Closure of the complete a=2 layer

Exact statement

The remaining odd-exponent type-II2 equation $9(4+3^v)(1+2^x3^y)+1=2^N$ , with $v,y\ge1$ and $x=v_2(37+3^{v+2})$ , has no solution. Modulo $73$ and the primary 2-adic gate force $v\equiv5\pmod{12}$ and $(x,y)\equiv(1,3),(4,11),(7,7)\pmod{(9,12)}$ . Modulo $577$ , where $3=2^{105}$ and $\operatorname{ord}_{577}(2)=144$ , the required logarithm classes $\{1,4,7\}\pmod9$ are disjoint from the available classes $\{2,3,5,6\}\pmod9$ . Together with the preceding canonical closures, this completes the $a=2$ layer of the active normalized branch.

StatusProved by displayed finite quotients and independently reconstructed

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

Context

The preceding packet had closed the other four canonical a=2 families. This odd-v type-II2 equation was the unique remaining family at the layer.

Proof or evidence

Modulo 27 fixes N=6 mod 18. A complete modulus-73 quotient plus the primary gate leaves three coupled x-y classes and v=5 mod 12. Writing 3=2^105 modulo 577 converts the last factor to 1+2^t; the quotient table contains only seven powers of two, none in an allowed t-class modulo 9.

Complete second-layer closure →

Verification notes

Independent code reconstructs the six-row modulus-73 quotient, the primary gate, the N=96 mod 162 3-adic lift, the 32-entry discrete-log table, a direct 162-by-8 residue separation, and 33153 bounded controls.

Limitations

This completes a=2 only inside one normalized proper three-addition branch.
FPRD-IC-T11 and FPRD-IC-T12 together close the complete a=3 positive-type-I layer.
No external review or novelty determination is recorded.

Open work

Use FPRD-IC-T11 for the a=3 entry table; do not reopen any entry with x at most 20.