The a=3 entry table is empty through x=20

Exact statement

In the active normalized branch $27P(1+2^x3^y)+1=2^N$ , where $P$ is one of the four negative-character canonical factors, the two $x=1$ families and every entry with $3\le x\le20$ have no solution. Modulo $81$ gives $N\equiv18$ or $36\pmod{54}$ . The exact gate $27P+1\equiv2^x3^y\pmod{2^{2x}}$ , followed by a complete quotient modulo $73$ , forces $(x,y)\equiv(0,6),(3,2),(6,10)\pmod{(9,12)}$ . Exact finite covers eliminate $x=3,6,9,12,15,18$ . Hence every survivor has $x\ge21$ and $3\mid x$ ; FPRD-IC-T12 closes this remote tail.

StatusProved by exact residue quotients and independently reconstructed

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

Context

After a=1 and a=2 were completed, the four canonical factors had to be re-entered because x=v_2(27P+1) changes with the exterior layer.

Proof or evidence

Modulo 81 fixes the two N-progressions. Modulo 73 forces three coupled x-y classes. Complete exact-gate covers close the multiples of three through x=18; the intervening values fall to the same modulus-73 quotient.

Third-layer entry theorem →

Verification notes

A second implementation reconstructs every fixed-x survivor count, both modulo-81 classes, the modulus-73 quotient, all three 2-adic roots, and 655872 bounded canonical presentations with no solution.

Limitations

This theorem alone stops at x=20; FPRD-IC-T12 supplies the unbounded tail closure.
The theorem concerns one normalized proper three-addition branch, not arbitrary expression trees.
No external review or novelty determination is recorded.

Open work

Use FPRD-IC-T12 for the remote tail; do not reopen the a=3 positive-type-I entry table.