Physics theoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)

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Pathwise energy balance for a finite driven protocol

Physical scope

An exact energy bookkeeping identity for every path of the stated finite-state, finite-stage model; it is not a claim that all heat or work processes are represented by this model.

Statement

For every path $ω = (i_{0}, \dots, i_{m})$ of Finite driven protocol and path work, the endpoint energy change is $E_{m} (i_{m}) - E_{0} (i_{0}) = W (ω) + Q_{i n} (ω)$ , with the stated sign convention.

Facts & Assumptions

Given: A nonempty finite state set, finite stage energies, a finite path, work positive into the system, and fixed-stage energy increment positive into the system, exactly as defined in Finite driven protocol and path work.

[F1]

Finite driven protocol and path work: at stage $k$ , add and subtract the pre-transition energy $E_{k} (i_{k - 1})$ between endpoint energies. The fixed-stage increment $Q_{i n}$ is interpreted as heat only on the supplier's reservoir-consistent subclass or under a separately supplied transfer model; its formal definition applies to every path.

Proof

Given: [F1].

1.1givenF1algebra

Insert the intermediate energies and telescope: $E_{m} (i_{m}) - E_{0} (i_{0}) = \sum_{k = 1}^{m} [E_{k} (i_{k}) - E_{k - 1} (i_{k - 1})]$ .

2.1givenF1step 1.1algebra∎

For each $k$ , split the summand as $E_{k} (i_{k - 1}) - E_{k - 1} (i_{k - 1}) + E_{k} (i_{k}) - E_{k} (i_{k - 1})$ . Summing the first terms gives $W (ω)$ and the second terms give $Q_{i n} (ω)$ .

Depends on

Finite driven protocol and path work

Pathwise energy balance for a finite driven protocol

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources