When Continuous Trajectories
Can and Cannot Compute

Fix state space \(\mathcal{X} = [-1,1]\), sampling interval \(\Delta > 0\), and the binary coarse-graining \(\Pi(x) = R\) if \(x \ge 0\), \(L\) otherwise. For a trajectory \(x(t)\), define \(A_n = \Pi(x(n\Delta))\). Three conditions on \((x, \Pi, \Delta)\) are necessary for the symbolic sequence to support emergent computation.

These three conditions are not redundant. Stability says the present symbol is durable; lumpability says it is predictive; decodability says it is legible. When all three hold with parameters \((\varepsilon, \eta, \lambda)\), the symbolic trajectory lies within total variation distance \(N(\lambda + 2p_\varepsilon + \eta)\) of a genuine Markov chain over \(N\) steps — so failure of any condition directly collapses the computation epoch \(\Tc\).

The argument \(1/t\) compresses the time axis near the origin. Zero crossings occur at \(t = 1/(n\pi)\), spacing \(\pi t_0^2\) apart near \(t_0\) — a gap that shrinks to zero as \(t_0 \to 0\). The number of crossings in any interval of length \(\Delta\) near \(t_0\) is approximately \(\Delta / (\pi t_0^2)\), which diverges.

Stability ✗

For any fixed \(\Delta > 0\), the flip probability \(\Pr(A_{n+1} \neq A_n) \to 1\) as \(t_n \to 0\). No choice of \(\Delta\) prevents this: the crossing density grows without bound faster than any fixed sampling rate can track.

Lumpability ✗

Near \(t_0 \approx 0\), two micro-states \(x = \sin(1/t_1)\) and \(y = \sin(1/t_2)\) can both sit in cell \(R\) yet have completely different next-symbol distributions — one is about to cross, the other just did. The lumpability error \(\lambda \approx 1\) for any partition boundary, since time-to-next-crossing varies enormously across micro-states within the same cell.

Decodability ✗

With stability and lumpability both failing, consecutive symbols become independent near the origin. The mutual information \(I(R_n; A_{n+1}) \to 0\): the observer's record is faithfully kept but conveys nothing about the future. Decodability collapses as a consequence, not independently.

Critically, this failure is thermodynamically irreparable. Entropy production can in general reduce \(\lambda\) via the dissipation-lumpability inequality \(\lambda \le \lambda_{\mathrm{eq}} - c\sqrt{\sigma}\) — but this requires a finite starting point. Here \(\lambda_{\mathrm{eq}} = 1\) and the root cause is not a blurry boundary but the absence of any dwell time within a cell. No non-equilibrium current can manufacture time the trajectory does not spend in a region. The computation epoch \(\Tc \approx 0\) regardless of power budget.

Replace \(1/t\) with \(t\). Zero crossings now occur at \(t = n\pi\), separated by exactly \(\pi \approx 3.14\) — a uniform gap, independent of \(t\). This single change repairs all three conditions.

Stability ✓

For \(\Delta < \pi\), each sampling interval contains at most one crossing. The flip probability is bounded uniformly: \[ \Pr(A_{n+1} \neq A_n) \;\le\; \frac{\Delta}{\pi} \;=:\; \eta(\Delta), \] and \(\eta(\Delta) \to 0\) as \(\Delta \to 0\). Stability is controllable by the observer's choice of sampling interval.

Lumpability ✓

The speed \(|\dot{x}| = |\cos t|\) is bounded, so no micro-state within a cell can be dramatically closer to the boundary than another. The spread in escape probabilities across micro-states in \(R\) is at most \(2\Delta/\pi\): \[ \lambda \;\approx\; \frac{2\Delta}{\pi}, \] which vanishes as \(\Delta \to 0\). The macro-symbol becomes increasingly predictive at finer sampling. Compare with \(\sin(1/t)\), where \(|\dot{x}| = |\cos(1/t)|/t^2\) grows without bound near \(t=0\) and \(\lambda \approx 1\) at all resolutions.

Decodability ✓

With stability and lumpability holding, the mutual information satisfies \(I(R_n; A_{n+1}) \ge \log 2 - H(\Delta/\pi)\), which approaches \(\log 2\) for small \(\Delta\). Knowing \(A_n = R\) reliably predicts \(A_{n+1} = R\). The observer's record is informative.

The computation epoch is now positive and grows as \(\Delta\) decreases. A small non-equilibrium drive \(\dot{W}\) reduces \(\lambda\) further via the dissipation-lumpability inequality \(\lambda \le \lambda_{\mathrm{eq}} - c\sqrt{\sigma}\), extending \(\Tc\) toward the critical point where \(\lambda = 0\) and the epoch diverges.

The two examples are identical in range, smoothness, and coarse-graining. Their difference is purely geometric: whether the zero-crossing density is bounded or accumulates without limit.