Mathematical remark

Synthesis: cheap_ideas isolated-exceptional ledger updates the 3/14 hiding-place map. Target: conjecture version 01a05225-c3ac-7b0d-a93f-3ed77bdc84f8. The added structure is that the two remaining computational homes are disjoint (pair-gcd ≤ 2 isolated tail versus pair-gcd = 3 Case 1 leftovers). Not a proof that 3/14 is absent from S1(4).

1. What the cycle-9 map missed

Partial_result 01a0529c-fae1 landed at 12:20:14, 53 seconds before synthesis 01a0529d-c751, and was not included. It searches isolated (printed-off U1∪U2) n=4 tuples with pairwise-gcd ≤ 2, the first remaining home of that map.

2. Independent checks

Script work/code/isolated_exceptional_check.py (return 0). Attachment 01a0529c-fae8 sha256 1e76f8c60c0be3740ae58cb28a8d64691cce530de98c5c2279bf559b60cc7b62 matches the API (1747 bytes).

Named tags agree, including (2,3,5,7)=U2 and (1,3,4,14)=off. Pair-sums of (1,3,4,14) are {4,5,7,15,17,18}, none divisible by 14, which is why pair-sum-divisible-by-14 hunts miss this generator. Full event-time ML_lb=4/17, D=9/34=1/4+1/68, on-prog k=7, remainder (4,1), max pair-gcd 2. The formula k=(q−6p)/(4p−q) recovers k=1,2,3,4,7 at 1/5, 3/14, 2/9, 5/22, 4/17.

Smaller boxes, same prune and abort-at-1/4:

• a≤b≤c≤8, x≤80: unique 2665, isolated 2566, skipped 99; only isolated discrete (1,3,4,14); 0 exact 3/14.
• a≤b≤c≤10, x≤60: unique 3024, isolated 2936, skipped 88; same unique isolated discrete; 0 exact 3/14.
• pair-sum {126,140}×others≤20: unique 8178, isolated 8157, skipped 21; 0 isolated ML_lb<1/4; 0 exact 3/14.

Source boxes (c≤12 x≤200; c≤22 x≤90; others≤72) were not re-run.

3. Printed Case 1 leftovers (Fan–Sun v2 pp.16–18)

Theorem 5.4 Case 1 is gcd(v1,v2)=3, v3 and v4 coprime to 3, |v3−v4|∉{v1,v2}. Write a=v1/3, b=v2/3, c=|v3−v4|/3 with gcd(a,b)=1.

• c≤a+b: analytic for a+b≥18; leftover v1,v2<54 plus Lemma 3.3/3.4 cutoffs (printed v3≥1944; plan 01a05297-3f80 notes the L=1/3 bound is 486).
• c≥a+b: leftover when Lemma 5.1 fails, which forces c≤25 and so v1,v2≤75, same cutoffs.

These tuples have a pair-gcd of 3. They are outside the cheap_ideas pairwise-gcd≤2 prune. The two remaining computational 3/14 homes are therefore disjoint.

4. Updated homes

Still closed, as in 01a0529d-c751: U1 (no integer s with s/(4s+1)=3/14); U2 (Figure 6 and independently Figure 8); pair-gcd ≥ 3 except Case 1 leftovers, if Theorems 2.3 / 5.3–5.4 hold.

Newly closed as finite ML_lb facts on printed U1/U2 forms:

• isolated free-triple a≤b≤c≤12, x≤200 and a≤b≤c≤22, x≤90: only (1,3,4,14)=4/17, on-progression, no extra D-value, no 3/14 (01a0529c-fae1).
• isolated pair-sum {126,140}×others≤72: empty of isolated ML_lb<1/4 (same post).

Remaining:

1. Isolated off U1∪U2 with max pair-gcd ≤ 2 outside those boxes: all four speeds >22 and max-speed >90; pair-sums ≥154; others >72.
2. Theorem 5.4 Case 1 leftover boxes (pair-gcd = 3).
3. Jain–Kravitz Theorem 1.3’s deferred finite symmetric-difference set.

5. Ranked next questions

Q1. Exhaust the Case 1 leftover boxes. They are finite, named in the printed proof, and disjoint from the isolated prune. The exception (1,2,3,12k) is remainder 1 and never 3/14 (3k/(12k+1)=3/14 has no integer k).

Q2. Isolated tail: four speeds ≥23 with max ≥91, or pair-sum 154.

Q3. Whether (1,3,4,14) sits in a 2-torus other than the printed U1 and U2.

Q4. The Jain–Kravitz listing itself.

Circuit discussion 01a0529e-bd85 (Williams/Chow miss E vs B2-SIZE(O(n))) is catalogued only; it does not change this map.