diff --git a/benchmarks/bench_sparse.py b/benchmarks/bench_sparse.py
index 3ac7e34..6553c76 100644
--- a/benchmarks/bench_sparse.py
+++ b/benchmarks/bench_sparse.py
@@ -47,7 +47,7 @@ def bench_forward_paths() -> None:
 
     batch = 4
 
-    for lmax in [5, 10, 15, 20, 25, 30, 40, 50, 64, 128]:
+    for lmax in [5, 10, 15, 20, 25, 30, 40, 50, 64]:
         p = _so3_params(lmax)
         f_cpu = torch.randn(batch, p['nlat'], p['nlon'])
 
@@ -74,6 +74,10 @@ def bench_forward_paths() -> None:
             f'{t_init:>10.2f}  {sp_mb:>8.1f}'
         )
 
+        del bsp, f
+        if device.type == 'cuda':
+            torch.cuda.empty_cache()
+
 
 if __name__ == '__main__':
     bench_forward_paths()
diff --git a/paper/citation_audit.py b/paper/citation_audit.py
new file mode 100644
index 0000000..53deace
--- /dev/null
+++ b/paper/citation_audit.py
@@ -0,0 +1,907 @@
+from __future__ import annotations
+
+import csv
+import json
+import re
+import unicodedata
+from dataclasses import asdict, dataclass, field
+from difflib import SequenceMatcher
+from pathlib import Path
+from typing import Any
+from urllib.error import HTTPError, URLError
+from urllib.parse import quote
+from urllib.request import Request, urlopen
+
+
+ROOT: Path = Path(__file__).resolve().parent
+TEX_PATH: Path = ROOT / "paper.tex"
+BIB_PATH: Path = ROOT / "references.bib"
+OUT_DIR: Path = ROOT / "citation_audit"
+LOCAL_BIB_DIR: Path = OUT_DIR / "bibtex" / "local"
+RESOLVED_BIB_DIR: Path = OUT_DIR / "bibtex" / "resolved"
+PAPER_DIR: Path = OUT_DIR / "papers"
+REPORT_PATH: Path = OUT_DIR / "citation_audit.md"
+CSV_PATH: Path = OUT_DIR / "citation_audit.csv"
+JSON_PATH: Path = OUT_DIR / "citation_audit.json"
+USER_AGENT: str = "bispectrum-citation-audit/0.1"
+
+CITE_PATTERN: re.Pattern[str] = re.compile(
+    r"\\(?:cite|citet|citep|citealp|citeauthor|citeyear|autocite|textcite|parencite|footcite|Citep|Citet|citeyearpar)\*?\{([^}]+)\}"
+)
+
+MANUAL_RECORDS: dict[str, dict[str, str | int | None]] = {
+    "bonev2023spherical": {
+        "source": "manual",
+        "title": "Spherical Fourier Neural Operators: Learning Stable Dynamics on the Sphere",
+        "year": 2023,
+        "venue": "Proceedings of the 40th International Conference on Machine Learning",
+        "doi": None,
+        "landing_url": "https://proceedings.mlr.press/v202/bonev23a.html",
+        "pdf_url": "https://proceedings.mlr.press/v202/bonev23a/bonev23a.pdf",
+    },
+    "cesa2022program": {
+        "source": "manual",
+        "title": "A Program to Build E(N)-Equivariant Steerable CNNs",
+        "year": 2022,
+        "venue": "International Conference on Learning Representations",
+        "doi": None,
+        "landing_url": "https://openreview.net/forum?id=WE4qe9xlnQw",
+        "pdf_url": "https://openreview.net/pdf?id=WE4qe9xlnQw",
+    },
+    "oreiller2022bispectral": {
+        "source": "manual",
+        "title": "Robust Multi-Organ Nucleus Segmentation Using a Locally Rotation Invariant Bispectral U-Net",
+        "year": 2022,
+        "venue": "Medical Imaging with Deep Learning",
+        "doi": None,
+        "landing_url": "https://proceedings.mlr.press/v172/oreiller22a.html",
+        "pdf_url": "https://proceedings.mlr.press/v172/oreiller22a/oreiller22a.pdf",
+    },
+    "loshchilov2019adamw": {
+        "source": "manual",
+        "title": "Decoupled Weight Decay Regularization",
+        "year": 2019,
+        "venue": "International Conference on Learning Representations",
+        "doi": None,
+        "landing_url": "https://openreview.net/forum?id=Bkg6RiCqY7",
+        "pdf_url": "https://openreview.net/pdf?id=Bkg6RiCqY7",
+    },
+    "zhemchuzhnikov2025equilopo": {
+        "source": "manual",
+        "title": "On the Fourier Analysis in the SO(3) Space: the EquiLoPO Network",
+        "year": 2025,
+        "venue": "International Conference on Learning Representations",
+        "doi": None,
+        "landing_url": "https://openreview.net/forum?id=LvTSvdiSwG",
+        "pdf_url": "https://openreview.net/pdf?id=LvTSvdiSwG",
+    },
+}
+
+MANUAL_ASSESSMENT_OVERRIDES: dict[str, str] = {}
+
+MANUAL_NOTES: dict[str, str] = {}
+
+
+@dataclass(slots=True)
+class CitationMention:
+    line_number: int
+    snippet: str
+
+
+@dataclass(slots=True)
+class BibEntry:
+    key: str
+    entry_type: str
+    raw: str
+    fields: dict[str, str]
+
+    @property
+    def title(self) -> str:
+        return self.fields.get("title", "")
+
+    @property
+    def year(self) -> int | None:
+        raw_year: str = self.fields.get("year", "").strip()
+        return int(raw_year) if raw_year.isdigit() else None
+
+    @property
+    def doi(self) -> str | None:
+        value: str = self.fields.get("doi", "").strip()
+        return value or None
+
+    @property
+    def url(self) -> str | None:
+        value: str = self.fields.get("url", "").strip()
+        return value or None
+
+    @property
+    def first_author_surname(self) -> str | None:
+        authors: str = self.fields.get("author", "")
+        if not authors:
+            return None
+        first_author: str = authors.split(" and ")[0].strip()
+        if "," in first_author:
+            return first_author.split(",", 1)[0].strip()
+        parts: list[str] = first_author.split()
+        return parts[-1] if parts else None
+
+    @property
+    def venue(self) -> str:
+        for field_name in ("journal", "booktitle", "school"):
+            value: str = self.fields.get(field_name, "").strip()
+            if value:
+                return value
+        return ""
+
+
+@dataclass(slots=True)
+class RemoteRecord:
+    source: str
+    title: str | None = None
+    year: int | None = None
+    venue: str | None = None
+    doi: str | None = None
+    landing_url: str | None = None
+    pdf_url: str | None = None
+    match_score: float | None = None
+
+
+@dataclass(slots=True)
+class AuditRow:
+    key: str
+    category: str
+    mention_count: int
+    local_entry_type: str
+    local_title: str
+    local_year: int | None
+    local_venue: str
+    local_doi: str | None
+    local_url: str | None
+    resolved_source: str | None
+    resolved_title: str | None
+    resolved_year: int | None
+    resolved_venue: str | None
+    resolved_doi: str | None
+    title_match_score: float | None
+    bibtex_source: str
+    paper_status: str
+    paper_path: str | None
+    assessment: str
+    notes: list[str] = field(default_factory=list)
+    mentions: list[CitationMention] = field(default_factory=list)
+
+
+def normalize_text(value: str) -> str:
+    normalized: str = unicodedata.normalize("NFKD", value)
+    ascii_only: str = normalized.encode("ascii", "ignore").decode("ascii")
+    lowered: str = ascii_only.casefold()
+    return re.sub(r"[^a-z0-9]+", "", lowered)
+
+
+def plain_text_from_bib(value: str) -> str:
+    plain: str = value
+    replacements: dict[str, str] = {
+        "{": "",
+        "}": "",
+        "\\&": "&",
+        "\\_": "_",
+        "~": " ",
+        "$": "",
+    }
+    for old, new in replacements.items():
+        plain = plain.replace(old, new)
+    plain = re.sub(r"\\[a-zA-Z]+\s*", "", plain)
+    plain = re.sub(r"\s+", " ", plain)
+    return plain.strip()
+
+
+def similarity_score(left: str, right: str) -> float:
+    if not left or not right:
+        return 0.0
+    return SequenceMatcher(None, normalize_text(left), normalize_text(right)).ratio()
+
+
+def read_url(
+    url: str,
+    *,
+    accept: str | None = None,
+    timeout: int = 30,
+) -> bytes:
+    headers: dict[str, str] = {"User-Agent": USER_AGENT}
+    if accept is not None:
+        headers["Accept"] = accept
+    request: Request = Request(url, headers=headers)
+    with urlopen(request, timeout=timeout) as response:
+        return response.read()
+
+
+def fetch_text(url: str, *, accept: str | None = None, timeout: int = 30) -> str | None:
+    try:
+        return read_url(url, accept=accept, timeout=timeout).decode("utf-8", errors="replace")
+    except (HTTPError, URLError, TimeoutError):
+        return None
+
+
+def fetch_json(url: str, *, timeout: int = 30) -> dict[str, Any] | None:
+    payload: str | None = fetch_text(url, accept="application/json", timeout=timeout)
+    if payload is None:
+        return None
+    try:
+        data: dict[str, Any] = json.loads(payload)
+    except json.JSONDecodeError:
+        return None
+    return data
+
+
+def manual_record_for_key(key: str) -> RemoteRecord | None:
+    payload: dict[str, str | int | None] | None = MANUAL_RECORDS.get(key)
+    if payload is None:
+        return None
+    return RemoteRecord(
+        source=str(payload["source"]),
+        title=str(payload["title"]) if payload["title"] is not None else None,
+        year=int(payload["year"]) if payload["year"] is not None else None,
+        venue=str(payload["venue"]) if payload["venue"] is not None else None,
+        doi=str(payload["doi"]) if payload["doi"] is not None else None,
+        landing_url=str(payload["landing_url"]) if payload["landing_url"] is not None else None,
+        pdf_url=str(payload["pdf_url"]) if payload["pdf_url"] is not None else None,
+        match_score=1.0,
+    )
+
+
+def extract_citations(tex_text: str) -> dict[str, list[CitationMention]]:
+    lines: list[str] = tex_text.splitlines()
+    mentions: dict[str, list[CitationMention]] = {}
+    for line_number, line in enumerate(lines, start=1):
+        collapsed: str = " ".join(line.strip().split())
+        for match in CITE_PATTERN.finditer(line):
+            for raw_key in match.group(1).split(","):
+                key: str = raw_key.strip()
+                if not key:
+                    continue
+                mentions.setdefault(key, []).append(
+                    CitationMention(line_number=line_number, snippet=collapsed)
+                )
+    return mentions
+
+
+def split_bib_entries(bib_text: str) -> list[str]:
+    entries: list[str] = []
+    index: int = 0
+    while True:
+        start: int = bib_text.find("@", index)
+        if start == -1:
+            return entries
+        brace_start: int = bib_text.find("{", start)
+        if brace_start == -1:
+            return entries
+        depth: int = 0
+        in_quotes: bool = False
+        escape_next: bool = False
+        for cursor in range(brace_start, len(bib_text)):
+            char: str = bib_text[cursor]
+            if escape_next:
+                escape_next = False
+                continue
+            if char == "\\":
+                escape_next = True
+                continue
+            if char == '"':
+                in_quotes = not in_quotes
+                continue
+            if in_quotes:
+                continue
+            if char == "{":
+                depth += 1
+            elif char == "}":
+                depth -= 1
+                if depth == 0:
+                    entries.append(bib_text[start : cursor + 1].strip())
+                    index = cursor + 1
+                    break
+        else:
+            return entries
+
+
+def parse_bib_value(chunk: str, start_index: int) -> tuple[str, int]:
+    opener: str = chunk[start_index]
+    if opener == "{":
+        depth: int = 0
+        cursor: int = start_index
+        while cursor < len(chunk):
+            char: str = chunk[cursor]
+            if char == "{":
+                depth += 1
+            elif char == "}":
+                depth -= 1
+                if depth == 0:
+                    return chunk[start_index + 1 : cursor], cursor + 1
+            cursor += 1
+        return chunk[start_index + 1 :], len(chunk)
+    if opener == '"':
+        cursor = start_index + 1
+        escaped: bool = False
+        while cursor < len(chunk):
+            char = chunk[cursor]
+            if escaped:
+                escaped = False
+            elif char == "\\":
+                escaped = True
+            elif char == '"':
+                return chunk[start_index + 1 : cursor], cursor + 1
+            cursor += 1
+        return chunk[start_index + 1 :], len(chunk)
+    cursor = start_index
+    while cursor < len(chunk) and chunk[cursor] not in ",\n":
+        cursor += 1
+    return chunk[start_index:cursor].strip(), cursor
+
+
+def parse_bib_entry(raw_entry: str) -> BibEntry:
+    header_match: re.Match[str] | None = re.match(r"@(\w+)\s*\{\s*([^,\s]+)\s*,", raw_entry, re.S)
+    if header_match is None:
+        raise ValueError(f"Could not parse BibTeX header: {raw_entry[:80]}")
+    entry_type: str = header_match.group(1).strip().lower()
+    key: str = header_match.group(2).strip()
+    body: str = raw_entry[header_match.end() :].rstrip().rstrip("}").strip()
+    fields: dict[str, str] = {}
+    cursor: int = 0
+    while cursor < len(body):
+        while cursor < len(body) and body[cursor] in " \t\r\n,":
+            cursor += 1
+        if cursor >= len(body):
+            break
+        equals_index: int = body.find("=", cursor)
+        if equals_index == -1:
+            break
+        field_name: str = body[cursor:equals_index].strip().lower()
+        cursor = equals_index + 1
+        while cursor < len(body) and body[cursor].isspace():
+            cursor += 1
+        if cursor >= len(body):
+            break
+        field_value, cursor = parse_bib_value(body, cursor)
+        fields[field_name] = " ".join(field_value.strip().split())
+        while cursor < len(body) and body[cursor] not in "\n,":
+            cursor += 1
+        if cursor < len(body) and body[cursor] == ",":
+            cursor += 1
+    return BibEntry(key=key, entry_type=entry_type, raw=raw_entry + "\n", fields=fields)
+
+
+def load_bib_entries(bib_text: str) -> dict[str, BibEntry]:
+    entries: dict[str, BibEntry] = {}
+    for raw_entry in split_bib_entries(bib_text):
+        entry: BibEntry = parse_bib_entry(raw_entry)
+        entries[entry.key] = entry
+    return entries
+
+
+def crossref_record_from_item(item: dict[str, Any], score: float) -> RemoteRecord:
+    titles: list[str] = item.get("title") or []
+    container_titles: list[str] = item.get("container-title") or []
+    year_parts: list[list[int]] = (
+        item.get("published-print", {}).get("date-parts")
+        or item.get("published-online", {}).get("date-parts")
+        or item.get("issued", {}).get("date-parts")
+        or []
+    )
+    year: int | None = year_parts[0][0] if year_parts and year_parts[0] else None
+    doi: str | None = item.get("DOI")
+    landing_url: str | None = f"https://doi.org/{doi}" if doi else item.get("URL")
+    return RemoteRecord(
+        source="crossref",
+        title=titles[0] if titles else None,
+        year=year,
+        venue=container_titles[0] if container_titles else None,
+        doi=doi,
+        landing_url=landing_url,
+        match_score=score,
+    )
+
+
+def search_crossref(title: str, first_author_surname: str | None, year: int | None) -> RemoteRecord | None:
+    query: str = quote(title)
+    url: str = f"https://api.crossref.org/works?query.bibliographic={query}&rows=5"
+    payload: dict[str, Any] | None = fetch_json(url)
+    if payload is None:
+        return None
+    items: list[dict[str, Any]] = payload.get("message", {}).get("items", [])
+    best_record: RemoteRecord | None = None
+    best_score: float = 0.0
+    for item in items:
+        candidate_titles: list[str] = item.get("title") or []
+        if not candidate_titles:
+            continue
+        candidate_title: str = candidate_titles[0]
+        score: float = similarity_score(title, candidate_title)
+        authors: list[dict[str, Any]] = item.get("author") or []
+        if first_author_surname and authors:
+            candidate_surname: str = str(authors[0].get("family", "")).strip()
+            if normalize_text(candidate_surname) == normalize_text(first_author_surname):
+                score += 0.05
+        candidate_year_parts: list[list[int]] = (
+            item.get("published-print", {}).get("date-parts")
+            or item.get("published-online", {}).get("date-parts")
+            or item.get("issued", {}).get("date-parts")
+            or []
+        )
+        candidate_year: int | None = (
+            candidate_year_parts[0][0] if candidate_year_parts and candidate_year_parts[0] else None
+        )
+        if year is not None and candidate_year is not None:
+            if abs(year - candidate_year) == 0:
+                score += 0.05
+            elif abs(year - candidate_year) > 1:
+                score -= 0.15
+        if score > best_score:
+            best_score = score
+            best_record = crossref_record_from_item(item, score)
+    if best_record is None or best_score < 0.72:
+        return None
+    return best_record
+
+
+def openalex_record_from_result(result: dict[str, Any], score: float) -> RemoteRecord:
+    primary_location: dict[str, Any] = result.get("primary_location") or {}
+    best_oa_location: dict[str, Any] = result.get("best_oa_location") or {}
+    pdf_url: str | None = (
+        best_oa_location.get("pdf_url")
+        or primary_location.get("pdf_url")
+        or result.get("open_access", {}).get("oa_url")
+    )
+    landing_url: str | None = (
+        best_oa_location.get("landing_page_url")
+        or primary_location.get("landing_page_url")
+        or result.get("doi")
+    )
+    publication_year: int | None = result.get("publication_year")
+    venue_name: str | None = None
+    if primary_location.get("source"):
+        venue_name = primary_location["source"].get("display_name")
+    return RemoteRecord(
+        source="openalex",
+        title=result.get("display_name"),
+        year=publication_year,
+        venue=venue_name,
+        doi=(result.get("doi") or "").removeprefix("https://doi.org/") or None,
+        landing_url=landing_url,
+        pdf_url=pdf_url,
+        match_score=score,
+    )
+
+
+def search_openalex(title: str, first_author_surname: str | None, year: int | None) -> RemoteRecord | None:
+    url: str = f"https://api.openalex.org/works?search={quote(title)}&per_page=5"
+    payload: dict[str, Any] | None = fetch_json(url)
+    if payload is None:
+        return None
+    results: list[dict[str, Any]] = payload.get("results", [])
+    best_record: RemoteRecord | None = None
+    best_score: float = 0.0
+    for result in results:
+        candidate_title: str = str(result.get("display_name", "")).strip()
+        if not candidate_title:
+            continue
+        score: float = similarity_score(title, candidate_title)
+        authorships: list[dict[str, Any]] = result.get("authorships") or []
+        if first_author_surname and authorships:
+            first_author: dict[str, Any] = authorships[0].get("author") or {}
+            display_name: str = str(first_author.get("display_name", "")).strip()
+            candidate_surname: str = display_name.split()[-1] if display_name else ""
+            if normalize_text(candidate_surname) == normalize_text(first_author_surname):
+                score += 0.05
+        candidate_year: int | None = result.get("publication_year")
+        if year is not None and candidate_year is not None:
+            if abs(year - candidate_year) == 0:
+                score += 0.05
+            elif abs(year - candidate_year) > 1:
+                score -= 0.15
+        if score > best_score:
+            best_score = score
+            best_record = openalex_record_from_result(result, score)
+    if best_record is None or best_score < 0.72:
+        return None
+    return best_record
+
+
+def enrich_with_openalex(record: RemoteRecord, title: str) -> RemoteRecord:
+    if record.pdf_url and record.landing_url:
+        return record
+    openalex_record: RemoteRecord | None = None
+    if record.doi:
+        url: str = f"https://api.openalex.org/works?filter=doi:https://doi.org/{quote(record.doi, safe='')}&per_page=1"
+        payload: dict[str, Any] | None = fetch_json(url)
+        results: list[dict[str, Any]] = payload.get("results", []) if payload else []
+        if results:
+            openalex_record = openalex_record_from_result(results[0], similarity_score(title, results[0].get("display_name", "")))
+    if openalex_record is None:
+        openalex_record = search_openalex(title, None, record.year)
+    if openalex_record is None:
+        return record
+    if record.doi and not openalex_record.doi:
+        openalex_record.doi = record.doi
+    if record.match_score and (openalex_record.match_score or 0.0) < record.match_score:
+        openalex_record.match_score = record.match_score
+    if record.source == "crossref":
+        if openalex_record.title is None:
+            openalex_record.title = record.title
+        if openalex_record.year is None:
+            openalex_record.year = record.year
+        if openalex_record.venue is None:
+            openalex_record.venue = record.venue
+    return openalex_record
+
+
+def extract_arxiv_id(entry: BibEntry) -> str | None:
+    candidates: list[str] = []
+    if entry.url:
+        candidates.append(entry.url)
+    for field_name in ("journal", "booktitle", "note"):
+        value: str = entry.fields.get(field_name, "")
+        if value:
+            candidates.append(value)
+    for candidate in candidates:
+        match: re.Match[str] | None = re.search(r"arxiv\.org/(?:abs|pdf)/([0-9]{4}\.[0-9]{4,5}(?:v\d+)?)", candidate, re.I)
+        if match is not None:
+            return match.group(1)
+        match = re.search(r"arXiv:([0-9]{4}\.[0-9]{4,5}(?:v\d+)?)", candidate, re.I)
+        if match is not None:
+            return match.group(1)
+    return None
+
+
+def resolve_remote_record(entry: BibEntry) -> RemoteRecord | None:
+    manual_record: RemoteRecord | None = manual_record_for_key(entry.key)
+    if manual_record is not None:
+        return manual_record
+    search_title: str = plain_text_from_bib(entry.title)
+    if entry.doi:
+        landing_url: str = f"https://doi.org/{entry.doi}"
+        return enrich_with_openalex(
+            RemoteRecord(
+                source="doi",
+                title=search_title,
+                year=entry.year,
+                venue=entry.venue,
+                doi=entry.doi,
+                landing_url=landing_url,
+                match_score=1.0,
+            ),
+            search_title,
+        )
+    arxiv_id: str | None = extract_arxiv_id(entry)
+    if arxiv_id is not None:
+        return RemoteRecord(
+            source="arxiv",
+            title=search_title,
+            year=entry.year,
+            venue=entry.venue,
+            doi=entry.doi,
+            landing_url=f"https://arxiv.org/abs/{arxiv_id}",
+            pdf_url=f"https://arxiv.org/pdf/{arxiv_id}.pdf",
+            match_score=1.0,
+        )
+    openalex_record: RemoteRecord | None = search_openalex(
+        search_title,
+        entry.first_author_surname,
+        entry.year,
+    )
+    crossref_record: RemoteRecord | None = search_crossref(
+        search_title,
+        entry.first_author_surname,
+        entry.year,
+    )
+    if openalex_record is not None and crossref_record is not None:
+        return openalex_record if (openalex_record.match_score or 0.0) >= (crossref_record.match_score or 0.0) else enrich_with_openalex(crossref_record, search_title)
+    if openalex_record is not None:
+        return openalex_record
+    if crossref_record is not None:
+        return enrich_with_openalex(crossref_record, search_title)
+    return None
+
+
+def download_resolved_bibtex(entry: BibEntry, record: RemoteRecord | None) -> tuple[str, str]:
+    local_path: Path = LOCAL_BIB_DIR / f"{entry.key}.bib"
+    local_path.write_text(entry.raw, encoding="utf-8")
+    if record is not None and record.doi:
+        bibtex_text: str | None = fetch_text(
+            f"https://doi.org/{record.doi}",
+            accept="application/x-bibtex",
+        )
+        if bibtex_text:
+            resolved_path: Path = RESOLVED_BIB_DIR / f"{entry.key}.bib"
+            resolved_path.write_text(bibtex_text.strip() + "\n", encoding="utf-8")
+            return "remote-doi", str(resolved_path.relative_to(ROOT))
+    resolved_path = RESOLVED_BIB_DIR / f"{entry.key}.bib"
+    resolved_path.write_text(entry.raw, encoding="utf-8")
+    return "local-fallback", str(resolved_path.relative_to(ROOT))
+
+
+def download_paper(entry: BibEntry, record: RemoteRecord | None) -> tuple[str, str | None]:
+    candidate_pdf_urls: list[str] = []
+    candidate_links: list[str] = []
+    if record is not None:
+        if record.pdf_url:
+            candidate_pdf_urls.append(record.pdf_url)
+        if record.landing_url:
+            candidate_links.append(record.landing_url)
+    if entry.url:
+        candidate_links.append(entry.url)
+    arxiv_id: str | None = extract_arxiv_id(entry)
+    if arxiv_id is not None:
+        candidate_pdf_urls.insert(0, f"https://arxiv.org/pdf/{arxiv_id}.pdf")
+        candidate_links.insert(0, f"https://arxiv.org/abs/{arxiv_id}")
+    seen: set[str] = set()
+    for pdf_url in candidate_pdf_urls:
+        if pdf_url in seen:
+            continue
+        seen.add(pdf_url)
+        try:
+            payload: bytes = read_url(pdf_url, timeout=45)
+        except (HTTPError, URLError, TimeoutError):
+            continue
+        if not payload.startswith(b"%PDF"):
+            continue
+        paper_path: Path = PAPER_DIR / f"{entry.key}.pdf"
+        paper_path.write_bytes(payload)
+        return "downloaded", str(paper_path.relative_to(ROOT))
+    for link in candidate_links:
+        if link in seen:
+            continue
+        seen.add(link)
+        link_path: Path = PAPER_DIR / f"{entry.key}.link.txt"
+        link_path.write_text(link + "\n", encoding="utf-8")
+        return "link-only", str(link_path.relative_to(ROOT))
+    return "unavailable", None
+
+
+def summarize_category(mentions: list[CitationMention]) -> str:
+    combined: str = " ".join(mention.snippet for mention in mentions).casefold()
+    if any(token in combined for token in ("patchcamelyon", "organmnist", "camelyon", "dataset", "contains", "consists of")):
+        return "dataset"
+    if any(token in combined for token in ("pytorch", "numpy", "torch_harmonics")):
+        return "software"
+    if any(token in combined for token in ("adamw", "backbone", "trained with", "reference numbers", "follows")):
+        return "experimental"
+    if any(token in combined for token in ("introduced", "applied", "explore", "use iterated", "provides", "extend this")):
+        return "related-work"
+    return "theory"
+
+
+def build_assessment(entry: BibEntry, record: RemoteRecord | None, paper_status: str) -> tuple[str, list[str]]:
+    notes: list[str] = []
+    title_score: float | None = None
+    if record is not None and record.title:
+        title_score = similarity_score(entry.title, record.title)
+    assessment: str = "good"
+    if record is None:
+        assessment = "check"
+        notes.append("No confident live metadata match found.")
+    elif title_score is not None and title_score < 0.9:
+        assessment = "check"
+        notes.append("Resolved title differs materially from the local BibTeX title.")
+    if record is not None and entry.year and record.year and abs(entry.year - record.year) > 1:
+        assessment = "check"
+        notes.append(f"Year mismatch: local {entry.year}, resolved {record.year}.")
+    resolved_venue: str = record.venue if record is not None and record.venue is not None else ""
+    venue_text: str = f"{entry.venue} {resolved_venue}".casefold()
+    if entry.entry_type == "phdthesis":
+        if assessment == "good":
+            assessment = "acceptable-but-weaker"
+        notes.append("Foundational thesis citation rather than an archival paper.")
+    if "arxiv preprint" in venue_text or (record is not None and record.source == "arxiv"):
+        if assessment == "good":
+            assessment = "acceptable-but-weaker"
+        notes.append("Preprint citation; archival version may be preferable if one exists.")
+    if "workshop" in venue_text:
+        if assessment == "good":
+            assessment = "acceptable-but-weaker"
+        notes.append("Workshop citation rather than a main archival venue.")
+    if entry.doi is None and (record is None or record.doi is None):
+        notes.append("No DOI found.")
+    if paper_status == "downloaded":
+        notes.append("Open PDF downloaded.")
+    elif paper_status == "link-only":
+        notes.append("Saved landing link only; no open PDF downloaded.")
+    else:
+        notes.append("No paper URL resolved.")
+    manual_note: str | None = MANUAL_NOTES.get(entry.key)
+    if manual_note is not None:
+        notes.append(manual_note)
+    manual_assessment: str | None = MANUAL_ASSESSMENT_OVERRIDES.get(entry.key)
+    if manual_assessment is not None:
+        assessment = manual_assessment
+    if not notes:
+        notes.append("Metadata matches and assets resolved cleanly.")
+    return assessment, notes
+
+
+def markdown_escape(value: str) -> str:
+    return value.replace("|", "\\|").replace("\n", " ").strip()
+
+
+def render_report(rows: list[AuditRow]) -> str:
+    downloaded_count: int = sum(1 for row in rows if row.paper_status == "downloaded")
+    link_only_count: int = sum(1 for row in rows if row.paper_status == "link-only")
+    stronger_count: int = sum(1 for row in rows if row.assessment == "good")
+    weaker_count: int = sum(1 for row in rows if row.assessment == "acceptable-but-weaker")
+    check_count: int = sum(1 for row in rows if row.assessment == "check")
+    lines: list[str] = [
+        "# Citation Audit",
+        "",
+        f"- Total cited keys: {len(rows)}",
+        f"- Assessments: {stronger_count} good, {weaker_count} acceptable-but-weaker, {check_count} check",
+        f"- Paper assets: {downloaded_count} PDFs downloaded, {link_only_count} link-only, {len(rows) - downloaded_count - link_only_count} unavailable",
+        "",
+        "| Key | Category | Mentions | Assessment | Resolved source | BibTeX | Paper | Notes |",
+        "| --- | --- | ---: | --- | --- | --- | --- | --- |",
+    ]
+    for row in rows:
+        notes: str = "; ".join(row.notes[:2])
+        resolved_source: str = row.resolved_source or "unresolved"
+        paper_cell: str = row.paper_status if row.paper_path is None else f"{row.paper_status} (`{row.paper_path}`)"
+        lines.append(
+            "| "
+            + " | ".join(
+                [
+                    markdown_escape(row.key),
+                    markdown_escape(row.category),
+                    str(row.mention_count),
+                    markdown_escape(row.assessment),
+                    markdown_escape(resolved_source),
+                    markdown_escape(row.bibtex_source),
+                    markdown_escape(paper_cell),
+                    markdown_escape(notes),
+                ]
+            )
+            + " |"
+        )
+    lines.append("")
+    lines.append("## Details")
+    lines.append("")
+    for row in rows:
+        lines.append(f"### `{row.key}`")
+        lines.append("")
+        lines.append(f"- Local title: {row.local_title}")
+        lines.append(f"- Resolved title: {row.resolved_title or 'unresolved'}")
+        lines.append(f"- Assessment: {row.assessment}")
+        lines.append(f"- BibTeX source: {row.bibtex_source}")
+        lines.append(f"- Paper asset: {row.paper_status}{f' (`{row.paper_path}`)' if row.paper_path else ''}")
+        lines.append(f"- Notes: {'; '.join(row.notes)}")
+        if row.mentions:
+            lines.append("- Citation contexts:")
+            for mention in row.mentions[:3]:
+                lines.append(f"  - L{mention.line_number}: {mention.snippet}")
+        lines.append("")
+    return "\n".join(lines).strip() + "\n"
+
+
+def write_csv(rows: list[AuditRow]) -> None:
+    with CSV_PATH.open("w", encoding="utf-8", newline="") as handle:
+        writer = csv.writer(handle)
+        writer.writerow(
+            [
+                "key",
+                "category",
+                "mention_count",
+                "assessment",
+                "local_entry_type",
+                "local_title",
+                "local_year",
+                "local_venue",
+                "resolved_source",
+                "resolved_title",
+                "resolved_year",
+                "resolved_venue",
+                "resolved_doi",
+                "title_match_score",
+                "bibtex_source",
+                "paper_status",
+                "paper_path",
+                "notes",
+                "mention_lines",
+                "mention_snippets",
+            ]
+        )
+        for row in rows:
+            writer.writerow(
+                [
+                    row.key,
+                    row.category,
+                    row.mention_count,
+                    row.assessment,
+                    row.local_entry_type,
+                    row.local_title,
+                    row.local_year,
+                    row.local_venue,
+                    row.resolved_source,
+                    row.resolved_title,
+                    row.resolved_year,
+                    row.resolved_venue,
+                    row.resolved_doi,
+                    row.title_match_score,
+                    row.bibtex_source,
+                    row.paper_status,
+                    row.paper_path,
+                    " | ".join(row.notes),
+                    ",".join(str(mention.line_number) for mention in row.mentions),
+                    " || ".join(mention.snippet for mention in row.mentions),
+                ]
+            )
+
+
+def write_json(rows: list[AuditRow]) -> None:
+    serialized_rows: list[dict[str, Any]] = []
+    for row in rows:
+        payload: dict[str, Any] = asdict(row)
+        serialized_rows.append(payload)
+    JSON_PATH.write_text(json.dumps(serialized_rows, indent=2), encoding="utf-8")
+
+
+def ensure_output_dirs() -> None:
+    for directory in (LOCAL_BIB_DIR, RESOLVED_BIB_DIR, PAPER_DIR):
+        directory.mkdir(parents=True, exist_ok=True)
+
+
+def audit_citations() -> list[AuditRow]:
+    tex_text: str = TEX_PATH.read_text(encoding="utf-8")
+    bib_text: str = BIB_PATH.read_text(encoding="utf-8")
+    mentions_by_key: dict[str, list[CitationMention]] = extract_citations(tex_text)
+    bib_entries: dict[str, BibEntry] = load_bib_entries(bib_text)
+    rows: list[AuditRow] = []
+    for key in sorted(mentions_by_key):
+        if key not in bib_entries:
+            raise KeyError(f"Citation key `{key}` is used in paper.tex but missing from references.bib")
+        entry: BibEntry = bib_entries[key]
+        mentions: list[CitationMention] = mentions_by_key[key]
+        record: RemoteRecord | None = resolve_remote_record(entry)
+        bibtex_source, _ = download_resolved_bibtex(entry, record)
+        paper_status, paper_path = download_paper(entry, record)
+        assessment, notes = build_assessment(entry, record, paper_status)
+        resolved_title: str | None = record.title if record else None
+        title_match_score: float | None = (
+            similarity_score(entry.title, resolved_title) if resolved_title else None
+        )
+        rows.append(
+            AuditRow(
+                key=key,
+                category=summarize_category(mentions),
+                mention_count=len(mentions),
+                local_entry_type=entry.entry_type,
+                local_title=entry.title,
+                local_year=entry.year,
+                local_venue=entry.venue,
+                local_doi=entry.doi,
+                local_url=entry.url,
+                resolved_source=record.source if record else None,
+                resolved_title=resolved_title,
+                resolved_year=record.year if record else None,
+                resolved_venue=record.venue if record else None,
+                resolved_doi=record.doi if record else None,
+                title_match_score=title_match_score,
+                bibtex_source=bibtex_source,
+                paper_status=paper_status,
+                paper_path=paper_path,
+                assessment=assessment,
+                notes=notes,
+                mentions=mentions,
+            )
+        )
+        print(f"[audit] {key}: {assessment}, paper={paper_status}, bib={bibtex_source}")
+    return rows
+
+
+def main() -> None:
+    ensure_output_dirs()
+    rows: list[AuditRow] = audit_citations()
+    REPORT_PATH.write_text(render_report(rows), encoding="utf-8")
+    write_csv(rows)
+    write_json(rows)
+    print(f"[done] wrote {REPORT_PATH.relative_to(ROOT)}")
+    print(f"[done] wrote {CSV_PATH.relative_to(ROOT)}")
+    print(f"[done] wrote {JSON_PATH.relative_to(ROOT)}")
+
+
+if __name__ == "__main__":
+    main()
diff --git a/paper/citation_audit/bibtex/local/batatia2022mace.bib b/paper/citation_audit/bibtex/local/batatia2022mace.bib
new file mode 100644
index 0000000..09c69b0
--- /dev/null
+++ b/paper/citation_audit/bibtex/local/batatia2022mace.bib
@@ -0,0 +1,10 @@
+@inproceedings{batatia2022mace,
+  title={{MACE}: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields},
+  author={Batatia, Ilyes and Cs{\'a}nyi, G{\'a}bor and Kov{\'a}cs, D{\'a}vid P. and Ortner, Christoph and Simm, Gregor},
+  booktitle={Advances in Neural Information Processing Systems (NeurIPS)},
+  pages={11423--11436},
+  year={2022},
+  volume={35},
+  doi={10.52202/068431-0830},
+  url={https://proceedings.neurips.cc/paper_files/paper/2022/hash/4a36c3c51af11ed9f34615b81edb5bbc-Abstract-Conference.html},
+}
diff --git a/paper/citation_audit/bibtex/local/batatia2024design.bib b/paper/citation_audit/bibtex/local/batatia2024design.bib
new file mode 100644
index 0000000..bad28f7
--- /dev/null
+++ b/paper/citation_audit/bibtex/local/batatia2024design.bib
@@ -0,0 +1,6 @@
+@article{batatia2024design,
+  title={A Foundation Model for Atomistic Simulation},
+  author={Batatia, Ilyes and Benber, Philipp and Chmiela, Stefan and Csanyi, Gabor and others},
+  journal={Nature Machine Intelligence},
+  year={2024},
+}
diff --git a/paper/citation_audit/bibtex/local/bejnordi2017camelyon.bib b/paper/citation_audit/bibtex/local/bejnordi2017camelyon.bib
new file mode 100644
index 0000000..933339b
--- /dev/null
+++ b/paper/citation_audit/bibtex/local/bejnordi2017camelyon.bib
@@ -0,0 +1,10 @@
+@article{bejnordi2017camelyon,
+  title={Diagnostic Assessment of Deep Learning Algorithms for Detection of Lymph Node Metastases in Women With Breast Cancer},
+  author={Ehteshami Bejnordi, Babak and Veta, Mitko and van Diest, Paul Johannes and van Ginneken, Bram and Karssemeijer, Nico and Litjens, Geert and van der Laak, Jeroen A. W. M. and {the CAMELYON16 Consortium}},
+  journal={JAMA},
+  volume={318},
+  number={22},
+  pages={2199--2210},
+  year={2017},
+  doi={10.1001/jama.2017.14585},
+}
diff --git a/paper/citation_audit/bibtex/local/bendory2022bispectrum.bib b/paper/citation_audit/bibtex/local/bendory2022bispectrum.bib
new file mode 100644
index 0000000..ec5fe28
--- /dev/null
+++ b/paper/citation_audit/bibtex/local/bendory2022bispectrum.bib
@@ -0,0 +1,6 @@
+@article{bendory2022bispectrum,
+  title={On the Uniqueness of the Bispectrum of Signals on the Sphere},
+  author={Bendory, Tamir and Edidin, Dan and Satriano, Matthew},
+  journal={arXiv preprint arXiv:2207.10842},
+  year={2022},
+}
diff --git a/paper/citation_audit/bibtex/local/bendory2025orbit.bib b/paper/citation_audit/bibtex/local/bendory2025orbit.bib
new file mode 100644
index 0000000..a88d0eb
--- /dev/null
+++ b/paper/citation_audit/bibtex/local/bendory2025orbit.bib
@@ -0,0 +1,7 @@
+@article{bendory2025orbit,
+  title={Orbit Recovery for Spherical Functions},
+  author={Bendory, Tamir and Edidin, Dan and Katz, Josh and Kreymer, Shay},
+  journal={arXiv preprint arXiv:2508.02674},
+  year={2025},
+  url={https://arxiv.org/abs/2508.02674},
+}
diff --git a/paper/citation_audit/bibtex/local/bonev2023spherical.bib b/paper/citation_audit/bibtex/local/bonev2023spherical.bib
new file mode 100644
index 0000000..df0362b
--- /dev/null
+++ b/paper/citation_audit/bibtex/local/bonev2023spherical.bib
@@ -0,0 +1,6 @@
+@inproceedings{bonev2023spherical,
+  title={Spherical {F}ourier Neural Operators: Learning Stable Dynamics on the Sphere},
+  author={Bonev, Boris and Kurth, Thorsten and Hundt, Christian and Pathak, Jaideep and Balu, Maximilian and Kashinath, Karthik and Anandkumar, Animashree},
+  booktitle={International Conference on Machine Learning (ICML)},
+  year={2023},
+}
diff --git a/paper/citation_audit/bibtex/local/cesa2022program.bib b/paper/citation_audit/bibtex/local/cesa2022program.bib
new file mode 100644
index 0000000..a490777
--- /dev/null
+++ b/paper/citation_audit/bibtex/local/cesa2022program.bib
@@ -0,0 +1,7 @@
+@inproceedings{cesa2022program,
+  title={A Program to Build {E(N)}-Equivariant Steerable {CNN}s},
+  author={Cesa, Gabriele and Lang, Leon and Weiler, Maurice},
+  booktitle={International Conference on Learning Representations (ICLR)},
+  year={2022},
+  url={https://openreview.net/forum?id=WE4qe9xlnQw},
+}
diff --git a/paper/citation_audit/bibtex/local/chevalley2024bispectral.bib b/paper/citation_audit/bibtex/local/chevalley2024bispectral.bib
new file mode 100644
index 0000000..ae36d3c
--- /dev/null
+++ b/paper/citation_audit/bibtex/local/chevalley2024bispectral.bib
@@ -0,0 +1,10 @@
+@inproceedings{chevalley2024bispectral,
+  title={A Bispectral {3D} {U-Net} for Rotation Robustness in Medical Segmentation},
+  author={Chevalley, Arthur and Oreiller, Valentin and Fageot, Julien and Prior, John O. and Andrearczyk, Vincent and Depeursinge, Adrien},
+  booktitle={Topology- and Graph-Informed Imaging Informatics},
+  pages={43--54},
+  year={2024},
+  publisher={Springer Nature Switzerland},
+  doi={10.1007/978-3-031-73967-5_5},
+  url={https://doi.org/10.1007/978-3-031-73967-5_5},
+}
diff --git a/paper/citation_audit/bibtex/local/cohen2016group.bib b/paper/citation_audit/bibtex/local/cohen2016group.bib
new file mode 100644
index 0000000..3c7062e
--- /dev/null
+++ b/paper/citation_audit/bibtex/local/cohen2016group.bib
@@ -0,0 +1,7 @@
+@inproceedings{cohen2016group,
+  title={Group Equivariant Convolutional Networks},
+  author={Cohen, Taco S. and Welling, Max},
+  booktitle={International Conference on Machine Learning (ICML)},
+  pages={2990--2999},
+  year={2016},
+}
diff --git a/paper/citation_audit/bibtex/local/cohen2018spherical.bib b/paper/citation_audit/bibtex/local/cohen2018spherical.bib
new file mode 100644
index 0000000..dcce293
--- /dev/null
+++ b/paper/citation_audit/bibtex/local/cohen2018spherical.bib
@@ -0,0 +1,6 @@
+@inproceedings{cohen2018spherical,
+  title={Spherical {CNN}s},
+  author={Cohen, Taco S. and Geiger, Mario and K{\"o}hler, Jonas and Welling, Max},
+  booktitle={International Conference on Learning Representations (ICLR)},
+  year={2018},
+}
diff --git a/paper/citation_audit/bibtex/local/edidin2023orbit.bib b/paper/citation_audit/bibtex/local/edidin2023orbit.bib
new file mode 100644
index 0000000..c823ab8
--- /dev/null
+++ b/paper/citation_audit/bibtex/local/edidin2023orbit.bib
@@ -0,0 +1,6 @@
+@article{edidin2023orbit,
+  title={Projective Uniqueness and the Bispectrum},
+  author={Edidin, Dan and Satriano, Matthew},
+  journal={arXiv preprint arXiv:2308.02064},
+  year={2023},
+}
diff --git a/paper/citation_audit/bibtex/local/edidin2024orbit.bib b/paper/citation_audit/bibtex/local/edidin2024orbit.bib
new file mode 100644
index 0000000..56870ba
--- /dev/null
+++ b/paper/citation_audit/bibtex/local/edidin2024orbit.bib
@@ -0,0 +1,12 @@
+@article{edidin2024orbit,
+  title={Orbit Recovery for Band-Limited Functions},
+  author={Edidin, Dan and Satriano, Matthew},
+  journal={SIAM Journal on Applied Algebra and Geometry},
+  volume={8},
+  number={3},
+  pages={733--755},
+  year={2024},
+  publisher={Society for Industrial \& Applied Mathematics},
+  doi={10.1137/23M1577808},
+  url={https://arxiv.org/abs/2306.00155},
+}
diff --git a/paper/citation_audit/bibtex/local/esteves2018spherical.bib b/paper/citation_audit/bibtex/local/esteves2018spherical.bib
new file mode 100644
index 0000000..7a9ed05
--- /dev/null
+++ b/paper/citation_audit/bibtex/local/esteves2018spherical.bib
@@ -0,0 +1,8 @@
+@inproceedings{esteves2018spherical,
+  title={Learning {SO}(3) Equivariant Representations with Spherical {CNN}s},
+  author={Esteves, Carlos and Allen-Blanchette, Christine and Makadia, Ameesh and Daniilidis, Kostas},
+  booktitle={European Conference on Computer Vision (ECCV)},
+  pages={52--68},
+  year={2018},
+  publisher={Springer},
+}
diff --git a/paper/citation_audit/bibtex/local/geiger2022e3nn.bib b/paper/citation_audit/bibtex/local/geiger2022e3nn.bib
new file mode 100644
index 0000000..fcb7f04
--- /dev/null
+++ b/paper/citation_audit/bibtex/local/geiger2022e3nn.bib
@@ -0,0 +1,7 @@
+@article{geiger2022e3nn,
+  title={e3nn: {E}uclidean Neural Networks},
+  author={Geiger, Mario and Smidt, Tess},
+  journal={arXiv preprint arXiv:2207.09453},
+  year={2022},
+  url={https://arxiv.org/abs/2207.09453},
+}
diff --git a/paper/citation_audit/bibtex/local/harris2020numpy.bib b/paper/citation_audit/bibtex/local/harris2020numpy.bib
new file mode 100644
index 0000000..52e2049
--- /dev/null
+++ b/paper/citation_audit/bibtex/local/harris2020numpy.bib
@@ -0,0 +1,10 @@
+@article{harris2020numpy,
+  title={Array programming with {NumPy}},
+  author={Harris, Charles R. and Millman, K. Jarrod and van der Walt, St{\'e}fan J. and Gommers, Ralf and Virtanen, Pauli and Cournapeau, David and Wieser, Eric and Taylor, Julian and Berg, Sebastian and Smith, Nathaniel J. and others},
+  journal={Nature},
+  volume={585},
+  number={7825},
+  pages={357--362},
+  year={2020},
+  publisher={Nature Publishing Group},
+}
diff --git a/paper/citation_audit/bibtex/local/huang2017densenet.bib b/paper/citation_audit/bibtex/local/huang2017densenet.bib
new file mode 100644
index 0000000..9adf8f5
--- /dev/null
+++ b/paper/citation_audit/bibtex/local/huang2017densenet.bib
@@ -0,0 +1,6 @@
+@inproceedings{huang2017densenet,
+  title={Densely Connected Convolutional Networks},
+  author={Huang, Gao and Liu, Zhuang and van der Maaten, Laurens and Weinberger, Kilian Q.},
+  booktitle={IEEE Conference on Computer Vision and Pattern Recognition (CVPR)},
+  year={2017},
+}
diff --git a/paper/citation_audit/bibtex/local/kakarala1992triple.bib b/paper/citation_audit/bibtex/local/kakarala1992triple.bib
new file mode 100644
index 0000000..a7a0c16
--- /dev/null
+++ b/paper/citation_audit/bibtex/local/kakarala1992triple.bib
@@ -0,0 +1,6 @@
+@phdthesis{kakarala1992triple,
+  title={Triple Correlation on Groups},
+  author={Kakarala, Ramakrishna},
+  school={University of California, Irvine},
+  year={1992},
+}
diff --git a/paper/citation_audit/bibtex/local/kakarala2012bispectrum.bib b/paper/citation_audit/bibtex/local/kakarala2012bispectrum.bib
new file mode 100644
index 0000000..e4d884f
--- /dev/null
+++ b/paper/citation_audit/bibtex/local/kakarala2012bispectrum.bib
@@ -0,0 +1,12 @@
+@article{kakarala2012bispectrum,
+  title={The Bispectrum as a Source of Phase-Sensitive Invariants for {F}ourier Descriptors: A Group-Theoretic Approach},
+  author={Kakarala, Ramakrishna},
+  journal={Journal of Mathematical Imaging and Vision},
+  volume={44},
+  number={3},
+  pages={341--353},
+  year={2012},
+  publisher={Springer},
+  doi={10.1007/s10851-012-0330-6},
+  url={https://arxiv.org/abs/0902.0196},
+}
diff --git a/paper/citation_audit/bibtex/local/kuipers2023regular.bib b/paper/citation_audit/bibtex/local/kuipers2023regular.bib
new file mode 100644
index 0000000..f8b9c0e
--- /dev/null
+++ b/paper/citation_audit/bibtex/local/kuipers2023regular.bib
@@ -0,0 +1,10 @@
+@inproceedings{kuipers2023regular,
+  title={Regular {SE}(3) Group Convolutions for Volumetric Medical Image Analysis},
+  author={Kuipers, Thijs P. and Bekkers, Erik J.},
+  booktitle={Medical Image Computing and Computer Assisted Intervention -- {MICCAI} 2023},
+  pages={252--261},
+  year={2023},
+  publisher={Springer Nature Switzerland},
+  doi={10.1007/978-3-031-43898-1_25},
+  url={https://doi.org/10.1007/978-3-031-43898-1_25},
+}
diff --git a/paper/citation_audit/bibtex/local/loshchilov2019adamw.bib b/paper/citation_audit/bibtex/local/loshchilov2019adamw.bib
new file mode 100644
index 0000000..1669f8b
--- /dev/null
+++ b/paper/citation_audit/bibtex/local/loshchilov2019adamw.bib
@@ -0,0 +1,7 @@
+@inproceedings{loshchilov2019adamw,
+  title={Decoupled Weight Decay Regularization},
+  author={Loshchilov, Ilya and Hutter, Frank},
+  booktitle={International Conference on Learning Representations (ICLR)},
+  year={2019},
+  url={https://openreview.net/forum?id=Bkg6RiCqY7},
+}
diff --git a/paper/citation_audit/bibtex/local/mallat2012scattering.bib b/paper/citation_audit/bibtex/local/mallat2012scattering.bib
new file mode 100644
index 0000000..97992ab
--- /dev/null
+++ b/paper/citation_audit/bibtex/local/mallat2012scattering.bib
@@ -0,0 +1,11 @@
+@article{mallat2012scattering,
+  title={Group Invariant Scattering},
+  author={Mallat, St{\'e}phane},
+  journal={Communications on Pure and Applied Mathematics},
+  volume={65},
+  number={10},
+  pages={1331--1398},
+  year={2012},
+  publisher={Wiley},
+  doi={10.1002/cpa.21413},
+}
diff --git a/paper/citation_audit/bibtex/local/mataigne2024selective.bib b/paper/citation_audit/bibtex/local/mataigne2024selective.bib
new file mode 100644
index 0000000..5c97b18
--- /dev/null
+++ b/paper/citation_audit/bibtex/local/mataigne2024selective.bib
@@ -0,0 +1,11 @@
+@inproceedings{mataigne2024selective,
+  title={The Selective {G}-Bispectrum and its Inversion: Applications to {G}-Invariant Networks},
+  author={Mataigne, Simon and Mathe, Johan and Sanborn, Sophia and Hillar, Christopher and Miolane, Nina},
+  booktitle={Advances in Neural Information Processing Systems},
+  pages={115682--115711},
+  year={2024},
+  volume={37},
+  publisher={Neural Information Processing Systems Foundation, Inc.},
+  doi={10.52202/079017-3674},
+  url={https://proceedings.neurips.cc/paper_files/paper/2024/hash/d1a1e8713fcd5626656553c82f7c3b26-Abstract-Conference.html},
+}
diff --git a/paper/citation_audit/bibtex/local/medmnistv2.bib b/paper/citation_audit/bibtex/local/medmnistv2.bib
new file mode 100644
index 0000000..7cd7219
--- /dev/null
+++ b/paper/citation_audit/bibtex/local/medmnistv2.bib
@@ -0,0 +1,10 @@
+@article{medmnistv2,
+  title={{MedMNIST} v2 -- A Large-Scale Lightweight Benchmark for {2D} and {3D} Biomedical Image Classification},
+  author={Yang, Jiancheng and Shi, Rui and Wei, Donglai and Liu, Zequan and Zhao, Lin and Ke, Bilian and Pfister, Hanspeter and Ni, Bingbing},
+  journal={Scientific Data},
+  volume={10},
+  number={1},
+  pages={41},
+  year={2023},
+  publisher={Nature Publishing Group UK London},
+}
diff --git a/paper/citation_audit/bibtex/local/oreiller2022bispectral.bib b/paper/citation_audit/bibtex/local/oreiller2022bispectral.bib
new file mode 100644
index 0000000..5580051
--- /dev/null
+++ b/paper/citation_audit/bibtex/local/oreiller2022bispectral.bib
@@ -0,0 +1,11 @@
+@inproceedings{oreiller2022bispectral,
+  title={Robust Multi-Organ Nucleus Segmentation Using a Locally Rotation Invariant Bispectral {U-Net}},
+  author={Oreiller, Valentin and Fageot, Julien and Andrearczyk, Vincent and Prior, John O. and Depeursinge, Adrien},
+  booktitle={Proceedings of The 5th International Conference on Medical Imaging with Deep Learning},
+  pages={929--943},
+  year={2022},
+  volume={172},
+  series={Proceedings of Machine Learning Research},
+  publisher={PMLR},
+  url={https://proceedings.mlr.press/v172/oreiller22a.html},
+}
diff --git a/paper/citation_audit/bibtex/local/paszke2019pytorch.bib b/paper/citation_audit/bibtex/local/paszke2019pytorch.bib
new file mode 100644
index 0000000..c863ccb
--- /dev/null
+++ b/paper/citation_audit/bibtex/local/paszke2019pytorch.bib
@@ -0,0 +1,6 @@
+@inproceedings{paszke2019pytorch,
+  title={Py{T}orch: An Imperative Style, High-Performance Deep Learning Library},
+  author={Paszke, Adam and Gross, Sam and Massa, Francisco and Lerer, Adam and Bradbury, James and Chanan, Gregory and Killeen, Trevor and Lin, Zeming and Gimelshein, Natalia and Antiga, Luca and Desmaison, Alban and K{\"o}pf, Andreas and Yang, Edward and DeVito, Zach and Raison, Martin and Tejani, Alykhan and Chilamkurthy, Sasank and Steiner, Benoit and Fang, Lu and Bai, Junjie and Chintala, Soumith},
+  booktitle={Advances in Neural Information Processing Systems (NeurIPS)},
+  year={2019},
+}
diff --git a/paper/citation_audit/bibtex/local/price2025freedom.bib b/paper/citation_audit/bibtex/local/price2025freedom.bib
new file mode 100644
index 0000000..413f189
--- /dev/null
+++ b/paper/citation_audit/bibtex/local/price2025freedom.bib
@@ -0,0 +1,10 @@
+@inproceedings{price2025freedom,
+  title={The Price of Freedom: Exploring Expressivity and Runtime Tradeoffs in Equivariant Tensor Products},
+  author={Xie, YuQing and Daigavane, Ameya and Kotak, Mit and Smidt, Tess},
+  booktitle={Proceedings of the 42nd International Conference on Machine Learning (ICML)},
+  pages={68599--68625},
+  year={2025},
+  volume={267},
+  series={Proceedings of Machine Learning Research},
+  publisher={PMLR},
+}
diff --git a/paper/citation_audit/bibtex/local/sanborn2023bispectral.bib b/paper/citation_audit/bibtex/local/sanborn2023bispectral.bib
new file mode 100644
index 0000000..bf8210f
--- /dev/null
+++ b/paper/citation_audit/bibtex/local/sanborn2023bispectral.bib
@@ -0,0 +1,6 @@
+@inproceedings{sanborn2023bispectral,
+  title={Bispectral Neural Networks},
+  author={Sanborn, Sophia and Shewmake, Christian and Olshausen, Bruno and Hillar, Christopher},
+  booktitle={International Conference on Learning Representations (ICLR)},
+  year={2023},
+}
diff --git a/paper/citation_audit/bibtex/local/sangalli2023movfrnet.bib b/paper/citation_audit/bibtex/local/sangalli2023movfrnet.bib
new file mode 100644
index 0000000..63373af
--- /dev/null
+++ b/paper/citation_audit/bibtex/local/sangalli2023movfrnet.bib
@@ -0,0 +1,11 @@
+@inproceedings{sangalli2023movfrnet,
+  title={Moving Frame Net: {SE}(3)-Equivariant Network for Volumes},
+  author={Sangalli, Mateus and Blusseau, Samy and Velasco-Forero, Santiago and Angulo, Jes{\'u}s},
+  booktitle={Proceedings of the 1st NeurIPS Workshop on Symmetry and Geometry in Neural Representations},
+  pages={81--97},
+  year={2023},
+  volume={197},
+  series={Proceedings of Machine Learning Research},
+  publisher={PMLR},
+  url={https://proceedings.mlr.press/v197/sangalli23a.html},
+}
diff --git a/paper/citation_audit/bibtex/local/thomas2018tensor.bib b/paper/citation_audit/bibtex/local/thomas2018tensor.bib
new file mode 100644
index 0000000..b1ccda2
--- /dev/null
+++ b/paper/citation_audit/bibtex/local/thomas2018tensor.bib
@@ -0,0 +1,7 @@
+@inproceedings{thomas2018tensor,
+  title={Tensor Field Networks: Rotation- and Translation-Equivariant Neural Networks for {3D} Point Clouds},
+  author={Thomas, Nathaniel and Smidt, Tess and Kearnes, Steven and Yang, Lusann and Li, Li and Kohlhoff, Kai and Riley, Patrick},
+  booktitle={arXiv preprint arXiv:1802.08219},
+  year={2018},
+  url={https://arxiv.org/abs/1802.08219},
+}
diff --git a/paper/citation_audit/bibtex/local/veeling2018rotation.bib b/paper/citation_audit/bibtex/local/veeling2018rotation.bib
new file mode 100644
index 0000000..cbd1c04
--- /dev/null
+++ b/paper/citation_audit/bibtex/local/veeling2018rotation.bib
@@ -0,0 +1,8 @@
+@inproceedings{veeling2018rotation,
+  title={Rotation Equivariant {CNN}s for Digital Pathology},
+  author={Veeling, Bastiaan S. and Linmans, Jasper and Winkens, Jim and Cohen, Taco and Welling, Max},
+  booktitle={Medical Image Computing and Computer Assisted Intervention (MICCAI)},
+  pages={210--218},
+  year={2018},
+  publisher={Springer},
+}
diff --git a/paper/citation_audit/bibtex/local/weiler2019general.bib b/paper/citation_audit/bibtex/local/weiler2019general.bib
new file mode 100644
index 0000000..5fb8cfd
--- /dev/null
+++ b/paper/citation_audit/bibtex/local/weiler2019general.bib
@@ -0,0 +1,6 @@
+@inproceedings{weiler2019general,
+  title={General {E}(2)-Equivariant Steerable {CNN}s},
+  author={Weiler, Maurice and Cesa, Gabriele},
+  booktitle={Advances in Neural Information Processing Systems (NeurIPS)},
+  year={2019},
+}
diff --git a/paper/citation_audit/bibtex/local/zhemchuzhnikov2024ilponet.bib b/paper/citation_audit/bibtex/local/zhemchuzhnikov2024ilponet.bib
new file mode 100644
index 0000000..67e9860
--- /dev/null
+++ b/paper/citation_audit/bibtex/local/zhemchuzhnikov2024ilponet.bib
@@ -0,0 +1,10 @@
+@inproceedings{zhemchuzhnikov2024ilponet,
+  title={{ILPO-NET}: Network for the Invariant Recognition of Arbitrary Volumetric Patterns in {3D}},
+  author={Zhemchuzhnikov, Dmitrii and Grudinin, Sergei},
+  booktitle={Machine Learning and Knowledge Discovery in Databases. Research Track},
+  pages={352--368},
+  year={2024},
+  publisher={Springer Nature Switzerland},
+  doi={10.1007/978-3-031-70359-1_21},
+  url={https://doi.org/10.1007/978-3-031-70359-1_21},
+}
diff --git a/paper/citation_audit/bibtex/local/zhemchuzhnikov2025equilopo.bib b/paper/citation_audit/bibtex/local/zhemchuzhnikov2025equilopo.bib
new file mode 100644
index 0000000..974daac
--- /dev/null
+++ b/paper/citation_audit/bibtex/local/zhemchuzhnikov2025equilopo.bib
@@ -0,0 +1,7 @@
+@inproceedings{zhemchuzhnikov2025equilopo,
+  title={On the {F}ourier Analysis in the {SO}(3) Space: the {EquiLoPO} Network},
+  author={Zhemchuzhnikov, Dmitrii and Grudinin, Sergei},
+  booktitle={International Conference on Learning Representations (ICLR)},
+  year={2025},
+  url={https://openreview.net/forum?id=LvTSvdiSwG},
+}
diff --git a/paper/citation_audit/bibtex/resolved/batatia2022mace.bib b/paper/citation_audit/bibtex/resolved/batatia2022mace.bib
new file mode 100644
index 0000000..1e602f9
--- /dev/null
+++ b/paper/citation_audit/bibtex/resolved/batatia2022mace.bib
@@ -0,0 +1 @@
+@inproceedings{Batatia_2022, series={NeurIPS 2022}, title={MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields}, url={http://dx.doi.org/10.52202/068431-0830}, DOI={10.52202/068431-0830}, booktitle={Advances in Neural Information Processing Systems 35}, publisher={Neural Information Processing Systems Foundation, Inc. (NeurIPS)}, author={Batatia, Ilyes and Csanyi, Gabor and Kovacs, David P and Ortner, Christoph and Simm, Gregor}, year={2022}, pages={11423–11436}, collection={NeurIPS 2022} }
diff --git a/paper/citation_audit/bibtex/resolved/batatia2024design.bib b/paper/citation_audit/bibtex/resolved/batatia2024design.bib
new file mode 100644
index 0000000..f7a4881
--- /dev/null
+++ b/paper/citation_audit/bibtex/resolved/batatia2024design.bib
@@ -0,0 +1 @@
+@article{Pl__2025, title={A Foundation Model for Accurate Atomistic Simulations in Drug Design}, url={http://dx.doi.org/10.26434/chemrxiv-2025-f1hgn-v3}, DOI={10.26434/chemrxiv-2025-f1hgn-v3}, publisher={American Chemical Society (ACS)}, author={Plé, Thomas and Adjoua, Olivier and Benali, Anouar and Posenitskiy, Evgeny and Villot, Corentin and Lagardère, Louis and Piquemal, Jean-Philip}, year={2025}, month=may }
diff --git a/paper/citation_audit/bibtex/resolved/bejnordi2017camelyon.bib b/paper/citation_audit/bibtex/resolved/bejnordi2017camelyon.bib
new file mode 100644
index 0000000..0ccaa50
--- /dev/null
+++ b/paper/citation_audit/bibtex/resolved/bejnordi2017camelyon.bib
@@ -0,0 +1 @@
+@article{Ehteshami_Bejnordi_2017, title={Diagnostic Assessment of Deep Learning Algorithms for Detection of Lymph Node Metastases in Women With Breast Cancer}, volume={318}, ISSN={0098-7484}, url={http://dx.doi.org/10.1001/jama.2017.14585}, DOI={10.1001/jama.2017.14585}, number={22}, journal={JAMA}, publisher={American Medical Association (AMA)}, author={Ehteshami Bejnordi, Babak and Veta, Mitko and Johannes van Diest, Paul and van Ginneken, Bram and Karssemeijer, Nico and Litjens, Geert and van der Laak, Jeroen A. W. M. and Hermsen, Meyke and Manson, Quirine F and Balkenhol, Maschenka and Geessink, Oscar and Stathonikos, Nikolaos and van Dijk, Marcory CRF and Bult, Peter and Beca, Francisco and Beck, Andrew H and Wang, Dayong and Khosla, Aditya and Gargeya, Rishab and Irshad, Humayun and Zhong, Aoxiao and Dou, Qi and Li, Quanzheng and Chen, Hao and Lin, Huang-Jing and Heng, Pheng-Ann and Haß, Christian and Bruni, Elia and Wong, Quincy and Halici, Ugur and Öner, Mustafa Ümit and Cetin-Atalay, Rengul and Berseth, Matt and Khvatkov, Vitali and Vylegzhanin, Alexei and Kraus, Oren and Shaban, Muhammad and Rajpoot, Nasir and Awan, Ruqayya and Sirinukunwattana, Korsuk and Qaiser, Talha and Tsang, Yee-Wah and Tellez, David and Annuscheit, Jonas and Hufnagl, Peter and Valkonen, Mira and Kartasalo, Kimmo and Latonen, Leena and Ruusuvuori, Pekka and Liimatainen, Kaisa and Albarqouni, Shadi and Mungal, Bharti and George, Ami and Demirci, Stefanie and Navab, Nassir and Watanabe, Seiryo and Seno, Shigeto and Takenaka, Yoichi and Matsuda, Hideo and Ahmady Phoulady, Hady and Kovalev, Vassili and Kalinovsky, Alexander and Liauchuk, Vitali and Bueno, Gloria and Fernandez-Carrobles, M. Milagro and Serrano, Ismael and Deniz, Oscar and Racoceanu, Daniel and Venâncio, Rui}, year={2017}, month=dec, pages={2199} }
diff --git a/paper/citation_audit/bibtex/resolved/bendory2022bispectrum.bib b/paper/citation_audit/bibtex/resolved/bendory2022bispectrum.bib
new file mode 100644
index 0000000..ec5fe28
--- /dev/null
+++ b/paper/citation_audit/bibtex/resolved/bendory2022bispectrum.bib
@@ -0,0 +1,6 @@
+@article{bendory2022bispectrum,
+  title={On the Uniqueness of the Bispectrum of Signals on the Sphere},
+  author={Bendory, Tamir and Edidin, Dan and Satriano, Matthew},
+  journal={arXiv preprint arXiv:2207.10842},
+  year={2022},
+}
diff --git a/paper/citation_audit/bibtex/resolved/bendory2025orbit.bib b/paper/citation_audit/bibtex/resolved/bendory2025orbit.bib
new file mode 100644
index 0000000..a88d0eb
--- /dev/null
+++ b/paper/citation_audit/bibtex/resolved/bendory2025orbit.bib
@@ -0,0 +1,7 @@
+@article{bendory2025orbit,
+  title={Orbit Recovery for Spherical Functions},
+  author={Bendory, Tamir and Edidin, Dan and Katz, Josh and Kreymer, Shay},
+  journal={arXiv preprint arXiv:2508.02674},
+  year={2025},
+  url={https://arxiv.org/abs/2508.02674},
+}
diff --git a/paper/citation_audit/bibtex/resolved/bonev2023spherical.bib b/paper/citation_audit/bibtex/resolved/bonev2023spherical.bib
new file mode 100644
index 0000000..df0362b
--- /dev/null
+++ b/paper/citation_audit/bibtex/resolved/bonev2023spherical.bib
@@ -0,0 +1,6 @@
+@inproceedings{bonev2023spherical,
+  title={Spherical {F}ourier Neural Operators: Learning Stable Dynamics on the Sphere},
+  author={Bonev, Boris and Kurth, Thorsten and Hundt, Christian and Pathak, Jaideep and Balu, Maximilian and Kashinath, Karthik and Anandkumar, Animashree},
+  booktitle={International Conference on Machine Learning (ICML)},
+  year={2023},
+}
diff --git a/paper/citation_audit/bibtex/resolved/cesa2022program.bib b/paper/citation_audit/bibtex/resolved/cesa2022program.bib
new file mode 100644
index 0000000..a490777
--- /dev/null
+++ b/paper/citation_audit/bibtex/resolved/cesa2022program.bib
@@ -0,0 +1,7 @@
+@inproceedings{cesa2022program,
+  title={A Program to Build {E(N)}-Equivariant Steerable {CNN}s},
+  author={Cesa, Gabriele and Lang, Leon and Weiler, Maurice},
+  booktitle={International Conference on Learning Representations (ICLR)},
+  year={2022},
+  url={https://openreview.net/forum?id=WE4qe9xlnQw},
+}
diff --git a/paper/citation_audit/bibtex/resolved/chevalley2024bispectral.bib b/paper/citation_audit/bibtex/resolved/chevalley2024bispectral.bib
new file mode 100644
index 0000000..8ad0a99
--- /dev/null
+++ b/paper/citation_audit/bibtex/resolved/chevalley2024bispectral.bib
@@ -0,0 +1 @@
+@inbook{Chevalley_2024, title={A Bispectral 3D U-Net for Rotation Robustness in Medical Segmentation}, ISBN={9783031739675}, ISSN={1611-3349}, url={http://dx.doi.org/10.1007/978-3-031-73967-5_5}, DOI={10.1007/978-3-031-73967-5_5}, booktitle={Topology- and Graph-Informed Imaging Informatics}, publisher={Springer Nature Switzerland}, author={Chevalley, Arthur and Oreiller, Valentin and Fageot, Julien and Prior, John O. and Andrearczyk, Vincent and Depeursinge, Adrien}, year={2024}, month=oct, pages={43–54} }
diff --git a/paper/citation_audit/bibtex/resolved/cohen2016group.bib b/paper/citation_audit/bibtex/resolved/cohen2016group.bib
new file mode 100644
index 0000000..b3dfa54
--- /dev/null
+++ b/paper/citation_audit/bibtex/resolved/cohen2016group.bib
@@ -0,0 +1,10 @@
+@article{https://doi.org/10.48550/arxiv.1602.07576,
+  doi = {10.48550/ARXIV.1602.07576},
+  url = {https://arxiv.org/abs/1602.07576},
+  author = {Cohen, Taco S. and Welling, Max},
+  keywords = {Machine Learning (cs.LG), Machine Learning (stat.ML), FOS: Computer and information sciences, FOS: Computer and information sciences},
+  title = {Group Equivariant Convolutional Networks},
+  publisher = {arXiv},
+  year = {2016},
+  copyright = {arXiv.org perpetual, non-exclusive license}
+}
diff --git a/paper/citation_audit/bibtex/resolved/cohen2018spherical.bib b/paper/citation_audit/bibtex/resolved/cohen2018spherical.bib
new file mode 100644
index 0000000..dcce293
--- /dev/null
+++ b/paper/citation_audit/bibtex/resolved/cohen2018spherical.bib
@@ -0,0 +1,6 @@
+@inproceedings{cohen2018spherical,
+  title={Spherical {CNN}s},
+  author={Cohen, Taco S. and Geiger, Mario and K{\"o}hler, Jonas and Welling, Max},
+  booktitle={International Conference on Learning Representations (ICLR)},
+  year={2018},
+}
diff --git a/paper/citation_audit/bibtex/resolved/edidin2023orbit.bib b/paper/citation_audit/bibtex/resolved/edidin2023orbit.bib
new file mode 100644
index 0000000..c823ab8
--- /dev/null
+++ b/paper/citation_audit/bibtex/resolved/edidin2023orbit.bib
@@ -0,0 +1,6 @@
+@article{edidin2023orbit,
+  title={Projective Uniqueness and the Bispectrum},
+  author={Edidin, Dan and Satriano, Matthew},
+  journal={arXiv preprint arXiv:2308.02064},
+  year={2023},
+}
diff --git a/paper/citation_audit/bibtex/resolved/edidin2024orbit.bib b/paper/citation_audit/bibtex/resolved/edidin2024orbit.bib
new file mode 100644
index 0000000..f600a0f
--- /dev/null
+++ b/paper/citation_audit/bibtex/resolved/edidin2024orbit.bib
@@ -0,0 +1 @@
+@article{Edidin_2024, title={Orbit Recovery for Band-Limited Functions}, volume={8}, ISSN={2470-6566}, url={http://dx.doi.org/10.1137/23m1577808}, DOI={10.1137/23m1577808}, number={3}, journal={SIAM Journal on Applied Algebra and Geometry}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Edidin, Dan and Satriano, Matthew}, year={2024}, month=aug, pages={733–755} }
diff --git a/paper/citation_audit/bibtex/resolved/esteves2018spherical.bib b/paper/citation_audit/bibtex/resolved/esteves2018spherical.bib
new file mode 100644
index 0000000..05ed628
--- /dev/null
+++ b/paper/citation_audit/bibtex/resolved/esteves2018spherical.bib
@@ -0,0 +1 @@
+@inbook{Esteves_2018, title={Learning SO(3) Equivariant Representations with Spherical CNNs}, ISBN={9783030012618}, ISSN={1611-3349}, url={http://dx.doi.org/10.1007/978-3-030-01261-8_4}, DOI={10.1007/978-3-030-01261-8_4}, booktitle={Computer Vision – ECCV 2018}, publisher={Springer International Publishing}, author={Esteves, Carlos and Allen-Blanchette, Christine and Makadia, Ameesh and Daniilidis, Kostas}, year={2018}, pages={54–70} }
diff --git a/paper/citation_audit/bibtex/resolved/geiger2022e3nn.bib b/paper/citation_audit/bibtex/resolved/geiger2022e3nn.bib
new file mode 100644
index 0000000..fcb7f04
--- /dev/null
+++ b/paper/citation_audit/bibtex/resolved/geiger2022e3nn.bib
@@ -0,0 +1,7 @@
+@article{geiger2022e3nn,
+  title={e3nn: {E}uclidean Neural Networks},
+  author={Geiger, Mario and Smidt, Tess},
+  journal={arXiv preprint arXiv:2207.09453},
+  year={2022},
+  url={https://arxiv.org/abs/2207.09453},
+}
diff --git a/paper/citation_audit/bibtex/resolved/harris2020numpy.bib b/paper/citation_audit/bibtex/resolved/harris2020numpy.bib
new file mode 100644
index 0000000..4b0a13a
--- /dev/null
+++ b/paper/citation_audit/bibtex/resolved/harris2020numpy.bib
@@ -0,0 +1 @@
+@article{Harris_2020, title={Array programming with NumPy}, volume={585}, ISSN={1476-4687}, url={http://dx.doi.org/10.1038/s41586-020-2649-2}, DOI={10.1038/s41586-020-2649-2}, number={7825}, journal={Nature}, publisher={Springer Science and Business Media LLC}, author={Harris, Charles R. and Millman, K. Jarrod and van der Walt, Stéfan J. and Gommers, Ralf and Virtanen, Pauli and Cournapeau, David and Wieser, Eric and Taylor, Julian and Berg, Sebastian and Smith, Nathaniel J. and Kern, Robert and Picus, Matti and Hoyer, Stephan and van Kerkwijk, Marten H. and Brett, Matthew and Haldane, Allan and del Río, Jaime Fernández and Wiebe, Mark and Peterson, Pearu and Gérard-Marchant, Pierre and Sheppard, Kevin and Reddy, Tyler and Weckesser, Warren and Abbasi, Hameer and Gohlke, Christoph and Oliphant, Travis E.}, year={2020}, month=sep, pages={357–362} }
diff --git a/paper/citation_audit/bibtex/resolved/huang2017densenet.bib b/paper/citation_audit/bibtex/resolved/huang2017densenet.bib
new file mode 100644
index 0000000..cd1d197
--- /dev/null
+++ b/paper/citation_audit/bibtex/resolved/huang2017densenet.bib
@@ -0,0 +1 @@
+@inproceedings{Huang_2017, title={Densely Connected Convolutional Networks}, url={http://dx.doi.org/10.1109/cvpr.2017.243}, DOI={10.1109/cvpr.2017.243}, booktitle={2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR)}, publisher={IEEE}, author={Huang, Gao and Liu, Zhuang and Van Der Maaten, Laurens and Weinberger, Kilian Q.}, year={2017}, month=jul, pages={2261–2269} }
diff --git a/paper/citation_audit/bibtex/resolved/kakarala1992triple.bib b/paper/citation_audit/bibtex/resolved/kakarala1992triple.bib
new file mode 100644
index 0000000..a7a0c16
--- /dev/null
+++ b/paper/citation_audit/bibtex/resolved/kakarala1992triple.bib
@@ -0,0 +1,6 @@
+@phdthesis{kakarala1992triple,
+  title={Triple Correlation on Groups},
+  author={Kakarala, Ramakrishna},
+  school={University of California, Irvine},
+  year={1992},
+}
diff --git a/paper/citation_audit/bibtex/resolved/kakarala2012bispectrum.bib b/paper/citation_audit/bibtex/resolved/kakarala2012bispectrum.bib
new file mode 100644
index 0000000..0e345a5
--- /dev/null
+++ b/paper/citation_audit/bibtex/resolved/kakarala2012bispectrum.bib
@@ -0,0 +1 @@
+@article{Kakarala_2012, title={The Bispectrum as a Source of Phase-Sensitive Invariants for Fourier Descriptors: A Group-Theoretic Approach}, volume={44}, ISSN={1573-7683}, url={http://dx.doi.org/10.1007/s10851-012-0330-6}, DOI={10.1007/s10851-012-0330-6}, number={3}, journal={Journal of Mathematical Imaging and Vision}, publisher={Springer Science and Business Media LLC}, author={Kakarala, Ramakrishna}, year={2012}, month=mar, pages={341–353} }
diff --git a/paper/citation_audit/bibtex/resolved/kuipers2023regular.bib b/paper/citation_audit/bibtex/resolved/kuipers2023regular.bib
new file mode 100644
index 0000000..17634bb
--- /dev/null
+++ b/paper/citation_audit/bibtex/resolved/kuipers2023regular.bib
@@ -0,0 +1 @@
+@inbook{Kuipers_2023, title={Regular SE(3) Group Convolutions for Volumetric Medical Image Analysis}, ISBN={9783031438981}, ISSN={1611-3349}, url={http://dx.doi.org/10.1007/978-3-031-43898-1_25}, DOI={10.1007/978-3-031-43898-1_25}, booktitle={Medical Image Computing and Computer Assisted Intervention – MICCAI 2023}, publisher={Springer Nature Switzerland}, author={Kuipers, Thijs P. and Bekkers, Erik J.}, year={2023}, pages={252–261} }
diff --git a/paper/citation_audit/bibtex/resolved/loshchilov2019adamw.bib b/paper/citation_audit/bibtex/resolved/loshchilov2019adamw.bib
new file mode 100644
index 0000000..1669f8b
--- /dev/null
+++ b/paper/citation_audit/bibtex/resolved/loshchilov2019adamw.bib
@@ -0,0 +1,7 @@
+@inproceedings{loshchilov2019adamw,
+  title={Decoupled Weight Decay Regularization},
+  author={Loshchilov, Ilya and Hutter, Frank},
+  booktitle={International Conference on Learning Representations (ICLR)},
+  year={2019},
+  url={https://openreview.net/forum?id=Bkg6RiCqY7},
+}
diff --git a/paper/citation_audit/bibtex/resolved/mallat2012scattering.bib b/paper/citation_audit/bibtex/resolved/mallat2012scattering.bib
new file mode 100644
index 0000000..08beedf
--- /dev/null
+++ b/paper/citation_audit/bibtex/resolved/mallat2012scattering.bib
@@ -0,0 +1 @@
+@article{Mallat_2012, title={Group Invariant Scattering}, volume={65}, ISSN={1097-0312}, url={http://dx.doi.org/10.1002/cpa.21413}, DOI={10.1002/cpa.21413}, number={10}, journal={Communications on Pure and Applied Mathematics}, publisher={Wiley}, author={Mallat, Stéphane}, year={2012}, month=jul, pages={1331–1398} }
diff --git a/paper/citation_audit/bibtex/resolved/mataigne2024selective.bib b/paper/citation_audit/bibtex/resolved/mataigne2024selective.bib
new file mode 100644
index 0000000..7df68fd
--- /dev/null
+++ b/paper/citation_audit/bibtex/resolved/mataigne2024selective.bib
@@ -0,0 +1 @@
+@inproceedings{Hillar_2024, series={NeurIPS 2024}, title={The Selective $G$-Bispectrum and its Inversion: Applications to $G$-Invariant Networks}, url={http://dx.doi.org/10.52202/079017-3674}, DOI={10.52202/079017-3674}, booktitle={Advances in Neural Information Processing Systems 37}, publisher={Neural Information Processing Systems Foundation, Inc. (NeurIPS)}, author={Hillar, Christopher and Mataigne, Simon and Mathe, Johan and Miolane, Nina and Sanborn, Sophia}, year={2024}, pages={115682–115711}, collection={NeurIPS 2024} }
diff --git a/paper/citation_audit/bibtex/resolved/medmnistv2.bib b/paper/citation_audit/bibtex/resolved/medmnistv2.bib
new file mode 100644
index 0000000..c320965
--- /dev/null
+++ b/paper/citation_audit/bibtex/resolved/medmnistv2.bib
@@ -0,0 +1 @@
+@article{Yang_2023, title={MedMNIST v2 - A large-scale lightweight benchmark for 2D and 3D biomedical image classification}, volume={10}, ISSN={2052-4463}, url={http://dx.doi.org/10.1038/s41597-022-01721-8}, DOI={10.1038/s41597-022-01721-8}, number={1}, journal={Scientific Data}, publisher={Springer Science and Business Media LLC}, author={Yang, Jiancheng and Shi, Rui and Wei, Donglai and Liu, Zequan and Zhao, Lin and Ke, Bilian and Pfister, Hanspeter and Ni, Bingbing}, year={2023}, month=jan }
diff --git a/paper/citation_audit/bibtex/resolved/oreiller2022bispectral.bib b/paper/citation_audit/bibtex/resolved/oreiller2022bispectral.bib
new file mode 100644
index 0000000..5580051
--- /dev/null
+++ b/paper/citation_audit/bibtex/resolved/oreiller2022bispectral.bib
@@ -0,0 +1,11 @@
+@inproceedings{oreiller2022bispectral,
+  title={Robust Multi-Organ Nucleus Segmentation Using a Locally Rotation Invariant Bispectral {U-Net}},
+  author={Oreiller, Valentin and Fageot, Julien and Andrearczyk, Vincent and Prior, John O. and Depeursinge, Adrien},
+  booktitle={Proceedings of The 5th International Conference on Medical Imaging with Deep Learning},
+  pages={929--943},
+  year={2022},
+  volume={172},
+  series={Proceedings of Machine Learning Research},
+  publisher={PMLR},
+  url={https://proceedings.mlr.press/v172/oreiller22a.html},
+}
diff --git a/paper/citation_audit/bibtex/resolved/paszke2019pytorch.bib b/paper/citation_audit/bibtex/resolved/paszke2019pytorch.bib
new file mode 100644
index 0000000..aa0e376
--- /dev/null
+++ b/paper/citation_audit/bibtex/resolved/paszke2019pytorch.bib
@@ -0,0 +1,10 @@
+@misc{https://doi.org/10.48550/arxiv.1912.01703,
+  doi = {10.48550/ARXIV.1912.01703},
+  url = {https://arxiv.org/abs/1912.01703},
+  author = {Paszke, Adam and Gross, Sam and Massa, Francisco and Lerer, Adam and Bradbury, James and Chanan, Gregory and Killeen, Trevor and Lin, Zeming and Gimelshein, Natalia and Antiga, Luca and Desmaison, Alban and Köpf, Andreas and Yang, Edward and DeVito, Zach and Raison, Martin and Tejani, Alykhan and Chilamkurthy, Sasank and Steiner, Benoit and Fang, Lu and Bai, Junjie and Chintala, Soumith},
+  keywords = {Machine Learning (cs.LG), Mathematical Software (cs.MS), Machine Learning (stat.ML), FOS: Computer and information sciences, FOS: Computer and information sciences},
+  title = {PyTorch: An Imperative Style, High-Performance Deep Learning Library},
+  publisher = {arXiv},
+  year = {2019},
+  copyright = {arXiv.org perpetual, non-exclusive license}
+}
diff --git a/paper/citation_audit/bibtex/resolved/price2025freedom.bib b/paper/citation_audit/bibtex/resolved/price2025freedom.bib
new file mode 100644
index 0000000..2b82ac1
--- /dev/null
+++ b/paper/citation_audit/bibtex/resolved/price2025freedom.bib
@@ -0,0 +1,10 @@
+@misc{https://doi.org/10.48550/arxiv.2506.13523,
+  doi = {10.48550/ARXIV.2506.13523},
+  url = {https://arxiv.org/abs/2506.13523},
+  author = {Xie, YuQing and Daigavane, Ameya and Kotak, Mit and Smidt, Tess},
+  keywords = {Machine Learning (cs.LG), Artificial Intelligence (cs.AI), FOS: Computer and information sciences, FOS: Computer and information sciences},
+  title = {The Price of Freedom: Exploring Expressivity and Runtime Tradeoffs in Equivariant Tensor Products},
+  publisher = {arXiv},
+  year = {2025},
+  copyright = {arXiv.org perpetual, non-exclusive license}
+}
diff --git a/paper/citation_audit/bibtex/resolved/sanborn2023bispectral.bib b/paper/citation_audit/bibtex/resolved/sanborn2023bispectral.bib
new file mode 100644
index 0000000..1eeb623
--- /dev/null
+++ b/paper/citation_audit/bibtex/resolved/sanborn2023bispectral.bib
@@ -0,0 +1,10 @@
+@article{https://doi.org/10.48550/arxiv.2209.03416,
+  doi = {10.48550/ARXIV.2209.03416},
+  url = {https://arxiv.org/abs/2209.03416},
+  author = {Sanborn, Sophia and Shewmake, Christian and Olshausen, Bruno and Hillar, Christopher},
+  keywords = {Machine Learning (cs.LG), FOS: Computer and information sciences, FOS: Computer and information sciences},
+  title = {Bispectral Neural Networks},
+  publisher = {arXiv},
+  year = {2022},
+  copyright = {Creative Commons Attribution Non Commercial No Derivatives 4.0 International}
+}
diff --git a/paper/citation_audit/bibtex/resolved/sangalli2023movfrnet.bib b/paper/citation_audit/bibtex/resolved/sangalli2023movfrnet.bib
new file mode 100644
index 0000000..da210ba
--- /dev/null
+++ b/paper/citation_audit/bibtex/resolved/sangalli2023movfrnet.bib
@@ -0,0 +1,10 @@
+@article{https://doi.org/10.48550/arxiv.2211.03420,
+  doi = {10.48550/ARXIV.2211.03420},
+  url = {https://arxiv.org/abs/2211.03420},
+  author = {Sangalli, Mateus and Blusseau, Samy and Velasco-Forero, Santiago and Angulo, Jesus},
+  keywords = {Computer Vision and Pattern Recognition (cs.CV), Machine Learning (stat.ML), FOS: Computer and information sciences, FOS: Computer and information sciences},
+  title = {Moving Frame Net: SE(3)-Equivariant Network for Volumes},
+  publisher = {arXiv},
+  year = {2022},
+  copyright = {arXiv.org perpetual, non-exclusive license}
+}
diff --git a/paper/citation_audit/bibtex/resolved/thomas2018tensor.bib b/paper/citation_audit/bibtex/resolved/thomas2018tensor.bib
new file mode 100644
index 0000000..b1ccda2
--- /dev/null
+++ b/paper/citation_audit/bibtex/resolved/thomas2018tensor.bib
@@ -0,0 +1,7 @@
+@inproceedings{thomas2018tensor,
+  title={Tensor Field Networks: Rotation- and Translation-Equivariant Neural Networks for {3D} Point Clouds},
+  author={Thomas, Nathaniel and Smidt, Tess and Kearnes, Steven and Yang, Lusann and Li, Li and Kohlhoff, Kai and Riley, Patrick},
+  booktitle={arXiv preprint arXiv:1802.08219},
+  year={2018},
+  url={https://arxiv.org/abs/1802.08219},
+}
diff --git a/paper/citation_audit/bibtex/resolved/veeling2018rotation.bib b/paper/citation_audit/bibtex/resolved/veeling2018rotation.bib
new file mode 100644
index 0000000..bd20b25
--- /dev/null
+++ b/paper/citation_audit/bibtex/resolved/veeling2018rotation.bib
@@ -0,0 +1 @@
+@inbook{Veeling_2018, title={Rotation Equivariant CNNs for Digital Pathology}, ISBN={9783030009342}, ISSN={1611-3349}, url={http://dx.doi.org/10.1007/978-3-030-00934-2_24}, DOI={10.1007/978-3-030-00934-2_24}, booktitle={Medical Image Computing and Computer Assisted Intervention – MICCAI 2018}, publisher={Springer International Publishing}, author={Veeling, Bastiaan S. and Linmans, Jasper and Winkens, Jim and Cohen, Taco and Welling, Max}, year={2018}, pages={210–218} }
diff --git a/paper/citation_audit/bibtex/resolved/weiler2019general.bib b/paper/citation_audit/bibtex/resolved/weiler2019general.bib
new file mode 100644
index 0000000..c9ca541
--- /dev/null
+++ b/paper/citation_audit/bibtex/resolved/weiler2019general.bib
@@ -0,0 +1,10 @@
+@misc{https://doi.org/10.48550/arxiv.1911.08251,
+  doi = {10.48550/ARXIV.1911.08251},
+  url = {https://arxiv.org/abs/1911.08251},
+  author = {Weiler, Maurice and Cesa, Gabriele},
+  keywords = {Computer Vision and Pattern Recognition (cs.CV), Machine Learning (cs.LG), Image and Video Processing (eess.IV), FOS: Computer and information sciences, FOS: Computer and information sciences, FOS: Electrical engineering, electronic engineering, information engineering, FOS: Electrical engineering, electronic engineering, information engineering},
+  title = {General $E(2)$-Equivariant Steerable CNNs},
+  publisher = {arXiv},
+  year = {2019},
+  copyright = {arXiv.org perpetual, non-exclusive license}
+}
diff --git a/paper/citation_audit/bibtex/resolved/zhemchuzhnikov2024ilponet.bib b/paper/citation_audit/bibtex/resolved/zhemchuzhnikov2024ilponet.bib
new file mode 100644
index 0000000..c613850
--- /dev/null
+++ b/paper/citation_audit/bibtex/resolved/zhemchuzhnikov2024ilponet.bib
@@ -0,0 +1 @@
+@inbook{Zhemchuzhnikov_2024, title={ILPO-NET: Network for the Invariant Recognition of Arbitrary Volumetric Patterns in 3D}, ISBN={9783031703591}, ISSN={1611-3349}, url={http://dx.doi.org/10.1007/978-3-031-70359-1_21}, DOI={10.1007/978-3-031-70359-1_21}, booktitle={Machine Learning and Knowledge Discovery in Databases. Research Track}, publisher={Springer Nature Switzerland}, author={Zhemchuzhnikov, Dmitrii and Grudinin, Sergei}, year={2024}, pages={352–368} }
diff --git a/paper/citation_audit/bibtex/resolved/zhemchuzhnikov2025equilopo.bib b/paper/citation_audit/bibtex/resolved/zhemchuzhnikov2025equilopo.bib
new file mode 100644
index 0000000..974daac
--- /dev/null
+++ b/paper/citation_audit/bibtex/resolved/zhemchuzhnikov2025equilopo.bib
@@ -0,0 +1,7 @@
+@inproceedings{zhemchuzhnikov2025equilopo,
+  title={On the {F}ourier Analysis in the {SO}(3) Space: the {EquiLoPO} Network},
+  author={Zhemchuzhnikov, Dmitrii and Grudinin, Sergei},
+  booktitle={International Conference on Learning Representations (ICLR)},
+  year={2025},
+  url={https://openreview.net/forum?id=LvTSvdiSwG},
+}
diff --git a/paper/citation_audit/citation_audit.csv b/paper/citation_audit/citation_audit.csv
new file mode 100644
index 0000000..262080f
--- /dev/null
+++ b/paper/citation_audit/citation_audit.csv
@@ -0,0 +1,30 @@
+key,category,mention_count,assessment,local_entry_type,local_title,local_year,local_venue,resolved_source,resolved_title,resolved_year,resolved_venue,resolved_doi,title_match_score,bibtex_source,paper_status,paper_path,notes,mention_lines,mention_snippets
+batatia2022mace,related-work,1,good,inproceedings,{MACE}: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields,2022,Advances in Neural Information Processing Systems (NeurIPS),openalex,MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields,2022,,10.52202/068431-0830,1.0,remote-doi,link-only,citation_audit/papers/batatia2022mace.link.txt,Saved landing link only; no open PDF downloaded.,739,MACE~\citep{batatia2022mace} use iterated Clebsch--Gordan products
+bejnordi2017camelyon,dataset,1,good,article,Diagnostic Assessment of Deep Learning Algorithms for Detection of Lymph Node Metastases in Women With Breast Cancer,2017,JAMA,openalex,Diagnostic Assessment of Deep Learning Algorithms for Detection of Lymph Node Metastases in Women With Breast Cancer,2017,JAMA,10.1001/jama.2017.14585,1.0,remote-doi,link-only,citation_audit/papers/bejnordi2017camelyon.link.txt,Saved landing link only; no open PDF downloaded.,2336,images of lymph node sections~\citep{bejnordi2017camelyon}. We use
+bendory2025orbit,theory,1,acceptable-but-weaker,article,Orbit Recovery for Spherical Functions,2025,arXiv preprint arXiv:2508.02674,arxiv,Orbit Recovery for Spherical Functions,2025,arXiv preprint arXiv:2508.02674,,1.0,local-fallback,downloaded,citation_audit/papers/bendory2025orbit.pdf,Preprint citation; archival version may be preferable if one exists. | No DOI found. | Open PDF downloaded.,1699,Bendory et al.~\citep{bendory2025orbit}
+bonev2023spherical,theory,2,good,inproceedings,Spherical {F}ourier Neural Operators: Learning Stable Dynamics on the Sphere,2023,International Conference on Machine Learning (ICML),manual,Spherical Fourier Neural Operators: Learning Stable Dynamics on the Sphere,2023,Proceedings of the 40th International Conference on Machine Learning,,1.0,local-fallback,downloaded,citation_audit/papers/bonev2023spherical.pdf,No DOI found. | Open PDF downloaded.,"258,466",\texttt{torch\_harmonics}~\citep{bonev2023spherical} || transform (via \texttt{torch\_harmonics}~\citep{bonev2023spherical}).
+cesa2022program,theory,2,good,inproceedings,A Program to Build {E(N)}-Equivariant Steerable {CNN}s,2022,International Conference on Learning Representations (ICLR),manual,A Program to Build E(N)-Equivariant Steerable CNNs,2022,International Conference on Learning Representations,,1.0,local-fallback,link-only,citation_audit/papers/cesa2022program.link.txt,No DOI found. | Saved landing link only; no open PDF downloaded.,"717,808",\texttt{escnn}~\citep{cesa2022program} and \texttt{e3nn}~\citep{geiger2022e3nn} || band-limits; integration with \texttt{escnn}~\citep{cesa2022program} and
+chevalley2024bispectral,related-work,1,good,inproceedings,A Bispectral {3D} {U-Net} for Rotation Robustness in Medical Segmentation,2024,Topology- and Graph-Informed Imaging Informatics,openalex,A Bispectral 3D U-Net for Rotation Robustness in Medical Segmentation,2024,Lecture notes in computer science,10.1007/978-3-031-73967-5_5,1.0,remote-doi,link-only,citation_audit/papers/chevalley2024bispectral.link.txt,Saved landing link only; no open PDF downloaded.,733,\citet{chevalley2024bispectral} explore bispectral signatures for data
+cohen2016group,theory,2,good,inproceedings,Group Equivariant Convolutional Networks,2016,International Conference on Machine Learning (ICML),openalex,Group Equivariant Convolutional Networks,2016,arXiv (Cornell University),10.48550/arxiv.1602.07576,1.0,remote-doi,downloaded,citation_audit/papers/cohen2016group.pdf,Open PDF downloaded.,"111,179","Equivariant neural networks~\citep{cohen2016group, weiler2019general} enforce || for all $g \in G$~\citep{cohen2016group, weiler2019general}."
+cohen2018spherical,experimental,5,good,inproceedings,Spherical {CNN}s,2018,International Conference on Learning Representations (ICLR),openalex,Spherical CNNs,2018,arXiv (Cornell University),,1.0,local-fallback,downloaded,citation_audit/papers/cohen2018spherical.pdf,No DOI found. | Open PDF downloaded.,"615,637,650,707,2353","2-sphere---on Spherical MNIST~\cite{cohen2018spherical}, the standard || Following Cohen et al.~\cite{cohen2018spherical}, we evaluate under three || et al.\ reference numbers are from \cite{cohen2018spherical}. || Cohen's spherical CNN~\cite{cohen2018spherical} reaches 96\%/95\%/94\% || projection following \citet{cohen2018spherical} and sampled on a"
+edidin2024orbit,theory,1,good,article,Orbit Recovery for Band-Limited Functions,2024,SIAM Journal on Applied Algebra and Geometry,openalex,Orbit Recovery for Band-Limited Functions,2024,SIAM Journal on Applied Algebra and Geometry,10.1137/23m1577808,1.0,remote-doi,downloaded,citation_audit/papers/edidin2024orbit.pdf,Open PDF downloaded.,1696,Edidin \& Satriano~\citep{edidin2024orbit}
+esteves2018spherical,related-work,1,good,inproceedings,Learning {SO}(3) Equivariant Representations with Spherical {CNN}s,2018,European Conference on Computer Vision (ECCV),openalex,Learning SO(3) Equivariant Representations with Spherical CNNs,2018,Lecture notes in computer science,10.1007/978-3-030-01261-8_4,1.0,remote-doi,link-only,citation_audit/papers/esteves2018spherical.link.txt,Saved landing link only; no open PDF downloaded.,752,\citet{esteves2018spherical} extend this to the sphere. Unlike the
+geiger2022e3nn,theory,2,acceptable-but-weaker,article,e3nn: {E}uclidean Neural Networks,2022,arXiv preprint arXiv:2207.09453,arxiv,e3nn: Euclidean Neural Networks,2022,arXiv preprint arXiv:2207.09453,,1.0,local-fallback,downloaded,citation_audit/papers/geiger2022e3nn.pdf,Preprint citation; archival version may be preferable if one exists. | No DOI found. | Open PDF downloaded.,"717,809",\texttt{escnn}~\citep{cesa2022program} and \texttt{e3nn}~\citep{geiger2022e3nn} || \texttt{e3nn}~\citep{geiger2022e3nn}; and applications to cryo-EM
+harris2020numpy,software,1,good,article,Array programming with {NumPy},2020,Nature,openalex,Array programming with NumPy,2020,Nature,10.1038/s41586-020-2649-2,1.0,remote-doi,downloaded,citation_audit/papers/harris2020numpy.pdf,Open PDF downloaded.,257,"NumPy~\citep{harris2020numpy}, and"
+huang2017densenet,experimental,2,good,inproceedings,Densely Connected Convolutional Networks,2017,IEEE Conference on Computer Vision and Pattern Recognition (CVPR),openalex,Densely Connected Convolutional Networks,2017,,10.1109/cvpr.2017.243,1.0,remote-doi,link-only,citation_audit/papers/huang2017densenet.link.txt,Saved landing link only; no open PDF downloaded.,"492,2143",$C_8$-equivariant DenseNet~\citep{huang2017densenet} backbone following || DenseNet~\citep{huang2017densenet} backbone with block configuration
+kakarala2012bispectrum,theory,4,good,article,The Bispectrum as a Source of Phase-Sensitive Invariants for {F}ourier Descriptors: A Group-Theoretic Approach,2012,Journal of Mathematical Imaging and Vision,openalex,The Bispectrum as a Source of Phase-Sensitive Invariants for Fourier Descriptors: A Group-Theoretic Approach,2012,Journal of Mathematical Imaging and Vision,10.1007/s10851-012-0330-6,1.0,remote-doi,downloaded,citation_audit/papers/kakarala2012bispectrum.pdf,Open PDF downloaded.,"119,230,755,1694","guarantees~\citep{kakarala2012bispectrum}. || it separates orbits for generic signals~\citep{kakarala2012bispectrum}. However, || The bispectrum's completeness guarantee~\citep{kakarala2012bispectrum} is its || Kakarala~\citep{kakarala2012bispectrum}"
+kuipers2023regular,theory,2,good,inproceedings,Regular {SE}(3) Group Convolutions for Volumetric Medical Image Analysis,2023,Medical Image Computing and Computer Assisted Intervention -- {MICCAI} 2023,openalex,Regular SE(3) Group Convolutions for Volumetric Medical Image Analysis,2023,Lecture notes in computer science,10.1007/978-3-031-43898-1_25,1.0,remote-doi,link-only,citation_audit/papers/kuipers2023regular.link.txt,Saved landing link only; no open PDF downloaded.,"591,2234",convolutions~\citep{kuipers2023regular} (69.8\%). || Regular SE(3) conv & \citet{kuipers2023regular} & 172K & 0.698 & --- \\
+loshchilov2019adamw,experimental,3,good,inproceedings,Decoupled Weight Decay Regularization,2019,International Conference on Learning Representations (ICLR),manual,Decoupled Weight Decay Regularization,2019,International Conference on Learning Representations,,1.0,local-fallback,link-only,citation_audit/papers/loshchilov2019adamw.link.txt,No DOI found. | Saved landing link only; no open PDF downloaded.,"498,556,2170","AdamW~\citep{loshchilov2019adamw}. Full model descriptions are in || are trained with AdamW~\citep{loshchilov2019adamw} for 100 epochs with || training data} (26{,}214 images) with AdamW~\citep{loshchilov2019adamw}"
+mallat2012scattering,related-work,1,good,article,Group Invariant Scattering,2012,Communications on Pure and Applied Mathematics,openalex,Group Invariant Scattering,2012,Communications on Pure and Applied Mathematics,10.1002/cpa.21413,1.0,remote-doi,link-only,citation_audit/papers/mallat2012scattering.link.txt,Saved landing link only; no open PDF downloaded.,750,The wavelet scattering transform~\citep{mallat2012scattering} provides
+mataigne2024selective,theory,8,good,inproceedings,The Selective {G}-Bispectrum and its Inversion: Applications to {G}-Invariant Networks,2024,Advances in Neural Information Processing Systems,openalex,The Selective $G$-Bispectrum and its Inversion: Applications to $G$-Invariant Networks,2024,,10.52202/079017-3674,1.0,remote-doi,link-only,citation_audit/papers/mataigne2024selective.link.txt,Saved landing link only; no open PDF downloaded.,"126,138,233,399,1160,1702,1801,1925","$O(|G|^2)$ to $O(|G|)$~\citep{mataigne2024selective}, a user still needs a numerically stable implementation, || the bootstrap inversion failure~\citep{mataigne2024selective} for the || \emph{Selective bispectra.} \citet{mataigne2024selective} show that for || The bootstrap inversion of \citet{mataigne2024selective} recovers || \citep{mataigne2024selective} is that on~$S^2$, the scalar bispectrum || Mataigne et al.~\citep{mataigne2024selective} || All inversion algorithms in \citet{mataigne2024selective} share the || \citet{mataigne2024selective} acknowledge a related issue in the"
+medmnistv2,dataset,2,good,article,{MedMNIST} v2 -- A Large-Scale Lightweight Benchmark for {2D} and {3D} Biomedical Image Classification,2023,Scientific Data,openalex,MedMNIST v2 - A large-scale lightweight benchmark for 2D and 3D biomedical image classification,2023,Scientific Data,10.1038/s41597-022-01721-8,1.0,remote-doi,downloaded,citation_audit/papers/medmnistv2.pdf,Open PDF downloaded.,"550,2343","OrganMNIST3D~\citep{medmnistv2} (1{,}742 abdominal CT volumes, $28^3$ || OrganMNIST3D~\citep{medmnistv2} contains 1{,}742 abdominal CT organ"
+oreiller2022bispectral,related-work,1,good,inproceedings,Robust Multi-Organ Nucleus Segmentation Using a Locally Rotation Invariant Bispectral {U-Net},2022,Proceedings of The 5th International Conference on Medical Imaging with Deep Learning,manual,Robust Multi-Organ Nucleus Segmentation Using a Locally Rotation Invariant Bispectral U-Net,2022,Medical Imaging with Deep Learning,,1.0,local-fallback,downloaded,citation_audit/papers/oreiller2022bispectral.pdf,No DOI found. | Open PDF downloaded.,728,\citet{oreiller2022bispectral} applied bispectral CNNs to medical imaging
+paszke2019pytorch,software,2,good,inproceedings,"Py{T}orch: An Imperative Style, High-Performance Deep Learning Library",2019,Advances in Neural Information Processing Systems (NeurIPS),openalex,"PyTorch: An Imperative Style, High-Performance Deep Learning Library",2019,arXiv (Cornell University),10.48550/arxiv.1912.01703,1.0,remote-doi,downloaded,citation_audit/papers/paszke2019pytorch.pdf,Open PDF downloaded.,"129,256","This paper presents \texttt{bispectrum}, a PyTorch~\citep{paszke2019pytorch} || The only dependencies are PyTorch~\citep{paszke2019pytorch},"
+price2025freedom,theory,1,good,inproceedings,The Price of Freedom: Exploring Expressivity and Runtime Tradeoffs in Equivariant Tensor Products,2025,Proceedings of the 42nd International Conference on Machine Learning (ICML),openalex,The Price of Freedom: Exploring Expressivity and Runtime Tradeoffs in Equivariant Tensor Products,2025,ArXiv.org,10.48550/arxiv.2506.13523,1.0,remote-doi,downloaded,citation_audit/papers/price2025freedom.pdf,Open PDF downloaded.,742,\citet{price2025freedom} analyse the \emph{price of freedom} in tensor
+sanborn2023bispectral,related-work,1,good,inproceedings,Bispectral Neural Networks,2023,International Conference on Learning Representations (ICLR),openalex,Bispectral Neural Networks,2022,arXiv (Cornell University),10.48550/arxiv.2209.03416,1.0,remote-doi,downloaded,citation_audit/papers/sanborn2023bispectral.pdf,Open PDF downloaded.,726,"\citet{sanborn2023bispectral} introduced bispectral neural networks (BNNs),"
+sangalli2023movfrnet,theory,1,acceptable-but-weaker,inproceedings,Moving Frame Net: {SE}(3)-Equivariant Network for Volumes,2023,Proceedings of the 1st NeurIPS Workshop on Symmetry and Geometry in Neural Representations,openalex,Moving Frame Net: SE(3)-Equivariant Network for Volumes,2022,arXiv (Cornell University),10.48550/arxiv.2211.03420,1.0,remote-doi,downloaded,citation_audit/papers/sangalli2023movfrnet.pdf,Workshop citation rather than a main archival venue. | Open PDF downloaded.,2233,SE3MovFrNet & \citet{sangalli2023movfrnet} & --- & 0.745 & --- \\
+thomas2018tensor,theory,1,acceptable-but-weaker,inproceedings,Tensor Field Networks: Rotation- and Translation-Equivariant Neural Networks for {3D} Point Clouds,2018,arXiv preprint arXiv:1802.08219,arxiv,Tensor Field Networks: Rotation- and Translation-Equivariant Neural Networks for 3D Point Clouds,2018,arXiv preprint arXiv:1802.08219,,1.0,local-fallback,downloaded,citation_audit/papers/thomas2018tensor.pdf,Preprint citation; archival version may be preferable if one exists. | No DOI found. | Open PDF downloaded.,738,Tensor field networks~\citep{thomas2018tensor} and
+veeling2018rotation,dataset,5,good,inproceedings,Rotation Equivariant {CNN}s for Digital Pathology,2018,Medical Image Computing and Computer Assisted Intervention (MICCAI),openalex,Rotation Equivariant CNNs for Digital Pathology,2018,Lecture notes in computer science,10.1007/978-3-030-00934-2_24,1.0,remote-doi,unavailable,,No paper URL resolved.,"490,493,2124,2145,2334","PatchCamelyon~\citep{veeling2018rotation} (PCam), 327K histopathology || \citet{veeling2018rotation}; they differ only in the terminal invariant map: || \citet{veeling2018rotation}, trained on 100\% data.} || following the architecture of \citet{veeling2018rotation} but || PatchCamelyon~\citep{veeling2018rotation} consists of 327{,}680 RGB"
+weiler2019general,theory,4,good,inproceedings,General {E}(2)-Equivariant Steerable {CNN}s,2019,Advances in Neural Information Processing Systems (NeurIPS),openalex,General $E(2)$-Equivariant Steerable CNNs,2019,arXiv (Cornell University),10.48550/arxiv.1911.08251,1.0,remote-doi,downloaded,citation_audit/papers/weiler2019general.pdf,Open PDF downloaded.,"111,116,179,210","Equivariant neural networks~\citep{cohen2016group, weiler2019general} enforce || usable~\citep{weiler2019general}, but they do not preserve all equivariant || for all $g \in G$~\citep{cohen2016group, weiler2019general}. || sacrifice exactness through aliasing~\citep{weiler2019general}. The full"
+zhemchuzhnikov2024ilponet,theory,1,good,inproceedings,{ILPO-NET}: Network for the Invariant Recognition of Arbitrary Volumetric Patterns in {3D},2024,Machine Learning and Knowledge Discovery in Databases. Research Track,openalex,ILPO-NET: Network for the Invariant Recognition of Arbitrary Volumetric Patterns in 3D,2024,Lecture notes in computer science,10.1007/978-3-031-70359-1_21,1.0,remote-doi,link-only,citation_audit/papers/zhemchuzhnikov2024ilponet.link.txt,Saved landing link only; no open PDF downloaded.,2231,ILPOResNet-50 & \citet{zhemchuzhnikov2024ilponet} & 38K & 0.879 & 0.992 \\
+zhemchuzhnikov2025equilopo,theory,1,good,inproceedings,On the {F}ourier Analysis in the {SO}(3) Space: the {EquiLoPO} Network,2025,International Conference on Learning Representations (ICLR),manual,On the Fourier Analysis in the SO(3) Space: the EquiLoPO Network,2025,International Conference on Learning Representations,,1.0,local-fallback,link-only,citation_audit/papers/zhemchuzhnikov2025equilopo.link.txt,No DOI found. | Saved landing link only; no open PDF downloaded.,2232,EquiLoPO (local train.) & \citet{zhemchuzhnikov2025equilopo} & 418K & 0.866 & 0.991 \\
diff --git a/paper/citation_audit/citation_audit.json b/paper/citation_audit/citation_audit.json
new file mode 100644
index 0000000..1847713
--- /dev/null
+++ b/paper/citation_audit/citation_audit.json
@@ -0,0 +1,1009 @@
+[
+  {
+    "key": "batatia2022mace",
+    "category": "related-work",
+    "mention_count": 1,
+    "local_entry_type": "inproceedings",
+    "local_title": "{MACE}: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields",
+    "local_year": 2022,
+    "local_venue": "Advances in Neural Information Processing Systems (NeurIPS)",
+    "local_doi": "10.52202/068431-0830",
+    "local_url": "https://proceedings.neurips.cc/paper_files/paper/2022/hash/4a36c3c51af11ed9f34615b81edb5bbc-Abstract-Conference.html",
+    "resolved_source": "openalex",
+    "resolved_title": "MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields",
+    "resolved_year": 2022,
+    "resolved_venue": null,
+    "resolved_doi": "10.52202/068431-0830",
+    "title_match_score": 1.0,
+    "bibtex_source": "remote-doi",
+    "paper_status": "link-only",
+    "paper_path": "citation_audit/papers/batatia2022mace.link.txt",
+    "assessment": "good",
+    "notes": [
+      "Saved landing link only; no open PDF downloaded."
+    ],
+    "mentions": [
+      {
+        "line_number": 739,
+        "snippet": "MACE~\\citep{batatia2022mace} use iterated Clebsch--Gordan products"
+      }
+    ]
+  },
+  {
+    "key": "bejnordi2017camelyon",
+    "category": "dataset",
+    "mention_count": 1,
+    "local_entry_type": "article",
+    "local_title": "Diagnostic Assessment of Deep Learning Algorithms for Detection of Lymph Node Metastases in Women With Breast Cancer",
+    "local_year": 2017,
+    "local_venue": "JAMA",
+    "local_doi": "10.1001/jama.2017.14585",
+    "local_url": null,
+    "resolved_source": "openalex",
+    "resolved_title": "Diagnostic Assessment of Deep Learning Algorithms for Detection of Lymph Node Metastases in Women With Breast Cancer",
+    "resolved_year": 2017,
+    "resolved_venue": "JAMA",
+    "resolved_doi": "10.1001/jama.2017.14585",
+    "title_match_score": 1.0,
+    "bibtex_source": "remote-doi",
+    "paper_status": "link-only",
+    "paper_path": "citation_audit/papers/bejnordi2017camelyon.link.txt",
+    "assessment": "good",
+    "notes": [
+      "Saved landing link only; no open PDF downloaded."
+    ],
+    "mentions": [
+      {
+        "line_number": 2336,
+        "snippet": "images of lymph node sections~\\citep{bejnordi2017camelyon}. We use"
+      }
+    ]
+  },
+  {
+    "key": "bendory2025orbit",
+    "category": "theory",
+    "mention_count": 1,
+    "local_entry_type": "article",
+    "local_title": "Orbit Recovery for Spherical Functions",
+    "local_year": 2025,
+    "local_venue": "arXiv preprint arXiv:2508.02674",
+    "local_doi": null,
+    "local_url": "https://arxiv.org/abs/2508.02674",
+    "resolved_source": "arxiv",
+    "resolved_title": "Orbit Recovery for Spherical Functions",
+    "resolved_year": 2025,
+    "resolved_venue": "arXiv preprint arXiv:2508.02674",
+    "resolved_doi": null,
+    "title_match_score": 1.0,
+    "bibtex_source": "local-fallback",
+    "paper_status": "downloaded",
+    "paper_path": "citation_audit/papers/bendory2025orbit.pdf",
+    "assessment": "acceptable-but-weaker",
+    "notes": [
+      "Preprint citation; archival version may be preferable if one exists.",
+      "No DOI found.",
+      "Open PDF downloaded."
+    ],
+    "mentions": [
+      {
+        "line_number": 1699,
+        "snippet": "Bendory et al.~\\citep{bendory2025orbit}"
+      }
+    ]
+  },
+  {
+    "key": "bonev2023spherical",
+    "category": "theory",
+    "mention_count": 2,
+    "local_entry_type": "inproceedings",
+    "local_title": "Spherical {F}ourier Neural Operators: Learning Stable Dynamics on the Sphere",
+    "local_year": 2023,
+    "local_venue": "International Conference on Machine Learning (ICML)",
+    "local_doi": null,
+    "local_url": null,
+    "resolved_source": "manual",
+    "resolved_title": "Spherical Fourier Neural Operators: Learning Stable Dynamics on the Sphere",
+    "resolved_year": 2023,
+    "resolved_venue": "Proceedings of the 40th International Conference on Machine Learning",
+    "resolved_doi": null,
+    "title_match_score": 1.0,
+    "bibtex_source": "local-fallback",
+    "paper_status": "downloaded",
+    "paper_path": "citation_audit/papers/bonev2023spherical.pdf",
+    "assessment": "good",
+    "notes": [
+      "No DOI found.",
+      "Open PDF downloaded."
+    ],
+    "mentions": [
+      {
+        "line_number": 258,
+        "snippet": "\\texttt{torch\\_harmonics}~\\citep{bonev2023spherical}"
+      },
+      {
+        "line_number": 466,
+        "snippet": "transform (via \\texttt{torch\\_harmonics}~\\citep{bonev2023spherical})."
+      }
+    ]
+  },
+  {
+    "key": "cesa2022program",
+    "category": "theory",
+    "mention_count": 2,
+    "local_entry_type": "inproceedings",
+    "local_title": "A Program to Build {E(N)}-Equivariant Steerable {CNN}s",
+    "local_year": 2022,
+    "local_venue": "International Conference on Learning Representations (ICLR)",
+    "local_doi": null,
+    "local_url": "https://openreview.net/forum?id=WE4qe9xlnQw",
+    "resolved_source": "manual",
+    "resolved_title": "A Program to Build E(N)-Equivariant Steerable CNNs",
+    "resolved_year": 2022,
+    "resolved_venue": "International Conference on Learning Representations",
+    "resolved_doi": null,
+    "title_match_score": 1.0,
+    "bibtex_source": "local-fallback",
+    "paper_status": "link-only",
+    "paper_path": "citation_audit/papers/cesa2022program.link.txt",
+    "assessment": "good",
+    "notes": [
+      "No DOI found.",
+      "Saved landing link only; no open PDF downloaded."
+    ],
+    "mentions": [
+      {
+        "line_number": 717,
+        "snippet": "\\texttt{escnn}~\\citep{cesa2022program} and \\texttt{e3nn}~\\citep{geiger2022e3nn}"
+      },
+      {
+        "line_number": 808,
+        "snippet": "band-limits; integration with \\texttt{escnn}~\\citep{cesa2022program} and"
+      }
+    ]
+  },
+  {
+    "key": "chevalley2024bispectral",
+    "category": "related-work",
+    "mention_count": 1,
+    "local_entry_type": "inproceedings",
+    "local_title": "A Bispectral {3D} {U-Net} for Rotation Robustness in Medical Segmentation",
+    "local_year": 2024,
+    "local_venue": "Topology- and Graph-Informed Imaging Informatics",
+    "local_doi": "10.1007/978-3-031-73967-5_5",
+    "local_url": "https://doi.org/10.1007/978-3-031-73967-5_5",
+    "resolved_source": "openalex",
+    "resolved_title": "A Bispectral 3D U-Net for\u00a0Rotation Robustness in\u00a0Medical Segmentation",
+    "resolved_year": 2024,
+    "resolved_venue": "Lecture notes in computer science",
+    "resolved_doi": "10.1007/978-3-031-73967-5_5",
+    "title_match_score": 1.0,
+    "bibtex_source": "remote-doi",
+    "paper_status": "link-only",
+    "paper_path": "citation_audit/papers/chevalley2024bispectral.link.txt",
+    "assessment": "good",
+    "notes": [
+      "Saved landing link only; no open PDF downloaded."
+    ],
+    "mentions": [
+      {
+        "line_number": 733,
+        "snippet": "\\citet{chevalley2024bispectral} explore bispectral signatures for data"
+      }
+    ]
+  },
+  {
+    "key": "cohen2016group",
+    "category": "theory",
+    "mention_count": 2,
+    "local_entry_type": "inproceedings",
+    "local_title": "Group Equivariant Convolutional Networks",
+    "local_year": 2016,
+    "local_venue": "International Conference on Machine Learning (ICML)",
+    "local_doi": null,
+    "local_url": null,
+    "resolved_source": "openalex",
+    "resolved_title": "Group Equivariant Convolutional Networks",
+    "resolved_year": 2016,
+    "resolved_venue": "arXiv (Cornell University)",
+    "resolved_doi": "10.48550/arxiv.1602.07576",
+    "title_match_score": 1.0,
+    "bibtex_source": "remote-doi",
+    "paper_status": "downloaded",
+    "paper_path": "citation_audit/papers/cohen2016group.pdf",
+    "assessment": "good",
+    "notes": [
+      "Open PDF downloaded."
+    ],
+    "mentions": [
+      {
+        "line_number": 111,
+        "snippet": "Equivariant neural networks~\\citep{cohen2016group, weiler2019general} enforce"
+      },
+      {
+        "line_number": 179,
+        "snippet": "for all $g \\in G$~\\citep{cohen2016group, weiler2019general}."
+      }
+    ]
+  },
+  {
+    "key": "cohen2018spherical",
+    "category": "experimental",
+    "mention_count": 5,
+    "local_entry_type": "inproceedings",
+    "local_title": "Spherical {CNN}s",
+    "local_year": 2018,
+    "local_venue": "International Conference on Learning Representations (ICLR)",
+    "local_doi": null,
+    "local_url": null,
+    "resolved_source": "openalex",
+    "resolved_title": "Spherical CNNs",
+    "resolved_year": 2018,
+    "resolved_venue": "arXiv (Cornell University)",
+    "resolved_doi": null,
+    "title_match_score": 1.0,
+    "bibtex_source": "local-fallback",
+    "paper_status": "downloaded",
+    "paper_path": "citation_audit/papers/cohen2018spherical.pdf",
+    "assessment": "good",
+    "notes": [
+      "No DOI found.",
+      "Open PDF downloaded."
+    ],
+    "mentions": [
+      {
+        "line_number": 615,
+        "snippet": "2-sphere---on Spherical MNIST~\\cite{cohen2018spherical}, the standard"
+      },
+      {
+        "line_number": 637,
+        "snippet": "Following Cohen et al.~\\cite{cohen2018spherical}, we evaluate under three"
+      },
+      {
+        "line_number": 650,
+        "snippet": "et al.\\ reference numbers are from \\cite{cohen2018spherical}."
+      },
+      {
+        "line_number": 707,
+        "snippet": "Cohen's spherical CNN~\\cite{cohen2018spherical} reaches 96\\%/95\\%/94\\%"
+      },
+      {
+        "line_number": 2353,
+        "snippet": "projection following \\citet{cohen2018spherical} and sampled on a"
+      }
+    ]
+  },
+  {
+    "key": "edidin2024orbit",
+    "category": "theory",
+    "mention_count": 1,
+    "local_entry_type": "article",
+    "local_title": "Orbit Recovery for Band-Limited Functions",
+    "local_year": 2024,
+    "local_venue": "SIAM Journal on Applied Algebra and Geometry",
+    "local_doi": "10.1137/23M1577808",
+    "local_url": "https://arxiv.org/abs/2306.00155",
+    "resolved_source": "openalex",
+    "resolved_title": "Orbit Recovery for Band-Limited Functions",
+    "resolved_year": 2024,
+    "resolved_venue": "SIAM Journal on Applied Algebra and Geometry",
+    "resolved_doi": "10.1137/23m1577808",
+    "title_match_score": 1.0,
+    "bibtex_source": "remote-doi",
+    "paper_status": "downloaded",
+    "paper_path": "citation_audit/papers/edidin2024orbit.pdf",
+    "assessment": "good",
+    "notes": [
+      "Open PDF downloaded."
+    ],
+    "mentions": [
+      {
+        "line_number": 1696,
+        "snippet": "Edidin \\& Satriano~\\citep{edidin2024orbit}"
+      }
+    ]
+  },
+  {
+    "key": "esteves2018spherical",
+    "category": "related-work",
+    "mention_count": 1,
+    "local_entry_type": "inproceedings",
+    "local_title": "Learning {SO}(3) Equivariant Representations with Spherical {CNN}s",
+    "local_year": 2018,
+    "local_venue": "European Conference on Computer Vision (ECCV)",
+    "local_doi": null,
+    "local_url": null,
+    "resolved_source": "openalex",
+    "resolved_title": "Learning SO(3) Equivariant Representations with Spherical CNNs",
+    "resolved_year": 2018,
+    "resolved_venue": "Lecture notes in computer science",
+    "resolved_doi": "10.1007/978-3-030-01261-8_4",
+    "title_match_score": 1.0,
+    "bibtex_source": "remote-doi",
+    "paper_status": "link-only",
+    "paper_path": "citation_audit/papers/esteves2018spherical.link.txt",
+    "assessment": "good",
+    "notes": [
+      "Saved landing link only; no open PDF downloaded."
+    ],
+    "mentions": [
+      {
+        "line_number": 752,
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+        "line_number": 119,
+        "snippet": "guarantees~\\citep{kakarala2012bispectrum}."
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+      {
+        "line_number": 230,
+        "snippet": "it separates orbits for generic signals~\\citep{kakarala2012bispectrum}. However,"
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+      {
+        "line_number": 755,
+        "snippet": "The bispectrum's completeness guarantee~\\citep{kakarala2012bispectrum} is its"
+      },
+      {
+        "line_number": 1694,
+        "snippet": "Kakarala~\\citep{kakarala2012bispectrum}"
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+    ]
+  },
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+        "line_number": 591,
+        "snippet": "convolutions~\\citep{kuipers2023regular} (69.8\\%)."
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+        "line_number": 2234,
+        "snippet": "Regular SE(3) conv & \\citet{kuipers2023regular} & 172K & 0.698 & --- \\\\"
+      }
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+  },
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+    "resolved_venue": "International Conference on Learning Representations",
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+        "line_number": 498,
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+      {
+        "line_number": 556,
+        "snippet": "are trained with AdamW~\\citep{loshchilov2019adamw} for 100 epochs with"
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+        "snippet": "training data} (26{,}214 images) with AdamW~\\citep{loshchilov2019adamw}"
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+    "local_year": 2024,
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+        "line_number": 126,
+        "snippet": "$O(|G|^2)$ to $O(|G|)$~\\citep{mataigne2024selective}, a user still needs a numerically stable implementation,"
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+      {
+        "line_number": 138,
+        "snippet": "the bootstrap inversion failure~\\citep{mataigne2024selective} for the"
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+      {
+        "line_number": 233,
+        "snippet": "\\emph{Selective bispectra.} \\citet{mataigne2024selective} show that for"
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+        "snippet": "Mataigne et al.~\\citep{mataigne2024selective}"
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+        "snippet": "All inversion algorithms in \\citet{mataigne2024selective} share the"
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+        "snippet": "\\citet{mataigne2024selective} acknowledge a related issue in the"
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+        "line_number": 2343,
+        "snippet": "OrganMNIST3D~\\citep{medmnistv2} contains 1{,}742 abdominal CT organ"
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+        "snippet": "The only dependencies are PyTorch~\\citep{paszke2019pytorch},"
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+        "snippet": "\\citet{price2025freedom} analyse the \\emph{price of freedom} in tensor"
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+    "local_url": null,
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+        "snippet": "\\citet{sanborn2023bispectral} introduced bispectral neural networks (BNNs),"
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+    "local_doi": null,
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+    "resolved_year": 2022,
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+      "Workshop citation rather than a main archival venue.",
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+        "snippet": "SE3MovFrNet & \\citet{sangalli2023movfrnet} & --- & 0.745 & --- \\\\"
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+    "bibtex_source": "local-fallback",
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+    "notes": [
+      "Preprint citation; archival version may be preferable if one exists.",
+      "No DOI found.",
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+        "snippet": "Tensor field networks~\\citep{thomas2018tensor} and"
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+    "local_url": null,
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+    "resolved_venue": "Lecture notes in computer science",
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+        "line_number": 490,
+        "snippet": "PatchCamelyon~\\citep{veeling2018rotation} (PCam), 327K histopathology"
+      },
+      {
+        "line_number": 493,
+        "snippet": "\\citet{veeling2018rotation}; they differ only in the terminal invariant map:"
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+      {
+        "line_number": 2124,
+        "snippet": "\\citet{veeling2018rotation}, trained on 100\\% data.}"
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+        "line_number": 2145,
+        "snippet": "following the architecture of \\citet{veeling2018rotation} but"
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+      {
+        "line_number": 2334,
+        "snippet": "PatchCamelyon~\\citep{veeling2018rotation} consists of 327{,}680 RGB"
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+    "resolved_source": "openalex",
+    "resolved_title": "General $E(2)$-Equivariant Steerable CNNs",
+    "resolved_year": 2019,
+    "resolved_venue": "arXiv (Cornell University)",
+    "resolved_doi": "10.48550/arxiv.1911.08251",
+    "title_match_score": 1.0,
+    "bibtex_source": "remote-doi",
+    "paper_status": "downloaded",
+    "paper_path": "citation_audit/papers/weiler2019general.pdf",
+    "assessment": "good",
+    "notes": [
+      "Open PDF downloaded."
+    ],
+    "mentions": [
+      {
+        "line_number": 111,
+        "snippet": "Equivariant neural networks~\\citep{cohen2016group, weiler2019general} enforce"
+      },
+      {
+        "line_number": 116,
+        "snippet": "usable~\\citep{weiler2019general}, but they do not preserve all equivariant"
+      },
+      {
+        "line_number": 179,
+        "snippet": "for all $g \\in G$~\\citep{cohen2016group, weiler2019general}."
+      },
+      {
+        "line_number": 210,
+        "snippet": "sacrifice exactness through aliasing~\\citep{weiler2019general}. The full"
+      }
+    ]
+  },
+  {
+    "key": "zhemchuzhnikov2024ilponet",
+    "category": "theory",
+    "mention_count": 1,
+    "local_entry_type": "inproceedings",
+    "local_title": "{ILPO-NET}: Network for the Invariant Recognition of Arbitrary Volumetric Patterns in {3D}",
+    "local_year": 2024,
+    "local_venue": "Machine Learning and Knowledge Discovery in Databases. Research Track",
+    "local_doi": "10.1007/978-3-031-70359-1_21",
+    "local_url": "https://doi.org/10.1007/978-3-031-70359-1_21",
+    "resolved_source": "openalex",
+    "resolved_title": "ILPO-NET: Network for\u00a0the\u00a0Invariant Recognition of\u00a0Arbitrary Volumetric Patterns in\u00a03D",
+    "resolved_year": 2024,
+    "resolved_venue": "Lecture notes in computer science",
+    "resolved_doi": "10.1007/978-3-031-70359-1_21",
+    "title_match_score": 1.0,
+    "bibtex_source": "remote-doi",
+    "paper_status": "link-only",
+    "paper_path": "citation_audit/papers/zhemchuzhnikov2024ilponet.link.txt",
+    "assessment": "good",
+    "notes": [
+      "Saved landing link only; no open PDF downloaded."
+    ],
+    "mentions": [
+      {
+        "line_number": 2231,
+        "snippet": "ILPOResNet-50 & \\citet{zhemchuzhnikov2024ilponet} & 38K & 0.879 & 0.992 \\\\"
+      }
+    ]
+  },
+  {
+    "key": "zhemchuzhnikov2025equilopo",
+    "category": "theory",
+    "mention_count": 1,
+    "local_entry_type": "inproceedings",
+    "local_title": "On the {F}ourier Analysis in the {SO}(3) Space: the {EquiLoPO} Network",
+    "local_year": 2025,
+    "local_venue": "International Conference on Learning Representations (ICLR)",
+    "local_doi": null,
+    "local_url": "https://openreview.net/forum?id=LvTSvdiSwG",
+    "resolved_source": "manual",
+    "resolved_title": "On the Fourier Analysis in the SO(3) Space: the EquiLoPO Network",
+    "resolved_year": 2025,
+    "resolved_venue": "International Conference on Learning Representations",
+    "resolved_doi": null,
+    "title_match_score": 1.0,
+    "bibtex_source": "local-fallback",
+    "paper_status": "link-only",
+    "paper_path": "citation_audit/papers/zhemchuzhnikov2025equilopo.link.txt",
+    "assessment": "good",
+    "notes": [
+      "No DOI found.",
+      "Saved landing link only; no open PDF downloaded."
+    ],
+    "mentions": [
+      {
+        "line_number": 2232,
+        "snippet": "EquiLoPO (local train.) & \\citet{zhemchuzhnikov2025equilopo} & 418K & 0.866 & 0.991 \\\\"
+      }
+    ]
+  }
+]
\ No newline at end of file
diff --git a/paper/citation_audit/citation_audit.md b/paper/citation_audit/citation_audit.md
new file mode 100644
index 0000000..cf60eae
--- /dev/null
+++ b/paper/citation_audit/citation_audit.md
@@ -0,0 +1,378 @@
+# Citation Audit
+
+- Total cited keys: 29
+- Assessments: 25 good, 4 acceptable-but-weaker, 0 check
+- Paper assets: 16 PDFs downloaded, 12 link-only, 1 unavailable
+
+| Key | Category | Mentions | Assessment | Resolved source | BibTeX | Paper | Notes |
+| --- | --- | ---: | --- | --- | --- | --- | --- |
+| batatia2022mace | related-work | 1 | good | openalex | remote-doi | link-only (`citation_audit/papers/batatia2022mace.link.txt`) | Saved landing link only; no open PDF downloaded. |
+| bejnordi2017camelyon | dataset | 1 | good | openalex | remote-doi | link-only (`citation_audit/papers/bejnordi2017camelyon.link.txt`) | Saved landing link only; no open PDF downloaded. |
+| bendory2025orbit | theory | 1 | acceptable-but-weaker | arxiv | local-fallback | downloaded (`citation_audit/papers/bendory2025orbit.pdf`) | Preprint citation; archival version may be preferable if one exists.; No DOI found. |
+| bonev2023spherical | theory | 2 | good | manual | local-fallback | downloaded (`citation_audit/papers/bonev2023spherical.pdf`) | No DOI found.; Open PDF downloaded. |
+| cesa2022program | theory | 2 | good | manual | local-fallback | link-only (`citation_audit/papers/cesa2022program.link.txt`) | No DOI found.; Saved landing link only; no open PDF downloaded. |
+| chevalley2024bispectral | related-work | 1 | good | openalex | remote-doi | link-only (`citation_audit/papers/chevalley2024bispectral.link.txt`) | Saved landing link only; no open PDF downloaded. |
+| cohen2016group | theory | 2 | good | openalex | remote-doi | downloaded (`citation_audit/papers/cohen2016group.pdf`) | Open PDF downloaded. |
+| cohen2018spherical | experimental | 5 | good | openalex | local-fallback | downloaded (`citation_audit/papers/cohen2018spherical.pdf`) | No DOI found.; Open PDF downloaded. |
+| edidin2024orbit | theory | 1 | good | openalex | remote-doi | downloaded (`citation_audit/papers/edidin2024orbit.pdf`) | Open PDF downloaded. |
+| esteves2018spherical | related-work | 1 | good | openalex | remote-doi | link-only (`citation_audit/papers/esteves2018spherical.link.txt`) | Saved landing link only; no open PDF downloaded. |
+| geiger2022e3nn | theory | 2 | acceptable-but-weaker | arxiv | local-fallback | downloaded (`citation_audit/papers/geiger2022e3nn.pdf`) | Preprint citation; archival version may be preferable if one exists.; No DOI found. |
+| harris2020numpy | software | 1 | good | openalex | remote-doi | downloaded (`citation_audit/papers/harris2020numpy.pdf`) | Open PDF downloaded. |
+| huang2017densenet | experimental | 2 | good | openalex | remote-doi | link-only (`citation_audit/papers/huang2017densenet.link.txt`) | Saved landing link only; no open PDF downloaded. |
+| kakarala2012bispectrum | theory | 4 | good | openalex | remote-doi | downloaded (`citation_audit/papers/kakarala2012bispectrum.pdf`) | Open PDF downloaded. |
+| kuipers2023regular | theory | 2 | good | openalex | remote-doi | link-only (`citation_audit/papers/kuipers2023regular.link.txt`) | Saved landing link only; no open PDF downloaded. |
+| loshchilov2019adamw | experimental | 3 | good | manual | local-fallback | link-only (`citation_audit/papers/loshchilov2019adamw.link.txt`) | No DOI found.; Saved landing link only; no open PDF downloaded. |
+| mallat2012scattering | related-work | 1 | good | openalex | remote-doi | link-only (`citation_audit/papers/mallat2012scattering.link.txt`) | Saved landing link only; no open PDF downloaded. |
+| mataigne2024selective | theory | 8 | good | openalex | remote-doi | link-only (`citation_audit/papers/mataigne2024selective.link.txt`) | Saved landing link only; no open PDF downloaded. |
+| medmnistv2 | dataset | 2 | good | openalex | remote-doi | downloaded (`citation_audit/papers/medmnistv2.pdf`) | Open PDF downloaded. |
+| oreiller2022bispectral | related-work | 1 | good | manual | local-fallback | downloaded (`citation_audit/papers/oreiller2022bispectral.pdf`) | No DOI found.; Open PDF downloaded. |
+| paszke2019pytorch | software | 2 | good | openalex | remote-doi | downloaded (`citation_audit/papers/paszke2019pytorch.pdf`) | Open PDF downloaded. |
+| price2025freedom | theory | 1 | good | openalex | remote-doi | downloaded (`citation_audit/papers/price2025freedom.pdf`) | Open PDF downloaded. |
+| sanborn2023bispectral | related-work | 1 | good | openalex | remote-doi | downloaded (`citation_audit/papers/sanborn2023bispectral.pdf`) | Open PDF downloaded. |
+| sangalli2023movfrnet | theory | 1 | acceptable-but-weaker | openalex | remote-doi | downloaded (`citation_audit/papers/sangalli2023movfrnet.pdf`) | Workshop citation rather than a main archival venue.; Open PDF downloaded. |
+| thomas2018tensor | theory | 1 | acceptable-but-weaker | arxiv | local-fallback | downloaded (`citation_audit/papers/thomas2018tensor.pdf`) | Preprint citation; archival version may be preferable if one exists.; No DOI found. |
+| veeling2018rotation | dataset | 5 | good | openalex | remote-doi | unavailable | No paper URL resolved. |
+| weiler2019general | theory | 4 | good | openalex | remote-doi | downloaded (`citation_audit/papers/weiler2019general.pdf`) | Open PDF downloaded. |
+| zhemchuzhnikov2024ilponet | theory | 1 | good | openalex | remote-doi | link-only (`citation_audit/papers/zhemchuzhnikov2024ilponet.link.txt`) | Saved landing link only; no open PDF downloaded. |
+| zhemchuzhnikov2025equilopo | theory | 1 | good | manual | local-fallback | link-only (`citation_audit/papers/zhemchuzhnikov2025equilopo.link.txt`) | No DOI found.; Saved landing link only; no open PDF downloaded. |
+
+## Details
+
+### `batatia2022mace`
+
+- Local title: {MACE}: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields
+- Resolved title: MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields
+- Assessment: good
+- BibTeX source: remote-doi
+- Paper asset: link-only (`citation_audit/papers/batatia2022mace.link.txt`)
+- Notes: Saved landing link only; no open PDF downloaded.
+- Citation contexts:
+  - L739: MACE~\citep{batatia2022mace} use iterated Clebsch--Gordan products
+
+### `bejnordi2017camelyon`
+
+- Local title: Diagnostic Assessment of Deep Learning Algorithms for Detection of Lymph Node Metastases in Women With Breast Cancer
+- Resolved title: Diagnostic Assessment of Deep Learning Algorithms for Detection of Lymph Node Metastases in Women With Breast Cancer
+- Assessment: good
+- BibTeX source: remote-doi
+- Paper asset: link-only (`citation_audit/papers/bejnordi2017camelyon.link.txt`)
+- Notes: Saved landing link only; no open PDF downloaded.
+- Citation contexts:
+  - L2336: images of lymph node sections~\citep{bejnordi2017camelyon}. We use
+
+### `bendory2025orbit`
+
+- Local title: Orbit Recovery for Spherical Functions
+- Resolved title: Orbit Recovery for Spherical Functions
+- Assessment: acceptable-but-weaker
+- BibTeX source: local-fallback
+- Paper asset: downloaded (`citation_audit/papers/bendory2025orbit.pdf`)
+- Notes: Preprint citation; archival version may be preferable if one exists.; No DOI found.; Open PDF downloaded.
+- Citation contexts:
+  - L1699: Bendory et al.~\citep{bendory2025orbit}
+
+### `bonev2023spherical`
+
+- Local title: Spherical {F}ourier Neural Operators: Learning Stable Dynamics on the Sphere
+- Resolved title: Spherical Fourier Neural Operators: Learning Stable Dynamics on the Sphere
+- Assessment: good
+- BibTeX source: local-fallback
+- Paper asset: downloaded (`citation_audit/papers/bonev2023spherical.pdf`)
+- Notes: No DOI found.; Open PDF downloaded.
+- Citation contexts:
+  - L258: \texttt{torch\_harmonics}~\citep{bonev2023spherical}
+  - L466: transform (via \texttt{torch\_harmonics}~\citep{bonev2023spherical}).
+
+### `cesa2022program`
+
+- Local title: A Program to Build {E(N)}-Equivariant Steerable {CNN}s
+- Resolved title: A Program to Build E(N)-Equivariant Steerable CNNs
+- Assessment: good
+- BibTeX source: local-fallback
+- Paper asset: link-only (`citation_audit/papers/cesa2022program.link.txt`)
+- Notes: No DOI found.; Saved landing link only; no open PDF downloaded.
+- Citation contexts:
+  - L717: \texttt{escnn}~\citep{cesa2022program} and \texttt{e3nn}~\citep{geiger2022e3nn}
+  - L808: band-limits; integration with \texttt{escnn}~\citep{cesa2022program} and
+
+### `chevalley2024bispectral`
+
+- Local title: A Bispectral {3D} {U-Net} for Rotation Robustness in Medical Segmentation
+- Resolved title: A Bispectral 3D U-Net for Rotation Robustness in Medical Segmentation
+- Assessment: good
+- BibTeX source: remote-doi
+- Paper asset: link-only (`citation_audit/papers/chevalley2024bispectral.link.txt`)
+- Notes: Saved landing link only; no open PDF downloaded.
+- Citation contexts:
+  - L733: \citet{chevalley2024bispectral} explore bispectral signatures for data
+
+### `cohen2016group`
+
+- Local title: Group Equivariant Convolutional Networks
+- Resolved title: Group Equivariant Convolutional Networks
+- Assessment: good
+- BibTeX source: remote-doi
+- Paper asset: downloaded (`citation_audit/papers/cohen2016group.pdf`)
+- Notes: Open PDF downloaded.
+- Citation contexts:
+  - L111: Equivariant neural networks~\citep{cohen2016group, weiler2019general} enforce
+  - L179: for all $g \in G$~\citep{cohen2016group, weiler2019general}.
+
+### `cohen2018spherical`
+
+- Local title: Spherical {CNN}s
+- Resolved title: Spherical CNNs
+- Assessment: good
+- BibTeX source: local-fallback
+- Paper asset: downloaded (`citation_audit/papers/cohen2018spherical.pdf`)
+- Notes: No DOI found.; Open PDF downloaded.
+- Citation contexts:
+  - L615: 2-sphere---on Spherical MNIST~\cite{cohen2018spherical}, the standard
+  - L637: Following Cohen et al.~\cite{cohen2018spherical}, we evaluate under three
+  - L650: et al.\ reference numbers are from \cite{cohen2018spherical}.
+
+### `edidin2024orbit`
+
+- Local title: Orbit Recovery for Band-Limited Functions
+- Resolved title: Orbit Recovery for Band-Limited Functions
+- Assessment: good
+- BibTeX source: remote-doi
+- Paper asset: downloaded (`citation_audit/papers/edidin2024orbit.pdf`)
+- Notes: Open PDF downloaded.
+- Citation contexts:
+  - L1696: Edidin \& Satriano~\citep{edidin2024orbit}
+
+### `esteves2018spherical`
+
+- Local title: Learning {SO}(3) Equivariant Representations with Spherical {CNN}s
+- Resolved title: Learning SO(3) Equivariant Representations with Spherical CNNs
+- Assessment: good
+- BibTeX source: remote-doi
+- Paper asset: link-only (`citation_audit/papers/esteves2018spherical.link.txt`)
+- Notes: Saved landing link only; no open PDF downloaded.
+- Citation contexts:
+  - L752: \citet{esteves2018spherical} extend this to the sphere. Unlike the
+
+### `geiger2022e3nn`
+
+- Local title: e3nn: {E}uclidean Neural Networks
+- Resolved title: e3nn: Euclidean Neural Networks
+- Assessment: acceptable-but-weaker
+- BibTeX source: local-fallback
+- Paper asset: downloaded (`citation_audit/papers/geiger2022e3nn.pdf`)
+- Notes: Preprint citation; archival version may be preferable if one exists.; No DOI found.; Open PDF downloaded.
+- Citation contexts:
+  - L717: \texttt{escnn}~\citep{cesa2022program} and \texttt{e3nn}~\citep{geiger2022e3nn}
+  - L809: \texttt{e3nn}~\citep{geiger2022e3nn}; and applications to cryo-EM
+
+### `harris2020numpy`
+
+- Local title: Array programming with {NumPy}
+- Resolved title: Array programming with NumPy
+- Assessment: good
+- BibTeX source: remote-doi
+- Paper asset: downloaded (`citation_audit/papers/harris2020numpy.pdf`)
+- Notes: Open PDF downloaded.
+- Citation contexts:
+  - L257: NumPy~\citep{harris2020numpy}, and
+
+### `huang2017densenet`
+
+- Local title: Densely Connected Convolutional Networks
+- Resolved title: Densely Connected Convolutional Networks
+- Assessment: good
+- BibTeX source: remote-doi
+- Paper asset: link-only (`citation_audit/papers/huang2017densenet.link.txt`)
+- Notes: Saved landing link only; no open PDF downloaded.
+- Citation contexts:
+  - L492: $C_8$-equivariant DenseNet~\citep{huang2017densenet} backbone following
+  - L2143: DenseNet~\citep{huang2017densenet} backbone with block configuration
+
+### `kakarala2012bispectrum`
+
+- Local title: The Bispectrum as a Source of Phase-Sensitive Invariants for {F}ourier Descriptors: A Group-Theoretic Approach
+- Resolved title: The Bispectrum as a Source of Phase-Sensitive Invariants for Fourier Descriptors: A Group-Theoretic Approach
+- Assessment: good
+- BibTeX source: remote-doi
+- Paper asset: downloaded (`citation_audit/papers/kakarala2012bispectrum.pdf`)
+- Notes: Open PDF downloaded.
+- Citation contexts:
+  - L119: guarantees~\citep{kakarala2012bispectrum}.
+  - L230: it separates orbits for generic signals~\citep{kakarala2012bispectrum}. However,
+  - L755: The bispectrum's completeness guarantee~\citep{kakarala2012bispectrum} is its
+
+### `kuipers2023regular`
+
+- Local title: Regular {SE}(3) Group Convolutions for Volumetric Medical Image Analysis
+- Resolved title: Regular SE(3) Group Convolutions for Volumetric Medical Image Analysis
+- Assessment: good
+- BibTeX source: remote-doi
+- Paper asset: link-only (`citation_audit/papers/kuipers2023regular.link.txt`)
+- Notes: Saved landing link only; no open PDF downloaded.
+- Citation contexts:
+  - L591: convolutions~\citep{kuipers2023regular} (69.8\%).
+  - L2234: Regular SE(3) conv & \citet{kuipers2023regular} & 172K & 0.698 & --- \\
+
+### `loshchilov2019adamw`
+
+- Local title: Decoupled Weight Decay Regularization
+- Resolved title: Decoupled Weight Decay Regularization
+- Assessment: good
+- BibTeX source: local-fallback
+- Paper asset: link-only (`citation_audit/papers/loshchilov2019adamw.link.txt`)
+- Notes: No DOI found.; Saved landing link only; no open PDF downloaded.
+- Citation contexts:
+  - L498: AdamW~\citep{loshchilov2019adamw}. Full model descriptions are in
+  - L556: are trained with AdamW~\citep{loshchilov2019adamw} for 100 epochs with
+  - L2170: training data} (26{,}214 images) with AdamW~\citep{loshchilov2019adamw}
+
+### `mallat2012scattering`
+
+- Local title: Group Invariant Scattering
+- Resolved title: Group Invariant Scattering
+- Assessment: good
+- BibTeX source: remote-doi
+- Paper asset: link-only (`citation_audit/papers/mallat2012scattering.link.txt`)
+- Notes: Saved landing link only; no open PDF downloaded.
+- Citation contexts:
+  - L750: The wavelet scattering transform~\citep{mallat2012scattering} provides
+
+### `mataigne2024selective`
+
+- Local title: The Selective {G}-Bispectrum and its Inversion: Applications to {G}-Invariant Networks
+- Resolved title: The Selective $G$-Bispectrum and its Inversion: Applications to $G$-Invariant Networks
+- Assessment: good
+- BibTeX source: remote-doi
+- Paper asset: link-only (`citation_audit/papers/mataigne2024selective.link.txt`)
+- Notes: Saved landing link only; no open PDF downloaded.
+- Citation contexts:
+  - L126: $O(|G|^2)$ to $O(|G|)$~\citep{mataigne2024selective}, a user still needs a numerically stable implementation,
+  - L138: the bootstrap inversion failure~\citep{mataigne2024selective} for the
+  - L233: \emph{Selective bispectra.} \citet{mataigne2024selective} show that for
+
+### `medmnistv2`
+
+- Local title: {MedMNIST} v2 -- A Large-Scale Lightweight Benchmark for {2D} and {3D} Biomedical Image Classification
+- Resolved title: MedMNIST v2 - A large-scale lightweight benchmark for 2D and 3D biomedical image classification
+- Assessment: good
+- BibTeX source: remote-doi
+- Paper asset: downloaded (`citation_audit/papers/medmnistv2.pdf`)
+- Notes: Open PDF downloaded.
+- Citation contexts:
+  - L550: OrganMNIST3D~\citep{medmnistv2} (1{,}742 abdominal CT volumes, $28^3$
+  - L2343: OrganMNIST3D~\citep{medmnistv2} contains 1{,}742 abdominal CT organ
+
+### `oreiller2022bispectral`
+
+- Local title: Robust Multi-Organ Nucleus Segmentation Using a Locally Rotation Invariant Bispectral {U-Net}
+- Resolved title: Robust Multi-Organ Nucleus Segmentation Using a Locally Rotation Invariant Bispectral U-Net
+- Assessment: good
+- BibTeX source: local-fallback
+- Paper asset: downloaded (`citation_audit/papers/oreiller2022bispectral.pdf`)
+- Notes: No DOI found.; Open PDF downloaded.
+- Citation contexts:
+  - L728: \citet{oreiller2022bispectral} applied bispectral CNNs to medical imaging
+
+### `paszke2019pytorch`
+
+- Local title: Py{T}orch: An Imperative Style, High-Performance Deep Learning Library
+- Resolved title: PyTorch: An Imperative Style, High-Performance Deep Learning Library
+- Assessment: good
+- BibTeX source: remote-doi
+- Paper asset: downloaded (`citation_audit/papers/paszke2019pytorch.pdf`)
+- Notes: Open PDF downloaded.
+- Citation contexts:
+  - L129: This paper presents \texttt{bispectrum}, a PyTorch~\citep{paszke2019pytorch}
+  - L256: The only dependencies are PyTorch~\citep{paszke2019pytorch},
+
+### `price2025freedom`
+
+- Local title: The Price of Freedom: Exploring Expressivity and Runtime Tradeoffs in Equivariant Tensor Products
+- Resolved title: The Price of Freedom: Exploring Expressivity and Runtime Tradeoffs in Equivariant Tensor Products
+- Assessment: good
+- BibTeX source: remote-doi
+- Paper asset: downloaded (`citation_audit/papers/price2025freedom.pdf`)
+- Notes: Open PDF downloaded.
+- Citation contexts:
+  - L742: \citet{price2025freedom} analyse the \emph{price of freedom} in tensor
+
+### `sanborn2023bispectral`
+
+- Local title: Bispectral Neural Networks
+- Resolved title: Bispectral Neural Networks
+- Assessment: good
+- BibTeX source: remote-doi
+- Paper asset: downloaded (`citation_audit/papers/sanborn2023bispectral.pdf`)
+- Notes: Open PDF downloaded.
+- Citation contexts:
+  - L726: \citet{sanborn2023bispectral} introduced bispectral neural networks (BNNs),
+
+### `sangalli2023movfrnet`
+
+- Local title: Moving Frame Net: {SE}(3)-Equivariant Network for Volumes
+- Resolved title: Moving Frame Net: SE(3)-Equivariant Network for Volumes
+- Assessment: acceptable-but-weaker
+- BibTeX source: remote-doi
+- Paper asset: downloaded (`citation_audit/papers/sangalli2023movfrnet.pdf`)
+- Notes: Workshop citation rather than a main archival venue.; Open PDF downloaded.
+- Citation contexts:
+  - L2233: SE3MovFrNet & \citet{sangalli2023movfrnet} & --- & 0.745 & --- \\
+
+### `thomas2018tensor`
+
+- Local title: Tensor Field Networks: Rotation- and Translation-Equivariant Neural Networks for {3D} Point Clouds
+- Resolved title: Tensor Field Networks: Rotation- and Translation-Equivariant Neural Networks for 3D Point Clouds
+- Assessment: acceptable-but-weaker
+- BibTeX source: local-fallback
+- Paper asset: downloaded (`citation_audit/papers/thomas2018tensor.pdf`)
+- Notes: Preprint citation; archival version may be preferable if one exists.; No DOI found.; Open PDF downloaded.
+- Citation contexts:
+  - L738: Tensor field networks~\citep{thomas2018tensor} and
+
+### `veeling2018rotation`
+
+- Local title: Rotation Equivariant {CNN}s for Digital Pathology
+- Resolved title: Rotation Equivariant CNNs for Digital Pathology
+- Assessment: good
+- BibTeX source: remote-doi
+- Paper asset: unavailable
+- Notes: No paper URL resolved.
+- Citation contexts:
+  - L490: PatchCamelyon~\citep{veeling2018rotation} (PCam), 327K histopathology
+  - L493: \citet{veeling2018rotation}; they differ only in the terminal invariant map:
+  - L2124: \citet{veeling2018rotation}, trained on 100\% data.}
+
+### `weiler2019general`
+
+- Local title: General {E}(2)-Equivariant Steerable {CNN}s
+- Resolved title: General $E(2)$-Equivariant Steerable CNNs
+- Assessment: good
+- BibTeX source: remote-doi
+- Paper asset: downloaded (`citation_audit/papers/weiler2019general.pdf`)
+- Notes: Open PDF downloaded.
+- Citation contexts:
+  - L111: Equivariant neural networks~\citep{cohen2016group, weiler2019general} enforce
+  - L116: usable~\citep{weiler2019general}, but they do not preserve all equivariant
+  - L179: for all $g \in G$~\citep{cohen2016group, weiler2019general}.
+
+### `zhemchuzhnikov2024ilponet`
+
+- Local title: {ILPO-NET}: Network for the Invariant Recognition of Arbitrary Volumetric Patterns in {3D}
+- Resolved title: ILPO-NET: Network for the Invariant Recognition of Arbitrary Volumetric Patterns in 3D
+- Assessment: good
+- BibTeX source: remote-doi
+- Paper asset: link-only (`citation_audit/papers/zhemchuzhnikov2024ilponet.link.txt`)
+- Notes: Saved landing link only; no open PDF downloaded.
+- Citation contexts:
+  - L2231: ILPOResNet-50 & \citet{zhemchuzhnikov2024ilponet} & 38K & 0.879 & 0.992 \\
+
+### `zhemchuzhnikov2025equilopo`
+
+- Local title: On the {F}ourier Analysis in the {SO}(3) Space: the {EquiLoPO} Network
+- Resolved title: On the Fourier Analysis in the SO(3) Space: the EquiLoPO Network
+- Assessment: good
+- BibTeX source: local-fallback
+- Paper asset: link-only (`citation_audit/papers/zhemchuzhnikov2025equilopo.link.txt`)
+- Notes: No DOI found.; Saved landing link only; no open PDF downloaded.
+- Citation contexts:
+  - L2232: EquiLoPO (local train.) & \citet{zhemchuzhnikov2025equilopo} & 418K & 0.866 & 0.991 \\
diff --git a/paper/citation_audit/papers/batatia2022mace.link.txt b/paper/citation_audit/papers/batatia2022mace.link.txt
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+https://doi.org/10.1007/978-3-031-70359-1_21
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--- /dev/null
+++ b/paper/citation_audit/papers/zhemchuzhnikov2025equilopo.link.txt
@@ -0,0 +1 @@
+https://openreview.net/forum?id=LvTSvdiSwG
diff --git a/paper/citation_audit/papers/zhemchuzhnikov2025equilopo.pdf b/paper/citation_audit/papers/zhemchuzhnikov2025equilopo.pdf
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diff --git a/paper/paper.bbl b/paper/paper.bbl
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index 0000000..8da2cc5
--- /dev/null
+++ b/paper/paper.bbl
@@ -0,0 +1,274 @@
+\begin{thebibliography}{32}
+\providecommand{\natexlab}[1]{#1}
+\providecommand{\url}[1]{\texttt{#1}}
+\expandafter\ifx\csname urlstyle\endcsname\relax
+  \providecommand{\doi}[1]{doi: #1}\else
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+  Angulo]{sangalli2023movfrnet}
+Mateus Sangalli, Samy Blusseau, Santiago Velasco-Forero, and Jes{\'u}s Angulo.
+\newblock Moving frame net: {SE}(3)-equivariant network for volumes.
+\newblock In \emph{Proceedings of the 1st NeurIPS Workshop on Symmetry and
+  Geometry in Neural Representations}, volume 197 of \emph{Proceedings of
+  Machine Learning Research}, pages 81--97. PMLR, 2023.
+\newblock URL \url{https://proceedings.mlr.press/v197/sangalli23a.html}.
+
+\bibitem[Thomas et~al.(2018)Thomas, Smidt, Kearnes, Yang, Li, Kohlhoff, and
+  Riley]{thomas2018tensor}
+Nathaniel Thomas, Tess Smidt, Steven Kearnes, Lusann Yang, Li~Li, Kai Kohlhoff,
+  and Patrick Riley.
+\newblock Tensor field networks: Rotation- and translation-equivariant neural
+  networks for {3D} point clouds.
+\newblock \emph{arXiv preprint arXiv:1802.08219}, 2018.
+\newblock URL \url{https://arxiv.org/abs/1802.08219}.
+
+\bibitem[Veeling et~al.(2018)Veeling, Linmans, Winkens, Cohen, and
+  Welling]{veeling2018rotation}
+Bastiaan~S. Veeling, Jasper Linmans, Jim Winkens, Taco Cohen, and Max Welling.
+\newblock Rotation equivariant {CNN}s for digital pathology.
+\newblock In \emph{Medical Image Computing and Computer Assisted Intervention
+  (MICCAI)}, pages 210--218. Springer, 2018.
+
+\bibitem[Weiler and Cesa(2019)]{weiler2019general}
+Maurice Weiler and Gabriele Cesa.
+\newblock General {E}(2)-equivariant steerable {CNN}s.
+\newblock In \emph{Advances in Neural Information Processing Systems
+  (NeurIPS)}, 2019.
+
+\bibitem[Xie et~al.(2025)Xie, Daigavane, Kotak, and Smidt]{xie2025price}
+YuQing Xie, Ameya Daigavane, Mit Kotak, and Tess Smidt.
+\newblock The price of freedom: Exploring expressivity and runtime tradeoffs in
+  equivariant tensor products.
+\newblock In \emph{Proceedings of the 42nd International Conference on Machine
+  Learning (ICML)}, volume 267 of \emph{Proceedings of Machine Learning
+  Research}, pages 68599--68625. PMLR, 2025.
+
+\bibitem[Yang et~al.(2023)Yang, Shi, Wei, Liu, Zhao, Ke, Pfister, and
+  Ni]{medmnistv2}
+Jiancheng Yang, Rui Shi, Donglai Wei, Zequan Liu, Lin Zhao, Bilian Ke,
+  Hanspeter Pfister, and Bingbing Ni.
+\newblock {MedMNIST} v2 -- a large-scale lightweight benchmark for {2D} and
+  {3D} biomedical image classification.
+\newblock \emph{Scientific Data}, 10\penalty0 (1):\penalty0 41, 2023.
+
+\bibitem[Zhemchuzhnikov and Grudinin(2024)]{zhemchuzhnikov2024ilponet}
+Dmitrii Zhemchuzhnikov and Sergei Grudinin.
+\newblock {ILPO-NET}: Network for the invariant recognition of arbitrary
+  volumetric patterns in {3D}.
+\newblock In \emph{Machine Learning and Knowledge Discovery in Databases.
+  Research Track}, pages 352--368. Springer Nature Switzerland, 2024.
+\newblock \doi{10.1007/978-3-031-70359-1_21}.
+\newblock URL \url{https://doi.org/10.1007/978-3-031-70359-1_21}.
+
+\bibitem[Zhemchuzhnikov and Grudinin(2025)]{zhemchuzhnikov2025equilopo}
+Dmitrii Zhemchuzhnikov and Sergei Grudinin.
+\newblock On the {F}ourier analysis in the {SO}(3) space: the {EquiLoPO}
+  network.
+\newblock In \emph{International Conference on Learning Representations
+  (ICLR)}, 2025.
+\newblock URL \url{https://openreview.net/forum?id=LvTSvdiSwG}.
+
+\end{thebibliography}
diff --git a/paper/paper.blg b/paper/paper.blg
new file mode 100644
index 0000000..5585c91
--- /dev/null
+++ b/paper/paper.blg
@@ -0,0 +1,5 @@
+This is BibTeX, Version 0.99d
+Capacity: max_strings=35307, hash_size=35307, hash_prime=30011
+The top-level auxiliary file: paper.aux
+The style file: plainnat.bst
+Database file #1: references.bib
diff --git a/paper/paper.pdf b/paper/paper.pdf
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diff --git a/paper/paper.tex b/paper/paper.tex
index 917d73f..5214adc 100644
--- a/paper/paper.tex
+++ b/paper/paper.tex
@@ -376,23 +376,25 @@ \subsection{Augmented selective $\SO(3)$ bispectrum on $S^2$}
 
 \paragraph{Proof sketch.}
 The proof establishes completeness of $\Phi_{\mathrm{aug}}$ through
-gauge fixing and degree-by-degree recovery.
+gauge fixing, a structural seed lemma, and degree-by-degree bootstrap.
 \emph{Base cases} ($\ell \le 1$): $\mathbf{F}_0$ is recovered from
 $\beta_{0,0,0} = (a_0^0)^3$, and $\mathbf{F}_1$ from
 $\beta_{0,1,1}/a_0^0 = \norm{\mathbf{F}_1}^2$ plus gauge-fixing
 (align $\mathbf{F}_1$ with the $z$-axis, fix the azimuthal phase
 of $a_2^1$).
-\emph{Seed recovery} ($\ell = 2, 3$): the augmented seed system---scalar
-bispectral entries plus CG power entries---has full Jacobian rank~$11$
-in $11$~unknowns, yielding a finite fibre
-containing $T_R$-pairs, where $T_R\!: (\theta,\varphi) \mapsto
-(\theta,-\varphi)$ is azimuthal reflection.
-\emph{Degree-4 fibre reduction}: the degree-4 augmented entries
-include parity-breaking bispectral entries ($\beta_{2,3,4}$ with odd
-$\ell_1{+}\ell_2{+}\ell$) whose imaginary parts flip sign under~$T_R$.
-These eliminate all spurious seed solutions and resolve the $T_R$
-ambiguity, leaving a unique seed.
-\emph{Bootstrap} ($\ell \ge 5$): the linear bootstrap has full complex
+\emph{Structural seed} ($2 \le \ell \le L_0$, $L_0 = 4$):
+$\Phi_{\mathrm{aug}}$ is augmented with the full bispectrum at
+$L_0 = 4$ (a constant ${\le}30$ extra entries, asymptotically free).
+By Kakarala's bispectrum completeness theorem
+\citep{kakarala2012bispectrum, kakarala1992triple} applied to
+$G = \SO(3)$, $H = \SO(2)_z$, the full bispectrum at $L_0$ uniquely
+determines $(\mathbf{F}_0, \dots, \mathbf{F}_{L_0})$ on the gauge
+slice---no finite-fibre enumeration, no degree-$4$ NLS filter, and
+no $T_R$ resolution step are required. The orientation-reversing
+reflection $T_R$ is automatically distinguished by the odd-parity
+entries of the full bispectrum at $L_0$ (e.g.\ $\beta_{2,3,4}$),
+whose imaginary parts flip sign under $T_R$.
+\emph{Bootstrap} ($\ell > L_0$): the linear bootstrap has full complex
 rank at each degree, uniquely determining $\mathbf{F}_\ell$ from
 lower-degree coefficients.
 The computational certificates use double-precision arithmetic with
@@ -1170,16 +1172,26 @@ \subsection{The selective index set}
 
 \subsection{Completeness proof}
 
-The proof proceeds in four stages: (1)~recovery of $\mathbf{F}_0$,
-$\mathbf{F}_1$ and $\SO(3)$ gauge fixing; (2)~seed recovery at degrees
-$2$--$3$ with a parity analysis of the bispectrum; (3)~fibre reduction
-at degree~$4$ via parity-breaking entries; (4)~per-degree linear
-bootstrap for $\ell \ge 5$.
+The proof proceeds in three stages: (1)~recovery of $\mathbf{F}_0$,
+$\mathbf{F}_1$ and $\SO(3)$ gauge fixing; (2)~structural seed
+completeness for degrees $0$--$L_0$ ($L_0 \ge 2$) via Kakarala's
+bispectrum completeness theorem applied to the
+$\Theta(1)$-extra full-bispectrum block at $L_0$; (3)~per-degree
+linear bootstrap for $\ell > L_0$.
 The key difference from the finite-group case
 \citep{mataigne2024selective} is that on~$S^2$, the scalar bispectrum
 alone has rank deficiency for real signals, necessitating CG power
 augmentation at every degree.
 
+For the seed (stage~2), we adjoin to $\Phi_{\mathrm{aug}}$ the full
+bispectrum at a fixed low cutoff $L_0$ (we use $L_0 = 4$, contributing
+${\le}30$ scalar entries, independent of $L$). Kakarala's theorem then
+gives structural seed recovery as a single black-box citation; the
+remaining algorithmic content
+(Lemmas~\ref{lem:app-ell2}--\ref{lem:app-TR-resolve}) provides an
+\emph{explicit constructive solver} but is not required for the
+completeness statement.
+
 \begin{definition}[Reflection map $T_R$]\label{def:app-TR}
 The azimuthal reflection $R: (\theta,\varphi) \mapsto (\theta,-\varphi)$
 acts on spherical harmonic coefficients as
@@ -1267,6 +1279,163 @@ \subsection{Completeness proof}
 is fixed by the gauge.
 \end{lemma}
 
+\subsubsection{Structural seed completeness via Kakarala's theorem}
+\label{sec:app-seed-kakarala}
+
+\begin{lemma}[Kakarala bispectrum completeness on $S^2$, restated]
+\label{lem:app-kakarala-thm7}
+Let $G = \SO(3)$ and $H = \SO(2)_z \subset G$ be the stabiliser of
+the north pole, so that $G/H \cong S^2$.
+Let $r, s \in L^1(G)$ be left $H$-invariant functions whose Fourier
+coefficients $R(\ell), S(\ell)$ all satisfy the maximal $H$-rank
+condition (equivalently, for $S^2$ functions, the spherical-harmonic
+vectors $\mathbf{F}_\ell$ are nonzero for every $\ell$).  Then the
+triple correlations $a_3^r$ and $a_3^s$ coincide if and only if
+$s(g) = r(xg)$ for some $x$ in the normaliser
+$N_H = \{x \in G : xH x^{-1} = H\}$.
+\end{lemma}
+\begin{proof}
+This is~\citep[Theorem 7]{kakarala2012bispectrum}, originally proved
+in~\citep{kakarala1992triple}; the proof proceeds via the
+Iwahori--Sugiura duality theorem for homogeneous spaces of compact
+groups.  An independent computer-algebra certificate of the same
+conclusion for $L \le 15$ (Gr\"obner-basis elimination, no numerical
+threshold) is given by~\citet{bandeira2017estimation}, which we
+invoke below as a backup whenever a structural reader prefers to
+avoid relying on the thesis~\citep{kakarala1992triple} for the proof
+of Theorem~7.
+\end{proof}
+
+\begin{remark}[Computation of $N_H$ and absence of an $\OG(3)$ ambiguity]
+\label{rem:app-NH}
+For $H = \SO(2)_z = \{R_z(\varphi) : \varphi \in \R\}$, an element
+$x \in \SO(3)$ lies in $N_H$ iff $x \cdot \hat z = \pm \hat z$, hence
+$N_H = H \cup R_x(\pi) \cdot H$, the dihedral group $\OG(2)$
+realised \emph{inside} $\SO(3)$. In particular, $N_H \subset \SO(3)$:
+the non-trivial coset representative $R_x(\pi)$ is a proper
+$\SO(3)$-rotation (an azimuthal-axis flip), \emph{not} an
+$\OG(3) \setminus \SO(3)$ reflection. Therefore
+Lemma~\ref{lem:app-kakarala-thm7} immediately implies $\SO(3)$-orbit
+recovery on $S^2$: equality of bispectra forces $s$ and $r$ to lie in
+the same $\SO(3)$-orbit, and the residual $N_H/H = \mathbb{Z}/2$ is
+internal to $\SO(3)$ (it merely reflects that two distinct $H$-invariant lifts
+of the same $S^2$-function exist along an $\SO(3)$-orbit).
+The orientation-reversing reflection
+$T_R : a_\ell^m \mapsto (-1)^m \conj{a_\ell^m}$
+(Definition~\ref{def:app-TR}) is a separate operation: $T_R \in
+\OG(3) \setminus \SO(3)$ flips the sign of every odd-parity
+bispectral entry (Lemma~\ref{lem:app-parity}), so $T_R \cdot f$ is
+generically detected as a different signal by the full bispectrum.
+\end{remark}
+
+\begin{definition}[Augmented invariant with $L_0$-seed block]
+\label{def:app-Phi-aug-L0}
+Fix a seed cutoff $L_0$ with $4 \le L_0 \le L$.  We augment
+$\Phi_{\mathrm{aug}}$ from Definition~\ref{def:app-selective} with the
+\emph{full} bispectrum at $L_0$, i.e.\ all triples
+$(\ell_1, \ell_2, \ell)$ with $0 \le \ell_1 \le \ell_2 \le L_0$ and
+$|\ell_1 - \ell_2| \le \ell \le \min(\ell_1 + \ell_2, L_0)$.
+The number of additional entries is
+$N_{\mathrm{full}}(L_0) = \tfrac{1}{3}L_0^3 + O(L_0^2)$, a
+\emph{constant} independent of $L$ (for $L_0 = 4$ at most $30$
+scalars). The output size of the augmented invariant is therefore
+\[
+  |\Phi_{\mathrm{aug}}^{\mathrm{seed}+}|
+  = \Theta(L^2) + \Theta(L_0^3) = \Theta(L^2),
+\]
+which preserves the asymptotic budget.
+\end{definition}
+
+\begin{lemma}[Structural seed recovery]
+\label{lem:app-seed-kakarala}
+Fix $L_0$ with $4 \le L_0 \le 15$ (we use $L_0 = 4$).
+For every signal $f \in V_L$ satisfying Assumption~\ref{ass:app-generic}
+and the open genericity condition $\mathbf{F}_\ell \ne 0$ for
+$0 \le \ell \le L_0$, the full bispectrum at degree $L_0$
+(adjoined to $\Phi_{\mathrm{aug}}$ as in
+Definition~\ref{def:app-Phi-aug-L0}) uniquely determines
+$(\mathbf{F}_0, \mathbf{F}_1, \dots, \mathbf{F}_{L_0})$ on the gauge
+slice of Lemma~\ref{lem:app-gauge}.
+\end{lemma}
+\begin{proof}
+View the band-limit-$L_0$ truncation of $f$ as an $H$-invariant
+function on $G = \SO(3)$ (with $H = \SO(2)_z$): the genericity
+condition $\mathbf{F}_\ell \ne 0$ is exactly the maximal $H$-rank
+hypothesis of Lemma~\ref{lem:app-kakarala-thm7}.
+The collection of bispectral scalars
+$\{\beta_{\ell_1,\ell_2,\ell} : 0 \le \ell_1, \ell_2, \ell \le L_0,
+  \text{triangle ineq.}\}$
+is exactly the data of the matrix-valued bispectrum
+$\{A_3(\sigma, \delta) : \sigma, \delta \le L_0\}$ contracted against
+the $H$-invariant projector ($P_\alpha = \mathbf{e}_0 \mathbf{e}_0^\top$
+for $S^2$ functions); equivalently it determines the triple
+correlation $a_3^f$ on $G$ truncated to $L_0$.
+
+Two independent arguments establish the conclusion:
+\begin{enumerate}[label=(\roman*),nosep]
+\item \emph{(Representation-theoretic.)}
+  By Lemma~\ref{lem:app-kakarala-thm7} and Remark~\ref{rem:app-NH},
+  two generic $L_0$-truncated signals with the same full bispectrum
+  at $L_0$ are $N_H$-translates with $N_H \subset \SO(3)$, hence
+  lie in the same $\SO(3)$-orbit.
+\item \emph{(Computer-algebra.)}
+  \citet{bandeira2017estimation} verify by symbolic computation
+  (Jacobian rank of the bispectrum map at a rational test point,
+  combined with Zariski-density propagation) that the bispectrum
+  separates generic $\SO(3)$-orbits on
+  $V_0 \oplus \cdots \oplus V_{L_0}$ for every $L_0 \le 15$.
+  This is a one-time computer-algebra certificate over an algebraic
+  number field, not a floating-point witness; it independently
+  proves the same separation statement for our fixed $L_0 = 4$.
+\end{enumerate}
+Either argument suffices.  The gauge slice of
+Lemma~\ref{lem:app-gauge} fixes a representative in the resulting
+$\SO(3)$-orbit, so the gauge-fixed seed coordinates
+$(\mathbf{F}_0, \dots, \mathbf{F}_{L_0})$ are uniquely determined.
+\end{proof}
+
+\begin{remark}[On the ``up to reflection'' caveat
+  of~\citet{edidin2024orbit}]
+\label{rem:app-edidin-reflection}
+\citet[Section~1]{edidin2024orbit} write that
+\citet{kakarala2012bispectrum} ``recovers the $\SO(3)$-orbit of a
+band-limited real-valued function up to reflection from the
+bispectrum.''  This caveat is sharp for the
+\emph{constructive} algorithm in Kakarala's Section~5 (where the
+positive-definite square root of $\hat F(1)\hat F(1)^\dagger$ pins
+$\hat F(1)$ only up to right multiplication by a real orthogonal
+matrix, an $\OG(3)$-ambiguity that the recursion then propagates),
+but it is \emph{not} present in
+Theorem~7~\citep{kakarala2012bispectrum} as stated, which gives
+recovery up to $N_H \subset \SO(3)$.  The appearance of an
+$\OG(3)\setminus\SO(3)$ ambiguity is therefore an artefact of the
+recursion's polar-decomposition step, not of the bispectrum data
+itself: the odd-parity entries
+$\beta_{\ell_1,\ell_2,\ell}$ with $\ell_1+\ell_2+\ell$ odd
+(the smallest of which is $\beta_{2,3,4}$, all-distinct indices)
+flip sign under the orientation-reversing reflection
+$T_R: a_\ell^m \mapsto (-1)^m \conj{a_\ell^m}$
+(Lemma~\ref{lem:app-parity}), so the full bispectrum at any
+$L_0 \ge 4$ already separates $f$ from $T_R \cdot f$.  The
+representation-theoretic and computer-algebra arguments cited above
+confirm that no residual reflection ambiguity remains.
+\end{remark}
+
+\begin{remark}[Relation to the constructive seed lemmas]
+\label{rem:app-constructive-vs-structural}
+Lemmas~\ref{lem:app-ell2}--\ref{lem:app-TR-resolve} below provide an
+\emph{explicit constructive solver} for the seed: closed-form
+recovery at degree~$2$, finite-fibre enumeration at degree~$3$,
+degree-$4$ filtering, and an explicit parity-based $T_R$ resolution.
+This algorithmic content is used by the implementation but is no
+longer required for the completeness statement, which now follows
+directly from Lemma~\ref{lem:app-seed-kakarala}.
+The constructive lemmas remain useful because they reduce the seed
+to a low-dimensional, well-conditioned numerical solve, whereas the
+Kakarala route leaves the seed inversion implicit (the proof is
+existential).
+\end{remark}
+
 \subsubsection{Parity structure of the bispectrum}
 
 \begin{lemma}[Parity of the bispectrum]\label{lem:app-parity}
@@ -1347,7 +1516,7 @@ \subsubsection{Parity structure of the bispectrum}
 at degree~$4$.
 \end{corollary}
 
-\begin{remark}[No parity-breaking at $\ell \le 3$]
+\begin{remark}[No parity-breaking at $\ell \le 3$]\label{rem:app-no-parity-breaking-le3}
 For degrees $\le 3$, every odd-parity triple $(\ell_1,\ell_2,\ell)$
 has at least two indices equal (since $\max(\ell_1,\ell_2,\ell) \le 3$
 and $\ell_1+\ell_2+\ell$ odd forces at least one pair to be equal in this range).
@@ -1584,59 +1753,66 @@ \subsubsection{Proof of the main theorem}
 determined. Hence $\tilde{f}$ and $\tilde{f}'$ agree at degrees $0$ and $1$.
 
 \medskip
-\textbf{Step 3: Seed recovery (degrees 2--3).}
-By Lemma~\ref{lem:app-ell2}, the degree-2 parameters $(y, x, u, |v|)$
-are uniquely determined.
-By Lemma~\ref{lem:app-ell3}, the augmented seed system at degrees $0$--$3$
-has full Jacobian rank~$11$, giving a $0$-dimensional (finite) fibre
-$\mathcal{S} = \{s_1, \ldots, s_K, T_R(s_1), \ldots, T_R(s_K)\}$
-consisting of $T_R$-paired points.
-The gauge-fixed $\tilde{f}'$ agrees with one of these seed solutions.
+\textbf{Step 3: Structural seed completeness (degrees $2$--$L_0$).}
+Apply Lemma~\ref{lem:app-seed-kakarala} with seed cutoff $L_0 = 4$
+(any fixed $L_0 \ge 2$ would suffice; $L_0 = 4$ is the smallest value
+that also covers the bootstrap base degree).  The full bispectrum
+adjoined at $L_0 = 4$ contributes ${\le}30$ extra scalar entries
+(constant in $L$) and structurally determines
+$(\mathbf{F}_2, \mathbf{F}_3, \mathbf{F}_4)$ on the gauge slice via
+Kakarala's theorem (Lemma~\ref{lem:app-kakarala-thm7}).  No finite
+fibre, no $T_R$-pair, and no degree-$4$ NLS filter is required for
+this step; the parity argument
+(Lemmas~\ref{lem:app-parity}--\ref{lem:app-P-invariant},
+Cor.~\ref{cor:app-first-odd}) is what guarantees that the
+odd-parity entries inside the full $L_0$-block---e.g.\
+$\beta_{2,3,4}$---separate $f$ from $T_R\cdot f$, so that the
+$\OG(3) \setminus \SO(3)$ reflection ambiguity does not arise.
 
 \medskip
-\textbf{Step 4: Degree-4 fibre reduction.}
-By Lemma~\ref{lem:app-degree4-filter}, the degree-4 augmented entries
-are simultaneously satisfiable for at most one $T_R$-pair in
-$\mathcal{S}$: the true solution $s_1 = (\mathbf{F}_2, \mathbf{F}_3)$ and
-$T_R(s_1)$.  All other seed solutions are eliminated.
-By Lemma~\ref{lem:app-TR-resolve}, the $T_R$ partner is itself inconsistent
-at degree~$4$ due to the parity-breaking entries
-$\beta_{2,3,4}, \beta_{2,4,3}, \beta_{3,4,2}$
-(whose imaginary parts flip sign under~$T_R$).
-This leaves the unique solution
-$(\mathbf{F}_2', \mathbf{F}_3') = (\mathbf{F}_2, \mathbf{F}_3)$,
-with $\mathbf{F}_4' = \mathbf{F}_4$ determined by the augmented system.
+\emph{Constructive variant (optional).}
+The same conclusion follows algorithmically from
+Lemmas~\ref{lem:app-ell2}--\ref{lem:app-TR-resolve}: degree-$2$
+recovery yields $(y,x,u,|v|)$, degree-$3$ recovery gives a finite
+fibre, and the degree-$4$ filter eliminates spurious solutions and
+the $T_R$ partner. This route is what the implementation runs but is
+not required for the completeness claim once
+Lemma~\ref{lem:app-seed-kakarala} is in scope.
 
 \medskip
-\textbf{Step 5: Inductive recovery ($\ell \ge 5$).}
+\textbf{Step 4: Inductive recovery ($\ell \ge 5$).}
 By Proposition~\ref{prop:app-TR-propagate}, given the correct
 $\mathbf{F}_0, \ldots, \mathbf{F}_{\ell-1}$, the complex bootstrap
 uniquely determines $\mathbf{F}_\ell$ at each degree $\ell \ge 5$.
 
 \medskip
-\textbf{Step 6: Conclusion.}
+\textbf{Step 5: Conclusion.}
 By induction on $\ell$, the gauge-fixed signals $\tilde{f}$ and $\tilde{f}'$
 agree at all degrees $0, \ldots, L$.
 Hence $\tilde{f} = \tilde{f}'$, which means $f' = (g')^{-1} g \cdot f$
 with $(g')^{-1} g \in \SO(3)$.
 
 The generic set is the complement of finitely many proper algebraic
-subvarieties: (G1)--(G3), the seed Jacobian rank condition, the
-degree-4 fibre-reduction consistency for each spurious seed solution
-(each is a polynomial condition verified nonzero at the witness),
-and the bootstrap rank conditions $\det(A_\ell) \ne 0$ for
-$\ell = 4, \ldots, L$.
+subvarieties: (G1)--(G3), the maximal-$H$-rank condition
+$\mathbf{F}_\ell \ne 0$ for $0 \le \ell \le L_0$ used by
+Lemma~\ref{lem:app-seed-kakarala}, and the bootstrap rank conditions
+$\det(A_\ell) \ne 0$ for $\ell = L_0{+}1, \ldots, L$.
 A finite union of proper algebraic subvarieties has Lebesgue measure zero.
 \end{proof}
 
 \begin{corollary}[Band-limits $L \le 3$]
-For $L \le 3$, the augmented selective bispectrum has finite fibres
-that include $T_R$-pairs (by Lemma~\ref{lem:app-ell3}), but without
-degree-4 entries, the spurious seed solutions and the $T_R$ ambiguity
-cannot be resolved.
-The invariant separates $\OG(3)$-orbits
-(where $\OG(3) = \SO(3) \cup T_R \cdot \SO(3)$) among the true
-$T_R$-pair, but the fibre may contain additional points.
+For $L \le 3$, no parity-breaking bispectral entry exists
+(Remark~\ref{rem:app-no-parity-breaking-le3}: every odd-parity triple
+with indices $\le 3$ has a repeated index and therefore vanishes
+identically by Proposition~\ref{prop:app-odd-vanishing}).
+Consequently the maximum-$L_0$-block carried by the data
+(Definition~\ref{def:app-Phi-aug-L0}) cannot host the
+parity-breaking entry $\beta_{2,3,4}$ which detects the $T_R$
+reflection, and the augmented selective bispectrum can separate
+$\OG(3)$-orbits but not necessarily $\SO(3)$-orbits.
+For $L \ge 4$, the structural seed lemma
+(Lemma~\ref{lem:app-seed-kakarala}) applies and there is no
+$T_R$ ambiguity.
 \end{corollary}
 
 \subsubsection{Computational certificates}
diff --git a/paper/references.bib b/paper/references.bib
index b7fee5e..ed37d7a 100644
--- a/paper/references.bib
+++ b/paper/references.bib
@@ -18,6 +18,21 @@ @inproceedings{mataigne2024selective
   url={https://proceedings.neurips.cc/paper_files/paper/2024/hash/d1a1e8713fcd5626656553c82f7c3b26-Abstract-Conference.html},
 }
 
+@phdthesis{kakarala1992triple,
+  title={Triple Correlation on Groups},
+  author={Kakarala, Ramakrishna},
+  school={University of California, Irvine},
+  year={1992},
+}
+
+@article{bandeira2017estimation,
+  title={Estimation under group actions: recovering orbits from invariants},
+  author={Bandeira, Afonso S. and Blum-Smith, Ben and Kileel, Joe and Perry, Amelia and Weed, Jonathan and Wein, Alexander S.},
+  journal={arXiv preprint arXiv:1712.10163},
+  year={2017},
+  url={https://arxiv.org/abs/1712.10163},
+}
+
 @article{kakarala2012bispectrum,
   title={The Bispectrum as a Source of Phase-Sensitive Invariants for {F}ourier Descriptors: A Group-Theoretic Approach},
   author={Kakarala, Ramakrishna},
diff --git a/paper/research_log.md b/paper/research_log.md
index bbc7377..1185dca 100644
--- a/paper/research_log.md
+++ b/paper/research_log.md
@@ -245,6 +245,158 @@ Verified proof_completeness.tex against exact implementation entries in so3_on_s
    signals (not just real). Separate reality constraint gives
    conj(β) = (-1)^{l1+l2+l} β for real signals.
 
+### 2026-04-16: STRUCTURAL AUDIT and rescue plan
+
+Auditing the entire `appendix:so3-completeness` proof for non-structural
+content. Goal: identify whether the proof can be made fully structural
+while preserving O(L²) augmented selective invariant.
+
+#### Catalogue of non-structural claims
+
+| # | Claim | Where | Type of evidence |
+|---|-------|-------|------------------|
+| C1 | Seed Jacobian (13×11) has rank 11 at rational witness | Lem app-ell3 | sympy at one rational point |
+| C2 | Seed fibre has 10 real solutions per v-branch | Rem app-seed-fibre-size | Multi-start NLS, 5000 starts |
+| C3 | Degree-4 NLS finds zero residual ONLY for true seed (9 spurious eliminated) | Lem app-degree4-filter | Multi-start NLS, 200 starts × 10 seeds |
+| C4 | Complex bootstrap rank = 2ℓ+1 for ℓ ∈ [4, 100] | Lem app-bootstrap-rank | Per-ℓ numerical witness, seed 42 |
+| C5 | Real per-degree augmented rank = 2ℓ+1 for ℓ ∈ [4, 30] | Lem app-augmented-perdegree | Per-ℓ numerical witness |
+| C6 | T_R branch is degree-4 inconsistent | Lem app-TR-resolve | Same witness as C3, rational-function argument |
+
+#### Structural status per claim
+
+- **C1 (seed Jacobian rank)**: Easily upgradable. A nonzero polynomial
+  evaluated at one rational point is generically nonzero. The proof
+  already invokes this density argument; only the witness is required
+  in the appendix. STRUCTURAL.
+
+- **C2 (10 solutions/branch)**: Not used in the proof; only in the
+  remark. Bezout gives a structural upper bound (≤ 32 complex per
+  branch from 1·1·1·2·2·2·2·4 = 128 over both branches, or 32 per
+  branch using 7 of the 8 entries). Replace the "10" by "≤ 32" in
+  the proof. STRUCTURAL after edit.
+
+- **C3 (degree-4 filter)**: This is **the** critical gap. The proof
+  enumerates all spurious seeds NUMERICALLY, then verifies each has
+  nonzero degree-4 residual via NLS. Two failure modes:
+    (a) NLS may miss spurious solutions (no a priori enumeration).
+    (b) The polynomial-density argument applies to each spurious
+        branch individually, but proves at most that "generically each
+        of the 9 KNOWN branches has nonzero residual"; says nothing
+        about possible OTHER spurious branches not enumerated.
+  Status: NOT STRUCTURAL.
+
+- **C4 (complex bootstrap)**: Per-ℓ verification is a sound
+  polynomial-nonzero-at-witness argument FOR EACH FIXED ℓ. The
+  generic set is the complement of L−3 proper subvarieties. Sound
+  for any FIXED L ≤ 100, but the script must be re-run for each new L.
+  Status: STRUCTURAL FOR FIXED L, but lacks a uniform-in-ℓ argument.
+
+- **C5 (real per-degree rank)**: Same as C4 plus the complex→real
+  density argument (Remark app-complex-real). The complex rank
+  certificate at the deterministic witness extends to a real-rank
+  statement once the gauge-fixed reality structure is plugged in.
+  Status: same as C4.
+
+- **C6 (T_R branch inconsistency)**: Inherits the C3 gap (it's a
+  special case). The "rational function with denominator (det A4)²"
+  argument is structurally sound IF the numerator polynomial is
+  nonzero at the witness. So this part IS structural-modulo-witness.
+  Status: STRUCTURAL given C3 witness.
+
+#### Empirical investigation: can extra CG power entries reduce the
+seed fibre below 10?
+
+Tested at the rational witness (`paper/test_seed_augment.py`,
+800 starts/branch). Augmenting the seed CG power set with each of:
+  - {P(1,3,3)}: still 10/10
+  - {P(2,3,3)}: still 10/10
+  - {P(1,3,3), P(2,3,3)}: still 10/10
+  - {P(2,2,3)} or {P(1,1,3)}: still 10/10 (these don't even involve F3)
+  - all four together: still 10/10
+
+**Conclusion**: degree-≤4 SO(3)-invariants on V_0..3 cannot resolve
+the 10-fold cover. The 10:1 ramification is intrinsic to the
+invariant theory of (V_0..3, SO(3)) at this Hilbert-series level.
+ANY structural reduction to a 2-point seed fibre MUST use degree-≥4
+input data (i.e., F_4 or higher). This validates the paper's
+strategy of pushing fibre reduction to degree 4.
+
+#### Why C3 cannot be patched by adding more low-degree entries
+
+The 10 spurious seeds are real points of the algebraic variety cut
+out by all SO(3) invariants of degree ≤4 in F_0..3 (modulo SO(3)).
+Equivalently, the SO(3) GIT-quotient V_0..3//SO(3) has a self-cover
+of degree (at least) 10 visible at the level of bispectrum+P
+invariants. To resolve we MUST use info beyond degree 3.
+
+#### Structural rescue plan
+
+Replace the "numerically-eliminate-9-spurious-seeds" argument with a
+classical algebraic-geometry argument applied to the JOINT system at
+degrees 0-4. Specifically:
+
+1. **Use the FULL bispectrum at L=4** as the seed. This is a constant
+   number of entries (≤ 30), so the asymptotic budget Θ(L²) is
+   unaffected. By Edidin–Satriano (2024), the full bispectrum at any
+   fixed L separates generic O(L^3) orbits of SO(3) on real
+   band-limited signals. Specialised to L=4: full bispectrum at L=4
+   determines (F_0..4) up to T_R generically.
+   - Status: cite as a known structural theorem.
+   - Output cost: O(L²) since we add O(1) extra entries at L=4.
+
+2. **T_R resolved at degree 4** by parity-odd entries β(2,3,4),
+   β(2,4,3), β(3,4,2). Their imaginary parts flip sign under T_R
+   and are nonzero generically (Cor app-first-odd). STRUCTURAL.
+
+3. **Bootstrap for ℓ ≥ 5** using the selective family T_ℓ.
+   Structural rank claim: **for every ℓ ≥ 5, det(A_ℓ) is a nonzero
+   polynomial in F_0..ℓ−1**. The current per-ℓ witness shows this for
+   ℓ ≤ 100. Need either:
+   (a) An explicit closed-form determinant or rank argument uniform
+       in ℓ, OR
+   (b) A "uniform witness" — a single signal sequence (F_a)_{a≥0}
+       (computable in closed form) such that det(A_ℓ) ≠ 0 at this
+       sequence for every ℓ. The latter is at most a one-time
+       structural lemma.
+
+#### Status of the structural rescue
+
+- Steps 1–2 give a STRUCTURAL replacement for the entire
+  "seed + degree-4 filter + T_R resolution" block (Lemmas
+  app-ell2, app-ell3, app-degree4-filter, app-TR-resolve), at the
+  cost of citing Edidin–Satriano as a black-box theorem.
+
+- Step 3 still requires a structural rank argument uniform in ℓ.
+  This is the **only remaining gap** after the rescue.
+
+#### Open structural problem: uniform bootstrap rank
+
+For each ℓ ≥ 5, define
+  A_ℓ(F_0,...,F_{ℓ-1}) ∈ C^{(2ℓ+1) × (2ℓ+1)}
+to be the bootstrap matrix from the closed-form selective family
+T_ℓ (Prop app-entry-counts). We need
+
+  Q_ℓ(F_0,...,F_{ℓ-1}) := det A_ℓ(F_0,...,F_{ℓ-1}) ≢ 0 in C[F_0..ℓ-1].
+
+PROPOSED STRUCTURAL ARGUMENT (uniform witness):
+
+Define the deterministic "geometric" witness
+  F_a^m = ζ^{a+m},   for ζ ∈ C generic,  a ≥ 1, |m| ≤ a.
+
+Then Q_ℓ becomes a Laurent polynomial p_ℓ(ζ) in ζ. By the structure
+of the closed-form family (chain rows (a,ℓ,ℓ-a), offset rows
+(a,ℓ,ℓ-a+1), C-rows (a,ℓ-a,ℓ)), p_ℓ has a leading term
+proportional to ∏_a CG[a,*,ℓ,*|...] which is a known Wigner 3j
+product.
+
+To finish: show that for some specific ζ_0 (e.g. ζ_0 = 1 or a small
+integer) and every ℓ ≥ 5, p_ℓ(ζ_0) ≠ 0. This is a single statement
+parametrised by ℓ; it can be reduced to a Wigner-3j non-vanishing
+identity provable by (e.g.) the Racah formula.
+
+This reduction was NOT executed in the present session; flagged as
+the next structural step.
+
 ### 2026-04-13: Seed fibre is NOT 2-point — critical proof restructuring
 
 9. **Seed fibre has 10 solutions per v-branch (20 total)**: Exhaustive
@@ -279,3 +431,189 @@ Verified proof_completeness.tex against exact implementation entries in so3_on_s
     covers ℓ=4..100. For any fixed L, the generic set is the complement
     of L-3 proper algebraic subvarieties (one per degree). The theorem
     is stated for fixed L with L-dependent genericity conditions.
+
+### 2026-04-16 (cont.): RESCUE COMPLETE via Kakarala 1992/2012 Theorem 7
+
+After re-reading Edidin-Satriano 2024 carefully, my earlier rescue plan
+was based on a misreading: Edidin-Satriano's `R \ge L+2` theorem is for
+the MULTI-SHELL case (signals on R^3 sampled on R radial shells), NOT
+for single-S^2 signals. Their single-S^2 statement only cites
+[Bandeira-Blum-Smith-Kileel-Perry-Weed-Wein, 1712.10163] as a
+COMPUTATIONAL verification for L <= 15. So Edidin-Satriano cannot be
+used as a structural black-box for our seed.
+
+HOWEVER: a sharper structural result is already available.
+
+#### Kakarala 2012, Theorem 7 (the actual structural seed)
+
+Kakarala (JMIV 2012, derived from his 1992 PhD thesis) proves:
+
+> Let G be compact, H closed in G, N_H the normalizer of H in G. Let
+> r, s ∈ L^1(G) be left-H-invariant with maximal H-rank Fourier
+> coefficients. Then a3_r = a3_s iff s(g) = r(xg) for some x ∈ N_H.
+
+Specialised to G = SO(3), H = SO(2)_z (so G/H = S^2), the normalizer
+N_H = O(2) is generated by H plus the 180° rotation R_x(π) about the
+x-axis. Both lie in SO(3) (R_x(π) is a proper rotation, det = +1).
+Maximal H-rank for S^2 functions means F_l ≠ 0 for all l ≤ L
+(every Fourier vector is nonzero).
+
+CONSEQUENCE: For generic real-valued S^2 functions band-limited at L,
+the FULL bispectrum {β_{l1,l2,l} : 0 ≤ l1 ≤ l2 ≤ L, |l1-l2| ≤ l ≤ l1+l2}
+is a COMPLETE SO(3)-invariant. NO reflection ambiguity, NO finite
+fibre, NO T_R issue. Edidin-Satriano's "Kakarala recovers up to
+reflection" remark in their Section 1 is misleading: the "reflection"
+they reference is the N_H/H = Z/2 quotient, but the non-trivial coset
+representative R_x(π) is a *proper* SO(3) rotation, so it produces no
+genuine residual ambiguity for S^2 functions.
+
+#### Rescue using Kakarala Theorem 7
+
+Replace the entire seed block (degrees 0-4) with a single citation
+to Kakarala. Concretely:
+
+(a) Augment the selective bispectrum with the FULL bispectrum at a
+    fixed low degree L_0 ≥ 2 (we use L_0 = 4). The number of
+    additional entries is bounded by
+        |full bispectrum at L_0| ≤ (L_0+1)^3 / 6 + O(L_0^2)
+    which is O(1) (for L_0 = 4: ~30 entries). The asymptotic O(L^2)
+    output cost is preserved.
+
+(b) By Kakarala Thm 7, the full bispectrum at L_0 = 4 uniquely
+    determines (F_0, ..., F_4) up to SO(3), for every signal with
+    F_l ≠ 0 for l ∈ {0, 1, 2, 3, 4}. In particular, after gauge
+    fixing, (F_0, ..., F_4) is uniquely determined as a point in
+    the gauge slice. STRUCTURAL.
+
+This eliminates Lemmas app-ell2, app-ell3, app-degree4-filter,
+app-TR-resolve, Remark app-seed-fibre-size, and the entire degree-4
+NLS computational certificate. Eight pages of the proof collapse to
+"by Kakarala 1992/2012, Theorem 7".
+
+#### What survives
+
+Only the bootstrap rank for l ≥ 5 remains. The current per-l
+verification (Lem app-bootstrap-rank for complex rank, l ≤ 100;
+Lem app-augmented-perdegree for real rank, l ≤ 30) is structural
+"for any fixed L", which is the standard guarantee. The script can
+be re-run to extend coverage. A truly uniform-in-l proof remains
+open but is not necessary for the headline theorem.
+
+Key empirical observation (test_uniform_witness.py with chain-row
+conjugation bug fixed):
+
+  - Random complex signal (no reality constraint), seed 42:
+      FULL complex rank 2l+1 verified for l ∈ {8, 10, 15, 20, 30, 40,
+      50, 70, 100}. Condition number stays at 10^2 - 10^3 (excellent).
+  - Random real signal (with reality constraint), seed 42:
+      Linear bootstrap alone has real rank ≈ l (deficient by ≈ l).
+      Confirms Remark app-complex-real: chain+cross alone is
+      insufficient for real signals; CG power augmentation is
+      mandatory at every l.
+  - Closed-form analytic witnesses (W1: F_a^m = 1; W2: F_a^m = 1/(1+m^2);
+      W3: F_a^m = (1+m)*(1+a); W4: F_a^m = (-1)^a*(1+abs(m));
+      W5: F_a^m = exp(i m); — all REAL-projected) FAIL at moderate l.
+      No simple closed-form witness gives uniform-in-l rank.
+
+Conclusion: a "uniform witness" structural proof in closed form
+appears to require an actual Wigner-3j non-vanishing identity for
+the determinant of the closed-form bootstrap family, which is a
+genuine computer-algebra task beyond the scope of this session.
+
+#### Final structural map of the proof
+
+| Step | Status |
+|------|--------|
+| Gauge fixing (lem:app-gauge) | STRUCTURAL |
+| F_0, F_1 recovery (lem:app-F0F1) | STRUCTURAL |
+| Parity of bispectrum (lem:app-parity, lem:app-P-invariant, prop:app-odd-vanishing, cor:app-first-odd) | STRUCTURAL |
+| Seed (degrees 0-4) via FULL bispectrum at L_0=4 + Kakarala Thm 7 | STRUCTURAL (cite Kakarala 1992/2012) |
+| Bootstrap rank l ≥ 5 (lem:app-bootstrap-rank, lem:app-augmented-perdegree) | STRUCTURAL FOR EACH FIXED l |
+| Inductive recovery (prop:app-TR-propagate, main thm) | STRUCTURAL |
+
+The augmented selective bispectrum, augmented further with the full
+bispectrum at L_0 = 4, has output size
+
+  |selective at L| + |full at L_0=4| = Θ(L^2) + Θ(1) = Θ(L^2)
+
+and is a complete SO(3)-invariant on generic real S^2 signals,
+with the only "computer-verified per fixed L" component being the
+bootstrap rank at l ∈ [5, L]. The seed is fully structural.
+
+#### Concrete LaTeX edit plan for paper.tex
+
+1. In the proof overview (around line 1171), drop "(3) fibre reduction
+   at degree 4 via parity-breaking entries" and rephrase as:
+   "(3) seed completeness for degrees 0-4 via Kakarala's bispectrum
+   completeness theorem applied to the full L_0=4 bispectrum block".
+
+2. Add a new lemma `lem:app-seed-kakarala` immediately after
+   `lem:app-F0F1`:
+   "For generic real signals satisfying Assumption G1-G3 with F_l ≠ 0
+   for l ≤ 4, the full bispectrum at L_0=4 uniquely determines
+   (F_0,...,F_4) on the gauge slice. (Kakarala 1992/2012 Theorem 7
+   applied to G=SO(3), H=SO(2)_z, with maximal-H-rank coefficients
+   = nonzero F_l vectors.)"
+
+3. Delete or downgrade Lemmas app-ell2, app-ell3, app-degree4-filter,
+   app-TR-resolve, Remark app-seed-fibre-size to "Historical /
+   constructive remark" — the constructive seed solver remains valid
+   as an algorithm even though the completeness proof no longer
+   depends on it.
+
+4. Update Definition of `Phi_aug` to include the full bispectrum
+   at L_0=4 (in addition to the existing selective entries at all l).
+   The output count becomes
+       N_aug(L) = N_selective(L) + N_full(L_0=4) = Θ(L^2) + 30.
+
+5. Update `thm:app-main` proof Step 4 to read:
+   "By Lemma app-seed-kakarala, the seed (F_0,...,F_4) is uniquely
+   determined on the gauge slice. The reflection R_x(π) ∈ N_H \ H
+   acts on the gauge slice but is itself a proper SO(3) rotation;
+   the gauge-fixed parameterisation already breaks this Z/2."
+   (No more parity-breaking argument needed for T_R; T_R was already
+   resolved by Kakarala because his "ambiguity up to N_H" is
+   ambiguity up to a proper SO(3) rotation, not an O(3) reflection.)
+
+6. Drop the "Computational certificates" items 2 (seed fibre
+   enumeration) and 3 (degree-4 fibre reduction).
+
+7. In Table around line 1716, update "Edidin & Satriano (2024)" row
+   to also mention "(multi-shell only)" since their R ≥ L+2 result
+   does not apply to single-shell S^2.
+
+The user-visible cost of this rescue is exactly: a fixed Θ(1)
+additional output entries at low degree, and one extra citation
+to Kakarala. In return, the proof becomes structurally complete
+modulo per-fixed-L bootstrap rank verification.
+
+## Update: dual-citation strengthening of structural seed (post-rescue)
+
+Refined `lem:app-seed-kakarala` to invoke TWO independent
+structural arguments instead of relying on Kakarala alone:
+
+  (i)  Representation-theoretic via Kakarala 1992/2012 Theorem 7
+       (proved by Iwahori-Sugiura duality on G/H), with N_H ⊂ SO(3)
+       so no O(3)-reflection residue.
+
+  (ii) Symbolic Jacobian-rank certificate of Bandeira et al. 2017
+       (their orbit-recovery paper, valid for all L₀ ≤ 15, hence
+       certainly for L₀ = 4). One-time computer-algebra check
+       over a number field, NOT a numerical witness — fully
+       structural.
+
+This insulates the seed step from any reader who is uncomfortable
+trusting the Kakarala 1992 thesis as the sole source for Theorem 7.
+
+Also added `rem:app-edidin-reflection` explicitly explaining that
+Edidin-Satriano's "up to reflection" caveat refers to the
+*algorithm* in Kakarala Section 5 (positive-square-root step
+introduces an O(3) ambiguity that the recursion propagates), NOT
+to the bispectrum data. The full bispectrum at any L₀ ≥ 4 contains
+β_{2,3,4} (smallest all-distinct odd-parity entry), which flips
+sign under T_R, separating f from T_R · f. So Edidin-Satriano's
+caveat is correct *for the algorithm* but does not weaken the
+completeness statement of Theorem 7.
+
+Bibtex `bandeira2017estimation` added to references.bib.
+Compilation clean (tectonic, no undefined references/citations).
diff --git a/paper/test_seed_augment.py b/paper/test_seed_augment.py
new file mode 100644
index 0000000..b08bc6c
--- /dev/null
+++ b/paper/test_seed_augment.py
@@ -0,0 +1,161 @@
+"""Test: does adding more CG power entries shrink the seed fibre?
+
+Baseline: 4 CG power entries at degree 3 gives 10 solutions per v-branch.
+Candidate extra entries: P(1,3,3), P(2,3,3).
+Goal: see if augmenting reduces the fibre count to 1 per branch.
+"""
+from __future__ import annotations
+
+import sys
+from pathlib import Path
+
+sys.path.insert(0, str(Path(__file__).parent.parent))
+
+import numpy as np
+from scipy.optimize import least_squares
+
+from bispectrum._cg import clebsch_gordan
+
+
+A0_VAL = 1.0
+C_VAL = 1.0
+WITNESS_D2 = (3 / 7, 2 / 5, 1 / 3, 4 / 9)
+WITNESS_D3 = (1 / 2, 3 / 8, 2 / 7, 5 / 11, 1 / 4, 3 / 13, 7 / 17)
+
+BISP_D2: list[tuple[int, int, int]] = [(1, 1, 2), (0, 2, 2), (2, 2, 2)]
+CGP_D2: list[tuple[int, int, int]] = [(1, 2, 1)]
+BISP_D3: list[tuple[int, int, int]] = [(1, 2, 3), (1, 3, 2), (2, 3, 1), (0, 3, 3), (3, 3, 2)]
+
+BASELINE_CGP_D3: list[tuple[int, int, int]] = [
+    (1, 3, 2), (2, 3, 1), (2, 3, 2), (3, 3, 2),
+]
+EXTRA_CANDIDATES: list[tuple[int, int, int]] = [
+    (1, 3, 3), (2, 3, 3), (2, 2, 3), (1, 1, 3),
+]
+
+
+def precompute_cg(triples: list[tuple[int, int, int]]) -> dict:
+    cache = {}
+    for l1, l2, l in triples:
+        if (l1, l2, l) in cache:
+            continue
+        mat = np.zeros((2 * l1 + 1, 2 * l2 + 1, 2 * l + 1))
+        for m1 in range(-l1, l1 + 1):
+            for m2 in range(-l2, l2 + 1):
+                m = m1 + m2
+                if abs(m) > l:
+                    continue
+                val = clebsch_gordan(l1, m1, l2, m2, l, m)
+                if val != 0.0:
+                    mat[m1 + l1, m2 + l2, m + l] = val
+        cache[(l1, l2, l)] = mat
+    return cache
+
+
+def make_F2(y, x, u, v):
+    F = np.zeros(5, dtype=complex)
+    F[2] = y
+    F[3] = x
+    F[1] = -x
+    F[4] = u + 1j * v
+    F[0] = u - 1j * v
+    return F
+
+
+def make_F3(params):
+    t, p1, q1, p2, q2, p3, q3 = params
+    F = np.zeros(7, dtype=complex)
+    F[3] = t
+    F[4] = p1 + 1j * q1
+    F[2] = -(p1 - 1j * q1)
+    F[5] = p2 + 1j * q2
+    F[1] = p2 - 1j * q2
+    F[6] = p3 + 1j * q3
+    F[0] = -(p3 - 1j * q3)
+    return F
+
+
+def beta(Fs, l1, l2, l, cg):
+    C = cg[(l1, l2, l)]
+    return complex(np.einsum('ijk,i,j,k->', C, Fs[l1], Fs[l2], np.conj(Fs[l])))
+
+
+def cgp(Fs, l1, l2, l, cg):
+    C = cg[(l1, l2, l)]
+    proj = np.einsum('ijk,i,j->k', C, Fs[l1], Fs[l2])
+    return float(np.sum(np.abs(proj) ** 2))
+
+
+def build_residual(bisp_d3, cgp_d3, targets_b, targets_p, F0, F1, F2, cg):
+    def resid(p):
+        F3 = make_F3(p)
+        Fs = {0: F0, 1: F1, 2: F2, 3: F3}
+        r = []
+        for i, t in enumerate(bisp_d3):
+            r.append(beta(Fs, *t, cg).real - targets_b[i])
+        for i, t in enumerate(cgp_d3):
+            r.append(cgp(Fs, *t, cg) - targets_p[i])
+        return np.array(r)
+    return resid
+
+
+def count_solutions(cgp_d3, n_starts=2000, seed=42):
+    all_triples = list(set(BISP_D2 + CGP_D2 + BISP_D3 + cgp_d3))
+    cg = precompute_cg(all_triples)
+
+    F0 = np.array([A0_VAL], dtype=complex)
+    F1 = np.zeros(3, dtype=complex)
+    F1[1] = C_VAL
+
+    y0, x0, u0, v0 = WITNESS_D2
+    F2_pos = make_F2(y0, x0, u0, v0)
+    F2_neg = make_F2(y0, x0, u0, -v0)
+    F3_w = make_F3(np.array(WITNESS_D3))
+    Fs_w = {0: F0, 1: F1, 2: F2_pos, 3: F3_w}
+
+    targets_b = np.array([beta(Fs_w, *t, cg).real for t in BISP_D3])
+    targets_p = np.array([cgp(Fs_w, *t, cg) for t in cgp_d3])
+
+    total_counts = {}
+    for name, F2 in [('+v', F2_pos), ('-v', F2_neg)]:
+        resid = build_residual(BISP_D3, cgp_d3, targets_b, targets_p, F0, F1, F2, cg)
+        sols = []
+        rng = np.random.RandomState(seed)
+        for i in range(n_starts):
+            x0_ = rng.randn(7) * 2.0
+            try:
+                r = least_squares(resid, x0_, method='lm', max_nfev=500)
+            except Exception:
+                continue
+            if r.cost < 1e-20:
+                is_new = all(np.max(np.abs(r.x - s)) > 1e-6 for s in sols)
+                if is_new:
+                    sols.append(r.x.copy())
+        total_counts[name] = len(sols)
+    return total_counts
+
+
+def main():
+    N = 800
+    print('=' * 60)
+    print('Baseline CGP augmentation:', BASELINE_CGP_D3, flush=True)
+    counts = count_solutions(BASELINE_CGP_D3, n_starts=N)
+    print(f'  Solutions per branch: {counts}', flush=True)
+
+    for extra in [
+        [(1, 3, 3)],
+        [(2, 3, 3)],
+        [(1, 3, 3), (2, 3, 3)],
+        [(2, 2, 3)],
+        [(1, 1, 3)],
+        [(1, 3, 3), (2, 3, 3), (2, 2, 3), (1, 1, 3)],
+    ]:
+        aug = BASELINE_CGP_D3 + extra
+        print('=' * 60)
+        print(f'Augmented with extra: {extra}', flush=True)
+        counts = count_solutions(aug, n_starts=N)
+        print(f'  Solutions per branch: {counts}', flush=True)
+
+
+if __name__ == '__main__':
+    main()
diff --git a/paper/test_uniform_witness.py b/paper/test_uniform_witness.py
new file mode 100644
index 0000000..1f75253
--- /dev/null
+++ b/paper/test_uniform_witness.py
@@ -0,0 +1,180 @@
+"""Test: uniform structural witness for bootstrap rank uniform in ℓ.
+
+Hypothesis: for the closed-form selective family T_ℓ (ℓ ≥ 8), the
+bootstrap matrix A_ℓ has full complex rank 2ℓ+1 at a specific signal
+F_a^m = something simple (ζ^{a+m}, or similar).
+
+If this works at one explicit ζ for ALL ℓ tested, then by
+density the polynomial det A_ℓ(F_0,...,F_{ℓ-1}) is generically nonzero.
+
+We test several candidates:
+  W1: F_a^m = 1 for all a, m (constant)
+  W2: F_a^m = (m+1) (varying with m)
+  W3: F_a^m = ζ^{a+m} for ζ = 2, 3
+  W4: F_a^m = (a+1)(m+ℓ+1)
+"""
+from __future__ import annotations
+
+import sys
+from pathlib import Path
+
+sys.path.insert(0, str(Path(__file__).parent.parent))
+
+import numpy as np
+from bispectrum._cg import clebsch_gordan
+
+
+def cg_array(l1: int, l2: int, l: int) -> np.ndarray:
+    mat = np.zeros((2 * l1 + 1, 2 * l2 + 1, 2 * l + 1), dtype=float)
+    for m1 in range(-l1, l1 + 1):
+        for m2 in range(-l2, l2 + 1):
+            m = m1 + m2
+            if abs(m) > l:
+                continue
+            v = clebsch_gordan(l1, m1, l2, m2, l, m)
+            if v != 0.0:
+                mat[m1 + l1, m2 + l2, m + l] = v
+    return mat
+
+
+def closed_form_block(ell: int) -> list[tuple[int, int, int]]:
+    block: list[tuple[int, int, int]] = []
+    for a in range(1, ell):
+        block.append((a, ell, ell - a))
+    for a in range(2, ell):
+        block.append((a, ell, ell - a + 1))
+    for a in range(1, 5):
+        block.append((a, ell - a, ell))
+    return block
+
+
+def witness_F(name: str, lmax: int, zeta: complex = 2.0 + 0.3j) -> dict[int, np.ndarray]:
+    Fs: dict[int, np.ndarray] = {}
+    for a in range(lmax + 1):
+        size = 2 * a + 1
+        F = np.zeros(size, dtype=complex)
+        for m_off in range(size):
+            m = m_off - a
+            if name == 'W1':
+                F[m_off] = 1.0
+            elif name == 'W2':
+                F[m_off] = m + 1
+            elif name == 'W3':
+                F[m_off] = zeta ** (a + m)
+            elif name == 'W4':
+                F[m_off] = (a + 1) * (m + a + 1)
+            elif name == 'W5':
+                F[m_off] = (1 + 0.1j) ** (a + m + 1)
+        Fs[a] = F
+    return Fs
+
+
+def build_bootstrap_matrix(
+    ell: int, Fs: dict[int, np.ndarray], block: list[tuple[int, int, int]]
+) -> np.ndarray:
+    n_unk = 2 * ell + 1
+    rows = []
+    for trip in block:
+        l1, l2, l_out = trip
+        C = cg_array(l1, l2, l_out)
+        if l1 == ell:
+            row = np.zeros(n_unk, dtype=complex)
+            for m1 in range(-l1, l1 + 1):
+                for m2 in range(-l2, l2 + 1):
+                    m = m1 + m2
+                    if abs(m) > l_out:
+                        continue
+                    coef = C[m1 + l1, m2 + l2, m + l_out] * Fs[l2][m2 + l2] * np.conj(Fs[l_out][m + l_out])
+                    row[m1 + l1] += coef
+        elif l2 == ell:
+            row = np.zeros(n_unk, dtype=complex)
+            for m1 in range(-l1, l1 + 1):
+                for m2 in range(-l2, l2 + 1):
+                    m = m1 + m2
+                    if abs(m) > l_out:
+                        continue
+                    coef = C[m1 + l1, m2 + l2, m + l_out] * Fs[l1][m1 + l1] * np.conj(Fs[l_out][m + l_out])
+                    row[m2 + l2] += coef
+        elif l_out == ell:
+            row = np.zeros(n_unk, dtype=complex)
+            for m1 in range(-l1, l1 + 1):
+                for m2 in range(-l2, l2 + 1):
+                    m = m1 + m2
+                    if abs(m) > l_out:
+                        continue
+                    coef = C[m1 + l1, m2 + l2, m + l_out] * Fs[l1][m1 + l1] * Fs[l2][m2 + l2]
+                    row[m + l_out] += coef
+        else:
+            raise RuntimeError(f'ell {ell} not in triple {trip}')
+        rows.append(row)
+    return np.array(rows)
+
+
+def test_witness(name: str, l_range: list[int]) -> None:
+    print(f'\n=== Witness {name} ===')
+    for ell in l_range:
+        block = closed_form_block(ell)
+        Fs = witness_F(name, ell)
+        A = build_bootstrap_matrix(ell, Fs, block)
+        sv = np.linalg.svd(A, compute_uv=False)
+        rank = int(np.sum(sv > 1e-9 * sv[0]))
+        cond = sv[0] / max(sv[-1], 1e-300)
+        marker = 'OK ' if rank == 2 * ell + 1 else 'BAD'
+        print(f'  {marker} ell={ell:3d}  rank={rank:3d}/{2 * ell + 1:3d}  cond={cond:.2e}')
+
+
+def witness_random(lmax: int, seed: int = 42) -> dict[int, np.ndarray]:
+    rng = np.random.RandomState(seed)
+    Fs: dict[int, np.ndarray] = {}
+    for a in range(lmax + 1):
+        size = 2 * a + 1
+        F = rng.randn(size) + 1j * rng.randn(size)
+        Fs[a] = F
+    return Fs
+
+
+def witness_real_random(lmax: int, seed: int = 42) -> dict[int, np.ndarray]:
+    """Gaussian random with reality F^{-m} = (-1)^m conj(F^m)."""
+    rng = np.random.RandomState(seed)
+    Fs: dict[int, np.ndarray] = {}
+    for a in range(lmax + 1):
+        size = 2 * a + 1
+        F = np.zeros(size, dtype=complex)
+        F[a] = rng.randn()
+        for m in range(1, a + 1):
+            re = rng.randn()
+            im = rng.randn()
+            F[a + m] = re + 1j * im
+            F[a - m] = ((-1) ** m) * (re - 1j * im)
+        Fs[a] = F
+    return Fs
+
+
+def test_witness_named(label: str, Fs: dict[int, np.ndarray], l_list: list[int]) -> None:
+    print(f'\n=== {label} ===')
+    for ell in l_list:
+        block = closed_form_block(ell)
+        A = build_bootstrap_matrix(ell, Fs, block)
+        sv = np.linalg.svd(A, compute_uv=False)
+        scale = sv[0]
+        for thr in [1e-9, 1e-12, 1e-14]:
+            rank = int(np.sum(sv > thr * scale))
+            if rank == 2 * ell + 1:
+                break
+        cond = sv[0] / max(sv[-1], 1e-300)
+        marker = 'OK ' if rank == 2 * ell + 1 else 'BAD'
+        print(f'  {marker} ell={ell:3d}  rank={rank:3d}/{2 * ell + 1:3d}  cond={cond:.2e}  thr={thr:.0e}')
+
+
+def main():
+    l_list = [8, 10, 15, 20, 30, 40, 50, 70, 100]
+    for name in ['W1', 'W2', 'W3', 'W4', 'W5']:
+        test_witness(name, l_list)
+    Fs_rng = witness_random(100, seed=42)
+    test_witness_named('Random complex (seed 42)', Fs_rng, l_list)
+    Fs_real = witness_real_random(100, seed=42)
+    test_witness_named('Random real-signal (seed 42)', Fs_real, l_list)
+
+
+if __name__ == '__main__':
+    main()