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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.

Two types on one meter scale

Example

Take two types with fixed totals $T_{1} = T_{2} = 1$ and a common meter satisfying the compatibility hypotheses of Finite-family calibration by a common meter, including its shared equal-mass comparison $⪯_{Y}$ . Let type-1 states $a_{0}, a_{1}$ have normalized values $0, 1$ , and type-2 states $b_{0}, b_{1}$ have normalized values $1 / 3, 3$ . Using the shared raw state tags of that theorem, let $C_{0} = e_{a_{1, a_{1}}} + e_{a_{2, b_{0}}}$ and $D_{0} = e_{a_{1, a_{0}}} + e_{a_{2, b_{1}}}$ . Their fixed-meter encodings are $C = C_{0} + 2 e_{ζ_{0}}$ and $D = D_{0} + 2 e_{ζ_{0}}$ , each of mass four. The calibrated values are $4 / 3$ for $C$ and $3$ for $D$ ; the common-meter comparison orders $C ⪯_{Y} D$ and, after common-meter cancellation, $C_{0} ⪯_{Y} D_{0}$ .

Facts & Assumptions

[A1]

The fixed type totals and values in the Example: one unit of each type, with $σ_{1} (a_{1}) = 1$ , $σ_{1} (a_{0}) = 0$ , $σ_{2} (b_{0}) = 1 / 3$ , and $σ_{2} (b_{1}) = 3$ .

[F1]

Finite-family calibration by a common meter: under its finite common-meter compatibility assumptions, including the defined shared relation on $Y$ , a compound comparison is equivalent to comparing the sum of its typewise calibrated values.

[F2]

Additive extensive representation of compound comparison: normalized values are real and may lie outside $[0, 1]$ for states other than the two references.

Verification

Given: [A1], [F1], and [F2].

1.1A1F1F2algebra

Both compounds have type-total vector $(1, 1)$ , so [F1] applies. Their calibrated values are $Σ (C) = 1 + 1 / 3 = 4 / 3$ and $Σ (D) = 0 + 3 = 3$ . Since $4 / 3 \leq 3$ , [F1] gives $C ⪯_{Y} D$ . The theorem also cancels their common $2 e_{ζ_{0}}$ , giving $C_{0} ⪯_{Y} D_{0}$ . The value $3$ is permitted by [F2]; the representation theorem imposes no unit-interval bound on arbitrary states.

2.1A1F1step 1.1algebra∎

Reversing the comparison would require $3 \leq 4 / 3$ , which is false; hence this particular comparison is strict. No separate per-type numerical reading is treated as a proof of common-meter compatibility; that remains the stated hypothesis.

Depends on

Dependency tree · 2 levels

Sources

Lieb and Yngvason, The Physics and Mathematics of the Second Law of Thermodynamics, §II.E