Final-addend and character restrictions for proper forks

Exact statement

Every normalized primitive proper fork $3^a(B+C)(D+E)+2^f3^g=2^N$ satisfies $f\le2$ . Moreover the odd parts of the two nonsmooth inner sums have opposite values of the quadratic character $\chi(x)=+1$ for $x\equiv1,3\pmod8$ and $-1$ for $x\equiv5,7\pmod8$ .

StatusProved and independently reconstructed within FPRD Lab

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

Context

The elementary valuation argument first gives f at most four. A complete periodic residue analysis excludes f=3 and f=4.

Proof or evidence

The opposite-character law follows modulo eight. Six complete periodic rows survive the first sieve for f=3,4; each is contradicted modulo 9, 27, 64, 81, or 271.

Thin-branch theorem →

Verification notes

An independent implementation reconstructs the residue rows and final contradictions; bounded searches are retained only as controls.

Limitations

The theorem excludes two valuation branches but does not prove the desired cost inequality for all proper forks.
Likely novelty has not been assessed externally.

Open work

Use the restriction to keep all later attacks inside the surviving f=0,1,2 branches.