Alphabeta MathHow 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.
Scope at phase boundaries and in the thermodynamic limit
Interpretive limits for the phase-branch, coexistence, Clapeyron, and compact-mixture statements on this page; this remark establishes no universal critical or thermodynamic-limit result.
Remarks
The branch theorems on this page require the stated smooth local charts. Here coexistence means an attained global entropy-maximizing endpoint on the exact feasible set of Phase branches and coexistence domains; equality of intensive phase data follows only for the bilateral exchanges assumed in the branch theorems and does not itself imply loss of chart regularity. The smooth-branch conclusions apply only on their specified C1 charts and interior composition domains. They supply no conclusion where a branch chart fails to be C1, a species amount reaches the composition boundary, or the specified coexistence-and-affinity Jacobian lacks its required nonsingular minor. In particular, Local Clapeyron slope on a regular pure-phase coexistence branch requires a nonzero molar-volume jump; when that jump vanishes, its local graph argument and quotient formula do not apply.
The compact-mixture result Attained entropy-maximizing mixtures on compact phase domains requires a compact phase-generator domain, a continuous entropy map, a fixed barycenter, and feasibility of all probability mixtures with that barycenter. It supplies no attainment theorem for noncompact domains. Likewise, Finite-state thermodynamic-limit contract specifies the sequence and normalization for a limiting claim but proves no limit exists and no derivatives commute with that limit.
A finite-state partition sum is finite and positive at each finite system size. A singular phase behavior therefore needs a separately specified limit or another model with a separately established domain and regularity analysis. This page does not assert a universal thermodynamic-limit theorem, ensemble equivalence, critical exponent, or physical applicability of the van der Waals equation.
The local homogeneity used by Local Clapeyron slope on a regular pure-phase coexistence branch may be independently stipulated in a phase model or supplied on the separately designated common-law bulk branch under the local Euler theorem's inverse-chart and common-reference conditions. A constructed local Gibbs model need not be embedded in a global bulk cone to use the general conditional result. The added branch supplies no universal finite-size, long-range or gravitational extensivity and does not remove any smoothness, exchange or nonzero-volume-jump condition.