Closure of the first a=2 boundary-lift family

Exact statement

The equation $9(1+4\cdot3^v)(1+2\cdot3^y)+1=2^N$ has no positive solution in the inherited $a=2$ type-I2 branch. Modulo $7$ and $13$ force $v\equiv5\pmod6$ , $y\equiv0\pmod6$ , and $N\equiv6\pmod{36}$ . An exact 3-adic comparison at $2^6$ sharpens this to $N\equiv42$ or $78\pmod{108}$ . Modulo $271$ , the resulting 25 left residues and 10 right residues are disjoint.

StatusProved by displayed valuation and finite residue certificates; independently reconstructed

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

Context

This was the first serious survivor after completion of a=1. Its formal rational boundary defeats every coarse coprime-modulus cover, but not the valuation-refined quotient.

Proof or evidence

Subtracting 64 gives 2^N-64=18(-3+3^y+2·3^v+4·3^{v+y}). On the coarse classes the bracket has 3-adic valuation one, so LTE yields v_3(N-6)=2. Since ord_271(2)=135 and ord_271(3)=30, the refined classes give 25 possible left residues and 10 possible right residues, with empty intersection.

First a=2 family closure →

Verification notes

Independent implementations reconstruct the entry quotients, the formal boundary, the valuation identity, both N-classes modulo 108, both multiplicative orders, and the complete modulus-271 residue sets.

Limitations

FPRD-IC-T10 closes the remaining odd-v type-II2 family and thereby completes a=2; the full three-addition grammar remains open.
FPRD-IC-T12 closes the remote a=3 classes isolated by FPRD-IC-T11.
No external review or novelty determination is recorded.

Open work

Use FPRD-IC-T10 for the final a=2 family; do not treat a=2 as an active survivor.