/wasserstein-audit — Audit Wasserstein Order Consistency

Scan all manuscript drafts and codebase for W₁ vs W₂ inconsistencies. Resolves the blocking Phase 0 deliverable: the legacy P01 manuscript uses W₁ in several places while project convention mandates W₂ as the primary metric.

Usage

/wasserstein-audit
/wasserstein-audit [papers|code|all]

Example: /wasserstein-audit all

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What this does

1. Reads papers/shared/notation.md (canonical notation standard, Wasserstein Audit section)
2. Scans manuscript .md files under papers/ for W₁, unsubscripted W, ambiguous W_p
3. Scans Python files under all domain packages for Wasserstein calls without explicit p=2
4. Cross-references findings against what notation.md already has "verified"
5. Produces a structured reconciliation report — does not fix anything without confirmation

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Manuscript patterns that trigger a flag

| Pattern | Issue | |---|---| | W_1 / $W_1$ / W_{1} | W₁ usage — needs justification or must change to W₂ | | $W$ (unsubscripted) | Always flag — violates notation standard | | bottleneck without alongside Wasserstein | Sole metric violation | | wasserstein without explicit order nearby | Ambiguous |

Code patterns that trigger a flag

| Pattern | Issue | |---|---| | wasserstein_distance(a, b) with no p= arg | Default may not be W₂ in all library versions | | p=1 or order=1 near Wasserstein call | Explicit W₁ usage | | bottleneck_distance(...) | Must not be used as sole metric | | gudhi.wasserstein.wasserstein_distance without order=2 | gudhi default not guaranteed |

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Report format

## Wasserstein Audit Report — YYYY-MM-DD

### Manuscript Findings
| File | Line | Found | Issue | Suggested Fix |

### Code Findings
| File | Line | Found | Issue | Suggested Fix |

### Status
- Blocking Phase 0: YES / NO
- Total issues: N (manuscript: N₁, code: N₂)

### Recommended Actions

Library defaults (reference)

• persim.wasserstein_distance — p=2 by default ✓
• gudhi.wasserstein.wasserstein_distance — requires explicit order= arg
• giotto-tda Wasserstein vectorisation — p=2 by default ✓
• W₁ in theory proofs is acceptable in Methods sections if clearly labelled and distinguished from the computational metric used