Proof. For (i), apply separation of an open convex set and a disjoint linear subspace in the real locally convex space \(C^\infty(X,I)\). The separating continuous functional vanishes on \(\mathcal Z_J\) and is nonnegative on its positive cone. Extend it to all two-forms by precomposing with \(1-R\). Adding a positive multiple of \(G\) and taking a limit proves nonnegativity on every semipositive invariant form.

For (ii), choose a smooth closed representative \(\beta\) of \(H\) and separate the open cone \(\mathcal P_J\) from the affine subspace \(\beta+\mathrm d\Omega^1(X)\) in \(\Omega^2(X)\). If \(N\) is the separating functional, the unbounded translation directions in this affine subspace give \(N(\mathrm d\alpha)=0\). Every positive form can be multiplied by any positive scalar, so the separation inequalities imply both \(N(\mathcal P_J)\ge0\) and \(N(\beta)\le0\). Adding an arbitrary real multiple of an anti-invariant form to an element of \(\mathcal P_J\) shows that \(N\) annihilates anti-invariant forms. The same limiting argument gives positivity on semipositive invariant forms.

For either functional, comparison of an invariant form with sufficiently large positive multiples of \(G\) gives \[|N(\xi)|\le C N(G)\|\xi\|_{C^0}.\] If \(N(G)\) were zero, this inequality and anti-invariance would make the functional zero. Normalize \(N(G)=1\). Locally, the functional is therefore represented by a positive Hermitian matrix of measures. Its trace is a positive measure \(\mu\); the Radon–Nikodym matrix relative to \(\mu\) is positive semidefinite with trace one. A measurable spectral decomposition of this \(2\times2\) matrix expresses it as a convex combination of rank-one Hermitian projections. Borel local frames and a Borel partition of \(X\) give the global measure \(\lambda\) in [eq:current-line-measure]. Finally \(H^3\hookrightarrow C^0\) in real dimension four, so every such measure coefficient belongs to \(H^{-3}\). ◻

Proof. On sections of \(E\) consider \(P=R\mathrm d\mathrm d^*\). If \(a\) is self-dual, then the self-dual projection of \(\xi\wedge\iota_\xi a\) is \(|\xi|^2a/2\): in a frame with first covector \(\xi/|\xi|\), the two self-dual halves of \(a\) have equal length. Hence \[\sigma_2(P)(x,\xi)=\tfrac12|\xi|_g^2\mathop{\mathrm{Id}}_E, \qquad \langle Pa,a\rangle_2=\|\mathrm d^*a\|_2^2.\] Thus \(P\) is elliptic, self-adjoint, and nonnegative. Its kernel consists of smooth sections with \(\mathrm d^*a=0\). Since \(a=*a\), these sections also satisfy \(\mathrm da=0\), and hence are harmonic. Conversely a closed anti-invariant form is self-dual and therefore harmonic and in the kernel.

Let \(H_0\) be the orthogonal projection onto this finite-dimensional kernel, and let \(L\) be the inverse of \(P\) on its orthogonal complement, extended by zero on the kernel. Elliptic Fredholm theory and the parametrix give \(L\) as a classical operator of order \(-2\), with \(H_0\) smoothing and \(PL=\mathop{\mathrm{Id}}-H_0\); see (Taylor, n.d.-c, secs. 4–5 and 10). Indeed, if \(B_0P=\mathop{\mathrm{Id}}+K\) is a parametrix identity with \(K\) smoothing, then \(B_0(\mathop{\mathrm{Id}}-H_0)=L+KL\). Elliptic regularity makes \(KL\) smoothing, which gives the asserted order of \(L\). Set \[ D=\mathrm d\mathrm d^*L+H_0. \tag{12}\] The image of both summands is closed, and \(RD=PL+H_0=\mathop{\mathrm{Id}}\). This proves [eq:current-lift-identities].

For closed \(\xi\), the form \(\xi-DR\xi\) is closed and invariant, so \(T(\xi)=N(\xi-DR\xi)=0\). The functional \(Q\) only depends on \(R\xi\), which proves its anti-invariance and self-duality. Its \(H^{-3}\) membership follows because an order-zero operator and its adjoint act boundedly on every Sobolev space (Taylor, n.d.-c, Proposition 5.5). The vanishing on exact forms makes \(T\) closed. Its pairings with all closed two-forms vanish, so Poincare duality, or the harmonic projection of \(*T\), gives \([*T]=0\).

For the family assertion, identify the varying bundles locally in the parameter and use a smooth unitary identification of the \(L^2\) spaces. The elliptic operators then have common domain \(H^2\) in a fixed Hilbert space. Choose a spectral contour enclosing only zero at one parameter value. The resolvent identity and elliptic regularity give a smooth family of resolvents on this contour after shrinking the parameter neighborhood. Its spectral projection has locally constant rank. Since the kernel dimension is constant with that rank, the enclosed spectral subspace consists exactly of the kernel. Thus \(H_0\) depends smoothly on the parameter, and so does \[L=(P+H_0)^{-1}(\mathop{\mathrm{Id}}-H_0)\] on every Sobolev scale. [eq:current-lift-definition] gives the required smooth lifts. ◻

Proof. We use two precise facts of the order-zero calculus: in local left quantization its symbol is bounded uniformly in the covariable, and its off-diagonal kernel satisfies \(|K_A(x,y)|\le C\mathop{\mathrm{dist}}(x,y)^{-4}\) for small positive distance. Smooth remainders have uniformly bounded kernels. These statements hold for bundle-valued classical operators, after finitely many coordinate localizations; see (Taylor, n.d.-c, sec. 2, Propositions 2.1–2.2).

For a fixed evaluation point \(x\), split the input using a smooth cutoff supported in \(B_{c\delta}(x)\) and equal to one on \(B_{c\delta/2}(x)\), with a fixed small \(c\). On its support \(s+\mathop{\mathrm{dist}}(p,y)\) is comparable to \(\delta\). After translation and dilation by \(\delta\), the localized input has derivatives through order six bounded by \(C\delta^{-a}\). Integration by parts in its Fourier transform therefore bounds its Fourier \(L^1\) norm by \(C\delta^{-a}\), since \((1+|\xi|)^{-6}\) is integrable in dimension four. The bounded symbol gives the same bound for the local contribution to \(Aa_{s,p}(x)\).

The remaining part has distance at least \(c\delta/2\) from \(x\). Its portion in \(B_{C\delta}(p)\) contributes at most \[C\delta^{-4}\int_{B_{C\delta}(p)} (s+\mathop{\mathrm{dist}}(p,y))^{-a}\,\mathrm dV_g(y) \le C\delta^{-a}.\] On a shell of radius \(r=2^j\delta\ge C\delta\) about \(x\), distance from \(p\) is comparable to \(r\), so its contribution is at most \(Cr^{-4}r^4r^{-a}=Cr^{-a}\). The shells up to the fixed coordinate radius sum to \(C\delta^{-1}\) if \(a=1\), and to \(C(1+|\log\delta|)\) if \(a=0\). The rest is uniformly bounded. If \(x\) stays away from \(p\), the singular input near \(p\) has uniformly bounded \(L^1\) norm and the kernel there is bounded; the remaining input is uniformly smooth. This proves the final assertion. ◻

Proof. Fix first a radius \(\rho_0>0\) for uniform normal coordinates and a smooth radial cutoff which is one on \(B_{2\rho_0}(p)\) and supported in \(B_{3\rho_0}(p)\). This cutoff will not change when a smaller inner radius is chosen below. The logarithmic potential will detect the mass of \(B_s(p)\), while a square-root potential will dominate the inverse-distance errors caused by varying \(J\) and by the closed lift. With \(t=\mathop{\mathrm{dist}}(p,\cdot)\) define the globally smooth cut-off functions \[u_s=\chi(t)\log\sqrt{s^2+t^2},\qquad h_s=\chi(t)\sqrt{s^2+t^2},\qquad \mathrm d_J^cu=-\mathrm du\circ J.\] Smoothness at \(p\) follows from the dependence on \(t^2\) and from \(s>0\). For a function \(f\), put \[\beta(f)= \begin{cases} (1-DR)\mathrm d\mathrm d_J^cf,&\text{in setup \textup{(A)}},\\ \mathrm d\mathrm d_J^cf,&\text{in setup \textup{(B)}}. \end{cases}\] In setup (A), \(\beta(f)\) is closed and invariant, and in setup (B) it is exact. Hence \(N(\beta(f))=0\) in both cases.

The principal symbol of \(\mathrm d\mathrm d_J^c\) on functions is a multiple of \(\xi\wedge J^*\xi\), which is \(J\)-invariant. Thus \(R\mathrm d\mathrm d_J^c\) is a first-order operator. Differentiating the radial functions gives, for \(0\le j\le6\), \[|\nabla^jR\mathrm d\mathrm d_J^cu_s|\le C_j(s+t)^{-1-j},\qquad |\nabla^jR\mathrm d\mathrm d_J^ch_s|\le C_j(s+t)^{-j}.\] On the cutoff annulus these inputs and their derivatives are uniformly bounded. 11 applied to \(D\) therefore bounds the two correction terms by \(C/(s+t)\) and \(C(1+|\log(s+t)|)\) respectively.

For clarity, the positive part of the test can be computed in a fixed Hermitian Euclidean space. If \(f=f(t)\) and \(\ell\) is a unit complex line, its Hessian trace is \[\mathrm d\mathrm d_{J_p}^cf(\ell) =2f'(t)/t+\bigl(f''(t)-f'(t)/t\bigr)a, \qquad 0\le a\le1,\] where \(a\) is the squared length of the radial unit vector projected onto the line. For the two functions without cutoff this gives \[\begin{align*} \mathrm d\mathrm d_{J_p}^c\log\sqrt{s^2+t^2}(\ell) &\ge \frac{2s^2}{(s^2+t^2)^2},\tag{15}\\ \mathrm d\mathrm d_{J_p}^c\sqrt{s^2+t^2}(\ell) &\ge \frac1{\sqrt{s^2+t^2}}. \tag{16}\end{align*}\] In normal coordinates \(g-g_p=O(t^2)\) and \(J-J_p=O(t)\), while the first derivatives of \(J\) are bounded. Replacing the frozen complex line and operator by the actual ones changes these estimates by at most \(C/(s+t)\) for \(u_s\) and \(C\) for \(h_s\). Indeed the Hessian errors are bounded by \(Ct|\nabla^2f|\) and the coefficient-derivative errors by \(C|\nabla f|\). With \(\delta=s+t\), it follows that on \(B_{\rho_0}(p)\) \[\begin{align*} \beta(u_s)(\ell)&\ge c_1s^2\delta^{-4}-C_1\delta^{-1}, \tag{17}\\ \beta(h_s)(\ell)&\ge c_2\delta^{-1} -C_2(1+|\log\delta|). \tag{18}\end{align*}\] The inequalities include setup (B), where the correction terms are absent.

All constants just obtained use the already fixed cutoff. Now choose \(0<\rho<\rho_0\) so small that \[C_2(1+|\log\delta|)\le c_2/(2\delta) \qquad(0<\delta\le2\rho).\] Next choose a fixed multiplier \(M\ge2C_1/c_2\). If \(s\le\rho\) and \(t\le\rho\), [eq:current-log-corrected,eq:current-root-corrected] give \(\beta(u_s+Mh_s)(\ell)\ge c_1s^2\delta^{-4}\). On \(X\setminus B_\rho(p)\) all terms are bounded independently of \(s\), including the nonlocal corrections, by the last assertion of 11. Since \(\delta\le2s\) on \(B_s(p)\), we conclude globally that \[\beta(u_s+Mh_s)(\ell)\ge c s^{-2}\mathbf1_{B_s(p)}-C.\] Apply \(N\) and use \(N(\beta(u_s+Mh_s))=0\) and \(\mu(X)=1\) to obtain [eq:current-quadratic-mass] for \(s\le\rho\). For larger \(s\) the same assertion follows from \(\mu(X)=1\). Compactness makes all choices uniform in \(p\). ◻

Proof. The spectral theorem gives the operator identity \[ S_s^2=\frac1{5!}\int_0^\infty h^5e^{-h}e^{-s^2h\Delta}\,\mathrm dh. \tag{24}\] We recall the precise heat-kernel facts being used for the nonnegative Hodge Laplacian on the closed manifold. Its local small-time kernel has a parametrix \[\chi(d)(4\pi t)^{-2}e^{-d^2/(4t)} \bigl(a_0(y,z)+ta_1(y,z)+\cdots+t^ma_m(y,z)\bigr), \qquad d=\mathop{\mathrm{dist}}(y,z),\] with arbitrarily high-order uniform remainder after increasing \(m\); see (Ludewig 2019, Theorems 1.1 and 3.1). Here \(\chi\) is supported inside the injectivity radius and equals one near the diagonal. For the Hodge Laplacian \(a_0(y,z)=j(y,z)^{-1/2}\tau_{z\to y}\), with \(j\) smooth, strictly positive, and equal to one on the diagonal (Ludewig 2018, sec. 3, equation (3.4), author version). After subtracting the leading term, for every prescribed \(L\) the kernel is bounded, for \(0<t<t_0\), by \[Ct^{-1}e^{-d^2/(Ct)}+C_Lt^L.\] The local expansion gives the Gaussian part of this bound; away from the diagonal, the global Gaussian estimate makes the cutoff error smaller than every power of \(t\) (Ludewig 2019, Theorem 3.5). For \(t\ge t_0\), the heat kernel is uniformly bounded: factor the heat operator through two fixed positive-time smoothing operators and the \(L^2\) contraction \(e^{-(t-t_0)\Delta}\). This last assertion uses the nonnegativity and self-adjointness of the Hodge Laplacian.

Take \(\chi\ge0\) and use its leading coefficient to define \[k_s(y,z)=\frac{\chi(d)j(y,z)^{-1/2}}{5!(4\pi)^2}s^{-4} \int_0^\infty h^3e^{-h-d^2/(4s^2h)}\,\mathrm dh.\] For \(d<s\), integration over \(1\le h\le2\) gives [eq:current-kernel-lower]. For every fixed \(k\), the elementary inequality \(e^{-r^2/(ch)}\le C_k(1+r)^{-k}(1+h^{k/2})\) shows that the last integral is bounded by \(C_k(1+d/s)^{-k}\). This proves [eq:current-kernel-upper].

Integrating the \(t^{-1}\) heat-kernel error in [eq:current-gamma-resolvent] gives \[Cs^{-2}\int_0^\infty h^4e^{-h-d^2/(Cs^2h)}\,\mathrm dh \le C_ks^{-2}(1+d/s)^{-k}.\] For the uniform remainder, integration over \(s^2h<t_0\) gives \(C_Ls^{2L}\). Since \(d\le\operatorname{diam}X\), this is bounded by the right side of [eq:current-kernel-error] when \(2L\ge k-2\). On \(s^2h\ge t_0\), the factor \(e^{-h}\) makes both the heat-kernel contribution and the extended leading term smaller than every power of \(s\). This proves the claimed error for each \(k\). ◻

Proof. Parallel transport preserves the Hodge star. From [eq:current-line-normalization], direct expansion gives \[ \langle\ell,*\tau_{z\to y}m\rangle =\tfrac12|\ell-\tau_{z\to y}m|^2 -\tfrac14|G_y-\tau_{z\to y}G_z|^2. \tag{29}\] For example, the right side equals \(-\langle\ell,\tau m\rangle+\tfrac12\langle G_y,\tau G_z\rangle\), which is the left side after splitting into self-dual and anti-self-dual parts. The last squared difference is bounded by \(Cd^2\), where \(d=\mathop{\mathrm{dist}}(y,z)\).

Express the pairing using the continuous kernel of \(S_s^2\) and [eq:current-line-measure]. Its leading scalar is nonnegative, so [eq:current-kernel-lower,eq:current-angular-identity] supply the positive term. The \(Cd^2\) term, using [eq:current-kernel-upper] with exponent at least eight, and the error kernel, using [eq:current-kernel-error] with exponent six, are bounded in absolute value by \[Cs^{-2}\iint(1+d/s)^{-6}\,\mathrm d\mu_1(y)\mathrm d\mu_2(z).\] This is the asserted error. The norm estimate follows by using the absolute kernel bound in the unstarred pairing: \[\|S_sN_i\|_2^2 \le Cs^{-4}\iint(1+d/s)^{-6}\,\mathrm d\mu_i(y)\mathrm d\mu_i(z) \le Cs^{-2}\mu_i(X)\le Cs^{-2}.\] The kernel formulas are legitimate for measure coefficients because the kernel is continuous at fixed \(s\); alternatively they follow by smooth approximation at that fixed scale. ◻

Proof. Set \(A_s=S_s^{-1}=(1+s^2\Delta)^3\) and \(U=S_sW\). Since \(\Pi A_sU=\Pi W=0\), the distributional identity \[\Pi U=S_s[A_s,\Pi]U\] holds. The commutator expands as a sum of \(\binom3j s^{2j}[\Delta^j,\Pi]\), \(1\le j\le3\). Because the principal symbol of \(\Delta\) is scalar, \([\Delta^j,\Pi]\) is a differential operator of order at most \(2j-1\). The spectral theorem and elliptic Sobolev equivalence give \[\|S_s\|_{L^2\to H^m}\le C_ms^{-m},\qquad 0\le m\le6.\] If \(B_j=[\Delta^j,\Pi]\), its formal adjoint has the same order, so \[\|S_s s^{2j}B_j\|_{L^2\to L^2} =\|s^{2j}B_j^*S_s\|_{L^2\to L^2} \le Cs^{2j}s^{-(2j-1)}=Cs.\] This proves [eq:current-projection-leakage], including its meaning for distributional \(W\) by the bounded extension of the displayed operator.

Apply this result to \(R N'=0\) and \((1-R)Z=0\). With \(u=S_sN'\) and \(v=S_sZ\), the orthogonal splitting gives \[|\langle u,v\rangle_2| \le\|Ru\|_2\|Rv\|_2+ \|(1-R)u\|_2\|(1-R)v\|_2 \le Cs\|u\|_2\|v\|_2.\] Now use [eq:current-submeasure-ltwo]. ◻

Proof. Let \(\mathcal A_s\) denote the double integral in [eq:current-angular-energy] and put \(x_s=\|S_sQ\|_2\). Since \(Q\) is self-dual and \(S_s\) commutes with \(*\), [eq:current-topological-pairing] applied to \(T=N+Q\) gives \[[*T]^2 =\langle S_sN,*S_sN\rangle_2 +2\langle S_sN,S_sQ\rangle_2+x_s^2.\] Use [eq:current-positive-pairing,eq:current-bracket-bound] for the first term and [eq:current-mixed-bound] for the second. The result is \[ cs^{-4}\mathcal A_s+x_s^2-Cx_s\le C+[*T]^2. \tag{35}\] The right side is independent of \(s\). In setup (A) its intersection term is zero. In setup (B) that term is bounded uniformly for mass-one \(N\): its harmonic coefficients are pairings with fixed smooth forms and are bounded by [eq:current-line-measure]. Completing the square in \(x_s\) bounds both \(x_s\) and \(s^{-4}\mathcal A_s\).

As \(s\downarrow0\), \(S_sQ\) converges to \(Q\) distributionally. A weakly convergent subsequence of the bounded \(L^2\) family therefore has limit \(Q\), proving \(Q\in L^2\). Its anti-invariance is preserved in this limit, and the spectral theorem now gives \(S_sQ\to Q\) strongly in \(L^2\).

Choose smooth anti-invariant \(Q_j\) with \(\|Q_j-Q\|_2\to0\); for example apply \(R\) to a smooth approximation. By [eq:current-mixed-bound] and the \(L^2\) contraction property of \(S_s\), \[|\langle S_sN',S_s(Q-Q_j)\rangle_2| \le C\|Q-Q_j\|_2.\] For fixed \(j\), \(S_s^2Q_j\to Q_j\) in \(C^\infty\), whereas \(N'(Q_j)=0\). Therefore \[\langle S_sN',S_sQ_j\rangle_2 =N'(S_s^2Q_j-Q_j)\longrightarrow0.\] The last convergence is uniform over the submeasures, because their trace masses are at most one. First let \(s\) tend to zero and then \(j\) to infinity to obtain [eq:current-mixed-vanishing]. ◻

Proof. Put \(\nu=\mu+|Q|\,dV_g\). Cauchy–Schwarz and the four-dimensional volume bound give \[\int_{B_s(p)}|Q|\,dV_g \le \mathop{\mathrm{vol}}_g(B_s(p))^{1/2}\|Q\|_{L^2(B_s(p))} \le Cs^2,\] so \(\nu(B_s(p))\le Cs^2\), uniformly in \(p\). Fix \(p\), identify a normal-coordinate ball with a Euclidean ball, and freeze \(J_p\) and \(G_p\) as constant tensors \(J_0,G_0\). If \(t=|\zeta|=\mathop{\mathrm{dist}}(p,\exp_p\zeta)\), define \[b_p(s)=s^{-2}T(\mathbf1_{B_s(p)}G_0).\] These evaluations are well-defined because \(T\) has measure coefficients. Since \(G(\ell)=1\), \(G-G_0=O(t)\), and \(Q\) annihilates the invariant form \(G\), we have \[ b_p(s)=s^{-2}\mu(B_s(p))+O(s). \tag{39}\] Indeed, both error integrals are bounded by \(Cs^{-2}s\nu(B_s(p))\).

Call a radius good if its sphere has zero \(\nu\)-measure. All but countably many radii are good. For good \(0<r<s\), closedness implies \[ b_p(s)-b_p(r) =\frac12 T\bigl(\mathbf1_{\{r<t<s\}}\,dd^c_{J_0}\log t\bigr), \qquad d^c_{J_0}u=-du\circ J_0. \tag{40}\] For completeness, replace \(\log t\) inside \(B_s\) by \[\phi_s(t)= \begin{cases} t^2/(2s^2)+\log s-1/2,&t<s,\\ \log t,&t\ge s. \end{cases}\] The difference \(\phi_s-\phi_r\) is compactly supported and \(C^{1,1}\). Inside \(B_s\), the quadratic part has \(dd^c_{J_0}(t^2/(2s^2))=2G_0/s^2\). Expanding \(T(dd^c_{J_0}(\phi_s-\phi_r))=0\) gives [eq:density-frozen-monotonicity]. This use of \(C^{1,1}\) tests is justified by smoothing: for fixed positive \(r,s\), their Hessians are uniformly bounded, converge away from the two spheres, and the spheres have zero \(\nu\)-measure.

The frozen form \(dd^c_{J_0}\log t\) is semipositive on \(J_0\)-complex lines and has norm at most \(Ct^{-2}\). A \(J\)-complex line at distance \(t\) is \(O(t)\) from a \(J_0\)-complex line. Its evaluation can therefore be negative only by \(C/t\). Likewise the projection of this form onto the actual anti-invariant summand has norm at most \(C/t\), because its frozen anti-invariant projection vanishes. Thus \[b_p(s)-b_p(r)\ge -C\int_{0<t<s}t^{-1}\,d\nu.\] Quadratic growth, or a sum over the annuli \(2^{-j-1}s<t\le2^{-j}s\), bounds this integral by \(Cs\). It follows that \[ b_p(r)\le b_p(s)+Cs\qquad(0<r<s,\ r,s\text{ good}). \tag{41}\] The bounded function \(b_p\) therefore has a limit along good radii. To see this directly, choose good \(s_j\downarrow0\) realizing its limit inferior and apply [eq:density-almost-monotonicity] to all smaller good \(r\); the limit superior is at most the same value. Equation (39) gives the same limit for the mass ratio. Squeezing an arbitrary radius between good radii with ratios tending to one gives [eq:density-definition].

Measurability follows by taking a fixed sequence of radii tending to zero, and boundedness follows from [eq:density-basic-bounds]. For the last assertion, work in a fixed relatively compact coordinate chart, localizing the measures before extending them to Euclidean space. Apply the differentiation theorem for Radon measures, with smooth four-dimensional volume as denominator (Kinnunen 2026, Theorem 4.19). At volume-almost every center, the measure-to-volume ratios of sufficiently small Euclidean coordinate balls are bounded. A geodesic ball of radius \(s\) is contained in a coordinate ball of comparable radius, whose smooth volume is \(O(s^4)\). Hence \(\mu(B_s(p))=O_p(s^4)\) at those centers, and \(s^{-2}\mu(B_s(p))\to0\). A finite chart cover proves the assertion on \(X\). ◻

Proof. Choose a positive-density point where differentiation of the bounded matrix \(M\) with respect to \(\mu\) holds in a fixed Euclidean coordinate chart and smooth local trivialization (Kinnunen 2026, Theorem 4.33 and Corollary 4.34). This excludes a set of zero \(\mu\)-measure. No doubling hypothesis is needed: for fixed \(R\), enclose \(B_{Rs}(p)\) in a coordinate ball \(E_s\) of radius comparable to \(Rs\). Quadratic growth bounds \(s^{-2}\mu(E_s)\), while differentiation makes the mean oscillation of \(M\) on \(E_s\) tend to zero. Therefore \[s^{-2}\int_{B_{Rs}(p)}|M(y)-M(p)|\,d\mu(y)\longrightarrow0.\] A smooth change of frame adds a vanishing \(O(s)\) error after this normalization. The quadratic mass bound gives subsequential locally weak limits \(\gamma\) of the rescaled scalar measures on all of \(T_pX\simeq\mathbb R^4\). The existence of the density, followed by approximation of balls from inside and outside, gives \[ \gamma(B_R)=\vartheta(p)R^2\qquad(R>0). \tag{42}\] In particular every centered sphere has zero \(\gamma\)-measure. The rescaled \(Q\) mass on that ball satisfies the sharper estimate \[ s^{-2}\int_{B_{Rs}(p)}|Q|\,dV_g \le CR^2\|Q\|_{L^2(B_{Rs}(p))}\longrightarrow0. \tag{43}\] Passing to the limit in closedness of the rescaled \(T\) therefore shows that the constant-coefficient current \(M(p)\gamma\) is closed.

Identify \(M(p)\) with a constant skew-symmetric matrix. The equation \(\partial(M(p)\gamma)=0\) says that all distributional derivatives of \(\gamma\) in the column space of that matrix vanish. Equivalently, \(\gamma\) is translation invariant along \(\mathop{\mathrm{im}}M(p)\). The nonzero positive Hermitian matrix \(M(p)\) has real rank two or four. Rank four would make \(\gamma\) a nonzero multiple of Lebesgue measure on \(\mathbb R^4\), which contradicts [eq:density-tangent-ball-mass]. Thus its rank is two. Trace-one normalization then makes \(M(p)=\ell_p\), the unit bivector of a complex line \(P_p\). The rank-one positive Hermitian matrix is an extreme point of the trace-one positive matrices: if its quadratic form vanishes on a vector, positivity forces almost every matrix in any convex decomposition to vanish there too. Hence \(\lambda_p\) is concentrated on \(\ell_p\).

A locally finite measure invariant under translations in \(P_p\) has the form \(\mathcal H^2\!\restriction P_p\otimes\sigma\), with \(\sigma\) a locally finite measure on \(P_p^\perp\). Equation (42) now becomes \[\vartheta(p)= \pi\int_{P_p^\perp} \bigl(1-|z_\perp|^2/R^2\bigr)_+\,d\sigma(z_\perp) \qquad(R>0).\] Each integrand is nondecreasing in \(R\), and is strictly increasing once \(R>|z_\perp|>0\). Constancy for all \(R\) forces \(\sigma\) to be supported at zero. Its mass is \(\vartheta(p)/\pi\). Every convergent subsequence has consequently the same limit, proving the asserted convergence. ◻

Proof. At the positive-density points covered by 20, divide the inner integral by \(s^2\) and pass to the planar tangent measure. The integrand vanishes on that plane, and the limiting sphere has zero measure. At zero-density points the same normalized integral is bounded by \(s^{-2}\mu(B_s(y))\), which tends to zero. These two arguments apply for \(\lambda\)-almost every \((y,\ell)\). The quadratic mass bound gives a uniform integrable bound, so dominated convergence proves [eq:density-transverse]. ◻

Proof. We first smooth the scalar density. Fix a nonnegative function \(h\in C_c^\infty((0,1))\), normalized by \(\int_0^1h(u)\,du=1\). For small \(s\), put \[ f_s(y)=s^{-2}\int_X h\bigl(\mathop{\mathrm{dist}}(y,z)^2/s^2\bigr)\,d\mu(z), \qquad M_s(z)=\mu(B_s(z)). \tag{46}\] The function \(f_s\) is smooth: the support lies inside the injectivity radius, and the cutoff vanishes near its boundary. The mass bound makes \(f_s\) uniformly bounded. Integration with respect to the radial distribution of \(\mu\) gives \[f_s(y)=-\int_0^1 h'(u)\, \frac{\mu(B_{s\sqrt u}(y))}{s^2}\,du \longrightarrow -\vartheta(y)\int_0^1u h'(u)\,du=\vartheta(y).\] There are no atoms by quadratic growth, so no endpoint term occurs at \(u=0\). Thus \(f_s\to0\) volume-almost-everywhere as well.

Fix a smooth one-form \(\alpha\). To prove that \(b(\vartheta)N\) is closed, apply closedness of \(T=N+Q\) before passing to the density limit: \[ 0=T\bigl(d(b(f_s)\alpha)\bigr) =T\bigl(b(f_s)d\alpha\bigr) +T\bigl(b'(f_s)\,df_s\wedge\alpha\bigr). \tag{47}\] The first term has the required limit. Indeed, dominated convergence gives \[b(f_s)N\longrightarrow b(\vartheta)N,\qquad b(f_s)Q\longrightarrow0\] as currents. The second convergence uses \(b(0)=0\), the volume-almost-everywhere convergence of \(f_s\), and domination by \(C|Q|\).

It remains to show that the derivative term in (47) vanishes. The derivative of \(b\) is bounded on the common bounded range of the \(f_s\). Write \(|df_s|_\ell\) for the norm of the restriction to the plane of \(\ell\). The \(N\) contribution is bounded by a constant times \(\|\alpha\|_{C^0}\int |df_s|_\ell\,d\lambda\), and the \(Q\) contribution is bounded by a constant times \(\|\alpha\|_{C^0}\int |Q|\,|df_s|\,dV_g\). We therefore need exactly two estimates: \[ \int_X |Q|\,|df_s|\,dV_g=o(1), \qquad \int |df_s|_\ell\,d\lambda(y,\ell)=o(1). \tag{48}\] The derivative against \(Q\). Differentiating the radial kernel gives \(|df_s(z)|\le Cs^{-3}M_s(z)\). Fubini and the volume bound give \[\int_X M_s\,dV_g =\int_X\mathop{\mathrm{vol}}_g(B_s(y))\,d\mu(y)\le Cs^4.\] Moreover \(M_s/s^2\) is uniformly bounded and tends to zero volume-almost-everywhere. Therefore \[\begin{align*} s^{-3}\int_X |Q|M_s\,dV_g &\le \left(\int_X |Q|^2\frac{M_s}{s^2}\,dV_g\right)^{1/2} \left(s^{-4}\int_X M_s\,dV_g\right)^{1/2}\\ &=o(1). \tag{49}\end{align*}\] The first factor tends to zero by dominated convergence against \(|Q|^2dV_g\), and the second is bounded. This proves the first estimate in [eq:density-two-gradient-estimates].

The tangential derivative. Closedness will convert tangential differentiation into normal derivatives. The radial kernel then supplies a transverse displacement factor, multiplied by the difference between nearby complex-line directions. The transverse estimate of 21 and the angular-energy bound will control these two factors together.

Fix a center and line \((y,\ell)\). Choose an orthonormal basis of \(T_yX\) whose first two vectors span the oriented plane of \(\ell\), and let \(\zeta_1,\ldots,\zeta_4\) be the corresponding normal coordinates. The estimates below are uniform in this choice, so a measurable choice of such bases suffices when integrating in \((y,\ell)\). Write \[k_s^y(z)=h(|\zeta|^2/s^2),\qquad q_i(z)=\partial_{\zeta_i}k_s^y(z),\qquad \omega_{12}=d\zeta_1\wedge d\zeta_2.\] First variation of squared distance yields \[ df_s(y)(e_i)=-s^{-2}\int q_i(z)\,d\mu(z), \qquad i=1,2. \tag{50}\] The sign comes from differentiating the center; the displayed \(q_i\) differentiates the radial expression in its coordinate variable.

We replace this scalar integral by \(T(q_i\omega_{12})\). On \(B_s(y)\), normal coordinates and parallel transport differ by \(O(s^2)\), and hence \[\begin{align*} \omega_{12}(m) &=\langle\ell,\tau_{zy}m\rangle+O(s^2),\\ |1-\omega_{12}(m)| &\le C\bigl(s^2+|\ell-\tau_{zy}m|^2\bigr). \tag{51}\end{align*}\] The second identity uses that both bivectors have unit length. Since \(|q_i|\le C/s\), the integrated replacement error is at most \[\begin{align*} &s^{-2}\int \left|\int q_i\,d\mu-T(q_i\omega_{12})\right|d\lambda(y,\ell) \\ &\hspace{1cm}\le Cs^{-3}\left( s^2\iint_{\mathop{\mathrm{dist}}(y,z)<s}d\mu(y)d\mu(z)+\mathcal A_s\right) +Cs^{-3}\int_X|Q|M_s\,dV_g =o(1). \tag{52}\end{align*}\] Here the double mass is \(O(s^2)\), the angular integral is \(O(s^4)\), and the last term tends to zero by [eq:density-q-gradient]. The normal-coordinate term is consequently \(O(s)\).

For \(i=1\), closedness of \(T\) on \(d(k_s^y\,d\zeta_2)\) gives \[T(q_1\omega_{12}) =-\sum_{a=3}^4T(q_a\,d\zeta_a\wedge d\zeta_2).\] For \(i=2\), using \(d(k_s^y\,d\zeta_1)\) gives the analogous formula, with an irrelevant overall sign. These are legitimate smooth compactly supported tests because the cutoff vanishes near the boundary of the coordinate ball. The mixed forms satisfy, for \(k=1,2\) and \(a=3,4\), \[ |(d\zeta_a\wedge d\zeta_k)(m)| \le C\bigl(s^2+|\ell-\tau_{zy}m|\bigr). \tag{53}\] Their value on the transported center line is zero, accounting for the linear angular bound. The normal derivative of the radial kernel has the sharper estimate \[ |q_a(z)|\le Cs^{-1} \left|\operatorname{pr}_{\ell^\perp}\zeta/s\right|. \tag{54}\] Replacing \(T\) by \(N+Q\) in these mixed terms, the \(Q\) contribution again tends to zero by [eq:density-q-gradient]; the \(s^2\) geometric error contributes \(O(s)\). Cauchy–Schwarz for the two remaining angular and transverse factors bounds the rest by \[ Cs^{-3}\mathcal A_s^{1/2}\mathcal T_s^{1/2} =s^{-3}O(s^2)o(s)=o(1). \tag{55}\] Combining [eq:density-first-variation,eq:density-replacement-error,eq:density-angular-transverse-product] proves the second estimate in [eq:density-two-gradient-estimates].

The closed weighted limit. The two derivative estimates now allow passage to the limit in (47). Its derivative term tends to zero, and its first term tends to \(b(\vartheta)N(d\alpha)\). Hence this last expression vanishes for every smooth one-form \(\alpha\), proving the first assertion.

The remaining weights are obtained in a separate limit, after this argument has been completed for each fixed smooth \(b\). Choose a smooth nondecreasing function \(\chi\colon\mathbb R\to[0,1]\) which is zero on \((-\infty,0]\) and one on \([1,\infty)\). The smooth bounded functions \(\chi(nt)\) converge pointwise to \(\mathbf1_{\{t>0\}}\) on \([0,\infty)\), and vanish at zero. For fixed \(a>0\), the functions \[b_{a,n}(t)= \begin{cases} \pi\,\chi(n(t-a))/t,&t>0,\\ 0,&t\le0 \end{cases}\] are smooth, vanish near zero, and are uniformly bounded by \(\pi/a\). They converge at every \(t\ge0\) to \(\mathbf1_{\{t>a\}}\pi/t\), with value zero at \(t=a\). Dominated convergence and closedness of each smooth-weight current prove [eq:density-weighted-restrictions]. No derivative bound uniform in \(n\) is used: the smoothing limit was taken first for each \(n\). Finally \(N^d+Q=T-N^+\) is closed. ◻

Proof. For each dyadic shell, quadratic growth gives the summable bound \[s^{-2}(1+2^j)^{-6}\mu(B_{2^{j+1}s}(z)) \le C\,2^{-4j}.\] The same estimate holds once the radius exceeds a fixed small radius, by increasing \(C\) and using the finite total mass. This proves uniform boundedness. If \(\vartheta(z)=0\), each fixed shell contribution tends to zero, because \(s^{-2}\mu(B_{2^{j+1}s}(z))\to0\). The summable bound then proves pointwise convergence. A second application of dominated convergence proves [eq:density-diffuse-error]. ◻

Proof of 17. Suppose that no compatible symplectic form exists. The separator in 9 gives a nonzero positive current \(N\), normalized to have trace mass one, and an anti-invariant \(Q\) such that \(T=N+Q\) annihilates all smooth closed two-forms. In particular \(T\) is closed and \(\mathfrak c(T)=0\). The preceding results apply to this separator.

By 22, \(T^d=N^d+Q\) is closed. Equation (57) gives \[\begin{align*} 0 &=\langle S_sT,*S_sT^d\rangle\\ &=\langle S_sN,*S_sN^d\rangle +\langle S_sN,S_sQ\rangle +\langle S_sQ,S_sN^d\rangle +\|S_sQ\|_2^2 . \tag{58}\end{align*}\] The self-duality of \(Q\) accounts for the signs in this expansion. Both mixed terms tend to zero by 16. The positive-current lower bound of 14, applied with second submeasure \(\lambda^d=\mathbf1_{\{\vartheta=0\}}\lambda\), gives \[\langle S_sN,*S_sN^d\rangle \ge -C\int_XB_s\,d\mu^d=o(1).\] Since \(S_sQ\to Q\) in \(L^2\), taking a limit inferior in [eq:density-compatibility-pairing] forces \(\|Q\|_2^2=0\). Thus \(T=N\) is a nonzero positive current annihilating all closed forms.

Let \(\eta\) be a taming symplectic form. Compactness of the bundle of unit complex lines and strict taming give \(\eta_y(\ell)\ge c_\eta>0\). Consequently \[N(\eta)\ge c_\eta\mu(X)>0,\] contradicting its annihilation property. This proves compatible-form existence. The proof has used neither rectifiability nor a theorem representing integral cycles by curves. ◻

Proof. Closedness follows from 22. By [eq:density-smoothed-intersection,lem:current-positive-pairing], \[\mathfrak c(N^d)^2 =\langle S_sN^d,*S_sN^d\rangle \ge -C\int_X B_s\,d\mu^d.\] The right side tends to zero by 23. ◻

Proof. Put \(E_+=\{\vartheta>0\}\) and \(\mu^+=\mathbf1_{E_+}\mu\). At \(\mu^+\)-almost every \(p\), Euclidean coordinate-ball differentiation of \(\mathbf1_{E_+}\) gives value one. The coordinate-ball enclosure argument in 20, applied to this bounded function, gives \[s^{-2}(\mu-\mu^+)(B_{Rs}(p))\longrightarrow0 \qquad(R<\infty\text{ fixed}).\] Thus \(\mu^+\) has the same positive finite two-density and the same planar tangent measure as \(\mu\) at almost every such point. Under a smooth coordinate change its limit is the linear image of the same plane measure. Thus, in any fixed coordinate chart, the Euclidean two-density exists and is positive and finite almost everywhere for the pushed-forward measure: every Euclidean sphere has zero mass for the limiting plane measure. Preiss’s rectifiability theorem therefore makes \(\mu^+\) a two-rectifiable measure (Preiss 1987); see also the theorem’s formulation in (Kenig et al. 2009, 771, opening paragraph). On a countably rectifiable carrier \(E\) it has the representation \[ \mu^+=\frac{\vartheta}{\pi}\, \mathcal H_g^2\!\restriction E. \tag{60}\] Here \(\pi\) is the area of the Euclidean unit two-disk. The density identification follows from differentiation on a rectifiable carrier. Its approximate tangent plane agrees almost everywhere with the unique plane in 20. Thus \(N^+\) is integration over \(E\) with that complex tangent orientation and multiplicity \(\vartheta/\pi\).

Multiplying by the weight defining \(I_a\) cancels this multiplicity: \(I_a\) is integration with integer multiplicity one over \(E\cap\{\vartheta>a\}\). Its mass is at most \(\pi\mu(X)/a\), and its boundary vanishes by 22. It is consequently an integral current: it is integer rectifiable, has finite mass, and has the zero integral current as its boundary.

Suppose now that \(J\) is tamed. By the already proved 17, a compatible symplectic form exists. In particular, the local compatibility hypothesis of the Rivière–Tian regularity theorem holds. That theorem applies to integer rectifiable almost complex cycles and represents them by \(J\)-holomorphic curves, with only isolated singularities (Rivière and Tian 2009, Theorem I.1). Integral multiplicity and complex orientation do not depend on the choice of Hermitian metric, so we may use the metric of this compatible symplectic form for that application. Normalize the resulting curve by separating its local branches at the isolated singularities; the local completion is part of the curve representation (Rivière and Tian 2009, sec. VIII). Its normalization components are closed. Compactness and local finiteness give only finitely many global components; equivalently, small-ball monotonicity gives a uniform positive lower area for each nonconstant closed component, whereas the total mass is finite. Factoring each component through its somewhere-injective image gives the asserted finite sum of irreducible closed curves with positive integer multiplicities.

Finally, pointwise on \(\{\vartheta>0\}\), \[\int_0^\infty \mathbf1_{\{\vartheta>a\}}\frac{\pi}{\vartheta}\,da=\pi.\] For a smooth two-form \(\alpha\), Fubini is justified by \[\int_0^\infty |I_a(\alpha)|\,da \le \pi\|\alpha\|_{C^0}\mu^+(X).\] This proves [eq:density-layer-cake] with its stated normalization. ◻

Proof of 18. Assume that the class \(H\) has no taming representative. The class separator of 9 gives a nonzero closed positive current \(N\), with \(Q=0\), satisfying \[ H\cdot\mathfrak c(N)\le0. \tag{61}\] Its positive-density and diffuse parts are separately closed. The anti-invariance of \(A,C\) implies \[\mathfrak c(N^d)\cdot[A]=N^d(A)=0,\qquad \mathfrak c(N^d)\cdot[C]=N^d(C)=0.\] Thus \(w=\mathfrak c(N^d)\) lies in \(V\), and \(w^2\ge0\) by 24.

If \(N^d\ne0\), choose a taming form \(\eta\) in \(U\). Strict taming on the compact bundle of unit complex lines gives \[U\cdot w=N^d(\eta)>0.\] In particular \(w\ne0\). Since \([A],[C]\) span a positive two-plane and \(b^+=3\), the intersection form on \(V\) has signature \((1,b^-)\). The conditions \(w^2\ge0\) and \(U\cdot w>0\) put \(w\) in the nonzero closed time cone determined by \(U\). Every timelike \(H\) in that component has \(H\cdot w>0\). Explicitly, take a unit timelike vector \(u\) in the direction of \(U\), and write \(H=h_0u+h_-\), \(w=w_0u+w_-\), with orthogonal negative components. Then \[h_0>|h_-|,\qquad w_0\ge|w_-|,\qquad w_0>0,\] so \[H\cdot w\ge h_0w_0-|h_-|\,|w_-|>0.\] If \(N^d=0\), its pairing is instead zero. This distinguishes the nonzero diffuse case required for strict positivity.

For every \(a>0\), 25 expresses \(I_a\) as a positive integral sum of irreducible closed \(J\)-holomorphic curves. The hypothesis on \(H\) therefore gives \[H\cdot\mathfrak c(I_a)\ge0,\] with strict inequality whenever \(I_a\ne0\). If \(N^+\ne0\), the layer identity has positive total mass, so the set of \(a\) for which \(I_a\ne0\) has positive Lebesgue measure. Pairing [eq:density-layer-cake] with any smooth representative of \(H\) then gives \(H\cdot\mathfrak c(N^+)>0\). The pairing is zero when \(N^+=0\). At least one of \(N^d,N^+\) is nonzero, and hence \[H\cdot\mathfrak c(N) =H\cdot\mathfrak c(N^d)+H\cdot\mathfrak c(N^+)>0.\] This contradicts [eq:density-cone-separator] and proves the theorem. A smooth closed taming representative is symplectic by pointwise nondegeneracy. ◻

Proof. Fix a metric \(g_0\) in the conformal class determined by the original triple. All \(J_v\) are orthogonal for \(g_0\), and the three closed self-dual forms \(\omega_i\) span its self-dual harmonic forms because \(b^+=3\). Denote by \(R_v\) the projection to the \(J_v\)-anti-invariant two-forms, and consider \[L_v^{\mathrm{ell}}=R_v\mathrm d\mathrm d^* \quad\text{on the anti-invariant forms},\] where the adjoint is taken with respect to \(g_0\). This is the nonnegative elliptic operator in 10. Its kernel consists exactly of the closed anti-invariant forms: its quadratic form is \(\|\mathrm d^*\xi\|_2^2\), and an anti-invariant form is self-dual, so coclosedness implies closedness. Its kernel is therefore precisely \[\left\{\sum_{i=1}^3 a_i\omega_i:a\cdot v=0\right\},\] a smooth vector bundle of rank two over \(S^2\).

Let \(H_v\) be the projection onto this explicit kernel. In local parameter coordinates, identify the varying anti-invariant bundles by their orthogonal projections. The kernel projection is then smooth, and \(L_v^{\mathrm{ell}}+H_v\) is invertible from \(H^{s+2}\) to \(H^s\). The inverse identity and elliptic regularity give smooth dependence on each Sobolev scale, as in the parameter statement of 10. Thus the generalized inverse is \[G_v=(L_v^{\mathrm{ell}}+H_v)^{-1}(\mathop{\mathrm{Id}}-H_v),\] and \[ D_v=\mathrm d\mathrm d^*G_v+H_v, \qquad R_vD_v=\mathop{\mathrm{Id}}, \qquad \mathrm dD_v=0 \tag{63}\] is a smooth family of closed lifts.

For any \(v_0\), 17 supplies a compatible form \(\eta\) for \(J_{v_0}\). For \(v\) near \(v_0\), set \(\eta_v^{(v_0)}=\eta-D_vR_v\eta\). These forms are closed and \(J_v\)-invariant, equal \(\eta\) at \(v_0\), and remain positive after shrinking the parameter neighborhood. A finite parameter partition of unity patches these local families. Since its coefficients depend only on \(v\), this operation preserves closedness on \(X\); positivity and invariance are preserved by convexity.

The class \([\eta_v]\) is orthogonal to the classes of the two anti-invariant forms, so its projection to \(P\) is a multiple of \([B_v]\). Moreover \(\int_X\eta_v\wedge B_v>0\): the invariant part of \(B_v\) is positive, and its anti-invariant part wedges trivially with \(\eta_v\). Since \([B_v]^2=1\), the multiple is positive. ◻

Proof. The critical wall-crossing theorem and its section-class version are (Li and Liu 2001, Theorem 4.10 and Proposition 4.11). We check the normalization and degree relevant here. A transverse homotopy between two families meets the harmonic wall at isolated parameters in \(S^2\times[0,1]\), because its normal rank is \(b^+=3\). Above such a parameter the reducibles form the torus \(\mathop{\mathrm{Pic}}^0(X)\). Represent the complex Dirac index on this torus as \(K-O\), with ranks \(k,o\), using a stabilization when necessary. Write \(x=c_1(\mathcal O(1))\) on the projective bundle of lines \(\pi:\mathbb P(K)\to\mathop{\mathrm{Pic}}^0(X)\). The obstruction bundle contributes \(e(\mathcal O(1)\otimes O)=\sum_i x^{o-i}c_i(O)\). Projective pushforward therefore gives \[\begin{align*} \pi_*\left(x^q\sum_i x^{o-i}c_i(O)\right) &=\sum_i s_{q+o-(k-1)-i}(K)c_i(O)\tag{68}\\ &=s_{b_1/2}(K-O), \end{align*}\] since [eq:families-topology,eq:families-dimensions] imply \(q+o-(k-1)=b_1/2\). The normal crossing signs sum to the degree difference, giving [eq:families-wall-formula].

The based-gauge normalization makes the universal flat line trivial at \(p\). Thus no Picard-base line is added to \(x\). Writing a determinant connection as a canonical connection plus twice a twisting connection does not alter this conclusion: the remaining circle acts on the spinor with weight one, as used in [eq:families-projective-pushforward].

For a constant regular family, negative ordinary expected dimension gives an empty moduli space. Otherwise the moduli space is the product of an ordinary \(E^2\)-dimensional moduli space with \(S^2\), while \(U_p^q\) is pulled back from the former and has degree \(E^2+2\). Its pairing is therefore zero.

Every maximal positive harmonic plane projects isomorphically to \(P\): the kernel would lie in the negative definite space \(P^\perp\). This identifies the harmonic-wall bundle with the fixed oriented vector space \(P\). For the perturbation in [eq:families-taubes-equations], its wall section has leading term \((r/4)[\eta_v]\), with a bounded shift from \(2E\). By 27, it is nonzero for all sufficiently large \(r\) and its normalization has degree one. The same is true after sufficiently small generic family perturbations. ◻

Proof. Since \(H^1(S^2)=0\), the Kunneth decomposition shows that \(c_1(\mathcal E_E)=E+eh\) for some integer \(e\). At \(p\), deform the compatible metrics to \(g_0\) through \(J_v\)-Hermitian metrics. The fixed spin-c extension over metric space identifies the positive spinor bundles along this homotopy. For \(g_0\), the first eigenspaces give the tautological line in the projective spinor sphere, pulled back by the coefficient sphere map. The latter has degree of absolute value one by 7, proving \(|e|=1\).

The evaluation transforms with the same gauge weight as the spinor. Its sign can also be checked on a reducible link: evaluation on a kernel line \(\ell\) is a map \(\ell\to\mathcal E_E|_p\), hence lies in \(\operatorname{Hom}(\ell,\mathcal E_E|_p) =\mathcal O(1)\otimes\mathcal E_E|_p\). On the quotient it is therefore a section of the tensor product of the based-gauge evaluation line with the pullback of \(\mathcal E_E|_{\{p\}\times S^2}\). Its Chern class is \(U_p+e\pi^*h\). Pairing \(\pi^*h\) with the family moduli cycle gives the ordinary dimension-zero invariant \(\operatorname{SW}_X(E)\); equivalently, restrict the family to a regular base point. A nowhere-zero evaluation section trivializes its line, giving [eq:families-evaluation-identity].

For the last assertion, suppose the exact family had no evaluation zero. Fixed-\(r\) compactness would give the same absence for all sufficiently small perturbations: otherwise a sequence of their zeros would limit to an exact zero. Apply the preceding identity to a regular such perturbation to obtain a contradiction. ◻

Proof. First take \(E=0\). Taubes’s canonical nonvanishing theorem (Taubes 1994) gives \(|\operatorname{SW}_X(0)|=1\), since \(b^+>1\). By 28, the exact symplectic family has no first-component zeros for all sufficiently large \(r\). Fixed-\(r\) compactness transfers this absence to sufficiently small regular perturbations. Applying [eq:families-evaluation-identity] gives \(|\operatorname{FSW}_0(U_p)|=1\). On the other hand, 30 and [eq:families-index-character,eq:families-segre-two] identify this number, up to sign, with \[\int_{\mathop{\mathrm{Pic}}^0(X)}\int_X\frac{u^4}{24},\] because \(\operatorname{ch}_1=0\) when \(E=0\). It follows that the quadruple cup product is nonzero.

Here is an explicit calculation of the remaining assertion. Choose a basis \(a_1,\ldots,a_4\) of \(H^1(X;\mathbb R)\), let \(t_1,\ldots,t_4\) be the corresponding Picard-torus basis, and write \[\delta=\int_Xa_1a_2a_3a_4\ne0, \qquad u=\sum_i a_it_i.\] The wedge pairing on \(\bigwedge^2H^1(X;\mathbb R)\) induced by this nonzero volume element is nondegenerate. Thus the cup-product map into \(H^2(X;\mathbb R)\) is injective. Both spaces have dimension six, by 6, so it is an isomorphism. We may consequently write \(E=\sum_{i<j}e_{ij}a_ia_j\). The graded-product signs give \[\begin{align*} \operatorname{ch}_2&=\delta\,t_1t_2t_3t_4,\tag{72}\\ \operatorname{ch}_1&=-\delta\bigl( e_{34}t_1t_2-e_{24}t_1t_3+e_{23}t_1t_4\\ &\hspace{40mm} +e_{14}t_2t_3-e_{13}t_2t_4+e_{12}t_3t_4\bigr),\\ E^2&=2\delta(e_{12}e_{34}-e_{13}e_{24}+e_{14}e_{23}),\\ \operatorname{ch}_1^2&=\delta E^2\,t_1t_2t_3t_4. \end{align*}\] Thus \(E^2=0\) implies \(\operatorname{ch}_1^2=0\), and the Segre integral equals the same nonzero integral of \(\operatorname{ch}_2\) found for \(E=0\).

Finally, a map \(S^2\to X\) pulls back every class in \(H^1(X;\mathbb R)\) to zero, and therefore pulls back every product of two such classes to zero. These products span \(H^2(X;\mathbb R)\), so its homology class is zero by Poincare duality. ◻

Proof of 26. The projection \(F^+\) is nonzero: otherwise \(F\) would be a nonzero square-zero vector in the negative definite space \(P^\perp\). Lift \(F\) to an integral class and use the fixed spin-c structure \(\mathfrak s_F\).

Its ordinary invariant is zero. Indeed, choose the opposite coefficient direction \(w=-F^+/|F^+|\), so that \(F[B_w]<0\). If \(\operatorname{SW}_X(F)\ne0\), ordinary invariance and fixed-parameter monopole compactness would give solutions of [eq:families-taubes-equations] for arbitrarily large \(r\) at this fixed background. Taubes compactness would give a nonempty positive \(J_w\)-holomorphic configuration of class \(F\), contradicting its negative \(B_w\)-area.

The spherical \(U_p\) pairing is nonzero. For \(b_1=0\), this follows directly from 30; for \(b_1=4\), use 32 as well. Since the ordinary invariant vanishes, 31 forces a first-component zero over \(p\) in the exact family for each sufficiently large \(r\). Extracting a convergent parameter subsequence and applying 28 gives a configuration through \(p\) in some \(J_v\), with total class \(F\).

The two closed forms whose coefficients are perpendicular to \(v\) are \(J_v\)-anti-invariant and restrict to zero on every component of this configuration. Thus \(F^+\) is parallel to \([B_v]\). Its \(B_v\)-area is positive, so the proportionality factor is positive, which proves the asserted direction. ◻

Proof. Lift \(E\) to an integral line-bundle class. Its ordinary expected dimension is \(-2\), and its spherical family has dimension zero. By 30, its large-parameter unmarked family invariant is \(\pm1\). For each sufficiently large \(r\), take small regular perturbations and then a fixed-\(r\) limit to obtain an actual solution of [eq:families-taubes-equations] at some parameter \(v_r\). Pass to a sequence with \(v_r\to v\), and apply 28. The limit is a nonempty positive \(J_v\)-holomorphic configuration of class \(E\). Every nonconstant component has positive \(B_v\)-area, since \(B_v\) tames \(J_v\). This contradicts \(E[B_v]=0\), which follows from \(E\perp P\). ◻

Proof of 28. For a fixed background, this is the compactness and incidence part of Taubes’s existence theorem. For the equations in [eq:families-taubes-equations], use (Taubes 1999, Theorem 2.2), with its zero-support conclusion as proved in Lemma 7.1 and Equation (7.5) for the first component defined in Equation (3.17) of that paper. The symplectic form has empty zero locus, so the transverse-zero hypothesis is vacuous. The class formula is \[2[\mathcal C]=c_1(L\otimes K_J).\] The splitting above gives \(L=K_J^{-1}\mathcal E^2\), so this is exactly \([\mathcal C]=E\) over \(\mathbb R\). The original fixed-background symplectic statement also includes incidence with prescribed closed sets (Taubes 1996, Theorem 1.3); the curve-recognition argument is understood in its corrected form (Taubes 2000).

Uniform estimates and support convergence. We explain the uniformity needed here. Smooth convergence provides a common positive normal-coordinate radius, uniform ellipticity, and uniform bounds on all derivatives of the metrics, forms, and induced canonical connections. It also bounds the topological terms \(c_1(L)[\eta_n]\). In the following estimates, constants are independent of \(n\) and \(r\) after these quantities have been bounded. For readability write \(w=1-|\alpha|^2\) and \(w_+=\max(w,0)\), and use the projected spinor covariant derivatives. The estimates used in the compactness proof include \[\begin{align*} |\psi|^2&\le1+Cr^{-1}, & |\beta|^2&\le Cr^{-1}w_++Cr^{-2},\tag{73}\\ |F_{\mathcal A}^+|&\le\frac{r}{2\sqrt2}w_++C, & |F_{\mathcal A}^-|&\le \frac{r}{2\sqrt2}(1+Cr^{-1/2})w_++C,\tag{74}\\ |\nabla_{\mathcal A}\alpha|^2 +r|\nabla_{\mathcal A}\beta|^2&\le Crw_++C, & r\int_X|1-|\psi|^2|\,\mathop{\mathrm{vol}}_g&\le C. \tag{75}\end{align*}\] These are the specialization to a nowhere-vanishing symplectic form of (Taubes 1999, Lemma 3.1, Proposition 3.1, Equation (3.27), and Propositions 3.2–3.3). The global pointwise bound follows directly from the spinor Weitzenbock formula and the maximum principle. The remaining bounds use its two eigenline projections, the curvature equation and its derivative, and the scalar Green kernel. All the geometric coefficients in those arguments are uniformly bounded on the chosen charts. In particular, \(\int w_+\le\int|1-|\psi|^2|+\int|\beta|^2\). Integrating the bound for \(\beta\) and absorbing \(Cr^{-1}\int w_+\) for large \(r\) therefore bounds \(r\int w_+\). The inequality \(w\ge-Cr^{-1}\), followed by [eq:families-taubes-curvature], then gives \[ r\int_X|w|\,\mathop{\mathrm{vol}}_g+\int_X|F_{\mathcal A}|\,\mathop{\mathrm{vol}}_g\le C. \tag{76}\]

For the nonnegative energy \(\mathscr E_r(U)=r\int_U|1-|\psi|^2|\,\mathop{\mathrm{vol}}_g\), the monotonicity estimate gives uniform constants \(c,C,s_0>0\) such that, whenever \(Cr^{-1/2}\le s\le s_0\), \[ \mathscr E_r(B_s(x))\le Cs^2, \qquad x\in\alpha^{-1}(0)\ \Longrightarrow\quad \mathscr E_r(B_s(x))\ge cs^2. \tag{77}\] This is (Taubes 1999, Proposition 4.1), with its universal normalizing factor absorbed into the constants. Consequently the zero sets have covers by at most \(Cs^{-2}\) such balls.

Here is also the scale and derivative dependence in the local compactness step. Use normal coordinates for each individual \(g_n\), rescale by \(r_n^{1/2}\), and compare on a ball of fixed radius \(R\). After identifying the center with Euclidean \(\mathbb C^2\), the rescaled coefficients satisfy \[\begin{align*} \|g_n'-g_{\mathrm E}\|_{C^0(B_R)}&\le C_Rr_n^{-1}, &\|\partial^kg_n'\|_{C^0(B_R)}&\le C_{R,k}r_n^{-k/2} &&(k\ge2),\tag{78}\\ \|\eta_n'-\eta_n'(0)\|_{C^0(B_R)}&\le C_Rr_n^{-1/2}, &\|\partial^k\eta_n'\|_{C^0(B_R)}&\le C_{R,k}r_n^{-k/2} &&(k\ge1). \end{align*}\] The first derivatives of \(g_n'\) are \(O_R(r_n^{-1})\). The scale in (Taubes 1999, Equation (5.1)) is \((r_n|\eta_n|)^{1/2}=2^{1/4}\sqrt{r_n}\). A fixed dilation therefore converts these coordinates to that paper’s flat-model normalization; all estimates below absorb this fixed factor, and curvature integrals are unchanged by it. These estimates follow by Taylor expansion in each metric’s own normal coordinates; compare (Taubes 1999, sec. 6, Step 4). For every fixed \(R,k\), the rescaled equations therefore have uniform coefficient bounds through order \(k\). Gauge fixing and interior elliptic estimates give the rescaled local approximation in any prescribed \(C^k\) norm by the flat model, as in (Taubes 1999, Proposition 5.2). Its quantifiers are: for each \(R,k,\varepsilon>0\), a common lower bound on \(r\) makes the approximation error smaller than \(\varepsilon\). We retain smooth background convergence, so every derivative order required in this argument is available; no assertion about a minimal finite regularity threshold is needed.

Let \(\mathcal A_{\mathrm{can},n}\) be the canonical connection on \(K_{J_n}^{-1}\). The first eigenline connection \(a_n\) has curvature \(F_{a_n}=\tfrac12(F_{\mathcal A_n} -F_{\mathcal A_{\mathrm{can},n}})\). The corresponding closed currents \[ T_n(\zeta)=\frac{i}{2\pi}\int_XF_{a_n}\wedge\zeta \tag{79}\] have uniformly bounded mass by [eq:families-taubes-mass]. Their periods on closed test forms are those of \(E\). On the complement of the first-component zero set, its unit section identifies the determinant with the canonical determinant. In that identification the further estimate is \[ |\mathcal A-\mathcal A_{\mathrm{can}}| +|F_{\mathcal A}-F_{\mathcal A_{\mathrm{can}}}| \le Cr^{-1}+Cr\exp\bigl(-\sqrt r\, \mathop{\mathrm{dist}}(x,\alpha^{-1}(0))/C\bigr) \tag{80}\] when the distance is at least \(r^{-1/2}\) (Taubes 1999, Proposition 6.1).

The ball covers, the lower bound at zeros, and [eq:families-taubes-off-zeros] now give a subsequence whose zero sets converge to the support \(Z_*\) of a weak limit of \(T_n\); the support has finite two-dimensional Hausdorff measure. These are the arguments of (Taubes 1999, Equations (7.1)–(7.5) and Lemma 7.1), using exactly the uniform estimates just listed.

We give the disk argument needed when the backgrounds vary. Write \[f_n=\frac{i}{2\pi}F_{a_n} =\frac{i}{4\pi}(F_{\mathcal A_n} -F_{\mathcal A_{\mathrm{can},n}}).\] The sharper estimate (Taubes 1999, Proposition 4.2, Equation (4.9)), specialized to \(|\eta_n|=\sqrt2\), is \[ |F_{\mathcal A_n}^-| \le \frac{r_n}{2\sqrt2}(1-|\alpha_n|^2)+C. \tag{81}\] Its constants have the same uniform geometric dependence as the estimates above. If \(\ell\) is a unit, positively oriented \(J_n\)-complex bivector, then \(\ell^+=\eta_n/2\), \(|\ell^-|=1/\sqrt2\), and the curvature equation gives \[iF_{\mathcal A_n}^+(\ell) =\frac{r_n}{4}(1-|\alpha_n|^2+|\beta_n|^2), \qquad iF_{\mathcal A_n}^-(\ell) \ge-\frac{r_n}{4}(1-|\alpha_n|^2)-C.\] The leading terms cancel. The canonical curvatures are uniformly bounded, so there is one constant \(C_0\ge1\) such that every \(J_n\)-holomorphic disk \(u\) satisfies \[ u^*f_n\ge-C_0\,\mathrm dA_u. \tag{82}\] Here \(\mathrm dA_u\) is the induced area density, counted on the source; the inequality also holds at critical points. This derives the disk bound directly for our normalization, including the factor one half in the twisting connection.

The relative integer assignment. First construct the integer assignment independently of positivity. A smooth disk \(u\) is admissible when its boundary misses \(Z_*\). For large \(n\), \(u^*\alpha_n\) is nonzero on a boundary collar and defines a relative Chern number \(k_n(u)\in\mathbb Z\). The unit first component trivializes the eigenline there, and [eq:families-taubes-off-zeros] gives \[ \int_Du^*f_n=k_n(u)+o(1). \tag{83}\] The error is uniform for disks with bounded first derivatives whose boundary collars have a common positive distance from \(Z_*\). For each sufficiently large \(n\), relative Chern numbers agree along any fixed admissible homotopy. The same conclusion holds for the varying homotopies below: pointwise short geodesics between uniformly close boundary maps stay in one fixed zero-free neighborhood of the original boundary. Thus their pulled-back unit sections extend over the boundary homotopy, uniformly in large \(n\).

For an immersed disk, take small normal translates, with a smooth averaging function in the two normal parameters. The normal bundle is taken over the source, so self-intersections cause no difficulty. All these disks have the same \(k_n\). Cut off the source near its boundary collar; the removed flux tends to zero by [eq:families-taubes-off-zeros]. The averaged remaining flux is the pairing of \(T_n\) with a fixed smooth two-form: the normal coordinate map is a local diffeomorphism, and a finite partition of unity pushes the compactly supported averaging forms to \(X\). Weak convergence of \(T_n\) proves convergence of these averages. Thus \(k_n\), being integer valued, is eventually constant. A general admissible smooth disk has an arbitrarily close immersed perturbation, connected to it by an admissible homotopy, and hence has the same conclusion. Denote the eventual integer by \(I(u)\). It is zero for disks missing \(Z_*\), invariant under admissible homotopy, additive under subdivision, and multiplied by the degree under proper disk maps; these are the corresponding properties of relative first Chern numbers. This is the Stokes and averaging construction of (Taubes 1996, Proposition 5.6 and Lemma 6.2). Only positivity for the limiting complex structure remains.

Disk deformations with prescribed incidences. We need the following elementary disk deformation fact. For a holomorphic map on a slightly larger disk, with no boundary condition, the linearized Cauchy–Riemann operator has a bounded right inverse even after prescribing values and complex-linear first derivatives at any finite set of distinct interior source points. Work, for example, in \(H^k\to H^{k-1}\), \(k\ge4\), so these evaluations are continuous. Extend the bundle and operator to a closed surface, and extend the target sections by a bounded Sobolev extension. The closed-surface operator augmented by the finite evaluations is Fredholm. Finitely many smooth forcing terms supported outside the larger disk make it onto. Indeed, an annihilator of all such exterior terms vanishes there. Away from the specified points it is smooth by elliptic regularity and vanishes everywhere by unique continuation for the adjoint real Cauchy–Riemann operator (Wendl 2014, Theorem 2.78 and Proposition 3.12). It is therefore a sum of distributions supported at the specified points. The invertible principal symbol eliminates derivatives of point masses of positive order: their highest derivative after applying the adjoint has order at least two, whereas the evaluation constraints have order at most one. The remaining term at each point is \(q\delta\). Testing independently the complex-linear and anti-complex-linear first derivatives eliminates both \(q\) and the first-derivative constraint covector; testing the value then eliminates its covector. The annihilator is zero, proving the assertion about exterior forcing. A bounded right inverse of this surjective Fredholm operator, restricted to the disk, proves the claimed right inverse.

The implicit function theorem and interior regularity consequently give nearby holomorphic disks for nearby almost-complex structures, with those finite values and complex-linear derivatives prescribed. All domain restrictions are chosen first, and all constraints lie in the smaller disk. The right-inverse bounds persist for nearby operators. Constants may depend on the chosen finite set of points and on a fixed neighborhood of the original map; this is harmless because these choices precede \(n\to\infty\). In particular, while retaining finitely many incident values, one may choose the tangent line at each of them independently in a nonempty open cap. At a critical point one first chooses a sufficiently small nonzero complex-linear derivative. This gives a fixed small \(J_\infty\)-holomorphic perturbation of the original disk; its \(J_n\)-holomorphic transfers converge smoothly to that perturbation on the smaller closed disk. They need not converge to the unperturbed disk, and admissible boundary homotopy is what identifies their integers.

Positive flux at a retained incidence. Here is the positive contribution at each retained incidence. Choose actual zeros \(p_{j,n}\) tending to the finitely many specified values in \(Z_*\). After a joint subsequence, rescaling at these zeros gives centered flat models on all fixed balls. Their first components vanish at the origin. The quadratic energy bound excludes the identically zero first component. Each limiting model has a polynomial divisor by (Taubes 1999, Proposition 5.1). For any nonempty open cap of complex lines through the origin, choose a direction where the highest homogeneous part of that polynomial is nonzero; only finitely many directions are excluded. Along the chosen line, distance from the divisor grows linearly at infinity. The exponential off-divisor estimate in (Taubes 1999, Proposition 5.1) makes the boundary connection term tend to zero. Stokes’ theorem therefore identifies the total line flux \((i/2\pi)\int F_{a_0}\) with the positive degree \(m_j\ge1\) of the restricted divisor, which contains the origin. The determinant connection in (Taubes 1999, Proposition 5.2) converges to \(2a_0\), so this is precisely the limiting normalization of \(f_n\), with the bounded canonical curvature disappearing under rescaling. Choose a fixed large radius \(R_j\) so that the boundary is zero-free and the flux differs from \(m_j\) by less than \(1/8\). These are the polynomial-model and line-selection steps in (Taubes 1996, Proposition 5.6, Steps 3–4), using the flat limits of (Taubes 1999, Proposition 5.2).

Prescribe those good tangent directions in the disk deformation just constructed. Choose a fixed approximation tolerance making the flux error on each model disk less than \(1/8\); only then take \(n\) large. Because the prescribed derivatives are nonzero and the disk maps have uniform higher derivative bounds, the rescaled maps on microscopic source disks converge to the chosen complex-linear maps. Each resulting source core therefore contributes more than \(m_j-1/4\ge3/4\) to the flux. The finitely many core radii tend to zero, so the cores are source-disjoint. Distinct source incidences may have the same target value: the flux is integrated on the source, and this does not affect the argument. The nonzero derivative sizes, the caps, the models, \(R_j\), and the approximation tolerances are all fixed before the final lower bound on \(n\) is imposed.

Positivity for the limiting complex structure. We now prove positivity for any admissible \(J_\infty\)-holomorphic disk \(u\) meeting \(Z_*\), allowing critical points and self-intersections. Shrink its domain inside a slightly larger disk so that every incidence lies in its interior and its boundary collar has a fixed positive gap from \(Z_*\). First fix a small \(C^1\) neighborhood in which this gap persists and all disk areas are bounded by a number \(A_*\). Put \(S=u^{-1}(Z_*)\).

If \(S\) is finite, retain all its points. Choose disjoint fixed neighborhoods of them whose total induced area, for all sufficiently close disks, is less than \(1/(4C_0)\). On the compact complement the original disk has a positive distance from \(Z_*\); shrink the allowed disk neighborhood to preserve half that distance. Choose the good tangent caps and the fixed small holomorphic perturbation within this neighborhood, and transfer it to \(J_n\), retaining the chosen zeros. The cores contribute at least \(3/4\) each. Their complement inside the selected neighborhoods contributes more than \(-1/4\) by [eq:families-disk-lower-bound]. On the remaining compact domain the flux tends to zero by [eq:families-taubes-off-zeros]. Thus the limiting integer in [eq:families-relative-degree] is positive.

If \(S\) is infinite, choose a finite number \(N\) of distinct source points in it with \(3N/4>C_0A_*+1\). The area bound was fixed before \(N\). Apply the same finite-constraint construction at these points, keeping the perturbed disks in the chosen area neighborhood. The \(N\) source-disjoint cores contribute more than \(3N/4\), whereas their full complement contributes at least \(-C_0A_*\). The flux is therefore bounded below by a positive number. Its limit is the unchanged integer \(I(u)\), so again \(I(u)>0\). Every choice was finite and fixed before the last passage to large \(n\); no rate relating \(J_n\to J_\infty\) and \(r_n\to\infty\) has been imposed.

Curve recognition and retention of the marked point. In particular the positivity axiom for embedded holomorphic test disks holds. The resulting positive cohomology assignment permits the corrected recognition theorem to be applied to the fixed structure \(J_\infty\), as in (Taubes 1999, Lemmas 7.2–7.3 and Proposition 7.1) and (Taubes 2000). The recognized normalization map is proper. Since \(X\) is compact, its source is compact and has finitely many components; its locally finite exceptional set is finite as well. The multiplicity of a resulting curve at a smooth point is the integer \(I\) of a small transverse disk there. The averaging construction identifies this same integer with the transverse coefficient of the weak curvature current: on each smooth branch, a closed current of order zero supported there is a constant multiple of its oriented integration current. The remaining difference is supported on the finite singular set, where a closed two-current of order zero must vanish. Thus the recognized configuration has current equal to the weak limit of \(T_n\), as in (Taubes 1999, Proposition 7.1 and Equation (7.12) in the proof of Lemma 7.4); in particular its class is \(E\).

The support convergence retains every limit of first-component zeros. If \(E=0\), a retained zero would give a nonempty positive configuration with \(\int_{\mathcal C}\eta_\infty=E[\eta_\infty]=0\), which is impossible. Failure of the final uniform assertion would supply a sequence with \(r_n\to\infty\); compactness of the parameter space would reduce it to the situation just proved. ◻

Proof. Let \(B_\infty\) be the limiting third form. It tames the limiting structure with a positive lower bound on the unit tangent bundle of a fixed metric. Smooth convergence therefore makes this same fixed form tame every sufficiently late structure. Its area on each sphere is \([B_\infty]F=[B]F\). The fixed-tamer hypotheses and uniform energy bound in the cited Gromov compactness theorem are consequently satisfied. The nonconstant components in the stable limit give a positive configuration of total class \(F\) for the limiting triple. By 37, there is exactly one simple image and its total multiplicity is one. In particular, the surviving map is not a nontrivial cover.

The domain of a genus-zero stable limit is a tree of spheres. With only one nonconstant vertex, a nonempty tree of constant vertices would have a constant leaf with at most one node and the one permitted mark. It would have at most two special points and hence would be unstable. There are therefore no constant components. Gromov convergence is now smooth convergence on the whole sphere after reparametrization, and the marked points converge as well. The same argument gives convergence in the corresponding finite differentiability orders when the backgrounds converge in a sufficiently high finite order.

If the limiting triple had no critical spheres but nearby triples had such spheres, mark a critical point on each. The conclusion just proved would give a critical point on a limiting sphere. This contradiction proves openness. This argument applies uniformly on any compact parameter set of positive triples with the stipulated periods. ◻

Proof. We prove universal surjectivity in the allowed space of perturbations. We reduce an annihilator to two constants arising from the pullback periods and a distribution supported at the marked point. The elliptic symbol and first-jet tests remove the point-supported part; variations of the coefficient sphere remove the two constants. Fix the complex structure \(j\) on \(S^2\). Write \(v_0=(0,0,1)\) for the coefficient direction of \(B\). For \(v\) close to \(v_0\), a convenient closed anti-invariant pair for the structure \(J_v\) is \[ A_v=A-\frac{v_1}{v_3}B,\qquad C_v=C-\frac{v_2}{v_3}B. \tag{95}\] For any map \(u\) in class \(F\), the two pullback periods are \[\left(\int_{S^2}u^*A_v,\int_{S^2}u^*C_v\right) =-[B]F\left(\frac{v_1}{v_3},\frac{v_2}{v_3}\right).\] Both pullbacks vanish when \(u\) is \(J_v\)-holomorphic, so every \(F\)-curve forces \(v=v_0\). Exact changes of \(A,C\) leave the two periods unchanged, whereas varying \(v\) supplies both period directions in the universal linearization. Thus this auxiliary parameter removes the period constraints without adding solutions in other coefficient directions. We retain it in the section \[ (u,z,v;A,C)\longmapsto \left(\overline\partial_{j,J_v}u, \partial_{j,J_v}u(z)\right). \tag{96}\] The second component is valued in the real rank-four bundle of complex linear maps \(T_zS^2\to T_{u(z)}X\). On the zero set of the first component, its vanishing is exactly \(\mathrm du(z)=0\).

Use \(W^{k,p}\) maps with \(p>2\), \(k\geq8\), and backgrounds of finite class \(C^r\), with \(r\geq k+8\). The first target is \(W^{k-1,p}\). Sobolev embedding gives \(W^{k,p}\subset C^{k-1,\alpha}\) for some \(\alpha>0\), so first-jet evaluation at the moving mark is at least \(C^{k-2}\), hence \(C^6\). The background regularity gives the same \(C^6\) regularity for the composition maps in the first component. Perturbing forms are \(\mathrm d\beta_A,\mathrm d\beta_C\), where the primitives belong to the \(C^{r+1}\) closure of smooth one-forms in the indicated support class. This gives separable Banach spaces. We use the ordinary Banach bundle construction for [eq:pencil-universal-section]; compare the first-jet framework in (Oh and Zhu 2009, secs. 2–3). The surjectivity argument for our restricted perturbations is given next.

Suppose \((\eta,\lambda)\) annihilates the derivative of [eq:pencil-universal-section] at a simple critical sphere. Here \(\eta\) is a distribution dual to the Cauchy–Riemann target and \(\lambda\) is a covector on the jet target. Testing map variations supported away from \(z\) gives \(D_u^*\eta=0\) there. Elliptic regularity makes \(\eta\) continuously differentiable away from \(z\), with the further finite regularity supplied by the coefficients. The choice of \(r\) gives more derivatives than are needed for all distributional products and integrations by parts below.

Removal of tangential coefficients. Although the index uses a fixed \(j\), we may test every infinitesimal variation \(\dot j\) of the domain complex structure. On \(S^2\) the map \(\xi\mapsto\mathcal L_\xi j\) is onto: its cokernel is \(H^{0,1}(TS^2)=0\), by the line-bundle surjectivity criterion since \(\deg TS^2=2>-2\) (Wendl 2014, Theorem 3.23). Naturality under a domain diffeomorphism, with the marked point moved by the inverse diffeomorphism, expresses the \(\dot j\) test as a combination of the already allowed map and mark tests. The direct variation of the jet with respect to \(j\) vanishes at \(z\), because \(\mathrm du(z)=0\). Thus \[\eta\left(\tfrac12J\mathrm du\,\dot j\right)=0 \quad\text{for every }\dot j.\] At an immersion point these variations span the tangential Cauchy–Riemann residuals. Hence \(\eta\) annihilates that tangential subbundle away from \(z\).

The two pullback coefficients. There are bundle maps \(L_A,L_C\), with the regularity of the coefficients, from Cauchy–Riemann residuals to real two-forms on the domain such that, at a solution, \[ \delta(u^*A_v)=L_A\,\delta(\overline\partial u),\qquad \delta(u^*C_v)=L_C\,\delta(\overline\partial u). \tag{97}\] These identities hold also for variations of the background pair and of \(v\). To see that they extend without loss of regularity at a critical point, choose \(j\partial_s=\partial_t\) and put \(e=(u_s+Ju_t)/2\). Anti-invariance gives the exact identity \[A_v(u_s,u_t)=-2A_v(u_s,Je),\] and the same identity holds for \(C_v\). Differentiation at \(e=0\) proves [eq:pencil-pullback-linearization]. In particular the maps \(L_A,L_C\) contain \(\mathrm du\), not its inverse, and extend continuously across every critical point. At an immersion point they vanish on tangential residuals and identify the real two-dimensional normal residual with the two pullback values: contraction with a nonzero complex tangent vector by a nonzero complex volume form pairs the complex normal line isomorphically with \(\mathbb C\).

Consequently, on the injective immersion locus away from \(z\), there are unique continuously differentiable real functions \(f_A,f_C\) such that \[ \eta(e)=\int_{S^2}\bigl(f_A L_Ae+f_C L_Ce\bigr) \tag{98}\] for tests supported in that locus. Let \(U\) be a relatively compact injective disk in the locus. A neighborhood of its image can be chosen to meet no other part of the curve. Restrictions of smooth ambient one-forms supported in this neighborhood are dense in \(\Omega^1_c(U)\) in the \(C^1\) topology. Indeed, the available regularity of \(u\) gives a compactly supported \(C^2\) ambient extension of any such source one-form in tubular coordinates. Approximate this extension in \(C^1\) by smooth ambient one-forms, keeping their supports in the same isolated neighborhood. Their pullbacks converge in \(C^1\). Testing the variations \(\mathrm d\beta_A\) for these smooth primitives with \(u\) fixed and passing to the limit gives \[\int_U f_A\,\mathrm d\theta=0 \quad\text{for every }\theta\in\Omega^1_c(U),\] since \(f_A\) is continuous on the compact source support. The jet contribution of this background variation is zero because \(\mathrm du(z)=0\). Integration by parts proves \(\mathrm df_A=0\) on \(U\); independent variations of \(C\) prove \(\mathrm df_C=0\). The complement of the finite exceptional set in \(S^2\) is connected, so these functions are global constants \(k_A,k_C\) on that locus.

Define on the whole source the distribution with continuous coefficients \[\eta_0(e)=k_A\int_{S^2}L_Ae+k_C\int_{S^2}L_Ce.\] Continuity of the \(L\)’s and elliptic regularity away from \(z\) show that \(\eta-\eta_0\) is supported at \(z\), including when there are other critical or double points. For every map variation \(w\), closedness of the two forms gives \[\eta_0(D_uw) =k_A\int_{S^2}\mathrm d\bigl(u^*\iota_wA\bigr) +k_C\int_{S^2}\mathrm d\bigl(u^*\iota_wC\bigr)=0.\]

Elimination of the point-supported remainder. Set \(R=\eta-\eta_0\). The annihilator equation for map variations is \[ D_u^*R=-\mathcal J_z^*\lambda, \tag{99}\] where \(\mathcal J_z\) is the linearization of the complex-linear first jet, with the mark fixed. The right side has distributional order at most one and is supported at \(z\). A point-supported distribution is a finite sum of derivatives of the Dirac delta. If its largest nonzero order were \(m\geq1\), its homogeneous coefficient polynomial \(R_m(\xi)\) would satisfy \[\sigma(D_u)^*(\xi)R_m(\xi)=0 \quad(\xi\in T_z^*S^2).\] There are enough coefficient derivatives for this comparison: a point-supported functional on \(W^{k-1,p}\) in real dimension two has order at most \(k-2\) when \(p>2\), while \(r\geq k+8\). The first-order Cauchy–Riemann symbol is invertible for every real \(\xi\ne0\), so this polynomial must vanish identically, a contradiction. Hence \(R=q\delta_z\) for a single covector \(q\). Test [eq:pencil-point-distribution] on variations with \(w(z)=0\). Since \(\mathrm du(z)=0\), the values of \(D_uw\) and \(\mathcal J_zw\) at \(z\) are respectively the complex-antilinear and complex-linear parts of \(\nabla w(z)\). These parts can be specified independently by a smooth variation supported near \(z\). It follows that \(q=0\) and \(\lambda=0\).

Finally, variation of \(v\) in [eq:pencil-coefficient-pair] gives \[ \delta\left(\int_F A_v,\int_F C_v\right) =-[B]F\,(\dot v_1,\dot v_2). \tag{100}\] This map is onto \(\mathbb R^2\). Applying the remaining annihilator \(\eta_0\) to these tests yields \(k_A=k_C=0\). Thus the universal derivative is onto. Its range is closed: the map-and-jet operator with background fixed is Fredholm and has finite-codimensional closed range, and adding parameter directions preserves this property.

Index and allowed genericity. The parametrized sphere operator has real index \(4\), since \(c_1(u^*TX)=0\) (Wendl 2014, Theorem 3.22). Adding \(v\) and the mark and imposing the first-jet constraint gives \[ 4+2+2-4=4. \tag{101}\] For a path, allowing its parameter adds one and gives index \(5\). The projection from the universal zero set to the space of perturbing primitives is therefore Fredholm of index \(4\), or \(5\) for paths. The projection is \(C^6\), and \(6>5\), so Sard–Smale applies with the chosen regularities (Smale 1965, Theorem 1.3). For a regular parameter its fiber would be a manifold of that dimension. But the six-dimensional group \(\operatorname{PSL}(2,\mathbb C)\) acts freely on simple parametrized spheres, with the marked point transformed contragrediently, and preserves the fiber. Freeness follows because a reparametrization fixing a simple map fixes its injective locus and hence is the identity. A manifold of dimension \(4\) or \(5\) cannot contain this six-dimensional orbit. The fiber is empty.

For a path, the local perturbations just used may be multiplied by arbitrary bumps in the path parameter, supported away from its endpoints. If finitely many immersed \(F\)-spheres are retained, a critical simple \(F\)-sphere cannot have one of those images. It is therefore disjoint from all of them by positivity of intersections, so every injective disk needed in the proof still admits the allowed local variations. Use the closure of smooth primitives supported off the retained images; this preserves them exactly without requiring a fixed forbidden tubular neighborhood. At a fixed model subdisk, every point already lies on a model \(F\)-curve. A competing critical sphere is disjoint from the whole subdisk for the same reason. The endpoint assertion follows with perturbations supported in its complement.

There are countably many map-space charts and homotopy components, so the regular parameter choices can be made simultaneously for all of them. The conclusions are open by 39. Approximation in the defining closures by smooth primitives therefore gives smooth perturbations with the same conclusion and the same support restrictions. For paths with immersed endpoints, openness allows fixed small parameter collars at both ends. Smallness ensures positivity with \(B\) fixed; the straight segment realizes the initial small perturbation by an exact positive-pair path. ◻

Proof. Represent nearby immersions by normal graphs over \(u\), removing the infinitesimal reparametrizations. If \(u\) is simple, its stabilizer under reparametrization is trivial: an automorphism fixing the map fixes its injective locus and hence is the identity. The normal-graph slice then also parametrizes the local unparametrized moduli. For a multiply covered immersion we make the smoothness assertion only for this slice. The symplectic form \(A\) identifies the real normal bundle with \(T^*L\) by \(w\mapsto\alpha=u^*\iota_wA\). Write along the source \[ H=\frac{C-xA}{y},\qquad \Omega=A+iH,\qquad x=\frac{A\wedge C}{A^2},\qquad y^2=\frac{C^2}{A^2}-x^2, \tag{102}\] where the nonzero sign of \(y\) is the one giving \(J\) tamed by \(B\). Let \(*_L\) be the star on one-forms for the induced conformal structure and orientation, with \(*_L\alpha=-\alpha\circ j\). The complex volume identity gives \[u^*\iota_wC=x\alpha+y*_L\alpha.\] Cartan’s formula now gives the normal derivative \[ \mathcal D\alpha =\left(\mathrm d\alpha,\mathrm d(x\alpha+y*_L\alpha)\right). \tag{103}\] The nonlinear pullbacks and their derivatives take values in pairs of exact two-forms: their integrals are the two fixed zero periods.

For completeness we solve [eq:pencil-normal-operator]. Given exact two-forms \(\zeta_1,\zeta_2\), choose \(\alpha_0\) with \(\mathrm d\alpha_0=\zeta_1\). Adding \(\mathrm df\) changes only the second equation, by \[ \mathcal P f =\mathrm d(x\mathrm df+y*_L\mathrm df) =\mathrm dx\wedge\mathrm df+\mathrm d(y*_L\mathrm df). \tag{104}\] After division by a positive area form its principal part is the nonzero signed multiple \(y\Delta f\) of the scalar Laplacian. Its sign is constant on connected \(L\), it has no zeroth-order term, and the strong maximum principle shows that its kernel consists exactly of constants (Kiselev et al. 2012, chap. 1, Theorem 1.6). Here one first adjusts the overall sign of the operator and then applies the principle at a maximum on the connected closed surface. It has index zero, by homotopy through scalar elliptic operators to the signed Laplacian, with the sign of \(y\) held fixed; the Fredholm index is constant along this homotopy (Taylor, n.d.-b, sec. 7, Propositions 7.3–7.4). Its image has integral zero by Stokes’ theorem. Its cokernel has dimension one, so its range is exactly the integral-zero two-forms. We can therefore solve \[\mathcal P f=\zeta_2-\mathrm d(x\alpha_0+y*_L\alpha_0).\] This proves surjectivity onto the asserted target, with bounded elliptic right inverses in the usual Sobolev or Hölder spaces.

If \(\mathcal D\alpha=0\), then \(\alpha\) is closed. Every class in \(H^1(L;\mathbb R)\) has a unique representative of this kind: start with any closed representative and solve [eq:pencil-scalar-operator] for an exact correction. Uniqueness follows from the constant kernel of \(\mathcal P\). Thus \[ \ker\mathcal D\longrightarrow H^1(L;\mathbb R),\qquad \alpha\longmapsto[\alpha] \tag{105}\] is an isomorphism. The implicit function theorem applied to the pullback equations, with exact two-forms as target, proves the first assertion and its parameter version. For \(L=S^2\) the derivative is an isomorphism, giving isolation and unique continuation.

For an embedded torus, the kernel has dimension two. In complex normal notation its elements solve the real-linear normal Cauchy–Riemann equation. Indeed, the normal projection of a linearized map equation is unchanged by variation of the source complex structure, and the two maps in [eq:pencil-pullback-linearization] identify that normal equation with [eq:pencil-normal-operator]. The complex normal line has degree \(F^2=0\). The similarity principle says that a nonzero solution has isolated positive zeros, whose total number with multiplicity is the degree; in the closed case this is the zero formula of (Wendl 2010, sec. 2.2, Equation (2.7)). Therefore every nonzero kernel section is nowhere vanishing. Evaluation at each point of the torus is consequently an isomorphism from the two-dimensional kernel to its real normal plane.

Let \(u_q\), \(q\in\mathbb R^2\) small, be the family supplied by the implicit function theorem. The derivative of \((x,q)\mapsto u_q(x)\) is an isomorphism at every \((x,0)\). Compactness of the torus gives a common small parameter disk on which this is a local diffeomorphism. The individual maps remain embeddings; different images cannot meet by \(F^2=0\) and positivity. The immersion slice distinguishes the images, so evaluation is injective after shrinking the disk. It is thus a diffeomorphism onto an open tube, foliated by the \(u_q(L)\). ◻

Proof. The unparametrized sphere space is compact by 39 and discrete by 41, so it is finite. Over a path, the total space of such spheres is compact and, by the parameter version of 41, its projection to the path interval is a local diffeomorphism, also in relative half-interval charts at the endpoints. Hence it is a finite covering of the interval. Equivalently, each isolated sphere has a unique local continuation, and compactness prevents any additional sphere from entering those local descriptions. The number is constant. ◻

Proof. Denote the node by \(p\) and its two preimages on the normalization by \(p_+,p_-\). We first identify the model with a neighborhood of \(K\) so that straight interpolation of the two pairs stays positive. We then localize this interpolation by two exact changes. At the node, equality of the forms as tensors makes the radial cutoff correction small. Along the rest of the sphere, the difference of the pairs need not be small; a logarithmic transverse cutoff makes its cutoff error small while retaining the positive interpolation.

Use \(x,y,\Omega\) from [eq:pencil-volume-normalization] on a neighborhood of \(K\). Choose the qualitative model of 44; write its normalization as \(u_Y:S^2\to Y\) with node preimages \(q_+,q_-\). Uniformization supplies a biholomorphism \(f\) of the normalized spheres carrying \(p_\pm\) to \(q_\pm\).

Matching the derivative and the complex volume. At \(p\), the two complex tangent lines of the branches are transverse. Their prescribed tangent maps, obtained from \(f\), define one complex linear isomorphism \[T:T_pX\longrightarrow T_{p_Y}Y.\] Choose the nonzero complex scale \(h\) of the model volume so that \(T^*\Sigma_Y=\Omega(p)\). Along each smooth point of the normalization, the tangent map and equality of complex volumes determine the map on the normal quotient. More explicitly, a nonvanishing complex volume identifies the complex normal line with the dual of the tangent line. The prescribed tangent map therefore gives the required normal-line isomorphism globally. The possible lifts to a map of the two ambient complex plane bundles form an affine bundle with fiber the complex-linear maps from the normal line to the target tangent line. A smooth section exists by partitions of unity. Its values at both preimages of the node can be chosen to equal the single map \(T\).

These prescribed first derivatives need only be realized to arbitrary accuracy along \(K\), with exact realization at the node. Here is a construction that addresses the crossing. First straighten the two transverse branches by smooth ambient charts in the source and target. Writing the restrictions of \(f\) in these charts as \(f_+(u)\) and \(f_-(v)\), the product map \(\Phi_0(u,v)=(f_+(u),f_-(v))\) is a genuine local diffeomorphism carrying the branches by \(f\) and having derivative \(T\) at \(p\). Its derivative along either branch and the prescribed complex-linear derivative have the same tangent restriction and agree at \(p\). They are consequently arbitrarily close on sufficiently short branch collars. On those collars interpolate their normal derivative data, keeping \(\Phi_0\) near \(p\) and the prescribed data away from the collars. The interpolated derivatives remain invertible when the collars are sufficiently short.

Away from the crossing the normal tubular construction realizes these jets: in normal coordinates send the zero section by \(f\) and use the prescribed normal derivative on the normal coordinate. This can be patched to \(\Phi_0\) where the jets already agree; a partition in normal coordinates preserves the prescribed first jets. Shrinking the tubes gives a diffeomorphism \(\Phi\) of neighborhoods of the nodal images. To check injectivity, any hypothetical pair of coincident images in arbitrarily shrinking neighborhoods would limit to two points of \(K\) with the same image. The map on the nodal image is a homeomorphism, and the map is a local diffeomorphism at every point, including the crossing. These facts rule out such a pair. We have arranged that \[ \Phi^*\Sigma_Y=\Omega\quad\text{at }p, \qquad \Phi^*\Sigma_Y\text{ is arbitrarily close to }\Omega \quad\text{along }K. \tag{106}\] Both volumes vanish when pulled back to the normalization.

Put \(x_p=x(p)\), \(y_p=y(p)\) and set on this neighborhood \[A_*=\operatorname{Re}\Phi^*\Sigma_Y,\qquad C_*=x_p\operatorname{Re}\Phi^*\Sigma_Y +y_p\operatorname{Im}\Phi^*\Sigma_Y.\] These forms are closed. If the volume matching along \(K\) were exact, their pointwise relation to the old pair would be \[ A_*=A,\qquad C_*=\left(x_p-\frac{y_px}{y}\right)A+\frac{y_p}{y}C. \tag{107}\] The ratio \(y_p/y\) is positive. Thus every straight interpolation of the two pairs in [eq:pencil-triangular-matching] has the same positive anti-invariant plane and remains in the same component relative to \(B\). In particular its triple Gram matrix has a uniform positive lower bound along \(K\), with the interpolation parameter in \([0,1]\). Choose the approximation in [eq:pencil-volume-matching] small compared with this bound. The actual straight interpolation between \((A,C)\) and \((A_*,C_*)\) is then positive along \(K\), and on a smaller neighborhood by compactness and continuity. The biholomorphism \(f\) uses the complex orientation selected by \(B\), so \(B\) is positive on the transported model’s complex tangent line at each smooth point of \(K\). Together with positivity of the triple, this selects the model complex structure as the sign tamed by \(B\). That sign persists on a smaller connected model neighborhood, independently of whether \(y_p\) is positive or negative.

The first exact correction, at the crossing. For either closed difference \(\xi=A_*-A\) or \(\xi=C_*-C\), one has \(\xi(p)=0\) as a tensor, and the pullback of \(\xi\) to each branch vanishes. In a crossing chart with \(p=0\), the radial homotopy primitive is \[ \beta_z(v)=\int_0^1 t\,\xi_{tz}(z,v)\,\mathrm dt, \qquad \mathrm d\beta=\xi. \tag{108}\] It satisfies \(|\beta(z)|\leq C|z|^2\). Since radial contraction preserves each straightened branch, \(\beta\) pulls back to zero on both branches. Choose \(\chi_r\) equal to one on the ball of radius \(r\), supported in the ball of radius \(2r\), and satisfying \(|\mathrm d\chi_r|\leq C/r\). The exact change \(\mathrm d(\chi_r\beta)\) has norm \(O(r)\): the two terms \(\chi_r\xi\) and \(\mathrm d\chi_r\wedge\beta\) both have that bound. For sufficiently small \(r\) its scalar multiples preserve positivity with \(B\) fixed. They preserve the common Lagrangian sphere because the primitive restricts to zero there. Do this for both differences. The resulting pair \((A',C')\) equals \((A_*,C_*)\) on the ball of radius \(r\) and differs from \((A,C)\) by \(O(r)\) near that ball.

The exact correction along the rest of the sphere. The remaining differences \(\xi'=A_*-A'\) and \(\xi'=C_*-C'\) are closed, vanish on a neighborhood of the crossing, and pull back to zero on the source. Outside a smaller crossing ball, take normal tubes of the embedded part of the source. Normal contraction \(h_t(s,n)=(s,tn)\) gives a relative primitive \(\beta'\) satisfying \[ \mathrm d\beta'=\xi',\qquad |\beta'|\leq C d, \qquad d=|n|. \tag{109}\] This is the homotopy identity \(\mathrm d\beta'=\xi'-\pi^*u^*\xi'\). Choose the tubes in the inner collars so that the contraction stays where \(\xi'=0\). Then \(\beta'=0\) on those collars and extends by zero into the crossing ball. The two branch tubes are disjoint outside that ball after shrinking. This gives a single smooth relative primitive on the required tubular region, without a compatibility condition left at the node.

Choose a transverse cutoff equal to one for \(d\leq r_{\rm in}\) and zero for \(d\geq r_{\rm out}\), where the outer tube is already small. Using a fixed smooth function of \(\log(d/r_{\rm in})/\log(r_{\rm out}/r_{\rm in})\) gives \[ d\,|\mathrm d\chi|\leq \frac{C}{\log(r_{\rm out}/r_{\rm in})}. \tag{110}\] The inner radius can be made sufficiently small that the right side is any prescribed \(\eta>0\). For \(0\leq t\leq1\), the exact change is \[A'+t\mathrm d(\chi\beta'_A) =A'+t\chi(A_*-A')+t\mathrm d\chi\wedge\beta'_A,\] and likewise for \(C'\). The last terms have norm at most \(C\eta\) by [eq:pencil-tubular-primitive,eq:pencil-log-cutoff]. The first terms are pointwise straight interpolation with parameter \(t\chi\in[0,1]\). Taking \(r\) small in the first correction preserves the positive margin already established along \(K\). Shrinking the outer tubes preserves that margin throughout them. Finally take \(\eta\) small compared with the margin. These choices prove positivity along the entire exact path.

The final pair agrees with the model on a neighborhood of the whole nodal image, and its primitives have compact support in the original chosen neighborhood. A sufficiently small complete model subdisk lies in this region: properness of \(p_Y\) and compactness of its central fiber imply that \(p_Y^{-1}(\Delta_\rho)\) lies in any chosen neighborhood of that fiber when \(\rho\) is sufficiently small. Every modification restricts to zero on the normalization, so the same sphere is retained. Disjoint supports prove the last assertion. ◻

Proof of 38. First apply 40 to make a small exact perturbation with \(B\) fixed so that all \(F\)-spheres are immersed. Connect it to the starting pair by a small straight segment of positive pairs. There is no assertion about immersion along this initial segment. By 42 the perturbed pair has finitely many nodal spheres \(K_1,\ldots,K_N\), all disjoint. If \(N=0\), the torus argument below already applies.

For \(N>0\), use 43 at all the \(K_i\), keeping \(B\) fixed and retaining every \(K_i\) along the resulting path. Choose smaller closed model subdisks at its endpoint. Make the endpoint generic by an arbitrarily small exact perturbation outside those subdisks. The relative assertion of 40 applies because any competing \(F\)-curve is disjoint from every model fiber. The endpoint now has no critical spheres, and the smaller model neighborhoods remain unchanged. Extend this endpoint perturbation a short distance into the path using a parameter cutoff; all retained \(K_i\) are still common Lagrangians.

The two ends of this latter path have only immersed spheres. Apply the path version of 40, keeping its endpoints and every \(K_i\) fixed, to remove all critical spheres from its interior. The resulting smooth path stays positive with \(B\) fixed. By 42, its endpoints have the same number of nodal spheres. All the original \(N\) spheres were retained, so they exhaust the endpoint spheres. This argument is needed to exclude additional immersed spheres outside the inserted model regions.

We now work at the endpoint. By 37, every point of \(X\) lies on a unique simple \(F\)-curve image. Each is an embedded torus or one of the \(K_i\). Define \(S\) as the set of these images, with quotient map \(p_X\). For a torus, 41 supplies an open tube diffeomorphic to \(T^2\times\Delta\). It is saturated: any other \(F\)-curve meeting a torus in this tube must equal it by intersection positivity. For a nodal sphere choose an open model subdisk. Its fibers also represent \(F\), since they are homologous to its central fiber. The same argument makes this open subset saturated. Its quotient is its base disk, since the proper holomorphic model projection is open and has connected fibers. Thus the torus tubes and the model subdisks give disk neighborhoods in \(S\) and cover it.

These saturated neighborhoods can be chosen arbitrarily small around an entire fiber. For tori this follows by shrinking the parameter disk in the tube; for nodes it follows from properness of the model projection. Two distinct fibers are disjoint compact subsets of \(X\), so they admit disjoint ambient neighborhoods. Choose saturated neighborhoods inside them. Their images are disjoint open neighborhoods of the two points of \(S\). The quotient is therefore Hausdorff.

The disk charts have smooth transition maps. To verify this even at a singular value, choose a smooth point of the corresponding fiber. Near that point both local quotient projections are smooth submersions with the same fibers. A local section of either submersion expresses the other base coordinate as a smooth function of it; interchanging the two gives its smooth inverse. The charts therefore make \(S\) a smooth surface without boundary and \(p_X\) a smooth map. They also make \(p_X\) open. Since \(X\) is second countable, images of a countable open basis under this open surjection form a countable basis for \(S\). Compactness and connectedness of \(X\) give compactness and connectedness of \(S\).

On regular fibers use their complex orientation and the complex orientation of the normal plane to orient the base. Evaluation along a connected torus preserves this choice, and the holomorphic model gives the same orientation on its regular part. Hence the orientation extends across each singular value. The map is proper because its domain is compact and its base is Hausdorff. Its singularities are precisely the ordinary holomorphic nodes of the models (locally \(s=z_1z_2\) after complex coordinate changes). The pair vanishes on every fiber by construction, and the prescribed model identities hold on the unchanged smaller neighborhoods.

All paths used above consist of exact changes to \(A,C\) and keep \(B\) fixed. Their finitely many joins can be made smooth by reparametrizations constant to infinite order at the join times. Concatenating them proves the theorem. ◻

Proof. We compare the families through the degree-three divisors given by three times their sections. Their line bundles embed the fibers as plane cubics. Laurent coefficients at the section, measured in the normalized Abel coordinate, will identify the three-dimensional spaces of sections across the puncture; taking closures will then identify the completed families.

Each \(p_\rho\) is a flat proper family of connected reduced curves. For flatness, the local ring of the smooth total surface has no torsion over the local ring of the one-dimensional smooth base, since \(p_\rho-s_0\) is not a zero divisor. Torsion-free modules over this discrete valuation ring are flat. The fibers have arithmetic genus one, and \(S_\rho\) is a Cartier divisor in the fiberwise smooth locus.

Set \(\mathcal L_\rho=\mathcal O_{Y_\rho}(3S_\rho)\). This line bundle is flat over the base, since it is locally free on the flat family. Its restriction to each fiber is a degree-three very ample line bundle, with \(h^0=3\) and \(h^1=0\). For the complex-linear Dolbeault operator on a line bundle of degree \(d\) over a closed connected Riemann surface of genus \(g\), the complex index is \(d+1-g\), and \(d>2g-2\) gives surjectivity (Wendl 2014, sec. 3.4, Theorems 3.22–3.23). On a smooth genus-one fiber this gives the asserted dimensions. After imposing any length-two subscheme, the remaining degree is one and the first cohomology still vanishes. This proves very ampleness on those fibers. For the nodal curve one can check the assertion explicitly. Normalize it by \(\mathbb P^1\), with the two preimages of the node at \(0,\infty\). A degree-three line bundle pulls back to \(\mathcal O_{\mathbb P^1}(3)\), with a nonzero gluing parameter \(\lambda\) for its fibers at these points. Its sections are degree-at-most-three polynomials with constant coefficient \(\lambda\) times the leading coefficient. The basis \(w,w^2,w^3+\lambda\) gives the map \[w\longmapsto[w:w^2:w^3+\lambda].\] It identifies exactly \(0\) and \(\infty\); away from these points the ratio of the second coordinate to the first is \(w\). At \(0\) and \(\infty\) the two tangent lines are distinct, as is seen in the affine chart about \([0:0:1]\). Thus it embeds the nodal curve as a nodal cubic. To compute its cohomology directly, let \(N\) denote the nodal fiber, \(\nu:\mathbb P^1\to N\) its normalization, and \(L=\mathcal L_\rho|_N\). The gluing condition gives the exact sequence \[0\longrightarrow L\longrightarrow \nu_*\mathcal O_{\mathbb P^1}(3) \xrightarrow{\operatorname{ev}_0-\lambda\operatorname{ev}_\infty} \mathbb C_{\mathrm{node}}\longrightarrow0,\] where the last sheaf is supported at the node and its fiber is \(\mathbb C\). In the polynomial trivializations just used, the map on global sections is the constant coefficient minus \(\lambda\) times the leading coefficient. It is surjective. The normalization is finite, so the cohomology of its direct image is that of \(\mathcal O_{\mathbb P^1}(3)\). The latter has \(h^0=4\) and \(h^1=0\). The cohomology exact sequence therefore gives \(h^0(N,L)=3\) and \(h^1(N,L)=0\).

We use the following proper holomorphic base-change theorem: for a proper holomorphic map to a reduced base and a coherent sheaf flat over that base, constancy of the fiber dimensions \(h^q\) implies that the \(q\)-th direct image is locally free and its restriction to each base point is the cohomology of that fiber; see (Grauert 1960, sec. 7, Satz 5) and the correction (Grauert 1963). Our map is proper and surjective, the disk is reduced, and \(\mathcal L_\rho\) is flat over it. Applying the theorem with \(q=0\) and the constant value \(h^0=3\) gives a holomorphic rank-three bundle \[E_\rho=(p_\rho)_*\mathcal L_\rho\] with those spaces as fibers. Evaluation gives a relative closed embedding in \(\mathbb P(E_\rho^*)\). Indeed the fiberwise embeddings and base change give the evaluation and separation properties locally over the base. At the nodal point, the fiber’s Zariski tangent space is the whole tangent space of the smooth total surface, so tangent separation there also gives an immersion of the total surface. Properness then gives the closed embedding.

Near the section \(p_\rho\) is a submersion. Write \(\Omega_\rho=f(s,w)\,ds\wedge dw\), with \(w=0\) on the section. The coordinate \[z(s,w)=\int_0^w f(s,u)\,du\] is holomorphic, has nonzero relative derivative, and is zero on the section. It extends the normalized local Abel coordinate also across \(s=0\). Regard sections of \(\mathcal L_\rho\) as meromorphic functions with poles of order at most three on \(S_\rho\). Their Laurent coefficients in degrees \(-3,-2,-1,0\) define \[ E_\rho\longrightarrow\mathcal O_{\Delta_R}^{\,4}. \tag{127}\] This map is fiberwise injective. A function in its kernel has no pole and hence is constant on the compact connected reduced fiber; its zero constant coefficient makes it zero. Thus its image is a rank-three subbundle, including over zero.

The images agree on the punctured disk by the regular elliptic identifications. Continuity of their Grassmannian-valued maps makes them agree also at zero, identifying the bundles \(E_\rho\). Their relative cubic images agree as well: each is the closure of its image over the punctured disk, and both are reduced analytic subspaces. This gives an isomorphism of the completed families. The holomorphic two-forms agree on the regular part and therefore everywhere. ◻

Proof. We use the two-cycle comparison from (Gross and Wilson 2000, proof of Lemma 4.3). The annular bundle retracts to the mapping torus of one shear. Its homology exact sequence is \[0\longrightarrow H_2(T^2;\mathbb R) \longrightarrow H_2(p^{-1}(K);\mathbb R) \longrightarrow\ker(T_*-1:H_1(T^2;\mathbb R)\to H_1(T^2;\mathbb R)) \longrightarrow0.\] The first term is generated by a fiber, and the last by the invariant \(a\)-circle. This circle has a sweep at \(b=0\). These two cycles thus test all real two-periods.

The fiber periods were computed above. The sweep is the connection circle bundle over a base circle in \(t=\rho/2\). The connection circle bundle over the full base disk in this slice is a solid torus bounding it in \(Y\). On the sweep \(dt=0\), and the base has only one tangent direction, so the restriction of \(dt\wedge\theta+V_\rho dx_1\wedge dx_2\) is zero. The invariant comparison form also has zero restriction, because \(db=0\) and the same dimensional observation applies. ◻

Completion of the proof of 44. The construction and 45 give the fixed holomorphic nodal disk. For general \(h=|h|e^{i\phi}\), rotate \(\gamma_1+i\gamma_2\) by \(e^{i\phi}\), scale the full triple and the metric by \(|h|\), and take \[\rho=\epsilon/|h|,\qquad D^{\mathrm{loc}}_\epsilon=|h|\gamma_3,\qquad g_\epsilon=|h|g_\rho.\] The holomorphic two-form becomes \(\Sigma=h\,ds\wedge dz\), the fiber area becomes \(\epsilon\), and the equal-square identities give [eq:nodal-exact-identities]. The scaled annular expression is precisely [eq:nodal-semiflat], with coefficient \(|h|^2/\epsilon\). The exponential comparison and both period identities follow from [eq:nodal-area-one-comparison,lem:nodal-annular-periods]. The polynomial estimates persist because \(h\) is fixed and nonzero. This proves all assertions. ◻

Proof. In a local translation chart define the average by integrating pullbacks against normalized Haar measure on \(\mathbb R^2/\mathbb Z^2\). The conjugacy in [eq:regular-angle-transition] and invariance of Haar measure make the definitions agree on overlaps. Each pullback commutes with \(\mathrm d\), which proves the assertion about differentiation.

For exactness, choose a translation-invariant horizontal splitting of \(TX^\circ\to p^*TS^\circ\) and a smoothly varying invariant flat metric on the vertical tori. Such choices exist: make them locally and patch horizontal lifts and vertical inner products by partitions of unity pulled back from the base. The transition rule above preserves invariance, and the relevant sets of choices are affine or convex.

Filter differential forms by horizontal degree. On the associated graded complex the leading differential is the vertical exterior derivative \(\mathrm d_v\). For each horizontal degree, fiberwise Hodge theory (Taylor 2010, Equations (1.11)–(1.22)) provides a homotopy \(H_v=\mathrm d_v^*G_v\) on the vertical complex, with the usual sign for its horizontal degree. Here \(G_v\) is the inverse of the vertical Laplacian on the complement of its kernel. Its harmonic forms are precisely the translation-invariant forms. On the zero-average subspace, therefore, \[ \mathrm d_v H_v+H_v\mathrm d_v=\mathop{\mathrm{Id}}. \tag{139}\] These operators act smoothly in the base: in a local torus chart they are the inverses on nonzero Fourier modes for a smoothly varying flat metric. They preserve zero average and use no values at other base points.

Suppose the smallest horizontal degree of the closed zero-average form \(\xi\) is \(r\), and denote its component of that degree by \(\xi_r\). Closedness gives \(\mathrm d_v\xi_r=0\). By [eq:regular-vertical-homotopy], subtracting \(\mathrm d(H_v\xi_r)\) removes \(\xi_r\), and the remainder is closed, has zero average, and has horizontal degree at least \(r+1\). Repeat this step. The horizontal degree is at most two, so the procedure terminates. A component of vertical degree zero that is vertically closed and has zero average is already zero. The sum of the primitives gives \(\beta\). Every operation is local in the base, including the differentiations used in the successive remainders. Hence the support assertion follows. Applying the construction to \(\xi-\operatorname{Av}\xi\) proves the remaining claims. ◻

Proof. Choose covectors \(e_1,e_2\) vanishing on \(L\) and complementary covectors \(f_1,f_2\) so that \[A=e_1\wedge f_1+e_2\wedge f_2, \qquad C=\sum_{i,j=1}^2 M_{ij}e_i\wedge f_j+c\,e_1\wedge e_2.\] Direct wedge multiplication yields \[A\wedge C=\tfrac12\mathop{\mathrm{tr}}(M)A^2, \qquad C^2=\det(M)A^2.\] Writing \(h=\mathop{\mathrm{tr}}(M)/2\) and \(T=M-h\mathop{\mathrm{Id}}\) gives \[ (rA+tC)^2=\bigl((r+th)^2+t^2\det T\bigr)A^2. \tag{140}\] Thus the pair is definite exactly when \(\det T>0\). Set \[T=\begin{pmatrix}u&v\\w&-u\end{pmatrix},\qquad q=\tfrac12(v-w),\qquad r_0=\tfrac12(v+w).\] Then \(\det T=q^2-r_0^2-u^2\). Its two positive components are \(q>\sqrt{r_0^2+u^2}\) and \(q<-\sqrt{r_0^2+u^2}\), both convex; \(h\) and \(c\) are unrestricted. This proves the first assertion. A quotient two-form wedges to zero with both members of a fiber-Lagrangian pair, and its own square vanishes. This proves the assertion about basic additions, also when both members change.

At a fixed point all translated pullbacks of \(C\) lie in one of the two components: translations preserve \(A\) and can be connected to the identity within the torus. Their compact average remains strictly inside that convex component, as does the segment to the average. The common Lagrangian plane is a complex line by 5. Continuing the choice of sign of the almost complex structure along the path keeps its orientation on this plane, and hence the quotient orientation, fixed. ◻

Proof. A fixed tamer for \(J_{t_0}\) remains a tamer on a parameter neighborhood of \(t_0\), by openness and compactness of \(X\). Choose finitely many such neighborhoods and corresponding fixed representatives. Near \(0\) and \(1\) use the prescribed representatives. Choose a subordinate smooth nonnegative partition of unity in the parameter, with the endpoint representative having weight one sufficiently near its endpoint. Their weighted sum is closed on \(X\) and represents \(H\), because all coefficients depend only on \(t\) and sum to one. It tames \(J_t\) by convexity of the taming cone. ◻

Proof. Let \(J_t\) be the structures defined by the pair, with the continued sign. First produce a convenient taming class. On the regular region, \(J_t\) preserves the vertical tangent planes and induces an oriented complex structure on each quotient \(T_xX/\ker\mathrm dp_x\). This quotient structure need not be independent of the point in the fiber. Nevertheless the positive base form satisfies \[p^*\nu_S(\zeta,J_t\zeta)>0 \quad\text{whenever }\mathrm dp(\zeta)\ne0,\] and it vanishes on vertical vectors. The restriction of \(B_0\) to the oriented vertical planes stays positive.

On a compact regular region containing the supports of the pair changes, use a smooth splitting \(\zeta=v+h\) into vertical and horizontal vectors and any fixed background norm. Compactness in space and parameter gives positive constants \(a,b\) and a finite \(M\) such that \[\begin{align*} B_0(\zeta,J_t\zeta)&\ge a|v|^2-M|v||h|-M|h|^2 \ge \tfrac a2|v|^2-M'|h|^2,\\ p^*\nu_S(\zeta,J_t\zeta)&\ge b|h|^2. \end{align*}\] Thus \(B_0+Kp^*\nu_S\) tames for one sufficiently large \(K>0\). On the unchanged nodal neighborhoods, \(B_0\) already tames and \(p^*\nu_S\) is semipositive. Indeed, the model matching in 43 chooses its complex structure with the sign tamed by \(B_0\), and its projection is holomorphic for that sign. The same \(K\) therefore works on all of \(X\) and at every parameter.

Put \(H_K=[B_0]+KF\) and \(V=[A_0]^\perp\cap[C_0]^\perp\). By [eq:regular-basic-class], \(H_K\) is the class just constructed. It belongs to \(V\), and \[H_K^2=1+2K[B_0]F>0,\qquad H_K[B_0]=1+K[B_0]F>0.\] Consequently \(H_K\) and \([B_0]\) lie in the same timelike component of \(V\). To apply 18, let \(d\) be the class, identified with its Poincaré dual, of a nonconstant irreducible \(J_t\)-curve. It has \(H_Kd>0\). By 36, either \(Fd\ne0\) and its sign is the sign of the coefficient \(c\) in \(d^+=cF^+\), or \(d=cF\). Also \([B_0]d=c[B_0]F\), with \([B_0]F>0\). If \([B_0]d\) were nonpositive, the alternatives would give \(c<0\) and \(Fd\le0\), so \(H_Kd=[B_0]d+KFd<0\), a contradiction. Thus every such curve pairs positively with \([B_0]\). The hypotheses of 18 hold with the already tamed class \(H_K\), and that theorem supplies a tamer in \([B_0]\).

When the initial pair is \((A_0,C_0)\), apply 50 to the family \(J_t\) and the fixed class \([B_0]\), using \(B_0\) at the initial endpoint and any prescribed endpoint tamer. This gives the asserted smooth representatives. By 5, their triples with \((A_t,C_t)\) are positive. ◻

Proof of 47. We first average the pair and construct a closed invariant form of unit fiber area. We then adjust its two total pairings so that exact basic changes of the pair can impose pointwise orthogonality. Throughout, disks may be replaced by smaller nested disks, with closures inside the original model disks.

Averaging the second form. On \(X^\circ\) set \(\overline C=\operatorname{Av}C_0\). It is closed and equals \(C_0\) on every model disk, so it extends smoothly over \(X\). The difference \(\overline C-C_0\) is supported over a compact subset of \(S^\circ\) and is exact on \(X\) by 48. The path \[(A_0,(1-t)C_0+t\overline C)\] is a definite pair path by 49, with the same periods and the same oriented fiber planes. It admits a completion in \([B_0]\) by 51. For the remaining construction write \(A=A_0\) and \(C=\overline C\). Both are invariant.

A closed form of unit fiber area. The positive number \(q_0=[B_0]F\) is the common integral of \(B_0\) over the regular fibers. Start with \[\Psi^{(0)}=\operatorname{Av}(B_0/q_0) \quad\text{on }X^\circ.\] This is closed and invariant, and its restriction to every fiber is the invariant two-form of integral one.

We next make it equal to \(\mathrm da\wedge\mathrm db\) near every puncture. On a punctured model disk, the two real two-homology generators are the fiber and the torus obtained by sweeping the invariant \(a\)-cycle around a base circle at \(b=0\). One can see this from the homology sequence for a torus mapping torus: the monodromy is one shear, so the fiber contributes one generator and the invariant part of its first homology contributes one more. The second generator bounds a three-chain in the full nodal neighborhood, as described in 44. The smooth closed form \(B_0/q_0\) consequently has periods \(1\) and \(0\) on these generators. Averaging does not change them, by 48. The form \(\mathrm da\wedge\mathrm db\) has the same periods: its fiber integral is one, and its restriction to the sweep at \(b=0\) is zero. It is a well-defined closed form under the shear transition of the angular coordinates.

It follows that \(\Psi^{(0)}-\mathrm da\wedge\mathrm db=\mathrm d\beta_j\) on the punctured neighborhood. Replace \(\beta_j\) by its average, so it is invariant. Choose a basic cutoff \(\chi_j\) that is one on a smaller punctured disk and zero near the outer boundary, and replace \(\Psi^{(0)}\) there by \[\Psi^{(0)}-\mathrm d\bigl((p^*\chi_j)\beta_j\bigr).\] These disjoint modifications give a smooth closed invariant form \(\Psi^{(1)}\) on \(X^\circ\), equal to \(\mathrm da\wedge\mathrm db\) near the punctures. Its fiber restriction is unchanged: the two invariant forms being compared already have the same restriction of integral one, and a differential of a basic cutoff has zero vertical restriction.

The obstruction to an exact orthogonality correction. The form \(\Psi^{(1)}\) has the required fiber restriction and nodal behavior. It remains to arrange orthogonality to the pair while preserving the pair’s cohomology classes. Invariance reduces these pointwise equations to a calculation on the base.

Let \(\Phi\) be an invariant two-form on \(X^\circ\) whose restriction to each fiber has integral one. For any invariant four-form \(\Xi\), we have the pointwise identity \[ \Phi\wedge p^*(p_*\Xi)=\Xi. \tag{141}\] Indeed, only the vertical restriction of \(\Phi\) contributes to its wedge with a basic two-form. Invariant top forms have constant vertical density on each fiber and are therefore determined by their fiber integral; both sides of [eq:regular-invariant-top-form] have the same integral on every fiber after fixing two base tangent vectors.

In particular, replacing \(A\) by \(A-p^*p_*(\Phi\wedge A)\) makes it orthogonal to \(\Phi\), and the same holds for \(C\). On the closed oriented surface \(S\), a smooth two-form is exact precisely when its integral is zero. We therefore first adjust \(\Psi^{(1)}\) so that both resulting base two-forms have zero integral. Their vanishing near the punctures will allow us to regard them as smooth forms on all of \(S\).

Removing the two total pairings. There are smooth closed invariant forms \(\widetilde A,\widetilde C\) on \(X\), representing \([A],[C]\), supported over a compact subset of \(S^\circ\), and vanishing on vertical planes. To construct them, observe that a full one-node neighborhood has the homotopy type of a torus with its primitive vanishing cycle collapsed; its real second homology is generated by the fiber. Hence \(A,C\) are exact there, since their fiber integrals vanish. Subtract differentials of local primitives multiplied by basic cutoffs equal to one near the nodal fiber. This gives global cohomologous forms that vanish on smaller model neighborhoods. Their restrictions to vertical planes remain zero, because both the original forms and the differentials of the cutoffs have that property in the required restriction. Averaging these forms, and extending by zero near the nodes, gives \(\widetilde A,\widetilde C\). Their cohomology classes are unchanged by 48.

On a punctured model disk, \(\Sigma_j=h_j\mathrm ds\wedge(\mathrm da+\tau\mathrm db)\): the term \(b\tau'(s)\mathrm ds\) in \(\mathrm dz\) disappears after wedging with \(\mathrm ds\). Thus \(A,C\) have only mixed base–fiber terms there, and \[(\mathrm da\wedge\mathrm db)\wedge A =(\mathrm da\wedge\mathrm db)\wedge C=0.\] Consequently the integrals \[I_A=\int_{X^\circ}\Psi^{(1)}\wedge A, \qquad I_C=\int_{X^\circ}\Psi^{(1)}\wedge C\] are well-defined: their integrands vanish over smaller punctured disks. The matrix of pairings of \(\widetilde A,\widetilde C\) with \(A,C\) is their cohomology Gram matrix, hence the identity under our normalization. Set \[ \Psi=\Psi^{(1)}-I_A\widetilde A-I_C\widetilde C. \tag{142}\] The form \(\Psi\) remains closed, invariant, of unit fiber area, and equal to \(\mathrm da\wedge\mathrm db\) near the punctures. By construction, \[ \int_{X^\circ}\Psi\wedge A =\int_{X^\circ}\Psi\wedge C=0. \tag{143}\] Without the normalization one would solve the same two equations using the positive invertible period Gram matrix.

Removing the pointwise pairings. Fiber integration of the invariant four-forms gives base two-forms \[\zeta_A=p_*(\Psi\wedge A),\qquad \zeta_C=p_*(\Psi\wedge C).\] They vanish near the punctures and so extend by zero to smooth forms on \(S\). Their integrals are zero by [eq:regular-zero-total-pairings]. Since \(S\) is a closed connected oriented surface, there are smooth one-forms \(\kappa_A,\kappa_C\) on \(S\) with \(\zeta_A=\mathrm d\kappa_A\) and \(\zeta_C=\mathrm d\kappa_C\).

Now make the simultaneous basic deformation \[ A_t'=A-tp^*\zeta_A, \qquad C_t'=C-tp^*\zeta_C, \qquad 0\le t\le1. \tag{144}\] Both changes are globally exact, since the primitives are the smooth pullbacks of \(\kappa_A,\kappa_C\). The changed forms are invariant, have zero vertical restriction, and agree with the original pair on smaller model neighborhoods. Their wedge-product matrix is unchanged by 49, so the path remains definite. [eq:regular-invariant-top-form] shows that its endpoint satisfies \[\Psi\wedge A_1'=\Psi\wedge C_1'=0.\] Moreover, contraction of a basic two-form with a vertical vector is zero. Hence the vertical Hamiltonian fields of base functions for \(A_t'\) are the same as those for \(A\), and their period lattice and torus translations are unchanged.

Finally concatenate the two pair paths, making their parameters flat at the join. They preserve the original pair periods, oriented fibers, and smaller nodal models. 51 gives a smooth completion by forms in \([B_0]\), with initial value \(B_0\). This proves all the assertions. We henceforth denote the endpoint pair again by \(A,C\) and its chosen third form by \(B\). ◻

Proof. We first check the three types of region in the construction. On a model core, \(A\) and \(C\) are fixed constant linear combinations of the two parallel real components of \(\Sigma\), and \(E_\varepsilon\) is the local parallel Kähler form. The model metric has volume \(A^2/2\). Consequently it is exactly \(g_\varepsilon\) there. The intrinsic charts and bounds of 44 therefore apply. This argument transports the metric, forms, and charts together through the holomorphic identification of the model disks. It requires no bound on derivatives of that identification in a separate fixed smooth atlas.

On the complement of smaller nodal disks, use finitely many fixed regular bundle charts. The coefficients in [eq:auxiliary-semiflat] and all their fixed-order derivatives grow at most polynomially in \(\varepsilon^{-1}\). Its products with \(A,C\) vanish and its square is \(A^2\). The remaining two-by-two wedge block is that of the fixed definite pair. It is uniformly positive on this compact region, and the same assertion holds on the cores because there its coefficients are fixed constants. The exponential annular gluing errors preserve these bounds on all of \(X\).

For completeness, the passage from these form bounds to metric bounds is algebraic. Normalize the frame by the positive square root of its wedge Gram matrix, obtaining \(\Theta_1,\Theta_2,\Theta_3\) with \(\Theta_i\wedge\Theta_j=2\delta_{ij}\nu\) and with the frame orientation chosen for a positive metric. In oriented coordinates, the identity \[ g_\varepsilon(v,w)\nu =\frac16\sum_{i,j,\ell=1}^3 \epsilon_{ij\ell} (\iota_v\Theta_i)\wedge(\iota_w\Theta_j)\wedge\Theta_\ell \tag{159}\] follows by evaluating on a standard self-dual orthonormal frame. In the fixed regular charts this gives polynomial coefficient and derivative bounds for \(g_\varepsilon\). Its determinant in these charts is fixed by \(\nu\), so the adjugate formula gives polynomial bounds for the inverse metric as well. The core metric and inverse bounds were already supplied by the transported model charts. Matrix square roots and inverses have bounded derivatives on the uniformly positive Gram matrices in question. This also proves the claimed frame bounds. On the fixed annuli all comparisons remain polynomial after the exponential changes. Curvature and any fixed number of its covariant derivatives are consequently polynomially bounded.

We spell out why local chart control gives a polynomial global cover. Shrink the charts uniformly by powers of \(\varepsilon\) so that every point has a chart containing an intrinsic ball of radius \(\rho_\varepsilon\ge c_1\varepsilon^{a_1}\). In that chart both the coordinate domain radius and the upper and lower metric eigenvalue bounds are controlled by powers of \(\varepsilon\). A coordinate ball of polynomial radius lies in a fixed smaller intrinsic ball, and its volume density has a polynomial lower bound. It follows that every intrinsic ball of radius, say, \(\rho_\varepsilon/8\) has volume at least \(c_2\varepsilon^{a_2}\), after a further common polynomial shrinkage if needed. A maximal disjoint collection of these balls has at most \((2c_2\varepsilon^{a_2})^{-1}\) members because the total volume is \(1/2\). Their enlarged balls cover \(X\) and still fit in controlled charts. Coordinate cutoff functions and a normalized partition of unity have polynomial derivative bounds. Even the crude overlap bound by the number of charts is polynomial. This proves (i) with the asserted global control.

Euclidean Sobolev estimates on this cover, followed by the polynomial coordinate and covariant-derivative comparisons, give polynomial intrinsic Sobolev constants. Since the dimension is four and \(k\ge6\), derivatives of order at most \(\lfloor k/2\rfloor\) are bounded in \(L^\infty\) by the \(H^k\) norm with such a constant. Apply the intrinsic product rule to each derivative of \(h\wedge\widetilde h\), placing the factor with fewer derivatives in \(L^\infty\) and the other in \(L^2\). This proves [eq:auxiliary-product].

Before normalization the residuals vanish on the cores and outside the gluing annuli. On the annuli their fixed-coordinate derivatives are exponentially small; all conversions just described cost only powers of \(\varepsilon^{-1}\). The constant corrections in [eq:auxiliary-normalization] introduce exponentially small coefficients multiplying forms with polynomial intrinsic bounds. The fixed total volume then gives [eq:auxiliary-residual-estimate] and proves (iv). ◻

Proof. Put \(E_1=A\), \(E_2=C\), and \(E_3=D^0_\varepsilon\), and let \(M\) be their pointwise inner-product Gram matrix for \(g_\varepsilon\). The equations uniquely specify the self-dual part \(\phi=h^+\): if \(f_i=t_i\nu\), then \[ \phi=\sum_{j=1}^3(M^{-1}t)_jE_j. \tag{162}\] Moreover \[ \langle\phi,E_i\rangle_{L^2}=\int_X f_i=0. \tag{163}\] Each \(E_i\) is closed and self-dual, hence harmonic. 6 gives \(b^+(X)=3\), so these three linearly independent forms are a basis of the harmonic self-dual forms. Thus [eq:auxiliary-cokernel-moments] is exactly the solvability condition for \(\mathrm d^+a=\phi\).

One may see existence directly from the Hodge Green operator \(G\) for \(g_\varepsilon\) (Taylor 2010, Equations (2.1)–(2.2)). Since \(\phi\) is orthogonal to the harmonic self-dual forms and the Hodge Laplacian commutes with the star, \(G\phi\) is self-dual and \(\Delta G\phi=\phi\). For a self-dual two-form \(\psi\), \(2(\mathrm d\mathrm d^*\psi)^+=\Delta\psi\). Therefore \[ h=2\mathrm d\mathrm d^*G\phi \tag{164}\] is exact and has self-dual part \(\phi\). This formula proves existence for each fixed metric; we will not estimate the Green operator itself.

If \(h\) is exact, Stokes’ theorem gives \[ 0=\int_Xh\wedge h =\|h^+\|_2^2-\|h^-\|_2^2, \qquad \|h\|_2=\sqrt2\,\|h^+\|_2. \tag{165}\] In particular an exact anti-self-dual two-form is zero, proving uniqueness. This identity is also the required \(L^2\) estimate, independent of any positive spectral gap.

For higher derivatives use \[ \mathrm dh=0,\qquad \mathrm d(*h)=2\mathrm d\phi,\qquad \mathrm d^*h=-2*\mathrm d\phi. \tag{166}\] The integrated Weitzenböck identity for two-forms gives \[\|\nabla h\|_2^2 \le 4\|\mathrm d\phi\|_2^2 +C\|\operatorname{Rm}\|_\infty\|h\|_2^2.\] To iterate, apply the same identity to the tensor-valued form \(\nabla^j h\). Commuting covariant differentiation with alternating derivative and divergence produces only curvature derivatives multiplied by lower derivatives of \(h\). Using [eq:auxiliary-divergence-identities], this yields, for fixed \(j\), \[\|\nabla^{j+1}h\|_2 \le C_j\|\phi\|_{H^{j+1}} +K_{j,\varepsilon}\|h\|_{H^j},\] where \(K_{j,\varepsilon}\) is a polynomial in finitely many suprema of curvature and its derivatives. Those suprema are polynomially bounded by 54. Induction starting with [eq:auxiliary-exact-ltwo] thus bounds \(\|h\|_{H^k}\) by a polynomial multiple of \(\|\phi\|_{H^k}\), without losing derivatives. Finally [eq:auxiliary-selfdual-data] and the frame and inverse-Gram bounds give the same estimate in terms of \(f\). Smooth approximation, or the fixed-metric Green formula, gives the Sobolev version. Uniqueness is always asserted for \(h\), rather than for its one-form primitive. ◻

Proof of 52. Fix once and for all an integer \(k\ge6\). Every constant and exponent in the following argument uses only finitely many derivatives at this order. Let \(\mathcal E^k_\varepsilon\) be the closed subspace of \(H^k\) two-forms consisting of exact forms. Equivalently, its elements are distributionally closed and have zero harmonic projection; they possess \(H^{k+1}\) primitives for this fixed metric.

Writing \(D_\varepsilon=D^0_\varepsilon+h\), the desired equations become the fixed-point equation \[ h=\mathcal T_\varepsilon(h):= \mathcal L_\varepsilon \left(-A\wedge D^0_\varepsilon,\, -C\wedge D^0_\varepsilon,\, \frac{A^2-(D^0_\varepsilon)^2-h\wedge h}{2}\right). \tag{167}\] This map is defined on all of \(\mathcal E^k_\varepsilon\). Indeed, the first two data have integral zero by [eq:auxiliary-exact-moments], as does the free term in the third slot. For each exact iterate, including Sobolev iterates, \(\int_Xh\wedge h=0\) by approximation and Stokes’ theorem. Thus every iteration satisfies precisely the three mean conditions of 55.

Let \(\delta_\varepsilon=\|\mathcal R_\varepsilon\|_{H^k}\), let \(P_\varepsilon\) be an inverse bound from [eq:auxiliary-inverse-bound], and let \(Q_\varepsilon\) be a multiplication bound from [eq:auxiliary-product]. Increasing them if necessary, write \[ P_\varepsilon\le C\varepsilon^{-p},\qquad Q_\varepsilon\le C\varepsilon^{-q},\qquad \delta_\varepsilon\le C\varepsilon^{-s}e^{-c/\varepsilon}. \tag{168}\] On the ball of radius \(R_\varepsilon=2P_\varepsilon\delta_\varepsilon\) in \(\mathcal E^k_\varepsilon\), \[\begin{align*} \|\mathcal T_\varepsilon(h)\|_{H^k} &\le P_\varepsilon\delta_\varepsilon+ \tfrac12P_\varepsilon Q_\varepsilon R_\varepsilon^2, \tag{169}\\ \|\mathcal T_\varepsilon(h)-\mathcal T_\varepsilon(\widetilde h)\|_{H^k} &\le P_\varepsilon Q_\varepsilon R_\varepsilon\, \|h-\widetilde h\|_{H^k}. \tag{170}\end{align*}\] Choose \(\varepsilon\) so small that \[ 2P_\varepsilon^2Q_\varepsilon\delta_\varepsilon<\frac12. \tag{171}\] This is possible because its left side is at most \(C\varepsilon^{-(2p+q+s)}e^{-c/\varepsilon}\), which tends to zero. The ball is invariant and the Lipschitz constant is less than \(1/2\). The contraction theorem gives an exact solution with \[ \|h\|_{H^k}\le 2P_\varepsilon\delta_\varepsilon. \tag{172}\] If the residual is zero, the same conclusion follows directly by taking \(h=0\). Polynomial losses can always be absorbed by replacing \(c\) with any fixed smaller positive exponent. Thus the correction is exponentially small also in \(C^0\), by the polynomial Sobolev embedding bound. The uniformly positive Gram matrix of the approximate triple stays positive, with the same choice of sign. [eq:auxiliary-contraction-map] gives [eq:auxiliary-exact-equations] exactly.

We finish by proving spatial smoothness; no simultaneous estimate over infinitely many derivative orders is needed. For the now fixed \(\varepsilon\), choose a Coulomb primitive \(a\in H^{k+1}\) of \(h\), so \(\mathrm da=h\) and \(\mathrm d^*a=0\). The three equations for \(D^0_\varepsilon+\mathrm da\), together with this gauge, form a first-order nonlinear system for the four components of \(a\). At the solution the principal symbol of its linearization sends a covector-valued unknown \(b\) at a nonzero covector \(\xi\) to the three wedge slots of \(\xi\wedge b\) against \(A,C,D_\varepsilon\), together with \(\iota_{\xi^\sharp}b\). If the three wedge slots vanish, \(\xi\wedge b\) belongs to the negative-definite wedge-orthogonal complement of the positive three-plane. But \((\xi\wedge b)^2=0\), so \(\xi\wedge b=0\). It follows that \(b\) is a multiple of \(\xi\), and the gauge slot then forces \(b=0\). The symbol is square, hence invertible.

Since \(k\geq6\) in real dimension four, the initial Sobolev order gives \(a\in C^{4,\alpha}\) and at least \(C^{3,\alpha}\) coefficients for this elliptic linearization, for some \(0<\alpha<1\). Differentiate the system in a smooth coordinate chart. The derivative of \(a\) satisfies the linearized elliptic system, with right-hand side involving only smooth background derivatives and already controlled first derivatives of \(a\). These coefficients and the right-hand side are \(C^{3,\alpha}\), so local first-order elliptic estimates with finitely differentiable coefficients (Taylor, n.d.-a, sec. 4, Theorem 4.3) give \(\partial a\in C^{4,\alpha}\) and hence \(a\in C^{5,\alpha}\). Repeating this argument gives smoothness. All constants in this bootstrap may depend on the fixed \(\varepsilon\). The resulting \(D_\varepsilon\) is therefore smooth and proves the theorem. ◻

Proof of 53. We identify integral curve classes with their Poincaré duals. Choose disjoint closed nodal neighborhoods \(N_i\) whose boundaries lie outside all gluing cutoffs, and put \(U=X\setminus\bigcup_i\operatorname{int}N_i\). The restriction of \(E_\varepsilon\) to \(U\) is \(D^{\mathrm{sf}}_\varepsilon\). If there is at least one node, \(U\) fibers over a compact surface with nonempty boundary, whose second real cohomology is zero. Thus the basic term in [eq:auxiliary-semiflat] is exact on \(U\), and \[ [D^0_\varepsilon]|_U =r_\varepsilon\bigl(\varepsilon[\Psi]|_U -a_\varepsilon[A]|_U-b_\varepsilon[C]|_U\bigr). \tag{173}\] In particular its restriction divided by \(\varepsilon\) is bounded.

When there is a node, the kernel of the restriction map \(H^2(X;\mathbb R)\longrightarrow H^2(U;\mathbb R)\) is exactly \(\mathbb RF\). To verify this assertion, excision and Poincaré–Lefschetz duality identify \[H^2(X,U;\mathbb R) \simeq\bigoplus_i H^2(N_i,\partial N_i;\mathbb R) \simeq\bigoplus_i H_2(N_i;\mathbb R).\] A one-node neighborhood has the homotopy type of a torus with its primitive vanishing cycle killed, and its second homology is generated by the fiber. Under the map to \(H^2(X;\mathbb R)\) these generators give the Poincaré dual fiber classes, all equal to the nonzero class \(F\). The long exact sequence of the pair gives the assertion. Hence restriction induces an injective map from \(H^2(X;\mathbb R)/\mathbb RF\) onto its image. Its inverse on that finite-dimensional image is bounded. [eq:auxiliary-restriction-class] proves that the class \([D^0_\varepsilon]\) modulo \(\mathbb RF\) is \(O(\varepsilon)\).

The nonlinear correction is exact, so \([D_\varepsilon]=[D^0_\varepsilon]\in V\). Choose a fixed linear complement of \(\mathbb RF\) in \(V\) and use it to lift the quotient class divided by \(\varepsilon\) to a bounded vector \(u_\varepsilon\). This gives \([D_\varepsilon]=\lambda_\varepsilon F+\varepsilon u_\varepsilon\). Choose any regular fiber outside the gluing region. The original glued form has integral \(\varepsilon\) there; the subtracted forms \(A,C\) have zero fiber integrals, and the correction is exact. Therefore \[ [D_\varepsilon]F=r_\varepsilon\varepsilon,\qquad u_\varepsilon F=r_\varepsilon=1+O(e^{-c/\varepsilon}). \tag{174}\] The square normalization and \(F^2=0\) now give \[ \lambda_\varepsilon =\frac{1-\varepsilon^2u_\varepsilon^2}{2\varepsilon(u_\varepsilon F)}. \tag{175}\] Boundedness of \(u_\varepsilon\) and [eq:auxiliary-fiber-area] prove [eq:auxiliary-class-expansion].

If there are no nodal fibers, \(\Psi\) is defined on all of \(X\) and the basic term in [eq:auxiliary-semiflat] is a multiple of \(F\) in cohomology. The form \(D^{\mathrm{sf}}_\varepsilon\) already solves the equations, so one takes it for \(D_\varepsilon\); the same conclusions follow with \(u_\varepsilon=[\Psi]\), \(r_\varepsilon=1\), and the coefficient computed by [eq:auxiliary-lambda].

It remains to establish the uniform implication. By 36, every curve class under consideration with \([D_\varepsilon]d>0\) satisfies \(d^2\ge-2\) and the following exhaustive alternatives: either \(m=Fd\ne0\) has the same sign as \([B]d\), or \(d=cF\) for a nonzero real number \(c\). The latter alternative has no sign difficulty, since both \([D_\varepsilon]F\) and \([B]F\) are positive.

Choose a null vector \(G_1\in V\) with \(FG_1=1\). Such a choice is obtained from any vector pairing to one with \(F\) by subtracting half its square times \(F\). The orthogonal complement of \(\operatorname{span}(F,G_1)\) in the Lorentzian space \(V\) is negative definite. Decompose \[d=mG_1+nF+z,\qquad u_\varepsilon=\alpha_\varepsilon G_1+\beta_\varepsilon F+w_\varepsilon,\] where \(z,w_\varepsilon\) lie in that complement, and use the Euclidean norm \(|z|^2=-z^2\) there. We have \(\alpha_\varepsilon=u_\varepsilon F\ge a_0\) and uniform bounds for \(\alpha_\varepsilon,\beta_\varepsilon,w_\varepsilon\).

Suppose that \(m<0\). Integrality of \(F\) and \(d\) means \(m=-a\) for an integer \(a\ge1\). The square bound gives \[ -2an-|z|^2=d^2\ge-2, \qquad n\le\frac{2-|z|^2}{2a}. \tag{176}\] Complete the square in the negative-definite component: \[\begin{align*} u_\varepsilon d &\le\frac{\alpha_\varepsilon}{a} -\frac{\alpha_\varepsilon|z|^2}{2a} +|w_\varepsilon||z|+|\beta_\varepsilon|a \\ &\le\frac{\alpha_\varepsilon}{a} +a\left(\frac{|w_\varepsilon|^2}{2\alpha_\varepsilon} +|\beta_\varepsilon|\right) \le Ka. \tag{177}\end{align*}\] The constant \(K\) is independent of \(a,z,d\), and of the future almost-complex structure. It follows that \[ [D_\varepsilon]d =-\lambda_\varepsilon a+\varepsilon u_\varepsilon d \le a\left(-\frac{c_0}{\varepsilon}+K\varepsilon\right)<0 \tag{178}\] whenever \(\varepsilon^2<c_0/K\) (with no restriction from this inequality if \(K=0\)). This contradicts \([D_\varepsilon]d>0\) and excludes every negative integer \(m\) simultaneously. The remaining alternative \(m>0\) gives \([B]d>0\), proving [eq:auxiliary-uniform-transfer].

Finally, both \([D_\varepsilon]\) and \([B]\) have positive square and pair positively with the same nonzero null vector \(F\). In a Lorentzian space the two components of the positive cone are distinguished by the sign of pairing with a fixed nonzero null vector: no positive vector is orthogonal to it. Thus the two classes are in the same component. All restrictions on \(\varepsilon\) in this proof depend only on fixed construction data and the original periods. They can therefore be imposed before choosing any later pair deformation. ◻