[Rafael-SOWNet]

Item 1 of the #1409 docket, and the redesign the book's notation remark asks for — "in mathematics, mod is a relation, not an operator or function". The remainder operator a mod n stays what it is, a number; the relation is a new node, Congruentf(Left, Right, Modulus), meaning n divides a - b.

Spelling (per the thread, and #1242): read as a = b (mod n) and as a ≡ b (mod n) — the modulus once at the end of the line, applying to every link of a chain, which is the conjunction of its links; printed in the ASCII spelling, as Stringize prints implies rather than the arrow; LaTeX a \equiv b \pmod{n}; SymPy Eq(Mod(a - b, n), 0). (mod n) after an order comparison, or ≡ without a modulus, is a parse error. One exception to implicit multiplication: a bracket that opens on mod is a modulus, so b (mod n) is not b * (mod n).

Decided where n | a - b is:

• integers outright: 5^6 = 1 (mod 7) → True (Example 6.5.12), -1 = 9 (mod 10) → True, 37457 = 38201 (mod 10) → False, 2^1 = 2^4 (mod 3) → False (Problem 6.7.11's spoof: exponents are not reduced); modulo 0 is equality; the modulus's sign is immaterial, as in n divides a - b;
• NaN off the integers (1/2 = 1 (mod 3), i = 1 (mod 3)), like divides;
• symbols where the difference is a polynomial every term of which the modulus divides: (n - a)^2 = a^2 (mod n) → True (§6.5.5 Try 2: n² − 2an), (x + y)^p = x^p + y^p (mod p) → True for p = 2, 3, 5, 7 (Problem 6.7.22), a + n = a (mod n), a n = 0 (mod n) — while (x + y)^4 = x^4 + y^4 (mod 4) stays as written, since 6x²y² is a multiple of 4 for some x, y and not others (the test pins both x = y = 1 → False and x = y = 2 → True).

Not in this PR: solving a x = b (mod n), inverses, CRT — item 3 of the docket, which sits on this node.

Tests: CongruenceTest — the chapter-6 rows above, chains, printing round trip in both spellings, LaTeX, SymPy, the entry points (MathS.NumberTheory.Congruent, Entity.CongruentTo), the operator untouched, the four parse errors. The reflective node lists (EveryNodeSurvivesEveryPipelineTest, EntitySerializationTest, StringizeRoundTrip) pick the node up on their own; PublicApi.txt +23; Syntax.md has a paragraph; BREAKING-CHANGES.md rows measured on a v2.5.0 build (every spelling was a parse error). Full suite green locally.

Part of #1409.

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