{"id":"dcb2f0d2-a3f8-4273-8c83-f9316eddfac1","arxiv_id":"1908.05650","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For the l1 unit cross-polytope, the maximum number of points with pairwise l1-distance at least 2r is exactly 2n for r in (1-1/n, 1], and in three dimensions is exactly 10 for r in (3/5, 2/3] and 12 for r in (4/7, 3/5], with an upper bound of 14 for r in (1/2, 4/7].","lead":"This paper finds the largest possible number of points that can be placed inside an n-dimensional cross-polytope while keeping any two points at least a chosen l1-distance apart, for several ranges of that distance. It proves exact counts in all dimensions for distances near 1, and exact counts of 10 and 12 in three dimensions for two distance ranges, with 14 as a ceiling in a third range.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lower-bound proofs assert distance upper bounds where packings need lower bounds, and Lemma 4.4 contains false case computations; direct recomputation supports the claimed values, so the results appear correct but the written proof is not.","rationale":"The reader's weakest assumption correctly identifies the lower-bound constructions: the printed inequality direction in Propositions 5.2-5.4 is backwards, so the proof as written does not demonstrate the packing property. I agree this is load-bearing for the lower-bound half of Theorem 1.2. However, I also found that the upper-bound proof for Theorem 1.2(b) has multiple false arithmetic equalities in Lemma 4.4; the reader did not flag these. Independent recomputation of the explicit 10-, 12-, and 13-point sets and of all 20 distances in the Lemma 4.4 grid shows that the intended inequalities do hold, so there is no indication that the mathematical results are false. The appropriate verdict remains CONDITIONAL: the exact values are credible and apparently correct, but the manuscript's written proofs contain sign errors and computational typos that must be corrected. I do not see a reason to move to ACCEPT or REJECT; the condition is that the proof exposition be repaired.","tokens_in":25909,"tokens_out":21200,"duration_ms":171354,"concrete_test":"Run a single exact-arithmetic script that (i) computes all pairwise l1 distances for the coordinate lists in Propositions 5.2, 5.3, and 5.4 and checks that the minima equal 4/3, 6/5, and 12/11 respectively, and (ii) recomputes all 20 entries of the Lemma 4.4 grid from the actual vectors and checks that each distance is below 2r for r in (4/7, 3/5]. If any check fails, the corresponding theorem or proposition needs revision; if all pass, the central claims are supported and only the proof exposition needs fixing.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The exact values in Theorem 1.2 rest on both the upper-bound arguments and the explicit lower-bound configurations. As printed, Propositions 5.2, 5.3, and 5.4 verify the packing condition by asserting ||x-y||_1 <= 4/3, <= 6/5, and <= 12/11. That is the wrong direction: the packing definition requires ||x-y||_1 >= 2r, i.e. at least those values at the endpoint radii. The written proofs therefore do not establish the lower bounds. Direct enumeration of the listed coordinate sets shows their true minimum distances are exactly 4/3, 6/5, and 12/11, so the constructions are valid and the defect is a sign typo. A second independent defect appears in Lemma 4.4: the 20-case table contains several false equalities (e.g. case 10 computes a distance as r, while the displayed vectors give 2-2r; case 17 gives 4/3-2r instead of 2-2r). Recomputing all 20 pairs with the actual vectors shows every distance is < 2r for r in (4/7, 3/5], so the lemma is true. Thus the central claims appear correct, but the printed proof is not sound as written and should be corrected before acceptance.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies gamma(C*_n,r), the maximum size of a subset D of the unit cross-polytope C*_n whose distinct points are pairwise at l1-distance at least 2r. It proves Theorem 1.1, that gamma(C*_n,r)=2n for r in (1-1/n,1], and Theorem 1.2, that in dimension three gamma(C*_3,r)=10 for r in (3/5,2/3], gamma(C*_3,r)=12 for r in (4/7,3/5], and gamma(C*_3,r)<=14 for r in (1/2,4/7]. The upper bounds are obtained by decomposing C*_3 into the vertex neighborhoods S_3(r) and eight regions conv(V(r,(sigma_1,sigma_2,sigma_3))), then showing that certain of these regions are blocked sets. The lower bounds are explicit configurations: V3 union Q10, V3 union Q+12 union -Q+12, and V3 union Q13, giving respectively 10, 12, and 13 points for the stated ranges of r. The paper also relates the problem to the packing density of the cross-polytope and to kissing numbers.","tokens_in":26175,"tokens_out":12437,"duration_ms":104936,"significance":"If the results are correct, they provide exact maximum cardinalities for finite l1 packings in cross-polytopes, a problem that complements known results on kissing numbers and on the quantities M(L,K,m). The upper-bound architecture is largely self-contained and convincing: Lemma 2.1 gives a useful containment, Lemmas 3.2 and 3.3 control the number of points near vertices, and the blocked-set arguments in Section 4 are conceptually sound. The explicit coordinate constructions are a strength and appear to be genuine packing sets. However, the printed proofs of the lower bounds contain reversed inequalities, and the proof of Lemma 4.4 contains arithmetically incorrect case computations, so the exact statements are not established by the manuscript as written. Direct recomputation of the listed configurations indicates that the intended inequalities hold, so a substantive revision should be able to repair the paper rather than change its conclusions.","major_comments":[{"comment":"The lower-bound proofs assert upper bounds on pairwise distances, which is the opposite of what the packing condition requires. For example, Proposition 5.2 states that any distinct x,y in V3 union Q10 satisfy ||x-y||_1 <= 4/3, but to be a packing set for r<=2/3 one needs ||x-y||_1 >= 2r, and in particular a lower bound of at least 4/3 at the endpoint r=2/3. The same reversed inequality appears in Proposition 5.3 as ||x-y||_1 <= 6/5 and in Proposition 5.4 as ||x-y||_1 <= 12/11, and Proposition 5.1 has the analogous statement ||x-y||_1 <= 2(1-1/n). As printed, these propositions do not verify the packing property and therefore do not establish the lower-bound halves of Theorem 1.2 or Proposition 1.3. Direct enumeration of the listed sets shows that their true minimum distances are exactly 4/3, 6/5, and 12/11 respectively, so replacing the upper bounds by lower bounds and checking the endpoint equalities repairs the proofs.","section":"Section 5, Propositions 5.1–5.4"},{"comment":"The 20-case table contains several false computations, and since Lemma 4.4 is load-bearing for the upper bound gamma(C*_3,r)<=12 in Theorem 1.2(b), these errors must be corrected. For instance, in case 10 the two displayed vectors differ by (r/2+1-r, 0, r/2-(2r-1)), whose l1-distance simplifies to 2-2r, not to r as printed; in case 17 the distance between (1/3,1/3,1/3) and (-(2r-1),2r-1,2r-1) is 2-2r, not 4/3-2r; and case 18 similarly miscomputes the first coordinate. These incorrect intermediate equalities do not immediately invalidate the lemma because the correct values are still below 2r on r in (4/7,3/5], but the written proof is not sound as it stands. The author should recompute all 20 cases and replace the erroneous expressions.","section":"Section 4.3, Lemma 4.4"}],"minor_comments":[{"comment":"Several lemma references are mismatched: the proof of Lemma 4.3 cites 'Lemma 4.3' where Lemma 4.2 is meant; the proof of Theorem 1.2(a) cites Lemma 4.4 where Lemma 4.3 is meant; the proof of Theorem 1.2(b) cites Lemma 4.6 where Lemma 4.5 is meant; and the proof of Theorem 1.2(c) cites Lemma 4.5 where Lemma 4.6 is meant. These should be corrected to avoid confusing the reader.","section":"Throughout Section 4"},{"comment":"The statement says Vn union {+-qn} subset C*_3, but this is the n-dimensional cross-polytope and should read C*_n; the same typo appears in the proof of Propositions 5.2 and 5.3 where C*_n is used instead of C*_3.","section":"Proposition 5.1"},{"comment":"The text says 'Theorem 1.3 is proved in Section 5', but the referenced result is Proposition 1.3, not Theorem 1.3.","section":"End of Section 2"},{"comment":"The notation S*_3 appears in the splitting argument of the proofs of Theorem 1.2(b) and (c), but only S_3 and C*_3 are defined in Section 4.1. Use C*_3\\S_3(r) consistently.","section":"Section 4.4 and proofs of Theorems 1.2(b) and 1.2(c)"},{"comment":"The proof writes 'p in V(r,(1,1,1))' but the hypothesis is p in conv(V(r,(1,1,1))). The intended convexity argument should be stated explicitly, using the fact that the l1 norm is convex.","section":"Lemma 4.2 proof"}],"recommendation":"major_revision","confidential_remarks":"The central claims appear likely correct after direct verification of the explicit configurations and the case computations, but the manuscript contains systematic sign errors in the lower-bound proofs and arithmetic errors in Lemma 4.4. These are fixable within the scope of the paper, so I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the exact values are probably right, but the printed lower-bound proofs do not prove what they claim. The upper-bound part is the real meat and looks largely sound. This is a fixable paper, not a broken one.\n\nWhat is new: Theorem 1.1 gives the sharp n-dimensional cutoff gamma(C*_n,r)=2n for r in (1-1/n,1]; Theorem 1.2 gives exact 3D values 10 and 12 on two intervals and an upper bound 14 on a third; Proposition 1.3 gives 13 points. The blocking-set method, adapted from Larman-Zong and Boroczky-Wintsche, is appropriate, and the upper-bound lemmas (2.1, 3.1-3.3, 4.1, 4.2, 4.6) are detailed and convincing. The connection to kissing numbers and packing density is honestly placed; the references look right and there is no self-citation inflation.\n\nSoft spots: Propositions 5.1-5.4 all state ||x-y||_1 <= 4/3 (or 6/5, 12/11) where the packing condition requires >=. As written, none of those proofs verifies the packing property. I checked the coordinate lists; the true minimum distances are exactly those values, so the constructions are valid and the defect is a sign typo repeated four times. Still, it must be fixed. Lemma 4.4 has a second defect: several entries in the 20-case table are wrong as printed (case 10 gives r, but the vectors give 2-2r; case 17 gives 4/3-2r instead of 2-2r). Recomputing all 20 cases shows every distance is below 2r for the stated r-range, so the lemma's conclusion is true, but the table cannot stand. There are also small cross-reference errors (e.g., 'Theorem 1.3' vs 'Proposition 1.3'; Lemma 4.3 used in place of Lemma 4.2/4.4). All are minor and repairable.\n\nWho it is for: people working on translative packing numbers, kissing numbers of polytopes, or finite packings in l1 geometry. It is a narrow but genuine exact computation. The paper also honestly flags its limits, including the unresolved interval (1/2,4/7] and the possibility that the 13-point configuration might be improved.\n\nRecommendation: send to peer review. The mathematical claims appear correct and the method is sound; the written proof needs a careful check and a corrected lower-bound section before acceptance.","headline":"Exact cross-polytope packing numbers are likely correct and worth a referee, but the printed lower-bound proofs assert the wrong inequality and the case table in Lemma 4.4 has errors.","tokens_in":26695,"tokens_out":7397,"would_cite":true,"duration_ms":64894,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["52C17"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper pins down the largest sets of $\\ell^1$-separated points in a unit cross-polytope: exactly $2n$ points at scales just below 1, and exactly 10 or 12 points in two three-dimensional ranges, with 14 as an upper bound below.","keywords":["cross-polytope","l1-distance","packing set","maximum cardinality","three-dimensional packing","kissing number","explicit construction"],"falsifier":"Check the smallest pairwise $\\ell^1$ distance in the ten-point set $V_3\\cup Q_{10}$; if it falls below $4/3$, the construction fails for $r=2/3$ and Theorem 1.2(a) loses its lower bound. Similarly, an exhaustive search for an 11-point packing at $r=2/3$ would disprove the upper bound of 10.","tokens_in":25702,"feed_emoji":"🔷","tokens_out":13581,"duration_ms":103671,"temperature":0.7,"pith_summary":"The paper asks how many points can be placed inside the unit cross-polytope—the $\\ell^1$ ball—so that every pair of distinct points is at least a given $\\ell^1$-distance apart. It proves that in any dimension $n\\ge 2$, when the required distance is close to the maximum possible, the answer is exactly the $2n$ vertices of the polytope. In three dimensions, it determines the exact maximum for two ranges of the distance parameter: 10 points when the scale lies in $(3/5,2/3]$ and 12 points when it lies in $(4/7,3/5]$, and it bounds the next range by 14 points. Each exact value is matched by an explicit set of points achieving it, so the bounds are sharp.","feed_headline":"Exact counts for 3D cross-polytope packings: 10, 12, at most 14","feed_subtitle":"Largest $\\ell^1$-separated point sets inside a cross-polytope, with explicit constructions.","key_machinery":"The working mechanism is a division of the cross-polytope into $S_n(r)$, the part lying within $\\ell^1$-distance $2r$ of a vertex, and, in three dimensions, eight leftover tetrahedra $\\operatorname{conv}(V(r,\\sigma))$ indexed by sign vectors. When $r>1-\\frac{1}{n}$, the first part covers the whole body, and a uniqueness lemma shows each vertex can serve at most one packing point, forcing the $2n$ bound. For smaller $r$ in dimension 3, each leftover tetrahedron has $\\ell^1$-diameter less than $2r$, so it holds at most one point; blocking lemmas then show that a point in one tetrahedron empties one or three neighbouring tetrahedra, which yields the upper bounds 10, 12, and 14 according to the interval.","core_discovery":"The central discovery is a complete evaluation of the packing number $\\gamma(C^*_n,r)$ on three intervals. Theorem 1.1 states that for $n\\ge 2$ and $r\\in(1-\\frac{1}{n},1]$, $\\gamma(C^*_n,r)=2n$, and that this interval is the largest on which the value remains $2n$. Theorem 1.2, for dimension 3, gives $\\gamma(C^*_3,r)=10$ when $r\\in(3/5,2/3]$, $\\gamma(C^*_3,r)=12$ when $r\\in(4/7,3/5]$, and $\\gamma(C^*_3,r)\\le 14$ when $r\\in(1/2,4/7]$. Proposition 1.3 supplies a 13-point packing for $r\\in(1/2,6/11]$, improving the lower bound within the last interval. The upper bounds come from a geometric decomposition into regions of diameter below $2r$, and the lower bounds come from explicit coordinate lists.","pith_inferences":["The same vertex-neighbourhood decomposition may extend to dimensions $n\\ge 4$: when the leftover regions have $\\ell^1$-diameter below $2r$, a similar blocking argument could yield exact values for other intervals of $r$.","The manuscript's lower-bound proofs state pairwise distances as upper bounds ($\\le 4/3$, $\\le 6/5$, $\\le 12/11$) where the packing condition requires lower bounds; direct computation of the listed coordinates shows the minimal distances equal those values, so the constructions are valid but the text appears to contain a sign typo.","The gap between 13 and 14 for $r\\in(6/11,4/7]$ suggests a possible 14-point configuration may exist near $r=6/11$; the paper's local argument rules out one natural family, but a different arrangement might close the gap.","At $r=1/2$, the bounds $19\\le\\gamma(C^*_3,1/2)\\le26$ could be tightened by combining the decomposition ideas here with the known kissing structure, potentially resolving a piece of the cross-polytope kissing problem."],"forward_implications":["For every $n\\ge 2$, any packing with $r\\in(1-\\frac{1}{n},1]$ has at most $2n$ points, and the vertex set achieves this bound, so the vertices are optimal throughout this interval.","In three dimensions, the maximum cardinality is exactly 10 on $(3/5,2/3]$ and exactly 12 on $(4/7,3/5]$, so no intermediate maximums occur in these ranges.","For $r\\in(1/2,4/7]$ the maximum is at most 14, and at least 12 everywhere; the 13-point construction for $r\\in(1/2,6/11]$ shows the lower bound is 13 on that subinterval.","Taking $r\\to 0$, the scaled packing numbers $\\gamma(C^*_n,r)$ are linked to the packing density of the cross-polytope, and the finite-$r$ results provide the first terms of that relationship.","The interval $(1-\\frac{1}{n},1]$ is maximal: for $r\\le 1-\\frac{1}{n}$ one can place $2n+2$ points by adding the centroids of two opposing facets to the vertices."],"supporting_citations":[{"why":"Supplies the kissing number of the cross-polytope and the vertex/midpoint decomposition idea that the paper adapts to bound packings.","marker":"[11]"},{"why":"Provides an alternative derivation of the cross-polytope kissing number and a structural description used as context for the $r=1/2$ boundary case.","marker":"[16]"},{"why":"Gives upper bounds for related packing quantities using midpoint-defined subsets, an approach the paper generalizes.","marker":"[1]"},{"why":"Hadwiger's upper bound on kissing numbers yields the initial 26-point cap for $\\gamma(C^*_3,1/2)$.","marker":"[9]"}],"fun_headline_variants":["Exact 3D cross-polytope packing counts: 10, 12, max 14","3D cross-polytope packing: exactly 10, 12, up to 14","10, 12, or 14: exact packing numbers for 3D cross-polytope","3D cross-polytope: 10, 12, and 14-point limits"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The paper's lower-bound constructions must genuinely be packing sets: every pair of listed points has to be separated by at least the claimed $\\ell^1$ distance, and the proofs state the inequality in the wrong direction (an upper bound where a lower bound is needed), although the coordinates themselves satisfy the intended condition.","fun_headline_variants_meta":{"raw":{"variants":["Exact 3D cross-polytope packing counts: 10, 12, max 14","3D cross-polytope packing: exactly 10, 12, up to 14","10, 12, or 14: exact packing numbers for 3D cross-polytope","3D cross-polytope: 10, 12, and 14-point limits"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001713,"raw_usage":{"total_tokens":6824,"prompt_tokens":1039,"completion_tokens":5785,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":655,"completion_tokens_details":{"reasoning_tokens":5684}},"tokens_in":655,"tokens_out":5785,"duration_ms":39705,"temperature":1.0,"reasoning_tokens":5684,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:08:12.683173+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check the smallest pairwise $\\ell^1$ distance in the ten-point set $V_3\\cup Q_{10}$; if it falls below $4/3$, the construction fails for $r=2/3$ and Theorem 1.2(a) loses its lower bound. Similarly, an exhaustive search for an 11-point packing at $r=2/3$ would disprove the upper bound of 10.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the kissing number of the cross-polytope and the vertex/midpoint decomposition idea that the paper adapts to bound packings."},{"cited_title":"Talata, On extensive subsets of convex bodies, Periodica Mathematica Hungarica, 38 (1999), pp","cited_arxiv_id":null,"evidence_quote":"Provides an alternative derivation of the cross-polytope kissing number and a structural description used as context for the $r=1/2$ boundary case."},{"cited_title":"Böröczky Jr, I","cited_arxiv_id":null,"evidence_quote":"Gives upper bounds for related packing quantities using midpoint-defined subsets, an approach the paper generalizes."},{"cited_title":"Hadwiger, Über treffanzahlen bei translationsgleichen eikörpern, Archiv der Mathematik, 8 (1957), pp","cited_arxiv_id":null,"evidence_quote":"Hadwiger's upper bound on kissing numbers yields the initial 26-point cap for $\\gamma(C^*_3,1/2)$."}],"review_version":1}