Formula Derivation

Use this skill when the task is not merely to prove a finished theorem, but to build the derivation itself:

• define the right object,
• decide what should be assumed,
• determine what is identity vs proposition vs approximation,
• connect simple and general regimes without splitting into two unrelated stories,
• and turn messy notes into a derivation line that can later be written into a paper.

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, stay focused on clarifying the derivation boundary, assumptions, and missing steps rather than pretending the proof is already complete.

Core Principle

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.

What to Produce

Prefer one of these outputs:

1. a mainline derivation note for internal alignment;
2. a paper-style theory draft with tighter narrative;
3. a blocker report if the current notes cannot support a coherent derivation.

Workflow

1. Freeze the Target

State explicitly:

• what phenomenon is being explained;
• what claim is being supported;
• whether the goal is:
  • identity / algebra,
  • local comparative statics,
  • approximation,
  • or mechanism interpretation.

Do not start symbolic manipulation before this is fixed.

2. Choose the Invariant Object

Find the single quantity that should remain meaningful across regimes.

Examples:

• objective / loss / utility
• total energy / cost / welfare
• state variable / conserved quantity / effective rate
• expected performance metric

If the current notes use a narrower quantity (c_i, throughput, delay, CW, etc.), decide whether it is:

• the true top-level object,
• or only a proxy / slice / approximation.

3. Put Assumptions and Notation First

Before deriving, list:

• assumptions;
• notation;
• regime boundaries;
• which quantities are fixed and which are state dependent.

Do not introduce hidden assumptions mid-derivation unless they are clearly marked as extra local assumptions.

4. Classify Every Step

Every nontrivial part of the derivation must be labeled mentally as one of:

• identity: exact algebraic reformulation;
• proposition: a claim requiring conditions;
• approximation: model simplification or surrogate;
• interpretation: prose-level explanation of what the formula means.

Never mix these without signaling the change.

5. Derive from the Global Quantity When Splitting Costs

If the goal is to split a quantity into components, start from the global quantity and then differentiate / decompose.

Pattern:

1. define the global quantity, e.g. W = \sum_j \Gamma_j;
2. perturb one local variable, e.g. c_i;
3. compute the marginal social effect;
4. split the result into:
   • direct term,
   • indirect term,
   • or private / external terms if that distinction is part of the model.

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.

6. Keep Special Cases and General Cases in One Line

If the theory must cover both a simplified regime and a more general regime, do not write two unrelated stories.

Use this pattern:

• same invariant object across all regimes;
• special case: some terms vanish or collapse;
• general case: the same object gains extra structure.

This prevents the simple case from looking like an exception and the general case from looking like a different theory.

7. Treat Simplified Parameters as Analysis Slices

If the true object is state dependent, adaptive, vector-valued, or otherwise complicated, but a theorem needs a simpler parameterization, write:

• the general object first;
• then define the simpler case as a tractable slice.

Use language such as:

• frozen-parameter approximation;
• constant-coefficient slice;
• local linearization;
• reduced-order case.

Do not let the simplified case silently replace the real conceptual object.

8. Separate Main Text from Remarks

For derivations intended for papers:

• the main derivation should contain only equations and immediate mathematical consequences;
• explanatory prose, intuition, scenario reading, and caveats should be moved to Remark / Discussion / Scope paragraphs.

If a section starts to read like an internal lecture note, split it into:

• derivation body;
• remark.

9. Write Boundaries Explicitly

At the end, state:

• what the derivation actually proves;
• what remains approximation;
• what should not be claimed.

Especially guard against:

• turning a local proposition into a universal theorem;
• letting an interpretation sound like a proof;
• hiding a proxy as if it were the true quantity.

Common Derivation Patterns

When the user is unsure how to start, try one of these common patterns:

1. Definition -> substitution -> simplification Use when the target formula is mostly algebraic.
2. Global quantity -> perturbation -> decomposition Use when the target needs direct / indirect, private / external, or local / global splitting.
3. Primitive law -> intermediate variable -> target expression Use when deriving from a physical principle, conservation law, or probabilistic identity.
4. Exact model -> approximation -> interpretable closed form Use when the exact formula is too heavy and a paper needs a usable surrogate.
5. 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.

Recommended Output Structure

For an internal derivation note:

1. Target
2. Invariant object
3. Assumptions and notation
4. Main derivation
5. Regime interpretation
6. Approximations and open risks

For a paper-style theory section:

1. Unified object
2. Formal proxy and assumptions
3. Special-case to general-case decomposition
4. Reward / objective reformulation
5. Local theorem or proposition
6. State-dependent extension
7. Scope and non-claims

Boundary With Finished Proofs

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”

The derivation is usually mature enough for a finished theorem-proof package only after:

• the exact claim is already fixed,
• the assumptions are stable,
• and the task is now to prove or refute that claim rigorously.