Complete f=0 triple-minimum classification

Exact statement

In the normalized complementary $f=0$ branch $RS+3^g=2^K$ , the quadratic gate leaves forty canonical factor placements and exactly twenty possible tied-minimum 3-adic cancellations. Only two can have $v=y=g$ . One has no solutions; the other has the unique solution $(2+3)(1+8\cdot3)+3=2^7$ , whose displayed construction costs $18=2K+4$ ones.

StatusProved; independently reconstructed

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

Context

The f=0 orientation was the remaining complementary branch after the first two minimum layers of f=1,2 were closed.

Proof or evidence

The first triple family has incompatible complete residue tables modulo 7 and 73. A 2-adic gate forces x=3 in the second, after which a difference-of-squares factorization of 65 gives one positive solution.

Complementary f=0 triple-minimum classification →

Verification notes

Independent Python and C++20 programs reproduce the 40/20/2 counts, both residue tables, the empty CRT intersection, the unique factorization survivor, and its four-one excess.

Limitations

The eighteen pair-minimum cancellation strata are classified by FPRD-IC-T30.
This is a theorem for the complementary f=0 branch of the normalized grammar only.
No external review or novelty determination is recorded.

Open work

Use FPRD-IC-T30 for the pair-minimum continuation; do not treat the eighteen strata as active survivors.