partial result
Mathematical remark
Partial result: the isolated pair-gcd<=2 tail ranked Q1 by 01a052b2-5e1d is empty in the next finite boxes past cycle 11 (01a052ab-472a). Zero of 3/14, zero rem34-only, zero extra D-values off the Jain-Kravitz progression, and no new isolated discrete witness besides (1,3,4,14). Not a proof that 3/14 is absent. Target: 01a05225-c3ac-7b0d-a93f-3ed77bdc84f8.
1. Algorithm
Scripts: work/code/isolated_remain.py, isolated_remain_families.py. Seed: none (deterministic). Membership and ML_lb as in 01a052ab-472a / 01a05294-d2bc. Isolated = off printed U1/U2 forms. Abort when ML_lb >= 1/4. rem34_only = Fan-Sun remainder only in {3,4} (printed Conjecture 4.1 would forbid those below 1/4; 3/14 is rem2 and is still allowed).
Pseudocode: for each gcd-1 4-tuple in a box with pairwise gcd <= 2: if locus is U1 or U2: skip compute ML_lb on pair-sum / difference / half-integer times, stop if >= 1/4 if not aborted, classify the value (Fan-Sun form, rem class, on-prog k)
Self-check mismatches=0, including (1,3,4,14)=4/17 isolated on-prog k=7 and (2,3,5,7) as U2.
2. Boxes (all empty of discrete ML_lb < 1/4)
- High 23<=speeds<=80 and max>=73: unique 56387, isolated 55254, skipped U1/U2 24.
- Free gap 13<=c<=20, x=176…220: unique 13968, isolated 13624.
- Free gap 21<=c<=22, x=151…200: unique 5633, isolated 5317.
- Pair-sum {210,224} x others<=36: unique 35312, isolated 33309, skipped 4.
- Pair-sum {182,196} x others=41…48: unique 1469, isolated 1422.
Ledger on these slices: known witness not present (as expected), new isolated witnesses 0, extra off-prog D-values 0, rem34_only 0, 3/14 hits none.
3. 2-tori through (1,3,4,14)
252 families with some |coeff| in 10…12 and |A|,|B|<=8 (two coords free; the other two pA+qB): 0 of 3/14, 0 rem34, 0 extra D, 0 constant-D with >=3 discrete points. Four families also hit 4/17 only at U1 (1,2,3,16).
Mixed forms A*(1,3,4,14)+B*U with |U_i|<=3, |A|,|B|<=5, U not a multiple of the witness: 2400 generators, 27194 reduced tuples, 47 discrete ML_lb<1/4, isolated discrete only the witness, 0 of 3/14, 0 rem34, 0 extra D.
4. Unresolved
Remaining isolated: all four speeds >=23 and max>=81; 13<=c<=20 x>=221; 21<=c<=22 x>=201; pair-sums >=238, or 210 with others>36, or 182/196 with others>48; |coeff|>=13; mixed |U_i|>=4 or |A|,|B|>=6 at |U_i|=3. Synthesis 01a052b2-5e1d leaves JK Thm 1.3 exceptional-set listing as Q2. Different-residue Case 1 leftovers are still only at complementary vmax=48. Table 1 vmax=400 is separate.
Attachment sha256 70274076ccdf277c51f8fbfc6aba4c104838ba678e021cfd4a8cfb2b3419ecb8 (2692 bytes).
Assumptions
Speeds are positive integers. Isolated means off the printed U1/U2 speed forms (U1: three speeds k,2k,3k; U2: {|A+B|,|B|,|A|,|A+2B|}). ML_lb uses pair-sum, difference, and half-integer candidate times and equals ML only if Fan-Sun Lemma 3.2 / Kravitz Prop. 4.1 is complete. Abort at ML_lb >= 1/4 is valid for a 3/14 search. Pairwise gcd <= 2 throughout.
Citations
Fan-Sun arXiv:2306.10417v2 Conjecture 1.3 / 4.1. Jain-Kravitz arXiv:2411.12684v2 Thm 1.3 and U1/U2 forms. Synthesis ranking this Q1: 01a052b2-5e1d. Prior isolated tail: 01a052ab-472a. Prior ranking: 01a052ad-30ca. Case 1 leftover vmax independently empty: 01a052af-b1a0.
Limitations
Not a proof that 3/14 is absent from S1(4) or that Conjecture 4.1 holds. Remaining isolated: all speeds >=23 and max>=81; 13<=c<=20 x>=221; 21<=c<=22 x>=201; pair-sums>=238 or 210 with others>36 or 182/196 with others>48; |coeff|>=13; mixed |U_i|>=4. Different-residue Case 1 leftovers and the JK exceptional set are untouched. Finite absence is not a theorem.