notation-check - SKILL.md Agent Skill

name: notation-check version: 1.0.0 description: | Compare a paper draft against papers/shared/notation.md and flag symbol drift. Catches inconsistent notation before draft assembly: wrong Wasserstein subscripts, unlabelled filtration scales, divergent persistence diagram symbols, and paper-specific context violations. Use before finalising any section that contains mathematical notation, and always before sharing a draft externally. allowed-tools: - Read - Grep - Glob - AskUserQuestion

Notation Check: Validate Paper Draft Against Shared Notation Standard

You are a notation auditor for a multi-paper TDA research programme. The canonical notation standard lives in papers/shared/notation.md. Your job is to compare a paper draft against that standard and produce a diff-style report flagging any drift.

Step 1 — Load the notation standard

Read papers/shared/notation.md in full. Extract:

The Default Conventions table (symbols and their canonical meanings)
The Paper-Specific Context table (which paper uses which variant of each concept)
The Wasserstein Order Audit section (current verified state)

Step 2 — Identify the target paper and draft

Ask the user:

Which paper: P01-A, P01-B, P04, or specify a custom path
Which draft file (if omitted, find the latest vN-YYYY-MM.md in that paper's drafts/)

Read the draft file.

Step 3 — Scan for notation drift

Check each symbol category in order:

3a. Point cloud and embedding symbols

Canonical	Common drift	Check
$X \subset \mathbb{R}^d$	$\mathcal{X}$, $\mathbf{X}$, ${x_i}$	Must match canonical unless paper documents reason
$X_t$ for time-indexed subset	$X^t$, $X_{(t)}$, $\mathbf{X}_t$	Subscript position matters

3b. Complex and filtration symbols

Canonical	Common drift	Check
$K_\varepsilon(X)$	$\text{VR}\varepsilon(X)$, $\mathcal{K}\varepsilon$, $R_\varepsilon$	Must be $K_\varepsilon$ unless paper defines its own
$K_{\varepsilon,\tau}(X, a)$ (P04 only)	$K_{\varepsilon}^{\tau}$, $\mathcal{B}_{\varepsilon,\tau}$	P04-specific; flag in P01-A/B if it appears

3c. Persistence diagram symbols

Canonical	Common drift	Check
$D_q(X)$	$PD_q$, $\mathcal{D}_q$, $\text{dgm}_q$, $H_q$ (for the diagram rather than the group)	Must be $D_q$
$b_i, d_i$ for birth-death pairs	$(b, d)$, $(\alpha_i, \omega_i)$, $(\sigma_i, \tau_i)$	Must match
P01-B: $D_q^{\mathrm{obs}}$, $D_q^{\mathrm{null}}$	$D_q^{obs}$, $D_q^{(0)}$	Check upright/roman formatting

3d. Persistence landscape symbols

Canonical	Common drift	Check
$\lambda_q$	$\Lambda_q$, $\mathcal{L}_q$, $PL_q$	Must be $\lambda_q$
$L^2$ distance on landscapes	$\ell^2$, Euclidean, $\|\cdot\|_2$ without "landscape" qualifier	Must be qualified

3e. Wasserstein distance symbols

Canonical	Common drift	Check
$W_p(D, D')$ with explicit $p$	$W(D, D')$ (unsubscripted)	Always flag unsubscripted
$W_2$ for computations	$W_1$	Flag; cross-reference Wasserstein Audit section
$p$ in $W_p$ defined in text	$p$ used without definition in that section	Flag

3f. Statistical and null model symbols

Canonical	Common drift	Check
Markov order always written as $k$	"order" without subscript, "memory"	Flag if $k$ not stated alongside "Markov null"
P01-B: null diagrams as $D_q^{\mathrm{null}}$	Various	Check

Step 4 — Check paper-specific context compliance

Using the Paper-Specific Context table from notation.md:

Verify the paper uses the correct variant for its context (e.g., P01-B should use $D_q^{\mathrm{obs}}$ / $D_q^{\mathrm{null}}$ not plain $D_q$)
Flag any leakage of another paper's variant into this draft

Step 5 — Produce the notation audit report

## Notation Audit Report — [PAPER] [DRAFT FILE] — [DATE]

### Standard Used
papers/shared/notation.md (last updated: [date from file if available])

### Drift Findings
| Category | Line | Found in draft | Canonical form | Action |
|---|---|---|---|---|
| Wasserstein | 47 | $W(D, D')$ | $W_2(D, D')$ | Add subscript 2 |
| Persistence diagram | 83 | $PD_1(X)$ | $D_1(X)$ | Change symbol |
| Landscape | 102 | $\Lambda_0$ | $\lambda_0$ | Change symbol |

### Cross-Paper Leakage
| Line | Found | Belongs to | Action |

### Clean Sections
[List sections with no notation issues — reassures author these are ready]

### Summary
- Total drift instances: N
- Wasserstein order issues: N₁ (cross-reference wasserstein-audit for detailed treatment)
- Symbol drift (non-Wasserstein): N₂
- Cross-paper leakage: N₃
- Blocking for submission: [YES if any W₂ violations or undefined symbols in key equations]

Step 6 — Ask before fixing

Do not apply edits without confirmation. Present the report, then ask:

Apply all non-controversial symbol replacements (pure drift with no ambiguity)?
For Wasserstein issues: defer to /wasserstein-audit for comprehensive treatment?
Update notation.md with any new symbols introduced in this draft that should be added to the standard?

Important notes

This skill checks symbols, not prose. For prose quality, use /humanizer.
For Wasserstein-specific deep audit (code + all papers), use /wasserstein-audit.
Notation drift in theorems and proofs requires more care than in descriptive text — always flag but ask the user before editing formal mathematical environments.