Watts-per-Intelligence Part II: Algorithmic Catalysis

Abstract: We develop a thermodynamic theory of algorithmic catalysis within the watts-per-intelligence framework, identifying reusable computational structures that reduce irreversible operations for a task class while satisfying bounded restoration and structural selectivity constraints. We prove that any class-specific speed-up is upper-bounded by the algorithmic mutual information between the substrate and the class descriptor, and that installing this information incurs a minimum thermodynamic cost via Landauer erasure. Combining these results yields a coupling theorem that lower-bounds the deployment horizon required for a catalyst to be energetically favourable. The framework is illustrated on an affine SAT class and situates contemporary learned systems within a unified information-thermodynamic constraint on intelligent computation.

• The paper introduces a thermodynamic framework defining algorithmic catalysts as reusable computational structures that lower energy costs and accelerate task performance.
• It formalizes key concepts such as pathway opening, bounded restoration, and structural selectivity with quantitative bounds based on mutual information.
• The work illustrates practical implications through an affine-SAT example, demonstrating exponential speed-ups and minimal adaptation cost relative to deployment gains.

Thermodynamic Theory of Algorithmic Catalysis in the Watts-Per-Intelligence Framework

Introduction and Motivation

"Watts-per-Intelligence Part II: Algorithmic Catalysis" (2604.20897) presents a formal, thermodynamic approach to reusable computational structures—termed algorithmic catalysts—that provide speed-ups for classes of computational tasks. The motivation builds on the analogy between chemical catalysis, which enables otherwise infeasible reactions by lowering thermodynamic barriers, and algorithmic catalysis, where persistent algorithmic structures enable or accelerate classes of computations by encoding class-specific structural information. The work connects the theory of catalytic computation and the thermodynamic constraints on intelligence, extending the watts-per-intelligence (WPI) framework to capture the energy costs and benefits of machine adaptation, deployment, and structural reuse.

Formalization: Definitions and Theorems

The core formalism distinguishes between:

• Pathway opening: Reduction of irreversible bit-operations (and thus energy) required for a computational task.
• Bounded restoration: The catalyst is only temporarily perturbed during use and must be restored close to its reference state.
• Structural selectivity: The catalytic structure encodes non-trivial mutual information about the computational task class, as quantified by algorithmic (Kolmogorov) mutual information.

A system is an algorithmic catalyst for a reference system on a task class $C$ if, at matched intelligence, it yields a lower deployment cost, supports repeated use with explicit restoration energy accounting, and realizes a speedup that extends across infinite augmentations of $C$—precluding class-memorisation. The class-specific speed-up $\Gamma$ is upper-bounded by the algorithmic mutual information between the substrate's description and a canonical descriptor of $C$.

Further, the cost to install the structural information that enables this speed-up is lower-bounded by the minimum number of logical erasures (via the Zurek–Bennett result on the physical cost of entropy reduction), leading to an energetic lower bound on adaptation.

The coupling theorem links the adaptation energy, mutual information installed, and achievable (logarithmic) speed-up $\log_2 \Gamma$, providing a lower bound on the deployment horizon $N_{\mathrm{inf}}^*$. This horizon is the minimal number of deployments over which the adaptation cost becomes energetically favorable relative to repeated baseline computation.

Algorithmic and Thermodynamic Bounds

Structural Selectivity Theorem: In the universal-search cost model (where the shortest program solving the class $C$ dominates average-case computational cost), the speed-up provided by a structure is upper-bounded by its mutual information with the task class descriptor:

$$\log_2\Gamma \leq I(\text{substrate} : \sigma(C)) + c_U$$

Non-structural accelerations (e.g., hardware or algorithmic optimizations generic to all classes) do not count toward class-specific speed-up, isolating the contribution due to encoded structure.

Thermodynamic Coupling: Installing mutual information $\mu$ in the catalyst costs at least $$F(H_{\text{adapt,cat}})c[\mu - (\mathcal{D} : \sigma(C)) - c_U]_+$$ in adaptation energy. Only information not supplied by the adaptation input ($C$0: such as training data, design, etc.) must be physically written, and the cost must be amortized over deployment.

Composition: Multiple catalyst stages compose multiplicatively in speed-up and (at minimum) sub-additively in installed structural information, depending on the independence of structures they encode.

Illustrative Case: Affine-SAT

On the class $C$1 of $C$2-SAT instances with solutions forming a $C$3-dimensional affine subspace, baseline exhaustive search is infeasible (requires $C$4 operations), while a catalyst encoding the subspace enables search in $C$5, yielding $C$6. The required structural information is $C$7. The adaptation cost quantitatively depends on how much of this structure is installed by adaptation versus provided in advance (e.g., by training data revealing affine structure).

For practical parameters, the work shows that the adaptation cost can be negligible compared to the deployment energy savings, even for single-use deployments, provided the catalyst encodes sufficient structure—a result made precise in the coupling theorem.

Theoretical and Practical Implications

The results unify thermodynamic and algorithmic information-theoretic limits on adaptive computation. They rigorously exclude non-structural caching or memorization from qualifying as catalysts and clarify that data-driven adaptation or model compilation incurs a fundamental energetic cost proportional to the mutual information to be installed, in line with the physical limits set by Landauer's principle. The framework provides a template for analyzing and engineering AI systems according to both energetic and structural efficiency.

Practically, this theory yields explicit design criteria for energetically efficient intelligent systems by:

• Quantifying the cost-benefit tradeoff in structural adaptation.
• Providing principled bounds on how much energy can be saved via architectural pre-structuring, transfer learning, or other catalyst-like mechanisms.
• Informing when and how complex modular catalyst systems (e.g., deep neural architectures with multi-level abstraction) confer compounding energy-efficiency gains.

Theoretically, these results ground aspects of intelligence and learning within the physics of computation, aligning with efforts in algorithmic thermodynamics and information-theoretic analysis of learning.

Future Directions

Successors to this work may extend the framework to:

• Richer, probabilistic or non-symbolic substrates and task descriptors typical of contemporary AI.
• Quantitative assessments of thermodynamic limits for foundation models and multi-modal reasoning architectures.
• Novel catalyst construction methods leveraging transfer learning and compositional program synthesis.
• Empirical measurements closing the gap between theoretical minima and actual energetic costs in deployed adaptive systems.

Conclusion

This work provides a robust, thermodynamics-grounded formalism for algorithmic catalysis, precisely characterizing the necessary and sufficient informational and physical preconditions for class-specific, reusable computational speed-ups. The presented theorems establish that genuine, reusable computational catalysts require encoding non-trivial structure about their task class, that the cost to install this structure sets limits on achievable efficiency, and that multi-stage systems yield multiplicative energetic gains subject to sub-additive structure accumulation. The theory constitutes a rigorous foundation for understanding and optimizing the energetic efficiency of intelligent computation in both abstract and engineered systems.