This is the first and main paper of a two-part series, in which we prove the $C^{2}$-formulation of the strong cosmic censorship conjecture for the Einstein--Maxwell-(real)-scalar-field system in spherical symmetry for two-ended asymptotically flat data. For this model, it is known through the works of Dafermos and Dafermos--Rodnianski that the maximal globally hyperbolic future development of any admissible two-ended asymptotically flat Cauchy initial data set possesses a non-empty Cauchy horizon, across which the spacetime is $C^{0}$-future-extendible. (In particular, the $C^{0}$-formulation of the strong cosmic censorship conjecture is false.) Nevertheless, the main conclusion of the present series of papers is that for a generic (in the sense of being open and dense relative to appropriate topologies) class of such data, the spacetime is future-inextendible with a Lorentzian metric of higher regularity (specifically, $C^{2}$).

In this paper, we prove that the solution is $C^{2}$-future-inextendible under the condition that the scalar field obeys an $L^{2}$-averaged polynomial lower bound along each of the event horizons. This, in particular, improves upon a previous result of Dafermos, which required instead a pointwise lower bound. Key to the proof are appropriate stability and instability results in the interior of the black hole region, whose proofs are in turn based on ideas from the work of Dafermos--Luk on the stability of the Kerr Cauchy horizon (without symmetry) and from our previous paper on linear instability of the Reissner--Nordstr\"om Cauchy horizon. In the second paper of the series, which concerns analysis in the exterior of the black hole region, we show that the $L^2$-averaged polynomial lower bound needed for the instability result indeed holds for a generic class of admissible two-ended asymptotically flat Cauchy initial data.

Assuming four strongly compact cardinals, it is consistent that all entries in Cichoń's diagram (apart from $\mathrm{add}(\mathcal{M})$ and $\mathrm{cof}(\mathcal{M})$, whose values are determined by the others) are pairwise different; more specifically, $\aleph_1 \lt \mathrm{add}(\mathcal{N}) \lt \mathrm{cov}(\mathcal{N}) \lt \mathfrak{b} \lt \mathrm{non}(\mathcal{M}) \lt \mathrm{cov}(\mathcal{M})\lt \mathfrak{d} \lt \mathrm{non}(\mathcal{N}) \lt \mathrm{cof}(\mathcal{N}) \lt 2^{\aleph_0}$.

We introduce the notion of a complex cell, a complexification of the cells/cylinders used in real tame geometry. For $\delta\in(0,1)$ and a complex cell $\mathscr{C}$, we define its holomorphic extension $\mathscr{C} \subset \mathscr{C}^\delta$, which is again a complex cell. The hyperbolic geometry of $\mathscr{C}$ within $\mathscr{C}^\delta$ provides the class of complex cells with a rich geometric function theory absent in the real case. We use this to prove a complex analog of the cellular decomposition theorem of real tame geometry. In the algebraic case we show that the complexity of such decompositions depends polynomially on the degrees of the equations involved.

Using this theory, we refine the Yomdin-Gromov algebraic lemma on $C^r$-smooth parametrizations of semialgebraic sets: we show that the number of $C^r$ charts can be taken to be polynomial in the smoothness order $r$ and in the complexity of the set. The algebraic lemma was initially invented in the work of Yomdin and Gromov to produce estimates for the topological entropy of $C^\infty$ maps. For analytic maps our refined version, combined with work of Burguet, Liao and Yang, establishes an optimal refinement of these estimates in the form of tight bounds on the tail entropy and volume growth. This settles a conjecture of Yomdin who proved the same result in dimension two in 1991. A self-contained proof of these estimates using the refined algebraic lemma is given in an appendix by Yomdin.

The algebraic lemma has more recently been used in the study of rational points on algebraic and transcendental varieties. We use the theory of complex cells in these two directions. In the algebraic context we refine a result of Heath-Brown on interpolating rational points in algebraic varieties. In the transcendental context we prove an interpolation result for (unrestricted) logarithmic images of subanalytic sets.

We prove global well-posedness of the Korteweg--de Vries equation for initial data in the space $H^{-1}(\mathbb{R})$. This is sharp in the class of $H^s(\mathbb{R})$spaces. Even local well-posedness was previously unknown for $s\lt -3/4$. The proof is based on the introduction of a new method of general applicability for the study of low-regularity well-posedness for integrable PDE, informed by the existence of commuting flows. In particular, as we will show, completely parallel arguments give a new proof of global well-posedness for KdV with periodic $H^{-1}$ data, shown previously by Kappeler and Topalov, as well as global well-posedness for the fifth order KdV equation in $L^2(\mathbb{R})$.

Additionally, we give a new proof of the a priori local smoothing bound of Buckmaster and Koch for KdV on the line. Moreover, we upgrade this estimate to show that convergence of initial data in $H^{-1}(\mathbb{R})$ guarantees convergence of the resulting solutions in $L^2_{\mathrm{loc}}(\mathbb{R}\times \mathbb{R})$. Thus, solutions with $H^{-1}(\mathbb{R})$ initial data are distributional solutions.

A countable discrete group $G$ is called Choquet-Deny if for every non-degenerate probability measure $\mu$ on $G$, it holds that all bounded $\mu$-harmonic functions are constant. We show that a finitely generated group $G$ is Choquet-Deny if and only if it is virtually nilpotent. For general countable discrete groups, we show that $G$ is Choquet-Deny if and only if none of its quotients has the infinite conjugacy class property. Moreover, when $G$ is not Choquet-Deny, then this is witnessed by a symmetric, finite entropy, non-degenerate measure.

In all dimensions, we prove that the marked length spectrum of a Riemannian manifold $(M,g)$ with Anosov geodesic flow and non-positive curvature locally determines the metric in the sense that two close enough metrics with the same marked length spectrum are isometric. In addition, we provide a new stability estimate quantifying how the marked length spectrum controls the distance between the isometry classes of metrics. In dimension 2 we obtain similar results for general metrics with Anosov geodesic flows. We also solve locally a rigidity conjecture of Croke relating volume and marked length spectrum for the same category of metrics. Finally, by a compactness argument, we show that the set of negatively curved metrics (up to isometry) with the same marked length spectrum and with curvature in a bounded set of $C^\infty$ is finite.