Landau 3 / Legendre, first computation

Outcome

There are three deliberately separate results.

1. Conditional improvement. Assume RH. For every real $x\geq1$ and every $\delta\geq901/4000=0.22525$, the interval $[x^{2+\delta},(x+1)^{2+\delta}]$ contains a prime. This tightens the printed $0.2253$ at the end of Chamberland--Straub, Section 3, by $1/20000$. It does not prove Legendre's conjecture.
2. Incomplete public-data reconstruction. The pinned public OLC worker logs contain five coverage components and four holes. Their maximum endpoint is $31{,}894{,}400{,}000{,}352$, below the published $70{,}500{,}000{,}000{,}000$. This is a statement about checked-in logs, not a challenge to the peer-reviewed computation.
3. Independent finite replay. Both Oppermann halves are certified for every $n$ from $2^{32}-100000$ through $2^{32}-1$. The CSV has 100,000 rows and 200,000 least-prime witnesses, ending immediately below square-height $2^{64}$. The largest normalized least-prime offsets in this slice are $489/4294952072$ on the left and $479/4294923874$ on the right.

Replay

From any directory:

problems/landau-legendre/compute/q1/run_all.sh

Dependencies are Python 3, Rust, and a standard C compiler. The command uses no network and writes only to a temporary directory. It takes about 30 seconds on the campaign machine.

To authenticate the OLC projection against a caller-supplied clone at the pinned commit:

problems/landau-legendre/compute/q1/replay_olc.sh /path/to/olc

Why the conditional exponent follows

Set

Chamberland--Straub Proposition 3.4 applies the finite Oppermann computation to every real $x\geq1$ with $x^{\alpha/2}<N$, hence to $x<X$. Under RH, their Theorem 3.2 gives a prime in the still smaller interval $[x^\alpha,(x+1)^\alpha-1)$ whenever

At $x=X$, this condition becomes

Taking logarithms leaves the exact sign checked by rh_delta.json:

The exact certified lower margin is greater than $0.0000797000675$. The same certificate proves $(901/8901)\log N>1$, so $x^{d/2}/\log x$ increases from the splice onward. The analytic side therefore covers $x\geq X$, including equality, and the finite side covers $x<X$.

For a larger exponent $\alpha'=r\alpha$ with $r\geq1$, put $y=x^r$. Then $y^\alpha=x^{\alpha'}$ and $y+1=x^r+1\leq(x+1)^r$, so the $\alpha$ interval at $y$ is contained in the $\alpha'$ interval at $x$. This proves the stated range of $\delta$.

The exact verifier encloses logarithms with a rational atanh series after power-of-two range reduction. The C program recomputes the signs with logl through an unrelated code path.

Certificates

• certs/rh_delta.json: exact logarithm intervals and positive margins.
• certs/olc_rows.tsv.gz: canonical projection of 322,073 cumulative rows from 2,560 tracked *.out worker files.
• certs/olc_public_audit.json: pinned commit, manifest and projection hashes, components, holes, duplicates, and invariant counts.
• certs/edge_witnesses.csv: least-prime offsets for both halves.
• certs/edge_summary.json: CSV hash and the top 20 exact ratios per half.

The Python edge verifier uses the first twelve prime Miller--Rabin bases. Sorenson and Webster computed the first composite passing those bases as $318665857834031151167461>2^{64}$ in Strong Pseudoprimes to Twelve Prime Bases, so the test is deterministic on every certificate candidate.

verify_gap_row.py uses the same deterministic range to replay the reported Oliveira e Silva--Herzog--Pardi row: both endpoints $1425172824437699411$ and $1425172824437700887$ are prime, and all odd interior values are composite, giving gap $1476$. This checks that row, not the exhaustiveness of their full table.