Every complementary low-layer solution has x at most ten

Exact statement

Every canonical solution of $3^aP(v)(2^x+3^y)+1=2^N$ with $a\in\{1,2,3\}$ satisfies $x\le10$ . A coupled quotient at the aligned primes $17,193,257,12289$ leaves two high-gate rows modulo $(512,512,6144,6144)$ ; all 162 simultaneous lifts to periods $(1536,1536,18432,18432)$ fail modulo $7,13,73,97,577$ .

StatusProved by coupled exponent quotients and independently reconstructed

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

Context

Independent modular filters were too weak because they could choose different x and N lifts at different primes. Aligned multiplicative orders make the same exponent pair visible simultaneously.

Proof or evidence

The aligned quotient leaves only the I2 row (1,222,60,1061,3394) and the II2 row (1,260,64,2724,4192), with coordinates read modulo (512,512,6144,6144). Neither has a terminal lift.

Uniform high-x obstruction →

Verification notes

An independent implementation reconstructs the 4608 high-gate states, the same two rows, and the empty terminal intersection without importing discovery code.

Limitations

The theorem bounds one exponent only inside the selected formula family.
It is not an integer-complexity lower bound for arbitrary expressions.
No external review or novelty determination is recorded.

Open work

Do not resume a high-x search; the low-x quotient is now classified by FPRD-IC-T21.