{"id":"ed8f65d6-7afe-4ceb-9d4d-979f7dce8b11","arxiv_id":"2501.18527","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A neural network relaxation of geometric coloring constraints produced new plane colorings, including an almost 5-coloring covering all but 3.74% of the plane, improving known bounds for Hadwiger-Nelson variants.","lead":"This paper uses neural networks to search for colorings of the plane that avoid monochromatic points at specified distances, a family of problems rooted in the Hadwiger-Nelson question. It reports new colorings for several variants and a method that turned network outputs into rigorous constructions.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Headline six-colorings are asserted but not proven in this text; the claimed [0.354,0.657] continuum rests entirely on a companion paper, so the paper's strongest claim is conditionally verified at best.","rationale":"The reader's weakest assumption identifies the same load-bearing gap: the finite-box numerical loss does not by itself establish a coloring of the entire plane, and the paper's formal verification for the headline six-colorings is externalized to a separate publication. This is the most important unresolved point because every other contribution in the paper is either presented with a formalization procedure (almost colorings), is a numerical insight explicitly labeled as such (triangle variant, polychromatic number), or reproduces known bounds. The six-colorings, by contrast, are presented as the paper's headline mathematical discovery, yet the manuscript contains no machine-checked proof, no explicit construction data sufficient to regenerate the formal coloring, and no argument that the low finite-box loss extends beyond [−R,R]^2. The concern is not that the result is false; it is that the current paper cannot be verified in isolation. The companion paper may well resolve this, but until it is checked, the conditional verdict is appropriate. I do not see a reason to move the reader's verdict.","tokens_in":18425,"tokens_out":5634,"duration_ms":63957,"concrete_test":"Obtain Mundinger et al. (2024a) and independently check the two constructions at the four endpoint distances d in {0.354, 0.553, 0.418, 0.657}. For each endpoint, verify that the described periodic coloring is explicitly defined (e.g., as a union of polygons or parallelograms with explicit inequalities) and that a finite check or proof rules out monochromatic pairs at the required distances. If all four endpoints verify, the continuum claim is supported; if the companion paper only reports numerical search or lacks endpoint certificates, the headline claim should be downgraded to conjectural in this paper.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim — that neural-network-guided search produced two novel six-colorings expanding the off-diagonal continuum to [0.354,0.657] — is not supported by any proof or certificate in this manuscript. Section 2 states the constructions and refers to Mundinger et al. (2024a) for 'a complete description,' while Figures 1, 3, and 11 show renderings rather than checkable definitions. The formal step from the finite-box numerical search to a coloring of all of R^2 is exactly the step that Proposition 3.1 does not cover: that proposition assumes exact zero loss, which training never achieves, and footnote 1 concedes that a box coloring need not extend. Since the paper itself says numerical results are only guidance and must be formalized, the existence of the six-colorings, the endpoint values 0.354 and 0.657, and the 'first improvement in thirty years' claim stand or fall with the companion paper. This is a verifiability/deferral gap rather than a demonstrated mathematical error, which is why the appropriate disposition is conditional rather than rejection.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper presents a machine-learning framework for exploring colorings of the Euclidean plane that avoid monochromatic unit-distance pairs (the Hadwiger-Nelson problem and its variants). The authors relax the discrete coloring constraint to probabilistic colorings and minimize a differentiable loss on a finite box [-R,R]^2, using neural networks as function approximators. They apply this framework to four variants: almost colorings, off-diagonal colorings of type (d1,...,dk), higher-dimensional spaces, and triangle-avoiding colorings. The paper claims two novel six-colorings that extend the known continuum of off-diagonal colorings to [0.354,0.657], a formalized almost 5-coloring covering all but 3.7356% of the plane, a formalized almost 14-coloring in R^3, and several improved triangle-avoidance bounds. The main mathematical claims for the six-colorings are deferred to a companion paper (Mundinger et al., 2024a), while the almost colorings are claimed to be verified by an automated pipeline (Algorithm 1).","tokens_in":18655,"tokens_out":9281,"duration_ms":98562,"significance":"The core relaxation (Section 3, Eq. (2)) is clean and Proposition 3.1 provides a correct statement of when an exact zero-loss probabilistic coloring yields a discrete coloring almost everywhere. The paper is honest about the distinction between numerical results and formal proofs, and it ships code that should facilitate reproduction. If the claimed colorings are valid, the results would be substantial improvements to long-standing variants of the Hadwiger-Nelson problem, particularly the extension of the off-diagonal continuum after thirty years. However, the headline six-coloring result is not verifiable from this manuscript, and the verification of the almost colorings is asserted rather than demonstrated. This limits the paper's significance as a mathematical contribution; as a machine-learning-for-discovery contribution, the framework is useful and well described.","major_comments":[{"comment":"The central claim of two novel six-colorings and the expanded continuum [0.354,0.657] is not supported in this manuscript. The colorings are only shown as renderings; no formal definition or proof is given, and Proposition 3.1 applies only to exact zero loss, while the training only approximates the loss on a finite box. Footnote 1 explicitly acknowledges that a box coloring need not extend to the plane. Since the text refers to Mundinger et al. (2024a) for 'a complete description,' the existence of these colorings and the 'first improvement in thirty years' assertion cannot be checked by a reader of this paper. The authors should include the formal constructions and proofs in an appendix, or provide a machine-checkable certificate, or re-scope the contribution statement to candidate colorings that are proven in the companion paper and clearly mark the numerical evidence as non-verifying.","section":"Section 2 (Variant 2), Section 4.2, Figures 1, 3, 11"},{"comment":"The paper states that Algorithm 1 yields an almost coloring that 'provably satisfies all unit-distance constraints,' but no correctness theorem is given. The algorithm's periodicity extraction (step 2) and discrete conflict resolution (step 5) are heuristics; the paper does not specify the measure of the set assigned to the additional color in the final output, nor the sense in which the periodic extension is conflict-free at tile boundaries. The 'formalized' values in Table 1 (e.g., 3.7356%) are stated without a precise definition of the computed measure or a verification certificate. Please provide a formal statement of the output, a proof of conflict-freeness, and a reproducible verification procedure.","section":"Section 3.3 (Algorithm 1) and Table 1"},{"comment":"The claimed formalized improvements for the triangle-avoidance variant are not described in the text; Figure 4 shows regions of the parameter space but the paper does not give the underlying colorings or the exact ranges of (a,b) for which the new bounds hold. Since these are listed as a contribution, the constructions need to be specified or referenced to a public source.","section":"Section 4.4 and Figure 4"}],"minor_comments":[{"comment":"The measure ν_k is defined as U(B_{d_k}(x)), but the integral is over the sphere ∂B_{d_k}(x); it should be the uniform distribution on the sphere, not the ball.","section":"Equation (6)"},{"comment":"'corresponding do the distances' should read 'corresponding to the distances.'","section":"Section 3.3, Variant 2"},{"comment":"The criterion for whether a point is 'achievable' with a given color count (top 3% of runs with less than 0.1% monochromatic sampled triangles) is arbitrary; please provide robustness information or justify the thresholds.","section":"Figure 8 and Section 4.4"},{"comment":"The reference Mundinger et al. (2024a) should state whether it is a published article, accepted manuscript, or preprint, so readers can access the formal constructions.","section":"References"},{"comment":"The phrase 'first improvement in thirty years' should be qualified (e.g., 'to our knowledge') or substantiated with a citation to a survey.","section":"Section 2, Variant 2"},{"comment":"The description 'the second coloring is constant' is ambiguous; what is constant is the coloring as d varies over [0.418,0.657].","section":"Section 4.2"},{"comment":"The negative results are phrased as 'additional evidence' for conjectures about the chromatic and polychromatic numbers; a failed search is a heuristic observation and should be described as such.","section":"Sections 4.2 and 4.3"}],"recommendation":"major_revision","confidential_remarks":"The paper is likely of interest to the ICML audience because of the method. However, the manuscript's strongest mathematical claim is entirely outsourced to a companion paper. If the authors can include the formal descriptions or certificates, or clearly separate the proven and conjectural parts, the paper would be acceptable. As it stands, the contributions section overstates what the reader can verify."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nPunchline: this is a solid ML-for-math paper, but the most advertised result is not in it. The two six-colorings that extend the off-diagonal continuum to [0.354, 0.657] are described only visually; their formal descriptions and proofs live in the authors' Geombinatorics note. What this paper actually contains is a clean probabilistic relaxation of the Hadwiger-Nelson problem, a neat Proposition 3.1 (zero loss implies a valid coloring almost everywhere), and a fully automated pipeline (Algorithm 1) that turns NN outputs into rigorous almost colorings. The formalized almost 5-coloring at 3.7356% of the plane removed, improving Parts' 4.0060%, is a real, self-contained result in this text, and so is the 14-coloring of R^3 covering all but 3.4622%. Code is shipped. That is real evidence.\n\nThe soft spots are mostly about what is not here. The headline six-colorings are not checkable from this manuscript; the numerical-to-formal bridge is asserted by reference to a companion paper. Same story for the triangle stripe bounds: Figure 4 shows regions, but the constructions are in preparation. The numerical conflict rates in Figure 6 and Table 1 come from finite-box training with no error bars, and some claims rely on post-hoc selection (top 3% of runs). The authors are honest that numerical results are guidance, and the footnote admitting a box coloring need not extend is the right caveat, but the gap between \"loss is small on a box\" and \"coloring of the whole plane\" is exactly where the six-colorings sit, and that gap is closed elsewhere.\n\nIs the central claim sound? The methodology is sound; the almost-coloring results are formally verified by construction and the code. The six-colorings are likely correct given they appear in a refereed journal note, but I can't vouch from this text. That warrants conditional, not accept.\n\nWho should read this: anyone interested in AI-assisted discovery or in HN variants. It's a good case study of ML producing candidate constructions that humans then formalize, and the automated pipeline for almost colorings is a genuine contribution.\n\nRecommendation: send it to referees. The paper deserves serious peer review, mostly to probe the deferral structure and the numerical claims. With the six-colorings down-weighted as \"discovered, proven elsewhere,\" the core of this paper stands.","headline":"A solid ML-for-math paper with one formalized improvement fully in-house, while the headline six-colorings are deferred to a companion paper and thus not checkable here.","tokens_in":19182,"tokens_out":3685,"would_cite":true,"duration_ms":39515,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C15","52C10"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that a neural-network-based continuous relaxation of the Hadwiger-Nelson problem discovers new plane colorings, extending the off-diagonal six-coloring continuum to $[0.354,0.657]$ and improving the almost 5-coloring…","keywords":["Hadwiger-Nelson problem","chromatic number of the plane","off-diagonal colorings","almost colorings","neural-guided mathematical discovery","probabilistic coloring relaxation","unit-distance graph","gradient-based optimization"],"falsifier":"Take the formal six-coloring from the companion paper and, over one fundamental period, check by exact computation such as interval arithmetic whether any two same-colored points are at distance 1 for colors 1 through 5 or at distance $d$ for color 6, with a value such as $d=0.354$ or $d=0.657$. A single monochromatic pair at a forbidden distance would refute the claimed continuum extension, because the formal construction is supposed to certify it.","tokens_in":18203,"feed_emoji":"🎨","tokens_out":12519,"duration_ms":128591,"temperature":0.7,"pith_summary":"This paper tries to establish that neural networks, trained with a differentiable, probabilistic relaxation of the Hadwiger-Nelson problem, can discover new mathematical constructions for coloring the Euclidean plane. The authors reformulate the search for colorings with prescribed forbidden distances as minimization of a loss that measures the probability that a random unit-distance pair shares a color, then use gradient descent on the network parameters. On this basis they report two novel six-colorings that extend the known continuum of off-diagonal colorings of type $(1,1,1,1,1,d)$ from $[0.415,0.447]$ to $[0.354,0.657]$, the first such improvement in thirty years, with formal proofs deferred to a companion paper. They also obtain a formalized almost 5-coloring covering all but $3.7356\\%$ of the plane, improving the previous $4.0060\\%$ bound, and a 14-coloring of $\\mathbb{R}^3$ covering all but $3.46\\%$. If these constructions hold, neural-network search becomes a viable source of rigorous or rigorously formalizable results in discrete geometry.","feed_headline":"Neural nets widen plane six-colorings to [0.354, 0.657]","feed_subtitle":"First improvement in 30 years for the off-diagonal Hadwiger-Nelson problem, plus an almost 5-coloring at 3.74%.","key_machinery":"The central object is the probabilistic coloring $p:\\mathbb{R}^2\\to\\Delta_c$ together with the conflict loss $L_R(p)=\\int_{[-R,R]^2}\\int_{\\partial B_1(x)} p(x)^T p(y)\\,d\\nu(y)\\,d\\mu(x)$, where $p(x)^T p(y)$ is the probability that independently sampled colors at $x$ and $y$ coincide. Minimizing this loss over a large finite box turns the hard unit-distance constraint into a continuous, differentiable objective that gradient descent can explore without assuming symmetry or periodicity. Two auxiliary mechanisms carry the formal results: a Lagrangian relaxation that adds a penalized 'bonus' color for almost colorings, and Algorithm 1, which extracts the periodicity of a trained network, retrains with exact periodicity, discretizes the fundamental parallelogram, and eliminates residual unit-distance conflicts to produce a provably valid almost coloring.","core_discovery":"The central discovery is a method, not a theorem: a hard combinatorial-geometric existence question about coloring the plane is replaced by a continuous optimization problem over probabilistic colorings $p:\\mathbb{R}^2\\to\\Delta_c$, with loss $L_R(p)=\\int_{[-R,R]^2}\\int_{\\partial B_1(x)} p(x)^T p(y)\\,d\\nu(y)\\,d\\mu(x)$, which equals the expected probability of a unit-distance conflict. Minimizing this loss by gradient descent on a neural network yields numerical colorings that the authors then interpret and formalize. For the off-diagonal variant where five colors must avoid distance 1 and the sixth must avoid distance $d$, this pipeline produced two new six-colorings that realize $(1,1,1,1,1,d)$ for all $d\\in[0.354,0.657]$, expanding the previously known interval, and an almost 5-coloring with uncovered fraction $3.7356\\%$. The paper emphasizes that numerical outputs are not proofs: the six-colorings are formally described in a companion paper, while the almost colorings are made rigorous by an automated periodicity-extraction and discrete-conflict-elimination procedure referred to as Algorithm 1.","pith_inferences":["Editorial inference: because the trained colorings vary smoothly with the free distance $d$, the realizable set of $d$ may be connected, and re-running the search on finer distance grids near the endpoints $0.354$ and $0.657$ could extend the continuum further.","Editorial inference: the paper's negative evidence on the polychromatic number is a numerical local minimum at roughly $4.9\\%$ conflict rate; a more conclusive test would apply the same pipeline with larger architectures or different loss schedules to see whether that minimum can be driven toward zero.","Editorial inference: the automated almost-coloring pipeline is the part of the framework that requires no human pattern-reading, so applying the same automated formalization to other geometric coloring problems with a bonus color, such as avoiding several distances at once, is a direct and testable next step."],"forward_implications":["For every $d$ in $[0.354,0.657]$, the plane admits a six-coloring of type $(1,1,1,1,1,d)$, a direct corollary of the two formalized constructions.","The almost 5-coloring bound of $3.7356\\%$ answers the explicit challenge, posed for the previous $4.0060\\%$ construction, to push the uncovered fraction below $4\\%$.","The same pipeline yields a 14-coloring covering all but $3.4622\\%$ of $\\mathbb{R}^3$, with numerical evidence that no conflict-free 14-coloring of $\\mathbb{R}^3$ exists.","The framework recovers known constructions, such as the pentagonal-rod six-color almost coloring, when applied to familiar variants, supporting its reliability as a discovery tool.","The negative results, including the absence of a five-color solution for the polychromatic number and the absence of a six-color solution for the original problem, are consistent with the conjectures that $\\chi(\\mathbb{R}^2)=7$ and $\\chi_p(\\mathbb{R}^2)=6$, though the paper does not claim to prove them."],"supporting_citations":[{"why":"Supplies the formal descriptions of the two six-colorings that realize the extended continuum; the paper's existence claim depends on these published constructions.","marker":"Mundinger et al. (2024a)"},{"why":"Established the previous lower endpoint of the known continuum around $\\sqrt{2}-1$, the baseline the new colorings improve upon.","marker":"Hoffman & Soifer (1996)"},{"why":"Established the previous upper endpoint of the known continuum around $1/\\sqrt{5}$, the baseline the new colorings improve upon.","marker":"Soifer (1994b)"},{"why":"Provided the previous best $4.0060\\%$ almost 5-coloring and the explicit challenge to go below $4\\%$ that the new $3.7356\\%$ construction answers.","marker":"Parts (2020b)"},{"why":"The known pentagonal-rod construction whose structure the neural networks recover, validating the method on a known solution.","marker":"Pritikin (1998)"},{"why":"Reference for the problem's history, the seven-coloring upper bound, and the variants used as test beds.","marker":"Soifer (2009)"},{"why":"Source of the unsupervised probabilistic-relaxation approach to graph optimization that the paper adapts to continuous geometric colorings.","marker":"Karalias & Loukas (2020)"},{"why":"Universal approximation theorem that justifies using neural networks as the parameterization of probabilistic colorings.","marker":"Cybenko (1989)"}],"fun_headline_variants":["Neural nets expand plane six-colorings to [0.354, 0.657]","First off-diagonal Hadwiger-Nelson gain in 30 years via neural nets","Two new six-colorings widen Hadwiger-Nelson interval","Almost 5-coloring plus wider six-colorings from neural nets"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole discovery depends on assuming that the clean, nearly conflict-free patterns a neural network finds inside a finite window really do continue to a coloring of the entire infinite plane; the paper does not prove that continuation for its main six-colorings, which are formalized in a separate companion paper.","fun_headline_variants_meta":{"raw":{"variants":["Neural nets expand plane six-colorings to [0.354, 0.657]","First off-diagonal Hadwiger-Nelson gain in 30 years via neural nets","Two new six-colorings widen Hadwiger-Nelson interval","Almost 5-coloring plus wider six-colorings from neural nets"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001262,"raw_usage":{"total_tokens":5162,"prompt_tokens":930,"completion_tokens":4232,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":546,"completion_tokens_details":{"reasoning_tokens":4148}},"tokens_in":546,"tokens_out":4232,"duration_ms":36177,"temperature":1.0,"reasoning_tokens":4148,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T23:06:20.098740+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the formal six-coloring from the companion paper and, over one fundamental period, check by exact computation such as interval arithmetic whether any two same-colored points are at distance 1 for colors 1 through 5 or at distance $d$ for color 6, with a value such as $d=0.354$ or $d=0.657$. A single monochromatic pair at a forbidden distance would refute the claimed continuum extension, because the formal construction is supposed to certify it.","supporting_citations":[{"cited_title":"and Soifer, A","cited_arxiv_id":null,"evidence_quote":"Established the previous lower endpoint of the known continuum around $\\sqrt{2}-1$, the baseline the new colorings improve upon."},{"cited_title":"All unit-distance graphs of order 6197 are 6-colorable","cited_arxiv_id":null,"evidence_quote":"The known pentagonal-rod construction whose structure the neural networks recover, validating the method on a known solution."},{"cited_title":"The mathematical coloring book: Mathematics of coloring and the colorful life of its creators","cited_arxiv_id":null,"evidence_quote":"Reference for the problem's history, the seven-coloring upper bound, and the variants used as test beds."},{"cited_title":"and Loukas, A","cited_arxiv_id":null,"evidence_quote":"Source of the unsupervised probabilistic-relaxation approach to graph optimization that the paper adapts to continuous geometric colorings."}],"review_version":1}