Combinatorics and Graph Theory

A Closed Distinction Engine: Derivation of a 137-Object Registry, a (81, 40, 16) Partition, and a K9 Bedrock Collapse from Two Primitives

Authors: Jason Merwin
Comments: The document contains 13 pages

We define a closed combinatorial process — a distinction engine — that starts from two primitive tokens and iteratively generates new objects by a single rule: any two existing objects that have not yet been distinguished produce a new object whose DAG contains the union of their DAGs plus itself. The process halts at step 6 with 2 598 062 total objects and a maximum DAG size of 19. At DAG size 7, exactly 137 objects exist [T1]; they decompose uniquely into sectors of sizes (81, 40, 16)under a partition controlled by two structural parameters [T1]. We show, by exhaustive sweep,that this partition is realized by exactly one combination of engine and partition hyperparameters out of 256 tested. The 16-element sector (hereafter G) consists of 16 nodes with identical (45, 21, 7) cross-sector degree and zero variance, realized by exactly 2 tree shapes, uniform depth 5, and an 8+8leaf-count split [T1]. Embedding these 137 objects into DAG layers 8—16 reveals a broadening—freeze-out—locking ladder that compresses the 81-node spatial sector into a persistent 612-edge graph. Its 95th-percentile weighted backbone is exactly K9 on the 9 spatial degree-136 bedrock nodes, matched bit-for-bit [T1]. A weight-permutation null at N = 10 000 recovers neither the K9 topology nor thebedrock identity in a single trial, with a mean Jaccard overlap of 0.24. The combinatorial facts in this paper are thus established from pure topology of the distinction rule, without appeal to any physical interpretation.
Category: Combinatorics and Graph Theory