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SystemSolveBudget

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Summary

Solves a system of equations by solving one after another with substitution, e.g.
let { x - y + a = 0, y + 2a = 0 } be a system of equations for variables { x, y }
Then we first find y from the first equation,
y = x + a
then we substitute it to all others
x + a + 2a = 0
then we find x
x = -3a
Then we substitute back
y = -3a + a = -2a

Summary

What a whole system solve is allowed to spend before it declines.

Remarks

Both ceilings, because neither is a bound on its own. A step here is one
candidate solution the elimination explores, and that is what compounds: each
elimination turns the next level's coefficients into nested radicals. Measured, the
systems that answer explore very few — a symbolic 2×2 takes 2, cyclic-4 takes 8,
and the largest that answers at all, five uncoupled quartics with 1024 solutions,
takes 341. Cyclic-5 passes 100 000 without finishing. So 10 000 admits everything
known to work with a factor of thirty to spare and still refuses the ones that run
away.
The clock is a backstop, not the bound, because a step can be arbitrarily expensive:
cyclic-4 spends five seconds in eight of them. It is set well above what any
answering case needs — the slowest takes about five seconds unloaded — since a
tight clock makes the same system answer or decline depending on what else the
machine is doing, and a flaky answer is worse than a slow one. That was measured
too: at five seconds the uncoupled case passed alone and failed inside the suite.
#896

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SystemSolveBudget