fix(stats): hold the Student-t tails past t-squared overflow

Assisted-by: GLM 5.3 Flash

stats/cdf.go

@@ -269,15 +269,18 @@ func StudentTCDF(t float64, df int) (float64, error) {
 	if df < 1 {
 		return 0, base.Errf("StudentTCDF: df must be ≥ 1, got %d", df)
 	}
-	z := float64(df) / (float64(df) + t*t)
-	upper, err := BetaIncomplete(z, float64(df)/2, 0.5)
+	// One house tail: the upper-tail helper carries the asymptotic forms
+	// the heavy df ≤ 2 laws keep past t²'s overflow, where the closed
+	// form's z = df/(df+t²) collapses to 0 and the lower tail answered a
+	// silent 0 for a tail the format still holds.
+	upper, err := studentTUpperTail(math.Abs(t), df)
 	if err != nil {
 		return 0, base.Errf("StudentTCDF: %w", err)
 	}
 	if t >= 0 {
-		return 1 - upper/2, nil
+		return 1 - upper, nil
 	}
-	return upper / 2, nil
+	return upper, nil
 }
 
 // PoissonCDF returns P(N ≤ k) for N ~ Poisson(lambda), through the
@@ -701,7 +704,10 @@ func studentTUpperTail(t float64, df int) (float64, error) {
 			// huge t keeps accurate.
 			return 1 / (math.Pi * t), nil
 		case df == 2:
-			return 1 / (2 * t * t), nil
+			// The tail 1/(2t²) divided one t at a time: the literal
+			// denominator overflows past √MaxFloat64, and the quotient
+			// would flush the still-representable tail to zero.
+			return 0.5 / t / t, nil
 		}
 	}
 	z := float64(df) / (float64(df) + t*t)