Generative Structural Theory v1.24 — Component-Level Minimality: Structural Decomposition and Counting
- 1SD Lab LLC
Abstract
For a finite loop-free relation z, GST v1.21 counted the sub-relations of z that keep its reflexive transitive closure, and v1.22 counted their inclusion-minimal members and found their least number of pairs, each as a product or sum over the strongly connected components of z and the links of its condensation. Inside a component C, three numbers were left without a formula: s(C), the number of subsets of the pairs of C that are strongly connected on C; mu(C), the number of inclusion-minimal ones; and mc(C), the least number of pairs of one. This manuscript reduces all three to the same numbers for the undirected blocks of C; when the undirected graph of C is a single block, the reduction is trivial. If the pairs of C are divided among strongly connected parts whose incidence graph with the shared nodes is a tree, then a subset of the pairs of C is strongly connected on C if and only if its restriction to every part is strongly connected on that part. This gives an order isomorphism (a bijection preserving inclusion in both directions) between the strongly connected subsets of C and the tuples of strongly connected subsets of the parts, under which the inclusion-minimal members correspond exactly. Hence mu(C) = PROD mu(C_i), mc(C) = SUM mc(C_i) and s(C) = PROD s(C_i), and the proper members number PROD s(C_i) - 1. The blocks of the undirected graph of C always give such a decomposition: each block, with the pairs on its edges, is itself strongly connected, and a bridge carries both of its pairs. So the three numbers of any finite strongly connected component, and through them the counts for the whole relation, reduce to the blocks. This manuscript gives no general formula for s, mu or mc inside a single block; for a bridge they are 1, 1 and 2. For the fixed cut z_t, the component {0, 2, 3, 4} splits at node 0 into a bridge and a triangle, which restates the published values 60, 59, 4 and 7 as products and sums over blocks; this is an application, not a new check. Three small examples show that the formulas are no longer guaranteed when parts share two nodes, form a cycle, or leave a pair outside every part. The proofs and counts are presented as symbolic arguments, without machine enumeration. AI assistance was used in drafting and review, as detailed in the disclosures. No novelty is claimed; the block facts are taken from the standard text.
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BibTeX
@misc{okino2026generative,
title = {{Generative Structural Theory v1.24 — Component-Level Minimality: Structural Decomposition and Counting}},
author = {Koji Okino},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23453355},
url = {https://zenodo.org/records/23453355},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Generative Structural Theory v1.24 — Component-Level Minimality: Structural Decomposition and Counting
AU - Koji Okino
PY - 2026
DA - 2026/10/18
PB - Zenodo
DO - 10.5281/zenodo.23453355
UR - https://zenodo.org/records/23453355
AB - For a finite loop-free relation z, GST v1.21 counted the sub-relations of z that keep its reflexive transitive closure, and v1.22 counted their inclusion-minimal members and found their least number of pairs, each as a product or sum over the strongly connected components of z and the links of its condensation. Inside a component C, three numbers were left without a formula: s(C), the number of subsets of the pairs of C that are strongly connected on C; mu(C), the number of inclusion-minimal ones; and mc(C), the least number of pairs of one. This manuscript reduces all three to the same numbers for the undirected blocks of C; when the undirected graph of C is a single block, the reduction is trivial. If the pairs of C are divided among strongly connected parts whose incidence graph with the shared nodes is a tree, then a subset of the pairs of C is strongly connected on C if and only if its restriction to every part is strongly connected on that part. This gives an order isomorphism (a bijection preserving inclusion in both directions) between the strongly connected subsets of C and the tuples of strongly connected subsets of the parts, under which the inclusion-minimal members correspond exactly. Hence mu(C) = PROD mu(C_i), mc(C) = SUM mc(C_i) and s(C) = PROD s(C_i), and the proper members number PROD s(C_i) - 1. The blocks of the undirected graph of C always give such a decomposition: each block, with the pairs on its edges, is itself strongly connected, and a bridge carries both of its pairs. So the three numbers of any finite strongly connected component, and through them the counts for the whole relation, reduce to the blocks. This manuscript gives no general formula for s, mu or mc inside a single block; for a bridge they are 1, 1 and 2. For the fixed cut z_t, the component {0, 2, 3, 4} splits at node 0 into a bridge and a triangle, which restates the published values 60, 59, 4 and 7 as products and sums over blocks; this is an application, not a new check. Three small examples show that the formulas are no longer guaranteed when parts share two nodes, form a cycle, or leave a pair outside every part. The proofs and counts are presented as symbolic arguments, without machine enumeration. AI assistance was used in drafting and review, as detailed in the disclosures. No novelty is claimed; the block facts are taken from the standard text.
ER -