Derive Theoretical Result

Rigorous step-by-step derivation from axioms/first principles/theorems. Every step justified. Limiting cases checked. Final result + notation glossary.

Use When

• Formula/theorem from first principles (e.g., Euler-Lagrange from action)
• Math proof by logic from axioms
• Re-derive textbook → verify/adapt
• Extend known → more general (flat → curved spacetime)
• Self-contained → paper/thesis/report

In

• Required: Target result (equation, inequality, theorem, relation)
• Required: Starting point (axioms, postulates, prior results, Lagrangian/Hamiltonian)
• Optional: Proof technique (direct, contradiction, induction, variational, constructive)
• Optional: Notation conventions
• Optional: Known intermediate results citable w/o re-deriving

Do

Step 1: State assumptions + target

Contract before calc:

1. Axioms + postulates: Every assumption listed. Physics: symmetry group, action principle, QM postulates. Math: axiom sys + prior lemmas.
2. Target: Precise notation. Equation → both sides. Inequality → direction + equality conds.
3. Scope: Domain of validity (e.g., "non-relativistic, spinless, 3D"). State what not covered.
4. Notation: Define every symbol. Self-contained.

## Derivation Contract
- **Starting from**: [axioms, postulates, or established results]
- **Target**: [precise mathematical statement]
- **Domain of validity**: [restrictions and assumptions]
- **Notation**:
  - [symbol]: [meaning and units]
  - ...

→ Complete unambiguous statement. Notation up front.

If err: Target ambiguous/assumptions incomplete → clarify before proceed. Hidden assumptions → unreliable.

Step 2: Math toolkit

Tools + applicability:

1. Algebra: Tensor, commutator, matrix, series. Verify prereqs (convergence, invertibility).
2. Calc/analysis: ODE/PDE, integration + domain, functional derivs, contour, distributions. Verify regularity (differentiability, integrability, analyticity).
3. Symmetry/group theory: Irreps, Clebsch-Gordan, character orthogonality, Wigner-Eckart.
4. Topology/geometry (if applicable): Manifolds, bundles, connections + topo constraints (boundary terms, winding, index).
5. Identities/lemmas: Specific ones invoked (Jacobi, Bianchi, integration by parts, Stokes). State explicitly, cite by name.

## Mathematical Toolkit
- **Algebra**: [techniques and prerequisites]
- **Analysis**: [calculus tools and regularity conditions]
- **Symmetry**: [group theory tools]
- **Identities to invoke**: [list with precise statements]

→ Checklist w/ applicability verified.

If err: Unverified prereqs (e.g., term-by-term diff w/o uniform convergence) → flag gap. Prove or state as additional assumption.

Step 3: Execute w/ justification

Every step labeled + justified:

1. One op per step: No combining.
2. Justification labels:
   • [by assumption] — stated axiom/assumption
   • [by definition] — prior definition
   • [by {identity name}] — named identity (e.g., "by Jacobi identity")
   • [by Step N] — prior step
   • [by {theorem name}] — external theorem (Step 2)
3. Checkpoints (every 5-10 steps):
   • Units/dimensions consistent
   • Symmetries preserved
   • Correct transformation props
4. Branches: Case analysis → each branch labeled sub-derivation, merge.

## Derivation

**Step 1.** [Starting expression]
*Justification*: [by assumption / definition]

**Step 2.** [Result of operation on Step 1]
*Justification*: [specific reason]

...

**Checkpoint (after Step N).** Verify:
- Dimensions: [check]
- Symmetry: [check]

...

**Step M.** [Final expression = Target result]
*Justification*: [final operation]  QED

→ Linear sequence, no logic gaps. Every step verifiable.

If err: Step doesn't follow → gap. Insert intermediates or identify new assumption. No "it can be shown" unless well-known identity listed Step 2.

Step 4: Limiting cases + special values

Validate vs known:

1. Limits (≥3): Simpler prior formula (non-rel limit), trivial case (coupling=0), extreme regime (high/low T).

2. Special values: Known independent (n=1 hydrogen, d=3).

3. Symmetry: Correct under group. Scalar → invariant. Vector → transforms right.

4. Consistency: Ward identities, sum rules, reciprocity.

## Limiting Case Verification
| Case | Condition | Expected Result | Derived Result | Match |
|------|-----------|----------------|----------------|-------|
| [name] | [parameter limit] | [known result] | [substitution] | [Yes/No] |
| ... | ... | ... | ... | ... |

→ All limits + special values match. Internally consistent.

If err: Failed limit → err in derivation. Trace to first step producing fail. Common: sign, missing 2/π, wrong combinatorial coeff, wrong order of limits.

Step 5: Final w/ notation glossary

Polished:

1. Narrative: Intro para → motivation, approach, main result.
2. Body: Steps from Step 3 cleaned. Group → logical blocks w/ headings.
3. Result box: Highlighted, separated.
4. Glossary: Every symbol + meaning + units + first occurrence.
5. Assumptions summary: All in one place, postulates vs technical (smoothness, convergence).

## Final Result

> **Theorem/Result**: [precise statement with equation number]

## Notation Glossary
| Symbol | Meaning | Units | First appears |
|--------|---------|-------|---------------|
| [sym] | [meaning] | [units or dimensionless] | [Step N] |
| ... | ... | ... | ... |

## Assumptions
1. [Fundamental postulate 1]
2. [Technical assumption 1]
3. ...

→ Self-contained doc, followable start to finish w/o external refs (except cited identities + theorems).

If err: Too long (>~50 steps) → break into lemmas. Derive each, assemble main result citing lemmas.

Check

• All starting assumptions stated before first calc
• Every step labeled justification (no unjustified leaps)
• Units/dimensions consistent at every checkpoint
• ≥3 limiting cases checked + match
• Special values match known
• Result transforms correctly under stated symmetry
• Glossary defines every symbol
• No deferred "it can be shown"
• Domain of validity stated w/ result

Traps

• Hidden assumptions: Analyticity, convergence, integral existence w/o stating. Every regularity cond = assumption, declare.
• Sign errs: Most common mech err. Track at every step. Cross-check dim analysis (sign err → dim inconsistent).
• Dropped boundary terms: Integration by parts / Stokes → boundary terms vanish only under conds. State why (e.g., "field decays > 1/r at infinity").
• Order of limits: Wrong order → diff results (thermodynamic before zero-T). State order explicit + justify.
• Circular reasoning: Using result as intermediate. Subtle for "obvious" formulas. Every step from stated start, not answer familiarity.
• Notation collisions: Same symbol for diff quantities (E = energy + E-field). Glossary prevents — IF written before derivation.

→

• formulate-quantum-problem — formulate QM framework before deriving
• survey-theoretical-literature — find prior derivations for comparison