Uniform finiteness of the normalized complementary family

Exact statement

The normalized complementary $f=0,1,2$ family has only finitely many presentations. More precisely, its eight fixed factor-shape patterns contain at most $8\exp(7\cdot30^{15})$ presentations in total. This is a direct corollary of the Evertse--Schlickewei--Schmidt bound for nondegenerate S-unit equations: after expansion and division by $2^{K+f}$ , five positive $\{2,3\}$ -units sum to one in a multiplicative group of rank at most six.

StatusProved from an established theorem; not an effective height bound

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

Context

The proposed uniform Matveev crossing requires a one-dimensional logarithmic form. Before a dominance pattern is frozen, the complementary equation expands into five positive S-units instead.

Proof or evidence

For each fixed factor shape, positivity makes every S-unit solution nondegenerate and the exponent-to-coordinate map is injective. The Evertse--Schlickewei--Schmidt theorem with n=5 and rank at most 6 gives the displayed count.

Uniform complementary finiteness →

Verification notes

The rank, injectivity, eight shape patterns, and exact specialization of the published bound are displayed in the proof packet.

Limitations

The theorem bounds the number of solutions, not their exponent heights.
The numerical bound is deliberately crude and is not used computationally.
This concerns a normalized restricted grammar, not arbitrary integer-complexity expressions.

Open work

Do not treat the solution-count bound as a height cutoff; obtain finite congruence quotients inside the remaining f=0 strata.