Dynamic Optimality in Physical and Arithmetic Systems

In the paper's words

The selection of an appropriate level of abstraction constitutes a critical and profoundly non-trivial challenge across the entire scientific enterprise. This fundamental problem of representational scale dictates not only the tractability of our models but also their ultimate predictive power in describing the world. From the coarse-grained variables of thermodynamics to the aggregated metrics of macroeconomics, scientific understanding is built upon a hierarchy of descriptions, none of which holds a universal claim to primacy.

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

1. This requires a normalization of the axes: physical time $t$ in Track A is aligned with system size $N$ in Track B, treating both as a measure of “system evolution.” The mechanism of comparison involves normalizing the timescales of both simulations to a percentage of their full evolution (0% to 100% complete).

   from 3.5 Comparative Analysis

2. The concept of efficiency as a guiding principle in science has deep roots, originating in the mid-20th century with the foundational work of Claude Shannon and later formalized by Jorma Rissanen.

   from 1.2 Predictive

3. Shannon’s rate-distortion theory established that for any given information source, there is an irreducible relationship between the fidelity of a representation (its “distortion”) and the complexity of that representation (its “rate”) (Shannon, 1959).

   from 1.2 Predictive

4. Information Efficiency The theoretical bedrock for quantifying the trade-off between representational fidelity and computational cost was established in the mid-20th century by the foundational works of Claude Shannon and Jorma Rissanen.

   from 2.1 Foundations of

5. When our predictive efficiency metric identifies a coarse-grained model (e.g., k=32) as optimal, it is detecting precisely this peak in Effective Information.

   from 2.2 Causal Emergence Theory

6. The work of Qi (2005) and others on the eigenvalues of supersymmetric tensors provides the necessary formalism to determine if a complex, multi-variable state corresponds to a stable local minimum, bridging the gap between our information-theoretic efficiency metric and physical stability criteria.

   from 2.5 Higher-Order Tensor

How it is done

1. This perspective suggests that coarse-graining is not merely a convenience for computationally-limited observers, but can be an objective method for isolating the true causal structure of a system.

   from 1.1 The Representational

2. The representation that maximizes this efficiency is, by definition, the optimal description of the system for a resource-bounded observer.

   from 1.2 Predictive

3. The context for this dual-track approach is the historical divide between physics and abstract mathematics.

   from 1.3 Dual-Track Investigation

4. Arithmetic Topology Model For the second track of our investigation, we construct a computational system based on the deep analogies of arithmetic topology.

   from 1.5 Track B: The

What would falsify it

1. Prior theoretical treatments of representation, while conceptually rich, have often been non-constructive, providing abstract principles without concrete, falsifiable algorithms.

   from 1.1 The Representational

2. While the microscopic state of a system is indeed fundamental in that it determines all higher-level properties, it does not follow that it is the most effective representation for prediction.

   from 1.1 The Representational

3. If the static analogy holds true, then the informational structure of the knot-like primes should manifest as similar “causal” patterns (like clustering or screening) when viewed through a predictive lens.

   from 1.5 Track B: The

Where the claim stops

1. The mechanism of limitation is structural.

   from 3.7 Methodological Limitations

2. A 1D system cannot exhibit the complex topological defects (like vortices) found in 2D or 3D systems.

   from 3.7 Methodological Limitations

3. A linear predictor cannot capture XOR-like interactions or chaotic attractors.

   from 3.7 Methodological Limitations

Where to read

1. Dynamic9,943 words · 43 min
2. PREDICTIVE_TROUGH) to the log.2,130 words · 9 min
3. CAUSAL_EMERGENCE, marking4,163 words · 18 min
4. MESOSCALE_OPTIMUM |6 words · 1 min
5. PREDICTIVE_TROUGH |6 words · 1 min
6. CAUSAL_EMERGENCE |6 words · 1 min
7. EQUILIBRIUM |453 words · 2 min
8. ARITHMETIC_GENESIS |6 words · 1 min
9. PREDICTIVE_TROUGH |6 words · 1 min
10. MACRO_DOMINANCE |6 words · 1 min
11. MACRO_DOMINANCE |693 words · 3 min