Core Definitions

Entity. Anything with causal presence. Entity and causal interaction are a co-definitional primitive pair — neither is prior to the other. The framework is silent on causally inert possibilia.

State. The posited referent that representations approximate. Epimechanics assumes states exist but does not claim direct access to them.

Computation. Producing a representation — discriminating, partitioning, compressing. Computation varies in depth (passive → active → adaptive → self-referential) and happens at every scale. This is pancomputational; the depth spectrum is the discriminator, the same way the entity definition is pan-entity and $\rho_{\text{ac}}$ is the discriminator.

Representation. The product of computation — an approximation of state. $X$ . A representation may be predictively effective or not. Some representations compress well (they track causal structure); others do not.

Information. Predictive information — the degree to which a representation's distinctions reduce uncertainty about future states, above background noise. Formally: mutual information between $X$ and dynamics.

Prediction. Reduction of uncertainty about future states given a representation's information content.

The Chain

Entity → Computation → Representation → Information → Prediction.

This chain is recursive, not linear. The computing entity is itself known only through representations computed by other entities (or itself). Every link presupposes the others.

The Depth Spectrum

Computation varies in depth. The following are landmarks on what we conjecture is a continuous spectrum, not discrete categories.

Level 0: Passive Encoding

Structure constrains but does not maintain itself against degradation. Examples: crystal lattice constraining phonon propagation, a canyon constraining water flow. $\rho_{\text{ac}}$ is low; the encoding degrades without external maintenance.

Level 1: Active Replication

Structure reproduces. Examples: DNA template-directed replication, autocatalytic chemical cycles. $\rho_{\text{ac}}$ is moderate — the system sustains its structure through reproduction, creating causal loops at the lineage level.

Level 2: Adaptive Modeling

Structure updates based on feedback. Examples: immune systems, neural networks, bacterial chemotaxis. $\rho_{\text{ac}}$ is high — multiple nested causal loops maintain and update internal structure.

Level 3: Self-Referential Modeling

Structure models its own modeling. Examples: conscious introspection, formal systems reasoning about their own provability, self-modifying code. $\rho_{\text{ac}}$ is high and recursive — the causal loops include the system's own loop structure as an object of representation.

Connections

Shannon

Predictive information is a specific application of mutual information. Shannon entropy measures uncertainty in a distribution; predictive information measures how much a representation reduces that uncertainty about dynamics specifically. Shannon gives the measure; Epimechanics specifies what is being measured — the predictive content of $X$ with respect to future states.

Landauer

Computation has thermodynamic cost because real-world representing involves irreversible steps. Landauer (1961) established that erasing one bit dissipates at least $k_B T \ln 2$ . The cost enters through irreversibility, not through representation per se. Updating a representation requires erasing the old one before writing the new one. For adaptive systems (Level 2+), continual model updating pays Landauer's cost at every cycle. Bennett (1973) showed that logically reversible computation can in principle avoid dissipation — whether higher depths are inherently more irreversible remains open.

Constructor Theory

Deutsch & Marletto (2013, 2015) define computation through constructors: systems that perform a task and retain the ability to perform it again. Constructors are high- $\rho_{\text{ac}}$ entities whose computation is reliable and repeatable.

Constructor Theory defines computation counterfactually (what transformations are possible); Epimechanics defines it dynamically (what happens). These are complementary perspectives.

The copyability gap. Constructor Theory requires that information be both distinguishable and copyable. Epimechanics requires only predictive content. A canyon's shape is not copyable as an object, but its causal constraining of water flow is real computation at Level 0. This is not a deficiency — it reflects a genuine difference in what "information" means.

The quantum case. Constructor Theory needs two tiers: classical information (distinguishable and copyable) and superinformation (distinguishable but not fully copyable, per the no-cloning theorem). Epimechanics handles both with one definition because causal encoding covers quantum states that constrain measurements without being copyable. A quantum state constrains measurement outcomes — one state's structure constraining another's trajectory — whether or not it can be cloned. This is a genuine structural advantage.

Church-Turing

The Church-Turing thesis concerns universal computation — the class of functions computable by an effective procedure. Turing computation is the algorithmic special case at the adaptive/self-referential depth. The framework is broader and does not claim that computational universality results apply across the whole spectrum. A crystal at Level 0 computes in Epimechanics' sense but is not Turing-complete. This is deliberate: restricting "computation" to Turing-complete systems would exclude most phenomena the framework addresses.

Kolmogorov Complexity

Good representations compress the data-generating process into fewer variables — short descriptions. Kolmogorov complexity formalizes minimal description length. Representations that compress well are precisely those that track causal structure, aligning Kolmogorov compression with high predictive information.

Autopoiesis

Maturana & Varela (1980) defined autopoietic systems as those that produce and maintain their own organization. $\rho_{\text{ac}}$ does similar work to organizational closure: it measures the degree to which a system's causal structure sustains itself. The difference is that Epimechanics quantifies it. Autopoietic closure corresponds roughly to the transition from Level 1 to Level 2 — where self-maintenance becomes active rather than merely replicative.

Connection to Self-Multiplication

The self-multiplication research note proposes that auto-causal structure emerges from states acting on themselves: $X_{t+1} = \mathbf{T}(X_t) \cdot X_t$ . If that proposal holds, computation has a natural origin story: a system computes when its self-multiplication dynamics produce stable internal structure ( $\rho_{\text{ac}} > 0$ ) that constrains other systems.

Eigenvectors of the transition operator with $|\lambda| = 1$ are persistent entities — stable causal constraints that survive the dynamics. The eigenvalue spectrum may correspond to the depth spectrum. This is speculative; the connection is suggestive but unproven.

Open Questions

Continuous parameterization of depth. The four-level spectrum is qualitative. A continuous parameterization (perhaps mutual information between encoder structure and constrained trajectory, normalized by complexity) has not yet been formalized.
Minimum $\rho_{\text{ac}}$ for non-trivial computation. Is there a threshold below which computation is negligible, or does the spectrum extend smoothly to zero?
Compositionality. How do computations compose? When do simple discriminations combine into complex representations? This likely connects to entity boundaries and coupling tensors.
Reversible computation and Landauer. Are higher depths inherently more irreversible? Bennett's reversible computation results suggest the answer is not obvious.
Relationship to Tononi's $\Phi$ . Integrated information (Tononi 2004) measures how much a system's parts constrain each other beyond what the parts do independently. This resembles $\rho_{\text{ac}}$ at the loop level. The formal relationship has not been established.

Research Note: Computation, Representation, and Information

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