v1.3.0 — two proved theorems in the note, the proof of one of them, and a referee pass that flagged everything

Released 2026-08-18. Previous: v1.2.0 (2026-08-17).

This release adds; it corrects nothing that was public. Every statement in v1.2.0 that this version touches was true when it was written. What changed is that two of the note's open questions became theorems.

Two proved theorems in the standalone note

paper/note1.tex — Two speeds at the boundary — grows from 11 to 12 pp., with paper-ja/note1_ja.tex moving in the same round.

thm:monotone — a lower bound for every non-increasing profile. Let w_0 ≥ … ≥ w_{k−1} ≥ 0 and t > 0. If arctan(2t) ≤ π/k then F_k(½+it) ≥ 1, so there is no zero on 0 < t ≤ ½tan(π/k). Two lines from an exact summation-by-parts identity, now displayed. The previous version had this for constant weights only. Monotonicity is doing work in the strong sense: the increasing ramp w_j = (j+1)/k drives F_k to −25.93 at k = 64 inside the window, so the conclusion fails and not merely the bound — and it is not necessary, since a bump and an alternating profile stay above 1 without being monotone. Proved, two routes.

thm:decayrate — the exact rate above the kink. Fix s > 1, w_j = (j+1)^−s. Then for all large k a first zero t₁(k) exists and 2k t₁(k)²/log k → s − ½. This is the first complete analytic theorem in the programme at the level of the rate, and it answers, for s > 1, the note's own section What a proof of the decaying case would have to supply. Proved, two routes.

rem:shape — the shape, statuses split rather than averaged. λ_∞(s) = max(s/2, s − ½), kink exactly where Σ_j w_j stops converging: above it the theorem, below it a measurement (s = 0.5, 0.75 at k = 32768, the two candidates separated by 22.2 and 20.5 times the observable's own quantum), at s = 1 derived by two routes. Near the kink the measurement cannot decide, and the remark now says so with the numbers.

The proof is in the deposit, because the status says proved

paper/proof_decayrate.md is new to this repository. It was written at r229, repaired at r231, and lived in a design document that .gitignore excluded — so the note carried STATUS{proved} on its headline theorem while a reader holding the note could not reach a line of the proof.

A statement that lives one artefact away from its reader has not been made.

The referee pass, reported in full because it did not go well

Before this release the new section went through the standing referee pass: a reader in fresh context, given exactly three jobs — restate the claim, name what would falsify it, flag any single word whose deletion changes it — and nothing else.

It flagged all nine units. The text was green on all twenty-two mechanical checks and built in both languages with zero undefined references.

what it caught	why no check could
"its limit is 2ζ(s)" — false as written; the sine series tends to 0, and the quantity with that limit is the series divided by t	valid mathematics, wrong claim
STATUS{proved} on a theorem whose proof the repository excluded	no check reads whether a cited proof is reachable
"appears to satisfy" hedging a branch the same status block calls proved	a status cannot fix a verb, and no check reads adverbs
"decisively", "twenty times the observable's resolution" — a tier not in the note's vocabulary, and a resolution the note never defines	the numbers were right; the reader could not find them
"Consequently the first zero…" — presupposing a zero the hypotheses do not give	quantifier presupposition, invisible to a grep
"drives F_k well below 1" — unquantified, and below 1 is the wrong threshold for the claim	the stronger, true statement was in a log we already had
"fix the shape", "has stopped being a table of measurements"	the statuses were right; the prose overtook them

Eight flags accepted, one word defended with its reason recorded and its secondary finding accepted. The pass and every adjudication are in lean/pnp/refpass_r233.log.

Grepping outward from one flag found a stale count in both editions: "the twenty-one mechanical checks". There are twenty-two. C22 was added the day after v1.2.0 was tagged — true when written, and the artefact kept asserting it.

Everything else

• The sweep before the tag — what did this release make false? — returned four artefacts, none of them a paper and so none covered by any check: the deposit's page count, the README's release line and log count, the homepage (which still called Part III's three headline theorems conditional, eight days after they stopped being so, and did not mention the note at all), and the check count above.
• New artefacts: paper/proof_decayrate.md, lean/pnp/refpass_r233.log.
• The Lean canon is unchanged, and still replays through an independent kernel that must first reject three deliberately corrupted modules.

What has not changed

Not peer-reviewed. Every statement declares its own status where it is stated. The s < 1 branch is still a measurement, and the note says so at the statement rather than in a footnote.

A DOI makes a version permanent; it does not make it true.

v1.2.0 — a standalone note, and four corrections

Released 2026-08-17. Previous: v1.1.1 (2026-08-16). (Correction, made minutes after publishing: the banner, CITATION.cff and this line first read 2026-08-18 — a date copied forward from a drifting round header and never once asked of the clock. Zenodo, which does not take our word for the date, is what caught it. Fixed here and in the repository; the copy inside the archive keeps the wrong date, because the tag now carries a DOI and is not moving. A fact every reader already knows is a fact nobody checks.)
This release adds a fourth manuscript and corrects four statements that were public in v1.1.1. Read the corrections first; they are the part a reader of the old version needs.

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What is new: a standalone note

paper/note1.tex — Two speeds at the boundary: zeros of sums of conjugate sections of power series, 11 pp., with a Japanese edition in paper-ja/note1_ja.tex.

The note takes one question out of Part III and states it in the standard language of power series, with no prerequisites from this repository and no citations to the other manuscripts at all. For non-negative reals w_j, with G(z) = Σ_j w_j z^j and G_k its sections, it studies the zeros of

    F_k(q) = 1 + G_k(2q) + G_k(2−2q)

— a sum of two sections under substitutions that are complex conjugates along the line Re q = ½. Jentzsch's theorem covers the zeros of a single section, and adding functions can cancel zeros; we found no result covering sums of sections, and the note says exactly where we looked so that one reference can overturn the claim.

The note contains: the reduction (on that line the object is real, so its zeros are sign changes, which cannot be cancelled); a closed form and an existence theorem for constant weights, with the zero set exact in one case; a two-regime rate law for decaying weights; what a proof of the decaying case would have to supply, including the attempts that failed; and one constant that remains unsettled, with the reason it will stay unsettled at computable sizes.

It carries no vocabulary of its own and cites nothing. That was a condition of its approval, and meeting it improved it: a note that cites its parents invites the reader to go and check whether the parents are any good, and this one does not need them.

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Corrections to statements that were public in v1.1.1

A DOI makes a version permanent; it does not make it true. Four statements in Part III are corrected here. Each is corrected at its own statement in the manuscript, with the reason, and none has been deleted.

1. rem:pinchformula — the k = 70 table

The measured row read 0.503895, 0.413403, 0.305147, 0.220690, 0.133982, 0.070741, 0.022306. Those values are wrong from the fourth significant digit, with mixed signs and a largest error of 3×10⁻⁴ at c = 2. The replacement is computed by two independent methods — one forming the polynomial and taking every root, one never forming it and locating the nearest zero by the argument principle — which agree to 94–120 significant digits, are stable between 60 and 120 digits of working precision, carry an argument-principle certificate per row, and reproduce a zero the paper proves exact.

We cannot reconstruct what produced the old row. The obvious hypothesis, an off-by-one in k, was tested and refused: the published values match neither k = 69 nor k = 71.

2. rem:nopinchreading — the displayed profile

The remark displayed m_j = ⌊2^j (j+1)^(−2)⌉ and, in the same sentence, the constant it generates, Γ_k → 5.230199559. The displayed formula gives 3.480199559. The missing ingredient is the clamp m_j ≥ 1, which is forced — m_j is half a gap between distinct odd numbers — and which was in the computation and not on the page.

3. rem:rateregimes — the dividing line

The dividing line between the two rates was conjectured to be Σ_j w_j < ∞. That is disproved. The harmonic profile w_j = (j+1)^(−1) is not summable and takes the √(s log k / 2k) rate: k·t₁ runs 11.9, 18.0, 27.3, 37.0, 55.6 at k = 64…1024, growing rather than approaching π/2. The line is w_j → 0.

The mechanism had said so all along: it uses w_k and never uses the sum. When a conjecture and its own mechanism disagree about what the hypothesis is, the mechanism is the one that was derived.

4. rem:rateregimes — a refutation of ours, withdrawn

We recorded the candidate λ_eff → s − ½ as refuted, on the grounds that |λ_eff − (s−½)| stays in [0.34, 0.48] over k ≤ 2048 without shrinking. That band is exactly what the candidate predicts at those k: the approach is −½ + (log log k)/(2 log k) + O(1/log k), and the correction term alone is 0.13966 at k = 1024. The refutation is withdrawn.

A refutation needs a resolution claim. "It did not converge" is evidence only if the hypothesis predicts convergence you could have seen.

Also corrected, without having been wrong

prob:converse was posed asking whether R = 1 forces the pinch at rate π/2k. The rate clause is answered no and has been withdrawn from the problem statement out loud; the existence question is untouched and remains open. And the "doubled ladder" fingerprint now carries its measured scope: the evenly spaced ladder is a constant-weight phenomenon, not a general one — measured gaps run 0.17 to 0.73 in units of π/k once the weights decay.

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How these were found, since it bears on the rest

Three of the four were found by building something new and noticing that it contradicted a sentence written elsewhere — not by any of the twenty-one mechanical checks. The fourth was found by a gate: a rule that the phase picture must postdict a known scale before it earned an instrument, which produced a model that then withdrew our own refutation.

The failure ledger in tools/ records all of this, with the general rule extracted from each. It now has 87 folded entries and 14 pending.

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Everything else in this release

• Part III: 51 → 54 pp., carrying the four corrections above and the two-regime rate law with its full record.
• Japanese editions track all four manuscripts; note1_ja.tex is new. C13 checks that every number in a translation occurs in its source (630 literals) and C19 that the skeletons match.
• New checks: C21 refuses any file in the tree whose name says it holds credentials. The check announces its own limit — it cannot see a secret that is innocently named.
• New scripts and logs: rate_r200, pinch_r202, trackm_r203, divide_r204, lambda_r206, envelope_r206b, note1tab_r208, rung0_r211, and the referee-pass logs refpass_r201, refpass_r209. Every number in every manuscript is copied from one of these.

What has not changed

Not peer-reviewed. Every statement still declares its own status where it is stated. The Lean canon is unchanged and still replays through an independent kernel that must first reject three deliberately corrupted modules.

v1.1.1 — correction release

This is a correction release. v1.1.0 carried false sentences — three about the Lee–Yang ladder of Part III, and one equivalence inside a proposition marked proved. This release says which ones rather than fixing them quietly.

A DOI makes a version permanent; it does not make it true.

What this release makes false

artefact	was	now
Part III, rem:leeyanglacunary	cos(k arctan 2t); rate 3π/2k; ladder (2n+1)π/2k	sin; rate π/2k; ladder θ_n = nπ/k, all integers
the same remark's STATUS	two measured series quoted under one symbol q_1	an erratum naming them as rung 1 and rung 3
prob:converse	rate (2n+1)π/2k, integer set by "the profile's first resonance"	rate π/2k — one ladder, one first rung, no profile-dependent integer
the fingerprint paragraph	"spacing π/k, offset by half", a signature of cos(kθ)	"twice as densely", a signature of sin(kθ)
prop:gqgen item 1	"R ≥ 1 is exactly the statement that Γ(A) is finite" — inside a proposition carrying STATUS proved	R > 1 forces it, R < 1 forbids it, R = 1 decides nothing
rem:nopinchreading	the same claim, in bold	corrected, with a profile of each kind named
.zenodo.json	Part III "45 pp."; version v1.1.0	51 pp.; v1.1.1; erratum in the description
README.md	"Archived release v1.0.0" — stale through the whole life of v1.1.0	v1.1.1, with the DOI caveat in bold under the badge
CITATION.cff	no version, no date-released, no doi at all	all three present

The first error, in one paragraph

On Re q = ½ write z = 1+2it, ρ = |z|, θ = arg z. Then z − 1 = 2it, so Σ_{j=1}^{k−1} z^j = (z^k − z)/(2it), and since ρ sin θ = Im z = 2t:

Γ^(q) = 2 + ρ^k sin(kθ) / (2t)      for a_i = 2^i + 1,  with no approximation
Γ^(q) =     ρ^k sin(kθ) / (2t)      for a_i = 2^i − 1,  an identity

v1.1.0 instead characterised the sum by its largest term, z^j + conj(z)^j = 2ρ^j cos(jθ). That is a modulus-level approximation, and it discards the factor 1/(z−1), whose argument is exactly −π/2.

A quarter turn of phase moves a ladder by half a rung. The half-offset that v1.1.0 wrote up as the discovery was the error itself.

The two measured series in that remark, |q_1−½| = 0.0520, 0.0253, 0.0167 and k·Im q_1 = 4.7736, 4.7457, 4.7299, are both correct measurements of genuine zeros — of the first and the third rung respectively, reported under one symbol. The agreement of the second with 3π/2 was read as confirmation because the search had been aimed where cos predicted a zero and found a real one there.

The second error

prop:gqgen item 1 said the radius of convergence decides whether Γ is finite. It does not decide the boundary case. With w_j ≥ 0, R > 1 forces G(1) < ∞ and R < 1 forbids it, but R = 1 allows either — and both occur among profiles the paper already uses:

profile	w_k^(1/k) at k=1024	R	Γ_k at k=1024
a_i = 2^i + 1	0.99932267	1	1026 (diverges)
m_j = round(2^j (j+1)^-2)	0.98654014	1	5.230199559 (converges)

The surviving direction is the one prop:nopinch actually uses (R > 1), so item 3 and the proposition's use are untouched; what was wrong was the equivalence.

What the corrections gained

Not weaker claims — stronger ones. For a_i = 2^i − 1 the zero set on the critical line is now exact and elementary: q = ½ + (i/2) tan(nπ/k), n = 1, 2, 3, …. All near-½ zeros were confirmed to lie on the line by an argument-principle count on |q−½| = r for r up to 0.25, matching twice the number of on-line sign changes in every case. And the boundary case the proposition folded away turns out to be the one carrying the phenomenon: at R = 1, Σ w_j = ∞ gives pinch rate π/2k while w_j ~ j^-s gives √(s log k / 2k).

Unchanged

The three headline results of Part III (prop:tiltlclt, thm:rigid, thm:transfer) remain unconditional. The main theorem, Part II's unconditional flatness, the coset identity, the character decomposition and the class-number corollary are untouched. prop:nopinch is untouched and its control passes. All twenty mechanical checks pass.

How the errors were found

By running the numerical rung of a proof plan — a script that scanned for sign changes from t = 0 upward and numbered them, rather than searching near where the wrong formula predicted a zero. The adjudication transcribed the paper's own displayed formula literally and summed it term by term; the two routes agreed to 2.8e-29, so the dispute was never about transcription. The second error was then found by a family built to test something else entirely.

A prediction that names a location will be confirmed by any zero near that location, and a dense ladder has one near everywhere. Count the zeros below the hit before calling it the first.

Full erratum: docs/RELEASE_v1.1.1.md. Scripts and logs: lean/pnp/leeyang6_r194, leeyang7_r194, leeyang8b_r194, figures_r194, radius_r197, audit2_r198, audit3_r198.

v1.1.0 — the three headline results of Part III are unconditional

The three headline results of Part III now stand without hypotheses.

prop:tiltlclt (tilted local limit theorem)	unconditional
thm:rigid (rigidity of the gap series)	a theorem, with no conditional clause
thm:transfer (the transfer function)	a theorem, with no conditional clause

All three shared a single missing ingredient, prob:R1 — the Edgeworth expansion of the tilted local limit theorem in region R1, with explicit constants. It is now written out in Appendix A with every constant proved rather than measured, and it has had the independent reading this project requires before an argument counts as proved.

What changed since v1.0.0

The reading is on record, and it is described precisely. It arrived in three parts, each covering the text as it then stood: the three lemmas line by line (r162), the three repairs landed against them (r164), and the restated proposition with the T* construction and all five explicit constants of ρ, each re-derived from scratch and found to agree symbolically (r171). No single reading has covered the appendix as a whole, because the text changed between the parts in response to the earlier ones — the appendix head says so, because a reader who wants the stronger statement is entitled to know it has not been supplied.

Nothing was deleted. prob:R1 stays in the paper, restated as CLOSED with what closed it. The honest-scope entry records in its own text that it read "proof skeleton, with the analytic ingredients in place" until r171. A status that improves is still a status change, and a reader who cannot see the old one cannot audit the new one.

The remaining caveat is louder, not quieter. The algorithmic reading used to attach two conditions to the sentence about restart counts. One was ours and is gone. The other is not, and now stands alone: the uniformity of the terminal distribution is an assumption about the search, not a fact about the landscape.

A referee pass was run before this release. Thirteen statements, read in a fresh context by a reader given three jobs and nothing else — restate the claim, name what would falsify it, flag any single word doing hidden work. Three came back clear. It found three defects that twenty mechanical checks and a typesetter had passed over, and every one of them was a claim about our own evidence rather than about the mathematics: three incompatible descriptions of one reading; an absolute "conditional on nothing" sitting where its qualification was not; and a miscount of which statements had been waiting. All are fixed here. Log: lean/pnp/refpass_r175.log; procedure: tools/referee_pass.md.

What this still is not

None of it is peer-reviewed. Every theorem, proposition and lemma declares its status — proved, derived, measured, or conjectured — at the place where it is stated. A DOI makes a version permanent. It does not make it true.

Tool and computational resource disclosure

Carried out with AI language models (Anthropic's Claude) as tools, under the author's direction, following recommendation 01 for individual mathematicians of the Leiden Declaration on Artificial Intelligence and Mathematics, which the author has signed. Each manuscript names the tools, the versions, and the computational resources — one personal computer, no cluster and no accelerator — and names the provision of the Declaration that this work does not meet.

Licences

Code (lean/, the Python scripts): Apache-2.0, matching Mathlib. Manuscripts (paper/): CC BY 4.0.

v1.0.0 — Arithmetic landscapes: three manuscripts and the apparatus that checks them

First archived release. Deposited in Zenodo to obtain a DOI.

What the mathematics says

Three manuscripts study the subset-sum landscape of a finite set A of odd integers. The ratio of strict local minima to ground states converges to a rational invariant Γ(A) read off the set by counting — and that invariant turns out to be the annealed (independence) prediction itself. So the main theorem says the annealed count of metastable states is asymptotically exact for this model, with an explicit condition that delimits it and an explicit family that breaks it. A lower bound that would normally cost a quantitative equidistribution estimate is obtained instead from the Kubert distribution relation, as an exact equality.

• Part I (26 pp.) — the gap series and the invariant.
• Part II (35 pp.) — asymptotic flatness for the odd primes, unconditionally.
• Part III (45 pp.) — deformed measures, random sequences, the coset identity and its decomposition into Dirichlet L-values. Appendix A writes out region R1 with every constant proved rather than measured.

What is in the deposit

• the three manuscripts;
• a Lean 4 development of the settled results, replayed through the kernel by an independent checker that must first reject three deliberately corrupted modules;
• the committed scripts and logs behind every number quoted in the text;
• a suite of mechanical checks (C1–C20) that runs before each commit;
• and a complete record of the mistakes this process has made.

What this is not

None of it is peer-reviewed, and no claim is made here beyond what each statement claims for itself. Every theorem, proposition and lemma declares its status — proved, derived, measured, or conjectured — at the place where it is stated, and three results of Part III remain explicitly conditional.

A DOI makes a version permanent. It does not make it true.

Tool and computational resource disclosure

Carried out with AI language models (Anthropic's Claude) as tools, under the author's direction, following recommendation 01 for individual mathematicians of the Leiden Declaration on Artificial Intelligence and Mathematics, which the author has signed. Each manuscript names the tools, the versions, and the computational resources — one personal computer, no cluster and no accelerator — and names the provision of the Declaration that this work does not meet. Responsibility for every claim and every citation is the author's alone.

Licences

Code (lean/, the Python scripts): Apache-2.0, matching Mathlib. Manuscripts (paper/): CC BY 4.0.