Closure of the complete a=3 positive-type-II layer

Exact statement

The normalized complementary family $27P(v)(2^x+3^y)+1=2^N$ , with $P$ any canonical negative-character factor, has no solution. Its exact gate quotient contains 524288 states; relaxed but solution-preserving filters modulo $6481,257,73,7,193$ leave respectively $16512,1268,390,124,0$ states.

StatusProved by a complete finite quotient and independently reconstructed

External reviewNo documented external or specialist review of these FPRD results is recorded.

Context

The paired-prime theorem had already removed a at least four, leaving three complementary low layers. This theorem eliminates the last of those layers completely.

Proof or evidence

Modulo 2048, the exact 2-adic gate has exponent period 512. A five-prime chain applied to all 524288 admissible a=3 gate states has empty intersection.

Complementary third-layer closure →

Verification notes

The primary C++ enumerator and an independent Python reconstruction reproduce every survivor count and the empty terminal set.

Limitations

The theorem concerns positive type II inside the normalized primitive f=g=0 proper fork.
The a=1,2 residual rows, other final valuations, and arbitrary expression trees remain open.
No external review or novelty determination is recorded.

Open work

Keep a=3 closed and concentrate on the exact residual quotient at a=1,2.