Closure of the complete a=3 positive-type-I layer

Exact statement

The remote branch $27P(1+2^x3^y)+1=2^N$ , with $x\ge21$ , $3\mid x$ , and $P$ one of the three surviving negative-character shapes, has no solution. Modulo $262657$ , where $\operatorname{ord}(2)=27$ and $\operatorname{ord}(3)=14592$ , the complete quotient leaves eight residue rows. Six fail modulo $7$ and two fail modulo $13$ . Together with FPRD-IC-T11, the complete $a=3$ positive-type-I layer is empty.

StatusProved by complete residue quotients and independently reconstructed

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

Context

The preceding entry theorem pushed all hypothetical a=3 solutions beyond x=20 and made each one cost-threatening. The remote quotient closes them uniformly rather than one x-value at a time.

Proof or evidence

The modulus-262657 quotient checks 623808 complete residue triples for each of I2, II1, and II2. It leaves one, four, and three rows respectively; displayed residue sets modulo 7 and 13 eliminate all eight.

Remote third-layer closure →

Verification notes

An independent solve-for-3^y implementation constructs a discrete-log table rather than enumerating y and reproduces exactly the same eight terminal rows and empty final intersections.

Limitations

This closes a=3 only for the positive-type-I orientation of the normalized f=g=0 proper fork.
The full three-addition grammar and arbitrary expressions remain open.
No external review or novelty determination is recorded.

Open work

Treat positive type I as closed and move to the three low positive-type-II layers.