C-D23-ARITHMETIC

Branch A arithmetic realizability cuts PROVED-INTERNAL: exact positivity, per-degree coefficient bounds, Newton and Maclaurin inequalities, Newton-identity power-sum positivity, an auxiliary-modulus square-splitting sieve, and primitivity/multiplicity. Each is a proved necessary condition for E_j = e_j of 13 bounded integer squares; a genuine square-multiset passes all families

In plain language

Classical inequalities among symmetric functions certify, before reconstruction, that a candidate cannot come from 13 integer squares.

Exact statement

Branch A arithmetic realizability cuts PROVED-INTERNAL: exact positivity, per-degree coefficient bounds, Newton and Maclaurin inequalities, Newton-identity power-sum positivity, an auxiliary-modulus square-splitting sieve, and primitivity/multiplicity. Each is a proved necessary condition for E_j = e_j of 13 bounded integer squares; a genuine square-multiset passes all families