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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.

Thermodynamic Framework — Examples

1 · Prerequisites

2 · Summary

These leaf examples apply the framework's boundary, SI-unit, and sign conventions and summarize a source-reported historical experiment. They are illustrative context; none supplies another page's proof or turns measurements into a deductive demonstration of a physical postulate.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

None yet.

5 · Examples, experiments and counterexamples

ExampleConstruction: AI-adaptedVerification: Not suppliedverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)Open item page →

Classifying closed, open, and isolated system boundaries

Physical scope

Illustrative boundary classifications in the macroscopic thermodynamic model; the setups below are idealized examples, not reported measurements.

Example

As defined in Thermodynamic systems and permitted exchanges, choose the gas inside a sealed cylinder with a conducting wall and movable piston as the system. If the outer boundary allows energy transfer but no matter transfer, it is closed. Choose instead a vessel whose outlet allows water vapour to cross the boundary; it is open because that named matter species crosses. An ideal rigid, impermeable, thermally insulating enclosure with no other external energy channel is isolated. These classifications depend on the selected boundary and stipulated channels; they do not claim that a real apparatus is perfectly sealed or insulated.

ExampleConstruction: AI-adaptedVerification: AI-generatedverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)Open item page →

Pressure-volume and force-distance units

Physical scope

Unit arithmetic for the SI quantities used in the thermodynamic framework; the example does not assert a pressure-volume work law.

Example

The SI units make both a pressure-volume product and a force-distance product have units of energy. In particular, one pascal times one cubic metre is one joule, and one newton times one metre is one joule.

Facts & Assumptions

Given: The SI unit relations specified for this framework.

[F1]

SI quantities and unit conventions: 1 Pa=1 J m−3, 1 N=1 kg m s−2, and 1 J=1 kg m2 s−2.

Verification

1.1F1algebra

Multiplying the pascal unit by the cubic-metre unit gives (J m−3)m3=J.

2.1F1algebra∎

Multiplying the newton unit by the metre unit gives (kg m s−2)m=kg m2 s−2=J.

ExampleConstruction: Literature-sourcedVerification: AI-generatedverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)Open item page →

Converting work-on and work-by conventions

Physical scope

Algebraic conversion between two signed-work conventions for the closed process class of the adopted first-law postulate.

Example

Some sources take work done by the system as positive and write the first law as ΔU=Q−Wby. The present framework takes work done on the system as positive. Setting Wby[P]:=−Won[P] converts one equation to the other.

Facts & Assumptions

Given: An admitted core closed process record P∈Pcl(s) with the equilibrium endpoints and energy exclusions of [F1], and the two declared sign conventions.

[F1]

First-law energy balance for closed process records: For an admitted closed record with the stated energy exclusions in the process-selected chart, Uc(xP+)−Uc(xP−)=Q[P]+Won[P].

[F2]

Thermodynamic process records and transfer conventions: Won[P]>0 means mechanical work is done on the system.

Verification

1.1givenF2algebra

Define the work-by-positive value by Wby[P]:=−Won[P], so Won[P]=−Wby[P].

2.1F1step 1.1algebra∎

Substituting the equality from step 1.1 into [F1] gives Uc(xP+)−Uc(xP−)=Q[P]−Wby[P].

ExampleConstruction: Literature-sourcedVerification: Not suppliedverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)Open item page →

Joule's 1850 paddle-wheel report

Physical scope

A source-based example of a historical calorimetry report and its apparatus-specific work-heat estimate; it is not a modern remeasurement or a proof of a universal law.

Example

Joule's 1850 report describes falling lead weights driving a paddle wheel in a sealed copper vessel of water. The water apparatus has rotating brass arms working between stationary vanes; a boxwood interruption reduces heat conduction along the shaft. The report describes thermometer calibration and resting trials arranged to estimate heat exchanged with the surroundings, alongside the friction trials. These are source-reported apparatus and procedure details, not measurements made for this library.

The water-series tabulation reports a nominal mechanical equivalent of 772.692 in the paper's historical convention: foot-pounds of falling-weight work compared with the heat needed to raise one pound of water by one Fahrenheit degree. The concluding proposition rounds that figure to 772 foot-pounds for raising one pound of water, weighed in vacuo and taken between 55°F and 60°F, by 1°F. Joule states that vibration and sound may make the estimate slightly high. This item adds no modern uncertainty interval or statistical inference. The historical result is empirical evidence from a particular apparatus and correction chain; it does not deductively establish the adopted first-law postulate for all systems.

Sources