Closure of the cyclotomic y=0 ladder

Exact statement

In the normalized $g=0$ , positive-type-I branch, the ladder $3^aP=(2^{2rx}-1)/(2^x+1)$ has no surviving canonical negative-character solution with $r\ge2$ . The remaining $r=1$ controls cost at least $2N+2$ .

StatusProved with complete periodic certificates and an independent evaluator

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

Context

Factoring 2^{2rx}-1 by 2^x+1 converts one fork branch into a cyclotomic quotient with exact order and valuation constraints.

Proof or evidence

Five fixed equations are excluded modulo 163, 37, 7, 757, and 271. The last periodic family forces its parameter to be odd modulo 7681 and even modulo 8641.

Cyclotomic ladder closure →

Verification notes

Rational-quotient and alternating-polynomial implementations independently reproduce the complete residue certificates.

Limitations

The theorem concerns one canonical y=0 branch, not arbitrary formulas for powers of two.
The Bennett dependency belongs to inherited boundary classification, not the final modular contradictions.

Open work

Keep this branch closed unless a defect is found in a displayed periodic certificate.