{"schema":"vela.problem-packet.v0.1","problem":488,"statement":"Let $A$ be a finite set and\\[B=\\{ n \\geq 1 : a\\mid n\\textrm{ for some }a\\in A\\}.\\]Is it true that, for every $m&#62;n\\geq \\max(A)$,\\[\\frac{\\lvert B\\cap [1,m]\\rvert }{m}&#60; 2\\frac{\\lvert B\\cap [1,n]\\rvert}{n}?\\]","status":"open","seam":"sealed","closureRoutes":[{"type":"formal_proof","verifierKind":"lean","note":"Lean patch building clean under the math CI profile (no sorry, no new axioms)"},{"type":"witness","verifierKind":"python:verify_488 (per-fixed-A deciders)","note":"a uniform argument is the open piece; per-A exact decisions are banked"},{"type":"obstruction_report","verifierKind":"review","note":"precise, artifact-backed reason a route cannot work"}],"obligations":[{"findingId":"vf_ef7066987d658f0e","banked":"per-fixed-A exact decision certificates (7500+ antichains, zero counterexamples)","open":"no uniform proof: the per-A certificates do not combine into a statement for all A.","dependents":1,"lease":null}],"attestations":[],"attempts":[{"id":"att_881114484d919f2b","kind":"partial_proof","claim":"Erdos #488 (OPEN): for finite A (2<=a), B={n: a|n some a in A}, is |B cap[1,m]|/m < 2|B cap[1,n]|/n for all m>n>=max(A)? Cheap exact verifier (inclusion-exclusion); zero counterexamples across ~16000 families + fresh probes. Likely TRUE & tight (constant 2 unattained). The |A|=2 case may admit an elementary Lean-checkable proof -- the one cheap partial worth attempting.","grade":"partial_proof","gateStatus":"needs_verification","superseded":false},{"id":"att_2a916c23408f4350","kind":"partial_proof","claim":"#488 |A|=2 extremal structure PINNED: F(x)=floor(x/a)+floor(x/b)-floor(x/lcm); the ratio F(m)/m / (F(n)/n) over m>n>=b has TIGHT sup = 2, approached by consecutive A={a,a+1}, n=2a-1, m=a^2 with exact ratio (2a-1)^2/(2a^2) -> 2^- (never reaching 2 since a>1/4). Holds strictly; constant 2 is tight even at |A|=2.","grade":"partial_proof","gateStatus":"needs_verification","superseded":false},{"id":"att_252a91af2aa946ff","kind":"reduction","claim":"#488 |A|=2 EXACT REDUCTION (Opus): with F(x)=floor(x/a)+floor(x/b)-floor(x/L), delta=1/a+1/b-1/L, theta(x)={x/a}+{x/b}-{x/L} in (-1,2), the target 2m F(n) > n F(m) is EXACTLY  mn*delta + n*theta(m) - 2m*theta(n) > 0. Separated one-sided bounds CANNOT work (ratio->2, zero slack); proof needs the joint m,n coupling.","grade":"verified_reduction","gateStatus":"needs_verification","superseded":false},{"id":"att_3da9f26a21f9901d","kind":"partial_proof","claim":"PROVEN (#488, |A|=2 case): for all integers 2<=a<b and all m>n>=b, F(m)/m < 2 F(n)/n, where F(x)=floor(x/a)+floor(x/b)-floor(x/lcm(a,b)) counts B={k: a|k or b|k} in [1,x]. The two-element case of the Erdos #488 doubling inequality holds. Elementary proof (GPT-Pro), independently verified by Opus.","grade":"partial_proof","gateStatus":"needs_verification","superseded":false},{"id":"att_4d4c25478e82c096","kind":"partial_proof","claim":"#488 |A|=3 extremal pinned (Opus): same shape as |A|=2 -- consecutive A={a,a+1,a+2}, n=2a-1 (F=3), m=a^2 (F=3a-3), ratio=(a-1)(2a-1)/a^2=2-3/a+1/a^2 -> 2^-. The tight constant 2 holds at every fixed |A|=k via consecutive {a,...,a+k-1}, n=2a-1, m=a^2.","grade":"partial_proof","gateStatus":"needs_verification","superseded":false},{"id":"att_46f132b2fb077951","kind":"partial_proof","claim":"PROVEN (#488, |A|=3 case): for any A={a<b<c} and all m>n>=c, F(m)/m < 2F(n)/n. Elegant elementary proof (GPT-Pro, Opus-verified 0 viol in 3.2M checks). #488 now holds for |A|<=3.","grade":"partial_proof","gateStatus":"verified","superseded":false},{"id":"att_98c7ccd8a1f8dbc5","kind":"partial_proof","claim":"PROVEN (#488, |A|=4 case): for any A={a<b<c<d} and all m>n>=d, F(m)/m<2F(n)/n. #488 now holds for |A|<=4. AND the natural general strategy is PROVABLY DEAD (GPT-Pro, Opus-verified): general |A|=k needs a fundamentally different bound.","grade":"partial_proof","gateStatus":"needs_verification","superseded":false},{"id":"att_298c624c91dfa318","kind":"known_result","claim":"#488 CRITICAL RE-ASSESSMENT from page comments: small-|A| is considered DONE by the field (our |A|<=4 is likely NOT novel), and the general case is a Tao-confirmed ANALYTIC WALL. Stop the general swing; verify |A|<=4 novelty before any write-up.","grade":"extends_prior_work","gateStatus":"needs_verification","superseded":false},{"id":"att_fe635c1c870f9c40","kind":"dead_end","claim":"#488 HYBRID ROUTE DISPROVEN (GPT-Pro, Opus-verified counterexamples): there is NO positive S-threshold for 2F(n)>(n+1)S, large S does NOT force F(n)/n>1/2 (can be <1/3 with S->inf), even asymptotic density delta(A)>1/2 fails, and a single injection cannot dominate the overlap mass. The exact threshold is the LOCAL overlap inequality N1 - sum_{j>=3}(j-2)Nj > sum (r_i+1)/a_i, which fails badly for natural primitive antichains. General #488 needs ratio-peak control of the periodic correction (analytic).","grade":"honest_null","gateStatus":"verified","superseded":false},{"id":"att_249bab45696b4b36","kind":"known_result","claim":"Erdős #488 small-|A|: CORRECT but NOT a novel contribution — do-not-claim verdict, Opus-confirmed. The |A|=2 inequality F(m)/m < 2 F(n)/n (F(x)=floor(x/a)+floor(x/b)-floor(x/lcm(a,b)), all 2<=a<b, m>n>=b) is independently VERIFIED: 0 violations over 306,802,291 (a,b,m,n) tuples; sup ratio -> 2 but never reached. Codex overnight re-derived it (elementary case proof + claimed Lean kernel check); we already had |A|=2,3,4 banked. NOVELTY VERDICT (prior-art gate): the LIVE #488 forum shows Terence Tao actively working it (Apr 2026: 'a=2 case proved, Chojecki reduction fails for a>=3') with Janson/Bonferroni machinery and periodic-correction ratio-peak analysis; our prior re-assessment already flagged small-|A| as 'considered done by the field'. So |A|<=4 is a correct-but-known result, NOT a resolution toward the mission five. The real #488 frontier is the general / all-primes case (analytic, hard). MISSION IMPLICATION: demote/drop #488 from the resolution lane.","grade":"extends_prior_work","gateStatus":"verified","superseded":false},{"id":"att_574b795d0b92fc4c","kind":"dead_end","claim":"Erdős #488 (for finite A, B=multiples of A, is F(m)/m < 2F(n)/n for all m>n>=max(A)?) remains OPEN — OBSTRUCTION MAP, not a settlement. |A|<=4 proven; general case a Tao-confirmed analytic wall. TWO verified obstructions: (1) the consecutive-integer EXTREMAL-FAMILY REDUCTION IS FALSE — A={14,15,16,17,19} has sup-ratio R=1593/980 > the same-maximum consecutive 5-set {15,16,17,18,19} with R=551/340, so #488 does NOT reduce to consecutive A. (2) NO uniform positive margin: mu({a})=1/(a(2a-1))->0, and consecutive blocks {a,...,a+k-1} give mu=O_k(a^-2)->0 for every fixed k; so NO finite per-A computation (the density-shadow grind) can close #488 via a uniform slack — it is PROVABLY FINITE-COVERAGE-ONLY. The exact missing input is a SUFFIX-ENVELOPE/DELAY theorem: for all primitive A and hard n>=max(A), max_{m>n} c_{m mod P}/m < E_A(n)=2F(n)/n - delta_A, where c_r=F(r)-delta_A*r and P=lcm(A); separated oscillation bounds lose the a^-2 scale at the extremal pair (n,m)~(2a-1, a^2).","grade":"obstruction_map","gateStatus":"verified","superseded":false},{"id":"att_74ca93a03adaf379","kind":"known_result","claim":"Erdős #488 public-counterexample AUDIT (Opus-verified): the erdosproblems.com/488 forum comments (30 posts; Cambie, van Doorn, Alexeev, Koizumi) contain NO counterexample to the current MULTIPLES statement (B={n: a|n some a∈A}, is F(m)/m < 2F(n)/n for all m>n≥max A). Of the public claims: 2 are counterexamples to the OLD `a∤n` typo version (irrelevant), the rest are lemma/reduction obstructions, not #488 counterexamples; 204 named-structure sets (2∈A, split-core tripod, prime-window) and a bounded frontier hunt yield 0. Independently re-verified by exact inclusion-exclusion. #488 is NOT already-falsified; it remains genuinely OPEN.","grade":"verified_reduction","gateStatus":"verified","superseded":false}],"velaLean":[],"oeis":[],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}