{"id":"34f813db-05e1-4b3f-9032-c3f716ce716c","arxiv_id":"2506.17888","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Persistent topological complexity and persistent zero-divisor-cup-length are defined, shown stable under homotopy interleaving and Vietoris-Rips perturbations, and used to recover a Gromov-Hausdorff lower bound of pi/6 for RP^n versus a wedge of spheres.","lead":"This paper introduces persistent versions of topological complexity, an invariant from robot motion planning, and its cohomological lower bound, and proves they are stable under perturbations of a space. For two explicit metric spaces, real projective space and a wedge of spheres, these new invariants separate the spaces better than persistent homology does.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 1.1's stability proof verifies only homotopy-equivalence invariance on CW complexes, but [MSZ24, Theorem 2.21] requires weak-equivalence invariance for the arbitrary spaces appearing in the homotopy interleaving distance; TC fails this condition in general.","rationale":"The reader's weakest_assumption identifies the same load-bearing gap in Proposition 1.1, and I agree. This gap matters because Proposition 1.1 underpins the second inequality of Theorem 1.2 for TC, and the paper's advertised stability and distinguishing-power claims depend on that inequality. I do not recommend REJECT: the persistent zcl version is likely sound because cohomology is weak-homotopy invariant; the example computations are plausible; and the TC stability gap may be repairable by a careful CW-approximation argument. The secondary issue in Lemma 4.5, where map-level homotopy equivalences are inferred from levelwise homotopy equivalences, is also real but probably repairable: for inclusions of CW subcomplexes in a Vietoris-Rips filtration, if both levels are homotopy equivalent to the same space, the long exact sequence of the pair implies the inclusion is a homotopy equivalence; the authors should state and prove that fact rather than leave it implicit. Until the weak-invariance gap is addressed, the paper should be accepted only conditionally on filling that proof hole.","tokens_in":21960,"tokens_out":25232,"duration_ms":269269,"concrete_test":"Compute (or locate in the literature) TC of the Warsaw circle and compare it with TC(point)=0. If TC(Warsaw)>0, then TC is not invariant under weak homotopy equivalences, directly falsifying the premise under which Proposition 1.1 invokes [MSZ24, Theorem 2.21]. The authors would then need either to prove a CW-relative version of Theorem 2.21 showing that dHI for persistent CW complexes can always be realized by persistent CW replacements, or to restrict the stability statement to invariants that are genuinely weak-homotopy invariant.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The proof of Proposition 1.1 invokes [MSZ24, Theorem 2.21], whose hypothesis is that the categorical invariant is invariant under pre- and post-composition with weak homotopy equivalences. The proof verifies only invariance under homotopy equivalences, using Whitehead's theorem to identify weak equivalences between CW complexes with homotopy equivalences. This does not cover the arbitrary compactly generated weakly Hausdorff spaces that may appear as X', Y', and W in the definition of dHI (Definition 2.4). TC is not invariant under weak homotopy equivalences of arbitrary path-connected spaces: a weakly contractible, non-contractible space such as the Warsaw circle is weakly equivalent to a point, yet is not contractible and therefore does not admit a global motion planner, so its TC differs from TC(point)=0. Thus the hypothesis of Theorem 2.21 is not verified for TC. The paper gives no argument that dHI between persistent CW complexes can be computed using only CW replacements; replacing X' and Y' by CW approximations does not automatically preserve the weak-equivalence spans required for the theorem, because the defined span relation is not shown to be compatible with such replacements. Consequently, Equation (1) for I=TC is unsupported as written. The same issue does not affect zcl, since cohomology rings are weak-homotopy invariant, but Proposition 1.1 states the result for both, and Theorem 1.2 relies on both.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces persistent analogues of two classical invariants in topological robotics: topological complexity (TC) and its cohomological lower bound, the zero-divisor-cup-length (zcl). For a persistent space X_•, these are defined by evaluating the map-level invariants on the structure maps X_a→X_b, yielding functors from the interval poset to the extended natural numbers. The main theoretical result is Proposition 1.1, which claims that, for persistent CW complexes, the erosion distance between these persistent invariants is bounded above by the homotopy interleaving distance, and hence, for Vietoris–Rips filtrations of compact metric spaces, by twice the Gromov–Hausdorff distance. The main application is Theorem 1.2, which gives a lower bound of π/3 on the erosion distance between the persistent TC (and zcl) of the Vietoris–Rips filtrations of RP^n and of the wedge sum ∨_{i=1}^n S^i, yielding d_GH(RP^n,∨_nS) ≥ π/6, matching the bound previously obtained via persistent Steenrod modules. The paper also compares the discriminative power of TC, zcl, cat, and cl on simple examples.","tokens_in":22307,"tokens_out":9361,"duration_ms":101499,"significance":"If the main stability statement is correct, the paper makes a useful contribution: it extends the categorical persistence framework of [MSZ24] to topological complexity, establishes a natural stability bound, and gives a concrete pair of metric spaces where persistent TC and zcl are strictly more discriminative than persistent homology. The example in Theorem 1.2 is worked out in detail and connects nicely to the existing literature on Gromov–Hausdorff lower bounds. The paper is clearly written and the definitions are reasonable. However, the central stability proof for TC has a gap that is load-bearing for the advertised claim, and a second gap appears in the verification that the structure maps in Lemma 4.5 are homotopy equivalences. These issues are potentially repairable, but they require substantive additional argument rather than cosmetic changes.","major_comments":[{"comment":"The proof invokes [MSZ24, Theorem 2.21], whose hypothesis is that the categorical invariant I is invariant under post- and pre-composition with weak homotopy equivalences. The proof verifies only invariance under homotopy equivalences, using Whitehead's theorem to identify weak equivalences between CW complexes with homotopy equivalences. This does not cover the arbitrary compactly generated weakly Hausdorff spaces that appear as W_• in the definition of the homotopy interleaving distance (Definition 2.4): the weak equivalences W_•→X_• and W_•→Y_• need not have CW domain, so Whitehead's theorem does not apply. TC is not invariant under weak equivalences with arbitrary domain: a weakly contractible non-contractible space such as the Warsaw circle is weakly equivalent to a point but has TC differing from TC(point), since it does not admit a global motion planner. Consequently, Equation (1) for I=TC is unsupported as written. The issue does not affect zcl, since singular cohomology is invariant under weak equivalences, but Proposition 1.1 and Theorem 1.2 state the result for both invariants, and the stability of persistent TC is a central advertised contribution.","section":"§4.2, proof of Proposition 1.1 (Eq. (1))"},{"comment":"The lemma concludes that the structure maps VR_s→VR_t are homotopy equivalences for s,t in the indicated ranges, based on the fact that each individual complex VR_t is homotopy equivalent to RP^n (respectively ∨_nS). The cited results — [AHP22, Theorem 4.5 and the following remark], [LMO24, Theorem 7.1], and [Ada+20, Proposition 1] — support the homotopy type of the individual levels and, in the wedge case, the homotopy equivalence of the natural inclusion VR_t(X)∨VR_t(Y)→VR_t(X∨Y) for fixed t. They do not explicitly establish that the inclusions VR_s→VR_t for varying scales are homotopy equivalences. This distinction matters because persistent invariants are defined on the structure maps f_a^b, so the assertion I(VR_•)(J)=I(RP^n) requires the map, not only the endpoint spaces, to induce the stated value. An inclusion between two spaces each homotopy equivalent to RP^n can fail to be a homotopy equivalence. The proof should either cite a result that explicitly gives the maps as homotopy equivalences or provide a direct argument.","section":"§4.2, Lemma 4.5"}],"minor_comments":[{"comment":"There are two typos in this definition: 'homotpy' should be 'homotopy', and 'or + inf' should read 'or +∞'.","section":"§2.2.2, Definition 2.13"},{"comment":"The word 'ϵ-erosed' should be 'ϵ-eroded'. Also, in the displayed definition the codomain of the functors is written as (R≥0,≤), while the persistent invariants take values in N∪{∞} with the opposite order; the intended meaning is clear but the notation should be made consistent.","section":"§2.3, Definition 2.19"},{"comment":"There is an inconsistency in the displayed definition of TC(f): the first sentence correctly says that an f-motion planner is a map U→Path(Y), but the displayed bullet writes s_i^f : U_i→Path(X) and then p_X∘s_i^f = id_{U_i}. The target should be Path(Y) and the fibration should be p_Y, matching the definition of the pullback topological complexity.","section":"§3.3, Definition 3.9"},{"comment":"The arrow in the displayed formula appears to have the wrong variance. For a map f_a^b : X_a→X_b, the induced map on cohomology is (f_a^b×f_a^b)^* : H^*(X_b×X_b)→H^*(X_a×X_a), so it sends ker Δ^*_{X_b} to ker Δ^*_{X_a}. As written, 'ker Δ^*_{X_a} → ker Δ^*_{X_b}' suggests the opposite direction.","section":"§4.1, Definition 4.3"},{"comment":"The notation ∨_nS is used in the abstract and introduction before being defined as ∨_{i=1}^n S^i in Section 4.2. A brief parenthetical definition on first use would improve readability.","section":"§1 and §4.2, notation"}],"recommendation":"major_revision","confidential_remarks":"The gap in Proposition 1.1 for TC is the main concern. It is not a matter of a missing citation; it is a genuine mismatch between the hypothesis of [MSZ24, Theorem 2.21] and what is verified. The paper could be salvageable by proving that the homotopy interleaving distance between persistent CW complexes can be computed using CW replacements, or by otherwise establishing the needed invariance for TC on the relevant weak-equivalence spans. I do not see a circularity problem in the reliance on [MSZ24] and [MZ25]; the new definitions and computations are independent. The contribution would be solid if the stability issue is resolved and the map-level homotopy equivalences in Lemma 4.5 are justified."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. The paper introduces persistent topological complexity and persistent zero-divisor-cup-length, proves a stability inequality against the homotopy interleaving distance, and shows these invariants separate RP^n from a wedge of n spheres with a GH lower bound of pi/6, matching what persistent Steenrod modules give. That is a genuinely useful addition to the TDA toolbox. The stability proof, however, has a load-bearing gap: Proposition 1.1 verifies invariance only under homotopy equivalences between CW complexes, while the MSZ theorem it invokes requires invariance under pre/post-composition with weak homotopy equivalences of arbitrary spaces. TC is not invariant under weak equivalences in general - a weakly contractible non-contractible space like the Warsaw circle is weakly equivalent to a point but has TC > 0. Since the homotopy interleaving distance is defined through spans of weakly equivalent persistent spaces that may be non-CW, the quoted theorem does not apply to TC as written. zcl is fine, because cohomology rings are weak-homotopy invariant. This is not a nitpick; the gap is exactly what lets the stability argument go through.\n\nWhat is good: the definitions are natural and the paper is the first to write them down; the categorical framework makes the lift to persistence almost automatic once the invariant is checked. The distinguishing example is worked carefully, and the authors are appropriately modest about the bound - they acknowledge the diameter bound pi/2 is stronger. Proposition 4.6, abstracting the erosion-distance estimate, is clean and reusable. Lemma 4.5 cites known results for the homotopy types of the Rips complexes, and the computation from those is sensible. One smaller soft spot: the lemma asserts the structure maps themselves are homotopy equivalences in the relevant scale ranges, but the cited sources support the levelwise homotopy type, not the map. That is likely true in these examples but needs a sentence or two.\n\nThe paper deserves a serious referee. It is a solid contribution to the growing literature on persistent invariants beyond persistent homology, and the main example is a meaningful comparison. The stability proof needs repair - either by formulating a version for CW complexes that respects the spans in dHI, or by proving TC is weak-equivalence invariant in the needed generality, which I suspect is false. The more plausible fix is to restrict the stability statement to a setting where TC is well-behaved and to adjust the proof accordingly. My recommendation: send to peer review, with the expectation of revision.","headline":"Useful new persistent invariants, but the TC stability proof leans on a weak-equivalence invariance that is not verified and is likely false in general.","tokens_in":22784,"tokens_out":5422,"would_cite":true,"duration_ms":55339,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["55M30","55N31"],"pacs":[],"model":"deepseek-v4-flash","headline":"Persistent topological complexity and zero-divisor-cup-length are stable, and they separate real projective space from a wedge of spheres more sharply than persistent homology.","keywords":["persistent topological complexity","zero-divisor-cup-length","Vietoris–Rips filtration","Gromov–Hausdorff distance","erosion distance","homotopy interleaving distance","motion planning","persistent invariants"],"falsifier":"Check whether each Vietoris–Rips inclusion $VR_s\\to VR_t$ in the ranges $0<s<t<2\\pi/3$ (for $\\mathbb{RP}^n$) and $0<s<t<\\zeta_n$ (for $\\vee^n S$) is actually a homotopy equivalence; if any one is not, Lemma 4.5's constant values fail and the $\\pi/3$ erosion lower bound collapses. A second check is to test Proposition 1.1 on a pair of persistent CW complexes with small homotopy interleaving distance but large erosion distance for TC.","tokens_in":21770,"feed_emoji":"🤖","tokens_out":9123,"duration_ms":81139,"temperature":0.7,"pith_summary":"The paper lifts topological complexity and its cohomological lower bound, zero-divisor-cup-length, from spaces to persistent families of spaces. It proves that these persistent invariants are stable: moving one persistent space to another by a homotopy interleaving of size $\\epsilon$ changes the invariant by at most $\\epsilon$ in erosion distance, and for Vietoris–Rips filtrations of compact metric spaces this gives a $2\\,d_{\\mathrm{GH}}$ bound. It then shows that for the pair consisting of real projective space $\\mathbb{RP}^n$ and the wedge of round spheres $\\vee^n S$, the erosion distance between the persistent invariants is at least $\\pi/3$, which forces the Gromov–Hausdorff distance between the two spaces to be at least $\\pi/6$. Persistent homology only yields $\\pi/8$ for this pair, so the new invariants are more discriminating in this case. The reason to care is that the invariants encode the complexity of continuous motion planning, not just homology.","feed_headline":"Persistent motion-planning invariants stable, sharper than homology","feed_subtitle":"For projective space versus a wedge of spheres, they push the distance lower bound from π/8 to π/6.","key_machinery":"The central machinery is the categorical invariant formalism for persistent objects: for a persistent space $X_\\bullet$, the persistent invariant $I(X_\\bullet)$ assigns to each interval $[a,b]$ the value $I(X_a\\to X_b)$ of the invariant on the structure map. Topological complexity and zero-divisor-cup-length qualify because neither increases under composition, making them functors from the interval poset to $(\\mathbb{N}\\cup\\{\\infty\\},\\ge)$. Stability is measured with erosion distance and homotopy interleaving distance. For the numerical lower bound, Lemma 4.5 pins down the scale intervals on which the two invariants are constant using known homotopy types of Vietoris–Rips complexes of projective space, spheres, and metric gluings, and Proposition 4.6 converts those intervals into an erosion-distance lower bound of $\\pi/3$ by comparing two step functions.","core_discovery":"The paper establishes that topological complexity and zero-divisor-cup-length, when lifted to persistent spaces by evaluating them on the structure maps $X_a \\to X_b$, are stable under the homotopy interleaving distance. Concretely, Proposition 1.1 gives $d_E(I(X_\\bullet),I(Y_\\bullet)) \\le d_{HI}(X_\\bullet,Y_\\bullet)$ for persistent CW complexes, and for Vietoris–Rips filtrations of compact metric spaces this becomes a $2\\,d_{\\mathrm{GH}}$ bound. The paper then computes the two invariants along the Vietoris–Rips filtrations of $\\mathbb{RP}^n$ and $\\vee^n S$: both take constant values $I(\\mathbb{RP}^n)>2$ and $I(\\vee^n S)=2$ on a scale interval and vanish after the complexes become contractible. Feeding these values into a two-step erosion-distance estimate yields $d_E(I(VR_\\bullet(\\mathbb{RP}^n)),I(VR_\\bullet(\\vee^n S))) \\ge \\pi/3$, and combining this with stability yields $d_{\\mathrm{GH}}(\\mathbb{RP}^n,\\vee^n S)\\ge \\pi/6$, improving the $\\pi/8$ lower bound that persistent homology alone provides.","pith_inferences":["The paper's template suggests a general recipe: any integer-valued homotopy invariant that is non-increasing under composition and takes distinct constant values on two spaces with known Vietoris–Rips homotopy types yields a stable lower bound on $d_{\\mathrm{GH}}$; the hard part is usually determining the scale interval where the structure maps are homotopy equivalences.","Because persistent Steenrod modules and these new invariants produce the same $\\pi/6$ bound for the same pair, one could test whether all such stable invariants are detecting a single geometric obstruction; if so, combining invariants would not increase the bound beyond $\\pi/6$ without a new idea.","An immediate testable extension would be to apply the same proof to lens spaces or other quotient metric spaces with known Vietoris–Rips homotopy types, and to see whether the erosion-distance gap grows with the size of the fundamental group."],"forward_implications":["Persistent TC and persistent zcl are quantitative shape signatures for compact metric spaces, with the same $2\\,d_{\\mathrm{GH}}$ stability guarantee as persistent homology.","For the pair $(\\mathbb{RP}^n,\\vee^n S)$, the persistent invariants give $d_{\\mathrm{GH}}\\ge\\pi/6$, strictly improving the persistent-homology bound of $\\pi/8$.","The same lower bound can be recovered from persistent LS-category and persistent cup-length, so the improvement is not specific to motion-planning invariants.","Neither persistent TC nor persistent zcl dominates the other: static examples (spheres and wedges) show each can separate spaces the other cannot."],"supporting_citations":[{"why":"Supplies the categorical-invariant stability theorem and the persistent-invariant construction that Proposition 1.1 applies.","marker":"[MSZ24]"},{"why":"Introduces homotopy interleaving distance and the bound $d_{HI}(VR_\\bullet(X),VR_\\bullet(Y)) \\le 2d_{GH}$ used in Proposition 1.1's consequence.","marker":"[BL23]"},{"why":"Defines erosion distance, the metric in which all stability statements are phrased.","marker":"[Pat18]"},{"why":"Defines topological complexity and zero-divisor-cup-length and provides their basic values and inequalities, including $zcl \\le TC$.","marker":"[Far03]"},{"why":"Extends TC and zcl to maps and proves $zcl(f)\\le TC(f)$ for ANRs, which makes the persistent lift well-defined and gives the lower-bound inequality.","marker":"[Sco22a]"},{"why":"Gives the homotopy type of Vietoris–Rips complexes of $\\mathbb{RP}^n$ used in Lemma 4.5 to bound the intervals where the invariants are constant.","marker":"[AHP22]"},{"why":"Describes homotopy types of Vietoris–Rips complexes of spheres at small scales, used in Lemma 4.5 for the wedge's factors.","marker":"[LMO24]"},{"why":"Shows Vietoris–Rips complexes respect metric gluings, letting the paper transfer sphere homotopy types to the wedge sum.","marker":"[Ada+20]"},{"why":"Establishes the persistent homology bound $d_I \\le \\pi/4$, hence the $\\pi/8$ lower bound that the paper's $\\pi/6$ improves.","marker":"[MZ25]"}],"fun_headline_variants":["Persistent motion-planning invariants: stable and sharper","Persistent complexity sharpens GH lower bound to π/6","Persistent invariants beat homology for distinguishing spaces","Persistent motion planning: stability and π/6 bound"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The stability theorem inherits a hypothesis that the invariants are unchanged when a space is replaced by a weakly homotopy equivalent one; the proof verifies invariance only under ordinary homotopy equivalences, so the full weak-equivalence condition is assumed rather than checked.","fun_headline_variants_meta":{"raw":{"variants":["Persistent motion-planning invariants: stable and sharper","Persistent complexity sharpens GH lower bound to π/6","Persistent invariants beat homology for distinguishing spaces","Persistent motion planning: stability and π/6 bound"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000798,"raw_usage":{"total_tokens":3489,"prompt_tokens":905,"completion_tokens":2584,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":521,"completion_tokens_details":{"reasoning_tokens":2518}},"tokens_in":521,"tokens_out":2584,"duration_ms":20278,"temperature":1.0,"reasoning_tokens":2518,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T19:01:43.702244+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check whether each Vietoris–Rips inclusion $VR_s\\to VR_t$ in the ranges $0<s<t<2\\pi/3$ (for $\\mathbb{RP}^n$) and $0<s<t<\\zeta_n$ (for $\\vee^n S$) is actually a homotopy equivalence; if any one is not, Lemma 4.5's constant values fail and the $\\pi/3$ erosion lower bound collapses. A second check is to test Proposition 1.1 on a pair of persistent CW complexes with small homotopy interleaving distance but large erosion distance for TC.","supporting_citations":[],"review_version":2}