P42 Prizes
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Problem № 9 · back to the register

PNT Sparse Mertens Construction

verifier 0.1.0 · pending admissionimage digest pending admissionrepo problems/pnt-sparse-mertens-construction

A queued sparse-support construction board. The local verifier now replaces live-board sampling with exhaustive exact constraints over all 960000 integer rows and an interval-certified logarithmic objective.

§1Statement

max⁡f −∑k=296000f(k)log⁡kks.t.∣supp⁡f∣≤2000, ∣f(k)∣≤10, ∑k≥1f(k)⌊x/k⌋≤10001/10000 ∀x≤960000\max_f\ -\sum_{k=2}^{96000} f(k)\frac{\log k}{k}\quad\text{s.t.}\quad |\operatorname{supp} f|\le 2000,\ |f(k)|\le 10,\ \sum_{k\ge1} f(k)\lfloor x/k\rfloor\le 10001/10000\ \forall x\le 960000

The verifier synthesizes the k = 1 balancing term, checks every integer row exactly, and accepts only a certified lower-bound decimal for the logarithmic objective; it is a finite reach-96000 construction board, not an asymptotic PNT theorem.

§2Verification

The chain does not trust this page.

Authority rests with the problem repo and the canonical verifier command. A revealed solution must reproduce the same exact VerdictReport for every honest runner; claimed scores are stripped and ignored. This board is not yet admitted — its verifier has not passed the gates below.

Conditions precedent to admission

  • R1rational support encoding
  • H3all 960000 rows checked
  • R6interval lower-bound objective
  • H5one-ULP bad-decimal fixture

§3Solution format

{
  "type": "object",
  "required": [
    "reach",
    "denominator",
    "printed_decimal",
    "support"
  ]
}
Plate 1. Canonical raw solution schema. Any claimed score fields a solver adds are treated as untrusted comments — the verifier recomputes everything from the raw artifact bytes.
{
  "reach": 96000,
  "denominator": "10000000000000000000000000000",
  "printed_decimal": "0.9974252022196793",
  "support": [
    {
      "k": 2,
      "value": "-10000989053719092000000000000"
    }
  ]
}
Plate 2. Sample solution shape — illustrative until the board’s repo is packaged at admission.

§4The record

no submissions

No award has yet been made. This board opens for submissions when its conditions precedent close.

ReproduceGET /prizes/api/leaderboard?problem_id=9