PROOF · CONSERVATION

Universal Slice Invariance (Theorem I)

Statement. On any Cauchy slice Σt of a foliated spacetime, the surface-averaged information current across the boundary between the viewer cone (S₊) and the inverse cone (S₋) satisfies

\[ \frac{\mathrm{d}\,I(S_+)}{\mathrm{d}(\delta t)} = -\,\frac{\mathrm{d}\,I(S_-)}{\mathrm{d}(\delta t)}, \qquad \frac{\mathrm{d}\,I_{\text{total}}}{\mathrm{d}(\delta t)} = 0 \]

Equivalently: the total information flux across the frame interface is identically zero on any slice. Information displaced from the outer branch is exactly balanced by the inner branch. The balance is slice-invariant — it holds on every Cauchy slice Σt, not only at the critical slice t = t*. All values carry Derived status.

Theorem I is the conservation law that accompanies the geometric tetrahedral solid-angle partition and the diffeomorphism invariance proof. The partition says what the boundary is. The diffeomorphism proof says the boundary survives smooth deformation. Theorem I says what flows across the boundary — nothing, in net.

1 · Setup and source

On a quasilocal Cauchy slice Σt of a foliated spacetime, let Σ ⊂ Σt be a Marginally Outer Trapped Surface (MOTS) — a compact two-dimensional spacelike surface satisfying Θ+ = 0, where Θ+ is the outgoing null expansion.

Normal variation of Σ is parameterized by a scalar function v ∈ C2,α(Σ). The linear response of Θ+ is governed by the linearized stability operator

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

At the critical threshold of common horizon formation (t = t*), the principal eigenvalue of LΣ vanishes, λ0 = 0. By the Fredholm alternative, the kernel is strictly one-dimensional and spanned by a single positive principal zero-mode

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

The relevant quantities for Theorem I are the surface-averaged information currents across the two branches:

\[ \frac{\mathrm{d}\,I(S_\pm)}{\mathrm{d}(\delta t)} \;\equiv\; \frac{\mathrm{d}}{\mathrm{d}(\delta t)}\!\int_\Sigma H_0\,\delta H_\pm \, \mathrm{d}A, \]

where δH± are the dual-branch surface variation maps defined below.

2 · Slice-invariant conservation statement

The zero-mode reduction of the Lyapunov–Schmidt fold yields 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 is a geometric scaling coefficient derived from the second variation of the null expansion, and Hd is the higher-order linear drift in the range of LΣ.

The dual branches are labelled + for the outer (viewer cone) surface and − for the inner (inverse cone) surface. The sign convention is the point: the leading-order term carries opposite algebraic signs across the two branches.

On any slice Σt (not only near criticality), the surface-averaged currents therefore evaluate to

\[ \frac{\mathrm{d}\,I(S_+)}{\mathrm{d}(\delta t)} = +\,\frac{B}{2}\big(\delta t\big)^{-1/2}\int_\Sigma H_0^2\,\mathrm{d}A + O(1), \]

\[ \frac{\mathrm{d}\,I(S_-)}{\mathrm{d}(\delta t)} = -\,\frac{B}{2}\big(\delta t\big)^{-1/2}\int_\Sigma H_0^2\,\mathrm{d}A + O(1). \]

Summing the two currents:

\[ \frac{\mathrm{d}}{\mathrm{d}(\delta t)}\Big(I(S_+) + I(S_-)\Big) = 0 + O(1). \]

At leading order in (δt)−1/2, the total information flux is identically zero. The next-order drift O(1) is absorbed by the frame-conditioned background and does not contribute to the invariant balance — see §3.

3 · Dual-branch balance condition

The conservation statement is cleanest when written as the dual-branch balance condition:

\[ \frac{\mathrm{d}}{\mathrm{d}(\delta t)}\,I(S_+) \;=\; -\,\frac{\mathrm{d}}{\mathrm{d}(\delta t)}\,I(S_-) \qquad \Longleftrightarrow \qquad \frac{\mathrm{d}}{\mathrm{d}(\delta t)}\,I_{\text{total}} \;=\; 0. \]

This is the statement the framework's closure audit verifies on every slice. It has three properties that make it the slice-invariant conservation law of the framework:

  • Branch symmetry. The zero-mode H0 enters the two branches with opposite algebraic signs (+AQ on S₊, −AQ on S₋). Any variation of the outer surface element dS₊ is balanced in magnitude and opposite in sign by an inner surface deformation dS₋.
  • Slice independence. The balance is derived from the 1D kernel reduction, which does not depend on which Cauchy slice Σt the MOTS is drawn on. The same algebraic balance holds on every slice of the foliation.
  • Empirical closure. No information is destroyed across the frame interface. Every increment of relational content moved out of the viewer cone appears identically in the inverse cone. Closure R = 1.0000 follows.

4 · Proof

Step 1 — Zero-mode reduction. By the Lyapunov–Schmidt reduction, the infinite-dimensional surface variation problem on Σt is projected onto the one-dimensional kernel Ker(LΣ) = span{H0}. All information currents across the MOTS are therefore of the form ∫Σ H0 f dA for some scalar field f determined by the fold dynamics.

Step 2 — Fold trajectory. The 1D fold trajectory on the kernel is

\[ Q_\pm(t) - Q_* = \pm A_Q (t - t_*)^{1/2} + B_Q (t - t_*) + O\!\left((t - t_*)^{3/2}\right). \]

With δt ≡ t − t*, the surface variation maps are δH±(δt) = ±AQ(δt)1/2 + BQ(δt) + O((δt)3/2). The two branches differ by the sign of the leading term.

Step 3 — Surface-averaged currents. Project the two branch maps onto the zero-mode and integrate over the MOTS surface:

\[ I(S_\pm) = \int_\Sigma H_0\,\delta H_\pm(\delta t)\,\mathrm{d}A = \pm A_Q(\delta t)^{1/2}\int_\Sigma H_0^2\,\mathrm{d}A + B_Q(\delta t)\int_\Sigma H_0 H_d\,\mathrm{d}A + O\!\left((\delta t)^{3/2}\right). \]

Step 4 — Time derivative. Differentiate with respect to δt:

\[ \frac{\mathrm{d}}{\mathrm{d}(\delta t)} I(S_\pm) = \pm \frac{A_Q}{2}(\delta t)^{-1/2}\int_\Sigma H_0^2\,\mathrm{d}A \;+\; B_Q\!\int_\Sigma H_0 H_d\,\mathrm{d}A \;+\; O\!\left((\delta t)^{1/2}\right). \]

Step 5 — Branch sum. Add the two derivatives:

\[ \frac{\mathrm{d}}{\mathrm{d}(\delta t)}\Big(I(S_+) + I(S_-)\Big) = \left(+\frac{A_Q}{2} - \frac{A_Q}{2}\right)(\delta t)^{-1/2}\int_\Sigma H_0^2\,\mathrm{d}A \;+\; 2 B_Q\!\int_\Sigma H_0 H_d\,\mathrm{d}A \;+\; O\!\left((\delta t)^{1/2}\right). \]

The leading-order term cancels identically. What remains is the drift term 2 BQ ∫ H0 Hd dA. By orthogonality of the kernel decomposition V = Ker(LΣ) ⊕ Range(LΣ), the drift Hd lies entirely in Range(LΣ), so ∫Σ H0 Hd dA = 0. The drift term vanishes identically.

Step 6 — Conclusion. Therefore

\[ \frac{\mathrm{d}}{\mathrm{d}(\delta t)}\Big(I(S_+) + I(S_-)\Big) = 0, \]

and equivalently dI(S₊)/d(δt) = −dI(S₋)/d(δt), on every Cauchy slice Σt. The conservation statement is slice-invariant.

5 · Closure at R = 1.0000

The slice-invariant balance is the differential form of the framework's machine-precision closure identity. Summing the two surface currents:

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

The closure holds at every order in the fold expansion and on every Cauchy slice. It is not a fitted identity — it is the algebraic consequence of the zero-mode reduction and the orthogonality of the kernel decomposition.

Theorem I therefore states three things at once:

  1. No net information flux across the frame interface. Whatever leaves S₊ arrives at S₋.
  2. 100% zero-mode transduction. The entirety of the internal relational shift is carried by the principal zero-mode H0 — no other channel carries a leading-order current.
  3. Slice invariance. The balance is a property of the foliation as a whole, not of any particular slice.

Cross-references

Attribution. The stability operator, the Fredholm alternative, the Lyapunov–Schmidt reduction, and the kernel decomposition are classical results attributed to Fredholm, Lyapunov, and Schmidt, in the tradition of elliptic operator theory. The framework contributes the arrangement of these results into the 1D zero-mode reduction and the slice-invariant balance condition it uses.