All three passed. A search of every integer up to ten million found no violation of the arithmetic criteria of Lagarias and of Robin. The 138 zeros of the zeta function up to height 300 were all found on the critical line, and all simple. A fine grid around the first 29 zeros found the real part of ξ′/ξ positive everywhere off the line. Page 2 sets out each test and the control that had to fire first.
A pass here is a statement about a finite range and nothing more. The published verification of the zeros reaches a height of 3×1012; this issue’s reaches 300. Page 2 puts the two side by side. One control, written into the plan before the run, turned out to be too weak to detect anything; Page 4 says what was done about it.
Page 5 is the open desk: the Li coefficients, Weil positivity and Jensen polynomials were not tested, because no instrument was built for them.
Page 3 carries the 55-minute hour, a clock at Yale’s founding date, and a headroom check on the 8/9 paper: how far 11/12 is from its method.
None of the Lean was rebuilt for this issue. The Reading Room, Page 6, carries Part 10 of the serial, set from its source.
138. The count of sign changes of Hardy’s Z-function on (0, 300] equals the argument-principle count of zeros in that range, 138. A gap would have meant a zero off the line or a double zero. There was none.
Every figure above comes from the run output in the geometry folder, docs/rh-criteria-run-2026-10-25.txt.
The plan was committed before any of the code ran: the data, the ranges, the pass rule and, for each test, a control. A control is a run on a world where the answer is known to be “fail”. If the control does not fail, the test cannot see what it is looking for, and its pass means nothing.
| Test | What is checked | Range | Control that had to fire | Result |
|---|---|---|---|---|
| T1 · Lagarias and Robin | Lagarias (2002): σ(n) ≤ Hn + eHn ln Hn for all n. Robin (1984): σ(n) < eγ n ln ln n for n > 5040. Each is equivalent to RH, as the papers state. | Every n up to 107, exact integer divisor sums. | Lagarias bound × 0.95: 25 violations, first at n = 12. Robin bound × 0.98: 2 violations, at n = 10080 and 55440. | No violation. Tightest margins 1.1% (n = 12) and 1.4% (n = 10080). |
| T2 · Zeros on the line | Count sign changes of Hardy’s Z on (0, 300] and compare with the number of zeros there from the argument principle. | Height 300, step 0.02, 20 digits. | One adjacent pair of zeros replaced by a complex pair: the count fell from 138 to 136, exactly 2. | 138 and 138. Equal. Numerical, not interval-certified. |
| T3 · Lagarias, analytic side | Re(ξ′/ξ)(s) > 0 for Re s > ½, which is equivalent to RH (Lagarias 1999, as cited). | σ from 0.5005 to 2, within ±0.5 of the first 29 zeros at step 0.01, elsewhere to t = 100 at step 0.05. | First zero moved to Re s = 0.52: the minimum fell to about −100 at t = 14.135. | Minimum +0.00203, at σ = 0.5005. Positive on the whole grid. |
On the critical line Re(ξ′/ξ) is exactly zero, so near the line the real part is small for a reason that has nothing to do with whether RH holds. At σ = 0.5005 it should be about 0.0005 divided by the squared distance to the nearest zero, and the minimum, 0.00203, is at the edge of the window half a unit from a zero. The test cannot see a zero closer to the line than the grid spacing of 0.0005 in σ.
| Source | What it claims | What was checked |
|---|---|---|
| Platt & Trudgian, arXiv:2004.09765 | The Riemann hypothesis is true up to 3×1012, in the paper’s own title. | Title seen in a search listing this week. The paper was not opened here. |
| Lagarias, Amer. Math. Monthly 109 (2002) | An elementary inequality on σ(n) and harmonic numbers, equivalent to RH. | Cited as the series cites it. The PDF was not opened this week. |
| Robin, J. Math. Pures Appl. 63 (1984) | The inequality on σ(n) and ln ln n, equivalent to RH for n > 5040. | Cited as the series cites it. The PDF was not opened this week. |
| This issue, T1–T3 | No counterexample to n = 107; zeros on the line to height 300. | Run by the desk, with controls. Output and script in the geometry folder. |
The title of the Platt–Trudgian paper puts the checked height at 3×1012. This issue checked up to 300, a ratio of ten billion to one. Read every pass here that way: the desk repeated a check in a small range that others have done in a vast one, with a control attached, and learned how its own instruments behave. The count of Z sign changes against the argument-principle count has no error bounds, so it is evidence in the range, not a certified verification.
Continued on Page 3 · The 55-Minute HourThe page The 55-Minute Hour · The Tomb keeps two clocks side by side. Seconds and minutes are the same; only the hour is re-cut, to 55 minutes. Everything follows from one ratio, 60/55 = 12/11 ≈ 1.0909.
| Quantity | Value | Why |
|---|---|---|
| Hour hand gains, per real hour | 60/11 ≈ 5 min 27 s | 1/11 of a turn per hour against 1/12; the gap is 1/132 of a turn. |
| Gain per day | 1440/11 ≈ 2 h 11 min | Same gap, 24 hours. |
| Hands realign every | 132 h, 5.5 days | 1 ÷ (1/11 − 1/12). Not six days: that figure divides 720 by the readout’s five minutes, which measures the wrong thing. |
| 55-minute hours in a day | 26.18… | 1440 has no factor of 11 and 55 = 5·11, so no whole number tiles a day. It can only be a rate. |
| New Haven against Eastern time | 8 min 17 s ahead | (75 − 72.9279) × 4 min per degree of longitude. |
| Charter, 9 Oct 1701 O.S. = 20 Oct 1701 N.S. | JDN 2,342,630 | One day count under both calendars; the eleven days drop out of the subtraction. |
The page’s own point is that the calendar and longitude corrections (eleven days, 8 m 17 s) are larger than the five-minute effect the clock was built to show, and that a clock set five minutes ahead and left alone is a constant, not a rate. Every figure in the table was recomputed for this issue; the page’s history of the Roman and English new year (Livy, 153 BC; Lady Day; Scaliger’s Julian Day) was not re-sourced here.
The editor asked whether the proportion of zeta zeros known to be simple or on the line can go from 0.88893 to 55/60 = 11/12 ≈ 0.9167, a gain of 0.0277. The Cobeli–Zaharescu paper was read in full and its constant reproduced from its own numbers: 0.88893.
Inside the paper’s formula, 11/12 needs the local remainder coefficient to be 2.3 to 3.1 times the paper’s. A proxy for the best that sharper local analysis could give with the same test function, built from lattice patches and not a proof, reaches about 0.8906, a gain of 0.0017. A remainder four times larger would give 0.9047, still short.
So 11/12 would need a different ingredient, not a tighter version of this one, and nothing here proves anything. The test function was not re-optimised. The clock’s 12/11 and the zero proportion share only that both are ratios: a shared functional form, not a shared invariant.
| Where | What changed |
|---|---|
| docs/prereg-rh-criteria.md | Registered a control for Robin’s inequality: the bound multiplied by 0.99 had to produce violations. In the first run it produced none. The tightest margin of Robin’s inequality below 107 is 1.4%, so a 1% reduction cannot be detected there. The control was vacuous and the first T1 result was discarded. |
| docs/prereg-rh-criteria-2.md | New dated file, written after seeing that and before the rerun, replacing the factor 0.99 with 0.98. The first file is not edited. After the change both T1 controls fire. |
| docs/audit-log.md | Entry for 25 October records the three results, the replaced control, and the limit on T3. |
The fault was in the control chosen, not in Robin’s inequality, and the result of the discarded run was the same as the rerun’s: no violation. It is recorded because a control that cannot fire is exactly the kind of check this series has caught before, and because the order matters: the weak control was found by running it, and was replaced in the open.
Nothing else in No. 14 changed. The RH paper’s §4.9, the six-check script and the dated correction boxes in Book 4 are as they were.
Not yet done. §4.7 of the RH paper has still not been updated for the Cobeli–Zaharescu bound. That section’s statement of the proven proportion remains out of date.
Continued on Page 5 · The Open Desk| Item | State |
|---|---|
| 11/12 (55/60) as a bound on the zero proportion | Headroom analysis run (page 3): the method as it stands tops out near 0.891 with the same test function. Needs a new ingredient; not derived. open |
| RH paper §4.7 | Needs the Cobeli–Zaharescu bound written in with its limits stated. open |
| Li coefficients, Weil positivity, Jensen polynomials | Not tested. No instrument has been built for them and none has been shown able to fail. open |
| T2 and T3 certification | Both are numerical, with no error bounds. An interval-arithmetic version would certify the range. open |
| T1 beyond 107 | Not run. The range tested is small enough that a pass carries no weight for RH. open |
| Lagarias (2002), Robin (1984), Lagarias (1999) | Cited as the series cites them. The PDFs were not opened this week, so the equivalence statements are unchecked here. open |
| ZeoliteCommutation.lean, the −6 commutator | To be checked that the file states it as an assumption. The finite-matrix trace argument shows a bounded commutator cannot equal −6 times the identity. open |
| Volume I: numbered Result to Lean name | 53 of 82 theorems unnamed in the foundational documents; no mapping yet. open |
| Volume I Lean port | A port-v4.32 branch exists; the claim of 82 rests on the pinned toolchain until it is done. open |
| The (c, κ) split in Vol I | Stated without derivation; unexplained; no erratum until found. open |
| Vacuity scan on geometry (31) and io | Not yet run; a probe file plus one CI step. open |
| Chapters 11–12 of Book 4 | Not checked for a proof of RH implying positivity. open |
You are invited to join a math study group that meets online, in the library. Everyone is welcome.
To join, or to ask when it meets, write to g6llc@proton.me.
The Reading Room follows on Page 6.
Continued on Page 6 · The Reading RoomThe buffalo slowed before either of them understood why, and it was another full minute before Fermat saw it: a shape low in the fork of a mangrove root, gold and black and entirely still, watching them with an attention that had nothing hurried in it at all.
Marie's hand found his arm without either of them deciding it should. "Pierre."
"I see it."
"What is it doing?"
It was, as far as Fermat could tell, doing nothing more alarming than a very large, very beautiful cat holding its own council, head tilted, nostrils working the humid air in slow, deliberate pulls, its golden eyes moving over the two of them with a curiosity that looked, absurdly, almost scholarly — the same unhurried attention Fermat himself might have given an unfamiliar equation before deciding whether it was worth the trouble of solving.
"I believe," he said, keeping his voice as level as six months of practiced calm could make it, "that we smell extremely strange to it."
"We smell extremely strange to everyone, Pierre, we have not bathed properly in five months."
"Precisely my point. To a creature that hunts by scent, two Europeans marinated in salt, mud, buffalo, and considerable personal despair do not smell like anything it has ever been taught to want. I suspect it is simply trying to decide what we are, in the way a man might stare at a coin from a country he's never traded with."
The jaguar's tail moved, once, an unhurried flick, and it did not come down from the root, and it did not look away, and for the length of perhaps four breaths the entire mangrove seemed to hold itself as still as the cat did, waiting to see which way the arithmetic of the moment would resolve.
It was the buffalo that resolved it. The great beast, which had not so much as broken stride at the sight of the jaguar, continued forward at exactly the pace it had been keeping since the sandbar, hooves finding the firm ground beneath the mud with the same placid indifference it had shown the crabs, the storm, the open sea — passing close enough beneath the mangrove root that Fermat could have reached out and touched the animal's paw, and never once turning its great head to acknowledge the danger at all.
The jaguar watched them go. It did not follow. Whatever question it had been turning over in that unhurried, golden-eyed way, it seemed, in the end, to decide the two strange-smelling creatures riding the indifferent giant were not worth the trouble of an answer.
"It's letting us pass," Marie whispered, once the shape had fallen behind them and the mangrove had swallowed it from sight.
"I don't believe it thought of it as letting anything," Fermat said. "I believe it simply couldn't find a category for us, and a creature that cannot categorize something is, in my experience — admittedly gained entirely from magistrates rather than jaguars — generally reluctant to commit to eating it."
Marie let out a breath that was not quite a laugh, but was closer to one than anything she'd managed since the wreck. "I have never in my life been grateful to smell this badly."
"Nor I," Fermat said, and meant it, and found himself thinking, not for the first time on this strange, drowned, buffalo-carried journey, that the whole of his life's careful arithmetic had never once prepared him for a proof this simple: that safety, out here, had nothing to do with cleverness at all, and everything to do with smelling like something the jungle had no name for.
Part 10 of 18 · Fiction, drawn from the documented world; the ledger below says which is which. Set from its source file, as written.
Rights notice. Pierre et Mademoiselle is not covered by this issue’s CC BY 4.0 licence. The serial and the work it belongs to — a limited series in development — are © 2026 Pablo Nogueira Grossi / G6 LLC, all rights reserved. No licence to reproduce, adapt, translate or perform it is granted here.
Documented: nothing in the series ledger (FACT_AND_FICTION.md) bears on this part. It has no entry for the jaguar, the mangrove or the encounter, so this part asserts nothing documented about them.
Invented: the jaguar on the mangrove root, the exchange between the two travellers about being unclassifiable by smell, and the buffalo passing beneath without turning its head. The buffalo carrying the travellers is the relocation of the Marajó legend that the ledger records for No. 8.
Each issue, when it goes out. No list bought, harvested, or lifted from a conference roster. Unsubscribe by replying with the word stop.
docs/rh-criteria-check.py — pre-registered tests of the Lagarias and Robin inequalities, zeros on the line to height 300, and Re(ξ′/ξ) near the zeros; four controls, all fired. Standard library, numpy and mpmath.tools/corpus_count.py · tools/self_reference.py — entity-aware counting, and the vicious-circle test. Both standard library only.sorry, standard axioms only.