{"id":"346b2df3-a918-497a-8648-d6f03ee7ef62","arxiv_id":"1908.05681","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For finite posets, compatibility and cancellation rules for the strong Hom-, G-, and I-scheme order relations are proved under sums, ordinal sums, and products, with several cases left open.","lead":"This paper studies when one finite poset has no more order-preserving maps into it than another, no matter which finite poset the maps come from. It proves rules for combining posets by direct sums, ordinal sums, and products, and gives conditions under which a common factor can be cancelled.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4's final step is under-justified: the proof does not show that the pointwise exponent n(αξ(x)) is constant, so the assembled τ is a patchwork and the required α-equality may fail.","rationale":"The paper's headline contribution is the unconditional direct-sum cancellation for the strongest notion, strong I-schemes (Theorem 4). The reader's weakest-assumption concerns an unpublished lemma in [3] used for Theorem 5 (ordinal-sum cancellation for G-schemes). That concern is legitimate, but it does not bear on the paper's strongest claim. The most load-bearing issue for Theorem 4 is the unproven constancy of n(αξ(x)). Without it, the final equality required by Theorem 1 is not derived. The gap is likely repairable: the order-preservation of α_{P,ξ} is a natural fact, and for connected P it would imply the needed constancy. It is nevertheless absent from the manuscript, so the proof as written is incomplete. I also note the reader's separate observation about Proposition 6: the claimed implication (14) requires τ_P(ξ) to be a homomorphism, which needs ρ_Q to be order-preserving; Definition 4 gives only injectivity. That is a real issue in a calculation rule, but it does not affect the cancellation theorems. Since the proof gap in Theorem 4 is substantial but not evidently fatal, the appropriate verdict remains CONDITIONAL, so the reader's verdict need not change.","tokens_in":17679,"tokens_out":22915,"duration_ms":192628,"concrete_test":"Prove or disprove the missing constancy lemma: for every connected P∈Pr and ξ∈H(P,R), the integer n(α_{P,ξ}(x)) is independent of x. The natural analytical route is to show that α_{P,ξ}:P→E(R) is order-preserving, which makes αξ(P) connected; the proof of Theorem 4 already shows n is constant on connectivity components of E(R), so constantness would follow. As a computational spot-check, take R,S to be small chains, Q a singleton, realize Q+R⊑_I Q+S via an embedding σ, build the corresponding ϵ from Theorem 1, and verify for every ξ∈H(P,R) that the pointwise-defined τ satisfies α_{τ(ξ)}(x)=E(αξ(x)) for all x. If a counterexample with non-constant n appears, Theorem 4's conclusion is at risk; if the check passes, the gap is fillable by adding the missing lemma.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The strongest claimed result, Theorem 4 (direct-sum cancellation for I-schemes), has a gap in its final step. After constructing E:E(R)→E(S) by iterating a one-to-one homomorphism ϵ until the iterates land in E(S), the proof defines, for ξ∈H(P,R), values η^{n(x)}(ξ)(x), where n(x)=n(αξ(x)) and x∈P. The induction shows only that, for each x, E(αξ(x)) = α_{η^{n(x)}(ξ)}(x). To apply Theorem 1 one must show α_{τ(ξ)}(x)=E(αξ(x)) for the pointwise-defined τ(ξ)(x)=E(αξ(x))_1. This requires G_{τ(ξ)}(x)=G_ξ(x) and τ(ξ)(y)=η^{n(x)}(ξ)(y) for all y in ↓°G_ξ(x)∪↑°G_ξ(x); but n(y) may differ from n(x). The paper never proves that n is constant on the α-image of each connected component of P. Without such a constancy lemma, τ is a patchwork of different iterates of η, and the equality α_{τ(ξ)}(x)=E(αξ(x)) is not established. The set-theoretic Lemma 5 and the construction of E are sound; the missing piece is order-theoretic: α_{P,ξ}:P→E(R) must be order-preserving, so that αξ(P) is connected and n is constant on each component. The paper does not state or prove this.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies three increasingly restrictive notions of strong Hom-schemes between finite posets—plain, G, and I—and asks whether the associated preorders R⊑S, R⊑G S, and R⊑I S are compatible with direct sums, ordinal sums, products, and duality, and whether cancellation rules hold. It proves duality and monotonicity results, a direct-sum cancellation rule for all three notions, ordinal-sum cancellation rules for G- and I-schemes under additional hypotheses, and product cancellation rules under additional hypotheses. It also explicitly states that a cancellation rule for Q⊕R⊑Q⊕S was not obtained. The main technical devices are the EV-system E(P), the maps α_{P,ξ}, and iterative applications of Hom-schemes to push embeddings from R into Q⊙R to embeddings from S into Q⊙S.","tokens_in":18162,"tokens_out":13626,"duration_ms":124066,"significance":"The intended results are nontrivial and, if correct, would provide a useful toolbox for comparing finite posets via homomorphism counts under regularity conditions. The set-theoretic lemmas (Lemma 5 and Lemma 8) are clean, and the iterative constructions in Sections 8 and 9 are inventive. The paper is also commendably explicit about the one cancellation rule it could not prove. However, the current version contains a false claim in Proposition 6, an unproved external dependence in Theorem 5, and an incomplete final step in Theorem 4; these issues need to be resolved before the results can be accepted as stated.","major_comments":[{"comment":"The final step of the proof of Theorem 4 is incomplete. After constructing E:E(R)→E(S), the proof defines, for each x∈P, the value η^{n(αξ(x))}(ξ)(x) and asserts that E fulfills the requirement of Theorem 1. But Theorem 1 requires a single homomorphism η(ξ) with α_{η(ξ)}(x)=E(αξ(x)) for every x. If n(αξ(x)) varies with x, the pointwise definition τ(ξ)(x)=E(αξ(x))_1 is a patchwork of different iterates of η, and the equality α_{τ(ξ)}(x)=E(αξ(x)) is not established. The gap can be repaired: since α_{P,ξ} is stated in Section 4 to be a homomorphism, αξ(K) is connected in E(R) for each connectivity component K of P, and the argument used to show that E is order-preserving implies that n is constant on each connectivity component of E(R). This constancy argument should be written out explicitly; without it, the proof as printed is not complete.","section":"Section 7, Theorem 4"},{"comment":"The proof of R⊑S ⇒ H(Q,R)⊑G H(Q,S) defines τ_P(ξ)=ρ_Q∘ξ. For this to be a well-defined element of H(P,H(Q,S)), the map ρ_Q:H(Q,R)→H(Q,S) must preserve the pointwise order on homomorphism sets: if ξ(x)≤ξ(y) in H(Q,R), then ρ_Q(ξ(x))≤ρ_Q(ξ(y)) in H(Q,S). Strong Hom-schemes as defined in Definition 4 are only required to be injective componentwise; no monotonicity with respect to the pointwise order is imposed. Thus the composition ρ_Q∘ξ need not be a homomorphism, and the statement (14) is not justified and appears false under the paper's definitions. This also affects the equivalence (16). Please either add an order-preservation hypothesis to the definition or to the statement, or withdraw (14) and repair the consequences for (16).","section":"Section 5, Proposition 6(14)"},{"comment":"The proof of the ordinal-sum cancellation rule for G-schemes depends entirely on [3, Lemma 4], quoted as 'As proven in [3, Lemma 4]', which asserts that R⊑G S is equivalent to #Γ_{P,R}(ξ)≤#Γ_{P,S}(ξ) for every P and every ξ∈H(P,R). This lemma is not stated or proved in the present paper, and reference [3] is listed as 'in preparation'. Since the counting argument in Theorem 5 (and part of the surrounding theory) rests on this characterization, the proof is not self-contained. The authors should include a complete statement and proof of this lemma, or clearly delimit Theorem 5 as conditional on an external result that is not yet available.","section":"Section 8, Theorem 5"}],"minor_comments":[{"comment":"There are numerous typographical errors and stylistic slips, including 'poests', 'containes', 'homomorphimss', and 'be setting' instead of 'by setting'. A careful proofreading pass is needed.","section":"Throughout"},{"comment":"Reference [3] is cited as 'in preparation' and is used for Theorem 2 and Lemma 4. The reader cannot verify these results; please update the reference status or include the needed statements in the paper.","section":"Section 1 and Section 4"},{"comment":"The notation ⊙∈{+,⊕,×} is used in the introductory paragraphs before it is formally introduced; please define all operators at first use.","section":"Section 6"}],"recommendation":"major_revision","confidential_remarks":"The reliance on a companion paper 'in preparation' for a load-bearing lemma is a significant concern for a journal submission; even if the companion eventually appears, the present manuscript should contain enough information to make the main proofs checkable. In addition, Proposition 6(14) appears to be a genuine error rather than a mere gap, since strong Hom-schemes are not defined to be order-preserving. If the author can remove or correct that proposition and patch Theorem 4 with the constancy argument, the cancellation theorems may still be salvageable. I would not recommend rejection solely on the basis of the unproved external lemma, but the current version needs substantial revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the cancellation theorems (Theorems 3–9) are genuine extensions of a Campo's own strong Hom-scheme framework, and the unconditional direct-sum cancellation for I-schemes in Theorem 4 is the pick of the lot. The paper is also honest about what it could not prove, including the missing ordinal-sum cancellation for plain Hom-schemes.\n\nThe stress-test note on Theorem 4 points at a missing justification, not a fatal hole. The proof needs to say explicitly that α_{P,ξ}:P→E(R) is order-preserving, so the exponent n(αξ(x)) is constant on each connected component of P and the pointwise iterate τ(ξ)(x)=η^{n(x)}(ξ)(x) is in fact a well-defined homomorphism with the right α-values. That fact is elementary from the definition of <+, and once written down the patchwork worry disappears. I would call it a one-paragraph fix.\n\nThe real soft spot is Proposition 6. The proof defines τ_P(ξ)=ρ_Q∘ξ and asserts this lands in H(P,H(Q,S)). For that, ρ_Q must preserve the pointwise order on H(Q,R). Definition 4 only requires ρ_P to be one-to-one; order-preservation is not part of a strong Hom-scheme. As written, the proposition does not follow, and the equivalence (16) leans on it. This is a substantive flaw in the calculation-rule section, even though it does not affect the main cancellation theorems.\n\nThe second obstacle is the dependence on [3, Lemma 4] and Theorem 2, both quoted from an unpublished manuscript “in preparation.” Theorem 5 and part of Proposition 4 rest on that lemma. A referee cannot verify the foundation without seeing [3], so this is more than a stylistic self-citation issue.\n\nWho is this for? Researchers working on poset homomorphism inequalities and on the author's strong Hom-scheme machinery. The cancellation rules are new relative to the cited literature and the technical core is coherent, but the paper is not fully verified as it stands. I would send it to a serious referee, with instructions to focus on Proposition 6 and on the [3] dependency. If the author supplies the missing order-preservation argument and either posts [3] or proves the needed lemma, the paper should be publishable after revision.","headline":"The cancellation theorems are real extensions of a Campo's own strong Hom-scheme program, with Theorem 4 (direct-sum cancellation for I-schemes) as the highlight; but Proposition 6 has a genuine well-definedness gap and the paper leans on an unpublished companion for a load-bearing lemma.","tokens_in":18532,"tokens_out":13216,"would_cite":false,"duration_ms":127758,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["06A07","06A06"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that, for finite posets, the strongest Hom-scheme relation cancels a common direct summand: comparability of Q+R and Q+S forces comparability of R and S.","keywords":["posets","homomorphism sets","strong Hom-schemes","G-schemes","I-schemes","cancellation rules","order arithmetic","EV-systems"],"falsifier":"Generate finite posets up to a few elements and, for every certified pair $Q+R \\sqsubseteq_I Q+S$, test whether the iterated action of the EV-homomorphism $\\epsilon$ on $\\mathcal{E}(R)$ lands in $\\mathcal{E}(S)$; Lemma 5 says it must, so any certified pair where the induced map $\\mathcal{E}(R) \\to \\mathcal{E}(S)$ is not one-to-one refutes Theorem 4. For the unproved ordinal-sum case, a single finite triple $Q,R,S$ with $Q\\oplus R \\sqsubseteq Q\\oplus S$ but $\\#\\mathcal{H}(P,R) > \\#\\mathcal{H}(P,S)$ for some $P$ would settle it.","tokens_in":17467,"feed_emoji":"🧮","tokens_out":11419,"duration_ms":98933,"temperature":0.7,"pith_summary":"This paper studies when a pointwise inequality between hom-sets, $\\#\\mathcal{H}(P,R) \\le \\#\\mathcal{H}(P,S)$, is compatible with summing, stacking, or multiplying posets, and when a common factor can be cancelled. The central unconditional result is the direct-sum cancellation rule: for finite posets $Q,R,S$, if there is a strong I-scheme from $Q+R$ to $Q+S$—the most tightly controlled of the three Hom-scheme notions—then a strong I-scheme from $R$ to $S$ already exists. The same cancellation works for ordinary strong Hom-schemes and for G-schemes, and the paper proves several calculation rules: the three relations are preserved under duality and direct sums, and the weaker two are preserved under ordinal sums and products. It also establishes ordinal-sum and product cancellation for G- and I-schemes under additional side conditions, while one case, $Q\\oplus R \\sqsubseteq Q\\oplus S$, is left open. A sympathetic reader would care because these rules turn a global counting relation between all homomorphism sets into a local, structural comparison between the original posets.","feed_headline":"Direct sums cancel in all three strong Hom-schemes","feed_subtitle":"If Q+R and Q+S have comparable homomorphism sets, then R and S already do—under the strongest regularity condition.","key_machinery":"The central object is the strong Hom-scheme: a choice, for every finite poset $P$, of a one-to-one map $\\rho_P$ from $\\mathcal{H}(P,R)$ to $\\mathcal{H}(P,S)$, with two refined versions. A G-scheme requires the connectivity classes $G_{\\rho(\\xi)}(x)$ to equal $G_\\xi(x)$ for every $x$; an I-scheme requires preservation of the order on the exploded-view system $\\mathcal{E}(R)$, built from triples $(x,\\downarrow^\\circ x,\\uparrow^\\circ x)$, and Theorem 1 from an earlier paper reduces I-schemes to one-to-one homomorphisms $\\epsilon : \\mathcal{E}(R) \\to \\mathcal{E}(S)$. The cancellation proofs push such an embedding of $\\mathcal{H}(P,R)$ (or $\\mathcal{E}(R)$) into the composite poset through the given scheme, and use two set-theoretic iteration lemmas: Lemma 5, which says an injective map on a finite set $A\\cup B$ into $A\\cup C$ must eventually send every $b\\in B$ into $C$, and Lemma 8, an analogous return-to-starting-point lemma for product constructions. These iterations are what turn membership in the larger hom-set into membership in the smaller one.","core_discovery":"The paper's core claim is Theorem 4: for finite posets $Q,R,S$ with pairwise disjoint carriers, $Q+R \\sqsubseteq_I Q+S$ implies $R \\sqsubseteq_I S$. Here $\\sqsubseteq_I$ is the relation \"there exists a strong image-controlled Hom-scheme\", meaning there is a one-to-one family of maps $\\mathcal{H}(P,R) \\to \\mathcal{H}(P,S)$ for every finite $P$ that additionally preserves the exploded-view (EV) order between homomorphisms. The same cancellation is proved for the weaker relations $\\sqsubseteq$ and $\\sqsubseteq_G$. For ordinal sums and products the situation is conditional: $Q\\oplus R \\sqsubseteq_G Q\\oplus S$ implies $R \\sqsubseteq_G S$, and $Q\\times R \\sqsubseteq_G Q\\times S$ implies $R \\sqsubseteq_G S$ provided the scheme satisfies a constancy-preservation condition, with analogous but stronger side conditions for I-schemes; the plain Hom-scheme case $Q\\oplus R \\sqsubseteq Q\\oplus S$ remains unproved.","pith_inferences":["Not pursued in the paper, but direct-sum cancellation suggests the finite-poset monoid under $+$ is cancellative for $\\sqsubseteq_I$; if so, $\\sqsubseteq_I$ could be studied componentwise on connected posets, simplifying any computational check of the relation.","The open case $Q\\oplus R \\sqsubseteq Q\\oplus S$ may well be false in general; the proof gap suggests looking for counterexamples where $R$ and $S$ differ only in strict-homomorphism counts, since the ordinal sum with a suitably tall $Q$ can hide those differences.","The same set-theoretic iteration used here would apply to other categories of finite structures with an \"exploded view\" construction, so a natural testable extension is to ask whether analogous cancellation rules hold for graphs or digraphs with defined neighbourhood systems."],"forward_implications":["Direct-sum cancellation is unconditional for all three relations: a common direct-summand $Q$ can be removed from both sides of $R \\preceq S$ for $\\preceq \\in \\{\\sqsubseteq, \\sqsubseteq_G, \\sqsubseteq_I\\}$.","The calculation rules make the three relations compatible with duality: $R^d \\preceq S^d$ whenever $R \\preceq S$, and with direct sums: $R_1+R_2 \\preceq S_1+S_2$ whenever $R_j \\preceq S_j$, in all three variants.","For ordinary Hom-schemes and G-schemes, ordinal sums and products are monotone: $R_1\\oplus R_2 \\preceq S_1\\oplus S_2$ and $R_1\\times R_2 \\preceq S_1\\times S_2$ follow from $R_j \\preceq S_j$.","The cancellation results turn a one-sided global inequality $Q\\odot R \\preceq Q\\odot S$ into the structural conclusion $R \\preceq S$: unconditionally for $\\odot = +$, and under the stated side conditions for $\\odot = \\oplus$ and $\\odot = \\times$.","The equivalence (16) says $H(Q,R) \\sqsubseteq H(Q,S)$ for every finite $Q$ is exactly equivalent to $R \\sqsubseteq S$, so the family of hom-posets carries no extra information beyond the original relation."],"supporting_citations":[{"why":"Supplies the classical theorem that operations with structures control homomorphism-set sizes, the starting point for the whole comparison relation.","marker":"[4]"},{"why":"Provides the earlier observation cited as [1, Theorem 5] that underpins the counting viewpoint on homomorphism sets.","marker":"[1]"},{"why":"Contains the theory of strong Hom-schemes, G-schemes, and I-schemes—definitions, theorems including Theorem 1, and the extension result used throughout.","marker":"[2]"},{"why":"Provides the characterization of $R \\sqsubseteq_G S$ by the fiber-counting inequality (18), quoted as [3, Lemma 4] and load-bearing for Theorem 5.","marker":"[3]"}],"fun_headline_variants":["Direct sums cancel in all strong Hom-schemes","Strong Hom-schemes: direct sums cancel","Direct-sum cancellation proved for strong Hom-schemes","If Q+R embeds, R embeds: cancellation for direct sums"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The ordinal-sum cancellation rule for G-schemes rests on a characterization of $R \\sqsubseteq_G S$ by a fiber-counting inequality that is quoted from a preprint listed as 'in preparation' and is not proved in this paper; if that characterization fails, Theorem 5 loses its foundation.","fun_headline_variants_meta":{"raw":{"variants":["Direct sums cancel in all strong Hom-schemes","Strong Hom-schemes: direct sums cancel","Direct-sum cancellation proved for strong Hom-schemes","If Q+R embeds, R embeds: cancellation for direct sums"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000763,"raw_usage":{"total_tokens":3350,"prompt_tokens":871,"completion_tokens":2479,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":487,"completion_tokens_details":{"reasoning_tokens":2413}},"tokens_in":487,"tokens_out":2479,"duration_ms":18927,"temperature":1.0,"reasoning_tokens":2413,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:10:57.209005+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Generate finite posets up to a few elements and, for every certified pair $Q+R \\sqsubseteq_I Q+S$, test whether the iterated action of the EV-homomorphism $\\epsilon$ on $\\mathcal{E}(R)$ lands in $\\mathcal{E}(S)$; Lemma 5 says it must, so any certified pair where the induced map $\\mathcal{E}(R) \\to \\mathcal{E}(S)$ is not one-to-one refutes Theorem 4. For the unproved ordinal-sum case, a single finite triple $Q,R,S$ with $Q\\oplus R \\sqsubseteq Q\\oplus S$ but $\\#\\mathcal{H}(P,R) > \\#\\mathcal{H}(P,S)$ for some $P$ would settle it.","supporting_citations":[{"cited_title":"Lov´ asz: Operations with structures","cited_arxiv_id":null,"evidence_quote":"Supplies the classical theorem that operations with structures control homomorphism-set sizes, the starting point for the whole comparison relation."},{"cited_title":"a Campo: Strong G-schemes and strict homomorphisms","cited_arxiv_id":null,"evidence_quote":"Provides the characterization of $R \\sqsubseteq_G S$ by the fiber-counting inequality (18), quoted as [3, Lemma 4] and load-bearing for Theorem 5."}],"review_version":1}