Infinitely many primes three modulo four

Restricting a successor-length prefix preserves every witnessed two-square factor representation.

The final decoded factor of a represented successor prefix has its own explicit two-square witnesses.

Induction on an arbitrary beta-coded product constructs a two-square representation from represented decoded factors.

Every concrete decoded entry in the canonical all-prime prefix is prime, by uniqueness of beta decoding.

Any finite product whose decoded factors are prime two or prime one modulo four is constructively a sum of two squares.

A positive natural number all of whose prime divisors are two or one modulo four has an explicitly constructed two-square representation via its canonical prime factorization.

Every witnessed natural of the form 4k+3 is nonzero.

Every witnessed natural of the form 4k+3 differs from the unit one.

A witnessed three-modulo-four natural cannot have a two-square representation.

A prime equal to two or one modulo four cannot also be three modulo four.

For every candidate and dividend, actual three-mod-four prime divisibility is constructively decidable.

Finite induction either excludes every three-mod-four prime divisor up to the supplied bound or returns an actual bounded prime-divisor witness.

Every natural congruent to three modulo four has an actual prime divisor congruent to three modulo four, obtained by constructive finite search and beta-coded prime factorization.

The subtraction-free Euclid number 4d+3 satisfies the exact identity (4d+3)+1=4(d+1).

Every actual subtraction-free Euclid number 4d+3 has a witnessed prime divisor congruent to three modulo four.

No prime dividing a nonzero common multiple can also divide its subtraction-free Euclid number 4c−1.

Every prime divisor of the exact Euclid number 4c−1 lies strictly above the bound encoded by its nonzero common multiple c.

For every natural bound, construct an actual strictly larger prime with an explicit residue witness p=4k+3.