Universal 3-adic valuation law in the complementary f=1 branch

Exact statement

In every normalized primitive proper solution in the three complementary $f=1$ residue rows, let $(1+3^b)/2$ be the odd part of the even factor and write the other factor as $W=2^h+2^q3^y$ . Then $\min(b,y,g)=1+v_3(K+1-h)$ . All six tied-minimum strata whose leading 3-adic digit could cancel are impossible modulo 7 and 73.

StatusProved without exponent bounds; independently reconstructed

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

Context

Multiplying the f=1 equation by two exposes a four-term 3-adic sum. Its lowest exponent is min(b,y,g), except when a tied leading coefficient vanishes modulo three.

Proof or evidence

Canonical residue parity reduces all possible cancellations to six families. Complete exponent periods modulo 7 and 73 have empty joint survivor sets in every family, including a separate treatment of the sole minimum-one exception.

Complementary f=1 valuation law →

Verification notes

Independent Python and C++ implementations reconstruct all residue intersections. A 312,300-presentation bounded census finds eleven controls and no violation of the valuation identity; it is not used for completeness.

Limitations

The theorem controls but does not exclude all complementary f=1 solutions.
The f=2 split (2,0) now has an analogous valuation law, but both controlled branches, the f=0 rows, and arbitrary expression trees remain open.
No external review or novelty determination is recorded.

Open work

Combine this law with the f=2 valuation theorem and classify the first two minimum layers m=1,2.