Generic Future Inextendibility with Square-Integrable Connection Near a Fixed Kerr Spacetime

Proof. These are [23, Proposition 7.4, Proposition 7.5, Proposition 7.6], with cylinder data supplied by Proposition  7.3 and exterior closure supplied by Proposition  6.4 there; see also [22, Proposition 4.2]. The low characteristic estimates give (2.5) and summable cell suprema. Proposition  7.5 includes the same-direction derivatives, which are needed in (2.6). 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. Proposition  7.6 provides (2.7) at every fixed order. The last statement follows by differentiating a=q\mathcal A and the relative shift identity in (2.3). None of these results assumes a bound for parallel-frame curvature on an observing geodesic.□

Proof. The full-lifetime classification is [23, Corollary 8.6]. Corner removal is [22, Proposition 3.10]; its section-volume argument applies to any smooth positive measure on T\Sigma. The geometric estimate used there is [23, Lemma 8.7]. Its hypotheses are the low interior bounds of Convention 2.2 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 [22, Lemma 4.3, Lemma 7.3]. Let \Xi=\dot\theta-b\dot u and P_A=\gamma_{AB}\Xi^B. Unit normalization gives

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\dd u<\infty and \int a\,\dd v<\infty. By (2.12),

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

Its angular Hamilton equation and the low angular coefficient bounds give

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 (2.10) has a uniform state-Lipschitz bound on a bounded momentum set. On this neighborhood, every fixed state derivative of (2.13) is bounded by Ce^{-\kappa v}e^{o(v)}. This follows from Convention 2.2, 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

Here \int_{-v}^{u}q\,\dd s\le q_0/\kappa and (2.4) 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\,\dd s\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 [22, Section 4.5, Corollary 4.8], using [23, Proposition 9.6, Proposition 9.7], 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 (2.11). Future causal vectors in (2.2) have \dd u,\dd v\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

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 (2.11).□

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, \dd s 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

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

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. Convention 3.1 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

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 Convention 3.1. 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

For each (s,y), the set of possible centers z has volume at most C_Kl^3. Tonelli’s theorem therefore gives

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 Convention 3.2. 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

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 Convention 3.3. For one of its boxes, write z_c for the center and set

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\R^3, |\eta|<\varepsilon_0:

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

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

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,

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

Their divergence in (z,\pi) is zero, so

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

The joint open domain and the smooth interior flow make this a measurable nonnegative function, possibly infinite. Using Equation (3.6), Tonelli’s theorem, and the fixed bound on the momentum fibers gives

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 Equation (3.7) 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

Consequently |\dd\eta/\dd s|\le C_2|\Gamma_M| before the stop. By Equations (3.5) and (3.7), 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

Here |\Delta z|\le(\rho_0+2\varepsilon_0)l. By Equation (3.3), 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 Convention 3.1.

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

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=\dd z/\dd s are bounded. The time component of the geodesic equation gives

Its right side is bounded and s has a finite endpoint. Thus \dd s/\dd\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 Equation (3.6). 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 Equation (3.9) 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 Equation (3.2) from proper time zero onward.□

Proof. Fix such a datum and extension. Convention 3.4 provides a measurable family S_* of positive \mu_d measure, finite full lifetimes, and one finite set of ambient chart witnesses. By Convention 2.3, 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 [23, Corollary 8.6] and the section-volume corner removal of [22, Proposition 3.10]; the latter uses only the low-order corner estimates of [23, Lemma 8.7]. 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 Convention 2.4 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 Convention 3.5(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 Convention 3.5(i).□

Proof. The inverse metric is

Linearization of g^{-1}(\dd u,\dd u)=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}(\dd u,\dd\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 (4.14), including its sign and factor.

To see the gain in (4.15), a typical term in the first equation has the form \amp e^{i\lambda v}f(u,v,\theta) with fixed coefficient jets e^{o(T)}. At fixed u,\theta,

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 (4.17) applies. The term a\gamma^{-1}\dd_\theta z^0 already has the inverse-frequency gain. Integrating its length costs only e^{o(T)}, also at differentiated orders. Finally,

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 (4.15) for the angles as well.

For explicit field formulas set W=z^\theta-z^0b, an angular vector field. A direct Lie derivative gives

Indeed (\Lie_Zg)_{AB}=2z^0\chi_0+2z^1\chi_1+\Lie_W\gamma, whereas (\Lie_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 (4.14). Hence k_H=-k(L_0,L_1)/a and k_b=-\gamma^{-1}k_{0\cdot}, which give (4.18). 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=\dd_\theta k_H. The linearized vacuum identities are available with arbitrary linear accuracy: diffeomorphism covariance on the vacuum background gives

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 (4.1) these powers are e^{C_sT} for a fixed C_s. After \zeta is fixed, increasing M in \amp\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 [23, Equation ( 7.12)]. Its linearization is

Here

and

The operator variation in the last term uses only k_\gamma and one angular derivative of k_\gamma. Thus the entire source in (4.19) has already been estimated using (4.18); it uses neither X_i nor an unknown relative shift derivative. Also

At the moving entry edge q=q_0, the initial X_0 is a finite-jet expression with no small denominator. It has size \amp\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 (4.19). This proves the finite-order assertion without a shifted transport containing an unknown k_b.

Finally linearize the defining shift relation. It gives exactly

The background relative derivative and all its fixed jets are subexponential. This proves (4.16). 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 [23, Proposition 7.6, proof], 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

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

The same argument controls the tail supremum. Together with the fixed summable g_i, this supplies all tail thresholds uniformly. The Lipschitz potential

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 \Sph 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

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

Inverse metrics and relative lapse factors through order M+1 are bounded by C_M\mathsf P: differentiate inversion tamely and use \dd\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

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 \divg S_i=\beta_i+\tfrac12\dd\tau_i, and the one-form data are \divg\xi_j=K_\gamma-m_i and \curl\xi_j=c_i.

Commute Equation (5.7) 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+\Lie_{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

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

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

This uses \nabla J=0 and the same derivative-slot contraction on both sides. Since \dd a=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 Equation (5.15) 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

Tame products, the low bounds for \dd Q_i, and |\beta_i|_{C^1}\le C h_i give, for r\le M,

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 \dd\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 \Lie_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

by Equation (5.14). These observations and Leibniz placement give, apart from favorable damping, paired squared-current costs at most C_M times

Their transport directions are j and i, respectively. The remaining currents, in directions i, 1, 1, and i, have costs

For the first line of Equation (5.18), 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 Equation (5.14) 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 Equations (5.17) and (5.18), after substitution of Equation (5.14), are

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

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

Uniform tail smallness from Equation (5.11) 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,

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

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 Equation (5.1) have the same outgoing signs. Since u\le u_2 and v\le C(T+1),

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

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 Equation (5.14) give

The homogeneous C_Mh_i factor has bounded directional integral. To see explicitly how the trace enters unweighted transport, fix y_j and integrate Equation (5.21) in its terminal variable. Reversing the integration triangle gives

The first transverse flux in Equation (5.20) 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

The norm inequalities for metric and torsion transport are

Substitution of Equation (5.22) and the transverse flux therefore gives

Multiply by e^{-C\Phi_T(y)} and take a supremum. Along either path,

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)}. Equation (5.22) 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

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 Equations (5.20) and (5.22). 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 Equation (5.20) 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

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

For k\ge1, Raychaudhuri, torsion transport, and angular curvature variation give the reductions

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

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 \divg\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 Equation (5.25) gives

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 Equation (5.25) 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 Equation (5.26) with D in increasing angular order. For r\ge4 the tame product and shift commutator inequalities give

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 Equation (5.24). For a mixed shape or lapse word, commute a cross derivative inward; the commutator in Equation (5.25) 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-\Lie_b on tensors, or \partial_u=L_0-b\cdot\dd 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

Thus |C|_0\le C|\Delta\gamma|_1. For a barred one-form f,

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

For the trace conversion, use the exact identity

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

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. Equation (5.31) and Equation (5.33) yield

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,

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:

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.

The paired currents. Writing k=\Delta K_\gamma and c=\Delta c_i, the primed principal part of the difference system is

There are no angular derivatives in the last principal equation. Use the currents with densities

The ratio of scalar to one-form weight is exactly a'/2. Since \dd and J'\dd have adjoints -\divg' and -\curl', their principal contribution, integrated on the full sphere, is

This has size C\sigma Y_iZ_i, because \dd a'=a'(\xi_0'+\xi_1') and a'\asymp q. The derivative of the scalar weight contains L_i'a'=a'\ell_i'=a'(-\kappa+O(h_i)), whose constant part is favorable damping. The area and tensor-norm derivatives contain only low shapes and shifts. They cost C(h_jY_i^2+h_iZ_i^2). Indeed, after the angular transport divergence is integrated out, their exact contributions to the two densities are

Here are all nonprincipal placements in F_i,G_i,G_i^m. The change of the angular principal coefficient on the barred field is

It costs CqP, since \Delta a=q\Delta\mathcal A and J'-\bar J is an algebraic metric difference. The divergence and curl coefficient differences against \bar\beta_i cost Ch_iP and therefore C\sigma h_iP after scalar weighting. Shift differences cost Ch_iD_b in the first equation and C\sigma D_b in the other two. The remaining a terms cost at most C\sigma(Z_0+Z_1+P) in the first equation. Indeed they consist of a'\Delta K, a'\nabla'\Delta\xi, bounded low factors times a'\Delta\xi, and coefficient differences on bounded barred factors. The shape products have bound

In particular the angular connection difference in \chi_i*\nabla\chi_j has the explicit factor \chi_i'*C*\bar\chi_j and costs Ch_ih_jP. In the scalar equations, the weighted remainder has bound

These lists exhaust the factors of Equation (5.7). Put Z=Z_0+Z_1. In particular all uses of the apparently dangerous second estimate in Equation (5.34) occur with one of the factors

There is no naked \sigma^{-1}Z in a current source. The combined remaining paired cost is consequently bounded by

apart from the defects paired with the current variables.

Trace, shape, lapse, and shift currents. Subtract Equation (5.6) with the actual L_i':

The defect \mathcal F_i^w has the input bound. Its one angular commutation has cost at most Ch_i(z_i+R_i+D_b+h_iP), in addition to the homogeneous damping: the differentiated quadratic shape has a low h_i factor, \Delta\mathcal A is controlled in H^1 by Equation (5.32), and the shift term uses \bar w_i through two low angular derivatives, all O(h_i). The commutator with L_i' uses only the low first angular derivative of B_i'. Thus the remaining z_i^2 current cost is CH_sz_i(V+R_i); the constant 2\kappa z_i^2 has favorable sign. In particular no \Delta\ell_i term is generated by this normalized trace equation.

For N_i use the shape and lapse cross equations in direction j. Their source factors are precisely

Their differences are bounded by C\{\sigma(Z_0+Z_1)+qP+h_iN_j+h_jN_i+h_ih_jP+h_iD_b\}. For example, using the connection difference in Equation (5.30),

This uses Equation (5.40) and Equation (5.35); the constant part of \ell_i does not contribute. The squared N_i currents therefore cost at most CH_sN_i(V+R_0+R_1). In particular there is no unsuppressed return from a lapse or metric error to N_i.

For the H^1 shift current in direction 1, subtract its exact equation:

One angular derivative uses Equation (5.32) and Equation (5.40), giving a source C(\sigma(Z_0+Z_1)+qP) and a squared-current cost CH_sD_bV. No differentiated shift is needed on its right side.

The unweighted recovery currents. Use direction 1 for \Delta\gamma in H^1 and \Delta H in L^2, where H=\log\mathcal A:

For comparison, an arbitrary-direction equation would be L_i'\Delta H=\Delta\ell_i-\Delta B_i\cdot\dd\bar H; the chosen direction has \Delta B_1=0. For \Delta\xi_j use direction i and

In its undifferentiated norm the trace difference costs C(N_i+h_iP) directly from the definition of \tau_i. The last Lie derivative involves bounded barred torsion and its first angular derivative, giving CD_b. Use fixed-background angular norms for these recovery currents to avoid any unnecessary time differentiation of their norm coefficients. Their total cost, including harmless low transport commutators, is

All occurrences of unweighted first angular torsion derivatives have already been confined to Equation (5.40).

The growth rate and the late support. Substitute Equation (5.34) in the preceding costs. At H_s=0, all currents except the P currents have zero unfavorable cost. The surviving directed interactions are

There is no reverse arrow. Give the component P currents a small fixed multiplier \delta_P^2 and every other current fixed positive multipliers, equal on each Bianchi pair. Let \mathcal E_T(t) be the resulting spatial slice energy. Its density is uniformly equivalent, with constants depending on \delta_P but not T, to

In particular it controls all the unweighted quantities claimed in Equation (5.28). The factors 1/2 from the time components of L_i' merely change this equivalence by constants.

Given a small rate \rho_*>0, first take \delta_P small so that the form in Equation (5.46), expressed in scaled variables, costs at most \rho_*\mathcal E_T/2. Next choose a fixed h_*>0 so that every term with factor H_s costs at most \rho_*\mathcal E_T/2 when H_s\le h_*. On the remaining bounded low coefficient set the rate is bounded by a constant C_* independent of T. This constant is fixed before \nu is chosen.

All difference fields vanish in v<(1-\nu)T by protected agreement, causal monotonicity and uniqueness of the optical transports. Fix a large R so that the background envelope suprema are small outside [-R,R], and a fixed t_0 so that both \sigma and q=\sigma^2 are below their required thresholds for t>t_0. For large T, h_1 is small on the support for t>t_0. At those times, the only possible failure of H_s\le h_* is where u\ge-R. At any such supported point,

Consequently the slice times containing a bad supported point lie in

after omitting any empty interval. Their total length is at most

The added e_T in the majorants is uniformly below the threshold for large T. Thus the integrated homogeneous rate is bounded by C\rho_*T+C_*\nu T/2+O(1), and is an arbitrarily small multiple of T after choosing \rho_* and then \nu.

Both optical directions have nonnegative outgoing flux on the tilted sides: the derivatives of u+c_0t/T and v+c_0t/T along L_0',L_1' are nonnegative, and are strictly positive where needed. There are no angular boundary terms because the sphere is complete. The defects have magnitude F_T on a spatial interval of length O(T) and therefore add at most C\operatorname{poly}(T)F_T\sqrt{\mathcal E_T} to the squared energy inequality. The initial discrepancy has the same bound after the finite spatial integration. The square-root inequality (or its regularization at zero) and Gronwall yield

Choose the several positive losses strictly below \eta_2 to absorb the fixed and polynomial factors. This proves the Proposition.□

Proof of Convention 5.1. 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,

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 Convention 5.2 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 Equation (5.50) retains a strict gap after adding all interpolation and later time-recovery losses. In Convention 5.3 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 Convention 4.1 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, Equation (5.28) and Convention 5.2 give, for each component error f,

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

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 Equation (5.5):

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+\Lie_b; for a scalar replace the Lie derivative by b\cdot\dd. 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 Convention 5.2; 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 Equation (5.2) with c<c_1. The relative shift conclusion follows by differentiating the algebraic identity

The template change at the fixed bootstrap orders is at most e^{-(\alpha-\eta_0)T} with \eta_0<\alpha/2. Together with Equation (5.2), this is strictly smaller than \tfrac12 e^{-\alpha T/2} for large T. It improves Equation (5.8), 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 [23, Proposition 7.7, proof] 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 \dd u,\dd v\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 Equation (5.1) 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 Convention 6.1. This proves the Theorem.□

Proof. The lower and exterior joins are the qualitative construction in [22, Section 7.1, Remark 7.2], based on [23, Lemma 8.1, Proposition 8.2, Proposition 9.10]. 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 [23, Lemma 8.1] 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 (4.7) contains a fixed-origin ball of radius

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 Convention 4.1, computed from these same partial-slice data, using the shrinking outer cutoff (4.10). 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 [22, Section 7.1, Remark 7.2]. 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

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|\dd r_{\rm co}/\dd t_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 Convention 5.1; 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,

Thus \dd u(V) and \dd v(V) have the same weak sign, and the future choice is the nonnegative sign. Also g_T^{-1}(\dd t,\dd t)=-1/(2a_T)<0; hence t is temporal. For a nonzero future causal vector, \dd t(V)>0, and

The strict side and terminal sublevels of (4.1) 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\dd u/\dd t,\dd v/\dd t\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; (6.1) 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 [23, Proposition 8.2]; 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

Since [\partial_v,\partial_A]=0, applying the curvature commutator and pairing with \partial_B gives

Use \partial_v\gamma=2\chi_1 to lower the differentiated index. This gives the exact formula

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 (6.5) is therefore

The angular Lie term in (4.18) contains only Z and angular derivatives of Z, without a longitudinal derivative of Z. Hence (4.15) gives

The formula for k_H has at most one longitudinal derivative of Z, so \partial_vk_H=O(\amp\lambda e^{o(T)}) as well. All other linear terms in (6.6) have this bound. Twice differentiating the real leading oscillation in (4.3), whose trace-free leading tensor is unchanged by trace reversal, yields

The last term comprises quadratic template products in the smooth relative-field expression (6.5). Choose \zeta small enough that this term is o(\amp). The exact-minus-template error from Convention 5.1, at the finite derivative orders in (6.5), is also o(\amp); multiplication by the individual e^{o(T)} coefficient bounds preserves this conclusion.

By (4.4), on the cosine-good set the leading norm is at least \tfrac14\amp\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 \amp also tends to infinity. This proves (6.4). 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

Thus the reduced v-Hamiltonian is

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 Convention 2.4. This bound is integrable over the late v interval. Differentiating (6.13) 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 Convention 4.1 and Convention 5.1. 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

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 (6.10) 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

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

independently of the shell root \pi_v. Its volume is \dd u\,\dd^2\theta\,\dd\pi_u\,\dd^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 (6.10) makes the map \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

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

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 (6.12).

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 [23, Lemma 2.2], also used in the closed-test proof of [22, Lemma 3.3, proof], 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

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 (6.9). 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

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

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

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

For a constant column V and W=rV^iF_i, the connection matrix on W is therefore

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)=\Id, k times in l:

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 (7.10).

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 \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

This proves (7.11) 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 Convention 7.1, 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,

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 (7.7) give

Consequently, for each fixed s,

Here \dd\nu_T=\dd x\,\dd m; 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, (7.15) 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 (7.16) in s\in[0,1] and sum the at most B charts. If

Cauchy–Schwarz and the chart L^2 bounds imply

Set R_T=r^{-1/2}e^{\kappa T/32}. Markov’s inequality applied to I_e^2 gives

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 (7.13). 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 (7.12), |\mathsf A|\le lQ_T. The connection transformation law along the smooth interior curve is the exact matrix equation

Here \widetilde\Gamma_c(Mf) means the chart connection matrix contracted with the chart velocity Mf. Before the stop,

Summing over all four stopped edges and using I_e\le R_T yields

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 \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

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 [22, Lemma 8.5]. Write

By Convention 7.2, 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

Both exponents have a strict gain over -\kappa/4. By (7.11) and boundedness of G, this controls D(X,X) for each X\in\mathcal X. Curvature pair symmetry gives D_{12}=D_{21}, and

These three estimates prove (7.14) for the entire symmetric 2\times2 block.□

Proof. Suppose instead that a nonempty relative open set V\subset\Ueps 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

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 Conventions 4.1, 5.1 and 6.1, 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 Convention 6.3 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,

and both metrics have the uniform hypotheses of Convention 7.3. 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

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 (7.4) 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 (7.25), 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,

The dependence on v is measurable by the smooth finite flow maps. The chart bounds from either witness apply at these observations because Convention 3.5 imposes them throughout the full future geodesic lifetime, and the matched observations are interior points of that geodesic.

Let

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 (7.26) over I_T gives

Apply Convention 7.3 separately to g_d and \widehat g_T. The union of the two additional exceptional sets has measure tending to zero. It cannot cover (7.27). At a remaining parameter both |D(g_d)| and |D(\widehat g_T)| are o(\amp), uniformly, whereas (7.24) says their difference exceeds \amp. This is a contradiction.

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

Proof of Convention 1.2. By Convention 3.6, every datum in \Ueps admitting the future extension in the statement belongs to some E(O,B). The seed basis and the positive integers B are countable. By Convention 7.4, their union is meagre in the full relative weighted smooth topology on \Ueps. Explicitly,

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 Convention 1.2, for this single p_{10} neighborhood.□