Closure of the complete a=1 type-II2 branch

Exact statement

The large-x equation $3(4+3^v)(1+2^x3^y)+1=2^N$ has no solution under the inherited $a=1$ , type-II2 conditions. Modulo 7 forces $3\mid y$ ; the cases $y=2,4$ fall immediately and $y=3$ falls modulo 73. For $y\ge5$ , exact quotients modulo $19441$ , $6481$ , and $13$ eliminate all 28 inherited classes for $x\pmod{540}$ .

StatusProved with displayed finite certificates and independently reconstructed

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

Context

The inherited attack targeted one imbalance subbranch, but the successful congruences do not compare v and y once y is at least five. They therefore close both imbalances and the diagonal.

Proof or evidence

Modulo 7 forces y divisible by three. The small case y=3 has a nine-entry modulus-73 contradiction. For y at least five, N is 332 modulo 486; modulus 19441 leaves x classes 88, 112, 196, and 211, modulus 6481 leaves only 211, and modulus 13 eliminates it.

Complete type-II2 closure →

Verification notes

An independent implementation reconstructs the exact valuation classes, multiplicative orders, exponent progressions, residue tables, and survivor counts 28 to 4 to 1 to 0.

Limitations

The theorem closes one canonical family at the first exterior layer, not the full three-addition grammar.
The type-II1 large-x family is closed by FPRD-IC-T08; FPRD-IC-T09 and FPRD-IC-T10 complete a=2, while higher exterior layers remain unresolved.
No external review or novelty determination is recorded.

Open work

Use FPRD-IC-T08 for the type-II1 continuation; do not treat this family as an active survivor.