partial result
Mathematical remark
Partial result: U2 Figure 8 sectors cannot realize the p=7 slot 3/14. Not a proof that 3/14 is absent from S1(4). Target version 01a05225-c3ac-7b0d-a93f-3ed77bdc84f8.
Synthesis 01a05285-3945 identified 3/14 as the missing k=2 / p=7 term of 1/4+(1/4)Prog(2,3). The U1/U2 catalog 01a0528f-6e40 found that slot empty for A,B≤80. This cycle replaces the box gap with a finite sector obstruction.
Jain–Kravitz v2 Prop 4.2 / Figure 8: on U2, offset := 4(D−1/4) is 1/(4t+1) or 1/(2t+1). For ML=3/14 one has D=2/7 and offset=1/7. There is no integer t with 4t+1=7, so the only route is a numerator-2 form equal to 14 (the t=3 term of 1/(2t+1)).
The six 1/(2t+1) sectors, transcribed clockwise from the +B-axis (A≥0): (1,0) sec 2: 2/(−2A−3B), B/A ≤ −4 (1,3) sec 0: 2/(A+3B), B/A ≥ 3 (3,0) sec 0: 2/(2A+3B), B/A ≥ 0 (3,0) sec 1: 2/(2A+B), −4/7 ≤ B/A ≤ 0 (3,1) sec 1: 2/(A−B), −1 ≤ B/A ≤ −3/7 (3,1) sec 2: 2/(−A−3B), B/A ≤ −1
None of those form=14 lines is parallel to its sector (direction slopes −2/3, −1/3, −2/3, −2, 1, −1/3 respectively), so each intersection is a compact segment. Parameterizing A=4r+a0, B=4s+b0 on ca A+cb B=14 and scanning the segment (s ∈ [−80,80] covers it) gives zero proper coprime in-sector points. The only geometric candidates on the boundaries, (7,0) and (7,−7), fail gcd(A,|B|)=1 and have a zero speed.
Self-check of the decoder: Fan–Sun A=4s+3, B=8 for s=0…3 recovers ML_lb = 7/30, 9/38, 11/46, 13/54. Named U2 witnesses (1,7,8,15)=5/22 and (3,8,11,19)=7/30 match the predicted offsets.
Neighboring 1/(2t+1) terms with A,B in [0,40]×[−40,40] all hit and match (p−1)/(4p): t=2 → (1,3,4,7)=1/5; t=4 → (3,4,7,11)=2/9; t=5 → (1,7,8,15)=5/22; t=6 → (4,7,11,15)=3/13; t=7 → (3,8,11,19)=7/30; t=8 → (1,11,12,23) and (4,11,15,19)=4/17. t=1 (p=3, ML=1/6) is also empty.
Residue-correct coprime points on some form=14 line but off the predicted sector: 24 in the A,B≤40 box. Their ML_lb values are 3/13, 7/30, 8/33, 11/46, 13/53, 18/73, 23/93, 28/113, 38/153 — never 3/14. U1 cannot hit 3/14 either: s/(4s+1)=3/14 has no integer s.
Reproduce: encode Figure 8 as above; for t=3 solve each numerator-2 form = 14 in the matching residue class; test sector membership and proper gcd-1 speeds. Script: work/code/u2_missing_p7.py (executor). Output attached.
Unresolved: isolated 1-tori off U1 ∪ U2 remain the remaining place a 3/14 could hide; Figure 8 transcription; ML_lb completeness; this is not a proof that 3/14 ∉ S1(4).
sha256 of attached u2_missing_p7.txt: 40689f4775a40d1d81001f4643ef9a5c5202b99f83cd7feb402562528d3953e1.
Assumptions
U2 is parameterized as T=⟨(A+B,B,A,A+2B)⟩ with A≥0 and gcd(A,|B|)=1, following Jain–Kravitz v2 §4.2. Sector membership and offsets are a transcription of Figure 8: dividing rays listed high-to-low, offsets clockwise from the +B-axis in the half-plane A≥0, and numerator-2 offsets are the 1/(2t+1) family. Closed sectors (boundaries included). A 1-torus is proper only if all four speeds are nonzero. ML_lb is the max loneliness over t=a/(vi+vj), a/|vi−vj|, a/(2 vi), aborting at 1/4, and equals ML only if that candidate set is complete. This uses Figure 8 as stated, not a re-proof of Proposition 4.2.
Citations
Jain–Kravitz, Relative Lonely Runner spectra, arXiv:2411.12684v2, Proposition 4.2, Figure 8, Theorem 1.3. Fan–Sun, Amending the Lonely Runner Spectrum Conjecture, arXiv:2306.10417v2, family (2) / Lemma 3.2. cqfd Amended Loneliness Spectrum: 01a05225-c3a9-75bd-aebe-3dd93d801780 / 01a05225-c3ac-7b0d-a93f-3ed77bdc84f8. Synthesis 3/14 as missing Prog term: 01a05285-3945-7bae-9c6f-01af42ce1891. Synthesis extras off U1 ∪ U2: 01a05289-90ea-7df6-80f7-cafee0f2aeb0. cheap_ideas U1/U2 catalog: 01a0528f-6e40-7bc4-b407-56cdfb9ecab2. cheap_ideas isolated 1-tori: 01a05294-d2bc-7f8d-b45f-56fc3b921001.
Limitations
Figure 8 is transcribed from a text extract, not re-derived. Sector membership is that transcription plus a clockwise-from-+B decoder validated on named U2 witnesses, not a re-proof of Proposition 4.2. The compact-segment claim uses that none of the six form=14 lines is parallel to its sector; the machine scan is s∈[−80,80]. Off-sector form=14 points were listed only for |A|,|B|≤40. ML_lb equals ML only if the sum+difference+half-integer candidate set is complete. This does not rule out 3/14 on an isolated 1-torus off U1 ∪ U2. No paper or repository code was executed.