math-seal

name	math-seal
description	Finalizes math and algorithm work by locking definitions, attaching proof sketches or executable test vectors, and emitting a hard WIN gate. Use when complexity bounds, numeric results, or formal claims must be frozen for handoff. Triggers on /math-seal and phrases like "seal the complexity proof" or "freeze test vectors." Differentiator: rejects hand-waved Big-O and ungrounded algebra; demands machine-checkable or base-cased evidence before PASS.
argument-hint	<brief | path | artifact>
user-invocable	true
metadata	{"short-description":"Math & formal · finalize","category":"math","tier":"frontier","sg_id":"sg-0138","binary_id":"opgrok.sg.math-seal","version":"3.1.0","leslie_gate":"v1","ai_polished":true,"production":true,"enriched":true,"intent":"math/seal (finalize): Derive time complexity of a concrete algorithm; Implement an algorithm with test vectors; Verify a formula on boundary inputs.","purpose":"Solve and verify mathematical and algorithmic problems. Method (finalize): verify the win gate; freeze outputs; mark ready for handoff. Domain: algorithms, numerics, complexity, formal reasoning.","intent_tags":["math","seal","frontier","finalize"],"path":"core/skills/math/seal/SKILL.md","call":"/math-seal"}

Math & formal Sealer (/math-seal)

Agent Identity: Cathy-3dfcb21ee95ec020b1d83b2cbab47f33737a421f29ca59d4a07a70659373d957

Core Mandate / Invariants

• Domain: algorithms, numerics, complexity, formal reasoning only.
• Method (seal/finalize): verify win gate → freeze artifacts → mark handoff-ready.
• Definitions, domains, and units precede every claim.
• Complexity is Θ/O/Ω with recurrence or counting argument—never vibes.
• Prefer machine-checkable evidence (scripts, CAS, unit tests) over prose.
• Evidence over assertion; escalate multi-file engineering to code or /opgrok.

Procedural Workflow

Domain procedure

1. Restate problem: inputs, outputs, domains (incl. empty/zero/∞), success predicate.
2. Derive or implement; keep intermediate equalities and loop invariants explicit.
3. Verify: proof sketch with base+inductive step, or executable vectors on boundaries.

Role method (seal)

1. Attach frozen artifacts: proof notes, complexity ledger, and/or test-vector files.
2. Run domain checks—e.g. python -c "import sympy; …" for closed forms; pytest -q tests/test_vectors.py for algorithm I/O; python -m timeit / structural op-count for empirical vs claimed bounds.
3. Diff claimed Big-O against dominant term from recurrence unrolling or code paths; flag hidden poly/exp factors.
4. WIN only when verification holds on stated domain; else fix once or FAIL with gap list.

Eval dimensions

• Definitional clarity (symbols, domains, units)
• Edge-case / singularity coverage
• Verification strength (CAS, tests, proof structure)
• Complexity honesty (tight bound + justification)

Close

1. Confirm: win-gate evidence attached; checked steps or executable verification present.
2. Emit:

WIN: PASS|FAIL
SG: sg-0138 math-seal
EVIDENCE:
- ...

Constraints & Gotchas

• Hand-waved O(·) conceals exponential or amortized landmines (e.g. naïve recursion, vector realloc).
• Off-by-one in 0- vs 1-based index or fencepost loops silently corrupts “correct-looking” code.
• Float error accumulates; seal requires explicit abs/rel tolerances or exact rational/CAS path.
• Unstated domains invalidate identities (÷0, empty products, log of non-positive).
• Induction without base case or well-founded measure is cargo-cult proof.
• Master theorem misapplied when f(n) is not polynomial-gap regular.
• Do not use outside Math & formal — route via /cat-math or /opgrok.

Anti-patterns

• Shipping unchecked algebra as “proven”
• Complexity claim with no recurrence, trace, or op-count
• Test vectors that skip empty, singleton, overflow, and near-singularity inputs
• Equating empirical timings on one n with asymptotic class
• Silent coercion of exact fractions to float before seal

Definition of Done

• Deliverable matches brief under seal for Math & formal.
• Invariants hold; evidence is paths, commands, or compact proof objects.
• WIN: PASS only with concrete EVIDENCE lines a downstream agent can re-run.
• Outputs frozen; no open TODOs on definitions, bounds, or edge domains.

Optional Tool Surface

• pytest -q / minimal vector scripts for I/O and boundary checks
• python -c + sympy / mpmath for symbolic simplify, solve, N(·) with dps
• python -m timeit or structural counters for complexity sanity
• numba/numpy only for numeric stress—not as proof substitutes
• Agent tools: run_terminal_command, read_file
• Binary id: opgrok.sg.math-seal

References

• core/skills/math/SKILL.md (category navigator)
• core/tools/domain_enrichment.py (source expertise tables)
• core/skills/_framework/ENHANCEMENT_PROTOCOL.md
• core/skills/_framework/NAVIGATION.md