[Rafael-SOWNet]

Part of #233. Fixes answers that held on only part of the real line.

Three places take a root as a nonnegative real u and attach radicand >= 0 to the answer:

• the factoring in SolveByLinearRadicalSubstitution;
• the "as the root" branch of SolveBySubstitution;
• the factored branch of SolveBySubstitution.

Where a second root is imaginary on the same side, the integrand is real there as a product of two imaginary roots, and the answer said nothing:

• Timofeev 926 and 927, asec(x) beside sqrt(x^2 - 1), are real on 0 < x < 1, where the arcsecant and the root are both imaginary.
• Rubi's sqrt(cot) (a + b tan)^(k/2) and its kin in 4.3.2.1 and 4.3.3.1 are real where tan < -a/b.

The harness now counts such an answer as Partial: right wherever it has a value, and without one at points where the integrand is real.

The change. ConditionedOnTheRadicand acts where the answer holds an even root besides. It asks PartialFractions.WhereTheDerivativeHoldsBeyondTheRadicand, which works in two stages:

1. It sorts 15 points per sign pattern by interval arithmetic on the integrand and the radicand.
2. Only where one of those points is real beyond the zero does it differentiate the answer, lazily, real points first.

The outcome sets the condition:

• holds at every real point there, beside exactly one other half-odd root: the answer carries the integrand's real domain, e.g. provided tan(c + d x) >= 0 or a + b tan(c + d x) <= 0;
• also holds at every complex point: no condition;
• anything else: the old condition, as it is for an integrand that is real at no point beyond the zero.

Results:

• 40 partial answers become whole: 26 of 4.3.3.1, 12 of 4.3.2.1, and Timofeev 926 and 927. Each was checked with mpmath on both sides.
• Only the condition changes, by 28–60 characters: Timofeev 926 goes 184 → 212, 4.3.3.1:740 goes 2,736 → 2,767.
• 2.5.0 declined every sample.

Chain against master a567c8b (this branch's base was a4e7656, one commit later, which touches 3.x only), with the harness at its current revision on both arms:

• pocket of 2,104 rows (all of 5.5.x, 5.6.x, Timofeev and 4.3.3.1, plus 510 tangent-root rows of family 4): exactly those 40 go Partial → Solved, 0 wrong;
• samples of families 0–8: unchanged in verdicts;
• the check costs 24–111 MB on the rows it changes;
• a first version re-answered Bondarenko 22 and 23 at 2.4× the allocation. Its dropped condition was right by mpmath, but the check is now bounded, and those rows are master's again, byte for byte, at 0.582 and 0.505 GB.

Rebased onto f705d99, which changes how constants differentiate (#2014): 2,312 related tests pass.

The BREAKING-CHANGES.md entry has 2.5.0's column measured.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura