1. The Observer's Sorrow and the Temporal Event Horizon
In RN-007, we rigorously defined the domain of validity for trajectory recurrence diagnostics: Soft Topological Recurrence (STR) measurements possess dynamical validity if and only if the underlying system satisfies Structured Connectivity Coherence (SCC > 0).
However, as an external observer in the physical world, we face a fundamental epistemological dilemma: we can never directly observe the true transition probability equations in state space. All we can access is a finite-time discrete segment of the trajectory of length T:
DT = { Xt } t=0T
This forces us to confront a deeper question: How long must we observe a trajectory (T) to assert, with sufficient statistical confidence, whether it was generated by a dynamical system with a weak attracting force (SCC > 0) or simply by a structureless random walk (SCC = 0)?
The answer is cold and unforgiving: under finite observation time T, some dynamical structures are mathematically forever unknowable. Long before a system reveals its true nature, the iron laws of statistics lock down our observational horizon.
2. Physical Limits of Statistical Identification: The Trial of Girsanov and Le Cam
To provide a rigorous quantitative answer, we formulate the identification of dynamical structure as a binary hypothesis testing problem.
Consider a hypothesis test between two systems:
- H0 (Null Hypothesis — Pure Random Walk, No Structure):
dXt = dWt(where Wt is a standard Wiener process; SCC = 0). The trajectory is unconstrained and drifts freely. - H1 (Alternative Hypothesis — Weak Attracting Manifold, Structured):
dXt = -θ Xt dt + dWt(an Ornstein-Uhlenbeck process with a weak attracting force toward the origin; SCC > 0). θ > 0 represents the signal strength of the attracting core.
Step A: Measure Equivalence and Girsanov's Theorem
By Girsanov's Theorem, since the two processes share the same diffusion coefficient, their path measures are mutually absolutely continuous. The Radon-Nikodym derivative (the log-likelihood ratio of the trajectory) is given by:
ln (dPθ / dP0)(X) = -θ ∫0T Xt dXt - ½ θ2 ∫0T Xt2 dt
Using Itô's Lemma, we directly compute the Kullback-Leibler (KL) divergence between the measures P0 and Pθ under the linear asymptotic limit:
DKL(P0 ∥ Pθ) ≈ θ2 T
Step B: Le Cam's Inequality and Indistinguishability
According to Le Cam's Inequality, the total error probability Perror (the sum of Type I and Type II errors) of any optimal decision maker is strictly bounded below by the KL divergence between the two measures:
Perror ≥ 1 - dTV(P0, Pθ) ≥ 1 - √( ½ DKL(P0 ∥ Pθ) ) ≥ 1 - θ √( T / 2 )
For the decision error rate to drop below a given acceptable threshold ε > 0, we must have Perror < ε, which mathematically requires:
T ≥ 2(1 - ε)2 / θ2
Core Theorem (Information-Theoretic Limits of Trajectory Recurrence Identification): For any weakly structured dynamical system with attraction strength θ, under a finite observation time T, the system is statistically identifiable if and only if:
T ≥ Ω( 1 / θ2 )
If the observation time is shorter than this limit, no physically realizable algorithm can distinguish the system from a pure, structureless random walk.
3. The Neural Network "Event Horizon" and the Security Dilemma
This information-theoretic limit yields a disruptive revelation for alignment, safety monitoring, and behavioral diagnostics in large neural networks (such as Large Language Models).
During LLM generation, the activation space trajectory behaves as a non-equilibrium stochastic flow. When a model begins to drift toward a "hallucination basin" or an "incoherent attractor" due to internal attention distortions or parameter drift, the initial deviation strength θ is typically extremely weak.
Our theorem predicts two critical security dilemmas:
- The Mathematical Inevitability of "Boiling the Frog": If the structural mutation of the model is highly stealthy (θ → 0), the sample length T required to catch the anomaly explodes quadratically. Over extremely long generation windows, the model's activation trajectory is probabilistically equivalent to a healthy path.
- The Observational "Event Horizon": There exists a physical boundary: within the window T < 1/θ2, no algorithm can warn of the impending failure. The model walks an identical probabilistic path until it crosses the horizon and suddenly erupts into severe hallucinations. This "silent failure" is explicitly forbidden by information theory.
4. Experimental Verification: Synthetic Flows and Real LLM Trajectories
To verify the universality of the T ∝ 1/θ2 scaling law, we performed hypothesis testing experiments on two fronts.
Experiment A: Precision Fitting on Synthetic Stochastic Flows
We generated trajectories of standard Brownian motion and OU processes with weak attracting cores (sweeping θ and T), classifying them using the optimal likelihood-ratio test (Neyman-Pearson tester, representing the absolute mathematical limit).
| Signal Strength (θ) | Critical Time Tcrit (1/θ2) | Empirical T for 80% Acc | Le Cam Bound Perror | Empirical Error Rate | Status |
|---|---|---|---|---|---|
| 0.50 | 4 | 4.2 | ≥ 0.29 | 0.31 | VALID |
| 0.20 | 25 | 26.1 | ≥ 0.29 | 0.30 | VALID |
| 0.10 | 100 | 104.5 | ≥ 0.29 | 0.29 | VALID |
| 0.05 | 400 | 412.0 | ≥ 0.29 | 0.30 | VALID |
The experiments precisely validate the quadratic scaling trend: when signal strength is halved, we must observe the system four times longer to detect its underlying dynamical structure.
Experiment B: Residual Layer Perturbation on Llama-3.2-3B
We injected a tiny, constant drift bias of strength θ (aligned with an incoherent representation direction) into the 16th residual layer of Llama-3.2-3B to simulate an emergent, stealthy trajectory collapse.
Using a Multi-Layer Perceptron (MLP) and a linear SVM to classify the trajectory within the first T tokens, we observed that at θ = 0.08, the classification accuracy hovered around 52% for the first 150 tokens (equivalent to random guessing, unable to break the Le Cam boundary). Only when the sequence length extended beyond T > 160 did the classification accuracy break the 80% threshold, perfectly matching our theoretical predictions.
5. Philosophical Footnote: The Observer's Limit and the Failure of Passive Safety
Centuries ago, thermodynamics used Carnot cycles to announce the death of perpetual motion machines; decades ago, quantum mechanics used Heisenberg's uncertainty principle to declare the end of Laplace's Demon.
Today, information theory draws an equally cold boundary in the sands of trajectory control: perfect passive safety monitoring is a myth.
If you sit back as a passive observer, reading activation trajectories in the hope of detecting all failures, you are guaranteed to be deceived by hidden drift signals that lie below the recurrence identification limit. They will lurk beneath the event horizon, only to overturn the system in the final few steps.
This tolls the death knell for passive safety. To break through this wall, we must abandon passive observation and transition to active perturbation — applying controlled energy inputs (interventions) to force hidden dynamical features to reveal themselves.
Key takeaway: Passive observation is bounded by a mathematical event horizon. If a drift is subtle enough, no monitor, regardless of its precision, can detect it before the system collapses. True safety cannot be found in the comfort of passive watching; it is forged in the fire of active intervention.
This note draws on theoretical and experimental results from "Fundamental Limits of Trajectory Recurrence Identification" (Haelio Tang, 2026).