Complete classification of the first two complementary minimum layers

Exact statement

In the normalized complementary $f=1$ and split- $(2,0)$ $f=2$ branches, every solution with $m=\min(b,y,g)\in\{1,2\}$ is one of exactly seventeen explicit proper presentations. All seventeen cost at least $2N+6$ . The eleven lift graphs rooted at improper rational identities are empty modulo $3^6$ ; 26 control-rooted graphs freeze completely; and the remaining three reduce to $2^{K+1}=11\cdot3^b+173$ or $2^K=3^g+295$ , whose nonzero lifts are excluded by Matveev's explicit lower bound and exact finite modular descent.

StatusProved with one classical linear-forms dependency; independently reconstructed

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

Context

The preceding exact quotient compressed the complete m=1,2 problem to forty periodic lift graphs, but did not determine whether any graph had a noncentered positive point.

Proof or evidence

Modulo 729 eliminates every boundary-rooted graph and leaves only three possible noncentered patterns. Two 2-adic gates reduce these to two exponential equations. Matveev bounds the lifted ternary exponent below 10^15; exact rational windows and moduli 29, 43, 59, 83, and 101 then eliminate every nonzero lift.

Complementary minimum-layer classification →

Verification notes

The primary Python certificate reconstructs all forty graphs and every finite descent count. A separately written C++20 audit starts from a frozen graph list and independently reproduces the 11/26/3 split and the terminal empty sets.

Limitations

This classifies only the first two least-ternary-exponent layers of two complementary branches.
Other normalized three-addition orientations and arbitrary expression trees remain open.
No external review or novelty determination is recorded.

Open work

Test whether the valuation period and an explicit linear-forms cutoff cross uniformly in m; otherwise open the complementary f=0 quotient.