Linearized Horizon Stability & Lyapunov–Schmidt Reduction

Statement. On a quasilocal Cauchy slice Σ_t, the dynamic stability of a Marginally Outer Trapped Surface (MOTS) is governed by a linearized stability operator L_Σ. At the critical threshold of common horizon formation, L_Σ loses invertibility. By the Fredholm alternative for elliptic operators of index zero, its kernel becomes strictly one-dimensional, spanned by a single positive principal zero-mode H₀. A one-dimensional Lyapunov–Schmidt reduction then projects the infinite-dimensional surface variation problem onto that kernel, yielding a single scalar bifurcation equation. Every value carries Derived status.

This page is the technical foundation for Theorem I and Theorem II. The slice-invariant balance and the universal square-root fold trajectory both rest on the kernel structure derived here.

1 · Marginally Outer Trapped Surfaces

Global event horizons are teleological: their definition requires complete knowledge of the future causal structure of the universe, and they cannot be tracked in real time. On a local Cauchy slice, they are replaced by quasilocal Marginally Outer Trapped Surfaces (MOTS).

A MOTS is a compact two-dimensional spacelike surface Σ ⊂ Σ_t whose outgoing null expansion vanishes identically:

\[ \Theta_+ = 0. \]

Equivalently, the surface is marginally trapped in the outward direction: the light rays emerging from Σ along the outgoing null normal have zero divergence. MOTS give a well-defined, slice-local notion of dynamical horizon that can be tracked forward in time without knowledge of the future.

2 · Linearized stability operator

Let v ∈ C^2,α(Σ) be a scalar function parameterizing a normal variation of the MOTS surface. The linear response of the outgoing null expansion Θ₊ to this variation is governed by the linearized stability operator L_Σ:

\[ L_\Sigma\, v = -\Delta v \;+\; 2 s^a d_a v \;+\; \left(\tfrac{1}{2} R_\Sigma - |\sigma|^2 - 8\pi T_{ab} k^a l^b - d_a s^a + |s|^2\right) v. \]

The geometric and tensor components of L_Σ are the standard quantities of the null geometry on Σ:

The operator L_Σ is a second-order elliptic operator of index zero acting on functions on a compact surface. Its spectral theory is well developed; in particular, the Fredholm alternative applies directly.

3 · Critical threshold and kernel

At the critical threshold of common horizon formation or surface bifurcation, t = t_*, the principal eigenvalue λ₀ of L_Σ vanishes:

\[ \lambda_0 = 0. \]

At this exact transition point, L_Σ loses invertibility under the Fredholm alternative, signaling a topological fold bifurcation.

By the Fredholm alternative for elliptic operators of index zero on a compact manifold, the kernel of L_Σ becomes strictly one-dimensional:

\[ \mathrm{Ker}(L_\Sigma) = \mathrm{span}\{H_0\}, \qquad H_0 > 0. \]

The single generator H₀ is the principal zero-mode of the stability operator. It is real, strictly positive on Σ, and unique up to scale. The positivity follows from the maximum principle for elliptic operators; the uniqueness from the index-zero condition.

The one-dimensionality of the kernel is the essential structural fact that the framework exploits. All of the framework's dynamic conservation laws reduce to the single channel H₀.

4 · Orthogonal decomposition

The Lyapunov–Schmidt reduction decomposes the function space into the kernel and its orthogonal complement:

\[ V \;=\; C^{2,\alpha}(\Sigma) \;=\; \mathrm{Ker}(L_\Sigma) \,\oplus\, \mathrm{Range}(L_\Sigma). \]

Two orthogonal projection operators act on V:

• P: V → Ker(L_Σ) — the projection onto the zero-mode kernel.
• Q = I − P: V → Range(L_Σ) — the complementary projection onto the range.

Every surface perturbation δH ∈ V therefore decomposes uniquely as

\[ \delta H = c\, H_0 + w, \qquad c \in \mathbb{R}, \quad w \in \mathrm{Range}(L_\Sigma). \]

The scalar c parametrizes the kernel component; the function w is the range component. Because L_Σ is invertible on Range(L_Σ) (its restriction has no vanishing eigenvalues), the range component w is determined by the kernel component c through the reduced equation. It does not evolve independently.

5 · The 1D reduced bifurcation equation

Substituting the decomposition δH = c H₀ + w into the full nonlinear surface variation problem and projecting onto the kernel gives a single scalar equation for the kernel coefficient c:

\[ G(c, \delta t) = 0. \]

Taylor expansion of G around the critical point (c, \delta t) = (0, 0) — with δt ≡ t − t_* — yields the normal form of a saddle-node bifurcation. Solving for the leading behaviour gives the surface variation map

\[ \delta H_\pm(\delta t) = \pm B\,(\delta t)^{1/2} H_0 + \delta t\, H_d + O\!\left((\delta t)^{3/2}\right), \]

where:

• B — the geometric scaling coefficient, determined by the second variation of the null expansion Θ₊ at the critical point.
• H_d ∈ Range(L_Σ) — the higher-order linear drift. It lies entirely in the range of L_Σ, so ∫_Σ H₀ H_d dA = 0 by orthogonality.
• The ± sign split distinguishes the two topological branches: + for the outer (expanding) boundary S₊, − for the inner (trapped) boundary S₋.

The (δt)^1/2 leading behaviour is the universal signature of a saddle-node fold. It does not depend on which scalar functional is tracked; it is forced by the structure of the reduced equation.

6 · Consequences

The 1D reduction has three immediate consequences for the framework:

• Single-channel dynamics. All internal relational phase shifts evolve along the 1D principal kernel span{H₀}. No other channel carries leading-order current.
• Universal square-root trajectory. For any quasilocal horizon scalar functional Q(t), the fold trajectory has the form Q_±(t) − Q_* = ±A_Q(t − t_*)^1/2 + B_Q(t − t_*) + O((t − t_*)^3/2). This is the foundation of Theorem II.
• Dual-branch balance. The kernel H₀ enters the two branches with opposite algebraic signs, so that normal variations of the outer surface are balanced in magnitude and opposite in sign by inner surface deformations. This is the foundation of Theorem I.

7 · Closure at R = 1.0000

The reduction is a structural identity, not a numerical procedure. It holds on every Cauchy slice, at every critical point of a MOTS bifurcation. The kernel dimension, the orthogonal decomposition, and the square-root fold normal form are the same algebraic facts on every realization. Combined with the framework's conservation theorems:

\[ I(S_+) + I(S_-) = I_{\text{total}}, \qquad R \equiv \frac{I(S_+) + I(S_-)}{I_{\text{total}}} = 1.0000. \]

The reduction is the technical bridge between the geometric partition of the framework and the conservation theorems that rest on it.

Cross-references

• Slice-invariant conservation (Theorem I). The dual-branch balance that follows from the kernel structure is in universal slice invariance.
• Zero-mode emission (Theorem II). The universal square-root trajectory that follows from the reduced bifurcation equation is in asymptotic zero-mode transduction.
• Zero-mode power law. The full three-term emission expansion is in zero-mode power law.
• Register. The kernel structure and the reduction results used here are indexed in register §02 · Conservation.
• Manual. The operational treatment in full — including the null geometric background and the geometric coefficients — is in manual 03 · Pure mathematical engine.
• Section. The reduction of which this page is the technical foundation is collected in Classical Maths.

Attribution. The null-geometric stability operator for MOTS is the standard construction of contemporary black-hole perturbation theory. The Fredholm alternative for elliptic operators is attributed to Fredholm (1903). The Lyapunov–Schmidt reduction is attributed to Lyapunov and Schmidt, developed in the early twentieth century and standard in bifurcation theory. The saddle-node fold normal form is classical (Poincaré). The framework contributes the arrangement of these results into the 1D zero-mode reduction and its use as the structural foundation of the framework's conservation theorems.