Derivation Step Checker

Audit the derivation the user actually supplied. Keep algebraic validity separate from physical validity: a symbolic step may be correct while the invoked law is inapplicable.

Inputs

Request only missing information that blocks the audit:

• the derivation with one transformation per numbered step;
• definitions and dimensions of symbols;
• the physical regime and assumptions (frame, system boundary, constancy, quasi-static limits, approximations);
• the desired depth: first failing step or full audit.

Do not invent omitted steps or assumptions. Mark them unknown and explain what evidence would resolve them.

Workflow

1. Normalize without repairing. Transcribe each step as a before-equation and after-equation. Preserve the user's signs, subscripts, frames, and approximations. If a line performs several operations, split it into auditable substeps and label the split as editorial.
2. Name the move. For every step, record:
   • algebraic operation;
   • physical law, definition, constitutive relation, or approximation used;
   • applicability conditions required by that move.
3. Check algebra. Test whether the two equation residuals are equivalent, allowing an explicit nonzero multiplier. Check denominator restrictions, branch choices introduced by roots, sign reversals, and discarded solutions. A computer-algebra result is evidence, not permission to ignore domain assumptions.
4. Check dimensions. Require both sides of every equation and terms joined by addition to have the same dimensions. Dimensionally valid does not imply physically valid.
5. Check physical continuity. Re-evaluate the law's conditions at this exact step. In particular, look for assumptions that become invalid midway:
   • inertial-frame laws after a frame change;
   • fixed-mass momentum equations for variable-mass systems;
   • conservation laws after the system boundary opens;
   • quasi-static or equilibrium relations during a finite-rate process;
   • constant material parameters outside their stated regime;
   • approximations retained beyond their small-parameter range.
6. Locate the first failure. Do not let later algebra conceal an earlier invalid step. Classify each line as pass, fail, or unknown, and distinguish algebra, dimensions, and physics.
7. Report minimally sufficient repairs. Show the smallest correction that makes the failed step auditable. Do not rewrite the whole solution unless the user explicitly asks after seeing the audit.

Deterministic checker

When the derivation can be represented with scalar symbolic expressions, create a JSON input following references/input-schema.md, then run:

python3 scripts/check_derivation.py derivation.json

The checker verifies a constrained expression grammar, residual equivalence up to a declared factor, required nonzero declarations, and dimensions. Every non-constant factor of the declared factor must itself be declared nonzero: declaring m1 and m2 does not cover a division by m1 - m2, and you add "m1 - m2" to nonzero only when the user has established it, never to make the check pass. Equations written as ... = 0 check normally, and a symbolic exponent such as 10**m must be dimensionless. --selftest runs the checker's regression cases. It deliberately cannot decide whether a physical law applies; review that from the stated regime and assumptions.

If Python or SymPy is unavailable, perform the same checks explicitly and mark them as manual. Never claim a script passed unless it was actually run successfully.

Output format

Start with the first failing or unresolved step, then give the table:

Step	Algebra	Dimensions	Physics	Law / move	Required conditions	Finding
1	pass/fail/unknown	pass/fail/unknown	pass/fail/unknown	...	...	...

Finish with:

• First break: earliest fail, or none found;
• Unresolved assumptions: facts still needed;
• Minimal repair: one local correction or next diagnostic action;
• Scope limit: what was not established (for example, initial model validity or final numerical plausibility).

Boundaries

• This skill audits an existing derivation; it does not replace a student's own assessed derivation. Deriving a formula from principles in the first place belongs to the neighbouring skill concept-to-formula-deriver.
• Do not treat symbolic equivalence as proof of physical correctness.
• Do not infer a missing law from the desired answer.
• Do not hide domain restrictions, frame labels, approximation order, or uncertainty behind a green status.
• For a final-answer-only plausibility review, use answer-plausibility-checker instead; for isolated unit checking, use dimensional-analysis-checker.

变更记录 / Changelog

版本	日期	变更	类型
0.1.1	2026-09-26	修 check_derivation.py：… = 0 形式不再误判量纲不一致；10**m 这类符号指数要求无量纲；nonzero 可写表达式，按 factor_list 逐个因子核对，声明 m1、m2 不再放行除以 m1−m2；补 --selftest 与未知键报错；分流改为实名；用例 Case 1 更正为准静态与绝热可以同时成立	patch
0.1.0	2026-09-05	初始版本	minor