Consecutive-gap constants on the circle

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clst25_theorem_2_reconstruction: Reconstructs the one-paragraph derivation of the upper bound on the ratio of the largest to the smallest r-span from the short-interval counting bound of Theorem 3, with the quantifiers made explicit; Theorem 3 itself is imported from the same source and not reconstructed, and the r = 1 boundary of both statements is recorded.

dber49_inequality_3_3_reconstruction: Reconstructs the 1949 count of intervals destroyed by the points inserted between stages rn and (r+1)n, giving the lower bound 1/log(1 + 1/r) for the upper limit of k times the largest r-span; the general-r case, which the note only sketches, is written out.

dber49_inequality_4_3_reconstruction: Reconstructs the 1949 cyclic-order count that bounds the lower limit of k times the smallest r-span by (r/(r+1))/log(1 + 1/r); the general-r case, which the note only sketches, is written out, and two printed slips are recorded.

dber49_inequality_5_7_reconstruction: Reconstructs the 1949 one-step inequality between the largest r-span before an insertion and the smallest after it, and the telescoping argument that turns it into the universal bound 1 + 1/r for the ratio constant; the small-n case of the one-step inequality is left as the note leaves it.

evidence/: Independent focused reviews and two distinct grades of the reconstruction pages for the de Bruijn--Erdős consecutive-gap constants, without executable evidence or accepted whole-proof credit.

ko26a_lemma_3_1_reconstruction: Reconstructs the monotonicity of the largest r-span under insertion (Lemma 2.1) and the protected-block lemma: when a split barely lowers the largest r-span and the ratio stays below rho, the 2r gaps around the split are short and none can be split again until the largest r-span has fallen by the factor (r-1)(rho-1+eta).

ko26a_proposition_4_1_reconstruction: Reconstructs the count of protected blocks over one epoch: if the ratio stays below rho after time N, the first time at which the largest r-span has fallen by the factor beta is at most (1 + 1/r) N plus a constant, since fast steps are boundedly many and the slow steps outside a bounded exceptional set are attached to initial gaps at mutual distance at least r.

ko26a_theorem_1_1_reconstruction: Reconstructs the epoch iteration that turns the protected-block count into the fixed-r bound 1 + r/(r^2 − 1) for the ratio of the largest to the smallest r-span over sequences of distinct points, improving the 1949 bound 1 + 1/r for each r at least 2.

ko26b_lemma_2_1_reconstruction: Reconstructs the comparison of the largest and smallest interval counts at scale D and time t with those at scale E and a nearby time, through injective compositions of forward and backward cyclic moves by kr places, under pointwise control of all r-spans.

ko26b_lemma_4_2_reconstruction: Reconstructs the transfer from a uniform short-interval counting bound B on intervals holding at most S points to a list of floor(S) points, in insertion order, all of whose prefixes have counting error at most B, and states the finite-prefix form of Schmidt's theorem, derived by the source from Larcher's proof, which then forces B ≥ (log floor(S))/16.

ko26b_lemma_6_1_reconstruction: Reconstructs the step from an eventual one-sided bound on the r-spans, in either direction, to a bound on the total absolute deviation of the kr-spans from their mean, through the zero-sum identity for the deviations of the r-spans at an integer time.

ko26b_lemma_6_2_reconstruction: Reconstructs the L^1 counterpart of the walk comparison: under a one-sided span bound, the positive spatial mass of the counting error at scale D and time t is bounded by a rescaled positive mass at scale E and time (1+q)t plus qD plus a transport error 8kA + 4kr/t.

ko26b_lemma_6_3_reconstruction: Reconstructs the terminal estimate of the averaged comparison: under a one-sided span bound, the positive spatial mass of the counting error on intervals of length r/t is at most A, by pairing each such interval count with the r-span arcs that cover the circle exactly r times.

ko26b_lemma_7_2_reconstruction: Reconstructs the localization step: a uniform L^1 short-interval counting bound B on intervals holding at most S points forces B ≥ c root log S, by reading the points of a moving arc, with insertion time as second coordinate, as planar point sets to which Halász's L^1 discrepancy lower bound applies.

ko26b_proposition_3_1_reconstruction: Reconstructs the iteration of the cyclic-walk comparison along doubling scales from r ± A down to the square root of A r, and the final comparison that bounds the counting error on every interval holding at most S points by 3A plus a smaller term, when the r-spans lie between (r - a_t)/t and (r + b_t)/t with a_t + b_t at most A.

ko26b_proposition_6_4_reconstruction: Reconstructs the iteration of the averaged comparison along doubling scales from r down to the square root of A r and the final descent to intervals holding at most S points, giving an L^1 counting error at most a constant times A at all late integer times under either one-sided span hypothesis.

ko26b_theorem_1_1_reconstruction: Reconstructs the two closing arguments of the preprint: the ratio bound 1 + log r/(100 r) from the pointwise short-interval count and the finite-prefix Schmidt bound, and the two one-sided bounds c root log r from the L^1 short-interval count and Halász's planar theorem; states which reading of Problem 1221 each part addresses and which inputs are imported unchecked.

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This folder holds author-recorded reconstructions of the proofs behind Problem 1221: whether the three de Bruijn--Erdős constants for the largest and smallest sums of $r$ consecutive gaps of a sequence on the circle, and for their ratio, deviate from their trivial values by more than any constant over $r$ as $r\to\infty$. Each page states the result with its hypotheses, writes out every essential deduction in the corpus's own words, cites imported theorems as imports, labels what is omitted, and says which reading of the problem's ambiguous statement the result addresses. The pages are named by the source key of the problem page (dBEr49, Ko26a, Ko26b, ClSt25) and the source's result label. None of them is an independent review; they change no status and assign no tier.

Where things stand

The problem stays open. The site's wording is defective in its first two parts, and the assessed question is the mean-normalized reading: whether $\Lambda_r-r$, $r-\lambda_r$ and $r(\mu_r-1)$ tend to infinity. The 1949 note proves $\Lambda_r-r\ge\tfrac12+o(1)$ (Section 3), $r-\lambda_r\ge\tfrac12+o(1)$ ((4.3)) and $r(\mu_r-1)\ge1$ ((5.7)) over all sequences, coincident points allowed; the reconstructions write out the general-$r$ cases the note only sketches and record three printed slips that do not affect the results. Korsky's 2026 note proves $r(\mu_r-1)\ge1+1/(r^2-1)$ for each fixed $r\ge2$ over sequences of distinct points (Theorem 1.1, with its protected-block lemma and epoch count), which is bounded in $r$. Clément and Steinerberger's preprint gives the upper bound $r(\mu_r-1)\le c'\log r$ for $r\ge2$ (Theorem 2 from Theorem 3; Theorem 3 itself is not reconstructed, and its literal $r=1$ case is recorded as false). Korsky's 2026 preprint claims all three parts of the mean-normalized reading over sequences of distinct points, with growth $c\sqrt{\log r}$ for the first two and $\log r/100$ for the third (Theorem 1.1); the claim is unrefereed and unreviewed, and the reconstruction changes that standing in no way.

What the reconstruction of the claim rests on. The chain Lemma 2.1 (cyclic-walk comparison), Proposition 3.1 (short-interval counts under pointwise span control), Lemma 4.2 (a short interval read as a finite list), and Section 5 gives the ratio part; the chain Lemma 6.1 ($L^1$ span control from a one-sided bound), Lemmas 6.2--6.3, Proposition 6.4 ($L^1$ short-interval counts) and Lemma 7.2 (localization to a planar point set), and Section 8 gives the two one-sided parts. Every step of these chains is written out on its page. Two inputs are imported and not checked: the finite-prefix form of Schmidt's discrepancy theorem with constant $1/16$, which the preprint derives from Larcher's 2015 proof (Larcher's paper is not held; the qualitative form with an unspecified constant follows from Schmidt's planar theorem, which would still give the growth $r(\mu_r-1)\to\infty$), and Halász's 1981 planar $L^1$ discrepancy theorem (not held). The constants $c$, $r_0$ are not explicit, and the ratio argument needs $r\ge e^{100}$ before its absolute constants enter. Distinctness of the points is used in Lemma 4.2; whether the constants over all sequences agree with those over distinct sequences is not settled in the sources read. The one omitted case in the 1949 material is (5.1) for $n<2r-1$, which (5.7) does not need.

Mechanism. The 1949 bounds and both Korsky arguments rest on the same two facts: inserting a point splits one gap and changes only the nearby $r$-blocks, and the $r$-spans at time $n$ average $r/n$; the 2026 preprint adds that forward and backward cyclic walks by $kr$ places, composed across nearby times, inject the points of a short interval at one time into a slightly longer interval at a later time, so that uniform span control transfers to counting control on intervals holding about $\sqrt{Ar}$ points, where a discrepancy lower bound (Schmidt's in one dimension for the ratio, Halász's $L^1$ planar bound for the one-sided parts) becomes a contradiction.

Reviewed. Each reconstruction page was independently reviewed as it stood on 2026-09-28T05:03:27Z by a focused review filed under evidence/verify/, with a distinct grade of the sixteen reviews. The graded verdicts, as the grade records them, are: the Clément and Steinerberger Theorem 2 page is faithful with corrections, and its argument was defective as written on $2\le r<2+c\log2$ and sound for $r\ge2+c\log2$, the range to which C1 restricts it; the 1949 Section 3 page is faithful with a correction and sound; the 1949 (4.3) page is faithful with a correction and sound, with the justification of its span step replaced; the 1949 (5.7) page is faithful and sound; the 2026 note's Lemma 3.1 page is faithful and sound; the 2026 note's Proposition 4.1 page is faithful and sound; the 2026 note's Theorem 1.1 page is faithful and sound; the 2026 preprint's Lemma 2.1 page is faithful and sound; the preprint's Proposition 3.1 page is faithful with a correction to its summary line and sound; the preprint's Lemma 6.1 page is faithful and sound; the preprint's Lemma 6.2 page is faithful and sound; the preprint's Lemma 6.3 page is faithful and sound; the preprint's Proposition 6.4 page is faithful with a correction and sound; the preprint's Lemma 7.2 page is faithful and sound relative to the imported Theorem 7.1; and the preprint's Theorem 1.1 page is faithful with corrections and sound relative to the two imported theorems and the input pages. A second independent review of the preprint's Lemma 4.2 page, graded in the second grade, passes, and its graded verdict is that the page is faithful with a correction and sound relative to the imported Theorem 4.1. The twelve corrections C1--C12 were applied, so the current text differs from the reviewed text at the places the grade names. No tier is assigned and the problem's status is unchanged. After the review, line wrapping was normalized on the reconstruction pages; no formula or sentence changed.