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Purpose and standing. This page draws the connections from the author-recorded reconstructions in the Problem 1221 folder to the rest of the corpus. Each section names its exact target with the target's current assessment, states what the reconstructed mechanism gives when transported, with a proof where one is given here, states what it does not give and why, and names the next investigation with what each outcome would establish. Every new statement below is author-recorded, and no status or tier changes. The statements were independently reviewed as they stood on 2026-09-28T13:50:48Z by four focused reviews, one per section, graded in the connections grade, which passes all four and accepts eight corrections, since applied; it assigns no tier. The Target sentences on the standings of Problems 255 and 1221 differ from that reviewed subject. In the Problem 1221 section, the current explanation specifies lifted span arcs, including full turns, the empty-list convention in Proposition 1221.1, and the vacuous separation restriction when a list has only one distinct value. Its standing and outcome distinguish the reviewed conditional extension from the separate whole composition, which has no independent review. The statements of Propositions 1221.1--1221.2 and the conditional deduction 1221.3 retain their mathematical conclusions. The notation is the folder's: a sequence $a=(a_1,a_2,\ldots)$ on the circle $\mathbb T$ of circumference $1$, the $n$ gaps cut by its first $n$ terms (of length $0$ between coincident terms), the $r$-spans, that is, the sums of $r$ cyclically consecutive gaps, with largest and smallest values $M_n^r(a)$ and $m_n^r(a)$, and the 1949 constants

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

For an arc $I$ and a stage $n$ the counting error is $D_n(I)=\#\{i\le n:\ a_i\in I\}-n|I|$, and $N_n(I)=\#\{i\le n:\ a_i\in I\}$ counts indices, so coincident terms count with multiplicity.

Problem 255: the 1949 implication toward the discrepancy theorem

Target. Problem 255 asks whether every sequence $z_1,z_2,\ldots\in[0,1]$ has an interval $I$ with $\limsup_N|D_N(I)|=\infty$. Its page records the problem as solved, with the question proved, on three accepted claim pages for Schmidt's papers of 1968 and 1972 and Tijdeman--Wagner 1980, and its Progress section records the implication proved below; the held source Schmidt 1972, part VI shows that the anchors $\alpha$ for which $D_n([0,\alpha))$ stays bounded form a countable set, so all but countably many anchored intervals answer the question. The connection named in the folder index is the remark on p. 14 of the 1949 note, recorded on the conjecture page: the third part of the conjecture would imply van Aardenne-Ehrenfest's "just distributions" theorem of 1945, that no sequence has $\sup_\alpha|D_n([0,\alpha))|$ bounded in $n$.

What transports. The remark is a theorem, each of the three parts of the mean-normalized reading suffices, and the proof is four lines of the folder's span bookkeeping.

Proposition 255.1 (author-recorded). Let $a$ be a sequence on the circle, coincident terms allowed, and let $K$ be a number with $|D_n(I)|\le K$ for every $n\ge1$ and every half-open arc $I$. Then for every $r\ge1$ and every $n>r$,

$$nM_n^r(a)\ \le\ r+K-1,\qquad nm_n^r(a)\ \ge\ r-K+1 .$$

Consequently $\Lambda_r-r\le\Lambda_r(a)-r\le K-1$ and $r-\lambda_r\le r-\lambda_r(a)\le K-1$ for every $r$, and for $r\ge K$

$$r(\mu_r-1)\ \le\ r\bigl(\mu_r(a)-1\bigr)\ \le\ \frac{2(K-1)\,r}{r-K+1} \ \xrightarrow[r\to\infty]{}\ 2(K-1),$$

so all three expressions of the mean-normalized conjecture stay bounded. Hence each part of the conjecture implies that no sequence has uniformly bounded counting error on arcs; a bound $K$ on the anchored intervals $[0,\alpha)$ gives the bound $2K$ on every arc and conversely, so this is the 1945 theorem in its original form.

Proof. The hypothesis passes from half-open arcs to open and closed arcs: for an open arc $(p,q)$ of positive length the half-open arcs $(p,q-\varepsilon]$ contain the same terms of stage $n$ once $\varepsilon$ is small, and their lengths tend to $|(p,q)|$; for a closed arc $[p,q]$ the arcs $(p-\varepsilon,q]$ contain the same terms once $\varepsilon$ is small. So $|D_n(I)|\le K$ for arcs of every endpoint type, and the complement of an arc, which has counting error $-D_n(I)$, covers the wrapping arcs. (At $n=1$ a short arc containing $a_1$ shows $K\ge1$.)

Fix $r\ge1$ and $n>r$, list the first $n$ terms in cyclic order, coincident terms in order of index, and let $J$ be an $r$-span: the clockwise distance $|J|$ from a term $p$ to the term $q$ that is $r$ places after $p$. The open arc $(p,q)$ contains only terms strictly between $p$ and $q$ in the cyclic order, since a term at the position of $p$ or of $q$ is excluded, so $N_n((p,q))\le r-1$; from $D_n((p,q))\ge-K$,

$$n|J|\ \le\ N_n((p,q))+K\ \le\ r-1+K ,$$

trivially so when $|J|=0$. The closed arc $[p,q]$ contains the $r+1$ terms $p,\ldots,q$, which are distinct terms because $n>r$; from $D_n([p,q])\le K$, $r+1-n|J|\le K$, that is, $n|J|\ge r+1-K$. The largest and the smallest span give the two displayed bounds. Letting $n\to\infty$ gives $\Lambda_r(a)\le r+K-1$ and $\lambda_r(a)\ge r-K+1$. For $r\ge K$ the lower bound makes every late span positive and

$$\frac{M_n^r(a)}{m_n^r(a)}\ \le\ \frac{r+K-1}{r-K+1}=1+\frac{2(K-1)}{r-K+1},$$

which gives the bound on $r(\mu_r(a)-1)$; the universal constants obey $\Lambda_r\le\Lambda_r(a)$, $\lambda_r\ge\lambda_r(a)$, $\mu_r\le\mu_r(a)$. The last display tends to $2(K-1)$, so each expression is bounded in $r$. Finally $D_n((\alpha,\beta])=D_n([0,\beta])-D_n([0,\alpha])$ up to the endpoint adjustments above, so anchored bounds give arc bounds with the factor $2$, and anchored intervals are arcs. $\square$

What it does not give.

• Not the fixed interval of Problem 255. The arcs whose counting error the proof exploits are the spans of stage $n$ and move with $n$; the conclusion is $\limsup_n\sup_I|D_n(I)|=\infty$, the 1945 form. Problem 255 asks for one interval $I$ with $\limsup_n|D_n(I)|=\infty$, which is the content of Schmidt's 1968 paper (the bounded anchors form a null set) and of part VI (they are countable); no page of the folder bears on the passage from a moving arc to a fixed one.
• No rate, in either direction. With a nondecreasing $K(n)\to\infty$ in place of $K$ the proof gives $nM_n^r(a)\le r+K(n)-1$ and $nm_n^r(a)\ge r-K(n)+1$, which bound nothing in the limit because $K(n)\to\infty$: the constants $\Lambda_r,\lambda_r,\mu_r$ are limits in $n$ at fixed $r$, so through Proposition 255.1 a growth rate of the discrepancy in $n$ neither follows from the conjecture nor feeds it. Coupling $r$ to a point count is exactly what Lemma 4.2 of the 2026 preprint adds; see the reverse edge.
• Circular through the only claimed proof of the premise. The fixed-$r$ bounds, (5.7) of 1949 and Theorem 1.1 of the 2026 note, are bounded in $r$ and do not reach the premise; the growth is claimed only by Theorem 1.1 of the 2026 preprint, whose ratio part consumes a finite-list form of Schmidt's theorem, itself a quantitative strengthening of the 1945 theorem, and whose one-sided parts consume Halász's planar theorem. Deriving the 1945 theorem from that claim through Proposition 255.1 proves nothing new. A proof of any one part that avoids discrepancy lower bounds would give a new proof of the 1945 theorem; none is in the sources read.
• Scope. The 2026 claim is stated over sequences of distinct points, so through Proposition 255.1 it would exclude only just-distributed sequences of distinct points; the extension to all sequences is the subject of the section on Problem 1221's own family below.

The reverse edge. A quantitative form of the 1945 theorem, the uniform statement that the conclusion of Problem 255 strengthens, is an input of the claimed resolution, not a consequence of it. Lemma 4.2 turns the short-interval counting bound of Proposition 3.1 into a list of $\lfloor S\rfloor$ points all of whose prefixes have counting error at most $B$, and the imported Theorem 4.1, the finite-prefix form $H_L\ge\frac1{16}\log L$ that the source derives from Larcher's 2015 proof, supplies the contradiction; Lemma 7.2 does the same with Halász's planar $L^1$ theorem. The Problem 255 source held here, part VI of Schmidt's series, is a different theorem (countability of the bounded anchors) from the planar theorem of part VII (Acta Arith. 21 (1972), 45--50) that the Lemma 4.2 page names as the source of the qualitative finite-list bound. Neither part VII, nor Larcher 2015, nor Halász 1981 is held, so the corpus holds neither of the two inputs the reconstruction imports.

Next investigation. Acquire and file Larcher 2015 (J. Complexity 31, 474--485; arXiv:1407.2094) and Halász 1981, and check the two imported statements as used: the finite-list bound with the constant $1/16$ for every list of length $\lfloor a^h\rfloor$, and the unnormalized planar $L^1$ bound $c_H\sqrt{\log M}$. A positive outcome closes the two import gaps of the reconstruction of the 2026 preprint and gives the Problem 255 library a hook to the theorem actually consumed; a negative one (for instance, Larcher's Section 3 bounding only infinitely many prefixes of an infinite sequence) leaves the growth statement of the ratio part standing through part VII with an unspecified constant and removes the warrant for the coefficient $1/100$. Neither outcome changes Problem 255.

Outcome. Transfers as an implication (Proposition 255.1); it yields nothing new for Problem 255, which is proved and stronger, and the mathematical dependency runs the other way.

Problem 480: the clustering constant and the Section 4 count

Target. Problem 480 asks whether every sequence $x_1,x_2,\ldots\in[0,1]$ satisfies

$$C(\bar x)=\inf_n\liminf_{m\to\infty}n\,|x_{m+n}-x_m|\ \le\ 5^{-1/2} .$$

Its Current assessment records the status proved: Theorem 1 of Chung and Graham (1984) gives $C(\bar x)\le\alpha=0.39441967\ldots$ for every sequence, their Theorem 2 shows that $\alpha$ is best possible, and erdosproblems.com showed PROVED (LEAN). The folder index names two links. First, the chapter's introduction attributes to the 1949 note the bound $1/\log4$ for its measure

$$\omega(\bar x)=\liminf_{n\to\infty}\ n\min_{1\le i<j\le n}|x_i-x_j| ,$$

which is the interval analog of $\lambda_1(a)=\liminf_nnm_n^1(a)$ on the circle; the card of the note points at Proposition 480.1 below for the passage from the circle to the interval. Second, the index says the Chung--Graham proof descends from the Section 4 count.

What transports. The Section 4 count, rerun on the interval, gives the chapter's display (1).

Proposition 480.1 (author-recorded). For every sequence $x_1,x_2,\ldots\in[0,1]$ and every $n\ge1$ there is an integer $k$ with $n<k\le2n$ such that

$$k\,g_k\ \le\ \tau'_n:=\Bigl(2\sum_{k=n+1}^{2n}\frac1k-\frac1{n+1}\Bigr)^{-1}, \qquad g_k=\min_{1\le i<j\le k}|x_i-x_j| .$$

Since $\tau'_n\to1/\log4$, every sequence in $[0,1]$ has $\omega(\bar x)\le1/\log4$, and the sequence $x_k=\{\log_2(2k-1)\}$ attains the bound.

Proof. Fix $n$, put $N=2n$, and list $x_1,\ldots,x_N$ in increasing order as $y_1\le y_2\le\cdots\le y_N$ with $y_i=x_{k_i}$, $(k_1,\ldots,k_N)$ a permutation of $(1,\ldots,N)$ (equal values in either order). For $1\le i\le N-1$ put $k_i^*=\max(k_i,k_{i+1},n+1)$, so $n<k_i^*\le2n$. The terms $x_{k_i}$ and $x_{k_{i+1}}$ are two of the first $k_i^*$ terms, so $y_{i+1}-y_i\ge g_{k_i^*}$. Suppose that $\varrho>0$ satisfies $kg_k>\varrho$ for every $k$ with $n<k\le2n$. Then $y_{i+1}-y_i>\varrho/k_i^*$ for every $i$, and summing the $N-1$ consecutive differences,

$$1\ \ge\ y_N-y_1\ >\ \varrho\sum_{i=1}^{N-1}\frac1{k_i^*} .$$

For $n+1<k\le N$ the equality $k_i^*=k$ forces $k\in\{k_i,k_{i+1}\}$, and $k=k_j$ for exactly one $j$, which occurs in the windows $i=j-1$ and $i=j$ only; so $k$ occurs $\varepsilon_k\le2$ times among the $k_i^*$, and $\varepsilon_{n+1}=(N-1)-\sum_{k\ge n+2}\varepsilon_k$. Expanding the coefficients as on the (4.3) page,

$$\sum_{i=1}^{N-1}\frac1{k_i^*}=\sum_{k=n+1}^{2n}\frac{\varepsilon_k}k =2\sum_{k=n+1}^{2n}\frac1k-\frac1{n+1}+\sum_{k=n+2}^{2n}(2-\varepsilon_k) \Bigl(\frac1{n+1}-\frac1k\Bigr)\ \ge\ \frac1{\tau'_n},$$

the coefficient of $1/(n+1)$ on the right being $1+\sum_{k\ge n+2}(2-\varepsilon_k)=2n-1-\sum_{k\ge n+2}\varepsilon_k=\varepsilon_{n+1}$. Hence $\varrho<\tau'_n$, and with $\varrho=\tau'_n$ the hypothesis fails: some $k\in(n,2n]$ has $kg_k\le\tau'_n$. Since $2\sum_{k=n+1}^{2n}1/k\to2\log2=\log4$, $\tau'_n\to1/\log4$, and choosing such a $k_n$ for each $n$ gives $\omega(\bar x)\le\lim\tau'_n=1/\log4$.

For the sharpness, the first $n$ terms of $x_k=\{\log_2(2k-1)\}$ are, modulo $1$, the numbers $\log_2k$ for $n\le k\le2n-1$ (the Section 2 page), so on the circle their $n$ gaps are $\log_2\frac{k+1}k$ for $n\le k\le2n-1$, and the $n-1$ interior gaps of the interval are these $n$ circle gaps with the wrap gap, from the largest term back to $x_1=0$, removed. The smallest interior gap is therefore at least the smallest circle gap and at most the second smallest: $\log_2\frac{2n}{2n-1}\le g_n\le\log_2\frac{2n-1}{2n-2}$ for $n\ge2$, and both bounds times $n$ tend to $1/(2\log2)=1/\log4$, so $ng_n\to1/\log4$. $\square$

The circle statement does not transfer by itself: reading $[0,1]$ as the circle adds the wrap gap $y_1+1-y_N$, which can be the smallest gap, so $m_n^1(a)\le g_n$ and (4.3) bounds the wrong side. The interval argument replaces the $N$ cyclic windows by the $N-1$ adjacent pairs and the total length $1$ by $y_N-y_1\le1$; its finite-$n$ bound $\tau'_n$ is weaker than the circle's $\tau_n^{(1)}=(2\sum_{k=n+1}^{2n}1/k)^{-1}$ by the term $1/(n+1)$ and has the same limit.

What it does not give.

• Nothing for the question of Problem 480, which is proved with the sharp constant $\alpha$ and concerns a measure of a different quantifier structure: $C$ reads pairs at a fixed lag $n$ in order of insertion, with $m\to\infty$, while $\omega$ and $\lambda_1$ read the closest pair among all the first $n$ terms. The two measures are not comparable. A sequence whose first two terms coincide has $\omega(\bar x)=0$, while $C$ ignores any finite set of indices, so prepending a repeated term to the extremal sequence of the chapter's Theorem 2 gives $\omega=0$ and $C=\alpha$; in the other direction the chapter's introduction, as the card of the chapter records, remarks that $C$ can be small while $\omega$ stays large, when the close pairs occur far out (for instance $|x_{m+1}-x_m|=m^{-1/2}$ at infinitely many $m$ gives $C=0$ without disturbing $\omega$). So neither bound transfers to the other measure.
• The "descent" is a resemblance of shape, not a dependency. Both arguments bound a sum of consecutive sorted gaps by $1$ and bound each gap below from the hypothesis, but the 1949 count uses all $N$ adjacent pairs, the bound $\varrho/k_i^*$ from the largest index in the window, and the multiplicity bound $2$, while the proof of Theorem 1 on p. 211 of the chapter uses only the pairs of an increasing subsequence of the order permutation of $N$ consecutive terms, the bound $(1+\epsilon)\alpha/|\pi(i_{k+1})-\pi(i_k)|$ from the lag hypothesis, and the exact extremal value $u_m$ of its Theorem 3, proved on pp. 188--210 and reconstructed nowhere in the corpus. No step of Section 4 enters Theorem 3.
• No reverse transfer is available either. Whether a permutation extremal problem in the style of Theorem 3 could sharpen (4.3) for $r\ge2$, where the 1949 windows read $r+1$ consecutive sorted terms, is addressed by no held source; for $r=1$ the bound is sharp, and the growth of $r-\lambda_r$ claimed by the 2026 preprint comes from the discrepancy route, not from a sharper count.

Next investigation. The Problem 480 page lists as an unread lead a later Chung--Graham paper on a sequence of points on a circle of circumference $1$. Reading it would settle whether its circle measure is the 1949 $\lambda_1$, a lag measure on the circle, or something else: a positive identification with the de Bruijn--Erdős constants or a variant would add a library hook for Problem 1221 and possibly small-$r$ information; a negative one leaves this connection at the background level of Proposition 480.1.

Outcome. Fails as a route to Problem 480 (proved, sharp constant, incomparable measure); the background transfer of the $1/\log4$ bound to the interval, which the card of the note left unreconciled, is proved above.

Problem 1221 itself: the site's family against sequences of distinct points

Target. Problem 1221, status claimed, claim proved, under its mean-normalized reading: whether $\Lambda_r-r$, $r-\lambda_r$ and $r(\mu_r-1)$ tend to infinity. The erdosproblems.com family, like Section 1 of the 1949 note, admits coincident terms, while the two 2026 sources state their lower bounds over sequences of distinct points; the sources do not establish equality of the constants over the two families. This section records what the reconstructed arguments give for the larger family.

The two families. Distinct sequences form a subfamily, so the infima over all sequences are at most those over distinct sequences and the supremum is at least: $\mu_r^{\mathrm{all}}\le\mu_r^{\mathrm{dist}}$, $\Lambda_r^{\mathrm{all}}\le\Lambda_r^{\mathrm{dist}}$, $\lambda_r^{\mathrm{all}}\ge\lambda_r^{\mathrm{dist}}$. An upper bound on $\mu_r$ proved by exhibiting a sequence of distinct points, Theorem 2 of the balanced-stick preprint, passes to the erdosproblems.com family at once; the 1949 bounds are stated over all sequences; the lower bounds of the 2026 note and preprint do not pass by inclusion, and they are the subject here. The note's zero-length intervals are the multiset model used below: the first $n$ terms are listed in cyclic order by position, coincident terms in order of index; the $n$ gaps, some possibly $0$, do not depend on the tie-break; an $r$-span is the sum of $r$ consecutive gaps. With cyclic coordinates lifted by $y_{i+n}=y_i+1$, its length is $y_{i+r}-y_i$, which can equal $1$ even when its two positions coincide on the circle. $N_n(I)$ counts indices.

Where distinctness enters the reconstructed arguments (author-recorded reading of the folder's pages).

• Positivity of the spans, used only to make ratios finite. Under the contradiction hypotheses, $R_n\le\rho$ for late $n$ in the note and the ratio hypothesis of Section 5 in the preprint, every late $r$-span is positive by the mean identity $m_n\ge M_n/\rho\ge r/(\rho n)$; where a span vanishes, the ratio conclusion holds trivially with $M/0=+\infty$. The one-sided hypotheses (6.1) do not need positive spans.
• The least distance $\delta$ between distinct points of $P_{n_0}$ in Lemma 4.2, used to keep early terms out of the short interval and to produce an interval with exactly $L$ terms. Proposition 1221.1 replaces it.
• Nowhere else. Checked page by page: Lemma 2.1 (the moves $F_s$, $B_s$ are bijections of index sets, the displacement bounds are positional, and the averaging counts with multiplicity); Proposition 3.1 (for the terminal bounds (3.2), among $r+1$ terms in an arc, ordered by cyclic rank from the arc's start, the first and the last are $r$ places apart and the terms between them lie in the arc); Lemma 6.1 (the zero-sum identity); Lemma 6.2; Lemma 6.3 (for $x$ at no term's position, $y_i$ is among the $r$ terms after $x$ exactly when $x$ lies in the lifted arc from $y_{i-r}$ to $y_i$, projected to the circle; zero spans give empty arcs and length-one spans give full turns, as the composition page's covering identity writes out); Proposition 6.4; Lemma 7.2 (the planar points carry the distinct second coordinates $i/N$, so the imported planar theorem meets a set of distinct points); and in the 2026 note Lemma 3.1 with Lemma 2.1, Proposition 4.1 and Theorem 1.1, where a term inserted at an occupied position comes last among the terms at that position and splits the gap that starts at the previous last term into a zero piece and the old gap, so "one insertion splits one gap" stays true, and every inequality on those pages uses only gaps $\ge0$.

Proposition 1221.1 (Lemma 4.2 without distinctness; author-recorded). Let $(x_n)$ be any sequence on $\mathbb T$, coincident terms allowed, let $B\ge1$ and $S\ge2$, and suppose that (4.1) holds for all integers $n\ge n_0$: $|N_n((x,x+D/n])-D|\le B$ for all $x\in\mathbb T$ and $0\le D\le S$. Put $L=\lfloor S\rfloor$. Then there are an integer $L'$ with $L-B\le L'\le L+B$ and a list $z_1,\ldots,z_{L'}\in(0,1)$, possibly with repeated entries, whose maximum prefix counting error satisfies $H_{L'}(z_1,\ldots,z_{L'})\le B$, with $H_0=0$ for the empty list. If the imported Theorem 4.1 holds for lists of distinct entries, it holds for all lists, and then $B\ge\frac1{16}\log(L-B)$ whenever $L-B\ge L_0$.

Proof. Choose an integer $N>n_0$ with $n_0L/N<1$ and put $w=L/N$. The positions of the first $n_0$ terms form a set $Q$ of at most $n_0$ points, and the set of $a\in\mathbb T$ for which $(a,a+w]$ meets $Q$ has measure at most $n_0w<1$; the finitely many $a$ for which $a$ or $a+w$ is the position of one of the first $N$ terms are excluded too. Fix such an $a$ and put $J=(a,a+w]$. By (4.1) at time $N$ with $D=L\le S$, the number $L'=N_N(J)$ satisfies $|L'-L|\le B$, and every term in $J$ has index greater than $n_0$. List the terms in $J$ by index, $m_1<\cdots<m_{L'}\le N$, and put $z_i=(x_{m_i}-a)/w\in(0,1)$. Fix $j$ and let $n=m_j\ge n_0+1$; the terms of $P_n$ in $J$ are exactly the first $j$ listed, so $\#\{i\le j:\ z_i\le u\}=N_n((a,a+uw])$ for $0\le u\le1$. Put $f(u)=N_n((a,a+uw])-nuw$. The arc $(a,a+uw]$ has length $D/n$ with $D=nuw\le Nw=L\le S$, so $|f(u)|\le B$ by (4.1); the complementary arc $(a+uw,a+w]$ has length $D'/n$ with $D'=n(1-u)w\le S$ and count $j-N_n((a,a+uw])$, so $|f(1)-f(u)|\le B$. Since $f(1)=j-nw$,

$$\#\{i\le j:\ z_i\le u\}-ju=f(u)-uf(1)=(1-u)f(u)-u\bigl(f(1)-f(u)\bigr),$$

of absolute value at most $B$; for the convention $z_i<u$ the count is the left limit, with the same supremum. So $H_{L'}(z_1,\ldots,z_{L'})\le B$.

For the last sentence let $z_1,\ldots,z_M$ be any list with $M\ge1$ and $\eta>0$, and let $\delta>0$ be smaller than $\eta/M^2$, than $(1-\max_iz_i)/M$, and than the least distance between distinct values of the list divided by $M$; omit the last restriction if the list takes only one value. For each value $v$ taken by $m\ge2$ entries, move those entries, in list order, to $v,v+\delta,\ldots,v+(m-1)\delta$; the new list $z'$ has distinct entries in $[0,1)$. For a prefix $j$ the function $F(u)=\#\{i\le j:\ z_i<u\}-ju$ is unchanged outside the intervals $(v,v+(m-1)\delta]$, and on such an interval the new function lies between $F(v)-j(m-1)\delta$ and $F(v+)$, where $F(v)$ and $F(v+)$ are the values of $F$ just before and just after its jump at $v$; so the new prefix error is at most $H_M(z)+M^2\delta<H_M(z)+\eta$, and $H_M(z')\le H_M(z)+\eta$. If Theorem 4.1 holds for distinct lists, $H_M(z)\ge H_M(z')-\eta\ge\frac1{16}\log M-\eta$ for every $\eta>0$, hence $H_M(z)\ge\frac1{16}\log M$. Applied to the list above with $M=L'\ge L-B$, this gives $B\ge\frac1{16}\log(L-B)$. $\square$

Proposition 1221.2 (the fixed-$r$ bound over all sequences; author-recorded). For every integer $r\ge2$ and every sequence on $\mathbb T$, coincident terms allowed, with $M/0=+\infty$,

$$\limsup_{n\to\infty}\frac{M_n^{(r)}}{m_n^{(r)}}\ \ge\ 1+\frac r{r^2-1}.$$

Proof. Suppose the upper limit is smaller and choose $\rho$, $\eta$, $N_1$ as on the Theorem 1.1 page; then $R_n\le\rho$ for $n\ge N_1$, so every $r$-block at those times is positive, while single gaps may vanish. The three pages are now read with gaps $\ge0$: Lemma 2.1 (a block containing a zero piece is compared with a block of the previous time as before), Lemma 3.1 (the blocks $W_i$, $U_i$ and the protected block of part 2 need only nonnegative gaps), Proposition 4.1 (the counting is on gaps as combinatorial objects; an insertion at an occupied position is a split with a zero piece, and a zero gap may later be split into two zero pieces, which the descendant accounting treats like any split), and Theorem 1.1 (the mean identity and the epoch recursion). The contradiction is reached as on those pages. $\square$

Conditional deduction 1221.3 (author-recorded). Let (P) be the premise that the chain reconstructed on the preprint's pages proves Theorem 1.1 for sequences of distinct points, with its two imported inputs as stated there (Theorem 4.1, stated there for every list in $[0,1)$, of which only the case of distinct entries is needed, and Halász's Theorem 7.1). Then the same chain, with Proposition 1221.1 in place of Lemma 4.2, proves the three bounds of Theorem 1.1, with the same $c$ and the same coefficient $1/100$, and an absolute threshold in place of $r_0$, for every sequence on $\mathbb T$, coincident terms allowed; in particular, under (P), all three parts of the mean-normalized conjecture hold for the erdosproblems.com family. Argument: Sections 2--3 and 6--8 use distinctness nowhere (the checklist above); Section 5 derives (2.1) from the ratio hypothesis through the mean identity alone; Proposition 1221.1 gives $B\ge\frac1{16}\log(\lfloor S\rfloor-B)$, and since $B=\frac3{100}\log r+O(1)$ while $\log\lfloor S\rfloor=\frac12\log r-O(\log\log r)$, subtracting $B$ inside the logarithm changes it by $o(1)$ and the comparison of Section 5 is unchanged; Lemma 7.2's planar sets are sets of distinct points, so Theorem 7.1 applies as before. Premise (P) is an explicit assumption: the source claim is unrefereed, and the focused review of this deduction did not check its two imports or independently verify the full chain. The separate composition below supplies written proofs for a variant with coefficient $1/200$; it does not establish (P) with the coefficient $1/100$. No status changes.

What it does not give. Equality of the constants over the two families is not proved: under (P) the extended lower bounds and the shared upper bound give $\mu_r-1$ the same order $\log r/r$ over both families, but $\mu_r^{\mathrm{all}}<\mu_r^{\mathrm{dist}}$ is not excluded, and nothing here needs it. The sources state their theorems for distinct points; the extension is the corpus's reading of the reconstructed arguments, not a statement of any source. The connections grade accepts the focused check of the extraction and the conditional extension, with (P) still an explicit premise. Its review took the reconstruction pages at their existing standing and did not certify their entire source chain. The filed reconstruction reviews cover their distinct-point subjects, relative to the imported inputs; neither review set covers the complete all-sequence composition.

Further scrutiny. The route recorded as the lead consecutive_gap_bounds_coincident_points has a focused conditional warrant as described above. The complete composition requires independent whole-conclusion scrutiny of its discrepancy inputs, the span-to-count chain, the lifted-arc convention and the closing quantifiers. A positive verdict would support that full argument at the scope its grade accepts; a defect must identify the affected branch and its first unsupported estimate.

Outcome. Transfers for the fixed-$r$ bound (Proposition 1221.2); transfers conditionally for the claimed growth bounds (deduction 1221.3, under the standing of the claim). The composition page instead proves the finite-list and planar estimates it uses and composes the all-sequence argument without assuming (P), with ratio coefficient $1/200$ in place of $1/100$. That whole argument remains author-recorded.

The span-to-count transfer has no second consumer

The mechanism at its actual scope. The reusable content of the 2026 preprint is the passage from span control to counting control for a nested family of point sets: the sets $P_t=\{x_1,\ldots,x_{\lfloor t\rfloor}\}$ grow by one term per step, and the hypothesis is an eventual bound on the $r$-spans at scale $1/t$, two-sided in Lemma 2.1 (every span in $[(r-a_t)/t,(r+b_t)/t]$ with $a_t+b_t\le A$) or one-sided in Lemma 6.1 (only $nM_n^{(r)}-r\le A$, or only $r-nm_n^{(r)}\le A$), with $A\ge1$ and $r\ge C_0A$ for an absolute $C_0$, that is, a span error small compared with $r$. The conclusion is a counting error $O(A)$, pointwise (Proposition 3.1) or in spatial $L^1$ (Proposition 6.4), on arcs holding at most $\sqrt{Ar}/\log^2(r/A)$ terms, with a threshold in $t$ depending on $r$, $A$ and the sequence. The nesting is essential: the cyclic walks compare $P_t$ with $P_s\supseteq P_t$ at nearby times, and the localization of Lemma 4.2 and Lemma 7.2 reads the insertion time as a coordinate. The converse passage, from counting control to span control, is the elementary direction and is what Theorem 2 of the balanced-stick preprint takes from its Theorem 3.

The search. The problems, research and theory trees were searched for sequences on the circle or in the unit interval, consecutive gaps and spacings, discrepancy, and the named imported theorems. No open problem consumes the mechanism:

• Problem 255 is an input, not a consumer (the section above).
• Problem 997 (well-distribution of $\{\alpha p_n\}$; proved) counts terms with indices in a block $n<m\le n+k$ of fixed length for every shift $n$, a statement about all blocks of one sequence rather than about the prefixes $P_t$ of a growing family; the mechanism's conclusion concerns prefixes on arcs of length $D/n$, and the problem is proved.
• Problem 987 (exponential sums of a sequence in $(0,1)$; proved) and Problem 998 (bounded remainder intervals of $\{\alpha m\}$; disproved as worded, with Kesten's length criterion the correct theorem) are resolved, and the Kronecker sequence that appears in the balanced-stick preprint enters the folder only through the imported Theorem 3, which is not reconstructed.
• Problem 992, Problem 994, Problem 995 and Problem 996 are metric statements for almost every $\alpha$; the mechanism is deterministic for one sequence and carries no metric information.
• Problem 988, Problem 989 and Problem 991 concern finite point sets on the sphere or a sequence in the plane without the one-insertion-per-step structure, and are resolved.
• The integer-sequence gap problems (for instance Problem 208 and Problem 153) concern the gaps of one fixed set, not of a growing family, and the theory tree holds no claim about sequences in the unit interval.

Outcome. Open: no second consumer in the corpus, so no method page is made. A future consumer would be recognized by a hypothesis that controls the $r$-th neighbor distances of a nested growing family uniformly in the position to within an error small compared with $r$, and a conclusion that counts terms in short arcs; the (4.3) count of 1949, rerun on the interval in Proposition 480.1, is the only transfer of the folder's mechanisms to another setting proved here.