Closure of the complete a=1 layer

Exact statement

The last $a=1$ type-II1 family $3(2+3^v)(1+2^x3^y)+1=2^N$ , with $v$ odd, $N\equiv16\pmod{18}$ , and $x$ in the sixteen inherited classes modulo $270$ , has no solution. Modulo $7$ forces $(v,y)\equiv(3,0)\pmod6$ ; modulo $13$ , the left residues are then $\{0,7\}$ and the right residues are $\{3,10\}$ . Together with the preceding canonical closures, this completes the full $a=1$ layer of the active normalized branch.

StatusProved by two displayed residue quotients and independently reconstructed

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

Context

All other canonical forms at a=1 had already been closed. The sixteen remote type-II1 classes were the sole remaining first-layer family, and every hypothetical solution would have saved at least one one in the displayed grammar.

Proof or evidence

All sixteen x-classes are 2 modulo 6. A complete modulus-7 table has one matching cell, forcing v=3 and y=0 modulo 6. Modulo 13 this makes both powers of 3 equal to one, and the two possible left residues are disjoint from the two possible powers of 2.

Complete first-layer closure →

Verification notes

Independent code reconstructs the full six-by-three table, checks every inherited x-class separately modulo 12, and confirms the empty residue intersection. The unused 2-adic gate independently sharpens the hypothetical lower bound on v to 137494267053.

Limitations

This completes one canonical branch at a=1, not the full three-addition grammar or arbitrary formulas.
The complete a=2 and a=3 positive-type-I layers are now closed by FPRD-IC-T09 through FPRD-IC-T12.
No external review or novelty determination is recorded.

Open work

Use FPRD-IC-B01 and FPRD-IC-T09 for the first a=2 family; do not reopen a=1.