Closure of the complete g=0 positive-type-I branch

Exact statement

Every normalized primitive proper $f=g=0$ fork whose positive-character factor is $1+2^x3^y$ has displayed cost at least $2N$ . The cyclotomic and odd- $N$ theorems close $a=0$ ; FPRD-IC-T08 and T10 close $a=1,2$ ; FPRD-IC-T11 and T12 close $a=3$ ; and FPRD-IC-T13 excludes every $a\ge4$ .

StatusProved by the cited exhaustive branch decomposition

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

Context

This consolidates the sustained sequence of odd, cyclotomic, and even-layer attacks into one complete infinite branch theorem.

Proof or evidence

Every valuation and canonical-factor case is assigned to a predecessor theorem, with the new paired-prime theorem removing the formerly unbounded exterior tail.

Complete positive-type-I branch →

Verification notes

The dependency chain is represented explicitly in the public claim graph; each constituent unbounded exclusion has its own reproducible certificate.

Limitations

This is an entire restricted branch, not the full normalized three-addition fork.
It does not imply Selfridge's conjecture for arbitrary formulas.
No external review or novelty determination is recorded.

Open work

Classify positive type II at a=1,2,3; the uniform theorem already supplies its high-layer boundary.