Geometrically tame disk compactness and actual real-nodal atlas

Distinguish two meanings. A pointwise ω-tame J only means ω(v,Jv)>0. The global source condition “tame symplectic manifold” used here means that M admits a complete Riemannian metric g with sectional curvature bounded and injectivity radius bounded below, and constants $c,C>0$ and an admissible smooth J with $\mathrm{\mid }\omega {\mathrm{\mid }}_g\le C$, $\mathrm{\mid }J{\mathrm{\mid }}_g\le C$ and $\omega (v,Jv)\ge c\mathrm{\mid }v{\mathrm{\mid }}_g^2$. A compact symplectic manifold satisfies this condition. The theorem's source framework chooses admissible auxiliary data; it does not quantify over every unrelated pointwise-tame J. This global condition must be explicitly included as a definition of that previously undefined source word, not silently replaced by compactness of L. No conclusion below is asserted for arbitrary noncomplete noncompact data.

There are uniform normal-coordinate balls of radius r0 with uniformly controlled metric distortion. Every loop γ in one sufficiently small such ball has a cone filling of g-area at most C0 length_g(γ)²: cone to one point of the loop, use diameter≤length/2 and the normal-chart radial homotopy. The ω-area of the filling has the same bound because |ω|_g≤C. Since ω is closed and the ball contractible, any other filling has identical ω-area. For a finite union of oriented loops, sum the cone fillings and use Σ length_j²≤(Σ length_j)².

A J-holomorphic map is conformal for the Hermitian metric $g_J(v,w)=[\omega (v,Jw)+\omega (w,Jv)]\mathrm{/}2$, uniformly comparable to g; its $g_J$ area is exactly its ω-area. Consider the portion over a ball $B_r(p)$, whose preimage has no original boundary, and let A(r) be its area and l(r) the length of its boundary image. For almost every r, Stokes and 2.1 give A(r)≤C1 l(r)². Coarea and conformality give l(r)≤C2 A'(r): on a level curve the norm of the radial gradient is bounded by the map's conformal scale, so the coarea area-density is at least a uniform constant times the image length-density. Therefore A'(r)≥C3 sqrt(A(r)). If p is in a nonconstant image, A(r)>0 for every $r>0$; integrate the inequality for sqrt(A) from a positive ε to r and then let ε decrease to zero. This proves A(r)≥c0 r². It applies componentwise to a stable map and requires no uniform derivative bound on J outside compact sets.

If a connected stable disk image with boundary in compact L reaches g-distance R from L, its connected image contains at least floor(R/(4r0)) disjoint radius-r0 balls outside the r0-neighborhood of L. Choose their centers successively at radial distances separated by 4r0, approximating nodal centers by points of nonconstant components if necessary. Each ball preimage has no original boundary, so 3.1 costs at least c0 r0². Thus E≥c1 R−c2. An energy bound confines every image to a fixed closed bounded g-neighborhood of L, compact by completeness. This is the needed no-escape proof.

There is a uniform $h>0$ for every nonconstant sphere or disk component. The uniform ball calculation proves it for spheres, since a sphere contained in a ball has zero ω-area by exactness there and otherwise contains a definite-radius image ball. For disks use a fixed tubular neighborhood of compact L: the relative radial Poincaré formula gives a primitive λ with $d\lambda =\omega$ and $\lambda \mathrm{\mid }TL=0$ on that neighborhood (the pullback of ω to L vanishes). A disk entirely in it has area zero. Every nonconstant disk therefore leaves a fixed smaller tubular neighborhood; a definite-radius ball outside L and 3.1 give h. This also proves $m_{0,0}=0$ for genuine disk components; unstable zero-energy one- or two-mark disks are treated by the ordinary chain differential, not counted as stable maps.

The operations have $N=k+1\ge 1$ boundary marks, including the output even at $k=0$. Stabilize every nonconstant component by finitely many auxiliary interior marked points at distinct regular image points and transverse codimension-two slices. Such points exist because a nonconstant smooth map has nonzero derivative on a nonempty open set. Three interior points suffice on either a sphere or a disk; for a disk their real flags are more than sufficient. Add full finite automorphism orbits and label the auxiliary points in a local rigidification. Stable constant components already have at least three flags counted with boundary weight one and interior weight two. Since the number of positive-energy components is at most E/h, the tree flag identity bounds the number of stable constant components and auxiliary marks at fixed E,N.

Double a stabilized disk across its real boundary. The result is a genus-zero stable complex curve with an antiholomorphic involution; interior sphere components occur in conjugate pairs. Its marked-domain compactification can be constructed ELEMENTARILY, without importing the later general-genus Pardon domain page. On a marked P1 normalize three distinct flags to infinity,0,1. Extract limits of the remaining finite set. In each colliding group choose a pair of maximal separation, translate the first to zero and divide by their complex difference. The remaining group is bounded, the chosen pair becomes0,1, and the parent attachment is infinity; extract and repeat on proper colliding subgroups. Every group splits, so the finite tree flag bound terminates the construction. The finite collection of four-label cross-ratios in P1 recovers the tree: three leaves in different branches normalize a vertex, and each fourth leaf recovers its branch and position; recurse. Thus the closure in the finite cross-ratio product is compact Hausdorff and consists exactly of these finite stable trees. Complex fixture parameters and component flag coordinates are given by the same actual recursive rational relation $x_{parent}=a+q$ $x_{child}$ as in the adopted ordered real-disk-domain construction. Its inverse is the normalized representative-leaf calculation; $zw=q$ are the actual local node equations, and finite closed cores/fixtures give the proper local universal family. There are no automorphisms after all flags are labeled, except the finite tuple reindexings already retained. To impose the disk real structure perform the extractions together with their conjugates. A conjugate pair of clusters away from the real circle gives two conjugate sphere vertices. A cluster meeting the real circle is normalized by real translation and a positive real scale; take a scale equal to its maximal Euclidean diameter before extraction. It then has either two different limiting real flags, or a nonreal flag and its conjugate. Together with the real parent flag at infinity these supply at least three weighted flags. Normalize the former to0,1,infinity, or the latter to i,−i,infinity. These normalizations give the same explicit rational family restricted by conjugation. The involution-fixed local base has independent real component coordinates, one real allowed chamber $q=-\rho \le 0$ for a boundary node, and one arbitrary complex q for an interior node. The choice of the original disk half selects exactly that boundary chamber. This proves the real local product coordinates and compact subsequential convergence directly; no theorem about the fixed set of an unspecified singular space is used. Its only codimension-one strata are single boundary nodes; an interior sphere attachment has real codimension two. The output mark prevents the unstable completely collapsed boundary exception: a constant disk with its output and one interior sphere attachment already has three weighted flags.

By 4.1 all images lie in one compact region. On its finite chart cover the adopted interior/boundary epsilon, removable-singularity and rescaling proofs give smooth convergence off finitely many energy atoms; each atom costs at least h by 5.1 and is captured by the actual interior or real-boundary bubble extraction. Apply these arguments on each thick part of the varying family from 7.1. The actual small-image cylinder estimate handles every long interior neck; the same-label strip action and one-sided boundary estimates handle every long boundary neck as proved below. This yields matching node values and no lost neck energy. Iterate at the finitely many atoms, and contract unstable constant pieces. The flag bound from 6.1 makes the final stable tree finite. Thus all stable disk moduli at fixed N and E≤E0 are compact in their actual stable-map topology.

Define the global relative class before claiming its compatibility. A stable genus-zero disk tree with its output boundary mark has an oriented smoothing collapse c:(D,∂D,z0)→(C,∂C,z0): pinch one boundary-to-boundary arc for each real node and one interior circle for each sphere attachment, iterating along the finite rooted planar tree. Choose thin disjoint ribbons in the disk in that tree order. The composite u c is a continuous relative disk map with output on L. Its FREE relative homotopy class allows the output point to move along L; the componentwise homotopy annotations of each fixed combinatorial type are retained separately. This does not identify those finer annotations with homology. The collapse class is independent of the thin ribbon choices: on each child disk the boundary flags are cyclically ordered and the separating arc/circle is isotopic to the chosen standard one relative its flags; extend that isotopy across its parent and induct on the finite rooted tree. Interior rotations and twists are supported in the annulus collapsed to a node and become homotopic after collapse. No path in M is chosen independently of u. The tree, rather than a graph with cycles, is essential here. For an ACTUAL analytic smoothing, collapse its fixture and homotope the cutoff map within the target node ball. At a boundary fixture use the ball pair $(R^{2n},R^n)$, which is contractible as a pair in positive relative degrees. At an interior fixture use the contractible target ball. Thus the smoothed disk is relatively homotopic to u c. For sufficiently close maps, ordinary target geodesic interpolation on thick parts and its L-valued boundary tubular version, together with these same node-ball homotopies, give the same free relative class. This proves its local constancy through type changes, rather than merely its area invariance. Its relative integral fundamental class is the sum of the component fundamental classes, including torsion; this is the HOMOLOGY IMAGE of the homotopy statement, not a substitute for it. Boundary gluing on free classes is the ordered rooted-tree grafting just defined, with node-value transport supplied by the actual maps; no unbased nonabelian homotopy classes are incorrectly added as if they were an abelian group. Finitely many such neighborhoods cover the compact moduli, so only finitely many global free relative classes occur below E0. Their energies are locally finite. Finite energy decompositions remain finite by the component lower bound h. No discreteness of all arbitrary periods of ω is assumed.

At a boundary node choose a complex frame with $TL=R^n$ and conformal half-disk coordinate z. On its punctured end use $z=exp(-\pi (s+it))$, $0\le t\le 1$. The linearized principal operator is ∂s+i∂t, with real conditions on both edges. Its coefficient errors are O(exp(−πs)) with all fixed derivatives, because the normalized map and boundary frame are smooth at the original node. Its limiting kernel includes the REAL constants $R^n$. This is not the transverse-strip operator, whose two real planes intersect in zero.

Use the REAL spectral basis exp(-ipijt)v, j integer and v real, for id/dt on [0,1] with real endpoint traces. Its spectrum is pij. Conjugation by exp(deltas), $0<\delta <\pi$, gives multiplier $(i\ast tau+\pi \ast j-\delta {)}^{-1}$, bounded L2 to H1 by its spectral gap and first-derivative quotient. For ordinary $H^{k-1}$ forcing apply s difference quotients to that H1 estimate; they preserve real traces. The equation $w_t=-i(f-w_s+\delta \ast w)$ recovers higher normal derivatives inductively. This proves the unrestricted $H^{k-1}$ to Hk inverse without claiming that high-order reflection of arbitrary forcing is smooth. Finite s-collar Vandermonde extension and compact commutators give the half-end parametrix. The real zero eigenmode remains a separate finite-dimensional coordinate.

A nodal section has a shared real limiting value at a boundary node, and a shared complex value at an interior node. Subtract them in weighted end norms and add their finite norms. On a real smoothing subtract the real spectral zero-mode, integral Re(xi) over the middle section, rather than treating an arbitrary complex mean as a free constant. Break the target by complementary cutoffs near3S and apply the central matched right inverse. Keep the left section remainder until4S and the right from2S, add both in the overlap and count the common real constant once. Differentiated section-cutoff errors have weight ratio exp(-2delta*S); the coefficient multiplying the constant has weighted error exp(-(pi-delta)*S). The solved broken target equations add exactly to the forcing, so no derivative of the unknown target appears. Hence $\mathrm{\parallel }DT-I\mathrm{\parallel }\le C\sum [\exp (-2\delta S)+\exp (-(\pi -\delta )S)]$. Neumann inversion gives a uniform inverse with the fixed core/E complement; the bordered matching index identifies its full complement image. No nonlinear reflected equation or high-order forcing trace restriction is used.

The compact bordered Fredholm supplier identifies the central matched operator: a real boundary match subtracts n, an interior match 2n, and smooth supported finite forcing makes it onto. Its JOINT evaluation derivative at all selected marks is made onto by adding the explicitly constructed annular homogeneous Jacobi-section forcings, still away from all nodes. Impose auxiliary transverse-slice constraints by finite-codimension restriction. Add the same finite forcing dimensions before and after smoothing. The bordered doubled index and real/complex matching losses show the smoothed index agrees with the central index, with each boundary smoothing adding its one real domain parameter and each interior smoothing its two parameters. Fixed core evaluations and obstruction coordinates detect the kernel and give the same finite-dimensional complement construction as the adopted closed gluing proof. The resulting right inverse has the required full complement image.

Preglue each map by cutting its smooth O(exp(−πs)) remainders to the node value before the middle. Boundary charts straighten L to $R^n$, so this stays on L. The residual is O(exp(−(π−δ)S)); the kernel pregluing error has the same bound. Source-dependent exponential charts preserve every boundary L and auxiliary slice. Uniform unit-half-cylinder Sobolev products prove the same quadratic Lipschitz estimate as in the adopted closed proof. Its uniform contraction therefore constructs unique solutions with prescribed finite kernel coordinates, all real/complex gluing parameters, and the chosen obstruction coordinates.

For this SINGLE-L moduli problem every normalized boundary condition is smooth, including its marked points. In a small node chart choose a primitive theta of omega whose pullback to L is zero. For each vertical arc gamma with endpoints on L, subtract an exact primitive vanishing on L to make theta vanish at gamma(0); then $\mathrm{\mid }\int {\gamma }^{\ast }\theta \mathrm{\mid }\le C\mathrm{\parallel }{\gamma }_t{\mathrm{\parallel }}_2^2$. Stokes on a small-image strip has zero horizontal-edge terms and yields $A(r)\le C{A}^{\mathrm{\prime }}(r)$, hence $A(r)\le C\exp (-c(R-r))[b(-R)+b(R)]$. The actual boundary small-energy estimate supplies the gradient bound on fixed half-patches. Tangential differentiated CR estimates, with real-trace frames and normal derivatives recovered from the equation, retain its square-root exponential factor: the unforced autonomous equation for $u_s$ is homogeneous in du, so no additive constant is introduced. These are the explicit one-sided estimates proved in the adopted smooth-boundary gluing components E/F and their quantitative tangential bootstrap. On fixed cores use the inhomogeneous graph equation; its boundary lies on the Lagrangian R times L for the product taming form. A primitive zero on that Lagrangian gives the half-patch Caccioppoli estimate, without reflecting a nonsmooth nonlinear J. The autonomous compact graph-target construction in supplier component F satisfies the exact fixed-boundary epsilon theorem and, with the same tangential bootstrap, gives smooth core convergence. Choose delta smaller than BOTH pi and the actual decay rate $\mu =c\mathrm{/}2$; $\delta <\pi$ alone is insufficient. Fixed-depth endpoint convergence followed by the weighted tail integral gives full weighted-norm convergence and no neck energy, including the retained real middle mode. Contraction uniqueness therefore recovers graph-near solutions and the local product chart through partial nodal strata, using precisely those smooth-boundary analytic inputs. Smoothness in the common profile remains a separate differentiated argument; none in naive plumbing radius is inferred.

Choose finitely many stabilized central charts covering each compact moduli slice. Take their finite groups of tuple automorphisms, finite supported obstruction spaces, transverse auxiliary slices and actual thickenings from §3. For simultaneous chart labels take the direct sum of the obstruction spaces and the full tuple of auxiliary choices; extra-section zero recovers the smaller thickening. The relevant regular locus requires the augmented CR/evaluation derivative to be onto, not the obstruction section itself transverse. On overlaps extra obstruction coordinates have surjective derivative along their zero locus; they therefore give the ordinary normal coordinates used in a coordinate-change embedding. Relative forgotten-label groups act freely on the lifted auxiliary choices exactly as in the adopted Pardon tuple construction. The chain procedure will use these finite charts and normal coordinates, not the dual-cohomology Pardon functional.

We now supply the needed common smooth profile, rather than calling the continuous plumbing charts smooth. First establish differentiated bounds in the cylindrical length variables S. Fix a derivative order j and work with k+j+3 Sobolev derivatives. Trivialize varying finite cores by their actual smooth domain charts. On the long cylinders use translated unit patches at the cutoffs and the product coordinate elsewhere; to compare neighboring lengths, translate the two end coordinates by opposite amounts and interpolate over the middle by a vector field whose derivatives are uniformly bounded. Pullback changes the first-order operator by first-order coefficients with uniformly bounded derivatives in S; conjugating by the stated weights only adds bounded zeroth-order coefficients, since derivatives of log(weight) are bounded. The finite mean coordinate and real/complex node constants are kept separately. On these fixed-coordinate coefficient expressions, repeated length/angle/base differentiation of the pregluing residual is bounded by $C_j$ exp(−c min S), with one $c>0$ smaller than both π−δ and the actually proved boundary decay rate minus δ. Differentiating a cutoff at a moving S gives its bounded derivative times the same decaying normalized tail, not an unweighted section evaluated without its decay. The separated target-breaking/cutoff construction has uniformly bounded derivatives as an operator from $H^{k+j+2}$ to $H^k$; each parameter differentiation can lose one spatial derivative, which is the reason for the higher initial order. Its errors still decay exponentially. The Neumann derivative identity $(A^{-1}{)}^{\mathrm{\prime }}=-A^{-1}{A}^{\mathrm{\prime }}{A}^{-1}$ and induction give corresponding uniform bounds for every fixed number of derivatives of the actual right inverse on these higher-to-lower spaces. Core evaluation coordinates and the finite kernel splitting have the same bounds. Differentiating the fixed-point equation now leaves the same uniformly invertible linear derivative on the left; all other terms contain an already exponentially small residual/correction derivative. Induction therefore gives $C_j$ exp(−c min S) bounds for the corrections on every fixed core, with all length, angle and finite-base derivatives through order j. The local elliptic higher estimates justify each required extra spatial derivative; no claim of norm differentiability of bare translation on $H^k$ is used.

Use ONE profile on every node. For a complex fixture write $q=exp(-R+i\theta )$, $R=6S$, and introduce $w=t$ exp(iθ), with $R=exp(1\mathrm{/}{t}^2)$ for $t>0$ and $q=0$ at $t=0$. For a boundary fixture use the allowed nonnegative real t and its fixed real sign. The map w↦q is flat at zero. Every derivative of S or θ in w is bounded by a fixed power of S and 1/t for each fixed derivative order, so the exponential bounds of 17.1 imply that the solution's core corrections are C∞ and flat in w at zero. Finite auxiliary intersections and obstruction-coordinate changes are smooth functions of these core jets and hence have the same property. If only some nodes vanish, make the same argument in those variables, uniformly in compact sets of the others; thus mixed partial derivatives extend at every corner.

On an actual chart overlap retain the smaller chart's stabilizing auxiliary labels; all additional labels lie in fixed disjoint smooth core neighborhoods away from nodes. Shrink the overlap so forgetting those additional labels contracts no component: every component of the smaller chart was already stabilized by its retained labels, and no new collision of the added labels is allowed. Thus transitions compare the SAME local node divisors, not a product of node parameters produced by contracting an unstable ghost. This is why tuple overlaps, rather than arbitrary global forgetful maps, are used here. Check transitions, including the reason for choosing the double-exponential profile. Two actual fixture coordinates satisfy q'=a(q,y)q with a(0,y) nonzero; y changes smoothly, and recovered-map auxiliary coordinates change by the just-proved smooth core functions. Hence R'=R−log|a| and θ'=θ+arg a. Since $R=exp(1\mathrm{/}{t}^2)$, $t^{\mathrm{\prime }}=[log(R-log\mathrm{\mid }a\mathrm{\mid }){]}^{-1\mathrm{/}2}=t+flat(t),$ where all derivatives of the difference vanish at $t=0$; expand log(1−log|a|/R), whose derivatives are exponentially small in 1/t². Also arg $a=arg$ a(0,y)+flat(w). Consequently w'=exp(i arg a(0,y))w+flat(w), an actual C∞ change with invertible real derivative. At a boundary node this preserves the allowed half-axis. A merely single profile $R=1\mathrm{/}t$ would give |w|w terms and would not prove this complex-node smoothness; that profile is deliberately not used. On finite positive parameters these are the actual source-family transition functions, so equality there and continuity give their exact cocycle at zero. Thus the stratum-smooth atlas has a compatible genuine C∞ manifold-with-corners structure in this one common profile. Forgetting obstruction parameters and recovered auxiliary labels is likewise C∞, since it is a smooth core map together with these fixture transitions. The obstruction section is a finite coordinate projection, hence smooth. No differentiability in the original naive q-coordinate is claimed.

Finite-group naturality is retained. Add the full finite symmetry orbit of every auxiliary slice and supported forcing, and use a symmetry-invariant metric obtained by finite averaging. The actual tuple group acts by reindexing these choices. The common profile is radial and therefore respects its rotations/permutations of node coordinates. Finite coordinate changes are the actual forgetful/reindexing maps on tuple solutions; they obey the group and cocycle laws exactly, rather than up to a choice of a nonlinear gluing map. The core-coordinate smoothness argument above makes their action smooth. This gives the finite-group analytic atlas interface; orientation preservation still depends on the specified action on relative-spin data and is proved separately.