advanced-mathematics-computation

Files (5)

• SKILL.md
• problem-families.md · references/
• vvuq-ladder.md · references/
• validate_experiment.py · scripts/
• experiment-plan.md · templates/

SKILL.md (first lines)

# Advanced Mathematics Computation

Use this skill when a task asks computation to investigate a deep theorem, conjecture, invariant, counterexample, or proof strategy.

## Core workflow

1. **State the exact mathematical target.** Write the domain, objects, quantifiers, assumptions, and what would count as a proof, counterexample, lemma, or calibration result.
2. **Separate evidence levels.** Label every output as calibration, finite empirical evidence, certified finite result, research lead, theorem candidate, or proof. Never promote a finite scan to a universal claim.
3. **Choose the computational lane.** Select exhaustive enumeration, symbolic algebra, interval/ball arithmetic, numerical simulation, optimization, formal verification, or a hybrid. Read [problem-families.md](references/problem-families.md) for domain-specific defaults.
4. **Design controls before running.** Specify seeds, bounds, precision, discretization, checkpoint format, negative controls, independent implementation, and failure conditions.
5. **Implement reproducibly.** Use deterministic metadata, atomic checkpoints, machine-readable summaries, and a manifest of source/data hashes. Prefer streaming algorithms when output size is large.
6. **Run verification, validation and uncertainty checks.** Separate implementation verification, external-model validation, and numerical/parameter/stochastic uncertainty. Read [vvuq-ladder.md](references/vvuq-ladder.md) for simulations or high-consequence computation.
7. **Run an adversarial audit.** Test precision sensitivity, resolution sensitivity, boundary effects, generator bias, degenerate cases, negative controls, and an alternative implementation or special case when feasible. Inspect the most surprising result first.
8. **Write the proof-gap ledger.** For each promising result, name the exact missing theorem obligation: universal quantifier, continuum limit, completeness count, independence proof, error bound, or model-transfer theorem.
9. **Report honestly.** Use `templates/experiment-plan.md`, preserve a reproducibility manifest and rerun command, and include a formal limitations section and references. Route large ensemble, design-space, or certification-like work to `eureka-computational-lab` when available.

## Domain routing

For RH or L-functions, prefer validated arithmetic, zero isolation, and a completeness count. For PDEs, require resolution and time-step refinement, conservation diagnostics, and a distinction between numerical concentration and singularity. For arithmetic geometry, pair point counts with certified rank or cycle calculations. For complexity, state the model and lower-bound target before collecting timing data. For gauge theory, track finite volume, lattice spacing, autocorrelation, renormalization, and continuum extrapolation.

## Quality gates

Do not claim a proof when any of the following is missing: a precise quantified statement, controlled numerical error, independent reproduction, complete coverage of the required domain, or a mathematical argument connecting the computation to the theorem. Do not call a simulation validated merely because it converges; record which VVUQ layers were completed or unavailable. Run `scripts/validate_experiment.py` to check an experiment plan before execution.

From minagayid/innovator, MIT.

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