{"id":"d327b3a2-2133-42e5-8a40-6072be1f87fd","arxiv_id":"2607.14668","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Cocompact lattice actions on flag boundaries of Euclidean buildings are topologically stable: every sufficiently small perturbation is semi-conjugate to the original action.","lead":"This paper proves that the natural actions of certain symmetry groups on the boundary flags of Euclidean buildings (including p-adic flag varieties) are rigid: every sufficiently small perturbation of the action is semi-conjugate to the original. It extends a known C0-stability phenomenon from hyperbolic and real symmetric spaces to higher-rank p-adic buildings.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central coding construction is contingent on the KLP higher-rank Morse lemma and its uniformity; a failure there would collapse Propositions 5.11, 5.15, and P4.","rationale":"The reader accepted with moderate confidence; my reading confirms the architecture is sound and the main risk is the importation of the heavy KLP machinery. I found only minor typos and indexing shifts, not fatal gaps. The proposed test targets the precise point where the argument is most contingent.","tokens_in":37058,"tokens_out":45978,"duration_ms":407605,"concrete_test":"Independently verify the exact statement of [KLP18, Theorem 1.3] for locally compact Euclidean buildings: (a) the constant D is uniform over all Θ-regular (K,A)-quasi-geodesic sequences; (b) the flag τ is unique. Then patch Corollary 3.16 by replacing the invocation of uniqueness with a direct argument using Lemma 3.8 (separated stars) to show the tail must have the same τmod-limit as the original sequence. If the patched proof goes through, the concern is resolved; if not, Propositions 5.11/5.15 lack the uniformity needed for P4.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 1.1 is built on Theorem 3.15 (Kapovich–Leeb–Porti [KLP18, Thm 1.3]) and the associated Corollary 3.16. This lemma is the only mechanism that turns a Θ-regular (K,A)-quasi-geodesic sequence into a unique τmod-flag at infinity, and conversely guarantees that every G-coding produces a regular quasi-geodesic with a well-defined flag (Prop. 5.11 and 5.15). It is also used again in the interpolation step (Prop. 6.1) through [KLP18, Lem. 3.11]. The authors do not reprove Theorem 3.15; the central claim is therefore conditional on the exact uniformity statement (D depending only on Θ,K,A, and uniqueness of τ). In particular, Corollary 3.16, which is proved in the paper, uses uniqueness of the Morse lemma to identify the flags associated to tails; this step is not fully justified because the tail may only track the same flag with a larger constant. The result is patchable via Lemma 3.8, but as written it reveals how much weight the argument places on the quoted theorem. If Theorem 3.15's uniformity fails in the needed form, the coding construction and (P4) collapse. No internal inconsistency is apparent; this is an external-input risk.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves Theorem 1.1: for a locally compact Euclidean building X and a group Γ acting properly and cocompactly by isometries on X, the induced action on the space M of τmod-flags in the visual boundary is topologically stable: every sufficiently small perturbation of ρ0 in Hom(Γ, Homeo(M)) is semi-conjugate to ρ0. The proof is modular. Section 2 develops an automata-theoretic stability criterion (Theorem 2.12) in terms of two proto-coders and four conditions (P1)–(P4). Section 4 proves a quantitative expansivity/contraction estimate (Theorem 4.1) for the action on flag boundaries. Section 5 constructs proto-coders from cocompact building actions, using the Kapovich–Leeb–Porti higher-rank Morse lemma (quoted as Theorem 3.15) to pass between regular quasi-geodesics and codings. Section 6 establishes an interpolation property (Proposition 6.1) and assembles the proof of Theorem 1.1. Section 7 gives an example showing that semi-conjugacy cannot in general be upgraded to conjugacy. Corollary 1.3 applies the theorem to parabolic quotients G/Q for semisimple p-adic Lie groups.","tokens_in":37371,"tokens_out":21690,"duration_ms":223908,"significance":"If correct, this is a substantial advance: it extends C0 boundary stability beyond hyperbolic/rank-one and real higher-rank settings to Euclidean buildings and totally disconnected flag spaces, where no smooth structure is available. The paper is careful with quantified constants, makes the logical architecture transparent, and includes a counterexample showing the optimality of semi-conjugacy. Its main strengths are the clean separation of the stability criterion from the geometric construction, the explicit use of the higher-rank Morse lemma, and the detailed verification of the coding properties. The principal caveat is the heavy reliance on the quoted Morse lemma and one under-justified step in the proof of Corollary 3.16; with that repaired, the result is significant and publishable.","major_comments":[{"comment":"The proof of Corollary 3.16 applies Theorem 3.15 to each tail and then asserts that uniqueness of the Morse-lemma flag gives τ_n = τ_0. As stated, Theorem 3.15 fixes the basepoint as the first term of the sequence and gives a constant D depending only on Θ,K,A; a tail with basepoint x_n need not satisfy the same D-neighborhood condition for τ_0, so uniqueness for the tail does not, by itself, identify its flag with τ_0. This identification is load-bearing: Corollary 3.16 is used in Proposition 5.11, Proposition 5.15, and the interpolation argument in Proposition 6.1, and hence for property (P4) in the proof of Theorem 1.1. Please fill this step, for example using Lemma 3.8/separated stars, or quote explicitly a stronger tail-independence statement from [KLP18].","section":"Section 3.8, Corollary 3.16"},{"comment":"The coding construction is contingent on the exact uniformity of the quoted higher-rank Morse lemma ([KLP18, Theorem 1.3], stated here as Theorem 3.15). Propositions 5.11, 5.15, and 6.1 each require that the constant D depend only on Θ,K,A, and that the associated flag be unique for the whole sequence and its tails. The manuscript does not prove this uniformity and does not isolate the precise statement it needs from [KLP18]. This is an external-input risk rather than an internal inconsistency, but it is load-bearing: if the required uniformity fails, the coding construction and (P4) collapse. Please add an explicit statement of the exact uniformity hypothesis used, with a precise reference to the corresponding statement in [KLP18].","section":"Sections 5–6 (proof of Theorem 1.1)"}],"minor_comments":[{"comment":"Many cross-references are mislabeled: Definition 2.3 is cited as Theorem 2.3, Lemma 2.5 as Theorem 2.5, Remark 2.7 as Theorem 2.7, Definition 2.9 as Theorem 2.9, Lemma 2.11 as Theorem 2.11, Lemma 2.14 as Theorem 2.14, Corollary 2.15 as Theorem 2.15, and Lemma 2.16 as Theorem 2.16. Please correct.","section":"Section 2"},{"comment":"The displayed line \"Φ(ϕ(ρ(γ)p))⊂W(z)\" appears to be a typo. The intended argument seems to be that ρ(γ)p ∈ ρ(γ)Φ(ϕ(p)) = Φ(ρ0(γ)ϕ(p)) ⊂ W(z). Please rephrase.","section":"Lemma 2.18 proof"},{"comment":"Near the end of the proof, d_X(u,v) is used for points u,v in the boundary flag space M; this should be d_o (or another metric on M), not the metric on X.","section":"Lemma 5.10 proof"},{"comment":"The proof begins by taking x,y∈B(γ^{-1}τ, ϵ0), but the statement uses ϵ. Unify the notation and make the dependence of ϵ on the constants explicit.","section":"Theorem 4.1 proof"},{"comment":"In the inductive construction of (w_k), the line \"y_n ∈ V(w_{n_k}, st_{Θ'}(τ))\" should read \"y_n ∈ V(w_k, st_{Θ'}(τ))\". Also, references to \"Theorem 5.5\" and \"Theorem 5.2\" should be to \"Remark 5.5\" and \"Proposition 5.2\" respectively.","section":"Section 6, proof of Proposition 6.1"},{"comment":"The assertion that every lattice in a semisimple p-adic Lie group is uniform is used without a reference or proof. If this is standard, please supply a citation; if it is not always true, the statement of the corollary needs adjustment.","section":"Corollary 1.3"}],"recommendation":"major_revision","confidential_remarks":"I am not recommending rejection. The architecture of the proof is sound and the dependence on [KLP18] is not circular. The main issue is the under-justified step in Corollary 3.16, which is patchable but load-bearing for the central theorem. I would be comfortable with acceptance once that step is either proved in the paper or its stronger version is explicitly quoted from [KLP18]."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main result is genuinely new: C0 topological stability for boundary actions of cocompact lattices on flag boundaries of locally compact Euclidean buildings, covering the p-adic flag variety case as a corollary. The architecture is clean: a general automata-based stability criterion (Section 2), a quantified expansivity estimate (Section 4), a proto-coder construction (Section 5), and an interpolation property (Section 6). The authors build honestly on MMW22/MMW24 and KLP18, and the semi-conjugacy optimality example in Section 7 is a nice touch.\n\nThe soft spots are real but not fatal. Corollary 3.16, which converts the higher-rank Morse lemma into a uniform statement about tails, has a gap: the proof says uniqueness of the flag in Theorem 3.15 forces τ_n = τ_0, but the basepoint changes from x_0 to x_n, so that uniqueness doesn't apply. The tail might track the same flag only with a larger constant. This is exactly what the stress-test flagged, and it matters because Propositions 5.11, 5.15 and the interpolation step rely on Corollary 3.16. I believe it's patchable—using the separated-stars lemma (Lemma 3.8) and the Θ-regularity of the sequence should force the tails to converge to the same flag—but as written it's a genuine gap in a load-bearing lemma.\n\nThe other caveat is the heavy black-box use of the KLP Morse lemma. The paper doesn't reprove it, and the uniformity statement (D depending only on Θ,K,A) is essential. That is a reasonable thing to rely on in a research paper, but a referee should check that the quoted form matches the needs here, especially the uniqueness clause.\n\nOverall, the paper is a substantial advance within the p-adic Zimmer program. I think the reader's ACCEPT verdict is about right, though confidence should stay moderate until the Corollary 3.16 gap is filled. I'd send it to a serious refereeing process, and I'd expect a revision to address that step. I'd also bring it to a reading group—it's a good example of how the automata/coding technology transfers to higher-rank geometric settings.","headline":"A genuinely new and likely correct C0 stability theorem for boundary actions of cocompact lattices in higher-rank Euclidean buildings, with one small but real gap in Corollary 3.16 that should be patchable.","tokens_in":37865,"tokens_out":2749,"would_cite":true,"duration_ms":27414,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C24","53C20","37D40"],"pacs":[],"model":"deepseek-v4-flash","headline":"The induced action of a cocompact lattice on every flag space of a locally compact Euclidean building is topologically stable: every sufficiently small perturbation is semi-conjugate to the original action.","keywords":["topological stability","Euclidean buildings","flag boundaries","semi-conjugacy","lattice actions","p-adic Lie groups","point coders","higher-rank Morse lemma"],"falsifier":"Look for a locally compact Euclidean building and a compact set Θ inside the open star of τmod for which some Θ-regular quasi-geodesic sequence does not lie in any bounded neighborhood of a single Weyl cone V(x0,st(τ)) over a unique τmod-flag. Finding one such sequence would refute the quoted higher-rank Morse lemma and remove the mechanism that converts coding data into regular quasi-geodesics, so the paper's argument would no longer go through.","tokens_in":36932,"feed_emoji":"","tokens_out":7765,"duration_ms":78601,"temperature":0.7,"pith_summary":"The paper proves a C0 rigidity statement for boundary actions of cocompact lattices on Euclidean buildings. It shows that the standard action of such a lattice on the space of τmod-flags in the visual boundary cannot be destroyed by small perturbations in the homeomorphism group: any nearby action is forced onto the original one by a continuous equivariant surjection. This extends rank-one boundary stability results to higher-rank Euclidean buildings, and it implies that actions of cocompact lattices in semisimple p-adic Lie groups on parabolic quotients G/Q are topologically stable. The proof works by encoding points of the flag space as infinite paths in finite directed graphs built from a quantitative expansivity estimate and the higher-rank Morse lemma.","feed_headline":"p-adic lattice boundary actions are topologically stable","feed_subtitle":"Nearby p-adic lattice actions are forced back to the original by a continuous surjection.","key_machinery":"The proof rests on a general stability criterion built from finitary point coders: finite directed graphs whose vertices are open subsets of the flag space and whose edges are labeled by group elements, so that an infinite path yields a nested intersection coding a point. Two such coders, related by an interpolation property, turn coding data into a semi-conjugacy. To build the coders, the paper proves a quantitative expansivity estimate: for each flag z, some group element moving the basepoint far out along a Weyl cone over z contracts nearby flags by a factor E in an ultrametric visual metric. A higher-rank Morse lemma, quoted as a black box, guarantees that uniformly regular quasi-geodesi","core_discovery":"The central claim is that the action ρ0 of a cocompact lattice Γ on a flag boundary M of a locally compact Euclidean building is topologically stable. Concretely, there is a neighborhood of ρ0 in the space of all homomorphisms Γ→Homeo(M) such that every action ρ in that neighborhood is semi-conjugate to ρ0: a continuous surjection M→M intertwines ρ with ρ0. Because M is a Cantor set for each choice of flag type, semi-conjugacy is the natural C0 equivalence; the paper shows in Section 7 that it cannot generally be improved to conjugacy, even though bi-Lipschitz perturbations would admit such an upgrade. A direct corollary is that for any semisimple p-adic Lie group G, any parabolic subgroup Q","pith_inferences":["A natural testable extension is to apply the same expansivity-plus-Morse-lemma combination to other CAT(0) spaces with a regular quasi-geodesic theory, such as higher-rank symmetric spaces, where no C0 stability result is currently known for nonuniform lattices.","The blowup construction suggests that topological stability, rather than conjugacy, is the correct expected rigidity in the totally disconnected setting; this may guide targeted results in the p-adic analogue of the general program on classifying actions of large groups.","Because the interpolation property is the only place where the higher-rank Morse lemma enters, one could test the machinery by replacing that lemma with a synthetic 'straightness' axiom and checking which groups acting on which flag boundaries satisfy it.","The paper's rank-two counterexample may generalize to all ranks at least two, indicating that the C0 stability theorem is sharp in a way that would not be visible from smooth or bi-Lipschitz rigidity alone."],"forward_implications":["Every cocompact lattice in a semisimple p-adic Lie group acts topologically stably on G/Q for every parabolic subgroup Q.","Semi-conjugacy is optimal in general: the paper constructs arbitrarily small perturbations of a rank-two p-adic flag action that are not conjugate to the original action, only semi-conjugate.","If attention is restricted to bi-Lipschitz perturbations, the semi-conjugacy upgrades to a genuine conjugacy.","The coding criterion is a general device: any action admitting two stable proto-coders satisfying contraction, separation, and interpolation is topologically stable, independent of the building geometry."],"fun_headline_variants":["Lattice actions on building boundaries are topologically stable","Small perturbations of p-adic lattice actions snap back semi-conjugately","Cantor-set boundary actions stay stable under small perturbations","p-adic lattice boundary actions: perturb and you get semi-conjugate"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing assumption is the higher-rank Morse lemma: uniformly regular quasi-geodesic sequences in a Euclidean building stay within a bounded distance of a Weyl cone over a unique boundary flag; if that statement failed, the coding construction and the interpolation property would collapse.","fun_headline_variants_meta":{"raw":{"variants":["Lattice actions on building boundaries are topologically stable","Small perturbations of p-adic lattice actions snap back semi-conjugately","Cantor-set boundary actions stay stable under small perturbations","p-adic lattice boundary actions: perturb and you get semi-conjugate"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000344,"raw_usage":{"total_tokens":1688,"prompt_tokens":665,"completion_tokens":1023,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":409,"completion_tokens_details":{"reasoning_tokens":952}},"tokens_in":409,"tokens_out":1023,"duration_ms":8286,"temperature":1.0,"reasoning_tokens":952,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T01:24:35.818520+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Look for a locally compact Euclidean building and a compact set Θ inside the open star of τmod for which some Θ-regular quasi-geodesic sequence does not lie in any bounded neighborhood of a single Weyl cone V(x0,st(τ)) over a unique τmod-flag. Finding one such sequence would refute the quoted higher-rank Morse lemma and remove the mechanism that converts coding data into regular quasi-geodesics, so the paper's argument would no longer go through.","supporting_citations":[],"review_version":1}