[Rafael-SOWNet]

A floor or a ceiling whose argument arrives at a jump asks which side of the jump the argument comes from. That side was read only from the first derivative that is not zero. sqrt(x) at 0 has none that is finite, so floor(sqrt(x)) at 0 from the right was not answered, though it is 0 all the way in.

When the derivative test gives no answer, the side is now also read from the slope g(x) / (x - a). If the slope tends to a non-zero L, infinities included, then g has the sign of L on the right and the opposite sign on the left. For sqrt(x) the slope is 1 / sqrt(x), which tends to +oo.

This only applies where the expression is real throughout the side approached. A complex value's slope can still tend to a real infinity. On the left of 0, x ^ tanh(pi) is |x| ^ tanh(pi) times e ^ (i pi tanh(pi)), and the first version of this change answered its sign as -1. The test for realness is IsRealValued, or WhereRealValued's condition holding near the destination. The real domain condition can't be used for this, because for x ^ tanh(pi) it is True.

Limit	2.6.0	Now
floor(sqrt(x)) at 0, right	unevaluated	0
ceil(sqrt(x)) at 0, right	unevaluated	1
floor(x - sqrt(x)) at 0, right	unevaluated	-1
floor(1 - sqrt(1 - x)) at 1, left	unevaluated	0
floor(sqrt(x)) - (x^3)^(x/2) at 0, right	NaN	-1

Left unanswered:

• sgn(x ^ tanh(pi)) at 0 from the left.
• floor(sqrt(x)) at 0 from the left.
• floor(cbrt(x)) at 0 from the left. With downcasting off, a cube root of a negative is the principal one, so it is not real there.

The entry is in BREAKING-CHANGES.md.

Tests: ASignFromTheSlopeTest has 11 cases. The 9 answering cases fail on master, and the two cases that must decline pass on both.

Measured:

• The unit, MCP, terminal, interactive and F# suites pass.
• AOT builds.
• The boundcheck, rootcheck, casbench, propcheck, simpsweep, rulecheck, canoncheck and confluence harnesses all pass.
• limitcheck at DEPTH=3 over 30 seeds, against master: no case lost. There is one timeout fewer, which is noise: that case takes about 6 s either way.

Not fixed: floor(x * ln(x)) at 0 from the right. The floor asks for its argument's limit through a path that does not handle 0 * oo.

Found by limitcheck at SEED=59. No issue was filed.

🤖 Generated with Claude Code

[codecov]

Codecov Report

❌ Patch coverage is 88.88889% with 1 line in your changes missing coverage. Please review.
✅ Project coverage is 86.76%. Comparing base (595d9b6) to head (73bf581).

Files with missing lines	Patch %	Lines
...ath/Functions/Continuous/Limits/Transformations.cs	88.88%	0 Missing and 1 partial ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #2375      +/-   ##
==========================================
- Coverage   86.77%   86.76%   -0.01%     
==========================================
  Files         312      312              
  Lines       68979    68987       +8     
  Branches    20107    20112       +5     
==========================================
+ Hits        59854    59857       +3     
- Misses       4920     4923       +3     
- Partials     4205     4207       +2     

Flag	Coverage Δ
cpp	37.77% <ø> (ø)
csharp	87.30% <88.88%> (-0.01%)	⬇️
fsharp	32.97% <ø> (ø)
interactive	40.72% <ø> (ø)

Flags with carried forward coverage won't be shown. Click here to find out more.

Files with missing lines	Coverage Δ
...ath/Functions/Continuous/Limits/Transformations.cs	88.52% <88.88%> (+0.14%)	⬆️

... and 4 files with indirect coverage changes

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