Exact 21-row quotient for the complementary low layers

Exact statement

Every remaining canonical solution of $3^aP(v)(2^x+3^y)+1=2^N$ , $a\in\{1,2,3\}$ , lies in one of 21 displayed periodic rows modulo $(v,y,N)=(1536,1536,18432)$ , all at $a=1,2$ , with $x\in\{1,3,4,5,6,8\}$ . These are necessary lift classes, not a classification.

StatusProved as a complete necessary quotient; every row is now classified by FPRD-IC-T21

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

Context

Once a=3 and every high-x state are removed, the complementary branch becomes a finite list of periodic lifting problems rather than an open-ended exponent search.

Proof or evidence

The exact quotient has 21 rows. A bounded control search through v,y at most 384 finds ten presentations representing four factorizations, all costing at least 2N+4.

Twenty-one residual lift classes →

Verification notes

The row list and four controls are stored in a machine-readable certificate; the page explicitly distinguishes residue zero from exponent zero.

Limitations

The row table alone did not exclude positive lifts such as y=1536; that obligation is discharged by FPRD-IC-T21.
The ten controls were initially bounded evidence and become a classification only after the full second-gate proof.
No external review or novelty determination is recorded.

Open work

Use this finite quotient only as the input to the completed second-gate classification FPRD-IC-T21.