Relative-spin disk determinants and the actual suspended boundary sign

Continuous cellular homotopies and actual Fredholm transport. Cellular approximation in [F2] is continuous, not smooth. First cellularly approximate the boundary map and extend that homotopy over the disk by the CW homotopy extension property used in that supplier; then approximate the disk relative to its now cellular boundary. This represents its pair class by $g:(D,\mathrm{\partial }D)\to (M^{(2)},L^{(1)})$. For a path or loop of disks do the same on the side boundary first, relative to prescribed cellular end disks when present, and then on the total finite CW source relative to that side and those ends. The resulting family lies in $M^{(3)}$ with its boundary family in $L^{(2)}$. The homotopy to it lies in $(M,L)$ and need not remain in any skeleton. We now supply its missing analytic bridge without asserting a smooth map into an arbitrarily triangulated skeleton. Pull back $(TM,J;TL)$ along this continuous homotopy. This is a continuous complex bundle $E$ with a continuous totally real real subbundle $F$ on the side boundary of its compact smooth source; that source is $D$ times finitely many interval or circle parameters. For this construction first use smooth analytic end maps or smooth finite-parameter families in the ordinary smooth-boundary configuration charts. A continuous path of smooth disks can be reduced to this case: subdivide into finitely many boundary-preserving exponential section charts, replace each track by the straight segment between its smooth endpoint sections, and interpolate between the old and new tracks inside that chart. These homotopies agree at the subdivision endpoints and preserve the boundary condition; flatten the new parameter tracks near their joints. Operator-norm continuity makes the determinant transport invariant under this relative path homotopy by the local augmentation construction below. Loops are treated with the same finite cyclic subdivision. At nodes use the separate matched/smoothing determinant construction in [F1]. Reparametrize cellular homotopies to be constant near their end faces. At an analytic end retain its actual smooth bundle pair. At a cellular end choose a smooth representative of its pulled-back bundle pair by the following explicit relative construction. A continuous bundle on a compact base embeds in a finite trivial bundle: choose finite local frames and a subordinate partition of unity and take their weighted coordinate embeddings. Choose the embedding smooth on prescribed collars where a smooth model has already been fixed, extending the collar embedding by the same partition construction in additional ambient coordinates. Its Hermitian projection $P$ is continuous and smooth on those collars. Approximate its matrix entries uniformly by smooth entries, relative to smaller such collars, using a finite coordinate cover, convolution and smooth cutoffs; at corners first extend coordinate functions by reflection. Symmetrize the approximation $A$ and take its spectral projection $P^{\mathrm{\prime }}$ for eigenvalues greater than $1\mathrm{/}2$. If the error is less than $1\mathrm{/}4$, the eigenvalues stay in disjoint neighborhoods of zero and one; the finite-dimensional resolvent formula around the latter gives a smooth rank-preserving projection and a homotopy through projections to $P$. The map $P^{\mathrm{\prime }}{\mathrm{\mid }}_{\mathrm{im}P}$ is an isomorphism close to identity. On the side boundary apply the same real-symmetric construction to the real orthogonal projection onto $F$, then project its image into $\mathrm{im}{P}^{\mathrm{\prime }}$. This gives a smooth real rank-$n$ bundle $F^{\mathrm{\prime }}$ inside $E^{\mathrm{\prime }}=\mathrm{im}{P}^{\mathrm{\prime }}$. Compactness gives a positive lower bound for the least singular value of $(v,w)\mapsto v+iw$ on $F\oplus F$, so sufficiently close approximation keeps $F^{\mathrm{\prime }}$ totally real. The resulting real isomorphism $a:F\to F^{\mathrm{\prime }}$ is close to identity and preserves any given orientation. Its complexification $a(v+iw)=a(v)+ia(w)$ identifies the entire boundary complex bundle. On a side collar interpolate this identification to $P^{\mathrm{\prime }}{\mathrm{\mid }}_E$; both are sufficiently close, so the interpolation remains invertible. This gives an isomorphism of bundle pairs, relative to every prescribed analytic collar. All required finite smooth derivatives exist on the compact source; this is not approximation of the target map with its image constrained to a skeleton. Two sufficiently close matrix approximations are compared by interpolating their matrices before spectral projection, keeping the prescribed collars fixed. Distinct finite embeddings are compared in their direct sum by $(\cos \theta j_0,\sin \theta j_1)$ and the same construction. Thus the smooth pair and its identification are unique up to the corresponding relative bundle-pair homotopy. A pulled-back oriented background bundle is smoothed by the real version of this construction. Transfer its orientation and the given spin double cover by the bundle isomorphism, rather than choosing a new spin structure. Continuous boundary frames can be approximated in the smoothed bundle and orthonormalized by the positive Gram matrix; the close-frame interpolation preserves their homotopy class, and its lift to the spin cover is uniquely determined by its initial lift. Hence the relative-spin boundary-frame class and its complex clutching class are unchanged. Over a disk a smooth metric connection, parallel transported along radial segments, supplies the oriented trivialization of the smoothed background bundle used below. To construct the required analytic family, choose a smooth complex connection on $E^{\mathrm{\prime }}$ which on the boundary collar complexifies a real connection on $F^{\mathrm{\prime }}$. Its vertical $(0,1)$ part on each disk is a real Cauchy–Riemann operator with that totally real boundary condition. Match the original analytic end operator by extending its zeroth-order term from its end collar; interpolate the remaining zeroth-order terms in their affine space. The actual smooth-boundary Fredholm realization in [F1] applies to these smooth abstract bundle pairs, not merely bundles arising from holomorphic maps. In local parameter trivializations their $H^k\to H^{k-1}$ realizations are operator-norm continuous for fixed $k\ge 1$. At any parameter choose finitely many smooth target sections spanning its cokernel. The resulting surjective augmentation remains surjective nearby by the Neumann inverse, and $(I-QD)$ identifies its finite kernel continuously. The exact sequence gives $\det D=\det \ker (D+W)\otimes \det W^{\ast }$ there. Enlarging $W$ cancels the identical added block, so these local determinant lines and their transport of orientation rays agree; no canonical scalar identification of two determinant lines is asserted. Interpolation of the smoothed pairs and zeroth-order terms proves independence of these auxiliary choices. Continuous paths in analytic charts give the same transport: subdivide into their operator-norm trivializing neighborhoods and use these same finite augmentations, or interpolate within the smooth boundary-preserving section charts. The skeleton is used only at the cellular end, to pull back $V$ and its spin lift. Two cellular disk representatives are compared by a cellular cylinder in $M^{(3)}$ with side boundary in $L^{(2)}$; the same bundle smoothing and determinant construction apply to that cylinder. A cellular loop of disks likewise has its total image in $M^{(3)}$ and its boundary torus in $L^{(2)}$. The homotopies from analytic families to these cellular representatives require no extension of $V$ over their unrestricted images. The spin-frame argument below therefore compares their orientation rays on exactly the skeleton where the datum exists. Finite matching flags may be treated simultaneously on the nodal normalization, with shared bundle models and frames at matched flags; the relative collar construction fixes those models and preserves their real or complex matching orientations.

At a cellular representative use the smoothed oriented pullback of V and the boundary bundle pair constructed above; choose an oriented real trivialization over the disk. Put $D_V$ equal to the standard real-boundary CR operator in its complexified trivialization. Its kernel is the constant real vectors, its cokernel is zero, so $\det D_V$ has the chosen orientation of V; changing the real trivialization through an oriented frame preserves it. If $N=\mathrm{rank}(TL\oplus V)$, set $q=\max (0,3-N)$ and replace V for this argument by $V^{\mathrm{\prime }}=V\oplus {\underset{‾}{\mathbb{R}}}^q$. Stabilize its boundary spin lift by the standard inclusion $\mathrm{Spin}(N)\to \mathrm{Spin}(N+q)$; the resulting bundle $TL\oplus V^{\mathrm{\prime }}$ has rank $N+q\ge 3$. Use auxiliary operator $D_{V^{\mathrm{\prime }}}=D_V\oplus D_{{\underset{‾}{\mathbb{R}}}^q}$, where each added real line has the standard real-boundary CR operator, kernel the positively oriented constant real sections, and zero cokernel. Dividing by $\det D_{V^{\mathrm{\prime }}}$ later cancels these added determinant factors in the same order. A different oriented trivialization of V' over the disk differs by a map $D\to \mathrm{SO}(\mathrm{rank}{V}^{\mathrm{\prime }})$ whose boundary loop is nullhomotopic, so induces the same determinant orientation. A section of the principal $\mathrm{Spin}(N+q)$ bundle along $\mathrm{\partial }D$ gives an oriented boundary frame. Two such sections differ by a loop in $\mathrm{Spin}(N+q)$, which is nullhomotopic: $\mathrm{Spin}(3)$ is the unit quaternion sphere $S^3$; the Clifford action gives the locally trivial bundle $\mathrm{Spin}(m-1)\to \mathrm{Spin}(m)\to S^{m-1}$, with local sections given by the normalized elementary rotation from the first vector to a nearby vector. For $m\ge 4$ the base and the inductively connected simply connected fiber are simply connected, so lifting a contracted projected loop proves ${\pi }_1(\mathrm{Spin}(m))=0$. The projected $\mathrm{SO}(N+q)$ frames are therefore homotopic. Hence the spin datum supplies the required homotopy class of boundary frame, not a frame asserted to be canonical pointwise.

Use the spin-compatible real boundary frame on a collar of the stabilized disk bundle. Pinching a concentric INTERIOR circle produces a real-constant disk and a closed sphere carrying the complex clutching; the matching fiber is ${\mathbb{C}}^{N+q}$. For a direct-sum normalization operator P and onto node-difference map Delta to V, the exact sequence is 0 -> ker(P restricted to ker Delta) -> ker P -> V -> coker(P restricted to ker Delta) -> coker P -> 0. Its connecting map is v -> [P Rv], with $\mathrm{\Delta }R=I$; a different lift changes this by the matched range. Cancelling ordered lifted wedges defines $\det Dmatch=(\otimes \det P_v)\otimes \det (V{)}^{\ast }$. This is an actual operator sequence, not a declaration of an orientation sign. To compare smoothing, add the same finite-dimensional smooth core forcing W spanning the central cokernel. The separated-cutoff smooth-boundary construction gives a uniform right inverse Qq and transports an augmented kernel element by $(I-QqDq){\kappa }_q$. Fixed core coordinates identify its kernel continuously through the pinch. The exact sequence 0 -> ker Dq -> ker(Dq+W) -> W -> coker Dq -> 0 identifies det Dq with det ker(Dq+W) $\otimes \det W^{\ast }$. Cancel the SAME ordered W basis on both sides. Enlarging W cancels the added block; changes of lifts/complements are triangular determinant-one or homotopies. This is smoothing-compatible determinant transport from the actual augmented-kernel construction and its uniform inverse; a determinant identification is not inferred merely from a dimension count. The real-constant reference disk kernel is ${\mathbb{R}}^{N+q}$ with the frame orientation, and the sphere determinant and ${\mathbb{C}}^{N+q}$ matching fiber have their complex orientations. Thus $\det D_{\text{stabilized}}=\det D_{\text{real-constant}}\otimes \det D_{\text{sphere}}\otimes \det ({\mathbb{C}}^{N+q}{)}^{\ast }$. Deform zeroth-order coefficients within their contractible affine space and continue this orientation, then divide by the oriented $\det D_{V^{\mathrm{\prime }}}$. This use is entirely within the smooth single-L boundary branch and does not require or certify changing-label corner theory.

Check all choices. Homotopic spin-compatible boundary frames give a homotopy of the clutching/pinching construction and hence the same orientation. Moving the pinched circle or its coordinate changes only a contractible collar choice and a complex-linear node factor. Interior complex frame changes preserve the complex orientations. The real auxiliary trivialization was dealt with in 2.1. The trivial summands introduced there, and any further oriented trivial stabilization, add a real constant determinant in numerator and denominator with identical order and orientation, so cancel. For a cellular path of disk representatives, the pulled-back spin datum on its boundary family and the smooth bundle-pair construction above carry the spin-compatible frame continuously after subdividing its compact parameter interval; the resulting orientation is then transported to the original analytic family by that construction. On a cellular loop of disk representatives the spin bundle over its boundary torus has a global section, by extending a section across its finite two-dimensional cells using connectedness and ${\pi }_1(\mathrm{Spin}(N+q))=0$, and provides the same class at both endpoints: possible changes of spin-frame section are loops in simply connected $\mathrm{Spin}(N+q)$. Thus the orientation has no monodromy and patches to an actual orientation local system trivialization on every disk chart. This is exactly where the relative-spin hypothesis is used; orientation of L alone would not provide the homotopy class of frame.

For a nodal disk do this on every disk component; spheres retain their canonical complex CR orientations. At an interior node divide by the complex-oriented $\det T_{p}M$, and at a boundary node divide by the oriented $\det T_{p}L$ using the difference map (v1,v2)↦v2−v1. Add obstruction-space orientations and divide by the auxiliary-slice normal orientations. For a finite sum of obstruction spaces choose a fixed order of labels; permutation acts by the actual graded determinant permutation. The auxiliary real operators themselves glue by $\det D_{V1}\otimes \det D_{V2}\otimes \det (V_{node}{)}^{\ast }=\det D_{Vglued}$, with the same diagonal matrix sign; dividing them out therefore cancels their background-node factor and leaves exactly $\det (T{L}_{node}{)}^{\ast }$, not an extra undeclared V orientation. These are precisely the determinant lines of the actual thickened equations and finite-codimension conditions, so forgetful chart embeddings carry these orientations by the ordinary exact sequence rather than by an independent arbitrary sign choice.

To compare the two sides of a boundary-node gluing, use the stabilization of step 2.1 and choose the spin-compatible frames to agree at the node by a constant adjustment in $\mathrm{Spin}(N+q)$. This group is connected because $N+q\ge 3$, so the adjustment preserves the frame homotopy class and orientation. The glued boundary frame is then the concatenation of the two stabilized frames; cancel the auxiliary determinant factors in the same order as in steps 4.1–5.1. Pinch each component as in 3.1. Complex sphere/node factors have even real degree and their orientations agree under the already proved complex-node determinant sequence. The remaining real comparison is the exact sequence $0\to R_d^{n}iag\to R^n\oplus R^n\to R^n\to 0,$ with last map v2−v1 and lift w↦(0,w). In the ordered bases $(\mathrm{diag}{e}_i,(0,e_i))$ its matrix in the two factor frames is [[I,0],[I,I]], determinant +1. Thus the orientation of the matching diagonal is exactly the L orientation divided out by the node normal; this settles the TL diagonal comparison. It does not yet compare the unstable/stable incidence normals or their order; that separate comparison is performed in7.2–9.2 below. Associativity of these determinant sequences follows by eliminating matched coordinates in either order in this same block matrix. Swapping the order of two odd determinant factors has exactly its Koszul sign.

The real domain orientation is explicit. Normalize output infinity, first and last inputs 0 and1. The stable k-input coordinate orientation is dx2 wedge … wedge $d{x}_{(k-1)}$. A block of s inputs collapsing after r and before $t,r+s+t=k$, has the actual plumbing coordinate ρ≥0 and wedge ${\mathrm{\Omega }}_k=(-1{)}^{r+st}{\rho }^{s-2}d\rho wedge{\mathrm{\Omega }}_{o}uterwedge{\mathrm{\Omega }}_{i}nner\mathrm{.}$ This is the adopted ordered-disk-domain supplier's direct Jacobian calculation, including the first- and last-block cases. The outward normal is −∂ρ. The smooth common gluing profile is increasing in ρ, so does not change that normal orientation. More general disks with interior sphere flags use this real Jacobian times their complex node/base orientations; interior nodes have two real normal directions and give codimension two, not additional boundary faces. Auxiliary incidence orientations cancel between the simultaneous chart descriptions.

We fix the ACTUAL incidence-normal convention. At a critical point x let $H_x=T_{x}L,U_x=T_x{W}^u(x),m_x=\dim U_x$ and $q_x=n-m_x$. Orient $U_x$ by its given Morse generator. Orient the $q_x$-dimensional stable space $N_x$ by $N_x$ followed by $U_x$ equals $H_x$; transport this orientation along its stable and unstable sheets. Thus the incoming normal to $W^u(x)$ is oriented normal-first, while the outgoing normal quotient to $W^s(x)$ has the given $U_x$ orientation. At an arity-k vertex let V be its disk/domain augmented tangent. Its incidence-normal map has the single ordered target $(N_1,\mathrm{.}\mathrm{.}\mathrm{.},N_k,U_0)$: in incoming coordinates $(z_i,u_i)$ with $z_i$ in $N_i$ it is projection to $z_i$, and in outgoing coordinates $(z_0,u_0)$ it is projection to $u_0$. Choose lifts of THIS whole ordered normal basis. If K is its kernel, kernel-first compares (basis K,lifted ordered normals) with V. Changes of lifts are triangular determinant-one operations. This defines the bare incidence orientation B without pretending that sequential elimination, or a product with another stable tangent factor, has the same order automatically. The real disk/domain virtual tangent has orientation ${\lambda }_{k}H$, times its even complex factors, by3.1–6.1; ${\lambda }_k$ has degree k−2, and H is the constant-disk factor, not a second domain quotient. For a pure-Morse unary component H instead means its actual fixed-time evaluation tangent, and ${\lambda }_1$ is the dual positive-flow quotient line of8.2; this introduces no fictitious constant disk. Include the same obstruction and auxiliary bases in every comparison and cancel each identical parameter/normal pair in its fixed order first. Thus all the calculations below are calculations of the actual augmented-kernel determinant sequence, also when an unstabilized virtual dimension is negative.

The internal edge comparison includes both its tangent and its quotient normal. For an actual finite downward-flow edge of length T the matching equation is $e{v}_{+}={\mathrm{\Phi }}_T(e{v}_{-})$; its normal map is ${\mathrm{\Delta }}_T(v_1,v_2)=v_2-D({\mathrm{\Phi }}_T)v_1$. Its kernel is the actual flow graph oriented by its first endpoint H. Temporarily choose a stable basis $S^0$ with $H=(U,S^0)$, and consider the split linear passage $A_R=diag(R{I}_U,R^{-1}{I}_S),R>0$. After dividing its unstable columns by the positive R, the oriented graph tends to $(U_{second},S_{first}^0)=(-1{)}^{(mq)}(S_{first}^0,U_{second})$. The stable orientation fixed in7.2 is $N=(-1{)}^{(mq)}{S}^0$, so the limiting graph is EXACTLY $(N_{first},U_{second})$. The complementary normal basis $(U_{first},S_{second}^0)$ has ${\mathrm{\Delta }}_T$ image determinant sign $(-1{)}^m$. The concatenation $(S_{first}^0,U_{second},U_{first},S_{second}^0)$ has ambient permutation sign $(-1{)}^{(m(q+m))}$; multiplying by the graph sign $(-1{)}^{(mq)}$ gives $(-1{)}^m$, the same quotient-normal sign. This checks the full exact-sequence orientation, not just the limiting tangent plane. In the quotient coordinates of7.2 use $H=(N,U)$: the limiting kernel is $(N_{first},U_{second})$, and the complement $(U_{first},N_{second})$ maps by ${\mathrm{\Delta }}_T$ to H with sign $(-1{)}^{(m(q+1))}$. Its kernel/normal permutation in ambient order $(N_{first},U_{first},N_{second},U_{second})$ has sign $(-1{)}^{(m(m+q))}$, the same sign since m squared equals m modulo2. Thus the comparison checks the full quotient determinant as well as the limiting tangent plane. Its explicit minus on the unstable normal will be retained in the full constraint matrix of10.1; it is not discarded by writing an abstract endomorphism composition. The formula includes $m=0$ or $q=0$ by empty wedges.

We check the low-arity domain and stabilization signs explicitly. With the output at infinity, use the virtual domain line ${\lambda }_k=(-1{)}^k\det (R_{marks}^k)\otimes \det (af{f}_{+}(R){)}^{\mathrm{dual}}$, with ordered affine basis (translation,dilation). This is the determinant of the marking/infinitesimal-automorphism complex even when the underlying marked domain is unstable; the actual map atlas retains its stabilizing labels. For $k\ge 2$, the derivatives of the affine action followed by the middle-mark directions have determinant sign $(-1{)}^{(k-2)}$ in the ordered mark coordinates, so $(-1{)}^k$ makes ${\lambda }_k$ exactly the normalized orientation ${\mathrm{\Omega }}_k$ of7.1. For $k=1$ the kernel of the affine action on its single input at0 is positive dilation; the ordered basis (dilation,translation) has sign minus. Thus ${\lambda }_1$ is the dual positive dilation line, which is the positive-flow quotient line in the pure-Morse strip convention. For $k=0{\lambda }_0$ is dual(translation,dilation), of even degree −2. No zero-energy nullary constant disk is introduced by this convention.

For a genuine metric gradient no smooth linearization is assumed. The actual local matching supplier first straightens only the stable and unstable disks; its mixed-end equations at a break have normal derivative the identity and passage endpoint substitutions flat in $\rho =1\mathrm{/}T$. Shrink the axis box and interpolate its nonlinear remainder to zero; the spectral gaps persist, the passage-end substitutions remain uniformly small, and their normal derivative at the broken stratum remains the identity throughout this compact interpolation. The parameter-dependent implicit-function theorem therefore gives the same determinant continuation. Deform the positive contraction/expansion spectral blocks through invertible positive blocks to those in8.1. The graph-normal determinant therefore has exactly that sign; the exterior fixed-time flow maps are orientation preserving, and elimination of the supplier's normal variables is continuous with positive determinant at $\rho =0$. Its coordinate changes have positive boundary-normal derivative. This transfers8.1, including its unstable-normal minus, to the actual broken-edge chart and its simultaneous independent-neck version. The comparison applies to any actual transverse disk/evaluation thickening with these edge equations. In particular, in a sewn disk/finite-edge family, the $T=0$ graph is the same diagonal as6.1; it is an internal seam with the disk smoothing half. Choose the signed seam coordinate negative on the disk-plumbing side and positive on the increasing-length side. Hence transporting the plumbing orientation across it reverses the increasing-length normal. At $T=\mathrm{\infty }$ the outward normal is increasing T, since $\rho =1\mathrm{/}T$ has outward $-{\mathrm{\partial }}_{\rho }$. The one common outward minus in7.1 is therefore retained at the genuine broken-flow boundary. This is determinant transport on the actual family, not the false equality of the diagonal with the union of $N_x$ times $U_x$. The global construction using such families must separately supply its compactification and product perturbations.

Here is the auxiliary-flag calculation extending the actual stable Jacobian to those virtual lines. Retain every original stabilizing interior label and its slice normal, so no component is contracted. Add free boundary flags away from all nodes on each component until the boundary-marked component is stable; the already stabilized local universal curve gives their one-dimensional free vertical coordinates. A local section of the forgetful submersion imposes each corresponding coordinate-normal equation with derivative +1. Its tangent/normal pair cancels by the actual exact sequence. This is a further local stabilization of the SAME chart, not a new global smooth-forgetting assertion. If a free flag a is added at position j, moving its mark differential to the last position in8.2 gives ${\lambda }_{(k+1)}=(-1{)}^{(k-j)}{\lambda }_k$ da. At a split with $N=r+1+t$, an added inner flag at relative position h has global position $j=r+h$; the component/global exponent difference is (s−h)−(k−j)=t modulo2. An outer prefix flag at position h contributes N−h in the outer line, and moving da past ${\lambda }_s$ contributes s, giving difference $N+s-k=1$. An outer suffix flag has global position $j=h+s-1$ and gives difference (N−h)+s−(k−j)=s. These are EXACTLY the changes of r+st caused by adding, respectively, an inner, prefix or suffix flag. Iterate one flag at a time; this includes all cross terms and descends the stable Jacobian of7.1 to the physical r,s,t, including $s=0,1$ and $N=1,2$. The positive inner coordinate scale contributes no sign. Original auxiliary interior flags and their real two-dimensional slice normals cancel with their complex orientations, of even degree. The free flag and its inverse normal are an even pair. In particular, if an inner auxiliary flag temporarily raises the parity of ${\lambda }_s$ by one, moving that flag direction across the physical prefix contributes Q and moving its inverse normal across the same prefix contributes Q again; they cancel. Equivalently eliminate this identity normal block first, before forming the physical incidence-normal map of7.2. Thus no low-arity face has been assigned a sign independently: the same mark/automorphism determinant and actual parameter/normal cancellations give all of them.

Here is the full geometric constraint-matrix comparison, including the arity normalization; no cap-word/direct-sum sign is assumed. Put $N=r+1+t,k=N+s-1$. Let B be the sum of inner input q degrees, Q the prefix sum, S the suffix sum, m the inner output's unstable dimension, $q=n-m$, and $m_0$ the final output's unstable dimension. Since the inner and outer factors are rigid, their unconstrained virtual dimensions satisfy $d_s=s-2+n=B+m$ and $d_N=N-2+n=Q+q+S+m_0$ modulo2. Maslov and complex factors have even degree: for oriented L its boundary frame has a complex determinant phase whose squared phase has even winding. Work first with bare kernel-first orientations of7.2. Order the unbroken ambient variables as $(T,V_{inner},V_{outer})$, and the disk/domain blocks as $(T,{\lambda }_N,{\lambda }_s,H_{inner},H_{outer})$. Moving ${\lambda }_N$ left across ${\lambda }_s$ and $H_{inner}$ gives $E=(s-2)(N-2)+n(N-2)$. In this order the zero-length match is exactly the inner-first difference $\mathrm{\Delta }(v_{inner},v_{outer})=v_{outer}-v_{inner}$ of6.1, with its determinant-one diagonal/lift matrix. At infinity the actual source normal blocks, using the rigid factors' lifted normal bases, have order (inner inputs [B],$U_y$ [m],prefix [Q],$N_y$ [q],suffix [S],$U_0$ [$m_0$]). The target for the WHOLE constrained family is (prefix,inner inputs,suffix,$U_0$,$H_{node}$), with $H_{node}$ ordered $(N_y,U_y)$. Indeed restriction to the graph match first and then to external incidences nests kernel-first as $ambient=K_{final}$ followed by external normals followed by $H_{node}$. Moving prefix across the first two source blocks contributes Q(B+m); moving $U_y$ to the final U part of $H_{node}$ contributes $m(q+S+m_0)$; moving $N_y$ across suffix and $U_0$ contributes $q(S+m_0)$; and the actual normal map ${\mathrm{\Delta }}_T$ sends $U_{first}$ to −R U, contributing m. Every other diagonal block is positive or the unchanged external-normal identity. Thus its determinant exponent is $F=Q(B+m)+m(q+S+m_0)+q(S+m_0)+m$. Off-diagonal derivatives are removed by triangular determinant-one operations;9.1 gives identity normal derivative in the actual broken chart. Substituting the two rigidity relations and $q+m=n$ gives $E+F=sQ+sN$ modulo2: after substitution the leftover is $n+nq+mq+m=(n+m)+(n+m)q=q+q^2=0$. The domain Jacobian and common outward normal from7.1,9.1–9.2 contribute r+st+1. Consequently the BARE incidence convention has geometric boundary exponent r+st+1+sQ+sN. This explicit sN term is why merely moving an abstract inner map would not have proved the result. Normalize the coefficient orientation ONCE on each arity-k constrained system of virtual dimension d by the fixed exponent f(k,d)=k(k−1)/2+kd+k−1, and use the critical-line ray comparison h(q)=q(q−1)/2 on every incoming and outgoing critical line. In concrete terms its orientation is the actual B orientation multiplied by $(-1{)}^{[f(k,d)+h(q_0)+{\sum }_{j}h(q_j)]}$. The spin Cauchy–Riemann determinant orientation constructed in1.1–6.1 stays fixed: these exponents specify only the finite Morse-constrained COEFFICIENT orientations, including the orientations used on their parameter families and the identification with the declared critical-generator lines. These formulas are part of the operation convention on every chart, not choices on individual faces. They are constant on every fixed-dimensional component and are unchanged by coordinate maps and oriented obstruction/auxiliary stabilization. On a one-dimensional system with rigid factors the internal h(q) appears twice and cancels. Because $k=N+s-1$, direct expansion gives f(k,1)+f(s,0)+f(N,0)=sN modulo2; the k−1 summands have zero composition coboundary. The NORMALIZED raw geometric boundary exponent is therefore r+st+1+sQ, equivalently r+st+1+(2−s)Q. This is the required actual incidence determinant comparison before suspension algebra. Check the fixed normalizations against unary and unit geometry. At a positive rigid pure-Morse trajectory write $U_p=(\text{positive flow }f,U_0)$, so $H=(N_p,f,U_0)$ and the total normal target is $(N_p,U_0)$. Its kernel-first basis $(f,N_p,U_0)$ has sign $(-1{)}^{q_p}$. Since f(1,0)=0 and $h(q_p+1)+h(q_p)=q_p$, the normalized coefficient is exactly the existing Morse epsilon, with no rescaling of declared unstable generators. For a one-dimensional pure trajectory space the corresponding bare sign is $(-1{)}^{(2q_p)}=+1$, while f(1,1)=1 and $h(q_p+2)+h(q_p)=1$ cancel, so its orientation is the published flow quotient and its boundary is $-{ϵ}_1{ϵ}_2$. For a metric gradient the same normal-basis comparison is transported by9.1, without assuming an exact linear metric chart. For a constant binary disk with a fundamental-class input and a repeated remaining generator the normal map is identity on $H=(N_x,U_x)$. Here f(2,0)=0 modulo2 and the two $h(q_x)$ phases cancel, so its raw left and right unit coefficients are +1; the suspension in11.1 supplies the usual right-unit sign. A positive-energy nullary vertex uses f(0,d)=1 modulo2, the even automorphism line ${\lambda }_0$, and the SAME formula. For a Maslov-two nullary disk with output degree0, the fixed convention gives minus its bare B coefficient, since f(0,0)=1 and h(0)=0 modulo2. This explicitly fixes the curvature sign; it asserts no equality with a numerical disk potential computed in a different convention. No zero-energy nullary constant disk is introduced; its geometric exponent is r+1. Thus unary, curvature and all stable arities have one globally fixed geometric convention.

Finally make the suspension convention explicit. Let $a_j$ have cohomological degree $q_j$, let mathsf s lower degree by one, and put $v_j=\mathsf{s}{a}_j$. Let ${\mu }_k$ be the normalized raw operations of10.1, of degree2−k including their even disk-class coefficient. Define $b_k(v_1,\mathrm{.}\mathrm{.}\mathrm{.},v_k)=(-1{)}^{{\beta }_k}\mathsf{s}{\mu }_k(a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_k)$, with ${\beta }_k={\sum }_{(j=1)}^k(k-j)q_j$. Equivalently this is $(-1{)}^{(k(k-1)\mathrm{/}2)}\mathsf{s}{\mu }_k({\mathsf{s}}^{-1}{)}^{\otimes k}$: the tensor sign is $\sum (k-j)(q_j-1)$, and the arity factor adds k(k−1)/2. The arity factor is fixed once on every operation, not chosen separately on faces. Empty sums give ${\beta }_0=0$, and every $b_k$ has degree1.

Put $N=r+1+t$ and $q_{\ast }={\sum }_{inner}q+2-s$, including an even coefficient degree if present, and let q' be the outer list with that replacement. Expanding the three finite sums gives ${\beta }_s+{\beta }_N(q^{\mathrm{\prime }})-{\beta }_k=st+(1-s)Q$ modulo2: preceding entries contribute (1−s)Q, the inner block and replacement contribute t(2−s)=st modulo2, and following entries have unchanged coefficients. Combining this with the PROVED raw exponent in10.1 gives the suspended outward-boundary exponent $r+st+1+(2-s)Q+{\beta }_k-{\beta }_s-{\beta }_N=1+{\sum }_{(j\le r)}(q_j-1)$. Removing the one common outward minus from the zero-boundary equation therefore gives exactly the stated insertion sign. The normal is the actual normal checked in9.1 and the pure unary test in10.1 fixes the same common normalization; no relative face orientation is changed. The exact domain and auxiliary-normal computation in8.2–9.2 applies also to $s=0,1$, so this cancellation includes all curvature and unary insertions.

A sphere bubble contributes its complex two-dimensional smoothing normal. Its possible intersection with a real boundary-face stratum has an additional two codimensions, so is never a new codimension-one face of a one-dimensional transverse perturbation. Iterated real boundary normals anticommute. Their determinant contraction orders have the identical Koszul interchange, so the codimension-two corners agree and their boundary-of-boundary terms cancel. Thus the orientation system is coherent on all stable disk tree faces, not merely each single split.

Actual atlas symmetries and determinant equivariance. Take the specified finite tuple action on the atlas in [F1], or its specified finite central spin extension, with its lift preserving L's orientation and the chosen relative-spin datum. An action here is on the domain, map, obstruction and auxiliary data and intertwines the full thickened Cauchy–Riemann equation and its gluing maps. This is the actual tuple-action interface of that supplier, not a claim that any symplectomorphism preserving L and its spin datum preserves a chosen J. For a fixed target J, a target-induced action must in particular satisfy $dg\circ J=J\circ dg$ and carry the forcing and incidence data to their stated translates. If auxiliary structures depend on free word or chart labels, the corresponding condition is $J_{ga}=g_{\ast }{J}_a$, with the same transport for every Hamiltonian, forcing, slice and gluing datum; a single invariant J is not required for such a labeled family. Write the local thickened equation as ${\mathcal{F}}_a(x)=0$. The specified tuple maps ${\mathrm{\Phi }}_g$ on configurations and ${\mathrm{\Psi }}_g$ on equation bundles satisfy ${\mathcal{F}}_{ga}({\mathrm{\Phi }}_{g}x)={\mathrm{\Psi }}_g(x){\mathcal{F}}_a(x)$. At an actual tuple solution, differentiating gives $D_{ga}{A}_g=B_g{D}_a$, where $A_g=d{\mathrm{\Phi }}_g$ and $B_g={\mathrm{\Psi }}_g(x)$ are invertible real maps of the declared Sobolev domains and targets; the derivative of ${\mathrm{\Psi }}_g$ multiplies ${\mathcal{F}}_a(x)=0$ and disappears. Thus there are actual maps on kernels and cokernels and the resulting exterior-power map on $\det D_a$. The exact tuple group law makes their compositions the map for the group product, not merely a homotopy action. Use the same constraint, obstruction and auxiliary-normal exact sequences before and after applying g. Their natural kernel/cokernel maps commute with quotient-basis cancellation; reindexing ordered factors has precisely its graded determinant permutation. For the spin comparison, transport the connections, frames and pinching collars as well. The specified spin lift carries every spin-compatible boundary-frame class to the corresponding class. Domain reparametrizations and the transported target derivative are complex-linear on sphere and complex-node factors, and the real matching map is carried by the orientation-preserving TL map. Hence the pinched reference orientations and their real matching contractions are carried to the same orientation rays. Independence of the smoothing, frame and connection choices proved above identifies these with the originally chosen chart orientations. The actual gluing maps intertwine the tuple action by hypothesis and by [F1]'s tuple construction, so this argument also applies through the nodal strata. It retains the specified spin/brane-line action on external factors and the actual determinant action on the ordered obstruction tuple. Finite orbit multisections consequently retain these oriented maps and the rational weights $1\mathrm{/}\mathrm{\mid }G\mathrm{\mid }$; no nonlinear operation is averaged. For clarity, a symplectomorphism not preserving J still carries the disk problem naturally to the one with $g_{\ast }J=dg\circ J\circ d{g}^{-1}$ and all data pushed forward: the Cauchy–Riemann equations are then intertwined by $dg$, and the same spin-frame argument gives oriented covariance between those two problems. This supplies no action on the original fixed atlas unless its actual data-intertwining maps have been specified.