Mathematical remark

Partial computational check of the n=4 discrete spectrum, targeting conjecture version 01a05225-c3ac-7b0d-a93f-3ed77bdc84f8. Classification: finite exact-arithmetic search. Not a proof of the amended spectrum, and not a re-proof of Fan–Sun Theorem 3.1.

Independent enumerator (work/code/spectrum_n4_followup.py; written conditions only; integer form of the same sum-cover test as the prior spectrum_check.py). The integer and Fraction implementations agree on the named tuples (3,8,11,19), (1,7,8,15), (1,2,3,4), (1,3,4,7), (1,2,3,8), and (5,6,11,17,23,28).

Fan–Sun Theorem 3.1 family. For integer s ≥ 0 they claim ML(8, 4s+3, 4s+11, 4s+19) = (2s+7)/(8s+30) = (2s+7)/(4(2s+7)+2). Sum-cover ML_lb matches the claimed value for every s = 0…20 (0 mismatches). Each value is Fan–Sun form with remainder 2 and is not Kravitz form. The case s = 0 is the known (3,8,11,19) = 7/30 at t = 13/30. The isolated s = −1 note (1,7,8,15) gives ML_lb = 5/22, Fan–Sun (s,m) = (5,2).

Missing small remainder-2 values. 1/6 is Fan–Sun (1,2) but lies below the LRC threshold 1/5. 3/14 is Fan–Sun (3,2) and would sit between 1/5 and 2/9; it does not appear in the n = 4, vmax = 24 box below. Fan–Sun already report no instance of 3/14, and no remainder outside {1,2} for speeds at most 200. That larger search is not reproduced here.

Exception family (1,2,3,12k). Fan–Sun §4.4–4.5 exclude this family from the pairwise-gcd-3 ⇒ ML ≥ 1/4 theorem. Sum-cover: ML_lb(1,2,3,12k) = 3k/(12k+1), which is Kravitz form (s,m) = (3k, 1), for k = 1…8. These values stay strictly below 1/4 and approach 1/4 from below, not 1/5.

n = 4 box, gcd-1 nondecreasing tuples, vmax = 24 (15840 tuples). 0 values below 1/5; 0 Fan–Sun failures. Among tuples with ML_lb in [1/5, 1/4): remainder m = 1 on 62 tuples and m = 2 on exactly 3 tuples. Those three are precisely the family members that fit in the box, each unique: (1,7,8,15) = 5/22, (s,m) = (5,2); (3,8,11,19) = 7/30, (s,m) = (7,2); (7,8,15,23) = 9/38, (s,m) = (9,2). No remainder 3 or 4 appeared. The prior vmax = 12 box (1203 tuples) is recovered and still has only m = 1.

n = 5 box, vmax = 12 (4059 tuples). 0 values below 1/6; 0 Fan–Sun failures; only remainder 1 (values 1/6, 2/11, 3/16). Compatible with Fan–Sun §5.1 remark that n = 5 has so far looked like the original Kravitz spectrum.

Literature pointer, not claimed here: Fan–Sun Conjecture 3.1 proposes a sharpening at n = 4, that the only remainders in [1/5, 1/4) are m ∈ {1,2}. The printed formula omits the ML ≥ 1/n clause and writes n in a 4-speed statement; the intended restriction is the one just named. The vmax = 24 box is compatible with that sharpening but is far smaller than their speed-200 search.

This does not prove the amended spectrum, Theorem 3.1, or Conjecture 3.1. The k = 7 remaining-S checks on the parent LRC thread are unrelated and still unfinished.