Formula Derivation

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.