Gao's typed product has a constant-template language compiler

Exact statement

Retain the ten fixed-length languages $I,B,R,Q,P^{\mathrm H},P^{\mathrm V},X,O,H,V$ of a Gao gadget. The product code is the disjoint union of seven concatenation rectangles, and all ten output types close under a constant union/concatenation template. Twelve base syntax nodes and 32 nodes for each of six product steps give an exact 204-node typed syntax DAG denoting Gao's full $M_{40}$ -word code without enumerating a product word.

StatusStructural induction proved; exact cardinality audit reproduces Gao's M40

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

Context

The question is whether Gao's recursive set construction can be represented compositionally as regular languages after the five-dimensional base is generated without a vector table.

Proof or evidence

Gao's set-valued maps unpack into seven code rectangles and nine further closed typed formulas. Structural induction gives exact denotation and imports independence from Gao's lemma.

Typed product language compiler →

Verification notes

The base contributes zero, epsilon, and ten generated-language nodes; each of six product steps adds 32 canonical syntax nodes. The 204-node DAG reproduces the published M40 integer exactly.

Limitations

The independent-set construction and its product proof are Gao's; the contribution is the explicit language-level compiler and resource account.
The compiler presents a known code and does not improve the current C7 lower bound.

Open work

Retain this resource-preserving translation as the input for any later semantic minimization or comparison.