Coprime-modulus obstruction at the a=2 boundary lift

Exact statement

For the $a=2$ type-I2 equation $9(1+4\cdot3^v)(1+2\cdot3^y)+1=2^N$ , the coarse necessary classes $v\equiv5\pmod6$ , $y\equiv0\pmod6$ , and $N\equiv6\pmod{36}$ cannot be excluded by any finite collection of moduli coprime to $6$ : the formal boundary $(v,y,N)=(-1,0,6)$ supplies compatible residues. The sharper 3-adic condition $N\equiv42$ or $78\pmod{108}$ breaks this obstruction and enables FPRD-IC-T09.

StatusProved algebraically and checked on independent finite quotients

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

Context

The first serious a=2 survivor behaves like a lift of a rational exponent boundary. Recognizing that obstruction prevents an unproductive repetition of the successful a=1 prime-cover strategy.

Proof or evidence

Modulo 7 and 13 give the coarse exponent classes. Before the valuation refinement, for every modulus coprime to 6 the substitutions 3^v=3^{-1}, 3^y=1, and 2^N=2^6 make the equation an identity. Subtracting 64 and applying LTE then gives v_3(N-6)=2, removing the formal boundary from the admissible N-classes.

Next-layer boundary →

Verification notes

Independent enumeration checks the entry quotients, the valuation bracket, the two N-classes modulo 108, and the formal boundary on representative coprime moduli.

Limitations

This is a restriction on congruence-cover methods, not evidence that an integer solution exists.
The obstruction applies only before the 3-adic refinement; the refined family is closed in FPRD-IC-T09.

Open work

Apply the 3-adic refinement before coprime order moduli; FPRD-IC-T09 shows that this closes the family.