First even-exponent layer and the 3-adic control lift

Exact statement

At the first even layer $a=1$ , all but two large-x residue families and one x=4 control lift are excluded. In the x=4 family, every positive solution of $3(4+3^v)(1+16\cdot3^y)+1=2^N$ must satisfy $(v,y,N)\equiv(0,0,8)\pmod{(576,144,360)}$ , $v_3(2^N-13)=\min(v,y)+1$ , and $v+y>10^{70}$ .

StatusProved and independently audited as an intermediate reduction; the lift is closed by FPRD-IC-T05

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

Context

The zero-exponent identity 15 times 17 plus 1 equals 2^8 survives every low-modulus test and lifts to one 3-adic branch. Any positive integer point would be extraordinarily remote.

Proof or evidence

A seven-modulus cover leaves one triple class. A simple Hensel root modulo 3^140 then supplies the displayed lower bound on N and hence on v+y.

Even-layer reduction →

Verification notes

The residue cover, 140-step lift, 1,600 valuation controls, and a fresh 655,872-presentation box were independently reconstructed. Two generator defects were corrected before freezing the statement.

Limitations

A huge lower bound was not itself a contradiction; the later modulus-19441 certificate is what closes this branch.
The type-II2 large-x family is now closed by FPRD-IC-T07; type-II1 remains unresolved.

Open work

Retain the Hensel class as the input to FPRD-IC-T05; do not treat the x=4 lift as an active survivor.