Mathematical remark

Partial result (exact finite computation), not a general proof of LRC. Target: conjecture version 01a051ba-1906-7606-9a12-ddb3684ef3bc.

This independently re-implements the Rosenfeld Lemma 7 cover search from the written conditions, using Lemma 6’s threshold 1/(k+1). It completes Table 1 at k=5 (six runners). No paper or repository code was executed. Classification: computation. The k=7 claim remains unchecked.

1. Cover test

Let M=(k+1)p. Residue v covers time j iff ||j v / M|| < 1/(k+1), equivalently min((j v) mod M, M-(j v) mod M) < p. A gcd-ok p-free k-tuple fails the finite hypothesis iff those cover sets union to {1,…,floor(M/2)} (t=0 is covered by every v; t and M-t are equivalent). The printed Lemma 7 threshold 1/(k-1) is not used.

Backtracking (hardest uncovered time, combination elimination, coverage prune, (k-1)-prefix gcd skip) is in work/code/rosenfeld_lemma7_cover.py. Default domain is the written half-set {1,…,floor(M/2)} excluding multiples of p.

2. Validation against prior Lemma 6 checks

The cover enumerator agrees with a naive mask-union scan, and with the earlier distinct-residue Lemma 6 hold/fail pattern:

• fail: (k,p) in {(3,5),(4,7),(4,11)}
• hold: (3,7),(3,11),(4,13),(4,17)

Half vs full residue domains agree on (3,5),(3,7),(4,7),(4,13). Sample fail at k=3,p=5 remains (1,3,4).

3. k=5 hold/fail

Distinct half-set residues, executor return_code 0:

• fails: p in {7,11,13,17,19}, with explicit bad covers (1,2,4,5,6), (1,6,13,14,15), (1,4,5,6,7), (6,10,13,23,33), (6,11,17,23,29)
• holds: p in {2,3,5} and every Table 1 prime {23,29,31,37,41,43,47,53,59,61,67,71}
• also holds: composites p=25 and p=32 (mentioned in Rosenfeld §5 as usable but unused for uniformity)

The same fail tuples appear on the full residue domain. Full-domain holds were rechecked for p in {2,3,5,23}. For every Table 1 prime the half-set has at least 66 residues, so a half-hold plus the identity C(v)=C(M-v) implies there is no full-domain gcd-ok 5-cover either (a tuple using both v and M-v has at most four distinct covers and would pad to a half 5-cover).

This matches the paper’s remark that for k=5 the finite test seems to hold for every prime larger than 23, and explains why Table 1 starts at 23: 7 through 19 independently fail.

4. Closing arithmetic at k=5

Corollary 3 bound = 106408215362548828125. Lemma 4 gives lcm(2,…,6)=60. With the checked S,

lcm(60, 2329313741434753596167*71) = 3451290691517145151860 > bound.

So the computer step of Rosenfeld’s k=5 argument reconstructs independently (six runners is classically known; the content here is the check). The initial segment of S through 67 is still below the bound, so 71 is required for that prefix. Dropping one prime from S still exceeds the bound for q in {23,29,31} and fails for q>=37.

5. What this does not do

No k=6 or k=7 check. Lemma 6/7’s implication from the finite check to a forced prime factor is taken from the written proof. Later range claims (arXiv:2511.22427, 2604.23906) are untouched.

Local artifacts: work/code/rosenfeld_lemma7_cover.py; work/notes/rosenfeld_lemma7_k5_2026-08-30.md.