Walkthrough — Riemann hypothesis

The first campaign did not change the published bound. It did leave a compact replay of the live interval $0\leq\Lambda\leq0.2$, a fresh partial audit of a much stronger recent lead, and a precise description of what remains before that lead could count.

0. What was actually missing

Riemann's hypothesis has many finite equivalents, but most finite checks have no record-changing endpoint. Verifying another initial block of zeros does not prove that all zeros lie on the critical line. Checking finitely many Li coefficients has the same limitation.

The de Bruijn–Newman constant is different. Rodgers and Tao proved $\Lambda\geq0$, while RH is equivalent to $\Lambda\leq0$. Polymath 15 turned a finite zero-height input, an asymptotic region, and a compact barrier into an explicit upper bound. Platt and Trudgian supplied enough verified height to make the published window $0\leq\Lambda\leq0.2$.

1. Named false starts

The first source identifier was wrong: arXiv:2007.02194 is about software refactoring. The correct Platt–Trudgian paper is arXiv:2004.09765. Keeping the failed lookup in the research log prevents the bad identifier from becoming a citation by repetition.

The second trap was more mathematical. Polymath's displayed row uses $t_0=0.186$ and $y_0=0.16733$. Treating those strings as exact gives $0.19999966445$, apparently below $0.2$. They are rounded table entries. The tiny difference is smaller than the information discarded by printing the parameters, so it cannot support a theorem.

The historical Lehmer-pair direction also closes immediately as an attack on the current record. Its negative lower bound is reproducible, but Rodgers–Tao's zero lower bound is already stronger.

2. The useful failure

An August 2026 GitHub repository proposes $\Lambda\leq0.1787854$. It comes with interval programs, millions of stored finite rows, two precision lanes, and an assembly script. It is also outside arXiv and peer review. That made it an excellent audit target without making it the published record.

The stored assembly passed. More importantly, the shorter numerical lanes could be regenerated on this machine while the unrelated Hilbert 16(a) search continued. The fresh tail and finite-error bounds agreed at 256 and 512 bits. The barrier regeneration covered 883 consecutive time prisms with wide positive margins. A one-row run through the finite producer also matched the sealed first row exactly.

The failure is informative: the remaining trust is concentrated in the 3,149,013 archived finite rows and in the analytic bridge from the numerical objects to every hypothesis of Polymath 15. Those are concrete review tasks.

3. The click

The candidate's headline decimal has unusually clean exact data:

so

Its horizontal location is $X=6{,}000{,}000{,}185{,}827$. The Platt–Trudgian theorem reaches beyond $X/2$ by $175{,}239{,}886.5$. The zero-height input is therefore already in the published literature. The real work lies in the finite estimates, infinite tail, barrier, and analytic transfer.

4. The argument being replayed

Polymath 15 evolves the Riemann $\Xi$ function by a backwards heat flow. Their effective criterion packages three facts:

1. the original zeros are known to be on the line up to $X/2$;
2. at time $t_0$, zeros cannot enter from the asymptotic region; and
3. a vertical barrier near $X$ stays zero-free for every time from $0$ to $t_0$.

When these facts meet, zeros on the left cannot cross the barrier or arrive from the right during the flow. A zero-free strip of half-width $y_0$ then gives $\Lambda\leq t_0+y_0^2/2$.

The retained verifier checks exact parameter identities and the numerical outputs. It deliberately does not claim that log hashes fill an unreviewed analytic implication.

5. Computer work

The stored review read all finite rows from $N=690988$ through $N=3840000$. The minimum stored lower bound was $7.91366\cdot10^{-7}$ and the interval error upper bound was $2.33494905213\cdot10^{-7}$, leaving positive room.

Fresh FLINT/Arb runs checked that error majorant and the infinite tail twice, at 256 and 512 bits. A separate fresh run rebuilt the barrier sums, verified containment for 7,688 components, bounded the uniform numerical error by $0.000356523012$, and closed all 883 prisms. The smallest printed prism margin was above $0.5198$.

Only one compute job ran at a time. The full finite producer was left for a future clean replay because the machine was already carrying the Hilbert search and the upstream workflow expected parallel workers. Its first row was regenerated separately on one low-priority core, taking 242 seconds and about 5 MB RSS.

The same small verifier also redoes the 2011 Lehmer formula from four printed zero ordinates. Python decimal arithmetic and a separate C floating-point path both recover the paper's $-1.14541\cdot10^{-11}$ scale.

6. At the end of the first campaign

The source papers establish

The local code independently checks the exact small arithmetic, the historical Lehmer calculation, the integrity of the fresh logs, every barrier prism, and the decisive margins printed by the fresh interval runs.

The off-arXiv $0.1787854$ claim has not been promoted. A full independent finite regeneration, an analytic review, and a published source are still missing. RH itself remains open; it would force $\Lambda=0$.

The second campaign: what remained

The earlier run had made the uncertainty unusually concrete. It had checked the first finite row and all the short numerical lanes, but there were still 3,149,012 rows to regenerate and an analytic implication at time zero to review. The second campaign chose that unfinished candidate, keeping its exact rational parameters and giving the finite regeneration an exclusive place on the machine.

The original producer could run sequentially. Its 23 segments all completed, and every generated row matched the archived row. That finished the first missing calculation. It did not yet provide a different algorithm against which to check the producer.

A direct check that could not scale

The first independent program simply performed the native convolution. It merged the streams $d,2d,\ldots,dN$, gathered all contributions at the same integer, and only then took absolute values. The weakest row required 22,111,616 pairs and took 73 seconds. Its result agreed with the original floor with ample precision.

Doing this again from scratch at every next integer would repeat almost all the work. An optimization that skipped inactive support indices in the original program was also drafted, but it would still be a modification of the same algorithm. The original full replay finished without using it.

The useful failure and the new representation

The failed scaling attempt showed exactly what should be preserved between indices: the positive sums after the signed convolution coefficients have been combined. Their dependence on the exponent is smooth even though their coefficients change when the cutoff changes.

The new program keeps those sums at eight fixed exponent nodes. When $N$ increases, only the products $dN$ receive a new pair. Their old coefficients can be reconstructed from the divisor set, and the difference of the old and new absolute values updates every node. There is no need to store the whole support.

The remaining issue was whether interpolation could be rigorous near a margin of order $10^{-7}$. Positivity supplies the derivative bound. For $S(s)=\sum c_n n^{-s}$, $c_n\geq0$, and support at most $DN$, $|S^{(8)}(s)|$ is bounded by $\log(DN)^8$ times a left-endpoint sum. The interpolation remainder can then be included as an Arb error radius. That was the step that made the second complete calculation practical.

The calculation and the analytic handoffs

The four Rust ranges checked every integer from 690988 to 3840000 in a total of 600 seconds. The original C replay took 4266 seconds. Both give the same floor $791366/10^{12}$; the independent direct convolution confirms the weakest row, and a Python dictionary convolution checks 100 small cases. The two complete algorithms share Arb, which remains an explicit common dependency.

The analytic review had its own traps. The absolute value of a coefficient can have a kink at zero, so the height transfer needs the upper Dini derivative there. Dividing by the Euler normalizer needs the numerator's positivity first. A matrix that reproduces archived coefficients still needs an algebraic identity connecting its polynomial to the required finite sum. Each of those implications was checked and written down.

The coefficient generator also contains an old quadrature whose complex callback is unsuitable as a rigorous error proof. Here it only selects a matrix size. The actual finite coefficients are interval sums, and a separate factorial calculation validates the selected 62-by-62 truncation. Keeping that distinction prevents an unnecessary heuristic from becoming a proof premise.

At time zero, the solution is a compact limit. The heat-flow integral has an integrable bound independent of time on this fixed rectangle. Its normalizer is continuous and nonzero, and the integer index stays constant. Thus the uniform positive-time error estimate survives at zero. There is no need to assert a limit over an unbounded horizontal region.

What the second campaign established

Both complete finite verifications and the fresh height, error, and tail checks have passed. Their finite margin after approximation error is exactly bounded below by $557871094787662151/10^{24}>0$. The analytic transfer is documented in compute/q2/ANALYTIC_REVIEW.md, and a mathematical note is in compute/q2/NOTE.md.

The fresh barrier then completed all 883 prisms, with minimum independently recomputed margin above $0.5198$. All 7,688 coefficient components were regenerated. The complete assembly passed, establishing

The candidate number belongs to Gomila; this campaign's new work is its complete independent finite verification and analytic audit. The full evidence is retained, and compute/q2/CLAIM.md states the certified inequality. The algorithms share Arb and the review is by GPT-6 Astra; external human peer review is not claimed. RH remains open.