{"id":"1e3641be-060c-4d80-837c-6bef439fdc05","arxiv_id":"2505.08495","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Lattice tilings of Z^n by error balls B(n,2,3,0) exist only for n=3, none exist for B(n,2,k,k-1), and for k1>k2 with k1+k2+1 composite no tiling exists in sufficiently high dimension.","lead":"What shapes made from small integer vectors can perfectly tile the infinite grid? This paper settles the question for a family of such shapes, showing exactly when these tilings exist, and proves that most parameter choices have no tiling in high dimensions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Section 5's central inequality (9) is undermined by an internal count error: with the displayed definitions, replacing Z by X adds zero S-sets when k2=0, so the claimed A/2·(n^2−n) leading term does not exist.","rationale":"The reader's weakest-assumption flag on Lemma 2.9(a) is reasonable: that lemma is compressed and would need a full verification. However, the more immediate and decisive problem is the counting that feeds inequality (9). The displayed definitions of X, Y, and Z, together with equation (5), do not support the leading term of (9); the discrepancy is concrete and independent of Lemma 2.9. The reader's conditional verdict is therefore retained, but for a sharper reason: the general non-existence theorem should be accepted only after the set counts are corrected and inequality (9) is re-derived. The special results in Theorem 1.1 and Theorem 1.2 are separate and are not affected by this particular critique.","tokens_in":21721,"tokens_out":28608,"duration_ms":262178,"concrete_test":"Recompute the set counts directly from the displayed definitions. Let d=k1−k2; verify |X1|=d(d+1)/2, |Z|=d(d+1)/2, |X2|=2Mk2, and count the cross pairs i∈[−k2,k2]^*, j∈[k2+1,k1]^* that appear in the original decomposition of G but in neither Y nor Z. Then re-derive the leading n^2 coefficient of |G'|−|G| using equation (5) and Lemma 5.2; if it is 2k2(M−d) rather than [k1+(4M−1)k2]/2, inequality (9) is invalid. As a minimal instance, take k2=0, M=1, k1=5, where the manuscript's left-hand side 5/2(n^2−n) is nonzero but the set replacement adds zero sets.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The proof of Theorem 1.3 reduces to comparing G' (built with X) with G (built with Z), culminating in inequality (9). Two counting errors occur before any use of Lemma 2.9. First, equation (5) subtracts ((k1−k2)^2/2)(n^2−n) for the sets Z={(i,j): k2+1≤i≤j≤k1}; these number d(d+1)/2 with d=k1−k2, not d^2/2. Second, with Y defined as pairs in [−k2,k2]^*×[−k2,k2]^*, the original partition of G also contains 2dk2 cross pairs (i,j) with i∈[−k2,k2]^*, j∈[k2+1,k1]^*, which are omitted from both Y and Z. Using the displayed definitions, |X1|=d(d+1)/2, the same as |Z|, and |X2|=2Mk2. Hence the replacement of Z by X changes the number of S-sets by 2Mk2−2dk2≤0, rather than the positive amount A/2=[k1+(4M−1)k2]/2 asserted by (9). For k2=0 the net change is exactly 0. Since (9) is the only step producing the n^2 lower bound for |G'|, the proof of Theorem 1.3 and Corollary 1.4 as written is unsupported.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies lattice tilings of Z^n by asymmetric limited-magnitude error balls B(n,2,k1,k2) for k1>k2. Using the Horak--AlBdaiwi characterization of lattice tilings by finite abelian groups, it develops a group-ring and counting framework and claims three main results: Theorem 1.1 completely classifies B(n,2,3,0) tilings (they exist only for n=3); Theorem 1.2 excludes all B(n,2,k,k-1) tilings for n>=3, k>=2; and Theorem 1.3, together with Corollary 1.4, shows that for composite k1+k2+1 no lattice tiling exists in all sufficiently high dimensions. The proofs proceed by assuming a tiling, passing to a finite abelian group G of the correct order, and deriving lower bounds on the size of a modified union of S(i,j) sets. The final dimension bound is obtained by comparing the set X with the set Z in Section 5.","tokens_in":22014,"tokens_out":23650,"duration_ms":207216,"significance":"If the results are correct, they settle two natural infinite families and provide a general non-existence mechanism for a class of asymmetric error balls, which is a substantive advance over the existing partial results. The reduction to finite group counting is elegant, and the explicit generator T={1,10,26} in Z_37 for the n=3, (3,0) tiling is a valuable concrete certificate. The paper also gives a clear roadmap for the general composite case. However, verification is currently hindered by an incomplete proof of the key collision bound (Lemma 2.9), by undocumented computer checks for several small cases, and by a very compressed algebraic derivation of the main inequality (9). These issues are local in nature but load-bearing for the stated theorems.","major_comments":[{"comment":"The bound C<=3*sqrt(p) in Lemma 2.9(a) is load-bearing: it enters Lemma 2.10(c) and, through inequality (9), the proof of Theorem 1.3. The proof as written is not complete. The step 'As |m-(|x|+y)ell|<2(ell-1), we conclude that |x|+y=p-1' requires an additional argument that k1+k2+1-(|x|+y)ell is a positive multiple of ell and hence equals ell; the displayed inequality alone does not imply the conclusion. More importantly, the inequality ((p-1 choose 2)|A_i|(|A_i|-1)<=p^2 appears to treat the displayed union as a subset of a coset of H with pairwise disjoint full-size contributions, but no proof of disjointness or of the stated size of the expressions A_i^{(alpha)}A_i^{(beta)}-A_i^{(alpha+beta)} is given. The same kind of unproved counting is used in part (b) for the bound on psi(m,0). Please provide a complete proof or a precise reference for this lemma.","section":"Section 2, Lemma 2.9"},{"comment":"The complete classifications in Theorems 1.1 and 1.2 depend on undocumented computational searches. Section 3 states 'For 3<=n<=6, we use a computational search and check that no tiling of Z^n exists by B(n,2,3,0) if n=4,5 and 6', and Section 4 states 'If k=2, a computational search confirms that no lattice tiling of Z^3 by B(3,2,2,1) exists.' No algorithm, source code, or verifiable certificate is provided, so the reader cannot check these finitely many cases. Because these checks are essential to the claimed full classifications, the authors should supply reproducible code or a complete hand-checkable mathematical verification for these instances.","section":"Section 3 and Section 4"},{"comment":"The derivation of the key inequality (9) from Lemmas 5.9, 5.10, 5.11 and inequalities (7)-(8) is a large unshown algebraic step. In particular, the assembly of the constant B and the transition to the final bound n>=floor(B/A)+1 are not displayed. Since Theorem 1.3 rests entirely on this inequality, the authors should present the intermediate inequalities and show explicitly how each term of B arises. Relatedly, equation (5) is correct only if one uses |S(i,i)|=n(n-1)/2 for the d diagonal terms; the text appears to count d(d+1)/2 pairs each of size n^2-n, so this step must be written out.","section":"Section 5, inequality (9)"}],"minor_comments":[{"comment":"I checked the specific count objection raised about the leading term of (9) and do not find it to be valid. Although the number of pairs in Z is d(d+1)/2 with d=k1-k2 rather than d^2/2, the displayed total (k1-k2)^2/2 (n^2-n) is correct because the d diagonal pairs S(i,i) have size n(n-1)/2, not n^2-n. Likewise, when k2=0 the replacement of Z by X does not add new pairs, but it replaces d diagonal half-size sets by d off-diagonal boundary sets of size roughly n^2-n, which produces the positive leading term k1/2(n^2-n). The text should spell this out, since the current wording invites the miscount.","section":"Section 5, equations (5) and (9)"},{"comment":"In Lemma 4.1(a), the equality '|S(-k,1)|=|S(-k,1)|=n^2-n' contains a self-referential typo; it should presumably compare |S(-k,1)| with |S(1,-k)| or another symmetric partner.","section":"Section 4, Lemma 4.1"},{"comment":"In the proof of Lemma 5.4, the summation 'Sum_{j in [k2+1,k2]*}' has an empty range and should presumably be '[k2+1,k1]'.","section":"Section 5, Lemma 5.4"},{"comment":"In the proof of Lemma 5.9, the symbol 'phi(ell,-i_1,j_2)' should be 'psi(ell,-i_1,j_2)'.","section":"Section 5, Lemma 5.9"},{"comment":"The displayed bound contains a stray period inside the formula: '2k2 + 8.' should read '2k2 + 8'.","section":"Corollary 1.4"}],"recommendation":"major_revision","confidential_remarks":"The stress-test concern about the leading count in inequality (9) does not survive a detailed accounting: the discrepancy between d(d+1)/2 and d^2/2 is compensated by the half-size of the diagonal S(i,i), and the k2=0 case still yields a positive leading term from the boundary pairs in X1. The real obstacles are the incomplete proof of Lemma 2.9 and the undocumented computational searches for several small cases. The main strategy appears coherent and the results are plausible, so I would not recommend rejection; the requested revisions are within the scope of the manuscript."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this paper has two genuinely new classification results that look like real progress, and one general theorem whose proof does not hold up. The classification of B(n,2,3,0) (tiling iff n=3) and the full resolution of B(n,2,k,k-1) are both substantive and appear plausible. The index-set counting technique used in those sections is clever and worth reading. I did not find a concrete flaw in Sections 3 and 4.\n\nThe problem is Section 5. The stress-test note flags a counting error, and I think the core of it is correct. Equation (5) treats the index set Z as having (k1-k2)^2/2 elements, but with the displayed definition Z={(i,j): k2+1 ≤ i ≤ j ≤ k1} the count is d(d+1)/2 where d=k1-k2. That is already a red flag. Then |X1|, the part of X described by i∈[-k1,-k2-1], j∈[k2+1,k1] with j-i≤k1+k2+1, is also d(d+1)/2—the same as |Z|. The extra set difference comes only from X2, which has 2Mk2 elements. So replacing Z by X changes the number of S-sets by 2Mk2, not by A/2 = [k1+(4M-1)k2]/2. For k2=0 the change is exactly zero. Inequality (9) claims a positive n^2 term on the left from this replacement, but the only n^2 term available from the definitions is 2Mk2(n^2-n). For k2=0 there is no n^2 term at all. The stress-test's secondary claim about omitted cross pairs is wrong—Y as defined does include the pairs with i∈[-k2,k2] and j∈[k2+1,k1]—but that does not rescue (9).\n\nThe O(n) intersection bounds in Lemmas 5.4–5.11 may be fine, but they cannot compensate for a missing n^2 term. As written, the proof of Theorem 1.3 and Corollary 1.4 is unsupported. Separately, the computational checks for n=4,5,6 and n=3,k=2 are undocumented; that is a verification gap, but minor compared with the Section 5 issue.\n\nWho this is for: people working on lattice tilings, perfect codes, and limited-magnitude error models. The first two theorems are worth publishing on their own if the Section 5 problem is fixed or the general theorem is withdrawn. I would send this to a serious referee, but ask the referee to focus on the counting in Section 5. If the authors can correct the miscount and still get a non-existence theorem, fine; if not, the paper should be split and Theorem 1.3 removed.","headline":"The two classification theorems look solid and interesting, but the general non-existence proof collapses on a counting error in equation (5), so Theorem 1.3 as written is unsupported.","tokens_in":22577,"tokens_out":13905,"would_cite":true,"duration_ms":118869,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["52C22","11H31","11H71"],"pacs":[],"model":"deepseek-v4-flash","headline":"For two-coordinate limited-magnitude error balls, lattice tilings of $\\mathbb{Z}^n$ exist only up to an explicit dimension bound when $k_1+k_2+1$ is composite; $\\mathcal{B}(n,2,3,0)$ tiles exactly at $n=3$, and $\\mathcal{B}(n,2,k,k-1)$…","keywords":["lattice tiling","perfect code","limited magnitude error ball","group ring","finite abelian group","two-coordinate errors","flash memory","non-existence threshold"],"falsifier":"Re-run the computational search claimed for $\\mathcal{B}(n,2,3,0)$ at $n=4,5,6$; any tiling found would refute Theorem 1.1, and because the proof's inequality (3) together with Lemma 2.9 rules out $n\\ge 7$, the search is the decisive check. For the general theorem, compute the collision count $C=\\max_g|\\{h:g^{k_1+k_2+1}=h^{k_1+k_2+1}\\}|$ in a group whose elementary abelian $p$-core has order exceeding $p^2$; a value above $3\\sqrt{p}$ would invalidate Lemma 2.9(a) and with it inequality (9).","tokens_in":21505,"feed_emoji":"🧩","tokens_out":9622,"duration_ms":85949,"temperature":0.7,"pith_summary":"The paper asks when the integer lattice $\\mathbb{Z}^n$ can be tiled by the limited-magnitude error ball $\\mathcal{B}(n,2,k_1,k_2)$ of vectors with at most two nonzero coordinates, each coordinate lying in $[-k_2,k_1]$. Because a lattice tiling is exactly a linear perfect code for this asymmetric error model, the question carries coding-theoretic weight, especially for flash memory where symbol values are integer vectors. The paper's main theorem states that whenever $k_1+k_2+1$ is composite, no such lattice tiling exists once $n$ exceeds an explicit threshold quadratic in $k_1-k_2$. It also completely settles two families: $\\mathcal{B}(n,2,3,0)$ tiles $\\mathbb{Z}^n$ if and only if $n=3$, and $\\mathcal{B}(n,2,k,k-1)$ tiles no $\\mathbb{Z}^n$ for any $n\\ge 3$. The proof translates tilings into decompositions of finite abelian groups and then bounds the overlaps of the resulting power-product sets.","feed_headline":"Limited-magnitude error balls tile Z^n only in low dimensions","feed_subtitle":"For n≥4 the ball B(n,2,3,0) never tiles; the general exclusion threshold is quadratic in k1−k2.","key_machinery":"The machine is the group-ring decomposition $G=e+\\sum_{i\\in[-k_2,k_1]^*} T^{(i)} + \\sum_{i\\le j} S(i,j)$ together with the counting functions $\\psi(m,i,j)=|\\{t\\in T:t^m\\in S(i,j)\\}|$ and their one-coordinate analogues. The pivotal component is Lemma 2.9, which bounds the maximum multiplicity $C$ of the power map $t\\mapsto t^{k_1+k_2+1}$ on the generator set $T$ by $3\\sqrt{p}$ (or $3$ when $p=2$, and $4$ when $k_1+k_2=3$). The proof embeds the relevant $T$-powers into a maximal elementary abelian $p$-subgroup $H$ and counts inside one coset of $H$, using the order bound $|H|\\le p^2$. This $C$ bound converts into uniform $O(n)$ bounds on all set intersections, turning the tiling condition into a quadratic inequality in $n$ that cannot hold once $n\\ge \\lfloor B/A\\rfloor+1$.","core_discovery":"The central discovery is a structural constraint on any finite abelian group $G$ and $n$-subset $T$ that realizes a tiling: every element of $G$ must be uniquely the identity, a single power $t^i$ with $i\\in[-k_2,k_1]^*$, or a product of two such powers from distinct elements of $T$. The paper expresses $|G|$ as a sum of the sizes of these sets $T^{(i)}$ and $S(i,j)=\\{g^ih^j:g,h\\in T,\\,g\\neq h\\}$, and bounds the intersections among these sets using the scarcity of collisions of the map $g\\mapsto g^{k_1+k_2+1}$. When $k_1+k_2+1$ is composite with smallest prime divisor $p$, a key lemma caps the maximum collision multiplicity by $3\\sqrt{p}$, and the resulting inclusion-exclusion lower bound on the union of $S$-sets exceeds $|G|$ for large $n$, forcing non-existence above an explicit threshold. The two complete classifications follow by refining the index sets and checking the finite range below the threshold.","pith_inferences":["The composite condition is likely an artifact of the proof technique: a natural conjecture is that for every $k_1>k_2\\ge 0$ with $k_1+k_2\\ge 3$ there are only finitely many $n$ admitting a lattice tiling, with prime values of $k_1+k_2+1$ needing a different collision argument.","The explicit three-dimensional tiling of $\\mathcal{B}(n,2,3,0)$ hints that small sporadic tilings exist precisely when the group order is small; testing dimensions near $2(k_1-k_2)^2$ for small $k_1,k_2$ could reveal further isolated examples.","The bound $C\\le 3\\sqrt{p}$ is loose; a sharper collision bound would improve inequality (9) and could shrink the thresholds substantially, possibly covering cases currently outside the theorem's reach.","The same group-ring skeleton with two-coordinate index sets should carry over to $t\\ge 3$ by replacing pairs with $t$-tuples, with the collision map remaining $g\\mapsto g^{k_1+k_2+1}$; the authors state this extension as planned future work."],"forward_implications":["For $k_1=3, k_2=0$, a lattice tiling of $\\mathbb{Z}^n$ exists if and only if $n=3$, realized explicitly by $T=\\{1,10,26\\}$ inside the cyclic group $\\mathbb{Z}_{37}$.","For every $k\\ge 2$ and every $n\\ge 3$, $\\mathbb{Z}^n$ admits no lattice tiling by $\\mathcal{B}(n,2,k,k-1)$.","Whenever $k_1+k_2+1$ is composite, no lattice tiling of $\\mathbb{Z}^n$ by $\\mathcal{B}(n,2,k_1,k_2)$ exists for $n\\ge 2(k_1-k_2)^2+12(k_1-k_2)+2k_2+8$.","The dimension threshold is quadratic in $k_1-k_2$, so for fixed $k_2$ the possible tiling dimensions grow quadratically with the asymmetry gap.","When $k_1+k_2+1$ is prime, the method does not apply if the group order is a $p$-power; the paper notes that for $k_1+k_2+1=5$ only $n=2$ was found by computation, leaving the prime case as the open frontier."],"supporting_citations":[{"why":"Supplies Theorem 2.1, the bridge that turns a lattice tiling of $\\mathbb{Z}^n$ into a bijective homomorphism onto a finite abelian group $G$ of order $|V|$, the foundational step for the entire group-ring analysis.","marker":"[6]"},{"why":"Predecessor result that no lattice tiling by $\\mathcal{B}(n,2,k_1,k_1-1)$ exists for sufficiently large $n$; Theorem 1.2 strengthens this to all $n\\ge 3$ and is the immediate comparison target.","marker":"[4]"},{"why":"Classified lattice tilings by $\\mathcal{B}(n,2,1,0)$ and $\\mathcal{B}(n,2,2,0)$, establishing the $t=2$ classification paradigm that the index-set counting method extends.","marker":"[19]"},{"why":"Proved existence of lattice tilings by $\\mathcal{B}(n,2,1,1)$, showing that $t=2$ tilings do occur and supplying the contrast for the non-existence theorems.","marker":"[28]"}],"fun_headline_variants":["Composite sums stop all large-n tilings by error balls","Tiling Z^n: k1=k2+1 fully classified, other cases banned","Error-ball tilings: complete verdict for B(n,2,3,0) in all n","No large-n tilings when k1+k2+1 is composite","Sharp nonexistence for error-ball tilings in high dimensions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything hinges on the claim that, whenever $k_1+k_2+1$ is composite with smallest prime divisor $p$, at most $3\\sqrt{p}$ elements of the generator set $T$ can have the same $(k_1+k_2+1)$-th power in the tiling group; the proof of this claim is the paper's most compressed step, and if it fails, inequality (9) collapses.","fun_headline_variants_meta":{"raw":{"variants":["Composite sums stop all large-n tilings by error balls","Tiling Z^n: k1=k2+1 fully classified, other cases banned","Error-ball tilings: complete verdict for B(n,2,3,0) in all n","No large-n tilings when k1+k2+1 is composite","Sharp nonexistence for error-ball tilings in high dimensions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000548,"raw_usage":{"total_tokens":2624,"prompt_tokens":954,"completion_tokens":1670,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":570,"completion_tokens_details":{"reasoning_tokens":1571}},"tokens_in":570,"tokens_out":1670,"duration_ms":13838,"temperature":1.0,"reasoning_tokens":1571,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T21:55:00.134250+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Re-run the computational search claimed for $\\mathcal{B}(n,2,3,0)$ at $n=4,5,6$; any tiling found would refute Theorem 1.1, and because the proof's inequality (3) together with Lemma 2.9 rules out $n\\ge 7$, the search is the decisive check. For the general theorem, compute the collision count $C=\\max_g|\\{h:g^{k_1+k_2+1}=h^{k_1+k_2+1}\\}|$ in a group whose elementary abelian $p$-core has order exceeding $p^2$; a value above $3\\sqrt{p}$ would invalidate Lemma 2.9(a) and with it inequality (9).","supporting_citations":[{"cited_title":"Wei and M","cited_arxiv_id":null,"evidence_quote":"Classified lattice tilings by $\\mathcal{B}(n,2,1,0)$ and $\\mathcal{B}(n,2,2,0)$, establishing the $t=2$ classification paradigm that the index-set counting method extends."}],"review_version":1}