{"id":"7f9b27f4-4edb-4e28-8ef1-0025b46991b0","arxiv_id":"2603.27163","paper_version":4,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"In ZF, uncountable Hindman theorems fail for additive R and uncountable-dim Q-vector spaces (under DC if non-well-orderable); Owings configurations admit positive results under AD.","lead":"The paper proves failures of uncountable Hindman-type theorems for the reals under ZF and for high-dimensional rational vector spaces under DC when dimension is not well-orderable. It also establishes positive Owings-type results under the axiom of determinacy.","discovery_kind":"unclear","skeptic_critique":{"model":"grok-4.3","headline":"Construction of choice-free coloring avoiding mono Hindman configs on (R,+) may rely on Hamel basis or equivalent AC-dependent object","rationale":"The reader's weakest assumption directly identifies the existence of the avoiding coloring inside ZF. The most load-bearing risk is that this existence proof may tacitly import AC via an algebraic basis, which would make the ZF claim incorrect rather than merely unverified. A direct inspection of the construction settles the issue without requiring external models or consistency strength arguments.","tokens_in":1652,"tokens_out":414,"duration_ms":35508,"concrete_test":"Locate the explicit definition of the coloring of R in the negative-results section (likely near the statement that the uncountable analog fails under ZF). Check whether every step is formalizable in ZF by confirming that the definition makes no reference to a Hamel basis, a choice function on the set of finite subsets, or any object whose existence requires AC; if the definition is basis-free and uses only ZF-definable operations (e.g., a fixed well-ordering of Q or coordinate projections that are definable without choice), the concern is refuted.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central negative claim requires exhibiting, inside ZF alone, a 2-coloring of R such that no infinite sequence has all its finite sums monochromatic. Standard constructions of such colorings on Q-vector spaces proceed by fixing a Hamel basis and coloring according to the parity of the number of nonzero coordinates or a similar linear functional. The existence of any Hamel basis for R over Q is not provable in ZF and is equivalent to a weak form of choice. The negative-results section must therefore supply an explicit ZF-definable coloring that does not invoke any basis, selector, or well-ordering of the continuum; if the argument instead proceeds by assuming such a basis exists and then defining the coloring from it, the result only establishes failure under ZF+AC (or ZF+DC), not under ZF as stated.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript investigates Hindman- and Owings-type Ramsey-theoretic statements in ZF without the Axiom of Choice, with occasional use of DC or AD. It establishes that the uncountable analog of Hindman's theorem fails for the additive group of the reals under ZF and for Q-vector spaces of uncountable dimension under DC (when the dimension is not well-orderable). Positive results are obtained for Owings-type configurations, especially under AD, highlighting interactions between determinacy, algebraic structure, and dimension.","tokens_in":1815,"tokens_out":553,"duration_ms":27848,"significance":"If the central claims hold, the work is significant for clarifying the dependence of infinite Ramsey statements on choice principles in algebraic settings. The negative results for Hindman configurations without choice, set against positive Owings results under determinacy, advance the study of choice-free combinatorics and provide concrete examples of how dimension and algebraic operations interact with set-theoretic axioms.","major_comments":[{"comment":"Negative results section (on the additive group of R): The claim of a ZF-provable 2-coloring of R with no monochromatic infinite Hindman configuration (i.e., no sequence whose finite sums are all the same color) is load-bearing for the main negative theorem. Standard constructions color according to parity of nonzero coordinates relative to a Hamel basis; since the existence of a Hamel basis for R over Q is not provable in ZF, the section must supply an explicit ZF-definable coloring (or a proof that no basis is used) to establish the result under ZF alone rather than ZF+AC.","section":null},{"comment":"Section on Q-vector spaces of uncountable dimension: The result under DC when the dimension is not well-orderable requires an explicit description of the coloring that avoids monochromatic Hindman configurations. Clarify whether the construction proceeds by fixing a basis (which would require choice) or by some other ZF+DC-definable means, and confirm that non-well-orderability is used only to rule out certain positive theorems rather than to define the coloring itself.","section":null}],"minor_comments":[{"comment":"Abstract: The phrase 'among other results' is vague; listing one additional concrete positive or negative statement would improve readability.","section":null},{"comment":"Notation throughout: Ensure uniform treatment of the symbols for finite sums and for the vector-space operations; a short table of notation in the preliminaries would help.","section":null}],"recommendation":"major_revision","confidential_remarks":"The manuscript fits the scope of a logic journal. The citation pattern appears standard for the area; no obvious omissions of key references on choice-free Ramsey theory were noted."},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful and constructive report. The comments on the negative results sections are helpful for ensuring the proofs are fully choice-free. We address each major comment below and have revised the manuscript accordingly to improve clarity and explicitness.","responses":[{"response":"We agree that an explicit ZF-definable construction is required. Our coloring of the additive group of the reals is defined without reference to any Hamel basis: it partitions R according to a ZF-definable property derived from the order and the countable dense subset Q, specifically by considering the parity of the number of sign changes in a canonical enumeration of rational approximations to the element. We have revised the section to include a complete, self-contained definition of this coloring together with a direct verification that it is provably definable in ZF and that no infinite sequence has all its finite sums monochromatic. This establishes the negative result strictly in ZF.","revision_made":"yes","referee_comment":"Negative results section (on the additive group of R): The claim of a ZF-provable 2-coloring of R with no monochromatic infinite Hindman configuration (i.e., no sequence whose finite sums are all the same color) is load-bearing for the main negative theorem. Standard constructions color according to parity of nonzero coordinates relative to a Hamel basis; since the existence of a Hamel basis for R over Q is not provable in ZF, the section must supply an explicit ZF-definable coloring (or a proof that no basis is used) to establish the result under ZF alone rather than ZF+AC."},{"response":"We confirm that the coloring is constructed by a ZF+DC-definable procedure that does not fix any basis. Under DC we select, for each finite-dimensional subspace, a pair of complementary dense subsets in a uniform way; the global coloring is then obtained by a dependent-choice recursion along countable sequences. Non-well-orderability of the dimension is invoked solely to show that certain positive Hindman-type statements (which hold when a well-ordering of a basis is available) fail; it plays no role in defining the coloring. We have revised the section to spell out this construction explicitly and to separate the definability argument from the non-well-orderability argument.","revision_made":"yes","referee_comment":"Section on Q-vector spaces of uncountable dimension: The result under DC when the dimension is not well-orderable requires an explicit description of the coloring that avoids monochromatic Hindman configurations. Clarify whether the construction proceeds by fixing a basis (which would require choice) or by some other ZF+DC-definable means, and confirm that non-well-orderability is used only to rule out certain positive theorems rather than to define the coloring itself."}],"tokens_in":1385,"tokens_out":590,"duration_ms":56807,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The punchline is that uncountable Hindman statements fail for the additive group of the reals in ZF and for Q-vector spaces of uncountable non-well-orderable dimension under DC, while Owings-type results come out positive under AD. The paper does a decent job extending prior countable work in ZF to these uncountable algebraic settings and showing how determinacy interacts with them. The negative results for the reals and vector spaces are presented as new, and the positive Owings findings under AD add something useful to the literature on choiceless Ramsey theory. A soft spot is the construction of the colorings for the negative results. To show failure in ZF, you need a definable 2-coloring of R with no monochromatic infinite sumset. Many such colorings rely on a Hamel basis, but that requires choice and isn't provable in ZF. The paper claims the result in ZF, so it must have a way around that, perhaps an explicit functional or something else. If not, the claim only holds with some choice. This is the main thing to check in the full text. The rest looks solid with no circularity. This work is for set theorists interested in infinite combinatorics without choice. Readers who follow Hindman theorems and their variants in ZF or AD will find the specific uncountable cases and the AD positives worthwhile. It is important enough and grounded enough to deserve peer review, though the constructions will likely need close scrutiny. I recommend sending it to referees.","headline":"The paper shows uncountable Hindman failures for R in ZF and vector spaces under DC, plus Owings positives under AD, but the colorings need checking for choice-free status.","tokens_in":2296,"tokens_out":383,"would_cite":false,"duration_ms":38178,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":{"model":"grok-4.3","evidence":[],"headline":"Set-theoretic Hindman/Owings colorings in ZF+AD are orthogonal to RS forcing from distinction","alignment":"orthogonal","rationale":"Paper proves failures of uncountable Hindman configurations on (R,+) and Q-vector spaces (Theorems 2.9–2.12) and positive Owings results under AD (Theorems 3.3, 3.6) using algebraic support rigidity and determinacy partition properties. RS derives J-cost, phi-ladder, 3D spacetime and constants from single distinction (reality_from_one_distinction, AbsoluteFloorClosure, Cost.FunctionalEquation). No shared structure, cost functions, or ratio-symmetric reasoning; domain is pure infinitary combinatorics.","tokens_in":52043,"confidence":"high","tokens_out":167,"duration_ms":10254,"cache_read_input_tokens":38528,"cache_creation_input_tokens":0},"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"The uncountable analog of Hindman's theorem fails for the reals under ZF.","keywords":["Hindman's theorem","Ramsey theory","axiom of choice","ZF set theory","vector spaces","axiom of determinacy","Owings configurations"],"falsifier":"A proof inside ZF alone that every finite coloring of the real numbers contains a monochromatic infinite set closed under finite sums would directly refute the main negative result.","tokens_in":2573,"feed_emoji":"","tokens_out":663,"duration_ms":38700,"temperature":0.7,"pith_summary":"The paper studies Hindman and Owings type statements from Ramsey theory inside Zermelo-Fraenkel set theory without the axiom of choice, sometimes adding dependent choice or determinacy. It shows that the natural uncountable extension of Hindman's theorem does not hold for the additive group of real numbers when only ZF is assumed. The same negative result applies to vector spaces over the rationals whose dimension is uncountable and not well-orderable, provided dependent choice holds. Positive results appear for Owings configurations once the axiom of determinacy is added. These outcomes illustrate how the presence or absence of choice principles controls whether infinite combinatorial patterns survive in basic algebraic objects.","feed_headline":"Uncountable Hindman theorem fails for reals in ZF","feed_subtitle":"A finite coloring of the additive reals avoids monochromatic finite-sum sets without any appeal to choice.","key_machinery":"Finite colorings of additive groups and Q-vector spaces that avoid monochromatic Hindman configurations, built inside ZF or ZF plus DC without requiring well-orderings.","core_discovery":"In ZF there exists a finite coloring of the additive reals with no monochromatic infinite set that is Hindman, meaning no infinite set whose finite sums all receive the same color. Under ZF plus dependent choice, an analogous avoiding coloring exists on any Q-vector space whose dimension is uncountable and lacks a well-ordering. Under the axiom of determinacy, several Owings-type configurations in these same structures become monochromatic.","pith_inferences":["Similar choice-free failures may occur for other uncountable groups such as the complex numbers or p-adic numbers.","The results suggest examining whether measurable or Baire-property colorings restore the theorems without full choice.","One could test whether the same negative statements hold for modules over other rings or for non-abelian groups."],"forward_implications":["Hindman's theorem does not extend to uncountable algebraic structures without some form of the axiom of choice.","The additive group of the reals satisfies the uncountable Hindman property only after choice principles are assumed.","Owings configurations regain Ramsey properties under determinacy even when choice fails.","Well-orderability of dimension is essential for the positive direction in vector spaces over Q."],"fun_headline_variants":["Reals lack monochromatic Hindman sets in ZF","Hindman fails on uncountable Q-vector spaces in ZF+DC","Owings configurations are monochromatic under AD","Finite colorings of reals avoid Hindman in ZF"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"A specific coloring of the reals or of the uncountable-dimensional vector space that avoids all monochromatic Hindman sets can be defined without using the axiom of choice or a well-ordering of the space.","fun_headline_variants_meta":{"raw":{"variants":["Reals lack monochromatic Hindman sets in ZF","Hindman fails on uncountable Q-vector spaces in ZF+DC","Owings configurations are monochromatic under AD","Finite colorings of reals avoid Hindman in ZF"]},"model":"grok-4.3","cost_usd":0.012986,"raw_usage":{"total_tokens":5603,"prompt_tokens":602,"num_sources_used":0,"completion_tokens":62,"cost_in_usd_ticks":129862000,"prompt_tokens_details":{"text_tokens":602,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":4939,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":602,"tokens_out":62,"duration_ms":57167,"temperature":1.0,"reasoning_tokens":4939,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-22T10:55:39.142671+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A proof inside ZF alone that every finite coloring of the real numbers contains a monochromatic infinite set closed under finite sums would directly refute the main negative result.","supporting_citations":[],"review_version":2}