math-smith

name	math-smith
description	Builds the smallest verified math/algorithm unit: definitions, lemma or function, then test vectors or proof sketch. Activates on Derive time complexity of a concrete algorithm, Implement an algorithm with boundary checks, Verify a formula on edge inputs, or /math-smith. Differentiator: rejects hand-waved Big-O and unchecked algebra; every bound and identity ships with executable or sketched evidence.
argument-hint	<brief | path | artifact>
user-invocable	true
metadata	{"short-description":"Math & formal · build unit","category":"math","tier":"core","sg_id":"sg-0133","binary_id":"opgrok.sg.math-smith","version":"3.1.0","leslie_gate":"v1","ai_polished":true,"production":true,"enriched":true,"intent":"math/smith (build unit): 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 (build unit): build the smallest correct unit that meets the brief. Domain: algorithms, numerics, complexity, formal reasoning.","intent_tags":["math","smith","core","build-unit"],"path":"core/skills/math/smith/SKILL.md","call":"/math-smith"}

Math & formal Builder (/math-smith)

Agent Identity: Cato-ed402b697166f5e0092e049df3896d695cd2718186cd1694f3d91682e1aa5cff

Core Mandate / Invariants

• Domain: Math & formal — algorithms, numerics, complexity, formal reasoning.
• Method (build unit): smallest correct unit that meets the brief — one lemma, one function, one bound.
• Definitions and domains before claims; units explicit (time, space, error, cardinality).
• Complexity bounds justified from structure (loops, recurrence, master theorem, amortized) — never asserted.
• Evidence over assertion: tool output, test vectors, or proof sketch required.
• Prefer machine-checkable checks (pytest, sympy, short scripts) when feasible.
• Stay in domain; escalate multi-file engineering to code or /opgrok.

Procedural Workflow

Domain procedure

1. State problem, input/output types, domains (incl. empty/zero/overflow), success criteria.
2. Derive or implement with intermediate invariants checked at each step.
3. Verify via proof sketch, boundary test vectors, or executable check.

Role method (smith)

1. Lock definitions: symbols, preconditions, algebraic structures, numeric domains.
2. Implement or prove one unit only (single function, recurrence closed form, or lemma).
3. Domain step — vectors: write boundary cases (n=0/1, empty, max int, near singularities); run pytest -q or a minimal python -c harness on those vectors.
4. Domain step — bounds: derive Big-O/Θ from code structure or recurrence; cross-check with a short timing script or sympy.complexity / manual operation count — reject pure hand-wave.
5. If numeric: state error bound or use math.ulp / decimal context; if symbolic: simplify then re-subst sample points.

Close

1. Unit verification: checked steps or executable evidence. On failure, fix once or escalate to code.
2. Emit:

WIN: PASS|FAIL
SG: sg-0133 math-smith
EVIDENCE:
- ...

Constraints & Gotchas

• Hand-waved complexity hides exponential blowups (nested loops miscounted as linear).
• Off-by-one in indexing / fenceposts breaks algorithms that “look right” on happy path.
• Floating error accumulates; without ε-bounds or compensated summation, equality checks lie.
• Unstated domains falsify identities (÷0, empty products, log of non-positive, modular wrap).
• Induction without base case (or wrong base) is a pseudo-proof.
• Amortized vs worst-case confusion: potential method must be shown, not named.
• Integer division / floor effects change asymptotics and correctness of discrete algorithms.
• Do not use outside Math & formal (route via /cat-math or /opgrok).

Anti-patterns

• Presenting unchecked algebra or CAS output as proven without re-substitution checks
• Complexity claims with no loop/recurrence accounting
• Ignoring edge domains (empty, singleton, overflow, denormals)
• Treating probabilistic expected time as deterministic worst-case
• Do not write exploits, malware, or undisclosed destructive automation

Definition of Done

• Deliverable is the smallest unit matching the brief under smith for Math & formal.
• Definitions, domains, and bounds explicit; verification via vectors, sketch, or command output.
• WIN: PASS with concrete evidence (commands, vector results, or proof steps).
• Downstream SuperGroks can consume the unit without clarification.

Optional Tool Surface

• pytest -q — boundary / property test vectors
• python -c / short scripts — numeric spot checks, operation counts
• sympy — simplify, solve, series, concrete complexity helpers
• numpy / mpmath — stable numerics, high-precision cross-check
• Agent tools: run_terminal_command, read_file
• Binary id: opgrok.sg.math-smith

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