Two-block reversible lowering is the distinct-successor-set quotient

Exact statement

For $D\subseteq A^2$ , let $D_x=\{y:(x,y)\in D\}$ and index intermediate states by the distinct nonempty sets $R\in\{D_x\}$ . The transitions $r\xrightarrow{x}p_{D_x}$ and $p_R\xrightarrow{y}r$ recognize $D^*$ . This residual quotient is reversible iff its distinct successor sets are pairwise disjoint, and it is a graph-language for $G$ iff $D$ is independent in $G^{\boxtimes2}$ .

StatusSelf-contained proof; exhaustive small-relation audit and corrected standard-alphabet C7 regression pass

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

Context

Sequential lowering turns a synchronized signature tuple into a regular language over the original graph alphabet.

Proof or evidence

First symbols have the same right residual exactly when their successor sets agree, so equal fibres merge. Reversibility then fails exactly when one second symbol sends two distinct residual states to the root. Fixed-length block concatenation preserves distinguishability exactly when the block set is independent.

Reversible two-block compiler →

Verification notes

All 530 two-block relations through three symbols pass both the unmerged and residual-quotient criteria; 4,130 graph-language slice checks pass. With C7 labelled by 0 through 6, Meiburg's ten-word code drops from an irreversible eight-state first-coordinate presentation to the six-state quotient by merging the 2/4 and 3/5 residuals; an independent audit also checks strong-square independence.

Limitations

This is an explicit formulation of the block-code quotient used by Meiburg and should not carry a novelty claim or be presented as a new capacity bound.

Open work

Generalize the distinct-successor-set quotient to the typed five-block compiler and test compositional residual minimization across Gao's product nodes.