The named Fermat graph lifts identify the actual vanishing cycles

Auxiliary critical points and positive real thimbles Eliminate $u_N$ by $\prod u_i=1$. For fixed $u_1=v$, the critical point equations of varpi on $W=0$ force $u_2=\cdots =u_N=a,v{a}^{(N-1)}=1$ and $v+(N-1)a=0$. Hence $v^N=(-1{)}^{(N-1)}(N-1{)}^{(N-1)},a=-v\mathrm{/}(N-1)$. The Hessian on the sum-zero variations among $u_2,\dots ,u_N$ is a nonzero multiple of the standard nondegenerate quadratic form; these N auxiliary critical points are Morse. The pair-of-pants quotient coordinates reduce the positive real chamber to $z=(z_2,\dots ,z_{(N-1)})\in (R_{>0}{)}^n$ and $f(z)=(1+\sum z_j{)}^{(N-1)}\mathrm{/}\prod z_j,v^N=(-1{)}^{(N-1)}f(z)$. Its complete topology follows from strict convexity, not merely from uniqueness of a critical point. Put $z_j=\exp (t_j),g=\log f$. With $w_j=\exp (t_j)\mathrm{/}(1+\sum \exp (t))$, the Hessian is $(N-1)(diag(w)-w{w}^T)$. For any nonzero real vector b its quadratic value is (N−1) times the variance of the values $0,b_2,\dots ,b_{(N-1)}$ with strictly positive weights $w_0,w_2,\dots ,w_{(N-1)}$; it is positive. The gradient vanishes exactly when $w_j=1\mathrm{/}(N-1)$, hence all $z_j=1$. The minimum is $f_{\min }=(N-1{)}^{(N-1)}$. The function is proper. For a unit direction b and $\rho \to \mathrm{\infty },g(\rho b)=\rho [(N-1)\max (0,b_j)-\sum b_j]+O(1)$. The coefficient is strictly positive for every nonzero b: if the maximum is zero, at least one $b_j$ is negative; if it is positive, there are only N−2 summands, each at most that maximum. Compactness of the unit sphere gives a uniform positive lower bound. Along each ray from its unique minimum, strict convexity makes g strictly increasing. Every level above the minimum is thus a smooth radial graph over $S^{(n-1)}$, while the minimum is one point. Each of the N choices of v above this chamber gives an $R^n$ thimble $U_{\zeta }$ projecting to the ray from its auxiliary critical value ζ outward. For the symplectic assertion use conjugation rather than a complex-linearity assertion for a real metric. Every point of $U_{\zeta }$ has first coordinate of phase arg ζ and other coordinates of phase arg ζ+π. The involution $u_i\mapsto \exp (2i\arg \zeta ){\bar{u}}_i$ preserves $\prod u_i=1$ and $W=0$, because exp(2Ni arg ζ)=1, and is anti-symplectic for the source real invariant Kähler data. $U_{\zeta }$ is a component of its fixed locus and hence is Lagrangian. The symplectic horizontal lift of the corresponding real base ray is invariant under the involution and therefore tangent to $U_{\zeta }$. Its fibre sections are the spheres just computed and collapse at ζ. This proves that $U_{\zeta }$ is the actual auxiliary Lefschetz thimble. The thimble is simply connected, so its closed Liouville restriction has a primitive.

Zonotope separation without diagrams Let $m=N-1,f_j=e_j-(1\mathrm{/}N)1$ in $R^m$. The relevant lifted zonotope $Z_N$ is the Minkowski sum of the segments $[0,\pi f_j]$ and $[0,-\pi \sum f_j]$. Thus its points have the form $q=\pi \sum c_j{f}_j,c_j=a_j-b,0\le a_j,b\le 1$. In particular max c−min c≤1. The argument of $U_{\zeta }$ is the single point ${\theta }_{\zeta }=(\alpha ,\alpha +\pi ,\dots ,\alpha +\pi )$ mod 2π, Nα=(N−1)π mod 2π. If a representative q of that point belongs to $Z_N$, then $q_j-q_1=\pi (c_j-c_1)$ is π modulo 2π. Since $\mathrm{\mid }{c}_j-c_1\mathrm{\mid }\le 1$, each difference is 1 or −1. They must all have the same sign σ, since using both would give range two. Hence $c_j=c_1+\sigma$ for $j>1$. Substituting in $q_1$ gives $q_1\mathrm{/}\pi =(c_1-(N-2)\sigma )\mathrm{/}N$. For $\sigma =1,c_1\in [-1,0]$, so $q_1\mathrm{/}\pi \in [-(N-1)\mathrm{/}N,-(N-2)\mathrm{/}N]$. For $\sigma =-1,c_1\in [0,1]$, giving the opposite interval. Among the N critical arguments, spaced by 2π/N, only the two extreme values α=±(N−1)π/N lie in these intervals. Every other thimble argument is a positive distance from the compact zonotope. The first coordinate of every point of $Z_N$ lies between −(N−1)π/N and (N−1)π/N: sum the positive and negative first-coordinate contributions of its defining segments. Thus its sufficiently close graph lift also has varpi-image inside the slit plane $S={\mathbb{C}}^{\ast }\setminus {\mathbb{R}}_{\le 0}$. Here is the required uniform phase estimate for the actual graph, rather than an assumption about its argument image. In the cotangent chart of [F2, proof 1.10–1.11], choose a unit homogeneous lift of its map near the real sphere $S=\{\sum x_i=0,\sum x_i^2=1\}$. Its expansion is $z_{ϵ}(x)=x+iϵ\mathrm{\nabla }f(x)+O({ϵ}^2)$ uniformly in $x$, with the positive normal normalization of that chart. A positive smooth scalar in this normalization makes no difference to the following argument. The cubic has $(\mathrm{\nabla }f{)}_i=3\mathrm{/}N-3x_i^2+3x_i{\sum }_j{x}_j^3$, so on $x_i=0$ this component is $3\mathrm{/}N>0$. Compactness gives a fixed neighborhood of that locus where $\mathrm{Im}(z_{ϵ}{)}_i>0$ for all sufficiently small positive $ϵ$. Away from that neighborhood the real part has the sign of $x_i$. Consequently every coordinate has a continuous argument ${\beta }_i\in (-{\delta }_{ϵ},\pi +{\delta }_{ϵ})$ with ${\delta }_{ϵ}\to 0$: use the upper-half-plane branch near $x_i=0$, and the branches near $0$ and $\pi$ on the positive and negative regions. They agree on overlaps, since the imaginary part is positive in the transition region. Put $\bar{\beta }=N^{-1}{\sum }_i{\beta }_i$. On the centred lift to the product-one cover the first $N-1$ arguments are $q_j={\beta }_j-\bar{\beta }$; the other sheets add their constant deck translate. To prove uniform closeness, take any sequence ${ϵ}_l\to 0$ and $x_l\in S$. A subsequence has $x_l\to x\in S$ and all ${\beta }_i(x_l)\to \pi a_i$ with $a_i\in [0,1]$. The limit $x$ has at least one positive and one negative coordinate because its sum is zero and its norm is one. At these coordinates respectively $a_i=0$ and $a_j=1$. Thus $q\mathrm{/}\pi$ is the projection of $a=(a_1,\dots ,a_N)$ by subtracting its mean. Writing $c_j=a_j-a_N$ gives exactly $q=\pi {\sum }_{j<N}{c}_j{f}_j$ in the displayed zonotope. Moreover the difference of its $i$th and $j$th coordinates has absolute value $\pi$, attaining the supporting bound of that zonotope; hence $q\in \mathrm{\partial }{Z}_N$. If the distances to $\mathrm{\partial }{Z}_N$ did not tend uniformly to zero, such a convergent subsequence would give a limit a fixed positive distance away, a contradiction. The explicit graph lift $L_N$ therefore has argument projection uniformly close to $\mathrm{\partial }{Z}_N$. Choose that parameter below the minimum of the finitely many positive separation distances. Then $L_N$ is disjoint from every $U_{\zeta }$ except the two extreme thimbles. This proves the geometric disjointness assertion used in the source orthogonality argument; it does not yet identify an auxiliary derived category.

Actual matching path for the distinguished W-vanishing cycle Consider the bifibration $(W,u_1)$. At $W=t$ the auxiliary critical points have $u_2=\cdots =u_N=a$ and satisfy $(N-1)a^N-t{a}^{(N-1)}+1=0,v=a^{(-(N-1))}$. Reduce a radial W-path to t∈[0,N] using the cyclic symmetry. Two roots initially have arguments ±π/N and radius $(N-1{)}^{(-1\mathrm{/}N)}$; they collide at $a=1$ when $t=N$. Their trajectory has the explicit parametrization $a=r\exp (\pm i\theta ),r^N=sin((N-1)\theta )\mathrm{/}((N-1)sin\theta ),t=(N-1)r$ sin(Nθ)/sin((N−1)θ), 0≤θ≤π/N. These equations follow by taking the imaginary and real parts of $t=(N-1)a+a^{(-(N-1))}$. They give $t=0$ at θ=π/N and $t=N$ at θ=0. There is no turning point inside: $dt\mathrm{/}da=(N-1)(1-a^{(-N)})$ is nonzero there, and da/dθ is nonzero because its argument changes strictly. Thus the real derivative dt/dθ never vanishes and has the sign fixed by the endpoints. Moreover r increases as θ decreases. Indeed the derivative of $\log r^N$ is (N−1)cot((N−1)θ)−cotθ<0: the function x cot x is strictly decreasing on (0,π), as differentiation gives $(\sin x\cos x-x)\mathrm{/}{\sin }^{2}x<0$. The auxiliary v-trajectories therefore lie strictly inside the critical-value circle except at their initial endpoints, with arguments ∓(N−1)θ. They join the two extreme auxiliary critical values through $v=1$ and lie entirely in $S={\mathbb{C}}^{\ast }\setminus {\mathbb{R}}_{\le 0}$. They avoid all other auxiliary critical values. Their concatenation is the matching path ${\mu }_N$ used for the distinguished W-vanishing sphere. The local matching identification follows from the parameter-dependent holomorphic Morse normal form at $(u_i)=(1,\dots ,1)$. The differential of $u_1$ there is nonzero. First use $u_1$ as a coordinate; applying the holomorphic Morse lemma in its remaining fibre variables makes W−N a nondegenerate quadratic form in those variables plus a one-variable function of $u_1$ with nonzero second derivative. Straightening that one-variable function gives $W-N=z_1^2+\cdots +z_{(N-1)}^2$ and $u_1$ a regular function of $z_1$. A local W-vanishing sphere is the real sphere in the appropriate rotated quadratic form. Its sections at fixed real $z_1$ are the auxiliary vanishing (n−1)-spheres collapsing at the two endpoints. Hence it is exactly the local matching sphere. Extend this description along t toward zero using symplectic transport and the compact two-critical-value trajectory above. Different connections and matching interpolations form a smooth family of compact exact spheres; the resulting Lagrangian isotopy is Hamiltonian because $H^1(S^n;R)=0$. Thus the named W-vanishing cycle $V_N$ is represented by the auxiliary matching cycle for ${\mu }_N$, up to Hamiltonian isotopy and its continuously transported grading. This identifies the source's chosen matching path geometrically, rather than assuming it from a diagram.

Algebraic two-exceptional-object core Let a strictly unital enhanced triangulated category be generated by two exceptional objects E,F with Hom*(F,E)=0 and Hom*(E,F)=C a ⊕ C b, |a|=0, |b|=n−1, where $n=2$ or3. Its minimal A∞ model is formal: any sequence of nonidentity arrows has length at most one, so all normalized higher operations vanish. Every perfect object has a finite two-term presentation Cone(V⊗E→W⊗F). Indeed cancel the internal complexes of E- and F-summands over C; the remaining off-diagonal differential is the single E→F block, since there is no reverse arrow. The same assertion for summands follows by the length-one projective resolution of a finite module over this directed two-vertex algebra. The arrow is α⊗a+β⊗b, with α degree zero and β degree 1−n. For $r=n-1>0$ the graded representation is a finite-support representation of a disjoint union of infinite paths … $W_{(j-1)}$ ← $V_j\to W_j$ ← $V_{(j+1)}\to$ …, where $V_j,W_j$ denote degree k+jr and the arrows are β and α. Finite path representations decompose into interval representations. One elementary basis construction is to begin at an endpoint, record the nested image/kernel subspaces in the adjacent vector space obtained by extending along successive edges, choose a basis simultaneously adapted to that finite flag, and propagate it along each edge until its image ends. At a backwards edge choose lifts of the already chosen image basis and extend by a basis of the kernel. The flag-adapted choice ensures that each new basis vector either continues one existing string or begins a new one. Induction on the number of vertices gives bases in which every arrow sends a basis vector to one next basis vector or to zero; each connected string is an interval, with all its nonzero maps rescaled to one. This algorithm uses only Gaussian elimination and finite flags. For completeness the endomorphism dimensions can be computed directly on an interval. If its numbers of V- and W-vertices differ by one, its endomorphism complex has one-dimensional kernel and zero cokernel: the arrow equations successively make every surviving diagonal scalar equal and eliminate off-diagonal entries. It is exceptional. If both numbers equal m, one family of alternating arrows is invertible; identify the V and W spaces through it. The other is a single nilpotent Jordan string. The two-term endomorphism differential has kernel of dimension m (the successive diagonal/Jordan-shift solutions) and cokernel of dimension m, since source and target have equal dimension. The graded signs only rescale those successive equations by ±1 and do not change either dimension. Thus its total self-Ext dimension is 2m. If an object L has $\mathrm{Hom}\ast (L,L)=H\ast (S^n;C)$, it is indecomposable because Hom⁰=C. The interval calculation forces $m=1$. There are two possibilities. Cone(a) has endomorphism cohomology C in degrees 0 and n: the differential kills the a-arrow and leaves b in degree n. The cone on b, after making b a degree-zero map by shifting its source, instead leaves a in degree 2−n. For $n=2$ that gives two degree-zero classes, and for $n=3$ it gives a negative-degree class; neither is $H\ast (S^n)$. Hence L ≅ Cone(a:E→F)[s] for some shift s. This is the precise Kronecker uniqueness implication used in the source. The hypothesis that L actually belongs to the two-object subcategory is essential.

Complete logarithmic geometry and proper thimbles without changing the finite compact data. Fix the finite compact probes, physical copies, matching sphere, critical charts and relevant outward real-thimble segments. The real-chamber properness calculation above makes those segments compact. Include the coordinate-maximum-principle image box for raw polygons on the compact branes, and call the resulting compact set Q. Write $r_i=\log \mathrm{\mid }{u}_i\mathrm{\mid }$, ${\theta }_i=\arg u_i$ locally; ${\sum }_i{r}_i=0$ and ${\sum }_{i}d{\theta }_i=0$ on the product-one torus. Normalize $d^c=-d\circ J$. The source Kähler potential descends from the Fermat cover as ${\phi }_0=c_{\mathrm{F}\mathrm{S}}N\log ({\sum }_i\exp (2r_i\mathrm{/}N))$, with the positive constant $c_{\mathrm{F}\mathrm{S}}$ fixed by the stated integral Fubini–Study normalization. Indeed $r_i=N\log \mathrm{\mid }{x}_i\mathrm{\mid }-{\sum }_j\log \mathrm{\mid }{x}_j\mathrm{\mid }$ and the omitted linear sum is pluriharmonic, so its $d{d}^c$ is precisely the descended source form. Its growth is at most linear in $\max \mathrm{\mid }{r}_i\mathrm{\mid }$. Put $\psi =\frac{1}{2}{\sum }_i{r}_i^2$. Then $d{d}^c\psi ={\sum }_{i}d{r}_i\wedge d{\theta }_i$, the flat logarithmic Kähler form, restricted to the complex submanifold $M^0$. For completeness define that smooth maximum by $\mathcal{M}(x,y)=(x+y+\rho (x-y))\mathrm{/}2$, where $\rho (t)=\int \mathrm{\mid }t-s\mathrm{\mid }\kappa (s)ds$ and κ is an even nonnegative smooth bump of integral one supported in (−1,1). Then $\mathrm{\mid }{\rho }^{\mathrm{\prime }}\mathrm{\mid }\le 1$, ${\rho }^{\mathrm{\prime }\mathrm{\prime }}=2\kappa \ge 0$, and $\rho (t)=\mathrm{\mid }t\mathrm{\mid }$ for $\mathrm{\mid }t\mathrm{\mid }\ge 1$. Take $\mathrm{\Psi }=\mathcal{M}({\phi }_0+C,\psi )$, which is exactly the selected argument where their difference has absolute value at least one. Choose C so ${\phi }_0+C$ wins on a neighborhood of Q. Quadratic growth makes ψ win outside a compact coordinate box. This Ψ is strictly plurisubharmonic: its Levi form is a nonnegative convex combination of the two positive Levi forms plus a nonnegative difference-gradient square. Hence ${\omega }_{\ast }=d{d}^c\mathrm{\Psi }$ is Kähler, equals the source form near Q and is EXACTLY the flat logarithmic form outside a compact set K. Take ${\lambda }_{\ast }=\lambda +d^c(\mathrm{\Psi }-{\phi }_0)$; this is the actual source primitive on Q. All potentials are invariant under the finite diagonal phases and coordinate permutations, and the construction is conjugation invariant. Thus it descends with the actual brane/group data and preserves the real thimble germs. No undeclared torus-collar regularization theorem is needed. Choose $L_1$ below every compact probe height and every critical height, to locate the finite low hooks; no flow cutoff is made at $L_1$. On the slit target put $\mathrm{\ell }=\log \mathrm{\mid }v\mathrm{\mid }=r_1$ and $h=\arg v$, $v=u_1$, and use the UNCUT Hamiltonian $H=h$. Holomorphicity gives $d\mathrm{\ell }(X_h)=-A$, $dh(X_h)=0$, where $A=\mathrm{\mid }\mathrm{\nabla }\mathrm{\ell }{\mathrm{\mid }}_{g_{\ast }}^2=\mathrm{\mid }{X}_h{\mathrm{\mid }}_{g_{\ast }}^2$. Here is the global estimate. In the flat logarithmic region a complex tangent vector $\xi =(d{u}_i\mathrm{/}{u}_i)$ satisfies $\sum {\xi }_i=0$ and $\sum u_i{\xi }_i=0$. The two Hermitian normal vectors $\mathbf{1}$ and $\bar{u}$ are orthogonal because $\sum u_i=0$. Projecting the coordinate covector of $\log v$ onto their orthogonal complement gives EXACTLY $A=1-1\mathrm{/}N-\mathrm{\mid }v{\mathrm{\mid }}^2\mathrm{/}S$, where $S=\sum \mathrm{\mid }{u}_i{\mathrm{\mid }}^2$, in the stated flat normalization. In particular $0\le A\le 1$. If $a\le \mathrm{\ell }\le b$ and $S\ge 2N{e}^{2b}\mathrm{/}(N-1)$, then $A\ge c_N=(N-1)\mathrm{/}(2N)>0$. On the compact remainder the vector field is smooth and bounded. The logarithmic metric is complete: the ambient product-one logarithmic torus has compact angular factors and a complete Euclidean radial hyperplane, $M^0$ is closed in it, and changing the metric only on K preserves completeness. Thus the globally bounded $X_h$ has a complete flow for every finite time. Equivalently its radial log coordinates have bounded speed outside K, so finite-time escape would force an unbounded radial coordinate in bounded time, which is impossible. It conserves h and remains in the chosen slit sector. More strongly, fix a closed finite height interval [a,b] and a compact set of starting points. Enlarge K to contain it and every point in that interval with $S\le 2N{e}^{2b}\mathrm{/}(N-1)$. This is compact: bounded S bounds every $\mathrm{\mid }{u}_i\mathrm{\mid }$ above, while $\prod \mathrm{\mid }{u}_i\mathrm{\mid }=1$ bounds each one below. While an inward trajectory stays in the interval, the total time it spends outside this K is at most $(b-a)\mathrm{/}{c}_N$, since $-\dot{\mathrm{\ell }}=A\ge c_N$ there and ℓ never increases. Every $r_i$ has speed at most one there. Its excursions from K are therefore bounded by that same constant; inside K all its radial coordinates are already bounded. The angular torus is compact, so this proves UNIFORM compact containment of the whole trajectory portion in the ORIGINAL affine target. The same argument for the reversed flow proves the outward bound. It is a finite-height statement, not a claim that an entire noncompact thimble is compact. Inward unstable thimbles start from the critical points at time minus infinity. Cut off a small local Morse cap at height ${\mathrm{\ell }}_i-\delta$. Its exit section is a compact sphere, including every outgoing direction. The preceding estimate applied to this compact section confines its entire forward flow over any [a,${\mathrm{\ell }}_i-\delta$]; the missing local cap is compact as well. Similarly a stable outward thimble is obtained from its compact cap section at ${\mathrm{\ell }}_i+\delta$ by reversed flow, with the same uniform bound up to any finite upper height. There is also no missing accumulation at infinite flow time in such a slab. If a trajectory stayed there for arbitrarily long time, compact containment and the identity $-\dot{\mathrm{\ell }}=\mathrm{\mid }{X}_h{\mathrm{\mid }}^2$ give a limiting critical point: the integral of $\mathrm{\mid }{X}_h{\mathrm{\mid }}^2$ is finite, and its derivative is bounded on that compact box, so $\mathrm{\mid }{X}_h\mathrm{\mid }$ tends to zero; any convergent sequence then has critical limit. All critical points have distinct h values, so at the conserved phase the only candidate is the original critical point. Its height is above the inward exit section (below the outward exit section for reversed flow), and strict height monotonicity excludes that candidate. Every exit direction therefore reaches each retained regular height in finite time. The hitting time is continuous by $d\mathrm{\ell }(X_h)\mathrm{\ne }0$ there, and compactness of the exit sphere makes it uniformly bounded. Together with the compact local cap this proves that every closed finite-height thimble portion is compact. The outward real loci are the actual ones already computed: conjugation invariance makes their horizontal lifts tangent to those loci; Q includes every segment that can meet the compact probes. Monotonicity of ℓ prevents a later part of an outward thimble from returning to that probe height range. Hence no compact intersection, grading, primitive or operation fixed above has changed. The finite low hooks used below also have compact fibre control. Choose their closed base annulus strictly below all critical heights. There $A$ is nonzero, since its zeros are exactly the already computed critical points; the preceding outside-box estimate and compactness of the remainder give a positive lower bound on A. The horizontal fields $-X_h\mathrm{/}A$ and $X_{\mathrm{\ell }}\mathrm{/}A$ project respectively to unit positive ℓ and unit positive h; their norms are bounded by $A^{-1\mathrm{/}2}$. Any finite collection of smooth base hooks and compact parameter families in this regular annulus has bounded base length, hence bounded logarithmic displacement from its compact starting sections. Their transport exists and has compact image. After the hook, the radial tails have the fixed new phases and the same radial argument-flow estimate. This proves finite-annulus properness also for all finitely many layered copies and their allowed hook/continuation families. It does not deduce fibre no-escape from a base diagram.

Stable and unstable critical charts with all direction parameters retained Near a critical point the logarithmic modification is absent, because its whole chart lies in Q; the original Kähler form is unchanged. The holomorphic Morse normal form for f makes its real and imaginary parts quadratic to leading order. The linear Hamiltonian system for h has paired nonzero real eigenvalues. After a unitary Takagi diagonalization and a linear symplectic change, write its stable coordinates q and unstable coordinates p as $qdot=-\mathrm{\Lambda }q+F(q,p),pdot=\mathrm{\Lambda }p+G(q,p)$, where Λ has positive diagonal entries and F,G have vanishing first jets. The stable and unstable manifolds are Lagrangian: their tangent planes are Lagrangian, and invariance plus exponential decay forces ${\omega }_{\ast }$ to vanish on their transported tangent planes. Their local existence and smoothness follow directly from the integral equations: solve the stable component forward with $e^{(-\mathrm{\Lambda }t)}$, and solve the unstable component by the integral from t to infinity with $e^{(\mathrm{\Lambda }(t-s))}$. On a small ball the nonlinear terms have arbitrarily small derivative, so the weighted sup-norm map is a contraction; differentiating that fixed-point equation gives the smooth dependence on the stable initial coordinates. Reverse time for the unstable manifold. This is also the usual elementary proof of the local stable/unstable theorem needed here. For the actual outward chamber $C_i$, the fixed-locus calculation shows that it is the stable manifold for this inward-height flow. Denote its unstable thimble by $D_i$. It projects inward from ${\zeta }_i$ toward the low-height region. Its outgoing tail may be bent toward a common small positive basepoint below $L_1$ without meeting compact probes; use disjoint ordered arcs there. These compactly supported end fixtures fix the ordinary directed-thimble convention. Make a compact input brane U transverse to every $C_i$. Intersections are finite in the range under consideration. Near $x\in U\cap C_i$, use p as coordinates on U, writing $q=q_x+O(p)$. In the exactly linear model its time-T image is $q_T=e^{(-\mathrm{\Lambda }T)}{q}_x+O(e^{(-2{\lambda }_{\min }T)}\mathrm{\mid }{p}_T\mathrm{\mid }),p_0=e^{(-\mathrm{\Lambda }T)}{p}_T$. Thus on a fixed compact unstable patch it converges to $D_i$ as a small graph. For the nonlinear model use the same stable/unstable coordinates and the integral equations above. The graph transform contracts the transverse slope by a factor at most $e^{(-{\lambda }_{\min }T)}+o(1)$ after shrinking the chart; the translation term has the same decay. Differentiate those integral equations for any fixed finite number of derivatives. On a fixed compact exit patch they give $C^k$ convergence to the unstable manifold, with an exponential rate that may depend on k. No uniform estimate over all k or all objects is asserted. ALL p-coordinates are retained. There is no replacement of an n-dimensional point handle by a single planar quadrant with its $S^{(n-1)}$ angular family discarded. If a compact test brane K meets $D_i$ transversely away from the critical point, each full vector p at that intersection fixes both the radius and the angular direction of the local emerging branch. The implicit-function theorem then gives exactly one intersection with that branch of ${\phi }_H^T(U)$, for sufficiently large T.

Exhaustion of the compact-probe generators Fix finitely many compact probes K, make them transverse to the $D_i$ and disjoint from the critical points and the low hook region. Their intersections with $D_i$ in the fixed high-region target compact set are finite. Every pair $x\in C_i\cap U,y\in K\cap D_i$ gives the nearby intersection just constructed. There are no other intersections for large T. Otherwise take $x_T\in U$ with ${\phi }_H^T(x_T)$ in a compact probe. The entire relevant orbit lies in the uniform compact finite-height set proved above and above $L_1$ until its endpoint, so $H=h$ is constant on it and ℓ strictly decreases away from critical points. Outside fixed small critical neighborhoods its rate of decrease has a positive lower bound. A trajectory of arbitrarily long duration with both endpoints in the fixed height range must therefore spend an arbitrarily long time in a critical neighborhood. The h-neighborhoods can be chosen disjoint because the critical values ${\alpha }_i$ are distinct. Conservation of h forbids a single orbit from being trapped near two different critical points. Compactness and the integral-equation description then show that the limiting incoming point lies on $C_i$ and the limiting outgoing point on $D_i$. Transversality gives exactly the branches already listed. Consequently, for every finite compact test collection, sufficiently large T gives the actual orientation-line generator identification $CF(K,{\phi }_H^{T}U)$ ≅ ${\oplus }_{i}CF(C_i,U)\otimes CF(K,D_i)$ as graded vector spaces. The grading of $D_i$ is normalized against its $\mathrm{dual}{C}_i$ in the local quadratic chart. Transport of the same intersection orientation caps then gives the tensor grading and its Koszul sign. This is a geometric generator statement; the differential and higher module operations still require the diagonal/continuation construction below.

Exact cluster actions and the energy issue Normalize the exact primitives of $C_i$ and $D_i$ to zero at their common critical point. For $x\in C_i\cap U$ set $c_x=f_U(x)-f_{(C_i)}(x)$, and let ${\alpha }_i=h(cri{t}_i)$. The transported primitive is exactly $f_{({\phi }_H^{T}U)}({\phi }_H^{T}x)=f_U(x)+{\int }_0^T({\lambda }_{\ast }(X_H)-H)({\phi }_H^{t}x)dt$. On the branch emerging at $y\in D_i$ the stable and unstable exponential estimates imply $A_{(K,{\phi }_H^{T}U)}(x,y)=-T{\alpha }_i+c_x+A_{(K,D_i)}(y)+o(1)$. The stable λ integral is $-f_{(C_i)}(x)$, and the outgoing unstable integral is $f_{(D_i)}(y)$; these are the two finite terms. The error tends to zero on every fixed finite transverse generator collection. Thus cluster centers are $-T{\alpha }_i$ and are separated linearly in T. This explicitly identifies why ordinary fixed-energy Floer compactness cannot simply be applied as T→∞. A strip changing clusters can have energy $T({\alpha }_i-{\alpha }_j)+O(1)$. The actual compact-probe diagonal comparison must subtract the calibrated long-strip contribution and prove bounded nonnegative residual energy. Generator convergence alone does not do so.

Directed algebra and its actual finite bar object Let A be a strictly unital A∞ category over C with objects $D_1,\mathrm{.}\mathrm{.}\mathrm{.},D_N,\mathrm{Hom}(D_j,D_i)=0$ for $j>i$, and $\mathrm{Hom}(D_i,D_i)=C{e}_i$. Assume finite-dimensional graded Hom complexes; use a central minimal model if necessary, by the already constructed objectwise contraction. Every nonunit arrow strictly increases its index. Hence a normalized composable string has length at most N−1. No convergence issue in length occurs. For an actual object U in a larger enhanced category C containing A, set $M_i={\mathrm{Hom}}_C(D_i,U)$. Its right A∞ module structure is the actual higher composition. Define the finite twisted bar object $P(M)={\oplus }_{(i_0<\mathrm{.}\mathrm{.}\mathrm{.}<i_k)}{M}_{(i_k)}\otimes \mathrm{Hom}(D_{(i_{(}k-1))},D_{(i_k)})[1]\otimes \mathrm{.}\mathrm{.}\mathrm{.}\otimes \mathrm{Hom}(D_{(i_0)},D_{(i_1)})[1]\otimes D_{(i_0)}$, including $k=0$. The suspension [1] lowers cohomological degrees by one. Its differential/twisting cochain is the suspended bar differential: compose any consecutive nonunit arrow block by the corresponding $m^r$, or apply a module operation to the terminal M block and adjacent arrows; the last Yoneda/object block is composed by the same actual operations. The original bar identities imply its square is zero, since each two-stage collapse occurs twice with opposite suspended signs. All matrix blocks are finite because index increases strictly. The actual evaluation/augmentation ε:P(M)→U uses the higher compositions of C. Its A∞ morphism equations are the same bar cancellation with the output rooted at U. For a finite Hom model M this is a genuine finite twisted complex of the $D_i$. No generation hypothesis for U was used to define it or the augmentation.

Exact thimble-test contraction, and its limitation Evaluate on $D_j$. A word in $\mathrm{Hom}(D_j,P(M))$ has a first unsuspended factor $a_0\in \mathrm{Hom}(D_j,D_{(i_0)})$, followed by strictly increasing suspended arrows and a terminal $M_{(i_k)}$. If $i_0>j$, the contracting homotopy inserts a new first object $D_j$ and replaces $a_0$ by the suspended arrow $s{a}_0$ followed by the initial unit $e_j$. If $i_0=j$, its initial factor is a scalar unit and that homotopy term is zero. On the augmentation term $M_j$, insert the length-zero $\otimes m\otimes e_j$. Give the insertion the usual suspended-bar sign so it has degree minus one. In d h+h d, the binary collapse of the inserted initial unit gives the original word. Collapses strictly after the inserted unit cancel in pairs with the corresponding insertion after a collapse. All higher operations involving that unit vanish by strict unitality. The remaining endpoint is the augmentation. Thus this is an explicit contraction of the AUGMENTED complex $\mathrm{Hom}(D_j,P(M))\to \mathrm{Hom}(D_j,U)$. A word can acquire the index j at most once; the maximum path length is still N−1. This proves the augmentation is a quasi-isomorphism on every $D_j$. It does NOT prove ε is an isomorphism in C. A nonzero object orthogonal to every $D_j$ could survive in its cone. Treating the $D_j$ as already generating C would beg the question.

The geometric costandard tests are genuinely dual Use the actual high-sheet Hamiltonian geometry supplied by the HMS author. In the auxiliary base the N critical values lie on a common circle $\mathrm{\mid }\pi \mathrm{\mid }=r_0$ and have distinct principal arguments ${\alpha }_i$. Let $C_i$ be the stable OUTWARD thimble over $r\exp (i{\alpha }_i):r\ge r_0$. Let $D_j$ be the unstable INWARD thimble over the corresponding ray from its critical point down to the chosen lower base region, with any end hooks confined to radii strictly below $r_0$. Stable/unstable names refer to the actual Hamiltonian flow; their existence/properness is part of that geometric construction, not a consequence of the following algebra. For i≠j their base projections are disjoint: an outward radius is at least $r_0$, an inward one at most $r_0$, and equality occurs only at distinct critical arguments. For $i=j$ the sole intersection is the critical point. In the holomorphic Morse chart, after rotating the quadratic form, the stable and unstable tangent spaces are the two transverse real planes $R^n$ and $i{R}^n$. Shift the costandard brane $C_i$ so the intersection has degree zero. Therefore the actual pair complex is C for $i=j$ and zero otherwise. The diagonal differential is zero because there is a single generator and exactness excludes a nonconstant strip from it to itself. Hence $HF\ast (C_i,D_j)={\delta }_{(ij)}C[0]$. This is not a statement that the original ordered type-(B) objects themselves satisfy that duality. The $C_i$ are the actual outward/costandard probes. Their identification with the appropriate dual modules is the point of this construction. For completeness, the mixed probe polygons can be defined with the same integrable zero-H raw model. On every sufficiently remote radial base end their log projection has boundary on a HORIZONTAL line Im logπ=constant. Its real part has normal derivative zero along that boundary, by the Cauchy–Riemann equations. The boundary Hopf lemma or Schwarz reflection prevents a maximum or minimum there unless the projection is constant. Every actual finite-energy solution extends to its finite chord ends, so its individual projection has a maximum/minimum. The nonconstant extrema must therefore lie on the fixed compact portion of the base paths or at fixed chord ends. This gives a uniform annular base bound, including bounds away from 0 and infinity. The finite-annulus properness just proved for the actual inward/outward thimbles and all their finite hooks gives a common compact target subset for those boundary images. The holomorphic coordinate functions $u_l$ and $u_l^{-1}$ then have bounded boundary and chord-end values; the maximum principle bounds them on the whole raw polygon. Their fixed product-one and sum-zero equations define a closed subset of this compact coordinate box. Thus the mixed raw problems have actual fibre no-escape, with a common compact region for their finite virtual charts. This uses the proved logarithmic flow estimate, not a base plot or an additional properness assumption. Finite virtual charts can be chosen in that common compact region. Thus $Q_i=\mathrm{Hom}(C_i,-){\mathrm{\mid }}_A$ is an actual A∞ module on A. Transfer it to its cohomology using the linear contraction. Its only nonzero value is at $D_i$. Any nonunit directed path from $D_i$ back to $D_i$ has positive length and cannot exist. Every higher transferred module operation would need its source and target module values at that same index, so it vanishes. The transferred module is precisely the simple coordinate test $L_i$, with $L_i(D_j)={\delta }_{(ij)}C$ and all nonunit actions zero. This establishes the geometric-to-algebraic dual test, including higher operations.

Only finitely many operation problems are needed Let A be the strictly unital directed thimble category with N objects $D_i$. Its normalized bar object $P(M_T)$, for $M_T(D_i)=CF(D_i,U_T)$ and $U_T={\phi }_H^{T}U$, has only strict chains $i_0<\cdots <i_k$. Hence k≤N−1. Its twisting equation and its canonical closed augmentation $\epsilon :P(M_T)\to U_T$ are the ordinary finite A∞ bar identities. Constructing ε does not require guessing a global geometric mutation map. Testing it on a single compact U and on the $D_i$ involves only finitely many morphism complexes and polygon operations of bounded arity. Once ε is an actual closed morphism of twisted objects, its full Yoneda module map is provided by the existing A∞ operations automatically; no action filtration on arbitrary-length words of compact probes is asserted. Use N+2 ordered physical Hamiltonian-copy layers for those finite chains and unit fixtures. The tails of every test copy run to a slit end with |arg v| larger than the phase range of U. Since the uncut argument flow preserves that range, these tail regions have no intersections with $U_T$. In the finite middle slab choose successive hook heights for the copy layers; target thimble hooks lie below the preceding probe layer. All intersections used in a chain are therefore in one fixed finite base slab. The thimble properness on finite base annuli makes its target portion compact. This construction uses a finite number of end fixtures, not an infinite wrapped-end limit. Ordinary continuation units identify the layers. Choose T large enough for the branch convergence simultaneously on that finite slab and on the compact probe U. It gives the actual generators (x,a) with $x\in C_l\cap U,a\in D_i^{probe}\cap D_l$, in $M_T(D_i)$, and (x,y) with $y\in U\cap D_l$ in $CF(U,U_T)$. Their actions are $-T{\alpha }_l+c_x+A_{(D_i,D_l)}(a)+o(1),-T{\alpha }_l+c_x+A_{(U,D_l)}(y)+o(1)$, respectively, with the actual grading/orientation lines from the hyperbolic branch model. Choose small primitive-value adjustments on the finite transverse generators to separate every non-forced finite action equality. Such adjustments are realized by Hamiltonians constant near the selected intersections, as in the existing evaluation-cone normal-form construction. There are finitely many relations because the bar chains have bounded length. The forced equalities are precisely the unit/evaluation identities retained below. The input actions of every fixed thimble-chain operation have a common finite bound. Since the ${\alpha }_l$ are distinct, sufficiently large T makes reverse critical-cluster terms have negative energy; they vanish. Within a fixed critical cluster the finite residual actions supply a second finite filtration. All positive-area terms raise total action in that filtration. Large-area off-diagonal terms are retained as the actual bar extension data.

One simultaneous TRANSVERSE unit model for the finite compact-probe problem. Fix ONE finite compact-probe/bar window of12.1. Include a compact disk supporting its finitely many selected intersection neighborhoods and Morse handle fixtures BEFORE choosing its sufficiently large finite T, then hold T fixed. For a stable branch label x write its high sheet of $U_T$ as the exact graph of $d{g}_x$ over that compact disk. The selected p are the ACTUAL sheet/probe intersections in the generator correspondence, not hypothetical intersections with every future probe. Extend the ACTUAL primitive $g_x$, including its transported constant, to an auxiliary graph $B_x$ with the previously chosen end fixtures. Enlarge this compact graph region to contain all finitely many selected probe intersections p and the finite Morse handles used below. The graph $B_x$ agrees with that sheet near those points, but it is NOT used as a Floer input against $U_T$: that coincident pair would be degenerate. On the parameter disk choose one proper $f_x\ge 0$ with each selected p a nondegenerate minimum of value zero, no other minima, finitely many remaining critical points of positive value, and the prescribed increasing end profile. Disjoint quadratic wells joined by compact one-handles and their cancelling higher handles give such a Morse function on the disk. Put all its critical points in the retained graph region. The actual physical copy is instead $D_l^{x,\delta }=\mathrm{graph}(d(g_x-\delta f_x))$ on the retained Morse/triangle region, with one FIXED sufficiently small $\delta >0$ for this whole finite window. Realize this displacement by a compactly supported Hamiltonian, equal to the stated function on a neighborhood of the finite Morse critical points and their retained compact sublevels, and cut it off farther out while keeping the already admissible end fixtures fixed. Arrange the auxiliary graph extension transverse to the other sheets and probes outside this protected region; in a collar use a radial difference with the same sign as the outgoing $-\delta d{f}_x$ before cutting off, so no coincident annulus is created. Generic compact choices off the protected cores retain every additional finite chord. Carry the actual Hamiltonian primitive throughout; near each chosen p it is exactly $g_x-\delta f_x$, not a separately reset constant. Hence $D_l^{x,\delta }$ and $U_T$ meet at p, and in the Weinstein chart relative to the sheet their tangent planes are $\mathrm{graph}(-\delta {\mathrm{Hess}}_p{f}_x)$ and the zero plane. The Hessian is positive definite, so this corner is transverse. Let P denote the EXTERNAL probe brane (the compact U or one of the layered thimble probes), not another name for the unit self-copy. Fix these probe-role copies first, using12.1, so each already meets the sheet transversely at p. If one underlying thimble occurs in both roles, use separate physical copy labels and their actual continuation comparison; never use the coincident auxiliary graph as its own probe. Only finitely many such role copies are needed for the bounded bar words. Their transverse probe/sheet gaps have a positive finite minimum. Openness and finiteness therefore make both $P\mathrm{/}{D}_l^{x,\delta }$ and $P\mathrm{/}{U}_T$ corners transverse simultaneously for this same δ. The isotopy from the sheet tangent to the displaced tangent stays transverse to P, and carries its actual chord cap, Spin lift and orientation line. Retain the graded branch coefficient line $o(x)$ from the actual hyperbolic cap matching of6.1–8.1; write its degree as $d_x$. The grading sign of the UNIT FACTOR is fixed explicitly: F1's graph convention assigns degree $n-\mathrm{index}(F)$ to $\mathrm{graph}(\delta dF)$ relative to equally graded base planes. Here $F=-f_x$, so a critical point of Morse index r for $f_x$ has reduced unit-factor degree $n-(n-r)=r$. Its FULL degree in $CF(D_l^{x,\delta },U_T)$ is $d_x+r$, because $o(x)$ is retained. At a selected minimum the unit factor has degree zero and the full branch input has degree $d_x$. A positive graph $+\delta d{f}_x$ would give the wrong reduced degree n. No integer shift of the original $D_l$ or $U_T$ is being made; the original coefficient complex $W_l$ carries $o(x)$ and its differential. Keep the whole proper Morse cochain factor, graded by r, and its one constant cocycle $e_{M,x}={\sum }_p{e}_p$; the two endpoints of every one-cell cancel in its coboundary. Proper downward trajectories stay in compact sublevels, and the finite handle cancellation contracts this disk Morse factor onto its constant class. Realize it at fixed positive δ by a Morse/PSS cap: one proper incoming gradient edge is matched to the boundary evaluation of a fixed-scale cap, whose only Floer end is the ACTUAL nondegenerate displaced pair. Keep the gradient/evaluation row in the joint operator, and add the supported core/evaluation forcing of the unit construction; the transverse end inverse and its finite kernel complement give the matched inverse and local contraction. The compact-image bound on this finite window and the proper Morse sublevels give precisely Floer and Morse breaking, so its oriented one-dimensional boundary is the chain-map equation. This is the existing unit-cap construction on these retained fixtures. Use its chain-map image, not the assertion that a generic Floer differential literally equals the Morse differential. Its minimum coefficient at the corresponding p is the constant real cap identity; other nonlocal terms and all coefficient/within-cluster differentials are retained in the actual comparison. Thus this is a unit FACTOR in the branch comparison, not a replacement of the full coefficient complex $W_l$ by a scalar. All corners used by these fixed-δ maps are nondegenerate; no passage through the pair $B_x=U_T$ at δ=0 is made. Compare the displaced copies to the original admissible ones by actual Hamiltonian-label transport followed by fixed-end DATA continuation, retaining its painted product/parameter faces. Keep the reverse cap/DATA comparison and its two-transition family as well: its separated end is the composite and its merged end the constant unit/copy fixture. Their actual boundary identities give both inverse homotopies on the leading unit factor. Thus the same finite Morse/PSS unit model can be used on both sides by ACTUAL forward/reverse chain comparisons; no target differential is identified from a graded generator bijection alone. The finite raw moduli and their supported reductions stay in the compact boxes of5.1/11.1, while the proper Morse edges have their compact-sublevel bound. These are precisely the unit/cap and regular continuation fixtures, with their actual finite matching rows; no all-arity adiabatic theorem is needed. Choose the same $f_x$, δ, caps and continuations on every shared face, and translate free label orbits. At each selected p all three ACTUAL transverse branes meet. Their primitive values still give the EXACT identity $[f_{U_T}(p)-f_D(p)]+[f_D(p)-f_P(p)]-[f_{U_T}(p)-f_P(p)]=0$, including $-T{\alpha }_l+c_x$, since $f_x(p)=d{f}_x(p)=0$. The selected triangle has raw energy zero for every retained positive δ. The copy perturbations and finite continuation windows change the remaining residual action budgets by at most $C_T\delta$ for a finite constant $C_T$; their leading action centers and these zero-energy identities do not change. Retain their actual corrections in the finite triangular comparison below.

The entire diagonal is the constant unit triangle. Use the integrable zero-Hamiltonian raw equation with the above physical copies. The constant map at$p$ is a solution of the selected triangle problem. Every other solution with those same generators would have exactly the same zero symplectic area by the primitive identity. A raw holomorphic map for the tamed almost-complex structure has nonnegative energy, with zero only for a constant map. Hence no nonconstant low bulb, long thin corridor or extra component can contribute to that same coefficient; constant at$p^{\mathrm{\prime }}\mathrm{\ne }p$ does not have its prescribed input/output. This avoids interchanging a large-$T$ limit with compactness. All three corner pairs are transverse by13.1, so the weighted operator is Fredholm with a positive gap at each actual corner; δ is held fixed, and no inverse uniform as δ tends to zero is claimed. Its kernel is zero: for a kernel section the primitive $\frac{1}{2}\omega (z,dz)$ vanishes on EACH of the three linear Lagrangian planes. Stokes and the decaying nondegenerate corner asymptotics give zero linear energy, so the section is constant zero. Write a for the probe generator from P to $D_l^{x,\delta }$ and $a_{\delta }$ for the corresponding generator from P to $U_T$. After retaining the external $o(x)$ factor, the probe-plane isotopy stays transverse and its actual cap matching gives $\mathrm{\mid }{a}_{\delta }\mathrm{\mid }=\mathrm{\mid }a\mathrm{\mid }+d_x$. The third input $e_p:D_l^{x,\delta }\to U_T$ has FULL degree $d_x$ and reduced Morse unit degree zero, by the explicit $F=-f_x$ index calculation. Thus the genuine triangle index is $\mathrm{\mid }a\mathrm{\mid }+d_x-(\mathrm{\mid }a\mathrm{\mid }+d_x)=0$, and its cokernel is zero as well. This is an actual regular constant chart with zero obstruction space. For its sign use the real unit-cap determinant convention: the maximum line of $F=-f_x$ is its oriented top tangent line, and the real Morse/holomorphic matching divides by the SAME oriented tangent factor. Their cancellation leaves the positive degree-zero line corresponding to the minimum of $f_x$. Sew the actual probe cap and its continuously transported copy cap in this fixed order. Use the SAME target-then-source cap order as the generator identification of7.1; here $o(a_{\delta }{)}_{\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{e}\mathrm{d}}=o(x{)}^{\vee }\otimes o(a_{\delta })$ is identified with $o(a)$ by that actual cap transport. The full map is $o(x)\otimes o(e_{\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}})\otimes o(a)\to o(x)\otimes o(a_{\delta }{)}_{\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{e}\mathrm{d}}$. The degree-zero unit factor cancels by that real cap identity, and the remaining probe cap is its actual transported reference. Thus the reduced map is identity with coefficient +1, including the original graded tensor/Koszul normalization; no odd $o(x)$ factor is silently commuted past a probe factor. This is the existing real unit reference, not an independent choice of a positive triangle sign. The finite relative perturbations keep this regular constant solution and the actual cap comparison fixed. This proves the WHOLE raw triangle coefficient for those three fixed chord inputs in the one model of13.1. It does not discard other Morse/PSS or continuation components: they remain in the actual leading unit map C and its counted unit homotopy below. Distinct minima have the same value0, but a zero-area map cannot connect them; in a Morse limiting description any such path would have zero total drop and is likewise constant. For the remaining finite generator words separate the explicit small Morse/copy-factor actions from the NONLOCAL residual actions fixed in12.1. The auxiliary extension outside the Morse region has fixed transverse generators; after its finite generic primitive adjustments, every non-forced nonlocal equality is separated. Let $g_T>0$ be the smallest absolute value of these nonzero nonlocal residual differences, when they exist. The small Morse-handle differences are kept in their finite factor and are not included in this gap. Terms of negative difference are absent. The identically zero differences are the prescribed constant-unit identifications; any other zero difference has no nonconstant raw map. All remaining nonlocal terms have positive residual action and remain in the actual differential, counit and bar extension. The proper Morse factor, its actual copy/PSS continuation terms and the displaced physical generators have a common residual-action loss bound $C_T\delta$ on this finite word collection. Here $g_T$ is the minimum positive NONLOCAL residual gap, excluding the explicit small Morse-handle differences which are canceled as that finite factor. Choose the common positive δ so $C_T\delta$ times the finite maximum word length is less than $g_T\mathrm{/}2$; if there are no nonlocal terms, no such restriction is needed. The unit contractions can then lower action only by this smaller error, while a nonlocal correction raises it by at least$g_T$. Thus after the finite Morse cancellations every such correction is strictly triangular. This is a finite fixed-$T$ argument; no energy gap uniform over$T$ or over all compact words is asserted.

Consequence for the restricted Yoneda counit and bar augmentation. For critical cluster l let $V_l=({\mathrm{gr}}_l{M}_T)(D_l)$ retain its ACTUAL unary differential. The canonical counit ${\mathrm{\Psi }}_l:V_l\otimes Y_{D_l}{\mathrm{\mid }}_A\to {\mathrm{gr}}_l{M}_T$ and its higher components are the existing Yoneda evaluation maps, not a new augmentation chosen per matrix entry. Separate its leading unit/Morse/copy part C from the strictly positive NONLOCAL residual-action terms. The raw constant triangle fixes the unit reference with coefficient +1, while every other Morse/PSS and copy-continuation term stays in C. Use the actual forward/reverse Morse/PSS and copy comparisons with their two-transition inverse homotopies from13.1 to put BOTH leading complexes in their common unit model. Then the actual unit/cap identity is $C-1=dK+Kd$, with the same external coefficient complex and all its differential retained. This is not an identification of differentials inferred from a graded leading matrix. Both that contraction and K have action-loss bound at most a fixed finite multiple of δ: the proper Morse factor has compact critical range, its geometric displacement/regular DATA windows are compact with Hamiltonian and curvature variation of size $O(\delta )$, and the action identity bounds each of their finite word components. Absorb all these finite constants into $C_T$. Here is the chain-level argument that does not require a strict geometric unit. For $C=1+dK+Kd$, the matrix $T=\left(\begin{array}{cc}1& -K\\ 0& 1\end{array}\right)$ conjugates the differential of $\mathrm{Cone}(1)$ to that of $\mathrm{Cone}(C)$. Transport its explicit identity-cone contraction by T and its inverse, and compose with the finite Morse contraction. This gives an ACTUAL contraction $h_0$ of the leading cone. The remaining cone differential Q, including the actual off-diagonal module/counit corrections, increases nonlocal residual action by at least $g_T$, whereas $h_0$ lowers it by at most $C_T\delta$. Choose the common δ so this loss, multiplied by the finite maximum word length, is less than $g_T\mathrm{/}2$. Then $Q{h}_0$ strictly raises the finite residual-action order and is nilpotent. The finite perturbation formula $h=h_0(1+Q{h}_0{)}^{-1}$ contracts the full cone. This proves that the ACTUAL ${\mathrm{\Psi }}_l$ is a module quasi-isomorphism, without asserting its uncontracted matrix is literally $1+Q$. Apply the finite directed bar construction to these same counits. Its unit-insertion cancellation preserves the critical cluster; after it, the compact-probe leading term is $V_l\otimes CF(U,D_l)$ with the SAME actual unit/cap map C and its K. The identical cone argument proves that $CF(U,\epsilon )$ is a quasi-isomorphism on each critical grade, hence on the whole finite critical filtration. All positive-area bottom and cross-cluster terms remain actual extension data. The algebraic bar contraction gives the same on each $D_i$ probe. Finally $R=\mathrm{Cone}(\epsilon )$ lies in $\mathrm{thick}(U_T,D_1,\dots ,D_N)$. Its Homs from every $D_i$ and from U vanish; actual Hamiltonian continuation identifies U with $U_T$, so the objects having zero Hom into R contain all its generators and hence R itself. Its identity is zero, so R is zero. Thus $U_T$, and hence U, is generated. This uses the complete retained unit/cap comparison and its finite-action homotopy, not a supposed transverse problem at the coincident δ=0 pair or a generator splitting alone.

Finite inputs, not an arbitrary compact-word filtration Let A be the directed N-object category and $P(M_T)\to U_T$ its actual normalized bar augmentation, $M_T(D_i)=CF(D_i,U_T),U_T={\phi }_H^{T}U$. Every nonunit A string has length≤N−1. Hence the twisting cochain of $P(M_T)$, its Maurer–Cartan equation, the closed augmentation, and the unary component obtained by testing it on U involve finitely many generator words of bounded length. The same is true for tests on the $D_i$. This permits ONE finite T, chosen beyond the action budgets of precisely those finitely many words. It does not imply that all arbitrary-length module words on every compact probe preserve a fixed-T action filtration. The algebraic bar equations make ${\epsilon }_T$ a closed object morphism before any additional geometric argument. After this is proved, its full A∞ Yoneda map comes from the already defined higher compositions; there is no need to establish an extra unbounded family of geometric word-length estimates to define it.

Layered inward probes and the finite critical filtration Use finitely many layered copies of inward $D_i$. Their tails turn in a fixed low slab into phases outside the compact input phase range $\mathrm{\mid }h\mathrm{\mid }\le h_0<\pi$. Since $H=h$ is conserved, $U_T$ has no intersections on those remote tails. The finitely many hooks of the copies used for successive directed inputs are at distinct lower heights. Choose enough layers for the maximum bar-chain length, plus the input/output comparisons. All relevant intersections lie in one fixed finite slab. The actual high-branch theorem, applied to this slab, gives generator decompositions $M_T(D_i)={\oplus }_l{V}_l\otimes CF(D_i,D_l),CF(U,U_T)={\oplus }_l{V}_l\otimes CF(U,D_l)$, where $V_l$ is the graded vector space on the stable intersections $C_l\cap U$, with the full orientation lines. The finite-annulus properness, compact hook transport and complete radial tails proved above apply to every one of these finitely many layers. Together with the conserved phase and the deliberately separated remote-tail phases, they ensure that no extra intersection lies on an ignored tail; they also place every boundary portion used in the finite bar comparisons in a common compact coordinate box. Actions on each l block have center $-T{\alpha }_l$ plus bounded generator actions. For the finite word collection of §1, choose T so every different-center gap is larger than the total bounded action/curvature budget. The energy identity then makes the actual restricted-module operations, bar differential, and tested augmentation triangular in the critical index. This produces a genuine finite filtration for THOSE finite chain equations. It is not an assertion that all off-diagonal curves have bounded energy as T→infinity. The equal-center terms may still contain low-bottom polygons. They are retained. A coarea estimate that excludes a phase-separated strip of length T does not exclude a finite-height bottom bulb, and does not justify deleting those terms.

Actual representable counit on each critical block. On the lth critical grade let $W_l=M^l(D_l)$ have its ACTUAL unary differential. The normalized directed category has $\mathrm{Hom}(D_l,D_l)=\mathbb{C}{e}_l$. The Yoneda counit ${\mathrm{\Psi }}_l:W_l\otimes Y_{D_l}{\mathrm{\mid }}_A\to M^l$ is given by existing module operations, whose bar identities give all its equations. Retain the finite proper Morse factor and actual positive-δ copy/PSS maps of13.1, not a guessed equality of their differential with a Floer differential. The transverse constant triangle of14.1 fixes its raw unit coefficient +1. Retain all further unit/cap and continuation terms in the leading map C and its actual counted homotopy $C-1=dK+Kd$. The cone conjugation and finite Morse contraction of15.1 give its leading contraction. The same finite action budget makes every remaining nonlocal term raise the residual order after this contraction; its finite perturbation therefore contracts the full counit cone. Reinsert all finite contractible Morse summands. This proves the module quasi-isomorphism while retaining the full coefficient differential on $W_l$. Higher components are the actual Yoneda components; no full module map is reconstructed from a unary matrix. In particular the original uncontracted map need not itself be a strict identity plus a nilpotent matrix: the actual unit homotopy is part of the proof.

Compact-probe augmentation with noncircular detection. Filter the cone of the bar map induced by ${\mathrm{\Psi }}_l$ by maximum strict directed path length. Each associated quotient is the counit component cone tensored with fixed finite arrow complexes and one $D_i$. Use its actual contraction from18.1; the other bar terms shorten the strict path, so their perturbation series has at most N terms. Thus $P({\mathrm{\Psi }}_l)$ is an equivalence of the actual finite twisted objects. The algebraic unit-insertion cancellation identifies $P(W_l\otimes Y_{D_l})$ with $W_l\otimes D_l$, and applying $\mathrm{Hom}(U,-)$ preserves that actual homotopy. On each critical grade the tested augmentation has the SAME leading unit/cap map C, whose raw constant triangle coefficient is +1 but whose remaining PSS/continuation terms and counted homotopy K are retained. The identity-cone conjugation and finite-action perturbation of15.1 contract its full cone. Consequently $\mathrm{Hom}(U,{\epsilon }_T)$ is a quasi-isomorphism on every grade and on the whole finite filtration. All $O(T)$-energy cross-cluster curves remain actual triangular extensions; none is deleted by an unproved limit. The directed bar contraction already proves the same for every $D_i$ probe. Put $R=\mathrm{Cone}({\epsilon }_T)$. It lies in $\mathrm{thick}(U_T,D_1,\dots ,D_N)$ and has zero Hom from U and all the $D_i$. The actual continuation from U to $U_T$ gives zero Hom from $U_T$ as well. The class of objects with zero Hom into R is closed under shifts, cones and summands, hence contains R. Its identity is zero and R is zero. This proves generation from the fixed actual closed augmentation without presupposing that the thimbles detect every compact object.

One coherent finite comparison, not a companion-file premise. The proof just given uses only the actual finite operations of the relative category on the listed physical thimbles, copies and compact$U$. Its maximum directed path length is$N-1$, so there are finitely many parameterized continuation, bar and unit problems. Choose their copy charts and the one family of multi-minimum units as in13.1. At each boundary of a marked-domain parameter space keep the already fixed product choices, extend their cutoff profiles over its compact core, and choose the remaining small Morse scale below the finite minimum action budget of14.1. This is possible by induction over the finite face poset and a finite partition of unity; the cap data agree on shared faces because the SAME unit sum is being used. Constants are regular and unchanged; supported reductions for the other raw compact moduli can be chosen relative to those constant charts and all faces. Parameterized boundaries are precisely the A∞ naturality and continuation equations. Their leading unit maps have the same counted unit/cap homotopies, and their remaining nonlocal terms raise the SAME finite residual order. The explicit cone contractions and finite perturbation of15.1 therefore give their comparison quasi-isomorphisms, without replacing a homological unit by a strict geometric identity. This binds the calculations to the fixed${\mathrm{\Psi }}_l$ and the original canonical CLOSED bar augmentation${\epsilon }_T$; it does not define a different augmentation for every matrix entry. No research companion is a logical input. The concrete fixed-$T$ construction of13.1–19.1 supplies both the restricted-module and the compact-$U$ diagonal duties, with all finite bottom contributions retained. The quantifier is one finite window at a time. The underlying branes $D_l$, U and their assigned primitive/graded/Spin data are fixed; $D_l^{x,\delta }$ and its Morse choices are auxiliary Hamiltonian-copy presentations. No single fixed $f_x$ is required to have minima at the intersections of arbitrary future probes. If the probe set is enlarged, choose another finite auxiliary model, put the old and new choices in a common finite colored-copy context, and use the actual Hamiltonian-label, fixed-end DATA and Morse/PSS comparisons with their painted higher faces. Keep every comparison end nondegenerate; never use the coincident δ=0 pair as an endpoint. These all-arity comparison equations intertwine the existing module actions and hence the finite bar construction and its canonical augmentation, up to their counted homotopies. A unit/counit contraction proved in the new auxiliary coordinates therefore transports back to the originally assigned category and original closed augmentation. Large fixed copy-transport work is included before the finite residual-action budget is chosen; only the final compact δ-displacement and its finite windows use the $C_T\delta$ bound. The triangular argument is needed in that one auxiliary window, not uniformly for the comparison between arbitrary future models. Old object data and old comparison maps are retained through these actual coherent identifications, rather than silently replaced by a globally probe-dependent graph.

Correct two-extreme projection after generation Suppose an object U has ALREADY been proved to belong to $thick(D_1,\mathrm{.}\mathrm{.}\mathrm{.},D_N)$. Present it by a finite twisted complex $\oplus V_i\otimes D_i$; diagonal endomorphism algebras are C, so the diagonal parts of its twisting differential are ordinary coefficient differentials on $V_i$. Applying $Q_i$ and then the transferred simple module gives exactly the coefficient complex $V_i$, with no hidden action of a nonunit path returning to i. Hence $HF\ast (C_i,U)\cong H\ast (V_i)$. If the middle geometric tests vanish, all corresponding $V_i$ are acyclic. Contract these coefficient complexes over C. Off-diagonal arrows strictly increase index; applying the tensor-length/triangular perturbation formula eliminates the contracted middle blocks after finitely many steps and produces an equivalent twisted complex on the remaining $D_1,D_N$. Higher transferred arrows through the removed vertices are retained as actual higher compositions; they are not silently set to zero. Thus U lies in $thick(D_1,D_N)$. This is the sound use of geometric middle-thimble disjointness. The assertion $\mathrm{Hom}(D_i,U)=0$ for middle i alone would not imply this for an arbitrary exceptional collection. Nor is $\mathrm{Hom}(D_i,D_N)=0$ assumed. The OUTWARD $C_i$, identified above with the dual simple modules, are the coefficient detectors. For the graph lift, zonotope separation makes its intersections with the middle outward $C_i$ empty. For the matching cycle, its auxiliary path lies inside the critical circle except at the two extreme endpoints, while every middle $C_i$ lies outside that circle on a different radial ray, so those tests also vanish. These are the actual geometry statements already constructed by the HMS author. They apply to the dual tests above; their use is conditional on the separate generation step.

Dual tests and extreme projection after actual generation. The compact-probe proof applies to each finite chosen compact$U$, and the geometric costandard probes of11.1 have$HF(C_i,D_j)={\delta }_{ij}\mathbb{C}[0]$. Their higher transferred actions vanish because a nonunit directed path cannot return to$i$. Thus their restriction is the actual simple coordinate module, not an informal rank test. After the generation of19.1, present$U$ by a finite twisted complex${⨁}_i{V}_i\otimes D_i$. Applying that simple module gives the coefficient complex$V_i$ with its actual differential, so$HF(C_i,U)=H^{\ast }(V_i)$. If the middle outward tests vanish, their$V_i$ are acyclic and may be canceled by finite Gaussian/complex contraction, leaving precisely the extreme two-object subcategory. The graph lift has those middle vanishings by its explicit zonotope separation, and the matching sphere has them by its actual matching path lying inside the critical circle away from the two extremes. Both are therefore objects of the same extreme span. This uses the compact probe as well as the thimble probes; without19.1 it would be circular. The matching cone and its chosen-character calculation below identify the two extreme objects and their degree-zero forward arrow, and the graded Kronecker argument then applies.

Exact product model and the actual two-sheet surgery Let $E_B,E_F$ be the type-(B) doubles of the two matching thimbles. Choose their paths to meet only at the regular branch basepoint, and identify their boundary vanishing spheres there by the actual matching data. Their clean intersection is $S=S^{(n-1)},n=2$ or3. The clean Lagrangian-neighborhood construction identifies a neighborhood of S with (T*S × $C,{\theta }_S+(xdy-ydx)\mathrm{/}2),E_B=S_{zero}\times iR,E_F=S_{zero}\times R,\sigma (q,p,z)=(q,p,-z)$. This normal form is obtained by taking the symplectic normal line to the branch fibre and the common real tangent TS; the remaining directions are paired with TS by ω. Relative Moser interpolation removes the terms vanishing on those two Lagrangians. Its radial primitive and vector field can be chosen σ-natural. Thus the displayed primitive is the actual local exact model, up to an exact function, not a complex-linear claim about a real Kähler metric. Fix a small ρ>0. Replace the crossing of the two normal axes by two smooth disjoint resolution arcs $c_{\rho }$ and $-c_{\rho }$ joining the positive and negative matching quadrants. Near the bounded comparison triangle take $c_{\rho }$ exactly as $x+y=\rho$; farther away bend it to the two axes, then glue to the unchanged thimble hemispheres. The bends can be chosen outside a larger normal core and are part of the ordinary continuation region, not additional local cap vertices. Both arcs are isotopic to the corresponding positive hyperbola resolution $xy>0$ without crossing zero. Choose the primitives on the two original thimbles to agree at S; the two arc primitive differences are equal by z↦−z. The local primitive therefore glues on both resolved components. The result is an exact Lagrangian $U_{\rho }$ with two components exchanged by σ. Topologically cut each doubled thimble along its equator S. Each is two n-balls. The matching resolution joins one B-hemisphere to one F-hemisphere and does the same to their deck translates. Thus $U_{\rho }$ is precisely the double of the auxiliary matching sphere, with two $S^n$ components. Varying the bend profile or radius produces a compact exact Lagrangian isotopy. Because the components have H1=0, its flux vanishes and the isotopy is Hamiltonian; this also follows directly by differentiating their glued exact primitives. The brane Spin structure is the unique sphere Spin structure. This identifies the geometric surgery object with the actual type-(U) matching object.

Chosen character from the cap determinant, not both equivariant resolutions. In the product normal model normalize$\theta (i\mathbb{R})=1\mathrm{/}2$, $\theta (\mathbb{R})=0$, with phase$\arg (\text{tangent})\mathrm{/}\pi$. Capping an intersection in the forward direction$i\mathbb{R}\to \mathbb{R}$ has degree0; reversing it has degree1. Tensoring with the tangential Morse complex therefore gives forward$C^{\ast }(S)$ with the constant cap$a$ in degree0 and top cap$b$ in degree$n-1$. The deck involution fixes$S$ and acts by$-1$ on its normal variations. For an equivariant real Fredholm normal operator its action on the determinant is$(-1{)}^{\dim \ker +\dim \mathrm{coker}}=(-1{)}^{\mathrm{index}}$: this follows directly on${\mathrm{\Lambda }}^{\mathrm{t}\mathrm{o}\mathrm{p}}\ker \otimes ({\mathrm{\Lambda }}^{\mathrm{t}\mathrm{o}\mathrm{p}}\mathrm{coker}{)}^{\ast }$. The forward cap consequently carries$+1$ and the reverse cap$-1$; the tangential determinant is unchanged because the action on$S$ is the identity. Fix the relative equivariant brane characters by the actual forward cap, meaning that its character product is chosen to make$a$ invariant. This fixes ONE character resolution: all forward$H^{\ast }(S)$ survives and all reverse$H^{\ast }(S)$ is anti-invariant. Changing only one character would instead select the opposite resolution, which is not identified with this cone. The reflection on each doubled$B$ sphere fixes the unit and negates its top cohomology, so its invariant endomorphism algebra is exactly$\mathbb{C}$. The surgery components are exchanged freely. On each output component the transported cap contributes the forward unit once; the regular/permutation type-$U$ structure therefore agrees with the chosen forward resolution, without summing two characters or inserting a factor2. The outward dual caps are assigned the dual character so their unique critical generator is invariant, giving the coordinate tests used in22.1 with the same normalization.

Compute the small normal triangle and its linearization Take the three local normal sides iR, $R,x+y=\rho$. The bounded region is the right triangle with vertices 0,ρ,iρ. A positive holomorphic triangle with the indicated corner limits has boundary on those lines. Its bounded image cannot occupy an unbounded complementary region of the three lines: the argument-principle degree is constant there and zero near infinity. Its image therefore lies in the bounded triangle. Its exact boundary action is ρ²/2, the area of that triangle. Holomorphic multiplicity is a nonnegative integer, so it has degree one, not a higher cover. It is the unique conformal map after fixing the three source corner flags. Equivalently its derivative is the elementary Schwarz–Christoffel integral with corner exponents −1/2,−3/4,−3/4; the integration constant fixes 0,ρ,iρ. Reversing the cyclic ordering would give action −ρ²/2 and has no holomorphic solution. The negative-quadrant cap is its σ-translate and has the same transported determinant orientation. The tangential projection to T*S has boundary on its exact zero section on every side. Its energy is the integral of ${\theta }_S$ on that boundary, zero. With the product compatible J it is therefore constant. The full local cap family is S, with evaluation the identity on S. It represents the cohomological unit a, geometrically the fundamental chain of S, rather than the top Morse cochain b. For regularity, the tangential real CR operator is onto with constant kernel TS, exactly the real-zero-mode calculation already supplied for a boundary node. In the normal factor the three line conditions and corner weights give index zero. A kernel variation divided by the conformal triangle map is a holomorphic vector field tangent to the disk boundary and vanishing at all three marked corners. Reflection extends it to a real infinitesimal Möbius field; three fixed points force it to vanish. The index-zero operator is therefore invertible by the actual bordered Fredholm theorem. Its determinant has the positive complex/conformal orientation; normalize a so this count is +1. The product orientation is then the ordinary orientation of S. There is no free deck multiplicity in an invariant coefficient: the + and − cap components are exchanged, and their invariant count is the diagonal coefficient +1, not their sum +2. For the bends outside the normal cap core use the fixed short continuation strips matching the original axis halves to their perturbed copies. They are the usual graph/zero-section continuation identity and carry coefficient +1. Their source/target evaluations are retained as separate gluing fixtures. Thus the cap calculation concerns the exact straight-line triangle just computed; it does not assume that every small-action curve on the entire thimble is confined to that triangle rather than following a continuation zero mode.

Neck inverse with the full clean zero mode retained Stretch the product neighborhood at S. The tangential limiting operator is ∂s+i∂t with equal real boundaries; its Fourier modes have gaps π away from its REAL zero mode TS. Retain that shared TS coordinate and impose the actual matching evaluation on S. The normal boundaries are the transverse planes iR and R. Its frequencies are π(k+1/2), so its gap is π/2 and it has no normal zero mode. Choose 0<δ<π/2. The weighted Fourier/convolution inverse on the complement of TS is bounded, exactly as in the existing transverse-strip and real-node suppliers. On the local triangle use the just-proved normal inverse and tangential constant kernel. On each outer core use the already constructed jointly augmented CR/evaluation inverse. Match its TS evaluations and cap evaluation before inverting, rather than discarding the clean kernel. Splice these inverses on length-T fixtures. The separated cutoff error is bounded by C exp(−cT), with c<π/2−δ; any finite number of fixtures contributes the sum of these errors. The small normal scale is included in the normal-coordinate Sobolev norm by dividing by ρ. The cap equation is then the fixed triangle equation above, so its nonlinear estimates have a bound independent of ρ. The actual product model has no mixed curvature term; coordinate changes outside it are on fixed cores and use the same supported finite augmentations. Neumann inversion gives the matched uniform right inverse, and the ordinary unit-strip Sobolev product estimate gives the quadratic contraction. Thus every transverse core/cap matching has a unique nearby glued solution. Conversely the raw exact energy bounds and the established local bubble-tree theorem give convergence on all fixed cores. At the normal cap the action fixes the forward triangle multiplicity to one. A reverse switch would be the negative-action normal cap excluded in §3. A boundary path cannot make two forward changes B→F without an intervening F→B change. Hence at most one local switching cap occurs. Tangential necks retain only the shared S coordinate, by the real-mode exponential decay. These facts give weighted-norm convergence to the matched configuration; the contraction uniqueness gives inverse gluing. Autonomous disk/sphere bubbles have zero area by exactness on a one-label boundary and do not create extra local caps. The continuation zero modes at the outer bends are included as the explicit fixtures described in §3. This is the specific clean-neck/exhaustion calculation needed here. A general multiple-stop wrapping or arbitrary thimble mutation is not inferred from it.

Identify the actual Yoneda module cone Let K be any finite compact probe brane. A small Hamiltonian perturbation makes it disjoint from S, because $\dim K+\dim S=n+(n-1)<2n$. Its compactness then gives positive distance from S. For sufficiently thin surgery neck, its intersections with $U_{\rho }$ are exactly its old intersections with the unchanged $E_B$ and $E_F$ hemispheres. The graded generator module is therefore $CF(K,U_{\rho })=CF(K,E_F)\oplus CF(K,E_B)[1]$. This statement can first be proved with nonequivariant probes. There is no claim that an invariant B probe can itself be perturbed equivariantly off its fixed equator. For invariant-category tests use the free induced probe K⊕σK after the nonequivariant perturbation; the original invariant brane and its opposite-character copy are summands of the corresponding induced object via the characteristic-zero idempotents. Thus these probes detect the required equivariant quasi-isomorphism too. Take any finite composable string of additional probes, also perturbed off S. The neck-limit configurations for their module operations have either no switch, giving the ordinary B or F Yoneda operation, or exactly the forward cap a followed by an ordinary outer polygon. The latter is precisely the all-arity Yoneda evaluation operation with an a input. The shift in §2 gives its cone sign. Product/domain-face degenerations use the same fixed cap and matched evaluations, so these identifications respect the entire A∞ module relation and its homotopies, not only its unary differential. One can choose the neck radius/length small/large enough for EACH finite operation problem by the preceding compactness and inverse gluing. To make the comparisons coherent, induct over the finite marked-domain faces: keep the already chosen product lengths on a face, extend its length function over the compact domain core and increase it there until the same gluing bound holds. Extra continuation fixtures compare to a fixed physical $U_{\rho }0$. The parameterized exact compactness/gluing proof supplies the corresponding continuation A∞ module maps. No single small radius uniform over all arities is asserted. At every fixed arity these are finite compact parameter spaces and the same one-cap inventory; the existing tuple/relative-section construction gives exact face compatibility. The resulting module comparison has generator-leading matrix identity on the F block and the shifted B block. Choose the local cap action smaller than the minimum nonlocal action-cluster gap for the finite probe problem. Any additional nonlocal configuration has a positive monotonicity cost on a fixed outer annulus. In the finite action-cluster order, the difference from the generator-leading matrix is strictly triangular. Its inverse is therefore the FINITE sum 1−N+N²−⋯, not an uncontrolled infinite series. Parameterized polygon boundaries give the module-map equations; the leading identity and triangular inverse make it a quasi-isomorphism. This is the same finite action-cluster mechanism already established for the geometric evaluation-cone supplier, now with the explicit clean normal cap of §§2–4. Consequently the right Yoneda module of the matching double is quasi-isomorphic to the actual module cone of evaluation by a. The cohomologically full Yoneda embedding identifies the objects: $U_{matching}$ ≅ $Cone(a:E_B\to E_F)$ with the grading normalization fixed in §2. The construction is σ-natural on free probe/data orbits, so exact invariants preserve this equivalence. On the matching U components the equivariant action is the required regular one.

A raw model with an actual holomorphic projection Use the fixed integrable complex structure on $M^0$ and ZERO Hamiltonian term. Raw polygon maps are therefore genuinely holomorphic, while their boundary branes need only be smooth Lagrangian copies. Regularity is supplied by the actual finite virtual atlas, not by falsely claiming that a fixed integrable J is generic. Self-Hom operations can be computed through the transverse copies and continuation caps; no projection-incompatible Hamiltonian forcing is inserted into the raw equation. For every coordinate $u_i$ and its inverse, composition with a raw map is a holomorphic nonvanishing function. Compact boundary values and asymptotic generators bound both $\mathrm{\mid }{u}_i\mathrm{\mid }$ and $\mathrm{\mid }1\mathrm{/}{u}_i\mathrm{\mid }$ by the maximum principle on an exhausted punctured disk. Thus all raw images lie in a fixed compact product of coordinate annuli, intersected with the closed affine equations of $M^0$. The same holds for fixed-word broken limits. Area is fixed by the exact boundary action formula, including the finite perturbation primitives. These give actual no-escape and energy bounds without assuming a general tame fibration theorem. Use the established exact disk/polygon compactness, supported finite reductions, node/chord inverses and relative-spin orientations on that compact region. The raw problem is allowed to be nonregular. Virtual forcing may cease to be projection holomorphic, but it is chosen in finite relatively compact chart cores around these raw moduli. Every barrier margin below is uniform on the raw compact slice; shrink the chart neighborhood in C0 so all its thickened maps remain within that margin. This is the same raw/virtual distinction as in the repaired mixed-pearl construction.

Split the auxiliary cover over every compact operation Choose real numbers $R<R\ast$ with Re π(K)<R, and place the regular branch basepoint at =R. The auxiliary double cover is ρ:Ẽ={ (x,y):x∈E, y²=π(x)−* }→E. It is smooth because * is a regular value. On the left half-plane $H=Rez<R,z-\ast$ has no zero and admits a single holomorphic square-root branch. Therefore ${\rho }^{-1}(E\cap {\pi }^{-1}(H))$ is the disjoint union of two holomorphic copies, labeled + and −. For a raw polygon in E, Re(π∘u) is harmonic and all its boundary/end values are $<R$. The maximum principle gives Re π(u)<R on its entire image. This holds for strips and continuation caps by exhaustion and their actual end convergence. Hence every such polygon lies in that split covering region. Its lift is uniquely determined by choosing + or − at one boundary point, and its lift has the same sign on every boundary segment. Define the type-(U) lift of $K_a$ as the disjoint union $K_a^{+}\bigsqcup K_a^{-}$, with the deck involution exchanging the two. Its brane structure is induced from $K_a$ and then transported to the other sheet. The invariant structure is the regular/permutation one; no nontrivial isotropy acts on a chosen component. Pairwise chord data are copied in the split region. Thus $CF(K_a^{+},K_b^{-})=CF(K_a^{-},K_b^{+})=0,CF(K_a^{+},K_b^{+})=CF(K_a,K_b)=CF(K_a^{-},K_b^{-})$. The raw cover moduli are precisely the two lifts of the base moduli. Their linearized CR/evaluation operators and determinant lines are identified by the local symplectic covering. A noncompact branch fibre needs an actual global form, not a uniform-smallness argument copied from a compact quadric. Here it is explicit. Choose a disk $D_{\ast }$ in S about * disjoint from π(K). Put $s=\mathrm{\mid }y{\mathrm{\mid }}^2$ and choose h(s)=s near zero and $h=0$ for s sufficiently large, with its support satisfying $y^2\in D_{\ast }-\ast$. Let f(s)=h'(s)+s h''(s), the coefficient of $d{d}^{c}h(\mathrm{\mid }y{\mathrm{\mid }}^2)$ in i dy∧dbar y, with the fixed positive normalization. Its negative part is supported in an annulus $a\le s\le b,a>0$. Choose a nonnegative base two-form β supported in $D_{\ast }$, equal to i dz∧dbar z on the image of that annulus. Then $\omega tilde=\rho \ast \omega +C(\pi \rho )\ast \beta +\epsilon d{d}^{c}h(\mathrm{\mid }y{\mathrm{\mid }}^2)$ is Kähler for C sufficiently large: since πρ=*+y², the added coefficient in the dy direction is 4Cs ${\beta }_{coefficient}+\epsilon f(s)$, nonnegative everywhere and positive at $s=0$. The pulled-back form is already positive in every remaining direction. Thus it is globally nondegenerate even along the noncompact ramification fibre. All added forms vanish near the fixed compact split region. Since S is contractible, β=dη, so this form is exact whenever ω is exact. On H the closed η is exact, and its primitive can be subtracted near the compact region; hence the brane primitives and actions agree with the base there. The same formula is valid with smooth normalizations of $d{d}^c$ after rescaling C. No global lower bound for the original metric at fibre infinity is needed for this construction. Choose the finite virtual charts on the + raw moduli by pullback from the base, and on the − moduli by exact deck translation. Their sufficiently small thickenings remain in the split region. All node/corner/continuation product charts are lifted in the same way, so their equations, orientations and rational branch weights agree exactly. Consequently the total type-(U) morphism complex has operations componentwise on A⊕A, with no mixed-sheet terms. Taking deck-invariant morphisms identifies a↦(a,a). Its kth operation is ${\mu }^k((a_k,a_k),\mathrm{.}\mathrm{.}\mathrm{.},(a_1,a_1))=({\mu }^k(a_k,\mathrm{.}\mathrm{.}\mathrm{.},a_1),{\mu }^k(a_k,\mathrm{.}\mathrm{.}\mathrm{.},a_1))$. There is NO factor two or division by two: this is an identity between invariant coefficient modules, not a sum of two counts into a single chosen output component. The target involution exchanges the two output components freely. The resulting functor is an A∞ isomorphism onto the full type-(U) subcategory in this shared model, hence cohomologically full and faithful in the auxiliary invariant category. A finite number of compact objects is all that is needed for the named object identification; different larger choices of * are related by the existing parameterized continuation construction outside their compact operation region. This proof concerns the unbranched compact image. It makes no assertion that the sphere-type-(B) objects have the same free deck representation or that their spherical twist decomposes into one directed mutation.

Full faithfulness for the slit inclusion $E\subset M^0$ Take a raw polygon in $M^0$ with all its boundary/end values on the chosen compact copies in E. Its punctured disk is simply connected. Since π∘u maps to C*, it has a holomorphic logarithm g. On the connected boundary, choose g to agree with the principal logarithm of π(K), possible because this boundary loop stays in S and has zero winding. Every end is in the same compact sector. Thus for some δ>0 and finite $a<b,a\le Reg\le b,-\pi +\delta \le Img\le \pi -\delta$ on the boundary and ends. The maximum principle gives the same bounds everywhere. In particular π(u)∈S. This is the actual reason a comparison polygon remains in E; one cannot simply assert that a disk whose boundary projects to S has already been given a map into S. The coordinate maximum principles from §1 make the image lie in one compact subset of E with a uniform margin from the removed ray and from 0 and infinity. The raw moduli in E and in $M^0$ are therefore IDENTICAL. Choose the same finite virtual atlas in their common compact region, with thickenings small enough to remain in E. All orientations, continuation caps and higher product collars coincide. The inclusion of the compact full subcategory is thus fully faithful in the actual A∞ sense, not merely on vector-space ranks. In particular an isomorphism produced in the auxiliary category between type-(U) objects pulls back to an isomorphism in F(E) by §2, and the very same inverse cycles and their product homotopies give an isomorphism in $F(M^0)$. This assertion is conditional only on having actually produced that auxiliary isomorphism; the generation and directed projection required to produce it are not supplied by this embedding theorem.

First construct the central Fermat-cover comparison, then lift with all q-obstructions retained. The double cover in29.1 is auxiliary and is NOT the Fermat cover. Write $p:X_N\setminus D\to M^0$, $u_i=x_i^N\mathrm{/}{\prod }_j{x}_j$, for the finite unramified cover constructed in F1, with deck group $\mathrm{\Gamma }$ of order $N^{N-2}$. Use its descended source symplectic and volume forms: the actual source form and CY grading pull back to those on $X_N\setminus D$, with the fixed residue normalization. For a finite family of compact base branes containing the graph and matching spheres, choose the same physical copies, unit caps and zero-Hamiltonian integrable raw model as above. Every coordinate $u_i$ and $u_i^{-1}$ has its boundary maximum, so all raw polygons on this finite family, their strips, caps and fixed-word limits lie in one compact coordinate box Q. Enlarge the Q of5.1 before making the logarithmic metric modification. Its form is exactly the original descended source form on a neighborhood of this box. The finite inverse cycles and product homotopies obtained downstairs thus use these source-form compact count problems. The covering is proper, so $p^{-1}(Q)$ is compact. A disk or genus-zero disk tree lifts uniquely after fixing the output sheet; tree nodes determine the next sheet with no new choice. The same holds on strip/cap limits. Lift the entire finite obstruction/slice tuple, its determinant and coefficient lines and its rational branch distribution by the local symplectic covering. Linearized equations and matching evaluations are identified, their finite automorphism groups are unchanged with the output sheet fixed, and all face restrictions commute with lifting. Thus each base count has coefficient ONE at each chosen output lift, not a deck-degree multiplier. Define the CENTRAL pullback functor on the additive category by $P(K)={⨁}_{A\text{ component of }{p}^{-1}(K)}A$ and by lifting each chord/coefficient to all its lifts; its higher functor components are zero. The unique-lift identity for every polygon says exactly $P^1{\mu }^k={\mu }^k(P^1,\dots ,P^1)$, with the same grading, orientation and local-system maps. It also carries the unit cap and every actual inverse/product homotopy to their lifted counterparts. This is a constructed central functor; no functor carrying positive-divisor polygons from $M^0$ is asserted. In the compact relative supplier, positivity and the degree-zero no-escape branch imply that its $q=0$ polygons avoid D. Their projections are precisely the base polygons with the sheet conditions just described, and conversely every such lifted base polygon is one of these degree-zero problems. Choose the degree-zero compact relative tuple data to be these lifted data, and choose all positive-degree data relative to them in the supplier's degree induction. Its bounding cochains have positive valuation, so their twist changes no degree-zero operation. Therefore this lifted central category is an actual full compact subcategory of ${\mathcal{C}}_R\mathrm{/}q{\mathcal{C}}_R$ in that auxiliary model. The supplier's coherent auxiliary-data comparisons carry it to any previously fixed model. This gives the required comparison with the relative category, not just an identification of two abstract Hom dimensions. There is a further component issue: P initially sends a sphere to its FULL inverse image. Since $n=2,3$, a sphere is simply connected, and its inverse image is a disjoint union of $\mathrm{\mid }\mathrm{\Gamma }\mathrm{\mid }$ spheres, each mapping diffeomorphically to it. Work with one physical component per sheet; duplicate free labels are restored by the already constructed continuation isomorphisms. Distinct components of the same inverse image are disjoint, so choose small copies still disjoint: their mutual central Hom complexes are zero. Each component has $H{F}^0(A,A)=\mathbb{C}$, by the full sphere self model. Thus the degree-zero endomorphism algebra of either full inverse image is ${\mathbb{C}}^{\mathrm{\mid }\mathrm{\Gamma }\mathrm{\mid }}$. A degree-zero base isomorphism between the graph sphere and matching sphere pulls back to an isomorphism of these sums. Conjugation by it permutes their primitive idempotents, and hence gives inverse degree-zero morphisms between each graph component and one matching component. To determine the required component, include the last outward disk $C_N$ as a CENTRAL auxiliary test. It is simply connected; its finitely many lifts have the finite-annulus properness of5.1 by properness of p. Mixed-test curves have the same lifted compact charts and no-escape as11.1. Anchor this test at a preimage of the actual last auxiliary critical point $u=(-(N-1)a,a,\dots ,a)$, $a^N=-1\mathrm{/}(N-1)$, with the root fixed by the named terminal cap. The graph point over the singleton cubic critical point has $df=0$, so its cotangent graph passes through this same point; the matching cap has this critical point as its endpoint. Their unique chosen lifted components are therefore the ones meeting that preimage, with their original grading/Spin transports. The lifted outward cap through it intersects exactly these two components, once in degree zero, and no other component of either inverse image. Small copy perturbations and their cap continuations retain these labels. Applying this test to the pulled-back isomorphism forces its primitive-idempotent permutation to match those two chosen components. This anchors the deck label; it is not a claim that a base isomorphism automatically lifts to every prescribed pair. Restricting the lifted matrix maps to those matched components gives actual closed degree-zero central inverse classes, with their product homotopies, in ${\mathcal{C}}_R\mathrm{/}q{\mathcal{C}}_R$. Transport the other named labels by the same actual graded deck maps. After the degree-zero comparison and absolute grading are proved in the final application below, these are the maps $f_0:L\to V$ and $g_0:V\to L$ used here. Now work in the ACTUAL uncurved complete compact relative category ${\mathcal{C}}_R$, $R=\mathbb{C}[\text{\hspace{0.17em}⁣}[q]\text{\hspace{0.17em}⁣}]$, on its supplied bounding objects and finite free Hom chain modules. No positive-order map has yet been presumed closed. Since these are actual unshifted central sphere isomorphisms and$n=2,3$, both central Hom groups in degree1 vanish. Suppose$f^{(r)}$ is closed modulo$q^r$. Its error${\mu }^1(f^{(r)})=q^r{e}_r<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>q^{r+1}$ is centrally closed by$({\mu }^1{)}^2=0$. Choose$h_r$ of degree0 with${\mu }_0^1{h}_r=e_r$, possible by that degree1 vanishing, and replace$f^{(r)}$ by$f^{(r)}-q^r{h}_r$. The inverse-limit sequence is Cauchy and its limit$f$ is genuinely closed by continuity of the complete operations. Do the same for$g$. This does not claim an arbitrary central morphism or isomorphism lifts: the explicit sphere$H^1$ obstruction calculation is essential. On the finite Hom complexes, kernels and images are finite$R$-modules and are closed. To see the asserted completeness concretely, reduce a finite matrix over the discrete valuation ring to diagonal form by successive lowest-valuation pivots; each summand is$R$, $q^{a}R$ or$R\mathrm{/}(q^a)$, all of which are complete and separated, as are their finite sums. The same holds for the relevant cohomology quotients. Reduction of a closed degree-zero class has kernel$q{H}^0$ by the exact sequence of complexes$0\to {\mathcal{C}}_R\stackrel{q}{\to }{\mathcal{C}}_R\to {\mathcal{C}}_0\to 0$. Thus$[gf]=1+qa$ and$[fg]=1+qb$ in the complete associative degree-zero cohomology category. Their inverses are the convergent$q$-adic geometric series. Replacing$g$ by$[gf{]}^{-1}g$ gives a left inverse, and invertibility of$fg$ makes it the right inverse too. Consequently the categorical isomorphism preserves the full compact relative higher-$q$ structure; it is not a specialization of the affine model. Finite group primitives may be chosen on free label-orbit representatives and transported, or averaged in the invariant LINEAR primitive equation; no nonlinear operation is averaged. The actual bounding-data and mixed-category supplier remain essential, and are not proved by this lifting argument.