Normalized proper three-addition fork

Exact statement

The active restricted grammar has value $M[A(B+C)(D+E)+F]$ , where displayed operands are positive $(2,3)$ -smooth constructions and all three displayed additions are paid. After cost-preserving normalization, $\gcd(B,C)=\gcd(D,E)=1$ , the inner equation is primitive, and a fork is proper when both inner sums are nonsmooth.

StatusPrecisely stated and proved within the restricted grammar

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

Context

The normal form isolates a substantial but narrow family of formulas with three additions. It is designed to retain factored construction costs while exposing exponential Diophantine structure.

Hypotheses and scope

Every displayed operand is a positive power of 2 times a nonnegative power of 3.
There is no reuse, subtraction, or division in the represented formula.

Proof or evidence

Common factors of either inner pair can be transferred to A without increasing cost, and the common exterior power of two can be transferred to M with a strict saving unless the inner equation is primitive.

Grammar and normalization →

Verification notes

The proof explicitly retains unit-operand charges, the principal place where a formal normalization can silently change the cost.

Limitations

The normal form does not cover arbitrary expression trees with three or more additions.
Smooth displayed costs are construction costs, not separate claims of optimal integer complexity.

Open work

Preserve the exact paid-atom cost and unit charges in every subsequent reduction.