{"id":"ceb572a6-f494-4aee-b617-106163216c9c","arxiv_id":"2506.21227","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For aligned interior systems of a poset, induction preserves interval covers and interval resolutions of persistence modules, and interval resolution dimensions are preserved.","lead":"This paper proves that certain special subsets of partially ordered sets, called aligned interior systems, allow a contraction and induction pair of operations that preserve the structure of persistence modules and their resolutions. This gives mathematicians and data scientists a way to shrink a large poset to a smaller one without losing the key algebraic features used in multiparameter persistence analysis.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 6.5's s=2 case cites a nonexistent Proposition 5.2; as written, the two-sink tilde-A formula is unproved, though the central Section 4 preservation theorem is unaffected.","rationale":"I read Sections 3 and 4 carefully as the central claim. The adjunction Cont_Q ⊣ Ind_Q, the full faithfulness of Ind_Q, the F G ≃ id identity, and the aligned-interior-system description of Cont_Q all cohere. The right-minimality step in Theorem 4.10(a) is acceptable because Ind_Q is fully faithful and F G ≃ id, so the map α ↦ F(α) is an isomorphism on endomorphism spaces of objects in the image. The interval-decomposability argument in Proposition 4.4 also checks out: T_Q is convex, and its connected components give an interval decomposition. I therefore find no flaw in the central preservation theorem. The paper's advertised computational claims are less secure. The most concrete defect is in Proposition 6.5: the proof of the s=2 case invokes a nonexistent Proposition 5.2, leaving that case unsupported as written. This is a genuine, locatable gap, but it is confined to the downstream classification and does not change the reader's CONDITIONAL verdict on the paper. A direct computation on the smallest two-sink example would settle whether the gap is merely expository or hides a substantive error.","tokens_in":34761,"tokens_out":46132,"duration_ms":518064,"concrete_test":"Enumerate the interval resolutions of the smallest two-sink ~A poset (the diamond with sources t1,t3 and sinks t2,t4) over F_2, computing int-res-gl.dim directly by the definition, or via the Proposition 5.5 formula sup_S int-res-dim Γ_S. If the value is 1, Proposition 6.5 survives and the missing proof can be patched; if the value is not 1, the classification in Section 6.2 is false.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Section 6.2, the proof of Proposition 6.5 opens: 'The claim for s = 2 can be shown by using Proposition 5.2.' The manuscript has no Proposition 5.2: Section 5 contains Theorem 5.2, Proposition 5.5, Proposition 5.8, etc., but no Proposition 5.2. If the intended pointer is Theorem 5.2, it does not apply: an ~A-type poset with s=2 sinks is a cycle, so a length-4 segment in it has endpoints of degree 2 rather than the leaf conditions in Definition 5.1, and Theorem 5.2's removals are unavailable. The s=2 case is therefore not established by the text. This is a real omitted proof in a stated classification result. It does not touch the paper's main preservation theorem (Theorem 4.10, Proposition 4.4) or the tree-type formula (Proposition 6.4); nevertheless, as written, the dichotomy '1 if s=2, 2 if s≥3' has a missing leg.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies Galois connections between posets whose left adjoint is the inclusion of a full subposet, called an interior system. It introduces the associated induction and contraction functors, proves that for aligned interior systems both functors preserve interval-decomposability, and uses this to show that induction preserves interval covers and interval resolutions. The paper then applies these ideas to finite posets, proving a stabilization theorem for interval resolution global dimensions, computing formulas for tree-type and tilde-A-type posets, and offering a partial classification of posets of interval resolution global dimension 2 under an assumption that the ground field has two elements.","tokens_in":34906,"tokens_out":5186,"duration_ms":61833,"significance":"If the main results hold, the paper gives a clean and broadly applicable reduction principle: interval resolutions over a large poset can be computed over a smaller aligned subposet and induced back. The authors are explicit about the new notion of aligned interior systems, they prove the central preservation statements in detail, and they provide parameter-free formulas for interval resolution global dimensions of tree-type and tilde-A-type posets. The proof of Theorem 4.10 is careful and does not depend on the finite-poset machinery used later, which is a genuine strength. The finite-poset sections are more computational in character and contain at least one missing case that needs to be repaired before the paper can be accepted.","major_comments":[{"comment":"The proof of the case s = 2 states that the claim 'can be shown by using Proposition 5.2', but there is no Proposition 5.2 in the manuscript; Section 5 contains Theorem 5.2, Proposition 5.5, and Proposition 5.8, but not Proposition 5.2. If the intended reference is Theorem 5.2, it does not apply to this situation: an ~A-type poset with two sinks is a cycle, so a length-four segment running through its vertices has internal vertices of degree 2 but its endpoints are not leaves in the sense of Definition 5.1. Thus the stated formula for s = 2 is not established as written. Please supply a direct proof for the two-sink case or a correct reference that covers it.","section":"§6.2, proof of Proposition 6.5"},{"comment":"The proof of Proposition 5.8 is written only for the case m = 2 and dismisses m = 1 with the sentence 'the case m = 1 can be shown similarly'. Since Theorem 5.2(b) relies on the m = 1 case when reflecting a leaf, this omitted case is load-bearing. The analogous argument is plausible, but the manuscript should either spell out the m = 1 verification or state explicitly that the m = 2 proof applies verbatim with the obvious notational changes.","section":"§5.2, proof of Proposition 5.8"}],"minor_comments":[{"comment":"The proof refers to 'Table 6.7', but the relevant statement is Example 6.7, not a numbered table. Please correct the cross-reference.","section":"§6.3, proof of Proposition 6.11(2)"},{"comment":"The completeness statement in Proposition 6.11 is proved under the standing assumption from the beginning of Section 6.3 that the ground field is F_2. This assumption should be recorded in the statement of the proposition itself, since the earlier results in the paper, such as Proposition 6.5, are not restricted to F_2.","section":"§6.3, Proposition 6.11"},{"comment":"The first sentence of Definition 4.9 reads 'For 0 ≠ M ∈ fprep M'; this should be 'M ∈ fprep P'.","section":"§4.2, Definition 4.9"},{"comment":"In the right-minimality part of the proof, the isomorphism Hom_P(G(V), G(V)) ≅ Hom_Q(FG(V), FG(V)) is asserted without explanation. This is valid because F is an equivalence when restricted to the essential image of the fully faithful functor G, but that fact should be stated explicitly for clarity.","section":"§4.2, proof of Theorem 4.10(a)"}],"recommendation":"major_revision","confidential_remarks":"The central preservation theorem, Theorem 4.10, appears sound and is the main contribution of the paper. The missing s = 2 case in Proposition 6.5 is localized and likely repairable, but as written it leaves a stated classification result unproved. The long case analysis in Section 6.3 is difficult to audit; the authors may want to add more computational detail or a reproducible verification for the tables."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis is a real advance on the contraction-functor program. The central result, Theorem 4.10, says that for an aligned interior system Q ⊆ P the induction functor preserves interval covers and interval resolutions, so you can compute interval resolutions over the smaller poset and push them back up. The proof is detailed and the setup is clean: aligned interior systems genuinely generalize the finite aligned subgrids in BBH25, and Proposition 4.4, showing that Cont_Q sends interval modules to interval-decomposable modules, is solid. The paper is also honest: it gives a counterexample for non-aligned systems and does not overclaim.\n\nI agree with the reader's conditional verdict. The soft spots are all in Sections 5 and 6. The stress-test note is correct: Proposition 6.5 opens its s=2 case with a citation to 'Proposition 5.2', and no such proposition exists. If the authors meant Theorem 5.2, the hypotheses do not line up—an tilde-A poset with two sinks is a cycle, so the A_n-type segment there need not have leaf endpoints and the removals in Theorem 5.2 are not available. As written, the '1 if s=2' leg of the formula is unproved. That is a real gap in a stated classification result, though it does not touch Theorem 4.10. Also, Proposition 5.8 defers the m=1 case with 'similarly', and Section 6.3 is a large hand-checked case analysis over F_2 that any referee would find hard to verify line by line. These are fixable, or at least properly flaggable, but they should not be waved through.\n\nThe tree-type formula (Proposition 6.4) is nice and the proof is short. The stabilization theorem 5.2(a) has a careful case analysis. No circularity in the reliance on AENY23 and BBH25; those are legitimate cited tools.\n\nFor whom: anyone in multiparameter persistence who wants computational reductions, and poset representation theorists. I would bring it to a reading group and would cite Theorem 4.10. The paper deserves a serious referee; the right outcome is likely major revision with the tilde-A gap fixed, not a desk reject.","headline":"Solid preservation theorem for interval resolutions over aligned subposets; the poset-dimension computations are less polished, with a broken reference in the s=2 tilde-A case.","tokens_in":35514,"tokens_out":2718,"would_cite":true,"duration_ms":26322,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16G20","55N31","18G25","16E10"],"pacs":[],"model":"deepseek-v4-flash","headline":"For aligned interior systems, induction preserves interval covers and interval resolutions, so interval resolution dimensions are unchanged when passing from a subposet to its ambient poset.","keywords":["persistence modules","interval modules","interval resolutions","Galois connections","interior systems","contraction functors","interval resolution global dimension","multiparameter persistent homology"],"falsifier":"Compute interval resolution global dimensions directly for a finite poset $P$ with an equioriented $A_4$-type segment and compare with $P'$ obtained by deleting $\\ell_4$; a single pair with different dimensions would falsify Theorem 5.2(a). Alternatively, rerun the Section 6.3 minimality classification over a field of characteristic different from $2$; a table entry whose interval resolution global dimension changes would show the low-dimensional classification is field-sensitive.","tokens_in":34487,"feed_emoji":"📐","tokens_out":8705,"duration_ms":86994,"temperature":0.7,"pith_summary":"The paper establishes that for a class of full subposets called aligned interior systems, the induction functor sends interval covers to interval covers and interval resolutions to interval resolutions, so the interval resolution dimension of an induced module equals that of the original. This matters for multiparameter persistent homology because interval resolutions are a relative homological invariant of persistence modules, and the result gives a way to compute them over a small aligned subposet and then extend the answer to the large poset. The argument rests on a contraction functor, the left adjoint to induction, which over aligned interior systems sends interval modules to interval-decomposable modules, yielding an adjunction between interval-decomposable categories. The paper then uses this to prove stabilization results for interval resolution global dimensions of finite posets and to compute those dimensions for tree-type posets, A-tilde-type posets, and low-dimensional minimal examples.","feed_headline":"Lift interval resolutions from aligned subposets unchanged","feed_subtitle":"Compute a module's interval resolution over a small aligned subposet, then induce it back: the dimension is unchanged.","key_machinery":"The load-bearing object is the aligned interior system: a full subposet $Q$ whose inclusion has a right adjoint (the floor function $\\lfloor \\cdot \\rfloor_Q$) and whose fibers $\\lceil y \\rceil_Q = \\{a \\in P \\mid \\lfloor a \\rfloor_Q = y\\}$ are filtered and satisfy $\\lceil y{\\downarrow} \\rceil_Q = (\\lceil y \\rceil_Q){\\downarrow}$. The contraction functor $\\mathrm{Cont}_Q$ is the left Kan extension along the floor function; alignment gives $\\mathrm{Cont}_Q M(y) = \\mathrm{colim}(M|_{\\lceil y \\rceil_Q})$, which makes it exact and lets it send interval modules to interval-decomposable modules. The induction functor $\\mathrm{Ind}_Q$, being pullback along the floor function, sends interval modules to interval modules. Together they restrict to an adjunction between the categories of interval-decomposable modules, and that adjunction is what transports interval covers and resolutions.","core_discovery":"On the paper's own terms, the central discovery is that an aligned interior system $Q \\subseteq P$ produces an adjoint pair $\\mathrm{Cont}_Q \\dashv \\mathrm{Ind}_Q$ that respects the interval-decomposable part of both representation categories: for every interval $S$ of $Q$, $\\mathrm{Ind}_Q I_S$ is the interval module $I_{\\lceil S \\rceil_Q}$, and for every interval $T$ of $P$, $\\mathrm{Cont}_Q I_T$ is the interval-decomposable module $I_{T_Q}$. Because both functors preserve interval decomposability, an interval cover or interval resolution of a module $M$ over $Q$ is carried by the exact, fully faithful induction functor to an interval cover or interval resolution of $\\mathrm{Ind}_Q M$, and the interval resolution dimension is unchanged. This is Theorem 4.10, and it holds without assuming the base poset is locally finite, an upper semilattice, or finite.","pith_inferences":["A practical algorithm suggested by the proof: given a finitely presentable module, choose a small aligned interior system containing the generators and relations of its presentation, compute the interval resolution of its contraction there, then induce back; the cost savings can be large when the small system is much smaller than the ambient poset.","The adjunction mechanism is not specific to intervals: any class of modules closed under $\\mathrm{Cont}_Q$ and $\\mathrm{Ind}_Q$ in the same way would be carried by the same adjunction, so analogous preservation statements may hold for other relative resolutions.","The stabilizing operation of Theorem 5.2 could plausibly be iterated to a normal form for finite posets, and the Section 6 computations suggest that interval resolution global dimension depends only on coarse graph data in many cases; this is an extension, not a claim of the paper."],"forward_implications":["For any finitely presentable module $M$ over $Q$, an interval cover $V \\to M$ induces an interval cover $\\mathrm{Ind}_Q V \\to \\mathrm{Ind}_Q M$, and the same holds for full interval resolutions.","Interval resolution dimensions are invariant under induction: $\\mathrm{int\\text{-}res\\text{-}dim}\\, M = \\mathrm{int\\text{-}res\\text{-}dim}\\, \\mathrm{Ind}_Q M$ for every finitely presentable module $M$ over $Q$.","Finite aligned subgrids of products of totally ordered sets are aligned interior systems, so the reduction to a subgrid applies to the usual finitely presentable multiparameter persistence modules.","For finite tree-type posets, the interval resolution global dimension is $\\#\\mathrm{leaf}(P)-2$, so it depends only on the underlying graph of the Hasse diagram.","For $\\widetilde{A}$-type posets, the interval resolution global dimension is $1$ when there are two sinks and $2$ when there are at least three sinks; Section 6.3 gives a partial classification of minimal posets of global dimension $2$ over the two-element field."],"supporting_citations":[{"why":"Supplies the formula $\\mathrm{int\\text{-}res\\text{-}gl.dim}\\, P = \\sup_S \\mathrm{int\\text{-}res\\text{-}dim}\\, \\Gamma_S$ used to compute global dimensions in Sections 5 and 6.","marker":"[AENY23]"},{"why":"Defines interval covers and interval resolutions, and gives the example where unrestricted induction fails to preserve interval decomposability.","marker":"[AET25]"},{"why":"Introduces finite aligned subgrids and the contraction functor for product posets that this paper generalizes to aligned interior systems.","marker":"[BBH25]"},{"why":"Provides the Auslander-Reiten, APR-tilting, and torsion-pair results used in the proofs for finite posets.","marker":"[ASS06]"},{"why":"Gives the decomposition theorem for pointwise finite dimensional persistence modules used throughout.","marker":"[BCB20]"},{"why":"Supplies the Kan-extension formalism behind the adjoint quadruple used in Section 2.","marker":"[Rie17]"},{"why":"Defines interior systems and the floor-function adjunction adopted here.","marker":"[EKMS93]"}],"fun_headline_variants":["Aligned subposets preserve interval resolution dimension","Interval resolutions lift unchanged from aligned subposets","Subposet alignment keeps interval resolution dimension","Resolutions survive induction from aligned subposets"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The finite-poset computations assume that representations of a finite poset are modules over its incidence algebra, that $\\mathrm{int\\text{-}res\\text{-}gl.dim}\\, P = \\sup_S \\mathrm{int\\text{-}res\\text{-}dim}\\, \\Gamma_S$ holds for every poset in the tables, and (in Section 6.3) that the ground field is the two-element field, so a failure of any of these would break Sections 5 and 6 even though Section 4 stands independently.","fun_headline_variants_meta":{"raw":{"variants":["Aligned subposets preserve interval resolution dimension","Interval resolutions lift unchanged from aligned subposets","Subposet alignment keeps interval resolution dimension","Resolutions survive induction from aligned subposets"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000589,"raw_usage":{"total_tokens":2779,"prompt_tokens":975,"completion_tokens":1804,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":591,"completion_tokens_details":{"reasoning_tokens":1746}},"tokens_in":591,"tokens_out":1804,"duration_ms":15073,"temperature":1.0,"reasoning_tokens":1746,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T22:31:45.283882+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute interval resolution global dimensions directly for a finite poset $P$ with an equioriented $A_4$-type segment and compare with $P'$ obtained by deleting $\\ell_4$; a single pair with different dimensions would falsify Theorem 5.2(a). Alternatively, rerun the Section 6.3 minimality classification over a field of characteristic different from $2$; a table entry whose interval resolution global dimension changes would show the low-dimensional classification is field-sensitive.","supporting_citations":[],"review_version":1}