Proof. Both \(g_d\) and \(U\) are parallel, so \(r_U\) is parallel. At the initial point choose an \(h\)-orthonormal spatial frame with first vector along \(w\), if \(w\ne0\), and write \(s=h(w,w)\) and \(\alpha=\sqrt{1+s}\). In the normal and first spatial directions, the matrix of \(r_U\) relative to \(k_{d,w}(0)\) is \[\begin{pmatrix}1+2s&-2\alpha\sqrt{s}\\ -2\alpha\sqrt{s}&1+2s\end{pmatrix}.\] Its eigenvalues are \((\alpha+\sqrt{s})^2\) and \((\alpha-\sqrt{s})^2\), which are reciprocal. The other two eigenvalues are one. This also covers \(s=0\), and parallel transport preserves the inequalities. ◻

Proof. We show that every failure persists in a neighborhood of the datum. If \(T_d(w)>B\) for one \(w\in O\), choose \(B<b<T_d(w)\). The geodesic segment \(\gamma_{d,w}([0,b])\) is a compact witness. Otherwise, failure is witnessed by one \(w\in O\), one \(0<\tau<T_d(w)\), and one loop with the two strict inequalities \[L(\ell)<B^{-1},\qquad \lVert P_\ell-\operatorname{Id}\rVert_{k_{d,w}(\tau)}>B L(\ell).\]

Smooth compact-subset vacuum Cauchy stability and geometric uniqueness apply to these witnesses: the data are smooth positive vacuum constraint data, and convergence in all \(p_m\) implies smooth convergence on every compact part of \(\Sigma\). This is precisely the compact-subset Cauchy statement used in (OpenAI 2026b, proof of companion@kind@foundation@cat:closed-tests companion@kind@foundation@cat:closed-tests ). To specify its use, enlarge the compact witness by finitely many local evolution neighborhoods and their causal shadows on the Cauchy hypersurface. The shadows of a compact subset of a globally hyperbolic spacetime on a Cauchy hypersurface are compact. Finite local Cauchy stability, followed by geometric uniqueness on the overlaps, therefore gives embeddings of a neighborhood of the witness into the nearby developments on which the metrics converge smoothly. Only finitely many finite orders are needed for any fixed witness.

For a loop witness also include compact timelike arcs from \(\Sigma\) to finitely many neighborhoods covering its image. Such neighborhoods exist because the original image is compact and contained in \(I^+(\Sigma)\). After a small change of data those arcs remain timelike; their final short segments can be varied within the neighborhoods. Consequently the nearby loop images are still contained in \(I^+(\Sigma)\).

The initial velocity in Equation (3), the geodesic on the chosen compact time interval, and parallel transport all vary continuously, by smooth ODE dependence in these common neighborhoods. For the loop, use a local diffeomorphism converging smoothly to the identity to move its base to the nearby geodesic at the same proper time; its support can be chosen in one of the preceding neighborhoods. Lengths and transport matrices then converge, so both strict inequalities persist. The duration witness similarly persists beyond \(B\). Thus the complement is open. No continuity assertion for \(T_d(w)\) at a maximal endpoint is used. ◻

Proof. Let \(E\) be the set of parameters whose whole paths are in \(P\). Compactness of \([0,1]\) and openness of \(P\) imply that \(E\) is open. For \(\lambda\in E\) the path lies in the intrinsic diamond \(D=J_P^+(p)\cap J_P^-(q)\), which is compact by global hyperbolicity. Its image in \(\widetilde M\) is compact and hence closed because the ambient manifold is Hausdorff. If \(\lambda_j\in E\) and \(\lambda_j\to\lambda\), continuity of \(H\) puts every \(H(\lambda,s)\) in that same closed image of \(D\), so \(\lambda\in E\). Thus \(E\) is nonempty, open, and closed in the connected interval. This argument does not assume ambient causal convexity or any ambient geodesic uniqueness. ◻

Proof. Suppose a unit geodesic has a position limit at an endpoint in the spacetime. Choose a precompact temporal coordinate chart \((t,x)\) about the limit. On a sufficiently late tail, the causal slopes \(V=(1,dx/dt)\) are uniformly bounded. With \(\dot t=dt/d\tau>0\), the geodesic equation reads \[ \frac{d}{dt}\log\dot t =-\Gamma^0_{\alpha\beta}V^\alpha V^\beta, \qquad \frac{dV^i}{dt} =-\Gamma^i_{\alpha\beta}V^\alpha V^\beta +\Gamma^0_{\alpha\beta}V^\alpha V^\beta V^i. \tag{8}\] The right sides are bounded on the finite chart-time tail. Hence \(\dot t\) is bounded above and below by positive constants and has a limit; \(V\) also has a limit. The remaining proper time is finite, and the limiting position and unit timelike tangent give smooth ODE continuation. This contradicts maximality. The same proof applies at the past end. Complete a maximal past geodesic by any future inextendible timelike continuation. Cauchyness gives a crossing of \(\Sigma\), and achronality places that crossing to the past of a point in \(I^+(\Sigma)\). ◻

Proof. This is the bounded-connection parallel-transport argument underlying (Sbierski 2022, Lemma 2.13), with a first-exit bootstrap to handle a loop specified only by its developed length. Related comparisons with coordinate length, including the small-developed-length case, appear in (Sbierski 2024, Lemma 4.2 and Corollary 4.12). Continue the initial frame \(F_0\) parallel along the loop, and write \(\dot\ell=Fz\). Up to the first departure from the selected chart, the coordinate parallel-transport equation implies, for a fixed constant \(C\) including the dimension-dependent matrix factors, \[|\dot F|\le C\lVert F\rVert^2|z|, \qquad |\dot\ell|\le\lVert F\rVert|z|.\] Use accumulated developed length \(s=\int|z|\) as the parameter on nonconstant pieces, or equivalently integrate these inequalities without reparametrizing. Comparison with \(f'=Cf^2\), \(f(0)=K\), gives \(\lVert F\rVert\le K/(1-CKs)\le2K\) for \(s\le(2CK)^{-1}\). Thus coordinate displacement is at most \(2Ks\). Choose \[l_*<\min\{(2CK)^{-1},\rho/(4K)\}.\] A first chart exit at developed length less than \(l_*\) would have displacement at least \(\rho\) and at most \(\rho/2\), a contradiction. The estimates consequently hold on the entire loop. They give \(\lVert F(1)-F_0\rVert\le4CK^2L\). Since the loop returns to its base point, its transport matrix in the initial orthonormal frame is \(F_0^{-1}F(1)\). Multiplication by the uniformly bounded inverse proves Equation (9), for example with \(C_*=4CK^3\). Piecewise smooth parametrizations cause no change in these integral estimates. ◻

Proof. Identify \(\mathcal M_d\) with its open image in the extension. Let \(c\) be the timelike curve reaching its boundary. Its tail has no interior future endpoint, since an interior limit and the given ambient boundary limit would coincide in the Hausdorff extension. Extending the interior part maximally to the past and using the Cauchy property, we find that a sufficiently late part lies in \(I^+_{\mathcal M_d}(\Sigma)\).

A uniformly timelike straight segment reaching the boundary. We first obtain a straight timelike segment that stays inside \(\mathcal M_d\) until its final point. The construction also covers an exiting curve whose timelike margin tends to zero at its endpoint. Take a precompact ambient chart \((t,x)\) containing the endpoint and a tail of \(c\), with \(dt\) temporal and \(\partial_t\) future timelike. In chart time the tail is \((t,x(t))\), \(t\le t_f\). On a smaller compact coordinate box, causal slopes \(v=dx/dt\) are bounded, and there is \(c_0>0\) such that \(\widetilde g((1,v),\partial_t)\le-c_0\) for all these slopes. The metric is Lipschitz there. Replace the curve’s time coordinate by \[\widetilde t(t)=t-\epsilon_1(e^{H(t_f-t)}-1), \qquad \epsilon_1>0.\] Put \(\Delta=t-\widetilde t\) and \(\eta=\widetilde t'-1=\epsilon_1H e^{H(t_f-t)}\). Then \(0\le\Delta\le\eta/H\). The change of the quadratic form from the displacement is bounded by \(C\Delta(1+\eta)^2\), whereas the addition of \(\eta\partial_t\) contributes at most \(-2c_0\eta\); its quadratic contribution is nonpositive. Choose \(H\) large and then \(\epsilon_1\) small enough that \(\eta\le1\). The new velocity is uniformly future timelike, with quadratic form at most \(-c_0\epsilon_1H\) after increasing \(H\) if necessary. The new curve has the same final point and begins inside \(I^+(\Sigma)\), since its initial displacement can be made arbitrarily small.

After reparametrization by its new chart time, take a sufficiently fine polygonal interpolation with the same endpoints. It remains uniformly future timelike. Indeed, on a short subinterval the velocities lie in a bounded convex cone of future timelike vectors with a fixed smaller margin for one frozen metric; averaging preserves that cone, and continuity of the metric preserves a smaller margin on the interpolating segment. A finite subdivision suffices. Follow this polygon to its first exit from \(\mathcal M_d\) and let \(z\) be that exit. The straight piece immediately before \(z\) has constant spatial slope \(v_*\), is interior before \(z\), and is uniformly timelike up to \(z\).

An open family of intrinsic geodesics has finite full duration. Choose \(l>0\) small and set \(t_0=t(z)-l\), \(t_1=t(z)+l\) so that the last length-\(l\) part of the straight piece is in the chart. Start intrinsic unit timelike geodesics at \(t=t_0\) with position within \(l^2\) of that piece and spatial slope within \(l\) of \(v_*\), restricting to a nonempty open subset of states inside \(I^+_{\mathcal M_d}(\Sigma)\). All these states form a six-dimensional open family. The Christoffel symbols of the ambient \(C^1\) metric are continuous and bounded. While an intrinsic geodesic remains in the box, Equation (8) therefore bounds its spatial acceleration by a constant \(C_0\), uniformly over the family. On the interval of chart-time length \(2l\), \[ |V^i-v_*^i|\le (1+2C_0)l, \qquad |x(t)-x_*(t)|\le C_1l^2, \tag{10}\] where \(x_*\) is the straight line extended through the box. Choose a compact box about that line with a fixed positive margin and shrink \(l\) first. These estimates, by a first-exit argument, prevent chart departure before \(t_1\). They also keep all slopes in a common uniformly timelike neighborhood of \(v_*\). Unit normalization gives uniform positive upper and lower bounds for \(dt/d\tau\).

If one such geodesic remains in \(\mathcal M_d\) through \(t_1\), Equation (10) places its point at \(t=t(z)\) within \(C_1l^2\) of \(z\). Add to its spatial path a smooth bump supported in \((t_0,t_1)\) that moves this point to \(z\) and leaves the endpoints fixed. A rescaled fixed bump has height \(O(l^2)\) and derivative \(O(l)\). Multiplying the bump by \(\lambda\in[0,1]\) gives a fixed-endpoint homotopy through future timelike curves inside the ambient box. Lemma 8, applied to the globally hyperbolic spacetime \(\mathcal M_d\), forces the final path to remain inside it, although it passes through \(z\notin\mathcal M_d\). This is impossible.

Thus every member of the family has maximal intrinsic chart-time strictly less than, or equal to, \(t_1\). The preceding bounds imply finite remaining proper time and limits of position and unit tangent in the ambient box. The position limit lies outside \(\mathcal M_d\): otherwise the smooth intrinsic geodesic equation would continue it. This is its maximal endpoint in the full \(\mathcal M_d\). Any intrinsic continuation would have an interior position whose image is the already determined ambient boundary limit, a contradiction. No geodesic equation has been solved outside \(\mathcal M_d\), and no uniqueness assertion for the merely continuous ambient Christoffel symbols has been made.

Continue one of the starting states backwards in \(\mathcal M_d\). Lemma 9 and its location in \(I^+(\Sigma)\) give a transverse crossing of \(\Sigma\). Smooth geodesic flow through this fixed compact earlier segment maps an open neighborhood in \(T\Sigma\) diffeomorphically onto an open part of the above starting-state family. Shrink it to a relatively compact open set with closure mapped into that family. Its earlier geodesic tube has compact closure and uniformly bounded proper duration. The final chart interval has uniformly bounded proper duration as well. Hence there is \(B_0<\infty\) such that \(T_d(w)\le B_0\) for every seed in this smaller neighborhood.

One transport constant for all base points and all short loops. Initial frames orthonormal for Equation (4) and their inverses are uniformly bounded on the compact seed closure. On the compact earlier tube, a finite cover by regular charts and the parallel-transport ODE give uniform bounds for their continued matrices and inverses. On the final part use the ambient chart: bounded Christoffel symbols and bounded chart-time slopes on an interval of length \(2l\) give the same bounds by the linear parallel-transport equation and its inverse equation. All final base points stay in a compact coordinate box strictly inside this chart, even as they approach the boundary of the image of \(\mathcal M_d\). A finite chart cover of the earlier compact tube, with smaller chart cores, provides a common positive chart margin there as well. Lemma 10 now supplies one \(l_*>0\) and one \(C_*<\infty\) for every base point on every geodesic in the seed neighborhood. In particular it applies to every loop required in Definition 6.

Choose an integer \(B\ge\max\{1,B_0,C_*,l_*^{-1}\}\) and a basis set \(O\in\mathcal O\) inside this neighborhood. The strict premise \(L<B^{-1}\) implies \(L<l_*\), so all the requirements for \(d\in\mathcal A(O,B)\) hold. ◻

Proof. The comparison construction is (OpenAI 2026b, companion@kind@foundation@cmp:development companion@kind@foundation@cmp:development ); the absence of finite exterior runout and the first-runout classification are respectively companion@kind@foundation@cmp:completeness companion@kind@foundation@cmp:completeness

  companion@equation@foundation@cmp:completeness ( companion@number@foundation@cmp:completeness companion@number@foundation@cmp:completeness

) companion@number@foundation@cmp:completeness companion@number@foundation@cmp:completeness

and companion@kind@foundation@cmp:classification companion@kind@foundation@cmp:classification

  companion@equation@foundation@cmp:classification ( companion@number@foundation@cmp:classification companion@number@foundation@cmp:classification

) companion@number@foundation@cmp:classification companion@number@foundation@cmp:classification

of that companion. Their hypotheses are exactly the ones imposed here: a fixed rotating subextremal center, the complete signed two-ended bridge, smooth vacuum data with finite weighted symbol seminorms, and smallness only in \(p_{10}\). The weighted seminorms and asymptotic powers agree with those of our phase space; equivalent reference atlases can be accommodated by reducing the radius. There is no requirement that asymptotic charges be fixed, that a curvature bound hold, or that an extension exist. The cylinder and wedge are furnished by the source’s finite bridge evolution, two exterior evolutions, and full-cylinder interior continuation.

To make explicit how the cited classification is used, let \(\tau_P\) be the supremum of the times for which the initial connected geodesic segment stays in \(\mathcal P_d\). Then \(\tau_P\le T_d(w)<\infty\). Viewed in \(\mathcal P_d\), this is a maximal future geodesic segment. The cited exterior completeness proposition rules out its avoiding \(\mathcal S\). Both \(u\) and \(v\) strictly increase on a future timelike wedge track. If they had finite limits, its position would remain in a compact regular coordinate range; the temporal-chart argument of Lemma 9 would give geodesic continuation there. Hence one or both limits are infinite, exactly as in Equation (12). If \(\tau_P<T_d(w)\), its endpoint is an ordinary point of \(\mathcal M_d\), although it is not a point of this wedge. The conclusion does not change in that case. ◻

Proof. This is the part of the corner-ratio lemma (OpenAI 2026b, companion@kind@foundation@cmp:corner-ratios companion@kind@foundation@cmp:corner-ratios ) that we need; we include its estimate to display the dependence on the individual track. Write \[U=\dot y_0L_0+\dot y_1L_1+\Xi, \qquad p_i=-g(U,L_i)=a\dot y_j>0\quad(j=1-i).\] The future unit condition and Equation (15) give \[\begin{align*} 1+|\Xi|_\gamma^2&=2a\dot y_0\dot y_1, \tag{16}\\ \frac{\dot p_i}{p_i} &=\ell_i\dot y_i+(\xi_i-\xi_j)(\Xi) -\frac{\chi_i(\Xi,\Xi)}{p_i}. \tag{17}\end{align*}\] For example, the last equation follows by differentiating \(-g(U,L_i)\), using \(\nabla_UU=0\) and the three terms in \(\nabla_UL_i\) from Equation (15). Since \(|\Xi|_\gamma^2/p_i\le2\dot y_i\), it follows that \[ \left|\frac{d}{d\tau}\log(e^{\kappa y_i}p_i)\right| \le C(q+g_i(y_i))\dot y_i+C|\Xi|_\gamma, \qquad e^{\kappa y_i}p_i=\frac{q_0}{\kappa}\mathcal A|\dot Y_j|. \tag{18}\] The positive functions \(Y_i\) decrease. On every initial part of the remaining track, Equation (16) implies \[\int|\Xi|_\gamma\,d\tau \le C\left(\int|\dot Y_0|\,d\tau \int|\dot Y_1|\,d\tau\right)^{1/2} \le C\sqrt{Y_0^{\mathrm{in}}Y_1^{\mathrm{in}}}.\] Moreover, \[\int q\dot y_i\,d\tau \le\frac{q_0}{\kappa}Y_0^{\mathrm{in}}Y_1^{\mathrm{in}}, \qquad \int g_i(y_i)\dot y_i\,d\tau \le\int_{y_i^{\mathrm{in}}}^{\infty}g_i(s)\,ds.\] Thus the logarithmic variation in Equation (18) is bounded independently of the terminal point of this track portion. Boundedness of \(\mathcal A^{\pm1}\) gives positive upper and lower bounds for each \(|\dot Y_i|\), proportional to its initial value. Finally \[1+|\Xi|_\gamma^2 =\frac{2q_0\mathcal A}{\kappa^2} |\dot Y_0|\,|\dot Y_1|\] bounds the screen velocity. These constants depend on the two initial \(Y\)-speeds; no bound uniform over all unit timelike tangents has been asserted. ◻

Proof. Use \(Y_i=e^{-\kappa y_i}\) as above, and consider the interior null sections \[\mathcal N_j=\{u=j,\ u+v>0\},\qquad j=1,2,\ldots.\] Every corner track hits all sufficiently large \(\mathcal N_j\) transversely before first runout. The tangential covariant momenta on these sections are \[ \pi_{Y_1}=-\frac{a\dot Y_0}{\kappa^2Y_0Y_1} =-\frac{q_0\mathcal A}{\kappa^2}\dot Y_0, \qquad P_A=\gamma_{AB}\Xi^B. \tag{19}\] The shift contributes to the \(Y_0\) covector component, but not to \(\pi_{Y_1}\), because \(\partial_{Y_1}\) is a multiple of \(L_1\). Lemma 13 and the uniform sphere-metric comparison show that, along each corner track, these momenta satisfy one finite bound on its entire sufficiently late tail. Also \(Y_1\to0\) along that tail.

Here is the exact volume statement used to transfer these bounds. On \(T^*\mathcal P_d\) take the canonical symplectic form \(\omega=d\pi_\alpha\wedge dx^\alpha\), with the corresponding Hamiltonian-flow sign convention, and set \(\mathcal H=\tfrac12g^{-1}(\pi,\pi)\). Restrict to the future unit mass shell \(\mathcal H=-\tfrac12\). Above a fixed \(\mathcal N_j\) the six variables \[(Y_1,\theta^1,\theta^2,\pi_{Y_1},P_1,P_2)\] give coordinates on the transverse section of this shell. To check the missing momentum explicitly, let \(\widetilde a=a/(\kappa^2Y_0Y_1)>0\) and \(\widetilde b=-b/(\kappa Y_0)\). Then \[g=-2\widetilde a\,dY_0dY_1+ \gamma_{AB}(d\theta^A-\widetilde b^A dY_0) (d\theta^B-\widetilde b^B dY_0),\] and the shell equation is \[ -\widetilde a^{-1} (\pi_{Y_0}+\widetilde b^AP_A)\pi_{Y_1} +\tfrac12\gamma^{AB}P_AP_B=-\tfrac12. \tag{20}\] Future timelikeness gives \(\dot Y_0<0\) and hence \(\pi_{Y_1}=-\widetilde a\dot Y_0>0\). Thus Equation (20) determines \(\pi_{Y_0}\) uniquely. Pulling back \(\omega\) to this section, where \(dY_0=0\), gives the nondegenerate form \[ \omega_j=d\pi_{Y_1}\wedge dY_1 +\sum_{A=1}^2dP_A\wedge d\theta^A. \tag{21}\] Its absolute symplectic volume is ordinary canonical coordinate volume; there is no factor involving \(a\) or \(Y_0\).

Let \(D_j\subset T\Sigma\) consist of seeds whose initial connected track in \(\mathcal P_d\) hits \(\mathcal N_j\) at a finite time. This is open: a given compact geodesic segment stays in the open comparison spacetime, and its transverse hit persists by smooth flow dependence and the implicit function theorem. The hitting map \(F_j:D_j\to\{\mathcal H=-\tfrac12\}\big|_{\mathcal N_j}\) is smooth. It is injective. A wedge trajectory has strictly decreasing \(Y_0\) and so hits that section at most once; a maximal trajectory in the globally hyperbolic \(\mathcal P_d\) meets its Cauchy hypersurface \(\Sigma\) once. If two hitting states were equal, uniqueness of the smooth Hamiltonian flow would therefore give the same initial seed.

On the shell above \(\Sigma\), restriction of \(\omega\) is the canonical form on \(T^*\Sigma\). Indeed the tangential initial covector is \(\pi|_{T\Sigma}=h(w,\cdot)\), so the map from seeds is a smooth fiberwise isomorphism onto \(T^*\Sigma\). Denote the pullback of its positive canonical volume by \(\mu\); this is a smooth positive measure on \(T\Sigma\).

The hitting map preserves the section symplectic forms and hence their volumes, including its variable hitting time. In detail, Hamiltonian flow preserves \(\omega\), and its generating vector \(X_{\mathcal H}\) belongs to the kernel of \(\omega\) restricted to the energy shell, since its contraction is, up to sign, \(d\mathcal H\). For a hitting time \(\tau(w)\), \[dF_j(V)=d\Phi_{\tau(w)}(V) +X_{\mathcal H}(F_j(w))\,d\tau(V).\] All terms involving \(d\tau\) therefore vanish when evaluated by \(\omega\) on shell-tangent vectors. This proves preservation of the section forms. Their nondegeneracy also shows that \(F_j\) is a local diffeomorphism. Its injectivity makes it a diffeomorphism onto its open image, so volume comparison for measurable subsets follows from change of variables, without any multiplicity factor.

Fix a smooth round norm on angular covectors. For \(r>0\) and an integer \(N\ge1\), the region on the target section defined by \[0<Y_1<r,\qquad |\pi_{Y_1}|\le N, \qquad |P|_{\mathrm{round}}\le N\] has canonical volume at most \(C_Nr\), uniformly in \(j\). A finite angular coordinate cover with precompact chart cores proves this directly from Equation (21); covector norms and coordinate components are uniformly comparable on each core. Restricting to the actual wedge \(u+v>0\) or to the future shell only reduces that volume. Let \(S_{j,r,N}\) be its inverse image under \(F_j\), including only the finite hits in \(D_j\). These are measurable and satisfy \[ \mu(S_{j,r,N})\le C_Nr. \tag{22}\] Fatou’s inequality applied to their indicator functions gives \[ \mu\!\left(\liminf_{j\to\infty}S_{j,r,N}\right) \le\liminf_{j\to\infty}\mu(S_{j,r,N})\le C_Nr. \tag{23}\] This use of Fatou does not require \(\mu(T\Sigma)\) to be finite.

Every corner seed is contained in \[ \bigcup_{N=1}^{\infty}\ \bigcap_{s=1}^{\infty} \liminf_{j\to\infty}S_{j,1/s,N}. \tag{24}\] Indeed first choose one integer \(N\) larger than the eventual momentum bound of its track. For each \(s\) all sufficiently late hits then have \(Y_1<1/s\) and that same momentum bound. For a fixed \(N\) the intersection in Equation (24) has measure at most \(C_N/s\) for every \(s\), and is therefore null. The countable union is null. All smooth positive measures on \(T\Sigma\) have the same null sets, proving the assertion. In particular no uniform momentum bound over the family of corner seeds has been inserted into the proof. ◻

Proof. The passing test gives finite full-development duration for every \(w\in O\), so Proposition 12 applies to all of them. The nonempty open set \(O\) has positive \(\mu\)-measure and cannot be contained in the null set of Proposition 14. Choose a noncorner seed. The source’s exchange lemma changes the coordinate preparation on the same cylinder and product spacetime and retains its estimates; it changes neither the geodesic nor its initial seed. ◻

Proof. Exterior evolution and cylinder data. The exterior input is the simultaneous closure in (OpenAI 2026b, companion@kind@foundation@far:closure companion@kind@foundation@far:closure ). It applies to both computing cylinders from the \(p_{10}\) input, with variable asymptotic parameters and arbitrary finite higher symbol seminorms. The higher estimates are (OpenAI 2026b, companion@kind@foundation@co2:higher companion@kind@foundation@co2:higher ). We describe the entry hypotheses so that the subsequent interior citations have a precise common domain of applicability.

For nearby physical Kerr parameters the exact profiles in (OpenAI 2026b, companion@kind@foundation@in:profiles companion@kind@foundation@in:profiles ) are smooth in \((q,\theta)\) and the parameters, with \(\mathcal A,\gamma\) and their inverses smooth at \(q=0\). They satisfy \[b=O(q),\quad \chi_i,\ell_i+\kappa,\beta_i=O(q), \qquad \xi_i,K_\gamma,\operatorname{curl}_\gamma\xi_i=O(1),\] after each fixed angular or parameter derivative; here \(K_\gamma\) is the Gaussian curvature of the angular metric. The common \(\kappa\) is obtained by rescaling the exact optical variables by \(\kappa_-/\kappa\), where \(\kappa_-\) is the positive absolute inner surface gravity of the current physical profile.

The moving preparation uses integrated Killing and rotation phases, whose derivatives are the instantaneous frozen frequencies. Its Jacobian errors contain positive parameter derivatives with bounded coefficients, without an elapsed-time factor. This is (OpenAI 2026b, companion@kind@foundation@in:moving-coordinates companion@kind@foundation@in:moving-coordinates ). The single physical profile on the wedge depends only on \(u\); a frozen differentiated profile means differentiation at fixed parameters followed by evaluation at this profile. The additional parameter defect is consequently confined to the \(u\) direction. After decreasing \(\epsilon\), the cylinder differences from these frozen profiles have arbitrarily small sums of unit-\(x\)-cell suprema in exactly the following norms: \[ \begin{array}{c|c} \mathcal A,\log\mathcal A,\gamma,b,\chi_i,\ell_i,\xi_i&H^6\text{ in angle}\\ \gamma&H^7\text{ in angle}\\ \partial_x(\chi_i,\ell_i,\xi_i,b)|_{\mathcal S}&H^2\text{ in angle}. \end{array} \tag{32}\] The unit-cell sum of the first profile derivative is small as well. Every fixed higher entry norm on \(|x|\le CT\) is subexponential. These are the full assumptions and conclusions of (OpenAI 2026b, companion@kind@foundation@in:entry companion@kind@foundation@in:entry ). Its trace estimate uses mixed spacetime metric derivatives through order eight and one additional time derivative outside the trapping region, supplied by companion@kind@foundation@co2:sliced companion@kind@foundation@co2:sliced

  companion@equation@foundation@co2:sliced ( companion@number@foundation@co2:sliced companion@number@foundation@co2:sliced

) companion@number@foundation@co2:sliced companion@number@foundation@co2:sliced

there. The entry supports lie between the horizons, separated from that trapping region. On the bounded middle the finite bridge evolution supplies the same derivative budget. Thus the trace step does not require a small high-order norm beyond the stated \(p_{10}\) input.

Interior continuation and derivative bounds. companion@kind@foundation@in:low companion@kind@foundation@in:low

  companion@equation@foundation@in:low ( companion@number@foundation@in:low companion@number@foundation@in:low

) companion@number@foundation@in:low companion@number@foundation@in:low

, companion@kind@foundation@in:pure-jets companion@kind@foundation@in:pure-jets

  companion@equation@foundation@in:pure-jets ( companion@number@foundation@in:pure-jets companion@number@foundation@in:pure-jets

) companion@number@foundation@in:pure-jets companion@number@foundation@in:pure-jets

, and companion@kind@foundation@in:higher companion@kind@foundation@in:higher

  companion@equation@foundation@in:higher ( companion@number@foundation@in:higher companion@number@foundation@in:higher

) companion@number@foundation@in:higher companion@number@foundation@in:higher

of (OpenAI 2026b) apply to precisely these profiles and entry norms. They give Equations (28)–(31). In particular the second line of Equation (29) includes the pure longitudinal derivatives \(\mathfrak L_i\chi_i\) and \(L_i\ell_i\); one has not inferred them by differentiating an uncontrolled limit. companion@kind@foundation@in:continuation companion@kind@foundation@in:continuation

  companion@equation@foundation@in:continuation ( companion@number@foundation@in:continuation companion@number@foundation@in:continuation

) companion@number@foundation@in:continuation companion@number@foundation@in:continuation

first uses finite-diamond high estimates, where \(a\) is bounded below, to continue and exhaust the full wedge. Only after that exhaustion does it use the tails of the summable low envelopes to prove the expanding-domain subexponential estimate. This order avoids assuming a global tail estimate to prove existence of the region carrying that tail. The factor assertions follow by differentiating \(a=q\mathcal A\) and \(\partial_vb=a\gamma^{-1}(\xi_1-\xi_0)\).

The other branch convention. Finally (OpenAI 2026b, companion@kind@foundation@in:label-swap companion@kind@foundation@in:label-swap ) negates the prepared \(x\) on \(\mathcal S\) and transports the newly prepared angles along the new shift-free direction. At each finite point this is a finite smooth sphere flow with a smooth inverse. The entry proof and finite-diamond estimates are reapplied in that gauge and exhausted. No uniform bound on an otherwise unestimated long angular coordinate change is used. These are estimates in the newly prepared coordinates on the same comparison spacetime. ◻

Proof. This is the observer-independent calculation in (OpenAI 2026b, companion@kind@foundation@pk2:branch-normalization companion@kind@foundation@pk2:branch-normalization ). Unit normalization gives \[ 1+|\Xi|_\gamma^2=2a\dot u\dot v=2p\dot u. \tag{34}\] For the unit tangent \(U\), differentiating \(p=-g(U,L_0)\) and using the geodesic equation gives the exact identity \[\frac{\dot p}{p}=\ell_0\dot u+(\xi_0-\xi_1)(\Xi) -\frac{\chi_0(\Xi,\Xi)}p.\] The null connection identities in Equation (15) therefore imply \(|\dot{\log p}|\le C(\dot u+|\Xi|_\gamma)\), using \(|\Xi|^2/p\le2\dot u\) from Equation (34). Moreover \[\int|\Xi|_\gamma\,d\tau \le\sqrt2\left(\int du\right)^{1/2} \left(\int a\,dv\right)^{1/2}<\infty.\] Indeed \(u\) has finite increasing variation and \(a\le Ce^{-\kappa v}\) on the bounded-\(u\) segment. Integration therefore bounds \(p\) above and below by positive constants. Since \(b\) is bounded, this also proves finite angular length and an angular limit.

Work near that limit in a sphere chart and introduce canonical momenta \(\pi_u=-p-b\cdot P\). The unit shell solved with \(v\) as time has Hamiltonian \[ -\pi_v=\frac{a(1+|P|_{\gamma^{-1}}^2)}{2p}, \qquad \frac{du}{dv} =\frac{a(1+|P|_{\gamma^{-1}}^2)}{2p^2}. \tag{35}\] At fixed canonical \((\pi_u,P)\), \(\partial_\theta p =-(\partial_\theta b)\cdot P\). Bounded first derivatives of \(\log a,\gamma,b\) in Hamilton’s equation for \(P\) consequently give \[\left|\frac{dP}{dv}\right| \le C(p+|P|)\frac{du}{dv}.\] Gronwall in the finite variation variable \(u\) bounds \(P\). Equation (34) bounds \(\Xi\) and \(\dot u\). Finally Equation (26) yields Equation (33). This argument uses only first metric derivatives and the unit shell, and no curvature bound along the geodesic. ◻

Proof. The timing and low exterior bounds are established in (OpenAI 2026b, companion@kind@foundation@sec:compact companion@kind@foundation@sec:compact ); the escape and all-subsegment statements are companion@kind@foundation@pk2:escape companion@kind@foundation@pk2:escape

  companion@equation@foundation@pk2:escape ( companion@number@foundation@pk2:escape companion@number@foundation@pk2:escape

) companion@number@foundation@pk2:escape companion@number@foundation@pk2:escape

there. We check why the launch geometry meets that proposition independently of the timelike observer. The original profile-depth choice makes the radial potential of these particular null rays strictly positive. For one of the nearby exact Kerr profiles, write \(M_K\) and \(\mathfrak a_K\) for its mass and spin length, \(r_K\) for the Boyer–Lindquist radius, \(\omega=(\vartheta,\varphi)\) for the angular variables, and \(|\cdot|_{\mathrm{rd}}\) for the unit round sphere norm. Thus \[\Delta_K=r_K^2-2M_Kr_K+\mathfrak a_K^2,\qquad r_\pm=M_K\pm\sqrt{M_K^2-\mathfrak a_K^2}.\] In aligned Kerr coordinates its optical correction \(F_K\) solves \[ 2(r_K^2+\mathfrak a_K^2)\partial_{r_K}F_K +\Delta_K(\partial_{r_K}F_K)^2 +|d_\omega F_K|_{\mathrm{rd}}^2 +\mathfrak a_K^2\sin^2\vartheta=0, \qquad F_K(r_-)=0. \tag{37}\] The finite root is noncharacteristic. In the original comparison preparation, \(q_0\) is chosen small enough for both angular transport conventions and for the uniform strict depth condition \[ \left(|\partial_\vartheta F_K|^2 +\mathfrak a_K^2\sin^2\vartheta\right)_{\rm entry} <4\mathfrak a_K^2. \tag{38}\] This is possible because the physical spins lie in a compact neighborhood inside \((0,1)\) and \(\partial_\vartheta F_K=O(q_0)\). The inner exit is fixed strictly deeper, with a positive margin before the inner horizon.

On fixed radial compact sets the metric and prepared entry covectors converge in the required low orders to a nearby frozen subextremal Kerr metric and to \(\lambda_-d(r_*+F_K+T_K)\) in aligned variables, with \(\lambda_-=\kappa_-/\kappa>0\). Here \(T_K\) is the signed Boyer–Lindquist Killing time and the tortoise coordinate \(r_*\) satisfies \(dr_*/dr_K=(r_K^2+\mathfrak a_K^2)/\Delta_K\). The integrated phase preparation used above gives this covector convergence even if accumulated azimuthal phases do not converge. The frozen axial momentum is zero and its radial potential, divided by the square of Killing energy, is \[ (r_K^2+\mathfrak a_K^2)^2-\Delta_K \left(|\partial_\vartheta F_K|^2 +\mathfrak a_K^2\sin^2\vartheta\right)_{\rm entry}. \tag{39}\] For \(r_K\ge r_+\), \(\Delta_K<r_K^2\) and \((r_K^2+\mathfrak a_K^2)^2/r_K^2\ge4\mathfrak a_K^2\); between the horizons \(\Delta_K<0\). The prior depth choice Equation (38) therefore makes this potential strictly positive. Its initial past radial direction is outward, since the finite Hamilton–Jacobi root gives \(r_K^2+\mathfrak a_K^2+\Delta_K\partial_{r_K}F_K>0\). It crosses each fixed compact radial band, including the future event horizon in regular coordinates, transversely in bounded regular time. The defining entry level is strictly monotone near its crossing; the other used compact portions are separated from it. These are uniform launch-patch facts, and persist by the compact convergence just stated.

In the far region the proof of companion@kind@foundation@pk2:escape companion@kind@foundation@pk2:escape

  companion@equation@foundation@pk2:escape ( companion@number@foundation@pk2:escape companion@number@foundation@pk2:escape

) companion@number@foundation@pk2:escape companion@number@foundation@pk2:escape

gives a summably small connection integral on the outward-to-the-past cone. More precisely, companion@kind@foundation@pk2:far-ray-coefficients companion@kind@foundation@pk2:far-ray-coefficients

  companion@equation@foundation@pk2:far-ray-coefficients ( companion@number@foundation@pk2:far-ray-coefficients companion@number@foundation@pk2:far-ray-coefficients

) companion@number@foundation@pk2:far-ray-coefficients companion@number@foundation@pk2:far-ray-coefficients

there gives the far-region lab-derivative estimate: it controls the bad first-derivative term on each radial dyad \(r\asymp R\) by an integrated bound \(O(R^{(e_f-(1-\delta_f)A_H)/2})\), with \(e_f-(1-\delta_f)A_H<0\), and the remaining term is summable. Here \(e_f,\delta_f,A_H\) are the fixed far-estimate exponents of that source. Thus the lab velocity stays close to a fixed outward unit direction and the frequency stays nondegenerate. Integration gives Equation (36); the same connection integral bounds tensor transport and its inverse. For the Hamiltonian first variation, only the momentum-to-position block need remain of order one at large radius. Rescaling its position variation by a small fixed constant makes that block small, and then taking the far radius large makes the other blocks small by the first and second metric derivative bounds. The complementary compact crossing has fixed length. This proves an arbitrarily small average growth rate on every subsegment. Higher variation equations have this same homogeneous propagator and sources involving lower jets and fixed high coefficient jets; induction gives the asserted subexponential bounds. The proof uses only the prepared launch covectors and the exterior geometry, so applies at the moving angular foot of any branch and after the separately prepared exchange of labels. ◻

Proof. The needed calculations are those of (OpenAI 2026b, companion@kind@foundation@pk2:coefficients companion@kind@foundation@pk2:coefficients ); we display them to identify every possible unsuppressed term. Koszul’s formula applied to Equations (26) and (27) gives, with \(j=1-i\), \[\begin{align*} \nabla_{L_i}\hat e_i&=\tfrac12\ell_i\hat e_i,& \nabla_{L_i}\hat e_j&=-\tfrac12\ell_i\hat e_j +\sqrt a\,\xi_i^Ae_A,\tag{44}\\ \nabla_{L_i}e_A&=(\chi_{i,A}{}^B-\partial_AB_i^B)e_B +\sqrt a\,\xi_{i,A}\hat e_i,\\ \nabla_{Z_A}\hat e_i&=\chi_{i,A}{}^Be_B +\tfrac12\sqrt a(\xi_i-\xi_j)_A\hat e_i,& \nabla_{Z_A}e_B&=\sqrt a\,\Gamma(\gamma)_{AB}^Ce_C +\chi_{0,AB}\hat e_1+\chi_{1,AB}\hat e_0. \end{align*}\] For example, differentiating \(L_i=\sqrt a\,\hat e_i\) gives \(\nabla_{L_i}L_i=\ell_iL_i\). The low estimates yield Equation (42). The exact brackets are \[ \begin{gathered} [L_0,L_1]=\sqrt a(\xi_0-\xi_1)^AZ_A,\\ [L_i,Z_A]=\tfrac12\ell_iZ_A-(\partial_AB_i^B)Z_B,\\ [Z_A,Z_B]=\tfrac12\sqrt a\bigl( (\partial_A\log a)Z_B-(\partial_B\log a)Z_A\bigr). \end{gathered} \tag{45}\] The radial trace of differentiated connection coefficients uses \(L_0V_1\) and \(L_1V_0\), and hence cross derivatives. In particular \(L_1\partial_Ab=\partial_A\partial_vb\) retains \(a\). Longitudinal derivatives of torsion reduce by the vacuum identities \[\mathfrak L_i\xi_j=\beta_i+\tau_i\xi_i, \qquad \mathfrak L_i\xi_i=d_\theta\ell_i-\beta_i-\tau_i\xi_i.\] The remaining angular derivatives carry \(\sqrt a\). These observations and Equation (29) give the boundedness and bulk smallness of the contracted differentiated connection terms.

Writing \(V_X\) for the connection matrix along \(X\), curvature obeys the exact matrix identity \[ aR(\hat e_\alpha,\hat e_\beta) =Z_\alpha V_\beta-Z_\beta V_\alpha +[V_\alpha,V_\beta]-V_{[Z_\alpha,Z_\beta]}. \tag{46}\] Besides the cross derivatives just considered, its radial-angular entries contain the pure derivatives \(L_i\chi_i\) controlled by Proposition 17. Two angular directions carry a \(\sqrt a\) derivative or a product of matrices tending to zero. The only nonzero limiting bracket coefficient multiplies \(V_A\to0\), while \([D_0,-D_0]=0\). Angular \(H^2\) embeds into \(C^0\) on the controlled spheres, so the stated low derivative budget suffices. Differentiating these identities and using Equation (30) proves the higher-jet assertions. The covariant tensor connection has the negative vector connection on each slot; its constant radial actions are therefore \(L_0+c_n\) and \(L_1-c_n\). Expanding their mixed trace gives Equation (43), since \(L_0x=-1/2\) and \(L_1x=1/2\). In particular \(2c_n^2\) has not been classified as a small coefficient. The angular metric and rescaled torsion need not tend to a prescribed constant or be small. ◻

Proof. We extract the linear construction of (OpenAI 2026b, companion@kind@foundation@pk2:deep-packet companion@kind@foundation@pk2:deep-packet ). The verification below records its amplitude independence and its uniformity at the actual frequency \(\lambda_T\).

Constrained transport in the interior. Set \(k_v=g^{-1}dv=-a^{-1}L_0\), \(K_vp=p(k_v,\cdot)\), and \(\mathcal T=2\nabla_{k_v}+\square_gv\). Since \(v\) is optical, the transport and divergence equations are \[ i\mathcal Tp_j+\mathcal Pp_{j-1}=0, \qquad iK_vp_j+\mathcal Cp_{j-1}=0, \qquad p_{-1}=0. \tag{55}\] Multiplication of the transport by \(-a/2\) gives \(\nabla_{L_0}+\tfrac12\tau_0\). On a normalized entry of weight \(c_n\) its same-weight constant part is \(L_0+c_n\), and the remaining same-weight coefficients have arbitrarily small bulk average. The connection formulas show that off-weight transport couples only from higher weights. At \(u=-v\) prescribe the slots with a \(0\) index using the gauge equation. The free angular entries vanish for \(j>0\); for \(j=0\) take a compact smooth angular and \(v\) bump times a trace-free polarization of unit size at the foot of the ray from \(p_T\). Initialize the negative-weight slots at \(u_f\) with zero terminal values and integrate them backwards. The constants \(c_n\) have the damping sign in these respective directions. The decreasing-weight triangular system therefore determines all coefficients.

Conjugate Equation (48) by the oscillatory phase and compare successive powers. Once preceding gauge defects vanish, the next defect in Equation (55) obeys homogeneous one-form transport and has zero initial value. Hence every gauge equation holds throughout the strip. The terminal choices impose no further gauge condition: contraction by \(k_v\) uses exactly the entries with a \(0\) slot.

For completeness, along each \(L_0\) transport the portion outside \(\min(v,-u,t)>R\) has length bounded in terms of \(R,u_f\) only. On its complement \(b\), its first angular derivatives, and the nonconstant diagonal coefficients become arbitrarily small. Their integrals are consequently \(o(T)\). The angular flow and its inverse have subexponential first jets. Every higher flow variation solves the same homogeneous equation with lower-jet sources and the high coefficient bounds of Equation (31); induction gives subexponential jets, including derivatives of the moving foot. The favorable constant signs just noted give the identical conclusion for each transport coefficient. The normalized sources \(a\mathcal Pp_{j-1}\) and \(\sqrt a\,\mathcal Cp_{j-1}\) have these bounds by Lemma 20. Integration over length \(O(T)\) preserves them. All these equations are local in \(v\) and along the angular flow, so preserve the asserted support, also in the terminal integrations.

The leading tensor remains trace-free and annihilates \(k_v\). Its screen class transports by an isometry of the positive screen metric times an area factor with logarithmic derivative of integral \(o(T)\). Its norm thus has two-sided subexponential size. A trace-free symmetric form on this two-dimensional screen has a unit eigenvector with a diagonal value comparable to its norm. Represent it by a purely angular \(X\). Adding a multiple of \(L_0\) would not change \(p_0(X,X)\), so no observer-dependent screen adjustment is required. Multiplication of every \(p_j\) by one constant unit complex number makes the diagonal contraction real at \(v=T\), proving Equation (53). Truncation of Equation (55) leaves only the last transport and gauge remainders. Taking \(J\) large relative to \(M_1\) and the fixed differentiated output orders proves the interior residual claim. Each ordinary differentiation has cost at most one positive power of \(\lambda_T\); the factors \(a\) and \(\sqrt a\) already account for the normalized wave and divergence operators.

We have obtained the interior packet, including its polarization, support, and linear residual bounds. It remains to realize its entry jets by an exterior field that returns to the original bridge.

Exterior complex phases and amplitudes. Complex Gaussian beam superposition follows the method developed in (Ralston 1982; Tanushev 2008; Liu et al. 2013). We retain the phase-space construction and finite matching below because they must also preserve the tensor gauge and give subexponential bounds on the growing time interval. Parameterize centers \(Y\) by the three coordinates \(w=(x,\theta)\) of the launch patch. Start a complex phase \(\varphi\) with spatial value \(v(0,w)+i|w-Y|^2/2\) and choose its temporal root to agree with \(dv\) at the center. Proposition  19 supplies the real rays and every subsegment propagator. Propagate finite complex phase jets along them. If \(P_c,Q_c\) are the position and momentum differentials of this complex gradient graph, symplecticity gives \[ P_c^*Q_c-Q_c^*P_c=2i\operatorname{Im}M_c, \qquad \operatorname{Im}(Q_cP_c^{-1}) =P_c^{-*}(\operatorname{Im}M_c)P_c^{-1}>0, \tag{56}\] where \(M_c\) is the initial Hessian. The initial positive imaginary part and its inverse are controlled on the noncharacteristic collar. The first identity bounds \(P_c^{-1}\) by the bound for \(Q_c\), and the second bounds the inverse of the propagated imaginary Hessian by the bound for \(P_c\) squared, times the initial inverse. Thus all these quantities are \(e^{o(T)}\), even when the real spatial projection has caustics. Finite Taylor composition and reversion use only smooth jets at real points.

Central amplitudes transport by tensor parallel transport times the inverse square root of the density ratio determined by \(\sqrt{|\det g|}\,k^{t_e}\det P_c\), where \(k=g^{-1}d\varphi\) is evaluated at the beam center. This propagator and its inverse are subexponential on every subsegment; higher jets have the same homogeneous propagator and lower-jet sources. Impose both wave and gauge transports in a sufficiently high finite Taylor quotient. The conjugated Bianchi identity is their defect compatibility identity. Contraction \(Kp=p(k,\cdot)\) has the explicit right inverse \[ R_kf=l\otimes f+f\otimes l-f(k)l\otimes l, \qquad l(k)=1. \tag{57}\] The nonzero pairing remains invertible in formal jets. This fixes the forced gauge jets and leaves the projected kernel jets free.

The complex phases now remain nondegenerate through real caustics, and the constrained amplitude transport has identified the free kernel jets. We use those jets to match the interior field at entry.

Matching the full spacetime jets. Superpose the beams with center measure \((\lambda/2\pi)^{3/2}\,dY\). On the unique entry collar the center projection is a diffeomorphism. Changing the center variable to its position and taking Gaussian moments produces \(e^{i\lambda v}\sum_j\lambda^{-j}\mathfrak p_j\) in integer inverse powers. At each order the new central value appears with coefficient one; the other terms have already been fixed at earlier orders. Derivatives of the new value first occur one inverse power later, because odd Gaussian moments vanish. For each fixed \(T\), an auxiliary limit \(\lambda\to\infty\) in the differentiated linear residual identities shows that the coefficients \(\mathfrak p_j\) satisfy exactly the real-phase wave and gauge transports. After matching orders below \(j\), the remaining launch discrepancy at order \(j\) is therefore in \(\ker K_v\). Choose the free central kernel value to cancel it, and extend its jets using \(\operatorname{Id}-R_kK\). This keeps the compact launch support. Transport is noncharacteristic at the entry cylinder, so the matched values and equations match all required normal jets as well; independent value and normal conditions are not being imposed on a single free datum.

For clarity, a finite normal-jet budget can be chosen explicitly. If coefficients through degree \(J\) must match to total order \(m\) at the last level, assign \(m_j=m+2(J-j)\) derivatives to coefficient \(j\). Differentiating its transverse first-order transport at most \(m_j-1\) times requires at most \(m_j+1\) derivatives of the preceding coefficient through the second-order wave forcing. These are available since \(m_{j-1}=m_j+2\). Tangential derivatives of the matched values and induction on transverse multiplicity then determine all mixed jets. The differentiated gauge forcing is only first order in the preceding coefficient and fits the same budget; Bianchi compatibility adds no independent normal datum. Increasing the finite \(J\) requests further fixed-order coefficient bounds, without changing the number of derivatives applied to the physical beam at the already fixed output order.

Remainders at the actual frequency. The matching argument has identified the coefficients and their normal jets. Its auxiliary-frequency limit alone does not estimate the error at the experiment’s frequency \(\lambda=\lambda_T\). For that estimate we use the following finite bounds. At each fixed Taylor and transport degree, all coefficient jets, center-change jets, Gaussian Hessians and their inverses, and the pairing in Equation (57) cost \(e^{o(T)}\). A Taylor error of transverse order \(d\) therefore gains \(\lambda_T^{-d/2}e^{o(T)}\) under Gaussian integration. Choose the finite degrees backwards from the last required differentiated residual and matching order. In the collar one may first restrict to transverse distances at most \(\lambda_T^{-1/2+\varepsilon_2}\), choosing \(\varepsilon_2>0\) small for these finite orders; the complementary Gaussian tail is negligible. These finite remainder estimates give \(A_T\lambda_T^{-M_1}e^{o(T)}\) at the actual diagonal frequency.

Realize the finite jets with smooth cutoffs at tube radius \(r_T=e^{-b_pT}\), where \(0<2b_p<\zeta\). On the cutoff annulus, the higher Taylor terms are smaller than half the positive quadratic imaginary part for large \(T\): their relative bound is \(r_Te^{o(T)}\to0\), using the inverse-Hessian bound as well as the finite higher-jet bounds. Positivity therefore implies \[ \exp(-\lambda_T\operatorname{Im}\varphi) \le \exp\left[-c\exp\{(\zeta-2b_p)T-o(T)\}\right]. \tag{58}\] Every fixed number of cutoff derivatives costs only \(e^{Cb_pT}\), and every fixed phase derivative costs a fixed power of \(\lambda_T\) times \(e^{o(T)}\). Equation (58) dominates all such losses and every prescribed inverse power. Uniqueness and separation of the entry crossing exclude additional entry traces. Tubes may be stopped after they pass through the strict deeper exit and, on the bridge, lie within arbitrarily thin margins of Equation (36).

Finally absolute integration over the fixed three-dimensional center patch costs \(\lambda_T^{3/2}\); \(N_b\) differentiations cost at most \(\lambda_T^{N_b}\) times subexponential factors. Each additional finite formal coefficient has subexponential fixed jets and an assigned nonpositive frequency power. Increasing formal accuracy therefore changes constants and the large-\(T\) threshold, but neither positive power in the proposition. Taking real parts, undoing trace reversal, and multiplying by \(A_T\) gives \(h_T^B\) and all the claimed bounds. Every step up to this point is linear and homogeneous in this common factor. ◻

Proof. Every transport, beam superposition, residual and matching operation in the proof of Proposition 21 is linear and homogeneous in the final factor \(A_T\). Coefficient estimates are independent of that factor. Multiplying the resulting unit-amplitude construction by \(e^{-\alpha T}\) proves the assertion. In particular, a requested bound \(A_T\lambda_T^{-M_1}e^{o(T)}\le e^{-4\kappa T}\) follows by choosing the finite accuracy with \(\alpha+M_1\zeta>4\kappa\) and then taking \(T\) large. ◻

Proof. Write \[N_D(v)=\Phi(D+v)-\Phi(D)-P_{\mathrm{flat}}v.\] The subtraction is essential: \(P_{\mathrm{flat}}\) sends a general \(X^s\) element to quantities of weight three, and we do not assert that the full constraint map has values in \(Y^s\) on an arbitrary \(X^s\) neighborhood. After its flat linear terms cancel, the scalar remainder has the product types \[a\,\partial^2a,\quad (\partial a)^2,\quad b^2,\] and the momentum remainder has types \[a\,\partial b,\quad (\partial a)b,\] with smooth bounded inverse-metric coefficients. Here factors may be from the background or the correction, and at least one is a correction factor. The weight budgets are \(1+3=4\) and \(2+2=4\). Unit-ball and rescaled-annulus Sobolev multiplication, valid for \(s\ge3\), therefore gives \[\begin{align*} \|N_D(v)-N_D(w)\|_{Y^s} &\le C_s(\|D-(\delta,0)\|_{X^s}+b) \|v-w\|_{X^s},\tag{67}\\ \|N_{D_+}(v)-N_{D_-}(v)\|_{Y^s} &\le C_s\|D_+-D_-\|_{X^s}\|v\|_{X^s}, \tag{68}\end{align*}\] when \(\|v\|_{X^s},\|w\|_{X^s}\le b\). For the second inequality, integrate the mixed second differential of \(\Phi\) along the segments in the datum and correction variables. The expression vanishes at \(v=0\), and the same displayed weight budgets bound this differential. Differentiation of the inverse-metric coefficients preserves these estimates on a uniformly positive small ball.

For both signs solve \[ z_\pm=-\phi_\pm-N_{D_\pm}(\mathcal Kz_\pm),\qquad v_\pm=\mathcal Kz_\pm. \tag{69}\] On the common ball \(\|z\|_{Y^s}\le2b_2\), the first inequality has Lipschitz constant at most \(1/2\) after shrinking the two base sizes. The map preserves this ball because \(N_D(0)=0\). Starting at zero preserves exterior support by Equation (65) and locality of \(N_D\). The limit consequently has that support, and \(P_{\mathrm{flat}}v_\pm=z_\pm\) proves the claimed exterior constraint equation. Subtract the two fixed-point equations, apply Equation (68) at \(v_-\), and absorb the \(\frac12\|z_+-z_-\|_{Y^s}\) term. Boundedness of \(\mathcal K\) gives Equation (66).

We include the all-order argument from the proof of (OpenAI 2026b, companion@kind@foundation@pk:constraint-nonlinear companion@kind@foundation@pk:constraint-nonlinear ) to specify that it regularizes these same fixed points. Let \(U_\vartheta,V_\vartheta\) be the actions on \(X^s,Y^s\) of a rotation and a positive argument dilation near the identity; rotations also act on tensor components. For either sign put \[\mathcal K_\vartheta=U_\vartheta\mathcal K V_\vartheta^{-1}, \quad \phi_\vartheta=V_\vartheta\phi, \quad N_\vartheta(w)=V_\vartheta N_D(U_\vartheta^{-1}w).\] The first family is norm-smooth by the cited inverse lemma. The all-order symbol bounds of \(D\) and smooth compact support of \(\phi\) make the other displayed expressions smooth in their parameters. In the explicit weighted products for \(N_\vartheta\), parameter differentiation differentiates background coefficients or the explicit component and scale factors; it does not require more unknown derivatives than the two metric and one second-form derivatives allowed by \(X^s\).

On the full space \(X^s\) consider \[ w+\mathcal K_\vartheta\phi_\vartheta +\mathcal K_\vartheta N_\vartheta(w)=0. \tag{70}\] Its derivative in \(w\) has the form \(\operatorname{Id}+\mathcal K_\vartheta DN_\vartheta(w)\) and is invertible by a Neumann series on a common small ball. The implicit-function theorem gives a smooth local solution. The already constructed \(U_\vartheta v\) satisfies the same equation algebraically. Choose \(b_2\) small enough that \(v\) lies in a smaller ball; uniform boundedness of \(U_\vartheta\) places all these transforms in the common uniqueness ball. Thus uniqueness identifies their orbit with the smooth implicit-function branch. This argument does not assume norm continuity of the unknown orbit. It uses the all-input definition of \(\mathcal K\), because dilation moves the support boundary.

Distributional differentiation of the orbit now gives every iterate of rotations \(\Omega_{ij}=y'_i\partial_j-y'_j\partial_i\) and the dilation \(E=y'\cdot\partial\) in \(X^s\). Constant component rotations contribute only removable zeroth-order matrices. Since \[|y'|^2\partial_i=y'_iE+\sum_j y'_j\Omega_{ji},\] these derivatives span the scaled coordinate derivatives on every exterior annulus and on a neighborhood of \(|y'|=1\). Scaled Sobolev embedding proves smoothness there and all differentiated symbol bounds. Inside the sphere the correction is zero. Hence it is smooth across the sphere as well. Higher bounds may depend on the finite experiment; their smallness has never been required. Finally, the small base norm preserves positive definiteness. ◻

Proof. The rescaled constraint solve and its odd difference estimates use only \(A_T\lambda_T^P\to0\) at finitely many fixed positive powers. Choose the internal losses smaller than both the prescribed \(\eta\) and a sufficiently small fixed fraction of \(\alpha\). The exterior bootstrap then improves because \(A_T^2e^{\eta_{\rm int}T}=o(A_T)\). Its common integrated rate in Equation (82) can be made arbitrarily small independently of \(\alpha\), by the same far-radius and first-arrival choices. The finite triangular propagator preserves that rate and the odd cubic gain. Choose \(\zeta\) so that every fixed positive frequency loss \(P\zeta\) fits its allocated part of the internal loss budget and is smaller than \(\alpha\). After fixing a positive margin \(\delta\) within that budget, choose the finite inverse accuracy \(M_1\) so that \[\alpha+M_1\zeta>\max\{3\alpha,4\kappa\}+\delta.\] Then \(A_T\lambda_T^{-M_1}e^{o(T)}\) is smaller than both the cubic amplitude scale and \(e^{-4\kappa T}\). These choices complete the proof, with all-order smoothness obtained from the same single base-order constraint solve. No interior comparison estimate is used in this corollary. ◻

Proof. Let \(\mathbf A(t)=(a_{j\ell}(t))\), with zero diagonal and entries zero unless \(\ell<j\). Conjugate the inequalities by \(e^{-\mathcal R_T(t)}\) and use comparison with the nonnegative linear system. Its time-ordered propagator is the finite sum \[\mathbf U(t,s)=\operatorname{Id} +\sum_{m=1}^{K} \int_{s<\tau_m<\cdots<\tau_1<t} \mathbf A(\tau_1)\cdots\mathbf A(\tau_m) \,d\tau_m\cdots d\tau_1.\] Every product of \(K+1\) such matrices is zero: a nonzero entry would require \(K+1\) strict descents among \(K+1\) levels. This does not require that the matrices at different times commute. If \(B_T=\int_0^L\lVert\mathbf A(\tau)\rVert\,d\tau\), then \(\lVert\mathbf U(t,s)\rVert\le\sum_{m=0}^K B_T^m/m!\). The right side is \(e^{o(T)}\), since \(K\) is fixed and \(B_T=e^{o(T)}\); for any desired exponential tolerance use that tolerance divided by \(K+1\) in each coefficient bound. Variation of constants and removal of the common conjugation give (109). The bound used the whole-interval \(B_T\), and is therefore uniform in both endpoints. ◻

Proof. Fix one sign. Perform each identification on enlarged solution regions; the retained regions will be chosen after these identifications.

The full lower block. The data converge to \(d\) in \(p_{10}\), are smooth, and solve the vacuum constraints. Hence the finite bridge lemma applies with the same fixed depths, time bands, and past collar. Its future artificial faces are strictly future-exiting for causal curves. At bounded radius its bounds are finite-time Cauchy bounds on compact causal shadows; on the ends they are uniformly local symbol bounds, with bounded causal coordinate speeds. Its smallness hypothesis is precisely the fixed \(p_{10}\) hypothesis, and no smallness at all smooth orders is needed.

Let \(t_T\) be the exterior launch time, let \(C_e\) be the exterior center, and let \(s_* =\sup|C_e'|<1-2\beta\). The protected ball from Proposition 24 contains the fixed-origin ball of radius \[ (1-2\beta)t_T-|C_e(t_T)| \ge (1-2\beta-s_*)t_T-|C_e(0)|\longrightarrow\infty. \tag{116}\] Thus every fixed compact causal shadow used in the lower joins has unchanged data for large \(T\). Domain of dependence identifies the bounded central pieces, the bent compact part of the selected partial slice, and the required pieces on the unselected end with the background ones. On the horizontal far part of the selected partial slice \(t_e=0\), however, the lower development carries the actual datum \(d_T^\pm\), including its entire tail. This is also the initial datum of the selected exterior evolution. Equality with the old datum is asserted only in the protected regions.

The fixed-time exterior identification. Choose the fixed matching time \(t_a>0\) and an enlarged band around it supplied by the comparison-development construction. Both the lower block and the selected exterior solution evolve the same geometric data on the partial slice. Solve the identifying wave map \(\Psi\) from the exterior solution to the lower solution with the slice identification as its value and the matching future unit normal as its normal derivative. The background identification may be used as the reference coordinate map. On the fixed slab from \(t_e=0\) through the matching band, the map remains a diffeomorphism on the smaller region.

Here is the quantitative input for this last assertion. The source construction gives metric \(H^{10}\) and second-form \(H^9\) control on compact shadows and the corresponding uniformly local end control. The differentiated wave-map equation at map order ten uses Christoffel derivatives through nine. In a bootstrap with bounded low map derivatives and inverse Jacobian, tame product and composition estimates bound its current-order coefficient by the uniformly bounded low metric and map norms. The known higher coefficient derivatives through the stated finite orders multiply lower map derivatives; their norms are controlled by the same fixed-time package. Thus the energy of the difference from the background map obeys an inequality of the form \[ E_{10}(s)\le C\eta_T+C\int_0^s E_{10}(\sigma)\,d\sigma, \qquad 0\le s\le t_a+\delta_a, \qquad \eta_T\longrightarrow0, \tag{117}\] where \(C\) and \(\delta_a>0\) do not depend on the growing outer cutoff. On the ends the estimate is taken as a supremum of energies in uniformly local charts, so its coefficient does not grow with their number or total volume. Uniqueness identifies the maps on chart overlaps. Fixed-time Gronwall, Sobolev embedding, and the strict spatial and deeper margins give the required \(C^1\) diffeomorphism control. To see global injectivity on the used region, compose with the inverse background identification. The resulting map \(\widehat\Psi\) is uniformly close to the identity in both \(C^0\) and \(C^1\). Two points with the same image must therefore be close in the reference bounded-geometry metric. They lie in one enlarged convex chart, where integration of \(D\widehat\Psi=\operatorname{Id}+o(1)\) along their joining segment rules out a collision. The inverse function theorem supplies the smooth inverse on the image. We also obtain at least \(H^9\) control after straightening the joining collar. This is exactly the finite-order map estimate in the cited comparison-development proof. Late high-order bounds of size \(e^{o(T)}\) are not used as its top Gronwall coefficient.

Write \(g_E,g_L\) for the exterior and lower metrics. Set \(h=\Psi^*g_L\) and now hold this smooth metric fixed as the target. The wave-map identity is \(\operatorname{tr}_{g_E}(\Gamma(g_E)-\Gamma(h))=0\). Thus \(g_E\) and \(h\) are vacuum metrics in the same target-\(h\) reduced system. Slice and future-normal matching give equal initial metric values and tangential derivatives. Equality of the second fundamental forms fixes their normal spatial-metric derivatives. More explicitly, let \(B_{\mu\nu}=\partial_0(g_E-h)_{\mu\nu}\) on the slice, so \(B_{ij}=0\). Lowering the relative gauge vector with the common initial metric gives \[\mathcal C_i=h^{00}B_{0i},\qquad \mathcal C_0=\tfrac12h^{00}B_{00}.\] Since \(\mathcal C=0\) and \(h^{00}\ne0\), the remaining normal derivatives agree as well. The two metrics have identical full first reduced jets. Smooth local uniqueness, with the strict outflow boundaries of the enlarged domain, proves that the map is an isometry throughout the used matching band. This uniqueness argument is for each fixed \(T\); its higher smooth norms do not enter the preceding uniform map estimate. The small coordinate straightening can be cut off farther back inside the available margin. The compact joins use the unchanged background identification.

Entry into the deep domain. The selected exterior exists through \(t_T+C_2\), with a shrinking outer cutoff and strict deeper exit boundaries, by Proposition 24. The used interval of \(\mathcal S=\{t=0\}\) and its collar lie inside these margins: its selected late endpoint has bounded co-radius and exterior time \(t_e=t_T+O(1)\), whereas its middle and unselected parts are supplied by the unchanged lower pieces. The spacelike conormal of this entry surface retains its strict sign under the low-order comparison. The deep evolution uses exactly the induced spacetime jets furnished by this exterior and the same target-\(g\) wave-map gauge. Consequently local reduced uniqueness identifies their actual entry collars. An approximate packet match alone would not suffice for this step.

The three metrics now agree under actual collar isometries. It remains to choose which portions to keep and to show that a past causal curve from any retained future point reaches the complete lower block.

Retained domains and their endpoints. Let \(r=|y|\) be the selected-end co-radius. Beyond the fixed join only the used exterior region is retained, with outer radius \(R_{\mathrm{out}}(t_e)\) and terminal time \(t_{\mathrm{end}}\). The speed chosen in Proposition 24 satisfies \[ \frac{d}{dt_e}\bigl(r-R_{\mathrm{out}}(t_e)\bigr) \ge V_{\mathrm{out}}-V_{\max}>0 \tag{118}\] on future causal curves near the active outer cut. The retained selected-end set near the matching face is \[ \{t_e<t_{\mathrm{end}}\}\cap \bigl(\{t_e<t_a\}\ \cup\ \{r<R_{\mathrm{out}}(t_e)\}\bigr), \tag{119}\] with the lower entry-surface cut imposed where active. The first member of the union contains all radii. In particular no physical initial tail is replaced by background data beyond the computing cutoff. Figure 2 depicts this local condition.

For a past curve initially in the added future part, \(f=r-R_{\mathrm{out}}\) is nonincreasing as long as that cut is active. It therefore keeps the strictly negative gap it had at its starting point until reaching \(t_e=t_a\). It reaches an actual matching collar there, then the unrestricted lower block. A curve already in \(t_e<t_a\) stays in that union member. In particular \((t_e,f)=(t_a,0)\) cannot be a past endpoint from the added part. The unmatched part of the \(t_a\) face is a future exit and is omitted.

For clarity, the retained deep domain has strict future inequalities \[ \begin{split} t&<L_T:=\frac{T+u_*}{2}+c_0,\qquad t>0,\\ F_0:=u+\frac{c_0t}{T}&<u_2, \qquad F_1:=v+\frac{c_0t}{T}<T+c_2, \end{split} \tag{120}\] where \(u_2>u_*\) and \(c_2>0\) are fixed, and \(c_0>0\) was chosen small relative to these margins. The lower face \(t=0\) is attached. Relative cone control from Equation (115) keeps \(t\) temporal and gives, for future causal vectors normalized by \(dt=1\), \[ du,dv\ge-C\varepsilon_T, \qquad dF_i\ge \frac{c_0}{T}-C\varepsilon_T>0. \tag{121}\] Indeed, after division of the causal inequality by \(a\), the null components are \(du,dv\) and the angular velocity is \((d\theta-b\,du)/\sqrt a\). On the section \(du+dv=2\), the background cone is a bounded set and has \(du,dv\ge0\); a small relative perturbation enlarges these lower bounds by at most \(C\varepsilon_T\). This also bounds the angular velocity on that section.

If a past curve starts at \(q\) inside this domain, then along it \(F_i\le F_i(q)\). Thus its gaps from both omitted sides stay strictly positive, rather than merely nonnegative. At a limiting point on \(t=0\) these same gaps put it strictly inside the used entry interval. This interval and its collar were chosen strictly inside the exterior interval available through \(t_T+C_2\), including its outer and deeper margins. Such a limit is therefore in the actual identified collar. It cannot be an entry/side or entry/outer-cut corner. The terminal time inequality has the same past-preservation property. The other active temporal and inner cuts in the lower matches have the source construction’s strict future-exit signs.

At an intersection of cuts, the retained prescription is a finite union or intersection of these past-preserved strict sublevels. For a union, one retains the member occupied by the curve until it enters an unrestricted lower member. For an intersection, every active gap is preserved. This locates any finite past endpoint in an ordinary retained chart, an actual matching collar, or the full lower block. No continuation across an omitted face is used. For each fixed \(T\) the added coordinate ranges are finite and the metric is smooth and nondegenerate on the enlarged ranges. A causal curve parametrized by the relevant temporal coordinate has bounded coordinate speeds there, hence a limit at a finite endpoint. The gap argument just given places this limit in a retained chart, where causal continuation is possible. In the lower end pieces, the fixed-time speed bound rules out escape to spatial infinity.

The gluing topology and the Cauchy property. Straighten each matching collar as a product with the matching hypersurface. Retain the lower side from the lower solution and the upper side from the added solution, with only one copy of the matching face. The interiors of these pieces are disjoint. Remove unused future faces and their edges before taking the quotient; the matching faces are then closed relative to their respective retained pieces. The identifications are homeomorphisms on those closed faces and isometries on their collars. A finite gluing of this kind is Hausdorff: saturating a closed subset adds its closed image in each matching face, so the quotient map is closed; its fibres are finite, and distinct fibres can be separated in the Hausdorff pieces and then by saturated neighborhoods. The product collars give the smooth charts across the faces. At intersecting joining prescriptions first join the two lower charts, then cross \(\mathcal S\); the uniqueness identifications agree on the overlap. Thus there is no duplicate open continuation of a single face. Second countability follows from the finitely many countable atlases.

Every past-inextendible causal curve starting above \(\Sigma\) now reaches the lower block and the complete bridge: it cannot stop at an omitted face, at a finite interior endpoint, or at infinity in bounded time. The joining faces are spacelike and have a fixed crossing orientation, so the finitely many joining layers cannot be revisited indefinitely. The lower block has the Cauchy control of the finite bridge lemma. A curve in the short past collar reaches \(\Sigma\) to the future by that same lemma. No causal curve crosses \(\Sigma\) twice, by lower-block Cauchyness and the one-way signs of the subsequent joining faces. Hence every inextendible causal curve meets \(\Sigma\) exactly once. The glued metric is smooth vacuum and induces exactly \((h_T^\pm,K_T^\pm)\) on this full hypersurface. It is therefore a globally hyperbolic development in the sense of (Sbierski 2016, Definition 2.1). The maximal-development theorem (Sbierski 2016, Definition 2.2 and Theorem 2.8) gives the claimed initial-data-preserving isometric embedding into \(\mathcal M_{d_T^\pm}\); an embedding of equal-dimensional manifolds is open. At \(p_T\), the gap from the time top is at least \(c_0\), and the two side gaps are at least \(u_2-u_*-c_0/2-O(T^{-1})\) and \(c_2-c_0/2-O(T^{-1})\). These are fixed positive numbers. The derivatives of the cut functions in \((u,v)\) are uniformly bounded. A fixed sufficiently small coordinate box therefore lies inside the retained deep domain for every large \(T\). It has positive time above the spacelike entry, so it lies in \(I^+(\Sigma)\). ◻

Proof. The state momenta and \(u\) are bounded by Equation (122) and Equation (123). Choose \(v_0\) so that the angular track is in an inner subchart about its limit, with a fixed coordinate margin. The bounded first derivatives of \(b\) in Proposition 17 imply, on a fixed bounded momentum range and at the same \(v\), \[|p(Y)-p(Y_0(v))| \le C\bigl(|u-u_0(v)|+|\theta-\theta_0(v)| +|\pi_u-\pi_{u,0}(v)|+|P-P_0(v)|\bigr),\] with \(C\) independent of \(v\). Choose \(r_0>0\) smaller than the angular margin and than \(\inf p(Y_0(v))/(2C)\). On every radius-\(r_0\) neighborhood of \(Y_0(v)\), the momentum \(p\) then stays uniformly positive. This is a pointwise tube choice and entails no multiplication of \(C\) by the length of the late interval.

Twice differentiating the Hamiltonian in Equation (124) uses a fixed finite number of ordinary background jets and bounded inverse powers of \(p\). Every derivative of \(a\) retains \(a\), and all remaining fixed jets cost \(e^{o(v)}\) on a bounded \(u\) interval by Proposition 17. Thus both the Hamiltonian vector field and its first state derivative retain \(a\asymp e^{-\kappa v}\), which proves Equation (125) on these constant-radius neighborhoods. In particular \(Y_0(v)\) has a limit \(Y_\infty\). Increase \(v_0\) until \(|Y_0(v)-Y_\infty|<r_0/8\) throughout the tail. A fixed ball \(\mathcal K\) of radius \(r_0/4\) about \(Y_\infty\) then contains the track with margin \(r_0/8\), and even its radius-\(r_0/2\) enlargement lies in the already controlled radius-\(r_0\) neighborhood of \(Y_0(v)\) for every \(v\). This gives a state domain independent of both \(v\) and \(T\). ◻

Proof. Put \(g'=g_T^\pm\). Choose a background angular orthonormal frame \(e_A\) in the fixed chart and write \(P_{(A)}=\pi(e_A)\). Let \(Q^{\alpha\beta}\) be the symmetric coefficients of \(g'^{-1}-g^{-1}\) in the vector basis \((a^{-1/2}L_0,a^{-1/2}L_1,e_A)\). They and the required ordinary jets are \(O(\varepsilon_T e^{o(T)})\). Multiplication of the new shell by \(a\) gives the exact equation \[ \mathcal F_T(v,Y,z) =2pz+a(1+\gamma^{AB}P_AP_B)+\Delta_T(v,Y,z)=0, \qquad \Delta_T=a(g'^{-1}-g^{-1})(\pi,\pi). \tag{129}\] Keeping all powers of \(a\), its last term is the polynomial \[\begin{align*} \Delta_T={}&Q^{00}p^2-2Q^{01}pz+Q^{11}z^2 -2\sqrt a\,Q^{0A}pP_{(A)} +2\sqrt a\,Q^{1A}zP_{(A)}\\ &\hspace{32mm}+aQ^{AB}P_{(A)}P_{(B)}. \tag{130}\end{align*}\] This follows from \(\pi(L_0)=-p\) and \(\pi(L_1)=z\). It displays the coefficients of sizes \(\varepsilon_T\), \(\sqrt a\,\varepsilon_T\), and \(a\varepsilon_T\) before their fixed-order losses. In particular the coefficient of \(z^2\) is allowed to be nonzero.

On bounded \((Y,z)\) sets, ordinary and state differentiation of this polynomial gives \[ |\partial_v^{j_0}D_Y^{j_1}\partial_z^{j_2}\Delta_T| \le C_j\varepsilon_T e^{o(T)} \quad (j_0+j_1+j_2\le3). \tag{131}\] A derivative of \(\sqrt a\) or \(a\) retains that factor; a derivative of \(p\) uses \(b\) and its ordinary derivatives; the angular orthonormalization uses only fixed-order background jets. Thus frame differentiation introduces no additional inverse power of \(a\). The two null covector factors have already been cancelled by the external \(a\) in Equation (129).

Fix a bounded \(z\) interval containing \(z_0\) with a fixed margin. For large \(T\), Equation (131) implies \[ \partial_z\mathcal F_T=2p+\partial_z\Delta_T\ge p_*>0. \tag{132}\] The values at \(z_0\pm C\varepsilon_T^{\mathrm{sh}}\) have opposite signs for \(C\) large, so there is a unique root in this neighborhood. Implicit differentiation, using the denominator in Equation (132), proves Equation (128); any finite background jet factors in the resulting products are absorbed in the displayed \(e^{o(T)}\). The reduced Hamiltonian is \(\mathcal H_T^\pm=-z_T^\pm\), which gives the asserted vector-field bound. This reasoning does not claim that \(z_T^\pm-z_0\) is small relative to \(a\).

To check both the parametrization and its time orientation, let \(\mathcal H_{\mathrm{aff}}=(g'^{-1}(\pi,\pi)+1)/2\) be the affine geodesic Hamiltonian. Since \(a\) is independent of \(z\), \[ \partial_z\mathcal F_T =2a\,\partial_z\mathcal H_{\mathrm{aff}} =2a\,\frac{dv}{d\tau}. \tag{133}\] Equation (132) is therefore exactly the positive-\(v\) branch condition. For the time orientation, the function \(t=(u+v)/2\) remains temporal under the small relative metric perturbation. At fixed position and angular momentum, \(\partial_{\pi_u}p=-1\), so the other momentum derivative gives \[4a\frac{dt}{d\tau} =\partial_{\pi_u}\mathcal F_T+\partial_z\mathcal F_T =2p-2z+\partial_{\pi_u}\Delta_T+\partial_z\Delta_T \ge 2p_*-C\varepsilon_T^{\mathrm{sh}}>0.\] Here \(z_0\le0\), \(|z-z_0|\le\varepsilon_T^{\mathrm{sh}}\), and Equation (131) controls both error derivatives. Thus the root is future pointing using only absolute root closeness. The other canonical equations, divided by \(dv/d\tau\), are those of \(-z_T^\pm\) by implicit differentiation of the shell. This remains true in the presence of the quadratic term in \(z\). ◻

Proof. Terminal state. The normalized components of \(U\) at \(p_T\) are \[\sqrt{a_p}\,\dot u,\qquad p/\sqrt{a_p},\qquad \Xi^{(A)}, \qquad a_p=a(p_T),\] and hence have size \(O(a_p^{-1/2})\). The new covector is \(\pi'=g_T^\pm(I_T^\pm U,\cdot) =g(U,(I_T^\pm)^{-1}\cdot)\). Equation (127) and the coordinate identities \(\partial_u=\sqrt a\,\hat e_0-b^A\partial_A\) and \(\partial_v=\sqrt a\,\hat e_1\) give \[ |Y_T^\pm(T)-Y_0(T)| \le C\varepsilon_T/\sqrt{a_p}=o(1), \qquad |\pi'_v-\pi_v|\le C\varepsilon_T, \qquad dv(U_T^\pm)=\frac{p+O(\varepsilon_T)}{a_p}>0. \tag{135}\] The terminal positions agree exactly, and the angular coordinate and orthonormal bases have bounded conversion matrices. The new covector satisfies the exact unit shell. It lies in the fixed bounded root interval and in the fixed state tube, so uniqueness in Lemma 32 places it on that lemma’s future root. The loss \(a_p^{-1/2}\) in the first estimate of Equation (135) is paid once, at this terminal conversion.

Backward comparison on the actual domain. Increase the fixed \(v_0\) if necessary so that \((v_0+\inf u)/2\) has a positive margin above zero. Choose the state tube with a smaller concentric tube around the background track, leaving a fixed boundary margin. For \(v_0\le v\le T\) the background spacetime track has the uniform gaps \[\begin{align*} u_2-u(v)-\frac{c_0t(v)}{T} &\ge u_2-u_* -c_0/2-O(T^{-1}),\\ T+c_2-v-\frac{c_0t(v)}{T} &\ge c_2-c_0/2-O(T^{-1}),\tag{136}\\ L_T-t(v)&\ge c_0, \qquad t(v)\ge t(v_0)>0. \end{align*}\] Here \(u\) and \(v\) increase along the background future timelike track. All four right-hand margins are positive after the original choice of \(c_0\) and for large \(T\).

Solve backward from the terminal state only as long as the solution lies both in the fixed state tube and in the actual domain of Equation (120), stopping no later than \(v_0\). On any such provisional interval split the field difference as \[V_T^\pm(v,Y_T^\pm)-V_0(v,Y_0) =\bigl(V_T^\pm(v,Y_T^\pm)-V_0(v,Y_T^\pm)\bigr) +\bigl(V_0(v,Y_T^\pm)-V_0(v,Y_0)\bigr).\] The first term is bounded by \(C\varepsilon_T^{\mathrm{sh}}\) and the second by \(\Lambda(v)|Y_T^\pm-Y_0|\). Only the background Lipschitz constant is integrated. Backward Gronwall gives, uniformly in the provisional stopping point, \[ \sup_{v\le s\le T}|Y_T^\pm(s)-Y_0(s)| \le \exp\!\left(\int_{v_0}^\infty\Lambda(s)\,ds\right) C\left(\frac{\varepsilon_T}{\sqrt{a_p}}+T\varepsilon_T^{\mathrm{sh}}\right) =:\delta_T\longrightarrow0. \tag{137}\] For large \(T\), this is smaller than half every state-tube margin and every spacetime margin in Equation [attach:track-margins]. It therefore excludes a first stop at the state boundary, either tilted side, the top time cut, or \(t=0\). At a finite remaining interior endpoint the smooth finite-\(T\) canonical ODE continues inside its actual domain. Thus the solution reaches \(v=v_0\). The argument has not extended a perturbed vector field beyond the constructed spacetime to justify its own estimate.

Return through the earlier compact segment. From the fixed section \(v=v_0\) to \(\Sigma\) the background observer runs along a fixed compact segment. Choose a compact flow neighborhood with strict margins in the entry and lower matching collars; the crossings of \(\mathcal S\) and \(\Sigma\) are transverse, since the tangent is timelike and both hypersurfaces are spacelike. For large \(T\) its bounded-\(v\) deep portion lies below \((1-\nu_0)T\), and all its lower causal shadows lie in the protected ball of Equation (116). Exact earlier entry agreement from Proposition 24, the support conclusion of Proposition 27, and the attachment therefore identify this entire neighborhood with the background one. Finite-time smooth geodesic flow dependence carries the convergent states at \(v_0\) through these collars to seeds \(w_T^\pm\to w\). This compact observer segment is distinct from the packet’s moving entry foot near \(v=T\).

The constructed path is a future unit geodesic of the attached vacuum metric. Equation (133) also gives \(d\tau/dv=2a/\partial_z\mathcal F_T\) on its late portion, so every finite experiment has finite proper duration. Its earlier portion has finite duration by smooth compact flow dependence. At \(p_T\) there is an open observation neighborhood in the attached spacetime, so local geodesic continuation extends it for a positive further proper time. Proposition 29 embeds this path and its continuation into the full MGHD. Its observation time is therefore strictly below the full-MGHD endpoint. Openness of the original seed set gives membership of \(w_T^\pm\) for large \(T\).

Comparison with the norm in the test. The frame in Equation (134) has the fixed Lorentz Gram matrix \[g_T^\pm(E_0^\pm,E_1^\pm)=-1, \qquad g_T^\pm(E_A^\pm,E_B^\pm)=\delta_{AB},\] with all other entries zero. By isometry, the components of \(U_T^\pm\) in it are precisely \((\dot u,p,\Xi^{(A)})\), bounded by Equation (122). They belong to a compact subset of the unit future timelike hyperboloid. It follows that the positive form \[r_{U_T^\pm}=g_T^\pm +2g_T^\pm(\cdot,U_T^\pm)\otimes g_T^\pm(\cdot,U_T^\pm)\] is uniformly equivalent at \(p_T\) to the Euclidean component form: it is positive definite for every unit timelike vector, and its largest and smallest eigenvalues vary continuously on this compact set. Both this form and the test form are parallel along the observing geodesic. At \(\Sigma\) their comparison is uniform because \(d_T^\pm\to d\) and \(w_T^\pm\to w\); equivalently use Lemma 5. Parallel transport preserves the comparison constants. This proves the norm assertion and, by applying it to a vector and its image, the induced operator-norm assertion. A smooth \(e_A\) with the stipulated central direction and fixed jets of cost \(e^{o(T)}\) is obtained by Gram–Schmidt in the angular chart followed by a constant orthogonal rotation. ◻

Proof. We give the scaling calculation behind the assertion. Set \(Z_\alpha=\sqrt a\,\hat e_\alpha\), and let \(V_\alpha\) be the matrices of \(\nabla^g_{Z_\alpha}\) on the \(\hat e\) frame. The coefficient estimates of Lemma 20 bound \(V\), the coefficients of \([Z_\alpha,Z_\beta]\) in \(Z\), their required ordinary jets, and the normalized entries of \(a\operatorname{Riem}(g)\) by \(e^{o(T)}\). Relative derivatives of \(a\) have the same bound. These are the coefficient inputs in (OpenAI 2026b, companion@kind@foundation@pk2:coefficients companion@kind@foundation@pk2:coefficients ).

For an arbitrary small relative change \(H\), write \(M=\eta+H\) and let \(k\) be the physical tensor with relative array \(H\). The connection difference \(K=\nabla^{g+k}-\nabla^g\) satisfies \[2(g+k)(K(V,W),Y) =(\nabla^g_Vk)(W,Y)+(\nabla^g_Wk)(V,Y) -(\nabla^g_Yk)(V,W).\] On normalized slots, multiplication by \(\sqrt a\) replaces each derivative on the right by a \(Z\) derivative. More explicitly, put \[\mathcal D_\alpha H_{\beta\delta} =Z_\alpha H_{\beta\delta} -(V_\alpha)^\rho{}_{\beta}H_{\rho\delta} -(V_\alpha)^\rho{}_{\delta}H_{\beta\rho}.\] The array \(W\) of \(\sqrt a K\) is therefore \[ W^\mu{}_{\alpha\beta} =\frac12(M^{-1})^{\mu\delta} \bigl(\mathcal D_\alpha H_{\beta\delta} +\mathcal D_\beta H_{\alpha\delta} -\mathcal D_\delta H_{\alpha\beta}\bigr). \tag{143}\] Thus \(V+W\) is the connection of \(g+k\), differentiated along \(Z\) and expressed in the unchanged normalized frame. If \([Z_\alpha,Z_\beta]=b^\rho{}_{\alpha\beta}Z_\rho\), its curvature matrix is \[ R^{g+k}(Z_\alpha,Z_\beta) =Z_\alpha(V_\beta+W_\beta)-Z_\beta(V_\alpha+W_\alpha) +[V_\alpha+W_\alpha,V_\beta+W_\beta] -b^\rho{}_{\alpha\beta}(V_\rho+W_\rho). \tag{144}\] Its entries are smooth functions of \(H,DH,D^2H\) with subexponential coefficients, since \(Z_i=L_i\) and \(Z_A=\sqrt a e_A\) have such coordinate jets. Contracting with \(M\) gives every normalized covariant entry of \(a\operatorname{Riem}(g+k)\). The isometry \(I(H)\) is likewise an analytic function of \(\eta^{-1}H\); hence including its four slots preserves these bounds.

At \(p_T\), Equation (141) equals \(a_p\) times the curvature on slots \(I(H)\hat e_1,I(H)e_2,I(H)\hat e_1,I(H)e_2\). Equations (143) and (144) therefore prove the claimed jet dependence and derivative bounds. This controls the scaled background coefficients; it does not assert bounded curvature in the boosted observer frame.

Taylor’s formula at zero, applied separately to the two signs, gives \[C_T^+-C_T^- =2D\mathcal C_T(0)[q_T] +D\mathcal C_T(0)[F_T^{\mathrm{odd}}]+\mathcal E_T, \qquad |\mathcal E_T|\leq e^{o(T)} (\|H_T^+\|_{C^2}^2+\|H_T^-\|_{C^2}^2).\] Equation (139) proves Equation (142). The common term \(\mathcal C_T(0)\) has canceled exactly. ◻

Proof. We compute the linear term in Equation (142). Up to the curvature sign convention, the principal second derivative part of the variation on the physical slots \(L_1,X_T,L_1,X_T\) is \[\frac12\bigl(\nabla_1\nabla_1 h_{XX} +\nabla_X\nabla_X h_{11} -2\nabla_1\nabla_X h_{1X}\bigr).\] The packet from Proposition 21 is \[h_T^{\mathrm p} =A_T\operatorname{tr.rev.}_g\operatorname{Re} \left(e^{i\lambda_T v} \sum_{j=0}^{N}\lambda_T^{-j}p_j\right).\] The finite \(N\) has already been fixed. Each amplitude has fixed ordinary normalized jets of size \(e^{o(T)}\); \(p_0\) is trace-free and annihilates \(L_0\). Since \(dv(L_1)=1\) and \(dv(X_T)=0\), only the two phase derivatives on the \(XX\) entry contribute order \(A_T\lambda_T^2\). The leading trace reversal has no effect. All terms with a derivative on an amplitude or connection, all \(j\geq1\) terms, and the variation of the four isometry slots on the scaled background curvature have at most order \(A_T\lambda_T e^{o(T)}\). The last assertion also follows directly from Equation (144): differentiation of \(I(H)\) is algebraic and so creates no phase derivative. We obtain \[ D\mathcal C_T(0)[q_T] =\pm\frac12 A_T\lambda_T^2 \operatorname{Re}\bigl(e^{i\lambda_TT}p_0(X_T,X_T)\bigr) +O(A_T\lambda_T e^{o(T)}). \tag{145}\] The choice of direction and constant complex phase in Proposition 21 makes the real contraction here equal to a positive \(\sigma_T\) with \(\sigma_T\) and \(\sigma_T^{-1}\) of size \(e^{o(T)}\). The source construction’s observer-orthogonal screen representative differs from the angular \(X_T\) by a multiple of \(L_0\); its contraction with \(p_0\) is identical because \(p_0(L_0,\cdot)=0\). Thus the polarization input precedes, and does not use, the bounded-curvature hypothesis in the tidal application of (OpenAI 2026b, companion@kind@foundation@pk2:deep-packet companion@kind@foundation@pk2:deep-packet ). Since \(\lambda_T=e^{\zeta T}\) with fixed \(\zeta>0\), \[\frac{A_T\lambda_T e^{o(T)}}{A_T\lambda_T^2\sigma_T} =e^{-\zeta T+o(T)}\longrightarrow0, \qquad \lambda_T^2\sigma_T\longrightarrow\infty.\] Equation (142) now gives \(|C_T^+-C_T^-|>2A_T\) for large \(T\), proving the claim by the triangle inequality. ◻

Proof. Write \(\widetilde e_\alpha=I\hat e_\alpha\) and let \(\widetilde V^\mu{}_{\alpha\beta}\) be defined by \[\nabla^{g'}_{\sqrt a\,\widetilde e_\alpha} \widetilde e_\beta =\widetilde V^\mu{}_{\alpha\beta}\widetilde e_\mu.\] Equation (143), the change of frame formula, and Equation (138) give \[ \|\widetilde V-V\|_{C^{20}} +\|I-\operatorname{Id}\|_{C^{21}} \leq e^{-(\kappa/2+c_1)T} \tag{147}\] for some \(c_1>0\), with \(I-\operatorname{Id}\) expressed in normalized entries. Indeed the change of frame formula uses \(I,I^{-1},Z I\) and the matrices \(V+W\); every difference has a factor \(H\) or \(DH\). Twenty derivatives use at most twenty-one metric derivatives; the thirty-derivative input leaves a buffer. Multiplication by any fixed collection of background \(e^{o(T)}\) factors only reduces \(c\) to a smaller positive \(c_1\).

Put \[s_0=\tfrac12,\qquad s_1=-\tfrac12,\qquad s_2=s_3=0, \qquad E_\alpha=a^{s_\alpha}\widetilde e_\alpha.\] The full conversion formula is \[ G^\mu{}_{\alpha\beta} =\frac{a_p}{a} a^{1/2+s_\alpha+s_\beta-s_\mu} \widetilde V^\mu{}_{\alpha\beta} +s_\beta(f_\alpha\log a)\delta^\mu_\beta. \tag{148}\] There are precisely seven triples with negative exponent in the first term. The following table lists them all; \(A=2,3\) in each of its last three rows.

For completeness, \(s_\alpha+s_\beta\) belongs to \(\{-1,-1/2,0,1/2,1\}\). If \(\mu=1\) the exponent is \(1+s_\alpha+s_\beta\geq0\). If \(\mu\) is angular it is negative only when \(\alpha=\beta=1\). If \(\mu=0\) it equals \(s_\alpha+s_\beta\), negative only for \((1,1),(1,A),(A,1)\). This proves exhaustiveness.

Nullness gives, for every \(\alpha\), \[0=\tfrac12\sqrt a\,\widetilde e_\alpha g'(\widetilde e_1,\widetilde e_1) =g'(\nabla^{g'}_{\sqrt a\widetilde e_\alpha} \widetilde e_1,\widetilde e_1) =-\widetilde V^0{}_{\alpha1}.\] Thus the first three individual entries vanish identically for the perturbed metric, together with every derivative. In the other four entries only the background vanishes. In fact \(L_1=\partial_v\) in the optical metric satisfies \(\nabla^g_{L_1}L_1=(L_1\log a)L_1\), so \(V^A{}_{11}=0\). Differentiating \(g(\hat e_1,e_A)=0\) along \(L_1\) then gives \(V^0{}_{1A}=0\). These are identities throughout the box, hence their ordinary derivatives also vanish. Equation (147) therefore bounds all four remaining perturbed entries. Their worst multiplier is \(a^{-1/2}\), leaving \[a^{-1/2}e^{-(\kappa/2+c_1)T}=O(e^{-c_1T})\] on this box. Derivatives of the boost factors retain these powers, because \(D^j a^r=a^r e^{o(T)}\) at every fixed order. Thus all twenty derivatives of the seven entries are controlled.

We also verify the coordinate bounds used in the diagonal term of Equation (148). A normalized radial basis vector has coordinate size \(a^{-1/2}e^{o(T)}\), whereas an angular one has size \(e^{o(T)}\). For a term with input label \(\alpha\) and normalized output label \(\rho\), the largest coordinate multiplier is \(a_pa^{s_\alpha-1/2}\) if \(\rho\) is radial, and \(a_pa^{s_\alpha}\) if it is angular. As \(a\asymp a_p\), their worst powers are respectively \(a^0\) and \(a^{1/2}\), both attained when \(\alpha=1\). Consequently the coordinates of \(f\) and their twenty derivatives are \(e^{o(T)}\). The same holds for \(f_\alpha\log a\) by the relative derivative bounds for \(a\). The nonnegative exponents in Equation (148) are now controlled as well.

Finally, torsion-freeness and constancy of \(a_p\) give the exact identity \[ [f_\alpha,f_\beta] =(G^\mu{}_{\alpha\beta}-G^\mu{}_{\beta\alpha})f_\mu. \tag{149}\] There is no factor \(a_p^{-1}\). This proves the bracket jet bounds without estimating the inverse coordinate matrix of \(f\). ◻

Proof. Finite products and sums of subexponential bounds remain subexponential; \(Q_T\) may be increased finitely many times below. Use \(r\in[0,1]\) as the flow variable. With \(c^\alpha=h(s\delta_1^\alpha+t\delta_2^\alpha)\), the flow solves \[ \partial_r\Psi=c^\alpha f_\alpha(\Psi),\qquad \Psi(0)=p_T. \tag{154}\] Its coordinate speed and its position-Lipschitz coefficient are bounded by \(hQ_T\). Since \(17h_TQ_T\to0\), the speed bound keeps the flow in the fixed strict box for the entire interval \([0,1]\); local existence and continuation give the stated time-one flow.

Coordinate parameter derivatives are also controlled. A derivative of total order \(n\geq1\) in \((h,s,t)\) satisfies a linear equation whose coefficient on its order-\(n\) unknown is \(c^\alpha Df_\alpha(\Psi)\), of size \(hQ_T\). Its other terms are finite products of derivatives of \(f\) through order \(n\), lower derivatives of \(\Psi\), and derivatives of the polynomial \(c\). The initial derivatives vanish because \(p_T\) is parameter independent. Induction through \(n=19\), using \(\exp(O(hQ_T))\leq2\) for large \(T\), bounds each such derivative by \(e^{o(T)}\). This uses at most nineteen coordinate derivatives of \(f\).

The coordinate bounds alone do not give the desired physical length. To keep the factor \(a_p\), consider the \(s\) variation in the \(f\) frame. Define \(J_s(r)\) by the linear system \[ \frac{dJ_s^\gamma}{dr} =h\delta_1^\gamma +c^\alpha J_s^\beta B^\gamma{}_{\beta\alpha}(\Psi(r)), \qquad J_s(0)=0. \tag{155}\] Then \(J_s^\beta f_\beta(\Psi)\) equals \(\partial_s\Psi\): differentiating this vector using Equation (155) and the definition of the bracket cancels the frame derivative terms and gives exactly the coordinate variation equation of Equation (154), with the same zero initial value. Uniqueness proves the equality. This construction never estimates an inverse coordinate frame. The equation for \(J_t\) has direct source \(h\delta_2^\gamma\) and zero initial value. In particular these are inhomogeneous equations for the parameter-dependent flow; their homogeneous matrix has size \(hQ_T\). Integration gives \[ |J_s|+|J_t|\leq hQ_T. \tag{156}\]

Differentiate Equation (155) directly in \(h\), including at \(h=0\). At derivative order \(n\), the coefficient on \(\partial_h^nJ\) is still \(c^\alpha B_{\beta\alpha}(\Psi)\). Every other term contains a lower \(h\) derivative of \(J\), already bounded coordinate derivatives of \(\Psi\), and derivatives of \(B\) of order at most \(n\). The direct source has first derivative \(\delta_i\) and all higher derivatives zero. Thus induction gives \[\sup_{r,s,t,h}|\partial_h^nJ_s| +|\partial_h^nJ_t|\leq e^{o(T)},\qquad 1\leq n\leq18.\] No division by \(h\) is used. The twenty-derivative bounds for \(f,B,G\) allow the nineteen coordinate variation derivatives and the eighteen transport derivatives with a spare derivative.

Parametrize each oriented boundary edge by \(z\in[0,1]\). Its tangent has the form \(J^\alpha f_\alpha\), where \(J\) is one of \(\pm J_s\), \(\pm J_t\) evaluated at \(r=1\) and at that edge. Parallel transport in the moving \(E\) frame solves \[ \frac{dP}{dz}=-\mathcal G(z,h)P,\qquad \mathcal G^\mu{}_{\beta}(z,h) =G^\mu{}_{\alpha\beta}(\Psi_h)J^\alpha. \tag{157}\] The undifferentiated coefficient is \(O(hQ_T)\) by Equation (156); its first eighteen \(h\) derivatives have size \(e^{o(T)}\). Fundamental matrices and their inverses are bounded by \(\exp(O(hQ_T))\) on each edge. Differentiating Equation (157) \(n\) times yields the same homogeneous coefficient on \(\partial_h^nP\) and source \[-\sum_{j=1}^n\binom nj (\partial_h^j\mathcal G)(\partial_h^{n-j}P).\] Induction through \(n=18\), followed by multiplication of the four edge matrices, gives the derivative bound in Equation (152).

At any point on the boundary the tangent has \(E\)-components \(a_pJ\). If \(P\) denotes accumulated parallel transport from the base, its norm in the loop-parallel positive metric is its base norm after applying \(P^{-1}\), hence at most \(C a_p|P^{-1}J|\leq a_phQ_T\) by the uniform base comparison and Equation (156). Summing the four unit edge intervals proves the length estimate in the actual test metric.

Finally consider the two-parameter surface \(\Phi(\sigma,\vartheta)= \operatorname{Fl}^1_{\sigma f_1+\vartheta f_2}(p_T)\). Our loops are its boundaries of \([0,h]^2\). For each fixed \(T\), pull back the connection and choose a smooth frame equal to \(E(p_T)\) at the origin whose connection one-form vanishes there. Such a frame is obtained by prescribing its first derivative at the origin; no uniform estimate on that auxiliary choice is used. Its connection one-form is \(O(|\sigma|+|\vartheta|)\). Therefore ordered products of two edge integrals have order at least \(h^4\), while Stokes’ formula gives the order-\(h^2\) term as minus the curvature at the origin. The differential of \(\Phi\) there sends the coordinate vectors to \(f_1(p_T),f_2(p_T)\). This proves Equation (153), with the sign accounting for orientation and curvature convention. The uniform derivative bound was proved separately in the original frame. ◻

Proof. Let \(L_j(x)=\prod_{0\leq r\leq17,\ r\ne j}(x-r)/(j-r)\). The coefficient of \(x^2\) in \(L_0\) is \(\sum_{1\leq r<s\leq17}(rs)^{-1}=c_0\). For \(j\geq1\), \(L_j\) has a factor \(x\), \(L_j'(0)=(-1)^{j-1}\binom{17}{j}/j\), and the logarithmic derivative at zero of its remaining product is \(-S_1+1/j\). The coefficient of \(x^2\) is therefore \(c_j\) as displayed. Lagrange interpolation gives the exact moment identities \[\sum_{j=0}^{17}c_jj^k= \begin{cases}1,&k=2,\\0,&0\leq k\leq17, k\ne2. \end{cases}\] Apply these identities to the Taylor polynomial \(\sum_{k=0}^{17}F^{(k)}(0)t^k/k!\), evaluated at \(t=jh\). Taylor’s integral remainder gives Equation (158). Bounding each remainder by \(M(jh)^{18}/18!\) gives Equation (159). ◻

Proof. Choose the sign in Proposition 35. Its adaptive seed \(w_T\) is in \(O\) for large \(T\) and reaches \(p_T\) in the full MGHD at a finite positive proper time, by Proposition 33. The loops in Lemma 37 lie entirely in \(I^+(\Sigma)\) of that development by Proposition 29. Their lengths tend to zero uniformly for \(0\leq h\leq17h_T\), so they satisfy the strict cutoff \(\mathcal L_T(h)<B^{-1}\).

Suppose the selected datum still belongs to \(\mathcal A(O,B)\). Its loop inequality in Definition 6, the uniform frame comparison, and Equation (152) then give \[\|\mathcal H_T(jh_T)-\operatorname{Id}\| \leq a_p jh_T e^{o(T)},\qquad 0\leq j\leq17.\] The fixed constants, including \(B\), have been included in \(e^{o(T)}\). Apply Lemma 38 with \(F=\mathcal H_T-\operatorname{Id}\), \(b=a_pe^{o(T)}\), and \(M=e^{o(T)}\). Equation (153) gives \[\begin{align*} \|a_p^2R^{g'}(E_1,E_2)(p_T)\| &\leq e^{o(T)}(a_p/h_T+h_T^{16}) \tag{160}\\ &\leq e^{o(T)} (e^{-15\kappa T/16}+e^{-\kappa T})=o(A_T). \end{align*}\] Indeed \(A_T=e^{-14\kappa T/16}\), so the two ratios have exponents \(-\kappa T/16+o(T)\) and \(-\kappa T/8+o(T)\), respectively. Both are strict, independent of the fixed interpolation constants.

Because the Lorentz Gram matrix of \(E\) is the constant matrix \(\eta\), the sectional component \(a_p^2\operatorname{Riem}(g')(E_1,E_2,E_1,E_2)\) is a fixed matrix contraction of the endomorphism in Equation (160). It is therefore \(o(A_T)\), contradicting \(|C_T^\pm|>A_T\). The chosen sign fails the test. The convergence in the prescribed finite seminorm and membership in the admissible data neighborhood are those of the signed construction; hence both data can be placed in any prescribed relative neighborhood of \(d\). ◻

Proof of Theorem 3. Work first at the fixed normalized center of Section 2.1. Lemma 7 makes each passing set \(\mathcal A(O,B)\) relatively closed in \(\mathcal U\).

It has empty relative interior. Indeed, given \(d\in\mathcal A(O,B)\) and any relative neighborhood \(V\) of \(d\) in \(\mathcal U\), Lemma 4 gives a sufficiently small finite \(p_m\) ball contained in \(V\). The strict \(p_{10}\) inequality defining \(\mathcal U\) has positive slack at \(d\), so the radius can be chosen to preserve it. Proposition 39 then gives a datum in that ball outside \(\mathcal A(O,B)\).

There are countably many basic seed sets \(O\) and integers \(B\). Consequently \[\mathcal G= \bigcap_{\substack{O\text{ in the chosen basis}\\ B\in\mathbb N,\ B\ge1}} \bigl(\mathcal U\setminus\mathcal A(O,B)\bigr)\] is a dense \(G_\delta\) in the nonempty Baire space \(\mathcal U\). By Proposition 11, every datum admitting a future \(C^1\) extension in Definition 2 belongs to at least one passing set. No datum of \(\mathcal G\) therefore admits such an extension. In particular the conclusion concerns the full maximal development and every future exit in the definition, without any vacuum assumption on the ambient extension. The rescaling in Section 2.1 proves the result for arbitrary fixed \(M>0\). ◻