{"id":"85f5db73-d192-495d-9e75-d1bf99332bb9","arxiv_id":"1908.00749","paper_version":2,"verdict":"CONDITIONAL","confidence":"LOW","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For any finite semimodular lattice L, this paper provides an algorithm that constructs all minimal covering-preserving embeddings of L into geometric lattices, and proves these embeddings have exactly |J(L)| atoms and the same length as L.","lead":"This mathematics paper finds all geometric lattices (special structures used in matroid theory) that contain a given semimodular lattice while preserving its covering relations and using the fewest possible elements. It gives an algorithm to list these minimal embeddings and proves that each uses exactly as many atoms as the original lattice's join-irreducible elements.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 4.3's unproved Fig.8 sublattice assertion is the load-bearing gap: cover-preservation of the algorithm's outputs, and hence the main classification, rests on it.","rationale":"The reader's weakest assumption identifies the same unproved structural assertion in Lemma 4.3 that I consider the most load-bearing gap. The paper is attempting to solve a genuine open problem with a plausible construction: the algorithm, the set-of-sets representation, the matroid-deletion flavor of Lemma 5.1, and the worked examples all suggest the central claim may be true. However, Lemma 4.3's cover-preservation proof depends on an unproved and nontrivial sublattice/independent-set fact. If that fact fails, the algorithm's outputs are not known to contain L cover-preservingly, so the completeness theorem and the characterization of best geometric lattices do not follow. The reader's verdict of CONDITIONAL with LOW confidence is appropriate: the paper should not be accepted unconditionally until the Fig.8 assertion is either proved from semimodularity or replaced by a rigorous argument. I do not see grounds, from the text alone, to move to REJECT, since the gap is a missing justification rather than an identified contradiction.","tokens_in":21619,"tokens_out":38787,"duration_ms":379690,"concrete_test":"Independently re-derive the Fig.8 step in Lemma 4.3 without citing the figure: for a finite semimodular lattice T_L^P and incomparable M,N with join Z, prove from the Jordan-Hölder chain property that there are chains N=N_0 ≺ ... ≺ N_k ≺ Z and M_0 ≺ ... ≺ M_k = M such that, in the atomistic lattice S_P, every maximal independent set η of N extends to an independent set η ∪ ρ of N_k with ρ ⊆ A_SP(M). If this step cannot be derived, the lemma is unsupported. As a complementary finite check, enumerate all semimodular lattices up to 8 elements with a lattice search and verify the exact extension property for all incomparable pairs; a counterexample would disprove Lemma 4.3.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central classification (Lemma 4.5 and Theorem 5.1) requires that every output Q of Algorithm 3.1 admits a cover-preserving embedding of L. That is exactly Lemma 4.3. Its proof contains an unproved structural assertion after showing M ∥ N: 'As T_L^P is a finite semimodular lattice, we know that T_L^P contains a sublattice lattice as presented in Fig.8.' No proof or citation is supplied. The subsequent argument uses Fig.8 in two nontrivial ways: to obtain the length identities ℓ(N_k)=t+k and ℓ(Z)=t+k+1, and to extend every independent set η ∈ I_SP(N) to a set π = η ∪ ρ ∈ I_SP(N_k) with ρ ⊆ A_SP(M). Neither follows from abstract semimodularity alone. Finite semimodular lattices are graded, but gradedness does not by itself produce the specific cover-chain structure of Fig.8, and the independent-set extension property is being asserted for S_P, an atomistic lattice that is not known to be geometric. If the Fig.8 assertion fails, the proof that T_L^P is a ∨-subsemilattice of Q collapses; then Q need not cover-preserve L, and Lemmas 4.5 and 5.1 lose their footing. This is the same gap the reader identified, and it is genuinely load-bearing.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the 2010 question of Czédli and Schmidt on the \"best\" cover-preserving embedding of a finite semimodular lattice L into a geometric lattice G. It introduces extending standard forms of L and an algorithm, Algorithm 3.1, intended to construct all finite geometric lattices Q that contain L as a cover-preserving sublattice with the same length. The main results are Lemma 4.5, which asserts that every finite geometric lattice Q with L as a cover-preserving sublattice and with ℓ(Q)=ℓ(L) is an output of the algorithm; Lemma 5.1, which asserts that every output with more than |J(L)| atoms can be strictly shrunk; and Theorem 5.1, which concludes that every best extending cover-preserving geometric lattice of L has exactly |J(L)| atoms and length ℓ(L). The paper also presents worked examples, in particular Examples 4.1 and 5.1, illustrating the algorithm and the claimed best lattices.","tokens_in":21930,"tokens_out":31549,"duration_ms":334056,"significance":"If the proofs were completed, the paper would resolve the Czédli-Schmidt question by providing a finite classification of the minimal geometric over-lattices and a constructive algorithm for obtaining them. The claimed equality |A(G)|=|J(L)| for every best extending geometric lattice is a clean and natural structural statement that would strengthen the existence theorem of Czédli and Schmidt. The manuscript is ambitious and contains instructive examples; the algorithm is described in full detail. However, the paper does not provide machine-checked proofs or reproducibility artifacts, and the central proof currently has gaps that are load-bearing for the classification. The significance is therefore conditional on completing the proof of Lemma 4.3 and formalizing the induction in Lemma 4.5.","major_comments":[{"comment":"The proof of Lemma 4.3 contains the unproved assertion that if M and N are incomparable elements of the finite semimodular lattice T_L^P, then T_L^P contains the sublattice presented in Fig. 8. No proof or citation is supplied, and this is not a direct consequence of semimodularity alone: finite semimodular lattices are graded, but gradedness does not by itself produce the displayed cover-chain structure. The subsequent length identities ℓ(N_k)=t+k and ℓ(Z)=t+k+1 in Eq. (4) depend on that structure. The later step \"there exists a subset ρ of A_SP(M) such that η∪ρ=π∈I_SP(N_k)\" is also asserted without justification; S_P is only known to be atomistic, so Lemma 2.4, which is stated for geometric lattices, does not apply, and the extension of an independent set is not automatic. Since Lemma 4.3 is exactly what ensures that every output of Algorithm 3.1 has L as a cover-preserving sublattice, this gap is load-bearing for Lemmas 4.4, 4.5, 5.1, and Theorem 5.1.","section":"§4, Lemma 4.3 (Fig. 8)"},{"comment":"The completeness proof in Lemma 4.5 is compressed into \"Repeating the process as above\" and \"Continuing as above.\" The induction must show that, at every stage, the newly added elements U (and U_1, W) satisfy conditions (i1)–(i3) or (j1)–(j3) with respect to the current set R_i, not merely with respect to Q; the manuscript verifies this explicitly only for the first addition from R_1=S_P. It also must prove that the sets R_i∪⋃{[0,M]_Q : ...} remain k-order normal subset lattices of Q. Without an explicit induction invariant, statements such as \"R_3 is a 4 order normal subset lattice of Q\" and the final conclusion Q=R_{m-1} are not established. This direction is essential because Lemma 4.5 is what converts the algorithm into a classification of all candidate geometric lattices.","section":"§4, Lemma 4.5 (pp. 19–21)"},{"comment":"In Lemma 4.1, Step A, after proving ℓ_Q(E∩X)≤t−1 the text asserts: \"there exists an element K∈ϕ_Q(t−1) such that E∩X⊆K, or E∩X∈A(Q).\" This assertion needs an argument: in a poset that is not yet known to be graded, an element of length at most t−1 need not lie below an element of exact length t−1. If this step fails, the application of condition (i1) and the conclusion E∧_R M_{t−1}=E∩M_{t−1} do not follow. Since Lemma 4.1 is the proof that Algorithm 3.1 outputs a lattice, this issue is also load-bearing; the authors should either prove gradedness of the intermediate structures or revise the argument.","section":"§4, Lemma 4.1 (Step A)"}],"minor_comments":[{"comment":"The notation for cover-preserving sublattice embedding, rendered as \"֒→≺\", is introduced inline; it would be clearer to define it in a displayed form or in the notation list.","section":"Throughout"},{"comment":"There is a typo: \"a sublattice lattice\" should read \"a sublattice\".","section":"§4, Lemma 4.3"},{"comment":"The interval notation [∅, M] is used for [0, M], which is inconsistent with the bottom element 0 used elsewhere in the paper.","section":"§4, Lemma 4.4"},{"comment":"\"a 2 order normal subset lattice of S\" should read \"of Q\".","section":"§4, Lemma 4.5, Case 2"},{"comment":"Reference [5] contains a typo: \"semimordular\" should be \"semimodular\".","section":"References"},{"comment":"The statement that Q is the unique best extending cover-preserving geometric lattice \"in the sense of isomorphism\" is not proved; it should either be proved or labeled as an observation based on the algorithm run.","section":"Example 5.1"}],"recommendation":"major_revision","confidential_remarks":"The main concern is the unproved structural assertion in Lemma 4.3. If the authors can supply a correct proof of that lemma, including the independent-set extension step, the paper may be suitable for publication; otherwise the classification theorem is not supported. I would also ask them to formalize the induction in Lemma 4.5 rather than using informal repetition. The paper is within the scope of a lattice theory journal, and the claimed result is significant, but the proofs are not yet complete."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here is my read. The paper takes the Czédli-Schmidt open question seriously and makes a genuine attempt: it gives an algorithm (Algorithm 3.1) that is supposed to enumerate every cover-preserving geometric over-lattice of a finite semimodular lattice with the same length, and it extracts the consequences |A(G)|=|J(L)| and ℓ(G)=ℓ(L) for the minimal ones. The idea of shrinking a larger geometric over-lattice by deleting an inserted atom (Lemma 5.1) is appealing, and the worked examples make the intended construction concrete. That is real work and worth engaging with.\n\nThe soft spots are not cosmetic. Lemma 4.3 is the load-bearing step: it must show that every output Q of the algorithm actually contains L cover-preservingly. The proof asserts that if M and N are incomparable in the finite semimodular lattice T_L^P, then T_L^P contains the sublattice of Fig.8. No proof or citation is given. This is not a minor omission; the statement is false as a general semimodularity fact. In the pentagon N5, take the two incomparable elements M=a and N=b. Their join is 1, with ℓ(b)=1 and ℓ(1)=3. No chain b<N_1<...<N_k<1 can produce the length identities ℓ(N_k)=1+k and ℓ(1)=2+k; there is no element between b and 1. So the configuration in Fig.8 simply does not exist. The subsequent extension of independent sets η∪ρ to N_k depends on that configuration, so the cover-preservation conclusion loses its footing. The stress-test note lands correctly.\n\nThere is a second, smaller gap: Lemma 4.5, the completeness direction, is a proof by repeated construction with phrases like \"Repeating the process as above\" and \"Continuing as above\". That may be fillable, but it is not a formal induction as written. Also Lemma 3.1 asserts that every extending standard form is an atomistic lattice; since E(L) is defined without a lattice requirement, that needs an argument too.\n\nWhat does this add up to? The paper identifies a real open question and has a plausible strategy, and the examples check out. But the main theorem depends on a false structural assertion, so the classification is not proven as it stands. I would not cite the result in the next year. I would still send it to a serious referee, because the problem is important and a repaired version could be a good paper. The referee should be asked to focus on Lemma 4.3 first.\n\nBring it to reading group? Maybe, as a case study in where semimodularity arguments go wrong.","headline":"Targets a real open problem and has a plausible algorithm, but Lemma 4.3's unproved Fig.8 assertion is false for N5; the classification is not established.","tokens_in":22415,"tokens_out":9725,"would_cite":false,"duration_ms":101921,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["06C10","06B15"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper answers the 2010 question of the smallest geometric lattices extending a semimodular lattice, giving an algorithm that produces them all and proving each has exactly $|J(L)|$ atoms and length $\\ell(L)$.","keywords":["finite atomistic lattice","semimodular lattice","geometric lattice","cover-preserving embedding","best extending cover-preserving geometric lattice","join-irreducible elements","atoms","lattice algorithm"],"falsifier":"Inspect finite semimodular lattices for a pair of incomparable elements $M,N$ with $M$ and $N$ both covering $M\\cap N$ whose interval $[M\\cap N,M\\vee N]$ lacks the matched covering chains of Fig. 8; if such a pair exists, Lemma 4.3's cover-preserving conclusion can fail for a suitable extending standard form, and the claimed classification would not hold as stated.","tokens_in":21403,"feed_emoji":"📐","tokens_out":10467,"duration_ms":101932,"temperature":0.7,"pith_summary":"This paper addresses a question left open by the 2010 embedding theorem for semimodular lattices: given a finite semimodular lattice $L$, which geometric lattices $G$ contain $L$ as a cover-preserving sublattice and have the smallest possible size? The authors introduce a construction, Algorithm 3.1, that starts from any extending standard form of $L$ and outputs all geometric lattices in which $L$ sits cover-preservingly with the same length. Their main theorem states that every best extending cover-preserving geometric lattice of $L$ is the best output of this algorithm, and that each such lattice has exactly $|J(L)|$ atoms and length $\\ell(L)$. This matters because it turns an existence result into a complete classification: the minimal geometric over-lattices of a finite semimodular lattice are finite in number and are governed by the join-irreducible elements of $L$.","feed_headline":"Minimal geometric over-lattices classified for all semimodular lattices","feed_subtitle":"An algorithm produces every smallest geometric super-lattice, each with |J(L)| atoms and original length.","key_machinery":"The machinery is the pair consisting of extending standard forms and Algorithm 3.1. Every finite lattice can be represented as a set system, a family of subsets of its atoms, and an extending standard form $P$ places new atoms below suitable non-atoms of $L$ so that $L$ becomes a cover-preserving sublattice of $P$ with $\\ell(P)=\\ell(L)$. The algorithm starts from the set-system copy $S_P$ of such a $P$ and repeatedly adds subsets $U$, and chains of subsets $W$, that pass three tests: $U$ is not contained in a previously present set of rank one less, $U$ does not contain the union of any independent set coming from the previous rank, and intersections with previous rank-one-less sets close inside $U$. These tests force the enlarged poset to remain closed under meets and to preserve joins inherited from $L$. The final condition (M), that adjoining any missing atom to a rank-$k$ element always raises the rank to $k+1$, guarantees semimodularity and hence that the output is geometric. Lemma 4.5 inverts the whole procedure, showing that any same-length geometric over-lattice must be obtainable in exactly this way.","core_discovery":"The central claim is Theorem 5.1: every best extending cover-preserving geometric lattice of $L$ is the best one in the family $\\mathcal{S}_{|J(L)|}$, where $\\mathcal{S}_k$ consists of the outputs of Algorithm 3.1 with $k$ atoms. The proof has two parts. Lemma 4.5 shows that every finite geometric lattice $G$ with $L$ as a cover-preserving sublattice and $\\ell(G)=\\ell(L)$ is an output of Algorithm 3.1; Lemma 5.1 shows that any such output with more than $|J(L)|$ atoms can be strictly shrunk by deleting an atom while remaining a geometric lattice that still contains $L$ cover-preservingly. Together these imply that a geometric extension is smallest exactly when it has $|J(L)|$ atoms, and that all smallest extensions have length $\\ell(L)$. Thus the paper's answer to the 2010 question is a full structural description rather than merely another existence proof.","pith_inferences":["This suggests the minimal geometric over-lattices of $L$ are exactly the flat completions of the atomistic poset $S_P$; if so, enumeration algorithms for matroid flats could be adapted to list them for moderate $|J(L)|$.","Because different choices of $U$ and $W$ inside Algorithm 3.1 yield different outputs, the number of best extensions is a computable invariant of $L$; comparing this count among lattices with the same $|J(L)|$ and $\\ell(L)$ might separate lattices that those two parameters cannot distinguish.","Condition (M) may be replaceable by a simpler exchange axiom, in which case the construction would translate directly into the language of single-element extensions of matroids, connecting the minimal over-lattice problem to matroid theory."],"forward_implications":["Every best extending cover-preserving geometric lattice of a finite semimodular lattice has exactly $|J(L)|$ atoms and length $\\ell(L)$, so the minimal size is determined by the join-irreducible elements of $L$.","Every finite geometric lattice that contains $L$ cover-preservingly and has the same length as $L$ is an output of Algorithm 3.1, so the algorithm lists the entire class of same-length geometric over-lattices.","Any same-length geometric extension with more than $|J(L)|$ atoms can be reduced by deleting one atom, so minimality is equivalent to having exactly $|J(L)|$ atoms.","The earlier existence result for cover-preserving embeddings into geometric lattices is sharpened from there being at least one such extension to a classification and construction of all minimal ones."],"supporting_citations":[{"why":"Poses the 2010 best-embedding question and supplies the existence theorem with |J(L)| = |A(G)| and equal lengths that the paper sharpens into a classification.","marker":"[4]"},{"why":"Supplies the algebraic background on semimodular lattices, the Birkhoff condition, and the maximal independent-set description of geometric lattices used in Definition 2.3 and Lemma 2.4.","marker":"[3]"},{"why":"Provides the earlier construction of a cover-preserving embedding of a finite semimodular lattice into a finite geometric lattice, which the present algorithm extends and makes exhaustive.","marker":"[5]"},{"why":"Gives the set-representation theorem for lattices that underlies the move from L to its extending standard forms and the set systems S_P and T_L^P.","marker":"[2]"}],"fun_headline_variants":["Smallest geometric over-lattices fully described","All minimal geometric extensions found for semimodular lattices","Algorithm yields all smallest cover-preserving geometric lattices","Optimal geometric extensions: length and atoms pinned down","Answer to 2010 question: minimal geometric covers classified"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof of Lemma 4.3 assumes, without proof or citation, that any two incomparable elements of a finite semimodular lattice whose meet they both cover force the specific sublattice pictured in Fig. 8; this is what makes the joins computed in the algorithm's output agree with joins in $L$.","fun_headline_variants_meta":{"raw":{"variants":["Smallest geometric over-lattices fully described","All minimal geometric extensions found for semimodular lattices","Algorithm yields all smallest cover-preserving geometric lattices","Optimal geometric extensions: length and atoms pinned down","Answer to 2010 question: minimal geometric covers classified"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000193,"raw_usage":{"total_tokens":1353,"prompt_tokens":952,"completion_tokens":401,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":568,"completion_tokens_details":{"reasoning_tokens":324}},"tokens_in":568,"tokens_out":401,"duration_ms":3934,"temperature":1.0,"reasoning_tokens":324,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T15:35:34.585643+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Inspect finite semimodular lattices for a pair of incomparable elements $M,N$ with $M$ and $N$ both covering $M\\cap N$ whose interval $[M\\cap N,M\\vee N]$ lacks the matched covering chains of Fig. 8; if such a pair exists, Lemma 4.3's cover-preserving conclusion can fail for a suitable extending standard form, and the claimed classification would not hold as stated.","supporting_citations":[{"cited_title":"Cz´ edli, E","cited_arxiv_id":null,"evidence_quote":"Poses the 2010 best-embedding question and supplies the existence theorem with |J(L)| = |A(G)| and equal lengths that the paper sharpens into a classification."},{"cited_title":"Crawley, R","cited_arxiv_id":null,"evidence_quote":"Supplies the algebraic background on semimodular lattices, the Birkhoff condition, and the maximal independent-set description of geometric lattices used in Definition 2.3 and Lemma 2.4."},{"cited_title":"Gr¨ atzer, E","cited_arxiv_id":null,"evidence_quote":"Provides the earlier construction of a cover-preserving embedding of a finite semimodular lattice into a finite geometric lattice, which the present algorithm extends and makes exhaustive."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the set-representation theorem for lattices that underlies the move from L to its extending standard forms and the set systems S_P and T_L^P."}],"review_version":1}