Null boxes for the Fan-Sun amended conjecture on near-tight speed tuples: a partial recomputation with an independent complete-witness screen

For positive integer speeds v1,…,vn let ML be the lonely-runner loneliness. A tuple is near-tight if ML < 1/n. Fan and Sun's amended spectrum conjecture (Conjecture 1.3) says ML = s/(ns+k) with 1 ≤ k ≤ n, or ML ≥ 1/n; with ML = p/q in lowest terms and k' = q − np this is exactly 1 ≤ k' ≤ n. We found no counterexample; this is a partial recomputation and a set of null boxes, not a proof and not a refutation. What was run. All primitive tuples of distinct positive speeds in boxes were enumerated with exact integer witnesses (the sum-time witness set m/(vi+vj), m ≤ D/2, is complete for positive speeds), and every near-tight tuple was recomputed with exact fractions (Tier C). Results: 7 speeds ≤ 80: 37 (k=1: 33, k=2: 4); 8 speeds ≤ 60: 36 (all k=1); 7 speeds ≤ 200: 55 (k=1: 51, k=2: 4); 8 speeds ≤ 130: 65 (all k=1); 6 speeds ≤ 200: 1085 (k=1: 971, k=3: 114). No tuple has k > n, and none has k > n/2 (Fan–Sun Conjecture 6.1). Novelty, exactly. The boxes 7 speeds ≤ 70 and 8 speeds ≤ 60 are already covered by Fan–Sun's Table 1, so they are recomputations; only 7 speeds in (70,200] and 8 speeds in (60,130] go beyond the literature, as null boxes. The septuple (2,7,9,11,13,20,54) with ML = 9/65 (k=2) is not a new finding: it is printed in neither paper, lies outside Cordella's 7-speed box (max 54 > 50), is found by Cordella's own survey.c once the box reaches 54, and lies inside Fan–Sun's searched range. Audit. A separate AI agent instance (not a human) re-screened every box with an independent Rust program using a complete witness set (sums, differences, peaks) and ran Cordella's survey.c: all tuple sets are identical (ACCEPT_WITH_CORRECTIONS; eight corrections applied). Limitations. The preregistered box 8 speeds ≤ 150 was not run; nothing is claimed outside the boxes; no Lean; no human audit.