Proof. Consider \(\mathcal L=Rdd^*\) on sections of \(\operatorname{im}R\). If \(\xi\ne0\), the projection of \(\xi\wedge\iota_{\xi^\sharp}\) from self-dual two-forms back to self-dual two-forms is \(|\xi|^2/2\) times the identity. By (3), the principal symbol of \(\mathcal L\) on \(\operatorname{im}R\) is therefore the same scalar. In particular \(\mathcal L\) is elliptic. For anti-invariant smooth forms \(a,b\), \[\left\langle \mathcal La,b\right\rangle_{L^2}=\left\langle d^*a,d^*b\right\rangle_{L^2}.\] It is nonnegative and self-adjoint. Every element of its kernel is smooth and coclosed; being self-dual, it is also closed.

Let \(H\) be the orthogonal projection onto this kernel and let \(G\) be the inverse of \(\mathcal L\) on its orthogonal complement, extended by zero on the kernel. The elliptic generalized-inverse theorem gives \(G\) of order \(-2\) and \(H\) a smoothing finite-rank operator; see (Melrose 2007, Proposition 6.4 and Theorem 6.3). Set \[B=dd^*G+H.\] Then \(dB=0\), while \(RB=\mathcal LG+H=\mathrm{id}\). ◻

Proof. In the Fréchet space of smooth invariant two-forms, the cone of positive definite forms is open and convex. It is disjoint from the linear subspace of closed invariant forms by hypothesis. Hahn–Banach separation gives a nonzero continuous functional nonnegative on the cone and zero on that subspace. Extend it to all two-forms by composing with \(1-R\), and call it \(P\). This is the positive-current separation framework of Sullivan and Harvey–Lawson (Sullivan 1976; Harvey and Lawson 1983); for its almost-complex formulation, see (Li and Zhang 2009, sec. 3).

Positivity extends to semidefinite invariant forms by adding \(\varepsilon F\) and taking \(\varepsilon\downarrow0\). Since \(C\|\alpha\|_\infty F\pm(1-R)\alpha\) are semidefinite for a fixed norm-comparison constant, \[|P(\alpha)|\le C\|\alpha\|_\infty P(F).\] Thus \(P\) has order zero, and \(P(F)>0\) because \(P\ne0\). Scale \(P\) so that \(P(F)=1\).

For completeness, the measure representation can be obtained in local unitary frames. Invariant two-forms identify with Hermitian symmetric forms by \(\alpha\mapsto\alpha(\,\cdot,J\,\cdot)\). The positive functional has a positive Hermitian matrix of coefficient measures. Its trace is a positive measure \(\mu\); the coefficient measures are absolutely continuous with respect to \(\mu\). Their Radon–Nikodym density is a positive semidefinite Hermitian matrix of trace one, or equivalently a convex combination of bivectors in \(\mathscr C_x\). Measurable spectral decomposition in measurable unitary frames gives \(\lambda\) in (4). The identity \(F(\ell)=1\) shows that its projection is \(\mu\) and that its total mass is one.

Define \[T(\alpha)=P(\alpha-BR\alpha),\qquad Q(\alpha)=-P(BR\alpha).\] For closed \(\alpha\), the form \(\alpha-BR\alpha\) is closed and invariant, proving (6). Moreover \(RP=0\) and \(RQ=Q\), so \(*Q=Q\) by (3). In particular \(T\) is a closed current.

A finite measure on a compact four-manifold defines an element of \(H^{-3}\), by the Sobolev embedding \(H^3\subset C^0\). The order-zero operator \(RB^*\) is bounded on this Sobolev space by the pseudodifferential Sobolev mapping theorem (Taylor 1991, Proposition 0.5.E), so the same conclusion holds for \(Q=-RB^*P\) and \(T\). ◻

Proof. The first-order property of \(Rdd_J^c\) gives \[|\partial^j(Rdd_J^cu_s)|\le C_j(s+t)^{-1-j},\qquad |\partial^j(Rdd_J^ch_s)|\le C_j(s+t)^{-j}\] near \(x\). Away from \(x\), all fixed-order derivatives are uniformly bounded, including terms from the cutoff.

At \(y\), put \(a=s+t\), assuming \(a\) small. Localize the input to a ball about \(y\) of radius a small fixed multiple of \(a\). On this ball \(s+d_g(x,\cdot)\) is comparable to \(a\), so (8) gives contributions \(C/a\) and \(C\). For the rest of the input within distance \(O(a)\) of \(x\), the kernel is bounded by \(Ca^{-4}\). The respective input \(L^1\) bounds on this region are \(Ca^3\) and \(Ca^4\), again giving \(C/a\) and \(C\).

On the more distant shells about \(x\), with radius \(\rho\ge Ca\), the kernel is \(O(\rho^{-4})\), the input bounds are \(C/\rho\) and \(C\), and the shell volume is \(O(\rho^4)\). Summation over dyadic shells gives \(C/a\) and \(C(1+|\log a|)\). The logarithmic input has uniformly bounded total \(L^1\) norm, since its local integral is at most \(C\int_0^{\rho_0}\rho^3/(s+\rho)\,d\rho\le C\rho_0^3\). The square-root input has bounded \(L^1\) norm as well. Thus the remaining smooth kernel terms are bounded uniformly. The same decomposition gives uniform bounds when \(y\) stays a fixed distance from the center. Compactness makes all chart and operator constants uniform in \(x\). ◻

Proof of Proposition 6. In the normal coordinates of Lemma 7, let \(J_0=J(x)\) be the constant complex structure, orthogonal for the Euclidean metric at the center. On the inner ball, the Hessians of the two radial functions are \[\frac{I}{s^2+t^2}-\frac{2zz^\top}{(s^2+t^2)^2}, \qquad \frac{I}{\sqrt{s^2+t^2}}-\frac{zz^\top}{(s^2+t^2)^{3/2}}.\] Since \((z\cdot v)^2+(z\cdot J_0v)^2\le t^2|v|^2\), their complex-line traces satisfy \[\begin{align*} dd^c_{J_0}u_s(v,J_0v) &\ge\frac{2s^2|v|^2}{(s^2+t^2)^2},\tag{10}\\ dd^c_{J_0}h_s(v,J_0v) &\ge\frac{|v|^2}{\sqrt{s^2+t^2}}. \tag{11}\end{align*}\] The first lower bound is nonnegative everywhere and at least \(c/s^2\) for \(t<s\) and \(g\)-unit \(v\).

Replacing \(J_0\) by \(J(y)\) both in the differential expression and in the second argument costs at most \(C/(s+t)\) for \(u_s\) and \(C\) for \(h_s\). To see this, use \(J(y)-J_0=O(t)\) on the Hessian terms and the bounded first derivatives of \(J\) on the gradient terms. Euclidean and \(g\)-norms are uniformly comparable.

Together with (9), this proves, for \(s+t\) sufficiently small, \[Du_s(v,Jv)\ge \frac{c}{s^2}\mathbf 1_{\{t<s\}}-\frac{C}{s+t}, \qquad Dh_s(v,Jv)\ge \frac{c'}{s+t}\] with \(c,c'>0\). In the second estimate the logarithmic correction is absorbed by the positive inverse-distance term on a sufficiently small fixed neighborhood. Choose a fixed \(A>0\) large enough to absorb the negative term in the first estimate. Outside that neighborhood all terms are uniformly bounded. We obtain globally \[D(u_s+Ah_s)(v,Jv)\ge \frac{c}{s^2}\mathbf 1_{B_s(x)}-C'\] for all unit \(v\) and small \(s\).

The form on the left is closed and invariant. Applying \(P\), which annihilates it, and using \(\mu(X)=1\), gives \(0\ge cs^{-2}\mu(B_s(x))-C'\). This proves (7) for small \(s\). Increasing the constant handles all larger radii. ◻

Proof. The operator \(S_r\) is self-adjoint, commutes with star in middle degree, and commutes across degrees with \(d\) and \(d^*\). It preserves smooth forms. Thus \(S_rT\) annihilates every smooth closed form by (6). Closedness of \(T'\) as a current means \(d^*T'=0\), so \(*S_rT'\) is a distributionally closed \(L^2\) form. Its heat regularizations are smooth closed forms and converge to it in \(L^2\). The asserted pairing follows by continuity. ◻

Proof. The current \(T\) has measure coefficients with variation bounded by a constant times \(\sigma=\mu+|Q|\,dV_g\). Cauchy–Schwarz and (26) give \[\sigma(B_s(p))\le Cs^2,\] uniformly in \(p\).

In normal coordinates \(z\) at \(p\), let \(J_0=J(p)\) and \(F_0=F_p\) be constant, and put \(t=|z|=d_g(p,y)\). Our sign convention gives \(dd^c_{J_0}(t^2/2)=2F_0\). Call a radius good if its sphere is \(\sigma\)-null. For good radii define \[b(r)=r^{-2}T(\mathbf 1_{B_r(p)}F_0).\] These pairings are well-defined using the coefficient measures. We claim that \[ b(s)-b(r)=\tfrac12 T\bigl(\mathbf 1_{\{r<t<s\}}dd^c_{J_0}\log t\bigr), \qquad 0<r<s. \tag{28}\]

To verify it, define \[\phi_a(t)= \begin{cases} \log a+(t^2-a^2)/(2a^2),&t<a,\\ \log t,&t\ge a. \end{cases}\] The function \(\phi_r-\phi_s\) is compactly supported and \(C^{1,1}\). Its \(dd^c_{J_0}\) is \(2(r^{-2}-s^{-2})F_0\) on \(B_r\), equals \(dd^c_{J_0}\log t-2s^{-2}F_0\) on the annulus, and is zero outside \(B_s\). For fixed \(r,s\), compactly supported smooth approximations have uniformly bounded Hessians and converge in second derivatives off the two spheres. Dominated convergence against \(\sigma\) permits testing closure on their exact two-forms and passing to the limit. The resulting identity is (28).

The form \(dd^c_{J_0}\log t\) is \(J_0\)-invariant and semipositive, with size \(O(t^{-2})\). Since \(J(y)-J_0=O(t)\), it evaluates on actual complex line bivectors to at least \(-C/t\), and its \(J\)-anti-invariant projection has size \(O(1/t)\). The latter estimate controls its pairing with \(Q\). Inward dyadic shells and the quadratic bound on \(\sigma\) give \[\int_{\{0<t<s\}}t^{-1}\,d\sigma\le Cs.\] It follows that \[ b(s)-b(r)\ge-Cs. \tag{29}\] Also \(T(\mathbf 1_{B_r}F)=\mu(B_r)\), since \(\left\langle Q,F\right\rangle_g=0\) almost everywhere. The bound \(F-F_0=O(t)\) therefore implies \[ |b(r)-r^{-2}\mu(B_r(p))|\le Cr. \tag{30}\]

In particular \(b\) is bounded. Fix a good radius \(s\) in (29) and let \(r\downarrow0\) through good radii; then let \(s\downarrow0\) along a sequence realizing the lower limit of \(b\). The upper and lower limits agree. Equation (30) gives the density limit along good radii.

Only countably many spheres at a fixed center can have positive \(\sigma\)-mass. Any radius \(r\) can therefore be bracketed by good radii in \(((1-\varepsilon)r,r)\) and \((r,(1+\varepsilon)r)\). Ball inclusion, followed by \(r\downarrow0\) and then \(\varepsilon\downarrow0\), proves the asserted limit along all radii. Its finiteness follows from (26). ◻

Proof. Write \(P=A\mu\) in a smooth local frame, so that \(A(p)\) is the conditional average of the line bivectors over \(p\). At \(\mu\)-almost every point, differentiation with respect to the Radon measure \(\mu\) gives \[ r^{-2}\int_{B_{Dr}(p)}|A(y)-A(p)|\,d\mu(y)\longrightarrow0 \qquad(D<\infty\text{ fixed}). \tag{32}\] For the precise differentiation input one may use (Kinnunen 2026, Theorem 4.33 and Corollary 4.34). To see the normalization, enclose \(B_{Dr}(p)\) in a Euclidean coordinate ball of radius \(CDr\). The mean oscillation on that ball tends to zero, while its mass divided by \(r^2\) is bounded by (26). This does not require \(\mu\) to be doubling. Smooth changes of frame add a vanishing \(O(r)\) error after this normalization.

Fix such a differentiation point \(p\) with \(\theta(p)>0\). The locally rescaled measures \[\nu_r=r^{-2}(z_p/r)_*\mu\] have uniformly bounded mass on each compact subset of \(T_pX\), so every sequence of scales has a vaguely convergent subsequence. Every limit \(\nu\) satisfies \[ \nu(B_D(0))=\theta(p)D^2\qquad(D>0). \tag{33}\] Initially the identity follows at continuity radii of \(\nu\). Increasing continuity radii recover the mass of each open ball, while decreasing continuity radii recover the corresponding closed ball. Both masses are \(\theta(p)D^2\), so the identity holds for every \(D>0\) and every centered sphere is \(\nu\)-null.

The current and measure rescalings have the same normalization. Indeed, put \(D_r(y)=z_p(y)/r\). For a compactly supported test two-form \(\beta=\sum_{i<j}\beta_{ij}(w)\,dw_i\wedge dw_j\) on \(T_pX\), \[D_r^*\beta=r^{-2}\sum_{i<j}\beta_{ij}(z_p/r)\,dz_i\wedge dz_j.\] Thus the pushforward \((D_r)_*T\), defined by testing \(T\) against \(D_r^*\beta\), already has the \(r^{-2}\) factor in its coefficient measures; no further normalization is used. Equation (32) makes its \(P\) part converge to \(A(p)\nu\). Its \(Q\) part vanishes in variation on each fixed ball, since \[ r^{-2}\int_{B_{Dr}(p)}|Q|\,dV_g \le C_D\|Q\|_{L^2(B_{Dr}(p))}\longrightarrow0. \tag{34}\] The last limit holds at every point by absolute continuity of the integral of \(|Q|^2\). Compactly supported tests in the rescaled coordinates pull back inside the original coordinate chart for small \(r\). Thus closure passes to the limit, and \(A(p)\nu\) is a closed constant-polarization current on \(T_pX\).

For a constant skew matrix \(A\), this closure says \(\sum_i A^{ij}\partial_i\nu=0\) for each \(j\). Consequently \(\partial_v\nu=0\) for every \(v\in\operatorname{im}A(p)\). For a compactly supported smooth function \(\phi\), the derivative of \(t\mapsto\int\phi(w+tv)\,d\nu(w)\) is therefore zero. This proves that \(\nu\) is invariant under translations along \(\operatorname{im}A(p)\). The normalization \(F_p(A(p))=1\) excludes rank zero. Rank four would imply that \(\nu\) is a constant multiple of four-dimensional Lebesgue measure, contrary to (33). Therefore \(A(p)\) has real rank two.

The associated positive Hermitian matrix has trace one and complex rank one. Thus \(A(p)\) is itself a unit complex line bivector. If \(\lambda_p\) is the conditional probability measure on \(\mathscr C_p\), then \[\int_{\mathscr C_p}|\ell-A(p)|^2\,d\lambda_p(\ell) =1-|A(p)|^2=0.\] Hence \(\lambda_p\) is concentrated on \(A(p)\). Denote its plane by \(L\).

Translation invariance along \(L\) implies \[\nu=\mathcal L^2_L\otimes\gamma\] for a positive Radon measure \(\gamma\) on \(L^\perp\). For example, testing against a nonnegative compactly supported function in the normal variables gives a translation-invariant measure in the tangential variables, hence a multiple of area; these multiples define \(\gamma\). Dividing the centered ball mass by \(D^2\) in (33) yields \[ \theta(p)=\pi\int_{L^\perp}(1-|w|^2/D^2)_+\,d\gamma(w). \tag{35}\] For \(D_2>D_1\), the difference of these integrands is nonnegative and strictly positive at every \(0<|w|<D_2\). Since the integrals are equal, \(\gamma\) is supported at the origin. Every tangent limit is therefore \((\theta(p)/\pi)\mathcal L^2_L\).

It follows that the normalized inner integral in (31) tends to zero at \(\lambda\)-almost every \((p,\ell)\) of positive density. Indeed, in the rescaled coordinates a continuous cutoff equal to one on \(B_1\), supported in \(B_2\), times \(|\operatorname{pr}_{L^\perp}z|^2\) dominates the integrand and vanishes on every tangent limit. Subsequential compactness gives convergence along all scales.

When \(\theta(p)=0\), the same conclusion follows directly by bounding the normalized inner integral by \(r^{-2}\mu(B_r(p))\). These normalized integrals are uniformly bounded by (26). Dominated convergence against the finite measure \(\lambda\) proves (31). ◻

Proof of Theorem 13. The existence of the density is Lemma 14. We approximate the positive-density restriction by smooth functions of a regularized density. Closedness of \(T\) then reduces the problem to controlling full derivatives against \(Q\) and tangential derivatives against \(P\). The tangential estimate combines the quantitative angular bound with the vanishing transverse moment from Lemma 15; their product will make the cutoff boundary tend to zero. Choose a nonnegative, smooth, nonincreasing function \(h:[0,\infty)\to[0,\infty)\), positive near zero and zero on a neighborhood of \([1,\infty)\). For \(r\) below the injectivity radius define \[ f_r(x)=r^{-2}\int_Xh(d_g(x,y)^2/r^2)\,d\mu(y). \tag{36}\] The kernel is smooth near the diagonal and vanishes before the cut locus, so \(f_r\) is smooth. It is uniformly bounded. Writing \(H(t)=h(t^2)\), integration of ball indicators gives \[f_r(x)=\int_0^1[-H'(t)]r^{-2}\mu(B_{rt}(x))\,dt \longrightarrow c_h\theta(x), \qquad c_h=\int_0^1h(u)\,du>0.\] The quadratic growth bound supplies domination for this limit. In particular \(\theta\) is Borel measurable. Differentiation of the finite measure \(\mu\) with respect to four-dimensional volume also shows that \[ \theta(x)=0\quad\text{for volume-almost every }x. \tag{37}\] Indeed, apply the differentiation theorem for Radon measures (Kinnunen 2026, Theorem 4.19) in a fixed coordinate chart, with smooth volume as the denominator measure. At volume-almost every center the measure-to-volume ratios of sufficiently small coordinate balls are bounded. Enclosing a geodesic ball of radius \(r\) in a coordinate ball of comparable radius therefore gives \(\mu(B_r(x))=O_x(r^4)\), and hence \(r^{-2}\mu(B_r(x))\to0\).

Proof of Theorem 1. Let \(\omega\) be a smooth closed form taming \(J\). Suppose for contradiction that there is no compatible symplectic form, and take \(P,Q,T,\lambda,\mu\) from Proposition 5. Propositions 6 and 12 verify all the hypotheses of Theorem 13. Consequently the current \(P_E=\mathbf 1_{\{\theta>0\}}P\) is closed. Set \[P'=\mathbf 1_{X\setminus E}P,\qquad T'=P'+Q=T-P_E.\] Then \(T'\) is closed. The current \(P'\) is represented by the restriction \(\lambda'\le\lambda\) to base points outside \(E\), with projection \(\mu'=\mathbf 1_{X\setminus E}\mu\). We pair \(T\) with this residual current rather than with itself: placing the second positive factor on zero-density centers makes the previously bounded negative error in the pairing tend to zero.

We first show that the negative term in (20) tends to zero for this choice of \(P'\). For each \(y\notin E\), \[ r^{-2}\int_X(1+d_g(x,y)/r)^{-6}\,d\mu(x) \longrightarrow0. \tag{46}\] On the inner ball and on every fixed dyadic shell this follows from \(s^{-2}\mu(B_s(y))\to0\). The quadratic mass bound dominates the shell contributions by \(C2^{-4j}\), a summable sequence, giving the full limit. The uniform bound (21) then permits dominated convergence against \(\mu'\). Discarding the nonnegative angular term in (20) yields \[ \liminf_{r\downarrow0}\left\langle S_rP,*S_rP'\right\rangle_{L^2}\ge0. \tag{47}\]

Apply Lemma 8 to \(T'\). Its regularizations are in \(L^2\), since \(P'\) has measure coefficients and \(Q\in L^2\). Expanding gives \[0=\left\langle S_rP,*S_rP'\right\rangle +\left\langle S_rP,*S_rQ\right\rangle +\left\langle S_rQ,*S_rP'\right\rangle +\|S_rQ\|_2^2.\] Both mixed terms tend to zero by (24), applied once to \(P\) and once to \(P'\). Since \(S_rQ\to Q\) strongly in \(L^2\), (47) implies \(0\ge\|Q\|_2^2\). Therefore \(Q=0\).

It follows that \(P=T\) annihilates all smooth closed forms. In particular \(P(\omega)=0\). On the other hand, \(\omega_x(\ell)>0\) on the compact bundle \(\mathscr C\); it has a positive minimum there. Equation (4) and \(\lambda(\mathscr C)=1\) give \(P(\omega)>0\), a contradiction.

The positive cone and the subspace of closed invariant smooth forms therefore intersect. Any form in their intersection is the required smooth compatible symplectic form for \(J\). ◻