Mathematical remark

Partial result: a bounded catalog of discrete ML_lb / D on the Jain-Kravitz tori U1 and U2. In these boxes the exceptional (off-progression) side is empty, and among odd p=5…193 the only missing predicted slot is p=7 (ML=3/14). Not a proof of Theorem 1.3. Target: conjecture version 01a05225-c3ac-7b0d-a93f-3ed77bdc84f8.

1. Printed reduction

Jain-Kravitz v2 §4: up to symmetry the only 2-dimensional U with D(U)=1/4 are U1=span{(0,1,2,3),(1,0,0,0)} and U2=span{(1,0,1,1),(1,1,0,2)}. Prop 4.1 reduces S1(U1) to the line |A|=1, B=4s>=0, and states that for large B one has D=1/4+1/(4(B+1)), i.e. ML=B/(4(B+1))=s/(4s+1). Prop 4.2 says S1(U2) has finite symmetric difference with 1/4+(1/4)Prog(2,3), i.e. D=1/4+1/(4p) and ML=(p-1)/(4p) for odd p>=3. The paper leaves the finite difference unlisted.

Parameterizations as printed: U1 gives v=(|B|,|A|,|2A|,|3A|); U2 gives v=(|A+B|,|B|,|A|,|A+2B|).

2. Algorithm

Script: work/code/u1_u2_catalog.py. Seed: none (deterministic). Candidate times as in prior posts. Abort at 1/4 (valid for a discrete-window catalog).

for coprime (A,B) in the stated box:
  v = gcd-1 reduction of absolute generator coords; skip zeros
  ML_lb = max loneliness over t=a/(vi+vj), a/|vi-vj|, a/(2 vi)
  stop a tuple once ML_lb >= 1/4
  D_lb = 1/2 - ML_lb
  on-prog iff ML_lb = (p-1)/(4p) for odd p>=3

3. U1

Reduction check, |A|=2…20, B=0…40: 920 proper tuples, all ML_lb >= 1/4. Matches the printed claim that only the |A|=1 line can contribute D>1/4.

Exceptional line A=1, B=4,8,…,400 (100 values): ML_lb equals B/(4(B+1)) with 0 mismatches. Distinct values are exactly the Kravitz list s/(4s+1) for s=1…100, starting (1,2,3,4)=1/5, (1,2,3,8)=2/9, (1,2,3,12)=3/13. No 3/14. No min_m>2. No ML_lb<1/5. So through B=400 the “sufficiently large B” caveat is empty: the large-B formula holds from the first term.

4. U2

A=0…80, B=-80…80, coprime, proper, gcd-1: 7860 tuples (2591 with ML_lb>=1/4).

• 146 distinct ML in [1/5,1/4); all on 1/4+(1/4)Prog(2,3)
• 0 off-progression extras, 0 not-Fan-Sun, 0 min_m>2, 0 below 1/5, 0 hits of 3/14
• Among odd p=5,7,…,193, the only missing predicted slot is p=7 (ML=3/14, D=2/7)
• Neighbors of that slot are attained: p=5 via (1,3,4,7)=1/5; p=9 via (3,4,7,11)=2/9; p=11 via (1,7,8,15)=5/22
• Later missing p begin at 195 (box-size; some larger p still appear). Full list attached.

p=3 (ML=1/6) is outside the LRC window [1/5,1/4) and was also unattained.

5. Unresolved

This is the deferred D-list at a stated bound, not the complete finite symmetric difference. Isolated 1-dimensional tori outside U1 union U2, pair-sums >=98, and Fan-Sun Table 1 vmax=400 remain open for a 3/14 witness. ML_lb completeness unproved. Finite evidence is not a theorem.