Task: Arithmetic-Relationship (source dataset: Auto-Relate)

You are given a table with numerical values in multiple columns. Your task is to analyze the data and identify semantic relationships, expressed in arithmetic relationships, that exist between columns.

These relationships may involve any combination of the following operations: addition (+), subtraction (−), multiplication (*), and division (/). The expressions can involve two or more columns and may be nested (e.g., B = (A + C) / D).

Please only identify arithmetic relationships that hold semantically based on the data, and do not return relationships that are likely spurious, e.g., (A = B * 0), where A is an all 0 column. Please only return minimal atomic relationships, e.g., if A = B + C and D = E + F, then there is no need to return A = B + C + E + F - D.

No explanation, return all the formula(s) in JSON format: {"Arithmetic-Relationship": [<list-of-formulas>]}. If no relationship is found, return {"Arithmetic-Relationship": []}.

Input Table:
| B   |   C |   D |   E |   F |   G |   H |   I |   J |   K |   L |   M |   N |        O |        P |          Q |         R |          S |          T |          U |   V |   W |   X |   Y |
|:----|----:|----:|----:|----:|----:|----:|----:|----:|----:|----:|----:|----:|---------:|---------:|-----------:|----------:|-----------:|-----------:|-----------:|----:|----:|----:|----:|
| AL  | 121 |  96 |   4 |  36 |   1 |   8 |   1 |   0 |   0 |   0 |   0 | 121 | 1        | 0.793388 | 0.0330579  | 0.297521  | 0.00826446 | 0.0661157  | 0.00826446 |   0 |   0 |   0 |   0 |
| AZ  | 110 | 102 |   3 |  35 |   0 |   6 |  11 |   0 |   0 |   0 |   0 | 111 | 0.990991 | 0.918919 | 0.027027   | 0.315315  | 0          | 0.0540541  | 0.0990991  |   0 |   0 |   0 |   0 |
| CO  |  23 |  23 |   1 |   8 |   0 |   1 |   0 |   0 |   0 |   0 |   0 |  28 | 0.821429 | 0.821429 | 0.0357143  | 0.285714  | 0          | 0.0357143  | 0          |   0 |   0 |   0 |   0 |
| CT  |  45 |  31 |   4 |  34 |   0 |   1 |   1 |   0 |   0 |   0 |   0 |  45 | 1        | 0.688889 | 0.0888889  | 0.755556  | 0          | 0.0222222  | 0.0222222  |   0 |   0 |   0 |   0 |
| DE  | 151 | 118 |   2 |  29 |   1 |   1 |   1 |   0 |   0 |   0 |   0 | 154 | 0.980519 | 0.766234 | 0.012987   | 0.188312  | 0.00649351 | 0.00649351 | 0.00649351 |   0 |   0 |   0 |   0 |
| DC  |  95 |  39 |   4 |  31 |   3 |  17 |   4 |   0 |   0 |   0 |   0 |  96 | 0.989583 | 0.40625  | 0.0416667  | 0.322917  | 0.03125    | 0.177083   | 0.0416667  |   0 |   0 |   0 |   0 |
| FL  | 150 | 109 |   4 |  61 |   0 |   1 |   1 |   0 |   0 |   0 |   0 | 154 | 0.974026 | 0.707792 | 0.025974   | 0.396104  | 0          | 0.00649351 | 0.00649351 |   0 |   0 |   0 |   0 |
| GA  |  39 |  40 |   2 |  21 |   0 |   2 |   7 |   0 |   0 |   0 |   0 |  55 | 0.709091 | 0.727273 | 0.0363636  | 0.381818  | 0          | 0.0363636  | 0.127273   |   0 |   0 |   0 |   0 |
| IL  |  98 |  58 |   3 |  81 |   3 |   3 |   5 |   0 |   0 |   0 |   0 | 141 | 0.695035 | 0.411348 | 0.0212766  | 0.574468  | 0.0212766  | 0.0212766  | 0.035461   |   0 |   0 |   0 |   0 |
| IA  |  59 |  31 |   0 |  21 |   1 |   3 |   0 |   0 |   0 |   0 |   0 |  61 | 0.967213 | 0.508197 | 0          | 0.344262  | 0.0163934  | 0.0491803  | 0          |   0 |   0 |   0 |   0 |
| KS  |  56 |  45 |   1 |  10 |   0 |   3 |   7 |   0 |   0 |   0 |   0 |  56 | 1        | 0.803571 | 0.0178571  | 0.178571  | 0          | 0.0535714  | 0.125      |   0 |   0 |   0 |   0 |
| LA  | 124 |  85 |   1 |  35 |   3 |   2 |   1 |   0 |   0 |   0 |   0 | 124 | 1        | 0.685484 | 0.00806452 | 0.282258  | 0.0241935  | 0.016129   | 0.00806452 |   0 |   0 |   0 |   0 |
| MD  | 131 | 109 |   3 |  49 |   3 |   1 |   0 |   0 |   0 |   0 |   0 | 133 | 0.984962 | 0.819549 | 0.0225564  | 0.368421  | 0.0225564  | 0.0075188  | 0          |   0 |   0 |   0 |   0 |
| MA  | 120 |  86 |   7 |  58 |   2 |   2 |   5 |   0 |   0 |   0 |   0 | 120 | 1        | 0.716667 | 0.0583333  | 0.483333  | 0.0166667  | 0.0166667  | 0.0416667  |   0 |   0 |   0 |   0 |
| MI  |  41 |  23 |   0 |  25 |   0 |   1 |   2 |   0 |   0 |   0 |   0 |  54 | 0.759259 | 0.425926 | 0          | 0.462963  | 0          | 0.0185185  | 0.037037   |   0 |   0 |   0 |   0 |
| MN  |  41 |  37 |   4 |  17 |   0 |   1 |   7 |   0 |   0 |   0 |   0 |  45 | 0.911111 | 0.822222 | 0.0888889  | 0.377778  | 0          | 0.0222222  | 0.155556   |   0 |   0 |   0 |   0 |
| MS  |  96 |  63 |   5 |  18 |  18 |   1 |   0 |   0 |   0 |   0 |   0 |  96 | 1        | 0.65625  | 0.0520833  | 0.1875    | 0.1875     | 0.0104167  | 0          |   0 |   0 |   0 |   0 |
| MT  |  51 |  44 |   1 |  12 |   1 |   3 |   0 |   0 |   0 |   0 |   0 |  52 | 0.980769 | 0.846154 | 0.0192308  | 0.230769  | 0.0192308  | 0.0576923  | 0          |   0 |   0 |   0 |   0 |
| NE  |  90 |  68 |   3 |  41 |   0 |   0 |   5 |   0 |   0 |   0 |   0 |  90 | 1        | 0.755556 | 0.0333333  | 0.455556  | 0          | 0          | 0.0555556  |   0 |   0 |   0 |   0 |
| NV  |  29 |  26 |   4 |  17 |   0 |   0 |   1 |   0 |   0 |   0 |   0 |  45 | 0.644444 | 0.577778 | 0.0888889  | 0.377778  | 0          | 0          | 0.0222222  |   0 |   0 |   0 |   0 |
| NH  | 188 | 140 |   5 |  59 |   3 |   2 |   4 |   0 |   0 |   0 |   0 | 188 | 1        | 0.744681 | 0.0265957  | 0.31383   | 0.0159574  | 0.0106383  | 0.0212766  |   0 |   0 |   0 |   0 |
| NJ  |  85 |  72 |   5 |  24 |   0 |   0 |   6 |   0 |   0 |   0 |   0 | 105 | 0.809524 | 0.685714 | 0.047619   | 0.228571  | 0          | 0          | 0.0571429  |   0 |   0 |   0 |   0 |
| NM  |  36 | 100 |   2 |  25 |   3 |   4 |   9 |   0 |   0 |   0 |   0 | 131 | 0.274809 | 0.763359 | 0.0152672  | 0.19084   | 0.0229008  | 0.0305344  | 0.0687023  |   0 |   0 |   0 |   0 |
| NY  |  86 |  30 |  18 |  39 |   4 |   9 |   6 |   0 |   0 |   0 |   0 |  95 | 0.905263 | 0.315789 | 0.189474   | 0.410526  | 0.0421053  | 0.0947368  | 0.0631579  |   0 |   0 |   0 |   0 |
| NC  |  93 |  85 |  14 |  20 |   4 |   1 |   3 |   0 |   0 |   0 |   0 | 101 | 0.920792 | 0.841584 | 0.138614   | 0.19802   | 0.039604   | 0.00990099 | 0.029703   |   0 |   0 |   0 |   0 |
| ND  |  65 |  53 |   2 |  21 |   2 |   4 |   0 |   0 |   0 |   0 |   0 |  65 | 1        | 0.815385 | 0.0307692  | 0.323077  | 0.0307692  | 0.0615385  | 0          |   0 |   0 |   0 |   0 |
| OK  | 114 |  73 |   9 |  25 |   1 |   5 |   1 |   0 |   0 |   0 |   0 | 114 | 1        | 0.640351 | 0.0789474  | 0.219298  | 0.00877193 | 0.0438596  | 0.00877193 |   0 |   0 |   0 |   0 |
| PA  |  17 |  50 |   0 |   3 |   0 |   0 |   2 |   0 |   0 |   0 |   0 |  62 | 0.274194 | 0.806452 | 0          | 0.0483871 | 0          | 0          | 0.0322581  |   0 |   0 |   0 |   0 |
| RI  |  29 |  43 |   0 |   7 |   1 |   2 |   1 |   0 |   0 |   0 |   0 |  62 | 0.467742 | 0.693548 | 0          | 0.112903  | 0.016129   | 0.0322581  | 0.016129   |   0 |   0 |   0 |   0 |
| SC  |  16 |  14 |   3 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |   0 |  23 | 0.695652 | 0.608696 | 0.130435   | 0         | 0          | 0          | 0          |   0 |   0 |   0 |   0 |
| SD  |  34 |  32 |   1 |   1 |   0 |   0 |   1 |   0 |   0 |   0 |   0 |  34 | 1        | 0.941176 | 0.0294118  | 0.0294118 | 0          | 0          | 0.0294118  |   0 |   0 |   0 |   0 |
| UT  |  61 |  44 |   4 |  19 |   2 |   1 |   0 |   0 |   0 |   0 |   0 |  61 | 1        | 0.721311 | 0.0655738  | 0.311475  | 0.0327869  | 0.0163934  | 0          |   0 |   0 |   0 |   0 |
| VT  |  36 |  19 |   1 |  17 |   1 |   0 |   1 |   0 |   0 |   0 |   0 |  36 | 1        | 0.527778 | 0.0277778  | 0.472222  | 0.0277778  | 0          | 0.0277778  |   0 |   0 |   0 |   0 |
| VA  |  33 |  40 |   1 |  10 |   0 |   0 |   3 |   0 |   0 |   0 |   0 |  69 | 0.478261 | 0.57971  | 0.0144928  | 0.144928  | 0          | 0          | 0.0434783  |   0 |   0 |   0 |   0 |
| WA  |  66 |  70 |  16 |  29 |  12 |   9 |  19 |   0 |   0 |   0 |   0 | 122 | 0.540984 | 0.57377  | 0.131148   | 0.237705  | 0.0983607  | 0.0737705  | 0.155738   |   0 |   0 |   0 |   0 |
| WI  |   6 |  10 |   0 |   6 |   0 |   1 |   0 |   0 |   0 |   0 |   0 |  12 | 0.5      | 0.833333 | 0          | 0.5       | 0          | 0.0833333  | 0          |   0 |   0 |   0 |   0 |



labeled_formulas: K=C*J
L=C*J
M=C*J
W=C*J
X=C*J
Y=C*J
L=C*K
M=C*K
V=C*K
K=C*W
X=C*K
Y=C*K
M=C*L
V=C*L
W=C*L
Y=C*L
V=C*M
W=C*M
X=C*M
M=C*Y
W=C*V
X=C*V
Y=C*V
X=C*W
Y=C*W
Y=C*X
K=D*J
L=D*J
M=D*J
V=D*J
W=D*J
X=D*J
Y=D*J
L=D*K
M=D*K
V=D*K
W=D*K
X=D*K
Y=D*K
M=D*L
V=D*L
W=D*L
X=D*L
Y=D*L
V=D*M
W=D*M
X=D*M
Y=D*M
W=D*V
X=D*V
K=(C+D)*J
L=(C+D)*J
M=(C+D)*J
V=(C+D)*J
J=(C-D)*W
X=(C+D)*J
Y=(C+D)*J
L=(C+D)*K
M=(C+D)*K
V=(C+D)*K
W=(C+D)*K
X=(C+D)*K
Y=(C+D)*K
M=(C+D)*L
V=(C+D)*L
L=(C-D)*W
X=(C+D)*L
Y=(C+D)*L
V=(C+D)*M
W=(C+D)*M
X=(C+D)*M
Y=(C+D)*M
W=(C+D)*V
X=(C+D)*V
Y=(C+D)*V
X=(C+D)*W
Y=(C+D)*W
Y=(C+D)*X
K=(C+E)*J
L=(C+E)*J
M=(C+E)*J
V=(C+E)*J
W=(C+E)*J
X=(C+E)*J
Y=(C+E)*J
L=(C+E)*K
M=(C+E)*K
V=(C+E)*K
W=(C+E)*K
X=(C+E)*K
Y=(C+E)*K
M=(C+E)*L
V=(C+E)*L
W=(C+E)*L
X=(C+E)*L
Y=(C+E)*L
V=(C+E)*M
W=(C+E)*M
X=(C+E)*M
Y=(C+E)*M
output: {"Arithmetic-Relationship": []}
source: $MMTU_HOME/resources/Relationship-finding/AR/table_28296___xlsx_c37e2860c3247b482d64df5c474a5a64b287bf5f.xlsx___line_708