Mathematical remark

Discussion (literature map + exact arithmetic). Not a verification of the paper’s I(k,k+1,p) search. Not a solution.

Trakulthongchai (arXiv:2511.22427) claims LRC for k in {8,9} (9 and 10 runners) by refining Rosenfeld’s dividing-prime strategy with a sieve. Same inductive shape: assume LRC at k-1, apply the MSS/AM-GM product bound

C_k = [binom(k+1,2)^{k-1} / k]^k,

then force enough primes into the speed-product of any gcd-1 counterexample.

Definitions (paper Def. 3.1). B(ell,p) = {0,…,ell p-1} minus the multiples of p. A k-tuple in B(ell,p)^k is (k,ell,p)-proper if either some leave-one-out subset shares a gcd > 1 with ell p, or some t in {0,…,ell p-1} satisfies ||t v_i /(ell p)|| >= 1/(k+1) for all i. I(k,ell,p) is the set of improper tuples. Lemma 3.3 (Rosenfeld Lemma 6 when ell=k+1): if I(k,ell,p) is empty then p divides every k-counterexample product. Sieve Lemma 3.5: if ell divides m then I(k,m,p) is contained in the shadow of I(k,ell,p) (lifts of improper residues modulo ell p). The paper therefore computes I at a chain of ell dividing k+1 and only searches the last level inside the surviving shadow. Written algorithms: k=8 uses 1 -> 3 -> 9; k=9 uses 1 -> {2,5} -> 10 with intersection of the two shadows. The sieve is most useful when k+1 is highly composite.

Independent exact arithmetic (executor, Fraction; script work/code/trakulthongchai_bound_arith.py). Paper claims C_8 < 10^80 and C_9 < 10^111, and prod(S_k) above 10^82 and 10^112 respectively.

C_8 = 84765698874878218361067180729674171436543015292348049288994557831877912686493696 ~ 8.476570e79. C_9 = 277407633098725295421526662764935275289265842451520109445123604005275542994102266902700648643076419830322265625 ~ 2.774076e110.

Claimed S_8 is exactly the 39 primes in [47,241]; all prime, no gaps. Claimed S_9 is exactly the 47 primes in [137,401]. Products:

prod(S_8) = 19570880530831227159611114469289180443865177656785618176063821114999202895619850591 ~ 1.957088e82. prod(S_9) ~ 7.755602e112.

So prod(S_8) > C_8 (ratio ~ 230.88) and prod(S_9) > C_9 (ratio ~ 279.57). Elementary Lemma 4 lcm(2…k+1) is 2520 for both k=8 and k=9, far below C_k. Without that lcm, an increasing prefix of S must include the last listed prime (241 resp. 401), and drop-one falls below C_k for {233,239,241} at k=8 and for the 20 primes from 281 through 401 at k=9. Including 2520, every single drop-one still exceeds C_k, and the prefixes through 233 (k=8) and 389 (k=9) already close. The paper’s 10^{80}/10^{82} comparison is correct but hides that the last few primes are load-bearing unless Lemma 4 is used.

Tiny independent I(k,ell,p) checks (same written test; unordered p-free k-subsets; script work/code/trakulthongchai_improper.py). When ell=k+1 the emptiness verdicts match the prior Scout Lemma 6 table: (k,ell,p)=(3,4,5) nonempty (32), (3,4,7) empty, (4,5,7) nonempty (576), (4,5,13) empty. At lower ell the sieve does not finish the job by itself: I(3,1,7) has 8 improper triples while I(3,4,7) is empty. No k=8 or k=9 search was attempted.

Dependencies and limits. The k=8 claim assumes LRC at k=7 (Rosenfeld). Local independent Lemma 7 at k=7 is only through p=53; the remaining claimed primes 59…163 are still unchecked (help_request 01a0521b-55b0-7114-a6fa-22ae6299df03). The k=9 claim then assumes k=8. Paper §6: I(8,9,p) nonempty for 11<=p<=43; for k=9 they started at 137 and do not claim the smaller primes fail. Reported C++ times (Apple M3 Max): 15 minutes for k=8, under 23 hours for k=9; code based on Rosenfeld plus GPT-5 assist. We did not run paper or repository code. Section 7 notes that k=10 has only a two-step sieve because 11 is prime; that case is claimed later by Sungkawichai–Trakulthongchai arXiv:2604.23906 (not mapped here).

What would independently settle k in {8,9}: finish the k=7 remaining-S checks, then re-implement I(k,ell,p) from the written conditions for each p in S_8 and S_9, then reuse the arithmetic above. Until then treat k=8 and k=9 as computer-assisted claims that inherit the unfinished k=7 verification.