{"id":"d6410579-53b9-46ef-8b3a-a58cbb55c9c9","arxiv_id":"2608.12749","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Domination of geometric data sets is closed under box convergence, and limit formulas recover observable diameter and multi-directed separation after one-sided perturbations of mass parameters.","lead":"This paper develops a one-sided version of Gromov and Shioya's box geometry for geometric data sets and quasi-metric measure spaces, proving when approximate maps between spaces imply domination in the limit. It also gives formulas for recovering concentration invariants such as observable diameter from weak limits of pyramids.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central transfer theorems inherit unverified compactness and weak-convergence criteria from companion preprints; if [11, Thm 3.19] or [11, Prop 7.2] fails, Theorem 3.11 and the limit formulas collapse.","rationale":"The strongest claim is the finite-measurement transfer to subsequential pyramid limits. The internal proofs of Lemmas 3.10, 3.13, 4.7, and the estimates in Theorems 4.5 and 4.9 are coherent and largely self-contained once the companion results are granted. I found no internal inconsistency in the main construction, and the manuscript's explicit list of upstream imports is a point in its favor. However, the decisive steps, namely compactness of measurement sets, weak-convergence detection, and reconstruction of an L-gd-set from bounded finite-feature models, are not derived in this manuscript. The reader's verdict of CONDITIONAL is therefore appropriate: acceptance should be conditional on independent verification of the companion theorems. My check does not change that verdict, and I agree with the reader's identification of the weakest assumption.","tokens_in":21110,"tokens_out":21708,"duration_ms":200973,"concrete_test":"Obtain arXiv:2603.23325 ([11]) and arXiv:2608.01145 ([12]); verify the exact statements and proofs of [11, Theorem 3.19], [11, Proposition 7.2], and [12, Theorem A.7]. In particular, instantiate [11, Theorem 3.19] with L=TB and X=R, F_X=Lip_1(R), mu=N(0,1) to test whether the theorem holds without extra boundedness or tightness assumptions; then check that the same hypotheses are satisfied by the qm-space representations Rep_+(X_n) used in Theorem 3.11. If any proof requires an additional condition not satisfied here, the main limit-transfer claims fail.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 3.11 depends on Lemma 3.14's reconstruction assertion and Lemma 3.12's compactness of M(P;N,R), both justified by citations to the unpublished same-author preprint [11]. The sentence before Lemma 3.12 asserts: 'By [11, Theorem 3.19], every L-closed gd-set is then L-compact, so pyramids in L circle D are the L-pyramids in D/L to which the results of [11] apply.' Lemma 3.14 then invokes [11, Proposition 7.2] to convert Hausdorff convergence of ordered finite measurements into weak convergence of pyramids, and [11, Lemmas 4.8 and 4.9] to reconstruct any L-gd-set from bounded finite-feature models. Appendix A additionally relies on [12, Theorem A.7 and Corollary A.12]. None of these results is proved or machine-checked here. If [11, Theorem 3.19]'s compactness claim fails for the TB-closed function families arising from Rep_+(X_n) (for example, if it requires a uniform bound on observable diameters that arbitrary qm-spaces need not satisfy), then Lemma 3.12's compactness conclusion fails, Lemma 3.14's equivalence and reconstruction fail, and the step 'Theorems 3.12 and 3.14 gives lambda in M(P;N,R)' in the proof of Theorem 3.11 is no longer valid. Theorems 4.5 and 4.9, which use the same finite-measurement transfer, would also not follow. The manuscript's explicit list of imported statements is honest but does not reduce the logical risk.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper develops a one-sided version of Gromov--Shioya box geometry for geometric data sets (gd-sets) and quasi-metric measure spaces (qm-spaces). It defines a one-sided box distance \\square^{\\preceq}, proves its basic order properties and closure under box convergence (Theorem 1.1(i), Propositions 3.2--3.5), and establishes a high-mass approximate domination transfer: Borel maps defined on asymptotically full-measure subsets, with pushforwards converging weakly and directed-distance error tending to zero, force the target representation into every subsequential weak limit of the source pyramids (Theorem 1.1(ii), Proposition 3.11). For weakly convergent pyramids, the paper proves limit formulas for observable diameter after a rightward mass perturbation (Theorem 4.5) and for multi-directed separation after a common leftward mass perturbation (Theorem 4.9). A final section gives elementary comparisons between observable diameter, nonnegative directed separation, and one-sided concentration functions, together with a characterization of function-family L\\'evy families (Corollary 5.6), illustrated by a discrete example. The paper is explicitly built on three companion papers by the same author, and it contains a self-declared ledger of every imported statement from those papers.","tokens_in":21475,"tokens_out":16652,"duration_ms":148319,"significance":"If the companion framework is correct, the results are a substantive extension of concentration and compactness theory to directed and asymmetric structures. The main formulas are parameter-free in the sense that no fitted constants appear: the perturbations in Theorems 4.5 and 4.9 are dictated by the limiting argument, not by normalization. The paper is also unusually transparent about its external dependencies, listing each imported result from [10], [11], and [12]. Several proofs are detailed and checkable, notably the zero-set argument in Proposition 3.4 and the squeeze estimates in Propositions 4.5 and 4.9. The discrete example in Section 5 cleanly illustrates the difference between function-family invariants and underlying-space invariants. The significance is conditional: the central transfer theorems 3.11, 4.5, and 4.9 rest on compactness, weak-convergence detection, and reconstruction results imported from unpublished same-author preprints, and those imports are not proved or machine-checked in this manuscript.","major_comments":[{"comment":"The central transfer theorems depend on unverified statements from the companion preprints [11] and [12]. In particular, the sentence before Lemma 3.12 asserts that by [11, Theorem 3.19] every L-closed gd-set is L-compact; Lemma 3.12 uses this to prove compactness of M(P;N,R), and Lemma 3.14 uses [11, Proposition 7.2] to convert Hausdorff convergence of finite measurements into weak convergence of pyramids and [11, Lemmas 4.8–4.9] for reconstruction. Proposition 3.11 then invokes Lemma 3.14 to place the target in the limit pyramid, and Propositions 4.5 and 4.9 use the same finite-measurement transfer. Appendix A additionally relies on [12, Theorem A.7, Proposition A.8, Corollary A.12]. None of these statements is proved here, and the manuscript does not state the precise hypotheses of [11, Theorem 3.19] nor verify, for instance, that the function families arising from Rep_+(X_n) satisfy any uniform boundedness condition that L-compactness might require. If the compactness or weak-convergence detection in [11] fails in this setting, then the step 'Theorems 3.12 and 3.14 gives λ in M(P;N,R)' in the proof of Proposition 3.11 is invalid, and Theorems 4.5 and 4.9 collapse as well. The explicit ledger of imported results is commendable, but it does not supply the missing verification. I recommend that the author either reproduce the needed statements and proofs in an appendix or ensure that the companion papers are accepted and publicly available in their final form, and that the manuscript explicitly verify that the hypotheses of [11, Theorem 3.19] hold for the function families used here.","section":"§3, Proposition 3.3"},{"comment":"In the triangle inequality proof, the set U = pr13({(x,y,z) | (x,y)∈S, (y,z)∈T}) need not be closed, since it is a projection of a closed subset of a noncompact product. The one-sided box relation in Definition 3.1 requires a closed witness set. The proof should replace U by its closure \\bar{U}; the mass estimate passes to \\bar{U} because \\theta(\\bar{U}) ≥ \\theta(U), and the approximate inequality for h and f passes to the closure by continuity of the functions involved. As written, the proof is incomplete at this point, though the repair is local.","section":"§3, Proposition 3.3"}],"minor_comments":[{"comment":"Several cross-references use inconsistent numbering: the introduction refers to 'Theorem 3.11' for the high-mass domination result, which is stated as Proposition 3.11; Proposition 3.8 refers to 'Theorem 3.1' where Definition 3.1 is meant; Proposition 3.11 refers to 'Theorem 3.10' where Lemma 3.10 is meant; and the proof of Proposition 3.11 says 'Theorems 3.12 and 3.14 gives' where both are lemmas.","section":"Throughout"},{"comment":"In the proof of Proposition 4.5, when applying Equation (20), the passage 'Letting R→+∞' could be made more explicit for the case of infinite observable diameter; the argument is standard but the monotone-convergence step is not spelled out.","section":"§4, Proposition 4.5"},{"comment":"The proof of the implication (ii)⇒(iii) handles κ>1/2 by saying that no admissible pairs exist; since Sep+ is then zero by Definition 5.1, this is correct, but it would be clearer to state explicitly that the vacuous case is covered by the definition.","section":"§5, Corollary 5.6"}],"recommendation":"major_revision","confidential_remarks":"The main risk is the dependence on three same-author preprints, two of which are unpublished. If the companion results are accepted and the authors verify that their hypotheses cover the function families and pyramids used here, the paper is likely publishable in its present mathematical direction. The small closure gap in Proposition 3.3 is easily fixed. I would not reject, but the current manuscript cannot be certified without access to or verification of the imported compactness and transport results."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a real extension, not a repackaging. Yokota introduces a one-sided box distance for gd-sets, proves it metrizes approximate directed domination, and shows that high-mass approximate 1-Lipschitz maps push the target into every weak subsequential limit of the source pyramids. Theorems 4.5 and 4.9, giving right-perturbation limits for observable diameter and left-perturbation limits for multi-directed separation, are the payoffs; the Section 5 characterization of function-family Levy families via two tail functions is a useful standalone result. Credit where due: the proofs carried out here are mostly detailed and checkable. Lemma 3.10, the one-sided McShane-Whitney extension with measure control, is nice, and the squeeze estimates in 4.5 and 4.9 are coherent. The paper's explicit list of every imported statement from [10]-[12] is good scholarly hygiene.\n\nThe soft spots are real but proportionate. First, Proposition 3.3: U is defined as the projection of a closed set, and projections of closed sets need not be closed. The line \"Continuity gives the corresponding non-strict estimate on U\" wants an argument or a slightly different definition. I think this is a small gap, not a fatal one. The larger issue is the one the stress-test flags: Lemmas 3.12-3.14 and Appendix A lean on [11, Thm 3.19, Prop 7.2, Lemmas 4.8/4.9] and [12, Thm A.7/Cor A.12], all same-author preprints whose proofs are not reproduced here. If the compactness theorem in [11] has hidden hypotheses that arbitrary qm-spaces do not satisfy, Theorems 3.11, 4.5 and 4.9 collapse. I did not find an independent reason to think the imports fail, and the author is unusually transparent about them, but transparent dependency is still dependency. A referee should be able to inspect [11] and [12] alongside, or the author should move the needed statements into this paper.\n\nThe citation pattern is fine: earlier symmetric work is credited, and self-citations point to the specific statements used.\n\nWho is this for? People working in metric measure geometry, concentration of measure, and irreversible Finsler spaces. It deserves a serious referee. I would not desk-reject. I would send it out and ask for the Proposition 3.3 closure fix and either inclusion or independent verification of the companion theorems before acceptance.","headline":"A serious directed analogue of Shioya's pyramid machinery with clean limit formulas; the load-bearing steps depend on same-author preprints, so the paper deserves refereeing but needs those imports made independently checkable.","tokens_in":21996,"tokens_out":1853,"would_cite":true,"duration_ms":17666,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C23","54E35","28A33"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper establishes that approximate domination is closed under box convergence, and that high-mass approximate directed maps force the target representation into every subsequential weak limit of the source pyramids.","keywords":["geometric data set","quasi-metric measure space","box distance","pyramid","observable diameter","separation distance","semi-Lipschitz functions","Levy family"],"falsifier":"A concrete way to settle the main claim is to find qm-spaces $X_n$ and a target $Y$ that satisfy every hypothesis of Theorem 3.11 but for which $\\mathrm{Rep}_+(Y)$ is absent from some weak limit of the pyramids of $\\mathrm{Rep}_+(X_n)$, or to compute an explicit weakly convergent pyramid sequence for which the right-perturbed upper limit in equation (22) differs from $\\mathrm{ObsDiam}(P;-\\kappa)$, which would disprove Theorem 4.5.","tokens_in":20912,"feed_emoji":"📐","tokens_out":12149,"duration_ms":99871,"temperature":0.7,"pith_summary":"This paper extends measured geometry to asymmetric and function-family settings, where a chosen family of functions, rather than the full Lipschitz class, defines the structure. Its central claim is that approximate domination is closed under box convergence: if each source dominates a target up to a vanishing additive error while both converge in box distance, then the limit source genuinely dominates the limit target. For directed spaces, approximate 1-Lipschitz maps defined only on Borel subsets of measure tending to one, with pushforwards converging weakly to the target measure, are shown to place the target representation in every subsequential weak limit of the source pyramids. For weakly convergent pyramids, observable diameter and multi-directed separation are recovered by perturbing their mass parameters and taking lower or upper limits, with both limits agreeing. This matters because finite measurement data from nearly all of the space, without exact global maps or symmetry, are enough to determine the limiting directed geometry.","feed_headline":"Approximate high-mass maps force the target into pyramid limits","feed_subtitle":"Nearly every point's worth of directed data carries a target representation into every subsequential weak limit.","key_machinery":"The load-bearing machinery is the one-sided box distance and the finite measurements attached to function families. The relation $Y\\preceq_{\\varepsilon} X$ means that a coupling $\\pi$ and a closed relation $S$ carry mass at least $1-\\varepsilon$, and every function of $Y$ is within $\\varepsilon/2$ in sup norm on $S$ of some function of $X$; the one-sided box distance $\\square_{\\preceq}(Y,X)$ is the infimum of such $\\varepsilon$. Limits are read through ordered finite measurements $M(X;N,R)$, the Prokhorov-closed sets of pushforwards of clipped $N$-tuples of functions, together with a reconstruction theorem: an $L$-gd-set belongs to a pyramid once its finite measurements are contained in the pyramid's. For qm-spaces the class $\\mathrm{Lip}_1^+(X)=\\{f:f(y)-f(x)\\le d_X(x,y)\\}$ represents the directed distance, and an inf-convolution extension lemma, a semi-Lipschitz version of McShane--Whitney, fills in maps defined only on high-mass subsets. The standing monoidal closure $TB\\subset L$ identifies closed gd-sets with the compact pyramid class to which the paper's compactness and weak-convergence criteria apply.","core_discovery":"At the center is a stability theorem for one-sided box geometry. The one-sided box distance $\\square_{\\preceq}(Y,X)$, the infimum of the additive error with which $Y$ is dominated by $X$, is shown to satisfy a triangle inequality and to vanish exactly when $Y\\preceq X$; therefore domination is closed under box convergence. For qm-spaces, the paper proves that Borel maps $p_n$ from sets $A_n\\subset X_n$ with $\\mu_n(A_n)\\to 1$ to $Y$, whose pushforwards $(p_n)_*(\\mu_n|_{A_n})$ converge weakly to $\\nu$ and whose directed distance distortion satisfies $d_Y(p_n(x),p_n(y))\\le d_n(x,y)+\\varepsilon_n$ with $\\varepsilon_n\\to0$, force $\\mathrm{Rep}_+(Y)$ to belong to every subsequential weak limit of the pyramids generated by $\\mathrm{Rep}_+(X_n)$. For weakly convergent pyramids, the exact limit formulas are established: $\\mathrm{ObsDiam}(P;-\\kappa)$ equals, as $\\varepsilon\\downarrow0$, both the lower and upper limits of $\\mathrm{ObsDiam}(P_n;-(\\kappa+\\varepsilon))$, and $\\mathrm{Sep}(P;\\kappa_0,\\ldots,\\kappa_N)$ equals both limits of $\\mathrm{Sep}(P_n;\\kappa_0-\\varepsilon,\\ldots,\\kappa_N-\\varepsilon)$. In the general unclosed setting, observable diameter, nonnegative separation, and the two one-sided median-tail masses are compared, and a sequence is a function-family Levy family exactly when both median-tail masses vanish at every positive radius.","pith_inferences":["As an extension beyond the paper, the same perturbation mechanism should apply to any directed invariant that is monotone in its mass parameters, not only observable diameter and separation; the proofs use only monotonicity plus finite-measurement approximation.","A testable prediction beyond the paper is that the one-sided concentration functions satisfy limit formulas under weak pyramid convergence parallel to Theorem 4.5, because they are suprema over the same one-sided Lipschitz observables.","The high-mass subset theorem could become a numerical recipe: sample growing random subsets of a candidate target, simulate maps on them, check the pushforward measure and the directed distance distortion, and certify pyramid membership; this recipe is my inference, not a claim of the paper.","For symmetric mm-spaces the same formulas should reduce to the classical limit statements, so the asymmetric perturbations are best read as a directed refinement of the existing theory; this comparison is my inference."],"forward_implications":["If Theorem 3.11 is right, a target can be certified as belonging to a weak limit by constructing approximate 1-Lipschitz maps on sets of measure tending to one; exact global maps are unnecessary.","If Theorems 4.5 and 4.9 are right, the limiting observable diameter and separation distance of any weakly convergent pyramid sequence are computable from bounded finite-tuple measurement distributions, with the directional mass perturbations built into the formulas.","Domination being box-closed means that convergence in box distance cannot lose the relation 'which function family approximates which'; approximate relations pass to exact ones in the limit.","Without closure assumptions on the function family, the Section 5 inequalities hold for every gd-set, requiring neither constants, truncations, nor inf-convolutions among the observables.","The equivalent description of function-family Levy families says that uniform closeness to constants around the whole family is the same as both one-sided median-tail masses vanishing at every positive radius."],"supporting_citations":[{"why":"This reference defines gd-sets, domination, the box distance, and the Prokhorov and partial-diameter lemmas used throughout the paper.","marker":"[10]"},{"why":"This reference supplies compactness of closed gd-sets and the finite-measurement criterion for weak convergence of pyramids on which Lemmas 3.12 through 3.14 depend.","marker":"[11]"},{"why":"This reference provides pyramid transport and the equivalence of weak convergence used to pull invariant limit formulas back along pyramidal adjunctions.","marker":"[12]"},{"why":"This reference supplies lower-semicontinuity of partial diameter under weak convergence, used in Proposition 4.4.","marker":"[4]"},{"why":"This reference supplies the ambient metric-measure geometry, the gluing and Strassen coupling tools, and compactness of the measurement spaces.","marker":"[6]"},{"why":"This reference introduces semi-Lipschitz functions on quasi-metric spaces, which underpin the one-sided Lipschitz class and the extension lemma.","marker":"[5]"}],"fun_headline_variants":["Box stability forces pyramid limits under approximation","New inequalities pin observables in weak limit pyramids","Domination closed under box convergence, pyramids follow","Median-tail masses vanish iff Levy family emerges"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the companion preprints' theorems are correct, namely that every closed gd-set is compact and that weak convergence of pyramids can be detected through finite measurements, and the proof transfers its results through those theorems without reproducing their proofs.","fun_headline_variants_meta":{"raw":{"variants":["Box stability forces pyramid limits under approximation","New inequalities pin observables in weak limit pyramids","Domination closed under box convergence, pyramids follow","Median-tail masses vanish iff Levy family emerges"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000124,"raw_usage":{"total_tokens":1159,"prompt_tokens":1053,"completion_tokens":106,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":669,"completion_tokens_details":{"reasoning_tokens":50}},"tokens_in":669,"tokens_out":106,"duration_ms":1861,"temperature":1.0,"reasoning_tokens":50,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T00:04:42.181301+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete way to settle the main claim is to find qm-spaces $X_n$ and a target $Y$ that satisfy every hypothesis of Theorem 3.11 but for which $\\mathrm{Rep}_+(Y)$ is absent from some weak limit of the pyramids of $\\mathrm{Rep}_+(X_n)$, or to compute an explicit weakly convergent pyramid sequence for which the right-perturbed upper limit in equation (22) differs from $\\mathrm{ObsDiam}(P;-\\kappa)$, which would disprove Theorem 4.5.","supporting_citations":[{"cited_title":"Pyramidal Compactification of Asymmetric Metric Measure Spaces via Adjoint Transport","cited_arxiv_id":"2608.01145","evidence_quote":"This reference provides pyramid transport and the equivalence of weak convergence used to pull invariant limit formulas back along pyramidal adjunctions."},{"cited_title":"Shioya.Metric measure geometry, volume 25 ofIRMA Lectures in Mathematics and Theoretical Physics","cited_arxiv_id":null,"evidence_quote":"This reference supplies the ambient metric-measure geometry, the gluing and Strassen coupling tools, and compactness of the measurement spaces."}],"review_version":1}