formula_img2latex/tests/test_pipeline.py

"""scan → (candidates) → verify → apply → revert on a synthetic epub2md output, without any model."""

from __future__ import annotations

from pathlib import Path

import pytest

from conftest import needs_latex, needs_pandoc
from formula_img2latex import pipeline
from formula_img2latex.cli import main
from formula_img2latex.engines.vlm import parse_answer, prepare_image
from formula_img2latex.sources.base import scan_markdown
from formula_img2latex.store import Workspace


def _inject(ws: Workspace, by_asset: dict[str, str]) -> None:
    """Pretend an engine answered (tests must not need a model server).

    The mechanics are tested here, not the calibrated thresholds (test_core.test_decision_rules),
    so the contrast threshold is set low enough for the synthetic images.
    """
    ws.settings["thresholds"] = {"min_contrast": 0.5}
    ws.settings["decide"] = "pixel"
    ws.save_meta()
    for f in ws.all():
        for name, tex in by_asset.items():
            if f.asset.endswith(name):
                f.candidates.append({"engine": "test", "model": "fixture", "tex": tex, "is_formula": True})
                ws.save(f)


def test_scan_markdown_skips_code_and_detects_display(epub2md_dir: Path) -> None:
    refs = scan_markdown(epub2md_dir / "chapters" / "001-kapitel.md")
    assert [(r.target.rsplit("/", 1)[1], r.display) for r in refs] == [
        ("0001-inline.jpg", False), ("0002-display.jpg", True), ("0003-figure.png", True)]


def test_scan_classifies(epub2md_dir: Path) -> None:
    source, ws = pipeline.open_all(epub2md_dir, None, create=True)
    pipeline.scan(source, ws, quiet=True)
    labels = {f.asset.rsplit("/", 1)[1]: (f.classify["label"], f.known_formula) for f in ws.all()}
    assert labels["0001-inline.jpg"] == ("formula", False)
    assert labels["0002-display.jpg"] == ("formula", True)       # epub2md warning
    assert labels["0003-figure.png"][0] == "figure"
    assert ws.root == epub2md_dir.parent / "buch.formulas"


@needs_latex
@needs_pandoc
def test_verify_apply_revert_roundtrip(epub2md_dir: Path) -> None:
    source, ws = pipeline.open_all(epub2md_dir, None, create=True)
    pipeline.scan(source, ws, quiet=True)
    _inject(ws, {"0001-inline.jpg": r"L = \{ - \sqrt{2} , \sqrt{2} \}",          # comma: wrong, repaired to ;
                 "0002-display.jpg": r"x_{2,k}=\arcsin\left(-\frac{3}{5}\right)+2k\pi"})
    counts = pipeline.verify_all(source, ws, preset="cm", workers=2, quiet=True)
    st = {f.asset.rsplit("/", 1)[1]: f for f in ws.all()}
    inline = st["0001-inline.jpg"]
    assert inline.verification["best"]["tex"].replace(" ", "") == r"L=\{-\sqrt{2};\sqrt{2}\}"
    assert inline.status == "auto_accepted", inline.reason
    assert sum(counts.values()) == 2
    md = epub2md_dir / "chapters" / "001-kapitel.md"
    before = md.read_text(encoding="utf-8")

    pipeline.apply(source, ws, quiet=True)
    after = md.read_text(encoding="utf-8")
    assert "lautet $L = \\{ - \\sqrt{2} ; \\sqrt{2} \\}$." in after or "lautet $L=\\{-\\sqrt{2};\\sqrt{2}\\}$." in after
    assert "```\n![kein](../media/images/0001-inline.jpg)\n```" in after      # code block untouched
    assert "0003-figure.png" in after
    pipeline.apply(source, ws, quiet=True)                                        # idempotent
    assert md.read_text(encoding="utf-8") == after

    pipeline.revert(source, ws, quiet=True)
    assert md.read_text(encoding="utf-8") == before


@needs_latex
def test_revert_refuses_hand_edited_files(epub2md_dir: Path) -> None:
    source, ws = pipeline.open_all(epub2md_dir, None, create=True)
    pipeline.scan(source, ws, quiet=True)
    _inject(ws, {"0001-inline.jpg": r"L=\{-\sqrt{2};\sqrt{2}\}"})
    pipeline.verify_all(source, ws, preset="cm", workers=1, quiet=True)
    pipeline.apply(source, ws, quiet=True)
    md = epub2md_dir / "chapters" / "001-kapitel.md"
    md.write_text(md.read_text(encoding="utf-8") + "\nHandnotiz\n", encoding="utf-8")
    assert pipeline.revert(source, ws, quiet=True)["skipped"] == 1
    assert "Handnotiz" in md.read_text(encoding="utf-8")


def test_cli_errors(tmp_path: Path, capsys) -> None:
    assert main(["status", str(tmp_path / "fehlt")]) == 1
    (tmp_path / "leer").mkdir()
    assert main(["scan", str(tmp_path / "leer")]) == 2
    assert "weder eine epub2md-Ausgabe" in capsys.readouterr().err


def test_engine_answer_parsing() -> None:
    assert parse_answer('{"is_formula": true, "latex": "x^2"}') == ("x^2", True)
    assert parse_answer('```json\n{"is_formula": true, "latex": "\\frac{a}{b}"}\n```') == ("\\frac{a}{b}", True)
    assert parse_answer('{"is_formula": true, "latex": "\\sqrt{2}"}') == ("\\sqrt{2}", True)   # unescaped backslash
    assert parse_answer('{"is_formula": false, "latex": ""}') == (None, False)
    assert parse_answer("x_1 + x_2") == ("x_1 + x_2", None)


def test_prepare_image_enlarges_small_formulas(epub2md_dir: Path) -> None:
    import io

    from PIL import Image
    data = (epub2md_dir / "media" / "images" / "0001-inline.jpg").read_bytes()
    with Image.open(io.BytesIO(prepare_image(data, "x.jpg"))) as im:
        assert im.height >= 96 and im.mode == "L"
    big = Image.new("L", (4000, 900), 255)   # 600-dpi box: shrunk so it fits the model's context
    big.paste(0, (100, 100, 3900, 800))
    buf = io.BytesIO()
    big.save(buf, "PNG")
    with Image.open(io.BytesIO(prepare_image(buf.getvalue(), "x.png"))) as im:
        assert max(im.size) <= 1600 * 1.4


def test_verify_refuses_huge_crops() -> None:
    import numpy as np

    from formula_img2latex.verify import verify
    v = verify(np.ones((2500, 2000), np.float32), "x", True, "cm")   # 5 MP: a figure, not a formula
    assert not v.ok and "zu groß" in v.reason


# ------------------------------------------------------------------------------- consensus

def _reading(engine: str, model: str, tex: str | None, p_mean: float = 0.99, p_min: float = 0.8,
             is_formula: bool = True) -> dict:
    return {"engine": engine, "model": model, "tex": tex, "is_formula": is_formula, "p_mean": p_mean, "p_min": p_min}


@needs_latex
@needs_pandoc
def test_consensus_mode(epub2md_dir: Path) -> None:
    source, ws = pipeline.open_all(epub2md_dir, None, create=True)
    ws.settings["consensus_rule"] = "vlm3"   # the earlier rule Qwen = Gemma = GLM
    pipeline.scan(source, ws, quiet=True)
    by = {f.asset.rsplit("/", 1)[1]: f for f in ws.all()}
    inline, display = by["0001-inline.jpg"], by["0002-display.jpg"]
    inline.candidates = [_reading("local", "A", r"L=\{-\sqrt{2};\sqrt{2}\}"),
                         _reading("local2", "B", r"L = \{ - \sqrt2 ; \sqrt2 \}")]          # same, other spelling
    display.candidates = [_reading("local", "A", r"x_{2,k}=\arcsin\left(-\frac{3}{5}\right)+2k\pi"),
                          _reading("local2", "B", r"x_{3,k}=\arcsin\left(-\frac{3}{5}\right)+2k\pi")]  # differ
    ws.save(inline)
    ws.save(display)
    pipeline.verify_all(source, ws, preset="cm", workers=2, quiet=True)
    st = {f.asset.rsplit("/", 1)[1]: f for f in ws.all()}
    assert pipeline.needs_third(st["0001-inline.jpg"]), st["0001-inline.jpg"].reason    # A = B, C missing
    assert st["0002-display.jpg"].status == "needs_review"
    assert "uneinig" in st["0002-display.jpg"].reason

    # the third reading confirms A = B -> accepted; for A != B a third reading does not decide 2:1
    i, d = st["0001-inline.jpg"], st["0002-display.jpg"]
    i.candidates.append(_reading("cloud", "C", r"L=\left\{-\sqrt{2};\sqrt{2}\right\}"))
    d.candidates.append(_reading("cloud", "C", r"x_{2,k} = \arcsin(-\frac35) + 2k\pi"))
    ws.save(i)
    ws.save(d)
    pipeline.verify_all(source, ws, preset="cm", workers=2, quiet=True)
    i, d = ws.load(i.id), ws.load(d.id)
    assert i.status == "auto_accepted" and "dritte Lesart bestätigt" in i.reason, i.reason
    assert d.status == "needs_review"

    # a disagreeing third reading blocks the acceptance
    i.candidates[-1] = _reading("cloud", "C", r"L=\{-\sqrt{2};\sqrt{3}\}")
    i.status = "pending"
    ws.save(i)
    pipeline.verify_all(source, ws, preset="cm", workers=2, quiet=True)
    i = ws.load(i.id)
    assert i.status == "needs_review" and "weicht ab" in i.reason, i.reason


@needs_latex
@needs_pandoc
def test_consensus_rule_p2t(epub2md_dir: Path) -> None:
    """Standard rule: pix2text first, Qwen confirms; where they differ Gemma may confirm pix2text.
    The reading used is always pix2text's."""
    source, ws = pipeline.open_all(epub2md_dir, None, create=True)
    pipeline.scan(source, ws, quiet=True)
    by = {f.asset.rsplit("/", 1)[1]: f for f in ws.all()}
    inline, display = by["0001-inline.jpg"], by["0002-display.jpg"]
    inline.candidates = [_reading("local", "Q", r"L=\{-\sqrt{2};\sqrt{2}\}"),
                         _reading("p2t", "P", r"L=\{-\sqrt{2};\sqrt{2}\}")]
    display.candidates = [_reading("local", "Q", r"x_{3,k}=\arcsin\left(-\frac{3}{5}\right)+2k\pi"),
                          _reading("p2t", "P", r"x_{2,k}=\operatorname{arcsin}(-\frac{3}{5})+2k\pi")]
    ws.save(inline)
    ws.save(display)
    pipeline.verify_all(source, ws, preset="cm", workers=2, quiet=True)
    i, d = ws.load(inline.id), ws.load(display.id)
    assert i.status == "auto_accepted" and "pix2text und Qwen einig" in i.reason, i.reason
    assert i.final_tex == r"L=\{-\sqrt{2};\sqrt{2}\}"                                    # pix2text's reading
    assert pipeline.needs_tiebreak(d), d.reason

    # Gemma confirms pix2text -> accepted with pix2text's reading
    d.candidates.append(_reading("local2", "G", r"x_{2,k} = \arcsin(-\frac{3}{5}) + 2k\pi"))
    ws.save(d)
    pipeline.verify_all(source, ws, preset="cm", workers=2, quiet=True)
    d = ws.load(d.id)
    assert d.status == "auto_accepted" and "Gemma bestätigt pix2text" in d.reason, d.reason
    assert d.final_tex == r"x_{2,k}=\operatorname{arcsin}(-\frac{3}{5})+2k\pi"

    # Qwen and Gemma together against pix2text -> a human decides
    d.candidates[-1] = _reading("local2", "G", r"x_{3,k}=\arcsin(-\frac{3}{5})+2k\pi")
    d.status = "pending"
    ws.save(d)
    pipeline.verify_all(source, ws, preset="cm", workers=2, quiet=True)
    d = ws.load(d.id)
    assert d.status == "needs_review" and "Mensch entscheidet" in d.reason, d.reason


def test_pair_tiebreak_rule() -> None:
    from formula_img2latex.decide import consensus_thresholds, decide_pair_tiebreak
    th = consensus_thresholds()
    q, p, g = _reading("local", "Q", "x^2"), _reading("p2t", "P", "x^{2}"), _reading("local2", "G", "x^2")
    veto = {"ok": True, "metrics": {"ncc": 0.9, "px_per_pt": 1.4}}
    assert decide_pair_tiebreak([q, p, None], [(0, 1, "normalform")], veto, True, 1.4, th)[::2] == ("auto_accepted", 1)
    bad = {"ok": True, "metrics": {"ncc": 0.5, "px_per_pt": 1.4}}
    assert decide_pair_tiebreak([q, p, None], [(0, 1, "normalform")], bad, True, 1.4, th)[0] == "needs_review"   # veto
    assert decide_pair_tiebreak([q, p, None], [(0, 1, "normalform")], veto, False, 1.4, th)[0] == "needs_review"  # pandoc
    assert "Stichwahl fehlt" in decide_pair_tiebreak([q, p, None], [], veto, True, 1.4, th)[1]
    assert decide_pair_tiebreak([q, p, g], [(1, 2, "normalform")], veto, True, 1.4, th)[::2] == ("auto_accepted", 1)
    assert decide_pair_tiebreak([q, p, g], [(0, 2, "normalform")], veto, True, 1.4, th)[0] == "needs_review"
    # image arbiter (off by default): Qwen = Gemma against pix2text, accepted only with enough image evidence
    assert decide_pair_tiebreak([q, p, g], [(0, 2, "normalform")], veto, True, 1.4, th, 99.0)[0] == "needs_review"
    arb = consensus_thresholds({"arbiter": 1, "arbiter_margin": 5.0})
    assert decide_pair_tiebreak([q, p, g], [(0, 2, "normalform")], veto, True, 1.4, arb, 4.0)[0] == "needs_review"
    assert decide_pair_tiebreak([q, p, g], [(0, 2, "normalform")], veto, True, 1.4, arb, 6.0)[::2] == ("auto_accepted", 0)
    assert decide_pair_tiebreak([q, p, g], [(0, 2, "normalform")], bad, True, 1.4, arb, 6.0)[0] == "needs_review"
    gb = _reading("local2", "G", r"\boldsymbol{x}^2")   # the agreeing reading with more bold markup is used
    assert decide_pair_tiebreak([q, p, gb], [(0, 2, "normalform")], veto, True, 1.4, arb, 6.0)[::2] == ("auto_accepted", 2)
    assert decide_pair_tiebreak([q, p, g], [], veto, True, 1.4, th)[0] == "needs_review"
    assert "fehlt" in decide_pair_tiebreak([q, None, None], [], veto, True, 1.4, th)[1]
    nq, ng = _reading("local", "Q", None, is_formula=False), _reading("local2", "G", None, is_formula=False)
    assert decide_pair_tiebreak([nq, p, ng], [], None, True, 1.4, th)[0] == "not_formula"
    assert decide_pair_tiebreak([nq, p, None], [], None, True, 1.4, th)[0] == "needs_review"   # ask Gemma first


def test_consensus_rules() -> None:
    from formula_img2latex.decide import consensus_thresholds, decide_consensus
    th = consensus_thresholds({"min_p_mean": 0.9, "min_p_min": 0.3, "require_third": 0})
    a, b = _reading("local", "A", "x^2"), _reading("local2", "B", "x^{2}")
    veto = {"ok": True, "metrics": {"ncc": 0.9, "px_per_pt": 1.4}}
    assert decide_consensus([a, b], [(0, 1, "normalform")], veto, True, 1.4, th)[0] == "auto_accepted"
    assert decide_consensus([a, b], [], veto, True, 1.4, th)[0] == "needs_review"                       # disagree
    low = _reading("local2", "B", "x^{2}", p_mean=0.7)
    assert decide_consensus([a, low], [(0, 1, "normalform")], veto, True, 1.4, th)[0] == "needs_review"  # unsure
    bad = {"ok": True, "metrics": {"ncc": 0.5, "px_per_pt": 1.4}}
    assert decide_consensus([a, b], [(0, 1, "normalform")], bad, True, 1.4, th)[0] == "needs_review"     # veto
    big = {"ok": True, "metrics": {"ncc": 0.9, "px_per_pt": 1.8}}
    sized = {**th, "veto_scale_dev": 0.15}
    assert decide_consensus([a, b], [(0, 1, "normalform")], big, True, 1.4, sized)[0] == "needs_review"  # size check on
    assert decide_consensus([a, b], [(0, 1, "normalform")], veto, False, 1.4, th)[0] == "needs_review"   # pandoc
    nf = [_reading("local", "A", None, is_formula=False), _reading("local2", "B", None, is_formula=False)]
    assert decide_consensus(nf, [], None, True, 1.4, th)[0] == "not_formula"
    assert decide_consensus([a], [], veto, True, 1.4, th)[0] == "needs_review"                          # one model
    strict = consensus_thresholds()                                                     # default: third must agree
    c = _reading("cloud", "C", "x^2")
    assert decide_consensus([a, b], [(0, 1, "normalform")], veto, True, 1.4, strict)[0] == "needs_review"
    assert decide_consensus([a, b, c], [(0, 1, "normalform"), (0, 2, "normalform"), (1, 2, "normalform")],
                            veto, True, 1.4, strict)[0] == "auto_accepted"
    assert decide_consensus([a, b, c], [(0, 2, "normalform")], veto, True, 1.4, strict)[0] == "needs_review"  # no 2:1


def test_calibration_rules_with_four_engines() -> None:
    from formula_img2latex.calibrate import _summarize_consensus
    from formula_img2latex.decide import consensus_thresholds, decide_consensus
    names = ["local", "local2", "cloud", "p2t"]
    veto = {"ok": True, "metrics": {"ncc": 0.9}}

    def res(agree: list[tuple[int, int]], ok: list[bool]) -> dict:
        rs = [_reading(e, e, "x") for e in names]
        pair = "normalform" if (0, 1) in agree else None
        return {"truth": "x", "readings": rs, "ok": ok, "agree_all": [(i, j, "normalform") for i, j in agree],
                "vetos": {0: veto, 1: veto, 3: veto}, "a": rs[0], "b": rs[1], "a_ok": ok[0], "b_ok": ok[1],
                "agree": pair, "veto": veto if pair else None}

    all4 = [(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)]
    results = [res(all4, [True] * 4),                                  # everyone agrees
               res([(0, 1), (0, 3), (1, 3)], [False, False, True, False]),  # Q = G = p2t, all wrong, GLM right
               res([(0, 2), (0, 3), (2, 3)], [True, False, True, True])]    # Gemma differs: 2:1 decided by p2t
    s = _summarize_consensus("t", results, names, consensus_thresholds, decide_consensus)
    got = [(r["accepted"], r["wrong"]) for r in s["rules"]]
    # Qwen = Gemma | + GLM | B (+ p2t) | GLM or p2t confirms | A (3 of 4) | C (all four) | standard rule p2t |
    # + image arbiter (no arbiter values here: same as the standard rule)
    assert got == [(2, 1), (1, 0), (2, 1), (2, 1), (3, 1), (1, 0), (3, 1), (3, 1)]


def test_canonical_form() -> None:
    from formula_img2latex.consensus import canonical
    same = [(r"x^{2}+y^{2}", "x^2 + y^2"), (r"\vec{b}", r"\overrightarrow{b}"), (r"\left(a\right)", "(a)"),
            (r"\mathrm{d}x", "dx"), (r"\frac{1}{2}", r"\dfrac12"), (r"{\mathrm{e}}^{x}", "e^x"), (r"x\le1", r"x \leq 1"),
            (r"\mathbf{v}", r"\boldsymbol{v}"),
            # Pix2Text spellings
            (r"\operatorname{l i m}_{x \to0} \frac{\operatorname{s i n} x} {x}", r"\lim_{x\to 0}\frac{\sin x}{x}"),
            (r"\mathrm{f \ddot{u} r ~ a l l e ~} x", r"\text{für alle } x"), (r"f^{\prime\prime}(x)", "f''(x)"),
            (r"\left( \begin{matrix} {1} & {2} \\ \end{matrix} \right)", r"\begin{pmatrix}1&2\end{pmatrix}"),
            (r"\begin{array}[t]{cc}a&b\end{array}", r"\begin{array}{cc}a&b\end{array}"),
            (r"\mathop{\mathbf{rot}}\boldsymbol{v}", r"\mathbf{rot}\,\boldsymbol{v}"),
            (r"{\int}_{0}^{\infty} \! e^{-x^{2}} \, d x \!=\! \frac{\sqrt\pi} 2", r"\int_0^\infty e^{-x^2}\,dx=\frac{\sqrt{\pi}}{2}")]
    differ = [(r"x_{2,k}", r"x_{2;k}"), ("x^{2}", "x_{2}"), (r"\mathbb{R}", "R"), ("a_{n+1}", "a_n+1"),
              (r"\frac{a}{b+c}", r"\frac{a}{b}+c"), (r"x \in t", r"x \int"), (r"\sqrt[3]{x}", r"\sqrt{x}"),
              (r"\ddot{u}", "ü"), (r"\mathrm{lin}", r"\lim"), (r"f^{\prime 2}", "f'"), (r"f^{\prime}", "f")]
    # a group after a group or after \Delta is no argument (only commands in _ARITY take one)
    same += [(r"t^{2}\,{\sqrt{2}}", r"t^2\sqrt{2}"), (r"\Delta{x_{n}}", r"\Delta x_n"),
             (r"a_1+\cdot\cdot\cdot+a_n", r"a_1+\cdots+a_n"), (r"1,2,...,n", r"1,2,\ldots,n"),
             (r"\root3\of{x}", r"\sqrt[3]{x}"),
             # multi-line: same rows, any frame
             (r"\begin{aligned}a&=b\\[2mm]c&=d\\\end{aligned}", r"a=b\\c=d"),
             (r"f(x)=\begin{cases}1,&x>0\\0,&x\le0\end{cases}", r"f(x)=\left\{\begin{array}{ll}1,&x>0\\0,&x\le0\end{array}\right."),
             (r"f(x)=\begin{cases}1,&x>0\\0,&x\le0\end{cases}",
              r"f(x)=\left\{\begin{aligned}{}&{{}1,}&{x>0}\\{}&{{}0,}&{x\le0}\\\end{aligned}\right."),
             (r"\begin{pmatrix}1&2\\3&4\end{pmatrix}", r"\left(\begin{array}{cc}1&2\\3&4\end{array}\right)"),
             (r"\begin{aligned}\displaystyle a&\displaystyle=0\,.\\ \displaystyle\end{aligned}", r"a=0."),
             # pix2text's \cfrac; model readings with literal \n line breaks (JSON escape gone wrong)
             (r"\cfrac{df}{dx}=\cfrac{df}{du}\cdot\cfrac{du}{dx}", r"\frac{df}{dx}=\frac{df}{du}\cdot\frac{du}{dx}"),
             (r"\begin{aligned}\n\lim_{x\to0} f &= 1 \\\nk_2 &= 2\n\end{aligned}", r"\lim_{x\to 0}f=1\\k_2=2")]
    differ += [(r"\begin{pmatrix}1&2\end{pmatrix}", r"\begin{pmatrix}12\end{pmatrix}"), (r"a\\b\\c", r"a\\bc"),
               (r"f(x)=\begin{cases}1\\0\end{cases}", r"f(x)=\begin{aligned}1\\0\end{aligned}"),
               (r"x^{2}y", r"x^{2y}"), (r"\frac{a}{b}c", r"\frac{a}{bc}"), (r"\cdot\cdot", r"\dots"),
               (r"\nu", "u"), (r"a\neq b", "a b"), (r"a\\\nu", r"a\\u")]   # real \n commands stay
    assert [p for p in same if canonical(p[0]) != canonical(p[1])] == []
    assert [p for p in differ if canonical(p[0]) == canonical(p[1])] == []


def test_structural_equal() -> None:
    pytest.importorskip("sympy")
    pytest.importorskip("lark")
    from formula_img2latex.consensus import same_reading
    from formula_img2latex.structure import structural_equal, structural_key
    same = [(r"\frac{a}{b}", "a/b"), (r"a\cdot b", "ab"), (r"a \div b", "a/b"), (r"\frac{1}{2}\cdot x", r"\frac12 x"),
            (r"\sin(x)", r"\sin x"), (r"(a+b)c", r"(a+b)\cdot c"), (r"2\pi r", r"2 \cdot \pi \cdot r"),
            (r"f'(x)>0", r"f^{\prime}(x) > 0"),
            (r"H_0:\mu\le\mu_0\quad A:\mu>\mu_0", r"H_{0} : \mu \leq \mu_{0} \qquad A : \mu > \mu_{0}"),
            (r"\lim_{x\to0}\frac{\sin x}{x}=1", r"\lim\limits_{x \to 0} \frac{\sin(x)}{x} = 1"),
            # Göllmann: canonical() takes {\sqrt{2}} after t^{2} for a second argument
            (r"|{\dot{\boldsymbol{y}}}(t)|=3t^{2}\,{\sqrt{2}}", r"|\dot{\boldsymbol{y}}(t)|=3t^{2}\sqrt{2}")]
    differ = [(r"v=\begin{pmatrix}3\\4\end{pmatrix}", r"\nu=\begin{pmatrix}3\\4\end{pmatrix}"),   # sympy: both False
              ("1+1=2", "2+2=4"), ("2x=4", "x=2"), ("1+1", "2"), ("a+b", "b+a"), ("a<b", "b>a"),
              (r"\frac{1}{2}x", r"\frac{x}{2}"), ("a-b", "a+(-b)"), ("(a)+b", "a+b"),       # no reordering, brackets count
              (r"\varepsilon_r", r"\epsilon_r"), (r"\varepsilon_r", r"\mathcal{E}_r"), (r"\mathbb{R}", "R"),  # glyphs
              ("X'", "x'"), ("v'''", "v''"), ("y_1", "y_i"), ("x^2", "x_2"), ("a_{n+1}", "a_n+1"),
              (r"\sin(x)", "sin(x)"), (r"\exp(x)", "exp(x)"), ("e^x", r"\exp(x)"), (r"\sqrt[3]{x}", r"\sqrt{x}"),
              (r"a \times b", r"a \cdot b"), ("0{,}5", "0.5"), ("(1,2)", "(1.2)"), (r"\frac{a}{b+c}", r"\frac{a}{b}+c")]
    assert [p for p in same if not structural_equal(*p)] == []
    assert [p for p in differ if structural_equal(*p)] == []
    assert not structural_equal(r"\boldsymbol{v}", "v", bold=True) and structural_equal(r"\boldsymbol{v}", "v")
    assert structural_equal("x=1", r"x=1\tag{3}") is False
    for tex in (r"\begin{pmatrix}1\\2\end{pmatrix}", r"a \pm b", r"[a,b]", r"\int_0^1 x\,dx", "x" * 400, r"\frac{"):
        with pytest.raises(Exception):   # not covered: not comparable, the caller falls back to render
            structural_key(tex)
    assert same_reading(r"\frac{a}{b}", "a/b", False, structural=True) == "strukturell"
    assert same_reading(r"\frac{a}{b}", "a/b", False, structural=False) != "strukturell"


def test_tidy_p2t() -> None:
    pytest.importorskip("pix2tex")
    from formula_img2latex.engines.p2t import tidy_p2t
    cases = {
        r"{\frac{\alpha} {3 6 0}}={\frac{x} {2 \pi}}": r"{\frac{\alpha}{360}}={\frac{x}{2\pi}}",
        r"\frac{\mathrm{d}} {\mathrm{d} x} | x |=\frac{1} {x} .": r"\frac{\mathrm{d}}{\mathrm{d}x}|x|=\frac{1}{x}.",
        r"\left( \begin{array} {c} {-y} \\ {0} \end{array} \right)": r"\left(\begin{array}{c}{-y}\\ {0}\end{array}\right)",
        r"a \cdot x \in t": r"a\cdot x\in t",                       # space between letters stays
        r"A \cdot\mathrm{\boldmath~ v ~}=0": r"A\cdot\boldsymbol{v}=0",  # Pix2Text's bold
        r"{\mathrm{{\bf~ r o t}}} \, v": r"{\mathbf{rot}}\,v",
        r"\mathrm{\bf~ r o t} \, E": r"\mathbf{rot}\,E",
        r"\cfrac{d f}{d x}": r"\dfrac{d f}{d x}",                     # Pandoc has no \cfrac
    }
    for raw, tidy in cases.items():
        assert tidy_p2t(raw) == tidy, (raw, tidy_p2t(raw))


def test_p2t_engine_is_lazy() -> None:
    from formula_img2latex.engines import make_engine
    e = make_engine("p2t")
    assert (e.name, e.model, e._ocr) == ("p2t", "pix2text-mfr-1.5", None)   # model loads only in check()
    from formula_img2latex.engines.p2t import ort_provider
    assert ort_provider("cpu") == ("CPUExecutionProvider", {})
    assert ort_provider("cuda") == ("CUDAExecutionProvider", {"device_id": 0})
    assert ort_provider("cuda:2") == ("CUDAExecutionProvider", {"device_id": 2})


def test_logprob_confidence() -> None:
    import math

    from formula_img2latex.engines.vlm import latex_token_probs, parse_confidence
    content = '{"is_formula": true, "latex": "x^{2}", "confidence": 97}'
    toks = ['{"', "is", "_formula", '": ', "true", ', "', "latex", '": "', "x", "^{", "2", "}", '", "', "confidence",
            '": ', "97", "}"]
    items = [{"token": t, "logprob": math.log(0.5 if t == "2" else 0.99)} for t in toks]
    p_min, p_mean = latex_token_probs(content, items)
    assert p_min == 0.5 and 0.8 < p_mean < 0.9          # only the four LaTeX tokens count
    assert parse_confidence(content) == 97


def test_equation_number_only_from_chosen_reading() -> None:
    from formula_img2latex.store import Formula
    f = Formula(id="f-x-d", sha256="x", asset="a.png", display=True)
    f.candidates = [{"engine": "local", "model": "old", "tex": r"a=b \tag{3.7}"}]    # an old, hallucinated number
    assert pipeline._with_tag(f, "a=b") == "a=b"
    assert pipeline._with_tag(f, r"a=b \tag{2.1}") == r"a=b\qquad\text{(2.1)}"
    f.display = False
    assert pipeline._with_tag(f, r"a=b \tag{2.1}") == "a=b"


def test_german_decimal_comma(epub2md_dir: Path) -> None:
    import json
    source, _ws = pipeline.open_all(epub2md_dir, None, create=True)
    assert source.format_math("3,6 + (1,2)", False) == "$3,6 + (1,2)$"          # no language: unchanged
    rep = json.loads((epub2md_dir / "conversion-report.json").read_text())
    rep["metadata"] = {"language": "de"}
    (epub2md_dir / "conversion-report.json").write_text(json.dumps(rep))
    source, _ws = pipeline.open_all(epub2md_dir, None)
    assert source.format_math(r"\frac{72}{3,6} = 20, x", True) == r"$$\frac{72}{3{,}6} = 20, x$$"


def test_verify_skips_formulas_without_current_reading(epub2md_dir: Path) -> None:
    source, ws = pipeline.open_all(epub2md_dir, None, create=True)
    pipeline.scan(source, ws, quiet=True)
    f = next(f for f in ws.all() if f.asset.endswith("0001-inline.jpg"))
    f.candidates = [{"engine": "local", "model": "A", "tex": "x", "is_formula": True, "prompt": 1},   # old reading
                    {"engine": "local", "model": "A", "tex": None, "is_formula": False, "prompt": 2}]  # current: none
    ws.save(f)
    pipeline.verify_all(source, ws, preset="cm", workers=1, quiet=True)    # must not crash