Is there a way to get dmatrix to drop all-zero columns?

I have an experiment design that does not include all combinations of its categorical variables, and ran into some difficulties getting a full-rank design matrix for statsmodels. I included a simplified version below.

import numpy as np
import numpy.linalg as la
import pandas as pd
import patsy

index_vals = tuple("abc")
level_names = list("ABD")
n_samples = 2


def describe_design_matrix(design_matrix):
    print("Shape:", design_matrix.shape)
    print("Rank: ", la.matrix_rank(design_matrix))
    print(
        "Approximate condition number: {0:.2g}".format(
            np.divide(*la.svd(design_matrix)[1][[0, -1]])
        )
    )


ds_simple = pd.DataFrame(
    index=pd.MultiIndex.from_product(
        [index_vals] * len(level_names) + [range(n_samples)],
        names=level_names + ["sample"],
    ),
    columns=["y"],
    data=np.random.randn(len(index_vals) ** len(level_names) * n_samples),
).reset_index()

print("All sampled")
simple_X = patsy.dmatrices("y ~ (A + B + D) ** 3", ds_simple)[1]
describe_design_matrix(simple_X)

print("Only some sampled")
simple_X = patsy.dmatrices(
    "y ~ (A + B + D) ** 3", ds_simple.query("A != 'a' or B == 'a'")
)[1]
describe_design_matrix(simple_X)

print("Reduced X")
simple_X = patsy.dmatrices(
    "y ~ (A + B + D) ** 3",
    ds_simple.query("A != 'a' or B == 'a'"),
    return_type="dataframe",
)[1]
reduced_X = simple_X.loc[
    :, [col for col in simple_X.columns if not col.startswith("A[T.b]:B")]
]
describe_design_matrix(reduced_X)

print("Only some sampled: alternate method")
simple_X = patsy.dmatrices(
    "y ~ (C(A, Treatment('b')) + B + D) ** 3", ds_simple.query("A != 'a' or B == 'a'")
)[1]
describe_design_matrix(simple_X)
print("Number of nonzero elements:", (simple_X != 0).sum(axis=0))
print("Number of all-zero columns:", np.count_nonzero((simple_X != 0).sum(axis=0) == 0))

print("Reduced X: alternate method")
simple_X = patsy.dmatrices(
    "y ~ (C(A, Treatment('b')) + B + D) ** 3",
    ds_simple.query("A != 'a' or B == 'a'"),
    return_type="dataframe",
)[1]
reduced_X = simple_X.loc[
    :,
    [
        col
        for col in simple_X.columns
        if not col.startswith("C(A, Treatment('b'))[T.a]:B")
    ],
]
describe_design_matrix(reduced_X)

produces as output

All sampled
Shape: (54, 27)
Rank:  27
Approximate condition number: 52
Only some sampled
Shape: (42, 27)
Rank:  21
Approximate condition number: 3.8e+16
Reduced X
Shape: (42, 21)
Rank:  21
Approximate condition number: 37
Only some sampled: alternate method
Shape: (42, 27)
Rank:  21
Approximate condition number: 3.4e+16
Number of nonzero elements: [42  6 18 12 12 14 14  0  6  0  6  2  6  2  6  4  4  4  4  0  2  0  2  0
  2  0  2]
Number of all-zero columns: 6
Reduced X: alternate method
Shape: (42, 21)
Rank:  21
Approximate condition number: 39

I don't mind spending the time to find the representation that produces all-zero columns, but there doesn't seem to be a way within patsy to say "I know some of these columns are going to be all zeros" or "These columns will be linear dependent on others". Since some statsmodels functions require the formula information from patsy.DesignInfo objects, I wanted to see what could be done within patsy.

matthewwardrop/formulaic#19 is a related issue, with some discussion of how to generalize the "Reduced X" method in the script.

Start with the reproducible script in the issue and inspect patsy.dmatrices and the resulting DesignInfo objects. Determine how zero or dependent columns could be identified while preserving the formula metadata needed by statsmodels. Done should include a defined behavior for reduced design matrices and coverage for the demonstrated categorical-interaction case.