Dissipation and the Emergence of Computation

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Overview

These three papers develop a unified framework for understanding when, and at what thermodynamic cost, a continuous physical process can genuinely support a finite symbolic computation. The central observation is that the world at its base may be neither discrete nor intrinsically lawful, but an evolving continuum glimpsed through the coarse lens of finite resolution.

The framework proceeds in three movements. The first derives the necessary and sufficient conditions for emergent symbolic dynamics and shows that dissipation is not incidental to computation but constitutive of it. The second grounds these conditions in concrete trajectories, showing that amenability to coarse-graining is a geometric property of crossing structure, not of smoothness. The third maps the geometry of the resulting partition of function space, revealing that computation is a measure-zero property of all possible trajectories, maintained against the natural drift toward generic continuous noise by a continuous thermodynamic expenditure.

The Papers

Framework & Foundations

Dissipation and the Emergence of Computation

Derives the three conditions (decodability, stability, and approximate lumpability) under which a continuous physical process genuinely supports finite symbolic dynamics. Proves a Markov approximation theorem bounding the symbolic trajectory's total variation distance from a finite-state machine. Introduces the dissipation-lumpability inequality and the computation epoch, establishing that entropy production is the mechanism by which physical systems sustain symbolic computation. Closes with a co-dissipation bound: system and observer jointly fund every computation.

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Examples & Contrasts

When Continuous Trajectories Can and Cannot Compute

Grounds the abstract framework through a minimal pair: sin(t) and sin(1/t), identical in range, smoothness class, and coarse-graining map, differing only in their zero-crossing structure. Shows that sin(1/t) fails all three conditions simultaneously near the origin. This failure is thermodynamically irreparable, since no non-equilibrium current can manufacture dwell time the trajectory does not possess. Establishes that amenability is a dwell-time property, not a smoothness property.

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III

Geometry & Structure

The Geometry of Computable Trajectories

Characterises the partition of continuous function space into amenable (𝒜) and non-amenable (𝒩) trajectories. Proves that 𝒜 is neither convex, closed, nor a cone, yet admits a natural stratification by minimum dwell time. Shows that 𝒜 has measure zero under Wiener measure: generic continuous processes do not compute. Connects the stratification to the spectral gap of the transfer operator, and reads the dissipation-lumpability inequality geometrically as a statement about motion in function space: entropy production is the mechanism that drives a system from 𝒩 into the interior of 𝒜.

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Separate Draft

A separate manuscript presents the core finite-horizon argument in single-paper form: readable, stable, approximately lumpable symbols define a computation epoch, and dissipation together with observation sets the physical cost of maintaining it.

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Central Results

Theorem 1

Markov Approximation

Decodability, stability, and approximate lumpability jointly bound the symbolic trajectory's TV distance from a finite-state machine over \(N\) steps.

\[\bigl|\mathcal{L}(\hat{A}_0,\ldots,\hat{A}_N) - \mathcal{L}(M_0,\ldots,M_N)\bigr|_{\mathrm{TV}}\] \[\le\; N(\lambda + 2p_\varepsilon)\]

Theorem 2

Dissipation-Lumpability Inequality

Entropy production reduces lumpability error as \(\sqrt{\sigma}\), directly extending the computation epoch toward a critical divergence at \(\sigma_c\).

\[\lambda(P,\Pi) = \lambda_{\mathrm{eq}}(\Pi) - c(\Pi)\sqrt{\sigma(P)} + O(\sigma)\]

\[\sigma_c(\Pi) := \!\left(\frac{\lambda_{\mathrm{eq}}(\Pi)}{c(\Pi)}\right)^{\!2}\]

Theorem 3

Causal Computation Gain

\(\CCG(\Pi,P) > 0\), meaning the macro description is causally superior to the micro, only when entropy production is nonzero. Equilibrium systems cannot exhibit causal emergence.

\[\CCG(\Pi,P) := \EI(K) - \EI(P,\Pi)\]

\[\sigma = 0 \;\Rightarrow\; \CCG(\Pi,P) \le 0\]

Theorem 4

Co-Dissipation Bound

Observer recording costs must enter the thermodynamic budget. System and observer jointly sustain every computation.

\[\dot{W}_{\mathrm{obs}} \ge \frac{k_BT\ln 2}{\Delta}\cdot H(p_\varepsilon)\]

\[W_{\mathrm{total}}(\Tc) = \bigl(\dot{W}_{\mathrm{sys}} + \dot{W}_{\mathrm{obs}}\bigr)\cdot \Tc\]

Theorem 5

Measure-Zero Computation

The amenable set \(\mathcal{A}\) has measure zero under Wiener measure. Generic continuous processes do not compute; engineering a system into \(\mathcal{A}\) is always a non-trivial imposition.

\[\mu_W(\mathcal{A}) = 0\]

\[\mathcal{A}_\tau = \bigl\{x \in \mathcal{A} : \tau_{\min}(x,\Pi^*) \ge \tau\bigr\}\]

Theorem 6

Spectral Gap Characterisation

The boundary \(\partial\mathcal{A}\) is exactly where the spectral gap of the transfer operator closes. Dissipation drives the system toward the interior of \(\mathcal{A}\).

\[x \in \mathcal{A} \;\iff\; \lambda_1 - \lambda_2 > 0\]

\[\sigma = \beta\int |g(x)|^2\,\pi(dx) \;\ge\; \sigma_c\]

Selected References

Shannon 1948. Claude E. Shannon. A Mathematical Theory of Communication. Bell System Technical Journal 27(3), 379-423; 27(4), 623-656.

Kemeny & Snell 1960. John G. Kemeny and J. Laurie Snell. Finite Markov Chains. Van Nostrand.

Buchholz 1994. Peter Buchholz. Exact and ordinary lumpability in finite Markov chains. Journal of Applied Probability 31(1), 59-75.

Landauer 1961. Rolf Landauer. Irreversibility and Heat Generation in the Computing Process. IBM Journal of Research and Development 5, 183-191.

Bennett 1982. Charles H. Bennett. The Thermodynamics of Computation: A Review. International Journal of Theoretical Physics 21, 905-940.

Hoel et al. 2013. Erik P. Hoel, Larissa Albantakis, and Giulio Tononi. Quantifying causal emergence shows that macro can beat micro. PNAS 110(49), 19790-19795.

Freidlin & Wentzell. Mark I. Freidlin and Alexander D. Wentzell. Random Perturbations of Dynamical Systems. Springer.

Schütte & Sarich 2013. Christof Schütte and Marco Sarich. Metastability and Markov State Models in Molecular Dynamics. American Mathematical Society.