Zero-Mode Power Law
Statement. For a quasilocal horizon scalar functional Q(t) whose dynamics are governed by the 1D reduced bifurcation equation of the Lyapunov–Schmidt reduction, the asymptotic boundary emission power across the frame interface follows the three-term expansion
\[ P(t) = \frac{\kappa A_Q^2}{4\,(t - t_*)} \;\pm\; \frac{\kappa A_Q B_Q}{\sqrt{t - t_*}} \;+\; \kappa B_Q^2. \]
Near the critical epoch t → t*+, the leading term dominates: the emission power exhibits an inverse-first-order divergence (t − t*)−1, modulated by a square-root interference term (t − t*)−1/2. Every term carries Derived status.
This page is the full derivation of the emission law that Theorem II states. Theorem II introduces the law; here it is derived from the fold trajectory step by step.
1 · Setup and notation
On a Cauchy slice Σt, let Q(t) be any quasilocal horizon scalar functional — the surface area, the integrated shear, a multipole moment, or any other scalar quantity integrated over the MOTS Σ. Its value at the critical epoch t = t* is denoted Q*.
From the Lyapunov–Schmidt reduction, the 1D reduced bifurcation equation yields the universal dual-branch fold trajectory
\[ Q_\pm(t) - Q_* = \pm A_Q \,(t - t_*)^{1/2} + B_Q \,(t - t_*) + O\!\left((t - t_*)^{3/2}\right), \]
where:
- AQ — the zero-mode expansion coefficient. Determined by the second variation of the null expansion Θ+ at t = t*.
- BQ — the linear background drift coefficient. Determined by the frame-conditioned background contribution to the reduced equation.
- the ± sign split distinguishes the outer branch S₊ from the inner branch S₋.
Let κ be the physical coupling constant that relates the horizon expansion rate to radiated power. Its value is frame-conditioned; its form enters the emission law below.
2 · The horizon expansion derivative
Differentiate the fold trajectory with respect to coordinate time t:
\[ \frac{\mathrm{d}Q}{\mathrm{d}t} = \pm \frac{A_Q}{2 \,(t - t_*)^{1/2}} + B_Q + O\!\left((t - t_*)^{1/2}\right). \]
The leading term diverges as (t − t*)−1/2 at the critical epoch. This is the asymptotic expansion rate of the horizon boundary along the principal zero-mode channel.
Each of the three terms in the derivative has a distinct physical interpretation:
| Term | Origin |
|---|---|
| ± AQ / (2(t − t*)1/2) | Zero-mode divergence — the leading saddle-node signature. |
| BQ | Frame-conditioned linear drift — background evolution of the horizon. |
| O((t − t*)1/2) | Sub-leading corrections from the range component of the reduction. |
3 · The emission power expression
The physical power P(t) radiated across the asymptotic boundary interface is proportional to the square of the horizon expansion rate, with proportionality constant κ:
\[ P(t) = \kappa \left(\frac{\mathrm{d}Q}{\mathrm{d}t}\right)^{\!2}. \]
Substituting the derivative from §2:
\[ P(t) = \kappa \left(\pm \frac{A_Q}{2\,(t - t_*)^{1/2}} + B_Q \right)^{\!2} + O\!\left((t - t_*)^{1/2}\right). \]
Expanding the square:
\[ P(t) = \kappa \left[\frac{A_Q^2}{4\,(t - t_*)} \pm \frac{A_Q B_Q}{(t - t_*)^{1/2}} + B_Q^2\right] + O\!\left((t - t_*)^{1/2}\right). \]
Collecting the three leading terms:
\[ \boxed{\,P(t) = \frac{\kappa A_Q^2}{4\,(t - t_*)} \;\pm\; \frac{\kappa A_Q B_Q}{\sqrt{t - t_*}} \;+\; \kappa B_Q^2\,} \]
The boxed expression is the zero-mode power law.
4 · Asymptotic behaviour of each term
Near the critical epoch, the three terms of the power law have distinct asymptotic scalings that determine the emission profile.
| Term | Scaling | Role |
|---|---|---|
| Leading singularity | (t − t*)−1 | Inverse-first-order divergence. Dominates as t → t*+. Carries the zero-mode channel's asymptotic emission. |
| Interference term | (t − t*)−1/2 | Square-root interference between zero-mode and linear drift. Sign set by branch. |
| Background term | (t − t*)0 | Frame-conditioned background emission. Independent of the bifurcation. Sets the asymptotic floor of the emission. |
The dominant asymptotic behaviour:
\[ P(t) \;\sim\; \frac{\kappa A_Q^2}{4\,(t - t_*)}, \qquad t \to t_*^+. \]
The power diverges as the inverse-first-order singularity. The total energy emitted over the interval [t*, t] is finite, however: the integral ∫ (t − t*)−1 dt = ln(t − t*) is logarithmically integrable on any finite interval.
5 · Branch sign and symmetry
The interference term ± κ AQ BQ/(t − t*)1/2 carries the sign of the branch:
- Outer branch S₊. Positive sign. The zero-mode and the linear drift add coherently, enhancing the emission by the interference amount.
- Inner branch S₋. Negative sign. The zero-mode and the linear drift subtract, suppressing the emission by the same magnitude.
The two branches are therefore not symmetric in their emission profile even though their leading (t − t*)−1 terms are identical. The asymmetry is carried entirely by the interference term, whose sign distinguishes the outward-directed emission from the inward-directed emission.
This is the algebraic origin of Theorem I's dual-branch balance: the leading emission terms on the two branches are equal in magnitude, and the difference between them is the interference term, which is antialigned across the branches. In the asymptotic limit t → t*+, the interference becomes negligible relative to the leading singularity, and the two branches merge into a single symmetric emission profile.
6 · Consequences
The zero-mode power law has three direct consequences for the framework:
- Universal signature. The (t − t*)−1 leading behaviour is independent of the choice of quasilocal scalar functional Q. Any scalar integrated over the MOTS produces the same asymptotic emission profile up to the frame-conditioned coefficients AQ and BQ.
- Zero-mode transduction. The entire leading contribution to the emission power is carried by the single zero-mode channel H0. No other channel enters at leading order. This is the exact content of Theorem II.
- Sub-threshold recovery. The inverse-first-order structure of the power law is what makes the framework's Mode B negative-space mining possible: the residual sub-threshold signal decays as a square-root fold, which is detectable via matched-filter correlation even when the raw signal is below the noise floor.
7 · Closure at R = 1.0000
The zero-mode power law is a positive-emission law: it describes what is radiated outward along the principal channel. Its partner is the slice-invariant conservation of Theorem I: what is radiated is exactly balanced by what is conserved in the inverse cone. Combined:
\[ \int_{t_*}^{t} P(t')\,\mathrm{d}t' = \Delta I(S_+), \qquad I(S_+) + I(S_-) = I_{\text{total}}, \qquad R \equiv \frac{I(S_+) + I(S_-)}{I_{\text{total}}} = 1.0000. \]
The emission law is the differential partner of the closure identity. What flows out is what is accounted for. No information is lost.
Cross-references
- Zero-mode emission (Theorem II). The theorem statement of the power law is in asymptotic zero-mode transduction.
- Lyapunov–Schmidt reduction. The fold trajectory that the power law is derived from is established in Lyapunov–Schmidt reduction.
- Slice-invariant conservation (Theorem I). The conservation partner of the emission law is in universal slice invariance.
- 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 emission law of which this page is the full derivation is collected in Classical Maths.
Attribution. The saddle-node fold normal form and the inverse-square-root derivative are classical results of bifurcation theory, attributed to Poincaré and developed through the twentieth century. The Lyapunov–Schmidt reduction that produces the fold trajectory is attributed to Lyapunov and Schmidt. The framework contributes the arrangement of these results into the physical emission law of the boundary interface.