formula-derivation

# 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: 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 t