diff --git a/crates/functions/src/stat/distributions.rs b/crates/functions/src/stat/distributions.rs
index 9bd519066..e3521592e 100644
--- a/crates/functions/src/stat/distributions.rs
+++ b/crates/functions/src/stat/distributions.rs
@@ -186,8 +186,14 @@ fn df_arg(a: &[Arg], i: usize) -> R<f64> {
 
 pub(super) fn t_dist(a: &[Arg], _: C) -> R<Value> {
     let x = num(a, 0)?;
-    let v = df_arg(a, 1)?;
+    let v = num(a, 1)?.trunc();
     let cum = boolean(a, 2)?;
+    if v == 0.0 {
+        return Err(if cum { CellError::Num } else { CellError::Div0 });
+    }
+    if v < 0.0 {
+        return Err(CellError::Num);
+    }
     num_val(if cum { t_cdf(x, v) } else { t_pdf(x, v) })
 }
 
diff --git a/crates/functions/src/stat/tests.rs b/crates/functions/src/stat/tests.rs
index 3e229262e..ee7639d6d 100644
--- a/crates/functions/src/stat/tests.rs
+++ b/crates/functions/src/stat/tests.rs
@@ -418,6 +418,17 @@ fn f_distributions_accept_ten_billion_degrees() {
     is_err(ev("F.DIST.RT", vec![n(0.0), n(1.0), n(10_000_000_001.0)]), CellError::Num);
 }
 
+#[test]
+fn t_dist_zero_degrees_error_depends_on_mode() {
+    for df in [0.0, 0.9, -0.1] {
+        is_err(ev("T.DIST", vec![n(0.0), n(df), b(false)]), CellError::Div0);
+        is_err(ev("T.DIST", vec![n(0.0), n(df), b(true)]), CellError::Num);
+    }
+    is_err(ev("T.DIST", vec![n(0.0), n(-1.0), b(false)]), CellError::Num);
+    close_tol(ev("T.DIST", vec![n(0.0), n(1.99), b(false)]), 1.0 / std::f64::consts::PI, 1e-15);
+    close_tol(ev("T.DIST", vec![n(0.0), n(1.0), b(false)]), 1.0 / std::f64::consts::PI, 1e-15);
+}
+
 #[test]
 fn binom_inv_huge_trials_returns() {
     // Above 2^53 neighbouring f64 values are more than 1 apart; the bisection must still end.
diff --git a/docs/function-parity.md b/docs/function-parity.md
index b92e3bd64..90d8344a4 100644
--- a/docs/function-parity.md
+++ b/docs/function-parity.md
@@ -1,6 +1,6 @@
 # Function parity: worksheet functions and formulas
 
-> **Last reviewed:** 2026-10-31 · **Last updated:** 2026-10-31 · **Change:** minor (F distribution functions accept degrees of freedom through 10 billion, and the extreme numerator-degree inverse uses a stable gamma-tail limit; earlier: DOLLAR keeps the negative layout when a negative input rounds to zero at nonnegative precision; negative precision that rounds to zero retains the positive layout, as measured in Windows Excel 16.0 build 20527, LCID 1033; earlier: ECMA.CEILING listed with USDOLLAR as not on Microsoft's list; IMSQRT reproduces the stray real part Excel gives for negative reals, and the other IM* differences from Excel are listed; the LAMBDA row lists the recursion limit and its `#NUM!`, and the error a defined name holding an error gives when called; number text with accounting parentheses coerces like Excel 365; INDEX with one index on a 2-D reference is #REF! as in Excel; SUBTOTAL and AGGREGATE cap the cells they visit, #2175) · **Target:** Microsoft Excel (Microsoft 365), Excel for Mac 16.113.4
+> **Last reviewed:** 2026-10-31 · **Last updated:** 2026-10-31 · **Change:** minor (T.DIST gives `#DIV/0!` for density mode and `#NUM!` for cumulative mode when truncated degrees of freedom are zero; earlier: F distribution functions accept degrees of freedom through 10 billion, and the extreme numerator-degree inverse uses a stable gamma-tail limit; earlier: ECMA.CEILING listed with USDOLLAR as not on Microsoft's list; IMSQRT reproduces the stray real part Excel gives for negative reals, and the other IM* differences from Excel are listed; the LAMBDA row lists the recursion limit and its `#NUM!`, and the error a defined name holding an error gives when called; number text with accounting parentheses coerces like Excel 365; INDEX with one index on a 2-D reference is #REF! as in Excel; SUBTOTAL and AGGREGATE cap the cells they visit, #2175) · **Target:** Microsoft Excel (Microsoft 365), Excel for Mac 16.113.4
 
 Excel's worksheet functions, category by category, against GridCraft's function library. Part of
 [target-app-parity.md](target-app-parity.md); gaps feed [gaps.md](gaps.md).
@@ -133,6 +133,7 @@ workbooks; results checked in as data, not Excel files) and fixing what it finds
 | 2026-10-31 | minor | FIXED accepts output through 255 characters; DOLLAR accepts output through 254. Windows Excel 16.0 build 20527 (LCID 1033) confirmed the distinct cutoffs with grouped and ungrouped values, signs, currency layout and decimal places |
 | 2026-10-31 | minor | DOLLAR uses the input sign for negative amounts that round to zero at nonnegative decimal places (`DOLLAR(-0.0001,0)` is `($0)`); when negative decimal places round to zero, Excel returns `$0`. Exact zero and negative zero stay unsigned; FIXED still returns `0` for `FIXED(-0.0001,0)` |
 | 2026-10-31 | minor | F.DIST, F.DIST.RT, F.INV and F.INV.RT accept degrees of freedom at the 10-billion boundary; F.INV uses the complementary tail, with a gamma-tail limit when numerator degrees dominate, to preserve inverse accuracy; values above the limit remain invalid |
+| 2026-10-31 | minor | T.DIST returns `#DIV/0!` in density mode and `#NUM!` in cumulative mode when degrees truncate to zero; negative integer degrees remain `#NUM!` |
 | 2026-10-27 | minor | ECMA.CEILING added to the functions that are in our library but not on Microsoft's list (with USDOLLAR), so the counts don't change |
 | 2026-10-26 | minor | IMSQRT uses Excel's polar route with its cosine reduction, so `IMSQRT("-4")` is `1.22514845490862E-16+2i` as in Excel (exact `2i` before); the rows list the IM* differences still open |
 | 2026-10-25 | minor | LAMBDA row: a LAMBDA that calls itself more than 140 times, and a formula nested past the depth limit, give `#NUM!` (#1809; nesting gave `#CALC!` before). `=Foo(1)` where the defined name Foo holds `#N/A` gives `#N/A`, not `#NAME?` |