Classification of the complete positive-type-II even tower

Exact statement

The normalized positive-type-II family $3^aP(v)(2^x+3^y)+1=2^N$ with $a\ge1$ has exactly ten canonical proper solutions, representing $3\cdot5\cdot17+1=2^8$ , $3\cdot31\cdot11+1=2^{10}$ , $9\cdot5\cdot91+1=2^{12}$ , and $9\cdot13\cdot35+1=2^{12}$ . Every presentation costs at least $2N+4$ .

StatusProved from the complete residual quotient, full 2-adic gate, and one aligned prime; independently reconstructed

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

Context

At e=22 the full gate leaves 20,480 states across the ten nonrigid rows. The prime 65537 sees their complete exponent periods while 2^N depends only on the inherited N residue.

Proof or evidence

Eleven rows are valuation-rigid. Each other row contributes 2,048 Hensel states modulo 3,145,728; ord_65537(3)=65,536 and ord_65537(2)=32, and none of the 20,480 states satisfies the equation modulo 65,537.

Positive-type-II classification →

Verification notes

Two independent exact implementations verify the row partition, Hensel counts, multiplicative orders, zero terminal count, ten controls, and minimum displayed excess four.

Limitations

This classifies one a≥1 branch of the normalized primitive proper three-addition grammar, not arbitrary expressions.
The complementary a=0<g and final f=1,2 orientations remain open.
No external review or novelty determination is recorded.

Open work

Move to the complementary a=0<g orientation and final 2-adic exponents f=1,2; do not reopen the classified tower.