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.
Boundary inviscid and nonlinear low-Mach limits require separate estimates
Remark
Assume Countable Choice. The inviscid-limit result in this batch assumes a fixed periodic domain and smooth solutions on a common local time interval; it gives no wall inviscid-limit or boundary-layer estimate. The acoustic result is an exact linear tangent-system theorem, and the scaling proposition is a chain-rule calculation. Neither result supplies a nonlinear low-Mach convergence theorem or a finite-amplitude error estimate. Those claims require separate hypotheses and estimates that are not established by these items.
Remark references
Facts & Assumptions
Given: The exact theorem scopes and conclusions of the three named batch items.
Controlled periodic smooth inviscid limit: the proved and estimates concern only smooth solutions on a fixed periodic domain and a common local interval; the statement explicitly excludes a wall inviscid limit and boundary-layer estimate.
Global smooth periodic linear acoustics: the result concerns an exact linear tangent system on the normalized periodic torus and does not assert equality with finite-amplitude nonlinear fluid evolution.
Fluid equation scalings and dimensionless numbers: the dimensionless coefficients and equation symmetries follow by chain rule; the proposition asserts no continuum or empirical similarity limit.
Countable Choice: every countable family of nonempty sets has a choice function (The Axiom of Countable Choice ()).
Proof
By [F1], the inviscid estimate is established on a fixed periodic domain and common local smooth interval, and the boundary case is outside its stated conclusion. Therefore it does not imply a wall inviscid-limit theorem or a boundary-layer estimate.
By [F2] and [F3], the acoustic result solves only the displayed linear tangent equations and the scaling result is an exact change of variables. Neither statement contains a nonlinear convergence conclusion or controls a finite-amplitude remainder. The remark consequently records these limits of coverage and requires separate hypotheses and estimates for such stronger claims.