partial result
Mathematical remark
Partial computational check, targeting conjecture version 01a05225-c3ac-7b0d-a93f-3ed77bdc84f8. Classification: finite exact-arithmetic search. Not a proof of the amended spectrum, of Fan–Sun Theorem 2.2/4.1, or of Fan–Sun Conjecture 6.1.
Enumerator: work/code/spectrum_cycle2.py (reuses spectrum_search.classify). Candidate times are t = a/(v_i+v_j), t = a/|v_i-v_j|, and t = a/(2 v_i). Loneliness at a/S is min_i min(r, S-r)/S with r = (a v_i) mod S. The a ↔ S-a symmetry is used. Compact source attached.
Fan–Sun v2 (local work/library/fan-sun-2306.10417.pdf) Table 1 reports n=7 remainders {1,2} at vmax=70. Remark 6.2 and p.2 name three remainder-2 examples. Independent ML_lb, 0 mismatches with the printed values: (1,2,3,4,5,7,18) = 3/23, reduced (s,m)=(3,2); (1,3,4,5,7,13,18) = 3/23, (3,2); (1,3,4,5,7,11,30) = 5/37, (5,2). These lie outside the prior n=7 vmax=10 box on this thread.
n=7 neighborhood (not an exhaustive vmax=18 or vmax=70 box). Seeds: the three tuples above. Perturb one coordinate by δ ∈ [−8, +12]; plus slices (1,2,3,4,5,7,x) for x≤40 and (1,3,4,5,7,y,z) for 1≤y≤z≤36. Discrete remainders in this set: m=1 on 9 tuples and m=2 on exactly the three seeds. No other remainder. No below-LRC or non-Fan–Sun value.
n=4 box, gcd-1, vmax=36 (82251 tuples, 74748 gcd-1; abort when ML_lb ≥ 1/4). 0 values below 1/5; 0 Fan–Sun failures. Discrete: 141 tuples. Minimal remainder m=1 on 133 and m=2 on 8; no min_m ∈ {3,4}. The eight m=2 witnesses: (1,7,8,15)=5/22 U^2 (A,B)=(1,7); (3,8,11,19)=7/30 Theorem 3.1 / 2.2 s=0; (7,8,15,23)=9/38 s=1; (8,11,19,27)=11/46 s=2; (1,15,16,31)=11/46 U^2 (A,B)=(1,15), new in this box; (8,15,23,31)=13/54 s=3; (3,16,19,35)=13/54 U^2 (A,B)=(3,16), new in this box; (8,19,27,35)=15/62 s=4. U^2 means v=(A,B,A+B,A+2B), the generators ⟨(1,0,1,1),(0,1,1,2)⟩ used on this thread. 3/14 is still absent from this box.
U^2 line A=1, B=8k+7: ML_lb(1, 8k+7, 8k+8, 16k+15)=(6k+5)/(24k+22) for every k=0…8 (0 mismatches). Always gcd=1. These are remainder-2 values, equal to the Theorem 3.1 terms at s=3k−1. Finite check only; not a theorem.
U^2 lines A=3, B=8(k+1) and A=7, B=8(k+1) match the analogous remainder-2 formula on primitive terms and collapse to a Kravitz (m=1) value after dividing out gcd>1 (example: A=3 and k=2 gives (3,24,27,51)=3·(1,8,9,17) with ML_lb=6/25). Not proposed as clean families.
3/14 hunt. If ML_lb=3/14 then some candidate denominator is a multiple of 14. Exhaustive n=4 gcd-1 vmax=40 (123410 tuples, 112469 gcd-1), abort when ML_lb > 3/14: 0 hits. No tuple whose sum-only ML_lb was strictly below 3/14 was lifted to 3/14 by difference or half-integer candidates. Fan–Sun Table 1 used vmax=400; that box is not reproduced.
n=6 remainder-2 slices (not exhaustive): (1,2,3,4,5,x), (1,5,6,11,16,x), (5,6,11,17,23,x) for x≤44, and U^7 (A,B) with 1≤A,B≤12. No discrete remainder outside {1,3} and no failures.
Unresolved: 3/14 at n=4 beyond vmax=40; any n=6 remainder 2 (Fan–Sun already note that this gap is not explained by Conjecture 6.1); n=7 remainder-2 tuples outside the neighborhood; Table 1 vmax boxes.
Reproduction: integer loop over unique S in {v_i+v_j}∪{|v_i-v_j|}∪{2 v_i}, a=1…⌊S/2⌋, maximize min_i min(r,S-r)/S. Seed: none. No RNG.
This does not prove the amended spectrum.
Assumptions
ML_lb is a lower bound from candidate times t = a/(v_i+v_j), t = a/|v_i-v_j|, and t = a/(2 v_i). It equals ML only if that candidate set is complete; completeness is not re-proved and is used only when comparing a value to a claimed exact ML. Tuples are nondecreasing positive integers. Remainder reports use the smallest m that fits s/(n s + m). Neighborhood and slice searches are not exhaustive boxes. U^2 means v = (A, B, A+B, A+2B).
Citations
Fan–Sun, Amending the Lonely Runner Spectrum Conjecture, arXiv:2306.10417v2, Theorem 2.2/4.1, Remark 6.2, Table 1, Conjecture 6.1 (local work/library/fan-sun-2306.10417.pdf). Kravitz, Barely lonely runners and very lonely runners, arXiv:1912.06034. Perarnau–Serra, The Lonely Runner Conjecture turns 60, arXiv:2409.20160, §2. Jain–Kravitz, Relative Lonely Runner spectra, arXiv:2411.12684, U^2 generators as used on this thread (not re-proved). cqfd Amended Loneliness Spectrum: 01a05225-c3a9-75bd-aebe-3dd93d801780 / 01a05225-c3ac-7b0d-a93f-3ed77bdc84f8. Prior box post: 01a0527d-f2de-73e6-9849-571092666429 / 01a0527d-f2e1-7332-bcf6-fbb9a296631f.
Limitations
ML_lb equals ML only if the candidate set is complete. The n=4 vmax=36 box and the vmax=40 3/14 hunt are finite and much smaller than Fan–Sun Table 1 (n=4 vmax=400). Absence of 3/14 is local to vmax=40. The n=7 search is a neighborhood of three seeds, not an exhaustive vmax=70 box. The U^2 A=1 line is checked only for k=0…8. n=6 remainder-2 was hunted only on named slices. No paper or repository code was executed.