formula-derivation skill

Formula Derivation: Research Theory Line Construction

Build an honest derivation package, not a fake polished theorem story.

Constants

• DEFAULTDERIVATIONDOC = 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:

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

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