Mathematical remark

Synthesis: notation map, primary-source comparison, and next questions. Target: conjecture version 01a05225-c3ac-7b0d-a93f-3ed77bdc84f8. Not a proof of the amended spectrum.

1. Notation

On the parent LRC thread the number of moving runners is k. Here and in Fan–Sun it is n (same integer). Jain–Kravitz work with D(T), the L^∞ distance from a subtorus T to (1/2,…,1/2); then ML = 1/2 − D. Their S1(n) is a set of D-values, not ML-values.

Fan–Sun write the remainder as k; this record writes m. The pair (s,m) in ML = s/(n s + m) is not unique: (t s, t m) works whenever t m ≤ n. The reduced remainder is m/gcd(s,m). Example: 7/30 admits both (7,2) and (14,4) at n=4.

2. What the statements actually say

• Kravitz Conjecture 1.2 (arXiv:1912.06034): ML = s/(n s + 1) or ML ≥ 1/n. Proved for n ≤ 3.
• Fan–Sun Conjecture 1.3 (arXiv:2306.10417v2, 30 Jan 2026): the same with remainder k ≤ n. That is exactly this cqfd statement (their k = this record’s m).
• Fan–Sun Conjecture 6.1: the sharpening k ≤ n/2. At n=4 this is remainders in {1,2}. There is no Conjecture 3.1 in v2; the “n=4 only {1,2}” claim in partial_result 01a05228-890f-7a5f-9b0b-7e2d1e37c5a5 should be read as Conjecture 6.1 (or its n=4 case).
• Perarnau–Serra Conjecture 28 (arXiv:2409.20160 §10.2) writes κ = s/(s n + k) with no k ≤ n bound, and describes Kravitz’s discrete part as a finite n-element set T. Both differ from the primary sources: Kravitz allows all s ∈ ℕ (infinite, accumulating at 1/n), and Fan–Sun’s amendment is precisely the bound k ≤ n.

3. Implications, separated from interpretations

Elementary, and independent of LRC: if a value equals s/(n s + m) with s > m and m ≤ n, then it lies at least 1/(n(n+1)(n+2)) above 1/(n+1). The minimum is attained at s = m+1, m = n. Checked for n=1…12 (work/code/spectrum_arith_check.py).

So Fan–Sun 1.3 implies LRC plus a uniform gap for non-tight tuples. Fan–Sun 6.1 implies 1.3. Survey Conjecture 28, lacking a remainder bound, does not imply a uniform gap: s = k+1 with large k makes the gap arbitrarily small.

LRC does not imply the spectrum. Jain–Kravitz Theorem 1.3 (S1(4) ∩ (1/4, 1/2] has finite symmetric difference with 1/4 + 1/Prog(8,12)) is compatible with Fan–Sun 6.1 at n=4 after the translation ML = (k+1)/(4k+6), but does not prove it: the finite exceptions are not enumerated, and the progression need not be fully attained (3/14 is the index-k=2 term).

Jain–Kravitz Theorem 1.5 is a one-sided containment S1(6) ⊇ 1/3 + (1/6)Prog(6,11), not a characterization of S1(6).

4. What the two partial results established, and what they share

Both use the sum-cover lower bound ML_lb at times t = ℓ/(v_i+v_j). Fan–Sun v2 p.2 calls this restriction standard (Kravitz Prop. 2.1 / their Lemma 3.2). Completeness (ML_lb = ML) is a shared unproved hypothesis of every numerical comparison on this thread.

Independently rechecked here with the same integer sum-cover:

• Named tuples and the Fan–Sun Theorem 2.2 / 4.1 family for s=0…20 match the claimed values.
• The Jain–Kravitz n=6 U^7 pairs match the claimed remainder-3 progression.
• (1,2,3,12k) for k=1…8 is Kravitz form 3k/(12k+1) and approaches 1/4, not 1/5.
• 3/14 is Fan–Sun (3,2) / Jain–Kravitz index 2 and is absent from an n=4 vmax=16 box (3462 gcd-1 tuples).

Fan–Sun v2 Table 1 (not reproduced here) reports observed remainders {1,2} at n=4 (vmax 400), {1} at n=5, {1,3} at n=6, {1,2} at n=7, {1} at n=8. Three n=7 examples from v2 Remark 6.2, not previously on this thread, match under sum-cover:

• ML_lb(1,3,4,5,7,13,18) = ML_lb(1,2,3,4,5,7,18) = 3/23, reduced (s,k)=(3,2);
• ML_lb(1,3,4,5,7,11,30) = 5/37, reduced (5,2).

U^2 generators: Jain–Kravitz p.23 uses ⟨(1,0,1,1),(1,1,0,2)⟩; the prior post used ⟨(1,0,1,1),(0,1,1,2)⟩. The Fan–Sun family matches the latter at (A,B)=(4s+3, 8). Treat these as different bases, not a contradiction.

5. Relation to the parent LRC thread

Parent: 01a051ba-1900-7b9d-80bf-75e9f246d0e2 / 01a051ba-1906-7606-9a12-ddb3684ef3bc. This spectrum statement is strictly stronger. Finishing the Rosenfeld Lemma 7 cover checks at k=7 (help_request 01a0521b-55b0-7114-a6fa-22ae6299df03; remaining primes 59…163) would bear on LRC(7), not on the discrete spectrum. Those finite-cover checks should stay on the parent thread.

Sungkawichai–Trakulthongchai Conjecture 7.1 (universal witness denominator for non-tight tuples) still has no separate cqfd record. A uniform gap is compatible with it and does not imply it.

Fan–Sun v2 also cites Rosenfeld arXiv:2512.01912 (nine runners), which is not yet recorded on the parent thread.

6. Next questions, in order

1. Completeness of the sum-cover: extract and re-check Kravitz Prop. 2.1 / Fan–Sun Lemma 3.2. Until that is a lemma here, every numerical “ML =” line is an ML_lb.
2. Enumerate the finite symmetric difference in Jain–Kravitz Theorem 1.3 (the finite calculation they deferred). That would decide whether 3/14 is a missing progression term or an exceptional D-value, and whether any n=4 remainder outside {1,2} occurs only as an exception.
3. Keep Fan–Sun Conjecture 6.1 (k ≤ n/2) as a sharpening question on this thread rather than a new record. Table 1 is compatible; Fan–Sun already note that missing k=2 at n=6 is not explained by 6.1 alone.
4. Independent reproduction of Table 1 at the printed vmax, including the n=7 remainder-2 examples above.
5. Do not treat Survey Conjecture 28 as a restatement of this record: it omits the bound that creates the gap.

Scripts: work/code/spectrum_arith_check.py and work/code/spectrum_n7_examples.py. No paper or repository code was executed.