formula-derivation¶
formal-modeling
Formula Derivation: Research Theory Line Construction¶
Build an honest derivation package, not a fake polished theorem story.
Constants¶
- DEFAULT_DERIVATION_DOC =
DERIVATION_PACKAGE.mdin project root - STATUS =
COHERENT AS STATED | COHERENT AFTER REFRAMING / EXTRA ASSUMPTION | NOT YET COHERENT
Context: $ARGUMENTS¶
Goal¶
Produce exactly one of: 1. a coherent derivation package for the original target 2. a reframed derivation package with corrected object / assumptions / scope 3. 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:
1. a file path explicitly specified by the user
2. a derivation draft already referenced in local notes
3. 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.