Proof. These are (OpenAI 2026b, companion@kind@foundation@in:low companion@kind@foundation@in:low ), with cylinder data supplied by companion@kind@foundation@in:entry companion@kind@foundation@in:entry

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

) companion@number@foundation@in:entry companion@number@foundation@in:entry

and exterior closure supplied by companion@kind@foundation@far:closure companion@kind@foundation@far:closure

  companion@equation@foundation@far:closure ( companion@number@foundation@far:closure companion@number@foundation@far:closure

) companion@number@foundation@far:closure companion@number@foundation@far:closure

there; see also (OpenAI 2026a, companion@kind@signed@bg:geometry companion@kind@signed@bg:geometry ). The low characteristic estimates give (8) and summable cell suprema. 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

includes the same-direction derivatives, which are needed in (9). After enlarging the envelopes to include these jets, replace each envelope on \([n,n+1)\) by its supremum on \([n,n+1]\). This gives a measurable step function whose sum of closed-cell suprema increases by at most a factor two. The frozen profile terms have the stated \(q\) bounds and are absorbed in the right sides. 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

provides (10) at every fixed order. The last statement follows by differentiating \(a=q\mathcal A\) and the relative shift identity in (6). None of these results assumes a bound for parallel-frame curvature on an observing geodesic. ◻

Proof. The full-lifetime classification is (OpenAI 2026b, companion@kind@foundation@cmp:classification companion@kind@foundation@cmp:classification ). Corner removal is (OpenAI 2026a, companion@kind@signed@test:corner-null ); its section-volume argument applies to any smooth positive measure on \(T\Sigma\). The geometric estimate used there is (OpenAI 2026b, companion@kind@foundation@cmp:corner-ratios companion@kind@foundation@cmp:corner-ratios ). Its hypotheses are the low interior bounds of 4 and a future unit timelike wedge track starting sufficiently far in both optical coordinates. It imposes no curvature bound or regular-extension assumption. These statements classify the first comparison runout; that event may precede the full geodesic endpoint in \(M_d\). ◻

Proof. We first recall the individual branch-state argument of (OpenAI 2026a, companion@kind@signed@bg:branch companion@kind@signed@bg:branch ). Let \(\Xi=\dot\theta-b\dot u\) and \(P_A=\gamma_{AB}\Xi^B\). Unit normalization gives \[ 2a\dot u\dot v=1+|\Xi|_\gamma^2, \qquad p=-g(\dot c,L_0)>0. \tag{15}\] The low connection identities imply \(|\dot{\log p}|\le C(\dot u+|\Xi|_\gamma)\): the shape term is controlled by \(|\Xi|_\gamma^2/p\le2\dot u\), and lapse and torsion coefficients are bounded. On a branch, \(\int\mathrm du<\infty\) and \(\int a\,\mathrm dv<\infty\). By (15), \[\int|\Xi|_\gamma\,\mathrm d\tau \le C\left(\int\mathrm du\int a\,\mathrm dv\right)^{1/2}<\infty.\] Hence \(p,p^{-1}\) are bounded, and the angular track has finite length because \(b\) is bounded and \(\dot\theta=\Xi+b\dot u\). It has a limiting point on the sphere and eventually stays in a fixed chart.

Solving the unit mass shell for \(\pi_v\) gives the reduced \(v\)-time Hamiltonian \[ \mathsf H_v=-\pi_v =\frac{a(1+\gamma^{AB}P_AP_B)}{2p}, \qquad \frac{\mathrm du}{\mathrm dv} =\frac{a(1+\gamma^{AB}P_AP_B)}{2p^2}. \tag{16}\] Its angular Hamilton equation and the low angular coefficient bounds give \[\left|\frac{\mathrm dP}{\mathrm dv}\right| \le C(p+|P|)\frac{\mathrm du}{\mathrm dv}.\] Gronwall in the increasing bounded coordinate \(u\) bounds \(P\). It follows that \(\Xi\) and \(\dot u\) are bounded. In particular \(\pi_u=-p-b^AP_A\) is bounded.

Choose a compact state neighborhood of a sufficiently late part of this track, with \(p\) bounded away from zero and with strict \(u\) and angular margins. Its radius can be fixed independently of \(v\): the low first derivatives of \(b\) are uniformly bounded, so (13) has a uniform state-Lipschitz bound on a bounded momentum set. On this neighborhood, every fixed state derivative of (16) is bounded by \(Ce^{-\kappa v}e^{o(v)}\). This follows from 4, since each derivative of \(a\) retains its factor and the state is in a compact set. The Hamiltonian field and its Lipschitz bound are therefore integrable in \(v\). Choose the initial late section so the remaining integrated field bound is smaller than the state margins. Gronwall then keeps all sufficiently close states in that neighborhood for every later \(v\). Their \(u\) coordinates converge, \(v\) is unbounded, and they have branch runout. Compact earlier smooth flow, followed by one further shrinking, transports this neighborhood to \(\mathcal N\subset T\Sigma\) with the stated closure margins. This proves (i).

For (ii), let \(\Phi_v(u,\theta)\) be the angular foot at \(u=-v\) of the \(L_0\) characteristic through \((u,v,\theta)\). The angular variational equation gives \[ \norm{D_\theta\Phi_v}+ \norm{(D_\theta\Phi_v)^{-1}} \le C\exp\left(C\int_{-v}^{u}(q+g_0(s))\,\mathrm ds\right) \le C. \tag{17}\] Here \(\int_{-v}^{u}q\,\mathrm ds\le q_0/\kappa\) and (7) controls the other integral. Differentiation in the terminal \(u\) contributes a bounded factor \(b\). Differentiation in \(v\) contributes the moving endpoint term \(b(-v,v,\cdot)\) and an integral of \(\partial_vb\) multiplied by the bounded angular variational maps. Since \(\partial_vb=O(a)\) and \(\int_{-v}^{u}a\,\mathrm ds\le C\), these terms are uniformly bounded as well. Thus the foot map is uniformly Lipschitz in \((u,\theta,v)\) on the tubes in question.

Shrink \(\mathcal N\) and the width of \(J\) so that the feet stay in an arbitrarily small fixed neighborhood of the central foot at \(v=T\). The central feet lie on the compact sphere. An unbounded subsequence therefore lies in one chart and one small patch with interior margins. The fixed-patch ray and collar estimates in (OpenAI 2026a, companion@kind@signed@bg:linear-packet-section companion@kind@signed@bg:linear-packet-section ), using (OpenAI 2026b, companion@kind@foundation@pk2:escape companion@kind@foundation@pk2:escape ), apply uniformly to all centers in a sufficiently small low-order patch. Their patch size is independent of later finite beam orders; higher accuracy changes those orders and constants. The integrable state-Lipschitz bound from (i) makes the hitting positions uniformly close to the central positions when \(\mathcal N\) is small; the Hamiltonian field also controls their variation across the fixed interval \(J\). Choose fixed product-box radii larger than these variations but small relative to the launch margin. The uniform foot-map Lipschitz bound then keeps the feet of a fixed enlargement inside that margin. These choices are made while shrinking \(\mathcal N,J\), before any later derivative order. This supplies \(\mathcal Q_T\) and proves (ii).

Finally choose \(c_0\) small relative to \(u_2-u_*\) and \(c_2\), then shrink the state neighborhood, \(J\), and box radii so that every point of the enlarged observation boxes has strict margins from all three upper faces in (14). Future causal vectors in (5) have \(\mathrm du,\mathrm dv\ge0\), and \(t\) is temporal. Hence each earlier point of its track from \(\mathcal S\) satisfies the same strict side inequalities: both \(u+c_0t/T\) and \(v+c_0t/T\) decrease toward the past. The slice interval has length \[T+c_2+u_2-2t-2c_0t/T.\] At \(t=L_T\) this tends to \(c_2+u_2-u_*-3c_0>0\) after the choice of \(c_0\); earlier intervals are longer. This proves (iii). This finite-domain fitting condition concerns only the track up to observation; its later portion need not remain in (14). ◻

Proof. Let \(A=\{r:H(r,[0,1])\subset M\}\). Compactness of the parameter interval and openness of \(M\) imply that \(A\) is relatively open. For \(r\in A\), the path is an intrinsic causal path in \(M\), so its image lies in the single compact intrinsic diamond \(K=J_M^+(p)\cap J_M^-(q)\). The image of \(K\) in the Hausdorff manifold \(\widetilde M\) is compact and hence closed. If \(r_n\in A\) and \(r_n\to r\), continuity gives \(H(r,t)\in K\) for every \(t\). Hence \(A\) is also closed, and it is nonempty. This proves the assertion. Global hyperbolicity has been used only inside \(M\). ◻

Proof. First the given exiting curve has a late segment in \(I_M^+(\Sigma)\). Extend it to the past by a past-inextendible timelike curve in \(M\). Its future end has no endpoint in \(M\), since its ambient limit lies on \(\partial M\). The resulting inextendible timelike curve meets the Cauchy surface, and its remaining future segment lies in \(I_M^+(\Sigma)\).

We can replace a late part by a straight timelike first-exit segment. To see this without any assertion about the ambient causal structure, choose a small chart about the given boundary point \(b\) and a point \(q\) slightly to its future on a fixed, uniformly timelike coordinate line. By continuity of \(\widetilde g\), for a sequence of late interior points \(x_n\to b\) all straight segments \([x_n,q]\) in this chart are uniformly future timelike. If one has a first exit, use it. Otherwise every such segment, including its endpoint \(q\), lies in \(M\). For a fixed earlier point \(p\) on the exiting curve, all sufficiently late \(x_n\) then lie in \(J_M^+(p)\cap J_M^-(q)\). Compactness of this intrinsic diamond would put their ambient limit \(b\) in \(M\), a contradiction. The starting point of the chosen first-exit segment is in \(I_M^+(\Sigma)\).

Translate its first exit to \((0,0)\) and make its direction vertical by a linear coordinate change. Choose the other coordinate vectors in the spacelike orthogonal hyperplane of that direction at the exit. After shrinking the chart, \(\partial_s\) is timelike, \(\mathrm ds\) is temporal, and a fixed cone \(|\eta|<c\) has the asserted strict timelike margin. The closure of the chart is taken inside a larger smooth coordinate chart; continuity and nondegeneracy give all the claimed uniform algebraic metric bounds.

Choose \(d_0>0\) so that the central segment \([-d_0,0)\times\{0\}\) is the final part of the selected straight segment. Its bottom has a neighborhood in \(I_M^+(\Sigma)\). Take \(0<\delta<d_0/4\). For \(z\) in a sufficiently small ball, define the first-exit height by \[f(z)=\inf\{s\in[-d_0,2\delta]:(s,z)\notin M\}, \qquad \inf\varnothing=2\delta.\] Compact openness along \([-d_0,-\delta]\times\{0\}\) permits shrinking the ball so that \(f(z)>-\delta\). Write its radius as \(R\) and shrink it further so that \(R/\delta<c/4\). If \(f(z)>\delta\), both \(p_z=(-d_0,z)\) and \(q_z=(\delta,z)\), and their entire vertical joining segment, are in \(M\). The broken path \[p_z\longrightarrow (0,0)\longrightarrow q_z\] has spatial slopes less than \(c/2\). Interpolate its spatial graph linearly with that of the vertical segment, using \(s\) as parameter. This is a fixed-endpoint homotopy of future timelike paths in the chart. 7 forbids the member through \((0,0)\notin M\). Thus \(f(z)\le\delta\); in particular the clipping at \(2\delta\) is inactive. Its definition now implies \((f(z),z)\in\partial M\), even if a vertical line reenters \(M\) at a later height.

Finally shrink \(R\) so that \(2R/(d_0-\delta)<c/2\), and fix \(L>4/c\). If \(f(z')>f(z)+L|z-z'|\), select a height \(b'\) with \[f(z)+2|z-z'|/c<b'<f(z').\] The path from \((-d_0,z')\) through \((f(z),z)\) to \((b',z')\) has first-leg slope less than \(2R/(d_0-\delta)<c/2\) and second-leg slope less than \(c/2\). Its endpoints and the reference vertical segment lie strictly below the first exit on the higher vertical. Linear interpolation of spatial graphs again contradicts 7. Interchanging \(z,z'\) proves \(|f(z)-f(z')|\le L|z-z'|\). Piecewise smooth timelike paths suffice in these homotopies; the strict slope inequalities also permit smoothing the corners. The common bottom height is what keeps the first leg uniformly inside the cone. ◻

Proof. On the compact chart region, \(L^2\) integrability implies \(L^1\) integrability. All the boxes with \(z\in D'\) and small \(l\) lie in a fixed compact region, and their union is contained in a layer \[\mathcal L_l=\{(s,y):y\in D'',\quad |s-f(y)|\le K(1+L)l\}, \qquad D'\Subset D''\Subset D.\] For each \((s,y)\), the set of possible centers \(z\) has volume at most \(C_Kl^3\). Tonelli’s theorem therefore gives \[\int_{D'} l^{-3}\int_{Q_l(z)}|\widetilde\Gamma| \,\mathrm ds\,\mathrm dy\,\mathrm dz \le C_K\int_{\mathcal L_l}|\widetilde\Gamma|.\] The graph of \(f\) has four-dimensional Lebesgue measure zero, by Fubini. Absolute continuity of the \(L^1\) integral makes the right side tend to zero. Choosing a center with no more than the average normalized mass proves the assertion along any sequence of sufficiently small scales. ◻

Proof. Use the graph and chart of 8. Constants in this proof depend on this fixed chart, but not on the small scale \(l\). Choose small numbers \(\rho_0,\varepsilon_0>0\) so that \[ L\rho_0<\tfrac14, \qquad 2\varepsilon_0+C_\beta(\rho_0+2\varepsilon_0)<c, \tag{20}\] where \(C_\beta\) bounds the derivative of a fixed smooth bump on \((-1,1)\) that equals one at zero. We may also require \(\rho_0+4\varepsilon_0<1\). Fix \(K>3\) in 9. For one of its boxes, write \(z_c\) for the center and set \[s_c=f(z_c),\qquad s_0=s_c-l,\qquad s_1=s_c+l.\] Every point \((s_0,z)\) with \(|z-z_c|<\rho_0l\) is strictly below \(f(z)\) and above the bottom, for small \(l\). It is in \(I_M^+(\Sigma)\), since its vertical segment from the bottom lies in \(M\) and is future timelike.

The initial phase volume. At such a point parameterize initial unit velocities by slopes \(\eta\in\mathbb R^3\), \(|\eta|<\varepsilon_0\): \[U=N(1,\eta),\qquad N=\bigl[-\widetilde g((1,\eta),(1,\eta))\bigr]^{-1/2}, \qquad \pi_i=N(\widetilde g_{i0}+\widetilde g_{ij}\eta^j).\] On \(|\eta|\le2\varepsilon_0\), \(N\) has positive uniform lower and upper bounds. Differentiation in the slope variables, which uses no derivative of the ambient metric, gives \[ \frac{\partial\pi_i}{\partial\eta^j} =N\widetilde g_{ij} +N^3(\widetilde g_{i0}+\widetilde g_{ik}\eta^k) (\widetilde g_{j0}+\widetilde g_{jk}\eta^k). \tag{21}\] The spatial block is uniformly positive definite. Thus this Jacobian is uniformly positive definite and has determinant bounded above and away from zero. Its positivity also makes \(\eta\mapsto\pi\) injective on the convex slope ball: integrate the quadratic form of its derivative along the segment between two slopes. The resulting open set \(A_l\) of canonical initial states \((z,\pi)\) on \(s=s_0\) therefore satisfies \[ c_1l^3\le |A_l|\le C_1l^3. \tag{22}\] Every initial and pre-stop momentum lies in a fixed bounded set, since both \(\widetilde g\) and \(U\) are bounded on the closed wider slope cone.

The stopped smooth flow and its exact Jacobian. From each state \(x\in A_l\) follow the maximal geodesic in the full \(M\), using \(s\) as parameter as long as it is in this chart and \(|\eta|<2\varepsilon_0\). Stop also at \(s=s_1\) or at a box face. Before these stops, \[|z(s)-z_c|\le(\rho_0+4\varepsilon_0)l<l,\] so no spatial box face can be reached; the time interval is also strictly inside the enlarged box. For \(s<s_1\) let \(D_s\subset A_l\) consist of the initial states whose arcs exist through \(s\) with all the pre-stop inequalities strict. Smooth dependence in the interior of \(M\) implies that \(D_s\) and the joint pre-stop domain in \((s_0,s_1)\times A_l\) are open.

The hitting map \(\Phi_s:D_s\to T^*\{s=\mathrm{constant}\}\), in coordinates \((z,\pi)\), is smooth and injective. Indeed the geodesic equations are smooth on \(M\), \(\dot s>0\), and backward uniqueness recovers the same initial state from a hit before any stop. Its canonical Jacobian is exactly one. For completeness, put \(\mathcal H=\tfrac12g^{\alpha\beta}\pi_\alpha\pi_\beta\). On the future unit shell \(\mathcal H=-\tfrac12\), \(\partial\mathcal H/\partial\pi_0=\dot s>0\); hence \(\pi_0=-h(s,z,\pi)\) is a smooth root. Implicit differentiation of the shell equation gives the reduced Hamilton equations \[\frac{\mathrm dz^i}{\mathrm ds}=\frac{\partial h}{\partial\pi_i}, \qquad \frac{\mathrm d\pi_i}{\mathrm ds}=-\frac{\partial h}{\partial z^i}.\] Their divergence in \((z,\pi)\) is zero, so \[ \Phi_s^*(\mathrm d^3z\,\mathrm d^3\pi) =\mathrm d^3z\,\mathrm d^3\pi. \tag{23}\] Equivalently, the restriction of the canonical symplectic form to the unit shell has the geodesic-flow direction as kernel; the additional derivatives of a variable hitting time disappear from its pullback. This proves the same volume statement for every transverse section used below.

Let \[I(x)=\int_{\text{before stop}} |\Gamma_M(\gamma_x(s))|\,\mathrm ds.\] The joint open domain and the smooth interior flow make this a measurable nonnegative function, possibly infinite. Using [ex:section-volume], Tonelli’s theorem, and the fixed bound on the momentum fibers gives \[\begin{align*} \int_{A_l} I(x)\,\mathrm dx &=\int_{s_0}^{s_1}\int_{\Phi_s(D_s)} |\Gamma_M(s,z)|\,\mathrm d^3z\,\mathrm d^3\pi\,\mathrm ds \\ &\le C\int_{Q_l(z_c)}|\widetilde\Gamma|\,\mathrm ds\,\mathrm dz =o(l^3). \tag{24}\end{align*}\] The equality of the weak and smooth Christoffel arrays is only an almost-everywhere equality on \(M\). This suffices: the same phase change of variables and Fubini’s theorem show that its exceptional chart-null set has zero time measure on almost every initial arc. Thus [ex:flow-average] does not prescribe a trace of a Sobolev function on a curve.

Small integrals prevent the slope stop. Put \(V=(1,\eta)\). The smooth geodesic equation in \(s\) time is \[ \frac{\mathrm d\eta^i}{\mathrm ds} =-\Gamma^i_{\alpha\beta}V^\alpha V^\beta +\eta^i\Gamma^0_{\alpha\beta}V^\alpha V^\beta. \tag{25}\] Consequently \(|\mathrm d\eta/\mathrm ds|\le C_2|\Gamma_M|\) before the stop. By [ex:initial-volume,ex:flow-average], for small enough \(l\) a set \(G_l\subset A_l\) of measure at least \(|A_l|/2\) satisfies \(I(x)<\varepsilon_0/(2C_2)\). On these arcs the slope remains less than \(3\varepsilon_0/2\). It cannot reach the wider threshold \(2\varepsilon_0\). The position estimate already excludes a chart or box-face stop.

A surviving arc would give a forbidden bump. Suppose such an arc exists inside \(M\) through \(s_1\). Choose \(\beta(s)\) supported in \((s_0,s_1)\), with \(\beta(s_c)=1\) and \(|\beta'|\le C_\beta/l\), and put \[\Delta z=z_c-z(s_c),\qquad z_r(s)=z(s)+r\beta(s)\Delta z,\qquad 0\le r\le1.\] Here \(|\Delta z|\le(\rho_0+2\varepsilon_0)l\). By [ex:cone-choice], every graph \((s,z_r(s))\) has spatial slope less than \(c\). The graphs stay in the ambient rectangle and have the same two endpoints in \(M\). Their first member is the assumed interior geodesic arc, whereas the last passes through \((s_c,z_c)\in\partial M\). This contradicts 7.

It follows that every arc in \(G_l\) has a maximal pre-stop height \(s_*\le s_1\), with no slope or chart stop. Unit normalization gives \[ 0<c_3\le\frac{\mathrm ds}{\mathrm d\tau}=N\le C_3. \tag{26}\] The elapsed proper time to \(s_*\) is therefore finite. The bounded spatial slope gives an ambient position limit in a fixed compact subrectangle of the chart. If this limit were in \(M\), smoothness of the metric there, the bounded unit velocity, and the geodesic equation would give a limiting velocity and smooth continuation. If \(s_*<s_1\), that contradicts maximality of the pre-stop interval. If \(s_*=s_1\), it gives the just-excluded surviving arc. Thus the limit lies in \(\partial M\). Any continuation in the full \(M\) would, under the continuous embedding, take this boundary value at the joining proper time, which is impossible. The finite lifetime just proved is therefore the full future lifetime. Nothing here solves an ambient geodesic equation at the boundary.

Returning to \(\Sigma\) and controlling the earlier arcs. We spell out why all the starting states have a past intersection with \(\Sigma\). A maximal timelike geodesic in a smooth spacetime has no interior causal endpoint. Indeed, in a relatively compact smooth temporal chart about a hypothetical endpoint, all causal slopes \(\eta=\mathrm dz/\mathrm ds\) are bounded. The time component of the geodesic equation gives \[\frac{\mathrm d}{\mathrm ds}\log\frac{\mathrm ds}{\mathrm d\tau} =-\Gamma^0_{\alpha\beta}(1,\eta)^\alpha(1,\eta)^\beta.\] Its right side is bounded and \(s\) has a finite endpoint. Thus \(\mathrm ds/\mathrm d\tau\) has positive upper and lower bounds. The affine duration to the endpoint is finite, the velocity is bounded, and the smooth geodesic equation gives a velocity limit and continuation. The same argument with reversed orientation handles a past endpoint. Hence maximal timelike geodesics are causally inextendible; the Cauchy property forces them to meet \(\Sigma\) exactly once. The starts here are in \(I_M^+(\Sigma)\), so the intersection is in their past.

For each initial state in \(A_l\), this past hit occurs in finite proper time and is transverse. Smooth dependence and the implicit function theorem give an open neighborhood on which the hit and its time are smooth. Shrink this to a precompact neighborhood on which the transverse-section hit map is a diffeomorphism onto its image and the complete earlier flow tube has compact closure in the interior unit shell. Local injectivity is sufficient: the original local time slice need not have a globally single-valued intersection with every geodesic. Second countability supplies countably many such smaller neighborhoods covering \(A_l\). One of them meets \(G_l\) in positive measure. Restrict to that intersection.

The backward hit preserves canonical section volume by the unit-shell argument following [ex:section-volume]. Identifying the future unit shell over \(\Sigma\) with \(T\Sigma\) by the velocity \(\sqrt{1+h(w,w)}\,n+w\), its tangential covector is exactly \(h(w,\cdot)\). The image \(S\) is measurable and has positive section measure. The compact earlier flow tube is covered by finitely many ambient charts with closures in \(M\). Taking smaller coordinate neighborhoods first gives a common positive coordinate margin. Compactness in the unit shell then bounds the velocity components in the selected charts, and bounds the earlier proper durations. Each chart can be taken precompact in a larger chart, so its metric, inverse, and Christoffel \(L^2\) norm are finite. Add the single exit chart, where [ex:proper-time] and the narrow cone supply the velocity bound and the compact subrectangle supplies the margin. Taking a maximum of the bounds and a minimum of the positive margins proves [ex:chart-witness] from proper time zero onward. ◻

Proof. Fix such a datum and extension. 10 provides a measurable family \(S_*\) of positive \(\mu_d\) measure, finite full lifetimes, and one finite set of ambient chart witnesses. By 5, every seed in \(S_*\) crosses the comparison cylinder and its first comparison runout is a branch or a corner. The corner subset is null. The inputs here are precisely the finite-full-lifetime classification of (OpenAI 2026b, companion@kind@foundation@cmp:classification companion@kind@foundation@cmp:classification ) and the section-volume corner removal of (OpenAI 2026a, companion@kind@signed@test:corner-null ); the latter uses only the low-order corner estimates of (OpenAI 2026b, companion@kind@foundation@cmp:corner-ratios companion@kind@foundation@cmp:corner-ratios ). These statements have neither a regular-extension hypothesis nor a curvature bound along the selected seed.

Choose a density point \(w\) of \(S_*\) outside that null subset, with \(w\in S_*\). It is a branch seed. Apply 6 and, if necessary, interchange the optical labels. This gives an open seed neighborhood \(\mathcal N\) of \(w\) with all the tube, footprint, and finite-domain properties in 11(ii). These are consequences of the background estimates and hold independently of the extension chart witnesses.

By density and the fine countable basis, choose \(O\ni w\) with \(\overline O\subset\mathcal N\) and \(\mu_d(S_*\cap O)>.99\mu_d(O)\). To justify using countably many balls, small balls centered at a density point can be enlarged by an arbitrarily small relative amount to balls with rational center and radius. Smooth positivity of the density keeps their measures comparable, so their relative complement measure still tends to zero. Set \(S=S_*\cap O\). Choose an integer \(B\) at least the number of the ambient charts, all their bounds, and the reciprocal of their common margin. This proves \(d\in E(O,B)\).

Every observation invoked here lies strictly before first runout from the comparison region. The finite full lifetime and ambient chart witnesses were obtained independently for the entire geodesic in \(M_d\). Thus first runout is used only to locate earlier observations; it need not be the ambient exit. Finally the reverse inclusion in the displayed union is part of 11(i). ◻

Proof. The inverse metric is \[g^{-1}=-a^{-1}(L_0\otimes L_1+L_1\otimes L_0) +\gamma^{AB}\partial_A\otimes\partial_B.\] Linearization of \(g^{-1}(\mathrm du,\mathrm du)=0\) gives \(-a^{-2}h^p_{11}-2a^{-1}\partial_vz^0=0\); the equation for \(v\) is analogous. For the angle condition \(g^{-1}(\mathrm du,\mathrm d\theta^A)=0\), the metric variation contributes \(a^{-1}\gamma^{AB}h^p_{1B}-a^{-2}b^Ah^p_{11}\). The coordinate variations contribute \(\gamma^{AB}\partial_Bz^0-a^{-1}b^A\partial_vz^0 -a^{-1}\partial_vz^A\). Substitution of the first eikonal equation gives the last equation in (41), including its sign and factor.

To see the gain in (42), a typical term in the first equation has the form \(A_Te^{i\lambda v}f(u,v,\theta)\) with fixed coefficient jets \(e^{o(T)}\). At fixed \(u,\theta\), \[ \begin{split} \int_{-u}^{v}e^{i\lambda s}f(u,s,\theta)\,\mathrm ds ={}&\frac{e^{i\lambda v}f(u,v,\theta) -e^{-i\lambda u}f(u,-u,\theta)}{i\lambda}\\ &-\frac1{i\lambda} \int_{-u}^{v}e^{i\lambda s}\partial_sf(u,s,\theta)\,\mathrm ds. \end{split} \tag{44}\] The interval has length \(O(T)\); its polynomial length is absorbed in \(e^{o(T)}\). Differentiation in \(u\) differentiates the lower endpoint and its coefficient as well as the integral. Each longitudinal derivative costs at most one additional \(\lambda\), including the phase \(e^{-i\lambda u}\) at that endpoint. Angular derivatives do not differentiate the phase. This proves the assertion for \(z^0\) with arbitrary fixed differentiated orders. The relative coefficient \(h^p_{11}/a\) is a normalized tensor entry, so no inverse lapse was lost.

For \(z^1\), the leading \(00\) coefficient is identically zero: \(p_0\) annihilates \(L_0\), and trace reversal leaves the same-null \(00\) component unchanged because \(g_{00}=0\). Thus \(h^p_{00}/a\) already has a factor \(\lambda^{-1}\). Integration along \(L_0\) keeps \(v\) fixed, with angular characteristic jets and inverse jets \(e^{o(T)}\). Differentiating its moving initial point \(u=-v\) has the same endpoint rule as above. Its interval length is again \(O(T)\). This proves the bound for \(z^1\) without requiring oscillation in the direction of that integration.

For the angular equation, \(h^p_{1A}\) is \(\sqrt a\) times a normalized oscillatory coefficient, so (44) applies. The term \(a\gamma^{-1}\mathrm d_\theta z^0\) already has the inverse-frequency gain. Integrating its length costs only \(e^{o(T)}\), also at differentiated orders. Finally, \[\int_{-u}^{v}b\partial_vz^0\,\mathrm dv =bz^0\big|_{-u}^{v}-\int_{-u}^{v}(\partial_vb)z^0\,\mathrm dv,\] where the lower value of \(z^0\) is zero. All derivatives of \(\partial_vb\) retain their factor \(a\), and the remaining coefficients have subexponential fixed jets. This proves (42) for the angles as well.

For explicit field formulas set \(W=z^\theta-z^0b\), an angular vector field. A direct Lie derivative gives \[ \begin{aligned} k_\gamma&=h^p_{AB}-2z^0\chi_0-2z^1\chi_1-\mathcal L_W\gamma,\\ k_H&=-a^{-1}h^p_{01}-Z\log a-L_0z^0-\partial_vz^1,\\ k_b&=-\gamma^{-1}h^p_{0\cdot} +\partial_uW+[b,W]-z^1\partial_vb -a\gamma^{-1}\mathrm d_\theta z^1. \end{aligned} \tag{45}\] Indeed \((\mathcal L_Zg)_{AB}=2z^0\chi_0+2z^1\chi_1+\mathcal L_W\gamma\), whereas \((\mathcal L_Zg)(L_0,L_1)=-Za-a(L_0z^0+\partial_vz^1)\). The \(0A\) component is \(\gamma_{AB}(\partial_uW+[b,W]-z^1\partial_vb)^B -a\partial_Az^1\). The linear double-null component conditions \(k_{11}=k_{00}=k_{1A}=0\) follow from (41). Hence \(k_H=-k(L_0,L_1)/a\) and \(k_b=-\gamma^{-1}k_{0\cdot}\), which give (45). All displayed divisions by \(a\) are relative factors: \(h^p_{01}/a\) is normalized and \(\partial a/a=\partial\log a\) has the given coefficient bounds. Thus these three field variations have the claimed fixed positive frequency powers. This argument alone does not estimate \(q^{-1}\partial_vk_b\); that relative derivative is obtained next.

Let \(X_i\) denote the exact linear variation of \(\xi_i\) under \(k\). The torsion sum identity gives \(X_0+X_1=\mathrm d_\theta k_H\). The linearized vacuum identities are available with arbitrary linear accuracy: diffeomorphism covariance on the vacuum background gives \[D\operatorname{Ric}_g(\mathcal L_Zg)=\mathcal L_Z\operatorname{Ric}(g)=0.\] The algebraic and differential steps turning Ricci components into the null equations can lose fixed powers of \(q^{-1}\), but only on the linear residual. On (28) these powers are \(e^{C_sT}\) for a fixed \(C_s\). After \(\zeta\) is fixed, increasing \(M\) in \(A_T\lambda^{-M}\) makes this residual smaller than any prescribed exponential at all finitely many required orders.

Use specifically the shift-free direction-\(1\) torsion equation of (OpenAI 2026b, companion@kind@foundation@in:vacuum companion@kind@foundation@in:vacuum ). Its linearization is \[ (\partial_v+\tau_1)X_0 =\delta\beta_1+\delta\tau_1\,\xi_1 +\tau_1\mathrm d_\theta k_H+\mathfrak r_{ \rm lin}. \tag{46}\] Here \[\delta\chi_1=\tfrac12\partial_vk_\gamma,\qquad \delta\tau_1=\operatorname{tr}_\gamma\delta\chi_1 -\gamma^{AC}k_{\gamma,CD}\gamma^{DB}\chi_{1,AB},\] and \[\delta\beta_1=\operatorname{div}_\gamma\delta\chi_1 -\mathrm d_\theta\delta\tau_1+(\delta\operatorname{div}_\gamma)\chi_1.\] The operator variation in the last term uses only \(k_\gamma\) and one angular derivative of \(k_\gamma\). Thus the entire source in (46) has already been estimated using (45); it uses neither \(X_i\) nor an unknown relative shift derivative. Also \[\int_{-u}^{v}|\tau_1(u,s,\theta)|\,\mathrm ds \le C\int_{-u}^{v}(q+g_1(s))\,\mathrm ds\le C.\] At the moving entry edge \(q=q_0\), the initial \(X_0\) is a finite-jet expression with no small denominator. It has size \(A_T\lambda^{P_s}e^{o(T)}\). The integrating-factor estimate therefore gives this bound for \(X_0\), and then for \(X_1\). Commutation with \(u\) and angular derivatives leaves the same top transport coefficient; differentiated coefficients multiply lower already estimated \(X_0\) jets. Entry terms from differentiating \(v=-u\) are controlled by the differentiated entry data, with normal jets also given by the equation. Derivatives in \(v\) are recovered from (46). This proves the finite-order assertion without a shifted transport containing an unknown \(k_b\).

Finally linearize the defining shift relation. It gives exactly \[ \begin{split} q^{-1}\partial_vk_b={}&\mathcal A\gamma^{-1}(X_1-X_0)\\ &+\bigl(k_H\mathrm{Id}-\gamma^{-1}k_\gamma\bigr) q^{-1}\partial_vb. \end{split} \tag{47}\] The background relative derivative and all its fixed jets are subexponential. This proves (43). In particular the proof never divides an ordinary error estimate for \(\partial_vk_b\) by a small \(q\). ◻

Proof. We give the finite-subslab argument, including the dependence needed here. We reproduce the finite-slab argument of (OpenAI 2026b, companion@kind@foundation@in:higher companion@kind@foundation@in:higher ), with uniformly controlled envelopes. Merely applying the fixed-datum conclusion of that Proposition would not establish the present assertion.

Choose smooth cutoffs \(0\le\psi_{i,T}\le1\) equal to one on the coordinate projection of \(\Omega_T\), supported in \([-C(T+1),C(T+1)]\). Extend only the scalar envelope, not the spacetime fields, and set \[ G_{i,T}(y)=g_i(y)+e_T\psi_{i,T}(y), \qquad h_i=C_0(q+G_{i,T}). \tag{61}\] The added \(L^1\) mass and sum of unit-cell suprema are \(O((T+1)e_T)\), uniformly bounded and tending to zero. Their tails are uniformly tight: if the added support reaches \(|y|>R\), then \(T+1\ge R/C\), and hence \[\sup_T\int_{|y|>R}e_T\psi_{i,T}(y)\,\mathrm dy \le C\sup_{T+1\ge R/C}(T+1)e^{-\alpha T/2}\longrightarrow0.\] The same argument controls the tail supremum. Together with the fixed summable \(g_i\), this supplies all tail thresholds uniformly. The Lipschitz potential \[ \Phi_T(u,v)=C_0\sum_{i=0}^1\int_{-\infty}^{y_i}G_{i,T}(r)\,\mathrm dr +\frac{C_0q_0}{\kappa}(1-e^{-\kappa(u+v)}) \tag{62}\] satisfies \(0\le\Phi_T\le C\) and \(L_i'\Phi_T=\partial_i\Phi_T=h_i\) almost everywhere. In particular both directional integrals of \(h_i\) are uniformly bounded on every past transport segment in the domain.

The weighted angular curvature pair will supply stable fluxes, while trace damping controls the shape trace. We will then recover unweighted metric and torsion fields by transport, retaining the small lapse in the curvature energy throughout.

Suppress primes in this proof only. Fix a smooth reference metric on \(\mathbb S^2\) and let \(D\) be its Levi-Civita connection, independent of \(u,v,T\) and of the solution. In this proof all angular Sobolev sizes use the tensors \(D^r f\); equivalently, they use component derivatives in the fixed finite atlas of the background. The low coefficient bounds are used in this same reference convention, together with the uniform positive upper and lower metric bounds in that atlas. This is the convention for the bounded low coefficient set below. The symbol \(\nabla\) in the equations continues to mean the connection of \(\gamma\).

For definiteness, put \[|f|_k^2=\sum_{r=0}^k\int_{\mathbb S^2}|D^r f|_\gamma^2\,\mathrm d\mu_\gamma.\] The contraction includes every angular tensor slot; finite scalar arrays have their Euclidean contraction. These norms are uniformly equivalent to the fixed-reference Sobolev norms at each fixed order. In particular \(D\gamma\) need not vanish, so \(|\gamma|_k\) records the coordinate coefficient jets. At a fixed high angular order \(M\ge5\), use the actual-field sizes \[ \begin{gathered} \mathsf Y_i=|\beta_i|_M,\quad \mathsf Z_i=\sigma|(Q_i,m_i)|_M,\quad \mathsf R_i=|\chi_i|_{M+1},\quad \mathsf z_i=|w_i|_{M+1},\quad \mathsf d_i=|B_i|_{M+1},\\ \mathsf d_1=0,\quad\mathsf D=\mathsf d_0,\quad \mathsf P_\gamma=1+|\gamma|_{M+1},\quad \mathsf P_i=|\xi_j|_M,\quad \mathsf P=\mathsf P_\gamma+\mathsf P_0+\mathsf P_1. \end{gathered} \tag{63}\] Inverse metrics and relative lapse factors through order \(M+1\) are bounded by \(C_M\mathsf P\): differentiate inversion tamely and use \(\mathrm d\log\mathcal A=\xi_0+\xi_1\). The undifferentiated mean of \(\log\mathcal A\) is already bounded by the low bounds for \(\mathcal A^{\pm1}\). The angular elliptic estimates are \[ \mathsf R_i\le C_M(\mathsf Y_i+\mathsf z_i+h_i\mathsf P), \qquad |\xi_j|_{M+1}\le C_M(\sigma^{-1}\mathsf Z_i+\mathsf P). \tag{64}\] To make the constant’s dependence explicit, freeze the principal coefficient in sufficiently small fixed charts, identify trace-free tensors by smooth metric-dependent bases, and use the first-order elliptic estimate there. The differentiated coefficient error is bounded by \(C_M(|f|_M+|\gamma|_{M+1}\|f\|_{C^1})\). Interpolation absorbs the penultimate derivative of \(f\) with a constant depending on \(M\) and the low Lipschitz/ellipticity bounds. Every high metric coefficient thus remains a variable in \(\mathsf P\); it does not enter \(C_M\). In the shape estimate the low factor multiplying that variable is \(h_i\), including conversion of \(\tau_i=\mathcal A w_i\). Equivalently, the trace-free tensor \(S_i=\chi_i-\tfrac12\tau_i\gamma\) satisfies \(\operatorname{div}S_i=\beta_i+\tfrac12\mathrm d\tau_i\), and the one-form data are \(\operatorname{div}\xi_j=K_\gamma-m_i\) and \(\operatorname{curl}\xi_j=c_i\).

Commute [dn:D3] with \(D^r\), \(0\le r\le M\). Transport the resulting angular tensor \(D^r f\) by \(\mathfrak L_i(D^r f)=\partial_{y_i}D^r f+\mathcal L_{B_i}D^r f\) on all of its slots. In particular, for a scalar \(f\) the commutator is \(D^rL_i f-\mathfrak L_i(D^r f)\), because its \(r\) derivative slots are now covariant tensor slots. In the commuted principal operators, \(\nabla\) acts on every slot. If \(I\) denotes the \(r\) derivative slots, the new gradient slot and the original one-form slot are paired by \[(\nabla F)_{IA}=\nabla_A F_I,\qquad (\operatorname{div}\mathcal B)_I=\gamma^{AB}\nabla_A\mathcal B_{IB},\qquad (\operatorname{curl}\mathcal B)_I=\epsilon_\gamma^{AB}\nabla_A\mathcal B_{IB},\] where \(\epsilon_\gamma\) is the oriented area bivector. Use the same contraction of the \(I\) slots in both members of the pair. Their densities are \[ \begin{aligned} e_{\mathsf Y_i} &=\frac12\sum_{r=0}^M|D^r\beta_i|_\gamma^2 &&\text{in direction }j,\\ e_{\mathsf Z_i} &=\frac a4\sum_{r=0}^M \bigl(|D^rQ_i|_\gamma^2+|D^rm_i|_\gamma^2\bigr) &&\text{in direction }i . \end{aligned} \tag{65}\] The scalar to one-form weight ratio is exactly \(a/2\), and the \(m_i\) current has no angular principal derivative. At each order, integration by parts on the full sphere cancels the two principal terms except for \[-\frac12\int_{\mathbb S^2} \big\langle D^r\beta_i,\, (\mathrm da)D^rK_\gamma+J(\mathrm da)D^rc_i \big\rangle_\gamma\,\mathrm d\mu_\gamma .\] This uses \(\nabla J=0\) and the same derivative-slot contraction on both sides. Since \(\mathrm da=a(\xi_0+\xi_1)\), the sum of these errors is bounded by \(C_M\sigma\mathsf Y_i\mathsf Z_i\). The sphere integrals of the densities in [dn:high-pair-densities] are uniformly equivalent to \(\mathsf Y_i^2,\mathsf Z_i^2\), because \(a=q\mathcal A\) and \(\mathcal A^{\pm1}\) are bounded.

Here are representative commutator bounds that specify the high coefficient dependence. The difference between the two angular connections is \[(\nabla-D)^C{}_{AB} =\tfrac12\gamma^{CD} (D_A\gamma_{BD}+D_B\gamma_{AD}-D_D\gamma_{AB}), \qquad |\,\nabla-D\,|_M\le C_M\mathsf P_\gamma.\] Tame products, the low bounds for \(\mathrm dQ_i\), and \(|\beta_i|_{C^1}\le C h_i\) give, for \(r\le M\), \[ \begin{aligned} \big|D^r\{a(\mathrm dK_\gamma+J\mathrm dc_i)\} -a\{\nabla D^rK_\gamma+J\nabla D^rc_i\}\big|_0 &\le C_M(\sigma\mathsf Z_i+q\mathsf P),\\ \big|D^r\{(\operatorname{div},\operatorname{curl})\beta_i\} -(\operatorname{div},\operatorname{curl})D^r\beta_i\big|_0 &\le C_M(\mathsf Y_i+h_i\mathsf P),\\ \big|[D^r,\mathfrak L_j]\beta_i\big|_0 &\le C_M(h_j\mathsf Y_i+h_i\mathsf d_j). \end{aligned} \tag{66}\] In the first line, every high derivative of \(a\) or \(J\) multiplies a low derivative of \(Q_i\) and costs \(q\mathsf P\); the other terms cost \(q|Q_i|_M\le\sigma\mathsf Z_i\). In the second line, a high derivative of \(\gamma^{-1}\) or \(\nabla-D\) multiplies low \(\mathrm d\beta_i\) or \(\beta_i\), giving \(h_i\mathsf P\); its scalar-current weight supplies one further \(\sigma\). The fixed reference curvature contributes only lower derivatives with these same factors.

For the last line, the identity \(\mathcal L_B f=B*Df+DB*f\) for a covariant tensor shows that the commutator has low \(DB_j\) against the top \(\beta_i\) jet, or a derivative of \(B_j\) through order \(M+1\) against a low \(\beta_i\) jet. Reference-curvature terms have the additional low factor \(B_j\). Likewise the scalar \(Q_i,m_i\) commutators cost \(C_M\{\sigma\mathsf d_i+h_i\mathsf Z_i\}\) after weighting. The order-\(M+1\) trace commutator costs \(C_Mh_i(\mathsf z_i+\mathsf d_i)\), whereas the order-\(M\) torsion commutator costs \(C_M(h_i\mathsf P_i+\mathsf d_i)\). The time-independent choice of \(D\) also gives \([\partial_v,D^r]b=0\) in the shift equation.

The remaining differentiated products have the same tame placement. A high angular connection in \(\chi_i*\nabla\chi_j\) costs \(h_ih_j\mathsf P\); in \(a\xi*\nabla\xi\) it costs \(q\mathsf P\). The estimate for a differentiated torsion in an \(a\) term is \[q\bigl(|\xi_0|_{M+1}+|\xi_1|_{M+1}\bigr) \le C_M\{\sigma(\mathsf Z_0+\mathsf Z_1)+q\mathsf P\}\] by [dn:high-hodge]. These observations and Leibniz placement give, apart from favorable damping, paired squared-current costs at most \(C_M\) times \[ \begin{aligned} \mathsf Y_i\big[&\sigma(\mathsf Z_0+\mathsf Z_1+\mathsf P) +h_j(\mathsf Y_i+\mathsf R_i) +h_i(\mathsf R_j+\mathsf d_j)+h_ih_j\mathsf P\big],\\ \mathsf Z_i\big[&\sigma(\mathsf Y_i+\mathsf R_i+ \mathsf d_i+\mathsf P) +h_i(\mathsf Z_0+\mathsf Z_1)\big]. \end{aligned} \tag{67}\] Their transport directions are \(j\) and \(i\), respectively. The remaining currents, in directions \(i\), \(1\), \(1\), and \(i\), have costs \[ \begin{aligned} &C_M\mathsf z_i h_i(\mathsf z_i+\mathsf R_i+ \mathsf d_i+h_i\mathsf P),\\ &C_M\mathsf D[\sigma(\mathsf Z_0+\mathsf Z_1+ \sigma\mathsf P)+h_1\mathsf D],\\ &C_M\mathsf P_\gamma\mathsf R_1+C_Mh_1\mathsf P_\gamma^2, \qquad C_M\mathsf P_i(\mathsf Y_i+\mathsf R_i+ \mathsf d_i+h_i\mathsf P). \end{aligned} \tag{68}\] For the first line of [dn:high-other-costs], the high relative lapse factor multiplies the two low shapes and therefore has coefficient \(h_i^2\). The shift source is \(q\mathcal A\gamma^{-1}(\xi_1-\xi_0)\), which gives the second line. The last two lines are the metric and torsion transports. Their unweighted sources explain why \(\mathsf R_1\) and \(\mathsf Y_i\) remain there. Area and norm differentiation of \(D^r\) tensors uses only \(\chi_i=O(h_i)\) on their contracted slots; the scalar weight uses \(L_i\log a=-\kappa+O(h_i)\). Thus the constant part is favorable damping and the remaining terms are among the displayed costs. These placements account for every same-order interaction.

For use on a fixed finite slab, boundedness of \(h_i\) already suffices: substitute [dn:high-hodge] and multiply all currents by \(e^{-2Kt}\), with \(K>C_M\) for the bounded low coefficient set. Both directions gain damping \(K\), giving finite flux bounds. Thus the estimates are a priori estimates on existing subslabs; their derivation does not require continuation or exhaustion.

These fixed-slab bounds ensure finiteness, but interpolation later needs more: high derivatives must cost only an arbitrarily small exponential rate in \(T\). We now obtain that rate by rescaling the one-way limiting couplings and assigning stronger damping to the bounded optical direction.

For the sharper rate fix \(0<\delta\le1\). The upper-bounded direction is \(d=0\), and the long direction is \(l=1\). When \(\sigma=h_0=h_1=0\), the only nonzero arrows in [dn:high-pair-costs,dn:high-other-costs], after substitution of [dn:high-hodge], are \[\mathsf Y_i,\mathsf z_i\longrightarrow\mathsf P_i, \qquad \mathsf D\longrightarrow\mathsf P_0, \qquad \mathsf Y_1,\mathsf z_1\longrightarrow\mathsf P_\gamma.\] There is also an asymmetric regime. Give the along-\(0\) variables \(\mathsf Y_1,\mathsf Z_0,\mathsf z_0,\mathsf P_0\) arbitrarily large damping and temporarily remove every product containing one of them. At \(\sigma=h_1=0\), with \(h_0\) bounded, the remaining arrows are exactly \[\mathsf z_1\longrightarrow\mathsf Y_0, \qquad\mathsf z_1\longrightarrow\mathsf P_1, \qquad\mathsf z_1\longrightarrow\mathsf P_\gamma.\] For example, the \(\mathsf P_1\) source contains \(\mathsf Y_1\), which has been removed, rather than \(\mathsf Y_0\); and \(\mathsf d_1=0\). The retained trace and shift equations have no unsuppressed source returning to \(\mathsf z_1\).

Give each paired current multiplier one, the trace currents multiplier \(\Lambda^2\), and the \(\mathsf P\) currents multiplier \(\epsilon_P^2\). Take \(\Lambda\) large and \(\epsilon_P\) small, depending only on \(M,\delta\) and the bounded low coefficients. Both displayed limiting quadratic forms then have norm at most \(\delta/4\) in the rescaled energy. After those choices, continuity on the bounded low coefficient set gives a fixed threshold \(\varepsilon_*\): the whole form has cost at most \(\delta/2\) when \(\sigma,h_0,h_1\le\varepsilon_*\), and the retained long-direction form has that cost when \(\sigma,h_1\le\varepsilon_*\).

Use the common Lipschitz weight \[ \mathfrak w_T= \exp\{-2\delta(u+v)-2K_0(u+B_{\mathrm{cut}})_+ -2K_0\min(t,S_0)\}. \tag{69}\] Uniform tail smallness from [dn:uniform-envelopes] permits \(B_{\mathrm{cut}},S_0\) independent of large \(T\) such that the all-small regime holds for \(t>S_0,u<-B_{\mathrm{cut}}\) and the asymmetric regime holds for \(t>S_0,-B_{\mathrm{cut}}\le u\le u_2\). Its directional damping is, almost everywhere, \[-L_k\log\mathfrak w_T =2\delta+2K_0\mathbf1_{\{k=0, u>-B_{\mathrm{cut}}\}} +K_0\mathbf1_{\{t<S_0\}}.\] If \(C_{\mathrm{bd}}\) bounds the complete rescaled form, the products containing an along-\(0\) factor are absorbed by \(K_0>C(C_{\mathrm{bd}}+C_{\mathrm{bd}}^2/\delta)\). In \(t<S_0\) both directions have this damping. The choice order is \[(\Lambda,\epsilon_P),\quad\varepsilon_*,\quad (B_{\mathrm{cut}},S_0),\quad K_0.\] In particular choosing \(K_0\) does not change a previously selected tail threshold. All constants use only the dependencies stated in the Lemma.

Integration gives positive null and slice fluxes bounded by the initial fluxes. The tilted sides of [dn:domain] have the same outgoing signs. Since \(u\le u_2\) and \(v\le C(T+1)\), \[1\ge\mathfrak w_T\ge \exp\{-C\delta T-2K_0(u_2+B_{\mathrm{cut}})_+-2K_0S_0-C\delta\}.\] Removing the weight loses an arbitrarily small exponential rate. Initial integration over intervals of length \(O(T)\) costs only a polynomial. In particular the transverse fluxes satisfy \[ \sup_{y_j}\|\mathsf Y_i+\mathsf d_i\|_{L^2(\mathrm dy_i)} +\sup_{y_i}\|\mathsf Z_i\|_{L^2(\mathrm dy_j)} \le e^{o(T)}. \tag{70}\] The intervals here are the available intersections with the initial partial slab. Every past coordinate segment stays inside it and reaches \(t=0\), because the side defining functions are nondecreasing along both \(L_0,L_1\).

The weighted angular fluxes are now controlled. Unweighted fields and ordinary longitudinal jets remain to be recovered. Recover the former by transport, rather than dividing the \(\mathsf Z\) estimate by \(\sigma\). The damped trace equation and [dn:high-hodge] give \[ \begin{split} \mathsf z_i(y)\le{}& e^{o(T)}e^{-\kappa(y_i+y_j)}\\ &+C_M'\int_{-y_j}^{y_i}e^{-\kappa(y_i-r)}h_i (\mathsf Y_i+\mathsf d_i+h_i\mathsf P)(r,y_j)\,\mathrm dr. \end{split} \tag{71}\] The homogeneous \(C_Mh_i\) factor has bounded directional integral. To see explicitly how the trace enters unweighted transport, fix \(y_j\) and integrate [dn:high-trace-convolution] in its terminal variable. Reversing the integration triangle gives \[\int_{-y_j}^{y_i}\mathsf z_i(s,y_j)\,\mathrm ds \le e^{o(T)}+\frac{C_M'}{\kappa} \int_{-y_j}^{y_i}h_i (\mathsf Y_i+\mathsf d_i+h_i\mathsf P)(r,y_j)\,\mathrm dr .\] The first transverse flux in [dn:high-transverse] bounds the \(\mathsf Y_i+\mathsf d_i\) integral by Cauchy–Schwarz, with only a polynomial interval factor. Since \(h_i\) is uniformly bounded, \(h_i^2\le C h_i\). The Hodge estimate for \(\mathsf R_i\) consequently gives \[ \int_{-y_j}^{y_i}(\mathsf z_i+\mathsf R_i)(r,y_j)\,\mathrm dr \le e^{o(T)} +C_M\int_{-y_j}^{y_i}h_i(r,y_j)\mathsf P(r,y_j)\,\mathrm dr . \tag{72}\] The norm inequalities for metric and torsion transport are \[\begin{aligned} \mathsf P_\gamma(y) &\le e^{o(T)} +C_M\int_{-u}^{v}(\mathsf R_1+h_1\mathsf P_\gamma)(u,r)\,\mathrm dr,\\ \mathsf P_i(y) &\le e^{o(T)} +C_M\int_{-y_j}^{y_i} (\mathsf Y_i+\mathsf R_i+\mathsf d_i+h_i\mathsf P)(r,y_j)\,\mathrm dr . \end{aligned}\] Substitution of [dn:high-pure-integrals] and the transverse flux therefore gives \[ \mathsf P(y)\le e^{o(T)}+ C_M''\sum_i\int_{-y_j}^{y_i}h_i(r,y_j)\mathsf P(r,y_j)\,\mathrm dr. \tag{73}\] Multiply by \(e^{-C\Phi_T(y)}\) and take a supremum. Along either path, \[\int_{-y_j}^{y_i}h_i(r,y_j) e^{-C(\Phi_T(y)-\Phi_T(r,y_j))}\,\mathrm dr\le C^{-1}.\] Choose \(C>4C_M''\) and absorb the two integrals. The uniform bound on \(\Phi_T\) proves \(\mathsf P,\mathsf z_i\le e^{o(T)}\). [dn:high-pure-integrals] now bounds the pure-direction integral of \(\mathsf R_i\) by \(e^{o(T)}\) as well.

We record the angular order obtained by the last transport step. The pointwise \(\mathsf P\) bound controls \(\gamma\) in \(H^{M+1}\) and \(\xi_i\) in \(H^M\). Tame differentiation of angular curvature and connection therefore gives \[|a\operatorname{Sym}\nabla\xi_j|_{M-1} +|aK_\gamma\gamma|_{M-1}+|a\xi_j^{\otimes2}|_{M-1} \le C_M q\mathsf P.\] Commuting the cross equations with \(D\) through order \(M-1\) gives top homogeneous coefficient \(C_Mh_j\) on the \(\chi_i,\ell_i\) norms. The other shape and shift terms are bounded by \(C_Mh_i(\mathsf R_j+\mathsf d_j)+C_Mh_ih_j\mathsf P\). Their integrals in direction \(j\) are \(e^{o(T)}\) by [dn:high-transverse,dn:high-pure-integrals]. Gronwall thus gives pointwise \(H^{M-1}\) bounds for \(\chi_i,\ell_i\). Finally \([\partial_v,D^r]b=0\) and the shift source has \(H^M\) norm at most \(C_Mq\mathsf P\); its norm derivative adds only \(C_Mh_1|b|_M\). Integration gives a pointwise \(H^M\) bound for \(b\). Thus the order-\(M\) current gives pointwise \(H^{M-1}\) bounds for all relative fields, and running it with \(M\ge A+1\) supplies an angular target order \(A\). The shift step uses the pointwise torsion bound: the transverse integration direction of the \(\mathsf Z_1\) flux in [dn:high-transverse] is \(0\).

We next specify the longitudinal induction and its finite angular buffer. Fix a largest longitudinal order \(n_{\parallel}\) and a desired final angular order \(A_{\rm out}\ge5\), and put \[A_k=A_{\rm out}+2(n_{\parallel}-k), \qquad 0\le k\le n_{\parallel}.\] At level \(k\) the assertion is an \(e^{o(T)}\) bound in reference \(H^{A_k}\) for every ordered transport jet of the relative fields listed in the Lemma with exactly \(k\) longitudinal derivatives: use \(L_i\) on scalars and \(\mathfrak L_i\) on angular tensors. For \(b\) we also keep its ordinary longitudinal jets at that order. All earlier levels retain their larger \(H^{A_\ell}\) bounds. Run the preceding angular argument with \(M\ge A_0+1\) to establish level zero. The first pure jets in the bootstrap supply low coefficient bounds; the high-angular induction begins at \(k=1\) from this level-zero conclusion.

For \(i\ne j\), the exact identities needed for the induction are \[ \begin{gathered} [\mathfrak L_j,\mathfrak L_i] =\mathcal L_{X_{ij}},\qquad X_{ij}=a(\xi_j-\xi_i)^\sharp,\\ \mathfrak L_i\gamma=2\chi_i,\qquad L_i\log\mathcal A=\ell_i+\kappa,\qquad \mathfrak L_i(\xi_0+\xi_1)=\mathrm d\ell_i . \end{gathered} \tag{74}\] For \(k\ge1\), Raychaudhuri, torsion transport, and angular curvature variation give the reductions \[ \begin{aligned} L_i^k\tau_i &=L_i^{k-1}(\ell_i\tau_i-|\chi_i|^2),\\ \mathfrak L_i^k\xi_j &=\mathfrak L_i^{k-1}(\beta_i+\tau_i\xi_i),\\ \mathfrak L_i^k\xi_i &=\mathrm dL_i^{k-1}\ell_i-\mathfrak L_i^k\xi_j,\\ L_i^kK_\gamma &=L_i^{k-1}(\operatorname{div}\beta_i-\tau_iK_\gamma),\\ [\mathfrak L_j,\mathfrak L_i^k] &=\sum_{r=0}^{k-1} \mathfrak L_i^r\mathcal L_{X_{ij}}\mathfrak L_i^{k-1-r}. \end{aligned} \tag{75}\] The scalar commutator has the same formula with \(X_{ij}\) acting by directional differentiation. Opposite shape and lapse derivatives use their cross equations. The variation of the angular Levi-Civita connection under \(\mathfrak L_i\gamma=2\chi_i\) is \[(\mathfrak L_i\nabla)^C{}_{AB} =\gamma^{CD} (\nabla_A\chi_{i,BD}+\nabla_B\chi_{i,AD} -\nabla_D\chi_{i,AB}),\] and every derivative of \(a\) retains its \(a\) factor times relative fields. Thus a reduction to a preceding longitudinal level adds at most two angular derivatives: the largest costs are \(\operatorname{div}\beta_i\) and the outer \(\nabla\) on the torsion reduction. The buffer \(A_k\) covers these costs.

Set \(U_{i,k}=\mathfrak L_i^k\chi_i\) and \(V_{i,k}=L_i^k\ell_i\). Commuting the cross equations and using [dn:longitudinal-reductions] gives \[ \begin{aligned} \mathfrak L_jU_{i,k} &=-\tfrac12\tau_jU_{i,k} +U_{i,k}\chi_j+\chi_jU_{i,k}+\mathcal F_{i,k},\\ L_jV_{i,k} &=-U_{i,k}\cdot\chi_j+\mathcal G_{i,k}. \end{aligned} \tag{76}\] Here \(\mathcal F_{i,k},\mathcal G_{i,k}\) contain only longitudinal levels below \(k\) and their buffered angular derivatives. In particular the first reduction in [dn:longitudinal-reductions] replaces the differentiated term \(-\tfrac12(L_i^k\tau_i)\chi_j\) by lower levels. This is why that term is absent from the displayed top system. The new lapse jet does not occur on its right side.

For completeness, commute the first equation of [dn:pure-top-system] with \(D\) in increasing angular order. For \(r\ge4\) the tame product and shift commutator inequalities give \[\begin{aligned} \frac{\mathrm d}{\mathrm dy_j}|U_{i,k}|_r &\le C_rh_j|U_{i,k}|_r +C_r\mathcal C_{j,r}\sum_{\ell<r}|U_{i,k}|_\ell +|\mathcal F_{i,k}|_r,\\ \mathcal C_{j,r} &=1+|\gamma|_{r+1}+|\gamma^{-1}|_{r+1} +|\chi_j|_r+|B_j|_{r+1}. \end{aligned}\] This norm inequality is understood in its regularized form at zero. At the first four angular orders the low \(H^4\) shape and \(H^5\) shift bounds give the same top coefficient directly. At higher orders \(\mathcal C_{j,r}\) multiplies already controlled lower angular orders of \(U_{i,k}\); it is a source in this subsidiary induction. It is \(e^{o(T)}\) by level zero. The lower longitudinal induction bounds \(\mathcal F_{i,k}\) using \(A_\ell\ge A_k+2(k-\ell)\). Consequently Gronwall exponentiates only the uniformly bounded directional integral of \(C_rh_j\). Finite products of earlier \(e^{o(T)}\) bounds and integration over length \(O(T)\) remain \(e^{o(T)}\). After \(U_{i,k}\) is controlled, the second equation has the same transport commutators and the known source \(U_{i,k}\cdot\chi_j\), so it controls \(V_{i,k}\) in the same way.

The reductions also cover mixed words. In an ordered word of length \(k\) acting on a torsion, apply its innermost transport derivative first: the torsion equation applies to the opposite torsion, and the torsion-sum identity gives the derivative of the same-index torsion. The remaining \(k-1\) operators then act on \(\beta_i,\ell_i\) and products of fields. Expanding their angular commutators uses the same two-derivative buffer. The metric and relative lapse words reduce in the same way by [dn:longitudinal-identities]. For a mixed shape or lapse word, commute a cross derivative inward; the commutator in [dn:longitudinal-reductions] has shorter words, and the cross equation reduces the resulting innermost derivative. This gives the already controlled pure or lower levels. For the shift, differentiate its exact direction-\(1\) equation \(k\) times in \(u\) and integrate to obtain \(\partial_u^kb\). Conversion of its source from transport to ordinary derivatives uses only longitudinal orders below \(k\) of \(b\): expanding \(\partial_u=\mathfrak L_0-\mathcal L_b\) on tensors, or \(\partial_u=L_0-b\cdot\mathrm d\) on scalars, differentiates \(b\) at most \(k-1\) times. Each replacement adds at most one angular derivative, which is covered by the same buffer. Mixed shift jets follow directly from its equation. Finally \(q^{-1}\partial_vb=\mathcal A\gamma^{-1}(\xi_1-\xi_0)\) is algebraic. This completes level \(k\) and the conversion to ordinary jets.

The required initial pure jets are finitely many entry jets. They can also be expressed using tangential \(x\) jets and the cross equations on the noncharacteristic cylinder, where \(q=q_0>0\). For an output order \(n\), take \(n_{\parallel}=n\) and \(A_{\rm out}=\max(n,5)\), then choose \(M\ge A_0+1\) as above. Include the derivatives defining \(K,\beta\) and the initial cross identities and take their finite maximum as \(r(n)\). All sources in the argument are polynomial in the finitely many earlier high norms; only low directional integrals enter an exponential. This gives the asserted uniform dependence and leaves \(r(n)\) independent of later frequency-accuracy choices. The Lemma follows. ◻

Proof. We give the operator differences first, to check derivative counts before estimating the currents. Put \(h_{AB}=\Delta\gamma_{AB}\) in the next two displays only. The difference of angular connections is exactly \[ C^C{}_{AB}:=(\Gamma')^C{}_{AB}-(\bar\Gamma)^C{}_{AB} =\tfrac12(\gamma'^{-1})^{CD} (\bar\nabla_Ah_{BD}+\bar\nabla_Bh_{AD}-\bar\nabla_Dh_{AB}). \tag{80}\] Thus \(|C|_0\le C|\Delta\gamma|_1\). For a barred one-form \(f\), \[ \begin{aligned} (\operatorname{div}'-\overline{\operatorname{div}})f &=((\gamma'^{-1})^{AB}-(\bar\gamma^{-1})^{AB})\bar\nabla_Af_B -(\gamma'^{-1})^{AB}C^D{}_{AB}f_D,\\ (\operatorname{curl}'-\overline{\operatorname{curl}})f &=\tfrac12((\epsilon')^{AB}-\bar\epsilon^{AB})(\mathrm df)_{AB}. \end{aligned} \tag{81}\] The analogous divergence formula on two-tensors has one connection difference for each covariant slot. No second derivative of \(h\) occurs. Against \(\bar\beta_i\) or \(\bar\chi_i\), these differences cost \(Ch_iP\), since their required low angular derivatives are \(O(h_i)\); against bounded barred torsion they cost \(CP\).

The exact torsion-sum identity gives \[ |\Delta\log\mathcal A|_1\le CP, \qquad |\Delta\mathcal A|_1\le CP. \tag{82}\] For the trace conversion, use the exact identity \[ \Delta\tau_i=\mathcal A'\Delta w_i+ (\Delta\mathcal A)\bar w_i, \qquad |\Delta\tau_i|_1\le C(z_i+h_iP). \tag{83}\] The trace of \(\Delta\chi_i\) in the primed metric is \(\Delta\tau_i-(\gamma'^{-1}-\bar\gamma^{-1})\bar\chi_i\). Writing \(C_{\rm inv}=\gamma'^{-1}-\bar\gamma^{-1}\), the full shape Hodge datum is exactly \[\operatorname{div}'\Delta\chi_i-\mathrm d\operatorname{tr}'\Delta\chi_i =\Delta\beta_i-(\operatorname{div}'-\overline{\operatorname{div}})\bar\chi_i +\mathrm d(C_{\rm inv}:\bar\chi_i).\] Apply the primed trace-free divergence estimate, including the undifferentiated tensor norm to cover its kernel, and then the primed one-form div–curl estimate. [dn:operator-differences] and [dn:trace-conversion] yield \[ R_i\le C(Y_i+N_i+z_i+h_iP),\qquad |\Delta\xi_j|_1\le C(\sigma^{-1}Z_i+P). \tag{84}\] For the latter, the divergence data are \(\Delta K_\gamma-\Delta m_i\) and the curl datum is \(\Delta c_i\), plus the displayed coefficient differences on \(\bar\xi_j\). Precisely, \[\operatorname{div}'\Delta\xi_j=\Delta K_\gamma-\Delta m_i -(\operatorname{div}'-\overline{\operatorname{div}})\bar\xi_j, \qquad \operatorname{curl}'\Delta\xi_j=\Delta c_i -(\operatorname{curl}'-\overline{\operatorname{curl}})\bar\xi_j.\] These first-order elliptic constants need only the uniform low Lipschitz geometry. The estimate does not assert an unweighted bound for the torsion derivative by \(Z_i\).

When subtracting a transport equation, keep the actual operator: \[ \begin{aligned} \mathfrak L_i'f'-\bar{\mathfrak L}_i\bar f &=\mathfrak L_i'\Delta f+\mathcal L_{\Delta B_i}\bar f,\\ L_i'f'-\bar L_i\bar f&=L_i'\Delta f+ \Delta B_i\cdot\mathrm d\bar f. \end{aligned} \tag{85}\] For a one-form, the last Lie term consists of \(\Delta B_i\cdot\partial\bar f+(\partial\Delta B_i)*\bar f\). It costs \(Ch_iD_b\) on \(\bar\beta_i,\bar\chi_i\), \(CD_b\) on bounded barred torsion, and \(CD_b\) on \(\bar Q_i,\bar m_i\) before their \(\sigma\) weight. On \(\bar\ell_i=-\kappa+O(h_i)\) it costs \(Ch_iD_b\), since the constant \(-\kappa\) has zero angular derivative.

Proof of 15. We first make explicit the finite order choices. Choose a spatial target order \(k\ge2s\), enlarged if necessary to improve every bootstrap and continuation norm. The time recovery below uses at most two spatial derivatives for each time derivative. Choose an integer \(M\) much larger than \(k+2\) so that, with \(\vartheta=(k+2)/M\), \[ 2\alpha(1-\vartheta)>(1+c_1)\alpha \quad\text{for some }0<c<c_1<1/2. \tag{100}\] The constant \(2\) in \(k+2\) supplies Sobolev embedding on the three-dimensional slices. Here \(M\) is the requested slice Sobolev order; the proof of 16 runs its angular currents at the larger order specified there. That Lemma determines a finite initial buffer \(r(M)\) and its finite template/exterior jet requirements. Include the finite order \(m\) in that base budget. Choose the rate tolerance \(\eta_2\) small enough that [dn:interpolation-choice] retains a strict gap after adding all interpolation and later time-recovery losses. In 17 choose the fixed current multipliers, the small-region threshold, the growth tolerance, then \(\nu\), and finally sufficiently small input/exterior losses.

Let \(P_{\mathrm{base}}\ge1\) be the maximum of the finitely many positive frequency exponents required at these fixed base orders. Choose \(\zeta>0\) so small that each has its allocated loss, in particular \(\zeta P_{\mathrm{base}}<\alpha/2\). Only now increase the finite hierarchy and beam accuracy orders to make all linear defects as small as required, including the finite inverse-lapse losses in deriving linearized double-null identities. The base-loss statement in 13 ensures that this last operation changes constants and required fixed background jets, but does not increase any previously budgeted positive power bounded by \(P_{\mathrm{base}}\) of \(\lambda\). Finally take \(T\) large. Every high input is a finite entry norm already supplied by the preparation. This order of choices has no unknown-high-norm smallness requirement.

Local smooth vacuum theory and noncharacteristic eikonal and angle transport start the primed solution at the actual cylinder data. On each available slice, [dn:low-conclusion] and 16 give, for each component error \(f\), \[\|f\|_{L^2}\le e^{-(2\alpha-\eta_2)T}, \qquad\|f\|_{H^M}\le e^{o(T)}.\] For the second inequality the barred fields have their individual high bounds directly from the linear construction. Sphere charts and extension in the interval variable give \[ \|f\|_{C^k(I_T(t)\times\mathbb S^2)} \le C\operatorname{poly}(T) \|f\|_{L^2}^{1-\vartheta}\|f\|_{H^M}^{\vartheta} \le e^{-(1+c_1)\alpha T} \tag{101}\] after reducing \(c_1\) within its fixed margin. The lower bound for the interval length ensures uniform local extension constants; covering a longer interval has at worst a polynomial cost. This argument is performed on each slice and needs no positive lower bound for the available time thickness.

To obtain time derivatives, use \(\partial_v=(\partial_t+ \partial_x)/2\) in [dn:recovery-equations]: \[\begin{aligned} \partial_t\gamma&=4\chi_1-\partial_x\gamma,\\ \partial_t\log\mathcal A &=2(\ell_1+\kappa)-\partial_x\log\mathcal A,\\ \partial_tb &=2q\mathcal A\gamma^{-1}(\xi_1-\xi_0)-\partial_xb . \end{aligned}\] Use the shape/lapse cross equations and the torsion equations for the other fields. For a tensor transported in direction \(0\) use \(\mathfrak L_0=(\partial_t-\partial_x)/2+\mathcal L_b\); for a scalar replace the Lie derivative by \(b\cdot\mathrm d\). Thus each time derivative is twice the corresponding right side, plus a spatial derivative and, when appropriate, an angular Lie term. A shape or lapse cross equation contains at most two angular derivatives of the metric and one of the torsion; the torsion equation contains one angular shape derivative. Thus \(r\) time derivatives and \(p\) spatial derivatives require at most spatial order \(p+2r\). The choice \(k\ge2s\) covers every requested ordinary derivative of total order at most \(s\). Subtract the equations with the actual principal operators, as above. All products use already controlled lower error jets and individual jets from 16; barred defects have the prescribed finite differentiated bounds. There is no division by \(q\). Finite product losses can be assigned arbitrarily small rates in that Lemma, leaving [dn:main-error] with \(c<c_1\). The relative shift conclusion follows by differentiating the algebraic identity \[q^{-1}\partial_vb=\mathcal A\gamma^{-1}(\xi_1-\xi_0).\]

The template change at the fixed bootstrap orders is at most \(e^{-(\alpha-\eta_0)T}\) with \(\eta_0<\alpha/2\). Together with [dn:main-error], this is strictly smaller than \(\tfrac12 e^{-\alpha T/2}\) for large \(T\). It improves [dn:bootstrap], preserves the nondegeneracy margins, and provides the finite Sobolev buffers required for restart.

We finish the existence argument, since an a priori comparison on a putative full domain would not suffice. On each fixed finite experiment, \(q\ge q_0e^{-2\kappa L_T}>0\). All requisite finite norms and inverse metrics are bounded up to a possible terminal slice. Use the finite-domain restart in (OpenAI 2026b, companion@kind@foundation@in:continuation companion@kind@foundation@in:continuation ) as follows. On an earlier slice take its induced metric and second fundamental form, initialize wave gauge, and extend the smooth slice jets across the interval endpoints on an enlarged margin. Their required finite Sobolev norms and positivity have controlled extensions. Constraints are needed only on the original margin. The reduced wave equation has a positive local existence time depending on those finite norms and the finite positive lapse. Solve the eikonal pair and angular transport from the slice with their inherited values. Transversality and invertibility persist on a smaller margin for a time bounded below at this fixed \(T\).

Restrict the new solution to the region with exiting sides. There \(\mathrm du,\mathrm dv\ge0\) on future causal vectors, so artificial extension data cannot enter the retained domain. Constraint and gauge propagation give the vacuum equations there. Local geometric uniqueness, followed by uniqueness for the eikonal and angle transports, identifies this solution with the existing one on their overlap. Restart before any proposed terminal time; the uniform bounds at that fixed \(T\) contradict termination. Slightly enlarged intervals and sphere patches give smooth endpoint jets throughout. The restart time need not be uniform as \(T\to\infty\).

Finally, smoothness at every order is preserved. At each fixed \(T\) the fixed-region version of the high-current proof, with the weight \(e^{-2Kt}\), propagates any larger finite order from the smooth entry data. Its constants may depend on that order and on \(T\); no simultaneous subexponential estimate at all orders is asserted or needed. The constructed solution therefore reaches all of [dn:domain] smoothly. On the initial collar it is the same vacuum solution as the prepared exterior solution in its newly initialized optical coordinates, by the same local uniqueness. The whole-bridge attachment is carried out in 18. This proves the Theorem. ◻

Proof. The lower and exterior joins are the qualitative construction in (OpenAI 2026a, companion@kind@signed@attach:whole-bridge-section companion@kind@signed@attach:whole-bridge-section ), based on (OpenAI 2026b, companion@kind@foundation@cmp:finite-range companion@kind@foundation@cmp:finite-range ). Its hypotheses are full-bridge \(p_{10}\) closeness, exact protected agreement on the compact causal shadows of the early joins, actual corrected partial-slice data, low exterior closeness on enlarged matching bands, and strict outflow and overlap margins. We verify these hypotheses and the new double-null join separately.

The finite lower-block construction in (OpenAI 2026b, companion@kind@foundation@cmp:finite-range companion@kind@foundation@cmp:finite-range ) applies to \(d_T\) since \(p_{10}(d_T-d)\to0\). It contains every spatial radius of the new bridge, a past collar, the two bent exterior initial slices, and fixed forward bands on them. On the ends the bent slices equal the original bridge outside a fixed compact set. The protected ball (34) contains a fixed-origin ball of radius \[(1-2\beta-s_*)t_T-O(1)\longrightarrow\infty.\] Consequently every fixed compact causal shadow needed for the bent pieces and bounded connecting pieces is exactly unchanged. Domain of dependence in a common early Cauchy gauge identifies these pieces with the background. At large radii the horizontal selected-end slice carries the actual new datum, including its tail; no replacement by the old datum is made there.

Choose a fixed exterior joining time \(t_a\) inside the enlarged lower block. On the selected end attach the exact exterior from 13, computed from these same partial-slice data, using the shrinking outer cutoff (37). The identifying wave map on an enlarged fixed band about \(t_a\) has the prescribed initial slice map and matching unit normals. Fixed-time wave-map stability gives a diffeomorphism on the smaller band, uniformly through the large outer cutoff: the exterior charts have uniform low regular geometry, the new metrics are low-order close, and artificial sides are strictly outgoing. The induced metric and second fundamental form agree; the wave gauge then determines the remaining normal metric jets. Reduced uniqueness therefore makes this map an isometry. These are exactly the finite-band hypotheses checked in (OpenAI 2026a, companion@kind@signed@attach:whole-bridge-section companion@kind@signed@attach:whole-bridge-section ). The unselected end uses the unchanged exterior. Straighten and identify inside the matching margins, without keeping a second copy of overlapping points.

It matters that the lower block has not been spatially truncated. In a selected-end overlap the retained region has, in addition to its finite future cuts, the condition \[ t_e<t_a\quad\hbox{or}\quad r_{\rm co}<R_{\rm out}(t_e). \tag{102}\] Both \(t_e\) and \(r_{\rm co}-R_{\rm out}(t_e)\) increase strictly along future causal curves at their active boundaries. The derivative of the second quantity with respect to \(t_e\) is at least \(V_{\rm out}-\sup|\mathrm dr_{\rm co}/\mathrm dt_e|>0\). Their strict sublevels and their union are preserved to the past. A past curve in the added piece stays inside its outer cutoff until reaching \(t_a\), after which all radii are present. A past curve already below \(t_a\) stays there. The omitted corner at \(t_e=t_a\), \(r_{\rm co}=R_{\rm out}(t_a)\) is consequently not a missing past endpoint.

The required interval of \(\mathcal S\) and a margin are present in this joined lower/exterior region. At its selected late end, \(t_e\le t_T+O(1)\) and the co-radius is bounded; its middle and unselected portions are unchanged. Low closeness preserves its spacelike character. Use the actual exterior induced data there. The nearby eikonal roots with the prescribed entry values, and the shift-free angular transport, define the new coordinates on a collar. They are noncharacteristic at this fixed depth. Local geometric vacuum uniqueness identifies this collar with the initial collar of the exact solution from 15; equivalently that solution may be started on the collar itself. The quadratic template comparison does not substitute for matching actual data. For every fixed finite \(T\) the collar widths can be chosen inside the available margins; no lower width uniform in \(T\) is required.

Above the join all cuts are made in the new double-null coordinates. For a causal vector \(V\) in their future cone, \[-2a_T\,\mathrm du(V)\mathrm dv(V) +\gamma_T\bigl(\mathrm d\theta(V)-b_T\mathrm du(V), \mathrm d\theta(V)-b_T\mathrm du(V)\bigr)\le0.\] Thus \(\mathrm du(V)\) and \(\mathrm dv(V)\) have the same weak sign, and the future choice is the nonnegative sign. Also \(g_T^{-1}(\mathrm dt,\mathrm dt)=-1/(2a_T)<0\); hence \(t\) is temporal. For a nonzero future causal vector, \(\mathrm dt(V)>0\), and \[ \frac{\mathrm d}{\mathrm dt}\left(u+\frac{c_0t}{T}\right) \ge\frac{c_0}{T},\qquad \frac{\mathrm d}{\mathrm dt}\left(v+\frac{c_0t}{T}\right) \ge\frac{c_0}{T}. \tag{103}\] The strict side and terminal sublevels of (28) are therefore preserved into the past. In particular a past-inextendible causal curve from its strict interior reaches \(t=0\) strictly inside the attached interval. Indeed its side functions never exceed their strictly smaller starting values. On each fixed experiment, the retained closure lies inside a compact enlargement on which smooth nondegeneracy bounds \(a_T,a_T^{-1},b_T,\gamma_T^{\pm1}\). In \(t\) parameter, \(0\le\mathrm du/\mathrm dt,\mathrm dv/\mathrm dt\le2\), and the displayed cone inequality then bounds the angular speed using the upper bound for \(a_T\) and the bounds for \(b_T,\gamma_T^{-1}\). A finite interior endpoint is therefore handled by ordinary continuation. This argument uses the exact new coordinates and does not require small relative cone error in a deep background wave chart.

Omit every unattached future face of a lower piece, including its edges. It is a future exit by the spacelike sign, so a past curve below it cannot end there. At intersections of cuts, the retained sets are finite unions and intersections of the same past-preserved strict sublevels; (102) handles the only change from unrestricted to truncated radii. Matching collars give product neighborhoods across the included faces. One first joins the lower bands and then the deep collar at their intersections, as in (OpenAI 2026b, companion@kind@foundation@cmp:development companion@kind@foundation@cmp:development ); this gives a Hausdorff smooth manifold with only one copy of each identified point.

Every past-inextendible causal curve from an added piece therefore reaches the complete initial bridge. In the lower pieces bounded speeds exclude escape to spatial infinity in bounded time, and their Cauchy construction gives the same conclusion. The past collar gives the future crossing for curves starting below the bridge. There is at most one crossing, by the lower-block Cauchy property and the one-way signs of all joins. Thus the complete \(\Sigma\) is Cauchy in the assembled vacuum region. Maximal-development uniqueness embeds this region in \(M_{d_T}\), proving the claim. All restrictions were made inside preassigned enlarged margins, so the earlier flow tubes and observation neighborhood remain interior. ◻

Proof. For a shift-free direction \(L_1=\partial_v\), the null connection identities are \[\nabla_{L_1}L_1=\ell_1L_1,\qquad \nabla_{\partial_A}L_1=\chi_{1,A}{}^B\partial_B+ \xi_{1,A}L_1.\] Since \([\partial_v,\partial_A]=0\), applying the curvature commutator and pairing with \(\partial_B\) gives \[D(g)_{AB}=\gamma_{BC}\partial_v(\chi_{1,A}{}^C) +(\chi_1\gamma^{-1}\chi_1)_{AB} -\ell_1\chi_{1,AB}.\] Use \(\partial_v\gamma=2\chi_1\) to lower the differentiated index. This gives the exact formula \[ D(g)=\partial_v\chi_1-\chi_1\gamma^{-1}\chi_1-\ell_1\chi_1 =\tfrac12\partial_v^2\gamma -\tfrac14(\partial_v\gamma)\gamma^{-1}(\partial_v\gamma) -\tfrac12(\partial_v\log a)(\partial_v\gamma). \tag{106}\] There is no inverse radial lapse in this formula.

For the template, \(\bar\chi_1=\chi_1+\tfrac12\partial_vk_\gamma\) and \(\bar\ell_1=\ell_1+\partial_vk_H\) exactly. The linear variation of (106) is therefore \[ \begin{split} \delta D={}&\tfrac12\partial_v^2k_\gamma -\tfrac14\bigl((\partial_vk_\gamma)\gamma^{-1}\partial_v\gamma +(\partial_v\gamma)\gamma^{-1}\partial_vk_\gamma\bigr)\\ &+\tfrac14(\partial_v\gamma)\gamma^{-1}k_\gamma \gamma^{-1}(\partial_v\gamma) -\tfrac12\ell_1\partial_vk_\gamma -\tfrac12(\partial_vk_H)\partial_v\gamma. \end{split} \tag{107}\] The angular Lie term in (45) contains only \(Z\) and angular derivatives of \(Z\), without a longitudinal derivative of \(Z\). Hence (42) gives \[\partial_v^2(\mathcal L_Zg)_{AB}=O(A_T\lambda e^{o(T)}).\] The formula for \(k_H\) has at most one longitudinal derivative of \(Z\), so \(\partial_vk_H=O(A_T\lambda e^{o(T)})\) as well. All other linear terms in (107) have this bound. Twice differentiating the real leading oscillation in (30), whose trace-free leading tensor is unchanged by trace reversal, yields \[ D(\bar g)-D(g) =-\tfrac12A_T\lambda^2\cos(\lambda v)(p_0)_{AB} +O(A_T\lambda e^{o(T)}) +O(A_T^2\lambda^{P}e^{o(T)}). \tag{108}\] The last term comprises quadratic template products in the smooth relative-field expression (106). Choose \(\zeta\) small enough that this term is \(o(A_T)\). The exact-minus-template error from 15, at the finite derivative orders in (106), is also \(o(A_T)\); multiplication by the individual \(e^{o(T)}\) coefficient bounds preserves this conclusion.

By (31), on the cosine-good set the leading norm is at least \(\tfrac14A_T\lambda^2e^{-o(T)}\) uniformly throughout \(\mathcal Q_T\). Its ratio to the linear lower-order error tends to infinity, since \(\lambda e^{-o(T)}\to\infty\), and its ratio to \(A_T\) also tends to infinity. This proves (105). The argument compares two curvature tensors at common numerical coordinate positions; the background curvature is never presumed small or bounded independently of \(T\). ◻

Proof. The unit mass shell in double-null variables reads \[p=-(\pi_u+b^AP_A)>0,\qquad 2p\pi_v+a(1+\gamma^{AB}P_AP_B)=0.\] Thus the reduced \(v\)-Hamiltonian is \[ \mathcal H=-\pi_v= \frac{a(1+\gamma^{AB}P_AP_B)}{2p}. \tag{114}\] The same expression with primed fields defines the new future unit root at the identical numerical terminal state. The background tube keeps \(p\) in a compact positive interval and the other momenta bounded. Since \(b_T-b\to0\) uniformly, the new \(p_T=-(\pi_u+b_T^AP_A)\) remains bounded away from zero. Choosing the root with \(\dot v=p_T/a_T>0\) fixes its future orientation.

Write \(V\) for the background Hamiltonian vector field. On the common late state tube this field and its first two state derivatives are bounded by \(a e^{o(v)}\), by 6. This bound is integrable over the late \(v\) interval. Differentiating (114) shows that the new field and its first two state derivatives differ by an absolutely exponentially small quantity, uniformly on that tube. Indeed \(a_T=q\mathcal A_T\) retains its factor \(q\) under ordinary differentiation; the denominators are powers of \(p_T\) separated from zero; and all relative coefficient differences and their needed jets are exponentially small by 13 and 15. Fixed higher background factors cost \(e^{o(T)}\), which is absorbed by that exponential.

The terminal state difference is exactly zero. Backward Gronwall on a late interval of length \(O(T)\) therefore gives state differences tending to zero. For first variations the equation is \(J'=D_zV\,J\). Its coefficient difference is bounded by the small difference of vector-field derivatives plus \(|D_z^2V|\) times the already small state difference. The latter background bound is integrable. A second Gronwall estimate therefore gives uniform \(C^1\) convergence of the backward maps. The background forward hitting map also has a uniform derivative bound: if \(v_0\) is the fixed late starting level, then \[\|D_w\mathcal H_v\| \le \|D_w\mathcal H_{v_0}\| \exp\!\left(\int_{v_0}^{v}\|D_zV(s)\|\,\mathrm ds\right)\le C.\] Composing the terminal-state estimates with \(\mathcal H_v\) therefore preserves their \(C^1\) convergence in the original seed variables. The strict tube and construction margins prevent exit during this comparison. Below any fixed late \(v\) the remaining flow tubes to \(\mathcal S\) and then \(\Sigma\) are compact and transverse. Smooth finite-flow dependence handles their hitting maps; protected agreement makes their metrics exactly the background metrics for sufficiently large \(T\). This proves (111) and, by the margin \(O_0\Subset O\), places the new seeds in \(O\).

Here is the exact volume statement. On the cotangent bundle, let \[\omega=\mathrm d\pi_u\wedge\mathrm du+\mathrm d\pi_v\wedge\mathrm dv +\mathrm dP_A\wedge\mathrm d\theta^A.\] On the unit Hamiltonian shell its restriction has the geodesic flow direction as kernel. A single-valued transverse hitting map therefore preserves the restrictions of \(\omega\) to two sections: flowing by a variable time only adds multiples of its kernel to tangent vectors. On \(v=\mathrm{constant}\) this restriction is exactly \[\omega_v=\mathrm d\pi_u\wedge\mathrm du+\mathrm dP_A\wedge\mathrm d\theta^A,\] independently of the shell root \(\pi_v\). Its volume is \(\mathrm du\,\mathrm d^2\theta\,\mathrm d\pi_u\,\mathrm d^2P\). Apply this to the background hit and to the new backward hit. It gives \(\Psi_{T,v}^*\mu_{d_T}=\mu_d\) exactly. The hits are single-valued on these compact transverse flow tubes; uniqueness also gives injectivity. Alternatively, in the ball coordinates (111) makes the map \(\mathrm{Id}+f\) with \(\|Df\|<1/2\), which is injective on the convex closed ball. Nothing in this argument assumes injectivity of the projection from seed states to observation positions.

At an observation state the unit tangent satisfies \[ \dot u=\frac{1+\gamma^{AB}P_AP_B}{2p},\qquad r_T\dot v=\frac{r_Tp}{a},\qquad \dot\theta=b\dot u+\gamma^{-1}P, \tag{115}\] and likewise for \(g_T\). On the whole observation box, \(u\) and \(v-T\) are bounded, so \(a/r_T=q_0\mathcal A e^{-\kappa(u+v-T)}\) is bounded above and below. These formulas prove the tangent bounds in \(F\). The Gram matrix has the block form \[ G_F=\begin{pmatrix} \gamma(b,b)&-a/r_T&-b^C\gamma_{CB}\\ -a/r_T&0&0\\ -\gamma_{AC}b^C&0&\gamma_{AB} \end{pmatrix},\qquad \det G_F=-(a/r_T)^2\det\gamma. \tag{116}\] The low field bounds give bounded entries and a determinant bounded away from zero, hence a bounded inverse. The same holds for \(g_T\). All ordinary derivatives of \(a/r_T\) retain this bounded relative factor, and the other required field jets are \(e^{o(T)}\) by the individual estimates. This proves the thirty-jet assertion on the larger box, not just on the hitting positions. Finally \(\sqrt{|\det g|}=a\sqrt{\det\gamma}\) proves (113).

We finish with the finite persistence assertion. For a fixed \(T\) the closures of the observation box, a small enlargement, and all backward flow tubes just used are compact subsets of the smooth attached development with an interior margin. The compact-subset Cauchy stability and geometric uniqueness of (OpenAI 2026b, companion@kind@foundation@cat:closed-tests companion@kind@foundation@cat:closed-tests ), also used in the closed-test proof of (OpenAI 2026a, companion@kind@signed@test:closed companion@kind@signed@test:closed ), supply corresponding compact coordinate witnesses for all sufficiently close smooth data. Require closeness in the finite \(C^k\) norms needed here. Since \(T\) is fixed, it can be arbitrarily accurate even in the rescaled frame \(F\). Preserve the Gram and inverse bounds, the thirty derivative bounds (for example allowing additive error at most one), the density comparison, and the curvature margin, so that \[ |D(\widehat g_T)-D(g)|>A_T \quad\hbox{when}\quad |\cos(\lambda v)|\ge1/2. \tag{117}\]

The common coordinates for \(\widehat g_T\) need not retain double-null form. To define its terminal state solve \(\tfrac12\widehat g_T^{-1}(\pi,\pi)=-\tfrac12\) for \(\pi_v\) with the other six numbers fixed as in (110). At the original terminal state the derivative with respect to \(\pi_v\) is \(U^v>0\). On the fixed compact state set it is separated from zero, so the implicit function theorem supplies the same smooth simple future root for close metrics. The level-\(v\) section remains transverse to this flow. Smooth backward-flow dependence lets us impose \(\sup_{v\in T+J_0}\|\widehat\Psi_{T,v}-\Psi_{T,v}\|_{C^1} <\min\{\eta_{\rm map},\eta_0\}\), where a fixed \(\eta_0>0\) preserves injectivity and the image margin in \(O\). The restriction of the canonical symplectic form to that section is still \(\omega_v\), regardless of optical gauge. Its transported volume is therefore exactly the new initial seed measure. All statements hold for this fixed experiment, with no assertion of a data-neighborhood radius uniform in \(T\). ◻

Proof. For any Lorentzian Gram matrix \(G\) and unit timelike column \(U\), set \[K(G,U)=G+2(GU)(GU)^{\mathsf t}.\] The associated form is positive definite: writing a vector as \(aU+Z\) with \(g(U,Z)=0\) gives \(a^2+g(Z,Z)>0\) unless the vector vanishes. The determinant lemma and \(U^{\mathsf t}GU=-1\) give the exact identity \[ \det K(G,U)=\det G\,(1+2U^{\mathsf t}GU)=-\det G. \tag{125}\] Bounded \(G,G^{-1},U\) give an upper eigenvalue bound for \(K\) and a positive lower bound for its determinant. Its smallest eigenvalue is therefore bounded below by the determinant divided by the cube of the upper eigenvalue bound. Apply this in both bases. If \(g_c\) is the chart matrix and \(U_c=MU_F\), then \[K(G,U_F)=M^{\mathsf t}K(g_c,U_c)M.\] The two uniform positive definite comparisons bound both \(M\) and \(M^{-1}\). In particular, bounded Lorentzian Gram matrices alone are not being used to bound a Lorentz boost. ◻

Proof. Since \(F\) commutes, its connection coefficients are given by Koszul’s formula \[C^k{}_{ij}=\tfrac12G^{kh} (F_iG_{jh}+F_jG_{ih}-F_hG_{ij}).\] For a constant column \(V\) and \(W=rV^iF_i\), the connection matrix on \(W\) is therefore \[ \mathsf A(W)^k{}_j =\tfrac12G^{kh}V^i \bigl((rF_i)G_{jh}+(rF_j)G_{ih}-(rF_h)G_{ij}\bigr). \tag{129}\] Each \(rF_i\) is one of \(r\partial_u,\partial_v,r\partial_{\theta^A}\). Their coefficients are constant and bounded. Thus the connection contracted with each of \(\partial_v\) and \(rX\), and its ordinary derivatives through order eighteen, are bounded by \(Q_T\). This proves the required cancellation directly, including all shift and lapse terms; uncontracted \(F\) connection coefficients need not have such a bound.

For an edge parametrized by \(s\in[0,1]\), its position has the form \(x+c_e(s,l)\), where \(c_e\) is linear in \(l\) and independent of momentum. Its transport equation in \(F\) has coefficient \(l\mathsf A(\pm W)(x+c_e(s,l))\) for \(W=\partial_v\) or \(rX\). Its integral norm is at most \(lQ_T=o(1)\), uniformly in the indicated range. Differentiating this coefficient at most eighteen times in \(l\) uses only ordinary derivatives in bounded coordinate directions, so each derivative is bounded by \(Q_T\). To see the corresponding assertion for transport, differentiate \(P'=-A_eP\), \(P(0)=\mathrm{Id}\), \(k\) times in \(l\): \[(\partial_l^kP)'=-A_e\partial_l^kP -\sum_{j=1}^k\binom{k}{j} (\partial_l^jA_e)(\partial_l^{k-j}P).\] Gronwall uses only the small integral of the undifferentiated \(A_e\). Induction in \(k\le18\) bounds every differentiated solution by finite products of \(Q_T\), hence by another subexponential bound. Multiplying the four edge transports proves (127).

For completeness, pull the smooth connection back by \((s,t)\mapsto x+s\partial_v+trX\). Choose a smooth frame agreeing with \(F\) at \((0,0)\) whose connection one-form vanishes there. This is possible by prescribing the first derivatives of a frame change at that point. On the boundary of \([0,l]^2\) its connection one-form has integral norm \(O(l^2)\). The ordered transport equals \(\mathrm{Id}\) minus its boundary integral up to \(O(l^4)\). Stokes’ theorem identifies the quadratic coefficient with minus its exterior derivative at the origin; since the connection form vanishes there, this is the curvature on \((\partial_v,rX)\), with the orientation sign. This auxiliary frame is used only to identify a Taylor coefficient, not to obtain a uniform estimate. Finally, multilinearity gives \[g\bigl(R(\partial_v,rX)(r^{-1}\partial_v),X\bigr) =r r^{-1}R(\partial_v,X,\partial_v,X).\] This proves (128) with no remaining factor of \(r\). ◻

Proof. At each good parameter select the least chart index for which the coordinate margin is at least \(1/B\) and \(|U_c|\le B\). The finite selector is measurable and may depend on all three momentum labels. Fix Borel representatives of the finitely many weak Christoffel arrays. By 21, the base frame matrices obey \(\|M(0)\|+\|M(0)^{-1}\|\le C_0\). Choose a fixed \(K>2C_0+1\).

Fix \(X\in\mathcal X\) and a sampled positive size \(l=j l_T\), \(1\le j\le17\). Use the selected base chart throughout the four concatenated edges. Define one first stop on this concatenation: stop when either \(\|M\|=K\) or the chart position has moved Euclidean distance \(1/(2B)\) from its base position. The latter displacement keeps the path in the chart because the base margin is \(1/B\). An edge after an earlier stop contributes nothing to any integral below. Before this global stop the smooth interior path, frame matrix, and chart coordinates are all defined. More precisely, a parameter is before the stop at a given time when both strict thresholds hold at every earlier time in the concatenation. In a fixed chart, continuity expresses this condition using suprema over rational earlier times, so it is Borel on the smooth state domain. Intersecting with the measurable selected parameter set gives the stopped subsets used below.

Write \(x=(v,u,\theta)\) and \(m=(\pi_u,P)\). At a fixed edge parameter \(s\) the position is the translation \(x\mapsto x+c_e(s,l)\), independent of \(m\). For the four edges the translations are, successively, \[sl\partial_v,\quad l\partial_v+slrX,\quad (1-s)l\partial_v+lrX,\quad (1-s)lrX.\] For a selected chart \(c\), restrict first to good parameters before the global stop. Enlarge this nonnegative integral to all \(m\in Q\) and all translated observation positions lying in that chart. On positions the translation, the smooth open isometric embedding, and the chart map compose to an injective smooth coordinate map. Its volume change is controlled without estimating its derivative matrix: isometry and (124) give \[ \mathrm dx\le C r^{-1}\mathrm d\operatorname{vol}_g =C r^{-1}\sqrt{|\det\widetilde g_c|}\,\mathrm dy_c \le C_B r^{-1}\mathrm dy_c. \tag{132}\] Consequently, for each fixed \(s\), \[ \int_{\substack{\mathcal G_T:\text{chart }c\text{ selected}\\ \text{before global stop}}} |\widetilde\Gamma_c(y_c(x+c_e(s,l)))|^2\,\mathrm d\nu_T \le \frac{C_B|Q|}{r} \int_{c}|\widetilde\Gamma_c(y)|^2\,\mathrm dy. \tag{133}\] Here \(\mathrm d\nu_T=\mathrm dx\,\mathrm dm\); no seed Jacobian remains. The bounded momentum fiber contributes only \(|Q|\). Momentum-dependent chart selection and stop conditions have only restricted the integral before it was enlarged. In particular, (132) has not used the unproved along-edge bound for \(M\).

The weak chart Christoffel array agrees almost everywhere on the open image with its smooth interior array. For each \(s\), the inverse image of the exceptional position null set under the preceding smooth coordinate map is null. Fubini in \(s,x,m\) shows that, outside a null set of parameters, the stopped trajectory integrals are unchanged by this choice of representative. We discard that null set for each of the finitely many sampled loops. This justifies the use of the smooth connection equation below, without asserting a Sobolev trace on an individually prescribed loop.

Integrate (133) in \(s\in[0,1]\) and sum the at most \(B\) charts. If \[I_e(x,m)=\int_0^1\boldsymbol 1_{\{\text{before global stop}\}} |\widetilde\Gamma_c(y_e(s))|\,\mathrm ds,\] Cauchy–Schwarz and the chart \(L^2\) bounds imply \[ \int_{\mathcal G_T} I_e^2\,\mathrm d\nu_T\le C/r. \tag{134}\] Set \(R_T=r^{-1/2}e^{\kappa T/32}\). Markov’s inequality applied to \(I_e^2\) gives \[ \nu_T\{I_e>R_T\} \le (C/r)R_T^{-2}=C e^{-\kappa T/16}. \tag{135}\] There are four edges, seventeen positive sizes, and three choices of \(X\). The union of their exceptional sets, together with the null sets already removed, satisfies (130). Only this finite collection is used: no common exceptional set for a continuum of positive loop sizes is asserted.

We now prove that no stop occurs for a remaining parameter. On one edge write its \(F\)-velocity as \(f=lrV\), with \(V\) a fixed bounded column, and let \(\mathsf A\) be its internal connection matrix. By (129), \(|\mathsf A|\le lQ_T\). The connection transformation law along the smooth interior curve is the exact matrix equation \[ M'=M\mathsf A-\widetilde\Gamma_c(Mf)M. \tag{136}\] Here \(\widetilde\Gamma_c(Mf)\) means the chart connection matrix contracted with the chart velocity \(Mf\). Before the stop, \[|M'|\le K lQ_T+C K^2lr|\widetilde\Gamma_c|, \qquad |y'|\le C K lr.\] Summing over all four stopped edges and using \(I_e\le R_T\) yields \[\begin{align*} \sup\|M-M(0)\|&\le 4K lQ_T+C K^2lrR_T=o(1),\\ \sup|y-y(0)|&\le C K lr=o(1). \tag{137}\end{align*}\] Indeed \(l\le17e^{-\kappa T/16}\) and \(lrR_T\le17e^{-17\kappa T/32}\). For large \(T\) these inequalities are strict improvements of both stop thresholds. Continuity rules out a first stop, including on any of the later three edges. Thus the entire loop stays in its selected base chart, with \(\|M\|<K\). Only an upper matrix threshold was needed; the inverse at the base is already bounded. If desired, inverse bounds along the loop also follow from \(\det(M)^2=\det G/\det\widetilde g_c\) and the cofactor formula.

Ordinary chart parallel transport now has total integrated coefficient at most \(C lr\sum_e I_e\le C lr^{1/2}e^{\kappa T/32}\). Its matrix differs from \(\mathrm{Id}\) by at most the exponential of this quantity minus one. Since the rectangle closes exactly, its final frame matrix equals its base matrix. Thus if \(P_c\) is the full chart transport, \(H=M(0)^{-1}P_cM(0)\). Conjugating by the bounded base matrix and its inverse gives \[ \|H_{x,X}(j l_T)-\mathrm{Id}\| \le C r^{1/2}l_T e^{\kappa T/32}, \qquad 1\le j\le17. \tag{138}\] At \(j=0\) the left side is zero. All transports in this argument have taken place in the smooth original spacetime.

It remains to extract the quadratic coefficient from these eighteen values. We include the elementary interpolation argument, as in (OpenAI 2026a, companion@kind@signed@holo:interpolation companion@kind@signed@holo:interpolation ). Write \[H_{x,X}(l)-\mathrm{Id}=\sum_{k=0}^{17}C_k l^k+\mathcal R_{18}(l).\] By 22, the Taylor integral remainder at \(l=j l_T\) is bounded by \(C Q_Tl_T^{18}\). The Vandermonde matrix \((j^k)_{0\le j,k\le17}\) is invertible and independent of \(T\). Therefore its inverse expresses \(l_T^2 C_2\) as a fixed linear combination of the values of the polynomial at these nodes, giving \[\begin{align*} \|C_2\| &\le C l_T^{-2}\max_{0\le j\le17} \|H_{x,X}(j l_T)-\mathrm{Id}\|+Q_Tl_T^{16}\\ &\le C r^{1/2}l_T^{-1}e^{\kappa T/32}+Q_Tl_T^{16}\\ &=C e^{-13\kappa T/32}+Q_Te^{-\kappa T}. \tag{139}\end{align*}\] Both exponents have a strict gain over \(-\kappa/4\). By (128) and boundedness of \(G\), this controls \(D(X,X)\) for each \(X\in\mathcal X\). Curvature pair symmetry gives \(D_{12}=D_{21}\), and \[2D_{12}=D(\partial_{\theta^1}+\partial_{\theta^2}, \partial_{\theta^1}+\partial_{\theta^2}) -D_{11}-D_{22}.\] These three estimates prove (131) for the entire symmetric \(2\times2\) block. ◻

Proof. Suppose instead that a nonempty relative open set \(V\subset\mathcal U_{\varepsilon}\) is contained in \(\overline{E(O,B)}\). Since \(E(O,B)\) is dense in \(V\), choose \(d\in E(O,B)\cap V\). The weighted seminorms are increasing, so there are a finite \(m\ge10\) and \(\eta>0\) such that \[ \{d'\in\mathcal U_{\varepsilon}:p_m(d'-d)<\eta\}\subset V. \tag{140}\] Fix the branch tube and the extension witness belonging to this datum. Apply the finite packet construction at order \(m\), along its unbounded sequence of admissible observation times. By [prop:packet-preparation,thm:dn-comparison,prop:experiment-attachment], the exact full-bridge data \(d_T\) satisfy \(p_m(d_T-d)\to0\); hence they lie in \(V\) for all sufficiently large \(T\).

The set \(E(O,B)\) is dense in \(V\), so 20 allows us, for every such fixed \(T\), to choose \(\widehat d_T\in E(O,B)\cap V\) as close to \(d_T\) as required by all the finite observation conditions. We also require \(p_0(\widehat d_T-d_T)\to0\). Write \(\widehat g_T\) for its observed metric. We then have, on the common position tube, \[ |D(\widehat g_T)-D(g_d)|>A_T \quad\text{if }|\cos(\lambda v)|\ge1/2, \tag{141}\] and both metrics have the uniform hypotheses of 23. In particular the constants there are independent of \(T\); the required finite closeness was chosen to retain them. Smooth dependence of the initial section density on \(h\) gives \[ \frac{\mu_{\widehat d_T}(O)}{\mu_d(O)}\longrightarrow1. \tag{142}\]

Let \(S\subset O\) and \(\widehat S_T\subset O\) be the good seed families in the two existential extension witnesses. Each has more than \(99\) percent of its respective measure of \(O\). Use (121) for the background hits of \(O_0\). At a fixed \(v\), the loss from parameters whose background seed is outside \(S\) is at most \(.01\mu_d(O)\). The matched backward map for \(\widehat g_T\) is injective into \(O\) and preserves its canonical section measure. Its loss from \(O\setminus\widehat S_T\) is therefore at most \(.01\mu_{\widehat d_T}(O)\). By (142), their combined loss is less than \(.04\mu_d(O)\) for large \(T\), uniformly on every \(v\) slice. Thus, if \(\mathcal G_T^{\cap}\) denotes parameters good for both metrics, \[ \operatorname{vol}_6(\mathcal G_T^{\cap}\cap\{v=\text{constant}\}) >(.95-.04)\mu_d(O)=.91\mu_d(O). \tag{143}\] The dependence on \(v\) is measurable by the smooth finite flow maps. The chart bounds from either witness apply at these observations because 11 imposes them throughout the full future geodesic lifetime, and the matched observations are interior points of that geodesic.

Let \[I_T=\{v\in T+J_0:|\cos(\lambda v)|\ge1/2\}.\] The condition occupies \(2/3\) of each full period. Only the two end periods of an interval can be incomplete, so \(|I_T|=\frac23|J_0|+O(\lambda^{-1})>\frac12|J_0|\) for large \(T\). Integration of (143) over \(I_T\) gives \[ \nu_T(\mathcal G_T^{\cap}\cap\{v\in I_T\}) >.455\mu_d(O)|J_0|. \tag{144}\] Apply 23 separately to \(g_d\) and \(\widehat g_T\). The union of the two additional exceptional sets has measure tending to zero. It cannot cover (144). At a remaining parameter both \(|D(g_d)|\) and \(|D(\widehat g_T)|\) are \(o(A_T)\), uniformly, whereas (141) says their difference exceeds \(A_T\). This is a contradiction.

We have excluded nonempty interior of \(\overline{E(O,B)}\). No closedness of \(E(O,B)\) was assumed. ◻

Proof of 2. By 12, every datum in \(\mathcal U_{\varepsilon}\) admitting the future extension in the statement belongs to some \(E(O,B)\). The seed basis and the positive integers \(B\) are countable. By 24, their union is meagre in the full relative weighted smooth topology on \(\mathcal U_{\varepsilon}\). Explicitly, \[\mathcal R=\bigcap_{O,B}\bigl(\mathcal U_{\varepsilon}\setminus\overline{E(O,B)}\bigr)\] is a countable intersection of relatively open dense sets and consists of future-inextendible data. The exit cover concerns all future exits of the full developments. Every background observation precedes the corresponding geodesic’s first comparison runout. The exact and persisted observations lie in finite interior regions of their full developments, where the full-lifetime chart witnesses apply. Thus the conclusion excludes every extension specified in 2, for this single \(p_{10}\) neighborhood. ◻