Compact rational geometric Dehn twist and actual evaluation-cone triangle

Quadratic twist, cap, complete local regularity and signed filling. The positive twist and its exact primitive

For the preliminary exact argument, let $M$ be an exact symplectic manifold with primitive ${\lambda }_M$ containing the framed cotangent tube, and use compact exact branes in $M$. The notation $E_S$ in those steps denotes the exact fibration with fibre $M$. This auxiliary calculation is applied to the compact $X$ only after the separate compact construction below; no exact form is asserted on a closed symplectic fibre. Write $T^{\ast }{S}^n=\{(p,q)\in {\mathbb{R}}^{n+1}\times {\mathbb{R}}^{n+1}:p\cdot q=0,\mathrm{\mid }q\mathrm{\mid }=1\}$, with ${\theta }_T=p\cdot dq$ and ${\omega }_T=d{\theta }_T$. The Hamiltonian convention is ${\iota }_{X_H}\omega =-dH$. On p nonzero, $\mu =\mathrm{\mid }p\mathrm{\mid }$ generates sigma_t(p,q)=(cos(t)p-sin(t)|p|q, cos(t)q+sin(t)p/|p|).

Direct differentiation gives ${\iota }_{X_{\mu }}\omega =-d\mu$. In particular ${\sigma }_{\pi }$ extends over the zero section by $(p,q)\to (-p,-q)$. Choose R(t), zero outside a small tube, with $R(-t)=R(t)-t$ near zero. Then R(t)-t/2 is even and hence a smooth function of $t^2$. Thus tau(p,q)=sigma_(2pi R'(mu))(p,q), tau|S=antipodal, tau*theta_T-theta_T=dK_tau, K_tau=2pi(mu R'(mu)-R(mu))

extends smoothly over S. The second identity follows from the Hamiltonian Lie derivative and integration along its flow; its derivative formula also shows smoothness since the expression is even near zero. This is our POSITIVE twist; its angle decreases from pi at the zero section to0 outside. Its inverse uses the opposite rotation. Any two such profiles are joined through profiles with the same germ condition and compact support, and their flow difference is a compactly supported Hamiltonian isotopy.

A useful explicit profile comes from the quadratic fibration. Let $q_{\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{d}}(z)={\sum }_{j=0}^n{z}_j^2$ and R_tilde,r(s)=s/2-(s^2+r^2/4)^(1/2)/2, R_r(s)=(1-g(s))R_tilde,r(s), where g is zero on a smaller tube and one near its outside boundary, with $g^{\mathrm{\prime }}\ge 0$. Near zero $R_r^{\mathrm{\prime }}(0)=1\mathrm{/}2$, $R_r(0)=-r\mathrm{/}4$ and $R_r^{\mathrm{\prime }\mathrm{\prime }}<0$. Outside the smaller tube its derivative is bounded by C $r^2$. Thus, for any fixed angular separation $\delta >0$, taking r small makes tau delta-wobbly: $R^{\mathrm{\prime }}\ge 0$ and $R^{\mathrm{\prime }\mathrm{\prime }}<0$ wherever $R^{\mathrm{\prime }}\ge \delta$. Also $K_{\tau }\mathrm{\mid }S=\pi$ r/2 is positive and tends to zero.

For two distinct cotangent fibres $F_x,F_y$ whose round distance alpha is at least2pi delta, $\tau (F_x)$ meets $F_y$ in exactly one transverse point. For $\alpha <\pi$ it is the endpoint of the UNIQUE shorter great-circle arc and has radius s determined by 2pi $R^{\mathrm{\prime }}(s)=\alpha$. The radial derivative R'' is negative and the angular derivative R'/s is nonzero, so its derivative matching matrix has no kernel. If $\alpha =\pi$ the point is the zero covector over $y=-x$. There the tangent planes of $\tau (F_x),S,F_y$ are symplectically equivalent to $R^n,\exp (2\pi i\mathrm{/}3)R^n,\exp (\pi i\mathrm{/}3)R^n$. This follows by linearizing the even Hamiltonian R(mu)-mu/2 at the zero point, whose fibre Hessian is R''(0)I, and continuously moving this negative quadratic shear to that ordered plane triple while retaining transversality. The $n=1$ antipodal case is a SINGLE zero covector, not two geodesics counted as different generators.

For $q_{\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{d}}$ the horizontal symplectic normal line is C times conjugate(z). Its vanishing spheres are ${\mathrm{\Sigma }}_w=\sqrt{w}{S}^n$. The function $h(z)=\mathrm{\mid }z{\mathrm{\mid }}^4-\mathrm{\mid }{q}_{\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{d}}(z){\mathrm{\mid }}^2$ is constant under horizontal transport. Away from these spheres, radial transport identifies the fibres with punctured $T^{\ast }{S}^n$. In polar base coordinates $w=r$ exp(i alpha), direct differentiation gives the primitive theta_T-R_tilde,r(mu) d alpha.

This identity can be verified on radial, angular and cotangent-fibre vectors; its angular component is the radial Hamiltonian generating the displayed normalized geodesic flow. Add the one-form $g(\mu )R_{\mathrm{t}\mathrm{i}\mathrm{l}\mathrm{d}\mathrm{e}},r(\mu )d$ alpha. It vanishes near the vanishing spheres and makes the connection product near the tube boundary. The resulting primitive is ${\theta }_T-R_r(\mu )d$ alpha, and its counterclockwise base monodromy is exactly tau, with the sign fixed by ${\iota }_X\omega =-dH$. Glue the product complement of the cotangent tube in M to obtain a one-critical-point exact fibration $E_S$. Its curvature density is (g(mu)-1) partial_r R_tilde,r(mu) dr wedge d alpha >=0.

Both factors have nonpositive sign. The local model is unchanged near the critical point. This constructs the fibration used later for the geometric module map, rather than postulating a monodromy equal to the desired algebraic twist.

The graded lift is normalized to be the identity outside the tube. In the quadratic residue trivialization, the volume of Sigma_w has phase w^((n-1)/2). A positive base loop changes the squared-phase lift by n-1. Our cohomological object shift [a] decreases the phase lift by a, so tau_tilde(S)=S[1-n], tau_tilde^-1(S)=S[n-1].

This derives the grading sign from the actual positive loop. It will be used explicitly in moving the [n-1] tensor shift in the Yoneda generator comparison.

Let $E_S$ be the standard fibration just constructed and choose a horizontal almost-complex structure equal to the standard one in a fixed quadratic chart of its critical point. Its vanishing-sphere boundary over $\mathrm{\mid }w\mathrm{\mid }=r$ is $Q_r=unio{n}_{(\mathrm{\mid }w\mathrm{\mid }=r)}Sigm{a}_w$. A section u of the base disk has integral u*Omega = pi r/2.

Indeed the primitive on $Q_r$ is (r/4)d arg(w); the sign agrees with the positive base orientation. Choose a fixed sufficiently large constant C so Omega+C $\pi \ast {\omega }_{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{e}}$ tames the total almost-complex structure on a fixed compact fibration region. This is possible because Omega is positive on vertical directions, its horizontal curvature is nonnegative, and it is already standard symplectic at the critical point. Its section area is pi r/2+C pi $r^2$, tending to zero.

The boundary images $Q_r$ are in a ball of radius $C_0$ sqrt(r) around the critical point. If a section escaped a fixed smaller quadratic chart, its image would cross an annulus at a fixed positive distance from those boundary images. Interior monotonicity on a fixed target ball in that annulus would give an area lower bound independent of r. Here the annulus and its small target balls are chosen INSIDE the fixed chart where J and Omega are standard. The lower bound has an elementary proof. For a regular target-ball radius rho put $e(\rho )=\int$ over its preimage of $u\ast {\omega }_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{a}\mathrm{r}\mathrm{d}}$. Stokes with the radial primitive, the coarea identity and the conformality of a complex curve give $e(\rho )\le (\rho \mathrm{/}2)e_{\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{m}\mathrm{e}}(\rho )$. At a regular image point $li{m}_{(\rho \to 0)}e(\rho )\mathrm{/}{\rho }^2$ is at least pi; integration gives $e(\rho )\ge \pi {\rho }^2$. A regular point can be chosen on the crossed annulus, and the fixed ball is disjoint from all source boundary images. Thus this is an actual interior area lower bound, without any estimate for the degenerating boundary conditions. This contradicts the area tending to zero. Thus every sufficiently small cap section is entirely in the quadratic chart. Degeneration of the boundary submanifolds creates no missing estimate, since the decisive target ball is disjoint from ALL boundary images.

Compactness and deformation of these cap spaces are obtained by the same local compactness proof as for the actual exact polygon supplier, applied to sections in the total space. A bubble projects to a constant base point. Its Omega area is zero by exactness, and a boundary bubble has boundary on the exact fibre sphere, so it too has zero area. The added base form contributes zero to either bubble. Hence there are no nonconstant bubbles. A limit principal section cannot pass through the critical point, since differentiating $q_{\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{d}}$ composed with $u=id$ would then give $0=1$. This also explains why local Fredholm theory on the regular part suffices for a fixed positive r.

Scale the disk to radius r. A holomorphic vector w on it with w(z) in sqrt(z) R^(n+1) on its boundary has the form w(z)=r^(-1/2) a z+r^(1/2) conjugate(a).

This follows by reflecting each scalar component across the circle: its boundary Fourier coefficients have only the two mutually conjugate modes allowed by the half rotation. More explicitly, exp(-i theta/2)w(r exp(i theta)) is real, so its nonnegative holomorphic Fourier modes pair only the frequencies0 and1. The condition $q_{\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{d}}(w(z))=z$ is exactly q_quad(a)=0, |a|^2=1/2.

Writing $a=x+iy$ gives $\mathrm{\mid }x\mathrm{\mid }=\mathrm{\mid }y\mathrm{\mid }=1\mathrm{/}2$ and $x\mathrm{.}y=0$. Thus the cap moduli is the Stiefel manifold, identified with $S(T^{\ast }{S}^n)$ by base $v=2x$ and unit covector $u=-2y$. Evaluation at the positive real boundary point r, followed by division by sqrt(r), is its projection to $S^n$.

Regularity is finite dimensional here. The unrestricted reflected holomorphic vector problem has real kernel $C^{(n+1)}$, of dimension2n+2, and no cokernel, by the scalar Fourier solution. Its scalar q constraint has a three-dimensional real holomorphic target: the coefficients of $z^2$ and1 are conjugate and that of z is real. The derivative of these three coefficients is the derivative of $q(a)=0$ and $\mathrm{\mid }a{\mathrm{\mid }}^2=1\mathrm{/}2$. It is onto because a and conjugate(a) are linearly independent over C at an isotropic nonzero a. Equivalently the three real constraint gradients are independent. The exact sequence of these two actual dbar realizations therefore gives zero vertical cokernel and kernel dimension2n-1, exactly the tangent space of the displayed Stiefel manifold. There is no unproved regularity hypothesis hidden in this cap computation.

The local determinant orientation is the constraint orientation of $C^{(n+1)}$ by the ordered real equations Re q(a), Im q(a), $\mathrm{\mid }a{\mathrm{\mid }}^2-1\mathrm{/}2$, up to one common dimension-dependent sign. This follows from the preceding exact kernel/cokernel sequence and its scalar Fourier coefficient ordering. Under $(x,y)\to (v,u)$ this is, with that same common sign, the boundary orientation of $D(T^{\ast }{S}^n)$. One can check the sign at one orthonormal frame; invariance under the real orthogonal group then fixes it on every component. For $n\ge 2$ this Stiefel space is connected, so one checked common orientation sign suffices. For $n=1$ the relative sign can be checked explicitly, rather than inferred from a spin-free symmetry: at $a=(1,\pm i)\mathrm{/}2$ the two boundary paths in the real vanishing circle are (cos(theta/2),minus or plus sin(theta/2)), $0\le \theta \le 2\pi$. Joining the first with the reverse of the second gives one full tangent-frame rotation on S1. Its lift to the connected Spin double cover changes sign. All nonzero Fourier-mode determinant factors occur in conjugate pairs with their fixed positive orientation, so this is exactly the relative minus sign between the two evaluation components. Hence their projection degrees are opposite, as for the two boundary circles of D(T*S1). This binds the cap orientation to the actual bounding/thimble Spin normalization; an independent Spin change would change this relative sign.

Consequently the actual evaluation chain S(T*S^n) -> S^n

has the explicit bounding chain $D(T^{\ast }{S}^n)\to S^n$, the projection. For $n=1$ this is the annulus with its opposite oriented boundary circles; for $n=2,3$ it is the oriented disk cotangent bundle. Multiply the whole bordism by the common orientation sign if necessary. Shrinking the original cap to this local one gives a compact evaluation bordism by the preceding area/no-bubble argument. Appending the disk bundle yields an ACTUAL bounding evaluation chain for the original cap. Pullbacks against transverse external evaluation maps therefore give a signed nullhomotopy, and not only a mod2 vanishing number or a grading-based assertion limited to $n>1$. The external maps and the all-arity pullback construction remain the next part of the shared triangle proof.

Exact geometric evaluation and its low matrix. Put each finite test collection in the required normal form

A framed exact sphere has an exact cotangent tube. One can construct it directly: a compatible metric identifies its normal bundle with T*S using omega. The normal exponential map has the canonical symplectic first jet on S. Its pulled-back two-form differs from omega_T by a closed form vanishing there; fibrewise dilation supplies a primitive of order |p|^2 for that difference. Solving the contraction equation for the interpolated forms and integrating its order-|p|^2 vector field corrects the map while fixing S. Compactness gives one small tube. The difference of the pulled-back primitives is closed and restricts to the exact form on S. The fibre retraction therefore makes its class zero, including for S1. Adjust the ambient primitive by the derivative of an extended function so it equals theta_T on the tube and the chosen primitive of S is zero.

Fix finitely many test branes W and the brane K. Small Hamiltonian copies make every needed pair transverse, make W intersection S disjoint from K intersection S, and remove triple intersections. Near any transverse point, W is a graph $q=q_0+dh(p)$ over the cotangent fibre. The Hamiltonian depending on p with this generating function straightens it to the fibre, fixing the zero section. Supports around the distinct footpoints can be made disjoint. Shrinking the tube excludes every other portion of these compact branes. This straightens ALL of the finite collection simultaneously to fibres near S.

Independently separate the relevant actions. Write $A_{(P,Q)}(x)=K_Q(x)-K_P(x)$, the target primitive minus the source primitive. A Hamiltonian which is constant near one intersection and supported away from the other intersections changes its primitive value by that prescribed small constant, while leaving that intersection point fixed. Thus avoid the finitely many equations saying that two old actions agree, that two sums $A_{(W,S)}(x)+A_{(S,K)}(y)$ agree, or that an old action equals such a sum. Choose $\eta >0$ so their distinct values are separated by the gaps3eta and5eta, respectively. These are finite open conditions. They do not change the brane isotopy classes. Choose $\delta >0$ smaller than every round distance between a W footpoint and a K footpoint divided by2pi, and choose the quadratic twist radius r so small that $a=K_{\tau }\mathrm{\mid }S=\pi r\mathrm{/}2<\eta$ and the twist is delta-wobbly. No uniform eta for infinitely many tests is claimed.

Put Y=tau_S^-1K. Applying tau identifies CF(W,Y) with CF(tau_S W,K), preserving the canonically transported primitive actions. Outside the tube every old intersection W intersection K is unchanged. Inside the tube the fibre calculation in the present construction gives exactly one new intersection p(x,y) for each x in W intersection S and y in S intersection K. There are no additional inside intersections. Let q(z)=z denote an old generator.

Let $V=CF(S,Y)$. The graded twist identity ${\tau }_{S}S=S[1-n]$ gives V ~= CF(S,K)[n-1].

Under this identification V's action is the action of CF(S,K) translated by -a. The source triangle domain CF(S,K) tensor $CF({\tau }_{S}W,S)$ therefore identifies with V tensor CF(W,S): the latter factor is CF(W,S)[n-1] with action translation -a, and moving its shift to the first factor has the precise Koszul sign $(-1{)}^{((n-1)\mathrm{\mid }v\mathrm{\mid })}$ on a homogeneous first factor v. This is the ordinary cohomological shift convention $C[s{]}^j=C^{(j+s)}$, $d_{[s]}=(-1{)}^s$ d; checking the two tensor differentials derives this sign. Transporting the triangle and then removing the shifted S gives EXACTLY the evaluation through S. No pairing or long-exact-sequence connecting map is being substituted for it.

The new-generator action is A_(tau W,K)(p(x,y))=A_(W,S)(x)+A_(S,K)(y)-K_tau(p(x,y)).

On the relevant wobbly region $0\le K_{\tau }\le a$, by the explicit primitive formula and $R^{\mathrm{\prime }\mathrm{\prime }}<0$. Thus its difference from the action in V tensor CF(W,S) is in [0,a). Old actions are unchanged. The chosen gaps imply: distinct generators of CF(W,Y) are separated by at least2eta; the tensor domain and CF(W,K) have their internal gaps3eta; their two action supports are at least4eta apart; an incorrect pair or incorrect old generator cannot differ by a nonnegative amount less than3eta. These statements follow by subtracting the displayed action formulas and using $a<\eta$. They also hold during the compact framing deformations used next.

At an antipodal pair the local ordered tangent planes are the three standard planes in the quadratic component. In one complex dimension their graded intersection degrees satisfy i(0,pi/3)=i(0,2pi/3)+i(2pi/3,pi/3); using the positive-angle formula gives1=1+0. Sum over the n diagonal directions. Thus the new generator has the degree of its tensor input. This determines the same grading as the canonical tau lift; arbitrary changes of input brane gradings add on both sides and do not alter it.

Use the right Yoneda convention: if $a_i$ belongs to $CF(W_i,W_{(i-1)})$, a right module operation takes a module element at $W_0$ and these inputs to an element at $W_d$. Put $E=V$ tensor $Y_S^r$ and $C=Y_Y^r$. The canonical evaluation module map $b:E\to C$ has d-th component the actual polygon operation composing $W_d\to \mathrm{.}\mathrm{.}\mathrm{.}\to W_0\to S\to Y$. In the unshifted reversed-input convention it is ${\mu }_{(d+2)}(v,x,a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_d)$, with its tensor/suspension Koszul signs. Equivalently on the shifted tensor bar module it is the degree-zero composition map. The A-infinity identities of the actual polygon operations give the FULL module-map equation for b.

For $d=0$ its low-action matrix beta sends the pair (x,y) to p(x,y) with an invertible orientation-line coefficient. Here is the local count. Deform the sphere framing by a diffeomorphism isotopic to the identity so that the chosen two footpoints become antipodes. Its cotangent lift preserves ${\theta }_T$ and fibres. Over this compact deformation shrink the radius and angle delta if necessary; all the original action gaps and wobbly uniqueness remain valid. Every triangle of the selected low area has area less than eta, whereas every nonconstant strip at an end costs at least2eta. Exactness excludes sphere or single-brane disk bubbles, and the actual local regular compactness argument supplies compactness of this parameterized low-area triangle space. It has no broken boundary. One can also localize it within the tube by fixed-radius interior/boundary monotonicity, taking r smaller than the annular crossing energy. The transverse fibre geometry is uniform along the compact framing deformation.

At the antipodal endpoint its area is zero. The energy identity forces the triangle to be the constant map at the triple point. The linear operator has zero kernel by the same zero-energy identity for the three linear planes and the transverse end constraints. Its cap-angle index is zero by the degree computation above, so its cokernel is zero. Its determinant identifies the two input orientation lines with the output line. This defines the coefficient +1 in that line identification. Regular one-parameter continuation of the low-area moduli preserves that orientation-line isomorphism back to the original pair. This proves the desired coefficient, rather than assuming an unoriented count. A different choice of numerical generators may change it to -1 but cannot make it zero. All other coefficients of $b^0$ have action order at least3eta, because negative area is impossible and the incorrect-generator gaps exclude [0,3eta).

The framing deformation can be chosen separately for each pair to COMPUTE its coefficient; the map b itself is the single fixed evaluation map. The computation does not make pair-dependent functors or perturb the meaning of fullness.

Actual monodromy section maps and full nullhomotopy. Construct the monodromy-cap module map

Let $D=Y_K^r$. Construct $c:C\to D$ from the one-critical-point fibration. Start with a strip, cut from its interior critical disk to its distinguished boundary arc, glue the two sides by tau, and insert the standard quadratic fibration in that disk. The complement is the product fibration. In the positive-end trivialization its boundary pair is (tau W,K); in the negative-end trivialization it is (W,K). Equivalently, transporting the whole boundary by ${\tau }^{-1}$ makes this a map from CF(W,Y) to CF(W,K). The primitive across the cut is theta-d(beta $K_{\tau }$), so the two primitives match exactly, not just their symplectic forms. Their end gradings match the canonical positive tau lift. This constructs $c^0$ of degree0.

For d extra category inputs use the disk with its d+2 input/output boundary marks and one additional marked seam point on the distinguished arc. Modulo its positive automorphisms this has dimension d. Choose the critical interior point in a normalized chart determined by output, first input and seam point; near a boundary stratum use the corresponding chart on the component carrying the seam. Such choices extend from the product collars to the interior because the available interior-point choices form a disk. Attach the same quadratic critical disk there and glue the cut to the seam. Its regular sections define $c^d$ of degree -d, with the inherited test-brane boundary labels. This is a genuine finite-dimensional family of base domains; an unconstrained extra complex interior-mark modulus has not incorrectly been counted as one real modulus.

All codimension-one ends are: a polygon splits off at a consecutive input block away from the seam, a module polygon splits off on the input side, or one splits off on the output side carrying the seam decoration. The real domain fixtures are zw=-rho on a boundary node, as in the actual disk-domain supplier. On each neck the fibrations and boundary conditions are products. The vertical operator is the actual transverse end operator, and the component inverses/finite evaluation obstruction theory give the same uniform cutoff right inverse and nonlinear contraction as in the supplied polygon gluing proof. No critical point is on a neck. Multiple necks use disjoint cutoffs and direct-sum matching. Conversely the actual compactness and exponential-neck extraction identify every limit with one such split, and the same inverse-gluing uniqueness gives its collar. Thus these ends yield exactly the full bar module equation for c, for EVERY d.

The analytic hypotheses have actual constructions here. Add a sufficiently large exact base-area form to Omega; it tames the horizontal almost-complex structure and has contact-type behaviour on the outer product boundary. Scalar Liouville no-escape keeps images in a fixed compact set. A bubble projects constantly to the base, so its added base area is zero and its exact fibre area is zero; hence it is constant. Local epsilon regularity, transverse strip extraction and compact domain fixtures apply in the total space. A limit principal section cannot hit the critical point because pi composed with it has derivative1. Horizontal sections of area0 have no linear kernel: differentiating the vertical-energy plus nonnegative-curvature identity gives a nonnegative integral of their parallel derivative and curvature Hessian; a kernel section is parallel and vanishes by a transverse end. For nonhorizontal sections, vertical almost-complex variations on a source-supported patch span the cokernel by the actual adjoint unique-continuation/evaluation supplier. Finite-dimensional Sard on the compact exhaustion gives regular data while preserving horizontal sections and every previously fixed product face. Higher-domain and parameterized versions use the same supported variations. This reconstructs the required section compactness, transversality and collars; it does not apply a closed-exact-brane theorem to a nonexact total boundary circle without checking vertical bubbles.

For $c^0$ the zero-area horizontal sections are exactly the old generators q(z), in the product complement of the tube. To see this, a horizontal section is constant in each cut trivialization; matching the two pieces makes it a tau-fixed common intersection of W and K. In the nontrivial tube the distinct fibre footpoints exclude such a common point. Outside the tube each old z gives one constant section. Its transverse linear strip operator has kernel zero and index0, so the count is +1 relative to the identical old orientation line. Hence c^0=gamma+(terms of action order at least3eta), where gamma sends old q(z) to z and every new p(x,y) to0. Nonnegative curvature and the exact primitive identity show that all coefficients have nonnegative action order. The action gaps give the stated stronger order for every nonhorizontal contribution.

We construct h:cb~0, not merely the fact that the cohomology composite vanishes. In a disk with evaluation inputs $W\to S\to Y$ and output in K, place one interior critical disk and a cut ending on the S-labelled arc. There is a one-parameter family moving the critical disk from the end carrying the c decoration to a small boundary cap attached on that S arc. At its first endpoint the base is the boundary connected sum of the evaluation polygon and the c domain, with a long product strip. At the other endpoint it is an ordinary exterior polygon with a boundary evaluation on S, glued to the standard critical cap with its whole boundary on S. An explicit construction is to fix the output/first input/seam as three real anchors in the upper half-plane, move a small round critical disk along an interior path ending at the selected S boundary point, and use real boundary plumbing $zw=-\rho$ at the two endpoint fixtures. The fibration outside that disk is flat with a tau cut; inside it is the standard quadratic fibration. At each stage its curvature is the pullback of the nonnegative standard curvature by an orientation-preserving base embedding. Along the S segment the primitive jump is the CONSTANT $K_{\tau }\mathrm{\mid }S=a$, so the exterior boundary form has integral -a and the cap contributes+a. These account for the total primitive exactly. Moving the cut along S changes no underlying boundary set, because $\tau (S)=S$; its graded change is the already retained S[1-n].

For d additional category inputs take the same family over the ordered disk-domain parameters, always keeping the critical decoration on the distinguished component. Its parameter dimension is d+1, so rigid sections define an $h^d$ of degree -1-d. Product collars are chosen first; extending the moving disk/path over the interior is possible in its simply connected half-plane region, relative to those fixed collars. The extra faces of this parameterized base family are exactly the category/module splitting faces already described. Their gluing uses the same actual regular inverses and contractions. The endpoint with a long c--b strip gives the module composite sum over all consecutive partitions, not only $c^0{b}^0$.

At the cap endpoint retain the actual exterior/cap matching correspondence with its framed determinant. For $n=2,3$ use one common finite Morse neck correction in the exact and compact regimes: insert a finite gradient edge at the anchored incidence, sew its zero-length side to the node collar, and use its infinite-length discrete-generator factors. The cap output-p dimension is $2n-1-\mathrm{ind}(p)>0$, so each rigid infinite endpoint has a negative-dimensional exterior factor, chosen empty relative to lower products. Rational zero/one-dimensional Stokes on the whole neck system supplies the correction $K$ with $\delta K=B$ and $h=h_{\mathrm{m}\mathrm{o}\mathrm{v}\mathrm{e}}+K$. The full constrained construction and its common-tuple and higher-face prescriptions are given in the compact extension below; the exact regime is its zero-area sector. The explicit smooth local filling above fixes the same thimble orientation but is not used to assert a raw high-dimensional virtual cap chain.

With the right-module bar differential $b_R$, the oriented boundary identity is b_D h+h b_E= c b, with the usual tensor signs. All determinants are oriented by the ordered input orientation lines, the domain coordinates and their dual matching lines. A node contracts an orientation line with its dual. Moving an inner degree-one operation past preceding shifted inputs gives $(-1{)}^{\sum \mathrm{p}\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{n}\mathrm{g}(\mathrm{\mid }{x}_i\mathrm{\mid }-1)}$; a homotopy parameter is oriented before the domain parameters and gives its two endpoint signs. The matching signs on the cap/exterior Morse neck are the same outward-normal rule. Thus the displayed identity is the actual graded one; no independent ungraded chain homotopy is substituted for the all-arity identity. This calculation also makes the multiple-face contractions agree, since permuting two real outward normals supplies exactly the anti-commuting corner sign.

The determinant orientations used here can be extended from the already fixed Floer orientation lines, rather than assumed for a new singular family. Cap every transverse end by its chosen real Cauchy--Riemann reference operator and remove the critical disk using its computed quadratic reference orientation. On the remaining bordered surface the stable boundary framings lift to the supplied relative Spin structures. The common background bundle restricts stably trivially to S in dimensions1,2,3: it is oriented, its w2 vanishes there because S is a brane and TS is spin, and the next stable obstruction on S3 is pi2(SO)=0. Thus its chosen stable Spin framing extends over the local vanishing thimble; no unasserted higher-dimensional background extension is being used. Extending those lifts and doubling gives a complex closed operator with its canonical determinant orientation; invert the reference-line contractions to orient the section operator. Stable changes of the Spin framing are homotopic, so this orientation is independent of the choices. Concretely the needed low homotopies are pi1(Spin)=0 and pi2(Spin)=0 after stabilization; the latter follows from Spin3=S3, Spin4=S3 times S3 and the successive sphere-bundle description Spin(m-1)->Spin(m)->S^(m-1). The critical seam reference carries the phase/orientation shift1-n, so no orientability of the uncut total mapping-torus boundary is incorrectly assumed. For circle branes the connected Spin structure is recorded by one marked transition with sign-1; concatenation of arcs adds those crossings. The two cap half-circle paths differ by one such crossing, giving exactly the sign already computed in the local component. Under neck gluing all reference lines occur with their duals and cancel, and the domain outward-normal rule gives the stated bar signs. This is also compatible with parameterized framings and cap bordisms.

For the $d=0$ filtration, the homotopy has nonnegative action order. Along the moving-disk part, the vertical energy plus nonnegative curvature equals the end-action difference, so that difference is nonnegative. The zero-area horizontal alternative is absent because it would force an original triple intersection W intersection S intersection K, excluded in normal form. On the exterior correction the boundary form is -a, so its nonnegative energy gives end-action difference at least a. The finite Morse edge used below stays on S with its fixed primitive0 and carries no symplectic area; it introduces no untracked action. This is the estimate needed below. Higher h components need no artificially uniform action gap: the quasi-isomorphism check is objectwise at $d=0$, while their already proved bar identity supplies the full naturality.

Finite exact cone contraction, units and represented comparison. Filtered acyclicity by an explicit finite contraction

For each test W set $E_W=CF(S,Y)$ tensor CF(W,S), $C_W=CF(W,Y)$, $D_W=CF(W,K)$. The preceding construction gives the module morphism f:Cone(b)->D, f(c,e[1])=c(c)+h(e), whose zero-th component is a chain map because $dh+hd=cb$. In the total cone $Cone(f)=D_W$ directsum $C_W[1]$ directsum $E_W[2]$ the differential has matrix [ d_D c h ] [ 0 -d_C -b ] [ 0 0 d_E].

The low-action part delta has only gamma and -beta. The generator decomposition says 0 -> E_W --beta--> C_W --gamma--> D_W ->0

is exact on the underlying graded orientation-line modules, with beta an isomorphism onto the new generators and gamma the identity on the old ones. Let sigma include the old generators and pi project the new generators by ${\beta }^{-1}$. A contraction of delta sends $D_W$ to $\sigma (D_W)$ in $C_W[1]$, and $C_W[1]$ to $-\pi (C_W)$ in $E_W[2]$. Checking the three summands gives delta $h_0+h_0\delta =id$. In particular this is an actual contractible low-order complex, not merely a dimension count.

Cluster action values within intervals of width eta around those tensor/old values. Their different clusters are separated by at least2eta. All differentials and the high parts of b,c strictly raise this finite cluster filtration. The homotopy h does too: its nonnegative action order, together with the support gap at least4eta between E and D, forces that stronger positive shift. Thus the full cone has a finite filtration whose associated graded is the explicitly contractible delta complex. A cocycle is killed first in its lowest occupied cluster using $h_0$; the remaining cocycle has strictly higher cluster support. Finitely many repetitions kill it. This proves that $f^0$ is a quasi-isomorphism for every W, with NO spectral-sequence convergence or infinite-series assumption. Equivalently its contraction can be written by the finite Neumann expansion in the strictly filtration-raising perturbation. The graded signs in the displayed cone matrix make its square zero exactly by the module homotopy equation.

This proves a quasi-isomorphism of the FULL right modules, since a module quasi-isomorphism is exactly an all-arity module morphism whose zero-th component is a quasi-isomorphism at every test object. There is no requirement, and no assertion, that all higher Taylor coefficients themselves be separately quasi-isomorphisms.

Coherent Hamiltonian copies supply cohomological units. Here is the local calculation needed even when the compact brane is not a flat base. In its cotangent neighborhood push it to graph(epsilon dh) for a Morse function h. Covariant coordinates along a gradient path turn the small-strip equation into its real average (the covariant gradient ODE for h) and its complementary Fourier modes of gap pi. Connection terms have a factor $p=O(ϵ)$; acting on the small solution derivative they give $O({ϵ}^2)$, so after slow time $\tau =ϵ$ s their correction is O(epsilon). The finite critical-point/trajectory compactness and the diagonal Hessian end inverses give the same uniformly bounded slow complement; Neumann inversion and the quadratic boundary remainder give a unique small strip for each rigid Morse trajectory. Nonzero modes and the covariant coefficient errors are controlled on finitely many charts of the compact brane, so flatness is not being assumed globally. Determinant reduction identifies its complex with Morse cochains with End(local-system) coefficients. The constant identity section represents its degree-zero class1. A shrinking copy arc at a polygon corner has exactly one local thin solution with that class; the zero-energy limit is the transverse constant triangle and its determinant coefficient is+1. The one-parameter corner-shrinking family has the original strip at its other endpoint and no bubbles by exactness, so its boundary gives mu2(e,-) homotopic to the continuation identity. Thus these are actual cohomological units, not an assumed transverse self-intersection. Near a copy this calculation identifies the distinguished degree-zero intersection class with the constant1 class. Its unit compositions are the constant/local thin polygons, with coefficient+1 in their determinant identifications. Insert such copies in any otherwise nontransverse word; generic supported perturbations and the same continuation domains define the module maps above on that unital pre-category. A unit input either gives the corresponding shorter operation or a one-parameter family whose two unit-placement faces cancel. Thus the geometric maps descend through the normalized unital bar resolution. This constructs compatibility with units, rather than treating a nontransverse self-intersection as a transverse generator.

Different cotangent embeddings, wobbly profiles, finite normal-form perturbations, domain data or small radius choices are connected by compact parameter families. Their section equations have uniformly bounded curvature and the same product boundary at infinity, so their Stokes energy bounds, compactness and supported regularization persist. Counts over an interval give the module continuation morphisms; counts over a square give their homotopies and compatibility with b,c,h. The same anchored finite Morse neck correction is carried along over those parameter families. These comparison maps have inverse up to homotopy from the reversed path, by the same square compactification. Consequently the module triangle is independent of all these computational choices. In particular it is legitimate to make a different finite normal-form/radius choice when checking a particular test W: that check is transported back to the single globally defined module morphism, and does not assume a uniform action gap for all objects.

The construction is also coherent in K, when that stronger comparison is needed. Define the two-sided modules $B_{\tau }(W,K)=CF(W,{\tau }^{-1}K)$, $\mathrm{\Delta }(W,K)=CF(W,K)$, and $E(W,K)=CF(S,{\tau }^{-1}K)$ tensor CF(W,S). For r inputs $b_j$ in $CF(K_{(j-1)},K_j)$, s inputs $a_i$ in $CF(W_i,W_{(i-1)})$, and the middle module element, use the same decorated disk with r additional marks on the K side and s on the W side. Its critical seam identifies the input K-side trivialization with ${\tau }^{-1}K$ and the output one with K; at an attached input component it therefore applies ${\tau }^{-1}$ to each $b_j$ exactly when that component lies on the input side of the seam. These are the actual geometric left actions on $B_{\tau }$. Its real parameter dimension is r+s: the ordinary r+s+1-input polygon has dimension r+s-1, and the marked seam supplies one. Count sections to define $c^{(r\mathrm{\mid }1\mathrm{\mid }s)}$, of degree -r-s. Evaluation defines $b^{(r\mathrm{\mid }1\mathrm{\mid }s)}$ with the two middle factors composed through S; the moving-critical-disk family has one additional parameter and defines $h^{(r\mathrm{\mid }1\mathrm{\mid }s)}$, of degree -1-r-s. The codimension-one faces now include all K-side as well as W-side consecutive splits, with the same inherited cuts and primitive jumps. They give precisely the two-sided bar equations and $b_D$ h+h $b_E=cb$, with the Koszul permutations of the two input lists. There is no additional analytic coefficient type: all necks, supported forcings, cap/exterior constrained Morse-edge systems and determinant references are those already constructed. Units are inserted on either side by the same copy normalization. The $(r,s)=(0,0)$ filtered contraction proves this BIMODULE comparison is a quasi-isomorphism. Thus it supplies naturality in K, rather than unrelated objectwise triangles, while requiring no action gap uniform in the word length.

The formal last passage can be constructed without assuming a geometric twist triangle. First make the cohomological-unit normalization explicit. Over the field split each finite morphism complex into its cohomology representatives and a contractible complement, choose the unit as one representative, and let the complement contraction kill it. The finite planar-tree transfer formulas give a minimal model, with strictly unital associative mu2 on its cohomology. Its higher operations need not initially be normalized at a unit. At the first unnormalized arity, the A-infinity equation says that its degenerate part is a cocycle for the Hochschild differential [mu2,-]. The ordinary unit-insertion degeneracy contracts this degenerate quotient: expanding alternating insertions shows adjacent terms cancel and leaves its identity. Subtracting that contracted cocycle as a bar-coordinate change kills the degenerate part and leaves all smaller arities unchanged. Repeat in increasing arity. Each finite word is affected finitely many times, so this is an actual arity-complete coalgebra isomorphism, not an unjustified infinite sum on a fixed word. This produces a strictly unital model and quasi-isomorphisms with the original category. The same finite-tree/bar formulas transfer its modules and their maps, preserving the actual evaluation map up to its explicit bar homotopy. No geometric triangle is assumed during this normalization.

In that strictly unital model form normalized bar modules on finite composable shifted words; the differential inserts the transferred operations. The representable module $Y_P$ has its usual Yoneda action. Evaluation at the strict unit and insertion of that unit give the inverse/homotopy to bar evaluation: along a word of positive length the contraction inserts it at the first position; the two adjacent insertions cancel and the length-zero term is evaluation. Higher products with a unit are zero by the normalization just constructed. Therefore the Yoneda embedding is cohomologically fully faithful, also after transferring back, on the actual twisted/split completion. On a finite tensor V tensor S, its module is V tensor $Y_S$ with the tensor differential; on its mapping cone, its module is the cone of the ALL-ARITY evaluation map b. These statements follow by expanding the twisted differential in its finite upper-triangular entries, or directly by the displayed bar differential. They are algebraic consequences of the constructed category, not an assumed identification with a symplectic twist.

Section6 now identifies that cone module with $Y_K$, for $Y={\tau }^{-1}K$. Fully faithful Yoneda gives Cone(CF(S,Y) tensor S --ev--> Y) ~= K.

Replace K with tau K. This is exactly the positive geometric triangle stated at the start. Its inverse-twist and shifted rotations follow by the same substitutions and the standard cone shifts. The proof has identified its FIRST map with actual Yoneda evaluation before proving acyclicity, so it has not inferred the evaluation triangle from an arbitrary long exact sequence.

Compact mixed-boundary extension. Fix one finite probe collection from the Statement, including $S$, its twists and Hamiltonian copies, with their given gradings, relative Spin structures and bounding cochains. Choose pair Hamiltonians with nondegenerate chords. Their chord sets are finite because their transverse endpoint intersections are compact. For a stable marked disk with a word of these labels, impose the indicated Lagrangian boundary condition on each open arc and the specified chord at each end. At a smooth point of an arc only its own smooth totally real tangent bundle enters the operator. At a chord end symplectic-flow conjugation gives the transverse-end operator of [F2]. Cap the ends by those operators; the remaining bordered operator is Fredholm by [F2], with smooth finite-dimensional cokernel. The supported forcings in [F2] make its joint node/mark evaluations surjective. Neither argument uses the topology of a boundary label or its being a sphere. For section domains away from the critical chart, use $u^{\ast }{T}^{\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{t}}{E}_S$ and the same arc conditions. The condition $\pi u=\mathrm{i}\mathrm{d}$ removes horizontal variations; its vertical equation has the same Cauchy–Riemann principal symbol and smooth lower-order Hamiltonian/connection terms. Thus exactly these Fredholm and supported-forcing arguments apply to sections. A principal section avoids the critical point since $d\pi \circ du=\mathrm{i}\mathrm{d}$. Its neighborhoods are therefore ordinary regular-fibration charts. For compactness at a fixed energy and word, the interior and open-arc rescaling arguments of [F2] are local and apply with that arc's compact label. A concentration at a puncture gives a broken finite-energy chord solution; the transverse weighted-end estimate supplies exponential convergence there. The fixed compact target, or the fixed compact section region over the compact base family, prevents escape. Nonconstant bubble energy has a positive lower bound for this finite boundary data by the same local rescaling argument. Hence only finitely many positive-energy components occur. Stabilize every nonconstant component by interior slices and keep all stable constant components. The compactness proof of [F2] now gives the same finite normalizations and tree-collapse class annotations, with the boundary label retained on each arc. An ordinary boundary bubble node has one label and a shared real tangent mode; an interior sphere node has a shared complex mode; a broken corner has its fixed chord and no continuous matching mode. These are the only local fixtures. Preglue supported obstruction spaces on the compact component cores. Splicing their core right inverses with the supplied end and node inverses gives an approximate inverse whose error tends to zero with the minimum neck length. The commutator terms are cutoff derivatives and exponentially decaying tails; the lower-order connection coefficients are smooth and bounded on the fixed cores. The geometric series corrects this approximate inverse once its norm error is less than $1\mathrm{/}2$. The local nonlinear remainder has the same uniform quadratic bound as in [F2], since exponential charts, compact boundary labels and their derivatives range over compact sets. The contraction argument then gives gluing and inverse gluing. Use the common flat gluing profile of [F2] for the real and complex fixtures; their disjoint supports give the tuple coordinate changes, normal/obstruction isomorphisms and compatible product collars exactly as for the supplied single-boundary charts. This constructs the mixed and section atlases rather than invoking a single-label Statement for a mixed problem. Orient them by capping each chord determinant line and applying the relative-Spin determinant construction to the resulting smooth boundary bundle. Real and complex matching orientations then give the suspended signs in [F2]. At each finite energy/arity cutoff the rational perturbation construction in [F2] uses only these finite groups, supported obstruction charts, evaluation-submersive thickenings and compatible face products. Its relative extension proof therefore gives compatible finite branches here as well. Retain all disk and sphere bubbles; prescribe their lower-face products before extending over the next chart. The oriented boundary equation is the full curved category/module identity. Inserting the supplied bounding cochains cancels its curvature faces. No unobstructedness of an arbitrary probe is inferred: it is a premise of this Statement. Sphere-specific bounding and cap models are used only on $S$, with the local sphere construction of [F1]. Thus the construction supplies precisely the mixed compact disk and decorated-section correspondences needed below for general probes.

RELATIVE REGIME. All branes lie in X\D and the primitive/action data satisfy $[\omega ]=\kappa \mathrm{PD}[D_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}]$. Because the local twist patch misses D, the vertical divisor is literally D×base outside that patch and extends globally through every decorated fibration. The prescribed near-divisor complex structures and vanishing inhomogeneous terms make the raw map equations ordinary holomorphic in the normal-crossing charts, including their product extensions in the section fibrations. Thus the nonnegative normal zero orders and the total-divisor argument of F1 give total intersection d nonnegative on the raw moduli and additive under bubbles and gluing. A component contained in D is retained; positivity of its TOTAL intersection follows from its positive area, not an assertion about individual SNC self-intersections. The energy identity is E_vertical=κd+Σ(input actions)−output action+controlled curvature/cap terms.

For every fixed word and d this gives the actual compactness bound. Assign weight $q^d$ with $degree(q)=0$. Finite q truncations give compact rational virtual correspondences and the common real/complex/chord collars. Positive-d disks and spheres are not discarded by affine exactness.

ABSOLUTE RATIONAL REGIME. A compact probe L may cross D, so β·D is not a relative intersection invariant and $q^d$ must NOT be used as its weight. Use the actual rational-brane prequantum/action-line convention. For each finite rational-brane test collection, including the finite Hamiltonian copies and the twist tube, choose its common period lattice $\mathrm{\Gamma }=(1\mathrm{/}m)Z$. This means that after choosing the declared one-dimensional action-coset lines at the generators, every closed scalar exponent difference of a polygon/section lies in Γ. The line at a generator need not have a rational REAL action origin: its formal action symbol belongs to its line, not necessarily to the scalar field. Never replace it by a fabricated scalar $q^A$ for arbitrary real A.

Work over the complete Laurent/Puiseux field Λ_m=C((q^(1/m))) with those action-coset lines and the actual energy filtration. Choose the local cap action a rational and sufficiently small compared with the original finite period/action gaps; enlarging the finite common denominator to include a is allowed. Count every actual stable section/bubble with its energy/action-line weight. At fixed energy bound, compact target virtual compactness gives finitely many class/branch contributions; the resulting coefficients are complete energy series in Λ_m after the line bases are fixed. Exactness is not used to delete positive-area bubbles. This finite common-denominator passage is the source Seidel section 9c rational-brane convention, not an assertion that all objects of an unrestricted universal-Novikov category share one denominator.

In both regimes insert the actual positive-valuation bounding cochains at every boundary label. The resulting $\mu 0^b$ vanishes. Twisting the compact virtual bar boundary identity then includes every disk curvature face and cancels it by that MC equation. Interior spheres have their real codimension two and virtual matching; they create no extra real codimension-one boundary term. The unmarked collapsed-boundary S² construction and the S³ degree-two exclusion supply the needed actual sphere bounding model, not merely a claim that $q=0$ curvature vanishes.

The map b is the all-arity evaluation defined by the actual compact operations. The section family with one critical disk and the product complement defines c in all arities and on both module sides if used. Its compact virtual codimension-one boundary consists of the category/module splitting products plus curvature insertions. After the MC twist, the latter vanish and the boundary equation is exactly $\delta c=0$. The local seam primitive and degree conventions are unchanged from the quadratic model; no critical point lies on a gluing neck.

The moving-critical-disk family defines $h_{\mathrm{m}\mathrm{o}\mathrm{v}\mathrm{e}}$. Its oriented boundary is $\delta h_{\mathrm{m}\mathrm{o}\mathrm{v}\mathrm{e}}=cb-B$, where $B$ is the actual full cap/exterior endpoint correspondence with its real $TS$ matching constraint. All positive-energy cap sections remain in $B$. Their areas need not tend to zero with the local radius, so the exact Stiefel filling is not used to localize them. We construct a finite Morse neck correction for this endpoint rather than postulate a high-dimensional singular evaluation chain.

The cap with its attaching flag has virtual dimension $2n-1$. The quadratic calculation fixes this index, and compact CY disk/sphere classes have Maslov index zero with degree-zero energy variables, so positive-energy stable classes have the same index. Its cohomological output degree would be $1-n$. Retain the critical-seam line $S[1-n]$; it is not an unshifted degree-zero endomorphism of $S$.

Fix the actual sphere Morse/external-edge model of F1 and F2, with $\mathrm{\mid }p\mathrm{\mid }=n-\mathrm{ind}(p)$ and $0\le \mathrm{ind}(p)\le n$. Fix the cap data independently of all exterior flags and parameter cells. At the anchored exterior incidence insert one gradient edge of length $\mathrm{\ell }\in [0,\mathrm{\infty }]$, from the cap evaluation to the exterior evaluation. Its finite equation is ${\mathrm{ev}}_{\mathrm{e}\mathrm{x}\mathrm{t}}={\mathrm{\Phi }}_{\mathrm{\ell }}({\mathrm{ev}}_{\mathrm{c}\mathrm{a}\mathrm{p}})$; at zero it is the original real-node matching equation.

Construct the constrained thickenings before perturbing. Sew the zero-length gradient collar to the real-node collar with their identical obstruction tuple, matching equation, determinant and common flat profile. This is an internal signed seam, not an intersection of independently perturbed component zero sets. At infinite length use the actual broken Morse fixtures and retain every independent neck and discrete critical label. All disk/sphere bubbles remain; disk-curvature insertions cancel only after the actual bounding twist.

Complete coefficient contraction. Prove the actual full map is a quasi-isomorphism

RELATIVE. Reduction of F modulo q is the already proved exact geometric evaluation-cone comparison. For each finite probe W its cone complex is bounded finite free, and that reduction has the explicit finite action-cluster contraction $k_0$. Write its full differential as $D_0+\mathrm{\Delta }$, with Δ q-positive. The series k=k_0(1+Δk_0)^−1

converges q-adically; the usual finite-order telescoping gives $Dk+kD=1$ (or normalize the original contraction before applying this formula). Hence the ACTUAL map F just constructed is a quasi-isomorphism on that probe. No claim of arbitrary $q=0$ equivalence lifting is used: the closed full map and nullhomotopy were constructed first.

ABSOLUTE RATIONAL. Use the local new/old generator and action-line matrix of the exact construction above. The local cap action is chosen below the finite common period/action gaps. The low matrix is its actual invertible β on new generators and identity γ on old generators; the neck correction retains the same permitted action sector. Normalize these finitely many orientation/action-line entries by their invertible scalar monomials. The associated leading cone has the same explicit finite contraction. Every remaining term either strictly raises a finite action-cluster order or carries strictly positive energy valuation in Γ. A finite number of cluster-raising steps is nilpotent; repeated positive-energy steps have valuation tending to infinity. The corresponding Neumann contraction therefore converges over the complete field Λ_m with its action-coset filtration. Equivalently group the finite nilpotent leading cluster changes first, then apply the strictly positive-valuation perturbation series. This proves the actual map is a quasi-isomorphism for EVERY finite rational compact-brane probe collection, including those crossing D.

Different finite collections may require different m and different small cap actions. Their actual continuation/parameterized comparison squares are the same ones used to define b,c,h. Thus they identify the single geometric triangle after extension to the stated algebraic closure Λ_Q. No uniform denominator for an infinite object set, no universal-real-Novikov replacement, and no claim that an arbitrary real action origin is a Puiseux scalar is introduced.

The resulting represented triangle is CF(S,K)⊗S --actual evaluation--> K → τ_S K → (CF(S,K)⊗S)[1] in the actual compact unobstructed rational category, for $n=2,3$. The inverse form and graded rotations follow with the previously fixed $\tau (S)=S[1-n]$. It therefore supplies the compact rational generation application of the negative-monodromy argument, rather than only its exact affine shadow. The full compact relative model and all positive-energy bubbles are retained.

Regular local section plumbing at a prescribed cap/exterior face. Suppose an actual stable decorated base has an exterior component and a critical-cap component joined at one boundary node, with critical value in a fixed compact interior subset of the cap component. At the node choose holomorphic half-disk coordinates $z,w$ and base plumbing $zw=-\rho$, $\rho \ge 0$. Trivialize the fibration there with fibre $X$ and boundary fibre $S$. Glue the two product fixtures by the identity on $X$, keeping the forms, compatible structures and connections unchanged outside them and product on their neck. This defines an actual local family of fibrations relative to the given nodal one; the critical point stays off its neck. The nodal constraint is ${\mathrm{ev}}_{\mathrm{e}\mathrm{x}\mathrm{t}}={\mathrm{ev}}_{\mathrm{c}\mathrm{a}\mathrm{p}}$ in $S$, with a shared real $TS$ mode, not a discrete chord match.

A specified global moving-critical family. Normalize a disk as the upper half-plane with input anchors $x=0$, $y=\mathrm{\infty }$ and output $o=1$. Its $S$ arc is the negative real axis. For $0<t<1$ take the critical value $p(t)=e^{\pi it}$ and the slit ${\gamma }_t=\{e^{\pi iu}:t\le u\le 1\}$ ending at $-1$ on that arc. At $t=0$ the real blowup at $p=o$ has exterior anchors $0,1,\mathrm{\infty }$, with $1$ now its node; on the critical component $w=(z-1)\mathrm{/}\sin (\pi t)$ puts output at $0$, node at $\mathrm{\infty }$ and critical value tending to $i$, since its real part is $-\tan (\pi t\mathrm{/}2)$. This is the evaluation component followed by the two-ended critical component. At $t=1$ use $v=(z+1)\mathrm{/}\sin (\pi t)$: the critical value tends to $i$, the cap has attaching flag $\mathrm{\infty }$, and the exterior has its additional node at $-1$. The slit collapses onto the cap. These coordinate limits specify both stable endpoints; no arbitrary embedded-disk contractibility premise is used.

Total-space Spin and the actual cap reference. Before attaching the quadratic handle let $A=X\times \mathbb{D}$. The relative handle has index $n+1\ge 3$, hence $H^1(E,A;{\mathbb{F}}_2)=H^2(E,A;{\mathbb{F}}_2)=0$. The restriction map on $H^2$ is injective. Since the CY $X$ is spin, $w_2(TE)\mathrm{\mid }A=0$ and therefore $w_2(TE)=0$; the fixed Spin structure extends, with the framed quadratic thimble fixing its identification. For a common oriented relative-spin background $B$, its restriction to the attaching $S^2$ has $w_2(B)=w_2(T{S}^2)=0$ and so is stably trivial by ${\pi }_1(SO)=\mathbb{Z}\mathrm{/}2$; it extends over the $3$-handle. At $S^3$ use ${\pi }_2(SO)=0$. This extends the background on the relevant skeleton and does not confuse it with ambient Spin.

Use one actual atlas on the new section faces and the mixed category. Extend the one-critical-point total space slightly beyond its base boundary by its monodromy mapping-torus cylinder. Choose the base metric cylindrical outside a compact annulus and use the fixed compact fibre metric and connection there; add a sufficiently large cylindrical base-area form. The resulting target has uniformly tame local geometry on its end and a compact critical core. Its raw almost-complex structure makes the base projection holomorphic. A degree-one raw section has boundary on the prescribed base boundary, so the maximum principle keeps its projection in the closed base disk; every sphere or disk bubble has degree-zero projection and the same compact image bound. Hence the raw degree-one stable-map set lies in a compact target region. Finite thickenings are chosen in a slightly larger compact region of this extended target. This supplies the target hypotheses for the actual total-space local map atlas without an unproved compact symplectic capping construction.

Use the SAME decorated base families that define b, the monodromy-cap map c, and the moving-critical-disk family $h_{\mathrm{m}\mathrm{o}\mathrm{v}\mathrm{e}}$ in the exact construction above, including their two-sided input version when needed. Replace the exact-fibre no-bubble deletion by the compact mixed-boundary and section atlases constructed in step 1.62. Near the quadratic critical point keep the exact local model; elsewhere the fibration is a product. The vertical almost complex structure, perturbations and divisor data are products on every domain face. Actual principal sections cannot pass through the critical point, since differentiating $\pi \circ u=id$ would give $0=1$ there. Vertical disk/sphere bubbles are retained with their actual classes and matching maps.

For a fixed word and compact exterior parameter cell $P$, the cap output constrained to the stable manifold of $p$ has dimension $c(p)=2n-1-\mathrm{ind}(p)\ge n-1>0$. All $P$ coordinates stay on the exterior since the cap datum was fixed independently. In a rigid infinite-length endpoint, matching is over discrete $p$, so the exterior factor has dimension $-c(p)<0$; there is no second subtraction of $n$. The same calculation holds for every coefficient of every higher parameterized operation: its zero-dimensional boundary has total factor dimension zero.

At a fixed nodal section the vertical linearizations are bordered real CR operators on the regular vertical bundles. Supported forcings from the bordered evaluation supplier make their CR/evaluation-difference derivative onto. Retain its finite kernel and shared real node mode. The real-node weighted inverse and cutoff construction then have the same bounded coefficients, exponentially small residual and quadratic estimate as the disk real-node construction, since this fixture is disjoint from the critical chart. Its contraction gives the local smoothed solutions; the common flat gluing profile gives smooth core/evaluation maps without claiming differentiability in raw $\rho$. For several prescribed nodes take the product of disjoint fixtures and ordered matching exact sequences. Their corner restrictions agree, with the sign for exchanging two real outward normals. This gives a local collar at an already specified cap/exterior stratum. It does not prove that this stratum belongs to a global moving-critical-disk family.

The raw regular-piece equations and compatible-structure extension have explicit constructions: use ${\mathrm{\Omega }}_K=\omega -dK$ and horizontal lifts $e+X_{K(e)}$ with the Hamiltonian sign fixed above. On compact regular cores add a base-area term larger than their bounded curvature; near the critical point retain the standard quadratic structure. On protected collars extend compatible vertical structures by extending their positive metrics and applying $J=A(-A^2{)}^{-1\mathrm{/}2}$, $g(Av,w)=\omega (v,w)$. This supplies actual data on a fixed smooth critical fibration and its regular-node collars. It is not a construction of its compactified global decorated-domain family.

For $r$ left and $s$ right inputs add ordered real flags $0<a_1<\cdots <a_s<1<b_1<\cdots <b_r<\mathrm{\infty }$. The open parameter dimension is $r+s+1$. Compactify by all nested flag-collision blowups, including the critical/output collision at $t=0$ and the critical/cap collision at $t=1$. In each finite real projective chart extract colliding groups, divide relative positions by a positive quadratic cluster scale and repeat on proper subgroups; include $p,\bar{p}$ in a critical cluster. Every extraction separates flags, so the process terminates at a finite ordered tree. On its distinguished branch retain $\sin (\pi t)\mathrm{/}\delta$ for each cluster scale $\delta$. A positive finite ratio puts the critical value on that component; zero puts it deeper. The added positive real flags stay away from the cap point $-1$, so none enters the cap at $t=1$.

On the cap circle let $\mathrm{\ell }$ be the Möbius line and $V={\mathrm{\ell }}^{\oplus (n+1)}$. The vanishing-sphere mapping torus is $Q=S(V)$, with $T^{v}Q\oplus \mathbb{R}={\pi }_Q^{\ast }V$ and $TQ$ stably ${\pi }_Q^{\ast }V$. Thus $w_1(TQ)=(n+1)\alpha$ and $w_2(TQ)=0$, since ${\alpha }^2=0$ for the pulled-back base class. In $n=2$ the first class is nonzero. Add the reference pair $E_{\mathrm{r}\mathrm{e}\mathrm{f}}={\mathbb{C}}^{n+1}$, $F_{\mathrm{r}\mathrm{e}\mathrm{f}}(\theta )=e^{i\theta \mathrm{/}2}{\mathbb{R}}^{n+1}$. Its transition is the same $-I$, so the stabilized boundary is orientable with zero $w_2$ and the paired quadratic Clifford Spin lift. The background has zero $w_2$ on this mapping torus because it is trivial on the sphere fibres, $H^1(S^n)=0$ and ${\alpha }^2=0$. Along other arcs use their given brane lifts and fixed capping lines. Do not assert that the odd-dimensional $-I$ lift itself lies in Spin or that ambient Spin makes the original $Q$ orientable.

Choose these fully constrained negative-dimensional exterior factors empty before extending the whole neck family. Order the finite types by energy, arity and virtual dimension as in the rational correspondence construction. Positive-energy zero-input factors have smaller energy; a zero-energy nonunary factor lowers the companion arity; the remaining pure unary Morse factor lowers dimension. A true boundary of a negative-dimensional exterior factor is a discrete Morse/chord product of total dimension at most $-2$, hence contains an earlier negative-dimensional factor. Continuous real-node matches are sewn internal seams, so no invalid inference from continuous fiber-product dimensions is used.

Given actual compact moduli of both pieces, first choose their joint evaluation-augmented thickenings. Their real-node constraint defines the finite-dimensional matching submanifold. Include the finite Morse-edge matching in the same tuple system and sew its zero-length side to this node collar before choosing the new distributions. Its endpoint is the full compact cap/exterior correspondence $B$, including all positive-energy orders; neither a high-dimensional singular cap chain nor its raw insertion is assumed.

Reconstruct from $z_{\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{t}}=c+\rho z_{\mathrm{c}\mathrm{h}\mathrm{i}\mathrm{l}\mathrm{d}}$ and $p=e^{\pi it}$. The distinguished height equals the product of its ancestor scales times a positive smooth normalized height. Since the derivative of $\sin (\pi t)$ is nonzero at each endpoint, these equations recover the endpoint parameter and give the independent nonnegative node scales and component positions. Different chart choices recover ratios of the same normalized differences. This proves the local corner products and the unique decorated tree on overlaps. The closure in the finite compact normalized cross-ratio/direction product is compact Hausdorff. Its real faces are exactly the $t=0$ ordered allocations between the evaluation and critical components, the $t=1$ cap/exterior product, and consecutive flag collisions giving category/input/output polygons. Intersecting clusters are nested or disjoint; their two restrictions are the same tree. Thus the two-sided composition endpoint includes all partitions, and no additional moving-critical endpoint is omitted.

The reference determinant is explicitly oriented. Its scalar disk operator has kernel $f(z)=a+\bar{a}z$, $a\in \mathbb{C}$, and zero cokernel by the boundary Fourier mode calculation. In rank $n+1$ its kernel is the complex-oriented ${\mathbb{C}}^{n+1}$. Pull it back by the raw degree-one base projection. Every vertical disk bubble contributes its constant real kernel and every sphere bubble its constant complex kernel, with zero cokernel. Removing a tree leaf cancels that constant determinant with the identical node-matching determinant. This is the canonical pairing $\det V\otimes (\det V{)}^{\ast }$, requiring no separate orientation of $V$. Induction leaves exactly the complex-oriented principal kernel. On a stable base split carry the half-rotation on the critical component and use constant reference factors on the other components. The same exact sequences give smoothing-compatible reference orientations on every tuple and face.

The prescribed lower products are therefore empty. Finite relative constrained Sard choices give empty zero sets on the negative-dimensional exterior interiors while retaining those germs. On each infinite cap face prescribe the product of this exterior section with the corresponding cap section; this product is zero-free even if the cap is positive-dimensional. Joint endpoint and section parameters, rather than unperturbed evaluation submersivity alone, give the needed constrained transversality.

The local cap/exterior collars by themselves do not construct the full compact cone argument. The subsequent construction must supply the global decorated base and universal fibration for the moving critical disk, give its critical decoration on every stable component, its c-composed-with-b and cap/exterior endpoint fixtures, all two-sided input faces and their intersecting plumbing maps, and prove that these exhaust its compactification. One must also construct the compact positive-energy section atlas for that family and identify its cap end with the same exterior operation. The local construction above gives collars only after a boundary stratum has been specified. Neither an interior disk of auxiliary choices nor an algebraic perturbation contraction supplies this global family. The ordinary compact-disk supplier has a single embedded fixed boundary label in its Statement; its application to a suspended mixed boundary requires the displayed additional geometric argument.

Insert the fixed cutoff quadratic fibration in a disk about $p(t)$ and use the flat $X$ bundle on its complement, clutched by the actual ${\tau }_S$ across ${\gamma }_t$. Match its annulus to the critical model's flat annulus with the same return map. On endpoint charts use the fixed critical model in the $w$ or $v$ component and glue the regular node fixtures by their product trivializations. This defines the universal fibration relative to all the product faces, with its unique critical point off every neck. The exact transition ${\tau }_S^{\ast }\theta -\theta =d{K}_{\tau }$ permits the local correction $-d(\beta K_{\tau })$; outside that tube the compact form is the product pullback of $\omega$, so no global primitive on compact $X$ is assumed. Near $D$ the construction is the literal product. Extend compatible vertical structures on protected regular collars by their positive metrics and the polar formula, retaining the standard critical chart. A large base form tames the total structure on every compact core.

Stabilize the TOTAL-space operator by this actual reference and the ordinary background reference, after capping all label-changing ends by their fixed operators. Its real boundary pair is oriented and has the specified Spin frame class. The frame-pinching proof of the determinant supplier is an operator-level argument, so applies to this bundle pair without positing a product Lagrangian $Q\times V$. Orient its real-constant disk and complex sphere factors, then divide by the oriented reference determinants of step 3.4. For tuple enlargements the same extra obstruction/normal pair cancels; augmented-kernel transitions are the actual determinant exact-sequence maps. Spin frames have no boundary-family monodromy by the Spin boundary-torus argument, and the reference has none by its complex principal kernel and canceled tree factors. Eliminating two node matches in either order is the same triangular sequence; odd-factor exchange gives the Koszul sign and exchange of real outward normals gives the corner sign. Therefore the resulting original determinant orientations agree on c/b, cap/exterior and every two-sided face, including critical bubbles, without using a vertical tangent bundle there. The scalar Fourier parameter here is the conjugate of the parameter in step 1.16. Its real orientation changes by $(-1{)}^{n+1}$ in rank $n+1$; include this fixed common factor when identifying it with the quadratic cap orientation in step 1.20. This fixes the common scalar normalization once, with all Spin-central external line actions retained.

At a pair of faces use the same fully cut decorated tree and simultaneous component tuples, with the same external Morse histories and stabilization slices. The two restrictions are literally the same distributions with the same branch masses. Exact normal retraction composites and relative finite extension preserve them on smaller protected neighborhoods; additional interior forcings vanish there. Increasing the budget leaves every earlier type fixed. This gives one all-arity, parameter-compatible neck system rather than separate nullhomotopies at each parameter.

Define boundary branes and end trivializations by that same slit transition. At $t=0$ the evaluation component has branes ${\tau }_{S}W,S,K$ and common end $CF({\tau }_{S}W,K)$; the critical component has that end and output $CF(W,K)$. The actual shift/action-line identification of steps 1.28 and 1.29 converts its input tensor to $CF(S,Y)\otimes CF(W,S)$, $Y={\tau }_S^{-1}K$. At $t=1$ trivialize the flat exterior: its two $S$ subarcs differ by the retained $1-n$ shift at their cap node. This is the line $\mathrm{Hom}(S,S[1-n])$, not an unshifted identity insertion. The local primitive constant $a=K_{\tau }\mathrm{\mid }S$ occurs with opposite signs on cap and exterior. These are consequences of applying the one fixed clutch transition on its two charts; no Hamiltonian isotopy from ${\tau }_{S}W$ to $W$ is asserted.

On each finite word/cutoff the suspension identity $E_{\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{t}}=\int U^{\ast }(\omega -dK)+\int R_K$ bounds section energy by its relative $\kappa d$ or rational action-line weight and the bounded core curvature/cap terms. Fibres are compact. On finite total-space cores the fibred boundary is totally real, namely the horizontal boundary tangent plus the vertical Lagrangian tangent. Energy-atom extraction there, transverse vertical end extraction and the compact stable-base charts above retain every positive-energy disk/sphere bubble. A principal section cannot hit the critical point because $d\pi dU=1$, but a vertical sphere bubble may hit it and is retained. This compactness argument uses total-space charts at such bubbles; a vertical tangent bundle through a critical point is not invoked.

Only fully constrained zero/one-dimensional whole pearl systems are counted. Their finite rational zero chains and oriented one-dimensional Stokes identities are the actual construction of F2; no arbitrary-dimensional weighted branch union is triangulated or pushed forward as a singular chain. The full-singular-chain model, when used, is the specified geometric Morse external-edge map $u$, retraction $p$ with $pu=1$, and extension through $u,p$ in F2 and F3. It transports these already constructed operations and homotopies.

Finite symmetry is retained by choosing sphere/label representatives and transporting the full tuples and Morse fixtures, with actual stabilizers and Spin-central lines. No nonlinear operation is averaged. The external-edge model is the same one on all cap, exterior and mixed-category factors; this is the geometric binding required before applying its algebraic chain comparison.

More precisely use the ordinary total-space CR map problem on $E$ with fibred boundary, projected degree one on the principal disk and zero on vertical bubbles, and the fixed principal source structure from the displayed base family. On the RAW zero set the projection is holomorphic; degree-zero bubbles project constantly, and the principal degree-one projection is a disk automorphism fixing the three anchors, hence identity. Thus the original total-map zero set is the actual section locus. For an endpoint component with too few anchors use a finite reparametrization slice at regular principal points, avoiding nodes and critical points; this removes its redundant base automorphisms. At a cap do not require a distinct critical-preimage mark when a bubble attaches there. Finite thickened maps may cease to project holomorphically; no infinite-dimensional projection constraint is imposed on the perturbation.

In the finite Morse model the cap cocycle has total degree $1-n<0$, whereas the sphere complex and nonnegative cellular parameter cochains have no negative degrees. Its coefficients and its compatible primitive are thus zero; the dimension calculation above proves the absence of the requisite rigid cap coefficients geometrically. The nontrivial correction is the neck comparison from the original real-node endpoint to this finite-model zero. Discarding that comparison by substituting a zero primitive into an unconstructed raw insertion would be incorrect.

Let $K$ denote the degree-minus-one two-sided module operation counted by this neck family. Order the increasing edge-length normal first, followed by the retained cap/exterior determinants and matching dual. At $\mathrm{\ell }=0$ the outward normal is $-{\mathrm{\partial }}_{\mathrm{\ell }}$, giving $-B$; at infinity the product is zero by step 4.1. The remaining boundaries are $\delta K$, with the degree-one inner operation moved past preceding suspended external lines. Rational Stokes therefore gives $0=\delta K-B$, hence $\delta K=B$, on every word and parameter face.

The total-space normalization operators have rank $n+1$ and smooth totally real boundary even at critical-fibre bubbles, since $TE$ is smooth. Supported forcings span their finite cokernels and joint node evaluations by the bordered/closed suppliers. Retain real total node modes at boundary bubbles and complex ones at sphere bubbles. The transverse-end and shared-mode node inverses, disjoint cutoffs, compact core inverses and quadratic contraction then give actual local smoothing charts. The common flat profile gives smooth core evaluations. Simultaneous obstruction tuples give the extra-section normal coordinate changes. On regular principal parts the base operator has three real automorphism modes and no cokernel before fixing the three anchors, and index zero afterward; base constants along attached genus-zero vertices cancel their matching modes. At critical bubbles use the total rank/index, not a false splitting by $d\pi$. The quadratic relative canonical line extends as $d{z}_0\wedge \cdots \wedge d{z}_n\otimes d{w}^{-1}$, so adding vertical CY classes changes no Chern/Maslov contribution.

The correction is consequently $h=h_{\mathrm{m}\mathrm{o}\mathrm{v}\mathrm{e}}+K$, and $\delta h=cb$. This is an identity in the ordinary unshifted module Hom complex. The cap/exterior endpoint and the zero-edge collar retain the same seam $S[1-n]$ and the same quadratic reference determinant, so no new overall sign is inserted when comparing them. The construction supplies an actual geometric comparison before any filtered or algebraic transfer.

Retain brane/capping orientation lines and the fixed quadratic reference at the critical component. Cancel the normalized base determinant on the regular principal part, use Spin pinching on its arcs, complex orientations on sphere factors and the ordered matching line/dual contractions at nodes. Do not claim that the cap mapping torus is orientable: for $n=2$ its antipodal $S^2$ mapping torus is nonorientable, so the seam reference is essential. The reference construction below gives its line-valued coherence over the total-space tuple transitions. Rational finite reduction/relative sections can be prescribed as the identical factor products on all the domain faces, retaining compact positive-energy bubbles. Their intersections are the same labeled tree products.

This defines the compact relative/rational-absolute nullhomotopy with its actual higher coefficients. The edge has no symplectic area and changes no action-coset line, primitive constant or divisor label, so it preserves the allowed exterior filtration sector. In particular the map $F:\mathrm{Cone}(b)\to Y_K$, $F(c_0,e[1])=c(c_0)+h(e)$, is defined before testing its quasi-isomorphism.

The ANCHORED exterior operation uses the $t=1$ exterior bases: anchors $0,1,\mathrm{\infty }$, incidence flag $-1$, and the ordered positive-arc W/K flags. Their dimension is $r+s$; at $r=s=0$ this is one fixed four-flag conformal disk, not the full space with an extra free input. At this incidence use the actual cap matching and its finite Morse edge, including rooted sphere disk trees, bounding insertions and the same geometric-to-finite-chain external edges as the mixed category. This defines a finite geometric family, not an operation on arbitrary singular simplices.

The map $F$ is an actual closed all-arity module morphism by $\delta h=cb$. Its two-sided version uses the same anchored neck system, with the same lower-face restrictions. The parameter-first orientation gives its homotopy degree and the Koszul corner signs. Finite chain-model transfer transports this existing identity; it does not construct an arbitrary raw singular insertion.

At the cap endpoint the universal fibration and boundary data are exactly the boundary connected sum of that anchored exterior and the fixed cap. Their node constraint is the evaluation difference in $TS$; the node/zero-edge seam has the same component tuple and branch data. Denote this full endpoint by $B$. At the other endpoint the shared transverse chord and all ordered allocations give $cb$. The remaining faces are the bimodule differential and actual MC curvature insertions, hence $\delta h_{\mathrm{m}\mathrm{o}\mathrm{v}\mathrm{e}}=cb-B$. The finite neck comparison supplies $\delta K=B$ and $h=h_{\mathrm{m}\mathrm{o}\mathrm{v}\mathrm{e}}+K$ without manufacturing geometry by an algebraic perturbation series.

The explicit decorated domains, their total-space operator and the anchored one-edge correction give the intended source construction with unchanged public Statement. The determinant and common-tuple constructions below bind these families to the retained mixed-category chain model. The proof uses the finite Morse zero/one-dimensional correspondence interface, not an unproved higher-dimensional raw branch-chain enhancement.

Auxiliary c comparisons on the constructed global family. Keep the actual slit-clutched fibration, critical chart, ends and product faces fixed. Two admissible auxiliary systems on this family have an affine path of Hamiltonian one-forms and a convex path of positive vertical metrics; polar decomposition gives the compatible structures, relative to their protected collars and zero near $D$. Extend the lower face paths first by relative partitions of unity. The compact parameter product has bounded curvature and the same total-space local inverses. On $P_c\times [0,1]$ its oriented boundary is the c comparison, with ordinary painted category comparisons on the input/output sides when those data change. On $P_h\times [0,1]$ use the same component paths: its composition face is the c/b comparison product and its cap face is the corresponding anchored exterior/cap path. On $A_{r,s}\times [0,1]$ it is the actual $J$ comparison with the conformal anchor fixed. Squares and ordered paths give higher compatibility with the determinant system of step 4.4.

This defines c on the explicit critical-height family and proves independence of actual auxiliary choices there. Earlier step 1.39 did not specify a separately constructed global c-map, so no equality to an independently authenticated old chain-level map is asserted. A different actually constructed decorated family would require its actual face-preserving base/fibration identification before this prism argument applies; equality of dimensions would not supply it. This proof-data replacement preserves the complete public Statement.

For common-atlas binding include the protected mixed/cap/exterior and finite Morse-edge tuples in one simultaneous finite system before freezing the new section data. On every true infinite-edge face use the identical fully constrained component distributions; at zero-edge/node seams choose the section jointly on the sewn chart. Exact normal retraction composites retain the product germs on smaller protected neighborhoods, and additional principal total-space forcings vanish there. The same mixed sphere external-edge model, stabilizations, branch masses and endpoint variables are retained on every factor.

Direct leading matrix for the newly defined c. For the finite normal-form probes of steps 1.25, 1.26, 1.27, 1.28, 1.29, 1.30 and 1.31 choose the SAME cutoff quadratic form and twist profile used there. On the critical disk it has nonnegative horizontal curvature; outside use the flat slit-clutched form. Its source changes are orientation-preserving embeddings, so pullback preserves the sign of its curvature two-form. The slit primitive correction is exact and changes no curvature. At a base degeneration use those same fixed component models and flat product necks. For this leading calculation the regular components have zero Hamiltonian forcing and static transverse normal-form branes; regularization uses supported virtual sections relative to the actual isolated horizontal solutions. Therefore the original energy-plus-nonnegative-curvature identity applies before virtual perturbation. General admissible computational choices compare to this one by the actual fixed-family prism in step 11.2.

The determinant construction and those common tuples identify the endpoint boundary with $\delta h_{\mathrm{m}\mathrm{o}\mathrm{v}\mathrm{e}}=cb-B$ and the finite neck boundary with $\delta K=B$. Their sum gives the full two-sided $\delta (h_{\mathrm{m}\mathrm{o}\mathrm{v}\mathrm{e}}+K)=cb$. These statements use actual constrained zero/one-dimensional families and the retained line coherence; algebraic transfer alone does not supply the geometry.

A horizontal c-section is constant in the flat cut charts and is fixed by the twist transition. Inside the nontrivial twist support the angle lies strictly between zero and a half-turn, with the antipodal half-turn on the sphere, and hence there is no fixed point. In the outer zero-angle portion, the distinct normal-form W/K fibre footpoints exclude a common intersection in the tube. Every remaining horizontal section is therefore the unique constant section at an old intersection $z$ outside the tube, with input $q(z)$ and output $z$. Its vertical linear problem is the product constant-strip operator with the two transverse planes and decaying ends: the quadratic primitive vanishes on both planes, so its linear energy identity forces any kernel vector to be constant and then zero. Its index is zero by the equal old-generator gradings, so its cokernel is zero. Identical end capping lines and the normalized base/reference cancellations identify its determinant with the identity line map, with coefficient $+1$.

At each cap/exterior and Morse-breaking face retain the earlier component obstruction tuples, endpoint variables and matching evaluations. On a principal section use the vertical operator; on a bubble use the full pullback tangent bundle, including $TE$ at a critical-fibre bubble. The integrable-neighborhood and zero-inhomogeneous-term conditions in the Statement concern the raw map equation. They do not restrict the auxiliary obstruction sections to vanish on $u^{-1}(D)$. In particular, for a sphere $u:C\to D$, choose smooth sections of ${\mathrm{\Lambda }}^{0,1}C\otimes u^{\ast }TX$, including normal directions to $D$, supported off the finite node and stabilization set. Such sections span the finite CR cokernel: their union is dense in the target $L^2$ space, since the omitted set is finite, and the projection to a finite-dimensional cokernel is continuous and onto. Choose a finite spanning list. For joint evaluation surjectivity, use the local homogeneous evaluation sections constructed in [F2], with arbitrary prescribed values at one selected node or mark and zero values at the others. Multiply each by a cutoff equal to one near its selected point and supported away from all the others. Its CR image is supported off every marked point, since the section solves the homogeneous equation on the smaller marked disk. Add these finitely many images to the obstruction space. The operator was already onto modulo the original obstruction space, and these additional extended-kernel sections adjust all evaluation values without changing that quotient equation. This gives surjectivity of the joint operator and evaluations without any assertion that a point-evaluation adjoint multiplier must lie in $L^2$. Take the span of the finite stabilizer translates and transport on the fixed cores. This also treats components whose entire image lies in the integrable neighborhood; no regularity of a divisor sphere is assumed. These are reductions of the ordinary stable-map problem with total class label $\beta$, without imposing a contact-order stratification. Step 1.63 proves nonnegativity of the total integer $d$ on the raw solutions; each small thickening, branch and gluing retains its original class and hence that same $d$, even when a thickened map is no longer holomorphic near $D$. Thus weight $q^d$ and its additivity require no positivity assertion for forced intersections. On protected faces use the direct sum of exactly the earlier unrestricted component tuples; add only interior summands and keep their cutoffs off the necks. The gluing and relative branch extensions of step 1.62 then retain these tuples and their branch masses on smaller collars.

For a nonhorizontal original c-section its scalar action difference is strictly positive by that same identity. For an old input/output pair with matching $z$ the difference is zero, so no such nonhorizontal section exists. Every other old pair has one of the finite separated old-action differences; for a new input, its action differs from its tensor action by less than the local cap budget $a<\eta$. The old/new support separation and gaps of step 1.31 therefore make every nonhorizontal c contribution have order at least $3\eta$. Keep zero virtual perturbation near the isolated regular horizontal sections. Elsewhere finite branches retain their original topological/action-line labels, so a perturbation cannot invent a coefficient in a forbidden lower-action slot. Consequently the newly defined map has $c^0=\gamma +$ terms of action order at least $3\eta$, with $\gamma (q(z))=z$ and $\gamma (p(x,y))=0$. This binds the actual new family directly to steps 1.42 and 1.76, without inventing a comparison to an unspecified former c-map.

For each finite word, energy budget and compact parameter cell take the complete simultaneous tuples of all charts and constraints, including Morse histories and stabilization slices. Work on noncontracting overlaps retaining the smaller stabilization; no smooth forgetting map through contracted flat-profile nodes is invoked. True infinite-edge distributions are products over a discrete generator. Node/zero-edge seams are instead sewn before choosing whole-tree distributions. Comparable reductions and the normal retraction induction give exact open-neighborhood triple composites and identical protected products; relative constrained extension changes only the remaining interior. At intersecting faces the fully cut tree supplies the same tuple and branch masses, with the exchange of outward normals and determinant Koszul factors. Increasing budgets preserves earlier types.

The leading moving-family estimate is retained for the same reason. Its only nonflat curvature is the pullback of that fixed nonnegative critical form, and its end actions and slit primitives are unchanged. A zero-action horizontal alternative would give the excluded triple intersection of the original finite normal-form branes. Thus $h_{\mathrm{m}\mathrm{o}\mathrm{v}\mathrm{e}}$ has nonnegative action order in the affine sector. On the exterior Morse-neck correction the boundary primitive is $-a$, the cap contributes $+a$, and the auxiliary Morse edge changes no primitive on $S$; the exterior energy identity gives scalar action difference at least $a$. The fixed E/D support gaps therefore give the positive shift used for the cone filtration. Compact positive-order caps are retained with their actual weights and are not localized by this zero-order estimate. This completes the direct leading-matrix/action-budget binding needed by the newly constructed comparison.

Keep the cap data independent of every anchored exterior parameter. Insert one sphere Morse edge at the fixed incidence $-1$; its finite matching is ${\mathrm{ev}}_{\mathrm{e}\mathrm{x}\mathrm{t}}={\mathrm{\Phi }}_{\mathrm{\ell }}({\mathrm{ev}}_{\mathrm{c}\mathrm{a}\mathrm{p}})$. The zero-length side is the identical real-node collar and is sewn by the common signed profile. Infinite-length compactification uses the actual Morse end fixtures, retaining all independent broken histories. The cap constrained to output $p$ has virtual dimension $c(p)=2n-1-\mathrm{ind}(p)\ge n-1>0$. In every rigid boundary product the complementary constrained exterior has dimension $-c(p)<0$, with all exterior parameters already included there. If extra cap parameters are used they increase $c(p)$, so the strict inequality persists. Thus no arbitrary-dimensional cap evaluation chain is needed.

Fix those negative-dimensional exterior factors first in the energy/arity/dimension ordering. Their true lower faces are discrete Morse/chord products of dimension at most $-2$, hence contain earlier empty negative factors; real-node matches are internal sewn seams. Positive-energy zero-input vertices lower energy, and the remaining pure-Morse vertex lowers dimension. Relative constrained Sard extends their empty products to empty interiors. Prescribe these zero-free products on the infinite cap faces before extending the whole edge family. MC twisting uses the actual mixed zero-input geometry and cancels curvature faces. The same fully cut tuples give compatible germs on every higher face, and the anchored exterior incidence never acquires a free-mark modulus.

Count only fully constrained zero/one-dimensional whole systems. Put the increasing edge-length normal first in the retained cap/exterior determinant order. Its zero-length boundary is $-B$, its infinite boundary is zero by step 16.1, and all other faces give the two-sided module differential with the suspended Koszul signs. Rational Stokes gives $0=\delta K-B$, so $h=h_{\mathrm{m}\mathrm{o}\mathrm{v}\mathrm{e}}+K$ has $\delta h=cb$ on every word and parameter cell. Its face prescription is the restriction of this same neck family, not an independently chosen primitive. The edge has no symplectic area and changes no primitive, action-coset line or divisor label; therefore the exterior action estimate and allowed filtration sector are retained. Every positive-energy cap remains in the compact family.

The seam shift is fixed by ${\tau }_S(S)=S[1-n]$. The cap endpoint and its zero-edge comparison contain the same reference determinant and the common conjugation normalization $(-1{)}^{n+1}$ of step 4.4; those are identical on both sides, not a second sign in $\delta K=B$. For comparison with the coefficient convention, put $j=n-1$: a separately defined degree-$j$ insertion $J:C\to M$ would satisfy $\delta J=(-1{)}^{j}JD$ because the differential on $M[j]$ is $(-1{)}^j\delta$, and a primitive correction would be $(-1{)}^{j}J(H)$. Here no such raw singular insertion is assumed. The actual finite Morse cap coefficient has total degree $1-n<0$ in a nonnegatively graded sphere/parameter complex, so it and its primitive are zero; the indispensable geometric correction $K$ compares the original node to this finite-model zero. The specified $u,p$ external-edge comparison transfers the already constructed identity and all its higher faces.

The compact decorated domains, total-space section charts, reference-oriented determinants, common constrained tuples and anchored finite Morse neck give the full two-sided moving-family identities with $h=h_{\mathrm{m}\mathrm{o}\mathrm{v}\mathrm{e}}+K$ and $\delta h=cb$. The correction uses genuine finite rational zero/one-dimensional counts and the same external-edge model as the mixed category; no high-dimensional raw cap-chain descent is presumed. The retained leading matrix and finite/filtered cone contraction, followed by Yoneda comparison, give precisely the unchanged compact relative and rational-absolute evaluation-cone Statement with its action lines and framed gradings. This is a draft source construction and asserts no native acceptance.