Attack log: Landau 3 / Legendre

Chronological attempts, newest last.

2026-08-27: record correction before search

The starting note said the computational record was $2^{64}$. The published record is much larger. Sorenson and Webster verified the stronger Oppermann conjecture through $n=7.05\cdot10^{13}$, which checks square-height $n^2=4.97025\cdot10^{27}$. A finite search near $2^{64}$ can be an independent replay, but it cannot move the record.

The proposed prime-gap formulation also needed correction. The inequality

is sufficient for Legendre, not equivalent to it. The exact finite predicate is that no interval between consecutive integer squares is prime-free.

Ten imagined end-states

These were written before implementation. Each is a concrete artifact that would improve, replay, or sharpen a documented result.

1. RH exponent $0.22525$. A rational-arithmetic certificate proves that the Chamberland--Straub overlap works for $\delta=901/4000=0.22525$, improving their printed $0.2253$. It might work because their value is rounded; it might die if the exact overlap or real-$x$ endpoint uses hidden slack in the other direction.
2. One new Oppermann chunk. Certify both half-intervals for every $n$ in a nonempty interval immediately beyond $7.05\cdot10^{13}$. It might work with their arithmetic-progression code; it might die on compute cost or unavailable production parameters.
3. Published endpoint replay. Recover the final checked Sorenson--Webster shard and independently validate its row counts and primality evidence. It might work if the repository contains the production logs; it might die if only a partial upload is public.
4. Public-log coverage certificate. Canonicalize every checked-in OLC log row, identify overlaps and holes, and certify the exact union of public intervals. It might expose a replayable documented slice; it might die as residue if the public logs do not reach the paper's endpoint.
5. Top-of-$2^{64}$ Oppermann slice. Give two prime witnesses for each of the final $100{,}000$ square intervals below $2^{64}$ and verify them with an implementation independent of the generator. It might work because deterministic 64-bit primality testing is cheap; it cannot beat the modern record.
6. Near-miss table at the same edge. Rank the largest least-prime offsets in each Oppermann half over that $100{,}000$-interval slice, with every row recoverable from the witness certificate. It might reveal a useful stress case; it might die as merely typical data.
7. Prime-gap-table replay through $10^{20}$. Hash and independently scan the current exhaustive gap data, deriving its exact Legendre prefix. It might give a clean provenance bridge; it cannot catch the published Oppermann record.
8. Explicit unconditional $x_0$. Extract all constants in the Baker--Harman--Pintz argument and certify an explicit threshold for some $\alpha<2.106$. It might turn an effective theorem into a finite statement; it might die because the paper explicitly leaves the constant extraction as substantial work.
9. Smaller-power exception extension. Extend and certify the OEIS list of prime-free intervals $[n^{3/2},(n+1)^{3/2}]$ far past its published search. It might be cheap and informative; it might not say anything new about exponent $2$.
10. Almost-prime bridge extension. Extend Campbell's explicit $\Omega\leq3$ square-interval range past $n^2=10^{31}$ with a checked multiplier and current gap bound. It might work by retuning one prime; it might remain adjacent to, rather than progress on, Legendre.

Pick three, then prune

The ranking criterion was published-record delta times checkability in one session.

1. Conditional exponent tightening: small but genuine theorem delta, and the only numerical step is an exact one-variable inequality.
2. Public OLC coverage audit: potentially reaches the published endpoint, and immediately says what the available evidence can actually replay.
3. $2^{64}$-edge slice plus near-miss table: no modern record delta, but fully checkable and supplies the requested independent fallback.

Items 2, 7, 8, 9, and 10 above were pruned on compute cost, missing source artifacts, or distance from Legendre. Item 3 was folded into the public-data audit, and item 6 was folded into the deterministic slice. No work starts on the pruned lines.

Certificates imagined before code

A. Conditional exponent

certs/rh_delta.json will contain $N$, the rational $\delta=901/4000$, the paper's overlap inequality, rational lower and upper bounds for every logarithm, and a positive rational lower bound for

The verifier must reconstruct the logarithm enclosures from rational series, not trust decimal strings in the certificate.

B. Public OLC data

certs/olc_public_audit.json will pin an upstream commit and list the number of parsed rows, unique starts, duplicates, covered endpoints, holes, and row invariant failures. A verifier will reparse a local clone when supplied. The committed certificate alone records provenance; it does not turn missing logs into a published-range replay.

C. Deterministic edge slice

certs/edge_witnesses.csv will have one row for every $n\in[2^{32}-100000,2^{32}-1]$ and the least prime in each open Oppermann half. certs/edge_summary.json will pin the range, row hash, maxima, and the largest normalized least-prime offsets. The generator and verifier will use different deterministic Miller--Rabin base sets and separate source files.

Execute A backwards: conditional exponent

The imagined certificate reduces cleanly. Put

At $x=X$, Chamberland--Straub inequality (5) is

The independent exact verifier reconstructs rational atanh-series bounds for both logarithms. It proves

It also proves $(901/8901)\log N>1$, so $x^{d/2}/\log x$ increases for $x\geq X$. The finite proposition covers $x<X$ and the RH interval theorem covers $x\geq X$, including the splice point. If $\alpha'\geq\alpha$, write $\alpha'=r\alpha$ and use $x^r+1\leq(x+1)^r$ to transfer the result to the larger exponent.

verify_rh_float.c independently recomputes the overlap margin as $0.0000797048500882\ldots$ with logl. The exact rational verifier is the authoritative check.

Dent: under RH, the printed threshold improves from $\delta=0.2253$ to

This is a tightening inside the published argument, not a proof of Legendre.

Execute B backwards: public OLC evidence

The published endpoint replay contradicted the available source artifacts, so that line was dropped immediately. The finite leftover was still checkable and became a public-data audit.

The first parser made a real mistake: it treated every cumulative worker row as an independent total. In an OLC *.out file, counters accumulate across rows. The correct invariant is

with previous count zero for the first row. Every raw row also satisfies found = R0 + R1 + R2 + R3 + R4. The upstream field named countfails counts use of a fallback, not a conjecture failure.

A second scope mistake included bigdawg/data14e12/bench.txt, inflating the row count from 322,073 to 322,357. The certificate now selects exactly the 2,560 tracked worker blobs matching ^(bigdawg|phi)/data[^/]*/[^/]+\.out$ at commit 5cdaa95f0a4b1428a05480cc1c69d556a8f9517a.

The authenticated replay gives 322,073 rows, 318,840 distinct intervals, 3,233 duplicate interval rows, no counter-invariant failures, and five union components. The four absent half-open ranges are

The public maximum endpoint is $31{,}894{,}400{,}000{,}352$, so these blobs cannot reconstruct the paper's endpoint $70{,}500{,}000{,}000{,}000$.

Residue: the pinned public projection is internally consistent but incomplete. Missing public rows do not contradict the published computation and do not create a lower bound.

Execute C backwards: $2^{64}$ edge

Rust generated the least prime offset in both open Oppermann halves for every

The independent Python verifier used the first twelve prime Miller--Rabin bases, retested every earlier odd candidate, and checked all 200,000 leastness claims. Sorenson--Webster's value of $\psi_{12}$ exceeds $2^{64}$, so this is a deterministic test on the whole certificate range.

The largest normalized least-prime offsets in this slice are

The maxima are only about $1.14\cdot10^{-7}$ and $1.12\cdot10^{-7}$ of their half widths. This is a descriptive near-miss ranking for the named slice, not a global closeness measure.

Independent replay: 100,000 consecutive square intervals and both halves verified immediately below square-height $2^{64}$. It does not move the modern record.

Stop

compute/q1/run_all.sh regenerates the two local certificates, byte-compares them with the committed copies, runs the exact and independent verifiers, and checks the public OLC projection offline. replay_olc.sh /path/to/olc additionally authenticates that projection against the pinned Git object.

The conditional exponent inequality is the dent. The public-log audit is residue. The edge slice is an independent replay with a near-miss table. Legendre's conjecture remains open.

2026-09-04: q3 starts from the exact q1 boundary

The starting conditional record was $d=901/4000=0.22525$. Merely solving the q1 splice equation more accurately gives the numerical root

That route could move the printed decimal, but it would still discard a positive term already visible one line earlier in the published proof. Chamberland--Straub explicitly describe their inequality (5) as sufficient, not necessary. Their preceding inequality (6) was therefore the useful place to restart.

The finite-extension route was pruned. The public OLC data audited in q1 do not reach the published endpoint, so they cannot certify even the first new integer above $7.05\cdot10^{13}$. No incomplete computation was promoted to a bound.

Retain the quadratic term

Write $\alpha=2+d$. Since $2<\alpha<3$, Taylor's theorem with positive third derivative gives

After subtracting $1$ and dividing by $\alpha x^{\alpha/2}$, the Chamberland--Straub short-interval inequality follows from

At $X=N^{2/\alpha}$, put $A=X^{d/2}=N^{d/\alpha}$ and $b=(\alpha-1)/2$. Relative to the first term, the correction is

The exact q3 certificate proves $2\cdot10^{12}<X<2.8\cdot10^{12}$ and $25<A<26$. It encloses every logarithm by rational atanh series. At

the upper bound for the old condition (5) log margin is $-2.0321572\cdot10^{-13}$, so that condition really fails. Using the exact lower bound on $t$ and $\log(1+t)\geq t/(1+t)$ changes the certified total margin to more than $1.5199\cdot10^{-14}$.

Finally,

The same coarse bounds give a rational lower bound greater than $1.9355$ for all $x\geq X$. Thus the analytic range includes $X$ and everything above it; the published finite proposition covers $x<X$.

compute/q3/run_all.sh first replays q1's exact $0.22525$ certificate. It then regenerates and byte-compares the new rational certificate, reconstructs every field independently, and checks the calculation through a separate C long double path. The full q1 replay, including all 200,000 edge witnesses, also passed unchanged.

Dent: assuming RH, for every real $x\geq1$ and every

the interval $[x^{2+\delta},(x+1)^{2+\delta}]$ contains a prime. The finite record remains $n=7.05\cdot10^{13}$, and Legendre's conjecture remains open.