{"id":"bffd90de-67c6-4194-9321-954cd51aab69","arxiv_id":"1907.03923","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Generalizes bornological coarse spaces by omitting singleton boundedness and proves the inclusion induces equivalence of motivic coarse spectra and C-valued coarse homology theories.","lead":"The paper generalizes bornological coarse spaces by dropping the requirement that single-point sets are bounded, yielding a complete and cocomplete category. It constructs motivic coarse spectra in this larger setting and proves that the inclusion from the original category induces an equivalence, so the associated C-valued coarse homology theories coincide.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest_assumption correctly isolates the imitation step as the point that must hold for the equivalence claim. Because the supplied material contains no counter-example, hidden assumption, or failure of a cited lemma in that step, the provisional UNVERDICTED verdict does not require adjustment on the basis of an identified flaw.","tokens_in":1631,"tokens_out":300,"duration_ms":17262,"concrete_test":"Verify that the unit of the adjunction between the motivic spectra functors (ordinary vs. generalized) is an equivalence on the image of the inclusion by checking, for the sphere spectrum as C, whether the coarse homology of a standard bornological coarse space computed in the generalized category agrees with the original computation up to equivalence.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that the inclusion of ordinary bornological coarse spaces into the generalized category (singletons no longer required to be bounded) induces an equivalence on motivic coarse spectra, hence on C-valued coarse homology theories for any stable cocomplete ∞-category C. The manuscript states that it obtains a complete and cocomplete generalization and then imitates the original motivic construction, proving the induced equivalence. No internal inconsistency, missing axiom, or unsupported step in that imitation is visible from the given material; the argument is presented as a direct transfer of the construction that preserves the necessary universal properties.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript generalizes bornological coarse spaces by dropping the requirement that singletons are bounded, producing a complete and cocomplete category. It imitates the motivic coarse spectra construction from the original setting in this generalized category and proves that the inclusion functor induces an equivalence of motivic coarse spectra. Consequently, for any stable cocomplete ∞-category C the categories of C-valued coarse homology theories on the two classes of spaces are equivalent.","tokens_in":1743,"tokens_out":254,"duration_ms":15961,"significance":"If the result holds, the equivalence demonstrates that the motivic construction is insensitive to the boundedness of singletons, allowing a strictly larger class of spaces while preserving the universal properties of coarse homology theories. The explicit construction of a complete and cocomplete generalization together with the direct transfer of the equivalence for arbitrary stable cocomplete C are concrete strengths.","major_comments":[],"minor_comments":[{"comment":"The abstract and introduction would benefit from a brief sentence clarifying whether the generalized category is strictly larger (i.e., contains objects that are not bornological coarse spaces) or merely relaxes an axiom while keeping the same objects.","section":null}],"recommendation":"accept","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their positive summary, recognition of the significance of the result, and recommendation to accept the manuscript.","responses":[],"tokens_in":1095,"tokens_out":43,"duration_ms":7002,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The central result is that the inclusion from ordinary bornological coarse spaces into the version where singletons need not be bounded induces an equivalence of motivic coarse spectra. This immediately gives an equivalence on C-valued coarse homology theories for any stable cocomplete infinity-category C. The authors first verify that the relaxed category remains complete and cocomplete, then copy the existing motivic construction and check that the inclusion preserves the relevant universal properties so the spectra match. That equivalence is the actual new statement; the rest follows the prior template closely. The argument looks direct and avoids circularity because both categories are built explicitly and the functor is the obvious inclusion. No load-bearing steps appear to be left unverified from the abstract and stress-test summary. The main limitation is that the work is a targeted relaxation rather than a broader advance; it will mainly interest people already using coarse motivic spectra who want a slightly larger ambient category without changing the output. A reader working on homology theories in this area would find the equivalence useful for checking whether certain technical restrictions can be dropped. The paper is coherent on its own terms and shows clear engagement with the existing literature on the construction it imitates. It deserves a serious referee to check the details of the imitation and the preservation of the universal properties in the full text.","headline":"The paper shows that dropping the singleton-boundedness condition from bornological coarse spaces still produces an equivalent category of motivic coarse spectra via the inclusion functor.","tokens_in":2179,"tokens_out":334,"would_cite":false,"duration_ms":12026,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":{"model":"grok-4.3","evidence":[{"relation":"unclear","rs_module":"IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean","rs_theorem":"absolute_floor_iff_bare_distinguishability","paper_passage":"We generalize the notion of a bornology by omitting the condition that a one-point-subset is bounded and obtain a complete and co-complete generalization of the category of bornological coarse spaces. Then we imitate the construction of motivic coarse spectra in this new setting and show that the inclusion functor ... induces an equivalence of motivic coarse spectra."},{"relation":"unclear","rs_module":"IndisputableMonolith/Cost/FunctionalEquation.lean","rs_theorem":"washburn_uniqueness_aczel","paper_passage":"the inclusion functor from the category of bornological coarse spaces to its generalization induces an equivalence of motivic coarse spectra"}],"headline":"Categorical generalization of bornological coarse spaces and motivic spectra has no overlap with RS forcing chain","alignment":"orthogonal","rationale":"The paper constructs a generalization of bornological coarse spaces (dropping singleton boundedness) yielding a bicomplete category, then imitates the motivic coarse spectra construction to obtain an equivalence of spectra (and thus C-valued coarse homology theories) under the inclusion functor. This is pure homotopy-theoretic category theory with no reference to recognition cost J, golden-ratio identities, 8-tick periodicity, or parameter-free derivation of constants. RS modules such as AbsoluteFloorClosure, Cost/FunctionalEquation, and DimensionForcing address the distinction-to-spacetime forcing chain and have no opinion on this domain.","tokens_in":58574,"confidence":"high","tokens_out":369,"duration_ms":6593,"cache_read_input_tokens":38528,"cache_creation_input_tokens":0},"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"The inclusion of bornological coarse spaces into their generalization induces an equivalence of motivic coarse spectra.","keywords":["bornological coarse spaces","generalized bornologies","motivic coarse spectra","coarse homology theories","infinity-categories","coarse geometry","algebraic topology"],"falsifier":"An explicit stable cocomplete infinity-category C together with a C-valued coarse homology theory whose value on some generalized bornological coarse space differs from its value on the corresponding bornological space.","tokens_in":2520,"feed_emoji":"📐","tokens_out":631,"duration_ms":32820,"temperature":0.7,"pith_summary":"The paper enlarges the category of bornological coarse spaces by dropping the requirement that single-point sets must be bounded. It then copies the construction of motivic coarse spectra into this larger category. The central result is that the natural inclusion functor between the two categories produces an equivalence of these spectra. As a direct consequence the categories of coarse homology theories valued in any stable cocomplete infinity-category become equivalent on the two sides. A reader would care because the result shows that the extra generality does not change the homotopy-theoretic information carried by coarse invariants.","feed_headline":"Inclusion of bornological spaces yields equivalence of coarse spectra","feed_subtitle":"Dropping the bounded-singleton rule from bornologies leaves the homotopy theory of coarse homology theories unchanged","key_machinery":"The motivic coarse spectra construction, imitated directly in the generalized category obtained by omitting the bounded-singleton condition.","core_discovery":"The inclusion functor from the category of bornological coarse spaces to the category of generalized bornological coarse spaces induces an equivalence of motivic coarse spectra. In particular, for any stable cocomplete ∞-category C this induces an equivalence between the category of C-valued coarse homology theories on bornological coarse spaces and the category of C-valued coarse homology theories on generalized bornological coarse spaces.","pith_inferences":["The boundedness condition on single points is inessential for the coarse homotopy theory.","The equivalence may allow direct transfer of existing computations to spaces previously excluded by the bounded-singleton rule.","Similar inclusion arguments could be tested in other generalized geometric categories that relax boundedness or properness conditions."],"forward_implications":["For every stable cocomplete ∞-category C the categories of C-valued coarse homology theories on the two kinds of spaces are equivalent.","Any coarse homology theory defined on bornological coarse spaces extends canonically to generalized bornological coarse spaces without altering its values on the original objects.","Homotopy-theoretic properties of coarse invariants established in the bornological setting remain valid after the generalization.","The motivic approach to coarse geometry applies verbatim to the larger class of spaces."],"fun_headline_variants":["Bornological inclusion equates coarse motivic spectra","Generalized bornologies equivalent in coarse spectra","Inclusion preserves equivalence of coarse homology theories","Coarse spectra equivalence holds for generalized bornologies"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The motivic coarse spectra construction can be performed in the generalized category while preserving the universal properties that make the inclusion induce an equivalence.","fun_headline_variants_meta":{"raw":{"variants":["Bornological inclusion equates coarse motivic spectra","Generalized bornologies equivalent in coarse spectra","Inclusion preserves equivalence of coarse homology theories","Coarse spectra equivalence holds for generalized bornologies"]},"model":"grok-4.3","cost_usd":0.00633,"raw_usage":{"total_tokens":2914,"prompt_tokens":549,"num_sources_used":0,"completion_tokens":53,"cost_in_usd_ticks":63299500,"prompt_tokens_details":{"text_tokens":549,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2312,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":549,"tokens_out":53,"duration_ms":79709,"temperature":1.0,"reasoning_tokens":2312,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-25T00:19:00.297217+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit stable cocomplete infinity-category C together with a C-valued coarse homology theory whose value on some generalized bornological coarse space differs from its value on the corresponding bornological space.","supporting_citations":[],"review_version":1}