Negative monodromy and full compact generation for the named Fermat pencils

The initial normal-crossing divisor is already orthogonal Identify tangent vectors at $[x]\in P^{N-1}$ with complex vectors v orthogonal to x for the standard Hermitian product. If $x_i=0$, $e_i$ is such a horizontal vector and spans the Hermitian normal of $C_i$. Thus all coordinate-divisor normal lines through the point are mutually orthogonal. At a point of $C_0$ the Hermitian gradient of F in homogeneous coordinates is $N({\bar{x}}_1^{N-1},\dots ,{\bar{x}}_N^{N-1})$. It is horizontal because its inner product with x is $N\overline{F(x)}=0$. At a point also belonging to $C_i$ its i-th coordinate is zero. Thus it is orthogonal to $e_i$ for every coordinate divisor through that point. It is nonzero. The Fermat divisor and all coordinate divisors therefore have mutually orthogonal complex normal lines along every intersection. Their union is a simple normal-crossing divisor: independence of these normal lines proves transversality, and each component is smooth. At most N−1 components meet. For every smooth finite pencil member $X_z$, its boundary divisor $B_z=X_z\cap P=0$ is likewise orthogonal. At a point where at least two coordinates vanish, $dP=0$ and the Hermitian normal to $X_z$ is the same vector as that to $\{F=0\}$. Hence the vectors $e_i$ for the vanishing coordinates lie in T $X_z$, remain mutually orthogonal there, and are the normals of the corresponding boundary components in $X_z$. At points of a single boundary component no crossing-angle condition is required. Consequently the named models do not require the generic-source deformation that first makes crossing components orthogonal. This is stronger than just observing that the ambient coordinate hyperplanes are orthogonal: the calculation also checks $C_0$ and the induced boundary divisor in $X_z$. For a generic polynomial replacing F the derivatives in the vanishing-coordinate directions need not vanish, and this simplification cannot be asserted.

Linear symplectic phase oscillation Let A be a real symplectic linear map on $C^m$ with its standard Hermitian structure. Write $Az=Pz+Q$ bar(z). The symplectic equations give $PP\ast -QQ\ast =I$, so P is invertible and $\mathrm{\mid }\mathrm{\mid }{P}^{-1}Q\mathrm{\mid }\mathrm{\mid }<1$: indeed $I-P^{-1}QQ\ast P^{-\ast }=P^{-1}{P}^{-\ast }$ is positive definite. Represent a real Lagrangian plane by the real span of the columns of a unitary U. Then ${\det }_C(AU)\mathrm{/}{\det }_C(U)=\det (P)$ $\det (I+P^{-1}Qbar(U)U^{-1})$. The matrix $K=P^{-1}Q$ $bar(U)U^{-1}$ has norm less than one. Every eigenvalue of I+K lies in the open right half-plane, so the imaginary part of its principal matrix-log trace lies between −mπ/2 and mπ/2. The logarithm exists by its norm-convergent power series and varies continuously with the plane. Changing its unitary real frame U by UO, O real orthogonal, leaves $bar(U)U^{-1}$ unchanged. Thus it is a well-defined continuous argument on the whole Lagrangian Grassmannian. The squared-phase displacement of A equals a plane-independent real constant plus (1/π)Im tr log(I+K). Consequently the difference of its displacement at ANY two planes has absolute value strictly less than m. This is independent of ||A||; the bound does not deteriorate under iteration. A specified grading lift only changes the common constant by an integer and does not change this oscillation bound. The same argument applies to different source/target compatible Hermitian frames and nonvanishing volume forms; their unitary-frame and volume-ratio contributions are plane independent. In particular if one explicit reference plane has displacement −c for A, all planes have displacement less than −c+m.

Ordinary product degeneration Take $C^n$ with its standard symplectic form and $p(y)=y_1\mathrm{.}\mathrm{.}\mathrm{.}{y}_k$, 2≤k≤n, away from the coordinate zero fibre. Put $S=su{m}_{j\le k}\mathrm{\mid }{y}_j{\mathrm{\mid }}^{-2}$. The clockwise horizontal lift of the base field −iζ is $Z_j=-i\mathrm{/}(bar(y_j)S)$ for j≤k; $Z_j=0$ otherwise. To check both assertions directly, $dp(Z)=p$ sum $Z_j\mathrm{/}{y}_j=-ip$, and Z is a complex multiple of the Hermitian normal vector $(overline(\mathrm{\partial }p\mathrm{/}\mathrm{\partial }{y}_j))$; it is therefore symplectically orthogonal to the complex kernel of dp. Each radius $\mathrm{\mid }{y}_j\mathrm{\mid }$ is constant along Z. Thus the flow exists for all time on each nonzero fibre, with angle velocities $-1\mathrm{/}(\mathrm{\mid }{y}_j{\mathrm{\mid }}^{2}S)$, and rotates p by $e^{-it}$. Use total volume $\eta =(d{y}_1\mathrm{/}{y}_1)\wedge \mathrm{.}\mathrm{.}\mathrm{.}\wedge (d{y}_k\mathrm{/}{y}_k)\wedge d{y}_{k+1}\wedge \mathrm{.}\mathrm{.}\mathrm{.}\wedge d{y}_n$ and relative volume obtained by dividing it by dζ/ζ². At y the real plane $L_y={\oplus }_{j\le k}R$ i $y_j$ ⊕ $R^{n-k}$ is Lagrangian, its total squared phase is $(-1{)}^k$, and $dp(L_y)=i\zeta R$. Its vertical part $L_y^v$ has dimension n−1. Dividing the volume by the base vector shows its relative squared phase is $(-1{)}^{k-1}{\zeta }^2\mathrm{/}\mathrm{\mid }\zeta {\mathrm{\mid }}^2$. This can also be checked by choosing k−1 independent angular differences and the remaining real vectors as a basis. The flow preserves these angular orbit planes: it preserves radii, and differentiating in the angle variables leaves the fixed angle velocities unchanged. Thus over the clockwise base loop their relative phase winds by −4π. The canonically graded monodromy h has displacement −2 on $L_y^v$, and $h^d$ has displacement −2d there. Apply §1 in the vertical dimension $m=n-1$. Every vertical Lagrangian plane has displacement strictly less than −2d+n−1 under $h^d$. This is a genuinely uniform negative bound, independent of the radii and of the nonzero ζ. It proves the complete negativity assertion for this exact local model without compactness or an unproved comparison between arbitrary tangent planes.

The rational base-locus model and its exact exceptional set Now take $p(y)=y_2\mathrm{.}\mathrm{.}\mathrm{.}{y}_k\mathrm{/}{y}_1$, 3≤k≤n, and total volume $\eta =d{y}_1\wedge (d{y}_2\mathrm{/}{y}_2)\wedge \mathrm{.}\mathrm{.}\mathrm{.}\wedge (d{y}_k\mathrm{/}{y}_k)\wedge d{y}_{k+1}\wedge \mathrm{.}\mathrm{.}\mathrm{.}\wedge d{y}_n$. For fixed ζ≠0 the closure of its graph fibre is the smooth $C^{n-1}$ with coordinates $y_2,\mathrm{.}\mathrm{.}\mathrm{.},y_n$ and $y_1=(y_2\mathrm{.}\mathrm{.}\mathrm{.}{y}_k)\mathrm{/}\zeta$. Its induced symplectic form has potential $\mathrm{\mid }{y}_2\mathrm{.}\mathrm{.}\mathrm{.}{y}_k{\mathrm{\mid }}^2\mathrm{/}\mathrm{\mid }\zeta {\mathrm{\mid }}^2+su{m}_{j\ge 2}\mathrm{\mid }{y}_j{\mathrm{\mid }}^2$. Dividing η by dζ/ζ² gives, up to an irrelevant constant sign, the ordinary volume $d{y}_2\wedge \mathrm{.}\mathrm{.}\mathrm{.}\wedge d{y}_n$. Put $r_j^2=\mathrm{\mid }{y}_j{\mathrm{\mid }}^2$, $P_j=produc{t}_{2\le l\le k,l\mathrm{\ne }j}{r}_l^2$, and $A=su{m}_{j=2}^k$ $P_j$. The horizontal clockwise field on the graph has base component −iζ and vertical components $Z_j=-i$ $y_j$ $P_j\mathrm{/}(\mathrm{\mid }\zeta {\mathrm{\mid }}^2+A)$, $j=2,\mathrm{.}\mathrm{.}\mathrm{.},k$, and zero components afterwards. On the dense open set where all coordinates are nonzero this is obtained by solving the horizontal equation and dividing numerator and denominator by their common product: equivalently it is $-i\mathrm{/}(bar(y_j)(\mathrm{\mid }\zeta {\mathrm{\mid }}^2\mathrm{/}\mathrm{\mid }{y}_2\mathrm{.}\mathrm{.}\mathrm{.}{y}_k{\mathrm{\mid }}^2+\sum \mathrm{\mid }{y}_l{\mathrm{\mid }}^{-2}))$. Explicitly, write the angular velocities of $y_j$ as ${\lambda }_j$ and note that $y_1$ has angular velocity 1+sum ${\lambda }_j$ because the base moves clockwise. Orthogonality to each vertical radial variation says $r_j^2$ ${\lambda }_j+r_1^2(1+\sum {\lambda }_i)=0$. Summing gives 1+sum ${\lambda }_i=1\mathrm{/}(1+\sum r_1^2\mathrm{/}{r}_i^2)$, and substitution is exactly the displayed formula. Orthogonality to vertical angular variations holds because the velocity is purely angular. This verifies the horizontal equation in every radial and angular direction. The polynomial formula extends smoothly to ALL graph points, since $\mathrm{\mid }\zeta {\mathrm{\mid }}^2+A>0$. It still solves the horizontal equations there by continuity. It preserves every radius and hence is complete. On the dense open set, the angular reference plane ${\oplus }_{j=2}^k$ R i $y_j$ ⊕ $R^{n-k}$ is Lagrangian for the induced graph symplectic form: varying only its angles varies $y_1$ by a purely angular multiple, so the standard product symplectic form vanishes on the whole real tangent plane. Its ordinary volume phase changes at rate −2 $su{m}_j$ $P_j\mathrm{/}(\mathrm{\mid }\zeta {\mathrm{\mid }}^2+A)$. Thus the canonically graded monodromy displacement is exactly −2a, where $a=A\mathrm{/}(\mathrm{\mid }\zeta {\mathrm{\mid }}^2+A)$, and the d-th iterate has displacement −2da. Since A≥0, its zero set is EXACTLY the set where at least two of $y_2,\mathrm{.}\mathrm{.}\mathrm{.},y_k$ vanish: if none or just one vanish, one of the products $P_j$ is positive; if two vanish every product is zero. On a compact subset K of the graph disjoint from that set, a has a positive minimum c. Section1 bounds the displacement at every vertical plane by −2dc+n−1 on the dense open subset of K. Continuity in the base point and plane extends the same weak bound to points with a single zero coordinate. For $d>(n-1)\mathrm{/}(2c)$, it is strictly negative everywhere over K.

Correctly normalized torus-invariant smooth perturbation The following extends the rational chart calculation and retains the Hamiltonian time factor. Assume the ambient symplectic form is standard and $p=(y_2\mathrm{.}\mathrm{.}\mathrm{.}{y}_k\mathrm{/}{y}_1)q(I_1,\mathrm{.}\mathrm{.}\mathrm{.},I_k,z)$, where $I_j=\mathrm{\mid }{y}_j{\mathrm{\mid }}^2\mathrm{/}2$, $z=(y_{k+1},\mathrm{.}\mathrm{.}\mathrm{.},y_n)$, q is smooth, nowhere zero, and $q(0)=1$. Assume $p$ is meromorphic for a smooth tame complex structure equal to the standard compatible structure at $0$: it is holomorphic on the complement of the polar divisor $\{y_1=0\}$, and $y_{1}p$ extends smoothly across that divisor with the displayed numerator. No holomorphic extension of $p$ across $y_1=0$ or across the base locus is assumed. Assume also that $(y_2\mathrm{.}\mathrm{.}\mathrm{.}{y}_k)\eta$ extends smoothly with value $d{y}_1\wedge \mathrm{.}\mathrm{.}\mathrm{.}\wedge d{y}_n$ at 0. These are precise chart hypotheses, not a claim that the named global pencils already have them. Fix an iterate d and an error ε. Shrink the chart to radius δ, starting in its radius-δ/2 subchart, and work where $y_1$ and p are nonzero. Put $s=\mathrm{\mid }{y}_2\mathrm{.}\mathrm{.}\mathrm{.}{y}_k\mathrm{/}{y}_1{\mathrm{\mid }}^2$ and $S=su{m}_{j=1}^k$ $\mathrm{\mid }{y}_j{\mathrm{\mid }}^{-2}$. All bounds below are initially on the dense torus; continuity will extend the final plane bound where only one numerator coordinate vanishes. The Hamiltonian $H=-\mathrm{\mid }p{\mathrm{\mid }}^2\mathrm{/}2=-s\mathrm{\mid }q{\mathrm{\mid }}^2\mathrm{/}2$ is invariant under the FULL torus, so all $I_j$ are conserved. Therefore s is conserved, and conservation of H makes |q| conserved along its flow. Its actual angular velocities satisfy ${\theta }_1$ $dot=s\mathrm{\mid }q{\mathrm{\mid }}^2\mathrm{/}\mathrm{\mid }{y}_1{\mathrm{\mid }}^2+O(s)$, ${\theta }_j$ $dot=-s\mathrm{\mid }q{\mathrm{\mid }}^2\mathrm{/}\mathrm{\mid }{y}_j{\mathrm{\mid }}^2+O(s)$, j≥2. Here the O(s) coefficients and all their derivatives in the residual z directions are uniformly bounded: they are derivatives of the smooth function |q|² with respect to $I_j$. The residual vector field and its z-linearization have norm at most C s, since s depends only on the conserved moments. The derivative of arg q along the flow is likewise O(s); q has no angular dependence. Hence (arg p) $dot=-s\mathrm{\mid }q{\mathrm{\mid }}^2$ S+O(s). The main coefficient s|q|² S is CONSTANT along the flow. Since $S\ge {\delta }^{-2}$ and |q| is bounded above and below, the base phase decreases strictly. For its d-fold clockwise loop time T, integration gives T s≤C d/S≤C d δ², T $s\mathrm{\mid }q{\mathrm{\mid }}^2=2\pi d\mathrm{/}S+O(d\mathrm{/}{S}^2)$, $\mathrm{\Delta }{\theta }_1=2\pi d\mathrm{/}(\mathrm{\mid }{y}_1{\mathrm{\mid }}^{2}S)+O(d{\delta }^2)$. The constants depend on the fixed smooth chart and d, not on the small individual numerator radii or on |p|. The residual displacement is O(dδ²), so the trajectory remains in the original radius-δ chart when δ is sufficiently small. Thus it exists for the whole loop even though its unnormalized time T may be very large. Its residual derivative differs from the identity by O(dδ²), by variation of constants and Gronwall applied to the bound C s T. These estimates use conserved moments; they do not make a false uniform Hessian estimate in the singular $y_1$ direction. Choose a residual Lagrangian plane containing the reduced Hamiltonian vector $X_{\mathrm{r}\mathrm{e}\mathrm{d}}$ (extend that one vector to a symplectic basis; if it is zero choose any plane), and take the full reference plane consisting of it and the torus angular directions. This full plane contains X. Therefore dH vanishes on it, dp has image the real tangent line i p R, and its intersection with ker dp is a genuine vertical Lagrangian plane of dimension n−1. Its transported residual plane continues to contain $X_{\mathrm{r}\mathrm{e}\mathrm{d}}$ because Hamiltonian flow transports its own vector field. Under the flow it remains of that form: moment variations are zero, residual variation is exactly the reduced flow derivative, and angular shears into the orbit directions do not change its phase determinant. With ${\eta }_0=d{y}_1\wedge (d{y}_2\mathrm{/}{y}_2)\wedge \mathrm{.}\mathrm{.}\mathrm{.}\wedge (d{y}_k\mathrm{/}{y}_k)\wedge dz$, its squared phase is a constant times $y_1^2\mathrm{/}\mathrm{\mid }{y}_1{\mathrm{\mid }}^2$ times the residual squared phase. The continuous phase drift of the latter is O(dδ²), because its derivative path stays O(dδ²) from the identity. Replacing ${\eta }_0$ by η changes the endpoint phases by O(δ): multiply BOTH by $y_2\mathrm{.}\mathrm{.}\mathrm{.}{y}_k$, obtaining smooth forms tending to the same nonvanishing standard volume form. On the compact unit Lagrangian Grassmannian their evaluations stay uniformly away from zero, so their ratio has an argument O(δ), with the unique small lift throughout the chart. Thus the total reference-plane grading drift is $2d\mathrm{/}(\mathrm{\mid }{y}_1{\mathrm{\mid }}^{2}S)+O(d{\delta }^2)+O(\delta )$. Dividing by the base form dζ/ζ² contributes exactly −2d on the vertical reference plane. For an arbitrary vertical plane adjoin the horizontal Hamiltonian direction X. This is an ambient Lagrangian plane. Apply §1 to $D{\phi }_T$ in ambient dimension n, then the uniform O(δ) comparison of total volumes. Changing the return time with the starting point only adds a multiple of X to the transported vectors; it does not change their full plane or the vertical quotient. The common base phase correction is therefore the same for every such plane. This bounds EVERY vertical plane by $-2d(1-1\mathrm{/}(\mathrm{\mid }{y}_1{\mathrm{\mid }}^{2}S))+n+\epsilon$ after choosing δ small. At any point with nonzero p and $y_1$, the numerator coordinates are also nonzero, so this estimate directly covers the punctured fibre. Its bounded-locus continuity limits can be used wherever the graph chart extends. The factor in parentheses is exactly A/(|ζ/q|²+A), with A as in §3, and hence retains the actual multiple-zero exceptional set. This proves the local smooth-q comparison rather than invoking a mislabeled normalized Hamiltonian field or integrating an O(1) error for an uncontrolled long time.

Finite relative collar induction Let $D={\bigcup }_{i=0}^N$ $C_i$ be the named simple normal-crossing divisor, with the complex structures and meromorphic pencil map $p=P\mathrm{/}F$ fixed. Write $C_I^{c}irc$ for an open stratum, and $e_0=-1$, $e_i=1$ for $i>0$. Near $C_I^{c}irc$ the divisor of p has exactly the multiplicities $e_i$, i∈I. We construct coordinates as smooth families indexed by the stratum point; for each fixed stratum point the fibre coordinates and their local defining germs are holomorphic. The family need not be a holomorphic tubular projection. The required relative extension is the following elementary coordinate-germ construction. Suppose a family of such fibre charts has already been prescribed over a closed collar A in the base stratum, and the prescribed germs agree on overlapping inherited collars. Choose a complex splitting of the normal exact sequence along the stratum, extending the prescribed splitting on A. Splittings form an affine bundle, so a partition of unity extends them without changing A. Cover the rest of the base by complex normal-crossing charts. At each basepoint x normalize each chart germ by a complex affine change of its tangential coordinates and a nonzero complex scalar on each normal coordinate, so that its value and first derivative are the fixed splitting identification $T_{x}X$ ≅ $T_x{C}_I$ ⊕ ${\oplus }_{i\in I}$ $N_{i,x}$. Tangential coordinates may have linear normal terms added to achieve this normalization. Normal coordinates are not mixed, so each continues to vanish on its particular divisor. Regard the normalized germs as maps into the displayed complex vector space, not into independently chosen scalar frames. Their linear transition on that target is then just change of vector-bundle frame. For fixed x, take their weighted sum with coefficients from a partition of unity on the BASE. Include the prescribed chart as one summand, with coefficient one on a smaller closed collar and zero outside its given collar. The sum is holomorphic in the ambient variable for fixed x, varies smoothly with x, and has the same invertible first derivative as every summand. Each normal component vanishes on its prescribed divisor, and its first derivative there is nonzero. In a local scalar frame it is the original local defining function times a holomorphic unit near x; shrinking excludes additional zeros. The parameter-dependent inverse-function theorem therefore gives a family of local coordinate germs. The inverse map at tangential coordinate zero gives the fibre over x. Its differential on the zero section is the prescribed normal splitting, so (x,normal coordinates) maps diffeomorphically onto a neighborhood of the zero section after shrinking. On noncompact open strata the radius is a positive smooth function. Relative equality holds on the smaller collar because there the prescribed summand has coefficient one. Prescribed fibres obtained by restricting a deeper fibre chart meet the hypotheses: they are holomorphic submanifolds of its holomorphic fibre. Their local defining functions extend holomorphically in an ambient complex chart by setting the additional tangential coordinates free. This extension can be made smoothly in the base parameter by the same inverse-function construction. Thus taking an inherited subfibre does not require the deeper tubular projection itself to be holomorphic. Now perform descending-depth induction on the finite index sets I. At maximal depth the strata are disjoint compact components, and ordinary normal-crossing coordinate germs give the initial families. At a later depth, the deeper data prescribe the new charts near ${\bigcup }_{J⊋I}{C}_J$: inside a J-chart the I-fibre is obtained by fixing the coordinates labelled by $J\setminus I$ and retaining the coordinates labelled by I. Its base consists of the old residual base together with those fixed omitted coordinates. Charts already prescribed by two deeper sets agree on their common deeper collar by the inductive hypothesis. Reserve finitely many concentric collar layers at each stage, more layers than the total number of remaining depths. Extend on the outer layer and impose exact equality on the next inner layer. After that stage use only the inner layer for the deeper tube. This makes the preceding relative construction applicable at every later stage, with no infinite limiting process. Here is the separation argument for incomparable tubes. Two closed strata $C_I,C_J$ intersect in $C_{I\cup J}$. Remove an open neighborhood of that union stratum whose closure lies inside its already fixed collar. The remaining closed cores of $C_I$ and $C_J$ are disjoint compact sets, so have positive distance in a fixed auxiliary metric. Their new tubular radii may therefore be chosen to have disjoint closures outside the prescribed collar. There are finitely many pairs. Taking the minimum of the finitely many permitted radii ensures that every incomparable overlap is inside an inherited union-stratum collar. The same argument ensures that a nested smaller tube meets the deeper tube only where its inherited chart is prescribed. Subsequent finite shrinking preserves these conditions. Thus all overlap compatibility is equality of prescribed coordinates on an OPEN collar, not merely agreement of their first derivatives along the stratum.

Normalize the pencil unit without breaking inherited collars On a fibre chart over $x\in C_I^{c}irc$, let $z_i$ denote its normal coordinates. Meromorphic normal-crossing factorization gives $p(z)=u_x(z)$ ${\prod }_{i\in I}$ $z_i^{e_i}$, where $u_x$ is holomorphic and nowhere zero, and varies smoothly with x. The factorization follows by successively dividing the numerator and denominator by the simple local defining functions; after those divisions neither zeros nor poles remain in a small fibre neighborhood. Fix one pivot a∈I and replace $z_a$ by $z_a$ $[u_x(z)\mathrm{/}{u}_x(0){]}^{1\mathrm{/}{e}_a}$, leaving the other normal coordinates unchanged. Since $e_a$ is 1 or −1, this uses multiplication or inversion, with no choice of a root. Its derivative at the zero section is the identity, and it is a local holomorphic coordinate change. In the new coordinates p is the product monomial times the base factor $u_x(0)$. A change of frame in a normal line changes $u_x(z)$ and $u_x(0)$ by the same base scalar; their ratio is frame-independent. Hence this normalization glues over the base. On an inherited collar, p is already a product monomial in the deeper coordinates. Restricting to an I-subfibre makes the omitted coordinates part of the base, so its unit is constant along that subfibre. The ratio $u_x(z)\mathrm{/}{u}_x(0)$ is therefore one there. The normalization does not change the prescribed collar. It can consequently be applied at each relative induction stage of §1. Fibrewise multiplication $z_i\mapsto \exp (i{t}_i)z_i$ now defines a smooth $T^I$ action. It satisfies $p({\rho }_{I,t}y)=\exp (i{\sum }_{i\in I}{e}_i{t}_i)p(y)$. On nested collars its label circles are exactly the corresponding circles of the deeper action, since the retained coordinates are literally the same coordinates there. On incomparable overlaps both actions are subactions of the prescribed union-stratum torus. In particular all such actions commute wherever both are used. At a point fixed by one of these circle subactions, its derivative is complex-linear for the original complex structure. Indeed the fixed divisor has complex tangent space and the circle is the identity there. The complementary normal direction lies in the holomorphic fibre and the circle acts holomorphically on that fibre, so its derivative on that complex line is multiplication by exp(it). These invariant complex subspaces span the tangent space. This observation is what permits tame averaging below without assuming a global holomorphic group action.

Averaging with a logarithmic cutoff The quadratic primitive estimate in the preceding note assumes that the initial form and its average agree as full tensors along the fixed stratum. The following version removes that extra requirement and is used in the relative induction. Let ω tame the original J, and average it over one of the compact tori to obtain $\bar{\omega }$. At each point of the fixed stratum S, the action derivative is complex-linear by §2, so every pullback of ω tames J there; their average does too. Uniform positivity on a compact stratum core and continuity give a tube on which both ω and $\bar{\omega }$ tame J, with a common positive lower bound. The difference $\mathrm{\Delta }=\bar{\omega }-\omega$ is closed and restricts to zero on TS, since the torus fixes S pointwise. Fibre radial homotopy therefore gives a primitive β with $d\beta =\mathrm{\Delta }$ and |β|≤Cr. Here β need only have order one. Choose an invariant radial cutoff χ satisfying $\chi =1$ at $r\le r_{i}n$, $\chi =0$ at $r\ge r_{o}ut$ and |rχ'(r)|≤ε. It exists with $r_{i}n$ much smaller than $r_{o}ut$: take χ to be a fixed smooth cutoff of $(\log r-\log r_{i}n)\mathrm{/}(\log r_{o}ut-\log r_{i}n)$, making the logarithmic ratio large. Then $\omega +d(\chi \beta )=(1-\chi )\omega +\chi \bar{\omega }+d\chi \wedge \beta$. The first two terms have the common taming lower bound, while |dχ∧β|≤Cε. Choose ε below half that bound. The displayed form, and the entire same linear surgery path, tame J. It equals the averaged form on the inner tube, equals ω outside the outer tube, and differs from ω by a globally exact compactly supported form. This proves positivity without the false assertion that shrinking alone makes an O(r) primitive cutoff arbitrarily small. If the original form is already invariant on an inherited collar and the contraction stays within that collar, $\mathrm{\Delta }=\beta =0$ there. Nested collars can be chosen invariant by using the coordinate radii before averaging. Thus perform the finite descending-depth surgeries with supports chosen as in §1. Surgeries on incomparable cores have disjoint supports. In inherited overlaps all participating actions are commuting subactions of the deeper torus, and both the existing form and the radial/cutoff data are invariant under that torus; later averaging and its primitive therefore preserve the previously imposed invariance. Shrinking to the inner tubes after each stage leaves a final closed form ${\omega }_{\ast }$ cohomologous to $N{\omega }_{F}S$, taming J and invariant under every active label circle on its final tube. Concatenating the explicit surgery paths gives the ambient path required for compact Moser.

Normalized moment functions glue exactly For a label circle with vector field $Z_i$, invariance gives $d(-{\iota }_{Z_i}{\omega }_{\ast })=0$. Its tube retracts to its fixed divisor. The closed one-form restricts to zero there; its periods vanish because every loop retracts to a loop in that divisor. Thus it has a primitive ${\mu }_i$, uniquely normalized by ${\mu }_i=0$ on the fixed divisor, with convention $d{\mu }_i=-{\iota }_{Z_i}{\omega }_{\ast }$. In a complex normal line the circle generator is rotation. The normal quadratic term of ${\mu }_i$ is a positive multiple of $\mathrm{\mid }{z}_i{\mathrm{\mid }}^2$, since ${\omega }_{\ast }$ tames J. Shrinking gives ${\mu }_i\ge 0$ with zero set exactly $C_i$ in its tube. On an overlap the label generators agree by §2, so the two differentials agree; the difference is constant. Every connected overlap component may be shrunk to meet the fixed divisor through its normal fibres, where both functions are zero. Hence those constants are zero. The normalized functions glue, including on nested and incomparable collars. Extend ${\mu }_i$ to a positive smooth function away from the divisor, equal to a fixed positive constant outside a slightly larger tube, with its small sublevel sets contained in the true moment-function tube. This extension does not claim to be a global moment map. It only ensures that $\{{\mu }_i<\delta \}$ lies in the region where the moment identity holds, for every sufficiently small δ. Since p transforms by a unitary character, $H=-\mathrm{\mid }p{\mathrm{\mid }}^2\mathrm{/}2$ is invariant under every active label circle. Therefore ${\mu }_i,H=0$. The horizontal lift is a positive scalar reparametrization of $X_H$ on the regular nonzero-p region, so it too preserves ${\mu }_i$. The identities extend to the smooth pencil graph over the compact regular base loop by continuity from its dense affine region. For a fixed regular fibre $Y=X_R$, define $W_{\delta }={\bigcup }_{1\le i<j\le N}$ $\{{\mu }_i<\delta ,{\mu }_j<\delta \}$ ∩Y. For small δ each pair sublevel lies inside the common two-divisor tube. Its boundary can be crossed only through ${\mu }_i=\delta$ or ${\mu }_j=\delta$, where the corresponding moment is conserved. Completeness on the compact fibre therefore makes each pair sublevel and their union invariant under the full return flow. The zero set of all these pair conditions in the δ→0 limit is exactly $B_R=Sing(X_{\mathrm{\infty }})\cap Y$. Compactness shows that the $W_{\delta }$ form arbitrarily small neighborhoods of $B_R$: otherwise points outside a fixed neighborhood would have a subsequence converging to a point with two μ's zero. This is the required invariant-neighborhood assertion, with no claim of negative drift on $B_R$ itself.

Binding the uniform local phase estimates The torus representation can be put in equivariant Darboux coordinates at every fixed-stratum point. To construct them, choose an invariant metric by Haar averaging and use its exponential map to linearize the action. The pulled-back form equals its constant value at the origin. Radial homotopy of their closed difference gives an invariant primitive; the straight interpolation is symplectic on a small ball, and its Moser vector field is invariant. Its flow gives the equivariant symplectic coordinates. The normal rotation planes are complex lines for J by §2 and are pairwise ${\omega }_{\ast }$-orthogonal by torus invariance. Residual tangent directions have a tame J; compatibility there is unnecessary, because a homotopy to a compatible structure changes the phase displacement by a bounded endpoint term on the compact Lagrangian Grassmannian. In these coordinates the exact torus character factors the pencil into a monomial times a smooth invariant nonzero factor q(I,z). The scalar factorization is checked one circle at a time: a smooth weight-one function vanishing simply on its zero normal plane is z times a smooth invariant function (write its restriction on $z=r$ real, use equivariance, and extend the even radial coefficient smoothly as a function of $r^2$). Negative weight is treated by the denominator. The factor remains nonzero after shrinking because the original meromorphic divisor is simple. Thus the ordinary and rational local models used in the preceding phase-rate calculation are the actual models of this constructed form, with I the conserved normal moments.

Finite cover with separated moment scales. Write $M=N+1$ for the number of divisor labels $0,\dots ,N$, and fix the form, compatible collar germs, normalized moment functions and torus actions constructed in 6.1, 7.1, 8.1 and 9.1. Choose $\eta >0$ so small that every level $0<{\mu }_i\le \eta$ is in its genuine moment tube. Crossings of any one of these levels are impossible for horizontal transport: in an open neighborhood of the level, its derivative is zero. In particular every condition ${\mu }_i<b$, with $0<b<\eta$, is invariant for the compact graph flow. The extension of a moment outside its tube need not be conserved. We construct $M+1$ disjoint closed intervals

Proceed downwards in $j$. Once $u_j$ is fixed, and for each subset $I\subset \{0,\dots ,N\}$, the core

is compact and is disjoint from every omitted divisor. For an empty core choose the small active-moment region to be empty; this is possible by compactness, since otherwise a sequence with all active moments tending to zero has a limit in that core. This also treats the forbidden all-active subset when its divisor intersection is empty. Ignore those regions thereafter. Over it, take a finite family of the equivariant symplectic charts constructed in 10.1, with a smaller residual cover whose closures lie in that family. The normal circle characters, meromorphic-volume model and their overlap germs are those already fixed by 6.1, 7.1 and 8.1. This finite construction also permits an arbitrary additional positive upper bound for $l_j$, specified from these fixed larger charts before $l_j$ is chosen. Choose $l_j>0$ small enough that (i) the region ${\mu }_i<l_j$ for $i\in I$, ${\mu }_b>u_j$ for $b\mathrm{\notin }I$, lies in this finite family, (ii) $l_j$ is below that specified positive upper bound, and (iii) it lies in the inner invariant collars. Thus this step establishes a construction parametrized by such stagewise bounds; it uses no phase estimate to choose them. There are finitely many $I$, so their minimum allowed positive value works. Next choose $u_{j-1}<l_j\mathrm{/}2$. Start with any sufficiently small $u_{M+1}<\eta$. This is a finite descending construction; no constant is estimated on a core which is subsequently moved towards an omitted divisor. Every vector of $M$ moments misses at least one of the $M+1$ disjoint closed intervals. Choose such a $j$, and set $I=\{i:{\mu }_i<l_j\}$. Then all omitted moments are $>u_j$. The label set and these inequalities persist throughout transport. After increasing $R$, every point of $Y=X_R$ has some coordinate-divisor moment $<l_1$. Indeed on the compact complement of those coordinate-divisor sublevels, $P$ is bounded away from zero and $F$ is bounded, whereas $P\mathrm{/}F=1\mathrm{/}R$. Thus $I$ contains a numerator label. All trajectories on $Y$ are covered by the finite families just chosen. This covers points on bin endpoints as well: the missed interval is closed, so no moment lies on either endpoint of that selected interval.

Compact-fibre Moser preserving the actual exceptional set. We construct the compact-fibre identification with $f_s(B_R)=B_R$. Preserving this smaller arrangement is sufficient; no identification of entire relative categories is required. Let ${\omega }_{\ast }$ be the final ambient form of step 8.1 and choose its actual primitive ${\beta }_1$ with ${\omega }_{\ast }-{\omega }_0=d{\beta }_1$. Use the straight ambient path ${\omega }_s=(1-s){\omega }_0+s{\omega }_{\ast }$ and initially ${\beta }_s=s{\beta }_1$. It is symplectic on every complex fibre $Y=X_R$, because both endpoints tame the fixed complex structure and taming is convex. Put ${\alpha }_s={\mathrm{\partial }}_s({\beta }_s\mathrm{\mid }Y)$. For $N=4$ the set $B_R$ is finite. For $N=5$ its components $C_{ij}=\{x_i=x_j=0,\sum x_l^5=0\}$ are smooth compact Fermat plane curves; two distinct components meet only at the finite triple-coordinate-zero set. Four zero coordinates give no projective solution. At a triple-zero point the three branches $C_{ij},C_{ik},C_{jk}$ are exactly the three coordinate axes in a holomorphic chart on $Y$: the two remaining nonzero coordinates make the derivative of the Fermat equation nonzero in one of them. Their real tangent planes are pairwise symplectically orthogonal for ${\omega }_0$ by step 1.1 and for ${\omega }_{\ast }$ by the independent normal-circle invariance of steps 7.1--9.1. Indeed each circle rotates one axis and fixes the others, so invariance forces every mixed two-form entry to vanish. They remain orthogonal for every straight-path ${\omega }_s$. First add $d{g}_s$ to ${\alpha }_s$, using disjoint cutoff linear functions at the finitely many crossing points, so that ${\alpha }_s(p)=0$ there. For $N=4$ make the same correction at every point of $B_R$ and set $F_s=0$; the resulting Moser field already fixes those points. For $N=5$, on each curve $C$ define the smooth tangent vector field $v_{C,s}$ by ${\iota }_{v_{C,s}}({\omega }_s\mathrm{\mid }C)=-{\alpha }_s\mathrm{\mid }TC$; its existence follows from symplectic nondegeneracy on this complex curve. It vanishes at the crossing points. The covector ${\rho }_{C,s}=-{\alpha }_s-{\iota }_{v_{C,s}}{\omega }_s$ along $C$ annihilates $TC$ and vanishes at those points. We now extend these covectors as the differential of a smooth function $F_s$ vanishing on the whole curve arrangement. Away from crossings, in tangent/normal coordinates $(x,y)$, the expression ${\rho }_{C,s}(x)(y)$ gives exactly the prescribed first jet. At a crossing use the three real two-dimensional axis coordinates $(x_1,x_2,x_3)$ write $C_i$ for its $i$th branch, and let $H_{ij}(x_i)$ be the restriction of ${\rho }_{C_i,s}$ to the $j$th coordinate plane, for $i\mathrm{\ne }j$. Each $H_{ij}(0)=0$. The bilinear forms $B_{ij}(u_i,u_j)=D{H}_{ij}(0)[u_i](u_j)$ obey $B_{ij}(u_i,u_j)=B_{ji}(u_j,u_i)$. To check this, differentiate ${\rho }_{C_i,s}$ at the crossing. The term involving $D{v}_{C_i,s}$ pairs a vector in the $i$th tangent plane with the orthogonal $j$th plane and is zero. The difference of the remaining two derivatives is $-d{\alpha }_s(u_i,u_j)=-{\mathrm{\partial }}_s{\omega }_s(u_i,u_j)=0$. The terms involving $v_{C_i,s}(0)$ also vanish. Thus the explicit smooth function $F_{p,s}(x)={\sum }_{i\mathrm{\ne }j}{H}_{ij}(x_i)(x_j)-{\sum }_{i<j}{B}_{ij}(x_i,x_j)$ vanishes on every axis and has differential ${\rho }_{C_i,s}$ on the $i$th axis: differentiating the first sum gives the desired $H_{ij}$ plus its transposed linear term, and the second sum subtracts exactly that extra term. These are all components of the required differential. Patch the node functions and the ordinary curve-tube functions with a smooth finite partition of unity equal to one near $B_R$, and cut off outside that neighborhood. Each local function vanishes on the arrangement, so the terms containing derivatives of the partition vanish there; their differentials therefore patch the prescribed covectors exactly. All choices and formulas are smooth in $s$. The modified one-form ${\widehat{\alpha }}_s={\alpha }_s+d{F}_s$ still has $d{\widehat{\alpha }}_s={\mathrm{\partial }}_s({\omega }_s\mathrm{\mid }Y)$ and satisfies ${\widehat{\alpha }}_s=-{\iota }_{v_{C,s}}{\omega }_s$ along every branch. Solve ${\iota }_{V_s}({\omega }_s\mathrm{\mid }Y)=-{\widehat{\alpha }}_s$, ${\mathrm{\partial }}_s{f}_s=V_s\circ f_s$, $f_0=\mathrm{id}$. Then $V_s\mathrm{\mid }C=v_{C,s}$, so its complete smooth compact-fibre flow preserves every branch and fixes their crossing points. Cartan's formula gives $f_s^{\ast }({\omega }_s\mathrm{\mid }Y)={\omega }_0\mathrm{\mid }Y$. Incorporate both exact corrections $d{g}_s,d{F}_s$ in the primitive used below: extend their scalar functions from the starting fibre to the compact pencil graph over the loop by a local bundle trivialization and a cutoff in the base, and integrate them in $s$. Replacing the original ${\beta }_s$ by ${\widehat{\beta }}_s={\beta }_s+d_E{\int }_0^s(g_t+F_t)dt$ leaves ${\mathrm{\Omega }}_s={\mathrm{\Omega }}_0+d_E{\beta }_s$ unchanged and restricts to ${\mathrm{\partial }}_s{\widehat{\beta }}_s\mathrm{\mid }Y={\widehat{\alpha }}_s$. Hereafter ${\beta }_s$ denotes this corrected primitive. Thus both the total-space Moser argument and the prequantum connection calculation below use the very same $f_s$. To compare actual monodromies, take the closure of the pencil graph in $X\times P^1$ and restrict to a compact regular base loop γ. Locally at a base-locus crossing its finite-z equation is $F-zP=0$, whose ambient derivative dF−z dP is nonzero at that crossing; hence this graph is smooth there along finite regular fibres. Away from the base locus it is the ordinary pencil graph. The projection to γ is therefore a smooth proper bundle with compact smooth fibres. The pullback of every closed ambient ${\omega }_s$ is nondegenerate on its vertical tangent bundle. It defines a connection uniquely: lift a base vector arbitrarily and subtract the unique vertical vector making the lift orthogonal to every vertical vector. Vertical nondegeneracy, rather than total-space nondegeneracy, is all that this linear equation requires. The horizontal lift is smooth, complete over γ by compactness, and its flow preserves vertical ${\omega }_s$ by closedness. Its return $h_s$ is consequently a smoothly varying family of fibre symplectomorphisms. This is the extension of the usual ambient pencil parallel transport: on the complement of the base locus the defining orthogonality equations are identical. At a base point the lift with constant ambient point satisfies them and fixes that point. Choose $f_s$ on the starting fibre as above. Then $g_s=f_s^{-1}$ $h_s$ $f_s$ is a smooth symplectic isotopy for the original fibre form, with $g_0=h_0$. This binds the modified monodromy to the original compact monodromy, with $f_s$ preserving the actual exceptional set $B_R$ as proved above. The same construction works for parallel transport along a fixed vanishing path truncated before its Morse endpoint; near the endpoint the parametrized nondegenerate holomorphic Morse model supplies its smoothly varying vanishing sphere. The sphere transported back by $f_s$ is a Lagrangian isotopy. Its infinitesimal one-form is closed by differentiating the Lagrangian condition, and is exact because $H^1(S^{N-2};R)=0$. Extending its primitive to a tubular neighborhood constructs a Hamiltonian vector field agreeing with the isotopy velocity modulo tangential reparametrization. Thus these transported spheres are Hamiltonian isotopic to the original vanishing spheres, not merely abstractly diffeomorphic. For rational-brane data use ${\mathrm{\nabla }}_s={\mathrm{\nabla }}_0-2\pi i{\beta }_s$ on the fixed prequantum line bundle, with curvature $-2\pi i{\omega }_s$. The pullback family connection on [0,1]×Y has mixed curvature component $\beta do{t}_s+{\iota }_{V_s}{\omega }_s=0$. It is consequently flat in the s direction relative to the unchanged fibre connection, and parallel transport in s identifies $f_s\ast {\mathrm{\nabla }}_s$ with ${\mathrm{\nabla }}_0$ up to unitary gauge. This transports rational multisections and spin data. Equality of symplectic cohomology classes alone would not give that connection assertion. A nonvanishing holomorphic volume form on the finite smooth hypersurface fibre is supplied by the residue construction. Along the tame path choose compatible complex structures continuously, for example by polar decomposition from a fixed positive auxiliary metric and ${\omega }_s$. Their squared-phase maps and $D{f}_s$ produce a continuous circle-valued phase ratio on the Lagrangian Grassmannian, equal to 1 at $s=0$. The covering homotopy property gives a unique real lift starting at 0. Apply the same construction to the connection transports along γ. It identifies the actual graded monodromies under the isotopy $g_s$; no integer shift can enter continuously once its initial lift has been fixed. Finally transfer the negativity conclusion. Suppose $h_1$ admits arbitrarily small invariant neighborhoods W of an exceptional Σ, with negative drift on $K=Y\setminus W$. Its conjugate $g_1$ preserves $f_1^{-1}W$. For a graded symplectic isomorphism f and every iterate d, the composition law gives $a_{f^{-1}{h}_1^{d}f}(\mathrm{\Lambda })$ $=$ $a_{h_1^d}(Df\mathrm{\Lambda })+a_f(\mathrm{\Lambda })-a_f(D(f^{-1}{h}_1^{d}f)\mathrm{\Lambda })$. The last two terms differ by a uniformly bounded quantity, because the Lagrangian Grassmannian over compact Y is compact. Negative drift over the invariant K decreases linearly after iteration, by the previous packet. It therefore dominates this fixed conjugation bound. The resulting graded $g_1$ is negative on $f_1^{-1}K$ after a sufficiently large iterate. Here $\sum =B_R$ and $f_1^{-1}(B_R)=B_R$. Thus the transported neighborhoods are arbitrarily small invariant neighborhoods of the original coordinate-zero set itself, and all its stratum dimensions are unchanged. Hamiltonian avoidance and the finite evaluation-cone argument apply to the original compact category. This proves the Moser/gradation binding needed for compact split-generation. The construction preserves the required smaller exceptional arrangement; it makes no additional claim of an equivalence of entire divisor-relative categories.

Quantitative bounds on each family. In a chart write $r_i^2=2{\mu }_i$, $I_i={\mu }_i$, and $s=\mathrm{\mid }\text{normal monomial}{\mathrm{\mid }}^2$. In the rational case label 0 is the pole and is renamed 1: $p=(y_2\cdots y_k\mathrm{/}{y}_1)q(I,z),S={\sum }_{i=1}^k{r}_i^{-2},a=S^{-1}{\sum }_{i=2}^k{r}_i^{-2}\mathrm{.}$ The case of one numerator ($k=2$) is included. In an ordinary chart omit the denominator, take $S=\sum r_i^{-2}$, and put $a=1$. Initially work on the dense set where the displayed normal coordinates are nonzero. For explicit constants choose $B\ge 1$ bounding $\mathrm{\mid }q\mathrm{\mid },\mathrm{\mid }q{\mathrm{\mid }}^{-1}$. Put $Q=\mathrm{\mid }q{\mathrm{\mid }}^2$. Choose $D\ge 1$ so that the angular error coefficients, reduced vector field $X_H^{\mathrm{r}\mathrm{e}\mathrm{d}}\mathrm{/}s$, its first residual derivatives, and $d_z\arg q$ all have norm at most $D$ in the fixed larger charts. These are maxima of finitely many continuous coefficient functions on compact sets. Choose a residual metric, a smaller-cover Lebesgue radius $\rho >0$, and a finite refinement of overlaps on which the squared-phase transition has a real lift of absolute value at most $A$. Such a refinement exists by local lifting and compactness. Let $L$ bound the normalized squared-phase rate of the reduced linear equation divided by $s$, including the chosen frame connection. This bound follows directly by differentiating the determinant in a unitary frame: the derivative is bounded by the dimension times the norm of the infinitesimal linear generator. It does not require the long-time derivative itself to be bounded. The normalized leading meromorphic volume has a bounded lifted comparison with the standard model in each such chart. To see this without assuming holomorphic Darboux coordinates, multiply it by the normal factors removing its poles, and use the fixed complex normal splitting on the zero section. The remaining residual volume is nonzero on every Lagrangian plane: a Lagrangian plane for a form taming $J$ contains no nonzero complex line and is therefore totally real. Deform the tame complex structure continuously to a compatible one. This supplies a real comparison lift on the Lagrangian Grassmannian by homotopy lifting. On the compact chart closures it is bounded, and shrinking normal radii preserves nonvanishing. Enlarge $A$ to bound these local comparison lifts and their overlap differences. No uniform bound on the argument of the original meromorphic expression across its poles is being asserted. With the harmless overestimate $E=kD+D^2$, the Hamiltonian $H=-sQ\mathrm{/}2$ obeys $\mathrm{\mid }\dot{\arg p}+sQS\mathrm{\mid }\le Es,\mathrm{\mid }{\dot{\theta }}_1-sQ\mathrm{/}{r}_1^2\mathrm{\mid }\le Ds$ in a rational chart. All moments, $s,S,Q$, are constant. Impose $S\ge 2E{B}^2$. For $d$ clockwise turns, with return time $T$, integration gives $sT\le 4\pi B^{2}d\mathrm{/}S,\mathrm{\mid }sQT-2\pi d\mathrm{/}S\mathrm{\mid }\le EsT\mathrm{/}S,$ and hence

The residual path has length at most $DsT$. Select charts successively while remaining in one chart until traveling at least $\rho \mathrm{/}2$; at most $1+2DsT\mathrm{/}\rho$ changes are necessary. This controls accumulated lifted transitions even for arbitrarily long time. The residual phase rate integrates to at most $LsT$; its comparison/transition contribution is bounded by $4A+(8AD\mathrm{/}\rho )sT$. Angular shears into the full torus orbit plane do not change its volume determinant. Choose the residual reference plane to contain the reduced Hamiltonian vector, so the full reference plane contains $X_H$ and its vertical intersection is genuinely Lagrangian. The rational model's full phase contributes $\mathrm{\Delta }{\theta }_1\mathrm{/}\pi$, and division by $d\zeta \mathrm{/}{\zeta }^2$ contributes $-2d$. The linear symplectic phase oscillation estimate of 2.1, in ambient dimension $m$, changes this bound by at most $m$. Changing the return time as the initial point varies only adds the Hamiltonian direction, which is already in the full reference plane. Consequently one permissible set of constants is $C=4\pi B^2((E+D)\mathrm{/}\pi +L+8AD\mathrm{/}\rho ),C_0=4A+m,$ with enlargement of $L,A$ if a local frame is refined. It yields, for every $d\ge 1$,

The ordinary case has the same estimate with $a=1$; it has no pole-angle contribution and can use the same overestimate $C$. This is an infinitesimal phase-rate estimate, not a Gronwall bound for an arbitrarily long derivative path. Now instantiate the finite construction of 11.1 with the quantitative upper bounds established here. At each descending stage, fix $u_j$ and its larger compact chart family first, compute the finite constants $B,D,E,L,A,\rho ,C$ on that family, and only then choose $l_j$ satisfying $2l_j\le (2E{B}^2{)}^{-1}$ and $2C{l}_j\le 1\mathrm{/}2$, as well as its chart-radius restrictions. If several charts or active subsets occur at that stage, use their finitely many bounds simultaneously. Next choose $u_{j-1}<l_j\mathrm{/}2$ and repeat. Each upper bound is positive and was computed before its radius is chosen, so 11.1 gives the resulting cover without using any bound from a later stage. Since $r_i^2<2l_j$, ordinary charts then have $C\mathrm{/}S\le 1\mathrm{/}2$; rational charts have $\frac{C\mathrm{/}S}{a}=\frac{C}{{\sum }_{i=2}^k{r}_i^{-2}}\le \frac{2C{l}_j}{k-1}\le \frac{1}{2}\mathrm{.}$ Thus the ordinary bound is at most $-d+C_0$, and the rational bound at most $-da+C_0$. The constants were fixed before this shrinking. Take the maximum $C_{\ast }$ of the finitely many $C_0$'s. Normal line frames preserve the normalized moments; residual chart changes have already been counted. Every point of every trajectory is included by 11.1, rather than by an assumption that it remains in a single coordinate ball.

The conjugated return is Hamiltonian isotopic, not merely symplectically isotopic. This verifies the continuation needed in the compact generation argument without an unstated vanishing of $H^1(Y)$. Write $E_{\gamma }\to S^1$ for the smooth compact pencil graph over the regular loop in 11.2, and pull all ambient forms and primitives back to it. Thus ${\mathrm{\Omega }}_s={\mathrm{\Omega }}_0+d_E{\beta }_s$, with ${\beta }_0=0$, and every vertical restriction is symplectic. Let $V_s$ be the VERTICAL vector field uniquely determined by ${\mathrm{\Omega }}_s(V_s,W)=-{\dot{\beta }}_s(W)$ for every vertical $W$. Its flow $F_s$ exists for the whole parameter interval because $E_{\gamma }$ is compact, covers the identity on $S^1$, and restricts on the starting fiber to precisely the Moser map $f_s$ in 11.2. Total-space nondegeneracy of ${\mathrm{\Omega }}_s$ is not required for this equation.

Refined local error needed near the exceptional locus The fixed-iterate estimate in local-monomial-monodromy-phase.md deliberately states an O(dδ²) residual error. That form cannot be used to prove negativity arbitrarily close to the multiple-zero locus at one fixed base radius: the negative coefficient a may tend to zero. The correct retained estimate is O(d/S), since its ratio to da is uniformly small in a small numerator tube. Here is the precise conditional calculation. Assume the rational torus tube $p=(y_2\cdots y_k\mathrm{/}{y}_1)q(I,z)$ has a compact residual atlas along its stratum, q is nowhere zero, all relevant reduced coefficients and their first derivatives are uniformly bounded, and the horizontal flow stays in this tube. Put $s=\mathrm{\mid }{y}_2\cdots y_k\mathrm{/}{y}_1{\mathrm{\mid }}^2$, $S={\sum }_{j=1}^k$ $\mathrm{\mid }{y}_j{\mathrm{\mid }}^{-2}$, $a=({\sum }_{j=2}^k\mathrm{\mid }{y}_j{\mathrm{\mid }}^{-2})\mathrm{/}S$. The torus moments are conserved, hence s and S are constant. The Hamiltonian $H=-s\mathrm{\mid }q{\mathrm{\mid }}^2\mathrm{/}2$ also conserves |q|. Its reduced vector field and reduced linearization are O(s), its angular errors are O(s), and its base angular velocity is −s|q|²S+O(s). All these are bounds on infinitesimal coefficients; none is a claim that a long-time derivative remains close to the identity. For its d-fold clockwise return time T, assuming S is sufficiently large, integration gives sT ≤ C d/S, s|q|²T $=$ 2πd/S + O(d/S²), $\mathrm{\Delta }{\theta }_1$ $=$ $2\pi d\mathrm{/}(\mathrm{\mid }{y}_1{\mathrm{\mid }}^{2}S)$ + O(d/S). The residual phase drift of any transported residual Lagrangian plane is O(sT). Indeed the derivative equation has generator of norm O(s); differentiating the determinant phase in a unitary frame bounds its instantaneous angular derivative by C s. This estimate remains valid after arbitrarily long time: it integrates the phase rate instead of using exponential bounds on the derivative norm. With a finite compact residual atlas, chart changes add O(1)+O(sT): choose a positive Lebesgue radius and change charts only after the trajectory travels that distance; its speed is O(s), so at most C(1+sT) changes are needed. In a finite good cover the lifted transition phase along a single overlap is bounded. The same accounting controls smooth leading-volume-form/frame factors. Normal angular shears do not affect the phase determinant on the reference plane made from the torus orbit directions and a residual plane containing the reduced Hamiltonian vector. Thus its total reference-plane grading is $2d\mathrm{/}(\mathrm{\mid }{y}_1{\mathrm{\mid }}^{2}S)$ + O(d/S) + O(1). Dividing by dζ/ζ² contributes −2d. The linear symplectic phase-oscillation lemma compares any vertical plane with this reference by a constant depending only on dimension. Consequently $a_{h^d}(\mathrm{\Lambda })$ ≤ −2da + C d/S + $C_0$. If every numerator radius is at most δ, then (1/S)/a $=$ $1\mathrm{/}({\sum }_{j=2}^k\mathrm{\mid }{y}_j{\mathrm{\mid }}^{-2})$ ≤ δ²/(k−1). Choose δ so that $C{\delta }^2\mathrm{/}(k-1)<1$. The bound becomes $a_{h^d}(\mathrm{\Lambda })\le -da+C_0$. On any compact invariant complement of a neighborhood of the multiple-zero locus, a has a positive minimum; hence a single sufficiently large d makes the displacement negative on every plane. This permits d to depend on that neighborhood, precisely as required by the definition of negativity.

Cartan's formula gives ${\mathrm{\partial }}_s(F_s^{\ast }{\mathrm{\Omega }}_s)=d_E{\alpha }_s$, where ${\alpha }_s=F_s^{\ast }({\dot{\beta }}_s+{\iota }_{V_s}{\mathrm{\Omega }}_s)$. By its defining equation ${\alpha }_s$ annihilates every vertical tangent vector. For the positively oriented base coordinate $t\in \mathbb{R}\mathrm{/}\mathbb{Z}$ it therefore has the global form $k_{s}dt$ for a smooth scalar function on $E_{\gamma }$. Put $K_s={\int }_0^s{k}_{r}dr$. Integration gives the EXACT identity $F_s^{\ast }{\mathrm{\Omega }}_s={\mathrm{\Omega }}_0+d_E(K_{s}dt)$, and its restriction to each fiber is ${\omega }_0$. The term $d_E(K_{s}dt)$ is not declared zero: its mixed component is what determines the change of horizontal transport.

Hamiltonian avoidance, with all dimension conditions Let (X,ω) be a compact smooth symplectic 2n-manifold, L a compact embedded smooth Lagrangian n-manifold, and Σ a closed subset with a finite smooth stratification $\sum =\bigcup S_a$, each stratum of real dimension $s_a<n$. Then arbitrarily small Hamiltonian perturbations of L avoid Σ. Strata need not be individually closed. Choose finitely many smooth compactly supported Hamiltonians $H_1,\dots ,H_N$ whose Hamiltonian vectors span $T_{x}X$ at every x. To construct them, take a finite coordinate-ball cover with smaller balls still covering X; multiply coordinate functions by cutoffs equal to one on those smaller balls. Their differentials span the cotangent space there, hence their Hamiltonian vectors span the tangent space by nondegeneracy of ω. Write ${\mathrm{\Phi }}_t$ as the composition of their $\text{time-}{t}_j$ flows, for t in a neighborhood P of 0 in $R^N$. The evaluation F:P×L→X, $F(t,x)={\mathrm{\Phi }}_t(x)$, is a submersion at {0}×L. Compactness of L makes it a submersion on a smaller P×L. For each stratum $S_a$, $F^{-1}(S_a)$ is a smooth manifold of dimension $N+n+s_a-2n=N+s_a-n<N$. Every point of its projection to P is a critical point of that projection. Sard's theorem on a countable atlas implies that its image has measure zero. The finite union of these images has measure zero; choose t arbitrarily close to 0 outside it. Then ${\mathrm{\Phi }}_t(L)\cap \mathrm{\Sigma }$ is empty. Since both ${\mathrm{\Phi }}_t(L)$ and Σ are compact, their distance is positive for any fixed Riemannian metric. Consequently ${\mathrm{\Phi }}_t(L)$ misses an open neighborhood of Σ. This uses Sard and the finite-dimensional regular-value theorem, not arbitrary smooth displacement of a Lagrangian by non-Hamiltonian maps. The gradings, spin structures and rational-brane flat data are transported by the Hamiltonian isotopy. For a Fukaya object with bounding data one must also transport its bounding cochain by the actual Hamiltonian continuation equivalence. The elementary differential-topological argument does not construct that continuation functor. In a normal-crossings degree-(n+2) pencil in $P^{n+1}$, put $B_z=Sing(X_{\mathrm{\infty }})\cap X_z$. Each coordinate double intersection has complex dimension n−1 in the ambient projective space. Its intersection with the generic pencil member has complex dimension n−2; higher intersections only lower the dimension. Stratification by coordinate intersections and their intersections with the generic member therefore has real dimensions at most 2n−4. In $n=3$ this is $2<3$. For the quartic source Proposition 7.22 uses the still smaller finite set Σ of triple/base-locus points, hence $0<2$. These are exactly the two dimension inequalities needed here. The argument does not extend automatically to n≥4, where $2n-4<n$ fails.

Uniform negative iterate on the actual compact complement. Fix a small moment-pair neighborhood $W_{\delta }$ from 9.1; its invariance does not depend on which missed gap is used for the phase cover. The extended rational coefficient on the graph is $a=\frac{A_0\mathrm{\mid }q{\mathrm{\mid }}^2}{R^{-2}+A_0\mathrm{\mid }q{\mathrm{\mid }}^2},A_0={\sum }_{j=2}^k{\prod }_{2\le b\le k,b\mathrm{\ne }j}\mathrm{\mid }{y}_b{\mathrm{\mid }}^2\mathrm{.}$ It vanishes exactly when two numerator coordinates vanish. Such points belong to $W_{\delta }$. On each closed member of the finite cover intersected with $Y\setminus W_{\delta }$, this continuous function therefore has a positive minimum. A cover member is closed here by replacing its defining strict inequalities with ${\mu }_i\le l_j$, ${\mu }_b\ge u_j$ inside the fixed larger atlas; the strict gap between these bounds is retained.

Let $H_0$ be the ${\mathrm{\Omega }}_0$-horizontal lift with $dt(H_0)=1$. Use ${\iota }_{X_H}{\omega }_0=-d_{\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{t}}H$. The horizontal lift for ${\tilde{\mathrm{\Omega }}}_s=F_s^{\ast }{\mathrm{\Omega }}_s$ is $H_0-X_{K_s}$: for vertical $W$, its pairing equals $-{\omega }_0(X_{K_s},W)-d_{\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{t}}{K}_s(W)=0$. If $T_0(t)$ and ${\tilde{T}}_s(t)$ are transports from the starting fiber to time $t$ for these two connections, then $U_s(t)=T_0(t{)}^{-1}{\tilde{T}}_s(t)$ satisfies $U_s(0)=1$ and ${\mathrm{\partial }}_t{U}_s=X_{-K_s(t)\circ T_0(t)}\circ U_s$. This follows by differentiating the two transports; the $H_0$ terms cancel, and symplectic conjugation pulls a Hamiltonian vector field back to the Hamiltonian composed with $T_0(t)$. Thus $U_s(1)$ is an ACTUAL Hamiltonian diffeomorphism, given by the displayed time-dependent Hamiltonian, not merely a zero-flux assertion.

Negativity off invariant neighborhoods is sufficient Let φ:X→X be a graded symplectomorphism. Assume that every sufficiently small neighborhood of a closed Σ contains an open W with $\phi (W)=W$ and with φ negative on $X\setminus W$. Here negative means that one iterate ${\phi }^r$ has strictly negative real grading displacement at every Lagrangian tangent plane over $X\setminus W$. The grading displacement satisfies $a_{\phi \psi }(\mathrm{\Lambda })=a_{\phi }(D\psi \mathrm{\Lambda })+a_{\psi }(\mathrm{\Lambda })$. Choose the perturbation $L^{\mathrm{\prime }}={\mathrm{\Phi }}_t(L)$ from §1 and then an invariant W disjoint from L'. Set $K=X\setminus W$. Since φ is bijective, $\phi (K)=K$. The Lagrangian Grassmannian bundle over compact K is compact; hence $a_{{\phi }^r}\le -\epsilon$ there for some $\epsilon >0$. Also put $C=\max (0,{\sup }_K{a}_{\phi })$. For $d=kr+j$, $0\le j<r$, the composition law gives $a_{{\phi }^d}(\mathrm{\Lambda })$ ≤ −kε+(r−1)C, Λ based in K. Both L' and ${\phi }^d(L^{\mathrm{\prime }})$ lie in K. Their initial lifted phases are bounded on L'; the usual transverse intersection index differs from their transported phase difference by a quantity bounded solely by n. Thus all generators of $CF(L^{\mathrm{\prime }},{\phi }^d{L}^{\mathrm{\prime }})$ have strictly negative degree when d is sufficiently large. Make the intersection transverse by a sufficiently small Hamiltonian perturbation of the first Lagrangian only. The displacement estimate is evaluated on the unperturbed ${\phi }^d{L}^{\mathrm{\prime }}$, so its base points stay in K; the phase lift of the perturbed first Lagrangian remains uniformly bounded. The strict degree margin is retained. No estimate on arbitrary planes over Σ is used. The resulting $H{F}^0$ is zero. If the actual geometric Dehn-twist/evaluation-cone theorem identifies φ with the ordered product $T_1\cdots T_m$, the finite cone proof in negative-monodromy-generation-algebra.md now shows that L' belongs to the thick closure of the vanishing spheres. Hamiltonian continuation identifies L and L', so the original object lies in the same thick closure. For a rational brane with transported bounding cochain, the differential is still of degree one on this same graded chain module; absence of degree-zero generators gives the same $H{F}^0$ vanishing once that actual chain model exists. This resolves the logical role of the multiple-zero locus: it need not acquire strictly negative displacement. What must be constructed is arbitrarily small invariant neighborhoods and negative drift on their complement. Deleting the exceptional locus from the theorem statement is unnecessary.

Since $F_s$ is defined over the WHOLE loop, its value on the starting and ending fiber is the same $f_s$. Hence the return of ${\tilde{\mathrm{\Omega }}}_s$ is $f_s^{-1}{h}_s{f}_s=g_s$, and $g_s=h_0{U}_s(1)$. At $s=0$, $K_0=0$ and $U_0(t)=1$. This proves the required Hamiltonian isotopy of $g_s$ relative to $h_0$, independently of any assertion about the first cohomology of the fiber. The grading lift is also fixed: the rectangle $(s,t)\mapsto U_s(t)$ carries the unique lifted phase starting from the identity at $t=0$. Its return lift and the lift of $g_s$ transported in 11.2 agree at $s=0$ and vary continuously. Their difference is integer valued and is therefore identically zero. No unknown shift is inserted when applying Hamiltonian continuation to the original monodromy iterate. The connection calculation of 11.2 transports the prequantum brane lines; this Hamiltonian path changes their end action origins through those lines and does not assert that arbitrary real actions are Puiseux scalars.

An explicit split-generation lemma Let C be an idempotent-complete triangulated category over a field k with finite-dimensional total graded Hom spaces. For finitely many objects $S_1,\mathrm{.}\mathrm{.}\mathrm{.},S_m$, suppose exact functors $T_i$ and natural transformations $u_i:Id\to T_i$ are equipped with distinguished triangles $\mathrm{Hom}\ast (S_i,Y)\otimes S_i$ → Y → $T_i(Y)$ → $(\mathrm{Hom}\ast (S_i,Y)\otimes S_i)[1]$. The first term means the finite direct sum of the shifts of $S_i$ prescribed by the finite graded vector space. In a dg/A-infinity model this is the actual evaluation cone. Fix the order $T=T_1\circ \mathrm{.}\mathrm{.}\mathrm{.}\circ T_m$. If for every Y there is d≥1 such that ${\mathrm{Hom}}^0(Y,T^{d}Y)=0$, then $S_1,\mathrm{.}\mathrm{.}\mathrm{.},S_m$ split-generate C. Here is the complete cone argument. List the m d elementary transformations along $Y\to T_{m}Y\to T_{m-1}{T}_{m}Y\to \mathrm{.}\mathrm{.}\mathrm{.}\to T^{d}Y$. At every step its cone is a shift of a finite direct sum of one $S_i$, so lies in the triangulated subcategory A generated by the $S_i$. For composable maps f:A→B and g:B→D, the octahedral axiom gives the triangle Cone(f)→Cone(gf)→Cone(g)→Cone(f)[1]. Thus induction on this finite list puts $Cone(Y\to T^{d}Y)$ in A. Rotate to the triangle $R\to Y\to T^{d}Y\to R[1]$, where R is that cone shifted by −1. The middle map is zero by the assumed degree-zero Hom vanishing. A distinguished triangle with zero middle map is split, giving $R\cong Y\oplus T^{d}Y[-1]$. One can verify the splitting without an extra theorem: apply Hom(Y,−) to the triangle to lift $I{d}_Y$ to a map Y→R; its composite R→Y is Id, so Y is a retract of R. Idempotent completeness therefore puts Y in the thick closure of A. This applies object by object, without a uniform d. For an A-infinity category it is enough to establish this for its original objects, since twisted finite cones and retracts are by definition their split-complete derived closure. No assertion that all objects with zero Hom from the $S_i$ vanish is needed, and no full faithfulness or mirror equivalence is assumed here.

Consequently any Floer vanishing obtained for $g_1^d$ after avoidance of the transported invariant neighborhood is transported to $h_0^d$ by the actual Hamiltonian continuation path, with its fixed bounding objects and grading. Compose the Hamiltonian isotopy $g_s=h_0{U}_s(1)$ finitely many times to obtain the path for the chosen iterate; symplectic conjugation preserves Hamiltonian isotopies. This proves the identification required before using the compact evaluation-cone triangles. It supplies no new evaluation-cone theorem and does not resolve the separate held compact-category or Dehn geometric inputs. The Statement remains the original full $N=4,5$, $n=2,3$ compact monodromy and generation claim, with all prior source scaling and coefficient conventions.

Uniform phase drift implies the needed Hom vanishing Let a compact graded symplectic manifold M have a chosen continuous squared-phase map on its Lagrangian Grassmannian. For a graded symplectomorphism φ denote its real grading displacement by $a_{\phi }(\mathrm{\Lambda })$; the convention is the additive composition law $a_{\phi \psi }(\mathrm{\Lambda })=a_{\phi }(D\psi \mathrm{\Lambda })+a_{\psi }(\mathrm{\Lambda })$. Assume one iterate $d_0$ has $a_{{\phi }^{d_0}}(\mathrm{\Lambda })<0$ at every point and every Lagrangian plane. This is an actual uniform hypothesis on the compact Grassmannian bundle, not pointwise asymptotic drift. Compactness supplies $\epsilon >0$ with $a_{{\phi }^{d_0}}\le -\epsilon$. Put $C=\max (0,\sup a_{\phi })$. For $d=k$ $d_0+r$, $0\le r<d_0$, the composition law gives $a_{{\phi }^d}\le -k$ ε+r C≤−k $\epsilon +(d_0-1)C$. If the phase map is replaced by $e^{2\pi i\chi }$ times itself, its displacement changes by $\chi \circ {\phi }^d-\chi$. That changes the displayed estimate by at most 2||χ||∞, independently of d. Thus negative drift is independent of this bounded change of phase convention. For compact graded Lagrangians $L_0,L_1$, their grading functions have finite upper/lower bounds $B_1,b_0$. At any transverse intersection of $L_0$ and ${\phi }^d(L_1)$, the difference of lifted phases is at most $B_1-b_0-k$ $\epsilon +(d_0-1)C$. The absolute intersection degree differs from this phase difference by a number in an interval of length $n={\dim }_R$ $L_0$: diagonalize the two transverse real Lagrangian planes in a unitary frame as $R^n$ and $\mathrm{diag}(e^{i{\theta }_j})R^n$ with $0<{\theta }_j<\pi$. The index is the phase-lift difference plus the sum of the normalized principal angles, with its fixed convention-dependent endpoint shift; each normalized angle lies in (0,1). In particular there is a uniform constant n+1 bounding the extra term, independently of d or of the two planes. This is all that is needed here; reversing the degree convention also reverses the definition of negative displacement. For sufficiently large d all intersection generators therefore have degree strictly less than zero, with a fixed margin. A sufficiently small Hamiltonian perturbation makes the pair transverse while retaining that margin. Its Floer cochain complex has zero degree-zero group and thus $H{F}^0(L_0,{\phi }^d{L}_1)=0$. This step uses a genuinely constructed graded Floer complex and continuation invariance for the pair; the phase estimate alone does not construct those analytic objects. Taking $L_0=L_1$ and an actual identification of T with the graded monodromy action then supplies precisely the objectwise vanishing in §1.