Closure of the odd-exponent branch

Exact statement

The complete odd- $N$ , $g=0$ , positive-type-I branch of the normalized proper fork has no displayed expression of cost at most $2N-1$ . Its only solution in the two nonsmooth canonical shapes is $7\cdot73+1=2^9$ , whose two presentations cost $2N+2$ and $2N+3$ .

StatusProved with congruence and primitive-divisor arguments; independently audited

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

Context

The surviving odd branch separates into two canonical exponential equations. Multiplicative orders and small moduli close every interior case while preserving the genuine 7 times 73 control.

Proof or evidence

Moduli 7, 19, 27, and 73 close the second shape; the first shape additionally uses Zsigmondy's theorem. Exact searches through exponent 2048 find only the proved control but are not used for completeness.

Odd-exponent branch →

Verification notes

Independent scripts reconstruct every periodic table and check the cost of the unique control.

Limitations

This is a substantial branch closure inside the normalized fork, not a solution of Selfridge's problem.

Open work

Do not reopen the odd-N branch; concentrate on the surviving even-N families.