Add tested examples, improvement plan, and updated docs for all Python use cases

[Copilot]

Summary

Add 6 tested example files covering every Python API surface, rewrite examples/README.md, and create a phased improvement plan based on hands-on exploration.

Type

• Bug fix
• New feature / enhancement
• Detection improvement (prior, oracle, heuristic)
• Refactor / cleanup
• Documentation
• CI / tooling

Description

The existing examples were generic and untested — simple_verification.py showed 3 hardcoded claims, and the README referenced verify_chain(llm_output) with the wrong call signature (missing steps param). New consumers had no way to understand the full API surface or validate their setup.

New examples (examples/)

• guard_verification.py — ReasoningGuard ECS scoring, threshold tuning, arithmetic repair, degradation tracking
• chain_of_thought.py — verify_chain multi-step reasoning with contradiction detection
• arithmetic_solver.py — solve_arithmetic, _safe_eval, _extract_numbers, _detect_operation
• syllogism_verification.py — verify_syllogism with Z3 + heuristic fallacy detection
• mcq_picker.py — pick_best_answer with strict/lenient tie handling
• arithmetic_repair.py — _repair_arithmetic_in_step, answer extraction, majority vote

Tests

• tests/test_examples.py — 36 tests covering all example use cases (mocked where Rust binary needed, direct for pure-Python)

Docs

• examples/README.md — Full rewrite: per-use-case sections with code, expected output, edge cases, install prerequisites per example group
• README.md — Fixed wrong verify_chain signature, added optional dependency install docs, linked new examples table
• docs/IMPROVEMENT-PLAN.md — Findings from exploration + 5-phase roadmap (robustness → test coverage → API ergonomics → performance)

Bugs fixed

• README showed verify_chain(llm_output) — correct signature is verify_chain(problem, steps)
• README referenced result.ecs on EpistemicChainReport (doesn't exist; it's result.chain_confidence)

CI fixes

• Fixed ruff F541 lint error: removed extraneous f prefix from f" ---" in examples/syllogism_verification.py
• Fixed ruff RUF059 lint error: renamed unused unpacked variable report → _report in tests/test_examples.py
• Fixed ruff formatting issues in examples/arithmetic_solver.py, examples/guard_verification.py, and tests/test_examples.py
• Added missing click dependency to CI install step in .github/workflows/lint.yml (required by spacy but not resolved transitively)

Testing

• Python unit tests pass: python3 -m unittest discover -s tests -p "test_*.py" -v
• Ruff clean: ruff check . && ruff format --check .
• Rust tests pass (if Rust code changed): cargo test
• Benchmark regression checked (if detection logic changed):
  python3 benchmarks/run_benchmarks.py

Benchmark impact (detection PRs only)

N/A — no detection logic changed.

Checklist

• No LLM, API key, or external model dependency introduced
• All Python files ≤ 400 lines (exceptions need justification)
• Public API documented (docstrings / /// for Rust)
• No commented-out code left in
• .github/CONTRIBUTING.md consulted for code style guidelines

Related issues

[sorunokoe]

@copilot Fix issues in CI.

[Copilot]

@copilot Fix issues in CI.

Fixed all CI issues:

1. Ruff F541 — removed extraneous f prefix from f" ---" in examples/syllogism_verification.py:109
2. Ruff RUF059 — renamed unused report to _report in tests/test_examples.py:311
3. Ruff format — auto-formatted 3 files that had formatting drift
4. Missing click — added click to the CI install step in .github/workflows/lint.yml (spacy requires it but wasn't resolving it transitively)