RESULTS: `skewes_enclosure` (hunt #152)

Status on 2026-10-27, three outcomes and one post-hoc finding:

• Target A (Chao-Plymen, 2e6 zeros): gate passed. pi(x) > li(x) for some x with log x in [727.9519119, 727.9521241] (section 1).
• Target B at the printed omega = 727.95134: gate failed. This is recorded permanently in section 6.
• The pre-registered omega-grid rerun (B2): hypothesis H not supported by its rule. At omega = 727.951335792, the sum misses Revers's S* by 3.37e-5, against a budget of 2.60e-5. The pre-declared localization at that omega holds anyway: pi(x) > li(x) for some x with log x in [727.951319892, 727.9513517] (section 7).
• Post-hoc review finding (2026-10-27, measured, not pre-registered). The 3.37e-5 gap is twice the second-order sum S2: Revers's printed S* for the Saouter-Demichel region equals Saouter-Demichel's printed first-order sum minus S2, to 1.0e-9 (S2 summed over the first 2e4 zeros; its tail is estimated, not bounded), where Theorem 3.1 adds S2. Our own first-order sum matches Saouter-Demichel's to 4.7e-8 (sections 3, 7 and 8). The pre-registered verdict stands.

Correction (2026-10-27, review of PR #360). The B interval was first quoted with lower end 727.9513199. That rounds the exact lower end, 727.951335792 - 0.0000159 = 727.951319892, inward, so the quoted interval was not the one proved. The quote above is exact at the lower end and rounded outward at the upper end (exact: 727.951351692). test_quoted_intervals_contain_the_exact_ones pins every quoted interval in this file and in the case log.

Nothing here is promoted: the grades below are this hunt's report, not a reviewer's.

1. The bounded result (target A, Chao-Plymen region)

Parameters (Revers 2025, subsection 6.2): omega = 727.952018, alpha = 1.34e11, A = 1.022e7, T = 1131944.4719, which Arb counts to exactly 2,000,000 zeros.

The claim, with its dependencies. (Corrected 2026-10-27: this heading first read "Measured claim", which contradicts the hardened grade given below.) With every computed step carried by Arb balls, I(727.952018, 1.061e-4) > 1.4747e-4 > 0. By the positivity argument in Revers's "Runs of successive integers" subsection, pi(x) > li(x) for some x with log x in [727.9519119, 727.9521241]. This rests on four inputs:

1. Revers 2025 Theorem 3.1, read in arXiv v2 (the version published in JNT 2025 was not read and may differ). It is cited, and this hunt does not re-derive it. It is the weakest link: the localization rests entirely on its constants, which this laboratory has not reviewed, and the hunt found three apparent slips in the same paper's section 6 (section 8). Its R1 to R6 are evaluated exactly as printed. They reproduce both of his resizing tables (13 rows, test_skewes_terms.py).
2. RH to height A = 1.022e7. Cited: Platt (2017, to 3.06e10) and Platt-Trudgian (2021, to 3e12).
3. The zeros and their completeness: rigorous on route 1 only. Arb's acb_dirichlet_zeta_zeros and Turing count. Every ball is disjoint from and below the next, both within and across shards. The last zero is below T, and N(T) = 2,000,000. (Corrected 2026-10-27: this item first said "route 1 only". The count N(T) also has a float-grade second route: mpmath nzeros, which shares no code with Arb and is not rigorous, gives N(T) = 2,000,000 at T = 1131944.4719 and 22,000,000 at T = 10379600, and one less at each of Revers's printed T; test_cutoffs_by_the_mpmath_route. That route checks the count only, not that each ball holds a zero.)
4. Route 2 is a sample. 1150 of the 2e6 Arb brackets (200 in shard 1 and 50 in each later shard, seeded) show a sign change of mpmath siegelz at +/- 1e-8. mpmath shares no code with Arb but is float grade, and it checks existence, not completeness.

Grade. The zero sum and the error terms are enclosure-carrying (the hardened rung by its ball-arithmetic clause). The zero layer has one rigorous route and a sampled, float-grade second route. The analytic inequality and RH to A are cited, not checked here. The composite claim is therefore a hardened computation on a cited derivation, with completeness resting on Arb alone. That is weaker than a fully independent two-route result.

What it is not. It is not a new record. This interval is wider than and sits above Saouter-Trudgian-Demichel 2015's [exp(727.951334747), exp(727.951336105)]. It reproduces, at ball grade, the localization Revers reported at float grade for the Chao-Plymen region. Our margin at eta = 1.061e-4 is his 1.28492e-4 plus his worst-case zero budget Delta S1 = 1.899e-5, as expected once the zero error is enclosed rather than bounded.

Novelty. Platt-Trudgian 2015 already carried the analogous theta(x) - x computation in MPFI interval arithmetic. The claim here is an enclosure-carrying localization of a pi(x) - li(x) sign change. It is novel only against the recorded search (MISSION.md section (a): three papers and one web search), which is not a literature search. No priority is claimed.

2. Gates and planted faults

• Gate: S* inside the ball. Passed. It was also within Revers's own zero budget (1.899e-5) by a factor of about 1e5.
• R4 dropped at eta = 1.06e-4. The lower bound moves by R4 = 3.380e-3, from -1.24e-4 to +3.256e-3: the sign flips. The error term is load-bearing and the pipeline sees it.
• gamma_1 shifted by 1e-6. S moves by 7.729e-5, and the faulted ball no longer contains S*. A one-zero error of 1e-6 is visible, and the gate catches it.
• Small-scale versions of both faults, plus an mpmath float sum over 120 zeros, are pinned in test_skewes_terms.py.

3. Observations about the source

• In Revers's Saouter-Demichel table, the row printed as eta = 0.0000155 with R = 5.15554e1 is the value at eta = 0.0000156. At 0.0000155, Theorem 3.1 gives R = 5473.12. Both rows are negative, so the paper's conclusion is unaffected.
• An observation about two printed values, not about the paper's results. arXiv:2501.04488 v2, subsection 6.2 ("Reworked Bays-Hudson region by Chao-Plymen, 2010"), prints "T=\gamma_{2000000}<1131944.4718". Subsection 6.3 ("The Saouter-Demichel region, 2010") prints "T=\gamma_{22000000}<10379599.7274". Both are quoted from the TeX source. Two routes place each zero just above its printed value:

(Updated 2026-10-27: the Arb cell for gamma_22000000 first read "not isolated yet". Production B isolated it; the value is last in shards_b2/shard_220.json, and it agrees with mpmath.)

So each printed decimal is the zero truncated to four places, a lower bound, where the text has "<". The same sentences also print "\gamma<1131945" and "\gamma<10379600", which are correct upper bounds. If T were taken literally as the printed decimal, the sum would omit one zero, a term of at most about 1.5e-8, far below either margin. This hunt sums to T = 1131944.4719 and 10379600, which both routes count to exactly 2e6 and 2.2e7.

• Added 2026-10-27 (post-hoc review, measured, not pre-registered): the subsection 6.3 S* subtracts the second-order sum. Split Revers's sum as S = S1 + S2, with S1 the first-order (Lehman) part over 1/rho and S2 the second-order part over 1/(omega rho^2), both with the same Gaussian weight. Subsection 6.3 quotes "the Saouter-Demichel machine value with size -1.002906086981405" (first order, S1 only, at their omega = 727.951335792, alpha = 6e12, 2.2e7 zeros). With S2 at omega = 727.951335792 from the first 2e4 zeros, 1.68592e-5 (section 7 has the tail estimate),

Saouter-Demichel's S1 - S2 = -1.0029229462, Revers's S* = -1.0029229472,

which agree to 1.0e-9 (with S2 truncated at 2e4 zeros; the tail is estimated, not bounded, section 7). Theorem 3.1 adds S2, and so do our sums. At the Chao-Plymen point Revers's S* agrees with ours, that is with +S2, to 1.7e-10 (section 1), and his quoted Chao-Plymen value -1.006569 is our S1 to its six printed decimals. So the 6.3 value appears to subtract S2 where the theorem adds it. Driver and data: s2_check.py, s2_first20000.json, s2_comparison.json, pinned by test_skewes_terms.py.

• Added 2026-10-27 (measured): an apparent digit slip in the 6.3 table. Every other positive row of both resizing tables equals -1 - S* - Delta S1 - Delta S2 - R, recomputed from the printed S*, R and Delta S values, to within 5e-9 (R is printed to six digits). The row eta = 0.0000159 prints 0.000015987, but the printed values give 6.5986e-5 there, 4.9999e-5 more. A "1" for a "6" in the leading digit fits. The row is positive either way, so the paper's conclusion is unaffected. Pinned by test_revers_saouter_demichel_row_1_59e_5_is_a_digit_slip.

4. Cost

Isolation of 2e6 zeros took 26,981 s, 7.5 core-hours: 12.5 to 17.5 ms/zero, 4 nice'd processes, 1.9 h wall. The estimate was 10 to 20 core-hours. Route 2 and the sums cost minutes. The shard JSONs total under 100 KB, and checksums are in shards/SHA256SUMS. No zero list is kept: each shard records the SHA-256 of its zeros as printed.

5. Reproduce

.venv/bin/python -m pytest -q -n0 hunts/skewes_enclosure/test_skewes_terms.py
hunts/skewes_enclosure/run_shards.sh <outdir> 1 2 ... 20      # ~7.5 core-hours
.venv/bin/python hunts/skewes_enclosure/combine.py --target chao_plymen --shards <outdir>

The committed shards/ are this run's outputs. combine.py on them reproduces shards/combined_chao_plymen.json. Shard 1 was run with --sample 200 --seed 152 and shards 2 to 20 with --sample 50 --seed 152000+k.

6. Production B, 2026-10-26: the gate failed (attempt unresolved)

All 220 shards (1 to 20 from target A, 216 on the box, 217 to 220 from the pilot, and 21 to 215 from Actions run 40363428968) are sound. Indices are contiguous, every ball is ordered and disjoint within and across shards, the last zero is below T = 10379600, and Arb's Turing count gives N(T) = 22,000,000 exactly. All 5150 route-2 brackets show a sign change. Isolation took 349,099 core-seconds = 97 core-hours in total: 87.4 on runners for shards 21 to 215 (88.6 by job wall time x 4 vCPU), plus 1.5 for the pilot, 0.6 for shard 216 and 7.5 for target A on the box. (An earlier version of this line said "about 64 on runners". That was an unchecked guess, corrected on 2026-10-26.)

By the kill condition, the gate failed, so this hunt stops target B and asks which side is wrong. What is known:

• The pipeline reproduces Revers at the Chao-Plymen parameters (section 1), with the same code and zero layer, to 1.7e-10. The runner shards reproduce the box bit for bit (smoke check), and target A's shards 1 to 20 feed both sums.
• Leading hypothesis: omega. Subsection 6.3 prints "\omega=727.95134". Revers's own history table gives the Saouter-Demichel omega as 727.951335792, and 727.951335792 rounded to five decimals is 727.95134. At alpha = 6e12 the Gaussian kernel has width about alpha^{-1/2} = 4e-7 in omega, so the 4.2e-6 difference is about 10 widths. Over the first 1e4 zeros (Arb, 128-bit sums) the two omegas give -0.977409433 and -0.977376629, which differ by only 3.3e-5, because gamma * Delta omega < 0.04 there. The phase shift grows like gamma and reaches 4 to 40 rad for gamma between 1e6 and 1e7. That is where the remaining -0.013 (ours) versus -0.026 (Revers) of the sum lives. This is consistent with the hypothesis but does not test it.
• Why it cannot be tested from the stored data. Shards store the zeros' SHA-256 and partial sums at the two fixed omegas only, so a new omega needs the zeros recomputed. Storing sums on an omega grid at almost no cost would have avoided this. That design gap is ours.
• Disposition (ALIGNMENT section 5): attempt unresolved for target B. It is not a refutation of Revers. The ball sum at omega = 727.95134 is itself a valid enclosure-carrying measurement, and at that omega the lower bound gives no conclusion.

7. The pre-registered omega-grid rerun (B2), 2026-10-27

Pre-registration: PREREG_B2.md and omega_grid.json, committed alone as e5b33dc90 before any rerun result existed. Shards 1 to 68 come from Actions run 40415010866, which the projection rule stopped (RUNS.md). Shards 69 to 220 come from run 40426999125. The computing code is byte-identical across both runs (SHA-256 table in RUNS.md). Evaluation: evaluate_b2.py, output shards_b2/evaluation.json.

Completeness: 220/220 shards are sound, every ball is ordered and disjoint within and across shards, N(T) = 22,000,000 by Arb's Turing count, all 4400 route-2 brackets show a sign change, and every shard carries all 83 omegas.

Control at 727.95134: [-0.99067672 +/- 1.89e-9], identical to the production B ball. The rerun reproduces production B.

The rule (|S - S*| + rad(S) <= B = 2.60111e-5):

H is not supported. The rounded-omega hypothesis accounts for most of the production-B gap (0.0122 at the printed omega, 3.4e-5 at 727.951335792), but not all of it, and the rule was fixed in advance. The remaining 3.4e-5 exceeds even Revers's worst-case zero budget, so it is not explained by zeros accurate to 1e-9. Candidate explanations, none tested here: further omega digits beyond those printed; double-precision reduction of gamma*omega ~ 7.6e9 mod 2 pi in a "machine computation" (Platt-Trudgian 2015 warn that this degrades relative accuracy to about 2^-17 at gamma = 1e9); or a different zero list or cutoff. Subsection 6.3 prints the sum as "S^(\alpha,\omega,T)\geq-1.002922947193156" with a "\geq", and our value at 727.951335792 does satisfy that inequality. The inequality was not the pre-registered rule, so this is noted, not used. Disposition (ALIGNMENT section 5): attempt unresolved for reproducing Revers's S. It is not a refutation of his result, whose positive conclusion holds at our value too (below).

Superseded in part, 2026-10-27 (post-hoc review of PR #360; measured; not pre-registered). The paragraph above is kept as recorded. Its three candidate explanations are replaced by one that was measured afterwards, and none of the three is it. Split S = S1 + S2 as in section 3 (S2 is the 1/(omega rho^2) part).

• S2 at omega = 727.951335792 over the first 2e4 zeros (gamma_20000 = 18046.46) is 1.685921553e-5, an Arb ball of radius 2e-18. The tail beyond gamma_20000 is not bounded rigorously. A main-term estimate, (2/omega)(log(Gamma/2 pi) + 1)/(2 pi Gamma) = 2.2e-7, bounds it if N(t) followed its main term. Going from 1e3 to 2e4 zeros moved S2 by 2.4e-8, about 1% of the same estimate at 1e3 zeros (2.0e-6). So the tail is probably of order 1e-9.
• The identity. Saouter-Demichel's printed first-order sum minus S2, -1.002906086981405 - 1.68592e-5 = -1.0029229462, equals Revers's S* = -1.0029229472 to 1.0e-9. With +S2, as Theorem 3.1 has it, the miss is 3.37e-5.
• The rule gap is 2 S2. S - S* = 3.3673e-5, and 2 S2 = 3.3718e-5. The 4.6e-8 difference is the next item.
• An end-to-end check of the 2.2e7-zero list. Our first-order sum, S - S2 = -1.0029061335, matches Saouter-Demichel's independently computed -1.0029060870 to 4.7e-8. That is about 1/1250 of the 5.8e-5 margin at eta = 1.59e-5. One zero at height gamma moves S by up to about 2 e^{-gamma^2/2 alpha}/gamma, depending on its phase. Below height about 5e6 that can exceed 4.7e-8, so a missing or spurious zero there could show at this level. Above 5e6 a single zero moves S by at most about 5e-8, and it would take over 1000 of them to erase the margin.
• Target A is consistent with +S2. There S2 = 1.595e-5. Revers's S* matches our S1 + S2 to 1.7e-10 (section 1) and misses S1 - S2 by 3.19e-5. Chao-Plymen's quoted -1.006569 is our S1 = -1.0065694 to its six printed decimals.
• The alternative omega is far worse. omega = 727.9513358 is 727.951335792 to seven decimals. Interpolating the pre-registered grid (degree-7 polynomial through the 8 nearest points) gives S(727.9513358) - S* = -8.0e-7. That is about 800 times the 1.0e-9 residual of the identity, and it uses +S2.

The pre-registered verdict stands: H is not supported by its rule. In substance, though, H's omega was right. The S* Revers printed is built from the sum at 727.951335792, and the rule failed on the sign of S2, not on omega. The grade is measured: it uses two printed decimals from the source and S2 truncated at 2e4 zeros with an estimated tail. The disposition for reproducing Revers's S* becomes explained at measured grade, post hoc. Driver: s2_check.py (compute writes s2_first20000.json in about 5 core-minutes; compare writes s2_comparison.json in seconds). Pinned in test_skewes_terms.py.

Descriptive grid (all 83 values are in evaluation.json, and no claim is made from them): S ranges from -1.00664805 to -0.99067672. No grid point meets the rule. The nearest, 727.951337542, is 2.88e-5 away.

Pre-declared localization at omega = 727.951335792 (alpha = 6e12, A = 6.85e7, T = 10379600):

So pi(x) > li(x) for some x with log x in [727.951319892, 727.9513517] (exact: [727.951319892, 727.951351692]). (Corrected 2026-10-27: the lower end was first quoted as 727.9513199, rounded inward; see the correction at the top.) The grade matches target A: a hardened computation on a cited derivation (Revers Theorem 3.1, RH to A = 6.85e7), with completeness on Arb alone and route 2 sampled. This is the Saouter-Demichel region at Revers's narrowest eta. It is not a new record: Saouter-Trudgian-Demichel 2015's interval is narrower and lies inside this one. (Corrected 2026-10-27: this sentence first said "narrower and lower". STD15's [727.951334747, 727.951336105] lies inside this interval, so its lower end is higher than ours, not lower. The localization here follows from theirs, if theirs holds.) At omega_0 = 727.951335426 every bound is negative (about -1.43e-3), so nothing is claimed there. That is expected, because STD15 used alpha = 1.61e15 and 5.25e8 zeros, not this family.

Planted faults at both decisive points: shifting gamma_1 by 1e-6 moves S by 7.79e-5 and out of its own ball, and dropping R4 moves the bound by 3.20e-5. Pinned in test_skewes_terms.py.

Cost: 106.2 runner core-hours by job wall for the whole rerun, against a final cap of 150 (RUNS.md has the cap history and watcher incidents). The zeros of all 220 shards are uploaded as 90-day artifacts, and each one's decompressed SHA-256 equals its shard's zeros_sha256. A new omega or alpha therefore no longer needs a recompute until they expire.

8. Literature note (2026-10-27)

M. Revers, New bounds in R.S. Lehman's estimates for the difference pi(x) - li(x), arXiv:2501.04488 v2 (J. Number Theory 2025), subsection 6.3, appears to subtract the second-order sum S2 where its Theorem 3.1 adds it. The printed S* = -1.002922947193156 equals the Saouter-Demichel first-order value it quotes, minus S2 at omega = 727.951335792, to 1.0e-9. With +S2, the miss is 3.37e-5. Subsection 6.2 uses +S2 and agrees with our ball sum to 1.7e-10. The evidence is in section 7. The grade is measured: two printed decimals, plus S2 truncated at 2e4 zeros with an estimated tail. The same subsection's table also has an apparent digit slip at eta = 0.0000159, and its eta = 0.0000155 row is the value at 0.0000156 (section 3).

The printed S* fits two readings: a sign slip, or S1 - |S2| used where the lower bound needs S1 + S2. Both give the same measured consequence, that the lower estimates for I in subsection 6.3 are too high by 2 S2 = 3.37e-5, and this note does not choose between them.

Whether that changes the paper's conclusion depends on the omega read. At omega = 727.951335792, the value this hunt inferred and that its pre-registered rule did not support (section 7), Theorem 3.1 with +S2 still gives a positive lower bound at eta = 1.59e-5: 5.83e-5 (section 7). The paper's printed S* and R give 9.20e-5 there before its Delta S. At the omega printed in subsection 6.3, 727.95134, the bound is negative at every eta in the table: I >= [-0.01211835 +/- 3.62e-9] at eta = 2.28e-5 (shards_b/combined_saouter_demichel.json). So the conclusion survives only at an omega this hunt inferred post hoc. (Scope corrected 2026-10-30: this paragraph first read "None of these changes the paper's conclusions", without the printed-omega case.)

Everything in this note is read against arXiv v2. The version published in J. Number Theory (doi 10.1016/j.jnt.2025.07.002) was not read, and may differ. This lab does not contact authors. The note is recorded here only.

The doors

Active constraints at the optimum. The binding quantity is the smallest eta with a positive lower bound, 1.061e-4. The binding constraints, ranked by how much of R they hold at eta = 1.061e-4 (R = 6.406e-3):

1. R4, the truncation at T (3.380e-3, 53%). It falls fast with T: 9.79e-4 at T = 1.25e6 and 5.3e-5 at T = 1.5e6.
2. R1, the prime-side floor 2/(omega - eta) + ... (2.763e-3, 43%). Theorem 3.1 fixes it, and it does not move with zeros or alpha.
3. R6, the RH-to-A and Gaussian-tail term (2.63e-4 at 1.061e-4, 5.35e-4 at 1.060e-4). It is the term that sets the eta floor: it grows like e^{-alpha eta^2/4 + omega/2}. Zero error, which Revers bounded at 1.9e-5, is no longer a constraint (1.3e-10).

The frozen-constant inventory. These numbers were chosen, not optimized:

• omega = 727.952018, Revers's choice. Re-optimizing omega reads the same zeros, so the trade shape is genuine and cheap.
• alpha = 1.34e11. Raising it narrows the eta floor (it scales like sqrt(omega/alpha)), but it raises R4 unless T grows: a genuine trade against zero count.
• T = gamma_2000000. More zeros cut R4 sharply. Cost is linear at about 13 ms/zero.
• A = 1.022e7, a cited input. RH is known to 3e12, so A could be raised for free inside R6. (Corrected 2026-10-27: false, see below.)
• The 64-bit isolation precision, and the route-2 sample of 1150. Neither binds.

Correction (2026-10-27, review of PR #360): A is a binding constant with a two-sided trade, not a free door. R6 has two terms. The first, (1 + 22/(A omega)) 8.283 log A e^{-alpha eta^2/4 + (omega+eta)/2}, grows like log A. The second, 7.152 eta A log A e^{-A^2/2 alpha + (omega+eta)/2}, falls off a cliff once A^2 is well above alpha omega, that is A above sqrt(alpha omega) = 9.9e6 here. So the best A sits just above the cliff, which is where Revers put it, and raising A toward the verified RH height costs margin. R6 and the lower bound at eta = 1.061e-4, with our S and every other parameter as in section 1 (all hypotheses of Theorem 3.1 hold at each A):

At A = 3e12 the bound is negative, so "raise A for free" would have destroyed the result. Raising alpha also forces A up like sqrt(alpha omega), so the alpha door pays this log A cost too. Pinned by test_doors_A_is_binding_not_free.

The information class of each door. Re-choosing omega and A stays inside the data the current run reads: the same 2e6 zeros and a cited RH height. Raising alpha or T needs more zeros, which means reading more of the same kind of data (target B is that door at 2.2e7 zeros). Lowering R1 needs new analysis on the prime side, not more data: Saouter-Trudgian-Demichel's weighted kernel (1 + 2u + 9.336u^2) is one such route, and it is not computed here. This is a measured optimum at one parameter family, not a proved ceiling. (Corrected 2026-10-27: re-choosing A needs no new data, but it is not an open door. A is already near its optimum, just above the R6 cliff, and moving it either way costs margin, per the table above.)

Doors at the Saouter-Demichel point (added 2026-10-27). At omega = 727.951335792, alpha = 6e12, the floor is eta = 1.59e-5. R = 2.831e-3 there: R1 = 2.763e-3 (98%), R6 = 3.6e-5 (which sets the eta floor, as before) and R4 = 3.2e-5. The constraint that dominates R is R1, which is prime-side and fixed by Theorem 3.1. More zeros could recover at most R4 = 3.2e-5, about half the 5.8e-5 margin at the floor, but they cannot move R1. That is a different balance from target A, where R4 was 53% of R. The frozen constants are the same, plus the grid spacing of 5e-8, which is descriptive only. Information class: reaching STD15's omega_0 needs alpha near 1.6e15 and about 5e8 zeros, which reads more of the same data. Lowering R1 needs new prime-side analysis (STD15's weighted kernel), not more data. The uploaded zero balls make any alpha <= 6e12 study on these 2.2e7 zeros free of isolation cost until the artifacts expire.

(Corrected 2026-10-27: "the frozen constants are the same" carried over the false "A for free" door. At B, A = 6.85e7 is again just above the R6 cliff, sqrt(alpha omega) = 6.6e7. At eta = 1.59e-5 and omega = 727.951335792:

Here the bound stays positive at A = 3e12, but the margin falls by 36%. A is binding at both points.)