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Standing. The ratio part is L163 and the two one-sided parts are L164, each at tier 1 on its own whole-statement examination and distinct passing grade. Their conjunction is L165, recorded by Plasma AI at tier 1 on its own whole-conclusion examination and distinct passing grade, examined on 2026-10-16 UTC. The accepted verdict is refutation-failed for the complete conjunction and supplied mean-normalized formulation. Catalog acceptance remains pending. The examination does not certify the original publications or their historical interpretation. The span-to-count mechanism and both closing arguments are S. Korsky's, from the unrefereed preprint arXiv:2609.07196v2, which states its theorem for sequences of distinct points; they are reconstructed on the folder's ko26b pages. The finite-list input follows Larcher's range splitting, and the planar input is Halász's theorem as Bilyk's survey presents it. The reading for coincident terms, the finite-list bound, the write-up of the planar bound with constants, and this composition are credited to Plasma AI.

Target

Problem 1221, in its mean-normalized reading, asks whether $\Lambda_r-r$, $r-\lambda_r$ and $r(\mu_r-1)$ tend to infinity. The constants are taken over all sequences on the circle, coincident terms allowed, as in the erdosproblems.com wording and Section 1 of the 1949 note.

The all-sequence circle model. Let $X=(x_n)_{n\ge1}$ be any sequence on $\mathbb T=\mathbb R/\mathbb Z$. At time $n$ the labels $1,\ldots,n$ are ordered cyclically by position, coincident terms by index. Let $y_1\le\cdots\le y_n$ be their positions in this order on a lift to $\mathbb R$, extended by $y_{i+n}=y_i+1$. The $n$ gaps $y_{i+1}-y_i\ge0$ sum to $1$, and for $n\ge r$ the $r$-span from the label in place $i$ is $y_{i+r}-y_i\in[0,1]$. Each gap lies in exactly $r$ spans, so the spans sum to $r$ (the mean identity) and $nm_n^{(r)}\le r\le nM_n^{(r)}$, where $M_n^{(r)}$ and $m_n^{(r)}$ are the largest and smallest $r$-spans. Counts $N_n(I)$ of arcs count labels, that is, terms with multiplicity. Counting intervals $(x,x+\ell]$ have $\ell<1$; the lifted span arcs in the covering identity below may also have length $1$. All logarithms are natural unless a base is displayed. The constants are

$$\Lambda_r=\inf_X\limsup_{n\to\infty}nM_n^{(r)},\qquad \lambda_r=\sup_X\liminf_{n\to\infty}nm_n^{(r)},\qquad \mu_r=\inf_X\limsup_{n\to\infty}\frac{M_n^{(r)}}{m_n^{(r)}},$$

with $M/0=+\infty$.

Quantifiers. An infimum is at least a number exactly when every member is. So $\Lambda_r-r\ge K$ holds when every sequence has $\limsup_n(nM_n^{(r)}-r)\ge K$; $r-\lambda_r\ge K$ when every sequence has $\limsup_n(r-nm_n^{(r)})\ge K$; and $r(\mu_r-1)\ge K$ when every sequence has $\limsup_nM_n^{(r)}/m_n^{(r)}\ge1+K/r$. Here $\limsup_n(r-nm_n^{(r)})=r-\liminf_{n\to\infty}nm_n^{(r)}$. No limit is exchanged with the infimum or the supremum.

Theorem (composed; L165, tier 1). There are absolute constants $c>0$ and $r_0$ such that, for every integer $r\ge r_0$ and every sequence on $\mathbb T$, coincident terms allowed,

$$\limsup_{n\to\infty}\bigl(nM_n^{(r)}-r\bigr)\ge c\sqrt{\log r},\qquad \limsup_{n\to\infty}\bigl(r-nm_n^{(r)}\bigr)\ge c\sqrt{\log r},\qquad \limsup_{n\to\infty}\frac{M_n^{(r)}}{m_n^{(r)}}\ge1+\frac{\log r}{200\,r}.$$

Consequently $\Lambda_r-r\ge c\sqrt{\log r}$, $r-\lambda_r\ge c\sqrt{\log r}$ and $r(\mu_r-1)\ge(\log r)/200$ for $r\ge r_0$, and all three expressions tend to infinity. The argument runs for an arbitrary sequence, so it uses no relation between the constants over all sequences and those over sequences of distinct points; whether those constants agree remains unsettled and is not needed.

The links and their standing

The ratio part runs through these links:

• Lemma 2.1, the cyclic-walk comparison of interval counts at nearby times;
• Proposition 3.1, which iterates it to bound the counting error on every short arc by $3A+C_1A/\log(r/A)$;
• Proposition 1221.1 of the connections page, which reads the terms of one short arc as a finite list, coincident terms allowed;
• Proposition FP of the finite-list page, $H_L\ge\lfloor\log_4L\rfloor/16$ for every list, in place of the imported Theorem 4.1;
• the comparison of Section 5, recomposed below with the coefficient $1/200$.

The one-sided parts run through these links:

• Lemma 6.1, Lemma 6.2 and Lemma 6.3, the averaged comparison under one one-sided span bound;
• Proposition 6.4, the resulting $L^1$ bound $C_3A$ on short arcs;
• Lemma 7.2, the localization to planar point sets with insertion time as second coordinate;
• the [[research/erdos_1221/halasz_planar_l1_bound|planar $L^1$ bound]], Halász's theorem written out with $c_H=1/41000$, in place of the imported Theorem 7.1;
• the comparison of Section 8, recomposed below.

The reconstruction pages were independently reviewed, as reconstructions of the distinct-point argument relative to its two imports, by the focused reviews and grades that the folder index lists; the statements of the connections page, Proposition 1221.1 among them, by the focused reviews graded in [[research/erdos_1221/evidence/verify/connections_grade|the connections grade]]. Those reviews cover their own subjects as they then stood. The complete ratio chain, including the finite-list page and its reading for coincident terms, is examined and graded in L163's record. The complete one-sided chain, including the planar page and its reading for coincident terms, is examined and graded in L164's record. Both records retain the exact mathematical subjects and their independence qualifications. They examine the supplied local deductions; the original publications remained unread by the reviewers and graders. The whole-conclusion report examines the conjunction and supplied formulation using those interfaces. Its distinct whole-contract grade is PASS at that full scope, supporting L165 at tier 1. The source attributions above and their original-publication reading limits remain explicit.

Reading the chain for coincident terms

The ko26b pages are written for distinct points. The following points make each step valid for an arbitrary sequence. The earlier checklist on the connections page made the same reading. The lower terminal bound and the full-turn case of the covering identity below are arguments that the checklist does not give.

Labels and lifts. The moves $F_s$ and $B_s$ by $kr$ places are bijections of labels. The displacement of $F_s$ at the label in place $i$ is the lifted difference $y_{i+kr}-y_i$, a sum of $k$ consecutive $r$-spans. Lemmas 2.1 and 6.2 use only these displacements as real numbers, positions modulo $1$ for membership in arcs shorter than $1$, injectivity of maps between label sets, and counts with multiplicity, so their proofs apply as written. Lemma 6.1 uses only the mean identity, the decomposition of a $kr$-span into $k$ consecutive $r$-spans, and the triangle inequality.

The terminal bounds (3.2) of Proposition 3.1. Let (2.1) hold at time $t$ with $a_t+b_t\le A$, and let $n=\lfloor t\rfloor$ be large.

• An arc $(x,x+(r-A)/t]$ holds at most $r$ labels. The labels in an arc form a contiguous block of the cyclic order. If there were $r+1$ of them, the first label of the block and the label $r$ places after it would both lie in the arc, and the $r$-span between them, which is the difference of their positions within the arc, would be less than $(r-A)/t\le(r-a_t)/t$, contrary to (2.1).
• An arc $I=(x,x+(r+A)/t]$ holds at least $r$ labels. Choose the lift so that $y_1\le\cdots\le y_n$ lie in $(x-1,x]$. The last label has position $y_n\le x$, and the next $r$ labels have lifted positions $y_1+1,\ldots,y_r+1$, which exceed $x$ and are at most $y_n+(r+b_t)/t\le x+(r+A)/t$, by (2.1) for the span from the last label. So these $r$ labels lie in $I$.

The covering identity of Lemma 6.3. For coincident terms a lifted span arc $(y_{i-r},y_i]$ can have length $0$, an empty arc, or length $1$, a full turn. Read each lifted arc modulo $1$. For $x$ at no position, with lift $\tilde x$,

$$\sum_{i=1}^n\mathbf 1_{(y_{i-r},y_i]}(x) =\sum_{i\in\mathbb Z}\mathbf 1\bigl[y_{i-r}<\tilde x\le y_i\bigr]=r,$$

since the translates by integers of the arcs with $1\le i\le n$ are the lifted arcs with $i\in\mathbb Z$, and if $j$ is the largest index with $y_j<\tilde x$, exactly the indices $i=j+1,\ldots,j+r$ qualify. In the pairing step, the arc of length $r/t$ ending at $y_i$ and the lifted arc of length $y_i-y_{i-r}\in[0,1]$ ending at $y_i$ have indicators differing on a set of measure $|y_i-y_{i-r}-r/t|$. So Lemma 6.3 holds as stated, without its assumption that every span is shorter than $1$. That assumption also holds at large times under either one-sided hypothesis: under $nM_n^{(r)}-r\le A$ every span is at most $(r+A)/n<1$; under $r-nm_n^{(r)}\le A$ with $r>A$ and $n\ge2r$, a span of length $1$ would leave $n-r\ge r$ consecutive zero gaps and hence a zero $r$-span, contrary to $m_n^{(r)}\ge(r-A)/n>0$.

Localization and extraction. In Lemma 7.2 the planar points $((x_i-a)/w,\,i/N)$ have distinct second coordinates, and the planar bound holds for lists in any case. The least-distance step of Lemma 4.2 is replaced by Proposition 1221.1, which chooses the short arc away from the positions of the first $n_0$ terms by a measure argument.

Positivity. The one-sided hypotheses need no positive spans. Under the ratio hypothesis every late $r$-span is positive, since a vanishing span makes the ratio $+\infty$.

The ratio part

Put $\alpha=1/200$ and $c_0=1/(16\log4)$, so that $3\alpha<c_0/2$ (already $\log4<2$ suffices). Let $r$ be large, put $A=\alpha\log r$, and suppose that some sequence has $\limsup_nM_n^{(r)}/m_n^{(r)}<1+A/r$. Then there is $C<A$ with $M_n^{(r)}/m_n^{(r)}\le1+C/r$ for all large $n$, and the mean identity gives $n(M_n-m_n)\le C$ and $nM_n\le r+C$. As on the Theorem 1.1 page, $a_t=r-nm_n$ and $b_t=tM_n-r$ with $n=\lfloor t\rfloor$ are nonnegative, bound every $r$-span of the time-$t$ set between $(r-a_t)/t$ and $(r+b_t)/t$, and satisfy $a_t+b_t\le C+(r+C)/n<A$ for all large $t$. This is hypothesis (2.1).

For large $r$ we have $A\ge1$ and $r\ge C_0A$, so Proposition 3.1 gives, at all late integer times and for every arc of length $D/n$ with $0\le D\le S$, counting error at most $B$, where

$$\Lambda=\log\frac rA,\qquad S=\frac{\sqrt{Ar}}{\Lambda^2},\qquad B=3A+\frac{C_1A}\Lambda .$$

Its hypotheses $B\ge1$ and $S\ge2$ hold for large $r$. Also $\lfloor S\rfloor>B$ eventually, because $B=O(\log r)$ and $S/(\log r)\to\infty$. Thus Proposition 1221.1 gives a nonempty list of length $L'\ge\lfloor S\rfloor-B$ with $H_{L'}\le B$, and Proposition FP gives

$$B\ \ge\ c_0\log L'-\frac1{16}.$$

As $r\to\infty$, $A/\Lambda\to\alpha$, so $B=3\alpha\log r+O(1)$, and

$$\log S=\tfrac12\log r-\tfrac32\log\log r+O(1).$$

Since $B=O(\log r)$ is much smaller than $S$, $L'\ge S/2$ for large $r$. Therefore

$$3\alpha\log r+O(1)\ \ge\ \frac{c_0}2\log r-\frac{3c_0}2\log\log r-O(1),$$

which fails for all large $r$ because $3\alpha<c_0/2$. The implied constants depend only on $\alpha$ and the absolute constants $C_0$, $C_1$, so the threshold $r_0$ is independent of the sequence. The time thresholds inside the argument depend on the sequence, but only their existence is used. This proves $\limsup_nM_n^{(r)}/m_n^{(r)}\ge1+\log r/(200r)$ for $r\ge r_0$.

The one-sided parts

Let $C_2,C_3$ be the absolute constants of Proposition 6.4 and $c_4,S_0$ those of Lemma 7.2 with the planar constant $c_H=1/41000$, and fix an absolute $c$ with $0<c<c_4/(2C_3\sqrt3)$. Let $r$ be large and suppose that some sequence has $\limsup_n(nM_n^{(r)}-r)<c\sqrt{\log r}=:A$. Then $nM_n^{(r)}-r\le A$ for all large $n$, the first alternative of hypothesis (6.1).

For large $r$: $A\ge1$, $C_3A\ge1$, $r\ge C_2A$, $S=\sqrt{Ar}/\log^2(r/A)\ge S_0$, and $\log S\ge(\log r)/3$, because $\log S=\frac12\log r+O(\log\log r)$. Proposition 6.4 gives hypothesis (7.1) of Lemma 7.2 with $B=C_3A$ at all late integer times, and Lemma 7.2 gives

$$C_3c\sqrt{\log r}=C_3A\ \ge\ c_4\sqrt{\log S}\ \ge\ c_4\sqrt{(\log r)/3},$$

that is, $c\ge c_4/(C_3\sqrt3)$, contrary to the choice of $c$. If instead $\limsup_n(r-nm_n^{(r)})<c\sqrt{\log r}$, the same argument runs with the second alternative $r-nm_n^{(r)}\le A$; Lemmas 6.1--6.3 and Proposition 6.4 use one alternative at a time, with the same constants. This proves the two one-sided bounds for $r\ge r_0$.

What remains

• The conjunction and formulation. No mathematical gap is identified in either separately examined part or in their composition. L165's whole-conclusion examination covers the common conventions, absolute quantifiers, separate limits, extremum directions, and exact match to the supplied mean-normalized statement, with verdict refutation-failed. Its distinct whole-contract grade is PASS for that entire scope. The conjunction's warrant is separate from the two input grades; catalog acceptance is still pending.
• Constants. $c$ is absolute but very small and is not computed; $r_0$ is very large, since the ratio part already needs $(\log r)/200\ge1$. The coefficient $1/200$ comes from the constant of Proposition FP. Korsky's coefficient $1/100$ would follow with the constant $1/16$ of his Theorem 4.1, which rests on Larcher's unchecked optimized lemmas.
• An independent qualitative route. The interface page derives a uniform qualitative finite-list bound from compactness and Schmidt's countability corollary. Used in place of Proposition FP, it gives $r(\mu_r-1)\to\infty$ without a rate.
• Standing of the problem. The proposed full conclusion has its independent whole-conclusion warrant at tier 1. The catalog record carries a pending full claim for the modified Statement; no accepted catalog solution is recorded. The printed first two expressions and the normalization issue remain explicitly distinguished on the problem page.