abs(x) = -1 returns a non-empty set whose members do not satisfy the equation #812

|x| = a is inverted to a parametric set without the condition that makes it valid, so for a negative right-hand side it returns solutions that are not solutions.

"abs(x) = -1".ToEntity().Solve("x")     // { -e ^ (i * r_1) provided r_1 in RR }

The correct answer is the empty set: |x| ≥ 0 for every complex x, so |x| = -1 has no solution.

It is wrong, not merely unhelpful

Take the member at r_1 = 0:

x = -e ^ (i * 0) = -1
abs(-1) = 1            <- measured

1 ≠ -1, so that member does not satisfy the equation it was returned for. The same holds for every member, since abs(a * e^(i*r)) = |a| regardless of r.

Measured on master (21f0d16)

Cause, and its relation to #318

#318 asks for exactly this inversion and states the missing piece in its own text:

|x| = a should return { a e ^ (i * r) : r in RR and a >= 0 }

The r in RR half is implemented; the a >= 0 half is not. So the parametric-set part of #318 is done, and what is left of it is the guard — which turns out not to be a cosmetic refinement but the difference between a right answer and a wrong one.

For a symbolic a the guard cannot be decided, so the honest result is a Providedf on a >= 0 rather than a decision. For a literal negative a it is decidable and the answer is the empty set.

Why the harnesses did not find it

rootcheck verifies that returned roots satisfy their equation, which is precisely the property violated here — but it generates polynomial and rational equations, not ones under abs. Extending it to absolute-value and other non-invertible-on-the-reals shapes would have caught this, and is worth doing whether or not this is fixed first.