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# Appendix

## A1. Constructions and validation

The after-tax-income invariance of Section 2: baseline after-tax income is Y − T. The corrected schedule replaces Y with Y + y·τ*/(1 − τ*) and T with T + y·τ*/(1 − τ*); the difference of the two is unchanged. The correction moves only the ratio.

The TAXSIM submissions run in chunks of one thousand tax units. Every chunk is prepended with a pinned self-check unit and the pipeline aborts if the calculator returns anything else for it. The unit is a 2021 single filer with $50,000 of wages and $100 of taxable interest, whose federal liability of $2,907.00 is verified by hand: the $12,550 standard deduction leaves $37,550 of taxable income, the 10 and 12 percent brackets give $4,307.00, and the $1,400 recovery rebate credit — full, at an income well below the phase-out — nets $2,907.00. The pin is deliberately on the analysis year and deliberately in the credit region: a check on a neighboring year would exercise the submission and response handling without exercising target-year parameter selection, and a check on wage tax alone would not touch the refundable credits that shape this paper's 2021 baseline schedule. The two-run interest margin uses an increment of $1,000 of taxable interest. Against the federal liabilities shipped with the NBER tax-unit files, our as-reported run matches {{N:taxsim_validation_within1_pct}} percent of units within one dollar; inspection of the residual cases shows differences concentrated in 2021 refundable credits, consistent with calculator revisions after those files were produced.

## A2. The expected-state-rate calculation

Every municipal-holding tax unit is run through all fifty-one TAXSIM state calculators (the fifty states and the District of Columbia) under 2021 law, with the same two-run interest differencing. The population-weighted expectation uses 2021 Census state population estimates. The dollar-weighted expected state marginal rate on interest across holders is {{N:exp_state_mtr_dollarwtd_pct}} percent; the maximum-state bound is {{N:exp_state_mtr_hightax_bound_pct}} percent. TAXSIM's state law is fixed at 2021, which matches the income year.

## A3. The robustness grid, extended

Table 4 reports the grid on both bases. Rescaling the survey's tax-exempt interest to the administrative record multiplies every aggregate by {{N:soi_scale_factor:.2f}}: the 2021 aggregate at the identified rate becomes ${{N:agg_implicit_tax_2021_soi_scaled_bn:.0f}} billion and the federal-component aggregate ${{N:fed_agg_soi_scaled_bn:.0f}} billion.

The percentile corrections scale too. A group's correction is B(Y − T)/[Y(Y + B)] with B = y·τ*/(1 − τ*), which at these magnitudes is very nearly linear in y, so a proportional rescaling of tax-exempt interest passes through almost exactly: the 99th-to-99.9th percentile's +{{N:top1_delta_pp_headline:.2f}} becomes +{{N:top1_delta_pp_soi_scaled:.2f}}, the 95th-to-99th's +{{N:p95_99_delta_pp:.2f}} becomes +{{N:p95_99_delta_pp_soi_scaled:.2f}}, and the top 0.1 percent's +{{N:top01_delta_pp_headline:.2f}} becomes +{{N:top01_delta_pp_soi_scaled:.2f}}. What proportional scaling leaves unchanged is the shape — the ordering of groups by correction size, and the confinement of any material correction above the 90th percentile — not the levels.

Ranks are held at the survey's on both bases. A uniform factor is a statement about the aggregate and carries no unit-level information about which respondents over-report, so re-ranking units on rescaled income would reshuffle them on the strength of an assumption rather than of data, and would move the thinnest cells most.

## A4. The survey-administrative gap

The SCF reports ${{N:agg_taxexempt_interest_bn}} billion of tax-exempt interest for 2021; SOI reports ${{N:soi_2021_taxexempt_interest_bn}} billion on 6.6 million returns. The ratio is {{N:scf_soi_ratio}}. The count dimension sharpens it: {{N:scf_holder_units_m}} million survey holder-units against {{N:soi_returns_m}} million administrative returns, with survey holders reporting ${{N:scf_per_holder_dollars:,}} each against ${{N:soi_per_holder_dollars:,}} — the survey misses small holders, inflates large ones, or both, which bears on the distributional shape and not just the level. A by-AGI-class shape comparison against SOI Table 1.4 (already in the replication pipeline) is the revision's first empirical task. The gap's candidate components, none of which the public data can size: non-filing holders (deflates SOI), under-reporting of line 2a information items (deflates SOI), respondent rounding and recall (inflates SCF), and gross reporting of fund distributions (inflates SCF). Both bases run through every aggregate in the paper.

## A5. Reproducibility

One command regenerates the paper's every number, table, and figure from public sources: the Federal Reserve Board's SCF files, the NBER SCF-to-TAXSIM tax-unit files, the TAXSIM35 web calculator, the municipal AAA curves from the MSRB's public displays, FRED (Treasury yields, the Bond Buyer historical index, the Financial Accounts, NIPA receipts), IRS SOI Table 1.4, Census state population estimates, and Treasury's tax-expenditure tables. Intermediate artifacts are cached; the TAXSIM runs carry the pinned self-check described in A1; and the text of the paper is compiled against a registry of pipeline outputs, so a quoted number cannot drift from the scripts that produce it. The replication package accompanies the draft.
