Structure and derive research formulas when the user wants to 수식 유도, derive a theory line, build equations from a problem statement, clarify assumptions, separate formal derivation from remarks, or turn messy theory notes into a paper-ready derivation skeleton. Use for research-style formula development, not for fully rigorous theorem proving once the claim is already fixed.
Use this skill when the task is not merely to prove a finished theorem, but to build the derivation itself:
Do not use this skill as a replacement for strict proof writing once the exact claim is already fixed and the user wants a theorem-proof package. In that case, hand off to proof-writer.
The derivation must be built around one invariant object. Do not start from scattered formulas. Start from the object that survives across regimes, then derive proxies, decompositions, and interpretations from it.
Prefer one of these outputs:
State explicitly:
Do not start symbolic manipulation before this is fixed.
Find the single quantity that should remain meaningful across regimes.
Examples:
If the current notes use a narrower quantity (c_i, throughput, delay, CW, etc.), decide whether it is:
Before deriving, list:
Do not introduce hidden assumptions mid-derivation unless they are clearly marked as extra local assumptions.
Every nontrivial part of the derivation must be labeled mentally as one of:
Never mix these without signaling the change.
If the goal is to split a quantity into components, start from the global quantity and then differentiate / decompose.
Pattern:
W = \sum_j \Gamma_j;c_i;Do not present the decomposition as if it appeared magically from one local variable itself. The split must come from the effect of changing that variable on the chosen global quantity.
If the theory must cover both a simplified regime and a more general regime, do not write two unrelated stories.
Use this pattern:
This prevents the simple case from looking like an exception and the general case from looking like a different theory.
If the true object is state dependent, adaptive, vector-valued, or otherwise complicated, but a theorem needs a simpler parameterization, write:
Use language such as:
Do not let the simplified case silently replace the real conceptual object.
For derivations intended for papers:
If a section starts to read like an internal lecture note, split it into:
At the end, state:
Especially guard against:
When the user is unsure how to start, try one of these common patterns:
Definition -> substitution -> simplification Use when the target formula is mostly algebraic.
Global quantity -> perturbation -> decomposition Use when the target needs direct / indirect, private / external, or local / global splitting.
Primitive law -> intermediate variable -> target expression Use when deriving from a physical principle, conservation law, or probabilistic identity.
Exact model -> approximation -> interpretable closed form Use when the exact formula is too heavy and a paper needs a usable surrogate.
General dynamic object -> frozen slice -> theorem -> return to general case Use when the real system is adaptive or state dependent, but the proof needs a simpler slice.
For an internal derivation note:
For a paper-style theory section:
proof-writerUse formula-derivation when the user says things like:
Use proof-writer only after: