedge V . \tag{20}\] The representatives composing \(V'\) need not be closed for ordinary \(\mathrm{d}\). Their classes are \(\delta_i\)-cycles, which is precisely what was used to obtain (19).

Lifts over a three-parameter cube

The replacement class has the same pairing with the source graph as the original target class. Each of its tensor summands has at least three active factors lying in both the multiplication kernel and the kernel of the induced differential. Those three factors supply the parameter lifts used to force that summand’s pairing to vanish.

The boundary complex and its pairing

Define a nonunital commutative differential graded algebra \(C\) by \[ C^j=\left\{a\in\Omega^j(Y): a|_T\in \begin{cases} 0,&j=0,\\ \mathbb C\xi,&j=1,\\ \mathbb C\eta,&j=2,\\ 0,&j\geq3. \end{cases}\right\}. \tag{22}\] In degrees at least three the boundary condition is automatic. Multiplication is closed because degree-zero boundary values vanish, \(\xi\wedge\xi=0\), and all other positive-degree products in question vanish on the two-dimensional boundary.

We keep \(C\) for the scalar complex. The tensor product \(C\otimes\mathfrak{g}\) is a dg Lie algebra, with bracket given by the wedge product combined with the Lie bracket. For homogeneous \(a,b\in C\otimes\mathfrak{g}\) whose degrees sum to three, put \[\langle a,b\rangle=\int_Y\mathop{\mathrm{tr}}(a\wedge b).\] This pairing satisfies cyclic Stokes. Indeed, when the degrees of \(a,b\) sum to two, their boundary product is zero: either one has degree zero or both boundary traces are multiples of \(\xi\). Thus \[ \langle da,b\rangle+(-1)^{|a|}\langle a,db\rangle=0. \tag{23}\] Invariance of the matrix trace gives the corresponding cyclic identity for the bracket. No assertion that integration of a single degree-two element commutes with the differential is needed; its boundary trace may be a multiple of \(\eta\).

Choose representatives \(i:H^*(C)\to C\), a projection \(\operatorname{pr}:C\to H^*(C)\), and a degree-minus-one map \(K\) such that \[ dK+Kd=\mathop{\mathrm{id}}-i\operatorname{pr},\qquad K^2=Ki=\operatorname{pr}K=0,\qquad \operatorname{pr}i=\mathop{\mathrm{id}}. \tag{24}\] These maps exist by splitting the vector spaces into boundaries, cohomology representatives, and complements on which the differential is an isomorphism onto boundaries. Extend them by the identity on \(\mathfrak{g}\). In particular, the degree-one representatives have zero boundary pullback. Since \(H^0(C)=0\), we have \(Kd=\mathop{\mathrm{id}}\) on \(C^0\).

All series below are formal. Applying one of these linear maps means applying it separately to each coefficient, which is an actual smooth form. No boundedness or analytic convergence of \(K\) is required.

The central slice and its potential

Set \[H=H^1(C)\otimes\mathfrak{g},\qquad N=\dim H=3n, \qquad S=\mathbb C[[h_1,\ldots,h_N]],\] and let \(h\) denote the universal element of \(H\otimes S\). Its coordinates generate the maximal ideal \(\mathfrak{m}\) of \(S\). The recursion \[ s=i(h)-\tfrac12K[s,s] \tag{25}\] has a unique solution in \(C^1\otimes\mathfrak{g}\widehat\otimes\mathfrak{m}\): its linear term is \(i(h)\), and each higher homogeneous term is determined by previous terms. It satisfies \[\operatorname{pr}s=h,\qquad Ks=0, \qquad K P(s)=0, \qquad P(a)=da+\tfrac12[a,a].\] Conversely these three slice conditions imply (25), by (24). Write \(s|_T=v(h)\xi\).

We use the Chern–Simons transgression functional (Chern and Simons 1974, sec. 3). For a Lie-algebra-valued one-form \(a\), define \[ \mathcal C(a)=\int_Y\mathop{\mathrm{tr}}\left( \tfrac12a\wedge da+\tfrac16a\wedge[a,a]\right), \qquad f(h)=\mathcal C(s(h)). \tag{26}\] For any coefficient variation \(\dot a\), integration by parts gives \[ \dot{\mathcal C}(a) =\int_Y\mathop{\mathrm{tr}}(\dot a\wedge P(a)) -\tfrac12\int_T\mathop{\mathrm{tr}}(a\wedge\dot a). \tag{27}\] In the central slice the boundary term is zero.

For a formal form \(P\), its curvature ideal means the ideal generated by all its scalar coefficient functions, equivalently by applying arbitrary linear functionals on the vector space of Lie-algebra-valued forms coefficientwise. The next calculation in particular shows that the ideals occurring here are finitely generated.

A prescribed first-order boundary holonomy

The central slice fixes the surviving peripheral holonomy to the identity. The vertex construction will vary that holonomy by a square-zero parameter. Its curvature may then have a boundary component, so the next calculation includes that component in the gradient ideal.

Fix \(T_0\in\mathfrak{g}\) and a scalar parameter \(\tau\) with \(\tau^2=0\). Put \[S_\tau=S[\tau]/(\tau^2),\qquad u=\tau T_0.\] The formal topology is the one at \((h,\tau)=0\). Choose any linear extension \[K_{\mathrm{ext}}:\Omega^2(Y)\longrightarrow \mathop{\mathrm{im}}\bigl(K:C^2\to C^1\bigr)\] of the degree-two part of \(K\), and tensor it with \(\mathfrak{g}\). Define \(s=s(h,\tau)\) by \[ s=i(h)+u\widehat\zeta-\tfrac12K_{\mathrm{ext}}[s,s]. \tag{29}\] The recursion again determines a unique formal series. It reduces to the central slice when \(\tau=0\), and \[s-u\widehat\zeta\in C^1\otimes\mathfrak{g}\widehat\otimes S_\tau, \quad \operatorname{pr}(s-u\widehat\zeta)=h, \quad K(s-u\widehat\zeta)=0.\] Its boundary value has the form \[s|_T=u\zeta+v(h,\tau)\xi.\] The recursion implies \(K_{\mathrm{ext}}P(s)=0\). To see this directly, write its correction as an element \(\ell\) of the original one-form image of \(K\). The contraction identities give \(Kd\ell=\ell\); also \(di(h)=d\widehat\zeta=0\). Applying \(K_{\mathrm{ext}}\) to the curvature then gives exactly the recursion.

The boundary curvature is generally nonzero off the critical scheme: \[ P(s)|_T=c\,\zeta\wedge\xi,\qquad c=[u,v]. \tag{30}\] In particular it need not satisfy the degree-two boundary condition defining \(C\). The following argument keeps this additional component.

Based gauge and marked holonomies

The curvature and Jacobian ideals have been identified for both slices. We now identify their quotient schemes with based representations and track the two torus holonomies. Working on Artinian coefficient algebras retains the nilpotent structure used by the ideal calculations.

We use the connection \(d+a\) and parallel transport solving \(\dot U=-a(\dot\gamma)U\). With the chosen boundary paths this gives \[ G_\alpha=\exp(-u),\qquad G_\beta=\exp(-v) \tag{35}\] for a flat boundary connection \(u\zeta+v\xi\). Reversing the transport convention changes the corresponding signs throughout.

Let \(A\) be a local Artinian complex algebra with residue field \(\mathbb C\) and maximal ideal \(\mathfrak{m}_A\). All connections and gauge transformations in this paragraph reduce to the trivial ones modulo \(\mathfrak{m}_A\). The based gauge group consists of transformations equal to the identity at the fixed point of \(T\). On a connection already in the strip boundary form, we subsequently use gauge transformations equal to the identity on all of \(T\).

We collect the conclusions, including the first-order coefficient needed later.

The punctured filling and its algebraic critical chart

Let \(M\) be a solid torus with an open ball removed. Glue its torus boundary to \(\partial Y\), with the meridian glued to \(\alpha\), and write \[Z=Y\cup_{T^2}M,\qquad \partial Z=S^2.\] Choose a point on this sphere and a path to the chosen torus basepoint. The filling kills \(\alpha\) in the fundamental group. Thus its based representations are the based representations of \(\pi_1Y\) with \(G_\alpha=1\), and the path identifies their torus markings. The sphere boundary will supply the relative orientation used below. Set \(G=SL_2\). For a finite CW complex \(K\), write \(\mathop{\mathrm{Loc}}_G(K)=\operatorname{Map}(K_B,BG)\); a superscript \(\mathrm{fr}\) means that the local system is trivialized at the chosen basepoint. These are derived mapping stacks. In particular, framing is a derived fiber of evaluation at the basepoint, not a quotient by conjugation.

The invariant pairing \(\operatorname{tr}(XY)\) is nondegenerate on \(\mathfrak g=\mathfrak{sl}_2\) and defines a \(2\)-shifted symplectic form on \(BG\). Betti transgression gives a \(0\)-shifted symplectic form on \(\mathop{\mathrm{Loc}}_G(S^2)\) (Pantev et al. 2013, Theorem 2.5 and Corollary 2.6(4)). Restriction \[\mathop{\mathrm{Loc}}_G(Z)\longrightarrow\mathop{\mathrm{Loc}}_G(S^2)\] is Lagrangian (Calaque 2015, Definition 2.6, Claim 2.7, and Theorem 2.9). Here the hypotheses of these constructions have concrete meanings. The manifolds have finite CW type, so the Betti sources are \(\mathcal O\)-compact and their mapping stacks are derived Artin stacks locally of finite presentation. Integration over the fundamental class is the orientation map. Poincaré–Lefschetz duality supplies the required nondegeneracy for every locally constant perfect complex \(E\) over every base cdga \(D\), including homotopy-coherent coefficient systems: \[C^*(Z,S^2;E)\simeq \mathbf R\!\operatorname{Hom}_D\bigl(C^*(Z;E^\vee),D\bigr)[-3].\] To see the coefficient scope, local derived duality on contractible stars identifies the oriented interior dualizing complex with \(D[3]\) and takes \(E\) to \(E^\vee\otimes D[3]\). These natural local identifications descend with all higher restriction coherences; relative boundary cells give the displayed equivalence. The finite constructions in perfect fibers commute with derived base change. This is the compact-manifold application of (Calaque 2015, secs. 3.1.1–3.1.2), and includes adjoint coefficients. The isotropic homotopy is integration over the relative fundamental chain of \((Z,S^2)\).

The second Lagrangian is \[\mathop{\mathrm{Loc}}_G(S^2)^{\mathrm{fr}}\longrightarrow\mathop{\mathrm{Loc}}_G(S^2).\] Indeed, the based sphere stack is \(\{1\}\mathbin{\times^{\mathbf R}_G}\{1\}\), the derived self-intersection of the identity point in the smooth group \(G\). It is the affine odd vector space with functions \(\mathop{\mathrm{Sym}}(\mathfrak g^*[1])\), whose generators have degree \(-1\). On this scheme a form of exterior degree \(2+j\) has internal degree at most \(-2-j\). Thus neither internal degree \(-j\), used by a \(0\)-shifted closed two-form, nor internal degree \(-j-1\), used by its nullhomotopy, occurs. The pulled-back form is zero, and its isotropic path is unique up to homotopy. At the unique classical point, the tangent complex of the unframed sphere stack is \(C^*(S^2;\mathfrak g)[1]\); the framed tangent retains exactly the degree-two cohomology summand. Integration and trace pair this summand perfectly with the omitted degree-zero summand. This proves the Lagrangian nondegeneracy condition. There are no other classical points at which to check it.

The shifted intersection theorem therefore equips \[ \mathcal X =\mathop{\mathrm{Loc}}_G(Z)\mathbin{\times^{\mathbf R}_{\mathop{\mathrm{Loc}}_G(S^2)}} \mathop{\mathrm{Loc}}_G(S^2)^{\mathrm{fr}} =\mathop{\mathrm{Loc}}_G(Z)^{\mathrm{fr}} \tag{38}\] with a \((-1)\)-shifted symplectic structure (Pantev et al. 2013, Theorem 2.9). This is a derived scheme. To see the finite-presentation assertion directly, a compact connected three-manifold with nonempty boundary has a finite spine of dimension at most two: a handle decomposition relative to a collar can be chosen with no absolute three-handles. Collapsing a maximal tree of its one-skeleton gives one vertex, say \(r\) edges, and \(s\) two-cells. Consequently its framed local-system scheme is the derived fiber \[G^r\mathbin{\times^{\mathbf R}_{G^s}}\{1\},\] where the map records the attaching words. This description also shows that every fixed based holonomy is a regular \(G\)-valued function on the classical scheme.

The algebraic Darboux theorem applies over \(\mathbb C\), at the trivial classical point of \(\mathcal X\). More precisely, (Brav et al. 2019, Theorem 5.18(i)) supplies a Zariski local algebraic Darboux chart minimal at that point; in shift \(-1\), (Brav et al. 2019, Example 5.15) presents its algebra as \[ A^0[p_1,\ldots,p_N],\qquad |p_i|=-1, \qquad Qp_i=\frac{\partial F}{\partial x_i}, \tag{39}\] where \(A^0\) is smooth and the \(x_i\) are \(\acute{e}\)tale coordinates. Minimality means that the cotangent differential vanishes at the point (Brav et al. 2019, Definition 2.13); in this presentation it is the Hessian of \(F\). Subtracting the constant term therefore gives \(F\in(x)^3\). In the completed \(x\)-coordinates this is an algebraic power series. Each of the finitely many regular holonomy entries on the classical critical chart lifts through the quotient \(A^0\to A^0/\mathop{\mathrm{Jac}}(F)\). After shrinking the smooth chart, the lifts are regular, and hence give algebraic power series in the same coordinates. This assertion concerns the matrix entries of holonomy; it makes no assertion that their formal logarithms are algebraic.

The Darboux chart supplies an algebraic potential and regular holonomy functions on its critical scheme. To identify that potential with the particular Chern–Simons function \(f\), we need more than an isomorphism of critical schemes or equality of tangent pairings. The full relative closed-form comparison below will give a difference of potentials in \(\mathop{\mathrm{Jac}}(f)^2\). A coordinate correction can then remove that difference while acting trivially on the marked critical scheme. We state the comparison here and give its proof in Appendix 8, so that we can first follow its application to the edge complex.

Comparison of the formal trace forms

The following comparison identifies the full closed form on the cochain model, including the relative integration homotopy. Write \(\kappa(u,v)=\operatorname{tr}(uv)\), and denote its \(2\)-shifted closed form on \(BG\) by \(\omega_\kappa\). For a nonpositive augmented completed semifree coefficient cdga \((R,Q)\), let \(\Omega_R^p\) mean continuous coordinate forms, with their internal grading. These compute the exterior powers of the cotangent complex. If \(A_X\) is a commutative cochain model of a finite Betti source, put \[\operatorname{Cl}^2_R(A_X)^n =\prod_{j\geq0} \bigl(A_X\widehat\otimes\Omega_R^{2+j}\bigr)^{n-j}.\] Its differential combines \(d_X+\mathcal L_Q\) and coordinate de Rham differentiation \(\delta_R\), with the total-complex signs. The factors of \(A_X\) have exterior weight zero. Coordinate differentiation is extended by \(\delta_R(c\otimes u)=(-1)^{|c|}c\otimes\delta_Ru\). In particular a strict representative has only its \(j=0\) component. Completion in this notation is in the coefficient augmentation ideal; the displayed product is the separate completion in exterior weight.

The proof is given in Appendix 8. It constructs the comparison on the formal group nerve, extends it to derived coefficient algebras, and checks source evaluation and relative integration in the full closed-form complex. Throughout the comparison, the marked holonomies are the forward transports fixed in Equation (35).

A model preserving the integrated trace

We identify the cochain model of (38) with the edge complex \(C\). On \(S^2\), choose a closed area form \(\nu\). The inclusion \(\mathbb C\nu\) into the augmented-kernel forms on the based sphere is a quasi-isomorphism. Thus the sphere boundary condition may be imposed by requiring its trace to lie in \(\mathbb C\nu\), with zero trace in the other degrees. Restriction of unrestricted forms to a boundary is surjective, so this strict fiber computes the required homotopy fiber.

The representatives on the punctured solid torus can be chosen as follows. Extend \(\xi\) using a bump density in its longitudinal circle coordinate. Put the puncture and a tube from the support disk of \(\eta\) to the puncture in the complement of that longitudinal strip. A transverse Thom form of the tube extends \(\eta\) to a closed two-form \(\widetilde\eta\) whose sphere trace is the appropriately oriented \(\nu\). Denote the extended one-form by \(\widetilde\xi\). These choices give \[ d\widetilde\xi=d\widetilde\eta=0, \qquad \widetilde\xi\wedge\widetilde\eta=0. \tag{41}\] The span of these two forms, with zero multiplication, is a quasi-isomorphic subalgebra of the forms on \(M\) with the sphere boundary condition. In fact that complex has one-dimensional cohomology in degrees one and two and no other cohomology, as follows from the relative sequence for \((M,S^2)\); the displayed forms represent the two generators.

For gluing, use forms smooth on each piece with matching tangential traces. This piecewise smooth de Rham model computes the cochains of \(Z\); its integration is the sum over the two pieces, and their torus boundary terms cancel by Stokes. Now glue the small model to the unrestricted forms on \(Y\). Surjectivity of restriction from \(Y\) to the torus shows that replacing the model on \(M\) by this quasi-isomorphic subalgebra preserves the homotopy pullback. The resulting algebra is exactly \(C\): its torus traces are zero in degree zero, multiples of \(\xi\) in degree one, and multiples of \(\eta\) in degree two. Moreover, (41) shows that every complementary-degree pairing on the added piece vanishes before integration. The full integrated pairing on the filling is consequently \[ \langle u,v\rangle =\int_Y\operatorname{tr}(u\wedge v) \tag{42}\] on \(C\otimes\mathfrak g\). The comparison of full closed forms and of the relative integration homotopy in Proposition 22 applies to this replacement. It identifies the shifted symplectic structure of \(\mathcal X\), rather than just its value on tangent cohomology, with this strict integrated form.

Choose the basing path through \(M\) outside the longitudinal strip. The degree-one connection on \(M\) is a multiple of \(\widetilde\xi\), so its transport along this path is the identity. The resulting comparison uses the fixed torus framing and preserves based holonomies as matrices, including the chosen \(\beta\)-holonomy.

Transfer with an arbitrary cochain contraction

The preceding model identifies the integrated trace form and the based holonomies. We next transfer it to the finite coordinate space used by the edge slice, keeping the original contraction. The required output is the Hamiltonian identity for the particular potential \(f\).

Write \(H^j=H^j(C)\otimes\mathfrak g\), and retain the splitting \((i,\operatorname{pr},K)\) from the edge construction, so \[dK+Kd=1-i\operatorname{pr},\qquad K^2=Ki=\operatorname{pr}K=0.\] No compatibility of \(K\) with the pairing is required. Choose bases of \(H^1,H^2\), and let \(h\) and \(q\) be their corresponding coordinates of degrees zero and \(-1\). In the completed semifree algebra \(R[q_1,\ldots,q_N]\), set \[ s=i_1h-\tfrac12K[s,s],\qquad P=ds+\tfrac12[s,s],\qquad r=\operatorname{pr}P, \qquad V=(1+K\operatorname{ad}_s)^{-1}i_2. \tag{43}\] The inverse exists in the \((h)\)-adic topology. The contraction identity and Bianchi identity give \(P=i_2r-K[s,P]\), and hence \[ P=Vr,\qquad a=s+Vq,\qquad Qh=0, \qquad Qq=-r. \tag{44}\] Here and below the suspension signs in the integrated symplectic form are fixed compatibly with the last convention.

For completeness, these formulas give the entire transferred Maurer–Cartan map. Put \(W=dV+[s,V]\). The recursion for \(V\) gives \(KV=0\), \(\operatorname{pr}V=1\), and \[KW=KdV+K[s,V]=V-i_2+K[s,V]=0.\] The columns of \(W\) are three-forms in \(C\otimes\mathfrak g\). They have zero differential by dimension and zero cohomology projection because \(H^3(C)=0\); the contraction identity therefore gives \(W=0\). The bracket of two columns of \(Vq\) is a physical four-form and is zero. Equations (44) now imply \[(d+Q)a+\tfrac12[a,a]=0.\] In particular there are no omitted terms quadratic in \(q\). The induced map of formal moduli problems has the identity on tangent cohomology, and is a formal equivalence. One can also see this directly by induction over nilpotent coefficient ideals: each square-zero extension has the same cochain deformation and obstruction groups, and the contraction identifies those groups.

Pull back the strict integrated closed form by this map and denote it by \(\omega\). It consists of \(dh\,dq\) terms and \(q\,dh\,dh\) terms. Its constant \(dh\,dq\) matrix is the perfect pairing \(H^1\otimes H^2\to\mathbb C\), so it is formally nondegenerate. Explicitly, give coordinate forms total parity equal to internal degree plus exterior degree, and extend coordinate differentiation across a physical form \(c\) by \(\delta(c\otimes u)=(-1)^{|c|}c\otimes\delta u\). The mixed part is then \[\omega_{\mathrm{mixed}} =-\sum_{i,\mu}\left\langle\partial_i s,V_\mu\right\rangle dh_i\,dq_\mu.\] The contraction \(\iota_Q\) is total-even. Since \(\iota_Qdq_\mu=-r_\mu\), contraction removes the \(dq\) and gives the pairing with \(Vr=P\); its value on the \(q\,dh\,dh\) terms is zero. The variation formula for the central Chern–Simons functional, whose boundary term vanishes, consequently gives the exact identity \[ \iota_Q\omega =\int_Y\operatorname{tr}(\delta s\wedge P) =\delta f. \tag{45}\] The symbol \(\delta\) differentiates only the formal coordinates. This proves the Hamiltonian identity for the chosen, possibly noncyclic, contraction. Reversing the global symplectic sign reverses both algebraic and Chern–Simons Hamiltonians and has no effect on the statement of Theorem 21.

A closed-form homotopy gives a squared-Jacobian error

We now have two potentials on equivalent minimal formal models: the algebraic Darboux potential and the Chern–Simons potential. The comparison already identifies their critical schemes and based holonomies. We use its additional full closed-form homotopy to place the difference of potentials in the square of the Jacobian ideal. This stronger conclusion permits a coordinate correction that fixes the critical scheme.

Compare the two minimal completed semifree models, the transferred model and the Darboux model (39), over the formal completion of \(\mathcal X\). A map between these models is represented by a formal power-series map: equivalently one uses homotopy transfer and maps of minimal Lie models, or lifts maps of completed semifree algebras order by order. An equivalence has invertible linear part on generators, since the cotangent differentials at the point are zero. It therefore has a formal inverse. Degree considerations make its shape particularly simple: \[ x=x(h),\qquad p=B(h)q, \tag{46}\] where the Jacobian of \(x(h)\) and the matrix \(B(h)\) are invertible. Let \(f'=F(x(h))\), and let \(\omega'\) be the pullback of the canonical Darboux form. Then \[\iota_Q\omega'=\delta f',\qquad (Qq_1,\ldots,Qq_N)=\mathop{\mathrm{Jac}}(f')=\mathop{\mathrm{Jac}}(f)=J\subset(h)^2.\] The first equality is naturality of the Darboux Hamiltonian identity; the equality of ideals follows from the two invertible matrices in (46) and from the edge curvature calculation. The comparison is over \(\mathcal X\), so it also identifies all the marked classical holonomy functions.

Both strict forms represent the same shifted closed-form class. We record explicitly the consequence for the Hamiltonians; equality of the ordinary leading two-form alone would not suffice.

Polynomial replacement of the active logarithm

Make the fixed coordinate change of the theorem separately on each edge, and again write \(f\) for its now algebraic central potential. For an active edge choose an algebraic matrix lift \(G_\beta\) of the based \(\beta\)-holonomy, with \(G_\beta(0)=1\). It suffices to lift its entries; all identities with the actual holonomy below are identities in \(R/\mathop{\mathrm{Jac}}(f)\). Set \[I_\beta=((G_\beta-1)_{ij})\subset R, \qquad L_d(G_\beta)=\sum_{k=1}^{d}\frac{(-1)^{k+1}}{k}(G_\beta-1)^k.\] With the connection convention \(d+a\) of the edge construction, \(G_\beta=\exp(-v)\) on the central critical scheme. Its active coefficient satisfies \[b\equiv-\epsilon\operatorname{tr} (T_0\log G_\beta)\pmod{\mathop{\mathrm{Jac}}(f)}.\] Define the algebraic germ \(b_{\mathrm{alg}}=-\epsilon\operatorname{tr}(T_0 L_d(G_\beta))\). For every \(d\geq2\), the \((h)\)-adically convergent logarithm series gives \[ b-b_{\mathrm{alg}}\in\mathop{\mathrm{Jac}}(f)+I_\beta^{d+1} \subset\mathop{\mathrm{Jac}}(f)+I_\beta^3. \tag{48}\] If \(b-b_{\mathrm{alg}}=\sum_i c_i\partial_i f+r\), with \(r\in I_\beta^{d+1}\), then the square-zero coordinate change \(x_i\mapsto x_i-\tau c_i\) transforms \(f+\tau b\) into \[f+\tau(b_{\mathrm{alg}}+r), \qquad \tau^2=0.\] Thus the Jacobian part is removed exactly. The remaining error has at least cubic order in the multiplicative holonomy entries; it is this explicit error, rather than an algebraicity assertion about the formal logarithm, that will be used in the vertex matching.

The linear restriction maps

Put \(\mathfrak{g}=\mathfrak{sl}_2(\mathbb C)\) and use the invariant pairing \(\kappa(U,V)=\mathop{\mathrm{tr}}(UV)\). The internal edge space is \[\mathcal H=\bigoplus_i H^1(Y_i,\partial Y_i;\mathbb C)\otimes\mathfrak{g}.\] For a connected oriented three-manifold with connected torus boundary, the natural map from relative degree-one cohomology identifies it with the kernel of restriction to the boundary. Thus these are precisely the internal spaces in Proposition 20.

At this linear stage, fixing an evaluation means requiring its tangent variation to vanish. For \(P=A,B\), let \(\mathcal V_P\) be the common kernel in \(H^1(P;\mathbb C)\otimes\mathfrak{g}\) of evaluation on the extra letters at \(P\) and on the \(\alpha_i\) whose plus endpoint is \(P\). The degree-one assertion below is the isomorphism \(\mathcal V_A\oplus\mathcal V_B\to\mathcal H\). Later prescribed holonomies need not be the identity; it is their variations that are fixed in this tangent-space statement.

The pairing \[ \bigoplus_i H^1(Y_i,\partial Y_i)\otimes\mathfrak{g} \ \times\ \bigoplus_i H^2(Y_i)\otimes\mathfrak{g} \longrightarrow\mathbb C, \qquad (u,v)\longmapsto\sum_i\int_{Y_i}\kappa(u\wedge v) \tag{53}\] is perfect by Poincaré–Lefschetz duality. We will use both the degree-one isomorphism and this degree-two pairing.

Marked formal backgrounds and vertex slices

The boundary laws will be tested at a finite family of conjugating matrices whose adjoint operators span every endomorphism of \(\mathfrak{g}\). Introduce a formal parameter \(\sigma\) and set \(K=\mathbb C((\sigma))\). Choose nine matrices in \(SL_2(\mathbb C[[\sigma]])\), one the identity, all equal to the identity at \(\sigma=0\), whose adjoint operators span \(\mathop{\mathrm{End}}_K(\mathfrak{g}\otimes K)\). Equation (60) gives an explicit family, and Lemma 29 verifies its spanning property. The family is fixed independently of the cuts. Its independent actions on the three slots will detect every triple coefficient in the boundary laws. Assign the eight nonidentity values to eight extra letters at each vertex and set the remaining extra holonomies to the identity. Write \(g_s(\sigma)\in SL_2(\mathbb C[[\sigma]])\), with \(g_s(0)=1\), for these assigned holonomies. These choices extend to a flat background on the whole ambient bipartite model with trivial meridian holonomies. To see the marking explicitly, choose the paths through the joins as edge paths of the free graph groupoid. The extra loops are its vertex generators, and edge transports can be chosen freely. Choose those transports so that the two torus frames at every join agree with the frames determined by the boundary basing paths. This simultaneously fixes the markings used for holonomy and for the boundary laws.

Initially all constructions take place jointly formally at \(\sigma=0\). The restricted flat backgrounds have coordinates of positive \(\sigma\) order in the central vertex and edge charts. We do not yet invert \(\sigma\).

Fix \(P=A\) or \(P=B\), with its basepoint in \(K_P\). In degree zero use smooth \(\mathfrak{g}\)-valued functions vanishing at that point, and in positive degrees use unrestricted forms on \(P\). Choose a contraction onto cohomology, denoted by inclusion \(\iota\), projection \(\pi\) and homotopy \(\mathsf K_P\), with \[\mathrm{d}\mathsf K_P+\mathsf K_P\mathrm{d}=1-\iota\pi, \qquad \mathsf K_P^2=\mathsf K_P\iota=\pi\mathsf K_P=0.\] Degree-zero cohomology is zero. For \(y_{\rm full}\in H^1(P;\mathbb C)\otimes\mathfrak{g}\), the recursion \[a_P=\iota y_{\rm full}-\tfrac12\mathsf K_P[a_P,a_P]\] defines a unique formal one-form. Write \[\mathcal F_P=\mathrm{d}a_P+\tfrac12[a_P,a_P],\qquad r_P=\pi\mathcal F_P.\] The list \(r_P\) has \(q_P=\dim(H^2(P;\mathbb C)\otimes\mathfrak{g})\) entries, of order at least two. The slice equation and Bianchi identity give \[ \mathcal F_P=\iota r_P-\mathsf K_P[a_P,\mathcal F_P] =(1+\mathsf K_P\mathop{\mathrm{ad}}_{a_P})^{-1}\iota r_P. \tag{54}\] In particular the curvature ideal is exactly the ideal generated by \(r_P\), and its leading coefficient forms are the selected representatives of \(H^2(P)\otimes\mathfrak{g}\). Successively removing exact one-form coordinates by based gauges shows that every based flat formal connection is represented by this slice modulo \(r_P\). These constructions are functorial over nilpotent coefficient rings and pass to their inverse limits. No regular-sequence assertion about \(r_P\) is made.

Fix nonzero \(T_i\in\mathfrak{g}\) for the active edges. We use the connection convention of Proposition 20, in which the strip connection \(u\zeta\) has holonomy \(\exp(-u)\). Impose on the vertex slice the exact logarithmic holonomies

• \(\log g_s(\sigma)\) on the extra letters;

• \(-t_iT_i\) on an active plus meridian at \(A\);

• zero on every plus meridian at \(B\).

There are no parameters on the other plus ends. Logarithmic holonomy along a fixed based loop is a formal function of the off-shell connection, whose linear term is minus integration of the connection along that loop. Lemma 26 therefore makes these conditions a smooth formal parameter slice. Keep its other coordinates \(y_P\) free. The resulting coefficient rings are \[R_A=\mathbb C[[\sigma,y_A]][t_i:i\text{ active}]/(t_i^2), \qquad R_B=\mathbb C[[\sigma,y_B]],\] completed at their displayed joint origins. Write \(I_P=(r_P)\) for the substituted residual ideal. It is contained in the square of the joint maximal ideal. Modulo \(I_P\) there is an actual based flat system with exactly the prescribed extra and plus holonomies. We keep this list \(r_P\) as functions on the parameter domain throughout the subsequent normalizations and section corrections. Thus the flat quotient \(R_P/I_P\) stays fixed even when its chosen lifts to forms and edge coordinates change.

For an active edge, the induced holonomies in the flat quotient \(R_A/I_A\) satisfy \[G_{\alpha_i}=\exp(-t_iT_i),\qquad G_{\beta_i}=1.\] In the flat quotient \(R_B/I_B\) they satisfy \(G_{\alpha_i}=1\), while the marked \(G_{\beta_i}\) varies. The central edge potential there contributes \(-f_i\) because the boundary orientation is reversed, where \(f_i\) is the central potential of Proposition 20. These holonomy identities concern the respective flat quotients, not the off-shell lifts.

Boundary normalization and lifts of the flat quotient

The quotient \(R_P/I_P\) carries the flat systems and their group laws. The tensor theorem, however, requires sections in the ambient edge coordinates, where the curvature need not vanish. We first normalize the restrictions on the flat quotient and then lift them to these ambient, or off-shell, coordinates. Changes in the lifts will be measured against the fixed list \(r_P\).

Recall that \(K_P\) is a connected sum of \(p\) copies of \(S^1\times S^2\) with tubes about an unlink removed. In the unlink exterior choose disjoint proper disks dual to the port meridians, with boundaries the unlink longitudes. In the connected-sum pieces choose disjoint nonseparating spheres dual to the extra \(S^1\) generators. Thin mutually disjoint collars of these disks and spheres carry closed Thom one-forms \(\rho_\nu\), normalized to evaluate to one on the corresponding generator. The disk strips meet the port tori in the coordinate densities dual to the port meridians; the sphere strips miss those tori. Choose the basing arcs off the strips. Since the port meridians and extra letters freely generate \(\pi_1K_P\), a flat system with their specified holonomies has the representative \[a_{K_P}=\sum_\nu M_\nu\rho_\nu,\] where the \(M_\nu\) are the corresponding signed logarithms. Each strip form is closed and their cross wedges vanish, so this connection is flat for arbitrary matrix coefficients. At a plus torus its trace is \(u_i\zeta_i\), and at a minus torus its trace is a multiple of \(\xi_i\). The two strip choices are the compatible torus choices of Proposition 20. The area representatives can be supported away from both strips. In particular, \[ \mathcal C_{K_P}(a_{K_P})=0. \tag{55}\]

Normalize first over \(R_P/I_P\). Based parallel transport gives a gauge from the restriction of \(a_P\) to this strip representative. After this gauge, the flat restriction on every \(Y_i\) can be put into the edge slice \(s_i(h_i,t_i)\), or \(s_i(h_i,0)\) as appropriate, by a gauge equal to the identity on its torus. This gives coordinates \(h_i^P(y_P)\) over \(R_P/I_P\). The normalizations are compatible on intersections of boundary faces: the second gauges are the identity on the tori.

To lift these quotient-ring data to actual formal forms, choose a power-series presentation of the Noetherian complete ring \(R_P\) and fix continuous linear division operators for the finite list \(r_P\) and for passage to \(R_P/I_P\). Such operators are obtained by formal division, or by choosing compatible filtered linear splittings. Applied coefficient by coefficient, they lift scalar series and write a form-valued discrepancy in \(I_P\) as \(\sum_a r_{P,a}\gamma_a\) with form-valued series \(\gamma_a\). Each output coefficient involves only finitely many input coefficients. Thus the same operators apply to the vector spaces of smooth functions and differential forms. This is the actual finite-list ideal membership used below.

Lift the near-identity gauges by first lifting their Lie-algebra-valued logarithms. Smooth collar extension extends each logarithm off the relevant boundary face; the compatibility at the tori allows the extensions to be combined. Lift the coordinates \(h_i^P\) as well. Finally change the connection by \(I_P\)-multiples of one-forms, extended over a collar of \(K_P\), so that its restriction to \(K_P\) is exactly the strip representative with the constrained plus coefficients. Denote this trial connection by \(\widehat a_P\). It has the following properties:

1. its curvature belongs to \(I_P\);

2. its restriction to \(Y_i\) equals the specified edge slice modulo \(I_P\);

3. to first order in the actual residual list, modulo the joint maximal ideal times that list, its curvature representatives are those in Equation (54), changed only by exact two-forms.

For the last statement, write an added one-form as \(\eta=\sum_a r_{P,a}\gamma_a\) and put \(\mathcal F(a)=\mathrm{d}a+\tfrac12[a,a]\) for an intermediate connection \(a\). The source differential treats the functions \(r_{P,a}\) of the formal parameters as constants. Hence \[\begin{aligned} \mathcal F(a+\eta) &=\mathcal F(a)+\sum_a r_{P,a}\,\mathrm{d}\gamma_a +[a,\eta]+\tfrac12[\eta,\eta]\\ &\equiv\mathcal F(a)+\sum_a r_{P,a}\,\mathrm{d}\gamma_a \pmod{\mathfrak m_P I_P}. \end{aligned}\] The congruence is coefficientwise, and \(\mathfrak m_P\) is the joint maximal ideal: \(a\) has positive order and \(I_P\subset\mathfrak m_P^2\). A gauge with identity value at the joint origin also changes curvature only modulo \(\mathfrak m_PI_P\). Thus the leading coefficient forms of the fixed residual list change by exact two-forms. This calculation uses a chosen expression for \(\eta\) and requires no independence of the \(r_{P,a}\). The lifts have the same first derivatives as the original quotient coordinates because \(I_P\) has order at least two.

The square-ideal and gradient identities

We have lifted the flat boundary restrictions to ambient edge coordinates without changing them modulo the fixed residual ideal. The next calculation measures the error in the sum of their potentials and identifies the restricted gradient ideal. These two statements will permit correction of the sections while preserving their flat-system markings.

Let \(f_i\) and \(b_i^{\rm form}\) be the edge functions before algebraizing the perturbations. With the orientations already chosen, put \[S_A=\sum_i f_i+\sum_{i\text{ active}}t_i b_i^{\rm form}, \qquad S_B=-\sum_i f_i.\] Write \(H_P\) for the lifted parametrization consisting of all \(h_i^P\).

We record the formal correction argument used at the joint origin.

Apply Lemma 28 to the identities of Lemma 27. The central edge functions have order at least three; the other terms carry a parameter \(t_i\). Hence the internal Hessian vanishes at the joint origin. We obtain corrected sections, still denoted by \(H_A,H_B\), with \[ S_A(H_A)=S_B(H_B)=0, \qquad ((\partial_h S_P)(H_P))=I_P. \tag{58}\] They retain their complementary tangents and their values modulo \(I_P\). In particular they retain exactly the marked edge holonomies on the flat residual schemes.

Algebraic central coordinates and the order of completion

The corrected sections already satisfy exact vanishing and retain the marked holonomies modulo their residual ideals. We now make the central edge potentials algebraic and move the common background to the origin. The order of this recentering and the subsequent passage to Laurent coefficients is part of the construction.

Apply Theorem 21 separately on the edges, initially at the trivial system. Thus, after separate invertible formal changes of the \(h_i\), the central functions are algebraic germs \(F_i\). On their critical schemes the marked \(\beta_i\) holonomies have algebraic lifts \(G_i\) in the smooth ambient germs. Choose these lifts with \(G_i(0)=1\). The functions \(b_i^{\rm form}\) satisfy \[ b_i^{\rm form}\equiv -\epsilon_i\mathop{\mathrm{tr}}(T_i\log G_i) \pmod{\mathop{\mathrm{Jac}}(F_i)}. \tag{59}\] Here the sign is the one fixed by \(\operatorname{Hol}_{\beta_i}=\exp(-v_i)\). Write the Jacobian-ideal difference as \(\sum_a c_{i,a}\partial_{h_{i,a}}F_i\). A separate coordinate change linear in \(t_i\) absorbs it: the identity \[F_i(h_i+t_i c_i(h_i)) =F_i(h_i)+t_i\sum_a c_{i,a}(h_i)\partial_{h_{i,a}}F_i(h_i)\] is exact because \(t_i^2=0\). Choosing the sign according to the direction of the change replaces the coefficient by \[\ell_i(G_i)=-\epsilon_i\mathop{\mathrm{tr}}(T_i\log G_i).\] The changes are separate in the edge blocks and preserve all equations and ideal memberships on the transformed sections. At parameter zero the changes linear in \(t_i\) are the identity.

Let \(y_P^0(\sigma)\) and \(h_i^0(\sigma)\) be the coordinates of the fixed flat background in these charts. They have positive \(\sigma\) order. Both vertex sections contain the same edge background, because all earlier corrections vanish on their residual schemes. Recenter by \(y_P=y_P^0(\sigma)+z_P\) and \(h_i=h_i^0(\sigma)+z_i\) before passing to Laurent coefficients. Substitution gives continuous maps \[\mathbb C[[\sigma,y_P]]\longrightarrow\mathbb C[[\sigma,z_P]], \qquad \mathbb C[[\sigma,h_i]]\longrightarrow\mathbb C[[\sigma,z_i]].\] For each coefficient in the new internal variables, the defining sum converges \(\sigma\)-adically. We then include the coefficient ring in \(K=\mathbb C((\sigma))\) and complete in the new internal variables. We rename them \(y_P,h_i\).

This order also describes the changes already used: if \(\phi\) is a separate original formal coordinate change, its recentered expression is \[z\longmapsto\phi(h_i^0(\sigma)+z)-\phi(h_i^0(\sigma)),\] formed over \(\mathbb C[[\sigma,z]]\) first. Later inverses over \(K\) may have coefficients with unbounded negative \(\sigma\) orders. They are only composed with centered series, or with series whose constant terms are nilpotent parameter multiples; these substitutions are well defined in \(K[[z]][t_i]/(t_i^2)\). In particular, an absorbed Jacobian term gives a separate square-zero change \[z_i\longmapsto z_i+t_i c_i(z_i),\qquad z_i\longmapsto z_i-t_i c_i(z_i) \quad\text{for its inverse}.\] Its constant displacement is nilpotent. No later substitution shifts an arbitrary Laurent-coefficient series by a nonzero \(\sigma\)-dependent constant, and no operation introduces relations among the external parameters. Artin approximation is applied only after the final graph identities have been obtained, over the field \(K\) as in Lemma 9.

The central functions and the \(G_i\) are still algebraic germs over \(K\): substitute the background values, which are elements of \(K\), into their finite algebraic presentations. The selected background is critical on every central edge and \(G_i(h_i^0)=1\). Moreover \(F_i(h_i^0)=0\). Indeed \(\partial F_i\) vanishes along the formal background curve, so its derivative with respect to \(\sigma\) is zero; its value at \(\sigma=0\) was zero. We retain these normalizations after recentering.

The complementary tangent determinant and the gradient minors have nonzero constant terms at the original joint origin. Their evaluations at the background are units of \(\mathbb C[[\sigma]]\), and hence are nonzero over \(K\). Thus the two recentered sections are still transverse with complementary dimensions, and their gradient ideals still equal the recentered residual ideals. The latter may now have linear terms; no later argument requires their original quadratic order.

Conjugated laws on the full residual schemes

The prescribed extra holonomies can be chosen explicitly. In the basis \((e,h,f)\) of \(\mathfrak{sl}_2\), put \[E=\mathop{\mathrm{ad}}(e)=\begin{pmatrix}0&-2&0\\0&0&1\\0&0&0\end{pmatrix}, \qquad F=\mathop{\mathrm{ad}}(f)=\begin{pmatrix}0&0&0\\-1&0&0\\0&2&0\end{pmatrix}.\] For \(a,b\in\{0,1,2\}\) prescribe \[ g_{ab}(\sigma)=\exp(a\sigma e)\exp(b\sigma f). \tag{60}\] The identity conjugator supplies \(g_{00}\) and eight extra letters supply the other eight matrices. These choices can be made at both vertices. They require only a finite list of conjugations in the geometric construction, chosen before the cuts are made.

Polynomial perturbations and the final graph families

The triple-ideal identities are now established on the full residual schemes. The remaining task is to replace the logarithmic perturbation by an algebraic function without losing exact vanishing or gradient generation. Algebraic lifts of the holonomy matrices suffice: a quadratic polynomial in those matrices has a controlled cubic error, which can be removed on the source graph by a nilpotent correction.

For an active edge replace the logarithmic function by the polynomial \[ b_i(h_i)=-\epsilon_i\mathop{\mathrm{tr}}\left( T_i\left((G_i(h_i)-1)-\tfrac12(G_i(h_i)-1)^2\right)\right). \tag{62}\] These are algebraic germs over \(K\), with zero constant term at the recentered background. We check the exact effects of this replacement.

On the plus section, setting \(t_i=0\) gives, modulo \(I_A\), a central critical system on edge \(i\) with both peripheral holonomies trivial: \(\alpha_i\) is constrained to be trivial and \(\beta_i\) bounds in \(K_A\). The marked algebraic holonomy therefore satisfies \[ G_i(H_A)-1\in (I_A,t_i) \tag{63}\] entry by entry. This is equality in the quotient ring, not just evaluation at its closed points. The coordinate change used to absorb the difference in Equation (59) is immaterial after setting \(t_i=0\).

Let \(R_i=\ell_i(G_i)-b_i\). It has order at least three in the entries of \(G_i-1\), and each internal derivative has order at least two in those entries. Since \(t_i^2=0\), Equation (63) gives \[\begin{align*} t_iR_i(H_A)&\in t_i(I_A,t_i)^3=t_iI_A^3,\tag{64}\\ t_i(\partial_hR_i)(H_A)&\in t_i(I_A,t_i)^2=t_iI_A^2. \tag{65}\end{align*}\] Thus the new total function on the existing plus section lies in \(\mathfrak t I_A^2\), where \(\mathfrak t=(t_i:i\text{ active})\), and its restricted gradient still has the form \((M_A+\mathfrak t Q)r_A\) with a unit full-column minor.

The correction at the joint origin used the vanishing internal Hessian and gained powers of the joint maximal ideal. After recentering, the Hessian may be nonzero. Here the error instead contains a factor from the nilpotent parameter ideal, and the following correction gains powers of that ideal while keeping the same residual list.

Apply Lemma 31 over \(T[[y_A]]\), with \(J=\mathfrak t\), \(\mathcal I=I_A\), and the fixed residual list \(r_A\). The preceding estimates give an error in \(\mathfrak t I_A^3\subset\mathfrak t I_A^2\) and a unit full-column gradient minor against that same list. The section is centered modulo \(\mathfrak t\): \(H_A(0,0)=0\), so \(H_A\in(y_A,\mathfrak t)^N\). Formal substitution is continuous even with its nilpotent constants: write \(H_A=H_A(y_A,0)+\eta\), where \(\eta\in\mathfrak t T[[y_A]]^N\); expansion in \(\eta\) is finite and substitution of \(H_A(y_A,0)\) is centered. The central tangent remains injective by Lemma 26 and the preceding coordinate changes. Thus the lemma gives a corrected source section \(H_L\) with exact vanishing for the polynomial perturbations. The corrections lie in \(\mathfrak t I_A\) and preserve the gradient ideal \(I_A\). The ideal \(\mathfrak t\) is nilpotent of exponent at most \(m+1\); its square is not set to zero. In particular, mixed parameter monomials are retained at every step. A nonzero Hessian of a recentered central potential causes no difficulty because the Taylor remainders gain powers of \(\mathfrak t\).

The minus function is still \(S_0\) up to its harmless overall sign, so the minus section needs no additional correction. Denote it by \(H_R\). Every weight in Equation (62) has zero constant term in the entries of \(G_i-1\). Lemma 30 therefore gives \((b_ib_jb_k)(H_R)\in I_B\) for distinct active indices. The source parameter products remain in \(I_A\). Since the final restricted gradients generate these same ideals, these are exactly Equations (50) and (51).

For completeness, the corrected source is a smooth graph over the full ring \(T\), rather than just on its reduced central fiber. Choose a linear projection \(\rho\) on the internal coordinate space which is invertible on the central tangent of the source. The Jacobian of \(y_A\mapsto\rho H_L(y_A,t)\) is invertible modulo \((t)\) and hence is a unit over \(T[[y_A]]\). The constant term of \(\rho H_L\) can lie in the nilpotent ideal \((t)\); translation by that constant is continuous for the \((y_A,t)\)-adic topology, which here agrees with the \(y_A\)-adic topology. The formal inverse theorem over \(T\) reparametrizes this map as \[H_L(x,t)=(x,\Psi(x,t)),\] while leaving the parameters fixed. All memberships pull back under this same \(T\)-algebra automorphism. At \(t=0\) the last corrections vanish, so complementarity with \(H_R\) is preserved. No flatness of \(T[[y_A]]/I_A\) over \(T\) is required.