Learn modulo a prime whether the elimination over the rationals declines (#2007)

TrySolveOverPolynomials' reading and Bareiss elimination now run first on
the system's image modulo 2^31 - 1: the same readings, pivots, products and
exact divisions, in residues that cost a machine multiplication where a
minor's rationals cost a greatest common divisor. The intermediates are
minors, so each image has no monomial the rational one lacks, and a product
past the term bound, an exponent past the degree bound, a refused entry or
a row reducing to 0 = c there is one over Q too. The solve over Q is then
declined at once. Anything else says nothing, and the solve runs as before;
nothing is answered on the strength of the images.

The Hermite reductions of Rubi's 1/(a + b cosh(x) + c sinh(x))^n under
u = e^x are twenty-row systems whose minors pass four thousand terms by the
sixth pivot, and they spent seconds each before declining.

Pocket of 1,951 rows: verdicts identical, 1,889 solved on both. Over the
1,886 solved on both, allocation 654 -> 429 GB, time 812 -> 733 s;
6.7.1:866 21.6 s -> 3.4 s (57.7 -> 1.3 GB), 880 16.0 s -> 1.6 s. Samples of
every family unchanged. The five rows that moved, re-run alone: 6.7.1:921
and 929 timeout -> 2.0 s and 2.8 s; the other three identical in
allocation. No row's allocation grew past 1.3x.

Part of #233.


Claude-Session: https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura

Co-authored-by: Claude Opus 5.5 (1M context) <noreply@anthropic.com>

Author: Rafael-SOWNet

Sources/AngouriMath/Functions/Algebra/Polynomials/MultivariatePolynomial.cs

@@ -225,6 +225,9 @@ private static void Accumulate(Dictionary<ulong, ERational> into, ulong monomial
             return result;
         }
 
+        /// <summary>The packed monomial of <paramref name="variable"/> to the first power.</summary>
+        internal static ulong MonomialOf(int variable) => Pack(variable, 1);
+
         internal MultivariatePolynomial ScaleBy(ERational factor)
         {
             if (factor.IsZero)