Closure of the exceptional even x=4 lift

Exact statement

The inherited $a=1$ , II2, $x=4$ equation $3(4+3^v)(1+16\cdot3^y)+1=2^N$ has no positive solution. Its first 3-adic Hensel digit forces $N\equiv3248\pmod{9720}$ ; modulo $19441$ , the right side is then $15812$ , while the nine possible left-side residues omit $15812$ .

StatusProved by a finite nine-residue certificate and independently reconstructed

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

Context

The previous attack left one extraordinarily remote 3-adic lift of the boundary identity 15 times 17 plus 1 equals 2^8. The first Hensel digit makes one small multiplicative-order quotient decisive.

Proof or evidence

Writing v=576r and y=144s, both powers of 3 have a three-element orbit modulo 19441. The nine resulting left residues are 256, 259, 18974, 7304, 3273, 6725, 12496, 12643, and 15951, whereas the forced right residue is 15812.

Exceptional-lift closure →

Verification notes

An independent implementation reconstructs the Hensel digit, all period identities, the nine-entry table, and 20,000 transcription controls without importing the discovery code.

Limitations

The theorem closes one branch of the normalized three-addition grammar, not arbitrary formulas for powers of two.
No external review or novelty determination is recorded.

Open work

Use FPRD-IC-T07 for the type-II2 continuation and concentrate on the remaining large-x type-II1 family.