A paired-prime obstruction closes every exterior layer a at least four

Exact statement

For every normalized primitive proper equation $3^aPQ+1=2^N$ with $f=g=0$ and $a\ge4$ , both canonical odd types $Q=1+2^x3^y$ and $Q=2^x+3^y$ are impossible. The valuation identity gives $27\mid N$ , so modulo $73$ and $262657$ both equations force $Q=0$ . Every canonical negative factor $P$ is nonzero at both primes, while the complete exponent classes making $Q=0$ at the two primes have no simultaneous lift.

StatusProved by displayed subgroup and incompatible-lift certificates

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

Context

The modulus selected for the remote a=3 attack has order 27 for the base 2. Once a is at least four, the valuation identity makes this a uniform rather than layer-specific obstruction.

Proof or evidence

Modulo 73 the two Q types each have three zero classes modulo (9,12). Modulo 262657 each has three zero classes modulo (27,14592). Projecting the latter classes modulo (9,12) gives an empty intersection in both cases.

Uniform high-layer obstruction →

Verification notes

Independent enumeration reconstructs both zero sets, verifies the nonvanishing of all four negative-factor shapes at both primes, and confirms that neither pair of zero sets has a simultaneous exponent lift.

Limitations

This theorem is confined to normalized primitive f=g=0 proper forks.
It does not settle positive type II at a=1,2,3, the a=0<g branch, f=1,2, or arbitrary expression trees.
No external review or novelty determination is recorded.

Open work

Do not resume exterior-layer induction above a=3; only the low positive-type-II layers remain in the g=0 branch.