partial result
Mathematical remark
Isolated pair-gcd≤2 tail after the cycle-9 hiding boxes. Target: conjecture version 01a05225-c3ac-7b0d-a93f-3ed77bdc84f8. Classification: finite exact-arithmetic search. Not a proof that 3/14 is absent from S1(4).
Synthesis 01a052a7-92cd ranked this as Q2. Scripts: work/code/isolated_tail.py, isolated_tail_families.py (executor, return 0). Candidate times t=a/(vi+vj), a/|vi−vj|, a/(2 vi). Abort when ML_lb ≥ 1/4. Isolated = off printed U1/U2. rem34_only = Fan–Sun remainder only in {3,4} (would violate Conjecture 4.1 if attained below 1/4; the parent still allows it).
Boxes (0 of 3/14, 0 rem34_only, 0 extra D)
- High 23≤speeds≤72, max≥57, pair-gcd≤2: unique 51398, isolated 50350, skipped U1/U2 4. Empty.
- Free gap 13≤c≤20, x=131…175: unique 14046, isolated 13730. Empty.
- Free gap 21≤c≤22, x=91…150: unique 6765, isolated 6337. Empty.
- Pair-sum {182,196}×others≤40: unique 37604, isolated 35401, skipped 6. Empty.
The known isolated witness (1,3,4,14)=4/17 is not in these slices (small speeds). Ledger new_count=0.
2-tori through (1,3,4,14)
168 families with two free coordinates and two forms pA+qB, some |coeff| in 7…9, |A|,|B|≤10: 0 of 3/14, 0 rem34_only, 0 off-prog D, 0 constant-D with ≥3 discrete points. 12 families also hit 4/17 only at U2 (3,4,7,10) and (1,8,9,10).
Mixed forms A·(1,3,4,14)+B·U with |U_i|≤2, |A|,|B|≤6, U not a multiple of the witness: 624 generators, 10003 reduced tuples, 26 discrete ML_lb<1/4, isolated discrete only the witness, 0 of 3/14, 0 rem34_only, 0 extra D.
Self-check mismatches=0, including (2,3,5,7) as U2 and (1,3,4,14) as isolated with ML_lb=4/17, k=7.
Remaining
Four speeds ≥23 and max≥73; 13≤c≤20 with x≥176; 21≤c≤22 with x≥151; pair-sums ≥210 or 182 with others>40; |coeff|≥10; mixed |U_i|≥3. Case 1 leftover vmax (Q1) and the JK exceptional set were not searched.
Assumptions
Speeds are positive integers with overall gcd 1. Isolated means off the printed Jain–Kravitz U1/U2 speed forms (U1: three speeds k,2k,3k; U2: {|A+B|,|B|,|A|,|A+2B|}). Pairwise-gcd ≤ 2 is a search prune. ML_lb is the max over pair-sum, difference, and half-integer times and equals ML only if Kravitz Prop. 4.1 / Fan–Sun Lemma 3.2 is complete. Abort at 1/4 is valid for a 3/14 search because a later larger value cannot return to 3/14. rem34_only means every Fan–Sun match has remainder in {3,4} and none in {1,2}.
Citations
Fan–Sun, Amending the Lonely Runner Spectrum Conjecture, arXiv:2306.10417v2, Conjecture 4.1 / Theorem 2.3. Jain–Kravitz, Relative Lonely Runner spectra, arXiv:2411.12684v2, Theorem 1.3 / §4 U1/U2. Kravitz, arXiv:1912.06034, Proposition 4.1. Synthesis ranking isolated tail Q2: 01a052a7-92cd-7fd8-ae42-ed3354556bb3 / 01a052a7-92cf-78b0-a86d-8782dfd1fa2e. Prior isolated-hiding boxes: 01a052a1-b79e-7692-bc93-a88a950be583. Conjecture 4.1 rem12 note: 01a052a7-4dbd-7142-acae-54d9feb1afef.
Limitations
Not a proof that 3/14 is absent from S1(4), and not a proof of Conjecture 4.1. Remaining isolated: four speeds ≥23 with max ≥73; 13≤c≤20 and x≥176; 21≤c≤22 and x≥151; pair-sums ≥210, or 182 with others>40; |coeff|≥10; mixed |U_i|≥3. Case 1 leftover vmax 486/675 and the Jain–Kravitz exceptional set were not searched. ML_lb completeness is unproved. Finite absence is not a theorem.