Proof. We use the standard unitary-invariant Gaussian ensemble of homogeneous polynomials; its covariance and conditional zero-density framework appear in Bleher et al. (2000, sec. 2.2 and 2.4). The proof below supplies the needed formulas and all additional quantitative grid properties. Randomness is used only to establish existence. A standard circular complex Gaussian has density \(\pi^{-1}e^{-|z|^2}\) on \(\mathbb C\). Choose the components of \(Q_N\) independently, with \[Q_N^{(a)}(z)= \sum_{|\beta|=N}\left(\frac{N!}{\beta!}\right)^{1/2} g_{a,\beta}z^\beta, \qquad \mathbb E\bigl[Q_N^{(a)}(z) \overline{Q_N^{(a)}(u)}\bigr]=(z\cdot\bar u)^N.\] Here all \(g_{a,\beta}\) are independent standard circular complex Gaussians. The covariance identity is the multinomial formula. In particular, at a unit vector the components of \(Q_N\) are independent standard complex Gaussians, and the law is invariant under unitary changes of coordinates.

We first prove (13) with probability tending to one. If \(A=\sup_{|z|=1}|Q_N(z)|\), homogeneity gives \(\sup_{|z|\le1+1/N}|Q_N(z)|\le eA\). Cauchy’s inequality on complex lines therefore bounds the first derivative on the unit sphere by \(CNA\), with an absolute constant \(C\). A sphere net of mesh \(c/N\), with a sufficiently small absolute \(c\), consequently contains a point where the norm is at least \(A/2\). Such a net has \(O_n(N^{2n-1})\) points. Gaussian tails show \[\mathbb P(A>N)\le C_n' N^{2n-1}e^{-c_n' N^2}\longrightarrow0.\] On the event \(A\le N\), the Lipschitz constant on the sphere is at most \(CN^2\). Use a second net of mesh \(cN^{-C_n-2}\). If the norm somewhere on the sphere is less than \(N^{-C_n}\), it is less than \(2N^{-C_n}\) at a point of this net. A standard complex Gaussian vector in \(\mathbb C^n\) has probability at most \(C_n' r^{2n}\) of lying in a ball of radius \(r\). The union bound is therefore at most \[C_n' N^{(C_n+2)(2n-1)-2nC_n} =C_n' N^{-C_n+4n-2}.\] Choose, for example, \(C_n=8n\). This proves both sphere bounds with probability tending to one.

Put \(d=n-1\). Almost surely the projective zeros of the \(d\)-component system \(H_N\) are transverse. To see this directly, let \(\mathcal A\) be its finite-dimensional complex coefficient space. Evaluation at any point of \(\mathbb P^d\) maps \(\mathcal A\) onto the fiber of \(\mathcal O(N)^{\oplus d}\). Hence the universal zero incidence manifold is smooth, and a coefficient vector is a regular value of its projection to \(\mathcal A\) precisely when the corresponding section has only transverse zeros. Sard’s theorem gives a Lebesgue-null exceptional set; the Gaussian law is absolutely continuous. Transversality and compactness imply that the zero set is finite. The projective Bézout theorem for a zero-dimensional intersection of \(d\) hypersurfaces of degrees \(N,\ldots,N\) gives total intersection number \(N^d\); every transverse intersection has multiplicity one (Eisenbud and Harris 2000, Theorem III-71). Thus the number of zeros is the deterministic quantity \(M_N=N^d\) almost surely.

We next justify the counting formulas needed for equidistribution. They are the one- and two-point, zero-dimensional cases of the correlation formula in Bleher et al. (2000, Theorem 2.1). Our subsequent off-diagonal argument proves the particular convergence needed here; the more general simultaneous-zero convergence theorem of Shiffman (2008, Corollary 1.3) supplies a broader context. In a holomorphic coordinate chart of \(\mathbb P^d\), regard the system as a holomorphic map \(H_a(z)\) depending linearly on its coefficients. Locally split those coefficients as \((a,b)\) so that \[H_{a,b}(z)=E(z)a+R(z,b),\qquad \det E(z)\ne0.\] This is possible because evaluation is surjective. The equation \(H_{a,b}(z)=0\) is equivalent to \(a=A(z,b):=-E(z)^{-1}R(z,b)\). At a zero, with \(J=D_zH_{a,b}(z)\), \[D_zA=-E(z)^{-1}J.\] For fixed \(b\), the change-of-variables formula with multiplicity for \(z\mapsto A(z,b)\) thus introduces the real Jacobian \(|\det E|^{-2}|\det J|^2\). Integrating in \(b\) against the smooth Gaussian coefficient density yields the one-point zero density \[ \rho_{1,N}(z)= p_{H_N(z)}(0)\, \mathbb E\bigl[|\det DH_N(z)|^2\mid H_N(z)=0\bigr]. \tag{16}\] The same argument works with a bounded measurable mark multiplying the zero count. It first applies to nonnegative marks, and the general bounded case follows by decomposition.

For two distinct projective points, joint evaluation onto the two fibers is surjective. Indeed, powers of linear forms that vanish at one point and not the other give independent prescribed values for each component. Solve now for \(2d\) coefficient variables and apply the same change of variables in the two point coordinates. The point Jacobian is block diagonal, giving \[ \begin{split} \rho_{2,N}(z,z')={}&p_{(H_N(z),H_N(z'))}(0,0)\\ &\quad\cdot \mathbb E\bigl[ |\det DH_N(z)|^2|\det DH_N(z')|^2 \mid H_N(z)=H_N(z')=0\bigr]. \end{split} \tag{17}\] Partitions of unity give these formulas on projective space and on the complement of its diagonal, respectively. This derivation uses only the coefficient density and ordinary change of variables; no independence of zeros has been assumed.

At a point represented by a unit vector \(u\), take affine tangent coordinates \(u+\sum_{j=1}^d z_j e_j\) with \((e_j)\) orthonormal in \(u^\perp\). Differentiating the covariance shows that \(H_N(u)\) is independent of \(N^{-1/2}DH_N(u)|_{u^\perp}\), and that the latter is a \(d\times d\) matrix of independent standard complex Gaussians. At two distinct projective points, the cross-covariances of these normalized value and first-derivative data tend to zero locally uniformly off the diagonal. More explicitly, on a compact set of pairs with \(|u\cdot\bar v|\le\rho<1\), they are bounded by a fixed power of \(N\) times \(\rho^{N-2}\). This follows by differentiating \((u\cdot\bar v)^N\) at most once at each point. The joint Gaussian covariance consequently converges uniformly there to the direct sum of its one-point covariances. Conditional Gaussian covariances given the values zero converge as well. Gaussian moments of the fixed polynomials \(|\det J|^2\) and their products therefore converge. Equations (16) and (17) imply that the two-point density, divided by \(M_N^2\), converges locally uniformly off the diagonal to the product of the normalized one-point densities.

The exact total number of zeros lets us pass from off-diagonal decorrelation to weak convergence without estimating the two-point density near the diagonal. Let \[\mu_N=\frac1{M_N}\sum_{H_N(\xi)=0}\delta_\xi.\] Unitary invariance and the deterministic count imply \(\mathbb E\mu_N=\sigma\). The positive measure \(\mathbb E(\mu_N\otimes\mu_N)\) has total mass one. Off the diagonal, its density converges to that of \(\sigma\otimes\sigma\), by the preceding calculation. This implies convergence on the whole product: given \(\varepsilon>0\), choose a compact subset of the off-diagonal region having \((\sigma\otimes\sigma)\)-mass greater than \(1-\varepsilon\). Its mass under \(\mathbb E(\mu_N\otimes\mu_N)\) is greater than \(1-2\varepsilon\) for all sufficiently large \(N\). At most \(2\varepsilon\) of the total mass can remain elsewhere. Since the diagonal has zero \((\sigma\otimes\sigma)\)-mass, there is no additional limiting mass there. Thus \[\mathbb E(\mu_N\otimes\mu_N) \rightharpoonup\sigma\otimes\sigma.\] For every continuous test function, its integral against \(\mu_N\) therefore has variance tending to zero. This proves weak convergence to \(\sigma\) in probability.

It remains to discard the bad zeros. Use (16) with marks. If \(G\) is a \(d\times d\) standard complex Gaussian matrix, then at a zero the derivative condition in (14) fails only when \(\operatorname{sing}_{\min}(G)<N^{-3/2}\). Its expected discarded fraction is \[\frac{\mathbb E\bigl[ |\det G|^2 \boldsymbol{1}_{\{\operatorname{sing}_{\min}(G)<N^{-3/2}\}} \bigr]}{\mathbb E|\det G|^2} \longrightarrow0.\] Indeed a Gaussian square matrix is nonsingular almost surely, \(|\det G|^2\) is integrable, and dominated convergence applies. The independent polynomial \(p_N\) has a standard complex Gaussian value at every unit representative. Its two forbidden tails have probability \[\mathbb P(|p_N|<N^{-1})+\mathbb P(|p_N|>N) =1-e^{-N^{-2}}+e^{-N^2}\longrightarrow0.\] Markov’s inequality shows that the total discarded fraction tends to zero in probability. Removing that fraction does not alter weak convergence. The modulus and singular-value conditions are unchanged on multiplying a unit representative by a phase, so they hold for every such representative.

We may now select deterministic systems. Take a countable uniformly dense family of continuous functions on projective space, and impose successively accurate conditions on successively longer finite subfamilies. Convergence in probability, the vanishing bad fraction, and the probability-one transversality statement show that these conditions and (13) have a nonempty intersection for every sufficiently large degree. Choose one system from that intersection. A diagonal choice gives all the asserted limits and works equally well along any prescribed sequence of degrees.

We have obtained the projective sampling properties. It remains to provide disjoint neighborhoods in which Section 8 can localize its shear coefficients. Fix a good direction and a lift \(v\) with \(p_N(v)=1\). Homogeneity and (14) give \(N^{-1/N}\le |v|\le N^{1/N}\). In the splitting of the domain into the radial complex line and its Hermitian orthogonal complement, \(DH_N(v)\) vanishes on the radial line, while \(Dp_N(v)\) has radial derivative \(N/|v|\). The transverse derivative of \(H_N\) has least singular value at least \(N^{-4}\) for large \(N\): the scaling from a unit representative costs at most one additional factor \(N\), so this is a deliberately weaker bound than necessary. Homogeneity, the sphere upper bound, and Cauchy’s inequality at scale \(c/N\) give first- and second-derivative bounds \(O_n(N^5)\) in the stated neighborhood. At the lift itself, the transverse first derivatives are \(O_n(N^4)\). Block inversion therefore gives \[\bigl\|D(H_N,p_N-1)(v)^{-1}\bigr\|\le N^9\] for all sufficiently large \(N\). Taylor’s formula, uniformly for \(|z-v|\le4N^{-30}\), now yields \[|(H_N(z),p_N(z)-1)| \ge N^{-9}|z-v|-C_n' N^5|z-v|^2 \ge\tfrac12 N^{-9}|z-v|.\] This proves (15). It also excludes every other zero of \((H_N,p_N-1)\) within distance \(4N^{-30}\), proving disjointness of the balls about all good lifts. ◻

Proof. Dimension gives \[\frac{\operatorname{rank}\Pi_N}{M_N} \le N^{1-n}\sum_{j=0}^{k-1} \binom{h+jN+n-1}{n-1}<\frac2{100}\] for all sufficiently large \(N\). Indeed its limit is the sum in (18).

There is a number \(t_h>0\) such that, on every sufficiently accurate grid, every polynomial with \(\|P\|=1\) satisfies \[ |P(v_\xi)|>t_h \quad\hbox{at at least }0.8M_N\hbox{ good directions}. \tag{20}\] For completeness, a nonzero homogeneous polynomial has a projective zero set of \(\sigma\)-measure zero. In affine charts this is the elementary fact that the zero set of a nonzero complex polynomial has Lebesgue measure zero, obtained by induction and Fubini’s theorem. Thus, for each norm-one polynomial \(P\), some positive threshold has a superlevel set of measure greater than \(0.9\). Uniform continuity in the coefficients makes the same superlevel estimate, with a smaller threshold, valid in a neighborhood of \(P\). The coefficient unit sphere is compact, so finitely many such neighborhoods suffice and their thresholds have a positive common lower bound. Weak convergence of the grid measures applied to these finitely many open superlevel sets gives the estimate for unit representatives, with a fixed margin in both threshold and counting proportion. Since \(|v_\xi|^h\to1\) uniformly, it gives (20). This proof requires only finitely many fixed approximation conditions for the given \(h\).

The projection has \(0\le(\Pi_N)_{\xi\xi}\le1\) and diagonal sum less than \(0.02M_N\). Consequently at most \(0.04M_N\) diagonal entries exceed \(1/2\). Choose the \(b_\xi\) temporarily as independent uniform phases. Writing \(Q_N'=1-\Pi_N\), independence gives \[\begin{split} &\mathbb E\left[ M_N^{-1}\left\|Q_N' \bigl(b_\xi P(v_\xi)\bigr)_\xi\right\|^2\right]\\ &\hspace{25mm}= M_N^{-1}\sum_\xi \bigl(1-(\Pi_N)_{\xi\xi}\bigr)|P(v_\xi)|^2 \ge0.38t_h^2 \end{split}\] when \(\|P\|=1\). The last inequality uses the intersection of the \(0.8M_N\) indices in (20) with the complement of the at most \(0.04M_N\) high-leverage indices.

We need a single phase choice for all polynomials, rather than a choice depending on \(P\). Let \((e_a)\) be a fixed orthonormal basis of the degree-\(h\) coefficient space, and put \(q_a(\xi)=e_a(v_\xi)\). These evaluations have a common bound depending only on \(n,h\) and the chosen norm. The Gram matrix of the random linear map in (19) has entries \[G_{ab}=M_N^{-1}\sum_{\xi,\eta} \overline{q_a(\xi)}\,\overline{b_\xi} (Q_N')_{\xi\eta}b_\eta q_b(\eta).\] Its diagonal-index terms are deterministic. For distinct indices, the products \(\overline{b_\xi}b_\eta\) are orthonormal in \(L^2\) of the phase probability space: two such ordered products have nonzero inner product only when their ordered pairs agree. In particular, reversal of the pair gives expectation \(\mathbb E(\overline{b_\xi}^{\,2}b_\eta^2)=0\). It follows that \[\begin{split} \mathbb E|G_{ab}-\mathbb EG_{ab}|^2 &\le C_{n,h}M_N^{-2} \sum_{\xi\ne\eta}|(\Pi_N)_{\xi\eta}|^2\\ &\le C_{n,h}M_N^{-2}\operatorname{rank}\Pi_N =O_{n,h}(M_N^{-1}). \end{split}\] The Gram matrix has fixed finite size. Chebyshev’s inequality therefore implies that its operator-norm distance from its mean tends to zero in probability. Its mean is bounded below by \(0.38t_h^2\) times the identity. A phase choice with least eigenvalue at least \(0.19t_h^2\) exists for every sufficiently large grid. Taking \(\delta_h=\sqrt{0.19}\,t_h\) proves the assertion. All estimates used only absolute values of lift evaluations, so the threshold and lower bound are uniform over their arbitrary representative phases. ◻

Proof. Cauchy’s inequality on the complex line through \(v\) gives \[|g_r(Rv)|\le K(R|v|/S)^r.\] The homogeneous expansion converges absolutely at the evaluation points. The finite Fourier average annihilates every degree not congruent to \(h\) modulo \(N\), giving the identity. Put \(\theta=R|v|/S\). The first tail is bounded by \[K\sum_{j=k}^{\infty}\theta^{h+jN} \le\frac{K e^{-0.99kc}}{1-e^{-0.99c}}.\] The case \(k=1\) gives the other estimate. The denominator is bounded away from zero for \(c\ge1\). ◻

Proof. For each stage choose the phases supplied by Lemma 5. Assign its phase to all \(N_j\) lifts of the same direction. We prove the assertion for a holomorphic function with \(|f|\le1\), which suffices by constant rescaling.

Fix a stage and temporarily omit its subscript. For each \(w\) with \(\operatorname{Im}w=D\), put \[A(s)=f_y(s,\zeta,w),\qquad I(s)=i f_w(s,\zeta,w).\] Both are holomorphic on a neighborhood of the closed spatial ball \(|s|\le S\), by the strict height margin and compactness. The maximum principle and (24) give \(|A|\le e^{p_0c}\) on the ball. The unit \(w\)-disk centered at any \((s,\zeta,w)\) in this ball lies in \(\mathcal T\), by (23); Cauchy’s inequality gives \(|I|\le1\). On a sheared disk, the derivative at \(t=0\) is \(A(Rv)+\alpha b_\xi I(Rv)\). Thus \[ |A(Rv)+\alpha b_\xi I(Rv)|\le e^{-m_0c}. \tag{26}\] Every estimate so far is uniform in the real part of \(w\).

For each good direction take the Fourier average in Lemma 7 over its cycle. Since \(b_\xi\) is constant on that cycle, it factors out of the averaged second term of (26). The lift radius bound gives \[\frac{R|v_\xi|}{S} \le\exp\left(\frac{\log N-c}{N}\right) \le e^{-0.99c/N}.\] Let \(a\) and \(i'\) be the resulting vectors of averaged \(A\) and \(I\) values, respectively. There is a vector \(a_0\in V_N\) such that \[\|a-a_0\|_\infty\le C e^{(p_0-0.99k)c}.\] If \(P\) is the homogeneous degree-\(h\) Taylor term of the function \(s\mapsto I(Rs)\), then \[\bigl\|i'-(P(v_\xi))_\xi\bigr\|_\infty \le C e^{-0.99c}.\] The averaged node inequalities still have supremum norm at most \(e^{-m_0c}\). Apply \(1-\Pi_N\) and use \(a_0\in V_N\). Since the number of good directions is at most \(M_N\), conversion from the supremum norm to \(M_N^{-1/2}\) times Euclidean norm costs at most one. Lemma 5 consequently gives \[\begin{split} \alpha\delta_h\|P\| &\le e^{-m_0c} +C e^{(p_0-0.99k)c} +C\alpha e^{-0.99c},\\ \|P\| &\le C_h\left( e^{(p_1-m_0)c} +e^{(p_1+p_0-0.99k)c} +e^{-0.99c}\right). \end{split}\] In particular, with the fixed positive number \[\varepsilon=\min\{m_0-p_1,\ 0.99k-p_0-p_1,\ 0.99\},\] we have \[ \|P\|\le C_h e^{-\varepsilon c}. \tag{27}\] The constants are independent of the stage once \(h\) is fixed. This is exactly the uniformity in Lemma 5; the Fourier tail denominator is \(1-e^{-0.99c}\), not \(1-e^{-0.99c/N}\). Since every degree-\(h\) coefficient of \(I(Rs)\) is \(R^h\) times the corresponding coefficient of \(I(s)\) and \(R\ge1\), (27) bounds each degree-\(h\) coefficient of \(f_w(s,\zeta,w)\) by the same exponential rate.

We have obtained exponential decay for every coefficient of the prescribed degree, uniformly along the entire horizontal \(w\)-line. It remains to bring that estimate down from the growing testing heights to a fixed open part of the tube.

Fix now a scheduled center \(\zeta\) and a multiindex \(\beta\) of length \(h\). The coefficient \[g_\beta(w)=\frac1{\beta!} \partial_s^\beta f_w(0,\zeta,w)\] is holomorphic on the whole connected half-plane \(\operatorname{Im}w>\ell(0,\zeta)\). At every point of this half-plane a spatial neighborhood is available, so the derivative is well-defined and holomorphic there. Choose once a small closed spatial polydisk about zero and a number \(t_0\) exceeding the supremum of \(\ell(s,\zeta)\) on that polydisk by more than two. Cauchy’s inequalities in \(s\) and on a unit \(w\)-disk give \[|g_\beta(w)|\le B_\beta \qquad (\operatorname{Im}w\ge t_0),\] where \(B_\beta\) is independent of every testing stage and of \(\operatorname{Re}w\). Along the stages assigned to \((\zeta,h)\), (27) gives \[\sup_{\operatorname{Im}w=D_j}|g_\beta(w)| \le C_h e^{-\varepsilon c_j}.\] For any fixed \(t>t_0\), apply Hadamard’s three-lines theorem (Tropp 2022, Theorem 7.13) to the horizontal strip between heights \(t_0\) and \(D_j\), once \(D_j>t\). Writing \(\theta_j=(t-t_0)/(D_j-t_0)\), it yields \[\sup_{\operatorname{Im}w=t}|g_\beta(w)| \le B_\beta^{\,1-\theta_j} (C_h e^{-\varepsilon c_j})^{\theta_j}.\] If \(B_\beta=0\) there is nothing to prove. Otherwise, the logarithm of the right side is \[\log B_\beta+ \frac{t-t_0}{D_j-t_0} \bigl(\log(C_h/B_\beta)-\varepsilon c_j\bigr) \longrightarrow-\infty\] by (22). Hence \(g_\beta\) vanishes in the upper half-plane and, by the identity theorem, throughout \(\operatorname{Im}w>\ell(0,\zeta)\). Repeating this argument for each order proves that all spatial Taylor coefficients of \(f_w\) at zero vanish at every scheduled center. Dependence of the constants and of the preliminary height \(t_0\) on the order causes no difficulty: each order has its own infinite sequence, and analytic continuation gives the same full half-plane afterwards.

To pass from these statements to a common open set, choose a spatial ball about zero and an open subdisk of the center disk, both with compact closures. Choose \(L\) larger than the supremum of \(\ell\) on their product. On the product with \(\operatorname{Im}w>L\), the function \(f_w\) is jointly holomorphic. At each scheduled center in the subdisk, all of its spatial Taylor coefficients vanish for every such \(w\). Taylor uniqueness and the identity theorem on the spatial ball give vanishing on that ball. Continuity in \(y\) and density of the scheduled centers then give vanishing on the whole product. The tube is connected: the coordinates \[(s,y,\operatorname{Re}w, \operatorname{Im}w-\ell(s,y))\] identify it with \(\mathbb C^{n+1}\times\mathbb R\times(0,\infty)\). The several-variable identity theorem therefore gives \(f_w=0\) everywhere on \(\mathcal T\).

For every fixed \((s,y)\) the \(w\)-fiber is a connected half-plane, so \(f\) is constant along that fiber. It descends to a bounded function \(F\) on \(\mathbb C^{n+1}\). This descended function is holomorphic: near any base point choose a single constant \(w_0\) whose imaginary part exceeds the supremum of \(\ell\) on a small base neighborhood, and write \(F(s,y)=f(s,y,w_0)\) there. These local holomorphic definitions agree on overlaps by fiber constancy. Liouville’s theorem, applied on complex lines in \(\mathbb C^{n+1}\), makes \(F\), and therefore \(f\), constant. ◻

Proof. The first three identities follow by changing the independent variable from \(\nu\) to \(u\). Since \(d\Psi/d\nu=t a_\psi\) and \(H=Y/(t a_\psi)\), the fourth follows as well. Differentiating \(\log H=\log Y-\nu-\log a_\psi\) gives the fifth. The lower bound on \(\kappa\) implies \[\frac{d\log(Y/t)}{d\nu}=\kappa(1+Y)-1\ge0.\] In \(u\) time, \(dY/du=\kappa(1+Y)\le1.1(1+Y)\), so \(Y\) remains finite and positive on each finite interval. The remaining variables are obtained by integrating the displayed equations there. In a cap, \(dY/d\nu=C_\kappa Y\), which gives the asserted global \(\nu\) continuation. Dividing the last equation of (81), with \(p=1\), by the second proves (82). Finally, \(\kappa\ge1\) gives \(Y(u)\ge Y(u_0)e^{u-u_0}\), and integration of \(d\nu/du=1/Y\) gives the last assertion. ◻

Proof. At \(s=(\sqrt t,0,\ldots,0)\) the metric has entries \[V_{i\bar j}=a\delta_{ij}+a'\bar s_i s_j,\] and its radial and transverse eigenvalues are \(a(1+Y)\) and \(a\). The unitary symmetry in the transverse variables and the Kähler symmetries leave only a transverse block, a radial–transverse bisectional block, and a pure radial entry. After removing the common factor \(Y/(ta)=H/\lambda\), their respective values in \(V\)-unit axes are \[ \delta_{ij}\delta_{kl}+\delta_{il}\delta_{kj},\qquad \kappa\delta_{kl},\qquad D=\kappa\bigl(1+\kappa+Y\kappa_Y\bigr). \tag{84}\] For completeness, the transverse coefficient is \(a'/a^2=Y/(ta)\). The radial–transverse coefficient is \[\frac{dY/d\nu}{t a(1+Y)}=\kappa\frac{Y}{ta}.\] For the pure radial entry, differentiate the logarithm of the radial metric along the radial complex line. Since \[\frac{d}{d\nu}\log\bigl(a(1+Y)\bigr)=Y(1+\kappa),\] its second \(\nu\) derivative divided by \(ta(1+Y)\) is exactly the last expression in (84) times \(Y/(ta)\). These calculations also follow directly by subtracting the cubic contractions in the Kähler curvature formula.

Write a Hermitian test matrix in radial/transverse blocks as \[T=\begin{pmatrix}r&b^*\\ b&S\end{pmatrix}.\] The contraction after removing \(Y/(ta)\) is \[Dr^2+2\kappa r\operatorname{tr}S+2\kappa|b|^2 +(\operatorname{tr}S)^2+|S|^2.\] The assumption on \(\kappa_Y\) gives \(D\ge\kappa\). Absorb the cross term with \(0.9Dr^2\). Its cost is at most \[\frac{\kappa^2}{0.9D}(\operatorname{tr}S)^2 \le\frac{1.1}{0.9}(\operatorname{tr}S)^2.\] Since \(\operatorname{rank}S\le2\), \(|\operatorname{tr}S|\le\sqrt2|S|\). The transverse reserve is therefore at least \(5|S|^2/9\), in addition to \(0.1Dr^2\) and \(2\kappa|b|^2\). As \(\kappa\le1.1\), these bound an absolute positive multiple of \(\kappa(r^2+2|b|^2+|S|^2)\). This proves (83). ◻

Proof. The eigenvalues of \(G_0\) are \(a_\psi\) transversely and \(a_\psi(1+pY)\) radially, giving (86). In these axes \(dt\) and \(d^2t\) have operator norms at most \(Ct/Y\), and higher derivatives of \(t\) vanish. The first bound includes the fact that the only nonzero first derivative is radial; thus its axis length includes the extra factor \((1+pY)^{-1/2}\).

Set a dot to mean \(d/d\nu\). We have \[\dot Y=\kappa Y(1+Y)=O(Y^2),\qquad \ddot Y=O(Y^3),\qquad \dddot Y=O(Y^4).\] The last two estimates use, respectively, the first and second indicated derivatives of \(\kappa\). Also \(\dot p=O(\delta Y)\) and \(\ddot p=O(\delta Y^2)\). Direct differentiation gives \[\begin{align*} a'&=aY/t,\\ a''&=a\bigl(Y^2-Y+\dot Y\bigr)/t^2,\\ a'''&=a\bigl(Y^3-3Y^2+2Y+3(Y-1)\dot Y+\ddot Y\bigr)/t^3. \end{align*}\] Consequently \(|a^{(j)}|\le Ca(Y/t)^j\) for \(0\le j\le3\). The same formulas for \(a_\psi\) use \(pY\) in place of \(Y\) and its derivatives. They imply the corresponding estimate for \(a_\psi\), and comparison at the point, where \(a=\lambda a_\psi\), improves their difference by the factor \(\delta\).

The chain rule for a radial potential in at most four spatial slots is a finite sum of products of its \(t\) derivatives and copies of \(dt,d^2t\). For a term containing its \(l\)th \(t\) derivative, the product of these latter factors is bounded by \(C(t/Y)^l\). Since \(B^{(l)}=a^{(l-1)}\), every such product is at most \(Cat/Y=C\lambda/H\), and the same argument gives \(C/H\) for \(\Psi\). The improved difference estimate proves the metric-jet assertion. For \(u\), its \(t\) derivatives through order four are \(O((Y/t)^j)\), by the displayed bounds on \(Y\) and its derivatives; the same chain rule proves the third assertion. All sums here have a number of terms bounded in terms of the derivative order, not the dimension.

Finally, the largest axis length in (86) is \(\sqrt{t/Y}\le1\). Applying two, three, or four such axes to the extra potential therefore gives at most \(CD_e t/Y=CD_e/(a_\psi H)\). ◻

Proof. Since \(q\) has no spatial dependence, \(\mathcal N=G\) and \(h=V+\tau G\). Lemma 19 shows that, through two metric jets, \(h\) differs from the constant multiple \((\lambda+\tau)V/\lambda\) by at most \(C\delta(\lambda+\tau)/H\). In particular \[h\asymp\frac{\lambda+\tau}{H}I,\qquad \Gamma^h=O(1).\] Lemma 18 gives a covariant curvature lower bound \(c\lambda|T|^2/H\) for \(V\) in these scaled axes. Multiplication of a metric by a positive constant multiplies its covariant curvature by that constant. The curvature difference is at most \(C\delta(\lambda+\tau)|T|^2/H\): indeed its fourth-derivative term has this bound, and its cubic contractions have the same bound by the inverse estimate \(|h^{-1}|\le CH/(\lambda+\tau)\) and the first-jet bounds. Lemma 13 makes this calculation explicit. Reducing \(\delta_0\) therefore gives \[ C^h(T,T)\ge c\frac{\lambda+\tau}{H}|T|^2 \quad(\operatorname{rank}T\le2). \tag{90}\]

Let \(\theta=\lambda/(\lambda+\tau)\). The identity \(\nabla^h(V+\tau G)=0\) implies \[\nabla^h G=\theta\bigl(\nabla^hG-\lambda^{-1}\nabla^hV\bigr), \qquad |\nabla^hG|\le C\delta\theta/H.\] The curvature table gives \(A=Pq^2D_q\), vanishing spin and \(A_i\) terms, and \(K=P\nabla^hG\). In \(G\)-unit normalization it gives \[ \frac{W}{Pq}=rI+\mathcal Y, \qquad \mathcal Y=V_G(V_G+\tau I)^{-1}, \qquad \mathcal Y\asymp\theta I. \tag{91}\] Here \(V_G\) denotes \(V\) expressed in \(G\)-unit axes; its eigenvalues are \((1+O(\delta))\lambda\). In particular \(W>0\). Spending a fixed small portion of \(W\) on the \(K\) coupling costs at most \[C\delta^2\frac{P\theta^2}{qH(r+\theta)}|T|^2 \le C\delta^2\frac{\tau\theta}{H}|T|^2,\] which is an arbitrarily small portion of (90) after reducing \(\delta_0\).

It remains to pay for \(2t_f(W:T)\). Spending \(0.98A t_f^2\) costs \((W:T)^2/(0.98A)\). First suppose the second alternative in (88) holds. Split the two terms in (91) using \((a+b)^2\le1.1a^2+11b^2\). The \(r\) term costs at most \[\frac{1.1}{0.98}P(\operatorname{tr}_G T)^2.\] The horizontal curvature contains the positive term \[P\bigl((\operatorname{tr}_G T)^2+|T|_G^2\bigr).\] As \(|\operatorname{tr}_G T|\le\sqrt2|T|_G\), their difference retains more than \(P|T|_G^2/2\). The remaining trace cost is at most \[ C\frac{P\theta^2}{D_qH^2}|T|^2. \tag{92}\] For \(\tau\le1\), (87) bounds \(P/D_q\) by \(C_E\). For \(\tau\ge1\), it bounds (92) by \[\frac{C_E}{H^2}\tau\theta^2(1+\log\tau)^2|T|^2 \le \frac{C_E(1+\log\lambda)^2}{H^2} (\lambda+\tau)|T|^2.\] For the last inequality, put \(z=\tau/\lambda\) and use boundedness of \(z(1+z)^{-3}\) and \(z(1+z)^{-3}(\log z)^2\) on \((0,\infty)\). Taking \(M_E\) sufficiently large absorbs this cost into a small portion of (90).

Now suppose \(\lambda=1\). The profile estimates give, for sufficiently large \(E\), \[ \|\mathcal Y\|\le r/18\qquad(\tau>2e^{-E}). \tag{93}\] To see the uniformity of this assertion, \(\|\mathcal Y\|\le(1+C\delta)/(1+\tau)\). On \(2e^{-E}\le\tau\le1\), the first summand of \(q_0\) gives a logarithmic slope bounded below by a large fixed multiple of one. On \(1\le\tau\le e^{3b_0E}\), it gives \(r\ge0.9b_0(1+\tau)^{-1/16}\) after increasing \(E\). Beyond this interval the logarithmic summand gives \(r\ge c/q\), which exceeds \(40/(1+\tau)\) by its linear-in-\(\log\tau\) upper bound and the starting size \(\tau\ge e^{3b_0E}\). These statements are unchanged by taking \(\delta_0\) smaller.

By (93), the trace expenditure outside the exceptional interval is at most \[\frac{P}{0.98} \left(|\operatorname{tr}_GT|+\frac{\sqrt2}{18}|T|_G\right)^2.\] For \(0\le s\le\sqrt2\), the ratio \((s+\sqrt2/18)^2/[0.98(1+s^2)]\) is strictly less than one, by an absolute amount. Thus a fixed portion of \(P|T|_G^2\) remains. On \(\tau\le2e^{-E}\), the explicit left-end profile gives \(q\le C\), \(r\le C\), and \(D_q\ge c\tau e^E\), where constants may depend on the fixed profile but not \(E\). Hence the trace cost is at most \(Ce^{-E}|T|^2/H^2\). Since \(H\ge\sigma\), this is a small portion of \((1+\tau)|T|^2/H\) for \(E\) sufficiently large in terms of \(\sigma\).

All cases retain fixed portions of \(A\), \(W\), (90), and the symmetrized \(PG^2\) term, proving (89). Lemma 12 gives nonpositive real sectional curvature off the fiber axis. The metric is positive there and \(h\ge cV\). Smoothness of the original affine fiber potential and its positive limiting metric extend both conclusions across the axis. ◻

Proof. A prime on \(\beta\) now means differentiation in \(\Psi\). Direct differentiation of \(\psi-F^\beta\) gives \[ \mathcal N=(1+D\beta')G_0+G_e +\bigl(D\beta''+D_\beta(\beta')^2\bigr) \partial\Psi\otimes\bar\partial\Psi. \tag{105}\] Here \(D_\beta\) is the derivative at fixed \(x\); thus all dependence of the primitive on the spatial cutoff is included. The last term and its first two spatial jets are \(O(C_f/H^2)\), because all positive derivatives of \(\Psi\) in scaled axes are \(O(1/H)\). The derivatives of the scalar \(D\beta'\) beyond its value are \(O(C_f/H)\). With \(d_f=1+D\beta'\ge1\), these facts show \[\mathcal N\asymp d_fG_0>0, \qquad h=V+\int_0^\tau\mathcal N(v)\,dv\ge cV.\] The constants in \(\asymp\) can be absolute by making \(H\) large enough; the value \(d_f\) itself is bounded by \(C_f\).

First consider \(x\le X\). Since \(\tau\le e^X\), the first two metric jets of \(h-V\) are \(O(C_f/H)\). Hence \[ \Gamma^h=\Gamma^{G_0}+O(C_f/\lambda),\qquad C^h(T,T)\ge c\lambda|T|^2/H \quad(\operatorname{rank}T\le2). \tag{106}\] The second estimate follows from Lemma 18, the scaled jet estimates, and the curvature perturbation identity, taking \(\lambda\gg C_f\).

The mixed form in Lemma 11 is \[\frac{W}{Pq}=(1+r)\mathcal N -\tau\mathcal N h^{-1}\mathcal N +q^{-1}\partial\bar\partial\log q.\] The last term equals \[-\mathfrak s_\beta\beta'G_0+O(C_f/H^2).\] The spin form, divided by \(Pq\), equals \[-\mathfrak s_\beta\beta'\nabla_h^{2,0}\Psi+O(C_f/H^2).\] In fact \(q_\beta/q^2=-\mathfrak s_\beta\); the remaining terms contain two first derivatives of \(\Psi\). Lemma 19 and (106) give \(|\nabla_h^{2,0}\Psi|\le C/H\) with absolute \(C\). Also \(|\tau\mathcal N h^{-1}\mathcal N|\le C_f/(\lambda H)\). Choose \(d_0\) first so that the domination in (104) is sufficiently strong in comparison with this absolute \(C\), and then choose \(\lambda,H\) large compared with \(C_f\). It follows that \[ W\ge c\frac{Pq}{H}d_f I, \qquad |\nabla_h^{2,0}P(b,b)|\le\tfrac14 W(b,\bar b). \tag{107}\] This argument also applies to the reverse mixture, using absolute values of \(\mathfrak s_\beta\).

By (101) and convexity of the mixture, \[D_q=(q_x+q_{xx})/q\ge c_f\min(1,\tau)\quad(x\le X).\] Left-end decay and boundedness on the rest of this interval give \[|\partial\log P|+|\partial P_\tau| \le C_f\min(1,\tau)/H.\] It follows from the curvature table that \[ A=(1+o(1))Pq^2D_q,\qquad |A_i|\le C_f P\min(1,\tau)/H, \tag{108}\] where the relative error tends uniformly to zero as \(\lambda,H\) increase. To verify uniformity at \(\tau=0\), the subtracted term in \(A\) is at most \(C_fP^2\min(1,\tau)^2/(\lambda H)\), while the leading term is \(Pq^2D_q\). The ratio tends to zero uniformly under the stated choices. Similarly (107) and (108) make the \(A_i\) coupling arbitrarily small in units \((AW)^{1/2}\).

Finally \(|K/P|\le C_f/H\). Spending fixed portions of \(A,W\) according to Lemma 12 costs at most \[C_f\frac{P}{qH}|T|^2\le\frac{C_f}{H}|T|^2 \quad\text{for the }K\text{ coupling},\qquad C_f\frac{P}{D_qH^2}|T|^2\le\frac{C_f}{H^2}|T|^2 \quad\text{for the trace coupling}.\] In these displayed bounds \(C_f\) is enlarged to include \(e^X\); also \(P/D_q\) is bounded on \(x\le X\), including the left endpoint. Both costs are small compared with (106). Thus fixed fractions of \(A\), \(W\), and \(\lambda|T|^2/H\) remain.

Now let \(x>X\). Since the two profiles coincide exactly here, \(F^\beta(x)-F^0(x)\) is independent of \(x\). Consequently \(\mathcal N=\mathcal N_\infty\) is independent of \(\tau\), and \[ h=V_f+\tau\mathcal N_\infty, \qquad V_f=V+\int_0^{e^X} \bigl(\mathcal N(v)-\mathcal N_\infty\bigr)\,dv. \tag{109}\] There is no infinite tail in this expression. Its first two spatial jets differ from those of \(V\) by \(O(C_f/H)\). Treating \(d_f=1+D\beta'\) at the point as constant, the jets of \(\mathcal N_\infty\) differ from those of \(d_fG_0\) by at most \(C\delta d_f/H\), after taking \(H\gg C_f/\delta\); similarly those of \(V_f\) differ from \(\lambda G_0\) by a sufficiently small relative amount after taking \(\lambda\gg C_f/\delta\).

The aligned argument can now be applied with the two spatial scales \(\lambda\) and \(d_f\). More explicitly, let \(\theta_f=\lambda/(\lambda+\tau d_f)\). The comparisons and the identity \(\nabla^h(V_f+\tau\mathcal N_\infty)=0\) give \[C^h(T,T)\ge c\frac{\lambda+\tau d_f}{H}|T|^2, \qquad |\nabla^h\mathcal N_\infty| \le C\delta d_f\theta_f/H.\] The \(K\) cost is bounded by \(C\delta^2(\lambda+\tau d_f)|T|^2/H\). The \(rI\) part of the trace cost uses the same rank-two symmetrized \(P\mathcal N_\infty^2\) reserve as before. The remaining cost is at most \[C\frac{\tau qd_f^2\theta_f^2}{D_qH^2}|T|^2 \le C_f\frac{(1+\log\lambda)^2(\lambda+\tau d_f)}{H^2}|T|^2.\] For the last step, use (87) and the scalar inequality from the proof of Proposition 20 with \(\lambda/d_f\) in place of \(\lambda\). We have \(1\le d_f\le C_f\) and may require \(\lambda\ge C_f\), so the logarithm is bounded by \(\log\lambda\). Taking \(H\ge M_f(1+\log\lambda)^2\) with sufficiently large \(M_f\) absorbs this last cost. Spin and \(A_i\) vanish in this region because \(q=q_0\) is spatially constant.

We have proved positivity and nonpositive curvature off the axis, with fixed fractional reserves. The profiles are smooth in \(\tau\) at zero, equal to one there, and have the prescribed left normalization. Thus the potential and metric are smooth in the original affine fiber coordinate, whose metric at the axis has entries \(V,e^\psi\). Curvature extends by continuity. This proves the lemma. ◻

Completion of Proposition 21. Choose \(d_0\) to dominate the absolute spin constants in the interpolation estimate, and choose \(K_0\) so that \(K_0/4-d_0/2>K_0/9\) and \(K_0/9>10\). The latter explicit exponent is larger than the exponent \(8\) in (117). For fixed old data and \(E\), choose \(L\) sufficiently large for (113) and for the moving-center metric and curvature errors to be less than the retained reserves. Next compute the finite constants \(C_f\) for both switches and choose \(\lambda\) and then \(H\) according to (95), with the larger thresholds needed in Lemma 22. This is the required noncircular order of choices.

Perform the forward switch with constant old center, the center interpolation with frozen profile \(q_1\) and offset, and finally the reverse switch with constant new center. Lemma 22 controls the two switches and the preceding argument controls the actual center motion. The cutoffs are flat-ended, so all joins are smooth. At the end the profile is \(q_0\), the center is the prescribed new constant, and the added offset is \(w^+(1)+w^-(1)\ge0\). All profiles are at least one, have smooth left endpoints with value one, and have normalized primitives tending to infinity on the right; thus they define smooth fiber potentials for all affine fiber values throughout. The spatial projection bound follows on each part, with a fixed fractional loss during the center motion. No bound on the final constant offset is required or asserted. ◻

Proof. Homogeneity and the spherical upper bound give \(|(\mathcal H,p_N)(s/R)|\leq Ne^l\). Cauchy estimates at scale \(c_0\sqrt t/N\) bound its derivatives of order at most four by this quantity times a fixed power of \(N\), since \(t\geq1\). In a differentiated logarithm there are only a fixed number of factors and denominator powers; the denominator in (123) is at least \(e^{-2c}\). This proves (124) with an absolute exponent; fixed-dimensional constants and \(d\) affect only its prefactor.

The spherical lower bound also gives \(|(\mathcal H,p_N)(s/R)|\geq e^lN^{-C_n}\). Once \(l\) exceeds a sufficiently large multiple of \(\log N\), subtraction of \((0,1)\) leaves norm at least \(e^lN^{-C_n}/2\). The factors \(e^l\) in derivatives of the logarithm can then be canceled. The remaining costs are fixed powers of \(N\), with exponents depending on the already fixed \(n\). Increase \(L_n\) to pay them and the degree threshold to pay the prefactor. ◻

Proof. On the spatial region following the previous center-change interval, the center is \(\zeta_j\) and \(\ell(s,\zeta_j)=B(s)\), including on shear supports. The same identity holds on the initial cap at the first stage. Thus the bound there follows from the fixed cap continuation up to \(\nu_B+2/Y_B\). The earlier spatial region is a fixed compact set \(K\) with already chosen profiles and centers; future stages do not change \(q(x,s)\) on \(K\). The reservation gives \(|\psi-\psi_{\rm old}|\leq1\) there, including the current and all later wells. Both centers are in the unit disk, so \(|\zeta_j-\zeta(s)|\leq2\). Choose \(A>\log4+\sup_K\psi_{\rm old}+1\). For each \(s\in K\), \(F(x,s)\to\infty\) as \(x\to\infty\). Continuity, a finite cover, and monotonicity in \(x\) give a common finite \(X\) with \(F(X,s)>A\) on \(K\). Inverting the fiber relation therefore gives \(\tau\leq e^X\). Since \(q\geq1\), the radial potential is bounded by \(\sup_K B+e^X\); the fixed shear adds at most \(2\sup_K|c_{\rm sh}|\). At center coincidences this argument is understood by the smooth limit \(\tau=0\). These bounds are independent of \(c\) and of the future wells. Add two and increase to at least \(j\). ◻

Proof. For raw estimates use Euclidean axes. Here \(t\geq1\), \(Y\asymp Y_B\), and \(\log Y_B\leq C u_s(\nu_B)\); the last estimate follows by integrating \(d\log Y/du_s=\kappa(1+Y^{-1})\leq2.2\) for \(Y\geq1\). Differentiation of \(a'=aY/t\) bounds the required radial metric jets by \(e^{C_{\rm rad}u_s}\). Their inverses cost no more. The radial curvature estimate gives \[ C^V(T,T)\geq c\kappa H|T|_V^2 \geq c(a/t)|T|^2, \tag{145}\] since \(V\geq aI\) and \(\kappa H=C_\kappa Y/((1+Y)ta)\geq c/(ta)\). The initial severity pays these bounds, the old data and the future allowance. The trigger cost \(e^{4E/k_b}\) fits the quiet raw exponent. The current well is small before saturation. Also \(G\geq V\), since all nonconstant extra twist terms in this interval are plurisubharmonic.

For the covariant condition, write \[\mathcal B(\nu)=B(e^\nu),\qquad \mathcal U(\nu)=\Psi(e^\nu)+Z_{j,\rm in}(e^\nu),\] and use primes for \(\nu\) derivatives within this proof. On the severity-varying interval, \[\begin{align*} \frac{\mathcal U''}{\mathcal U'},\ \frac{|\mathcal U'''|}{\mathcal U''},\ \frac{\mathcal B''}{\mathcal B'},\ \frac{|\mathcal B'''|}{\mathcal B''} &\leq C\sqrt{\mathcal U''}, \tag{146}\\ \frac{|E'|}{\mathcal U'},\ \frac{|E'|}{\sqrt{\mathcal U''}},\ \frac{|E''|}{\mathcal U''},\ \frac{|E'''|}{(\mathcal U'')^{3/2}} &\leq\varepsilon_b, \tag{147}\end{align*}\] where \(\varepsilon_b\) can be made as small as required. Away from the pole, \(h_b\) and its relevant derivatives are bounded, whereas \(\mathcal B'=Y/H\) and \(\mathcal B''=Y(1+Y)/H\). The choice \(\sigma M_bk_b^2\ll1\) makes these background terms dominate. Near the pole let \(D_b=Y_B/(1-\xi)\). Then \[Z_{j,\rm in}'\asymp M_bk_b^2D_b,\quad Z_{j,\rm in}''\asymp M_bk_b^2D_b^2,\quad |Z_{j,\rm in}'''|\leq C M_bk_b^2D_b^3,\] and \(|E^{(r)}|\leq Ck_b^rD_b^r\) for \(1\leq r\leq3\), including saturation. These prove (146) and make (147) small by first enlarging \(M_b\) and then decreasing \(\sigma\). The fixed intervening interval of \(\xi\) is covered by the first estimates. We also have \(\mathcal U''\geq1\).

At fixed \(\tau\), the radial part of the Schur metric is the Hessian of \[J_h=\mathcal B+\tau(\mathcal U-\bar F(x,E)),\] where the bar denotes averaging from zero to \(\tau\) in the \(\tau\) variable. Bounded severity derivatives of \(\bar F\) and (147) show that the first two derivatives of \(\mathcal U-\bar F\) are positively comparable with those of \(\mathcal U\) and its third derivative is bounded by \(C(\mathcal U'')^{3/2}\). Use affine axes normalized by the Hessian of \(\mathcal U\), with radial radius \(\sqrt{t/\mathcal U''}\) and transverse radii \(\sqrt{t/\mathcal U'}\). At a radial point, the only nonzero holomorphic connection types for a radial Hessian metric are radial–radial to radial and radial–transverse to transverse. Differentiation gives the bounds \[ \frac{|J_h'''/J_h''-1|}{\sqrt{\mathcal U''}}, \qquad \frac{|J_h''/J_h'-1|}{\sqrt{\mathcal U''}}. \tag{148}\] They are bounded by (146). No estimate of a radial/transverse condition number is used.

Write the remaining Hessian as \(\tau G_e\), with \(G_e\geq0\). Its size and first jets in these axes are bounded: old Euclidean bounds are dominated by the large initial \(a\), and the current and future wells have the reserved absolute bounds. The normalizing radii are at most \(a^{-1/2}\) because this background is at least \(V\geq aI\). If \(h_r\) denotes the radial Schur metric, the exact connection difference is, with the usual index placement, \[ \Gamma^h-\Gamma^{h_r} =\tau(\partial G_e-\Gamma^{h_r}G_e)(h_r+\tau G_e)^{-1}. \tag{149}\] Since \(h_r\geq c\tau I\), \(\tau h^{-1}\) is uniformly bounded for all \(\tau>0\). Thus the full connection is uniformly bounded as well.

In these axes the first derivative of \(E\) has norm \(|E'|/\sqrt{\mathcal U''}\); the complex Hessian entries are bounded by \(|E'|/\mathcal U'\) and \(|E''|/\mathcal U''\), and the only pure second derivative has size \(|E''-E'|/\mathcal U''\). Consequently subtracting the connection times the first derivative gives a small \(\nabla_h^{2,0}E\). Since the actual \(G\) is at least the normalizing background, the same smallness holds in \(G\) units. This proves all quiet smallness conditions uniformly in \(\tau\).

After saturation, severity derivatives vanish. Through \(l=4c\) the well’s raw costs are bounded by \(e^{C(c+\log N)}\); the post term below has cost at most a fixed multiple of \(N^4e^{2m_zc}\). Both fit \(\epsilon E_*\) by (128), while the inverse and radial curvature estimates persist. ◻

Proof. Each derivative of a function of \(l\) costs a fixed power of \(N\). Thus \(Z_{j,\rm out}\) has positive-order bounds through order four of size \(C N^4e^{m_z\max(0,l-2c)}\). These and (124) fit the raw quiet exponent by (128) and (155). Other radial and old-data bounds remain fixed on the cap continuation. Through \(l=4c\), severity is constant and the preceding lemma applies.

Where severity varies, take \(\mathcal U=\Psi+Z_{j,\rm out}\). Before division by \(t\), its transverse and radial Hessian entries have sizes \[m_zN e^{m_z(l-2c)},\qquad (m_zN)^2e^{m_z(l-2c)}.\] For \(l\geq4c\), \(m_z(l-2c)\) exceeds \(C_{\rm pol}(l+c+\log N)\) by a fixed positive multiple of \(c+l\). Thus this Hessian dominates the current well and its first Hessian jets in normalized axes. The affine \(Z_{j,\rm in}\) tail has coefficient at most a fixed factor times \(e^{E_*/k_b}\ll N\); old terms and future tails are covered as before. Moreover \(\mathcal U''/\mathcal U',|\mathcal U'''|/\mathcal U''\leq Cm_zN\), while \(\sqrt{\mathcal U''}\gtrsim m_zN e^{m_z(l-2c)/2}\). Severity derivatives of orders \(r\leq3\) are at most \(CB_*N^r\). Thus the connection ratios hold and all severity ratios are small. Equations (148) and (149) again prove the pure Hessian condition uniformly for all \(\tau\). Every extra Hessian is positive semidefinite in this interval.

Past \(T_*+4\) the new well satisfies (125), and the \(Z\) terms are affine in \(\nu\). Their positive-order Euclidean jets are bounded on the entire outer range by their endpoint bounds up to fixed constants. Their logarithmic costs are at most \(C(1+m_z)T_*+4\log N+C_{\rm old}\) with an absolute \(C\). The absolute constant \(C_*\) in (128) was chosen to pay this inequality too. Equation (155) then implies (156), after increasing \(c\) for the finite sum of old bounds, the number of generated terms, and dimensional prefactors. The dimension-dependent \(L_n\) multiplies \(T_*\) on both sides and changes no leading comparison. ◻

Proof. Initially grow \(H\) with \(\kappa=\kappa_*\) and \(p=\lambda=1\). Use the \(\lambda=1\), \(H\geq\sigma\) alternative of Proposition 20. Ultimately we will lower \(p\) slightly below one for a bump, thereby increasing \(\lambda\), and then restore \(p=1\) and return smoothly to \(\kappa=1\).

We first justify the order in which the targets are chosen. During the growth and the bump, \(p\leq1\) and \(\kappa=\kappa_*\), so \(d\log H/du_s\geq.08\). Thus \[\int_{u_{\rm entry}}^\infty \frac{du_s}{H} \leq \frac{1}{.08H_{\rm entry}}.\] This bounds the total increase of \(\Psi\) before the final transition independently of the size of either target. The transition from \(\kappa_*\) to one takes a fixed \(\log Y\) interval and leaves \(H\) nondecreasing, adding another fixed bound to \(\Psi\). Moreover \(\kappa\geq1\) gives \(d\log Y/du_s\geq1\), hence \[0\leq\nu-\nu_{\rm entry} =\int \frac{du_s}{Y}\leq Y_{\rm entry}^{-1}\] through the growth and holding intervals. The old nonradial twists therefore have bounded values on one fixed compact spatial interval, independently of the targets. Their scaled derivatives are bounded by the fixed Euclidean bounds, since the scaled coordinate radii are at most one. Derivatives of \(\Psi\) in these axes are \(O(1/H)\leq O(1/\sigma)\). The forward switch and center move each occupy a fixed amount of the variable \(\Psi\); these amounts can be included in the same bounds. Include the reserved future value and jet allowance as well. These are exactly old-twist bounds independent of \(\lambda,H\), as required by Proposition 21.

Choose its shift \(L_0\) from these bounds and \(E_{\rm post}\). This fixes its profile constants \(C_f\) and a sufficient target \(\lambda\), and that target determines a sufficient target \(H\). This is the order of parameter selection in Proposition 21. The radial path reaches the targets in the other order: grow \(H\) first until it exceeds every required multiple of \[C_{E_{\rm post}}(1+\log\lambda)^2 \quad\hbox{and}\quad C_f(1+\log\lambda)^2.\] Choose a smooth bump of \(1-p\geq0\) with small amplitude and small derivatives in \(\log Y\), and with integral in \(u_s\) equal to the desired \(\log\lambda\). Such a bump exists by making its middle plateau arbitrarily long; the endpoint ramps can have fixed small amplitude and sufficiently long fixed length, while the plateau length supplies the remaining integral. The conversion \(d\log Y/du_s=\kappa(1+Y^{-1})\) and its bounded derivatives show that bounds through order two in \(\log Y\) follow from the same bounds in \(u_s\). Throughout this bump \(H\) increases, and the already chosen lower bound for \(H\) pays the entire range \(0\leq\log\lambda(s)\leq \log\lambda\). Apply the large-\(H\) alternative of Proposition 20.

Restore \(p=1\) and lower \(\kappa\) smoothly to one. The downward transition can be made slowly enough in \(\log Y\) that \(Y\kappa_Y\geq-\kappa\), with absolute derivative bounds. Since \(\kappa\geq1\) here, \(H\) stays large. Now \(\kappa=p=1\), so both \(H\) and \(\lambda\) are constant. Perform the forward profile switch, the center move, and the reverse switch of Proposition 21. Interpolate the two center values by a flat-ended convex combination, keeping the center in the unit disk. The old bounds and targets have been chosen in the order just proved. The changes take a bounded interval of \(\Psi\), hence a finite interval of \(u_s\) because \(d\Psi/du_s=1/H\) and \(H\) is fixed, even though this interval can be very long. Restore \(q_0\) and retain the new constant offset. The center is now \(\zeta_{j+1}\). The offset may be arbitrarily large, but it is now constant and changes no derivative of the twist. Its value becomes part of the old data when the next testing height is chosen, before the next accuracy parameter \(c\).

Raise \(\kappa\) to \(\kappa_*\) again and use a bump with \(1<p<\kappa_*\) to bring \(\log\lambda\) down to zero. A small positive plateau and fixed ramps give any prescribed finite integral of \(p-1\); stop precisely when that integral equals the existing \(\log\lambda\). Thus \(\lambda\) never falls below one. Also \[\frac{d\log H}{du_s} =\kappa_*-p+\frac{\kappa_*-1}{Y}>0,\] so the large-\(H\) aligned condition remains valid. Restore \(p=1\).

Finally reduce \(H\) to \(\sigma\). Transition \(\kappa\) to \(1/2\), hold there for an adjustable interval, then return to one from below. For \(p=1\) the exact relation is \[ \frac{d\log H}{d\log Y}=1-\kappa^{-1}. \tag{159}\] Its value is \(-1\) on the middle plateau. The endpoint transitions have fixed finite effects on \(\log H\); the initial \(H\) can be made as large as needed. Choose the plateau length so that the final value is exactly \(\sigma\). Throughout this descent \(H\geq\sigma\), \(\lambda=1\), and \(1/2\leq\kappa\leq\kappa_*\), so the \(\lambda=1\) aligned alternative applies. Choose the downward transitions slowly enough to preserve the same derivative inequality as above.

All these intervals have finite \(u_s\) or finite \(\log Y\) length. Equations (120) show that \(Y\) remains finite on each finite \(u_s\) interval and that the radius advances by a finite positive amount. A putative pole of an indefinitely continued accelerated equation is therefore never reached during an episode. This proves the claim. ◻

Proof. The spatial scale \(u_s\) may now be enormous, and \(E_{\rm post}\) need not dominate it. Proposition 15 permits any fixed holomorphic affine frame at the evaluation point. We first use \(V\)-unit affine axes, where the curvature scale is the fixed \(\sigma\) and the relevant jets are bounded independently of \(u_s\). This will justify the initial severity increase before arranging the size inequality needed for the next trigger.

First hold \(\kappa=p=\lambda=1\) and \(H=\sigma\). Let \(E\) increase smoothly as a function of \(u_s\), starting at \(E_{\rm post}\), with a fixed flat-ended transition to slope \(4C_\uparrow\). Its derivatives through order four in \(u_s\) are bounded by constants depending on the fixed parameters, not on \(\sigma\) or the possibly enormous \(u_s\) at entry. Continue long enough that \[ E\geq2C_\uparrow u_s \tag{160}\] and any desired fixed threshold. The greater slope ensures that this occurs after a finite interval.

We verify the quiet criterion even before (160). Use \(V\)-unit affine axes. The scaled-jet radial estimates give metric jets of orders \(r=1,2\) bounded by \(CH^{r/2}\) in these axes, and scaled derivatives of \(u_s\) through order four are bounded by an absolute constant. Therefore \[|\partial^{[r]}E|_{V}\leq C H^{r/2},\qquad 1\leq r\leq4.\] The extras in (158) satisfy the same bounds after this normalization, and \(G\geq V\). For \(\sigma\) small, the severity terms in \(\mathcal N\) are small relative to \(G\), so \(h\asymp(1+\tau)V\), with first derivatives bounded in these axes by \(C(1+\tau)\sqrt H\). Its connection is consequently \(O(\sqrt H)\) uniformly in \(\tau\). The first, complex second, and pure covariant second severity derivatives are small for every \(\tau\).

The raw curvature margin on this hold is \(C^V(T,T)\geq c\sigma|T|^2\) in \(V\)-unit axes, because \(\kappa=1\) and \(H=\sigma\). All other raw data are bounded there by fixed constants. The previously chosen \(E_{\rm post}\) exceeds the fixed profile and \(\sigma\) thresholds, so it pays this margin even when it is much smaller than the current \(u_s\). Thus there is no gap before severity catches up.

After (160), keep the same slope while turning \(\kappa\) into a cap. Here is an explicit admissible transition. If \(Y_0\) is its starting clock value, set \[A(Y)=1+Y_0+\int_{Y_0}^Y \eta_0(v/Y_0)\,dv,\qquad \kappa(Y)=\frac{A(Y)}{1+Y},\] where \(\eta_0\) is smooth, equals one near \(1\), decreases nonnegatively to zero, and is zero for arguments at least two. For \(Y\leq Y_0\) use \(A=1+Y\). Then \(1\leq A\leq1+Y\), so \((1+Y)^{-1}\leq\kappa\leq1\). Moreover \[Y\kappa_Y+\kappa =\frac{YA'(Y)}{1+Y}+\frac{A(Y)}{(1+Y)^2}\geq0.\] The first two derivatives of \(\kappa\) in \(\log Y\) have absolute bounds, because \(Y/Y_0\) ranges over a fixed interval while \(\eta_0\) changes. For example, writing \(b=Y/(1+Y)\) and \(D=Y\partial_Y\), one has \(D\kappa=b(\eta_0-\kappa)\) and \(D^2\kappa=b(1-2b)(\eta_0-\kappa)+bD\eta_0\). At the end \(A\) is the new constant \(C_\kappa\). Since \(\kappa\leq1\) and \(p=1\), \(H\) is nonincreasing and stays at most \(\sigma\).

The same normalized jet and connection estimates continue through the transition and the cap. To check the raw curvature margin there, note that \(Y\geq t\): it holds initially, and \(d\log Y/d\nu=\kappa(1+Y)\geq1\) preserves it. Also \(u_s\geq t\) after choosing the initial constant large enough, since \(du_s/dt=Y/t\geq1\). Hence \[\kappa H\geq c/(ta)\geq c e^{-2u_s} \quad(t\geq1),\] and (160) with large \(C_\uparrow\) pays the quiet lower bound \(e^{-\epsilon E}\). The positive-order extras stay bounded relative to \(V\) on an arbitrarily long cap. More explicitly, a fixed Euclidean jet bound \(D_{\rm out}\) costs at most \(C(D_{\rm out}/a)H^{r/2}\) in \(V\)-unit axes for Hessian jets of order \(r\leq2\), since \(aH=Y/t\geq1\). The ratio only decreases as \(a\) grows. Future wells are included in the fixed allowance until their own activations.

Coast as far in \(\nu\) as desired. Shortly before the next chosen trigger, turn off the severity slope smoothly over a fixed-length \(u_s\) interval. The margin in (160) is sufficient to retain \(E_0\geq C_\uparrow u_s(\nu_B)\) after this bounded final increment, by making the cap endpoint large enough. The existing raw jet bounds can also be absorbed by making that endpoint large. This produces (136) and the next cap data. ◻

Proof of Proposition 23. The recursion gives a smooth strictly plurisubharmonic potential on \(\mathbb C^{n+1}\), with nonpositive real sectional curvature by the local lemmas and completeness by the preceding argument. Every required pair \((\zeta,h)\) is tested infinitely often. The stages satisfy \(R\geq1\), \(N\to\infty\), \(c\to\infty\), \(\log N\leq.005c<c/100\), \(D_j\to\infty\), and \(D_j/c_j\to0\). Lemma 25 supplies the height bound. The node disks, sphere derivative bound, and shear amplitude have the fixed exponents (131). Their phases were chosen as permitted by Proposition 8, and their geometric realization allows all those choices. That proposition proves the asserted Liouville property. ◻

Completion of the proof of Theorem 1. Apply Lemma 2 to the base just constructed. Its tube is connected, complete, and diffeomorphic to Euclidean space, hence simply connected. Its real sectional curvatures are at most \(-1\) by that lemma, and its bounded holomorphic functions are constant by Proposition 23. The complex dimension is the fixed finite \(m=n+2\).

There is no finite lower sectional-curvature bound. At \(s=0,y=\zeta_1\), the center and profile are constant nearby and there is no shear. On the center axis the spatial curvature tensor of the base equals that of \(V\): the added radial-fiber potential vanishes to second order in \(y-\zeta_1\), and its cubic derivatives with two spatial indices and one fiber index vanish there. The initial radial metric has a strictly negative holomorphic-plane curvature in a spatial direction. Corollary 3 therefore shows that the tube’s sectional curvatures are unbounded below. This proves every assertion of Theorem 1. ◻