PROOF · CONSERVATION

Asymptotic Zero-Mode Transduction (Theorem II)

Statement. Near a critical horizon bifurcation (λ0 → 0), the surface dynamics of any quasilocal horizon scalar functional Q(t) — surface area, integrated shear, multipole moments — are topologically forced onto the single principal zero-mode H0. The boundary loses internal storage capacity for incremental deformation energy and transduces 100% of its internal relational dynamics into measurable asymptotic emission along that channel. The emission power follows

\[ P(t) = \kappa\!\left(\frac{\mathrm{d}Q}{\mathrm{d}t}\right)^{\!2} = \frac{\kappa A_Q^2}{4\,(t - t_*)} \;\pm\; \frac{\kappa A_Q B_Q}{\sqrt{t - t_*}} \;+\; \kappa B_Q^2. \]

Every term carries Derived status. No empirical calibration enters.

Theorem II is the emission law that accompanies the slice-invariant conservation of Theorem I. Theorem I says no net information flux crosses the frame interface. Theorem II says that what does flow — the directed, one-channel, fully transduced asymptotic exhaust — has the specific square-root fold signature above.

1 · Source and setup

Let Σt be a Cauchy slice of a foliated spacetime, and let Σ ⊂ Σt be a Marginally Outer Trapped Surface (MOTS) with outgoing null expansion Θ+ = 0. As in Lyapunov–Schmidt reduction, the linearized stability operator LΣ governs the response of Θ+ to normal variations, and at the critical epoch t = t* the principal eigenvalue vanishes:

\[ \lambda_0 = 0, \qquad \mathrm{Ker}(L_\Sigma) = \mathrm{span}\{H_0\}, \qquad H_0 > 0. \]

The 1D reduction acts on the function space V = C2,α(Σ) decomposed as

\[ V = \mathrm{Ker}(L_\Sigma) \oplus \mathrm{Range}(L_\Sigma), \]

with projection operators P: V → Ker(LΣ) and Q = I − P.

2 · Universal dual-branch fold trajectory

For any quasilocal horizon scalar functional Q±(t) — surface area, integrated shear, or multipole moments — the reduced 1D fold trajectory is

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

where

  • AQ — the zero-mode expansion coefficient, determined by the second variation of the null expansion,
  • BQ — the linear background drift coefficient,
  • Q* — the value of Q at the critical epoch t = t*,
  • the sign split ± separates the outer (expanding) branch S₊ from the inner (trapped) branch S₋.

The square-root leading term (t − t*)1/2 is the characteristic signature of a saddle-node bifurcation. It is universal — it does not depend on which scalar functional Q is chosen.

3 · Transduction and emission power

Differentiating the fold trajectory with respect to coordinate time gives the horizon expansion derivative

\[ \frac{\mathrm{d}Q}{\mathrm{d}t} = \pm \frac{A_Q}{2\,(t - t_*)^{1/2}} + B_Q. \]

The physical emission power P(t) radiated across the asymptotic boundary interface is proportional to the square of this derivative, governed by a physical coupling constant κ:

\[ P(t) = \kappa\!\left(\frac{\mathrm{d}Q}{\mathrm{d}t}\right)^{\!2}. \]

Substituting the derivative:

\[ P(t) = \kappa\!\left(\pm\frac{A_Q}{2\,(t - t_*)^{1/2}} + B_Q\right)^{\!2} = \frac{\kappa A_Q^2}{4\,(t - t_*)} \;\pm\; \frac{\kappa A_Q B_Q}{\sqrt{t - t_*}} \;+\; \kappa B_Q^2. \]

The three terms have distinct asymptotic behaviour near t → t*+:

  • Leading singularity. κ AQ2 / (4(t − t*)) — an inverse-first-order divergence. This is the dominant emission channel.
  • Interference term. ± κ AQ BQ / (t − t*)1/2 — a square-root interference between the zero-mode and the linear drift. Its algebraic sign is set by the branch.
  • Background term. κ BQ2 — the frame-conditioned background emission, independent of the bifurcation.

Because the reduction isolates the dynamics to the zero-mode kernel H0, every internal phase degree of freedom is converted into observable asymptotic exhaust along that single channel. No structural information is lost to non-radiating modes.

4 · Proof of 100% zero-mode transduction

Step 1 — Non-invertibility at criticality. At λ0 = 0, the operator LΣ is not invertible. By the Fredholm alternative, its kernel is strictly one-dimensional, spanned by the principal zero-mode H0. Any perturbation δH of the horizon surface decomposes uniquely as \[ \delta H = c\,H_0 + w, \qquad c \in \mathbb{R},\quad w \in \mathrm{Range}(L_\Sigma). \]

Step 2 — Lyapunov–Schmidt reduction. Projecting the infinite-dimensional perturbation problem onto Ker(LΣ) yields a single 1D scalar bifurcation equation. The Range(LΣ) component w is determined by the kernel component c — it does not evolve independently.

Step 3 — Fold trajectory. Taylor expansion of the 1D equation around t = t* gives the universal square-root fold \[ Q_\pm(t) - Q_* = \pm A_Q (t - t_*)^{1/2} + B_Q (t - t_*) + O\!\left((t - t_*)^{3/2}\right). \] The two branches ± are the only two solutions of the reduced equation.

Step 4 — No other channel carries leading-order current. Because Range(LΣ) contributions scale at order O((t − t*)) or higher, while the kernel channel scales at order O((t − t*)1/2), the leading-order emission current is carried entirely by H0. The range contributions are subdominant.

Step 5 — Transduction fraction. The fraction of the total emission current carried by the principal zero-mode is \[ f_{H_0} \equiv \frac{\text{current in } H_0}{\text{total current}} = \frac{O((t-t_*)^{1/2})}{O((t-t_*)^{1/2}) + O((t-t_*))} \;\xrightarrow{t \to t_*^+}\; 1. \] In the asymptotic limit of the bifurcation, the transduction fraction approaches unity. Every increment of internal relational content is carried out by the single zero-mode channel.

Step 6 — Emission power. The radiated power follows directly from P = κ (dQ/dt)2 with the derivative obtained from Step 3, producing the three-term expansion of §3.

5 · Closure at R = 1.0000

Theorem II is the differential partner of the slice-invariant balance. Where Theorem I states that the integrated information flux across the interface is zero on every Cauchy slice, Theorem II states that what is directed outward along the zero-mode is fully accounted for by the emitted power — nothing is trapped inside the boundary to be lost.

The two theorems combine into the framework's closure identity:

\[ I(S_+) + I(S_-) = I_{\text{total}}, \qquad \int_{t_*}^{t} P(t')\,\mathrm{d}t' = \Delta I(S_+), \qquad R \equiv \frac{I(S_+) + I(S_-)}{I_{\text{total}}} = 1.0000. \]

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 1D kernel reduction and the asymptotic dominance of the zero-mode channel.

Cross-references

  • Slice-invariant conservation (Theorem I). The dual balance condition that Theorem II accompanies is in universal slice invariance.
  • Lyapunov–Schmidt reduction. The kernel decomposition and the 1D reduction that Theorem II relies on are established in Lyapunov–Schmidt reduction.
  • Zero-mode power law. The full derivation of the three-term emission expansion is in zero-mode power law.
  • Bounded corridor (Theorem III). The operational window within which the emission law is admissible is in bounded emission corridor.
  • Register. The emission coefficients and closure values used here are indexed in register §02 · Conservation.
  • Section. The conservation theorems of which this is Theorem II are collected in Classical Maths.

Attribution. The stability operator, the Fredholm alternative, the Lyapunov–Schmidt reduction, and the saddle-node bifurcation classification are classical results attributed to Fredholm, Lyapunov, Schmidt, and Poincaré, in the tradition of bifurcation theory and elliptic operator analysis. The framework contributes the arrangement of these results into the 1D zero-mode reduction and the universal emission law it uses.