Large-x remoteness and the first imbalance filter

Exact statement

Every remaining $a=1$ large-x solution would already have displayed cost at most $2N-1$ . The 2-adic gates force $v\ge55313581$ in II1 and $v\ge60254788$ in II2. In the II2 subbranch $y<v$ , $y\ge5$ , an exact modulus- $19441$ quotient eliminates 17 of the 28 inherited classes for $x\pmod{540}$ , leaving 11 necessary classes.

StatusProved and independently audited as an intermediate reduction; II2 is closed by FPRD-IC-T07

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

Context

At this intermediate stage, closing x=4 left two large-x families. Their exact 2-adic gates make them remote, while the equation itself makes every solution cost-threatening.

Proof or evidence

Compatible 2-adic logarithm lifts give the two lower bounds on v. In II2 with y<v and y at least 5, the valuation identity forces N congruent to 332 modulo 486; combining this with periods modulo 19441 leaves exactly 11 of 28 x classes.

Large-x frontier →

Verification notes

Independent code reconstructs the full finite quotient, both 2-adic lifts through precision 160, the 17/11 class split, and the exact cost thresholds 29 and 41.

Limitations

The eleven congruence classes were only necessary conditions and are all eliminated by the later finite cover.
The unrestricted problem remains open.

Open work

Retain the 17/11 split as the input to FPRD-IC-T07; do not treat the eleven classes as active survivors.