formula-derivation

name formula-derivation description Structures and derives research formulas when the user wants to 推导公式, build a theory line, organize assumptions, turn scattered equations into a coherent derivation, or rewrite theory notes into a paper-ready formula document. Use when the derivation target is not yet fully fixed, the main object still needs to be chosen, or the user needs a coherent derivation package rather than a finished theorem proof. argument-hint [problem-goal-current-formulas-or-notes] allowed-tools Read, Write, Edit, Grep, Glob Formula Derivation: Research Theory Line Construction Build an honest derivation package, not a fake polished theorem story. Constants DEFAULT_DERIVATION_DOC = DERIVATION_PACKAGE.md in project root STATUS = COHERENT AS STATED | COHERENT AFTER REFRAMING / EXTRA ASSUMPTION | NOT YET COHERENT Context: $ARGUMENTS Goal Produce exactly one of: a coherent derivation package for the original target a reframed derivation package with corrected object / assumptions / scope a blocker report explaining why the current notes cannot yet support a coherent derivation Inputs Extract and normalize: the target phenomenon, formula, relation, or theory line the intended role of the derivation: exact identity / algebra proposition / local theorem approximation mechanism interpretation explicit assumptions notation and definitions any user-provided formula chain, sketch, messy notes, or current draft nearby local theory files if the request points to them desired output style if specified: internal alignment note paper-style theory draft blocker report If the target, object, notation, or assumptions are ambiguous, state the exact interpretation you are using before deriving anything. Workflow Step 1: Gather Derivation Context Determine the target derivation file with this priority: a file path explicitly specified by the user a derivation draft already referenced in local notes DERIVATION_PACKAGE.md in project root as the default target Read the relevant local context: the chosen target derivation file, if it already exists any local theory notes, formula drafts, appendix notes, or files explicitly mentioned by the user Extract: target formula / theory goal current formula chain assumptions notation known blockers desired output mode Step 2: Freeze the Target State explicitly: what is being explained, derived, or supported whether the immediate goal is: identity / algebra proposition approximation interpretation what the derivation is expected to output in the end Do not start symbolic manipulation before this is fixed. Step 3: Choose the Invariant Object Identify the single quantity or conceptual object that should organize the derivation. Typical possibilities include: objective / utility / loss total cost / energy / welfare conserved quantity / state variable expected metric / effective rate / effective cost If the current notes start from a narrower quantity, decide explicitly whether it is: the true top-level object a proxy a local slice an approximation Do not let a convenient proxy silently replace the actual conceptual object. Step 4: Normalize Assumptions and Notation Restate: all assumptions all symbols regime boundaries or special cases which quantities are fixed, adaptive, or state dependent Identify: hidden assumptions undefined notation scope ambiguities whether the current formula chain already mixes exact steps with approximations Preserve the user's original notation unless a cleanup is necessary for coherence. If you adopt a cleaner internal formulation, keep that as a derivation device rather than silently replacing the user's target. Step 5: Classify the Derivation Steps For every nontrivial step, determine whether it is: identity : exact algebraic reformulation proposition : a claim requiring conditions approximation : model simplification or surrogate interpretation : prose-level meaning of a formula Never merge these categories without signaling the transition. If one part is only interpretive, do not present it as if it were mathematically proved. Step 6: Build a Derivation Map Choose a derivation strategy, for example: definition -> substitution -> simplification primitive law -> intermediate variable -> target expression global quantity -> perturbation -> decomposition exact model -> approximation -> interpretable closed form general dynamic object -> simplified slice -> local theorem -> return to general case Then write a derivation map: target formula or theory line required intermediate identities or lemmas which assumptions each nontrivial step uses where approximations enter where special-case and general-case regimes diverge or collapse If the derivation needs a decomposition, derive it from the chosen global quantity. Do not make a split appear magically from one local variable itself. Step 7: Write the Derivation Document Write to the chosen target derivation file. If the target derivation file already exists: read it first update the relevant section do not blindly duplicate prior content If the user does not specify a target, default to DERIVATION_PACKAGE.md in project root. Do NOT write directly into paper sections or appendix .tex files unless the user explicitly asks for that target. The derivation package must include: target status invariant object assumptions notation derivation strategy derivation map main derivation steps remarks / interpretations boundaries and non-claims Writing rules: do not hide gaps with words like "clearly", "obviously", or "similarly" define every symbol before use mark approximations explicitly separate derivation body from remarks if the true object is dynamic or state dependent but a simpler slice is analyzed, say so explicitly if a formula line is only heuristic, label it honestly Step 8: Final Verification Before finishing the target derivation file, verify: the target is explicit the invariant object is stable across the derivation every assumption used is stated each formula step is correctly labeled as identity / proposition / approximation / interpretation the derivation does not silently switch objects special cases and general cases still belong to one theory line boundaries and non-claims are stated If the derivation still lacks a coherent object, stable assumptions, or an honest path from premises to result, downgrade the status and write a blocker report instead of forcing a clean story. Required File Structure Write the target derivation file using this structure: # Derivation Package ## Target [what is being derived or explained] ## Status COHERENT AS STATED / COHERENT AFTER REFRAMING / NOT YET COHERENT ## Invariant Object [top-level quantity organizing the derivation] ## Assumptions - ... ## Notation - ... ## Derivation Strategy [chosen route and why] ## Derivation Map 1. Target depends on ... 2. Intermediate step A uses ... 3. Approximation enters at ... ## Main Derivation Step 1. ... Step 2. ... ... ## Remarks and Interpretation - ... ## Boundaries and Non-Claims - ... ## Open Risks - ... Output Modes If the derivation is coherent as stated Write the full structure above with a clean derivation package. If the notes are close but not coherent yet Write: the exact mismatch the corrected invariant object, assumption, or scope the reframed derivation package If the derivation cannot be made coherent honestly Write: Status: NOT YET COHERENT the exact blocker: missing object unstable assumptions notation conflict unsupported approximation theorem-level claim without enough conditions what extra assumption, reframe, or intermediate derivation would be needed Relationship to proof-writer Use formula-derivation when the user says things like: “我不知道怎么起这条推导主线” “这个公式到底该从哪个量出发” “帮我把理论搭顺” “把说明文档变成可写进论文的公式文档” “这几段公式之间逻辑不通” Use proof-writer only after: the exact claim is fixed the assumptions are stable the notation is settled and the task is now to prove or refute that claim rigorously Chat Response After writing the target derivation file, respond briefly with: status whether the target survived unchanged or had to be reframed what file was updated Key Rules Never fabricate a coherent derivation if the object, assumptions, or scope do not support one. Prefer reframing the derivation over overclaiming. Separate assumptions, identities, propositions, approximations, and interpretations. Keep one invariant object across special and general cases whenever possible. Treat simplified constant-parameter cases as analysis slices, not as the conceptual main object. If uncertainty remains, mark it explicitly in Open Risks ; do not hide it in polished prose. Coherence matters more than elegance.