DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Finite memory and retained-register reset model

Physical scope

A finite-state memory and retained register with a specified probabilistic reset map; the definition does not assert a particular apparatus, reset dynamics, or measured work.

Definition

Let $M$ be a finite memory-state set and $R$ a finite retained-register set. A joint preparation is a probability law $p (m, r) \geq 0$ with $\sum_{m, r} p (m, r) = 1$ . Write $a_{r} = \sum_{m} p (m, r)$ and, for $a_{r} > 0$ , $q (m ∣ r) = p (m, r) / a_{r}$ . Use the finite Shannon entropy in nats and the conditional entropy $H (M ∣ R)$ exactly as defined by Conditional entropy of finite memory given side information. For a fixed target memory state $m_{0}$ , a reset is the endpoint map $p^{'} (m, r) = 1_{{m = m_{0}}} a_{r}$ : it places the memory at $m_{0}$ while leaving the register marginal unchanged. The definition alone assigns no physical entropy, work cost, heat transfer, or implementation.

Depends on

Conditional entropy of finite memory given side information

Finite memory and retained-register reset model

Definition

Depends on

Dependency tree · 2 levels

Sources