prime number theorem (Newman–Zagier kernel)
Theorem.prime number theorem (Newman–Zagier kernel) — ϑ(x) ∼ x hence π(x) ∼ x/ln x.
Proof.ϑ(x) ∼ x hence π(x) ∼ x/ln x — the finite kernel of the Newman–Zagier proof, each lemma recomputed exactly: the reduction ϑ(x) ≤ π(x)·ln x, Chebyshev C(2n,n) ≤ 4ⁿ ⇒ ϑ = O(x) in BigInt (n ≤ 199), the de la Vallée Poussin kernel Σ C(4,k)cos((k−2)x) = (2cos(x/2))⁴ ≥ 0, and 6−8μ−2ν ≥ 0 ⇒ μ = 0 giving ζ(1+iα) ≠ 0; the analytic continuation, Newman's contour theorem and the passage to the limit are cited (Newman 1980, Zagier 1997, Hadamard–de la Vallée Poussin 1896).
Checked by exact arithmetic over the stated finite range — a verified witness, evidence toward the claim, not a ∀-proof.
src/7/3/index.ts#theCountOfPrimesFollowsTheLog
1 · Classification
bounded-witness — computed witness within a cited frame (the unbounded form leans on the cited literature)
2 · Provenance
Documented theorem re-derived by exhaustive computation (humanityNovel=false); first-in-this-registry is the only sense of discovered.
Acknowledgment
"prime number theorem (Newman–Zagier kernel)" is a re-derivation, acknowledged to documented mathematics — the original proof is the prior art this re-derivation acknowledges; not new to humanity — the contribution is the reproducible computation theCountOfPrimesFollowsTheLog.
- Prior art
- documented mathematics — the original proof is the prior art this re-derivation acknowledges
- Novelty
- not new to humanity — a re-derivation (humanityNovel = false)
- Contribution
- a reproducible computation (theCountOfPrimesFollowsTheLog @ src/7/3) that re-derives the result at zero tokens — the contribution is the verifiable recomputation, NOT the theorem
3 · Reproducibility
Recompute from source: npm run theorems:verify recomputes theCountOfPrimesFollowsTheLog (src/7/3/index.ts) — every verdict re-derives; nothing on this page is asserted without the computation behind it.