The Geometry of
Computable Trajectories

Every continuous trajectory \(x(t)\) either can or cannot support symbolic computation under some choice of coarse-graining \(\Pi\) and sampling interval \(\Delta\). This gives a natural partition of the space of continuous functions into two sets. Call them \(\mathcal{A}\) (amenable) and \(\mathcal{N}\) (non-amenable):

The question is: what is the geometry of this partition? Where is the boundary? What are the structural properties of each set? And what does it cost, in thermodynamic terms, to move from \(\mathcal{N}\) into \(\mathcal{A}\)?

The answer turns on a single geometric property of a trajectory: the structure of its zero set — or more generally, the set of times at which it crosses a cell boundary.

The set \(\mathcal{A}\) has a number of surprising non-properties. It is tempting to assume that the amenable trajectories form a well-behaved subspace — a cone, a convex set, a closed subspace. None of these hold.

\(\mathcal{A}\) is not a vector space

The sum of two amenable trajectories need not be amenable. Take \(x_1(t) = \sin(t)\) and \(x_2(t) = \sin(1/t) - \sin(t)\). Both are in \(\mathcal{A}\): \(x_1\) trivially, and \(x_2\) because its crossings, while irregular, remain locally finite away from the origin. Yet \(x_1 + x_2 = \sin(1/t) \in \mathcal{N}\). Amenability is not preserved by superposition.

\(\mathcal{A}\) is not closed under uniform limits

One can construct a sequence \(x_n \in \mathcal{A}\) converging uniformly to \(x \in \mathcal{N}\). Consider \(x_n(t) = \sin(t) \cdot \mathbf{1}_{t > 1/n} + \sin(1/t) \cdot \mathbf{1}_{t \le 1/n}\): each \(x_n\) is amenable (the non-amenable region is pushed toward zero), but the limit is \(\sin(1/t)\). The boundary \(\partial\mathcal{A}\) is not closed — amenability can be lost in the limit.

\(\mathcal{A}\) is not convex

A convex combination \(\alpha x + (1-\alpha)y\) with \(x, y \in \mathcal{A}\) can exit \(\mathcal{A}\) by the same superposition argument. The set has no convex structure to exploit.

What \(\mathcal{A}\) does have is a natural stratification by minimum dwell time. For each \(\tau > 0\), define the level set

where \(\Pi^*\) is the optimal coarse-graining for \(x\). These level sets are nested — \(\mathcal{A}_{\tau'} \subset \mathcal{A}_\tau\) for \(\tau' > \tau\) — and their union is \(\mathcal{A}\). Each \(\mathcal{A}_\tau\) is closed under time-reparametrisations that preserve minimum gap structure, and the deeper a trajectory sits within \(\mathcal{A}\), the longer its computation epoch and the less dissipation required to maintain it.

The most striking geometric fact about this partition is what happens when one asks about the size of \(\mathcal{A}\) in a probabilistic sense. The natural measure on continuous trajectories is Wiener measure — the law of Brownian motion. Under this measure, the set \(\mathcal{A}\) has measure zero.

This is a deep result about what computation means physically. Generic continuous processes — those sampled from the natural measure on path space — do not compute. The set of processes that can support symbolic dynamics is a negligible subset of all possible trajectories. This means that engineering a physical system into \(\mathcal{A}\) is always a non-trivial imposition of structure on what would otherwise be generic continuous noise.

The Hausdorff dimension of the zero set provides a quantitative measure of how far a trajectory is from \(\mathcal{A}\). For \(\sin(t)\), the zero set is countable — dimension zero. For Brownian motion, it is dimension \(\frac{1}{2}\). For \(\sin(1/t)\), the zero set near the origin is countably infinite but has an accumulation point, placing it strictly between the two. The dimension of the zero set is a natural distance from \(\mathcal{A}\): the further a trajectory is from amenability, the richer its crossing structure.

The cleanest characterisation of the partition uses the transfer operator \(\mathcal{P}\), which propagates probability densities forward under the dynamics. For a trajectory arising from a stochastic differential equation \(dx = f(x)\,dt + \sqrt{2\beta^{-1}}\,dW\), the transfer operator is the Fokker-Planck semigroup, and its spectral properties encode everything about the long-run behaviour of the system.

The stratification of \(\mathcal{A}\) by \(\tau_{\min}\) corresponds directly to the stratification by spectral gap magnitude. Large \(\tau_{\min}\) means a large gap — the dynamics rapidly equilibrates within each cell. Small \(\tau_{\min}\) means the gap is barely open. The boundary \(\partial\mathcal{A}\) is exactly where the spectral gap closes: \(\lambda_2 \to \lambda_1 = 1\).

This closing of the spectral gap is a phase transition. The computation epoch \(\Tc\) depends on the lumpability error \(\lambda \approx 1 - (\lambda_1 - \lambda_2)\Delta^{-1}\), so as the gap closes, \(\lambda \to 1\) and \(\Tc \to 0\). The boundary \(\partial\mathcal{A}\) is therefore the set of trajectories for which computation is possible in principle but not in practice — an infinitely thin region where the epoch has collapsed to zero.

The dissipation-lumpability inequality \(\lambda \le \lambda_{\mathrm{eq}} - c\sqrt{\sigma}\) can be read geometrically as a statement about how entropy production moves a trajectory in function space. A system at equilibrium (\(\sigma = 0\)) sits at a fixed point in the \((\lambda, \sigma)\) plane. Turning on dissipation moves it along a curve toward smaller \(\lambda\), which in the function-space picture corresponds to moving toward the interior of \(\mathcal{A}\).

Formally, for a trajectory governed by \(dx = f(x)\,dt + \sqrt{2\beta^{-1}}\,dW\) with non-equilibrium driving \(f = -\nabla V + g\) (where \(g\) is the non-gradient part generating currents), the entropy production rate is

where \(\pi\) is the stationary measure. The dissipation-lumpability inequality then gives a lower bound on how far the driven trajectory has moved into \(\mathcal{A}\) relative to its equilibrium position: the distance travelled is at least \(c\sqrt{\sigma}\) in the \(\lambda\)-coordinate. The critical dissipation \(\sigma_c = (\lambda_{\mathrm{eq}}/c)^2\) is the minimum cost of reaching \(\partial\mathcal{A}\) from the equilibrium position — the thermodynamic price of entry.

This gives a precise answer to the question of what it costs to compute. It is not merely the cost of a specific operation (erasing a bit, flipping a state) but the ongoing cost of maintaining membership in \(\mathcal{A}\) against the natural tendency of physical processes to drift toward \(\mathcal{N}\). Every living system — every cell, every neural circuit — is paying \(\sigma \ge \sigma_c\) continuously, just to remain in the set where computation is possible at all.

The following table collects the structural properties of the two sets and their boundary.

The boundary \(\partial\mathcal{A}\) — the set of trajectories where \(\tau_{\min} = 0\) and the spectral gap just closes — is the most physically significant region. It is where computation is balanced on a knife-edge: any reduction in dissipation tips a system from \(\mathcal{A}\) into \(\mathcal{N}\), and any reduction in power budget causes the computation epoch to collapse. Living systems near metabolic stress, neurons near firing threshold, and physical computers at minimum operating voltage all inhabit this boundary region. The cost of moving away from it — deeper into \(\mathcal{A}\) — is exactly what the dissipation-lumpability inequality quantifies.