Alphabeta MathMaximal slices, causal cuts and stability
Assemble the analytic and geometric inputs for maximal-slice continuation, causal completeness, null-boundary cuts, area comparison and horizon flux rigidity. Use the common data and the preceding field inputs while retaining every local argument and imported-premise qualification.
1 · Prerequisites
- Spacelike sections, maximality and lapse–shift data
- Causal exterior, chronological boundaries and Cauchy data
- The Axiom of Choice
- The inverse metric, musical isomorphisms and index raising
- The Riemannian gradient is the metric dual of the differential
- Riemannian divergence
- Laplace–Beltrami operator as the trace of the Hessian
- A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism
- Formal adjoint of a real drift Laplacian on a closed manifold
- Positive right and adjoint eigenfunctions have the same eigenvalue
- A smooth function with zero differential is constant on each connected component
- Eigenvalues, eigenvectors, eigenspaces E_λ(T)=ker(T-λ I), and the spectrum σ_F(T) of an endomorphism
- Orientation-free density integration and its properties
- Smooth maps are continuous
- General spacelike two-surfaces, null gauges and fluxes
- Normal deformation families and an outgoing null gauge
- The Axiom of Countable Choice (AC_ω)
- Riemannian metric and riemannian manifold
- Gradient hessian and divergence connection formulas
- For a topological space the following agree: no separation exists, the only clopen subsets are ∅ and X, and every continuous map to the two-point discrete space is constant
- Heine-Borel in ℝⁿ: with the Euclidean metric a subset of ℝⁿ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line
- A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value
- A variable-coefficient annular barrier and inward boundary derivative
- Weighted exterior elliptic domains and admissible weights
- Collar neighborhood theorem
- Closed real intervals are compact by completeness
- A product of finitely many compact spaces is compact in the product topology
- Mean value theorem on a closed real interval
- Continuous functions have an oriented Riemann integral
- Dedekind-complete ordered real field
- Finite Harnack chains on fixed compact connected interior sets
- Sum, product, quotient, and chain rules from difference quotients
2 · Logical flowchart
Arrows show item dependencies. A grouped prerequisite node points to items that use one or more of its members; their exact links are listed above.
3 · Items
Definition
Temporal normal angle and compact positive-height portions
Lemma
Ground-state identity, spectral ordering and eigenspace uniqueness
Lemma
Nonnegative supersolutions detect the principal spectral sign
Lemma
Outgoing expansion variation at a general MOTS
Lemma
Strong minimum principle from the annular barrier
Lemma
Hopf positivity and uniform linear growth on a fixed compact collar
Lemma