Axiom-Free Invariants of the Cut-Set Lattice in the Chrysene Formalism: Admissible Maximal Chains, Obstruction Functionals, and Disjoint Optima on Hamiltonian Paths

This paper characterizes the combinatorial and redox invariants governing the stepwise linearization of the chrysene skeleton, defining the cleavage poset of a chain-forming cut set — whose maximal chains represent the cleavage orderings — to establish how substituent placement and ring-opening sequence determine synthetic pathway properties. Two functionals on this poset structure the analysis: an activation index that counts cleavage steps at unhydrogenated carbons, and the oxidation number, which acts as a path-independent state function that fixes total redox demand directly from the endpoint structures. Because the chrysene skeleton has an automorphism group of order 2, its two K-regions lie in a single symmetry orbit that is preserved under 6,12-disubstitution, meaning that ring opening at positions 11–12 and 5–6 represents a single experimental equivalence class. Of the 27 chain-forming cut sets, 24 admit a clean ordering, while the remaining sets are characterized precisely as those in which every bond joins two ring-fusion carbons. Across all cut sets and clean orderings, the activation index has an exact lower bound of 2, which is attained on the poset. Evaluating where the substituted carbons land along the product chain reveals a structural trade-off between activation count and terminal substituent placement. Exactly 1 of the 7 realized substituent patterns places a substituted carbon at a chain terminus as required for the target 1-yl membrane lipid, and every cut set realizing this terminal pattern carries an activation index of 3. Consequently, minimizing the activation index and delivering the terminal-linkage product are optimized by disjoint families of cut sets separated by 1 step, confirming that the 3-step sequence is optimal for the target lipid architecture. Extending the enumeration across all 5 four-ring cata-condensed isomers — including the branched isomer — shows that all 5 share the minimum activation index of 2 and the clean cut-set count of 24, while chrysene uniquely minimizes the total number of clean orderings. Finally, because the oxidation number is a state function, reaching the target ether lipid chain requires 52 reducing equivalents per dimer independently of mechanism, confirming the route-based material ledger directly from the molecular endpoints.