12 September 2026 · a record · v1.0
Between 17 August and 12 September 2026 this programme gave detailed written feedback, in public and on dated occasions, on four notes by Kazushi Mizutani. The revisions that followed take up a number of its points. They carry no reference to it. A reader of those notes cannot see that history, and this page makes it checkable.
This is a record, not an allegation. No theorem, no measurement and no result of this programme appears in anyone else's work under their name. What was used was criticism — objections, a diagnosis, a direction and a reading list. That is a real debt of a different and lesser kind, and its remedy is one sentence in an acknowledgement. No plagiarism is alleged, none is implied, and the word is not applicable here.
Criticism that demonstrably reshapes a document is a contribution to it, and normal practice records that in a line. That is the whole of the claim made here — there is no other, and nothing on this page asks for more than a sentence and a reference entry.
| date | what was given |
|---|---|
| 17 Aug 2026 | Detailed written feedback on A Geometric Reinterpretation of the Riemann Hypothesis written, and dated as v1.0 of the page linked above. Nothing on this date is claimed to have reached him — see the note under this table |
| 18 Aug | He wrote: "I have made revisions of my essay about Riemann Hypothesis. Especially, I mention the Euler product. Would you check it, please?" — and was answered "I will." |
| 19 Aug | An expanded review — Parts VIII–XII of that page, covering the note of 18 August — and the page's address posted to him in the public thread |
| 19 Aug | An interactive explorer sent, which draws ζ and Dirichlet L-functions in the critical strip and reads function shapes from LMFDB |
| 19 Aug, 13:54 | He wrote: "Thank you for your rigorous reply." |
| 21 Aug | He posted a further revision and asked for a check; a reply was given that it would follow |
| 12 Sept | A third round of feedback completed on the notes of 21 August and 8 September — Parts XIII–XVII of that page |
Delivery and reading are established by his own words on a public timeline, not inferred from anything — and they are established for 19 August. On that date the address of the review page was posted to him in the thread; he replied "Thank you very much. I will check it, later. I'm working now.", and at 13:54 the same day, "Thank you for your rigorous reply."
There is no evidence that anything reached him before 19 August, and nothing here rests on the assumption that it did. The first review was written on 17 August, but this programme's own records do not establish that it was posted to him or that its page was publicly reachable on that date, and the surviving captures of the discussion thread failed to load their comments. His note of 18 August is therefore treated as his own work, arrived at independently, and every item in Part II below is dated to the 19 August review, which he acknowledged reading in writing.
1. The distinction between positivity and vanishing. Given on 19 August: "Weil and Li never ask a non-negative quantity to vanish — they ask it to be non-negative. Vanishing of a non-negative quantity is the strongest possible demand and gains nothing over the hypothesis itself." The note of 21 August carries §15, titled "A Critical Distinction: Positivity versus Vanishing", making that case in that order — that the inequality "does not by itself imply the Riemann Hypothesis", that the programme therefore "contains two logically distinct tasks", and that "the second task is substantially stronger". Two days separate the two documents.
His note of 18 August already carried the two stages as a sequence — a Stage V headed "Positivity and vanishing", and an open-problems section recording that the vanishing "has not been established in this note". The raw material was his. What the review supplied is the sharpening that became his section title: that the inequality is free and carries no information, that the vanishing is the hypothesis itself, and that asking a non-negative quantity to vanish is the strongest demand available and gains nothing.
2. The Weil–Li framing. The 19 August review's central structural finding was that the programme is the Weil–Li programme; that its fourth stage was completed by Bombieri and Lagarias in 1999; and that Bombieri's Clay Institute problem description states Weil's criterion with negativity rather than positivity. The previous note contained no Weil form and no Li. The 21 August note has §5 "From the Explicit Formula to a Quadratic Form", §13 "Relation to the Weil Criterion", a central conjecture stated in terms of the Weil quadratic form, and references to Li (1997), Weil (1952) and Bombieri's Clay article.
Weil and Li are standard literature in this area, and his own reference list includes Conrey's Notices survey, which would put any reader within reach of both. This row is recorded with that qualification attached rather than without it.
The sentence the first review was built to deliver was "The symmetry gives the axis. It does not give the confinement." The note that followed on 18 August was titled … and a Proposed Arithmetic Confinement Principle, added a section called Why Symmetry Alone Is Not Sufficient, turned to the Euler product for the first time, and was announced with the words "Especially, I mention the Euler product." None of that vocabulary appears in his previous note.
It is nevertheless withdrawn as a credit claim, because the dates do not support it. The first review is dated 17 August and the note is dated 18 August, but there is no record that the review reached him in between. On the evidence that exists, he reached that turn on his own, and the convergence is recorded here only because leaving it out would look like concealment once someone else noticed it.
4. The obstruction that the squared transverse displacement involves the conjugate and is therefore nowhere holomorphic, so it cannot pass through a linear explicit formula — given on 19 August with a measurement of the failure. It appears as his §9, labelled central. Recorded as citable rather than structural: it is elementary once the derivative is taken, and he may well have reached it independently — the feedback page credits him for doing so.
5. The distinction between the squared transverse displacement and the full complex one — his §14.3, in the review's own terms.
6. The ruling that one-point compactification is automatic for every discrete unbounded set and therefore distinguishes nothing — conceded in his §1, §12 and §18, in close to the review's wording. That concession was credited on the public page when he made it.
7. The scope correction from implied by the hypothesis to equivalent to it — the review's objection to his §7, which produced his largest technical improvement.
The review supplied, by name and with its constants, the Davenport–Heilbronn
function: the object that decides the question. It has the critical strip, the functional-equation
symmetry, the discrete zeros running to infinity — and zeros off the critical line. Supplied
with it: the period-five coefficient list, the constant, a verified off-line zero at
σ = 0.8085171824566374, t = 85.69934848537759, a comparative census, and on
19 August a working viewer that draws these functions so that no code needed writing.
The words Davenport and Heilbronn appear in neither new note. This is recorded not as uncredited use — it is the opposite. It is the more serious technical point, and it costs its author more than it costs anyone else: the instrument that decides his question was in his hands for three weeks, and the notes neither use it nor say why not.
The transverse displacement coordinate is his. It coincides with this programme's own displacement coordinate. He reached it unaided, and on the record set out above he reached it with nothing of this programme's in front of him. That was credited in public, on the feedback page, before any of this was written.
The decision to work with a quadratic rather than a linear functional is his.
The conditioning advantage is his, and was measured here for him: on the Davenport–Heilbronn ledger his quadratic observable reads 5.03810655514×10−5 against the classical linear criterion's −4.04982244779×10−9 — better conditioned by a factor of 12,440, both exact, both firing correctly.
193 certified quartets to height 4000, above that census's detection floor; a strict lower bound, not a value. It forbids nothing and says nothing whatever about where a zero of ζ is.
The note of 8 September owes this programme nothing. It is in a different subject — algebraic and transcendental numbers — it takes no material of ours, and nothing in it is claimed. Verified mechanically on 12 September: a search of its complete text for this programme's name, its page, and every feedback-related term returns no reference of any kind, and the note carries no bibliography at all. The feedback given on it is a gift, not a debt.
The European Mathematical Society's Code of Practice (approved 29 October 2012), Responsibilities of authors, item 1, quoted whole — including the half that protects the author:
"An aspect of good practice is the granting of proper credit, and the referencing of the work of others, with appropriate bibliographic references. It is important to note that it is not unethical to be mistaken in the attribution, or lack of attribution, of results, provided that authors have carefully sought to determine whether their claimed results are new, and provided that errors of attribution are corrected in a timely and appropriate manner, as they are discovered or pointed out."
The omission is therefore not an ethical finding. It is pointed out; the code names the remedy. The same item's plagiarism clause is expressly disclaimed here.
(a) One of the sentences given to him was withdrawn. The review stated that a proof must use the Euler product. That asserts more than the 1936 theorem supports: Davenport and Heilbronn exhibit what a functional equation cannot do, not what a proof must use. This programme's register now reads that the missing object must draw on more than the functional equation and the analytic axioms, with the Euler product as an identified candidate rather than an established necessity. The correction was volunteered to him in writing.
(b) This programme made the same species of omission, pointed inward. The measurements produced in the second round — including the figure quoted in Part IV — were handed back to this programme's own record on 19 August and were never entered into it, surviving in two review files and a probe log and nowhere else until 12 September. On the same date, this programme's own site record was found to have described the entire site as not uploaded for twenty-four days after it was deployed and serving. Both are corrected, and both are minted as errata in this programme's own register.
One sentence in the acknowledgements of the 21 August note, or of whatever supersedes it — for example: "The author thanks Ondřej Dvořák, whose detailed feedback on the previous note prompted the revisions in §§5, 13, 14 and 15." — and, since that note already carries a reference list, one entry in it naming the feedback page and its address.
Nothing further. No change to his mathematics, no withdrawal, no public correction, and nothing at all regarding the note of 8 September. The request has been made privately and once; this page exists because the record should be dated and checkable, not because anything further is being sought.
This programme is openly AI-assisted and says so on every page it publishes: the arguments are model-drafted and model-checked, and the author has not independently verified every proof. That disclosure applies to the feedback described here as much as to anything else published on this site. What the record has is a name, a date and a public address, so that any reader can check what was said and when.
No zero is located, excluded or constrained by anything on this page or in any document it describes, and nothing here bears on whether the Riemann hypothesis is true. No RH claim is made or implied — not by this programme, and not by the notes discussed, each of which states its own ceiling. The Davenport–Heilbronn figures quoted certify a finite window above a finite detection floor, are a strict lower bound rather than a value, and forbid nothing; they decide what a symmetry argument cannot do, and say nothing whatever about where a zero of ζ is.