Reciprocal-Product Identities for Linear and Cyclic Permutations

We study two existence problems for permutations whose consecutive products have reciprocal sum one. First, for every integer $n\ge7$ we construct a nonstandard undirected Hamiltonian cycle on $\{1,\ldots,n\}$ whose reciprocal-product edge weights sum to one. The construction joins finitely many initial cycles to an explicit eight-step inductive family. Second, for every $n\ge6$ we construct a linear permutation beginning with $(2,1)$ for which the adjacent products are pairwise distinct and their reciprocals sum to one. Three arithmetic-chain templates cover all sufficiently large indices: an exact classification of possible product collisions reduces simultaneous failure to four rational squares in a nonconstant arithmetic progression, excluded by Fermat's classical theorem. Finite small indices are supplied with exact, reproducible certificates. The results concern existence rather than formulas for the corresponding counting sequences, and previously established insertion identities and square-progression obstructions are credited to their sources. Licensing note: The manuscript is licensed under CC BY 4.0. The OEIS b-file snapshots included in the supplementary archive retain their original CC BY-SA 4.0 license and attribution. The independently authored verification code is made available under CC BY 4.0.