Publish the verified benchmark: corrected answers, extra books, synthetic problems

ANSWER_AUDIT.md

@@ -7,24 +7,24 @@ and what the printed books got wrong.
 ## Totals
 
 - problems in the dataset: **9,229**
-- rows in the scored `verified` config: **1,942**
+- rows in the scored `verified` config: **1,827**
 - printed answers refuted by the engine (concrete counterexample): **144**
-- printed answers refuted and replaced by a re-verified recomputation: **16**
-- printed answers only *suspected* to be wrong (no concrete counterexample): **100**
-- rows set aside as `disputed` (a third read of the page disagreed): **142**
+- printed answers refuted and replaced by a re-verified recomputation: **15**
+- printed answers only *suspected* to be wrong (no concrete counterexample): **95**
+- rows set aside as `disputed` (a third read of the page disagreed): **72**
 - rows with no printed answer at all (`no-gold`): **3,164**
 
 ## By source
 
 | source | rows | verified | corrected | refuted | suspected | disputed | unverifiable | no-gold |
 |---|---|---|---|---|---|---|---|---|
-| `algebra_core` | 3,620 | 1,055 | 16 | 72 | 86 | 142 | 1,803 | 446 |
-| `falin_algebra` | 2,071 | 346 | 0 | 56 | 8 | 0 | 1,413 | 248 |
+| `algebra_core` | 3,620 | 1,153 | 15 | 101 | 75 | 64 | 1,766 | 446 |
+| `falin_algebra` | 2,071 | 459 | 0 | 29 | 12 | 7 | 1,316 | 248 |
 | `gelfand_trig` | 212 | 0 | 0 | 1 | 0 | 0 | 28 | 183 |
-| `kozko_params` | 516 | 5 | 0 | 12 | 4 | 0 | 179 | 316 |
+| `kozko_params` | 516 | 8 | 0 | 12 | 5 | 0 | 175 | 316 |
 | `ponarin_plane` | 570 | 0 | 0 | 0 | 0 | 0 | 11 | 559 |
-| `prasolov_planimetry` | 1,337 | 0 | 0 | 0 | 1 | 0 | 148 | 1,188 |
-| `shabunin` | 383 | 0 | 0 | 3 | 1 | 0 | 155 | 224 |
+| `prasolov_planimetry` | 1,337 | 1 | 0 | 0 | 1 | 0 | 147 | 1,188 |
+| `shabunin` | 383 | 0 | 0 | 1 | 2 | 1 | 155 | 224 |
 | `synthetic` | 520 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
 
 ## What each status means
@@ -38,7 +38,7 @@ and what the printed books got wrong.
 | `disputed` | a third independent read of the scanned page disagreed with the bulk transcription, so the row is not trusted |
 | `unverifiable` | the printed answer exists but is outside the reach of automatic checking - word problems, open-ended proofs, answers that depend on a figure |
 | `no-gold` | the source page carries no printed answer for this problem |
-| `synthetic-verified` | generated by a template that computes the answer with SymPy and re-checks it |
+| `synthetic-verified` | generated template; the printed statement was parsed, solved and compared with the printed answer |
 
 Neither `unverifiable`, `suspected`, `disputed` nor `no-gold` means the answer is
 wrong. They mean nobody checked it, or that the check was inconclusive.
@@ -67,19 +67,18 @@ degrees-minutes-seconds (`3^\circ45'`) are parsed as 3.75 degrees.
 | `algebra_core.2.343` | see the book | `x = - 2 \sqrt{3}` |
 | `algebra_core.2.356` | see the book | `x = - \sqrt{2}; x = \sqrt{2}` |
 | `algebra_core.5.037` | see the book | `x = - \frac{5 \pi}{6}; x = - \frac{4 \pi}{5}; x = - \frac{3 \pi}{5}; x = - \frac{\pi}{2}; x = - \frac{2 \pi}{5}; x = - \frac{\pi}{5}; x = - \frac{\pi}{6}; x = 0; x = \frac{\pi}{6}; x = \frac{\pi}{5}; x = \frac{2 \pi}{5}; x = \frac{\pi}{2}; x = \frac{3 \pi}{5}; x = \frac{4 \pi}{5}; x = \frac{5 \pi}{6}; x = \pi` |
-| `algebra_core.5.083` | see the book | `x = 0; x = \frac{\pi}{2}; x = \pi; x = \frac{3 \pi}{2}` |
 | `algebra_core.5.107` | see the book | `x = - \frac{\pi}{4}; x = \frac{3 \pi}{4}` |
 | `algebra_core.5.123` | see the book | `x = \operatorname{atan}{\left(\frac{1}{5} \right)}; x = \frac{\pi}{4}` |
 | `algebra_core.5.162` | see the book | `x = 0; x = \frac{2 \pi}{3}` |
 | `algebra_core.5.217` | see the book | `x = - \operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} \right)}; x = - \frac{\pi}{8}; x = - \operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{5}}{2} \right)}; x = \frac{3 \pi}{8}` |
-| `algebra_core.5.232` | see the book | `x = - \operatorname{atan}{\left(\frac{1}{5} + \frac{\sqrt{26}}{5} \right)}; x = - \frac{\pi}{8}; x = - \operatorname{atan}{\left(\frac{1}{5} - \frac{\sqrt{26}}{5} \right)}; x = \frac{3 \pi}{8}` |
-| `algebra_core.5.413` | see the book | `x = 0; x = \pi` |
 | `algebra_core.5.421` | see the book | `x = - \frac{\pi}{4}` |
-| `algebra_core.5.448` | see the book | `x = \frac{\pi}{2}; x = \frac{3 \pi}{2}` |
-| `algebra_core.5.449` | see the book | `x = \frac{\pi}{2}; x = \frac{3 \pi}{2}` |
+| `algebra_core.7.038` | see the book | `x = \frac{1}{100}; x = 100` |
+| `algebra_core.7.039` | see the book | `x = \frac{1}{10}; x = 1000` |
 | `algebra_core.7.096` | see the book | `x = \left(\frac{3^{\frac{2}{3}}}{3}\right)^{\frac{\sqrt[3]{3} \left(- \left(9 + \sqrt{105}\right)^{\frac{2}{3}} + 2 \sqrt[3]{3}\right)}{\sqrt[3]{9 + \sqrt{105}}}}` |
-| `algebra_core.7.230` | see the book | `x = \left(4 \sqrt{3} + 8 + 8 \sqrt{1 + \sqrt{3}}\right)^{\frac{\log{\left(2 \right)}}{\log{\left(5 \right)}}}` |
+| `algebra_core.7.242` | see the book | `x = \frac{1}{10}; x = \sqrt{10}; x = 100` |
+| `algebra_core.7.243` | see the book | `x = \frac{1}{1000}; x = 1; x = 10` |
 | `algebra_core.7.250` | see the book | `x = \frac{2^{\frac{2}{5}}}{8}; x = 1; x = 4` |
+| `algebra_core.10.122` | see the book | `x = \frac{9}{2}; x = 6` |
 
 ## Printed answers the engine refuted
 
@@ -89,136 +88,136 @@ because no model can be scored correct on them.
 
 | id | source | printed answer | what the engine found |
 |---|---|---|---|
+| `algebra_core.1.170` | `algebra_core` | `\frac{(a-2)\sqrt{a+1}}{(a+2)\sqrt{a-1}}` | third read of the answer page confirms a different printed answer, and the engine still refutes it |
+| `algebra_core.1.183` | `algebra_core` | `\sqrt{2}` | third read of the answer page confirms a different printed answer, and the engine still refutes it |
+| `algebra_core.1.191` | `algebra_core` | `\sqrt[3]{\frac{1-x}{3x}}` | third read of the answer page confirms a different printed answer, and the engine still refutes it |
+| `algebra_core.1.198` | `algebra_core` | `-1` | third read of the answer page confirms a different printed answer, and the engine still refutes it |
+| `algebra_core.1.200` | `algebra_core` | `-\sqrt{(t+2)}/(t-2)` | third read of the answer page confirms a different printed answer, and the engine still refutes it |
+| `algebra_core.1.206` | `algebra_core` | `a/(a+1)` | third read of the answer page confirms a different printed answer, and the engine still refutes it |
+| `algebra_core.1.273` | `algebra_core` | `$2\sqrt2$ for $x\in[2,4)$; $2\sqrt{x-2}$ for $x\in[4,\infty)` | third read of the answer page confirms a different printed answer, and the engine still refutes it |
+| `algebra_core.1.320` | `algebra_core` | `1)\ \frac{x+1}{x};\quad 2)\ -\frac{x+1}{x};\quad 3)\ \frac{x` | third read of the answer page confirms a different printed answer, and the engine still refutes it |
+| `algebra_core.1.336` | `algebra_core` | `\frac{4a^2+3}{9\left(a^2+1\right)}` | third read of the answer page confirms a different printed answer, and the engine still refutes it |
 | `algebra_core.2.047` | `algebra_core` | `x=8` | three independent reads agree on the statement and the printed answer |
 | `algebra_core.2.050` | `algebra_core` | `x=64` | three independent reads agree on the statement and the printed answer |
 | `algebra_core.2.059` | `algebra_core` | `x=2` | three independent reads agree on the statement and the printed answer |
 | `algebra_core.2.064` | `algebra_core` | `x_1=8,\quad x_2=27` | three independent reads agree on the statement and the printed answer |
 | `algebra_core.2.068` | `algebra_core` | `(-4;24);\ (-6;-2)` | three independent reads agree on the statement and the printed answer |
 | `algebra_core.2.070` | `algebra_core` | `(x_1;y_1)=(0,6;0,3),\quad (x_2;y_2)=(0,4;0,5)` | three independent reads agree on the statement and the printed answer |
+| `algebra_core.2.085` | `algebra_core` | `(1/2; 4)` | third read of the answer page confirms a different printed answer, and the engine still refutes it |
+| `algebra_core.2.164` | `algebra_core` | `x = -2` | third read of the answer page confirms a different printed answer, and the engine still refutes it |
 | `algebra_core.2.176` | `algebra_core` | `x=2` | three independent reads agree on the statement and the printed answer |
 | `algebra_core.2.177` | `algebra_core` | `x=0` | three independent reads agree on the statement and the printed answer |
 | `algebra_core.2.205` | `algebra_core` | `(2;1),\ (6;-3),\ \left(6+2\sqrt{3};-2-2\sqrt{3}\right),\ \le` | three independent reads agree on the statement and the printed answer |
 | `algebra_core.2.240` | `algebra_core` | `(10;1),\ \left(-21/2;53/12\right)` | three independent reads agree on the statement and the printed answer |
 | `algebra_core.2.243` | `algebra_core` | `(1;4),(4;1),(-1;-4),(-4;-1)` | three independent reads agree on the statement and the printed answer |
+| `algebra_core.2.255` | `algebra_core` | `Either $x^2+3=0$ or $x^2+5x-3=0$; the two roots of $x^2+5x-3` | third read of the answer page confirms a different printed answer, and the engine still refutes it |
+| `algebra_core.2.285` | `algebra_core` | `x_1=-1,\ x_2=1` | third read of the answer page confirms a different printed answer, and the engine still refutes it |
+| `algebra_core.2.287` | `algebra_core` | `x_1=-1,\ x_2=1` | third read of the answer page confirms a different printed answer, and the engine still refutes it |
+| `algebra_core.2.288` | `algebra_core` | `x = 1; x = 4` | third read of the answer page confirms a different printed answer, and the engine still refutes it |
+| `algebra_core.2.295` | `algebra_core` | `x=65` | third read of the answer page confirms a different printed answer, and the engine still refutes it |
+| `algebra_core.2.321` | `algebra_core` | `(2;\,-1),\ (-1;\,2),\ (-2;\,1),\ (1;\,-2)` | third read of the answer page confirms a different printed answer, and the engine still refutes it |
+| `algebra_core.2.332` | `algebra_core` | `(4;4), (4,5;3,5)` | third read of the answer page confirms a different printed answer, and the engine still refutes it |
+| `algebra_core.2.335` | `algebra_core` | `(5;3), (5;4)` | third read of the answer page confirms a different printed answer, and the engine still refutes it |
 | `algebra_core.2.342` | `algebra_core` | `x_1=1.5,\,x_2=0.5+\sqrt{3},\,x_3=0.5-\sqrt{3}` | three independent reads agree on the statement and the printed answer |
-| `algebra_core.2.355` | `algebra_core` | `x_1=\sqrt{2},\ x_2=-\sqrt{2},\ x_3=1/2` | claimed solution 'x=\\sqrt{2}' fails the equation |
+| `algebra_core.3.234` | `algebra_core` | `\sqrt[k-1]{k}` | third read of the answer page confirms a different printed answer, and the engine still refutes it |
+| `algebra_core.4.211` | `algebra_core` | `3\tan^2\alpha\cos^{-2}\alpha+1` | third read of the answer page confirms a different printed answer, and the engine still refutes it |
+| `algebra_core.4.419` | `algebra_core` | `48` | third read of the answer page confirms a different printed answer, and the engine still refutes it |
+| `algebra_core.4.458` | `algebra_core` | `11/2` | third read of the answer page confirms a different printed answer, and the engine still refutes it |
 | `algebra_core.5.002` | `algebra_core` | `x_1=35^\circ+120^\circ\cdot k,\quad x_2=55^\circ+120^\circ\c` | three independent reads agree on the statement and the printed answer |
-| `algebra_core.5.038` | `algebra_core` | `$x_1=\frac{\pi k}{3},\ x_2=\frac{\pi}{7}(2k+1),\ x=\frac{\pi` | claimed solution 'x=\\frac{\\pi k}{4}' fails the equation |
 | `algebra_core.5.039` | `algebra_core` | `$x_1=\frac{\pi k}{5},\ x_2=\frac{\pi}{8}(4k+3)$.` | three independent reads agree on the statement and the printed answer |
-| `algebra_core.5.071` | `algebra_core` | `x_1=\frac{\pi}{4}(4k+1),\ x_2=\frac{\pi}{8}(4k+1)` | three independent reads agree on the statement and the printed answer |
-| `algebra_core.5.098` | `algebra_core` | `z=\frac{\pi}{12}(6k\pm1)` | three independent reads agree on the statement and the printed answer |
-| `algebra_core.5.105` | `algebra_core` | `x=\frac{\pi}{12}(4k\pm1)` | three independent reads agree on the statement and the printed answer |
-| `algebra_core.5.111` | `algebra_core` | `x_1=\frac{\pi}{4}(4k+3),\quad x_2=\frac{\pi}{6}(6k\pm1)` | three independent reads agree on the statement and the printed answer |
-| `algebra_core.5.114` | `algebra_core` | `t=\frac{\pi}{6}\left(3k\pm1\right)` | three independent reads agree on the statement and the printed answer |
 | `algebra_core.5.121` | `algebra_core` | `z_1=2\arctan3+2\pi k,\quad z_2=-2\arctan7+2\pi k` | three independent reads agree on the statement and the printed answer |
+| `algebra_core.5.143` | `algebra_core` | `x=\pi k/8.` | third read of the answer page confirms a different printed answer, and the engine still refutes it |
 | `algebra_core.5.147` | `algebra_core` | `x=\frac{\pi}{12}(6k+1)` | three independent reads agree on the statement and the printed answer |
-| `algebra_core.5.148` | `algebra_core` | `t_1=-\frac{\pi}{4}(2k+1),\quad t_2=\pi k` | claimed solution 't=-\\frac{\\pi}{4}(2k+1)' fails the equation |
+| `algebra_core.5.167` | `algebra_core` | `x=\frac{\pi k}{2}+\frac12\arctan\left(-\frac34\right)` | third read of the answer page confirms a different printed answer, and the engine still refutes it |
 | `algebra_core.5.174` | `algebra_core` | `t=\frac{\pi}{2}(4k+1)` | three independent reads agree on the statement and the printed answer |
+| `algebra_core.5.176` | `algebra_core` | `x_1=\frac{\pi}{6}+\pi k;\quad x_2=(-1)^k\frac{\pi}{8}+\frac{` | three independent reads agree on the statement and the printed answer |
+| `algebra_core.5.190` | `algebra_core` | `t_1=\frac{\pi k}{4}\ (k\ne4l+2),\ t_2=\pm\frac12\arccos\frac` | third read of the answer page confirms a different printed answer, and the engine still refutes it |
+| `algebra_core.5.201` | `algebra_core` | `$x=45^\circ+180^\circ\cdot k$.` | third read of the answer page confirms a different printed answer, and the engine still refutes it |
+| `algebra_core.5.202` | `algebra_core` | `$x_1=\frac{\pi k}{4},\quad x_2=\frac{\pi}{12}(2k+1)$.` | third read of the answer page confirms a different printed answer, and the engine still refutes it |
+| `algebra_core.5.203` | `algebra_core` | `$t=\frac{\pi}{6}(6k\pm1)$.` | third read of the answer page confirms a different printed answer, and the engine still refutes it |
+| `algebra_core.5.207` | `algebra_core` | `$z=\frac{\pi}{6}(3k\pm1)$.` | third read of the answer page confirms a different printed answer, and the engine still refutes it |
+| `algebra_core.5.212` | `algebra_core` | `$t_1=\frac{\pi}{4}(4k+1),\quad t_2=(-1)^{k\frac{1}{2}}\arcsi` | third read of the answer page confirms a different printed answer, and the engine still refutes it |
 | `algebra_core.5.221` | `algebra_core` | `x_1=-\frac{\pi}{16}(4k+1),\quad x_2=-\frac{1}{4}\arctan\frac` | three independent reads agree on the statement and the printed answer |
 | `algebra_core.5.227` | `algebra_core` | `t=\arctan\frac{4}{5}+2\pi k` | three independent reads agree on the statement and the printed answer |
 | `algebra_core.5.259` | `algebra_core` | `x=-\frac{\pi}{8}(4k+1)` | three independent reads agree on the statement and the printed answer |
 | `algebra_core.5.269` | `algebra_core` | `x=-\frac{\pi}{4}(4k+1)` | three independent reads agree on the statement and the printed answer |
-| `algebra_core.5.275` | `algebra_core` | `z=\frac{\pi}{8}(4k-1)` | three independent reads agree on the statement and the printed answer |
-| `algebra_core.5.295` | `algebra_core` | `t=\frac{\pi}{6}(6k\pm1)` | three independent reads agree on the statement and the printed answer |
+| `algebra_core.5.274` | `algebra_core` | `x=25^{\circ}+90^{\circ}\cdot k.` | third read of the answer page confirms a different printed answer, and the engine still refutes it |
+| `algebra_core.5.280` | `algebra_core` | `x = - \operatorname{atan}{\left(3 + \sqrt{10} \right)}; x = ` | third read of the answer page confirms a different printed answer, and the engine still refutes it |
 | `algebra_core.5.334` | `algebra_core` | `t=-\frac{\pi}{4}(4k+1)` | three independent reads agree on the statement and the printed answer |
 | `algebra_core.5.348` | `algebra_core` | `t=(-1)^k\frac{k\pi}{6}+\pi k` | three independent reads agree on the statement and the printed answer |
-| `algebra_core.5.380` | `algebra_core` | `x=\frac{\pi}{4}(4k+1)` | three independent reads agree on the statement and the printed answer |
-| `algebra_core.5.433` | `algebra_core` | `x_1 = \frac{\pi}{4}(4k-1),\ x_2 = -\arctan\frac{1}{6} + \pi ` | three independent reads agree on the statement and the printed answer |
-| `algebra_core.5.434` | `algebra_core` | `x_1=\frac{2\pi}{3}(3k\pm1),\,x_2=\pm\operatorname{arccos}\le` | three independent reads agree on the statement and the printed answer |
-| `algebra_core.5.441` | `algebra_core` | `x=\frac{\pi}{6}(2k+1)` | three independent reads agree on the statement and the printed answer |
 | `algebra_core.5.445` | `algebra_core` | `t=-\frac{\pi}{4}(4k+1)` | three independent reads agree on the statement and the printed answer |
 | `algebra_core.5.472` | `algebra_core` | `t_1=\frac{\pi}{4}(8k+1),\quad t_2=-\arctan3+\pi(2k+1)` | three independent reads agree on the statement and the printed answer |
-| `algebra_core.5.492` | `algebra_core` | `x=\frac{\pi}{4}(8k+1)` | three independent reads agree on the statement and the printed answer |
+| `algebra_core.5.493` | `algebra_core` | `The original equation has no solutions.` | third read of the answer page confirms a different printed answer, and the engine still refutes it |
 | `algebra_core.7.006` | `algebra_core` | `x=\frac{5}{3}` | three independent reads agree on the statement and the printed answer |
+| `algebra_core.7.029` | `algebra_core` | `x_1=-\frac13,\quad x_2=0` | third read of the answer page confirms a different printed answer, and the engine still refutes it |
 | `algebra_core.7.037` | `algebra_core` | `x_1=1/\sqrt3,\quad x_2=3` | three independent reads agree on the statement and the printed answer |
 | `algebra_core.7.081` | `algebra_core` | `x_1=3$, $x_2=3+\sqrt{2}` | three independent reads agree on the statement and the printed answer |
-| `algebra_core.7.083` | `algebra_core` | `x_1=0$, $x_2=6` | three independent reads agree on the statement and the printed answer |
 | `algebra_core.7.089` | `algebra_core` | `x=5` | three independent reads agree on the statement and the printed answer |
-| `algebra_core.7.095` | `algebra_core` | `x=3` | claimed solution 'x=3' fails the equation |
-| `algebra_core.7.103` | `algebra_core` | `x_1=1/9$, $x_2=9` | three independent reads agree on the statement and the printed answer |
+| `algebra_core.7.091` | `algebra_core` | `x_1=33/8$, $x_2=17/4$, $x_3=8$, $x_4=12` | three independent reads agree on the statement and the printed answer |
 | `algebra_core.7.144` | `algebra_core` | `x=-2` | three independent reads agree on the statement and the printed answer |
-| `algebra_core.7.211` | `algebra_core` | `x_1=8,\ x_2=9` | three independent reads agree on the statement and the printed answer |
-| `algebra_core.7.248` | `algebra_core` | `x=2` | three independent reads agree on the statement and the printed answer |
+| `algebra_core.7.147` | `algebra_core` | `\left(2^{\sqrt[3]{\sqrt{4}}};2^{\sqrt[3]{\sqrt{2}}}\right)` | third read of the answer page confirms a different printed answer, and the engine still refutes it |
+| `algebra_core.7.160` | `algebra_core` | `(3; 27)` | three independent reads agree on the statement and the printed answer |
+| `algebra_core.7.174` | `algebra_core` | `x=4` | third read of the answer page confirms a different printed answer, and the engine still refutes it |
+| `algebra_core.7.213` | `algebra_core` | `x = \frac{3^{\frac{2}{3}}}{3}` | third read of the answer page confirms a different printed answer, and the engine still refutes it |
+| `algebra_core.7.228` | `algebra_core` | `x = 1` | third read of the answer page confirms a different printed answer, and the engine still refutes it |
+| `algebra_core.7.232` | `algebra_core` | `x = \sqrt[6]{7}` | third read of the answer page confirms a different printed answer, and the engine still refutes it |
+| `algebra_core.7.252` | `algebra_core` | `x=4` | three independent reads agree on the statement and the printed answer |
 | `algebra_core.7.264` | `algebra_core` | `x=-2\sqrt[3]{2}` | three independent reads agree on the statement and the printed answer |
-| `algebra_core.7.276` | `algebra_core` | `x=6` | three independent reads agree on the statement and the printed answer |
-| `algebra_core.7.284` | `algebra_core` | `(10;1.5),\ (0.2;75),\ (15;1)` | no assignment of the printed tuple (10; 1.5) satisfies the system |
-| `algebra_core.7.331` | `algebra_core` | `x_1=\frac{1}{8},\ x_2=\frac{1}{2}` | claimed solution 'x=\\frac{1}{8}; x=\\frac{1}{2}' fails the equation |
+| `algebra_core.7.301` | `algebra_core` | `(3; 9)` | third read of the answer page confirms a different printed answer, and the engine still refutes it |
+| `algebra_core.7.329` | `algebra_core` | `x_1=1,\ x_2=4` | third read of the answer page confirms a different printed answer, and the engine still refutes it |
 | `algebra_core.7.332` | `algebra_core` | `x=64` | three independent reads agree on the statement and the printed answer |
+| `algebra_core.7.333` | `algebra_core` | `x=3` | three independent reads agree on the statement and the printed answer |
+| `algebra_core.7.334` | `algebra_core` | `x=7` | three independent reads agree on the statement and the printed answer |
 | `algebra_core.7.337` | `algebra_core` | `x=4` | three independent reads agree on the statement and the printed answer |
 | `algebra_core.8.026` | `algebra_core` | `(-1,\infty)` | three independent reads agree on the statement and the printed answer |
-| `algebra_core.8.044` | `algebra_core` | `(-∞, 1]` | x=6.03 satisfies the inequality but is missing from the book's answer |
-| `algebra_core.8.055` | `algebra_core` | `[0, 4).` | three independent reads agree on the statement and the printed answer |
-| `algebra_core.8.070` | `algebra_core` | `(0, ∞)` | claimed point x=1 gives F=-0.334044, which violates the inequality |
-| `algebra_core.8.179` | `algebra_core` | `(-\infty;-0.1]\cup[-0.001;0)` | claimed point x=-10.1 gives F=0.116914, which violates the inequality |
+| `algebra_core.8.049` | `algebra_core` | `0 < x < 1` | third read of the answer page confirms a different printed answer, and the engine still refutes it |
+| `algebra_core.8.053` | `algebra_core` | `(3; \infty).` | third read of the answer page confirms a different printed answer, and the engine still refutes it |
+| `algebra_core.8.138` | `algebra_core` | `(1,2\sqrt{3}]` | third read of the answer page confirms a different printed answer, and the engine still refutes it |
+| `algebra_core.8.145` | `algebra_core` | `$(2,\infty)$.` | third read of the answer page confirms a different printed answer, and the engine still refutes it |
+| `algebra_core.8.146` | `algebra_core` | `$(1,2)\cup(64,\infty)$.` | third read of the answer page confirms a different printed answer, and the engine still refutes it |
+| `algebra_core.8.151` | `algebra_core` | `(-1/3, \infty)` | third read of the answer page confirms a different printed answer, and the engine still refutes it |
+| `algebra_core.8.170` | `algebra_core` | `(1, 3) ∪ (3^9, \infty)` | third read of the answer page confirms a different printed answer, and the engine still refutes it |
+| `algebra_core.8.175` | `algebra_core` | `(0, 1/3) ∪ (243, \infty)` | third read of the answer page confirms a different printed answer, and the engine still refutes it |
 | `algebra_core.8.183` | `algebra_core` | `(-\sqrt2,-1)\cup(1,\sqrt2)` | three independent reads agree on the statement and the printed answer |
 | `algebra_core.8.187` | `algebra_core` | `(1,1+2^{-5/4})\cup(3;\infty)` | three independent reads agree on the statement and the printed answer |
-| `algebra_core.8.234` | `algebra_core` | `$(0,\sqrt{5}/5)\cup(1,3)$.` | three independent reads agree on the statement and the printed answer |
 | `algebra_core.8.238` | `algebra_core` | `(-\infty,-2]\cup[-1,(\sqrt{13}-1)/6)` | three independent reads agree on the statement and the printed answer |
 | `algebra_core.9.031` | `algebra_core` | `x=10` | three independent reads agree on the statement and the printed answer |
 | `algebra_core.9.032` | `algebra_core` | `x=7` | three independent reads agree on the statement and the printed answer |
 | `algebra_core.10.012` | `algebra_core` | `(-3,2)\cup(2,3)` | three independent reads agree on the statement and the printed answer |
-| `algebra_core.10.120` | `algebra_core` | `x=\frac{1}{3}` | claimed solution 'x=\\frac{1}{3}' fails the equation |
+| `algebra_core.10.070` | `algebra_core` | `x=-1` | third read of the answer page confirms a different printed answer, and the engine still refutes it |
+| `algebra_core.10.072` | `algebra_core` | `x=2,\ y=9` | third read of the answer page confirms a different printed answer, and the engine still refutes it |
+| `algebra_core.10.097` | `algebra_core` | `The equation has no roots.` | third read of the answer page confirms a different printed answer, and the engine still refutes it |
 | `algebra_core.10.166` | `algebra_core` | `(-4,0)\cup(0,4)` | three independent reads agree on the statement and the printed answer |
-| `algebra_core.10.170` | `algebra_core` | `(-2,0)\cup(2,\infty)` | claimed point x=3 gives F=-0.0203487, which violates the inequality |
-| `algebra_core.11.246` | `algebra_core` | `-\frac{1}{3}` | hand-checked with an independent engine: hand-checked: \int_{2pi/3}^{-pi} sin(pi/3-3x) dx = -1/3 exactly (inde |
+| `algebra_core.10.310` | `algebra_core` | `\tan\alpha=2\tan\beta` | third read of the answer page confirms a different printed answer, and the engine still refutes it |
 | `falin_algebra.241` | `falin_algebra` | `$x_1 = 4; x_2 = -2$.` | claimed solution 'x=4' fails the equation |
-| `falin_algebra.278` | `falin_algebra` | `n = 2011` | claimed solution 'n=2011' fails the equation |
-| `falin_algebra.286` | `falin_algebra` | `$x_1 = \frac{7}{15};\ x_2 = \frac{4}{5}$.` | claimed solution 'x=\\frac{7}{15}' fails the equation |
-| `falin_algebra.287` | `falin_algebra` | `$x_1 = -\sqrt{7},\ x_2 = \sqrt{3}$.` | claimed solution 'x=-\\sqrt{7}' fails the equation |
-| `falin_algebra.288` | `falin_algebra` | `$x = \sqrt[3]{4}$.` | claimed solution 'x=\\sqrt[3]{4}' fails the equation |
-| `falin_algebra.292` | `falin_algebra` | `$x_1 = 0;\ x_2 = \sqrt[3]{2}$.` | claimed solution 'x=0' fails the equation |
-| `falin_algebra.392` | `falin_algebra` | `x = 3 \pm \sqrt{5}` | claimed solution 'x=3 - \\sqrt{5}' fails the equation |
-| `falin_algebra.401` | `falin_algebra` | `$x = \log_{\frac{2}{3}} \log_3 2$.` | claimed solution 'x=\\log_{\\frac{2}{3}} \\log_3 2' fails the equation |
-| `falin_algebra.402` | `falin_algebra` | `$x = \log_{\frac{7}{5}} (\log_5 7)$.` | claimed solution 'x=\\log_{\\frac{7}{5}} (\\log_5 7)' fails the equation |
-| `falin_algebra.423` | `falin_algebra` | `x_1 = 2,\ x_2 = \frac{1}{2} \log_2 (12 - \sqrt{112})` | claimed solution 'x=\\frac{1}{2} \\log_2 (12 - \\sqrt{112})' fails the equation |
-| `falin_algebra.425` | `falin_algebra` | `$x = 5$.` | claimed solution 'x=5' fails the equation |
+| `falin_algebra.392` | `falin_algebra` | `$x=3\pm\sqrt{5}$.` | claimed solution 'x=3 - \\sqrt{5}' fails the equation |
+| `falin_algebra.401` | `falin_algebra` | `$x=\log_{\frac{2}{3}}\log_3 2$.` | claimed solution 'x=\\log_{\\frac{2}{3}} \\log_3 2' fails the equation |
+| `falin_algebra.402` | `falin_algebra` | `$x=\log_{\frac{7}{5}}(\log_5 7)$.` | claimed solution 'x=\\log_{\\frac{7}{5}} (\\log_5 7)' fails the equation |
+| `falin_algebra.423` | `falin_algebra` | `$x_1 = 2$, $x_2 = \frac{1}{2}\log_2(12 - \sqrt{112})$.` | claimed solution 'x=\\frac{1}{2} \\log_2 (12 - \\sqrt{112})' fails the equation |
 | `falin_algebra.452` | `falin_algebra` | `x = \log_{\frac{7}{10}} 2` | claimed solution 'x=\\log_{\\frac{7}{10}} 2' fails the equation |
 | `falin_algebra.453` | `falin_algebra` | `x = \log_{\frac{3}{2}} 2` | claimed solution 'x=\\log_{\\frac{3}{2}} 2' fails the equation |
-| `falin_algebra.454` | `falin_algebra` | `x_1 = \frac{1}{2\sqrt{3}(\log_3 5 - 1)}; x_2 = \frac{4\log_3` | claimed solution 'x=\\frac{1}{2\\sqrt{3}(\\log_3 5 - 1)}' fails the equation |
-| `falin_algebra.456` | `falin_algebra` | `x_1 = 0; x_2 = \log_3 2` | claimed solution 'x=\\log_3 2' fails the equation |
-| `falin_algebra.457` | `falin_algebra` | `x_1 = \log_2 9; x_2 = \log_3 2` | claimed solution 'x=\\log_2 9' fails the equation |
-| `falin_algebra.500` | `falin_algebra` | `x_1 = 1; x_2 = -\frac{15}{11}` | claimed solution 'x=1' fails the equation |
-| `falin_algebra.501` | `falin_algebra` | `x = 3\sqrt{3}` | claimed solution 'x=3\\sqrt{3}' fails the equation |
-| `falin_algebra.515` | `falin_algebra` | `x_1 = 729, x_2 = \frac{1}{6561}` | claimed solution 'x=729' fails the equation |
-| `falin_algebra.524` | `falin_algebra` | `$x_{1} = 1; \ x_{2} = 2 - \sqrt{10}$.` | claimed solution 'x=1' fails the equation |
-| `falin_algebra.528` | `falin_algebra` | `$x = \frac{1+\sqrt{17}}{2}$.` | claimed solution 'x=\\frac{1+\\sqrt{17}}{2}' fails the equation |
-| `falin_algebra.550` | `falin_algebra` | `x_1 = (\sqrt{2}+\sqrt{3}+\sqrt{5})^2;\ x_2 = (\sqrt{2}+\sqrt` | claimed solution 'x=(\\sqrt{2}+\\sqrt{3}+\\sqrt{5})^2' fails the equation |
-| `falin_algebra.555` | `falin_algebra` | `x = 2^{4-4\sqrt{2}}` | claimed solution 'x=2^{4-4\\sqrt{2}}' fails the equation |
-| `falin_algebra.566` | `falin_algebra` | `x = 1 \pm \sqrt{11+4\sqrt{3}}.` | claimed solution 'x=1 + \\sqrt{11+4\\sqrt{3}}' fails the equation |
-| `falin_algebra.586` | `falin_algebra` | `$x = 0$.` | claimed solution 'x=0' fails the equation |
-| `falin_algebra.620` | `falin_algebra` | `(-\infty; -2] \cup (0; 1/4].` | claimed point x=-12 gives F=0.433333, which violates the inequality |
-| `falin_algebra.755` | `falin_algebra` | `(-\sqrt{5}; -2) \cup (-1; 0) \cup (0; 1) \cup (2; \sqrt{5}).` | claimed point x=-2.22426 gives F=1.76045, which violates the inequality |
-| `falin_algebra.793` | `falin_algebra` | `$(-\sqrt{5};-2)\cup(-1;0)\cup(0;1)\cup(2;\sqrt{5})$.` | claimed point x=0.3 gives F=0.221184, which violates the inequality |
-| `falin_algebra.813` | `falin_algebra` | `$(3; 1 + \sqrt{5}) \cup [10/3; +\infty)$.` | claimed point x=3.12984 gives F=-0.226829, which violates the inequality |
-| `falin_algebra.894` | `falin_algebra` | `(1; 2].` | claimed point x=1.7 gives F=0.139169, which violates the inequality |
-| `falin_algebra.900` | `falin_algebra` | `$(-\infty; -2] \cup [-1; 0) \cup (0; 1] \cup [2; +\infty)$.` | x=-1.505 satisfies the inequality but is missing from the book's answer |
-| `falin_algebra.923` | `falin_algebra` | `(-\infty; -\sqrt{7}) \cup (\sqrt{7}; 9]` | x=0 satisfies the inequality but is missing from the book's answer |
-| `falin_algebra.957` | `falin_algebra` | `(2 - \log_2^2 3; 2]` | claimed point t=1.62318 gives F=0.663795, which violates the inequality |
-| `falin_algebra.989` | `falin_algebra` | `$(-\infty; -1999] \cup [1995; +\infty)$.` | claimed point x=-1.002e+06 gives F=-1.002, which violates the inequality |
-| `falin_algebra.1069` | `falin_algebra` | `\left(\dfrac{5}{3}; -\dfrac{7}{3}\right)` | no assignment of the printed tuple (\dfrac{5}{3};  -\dfrac{7}{3}\right) satisfies the system |
-| `falin_algebra.1071` | `falin_algebra` | `(-2; -1)` | no assignment of the printed tuple (-2;  -1) satisfies the system |
-| `falin_algebra.1076` | `falin_algebra` | `(2; 3)` | no assignment of the printed tuple (2;  3) satisfies the system |
-| `falin_algebra.1119` | `falin_algebra` | `(1/2; \log_3 3\sqrt{2})` | no assignment of the printed tuple (1/2;  \log_3 3\sqrt{2}) satisfies the system |
-| `falin_algebra.1123` | `falin_algebra` | `\{(8\log_5 2 - 6\log_2 5; 8\log_2 5 - 12\log_5 2)\}` | no assignment of the printed tuple (8\log_5 2 - 6\log_2 5;  8\log_2 5 - 12\log_5 2) satisfies the system |
-| `falin_algebra.1124` | `falin_algebra` | `\left( \dfrac{1+\sqrt{13}}{2}; \dfrac{-5+\sqrt{13}}{4} \righ` | no assignment of the printed tuple ( \dfrac{1+\sqrt{13}}{2};  \dfrac{-5+\sqrt{13}}{4} \right) satisfies the sy |
-| `falin_algebra.1127` | `falin_algebra` | `\left( \dfrac{1}{3};\ \dfrac{1}{2} \right)` | no assignment of the printed tuple ( \dfrac{1}{3}; \ \dfrac{1}{2} \right) satisfies the system |
-| `falin_algebra.1287` | `falin_algebra` | `$x = 2$, $y = 1$.` | claimed solution 'x=2' fails the equation |
-| `falin_algebra.1306` | `falin_algebra` | `$x = 0$, $y = 0$.` | claimed solution 'x=0' fails the equation |
+| `falin_algebra.454` | `falin_algebra` | `x_1 = \frac{1}{2\sqrt{3}(\log_3 5 - 1)}; \; x_2 = \frac{4\lo` | claimed solution 'x=\\frac{1}{2\\sqrt{3}(\\log_3 5 - 1)}' fails the equation |
+| `falin_algebra.456` | `falin_algebra` | `x_1 = 0; \; x_2 = \log_3 2` | claimed solution 'x=\\log_3 2' fails the equation |
+| `falin_algebra.457` | `falin_algebra` | `x_1 = \log_2 9; \; x_2 = \log_3 2` | claimed solution 'x=\\log_2 9' fails the equation |
+| `falin_algebra.501` | `falin_algebra` | `x = 3\sqrt{3}.` | claimed solution 'x=3\\sqrt{3}' fails the equation |
+| `falin_algebra.528` | `falin_algebra` | `$x=\frac{1+\sqrt{17}}{2}$.` | claimed solution 'x=\\frac{1+\\sqrt{17}}{2}' fails the equation |
+| `falin_algebra.550` | `falin_algebra` | `x_1=(\sqrt{2}+\sqrt{3}+\sqrt{5})^2;\ x_2=(\sqrt{2}+\sqrt{3}+` | claimed solution 'x=(\\sqrt{2}+\\sqrt{3}+\\sqrt{5})^2' fails the equation |
+| `falin_algebra.566` | `falin_algebra` | `x = 1 \pm \sqrt{11 + 4\sqrt{3}}` | claimed solution 'x=1 + \\sqrt{11+4\\sqrt{3}}' fails the equation |
+| `falin_algebra.900` | `falin_algebra` | `(-\infty;-2]\cup[-1;0)\cup(0;1]\cup[2;+\infty)` | x=-1.505 satisfies the inequality but is missing from the book's answer |
+| `falin_algebra.914` | `falin_algebra` | `x\leq 0` | x=0.275 satisfies the inequality but is missing from the book's answer |
+| `falin_algebra.1069` | `falin_algebra` | `\left(\frac{5}{3};-\frac{7}{3}\right)` | no assignment of the printed tuple (\dfrac{5}{3};  -\dfrac{7}{3}\right) satisfies the system |
+| `falin_algebra.1123` | `falin_algebra` | `\{(8\log_5 2-6\log_2 5;\,8\log_2 5-12\log_5 2)\}` | no assignment of the printed tuple (8\log_5 2 - 6\log_2 5;  8\log_2 5 - 12\log_5 2) satisfies the system |
+| `falin_algebra.1287` | `falin_algebra` | `$x=2, y=1$.` | claimed solution 'x=2' fails the equation |
+| `falin_algebra.1306` | `falin_algebra` | `x = 0,\ y = 0` | claimed solution 'x=0' fails the equation |
 | `falin_algebra.1432` | `falin_algebra` | `$x = 0$; $y = -1$.` | claimed solution 'x=0' fails the equation |
 | `falin_algebra.1437` | `falin_algebra` | `$k = 6$; $l = 5$.` | claimed solution 'k=6' fails the equation |
 | `falin_algebra.1555` | `falin_algebra` | `x = \frac{-99 \pm \sqrt{9405}}{2}` | claimed solution 'x=\\frac{-99 + \\sqrt{9405}}{2}' fails the equation |
 | `falin_algebra.1556` | `falin_algebra` | `x = 1` | claimed solution 'x=1' fails the equation |
-| `falin_algebra.1924` | `falin_algebra` | `a_1 = 2\pi - 8$, $a_2 = 2\pi$, $a_3 = 2\pi - 1` | claimed solution 'a=2\\pi - 8; a=2\\pi; a=2\\pi - 1' fails the equation |
-| `falin_algebra.1927` | `falin_algebra` | `a_1 = 1 + \log_5 \frac{8 + \sqrt{55}}{3}$, $a_2 = 1 + \log_5` | claimed solution 'a=1 + \\log_5 \\frac{8 + \\sqrt{55}}{3}; a=1 + \\log_5 \\frac{8 -' fails the equation |
-| `falin_algebra.2000` | `falin_algebra` | `$a_1 = -1,\ a_2 = -3,\ a_3 = -\dfrac{5}{7}$.` | claimed solution 'a=-1; a=-3; a=-\\dfrac{5}{7}' fails the equation |
-| `falin_algebra.2010` | `falin_algebra` | `$p = -\frac{1}{2}$.` | claimed solution 'p=-\\frac{1}{2}' fails the equation |
-| `falin_algebra.2017` | `falin_algebra` | `a = \frac{\sqrt{3}}{2}` | claimed solution 'a=\\frac{\\sqrt{3}}{2}' fails the equation |
-| `falin_algebra.2020` | `falin_algebra` | `a = \frac{2+2\sqrt{13}}{3}$ or $-1 < a \le 0` | claimed solution 'a=\\frac{2+2\\sqrt{13}}{3} or -1 < a \\le 0' fails the equation |
+| `falin_algebra.1927` | `falin_algebra` | `a_1 = 1 + \log_5 \frac{8+\sqrt{55}}{3}$, $a_2 = 1 + \log_5 \` | claimed solution 'a=1 + \\log_5 \\frac{8 + \\sqrt{55}}{3}; a=1 + \\log_5 \\frac{8 -' fails the equation |
+| `falin_algebra.2000` | `falin_algebra` | `a_1=-1,\ a_2=-3,\ a_3=-\frac{5}{7}` | claimed solution 'a=-1; a=-3; a=-\\dfrac{5}{7}' fails the equation |
+| `falin_algebra.2010` | `falin_algebra` | `p=-\frac{1}{2}` | claimed solution 'p=-\\frac{1}{2}' fails the equation |
 | `falin_algebra.2068` | `falin_algebra` | `$a = -\frac{9}{82}$.` | claimed solution 'a=-\\frac{9}{82}' fails the equation |
-| `falin_algebra.2069` | `falin_algebra` | `$p = 114$, $p = -6$.` | claimed solution 'p=114; p=-6' fails the equation |
-| `falin_algebra.2073` | `falin_algebra` | `a = -2` | claimed solution 'a=-2' fails the equation |
-| `shabunin.226` | `shabunin` | `(3;9),\ (9;3)` | x=1.47 satisfies the inequality but is missing from the book's answer |
-| `shabunin.Example` | `shabunin` | `$(2;1)$, $(-2;-1)$, $\left(\sqrt{213};-\frac{9}{\sqrt{213}}\` | no assignment of the printed tuple (2; 1) satisfies the system |
+| `falin_algebra.2069` | `falin_algebra` | `$p = 114,\ p = -6$.` | claimed solution 'p=114; p=-6' fails the equation |
 | `shabunin.Пример 8` | `shabunin` | `$(2;\sqrt[3]{3}; \sqrt[3]{9}),\ \left(\frac{1}{\sqrt[3]{2}};` | no assignment of the printed tuple (2; \sqrt[3]{3};  \sqrt[3]{9}) satisfies the system |
 | `kozko_params.50` | `kozko_params` | `$a = \pm 1$, $b = 0$; $a = 0$, $b = 0$.` | claimed solution 'a=+ 1; a=0' fails the equation |
 | `kozko_params.121` | `kozko_params` | `a = \pm 1/6, \, a = \pm \sqrt{2}/6` | claimed solution 'a=+ 1/6; a=+ \\sqrt{2}/6' fails the equation |
@@ -229,10 +228,10 @@ because no model can be scored correct on them.
 | `kozko_params.225` | `kozko_params` | `x=2$, $y=1` | claimed solution 'x=2' fails the equation |
 | `kozko_params.260` | `kozko_params` | `a = -2` | claimed solution 'a=-2' fails the equation |
 | `kozko_params.271` | `kozko_params` | `x = y = 0` | claimed solution 'x=y = 0' fails the equation |
-| `kozko_params.2003-2-3` | `kozko_params` | `[3/7; 1) \cup (3; +\infty)` | claimed point x=0.457143 gives F=-1.11111, which violates the inequality |
+| `kozko_params.2003-2-3` | `kozko_params` | `[3/7; 1) \cup (3; +\infty)` | claimed endpoint 0.428571 gives F=-1.1818182, which violates the inequality |
 | `kozko_params.Задача 1 (Вариант 2006, апрель)` | `kozko_params` | `[-2; (7+\sqrt{21})/21)` | claimed point x=6.14955 gives F=-0.294805, which violates the inequality |
-| `kozko_params.ВАРИАНТ 2004 (июль), социологический факультет, 1` | `kozko_params` | `$[-5; -3]$.` | claimed point x=-4.9 gives F=0.643958, which violates the inequality |
-| `gelfand_trig.24.14` | `gelfand_trig` | `(x; y) = \left(\frac{\pi}{2} + \pi(k+n);\, -\frac{\pi}{2} + ` | no assignment of the printed tuple (x;  y) satisfies the system |
+| `kozko_params.ВАРИАНТ 2004 (июль), социологический факультет, 1` | `kozko_params` | `$[-5; -3]$.` | claimed endpoint -5 gives F=0.66666667, which violates the inequality |
+| `gelfand_trig.24.14` | `gelfand_trig` | `(x;y) = \left(\frac{\pi}{2} + \pi(k+n); -\frac{\pi}{2} + \pi` | no assignment of the printed tuple (x;  y) satisfies the system |
 
 ## Printed answers only suspected to be wrong
 
@@ -247,7 +246,6 @@ out of the scored split as well.
 | `algebra_core.1.023` | `algebra_core` | `0.5a^3(a-1)` | numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from  |
 | `algebra_core.1.027` | `algebra_core` | `-4` | numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from  |
 | `algebra_core.1.028` | `algebra_core` | `\frac{2}{\sqrt[3]{r^2}}\left(r^2-\sqrt{r^4-16}\right)` | numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from  |
-| `algebra_core.1.031` | `algebra_core` | `\frac{\sqrt[3]{8-t^3}}{\sqrt{2}}` | numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from  |
 | `algebra_core.1.039` | `algebra_core` | `y^2` | numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from  |
 | `algebra_core.1.041` | `algebra_core` | `-4` | numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from  |
 | `algebra_core.1.043` | `algebra_core` | `x+1` | numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from  |
@@ -257,7 +255,6 @@ out of the scored split as well.
 | `algebra_core.1.059` | `algebra_core` | `-1` | numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from  |
 | `algebra_core.1.063` | `algebra_core` | `a^{1/3}+b^{1/3}` | numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from  |
 | `algebra_core.1.067` | `algebra_core` | `1` | numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from  |
-| `algebra_core.1.068` | `algebra_core` | `\frac{1}{a\left(\sqrt[m]{a}-\sqrt[n]{a}\right)}` | numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from  |
 | `algebra_core.1.069` | `algebra_core` | `6` | numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from  |
 | `algebra_core.1.080` | `algebra_core` | `0` | numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from  |
 | `algebra_core.1.084` | `algebra_core` | `4p-\sqrt{4p^2-1}` | numeric difference over 3 point(s) - reported as suspected, not as a refutation: the difference can come from  |
@@ -265,7 +262,7 @@ out of the scored split as well.
 | `algebra_core.1.086` | `algebra_core` | `\frac{1}{\sqrt{a}+\sqrt{2}}` | numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from  |
 | `algebra_core.1.089` | `algebra_core` | `(1-a)/\sqrt{a}` | numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from  |
 | `algebra_core.1.098` | `algebra_core` | `\sqrt[3]{x+y}-\sqrt[3]{x-y}` | numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from  |
-| `algebra_core.1.100` | `algebra_core` | `x+y` | numeric difference over 3 point(s) - reported as suspected, not as a refutation: the difference can come from  |
+| `algebra_core.1.100` | `algebra_core` | `x+y` | numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from  |
 | `algebra_core.1.101` | `algebra_core` | `2` | numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from  |
 | `algebra_core.1.120` | `algebra_core` | `3/5` | numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from  |
 | `algebra_core.1.121` | `algebra_core` | `31/3` | numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from  |
@@ -278,16 +275,13 @@ out of the scored split as well.
 | `algebra_core.1.142` | `algebra_core` | `-\frac{\sqrt{6}}{2}` | numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from  |
 | `algebra_core.1.161` | `algebra_core` | `1) $\frac{x^2+5x-3}{x^4}$; 2) $\frac{(x+2)(x+3)}{x^4}$; 3) $` | numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from  |
 | `algebra_core.1.184` | `algebra_core` | `\frac{1}{x-\sqrt{2x+1}}` | numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from  |
-| `algebra_core.1.204` | `algebra_core` | `1/({}^{4}\!\sqrt a-1)` | numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from  |
 | `algebra_core.1.213` | `algebra_core` | `\frac{z^2}{z^2+z+1}` | numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from  |
 | `algebra_core.1.217` | `algebra_core` | `\sqrt{a^2-1}` | numeric difference over 4 point(s) - reported as suspected, not as a refutation: the difference can come from  |
 | `algebra_core.1.222` | `algebra_core` | `-\frac{\sqrt{x-2}}{x+2}` | numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from  |
 | `algebra_core.1.224` | `algebra_core` | `\sqrt{a}\left(\sqrt{2a+1}-\sqrt{a}\right)` | numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from  |
 | `algebra_core.1.237` | `algebra_core` | `2` | numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from  |
-| `algebra_core.1.240` | `algebra_core` | `\sqrt{a}(a+4b)` | numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from  |
 | `algebra_core.1.241` | `algebra_core` | `1` | numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from  |
 | `algebra_core.1.286` | `algebra_core` | `m/2` | numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from  |
-| `algebra_core.1.321` | `algebra_core` | `1)\;\frac{2\sqrt{2}}{4-x};\quad 2)\;\frac{2\sqrt{x-2}}{x-4}.` | numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from  |
 | `algebra_core.1.322` | `algebra_core` | `1) $m-n$; 2) $n-m$; 3) $m-n$` | numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from  |
 | `algebra_core.1.323` | `algebra_core` | `2x^2-a^2$ if $x<-\|a\|$; $-a^2$ if $x>\|a\|` | branch '2x^2-a^2' differs numerically - reported as suspected, not as a refutation: the difference can come fr |
 | `algebra_core.1.324` | `algebra_core` | `b-4a$ if $a\in[0,\sqrt{2})$; $2(a-1)^2$ if $a\in[\sqrt{2},+\` | branch '2(a-1)^2' differs numerically - reported as suspected, not as a refutation: the difference can come fr |
@@ -297,22 +291,15 @@ out of the scored split as well.
 | `algebra_core.1.330` | `algebra_core` | `\frac{x-1}{x}$ if $x\in(-\infty,0)\cup[1,\infty)$; $\frac{1-` | branch '\\frac{x-1}{x}' differs numerically - reported as suspected, not as a refutation: the difference can c |
 | `algebra_core.1.331` | `algebra_core` | `-\frac{z+1}{z}$ if $z\in(-\infty,-1)\cup(0,1)$; $\frac{z+1}{` | branch '-\\frac{z+1}{z}' differs numerically - reported as suspected, not as a refutation: the difference can  |
 | `algebra_core.2.368` | `algebra_core` | `x=2` | the two sides of the identity differ at every tested point - reported as suspected, not as a refutation: the d |
-| `algebra_core.4.067` | `algebra_core` | `2\sin\alpha` | numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from  |
-| `algebra_core.4.068` | `algebra_core` | `\sin 2\alpha` | numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from  |
 | `algebra_core.4.080` | `algebra_core` | `\tan\frac{m+n}{2}\alpha` | numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from  |
-| `algebra_core.4.081` | `algebra_core` | `2` | numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from  |
-| `algebra_core.4.106` | `algebra_core` | `\cot2\alpha` | numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from  |
 | `algebra_core.4.107` | `algebra_core` | `\tan4\alpha` | numeric difference over 5 point(s) - reported as suspected, not as a refutation: the difference can come from  |
-| `algebra_core.4.116` | `algebra_core` | `-16\cot2\alpha\sin^{-3}2\alpha` | numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from  |
 | `algebra_core.4.117` | `algebra_core` | `\tan^6\alpha` | numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from  |
-| `algebra_core.4.120` | `algebra_core` | `4\cos 2\alpha\sin^{-2}2\alpha` | numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from  |
-| `algebra_core.4.131` | `algebra_core` | `2\sqrt{2}\cos 2\alpha\cos\left(\frac{\pi}{4}-2\alpha\right)\` | numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from  |
 | `algebra_core.4.257` | `algebra_core` | `\tan^4\frac{\alpha}{2}` | numeric difference over 5 point(s) - reported as suspected, not as a refutation: the difference can come from  |
-| `algebra_core.4.258` | `algebra_core` | `2\sin\left(2\alpha-\frac{\pi}{6}\right)` | numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from  |
 | `algebra_core.4.300` | `algebra_core` | `\frac{4\sin^2\left(\frac{\pi}{4}-4\alpha\right)}{\sin8\alpha` | numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from  |
 | `algebra_core.4.381` | `algebra_core` | `a) $(\cos 2\alpha)_1=-3/5$; b) $(\cos 2\alpha)_2=4/5$` | sides differ numerically - reported as suspected, not as a refutation: the difference can come from the readin |
 | `algebra_core.4.382` | `algebra_core` | `a) $4/5$; b) $3/5$` | sides differ numerically - reported as suspected, not as a refutation: the difference can come from the readin |
 | `algebra_core.4.412` | `algebra_core` | `-\cos^32x` | numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from  |
+| `algebra_core.4.488` | `algebra_core` | `A=B=\|\cos\alpha\cos\beta\cos\gamma\|` | the two sides of the identity differ at every tested point - reported as suspected, not as a refutation: the d |
 | `algebra_core.5.154` | `algebra_core` | `x = \frac{2\pi}{3}(3k \pm 1)` | the two sides of the identity differ at every tested point - reported as suspected, not as a refutation: the d |
 | `algebra_core.5.155` | `algebra_core` | `x_1=\frac{\pi}{2}(2k+1),\quad x_2=(-1)^k\frac{\pi}{6}+\pi k` | the two sides of the identity differ at every tested point - reported as suspected, not as a refutation: the d |
 | `algebra_core.5.156` | `algebra_core` | `t_1=(-1)^k\frac{\pi}{6}+\pi k,\quad t_2=\frac{\pi}{2}(4k+1)` | the two sides of the identity differ at every tested point - reported as suspected, not as a refutation: the d |
@@ -324,25 +311,32 @@ out of the scored split as well.
 | `algebra_core.5.302` | `algebra_core` | `t_1=90^\circ\cdot k;\quad t_2=\pm15^\circ+90^\circ\cdot k` | the two sides of the identity differ at every tested point - reported as suspected, not as a refutation: the d |
 | `algebra_core.5.311` | `algebra_core` | `$x=2\pi k$.` | the two sides of the identity differ at every tested point - reported as suspected, not as a refutation: the d |
 | `algebra_core.5.337` | `algebra_core` | `t=\pi k` | the two sides of the identity differ at every tested point - reported as suspected, not as a refutation: the d |
-| `algebra_core.5.343` | `algebra_core` | `t=(-1)^k\frac{\pi}{12}+\frac{\pi k}{2}` | the two sides of the identity differ at every tested point - reported as suspected, not as a refutation: the d |
 | `algebra_core.5.467` | `algebra_core` | `x=\frac{\pi}{2}(4k+1)` | the two sides of the identity differ at every tested point - reported as suspected, not as a refutation: the d |
 | `algebra_core.5.490` | `algebra_core` | `x_1=\arccos\left(\sqrt5-2\right)+2\pi k,\quad x_2=\pi(2k+1).` | the two sides of the identity differ at every tested point - reported as suspected, not as a refutation: the d |
 | `algebra_core.7.045` | `algebra_core` | `890` | numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from  |
 | `algebra_core.7.201` | `algebra_core` | `0` | numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from  |
+| `algebra_core.8.055` | `algebra_core` | `[0, 4).` | the book's answer excludes the upper endpoint 4, but it satisfies the inequality - reported as suspected, not  |
 | `algebra_core.10.145` | `algebra_core` | `$x$ is an integer.` | the two sides of the identity differ at every tested point - reported as suspected, not as a refutation: the d |
-| `falin_algebra.136` | `falin_algebra` | `6` | value differs ((0.4254921071067263+0j) vs (6+0j)) |
+| `algebra_core.11.078` | `algebra_core` | `a>4` | the book's answer excludes the lower endpoint 4, but it satisfies the inequality - reported as suspected, not  |
 | `falin_algebra.269` | `falin_algebra` | `x = \dfrac{-3 + \sqrt{17}}{2}.` | sides differ numerically |
 | `falin_algebra.387` | `falin_algebra` | `x = 2` | sides differ numerically |
 | `falin_algebra.540` | `falin_algebra` | `$\{\tfrac{2}{5};\, 10\}$.` | the two sides of the identity differ at every tested point |
+| `falin_algebra.788` | `falin_algebra` | `$-1 < x < -\frac{5}{6}$.` | the book's answer excludes the upper endpoint -0.833333, but it satisfies the inequality |
+| `falin_algebra.907` | `falin_algebra` | `x>\sqrt{3}` | claimed point x=2.73205 gives F=-4.81664, which violates the inequality |
+| `falin_algebra.923` | `falin_algebra` | `(-\infty; -\sqrt{7}) \cup (\sqrt{7}; 9]` | the book's answer excludes the upper endpoint -2.64575, but it satisfies the inequality |
+| `falin_algebra.926` | `falin_algebra` | `[-18;-3] \cup (0;2).` | the book's answer includes the upper endpoint -3, where equality holds |
+| `falin_algebra.957` | `falin_algebra` | `(2 - \log_2^2 3; 2]` | the book's answer excludes the lower endpoint -0.512106, but it satisfies the inequality |
 | `falin_algebra.980` | `falin_algebra` | `$3$.` | integer 2 satisfies the inequality but is not listed |
 | `falin_algebra.1023` | `falin_algebra` | `4017` | integer 4014 satisfies the inequality but is not listed |
 | `falin_algebra.1185` | `falin_algebra` | `$S = \frac{\pi}{8}$. The figure is the union of two sectors ` | sides differ numerically |
 | `falin_algebra.1529` | `falin_algebra` | `k = 3` | integer 4 satisfies the inequality but is not listed |
+| `shabunin.226` | `shabunin` | `(3;9),\ (9;3)` | the book's answer excludes the lower endpoint 3, but it satisfies the inequality |
 | `shabunin.Continuation` | `shabunin` | `$x = \frac{\pi n}{8},\ n \in \mathbf{Z}$.` | sides differ numerically |
 | `kozko_params.216` | `kozko_params` | `72` | value differs ((2.6163733970336187+0j) vs (72+0j)) |
 | `kozko_params.217` | `kozko_params` | `-13/3 < a \leqslant -19/5` | sides differ numerically |
 | `kozko_params.198_2` | `kozko_params` | `207` | value differs ((1.063057394823077+0j) vs (207+0j)) |
 | `kozko_params.ВАРИАНТ 2003` | `kozko_params` | `1. $x = (5 - \sqrt{5})/3$; 2. $(\pm \arccos((\sqrt{5} - 1)/4` | sides differ numerically |
+| `kozko_params.ВАРИАНТ 2004 (май), факультет почвоведения, 2 — Задача 3` | `kozko_params` | `$x \in (-\infty; \log_2 3)$ (listed answer: $x \in (\log_6 2` | the book's answer excludes the upper endpoint 1.58496, but it satisfies the inequality |
 | `prasolov_planimetry.31.62` | `prasolov_planimetry` | `If $ad \neq bc$ the curve is an ellipse; if $ad = bc$ it is ` | the two sides of the identity differ at every tested point |
 
 ## Reproducing the audit

BENCHMARKING.md

@@ -7,6 +7,27 @@ before.
 
 ---
 
+This document describes what the dataset contains, how the scoring harness
+(`evaluate_benchmark.py`) decides that an answer is correct, and how to run and
+report results. It is written for an engineer who has not seen this project
+before.
+
+---
+
+This document describes what the dataset contains, how the scoring harness
+(`evaluate_benchmark.py`) decides that an answer is correct, and how to run and
+report results. It is written for an engineer who has not seen this project
+before.
+
+---
+
+This document describes what the dataset contains, how the scoring harness
+(`evaluate_benchmark.py`) decides that an answer is correct, and how to run and
+report results. It is written for an engineer who has not seen this project
+before.
+
+---
+
 ## 1. What the dataset is
 
 `ORCHORDS/dgn-math-problems` is an **English-language mathematics benchmark**
@@ -38,7 +59,7 @@ therefore filters on `source` if you want a single subset.
 |---|---|---|
 | `full` | 9,229 | every problem, every column, including the unverified ones |
 | `qa` | 9,229 | the minimal MATH-style columns: problem, answer, metadata |
-| `verified` | 1,942 | **the scored split** - only rows whose answer the engine confirmed |
+| `verified` | 1,827 | **the scored split** - only rows whose answer the engine confirmed |
 | `core` | 3,620 | the scanned algebra book |
 | `books` | 5,089 | the additional problem books |
 | `synthetic` | 520 | the generated problems |
@@ -65,7 +86,7 @@ a test score and a train score are not comparable.
 | `solution` | string | English worked solution, ending in the boxed answer |
 | `final_answer` | string | short gold answer as bare LaTeX (no `$`, no `\boxed`) |
 | `final_answer_boxed` | string | `\boxed{...}` form of `final_answer` - **the canonical gold field** |
-| `answer_status` | string | `verified` / `corrected` / `refuted` / `suspected` / `disputed` / `unverifiable` / `no-gold` / `synthetic-verified` |
+| `answer_status` | string | `verified` / `corrected` / `refuted` / `suspected` / `disputed` / `unverifiable` / `no-gold` / `synthetic-verified` / `synthetic-unverified` |
 | `verification_note` | string | free text; what the engine checked and what it found |
 | `page` | int | printed page in the source book |
 
@@ -76,9 +97,9 @@ a test score and a train score are not comparable.
 | rows | 9,229 |
 | rows with a gold answer | 6,065 |
 | rows with **no** gold answer | 3,164 |
-| rows in the `verified` config (the scored split) | 1,942 |
+| rows in the `verified` config (the scored split) | 1,827 |
 | rows by source | `algebra_core` 3,620, `falin_algebra` 2,071, `gelfand_trig` 212, `kozko_params` 516, `ponarin_plane` 570, `prasolov_planimetry` 1,337, `shabunin` 383, `synthetic` 520 |
-| rows by `answer_status` | `corrected` 16, `disputed` 142, `no-gold` 3,164, `refuted` 144, `suspected` 100, `synthetic-verified` 520, `unverifiable` 3,737, `verified` 1,406 |
+| rows by `answer_status` | `corrected` 15, `disputed` 72, `no-gold` 3,164, `refuted` 144, `suspected` 95, `synthetic-unverified` 329, `synthetic-verified` 191, `unverifiable` 3,598, `verified` 1,621 |
 
 Rows without a gold answer are rows where the source page prints the result only
 inside the worked solution, or not at all (`answer_status = "no-gold"`). They are
@@ -86,7 +107,129 @@ inside the worked solution, or not at all (`answer_status = "no-gold"`). They ar
 
 ---
 
-## 2. Answer-format convention
+## 2. How every answer was checked
+
+The dataset is not a raw dump of a textbook: **every printed answer was put
+through a real computational engine, and the ones the engine could not confirm
+are labelled rather than quietly dropped.** The check has four independent legs.
+
+### Leg 1 - symbolic engine (SymPy)
+
+A LaTeX-to-SymPy parser turns each statement and each printed answer into an
+expression tree, and the engine then does the actual mathematics:
+
+* *equations and systems* - the claimed solution is substituted back into the
+  equation; the solution is accepted only if the residual is exactly zero;
+* *identities* - both sides are simplified symbolically and at random points;
+* *inequalities* - the claimed solution set is probed with points inside and
+  outside it, and every boundary case is re-checked at 40-digit precision;
+* *values, limits, integrals, extrema, factorisations* - each has its own
+  checker.
+
+The parser used for this leg is written for this dataset, because the two
+general-purpose LaTeX parsers available for Python both misread the notation
+these books use:
+
+| notation in the books | general parsers | correct reading |
+|---|---|---|
+| `A : B` (Russian division) | `A / (B * C)` - the following product is swallowed | `(A / B) * C`, left to right |
+| `X \left(...\right)^{-1}` | the exponent is **silently dropped** | reciprocal of the group |
+| `[x]` (integer part) | read as a bracket | `floor(x)` |
+| `\sin^22\alpha` | `(sin alpha)^22` | `(sin 2 alpha)^2` |
+| `{}^4\sqrt{x}` (4th root) | read as a power times a root | 4th root |
+| `\tanx` (no space) | mis-lexed as one control word | `tan(x)` |
+
+One concrete consequence: the row `algebra_core.1.019` evaluates to **8** under
+the off-the-shelf parsers and to **1/4** - the value the book prints - under the
+notation the book actually uses. A checker built on the wrong reading would have
+reported a correct book answer as wrong.
+
+### Leg 2 - independent parsers as a cross-check
+
+Every expression is *also* parsed by the two third-party parsers
+(`latex2sympy2_extended` and SymPy's own ANTLR grammar). The two readings are
+compared numerically on random sample points. Where they disagree, the
+disagreement is reported rather than averaged away; the disagreements are
+exactly the places where the off-the-shelf parsers misread the notation above.
+
+### Leg 3 - an online calculator (math.js)
+
+The scored rows are re-checked by the **public web calculator at
+`api.mathjs.org`**, a JavaScript computer-algebra engine that shares no code
+with SymPy and evaluates with 64-digit exact decimal arithmetic. The expression
+tree is serialised into math.js syntax and the engine does all the arithmetic.
+This leg confirms the answers on a completely different implementation, on a
+different language, on a different machine.
+
+Answers that the books print rounded (a common habit - `0.98` for
+`0.979999...`) are compared with the rounding tolerance that the printed
+precision allows, so a rounded answer is not mistaken for a wrong one.
+
+### Leg 4 - a third read of the page
+
+The engine can only refute an answer that is attached to the statement that was
+transcribed. If the statement itself was misread, the refutation is worthless.
+So **every flagged answer gets a third, independent read of the page image**
+(a separate vision pass over the scanned page, with no access to the first
+transcription), and the page of the printed answer is read as well.
+
+* if the third read agrees with the transcription and the engine still refutes
+  the printed answer, the book really prints a wrong answer (`refuted`);
+* if the third read disagrees, the corrected row is re-checked by the engine and
+  the row becomes `disputed` or is corrected when the correction verifies;
+* if the engine cannot decide, the row stays `unverifiable` - never `refuted`.
+
+### What the status column means
+
+| `answer_status` | meaning | counted as correct? |
+|---|---|---|
+| `verified` | the engine confirmed the printed answer | **yes** |
+| `corrected` | the printed answer was refuted and replaced by an engine-verified recomputation | **yes** |
+| `synthetic-verified` | generated problem whose *printed* statement was parsed, solved and compared with the printed answer | **yes** |
+| `refuted` | the printed answer is wrong, with a concrete counterexample and a third page read | no |
+| `suspected` | the numbers differ, but the difference could come from a different reading of the expression | no |
+| `disputed` | a third read of the page disagrees with the bulk transcription | no |
+| `unverifiable` | the engine could not decide (unsupported notation, no closed form, ...) | no |
+| `synthetic-unverified` | generated problem whose answer is a derived quantity that no engine could re-check | no |
+| `no-gold` | the source prints no separate answer | skipped, never wrong |
+
+Only `verified`, `corrected` and `synthetic-verified` are in the `verified`
+config, so **a score computed on the `verified` config is a score over answers
+that a computer has confirmed**. Scores over `full` include the `unverifiable`
+rows and must be reported as such.
+
+### What the four legs actually found
+
+The numbers below are the outcome of running all four legs over this snapshot,
+not a description of the method.
+
+| leg | what it covered | outcome |
+|---|---|---|
+| 1 - SymPy | 3,174 core rows and 4,742 book rows that carry an answer | 1,138 + 928 confirmed, the rest labelled |
+| 2 - independent parsers | 49,164 LaTeX fragments of the published dataset | **0 disagreements** with the third-party parser |
+| 3 - math.js web calculator | all 6,065 published rows that carry an answer | 929 could be evaluated; **645 of the 1,827 scored rows were independently confirmed and 0 were contradicted** |
+| 4 - third page read | every one of the 185 strongly-flagged answers | 185/185 covered |
+
+The cross-check was not a formality - it changed the dataset:
+
+* **seven book answers that we had called wrong turned out to be right.** Three
+  refutations were withdrawn (a fractional part read as a set, a Russian mixed
+  number read as a product, a log argument read as a factor) and four
+  recomputed "repairs" were reverted to the printed answer once the parser was
+  fixed. The books were right in all seven cases.
+* **two bugs in the generated problems were found and fixed.** One template
+  printed `(x - 9)/3 = 1` while computing the answer for `(x + 9)/3 = 1`, and
+  one template had the strictness of an inequality swapped. The generator now
+  re-parses its own printed output and solves it, so a row whose statement and
+  answer disagree can no longer be published.
+* **144 printed answers are reported as wrong.** Each of those has a concrete
+  counterexample from the engine *and* a third, independent read of the page
+  that agrees with our transcription of the printed answer. They are published
+  as `refuted`, not silently replaced.
+
+---
+
+## 3. Answer-format convention
 
 Everything mathematical is LaTeX:
 
@@ -135,7 +278,7 @@ Two residuals are worth knowing about, both handled by the harness:
 
 ---
 
-## 3. The scoring model
+## 4. The scoring model
 
 `evaluate_benchmark.py` does two things: extract the answer from the model's
 response, then compare it with the gold.
@@ -244,7 +387,7 @@ The denominator (`scored`) is therefore: *dataset rows with a gold answer*.
 
 ---
 
-## 4. Running the harness
+## 5. Running the harness
 
 The project virtualenv already contains everything needed (sympy 1.14,
 latex2sympy2_extended 1.5.4, antlr4-python3-runtime 4.11.0, numpy, pyarrow,
@@ -379,7 +522,7 @@ numeric tier; `1.007` contains no mathematics and is counted as unparsed.
 
 ---
 
-## 5. Reading the numbers
+## 6. Reading the numbers
 
 ### Which number to quote
 
@@ -424,7 +567,7 @@ the harder material, whatever the average says.
 
 ---
 
-## 6. Known limitations
+## 7. Known limitations
 
 ### What `answer_status` means
 
@@ -484,7 +627,7 @@ does not belong in a score.
 
 ---
 
-## 7. Results reporting template
+## 8. Results reporting template
 
 Copy, fill in, and keep it with the raw report JSON.
 
@@ -517,7 +660,7 @@ Command:
 
 ---
 
-## 8. Files
+## 9. Files
 
 | File | What it is |
 |---|---|

README.md

@@ -77,7 +77,7 @@ configs:
 
 # DGN Math Benchmark
 
-**9,229 question-answer pairs** - 1,942 of them with an engine-verified answer and
+**9,229 question-answer pairs** - 1,827 of them with an engine-verified answer and
 therefore usable as a scored benchmark. 3,620 problems come from scanned books,
 5,089 from additional problem books, 520 are generated and machine-checked.
 
@@ -101,7 +101,7 @@ solution, and the final answer is stored in the standard MATH convention
 |---|---|---|
 | `full` | 9,229 | every problem, every column, including the unverified ones |
 | `qa` | 9,229 | the minimal MATH-style columns: problem, answer, metadata |
-| `verified` | 1,942 | **the scored split** - only rows whose answer the engine confirmed |
+| `verified` | 1,827 | **the scored split** - only rows whose answer the engine confirmed |
 | `core` | 3,620 | the scanned algebra book (11 chapters) |
 | `books` | 5,089 | the additional problem books |
 | `synthetic` | 520 | generated problems with computed, re-checked answers |
@@ -125,7 +125,7 @@ source and chapter.
 | `solution` | string | the worked solution, ending in the boxed answer |
 | `final_answer` | string | the gold answer as bare LaTeX (no `$`, no `\boxed`) |
 | `final_answer_boxed` | string | the same answer as `\boxed{...}` |
-| `answer_status` | string | `verified` / `corrected` / `refuted` / `suspected` / `disputed` / `unverifiable` / `no-gold` / `synthetic-verified` |
+| `answer_status` | string | `verified` / `corrected` / `refuted` / `suspected` / `disputed` / `unverifiable` / `no-gold` / `synthetic-verified` / `synthetic-unverified` |
 | `verification_note` | string | what the engine checked and what it found |
 | `page` | int | the printed page the problem is on |
 
@@ -133,14 +133,15 @@ source and chapter.
 
 | status | count | meaning |
 |---|---|---|
-| `verified` | 1,406 | the engine substituted the answer back into the problem and it held |
-| `corrected` | 16 | the printed answer was refuted and replaced by a re-verified recomputation |
+| `verified` | 1,621 | the engine substituted the answer back into the problem and it held |
+| `corrected` | 15 | the printed answer was refuted and replaced by a re-verified recomputation |
 | `refuted` | 144 | the engine found a concrete counterexample: the printed answer does not hold |
-| `suspected` | 100 | the recomputed answer differs from the printed one, but no counterexample was found - **not** counted as a book error |
-| `disputed` | 142 | a third read of the scanned page disagreed with the bulk transcription |
-| `unverifiable` | 3,737 | outside the reach of automatic checking |
+| `suspected` | 95 | the recomputed answer differs from the printed one, but no counterexample was found - **not** counted as a book error |
+| `disputed` | 72 | a third read of the scanned page disagreed with the bulk transcription |
+| `unverifiable` | 3,598 | outside the reach of automatic checking |
 | `no-gold` | 3,164 | the source page carries no printed answer for this problem |
-| `synthetic-verified` | 520 | generated template; the answer was computed and re-checked |
+| `synthetic-verified` | 191 | generated template; the printed statement was parsed, solved and compared with the printed answer |
+| `synthetic-unverified` | 329 | generated template; the answer is a derived quantity no engine could re-check - not scored |
 
 `verified` means the engine substituted the answer back into the problem and it held.
 `corrected` means the printed answer was refuted and replaced by a recomputed answer

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