Greedy superstring has worst-case ratio at least 23/12, 53/27, 25/12 and 29/14 on strings of length 4, 5, 8 and 9

The greedy shortest-superstring algorithm repeatedly merges two strings with the largest overlap. Let ρ_k be its approximation ratio on strings of length k, in the worst case over inputs and tie-breaking, as defined by Chukhin, Kulikov, Mihajlin and Smal, who proved ρ_3 = 9/5 and ρ_k ≥ 2 for k ≥ 6. We prove ρ_4 ≥ 23/12 and ρ_5 ≥ 53/27, improving 2018 bounds, and ρ_k > 2 for every k ≥ 8, with ρ_8 ≥ 25/12, ρ_9 ≥ 29/14 and ρ_k ≥ (9k + 5)/(4k + 6) for odd k ≥ 9; for k = 8, k = 9 and odd k ≥ 11 these are, to our knowledge, the first bounds above 2. Each construction copies a few cyclic words, with a fresh letter per copy: greedy may cut every cycle open, after which the pieces barely overlap, while the optimum tours all cycles. Above 2 our constructions need worst-case ties throughout; for k = 4, 5 only the choice of cuts does. We explain why the gadget behind ρ_k ≥ 2 cannot exceed 2 for k < 6, and conjecture that 23/12 and 53/27 are optimal for constructions of this kind. A computer-assisted proof confirms this for blocks of at most 12 strings in which every string contains the fresh letter; at k = 4 the bound is attained. Version 1.1. New: Theorem 4 (rho_k >= 2 + 1/(2k-4) for every k >= 8, padding the k=8 gadget); Theorem 5 (rho_k >= (9k+5)/(4k+6) for every odd k >= 9, padding Shibata's construction with one shared letter; credited); best-bounds table showing rho_k > 2 for every k >= 8; Theorem 6 (computer-assisted: Conjecture 1 for blocks of at most 12 strings in which every string contains the fresh letter, and for the classes H_4(10), H_5(10); 273,975 certificates verified by two programs and a third independent implementation) and Corollary 1 (exact family limits 23/12, 53/27, 39/20 over all greedy runs); a remark on exact searches at k = 6, 7 (none above 2 found); full proof of Proposition 3(d); summary table of constructions; tie-breaking analysis extended to all new constructions; three authors (creators of 1.0 were corrected in place). The 1.0 results are unchanged. The record includes the paper, its LaTeX source, and a supplement with all instance certificates, the verification programs and their outputs.