RadonBench: A Benchmark of Classical and Learning-Guided Search Techniques for Problems in Combinatorial Geometry

# RadonBench: A Benchmark of Classical and Learning-Guided Search Techniques for Problems in Combinatorial Geometry
Source: [https://arxiv.org/html/2610.33259](https://arxiv.org/html/2610.33259)
Ethan X\. Fang††thanks:Department of Biostatistics and Bioinformatics, Duke University\.Jesus A\. De Loera††thanks:Department of Mathematics, University of California, Davis\.Junwei Lu††thanks:Department of Biostatistics, Harvard T\.H\. Chan School of Public Health\.Luze Xu††thanks:Department of Mathematics, The Hong Kong University of Science and Technology\.

###### Abstract

Classical optimization and learning\-guided search offer complementary approaches to open problems in discrete mathematics, but their outputs are difficult to compare when search spaces, validity checks, and stopping rules differ\. We present*RadonBench*, a benchmark for constructing maximum\-cardinality Radon\-free lattice configurations: finite subsets ofℤd\\mathbb\{Z\}^\{d\}admitting no integral Radon partition\. It remains open whether Radon\-free configurations with1111points inℤ3\\mathbb\{Z\}^\{3\}or2121points inℤ4\\mathbb\{Z\}^\{4\}are possible\. If they exist, they would improve the corresponding lower bounds on the integer Radon numbers\. We compare exhaustive extension, greedy local search, PatternBoost, AlphaZero, and AlphaEvolve\. All outputs use one success predicate: a convex target\-sized configuration succeeds exactly when its number of integral Radon points is zero\. PatternBoost and AlphaZero construct verified Radon\-free configurations of sizes1010and2020, one point short of the respective targets; AlphaZero’s best convex target terminals have two integral Radon points in both dimensions\. Exhaustive extension over fixed seeds and declared finite search boxes likewise reaches a minimum of two integral Radon points\. Greedy local search and AlphaEvolve each produce target\-sized near misses with one integral Radon point\. No audited run reaches zero\. These results establish reproducible near\-miss baselines over explicitly scoped search domains\.

## 1Introduction

Mathematical discovery is rapidly changing with tools from AI\. Classical enumeration and local optimization, neural construction, search with learned policies, and evaluator\-guided program evolution have all proved useful for combinatorial discovery\([Wagner, 2021](https://arxiv.org/html/2610.33259#bib.bib12);[Fawzi et al\., 2022](https://arxiv.org/html/2610.33259#bib.bib17);[Charton et al\., 2024](https://arxiv.org/html/2610.33259#bib.bib3);[Novikov et al\., 2025](https://arxiv.org/html/2610.33259#bib.bib4)\)\. Despite this progress, many basic combinatorial quantities, including numerous Ramsey numbers, remain beyond the reach of both conventional computation and current AI\-assisted methods\([Radziszowski, 1994](https://arxiv.org/html/2610.33259#bib.bib7);[McKay and Radziszowski, 1995](https://arxiv.org/html/2610.33259#bib.bib8);[Exoo, 1989](https://arxiv.org/html/2610.33259#bib.bib9)\)\. Here we investigate another equally difficult problem and use it to benchmark modern search techniques\.

Radon’s theorem is a fundamental result in convex and discrete geometry: anyd\+2d\+2points inℝd\\mathbb\{R\}^\{d\}admit a bipartition whose convex hulls intersect\([Bárány, 2021](https://arxiv.org/html/2610.33259#bib.bib15);[Barvinok, 2002](https://arxiv.org/html/2610.33259#bib.bib14);[Gruber, 2007](https://arxiv.org/html/2610.33259#bib.bib16)\)\. Motivated by applications in integer programming and combinatorics, researchers have studied integer versions of Radon’s theorem\([Onn, 1991](https://arxiv.org/html/2610.33259#bib.bib11);[Eckhoff, 1993](https://arxiv.org/html/2610.33259#bib.bib25);[De Loera et al\., 2017](https://arxiv.org/html/2610.33259#bib.bib23);[De Loera et al\., 2019](https://arxiv.org/html/2610.33259#bib.bib27)\)\. Here the points have integer coordinates, and the intersection of the two convex hulls must contain a lattice point\. This gives an*integral Radon partition*\. A finite set inℤd\\mathbb\{Z\}^\{d\}with no such partition is*Radon\-free*, and the*integer Radon number*r⁡\(ℤd\)r\(\\mathbb\{Z\}^\{d\}\)is the smallest cardinality for which every lattice point configuration has an integral Radon partition\. The exact value in the plane isr⁡\(ℤ2\)=6r\(\\mathbb\{Z\}^\{2\}\)=6, but it remains unknown in higher dimensions\([Onn, 1991](https://arxiv.org/html/2610.33259#bib.bib11);[Bezdek and Blokhuis, 2003](https://arxiv.org/html/2610.33259#bib.bib1)\)\. We study whether there is a Radon\-free configuration with1111points inℤ3\\mathbb\{Z\}^\{3\}or one with2121points inℤ4\\mathbb\{Z\}^\{4\}\.

Figure 1:The search problem\.\(a\) Onn’s five\-point configuration inℤ2\\mathbb\{Z\}^\{2\}is Radon\-free\([Onn, 1991](https://arxiv.org/html/2610.33259#bib.bib11)\)\. \(b\) An1111\-point near\-miss inℤ3\\mathbb\{Z\}^\{3\}found by the declared local\-search experiment\. The displayed split has the single shared lattice pointp=\(3,2,0\)p=\(3,2,0\), hencenrp=1n\_\{\\mathrm\{rp\}\}=1; success requiresnrp=0n\_\{\\mathrm\{rp\}\}=0\.The search for Radon\-free configurations offers a compact comparison setting: candidate verification is practical, yet the configuration space is already too large for naive enumeration at the open targets\. Prior work established structural bounds and recognition complexity for integral Radon partitions\([Onn, 1991](https://arxiv.org/html/2610.33259#bib.bib11);[Bezdek and Blokhuis, 2003](https://arxiv.org/html/2610.33259#bib.bib1)\), but those authors did not systematically study constructive search near the open lattice frontiers\.

We introduce*RadonBench*, a benchmark of five method families: exhaustive extension, greedy local search, a task\-adapted PatternBoost solver, AlphaZero\-Radon \(our AlphaZero\-style solver\), and AlphaEvolve\-style program evolution\. These methods instantiate two complementary search paradigms\.*Cardinality constructors*grow configurations point by point while maintaining Radon\-freeness at every proper prefix; PatternBoost and AlphaZero\-Radon belong primarily to this group\.*Fixed\-target optimizers*instead evaluate completeNN\-point configurations and minimize their number of integral Radon points; this group comprises exhaustive extension, greedy local search, and AlphaEvolve\-style program evolution\. AlphaZero\-Radon is construction\-oriented but also scores convexNN\-point terminals, so it contributes to the fixed\-target comparison whenever it reaches the horizon\. The methods do not search one common finite domain and their computational units are not matched\. We therefore compare verified outputs rather than claim a ranking of search efficiency\. In particular, we separate two questions:

1. RQ1Does any declared protocol produce a verifier\-accepted configuration at an open target?
2. RQ2If a run is unsuccessful, what does it nevertheless certify: the largest verified Radon\-free cardinalitykmaxk\_\{\\max\}, or the minimum target\-sized Radon\-point countrmin​\(N\)r\_\{\\min\}\(N\)?

This separation prevents a legal\(N−1\)\(N\-1\)\-point prefix from being treated as a scoredNN\-point candidate\. Explicit1010\-point and2020\-point witnesses provide finite\-box positive controls immediately below the two frontiers; smaller cardinalities need not be benchmarked separately because Radon\-freeness is hereditary\.

Our contributions are twofold\. First, we define a common success predicate, graded score, and reporting contract for search methods with different proposal spaces\. Second, we give an auditable outcome comparison across five classical and learning\-guided method families inℤ3\\mathbb\{Z\}^\{3\}andℤ4\\mathbb\{Z\}^\{4\}, while making each method’s reachable configurations and stopping conditions explicit\. Verifier derivations, hyperparameter tables, search\-space diagnostics, and additional experiments are provided in the appendices\.

No audited run reported here resolves either open frontier\. All negative observations are limited to the candidates or finite domains searched by the declared protocols; they are not non\-existence results forℤ3\\mathbb\{Z\}^\{3\}orℤ4\\mathbb\{Z\}^\{4\}\.

## 2Problem and Evaluation

LetX⊂ℤdX\\subset\\mathbb\{Z\}^\{d\}be a finite set of distinct points\. An*integral Radon partition*is a splitX=A⊔BX=A\\sqcup Binto nonempty parts satisfyingconv⁡\(A\)∩conv⁡\(B\)∩ℤd≠∅\\operatorname\{conv\}\(A\)\\cap\\operatorname\{conv\}\(B\)\\cap\\mathbb\{Z\}^\{d\}\\neq\\varnothing\. We call every lattice point in such an intersection an integral Radon point and write

nrp​\(X\)=\#⁡\{p∈ℤd:p​is an integral Radon point of​X\}\.n\_\{\\mathrm\{rp\}\}\(X\)=\\\#\\\{p\\in\\mathbb\{Z\}^\{d\}:p\\text\{ is an integral Radon point of \}X\\\}\.ThusXXis Radon\-free exactly whennrp​\(X\)=0n\_\{\\mathrm\{rp\}\}\(X\)=0\. Every such configuration is in convex position: ifx∈conv⁡\(X∖\{x\}\)x\\in\\operatorname\{conv\}\(X\\setminus\\\{x\\\}\), then\{x\}⊔\(X∖\{x\}\)\\\{x\\\}\\sqcup\(X\\setminus\\\{x\\\}\)already has the shared lattice pointxx\. Known global bounds establish the existence of Radon\-free sets through sizes1010inℤ3\\mathbb\{Z\}^\{3\}and2020inℤ4\\mathbb\{Z\}^\{4\}, whileN=11N=11andN=21N=21are the respective open targets\([Onn, 1991](https://arxiv.org/html/2610.33259#bib.bib11);[Bezdek and Blokhuis, 2003](https://arxiv.org/html/2610.33259#bib.bib1)\)\. Separate explicit witnesses certify those two feasible cardinalities immediately below the open targets\.

#### Shared verification\.

Every reported output is judged by the same two\-stage evaluator: it first checks convex position and then enumerates lattice points inconv⁡\(X\)\\operatorname\{conv\}\(X\)and tests whether each supports an integral Radon partition\. A candidate succeeds only if it has the target cardinality, is in convex position, and hasnrp=0n\_\{\\mathrm\{rp\}\}=0\. The full evaluator and audit procedure appear in Appendix[C\.1](https://arxiv.org/html/2610.33259#A3.SS1)\.

For each lattice pointp∈\(conv⁡\(X\)∩ℤd\)∖Xp\\in\(\\operatorname\{conv\}\(X\)\\cap\\mathbb\{Z\}^\{d\}\)\\setminus X, the evaluator tests whether there are weightsλi≥0\\lambda\_\{i\}\\geq 0and binary assignmentsbi∈\{0,1\}b\_\{i\}\\in\\\{0,1\\\}satisfying

∑ibi​λi​xi=p,∑i\(1−bi\)​λi​xi=p,∑ibi​λi=1,∑iλi=2\.\\textstyle\\sum\_\{i\}b\_\{i\}\\lambda\_\{i\}x\_\{i\}=p,\\quad\\sum\_\{i\}\(1\-b\_\{i\}\)\\lambda\_\{i\}x\_\{i\}=p,\\quad\\sum\_\{i\}b\_\{i\}\\lambda\_\{i\}=1,\\quad\\sum\_\{i\}\\lambda\_\{i\}=2\.Herebi=1b\_\{i\}=1assignsxix\_\{i\}to one side of the partition andbi=0b\_\{i\}=0assigns it to the other\. The first two equalities express the same pointppas a convex combination of the two sides\. The third equality normalizes the first combination; together with∑iλi=2\\sum\_\{i\}\\lambda\_\{i\}=2, it also makes the coefficients of the second combination sum to one\. Thus both sides are nonempty, and feasibility holds exactly whenppis an integral Radon point for some partition\. PyNormaliz enumerates the candidate lattice points and Gurobi performs these split tests\([Bruns and Ichim, 2010](https://arxiv.org/html/2610.33259#bib.bib6);[Gurobi Optimization, LLC, 2026](https://arxiv.org/html/2610.33259#bib.bib18)\)\. Each feasibleppis counted once, irrespective of how many partitions witness it, and their number isnrpn\_\{\\mathrm\{rp\}\}\. The input points are omitted because the preceding convex\-position check guarantees that none lies in the convex hull of the others\.

Table 1:Open targets, known feasible configurations immediately below them, and the coordinate domain used by each method\. Box dimensions are side lengths\. “No finite box” means that the evaluator accepts arbitrary finite integer coordinates\. The method\-specific domains are not box\-matched\.
#### Output semantics\.

A single leaderboard would conflate variable\-length construction with fixed\-target repair\. For audited outputs𝒪\\mathcal\{O\}of a declared run, we therefore report

kmax\\displaystyle k\_\{\\max\}=max\{\|X\|:X∈𝒪,Xconvex,nrp\(X\)=0\},\\displaystyle=\\max\\\{\|X\|:X\\in\\mathcal\{O\},\\ X\\text\{ convex\},\\ n\_\{\\mathrm\{rp\}\}\(X\)=0\\\},rmin​\(N\)\\displaystyle r\_\{\\min\}\(N\)=min\{nrp\(X\):X∈𝒪,\|X\|=N,Xconvex\}\.\\displaystyle=\\min\\\{n\_\{\\mathrm\{rp\}\}\(X\):X\\in\\mathcal\{O\},\\ \|X\|=N,\\ X\\text\{ convex\}\\\}\.The second quantity is undefined if no convex target candidate is emitted\. Method\-native counts and layer profiles describe outputs but are not pooled into cross\-method success rates\. Search boxes, proposal domains, stopping rules, and budgets are stated per method; shared verification makes certificates and scores comparable, not search coverage or efficiency\. Appendix[B\.1](https://arxiv.org/html/2610.33259#A2.SS1)gives the finite\-box witnesses, and Appendix[C](https://arxiv.org/html/2610.33259#A3)records the per\-method configurations\.

## 3Methods

All five methods use the target cardinality and verifier from[Section2](https://arxiv.org/html/2610.33259#S2)\. For a method with an origin\-anchored finite coordinate domain of side lengthsL=\(L1,…,Ld\)L=\(L\_\{1\},\\ldots,L\_\{d\}\), we writeΛ⁡\(L\)=∏j=1d\{0,…,Lj−1\}\\Lambda\(L\)=\\prod\_\{j=1\}^\{d\}\\\{0,\\ldots,L\_\{j\}\-1\\\}\. No single box is shared across methods: each protocol has the method\-specific coordinate domain listed in[Table1](https://arxiv.org/html/2610.33259#S2.T1), and the program\-evolution method has no finite coordinate bound\. Box membership is part of success only when the method itself enforces it\.

Exhaustive extension and greedy local search are classical methods; PatternBoost, AlphaZero\-Radon, and AlphaEvolve\-style program evolution are learning\-guided\. By search objective, PatternBoost and AlphaZero\-Radon are primarily cardinality constructors, while exhaustive extension, local search, and program evolution optimize completeNN\-point configurations\. AlphaZero\-Radon also scores the terminal configuration when it reachesNN\. The methods differ in their one\-step proposals and reachable states, summarized in[Table2](https://arxiv.org/html/2610.33259#S3.T2); these restrictions scope every negative result\. In[Table2](https://arxiv.org/html/2610.33259#S3.T2), “Starts from” denotes an externally supplied configuration; deterministic protocol anchors are specified in the method descriptions\.

Table 2:High\-level view of the five methods\. Gray bands separate the two search objectives; implementation details appear in the subsections below\.### 3\.1Shared interface

Every method calls the shared convex\-position and Radon\-point evaluator described in[Section2](https://arxiv.org/html/2610.33259#S2)\. A proposal may emit a complete configuration or build/edit one sequentially\. Reported outputs use the same success predicate, and every retained coordinate array is replayed with the reference evaluator\. The exact implementation and the exhaustive driver’s equivalent convex\-position test are documented in[SectionsC\.1](https://arxiv.org/html/2610.33259#A3.SS1)and[C\.2](https://arxiv.org/html/2610.33259#A3.SS2)\.

### 3\.2Exhaustive add\-one\-point

For each of 50 fixed Radon\-free\(N−1\)\(N\-1\)\-point seedsSS, this baseline tests every unoccupied positionqqin a declared finite boxBdB\_\{d\}and retains convex extensions\. Its per\-seed objective is

mBd\(S\)=minq∈Bd∖S:S∪\{q\}​in convex positionnrp\(S∪\{q\}\)\.m\_\{B\_\{d\}\}\(S\)=\\min\_\{\\begin\{subarray\}\{c\}q\\in B\_\{d\}\\setminus S:\\\\ S\\cup\\\{q\\\}\\text\{ in convex position\}\\end\{subarray\}\}n\_\{\\mathrm\{rp\}\}\\\!\\left\(S\\cup\\\{q\\\}\\right\)\.\(1\)“Exhaustive” therefore means exhaustive over the final point after fixingSSandBdB\_\{d\}, not over all seeds or allNN\-point configurations\. Exact boxes, coverage counts, and pseudocode are given in[Tables1](https://arxiv.org/html/2610.33259#S2.T1)and[C\.2](https://arxiv.org/html/2610.33259#A3.SS2)\.

### 3\.3Greedy local search

This deterministic\-after\-initialization repair baseline starts from a convexNN\-point extension of a stored Radon\-free\(N−1\)\(N\-1\)\-point configuration\. A sweep visits points in stored order and evaluates all full\-coordinate radius\-one moves

xi\+δ,δ∈\{−1,0,1\}d∖\{𝟎\},x\_\{i\}\+\\delta,\\qquad\\delta\\in\\\{\-1,0,1\\\}^\{d\}\\setminus\\\{\\mathbf\{0\}\\\},that remain distinct, convex, and inside the method’s box\. It accepts the admissible move with the lowest strictly smallernrpn\_\{\\mathrm\{rp\}\}, using a fixed direction order for ties, and stops atnrp=0n\_\{\\mathrm\{rp\}\}=0, after a non\-improving sweep, or after 100 sweeps\. Because it changes one point at a time and rejects plateaus and temporary increases, it cannot make coordinated moves or cross score barriers\. Initialization and ordering details are in[SectionC\.3](https://arxiv.org/html/2610.33259#A3.SS3)\.

### 3\.4PatternBoost generation

Our PatternBoost track is an iterative construction–training loop adapted from[Charton et al\. \(2024\)](https://arxiv.org/html/2610.33259#bib.bib3)\. Configurations inΛ⁡\(L\)\\Lambda\(L\)are encoded as mixed\-radix point\-token sequences\. Generation 0 greedily grows empty states by accepting an addition only when the new prefix is convex and hasnrp=0n\_\{\\mathrm\{rp\}\}=0\. The largest terminal states are retained, symmetry\-augmented, and used to train a decoder\-only Transformer\([Vaswani et al\., 2017](https://arxiv.org/html/2610.33259#bib.bib20)\); sampled sequences seed the next generation and are completed by the same greedy legality test\. The model is therefore trained as an ordering prior, while the verifier enforces every constraint\. Herenrpn\_\{\\mathrm\{rp\}\}is a gate rather than a graded objective: an illegal addition is rejected, and construction stops when no tested point extends the prefix\. Tokenization, selection, and training details are in[SectionC\.4](https://arxiv.org/html/2610.33259#A3.SS4)\.

### 3\.5AlphaZero\-style construction: AlphaZero\-Radon

AlphaZero\-Radon is a task\-specific adaptation of the AlphaZero algorithmic template\([Silver et al\., 2018](https://arxiv.org/html/2610.33259#bib.bib5)\), not a reuse of its chess, shogi, or Go model\. It retains policy–value\-guided PUCT and an iterative search–training loop, but replaces the original game representation and residual convolutional network\. Our single\-agent construction task uses unordered point\-set states, a permutation\-invariant set\-attention policy–value network\([Lee et al\., 2019](https://arxiv.org/html/2610.33259#bib.bib19)\), lattice\-cell actions, and verifier\-defined outcomes\. Each trajectory starts at the fixed origin and adds the remaining points inΛ⁡\(L\)\\Lambda\(L\)\. The network scores lattice locations, and Monte Carlo tree search plans with the shaped edge reward

r⁡\(s′\)=\{0if​s′​is not in convex position,CRN⁡\(1\+nrp​\(s′\)\)if​s′​is convex and​nrp​\(s′\)\>0,CSNif\|s′\|=N,s′is convex, andnrp\(s′\)=0,CstepNif\|s′\|<N,s′is convex, andnrp\(s′\)=0\.r\(s^\{\\prime\}\)=\\begin\{cases\}0&\\text\{if \}s^\{\\prime\}\\text\{ is not in convex position\},\\\\\[3\.0pt\] \\dfrac\{C\_\{\\mathrm\{R\}\}\}\{N\(1\+n\_\{\\mathrm\{rp\}\}\(s^\{\\prime\}\)\)\}&\\text\{if \}s^\{\\prime\}\\text\{ is convex and \}n\_\{\\mathrm\{rp\}\}\(s^\{\\prime\}\)\>0,\\\\\[7\.0pt\] \\dfrac\{C\_\{\\mathrm\{S\}\}\}\{N\}&\\text\{if \}\|s^\{\\prime\}\|=N,\\ s^\{\\prime\}\\text\{ is convex, and \}n\_\{\\mathrm\{rp\}\}\(s^\{\\prime\}\)=0,\\\\\[7\.0pt\] \\dfrac\{C\_\{\\mathrm\{step\}\}\}\{N\}&\\text\{if \}\|s^\{\\prime\}\|<N,\\ s^\{\\prime\}\\text\{ is convex, and \}n\_\{\\mathrm\{rp\}\}\(s^\{\\prime\}\)=0\.\\end\{cases\}\(2\)withCstep=1C\_\{\\mathrm\{step\}\}=1,CR=2C\_\{\\mathrm\{R\}\}=2, andCS=25C\_\{\\mathrm\{S\}\}=25\. A non\-convex or Radon\-positive state terminates the simulated branch; convex Radon\-positive states receive the displayed graded terminal reward at every cardinality\. We report fixed\-target quality only for convex terminals at the horizon\.

To control branching, each expanded node retains theK=64K=64unoccupied actions with the largest network probabilities\. A child found terminal during search is marked and skipped by subsequent tree selections\. Each decision uses 500 simulations, after which the trajectory generator samples from the root visit distribution at temperature one\. Training, architecture, and the exact Plan\-D search procedure are specified in[SectionC\.6](https://arxiv.org/html/2610.33259#A3.SS6)\.

### 3\.6AlphaEvolve\-style program evolution

AlphaEvolve\([Novikov et al\., 2025](https://arxiv.org/html/2610.33259#bib.bib4)\)treats executable programs, rather than point sets, as the population under search\. We instantiate this with OpenEvolve\([Sharma, 2025](https://arxiv.org/html/2610.33259#bib.bib22)\): an LLM proposes exact search\-and\-replace edits to a Python block, a fixed evaluator executes the edited constructor, and an island\-based MAP\-Elites archive\([Mouret and Clune, 2015](https://arxiv.org/html/2610.33259#bib.bib21)\)retains programs by both fitness and behavioural features\. Thus PatternBoost models a distribution over point sequences, whereas the AlphaEvolve\-style method evolves the algorithm that constructs or repairs point sets\.

The immutable wrapper gives each program sixteen fixed Radon\-free\(N−1\)\(N\-1\)\-point references and the verifier\. One evaluation permits at most1,0001\{,\}000find\_rpscalls and returns at most 32 integerN×dN\\times dcandidates\. Only four constructor functions may evolve; the wrapper, resource accounting, geometry code, and external evaluator remain fixed\. The initial program implements the interface but emits no candidate, so it contributes no hand\-written search heuristic\.

After the allowance closes, the evaluator validates shapes, finiteness, integrality and distinctness, then recomputes convexity andnrpn\_\{\\mathrm\{rp\}\}for every returned candidate\. Program fitness is the maximum score of one candidate; statistics from different candidates are never mixed\. A convex target withk\>0k\>0Radon points scores0\.50\+0\.49/\(1\+k\)0\.50\+0\.49/\(1\+k\); a convex Radon\-free target scores11\. The full fitness, evolution loop, and audit fields are given in[SectionC\.5](https://arxiv.org/html/2610.33259#A3.SS5)\.

## 4Results

The five methods share one headline question: did the declared run produce a verifier\-accepted configuration at the open target? Table[3](https://arxiv.org/html/2610.33259#S4.T3)therefore reports target recovery first\. When a run does not recover the target, it keeps two secondary outcomes in separate columns:kmaxk\_\{\\max\}, the largest fully Radon\-free configuration emitted at any cardinality, andrmin​\(N\)r\_\{\\min\}\(N\), the smallest verifier score among emitted convex target\-sized configurations\. These quantities describe different output types and are not combined into a cross\-method ranking\.

Table 3:Verified outcomes at the two open targets\.Founddenotes a verifier\-accepted target\-sized output; a dash denotes an inapplicable output type\.MethodFound?kmax↑k\_\{\\max\}\\uparrowrmin​\(N\)↓r\_\{\\min\}\(N\)\\downarrowℤ3\\mathbb\{Z\}^\{3\}, targetN=11N=11PatternBoostNo1010–AlphaZero\-RadonNo101022Exhaustive extensionNo–22Greedy local searchNo–11AlphaEvolve\-styleNo–11ℤ4\\mathbb\{Z\}^\{4\}, targetN=21N=21PatternBoostNo2020–AlphaZero\-RadonNo202022Exhaustive extensionNo–22Greedy local searchNo–11AlphaEvolve\-styleNo–11Figure 2:Method\-native construction progress\.\(a\) For PatternBoost, solid curves show the mean cardinality among the 50 retained configurations and dotted curves show the retained maximum at each generation\. \(b\) For AlphaZero\-Radon, curves show the mean number of post\-origin placements per self\-play trajectory, smoothed with a 50\-iteration moving average\. Values are normalized by the open target cardinality in \(a\) and by the post\-origin horizonN−1N\-1in \(b\)\. The dashed line marks target completion\. Horizontal axes are method\-specific and do not represent matched compute; each curve summarizes one complete declared run\.#### RQ1: Did a protocol recover an open\-target counterexample?

No completed PatternBoost run, exhaustive add\-one\-point experiment, AlphaZero\-Radon run, 1,000\-start greedy\-local\-search run, or AlphaEvolve\-style program\-evolution run produces a verified open\-target counterexample\.

#### RQ2: What certified output did an unsuccessful run produce?

PatternBoost reacheskmax=10k\_\{\\max\}=10inℤ3\\mathbb\{Z\}^\{3\}and2020inℤ4\\mathbb\{Z\}^\{4\}\. Across its complete 2,000\-iteration runs, AlphaZero\-Radon reacheskmax=10k\_\{\\max\}=10in 3D and2020in 4D; among its convex target terminals, the minimum score is22in both dimensions\. Exhaustive add\-one\-point search also attains minimum score22in both dimensions after covering every extension position in each of its 50 fixed\-seed boxes\. Greedy local search attains score11in both dimensions \(on 27 of 1,000 3D terminals and 25 of 1,000 4D terminals\), but never reaches zero\. The AlphaEvolve\-style method likewise attainsrmin=1r\_\{\\min\}=1in both dimensions: its best programs return convex 11\- and 21\-point batches whose best individual candidates each have one Radon point\. The two output types answer different questions: the former certifies a fully valid shorter configuration, whereas the latter measures a target\-sized near miss\. We therefore neither order them against one another nor pool method\-native candidate counts into a success rate\. Figure[2](https://arxiv.org/html/2610.33259#S4.F2)shows within\-run progress for the two cardinality constructors; Figure[3](https://arxiv.org/html/2610.33259#S4.F3)gives the complementary fixed\-target outcomes for exhaustive extension, greedy local search, and AlphaEvolve\-style program evolution\. Both use method\-native axes and do not compare computational efficiency\. Appendix[B\.2](https://arxiv.org/html/2610.33259#A2.SS2)provides representative verified constructions\.

The experiments are not compute\- or verifier\-query\-matched\. Method\-native run lengths and budgets are reported for reproducibility, but Table[3](https://arxiv.org/html/2610.33259#S4.T3)compares verified outputs rather than search efficiency\. Appendix[A](https://arxiv.org/html/2610.33259#A1)records the completed runs and retained outputs\.

Figure 3:Fixed\-target search outcomes under the current declared protocols\.\(a\) For exhaustive extension, bars show the distribution of the minimumnrpn\_\{\\mathrm\{rp\}\}obtained after searching the full declared box for each of 50 fixed seeds\. \(b\) For greedy local search, curves show empirical CDFs of the initial and terminal scores over 1,000 selected starts\. \(c\) For AlphaEvolve\-style program evolution, curves show the best\-so\-far score over 20 completed mutations; the first score\-one program appears at mutation 7 in 3D and 11 in 4D\. Colors identify the dimension\. Seeds, starts, and mutations are protocol\-native units and do not represent matched compute\.

## 5Related Work

Onn formalized integral Radon partitions in the lattice setting, provedr⁡\(ℤ2\)=6r\(\\mathbb\{Z\}^\{2\}\)=6, established general bounds, and studied recognition complexity\([Onn, 1991](https://arxiv.org/html/2610.33259#bib.bib11)\)\. Bezdek and Blokhuis proved the upper boundr⁡\(ℤ3\)≤17r\(\\mathbb\{Z\}^\{3\}\)\\leq 17\([Bezdek and Blokhuis, 2003](https://arxiv.org/html/2610.33259#bib.bib1)\); integral Radon partitions are also integer22\-Tverberg partitions\([De Loera et al\., 2017](https://arxiv.org/html/2610.33259#bib.bib23)\)\. These works motivate the open frontiers but do not provide a multi\-method empirical study of constructive search\.

Relevant ingredients in machine\-assisted mathematical discovery include neural construction\([Wagner, 2021](https://arxiv.org/html/2610.33259#bib.bib12)\), policy–value\-guided tree search from AlphaZero\([Silver et al\., 2018](https://arxiv.org/html/2610.33259#bib.bib5)\), its task\-specific adaptation to algorithm discovery in AlphaTensor\([Fawzi et al\., 2022](https://arxiv.org/html/2610.33259#bib.bib17)\), Transformer\-guided search\([Charton et al\., 2024](https://arxiv.org/html/2610.33259#bib.bib3)\), and LLM program evolution\([Romera\-Paredes et al\., 2024](https://arxiv.org/html/2610.33259#bib.bib13);[Novikov et al\., 2025](https://arxiv.org/html/2610.33259#bib.bib4)\)\. A recent comparative precedent studies classical, PatternBoost, and reinforcement\-learning methods for the no\-three\-in\-line problem\([Ramanathan et al\., 2025](https://arxiv.org/html/2610.33259#bib.bib2)\)\. RadonBench instead uses a lattice\-convexity verifier and explicitly separates constructive from fixed\-cardinality outputs\. Complete computational resolutions of combinatorial existence problems require exhaustive search and checkable certificates\([Heule et al\., 2016](https://arxiv.org/html/2610.33259#bib.bib24);[Heule, 2018](https://arxiv.org/html/2610.33259#bib.bib10);[Brakensiek et al\., 2022](https://arxiv.org/html/2610.33259#bib.bib26)\); our bounded searches do not provide such a non\-existence certificate\.

## 6Discussion

Neither PatternBoost, exhaustive extension, nor AlphaZero\-Radon reaches an open target; greedy local search likewise finds no target across 1,000 starts per dimension but reaches score\-one near misses in both\. Two explanations remain inseparable: the declared searches may cover too little of their method\-specific domains, or a Radon\-free configuration may not exist at that cardinality\. These observations support only search\-space diagnoses\. A constructor’s legal\-prefix frontier reflects its proposal distribution and cutoff, not a global obstruction\. After 20 mutations, the AlphaEvolve\-style runs in both dimensions reach score\-one near misses\.

#### Reading the results through each search space\.

Exhaustive extension provides a finite\-domain completeness statement only after fixing a legal\(N−1\)\(N\-1\)\-point seed and a finite box for the last point\. The completed experiment visits every position in each declared box, but its negative result concerns only those 50 fixed seeds per dimension, not configurations that require changing a seed or adding a point outside the box\.

Greedy local search has the opposite search\-space design\. It can modify every coordinate of every point of anNN\-point configuration over successive sweeps, but each proposal moves only one point by Chebyshev radius one, remains inside the fixed box, and must immediately lower the integer\-valued score\. All 2,000 reported runs stall at positive score, including 27 of 1,000 in 3D and 25 of 1,000 in 4D at score one\. A missing path may therefore require coordinated changes, a neutral or uphill intermediate state, or leaving the declared box\.

PatternBoost and AlphaZero\-Radon both build point sets sequentially\. PatternBoost assigns no graded score to a failing prefix, whereas AlphaZero\-Radon gives a larger terminal reward to a convex Radon\-positive state with fewer integral Radon points\. PatternBoost fits an ordering distribution from previously retained configurations and then uses the verifier to accept or skip sampled points\. AlphaZero\-Radon uses its policy to retain a top\-KKaction set at each expanded node and spends MCTS simulations only within that restricted support\. A frontier atN−1N\-1therefore means that the learned proposal mechanism did not produce a verified continuation under the declared run; it is not evidence that no continuation exists elsewhere in the box\.

AlphaEvolve\-style program evolution has the broadest coordinate interface because an evolved program may emit arbitrary finite integer coordinates\. Its practical search space is nevertheless the set of programs reached by its declared mutation lineage and the bounded batch of candidates each program returns\. Its 3D and 4D lineages both attain one\-Radon\-point target configurations, but neither returns a Radon\-free target\. These distinctions are why we keep target recovery, valid\-prefix cardinality, and target\-sized score in separate table columns rather than treating the five methods as an efficiency leaderboard\.

#### What defines a reproducible search\-space claim\.

A finite\-box exhaustive search must specify its fixed seeds and coordinate box; a repair method must specify its initialization, move radius, number of points changed together, and acceptance rule\. A learned constructor additionally needs its action vocabulary, candidate truncation, and terminal handling, while program evolution needs its evaluator batch and number of mutations\. Omitting any of these choices changes the reachable candidates\. We expose them so future runs can vary one restriction at a time without changing the success predicate\. The same accounting explains why a grid position, a repair trajectory, an MCTS game, and a program\-returned candidate are not interchangeable trials\.

A complete negative result would require more than exhaustive evaluation inside one coordinate box\. One would need a proved finite representative family, or an invariant coordinate bound, for lower\-cardinality Radon\-free configurations and a proved finite extension region for every representative\. Affine unimodular transformations preserve Radon\-freeness but need not preserve a box, so bounded\-box exhaustion alone does not cover all equivalence classes\. The appendices state the exact scope of every filled result\.

#### Implications for search design\.

The framework suggests three protocol\-level choices for future runs\. First, repair methods should test coordinated moves or permit neutral and uphill transitions instead of enlarging the coordinate radius alone\. Second, initialization and box constraints deserve the same care as the stopping rule because both restrict reachable basins\. Third, evaluation should retain the objective each algorithm actually optimizes\. Constructors should report the largest verified legal prefix, whereas fixed\-target methods should report the best verified target score; neither replaces the common binary target\-recovery outcome\.

#### Limitations\.

The methods use different proposal domains, move sets, and run lengths; hardware and verifier calls are not matched\. PatternBoost has one 20\-generation run per dimension, and AlphaZero\-Radon has one complete 2,000\-iteration cold\-start run per dimension\. AlphaEvolve\-style program evolution has one 20\-mutation lineage per dimension rather than repeated independent trials\. The greedy\-local\-search results are limited to 1,000 fixed starts per dimension, strict single\-point descent, and dimension\-specific fixed boxes\. The exhaustive experiment fixes only 50 catalog seeds per dimension and cannot exclude extensions of other seeds or points outside its declared boxes\. Consequently, Table[3](https://arxiv.org/html/2610.33259#S4.T3)compares verified outputs, not sample or computational efficiency\. Although the two constructive methods share the6×6×6×66\\times 6\\times 6\\times 6box in 4D, their search distributions and budgets are not matched\.

## 7Conclusion

RadonBench organizes five classical and learning\-guided searches around a common target\-recovery criterion while preserving their different output semantics\. PatternBoost reaches the known calibration cardinalities1010and2020; across its complete 2,000\-iteration runs, AlphaZero\-Radon reaches the same cardinalities and target score22in both dimensions, while exhaustive add\-one\-point search obtains target score22but no target recovery after completely visiting the declared box for each of 50 fixed seeds in both dimensions\. Full\-coordinate greedy local search reaches score11but no target across 1,000 starts per dimension\. AlphaEvolve\-style program evolution also reaches score11in both dimensions after 20 mutations, without recovering either target\. These results do not resolve the integer Radon numbers, but the success\-first reporting scheme provides a reproducible framework for adding future searches without conflating solution quality with compute efficiency\.

## Acknowledgments

This work was supported in part by NVIDIA through the NVIDIA Academic Grant Program under the project “AI Foundation Models for Reasoning in Polyhedral Geometry\.”

## References

- Bárány \(2021\)I\. BárányCombinatorial convexity\.University Lecture Series, Vol\.77,American Mathematical Society,Providence, RI\.Cited by:[§1](https://arxiv.org/html/2610.33259#S1.p2.1)\.
- Barvinok \(2002\)A\. BarvinokA course in convexity\.Graduate Studies in Mathematics, Vol\.54,American Mathematical Society,Providence, RI\.Cited by:[§1](https://arxiv.org/html/2610.33259#S1.p2.1)\.
- Bezdek and Blokhuis \(2003\)K\. Bezdek and A\. BlokhuisThe radon number of the three\-dimensional integer lattice\.Discrete & Computational Geometry30\(2\),pp\. 181–184\.Cited by:[§1](https://arxiv.org/html/2610.33259#S1.p2.1),[§1](https://arxiv.org/html/2610.33259#S1.p3.1),[§2](https://arxiv.org/html/2610.33259#S2.p1.2),[§5](https://arxiv.org/html/2610.33259#S5.p1.1)\.
- Brakensieket al\.\(2022\)J\. Brakensiek, M\. Heule, J\. Mackey, and D\. NarváezThe resolution of Keller’s conjecture\.Journal of Automated Reasoning66,pp\. 277–300\.Cited by:[§5](https://arxiv.org/html/2610.33259#S5.p2.1)\.
- Bruns and Ichim \(2010\)W\. Bruns and B\. IchimNormaliz: algorithms for affine monoids and rational cones\.Journal of Algebra324\(5\),pp\. 1098–1113\.Cited by:[§C\.1](https://arxiv.org/html/2610.33259#A3.SS1.p2.1),[§2](https://arxiv.org/html/2610.33259#S2.SS0.SSS0.Px1.p2.2)\.
- Chartonet al\.\(2024\)F\. Charton, J\. S\. Ellenberg, A\. Z\. Wagner, and G\. WilliamsonPatternboost: constructions in mathematics with a little help from ai\.arXiv preprint arXiv:2411\.00566\.Cited by:[§1](https://arxiv.org/html/2610.33259#S1.p1.1),[§3\.4](https://arxiv.org/html/2610.33259#S3.SS4.p1.1),[§5](https://arxiv.org/html/2610.33259#S5.p2.1)\.
- De Loeraet al\.\(2019\)J\. A\. De Loera, X\. Goaoc, F\. Meunier, and N\. H\. MustafaThe discrete yet ubiquitous theorems of Carathéodory, Helly, Sperner, Tucker, and Tverberg\.Bulletin of the American Mathematical Society56\(3\),pp\. 415–511\.External Links:[Document](https://dx.doi.org/10.1090/bull/1653)Cited by:[§1](https://arxiv.org/html/2610.33259#S1.p2.1)\.
- De Loeraet al\.\(2017\)J\. A\. De Loera, T\. A\. Hogan, F\. Meunier, and N\. H\. MustafaTverberg theorems over discrete sets of points\.arXiv preprint arXiv:1706\.01178\.Cited by:[§1](https://arxiv.org/html/2610.33259#S1.p2.1),[§5](https://arxiv.org/html/2610.33259#S5.p1.1)\.
- Eckhoff \(1993\)J\. EckhoffHelly, Radon, and Carathéodory type theorems\.InHandbook of Convex Geometry,P\. M\. Gruber and J\. M\. Wills \(Eds\.\),Vol\.A,pp\. 389–448\.Cited by:[§1](https://arxiv.org/html/2610.33259#S1.p2.1)\.
- Exoo \(1989\)G\. ExooA lower bound for r\(5,5\)\.Journal of Graph Theory13\(1\),pp\. 97–98\.Cited by:[§1](https://arxiv.org/html/2610.33259#S1.p1.1)\.
- Fawziet al\.\(2022\)A\. Fawzi, M\. Balog, A\. Huang, T\. Hubert, B\. Romera\-Paredes, M\. Barekatain, A\. Novikov, F\. J\. R\. Ruiz, J\. Schrittwieser, G\. Swirszcz, D\. Silver, D\. Hassabis, and P\. KohliDiscovering faster matrix multiplication algorithms with reinforcement learning\.Nature610\(7930\),pp\. 47–53\.External Links:[Document](https://dx.doi.org/10.1038/s41586-022-05172-4)Cited by:[§1](https://arxiv.org/html/2610.33259#S1.p1.1),[§5](https://arxiv.org/html/2610.33259#S5.p2.1)\.
- Gruber \(2007\)P\. M\. GruberConvex and discrete geometry\.Grundlehren der mathematischen Wissenschaften, Vol\.336,Springer,Berlin\.Cited by:[§1](https://arxiv.org/html/2610.33259#S1.p2.1)\.
- Gurobi Optimization, LLC \(2026\)Gurobi Optimization, LLCGurobi Optimizer Reference Manual\.External Links:[Link](https://www.gurobi.com/)Cited by:[§C\.1](https://arxiv.org/html/2610.33259#A3.SS1.p2.2),[§2](https://arxiv.org/html/2610.33259#S2.SS0.SSS0.Px1.p2.2)\.
- Heuleet al\.\(2016\)M\. J\. H\. Heule, O\. Kullmann, and V\. W\. MarekSolving and verifying the boolean pythagorean triples problem via cube\-and\-conquer\.InTheory and Applications of Satisfiability Testing \(SAT\),pp\. 228–245\.Cited by:[§5](https://arxiv.org/html/2610.33259#S5.p2.1)\.
- Heule \(2018\)M\. HeuleSchur number five\.InProceedings of the AAAI Conference on Artificial Intelligence,Vol\.32\.Cited by:[§5](https://arxiv.org/html/2610.33259#S5.p2.1)\.
- Leeet al\.\(2019\)J\. Lee, Y\. Lee, J\. Kim, A\. Kosiorek, S\. Choi, and Y\. W\. TehSet transformer: a framework for attention\-based permutation\-invariant neural networks\.InProceedings of the 36th International Conference on Machine Learning,Proceedings of Machine Learning Research, Vol\.97,pp\. 3744–3753\.External Links:[Link](https://proceedings.mlr.press/v97/lee19d.html)Cited by:[§3\.5](https://arxiv.org/html/2610.33259#S3.SS5.p1.1)\.
- McKay and Radziszowski \(1995\)B\. D\. McKay and S\. P\. RadziszowskiR\(4,5\) = 25\.Journal of Graph Theory19\(3\),pp\. 309–322\.Cited by:[§1](https://arxiv.org/html/2610.33259#S1.p1.1)\.
- Mouret and Clune \(2015\)J\. Mouret and J\. CluneIlluminating search spaces by mapping elites\.arXiv preprint arXiv:1504\.04909\.External Links:1504\.04909,[Link](https://arxiv.org/abs/1504.04909)Cited by:[§3\.6](https://arxiv.org/html/2610.33259#S3.SS6.p1.1)\.
- Novikovet al\.\(2025\)A\. Novikov, N\. Vũ, M\. Eisenberger, E\. Dupont, P\. Huang, A\. Z\. Wagner, S\. Shirobokov, B\. Kozlovskii, F\. J\. Ruiz, A\. Mehrabian,et al\.Alphaevolve: a coding agent for scientific and algorithmic discovery\.arXiv preprint arXiv:2506\.13131\.Cited by:[§1](https://arxiv.org/html/2610.33259#S1.p1.1),[§3\.6](https://arxiv.org/html/2610.33259#S3.SS6.p1.1),[§5](https://arxiv.org/html/2610.33259#S5.p2.1)\.
- Onn \(1991\)S\. OnnOn the geometry and computational complexity of Radon partitions in the integer lattice\.SIAM Journal on Discrete Mathematics4\(3\),pp\. 436–447\.External Links:[Document](https://dx.doi.org/10.1137/0404039)Cited by:[Figure 1](https://arxiv.org/html/2610.33259#S1.F1),[§1](https://arxiv.org/html/2610.33259#S1.p2.1),[§1](https://arxiv.org/html/2610.33259#S1.p3.1),[§2](https://arxiv.org/html/2610.33259#S2.p1.2),[§5](https://arxiv.org/html/2610.33259#S5.p1.1)\.
- Radziszowski \(1994\)S\. P\. RadziszowskiSmall ramsey numbers\.Electronic Journal of Combinatorics1\(DS1\)\.Note:Dynamic SurveyExternal Links:[Document](https://dx.doi.org/10.37236/21)Cited by:[§1](https://arxiv.org/html/2610.33259#S1.p1.1)\.
- Ramanathanet al\.\(2025\)P\. Ramanathan, T\. Prellberg, M\. Lewis, P\. D\. Joshi, R\. A\. Dandekar, R\. Dandekar, and S\. PanatThree methods, one problem: classical and ai approaches to no\-three\-in\-line\.arXiv preprint arXiv:2512\.11469\.Cited by:[§5](https://arxiv.org/html/2610.33259#S5.p2.1)\.
- Romera\-Paredeset al\.\(2024\)B\. Romera\-Paredes, M\. Barekatain, A\. Novikov, M\. Balog, M\. P\. Kumar, E\. Dupont, F\. J\. R\. Ruiz, J\. S\. Ellenberg, P\. Wang, O\. Fawzi, P\. Kohli, and A\. FawziMathematical discoveries from program search with large language models\.Nature625\(7995\),pp\. 468–475\.External Links:[Document](https://dx.doi.org/10.1038/s41586-023-06924-6)Cited by:[§5](https://arxiv.org/html/2610.33259#S5.p2.1)\.
- Sharma \(2025\)A\. SharmaOpenEvolve: an open\-source evolutionary coding agent\.GitHub\.External Links:[Link](https://github.com/algorithmicsuperintelligence/openevolve)Cited by:[§3\.6](https://arxiv.org/html/2610.33259#S3.SS6.p1.1)\.
- Silveret al\.\(2018\)D\. Silver, T\. Hubert, J\. Schrittwieser, I\. Antonoglou, M\. Lai, A\. Guez, M\. Lanctot, L\. Sifre, D\. Kumaran, T\. Graepel,et al\.A general reinforcement learning algorithm that masters chess, shogi, and go through self\-play\.Science362\(6419\),pp\. 1140–1144\.Cited by:[§3\.5](https://arxiv.org/html/2610.33259#S3.SS5.p1.1),[§5](https://arxiv.org/html/2610.33259#S5.p2.1)\.
- Vaswaniet al\.\(2017\)A\. Vaswani, N\. Shazeer, N\. Parmar, J\. Uszkoreit, L\. Jones, A\. N\. Gomez, L\. Kaiser, and I\. PolosukhinAttention is all you need\.InAdvances in Neural Information Processing Systems,Vol\.30,pp\. 5998–6008\.External Links:[Link](https://proceedings.neurips.cc/paper/7181-attention-is-all-you-need)Cited by:[§3\.4](https://arxiv.org/html/2610.33259#S3.SS4.p1.1)\.
- Wagner \(2021\)A\. Z\. WagnerConstructions in combinatorics via neural networks\.arXiv preprint arXiv:2104\.14516\.External Links:2104\.14516,[Link](https://arxiv.org/abs/2104.14516)Cited by:[§1](https://arxiv.org/html/2610.33259#S1.p1.1),[§5](https://arxiv.org/html/2610.33259#S5.p2.1)\.

## Appendix ADetailed Experimental Outcomes

All declared runs use the stopping rules and sample counts specified in the method appendix\. The headline outcomes appear in Table[3](https://arxiv.org/html/2610.33259#S4.T3); this section records only method\-native evidence that is not shown there\.

### A\.1Constructive methods

#### PatternBoost\.

Both 20\-generation loops completed\. In the20×20×520\\times 20\\times 5box, the largest verified Radon\-free cardinality is1010; generation 20 retains 47 ten\-point and three nine\-point states\. In the6×6×6×66\\times 6\\times 6\\times 6box, the largest cardinality is2020, reached in generations 12, 13, 15, and 18; generation 20 retains eight nineteen\-point, 19 eighteen\-point, and 23 seventeen\-point states\. Neither run reaches its open target\.

#### AlphaZero\-Radon\.

Each 2,000\-iteration run generates 200,000 trajectories\. In 3D, 157,485 trajectories reach the 11\-point horizon and 8,651 of those terminals are convex; their minimum score isrmin​\(11\)=2r\_\{\\min\}\(11\)=2\. In 4D, 120,973 trajectories reach the 21\-point horizon and 2,302 are convex; their minimum score isrmin​\(21\)=2r\_\{\\min\}\(21\)=2\. The largest verified Radon\-free prefixes have 10 and 20 points, respectively, and neither run reaches its open target\.

### A\.2Fixed\-target methods

#### Exhaustive extension\.

For each dimension, the experiment fixes 50 Radon\-free seeds and tests every unoccupied lattice position in the declared box\. The 3D run tests 159,500 additions and scores 85,913 convex targets; the 4D run tests 499,000 additions and scores 173,355 convex targets\. The minimum score is two in both dimensions\. This is exhaustive only for the fixed seed–box pairs\.

Table 4:Coverage of the declared exhaustive add\-one\-point domains\. Tested excludes positions occupied by the fixed seed; convex targets are the extensions scored by the verifier\.
#### Greedy local search\.

Both dimensions use fixed 1,000\-start evaluation sets\. In 3D, 990 starts improve and 27 terminate at score one; the median score falls from 10 to 4\. In 4D, 991 starts improve and 25 terminate at score one; the median falls from 15 to 4\. Neither run reaches zero\. The reported 4D aggregate is fixed to indices 0–999\.

#### AlphaEvolve\-style program evolution\.

Each dimension has one 20\-mutation lineage\. Every completed mutation returns 32 convex target\-sized configurations under a shared allowance of 1,000 internalfind\_rpscalls, for 640 externally scored candidates per dimension\. Neither lineage reaches zero\. The first score\-one program appears at mutation 7 in 3D and mutation 11 in 4D\.

## Appendix BAdditional Experiment Configurations

#### Software\.

Python 3\.11\.15, NumPy 2\.4\.6, PyNormaliz 2\.23, Gurobi 13\.0\.2\. The structural analyses use no GPU; all analysis runs on CPU on a shared machine, so wall\-clock figures are indicative rather than compute\-matched\.

### B\.1Finite\-box calibration witnesses

The following ordered coordinate arrays are finite\-box positive controls\. The 3D witness has been translated so that its coordinate\-wise minima are zero; the 4D witness is a verified 20\-point prefix emitted by the reported AlphaZero\-Radon run at iteration 190\. The shared verifier confirms row distinctness, convex position, andnrp=0n\_\{\\mathrm\{rp\}\}=0for both arrays\.

#### Three\-dimensional witness\.

W3=\[350840620520750661681211301052\]⊂Λ⁡\(20,20,5\)\.W\_\{3\}=\\begin\{bmatrix\}3&5&0\\\\ 8&4&0\\\\ 6&2&0\\\\ 5&2&0\\\\ 7&5&0\\\\ 6&6&1\\\\ 6&8&1\\\\ 2&1&1\\\\ 3&0&1\\\\ 0&5&2\\end\{bmatrix\}\\subset\\Lambda\(20,20,5\)\.This array is theℤ3\\mathbb\{Z\}^\{3\}positive\-control witness used atn=10n=10\.

#### Four\-dimensional witness\.

W4=\[00001000101520102015100505002513131015050505121515142511221515130105140000041501\]⊂Λ⁡\(6,6,6,6\)\.W\_\{4\}=\\begin\{bmatrix\}0&0&0&0\\\\ 1&0&0&0\\\\ 1&0&1&5\\\\ 2&0&1&0\\\\ 2&0&1&5\\\\ 1&0&0&5\\\\ 0&5&0&0\\\\ 2&5&1&3\\\\ 1&3&1&0\\\\ 1&5&0&5\\\\ 0&5&0&5\\\\ 1&2&1&5\\\\ 1&5&1&4\\\\ 2&5&1&1\\\\ 2&2&1&5\\\\ 1&5&1&3\\\\ 0&1&0&5\\\\ 1&4&0&0\\\\ 0&0&0&4\\\\ 1&5&0&1\\end\{bmatrix\}\\subset\\Lambda\(6,6,6,6\)\.This array is theℤ4\\mathbb\{Z\}^\{4\}positive\-control witness used atn=20n=20\.

Every subset of a convex\-position configuration remains in convex position, and Radon\-freeness is downward closed under taking subsets\. Consequently, auditing the two displayed arrays also certifies feasibility at every smaller cardinality in the same boxes\. The main experiments nevertheless use onlyn=10n=10andn=20n=20as positive controls, because smaller subsets add no independent feasibility question\. The witnesses certify feasibility only; they do not imply that a particular solver can reach them from its initialization\.

### B\.2Three\-dimensional diversity of learning\-guided 10\-point configurations

[Figure4](https://arxiv.org/html/2610.33259#A2.F4)gives eight verified 10\-point configurations inΛ⁡\(20,20,5\)\\Lambda\(20,20,5\)\. PB\-A–PB\-D are retained outputs from PatternBoost generation 20; AZ\-A–AZ\-D are retained AlphaZero\-Radon construction prefixes from iterations 1961, 1962, and 1999\. The panel headings report the layer profiles as counts onz=0,…,4z=0,\\ldots,4\. All eight profiles are distinct:\(0,0,5,5,0\)\(0,0,5,5,0\),\(0,5,5,0,0\)\(0,5,5,0,0\),\(0,1,4,5,0\)\(0,1,4,5,0\),\(0,5,4,0,1\)\(0,5,4,0,1\),\(4,2,0,4,0\)\(4,2,0,4,0\),\(3,3,0,4,0\)\(3,3,0,4,0\),\(4,2,0,1,3\)\(4,2,0,1,3\), and\(4,2,1,1,2\)\(4,2,1,1,2\)\. The examples were deliberately selected to expose different layerwise structures, not to estimate how frequently either method produces a profile\. The reference verifier independently confirms convex position andnrp=0n\_\{\\mathrm\{rp\}\}=0for every displayed configuration\.

Figure 4:Representative learning\-guided 10\-point configurations in 3D\. PB\-A–PB\-D \(circles\) are PatternBoost outputs; AZ\-A–AZ\-D \(squares\) are AlphaZero\-Radon construction prefixes\. Color encodes heightzz, and each heading gives the layer\-count vector fromz=0z=0toz=4z=4\. Gray edges and translucent faces show the convex\-hull scaffold\. To keep both compact AlphaZero\-Radon configurations and box\-spanning PatternBoost configurations legible, panels use their own displayedxx–yylimits, with the true limits shown by the ticks; thezzaxis is visually expanded\. Every panel is a distinct Radon\-free configuration\. The examples illustrate structural diversity rather than empirical profile frequencies\.
### B\.3Result aggregation and audit

For constructive protocols, we replay retained serialized terminal states and take the largest cardinality passing convexity andnrp=0n\_\{\\mathrm\{rp\}\}=0; this produceskmaxk\_\{\\max\}in[Table3](https://arxiv.org/html/2610.33259#S4.T3)\. For fixed\-target protocols, we discard repeated\-row or non\-convex arrays and take the minimumnrpn\_\{\\mathrm\{rp\}\}among the remainingNN\-point candidates; this producesrmin​\(N\)r\_\{\\min\}\(N\)in[Table3](https://arxiv.org/html/2610.33259#S4.T3)\. Ties may be resolved by lexicographic order of the sorted integer rows when an exact representative is needed\. Each reported result records its source run, iteration or generation, verifier output, and artifact path\. AlphaZero\-Radon aggregation includes all 2,000 completed iterations in each dimension\.

## Appendix CImplementation details

This appendix provides pseudocode for the reference verifier and every evaluated search method\. The complete algorithmic specifications appear in[Algorithms1](https://arxiv.org/html/2610.33259#alg1),[2](https://arxiv.org/html/2610.33259#alg2),[3](https://arxiv.org/html/2610.33259#alg3),[4](https://arxiv.org/html/2610.33259#alg4),[5](https://arxiv.org/html/2610.33259#alg5),[6](https://arxiv.org/html/2610.33259#alg6)and[7](https://arxiv.org/html/2610.33259#alg7)\.

### C\.1Reference verifier

The benchmark verifier consists of the functionsis\_conv\_position\_all\(\)andfind\_rps\(\)\. ForX=\{x1,…,xn\}⊂ℤdX=\\\{x\_\{1\},\\ldots,x\_\{n\}\\\}\\subset\\mathbb\{Z\}^\{d\}, the first routine performs one leave\-one\-out linear\-program feasibility test per input point\. For eachii, it asks whether coefficientsαj≥0\\alpha\_\{j\}\\geq 0exist such that

∑j≠iαj=1,∑j≠iαj​xj=xi\.\\sum\_\{j\\neq i\}\\alpha\_\{j\}=1,\\qquad\\sum\_\{j\\neq i\}\\alpha\_\{j\}x\_\{j\}=x\_\{i\}\.\(3\)Feasibility meansxi∈conv⁡\(X∖\{xi\}\)x\_\{i\}\\in\\operatorname\{conv\}\(X\\setminus\\\{x\_\{i\}\\\}\)and the candidate is rejected;is\_conv\_position\_all\(\)returnsTrueexactly when every leave\-one\-out model is infeasible\.

The functionfind\_rps\(\)first passes the homogenized polytope to PyNormaliz\[[Bruns and Ichim, 2010](https://arxiv.org/html/2610.33259#bib.bib6)\]and readsDeg1Elements, which enumerates the lattice points inconv⁡\(X\)\\operatorname\{conv\}\(X\)\. Input vertices are removed\. For every remaining lattice pointpp, the verifier asks whether there areλi≥0\\lambda\_\{i\}\\geq 0and binarybi∈\{0,1\}b\_\{i\}\\in\\\{0,1\\\}such that

∑ibi​λi​xi\\displaystyle\\sum\_\{i\}b\_\{i\}\\lambda\_\{i\}x\_\{i\}=p,\\displaystyle=p,∑i\(1−bi\)​λi​xi\\displaystyle\\sum\_\{i\}\(1\-b\_\{i\}\)\\lambda\_\{i\}x\_\{i\}=p,\\displaystyle=p,∑ibi​λi\\displaystyle\\sum\_\{i\}b\_\{i\}\\lambda\_\{i\}=1,\\displaystyle=1,∑iλi\\displaystyle\\sum\_\{i\}\\lambda\_\{i\}=2\.\\displaystyle=2\.\(4\)The variablebib\_\{i\}assignsxix\_\{i\}to the first part whenbi=1b\_\{i\}=1and to the second part whenbi=0b\_\{i\}=0; the corresponding products select the weights used by each convex combination\. The first coefficient sum is constrained to one, and the second also sums to one because∑i\(1−bi\)​λi=∑iλi−∑ibi​λi=2−1\\sum\_\{i\}\(1\-b\_\{i\}\)\\lambda\_\{i\}=\\sum\_\{i\}\\lambda\_\{i\}\-\\sum\_\{i\}b\_\{i\}\\lambda\_\{i\}=2\-1\. Thus a feasible solution placesppin the convex hulls of two nonempty disjoint parts\. Conversely, any integral Radon partition and an integral Radon point give a feasible assignment\. The implementation solves this model with Gurobi\[[Gurobi Optimization, LLC, 2026](https://arxiv.org/html/2610.33259#bib.bib18)\], setsNonConvex=2for the binary–continuous products, and tests each candidatepp\. It returns each feasible lattice point once, irrespective of the number of partitions that witness it; hence the benchmark score isnrp​\(X\)=\|find\_rps​\(X\)\|n\_\{\\mathrm\{rp\}\}\(X\)=\\lvert\\texttt\{find\\\_rps\}\(X\)\\rvert\. A candidate is a verified Radon\-free configuration exactly whenis\_conv\_position\_all\(\)returnsTrueandfind\_rps\(\)is empty\.

Algorithm 1Reference convexity and integral\-Radon\-point evaluator\.1:finite set of distinct lattice points

X=\{x1,…,xn\}⊂ℤdX=\\\{x\_\{1\},\\ldots,x\_\{n\}\\\}\\subset\\mathbb\{Z\}^\{d\}
2:convex\-position flag

ccand set

RRof integral Radon points

3:for

i=1,…,ni=1,\\ldots,ndo

4:if

xi∈conv⁡\(X∖\{xi\}\)x\_\{i\}\\in\\operatorname\{conv\}\(X\\setminus\\\{x\_\{i\}\\\}\)according to[Equation3](https://arxiv.org/html/2610.33259#A3.E3)then

5:return

\(False,∅\)\(\\textsc\{False\},\\varnothing\)
6:endif

7:endfor

8:

P←\(conv⁡\(X\)∩ℤd\)∖XP\\leftarrow\(\\operatorname\{conv\}\(X\)\\cap\\mathbb\{Z\}^\{d\}\)\\setminus Xusing PyNormaliz

9:

R←∅R\\leftarrow\\varnothing
10:for

p∈Pp\\in Pdo

11:ifthe feasibility system in[Equation4](https://arxiv.org/html/2610.33259#A3.E4)accepts

ppthen

12:

R←R∪\{p\}R\\leftarrow R\\cup\\\{p\\\}
13:endif

14:endfor

15:return

\(True,R\)\(\\textsc\{True\},R\)

Following the canonical implementation,Infeasiblerejects a candidate Radon point and every other returned status counts it\. All complete Radon\-count evaluations use this same rule\.

#### Cross\-track accounting\.

The five tracks are not compute\-matched\. Method\-native stopping rules are part of each protocol, but raw verifier\-call totals are not used as a common budget or efficiency metric\. Reported local\-search aggregates are computed from retained JSON/NPZ artifacts for both the 3D and 4D instances; retained outputs likewise support both dimensional instances of the other tracks\. A stochastic run is successful only if a serialized candidate passes the common verifier\.

The available implementations use the software versions listed below and in[SectionC\.3](https://arxiv.org/html/2610.33259#A3.SS3)\. Their evaluator implementations share the formulation and status rule above\. Retained configurations and aggregate logs support the reported audits\.

#### PatternBoost and AlphaZero\-Radon environment\.

PatternBoost and AlphaZero\-Radon use Python 3\.11\.15, PyTorch 2\.12\.0 with CUDA 13\.0, NumPy 2\.4\.6, SciPy 1\.17\.1 \(including Qhull\), PyNormaliz 2\.23, and Gurobi 13\.0\.2\. Neural\-network training and generation use one NVIDIA A100\-SXM4 GPU with 80 GB memory; CPU processes execute the geometric oracle\. We do not request PyTorch’s bitwise\-deterministic GPU kernels\.

#### Local and exhaustive environment\.

These CPU\-only searches use Python 3\.12\.11, NumPy 1\.26\.4, SciPy 1\.16\.2, PyNormaliz 2\.19, and Gurobi 12\.0\.3; the local\-search environment also contains PyTorch 2\.8\.0 and trimesh 4\.8\.3\. They run on an Intel Xeon Gold 5320 node \(two 26\-core sockets, SMT2, 503 GiB RAM\)\. Local tasks request 32 CPUs and 64 GB with at most three concurrent tasks; exhaustive tasks request 8 CPUs and 24 GB and run one at a time\. Neither requests a GPU\.

#### Program\-evolution environment\.

OpenEvolve uses Python 3\.12\.11, OpenEvolve 0\.2\.27, NumPy 1\.26\.4, SciPy 1\.16\.2, PyNormaliz 2\.19, and Gurobi 12\.0\.3 on the same CPU node\. Each job requests four CPUs, 64 GB RAM, and no GPU\. The method\-specific subsections give the remaining seeds, stopping rules, provenance, and run settings\.

### C\.2Exhaustive add\-one\-point experiment configuration

#### Dimension\-general procedure\.

For a fixed seedS⊂ℤdS\\subset\\mathbb\{Z\}^\{d\}and integer boxB=∏j=1d\{ℓj,…,uj\}B=\\prod\_\{j=1\}^\{d\}\\\{\\ell\_\{j\},\\ldots,u\_\{j\}\\\}, the driver visits everyq∈Bq\\in Bin deterministic Cartesian\-product order\. It skipsq∈Sq\\in S, rejects extensions outside convex position, and evaluatesnrp​\(S∪\{q\}\)n\_\{\\mathrm\{rp\}\}\(S\\cup\\\{q\\\}\)for every remaining extension\. It does not stop when a zero or a new minimum is found\. Therefore every unoccupied position inBBis either rejected by the convex\-position predicate or assigned a Radon\-point count\. This is the sense in which the search is exhaustive\. The seed is never modified, and no claim is made about other seeds or positions outsideBB\.

Algorithm 2Exhaustive add\-one\-point search within a fixed box1:verified Radon\-free seed

S⊂ℤdS\\subset\\mathbb\{Z\}^\{d\}with

\|S\|=N−1\|S\|=N\-1; bounds

\(ℓj,uj\)j=1d\(\\ell\_\{j\},u\_\{j\}\)\_\{j=1\}^\{d\}
2:minimum score

mm, minimizers

QQ, score histogram

HH, coverage counts

3:

B←∏j=1d\{ℓj,…,uj\}B\\leftarrow\\prod\_\{j=1\}^\{d\}\\\{\\ell\_\{j\},\\ldots,u\_\{j\}\\\};

m←\+∞m\\leftarrow\+\\infty;

Q←∅Q\\leftarrow\\varnothing;

H←∅H\\leftarrow\\varnothing;

cconvex←0c\_\{\\rm convex\}\\leftarrow 0
4:for

q∈Bq\\in Bin Cartesian\-product orderdo

5:if

q∈Sq\\in Sthen

6:continue

7:endif

8:

X←S∪\{q\}X\\leftarrow S\\cup\\\{q\\\}
9:if

XXis not in convex positionthen

10:continue

11:endif

12:

cconvex←cconvex\+1c\_\{\\rm convex\}\\leftarrow c\_\{\\rm convex\}\+1
13:

r←nrp​\(X\)r\\leftarrow n\_\{\\mathrm\{rp\}\}\(X\);

H⁡\[r\]←H⁡\[r\]\+1H\[r\]\\leftarrow H\[r\]\+1
14:if

r<mr<mthen

15:

m←rm\\leftarrow r;

Q←\{q\}Q\\leftarrow\\\{q\\\}
16:elseif

r=mr=mthen

17:

Q←Q∪\{q\}Q\\leftarrow Q\\cup\\\{q\\\}
18:endif

19:endfor

20:return

\(m,Q,H,\|B\|−\|S\|,cconvex\)\(m,Q,H,\|B\|\-\|S\|,c\_\{\\rm convex\}\)

#### Frozen seeds and sampling\.

Before evaluation, the experiment fixes 50verified Radon\-free seeds per dimension using NumPy seed2026081920260819\. The 3D seeds lie in\{−10,…,9\}2×\{0,…,7\}\\\{\-10,\\ldots,9\\\}^\{2\}\\times\\\{0,\\ldots,7\\\}; the 4D seeds lie in\{0,…,19\}2×\{0,…,4\}2\\\{0,\\ldots,19\\\}^\{2\}\\times\\\{0,\\ldots,4\\\}^\{2\}\. Seed selection is independent of extension scores, and no sampling occurs during a task\.

#### Boxes and coverage\.

The 3D box isB3=\{−10,…,9\}×\{−10,…,9\}×\{0,…,7\}B\_\{3\}=\\\{\-10,\\ldots,9\\\}\\times\\\{\-10,\\ldots,9\\\}\\times\\\{0,\\ldots,7\\\}, with3,2003\{,\}200positions\. Each 10\-point seed is contained inB3B\_\{3\}, leaving3,1903\{,\}190attempted additions per seed and159,500159\{,\}500overall\. The 4D box isB4=\{0,…,19\}×\{0,…,19\}×\{0,…,4\}2B\_\{4\}=\\\{0,\\ldots,19\\\}\\times\\\{0,\\ldots,19\\\}\\times\\\{0,\\ldots,4\\\}^\{2\}, with10,00010\{,\}000positions\. Each 20\-point seed is contained inB4B\_\{4\}, leaving9,9809\{,\}980attempted additions per seed and499,000499\{,\}000overall\.

#### Convex\-position and Radon tests\.

Before enumeration, the driver checks that the seed is row\-distinct, in convex position, and hasnrp​\(S\)=0n\_\{\\mathrm\{rp\}\}\(S\)=0\. For each extensionXX, it constructs the homogenized polytope with PyNormaliz and readsExtremeRays\. The extension passes exactly when the returned rays equal\{\(x,1\):x∈X\}\\\{\(x,1\):x\\in X\\\}, so every input point is a vertex\. This exact rational extreme\-ray test is mathematically equivalent to the reference leave\-one\-out convex\-position predicate, although it is a different implementation\. For every passing extension,find\_rps\_nd\(\)uses PyNormalizDeg1Elementsand the same Gurobi partition\-feasibility test in[SectionC\.1](https://arxiv.org/html/2610.33259#A3.SS1)\. Gurobi uses one thread per model; up to eight lattice points are tested concurrently within a task\.

The supplementary manifest records the fixed seed coordinates, declared boxes, and coverage counts needed to reproduce the reported aggregate\.

### C\.3Greedy local\-search experiment configuration

#### Dimension\-general procedure\.

For an instance\(d,N,L\)\(d,N,L\), the descent moves all coordinates\. Its ordered neighbourhood is

𝒟d=\{−1,0,1\}d∖\{𝟎\}\.\\mathcal\{D\}\_\{d\}=\\\{\-1,0,1\\\}^\{d\}\\setminus\\\{\\mathbf\{0\}\\\}\.For a boxΛ⁡\(L\)\\Lambda\(L\), proposals outside the box are rejected\. The evaluated 3D and 4D neighbourhoods contain 26 and 80 offsets, respectively, before boundary and geometry filtering\. The method uses a strict\-improvement rule, stored point order, lexicographic direction order, a 100\-sweep cap, and early stopping at score zero or after a full non\-improving sweep\.

#### Initial configurations\.

In 3D, initialization adds one uniformly sampled point in\{−10,…,9\}2×\{0,…,4\}\\\{\-10,\\ldots,9\\\}^\{2\}\\times\\\{0,\\ldots,4\\\}to a verified 10\-point seed and accepts the first distinct, convex extension within 100 draws\. From the eligible pool, 1,000starts are sampled with NumPy seed2026081920260819and translated into the20×20×520\\times 20\\times 5box\. In 4D, one uniformly sampled in\-box point is added to each verified 20\-point seed, again requiring distinctness and convex position; the reported set is the fixed first 1,000starts generated with NumPy seed2026081520260815\. Initialization does not filter or rank candidates bynrpn\_\{\\mathrm\{rp\}\}\.[Algorithm3](https://arxiv.org/html/2610.33259#alg3)gives the subsequent deterministic descent\. The point order is the stored array order\. Directions are the lexicographically ordered Cartesian product\{−1,0,1\}d\\\{\-1,0,1\\\}^\{d\}with the zero vector removed\. Only strict improvement is admissible, so the first direction attaining a given best score wins a tie and equal\-score plateaus are never traversed\. No anchoring translation occurs during descent\. All accepted 3D configurations remain inside\{0,…,19\}2×\{0,…,4\}\\\{0,\\ldots,19\\\}^\{2\}\\times\\\{0,\\ldots,4\\\}, and all accepted 4D configurations remain inside\{0,…,19\}2×\{0,…,4\}2\\\{0,\\ldots,19\\\}^\{2\}\\times\\\{0,\\ldots,4\\\}^\{2\}\.

Algorithm 3Dimension\-general ordered greedy local search\.1:ordered convex configuration

X=\(x1,…,xN\)⊂ℤdX=\(x\_\{1\},\\ldots,x\_\{N\}\)\\subset\\mathbb\{Z\}^\{d\}, sweep cap

T=100T=100, ordered directions

𝒟d\\mathcal\{D\}\_\{d\}, score

nrpn\_\{\\mathrm\{rp\}\}
2:for

t=1,…,Tt=1,\\ldots,Tdo

3:

sstart←nrp​\(X\)s\_\{\\mathrm\{start\}\}\\leftarrow n\_\{\\mathrm\{rp\}\}\(X\)
4:for

i=1,…,Ni=1,\\ldots,Ndo

5:

s⋆←nrp​\(X\)s^\{\\star\}\\leftarrow n\_\{\\mathrm\{rp\}\}\(X\);

X⋆←XX^\{\\star\}\\leftarrow X
6:for

δ∈𝒟d\\delta\\in\\mathcal\{D\}\_\{d\}in lexicographic orderdo

7:

Y←XY\\leftarrow Xwith

xi←xi\+δx\_\{i\}\\leftarrow x\_\{i\}\+\\delta
8:if

Y⊆Λ⁡\(L\)Y\\subseteq\\Lambda\(L\), has distinct points, and is in convex positionthen

9:

s←nrp​\(Y\)s\\leftarrow n\_\{\\mathrm\{rp\}\}\(Y\)
10:if

s<s⋆s<s^\{\\star\}then

11:

\(X⋆,s⋆\)←\(Y,s\)\(X^\{\\star\},s^\{\\star\}\)\\leftarrow\(Y,s\)
12:endif

13:if

s⋆=0s^\{\\star\}=0then

14:break

15:endif

16:endif

17:endfor

18:

X←X⋆X\\leftarrow X^\{\\star\}
19:endfor

20:if

nrp​\(X\)=0n\_\{\\mathrm\{rp\}\}\(X\)=0or

nrp​\(X\)≥sstartn\_\{\\mathrm\{rp\}\}\(X\)\\geq s\_\{\\mathrm\{start\}\}then

21:break

22:endif

23:endfor

24:return

XX

#### Convexity and score implementation\.

For every candidate, convex position is checked point by point\. To test pointxix\_\{i\}, a Gurobi feasibility LP asks whether nonnegative coefficients onX∖\{xi\}X\\setminus\\\{x\_\{i\}\\\}sum to one and reproducexix\_\{i\}; feasibility rejects the candidate\. For the objective, PyNormaliz enumerates the degree\-one lattice points of the homogenized polytope and the implementation removes the input vertices\. For each remaining lattice pointpp, Gurobi solves the feasibility model

∑ibi​λi​xi\\displaystyle\\sum\_\{i\}b\_\{i\}\\lambda\_\{i\}x\_\{i\}=p,\\displaystyle=p,∑i\(1−bi\)​λi​xi\\displaystyle\\sum\_\{i\}\(1\-b\_\{i\}\)\\lambda\_\{i\}x\_\{i\}=p,\\displaystyle=p,∑ibi​λi\\displaystyle\\sum\_\{i\}b\_\{i\}\\lambda\_\{i\}=1,\\displaystyle=1,∑iλi\\displaystyle\\sum\_\{i\}\\lambda\_\{i\}=2,\\displaystyle=2,withλi≥0\\lambda\_\{i\}\\geq 0continuous andbi∈\{0,1\}b\_\{i\}\\in\\\{0,1\\\}; the model usesNonConvex=2\. The number of feasibleppis the descent score; any solver status other thanInfeasibleis treated as feasible\. No Gurobi feasibility, integrality, or optimality parameters are overridden\. Radon points are memoized within a configuration after its first score evaluation\. The fixed selection manifests identify the 1,000 reported starts in each dimension, and the retained trajectories reproduce the aggregate outcomes in Appendix[A](https://arxiv.org/html/2610.33259#A1)\. The CPU implementation uses Python 3\.12\.11, PyNormaliz 2\.19, Gurobi 12\.0\.3, and SciPy 1\.16\.2\.

Table 5:Greedy local\-search hyperparameters shared across dimensions\.

### C\.4PatternBoost experiment configuration

For an instance\(d,N,L\)\(d,N,L\), PatternBoost uses variable\-length add\-point sequences inΛ⁡\(L\)\\Lambda\(L\)\. Its vocabulary has1\+∏j=1dLj1\+\\prod\_\{j=1\}^\{d\}L\_\{j\}entries: one sharedStart/Stoptoken and one token per lattice point\. A sequence has the form\[0,τL​\(p1\),…,τL​\(pk\),0\]\[0,\\tau\_\{L\}\(p\_\{1\}\),\\ldots,\\tau\_\{L\}\(p\_\{k\}\),0\], where the points are sorted lexicographically before supervised training; losses after the first stop token are masked\. The reference incremental legality test accepts a prefix only when it is in convex position and has no integral Radon point\. This is stricter than merely filtering complete model outputs: every training configuration and every saturated output is legal at each accepted prefix\. All dimension\-independent architectural, optimization, selection, generation, and saturation hyperparameters are shared across dimensions\.

The completed three\-dimensional run instantiatesL=\(20,20,5\)L=\(20,20,5\)andN=11N=11, giving a2,0012\{,\}001\-token vocabulary\. The completed four\-dimensional run usesL=\(6,6,6,6\)L=\(6,6,6,6\)andN=21N=21, giving a1,2971\{,\}297\-token vocabulary\.

Algorithm 4PatternBoost with verifier\-gated incremental saturation\.1:box

LL, maximum size

NN, generations

GG, state\-batch size

BB, keep fraction

qq, target retained\-set size

TT, reference predicateLegal\(S\)

2:form

T/qT/qindependent empty seeds; saturate each seed withLegal

3:retain the top

q​BqBterminal states within each batch to obtain

𝒟0\\mathcal\{D\}\_\{0\}
4:for

i=0,…,G−1i=0,\\ldots,G\-1do

5:split

𝒟i\\mathcal\{D\}\_\{i\}by complete symmetry orbit into train/validation sets

6:augment both splits by box\-preserving reflections and axis permutations

7:train a next\-token Transformer; retain the lowest\-validation\-loss checkpoint

8:sample

T/qT/qunconstrained token sequences fromStart

9:decode each sequence in order, skipping a token unless it denotes an unused lattice point whose addition satisfiesLegal

10:foreach decoded seed state

SSdo

11:while

\|S\|<N\|S\|<Nand untested lattice points remaindo

12:traverse the remaining points in a fresh random order

13:append the first

ppwithLegal\(

\(S∪\{p\}\)\(S\\cup\\\{p\\\}\)\); permanently remove every tested point, including

pp, from the remaining pool

14:stop if no tested point is legal

15:endwhile

16:endfor

17:in each batch of

BBterminal states, retain the top

q​BqBstates by cardinality

18:

𝒟i\+1←\\mathcal\{D\}\_\{i\+1\}\\leftarrowretained states; save their unaugmented and augmented forms

19:endfor

Table 6:PatternBoost hyperparameters\.Teacher forcing pads every sequence to context lengthN\+1N\+1\(12 in 3D and 22 in 4D\)\. For an ordered point\-token sequence\(τ1,…,τk\)\(\\tau\_\{1\},\\ldots,\\tau\_\{k\}\), the input is\[0,τ1,…,τk\]\[0,\\tau\_\{1\},\\ldots,\\tau\_\{k\}\]and the target is\[τ1,…,τk,0\]\[\\tau\_\{1\},\\ldots,\\tau\_\{k\},0\]; target positions after the first stop are set to−1\-1and ignored by cross\-entropy\. Minibatches are sampled uniformly with replacement\.

Letmℓm\_\{\\ell\}be the multiplicity of side lengthℓ\\ellinLL\. The augmentation group has

\|ℋ⁡\(L\)\|=2d​∏ℓmℓ\!\|\\mathcal\{H\}\(L\)\|=2^\{d\}\\prod\_\{\\ell\}m\_\{\\ell\}\!box\-preserving transformations: independent reflections and permutations among axes with equal side lengths\. All transforms are emitted without deduplicating coincident arrays\. ForL=\(20,20,5\)L=\(20,20,5\)this gives1616transforms; forL=\(6,6,6,6\)L=\(6,6,6,6\)it gives24​4\!=3842^\{4\}4\!=384\. With5050retained configurations, the nominal45/545/5orbit split yields720/80720/80augmented train/validation sequences in 3D and17,280/1,92017\{,\}280/1\{,\}920in 4D when all orbits are distinct\. Complete symmetry orbits are assigned wholly to one split before augmentation\. Validation evaluates at most ten batches of100100\. Training runs for500500updates and evaluates every2525updates; the minimum validation\-loss checkpoint among steps25,50,…,50025,50,\\ldots,500is used both for generation and to initialize the next round\. Exact split sizes and selected steps are recorded per generation\.

The Transformer uses learned token and absolute\-position embeddings, untied output weights, causal self\-attention, and PyTorch’s default parameter initialization\. Its dropout rate is zero\. At the start of a new round, model weights come from the preceding round’s selected checkpoint but AdamW is reinitialized; optimizer state is restored only when resuming an interruption within the same round\. AdamW uses the fixed learning rate in[Table6](https://arxiv.org/html/2610.33259#A3.T6)\.

Generation always drawsNNtokens after the initialStart, without constraining the vocabulary, in minibatches of1010\. Decoding stops at the first sampledStop; before that stop, duplicate or illegal point tokens are removed from the remaining\-point pool and skipped while later tokens are still attempted\. Saturation repeatedly draws a fresh permutation of the remaining pool, accepts the first legal point, and permanently removes all points tested in that pass\. The250250terminal states form one selection batch; the largest20%20\\%, namely5050states, survive\. Stable input order breaks cardinality ties\. Generation and saturation use1616CPU workers\. A master seed of2026080920260809initializes Python, NumPy, and PyTorch randomness; per\-state seeds are assigned before parallel dispatch\.

### C\.5AlphaEvolve\-style program\-evolution experiment configuration

#### Dimension\-general task interface\.

For every instance\(d,N\)\(d,N\), the immutable wrapper receives the same typed inputs: sixteen cataloged\(N−1\)×d\(N\-1\)\\times darrays encoding Radon\-free configurations, the dimension\-independent geometry routines, target sizeNN, output cap, and remaining oracle allowance\. Returned candidates must have shapeN×dN\\times d\. The mutable four\-function interface, validation order, fitness, oracle allowance, output cap, and MAP\-Elites mechanism are shared across dimensions\. The dimension, target, reference catalog, feature\-bin count forbest\_size, and number of evolution mutations are recorded separately below\. No lattice side lengths are passed to the evolved program or enforced by the evaluator\.

#### Dimension\-specific inputs\.

The 3D run samples eight references from each of two catalogs of verified 10\-point configurations; the 4D run samples sixteen references from a catalog of verified 20\-point configurations\. Both use selection seed20262026, run for 20mutations, and accept arbitrary finite integer output coordinates\. The completed runs take 4\.88 and 47\.58 hours in 3D and 4D, respectively\.

Each call togenerate\_candidatesinstalls process\-local wrappers around the geometry functions\. Calls tofind\_rpswhile the candidate program is active increment a counter and are limited jointly to1,0001\{,\}000; after exhaustion, a further call returns a nonempty sentinel rather than permitting a false Radon\-free result\. Convexity calls are uncharged\. The internal allowance closes before the external evaluator begins, so rechecking the returned batch neither consumes nor enlarges the program’s search budget\. Every program receives a newly initialized NumPydefault\_rng\(42\)and may return at most3232arrays\.

#### Candidate validation and fitness\.

The evaluator imports the generated module and invokesgenerate\_candidates\(N,32\); the declared three\-dimensional instance instantiates this call withN=11N=11\. Exceptions receive score00and an error flag\. Each returned object is converted to a NumPy array and rejected unless it is finite, two\-dimensional withddcolumns, integer\-valued, and row\-distinct\. More than3232objects are truncated; arrays with a row count other thanNNreceive candidate score00\. For a target\-sized arrayX=\(x1,…,xN\)X=\(x\_\{1\},\\ldots,x\_\{N\}\), letP=\(x1,…,xN−1\)P=\(x\_\{1\},\\ldots,x\_\{N\-1\}\)denote its ordered prefix\. The exact candidate score is

s⁡\(X\)=\{1,X​is convex and​nrp​\(X\)=0,0\.50\+0\.49/\(1\+nrp​\(X\)\),X​is convex and​nrp​\(X\)\>0,0\.40\+0\.09​\(\(N−1\)/N\),X​is not convex, but​P​is convex and​nrp​\(P\)=0,0,otherwise\.s\(X\)=\\begin\{cases\}1,&X\\text\{ is convex and \}n\_\{\\mathrm\{rp\}\}\(X\)=0,\\\\ 0\.50\+0\.49/\(1\+n\_\{\\mathrm\{rp\}\}\(X\)\),&X\\text\{ is convex and \}n\_\{\\mathrm\{rp\}\}\(X\)\>0,\\\\ 0\.40\+0\.09\(\(N\-1\)/N\),&X\\text\{ is not convex, but \}P\\text\{ is convex and \}n\_\{\\mathrm\{rp\}\}\(P\)=0,\\\\ 0,&\\text\{otherwise\.\}\\end\{cases\}\(5\)The program fitnesscombined\_scoreismaxX⁡s⁡\(X\)\\max\_\{X\}s\(X\)over the returned batch\. The evaluator records the best candidate and summary batch statistics; no metric combines properties from different candidates\.

Algorithm 5Dimension\-general AlphaEvolve\-style program\-evolution search\.1:instance

\(d,N\)\(d,N\), initial program

p0p\_\{0\}, fixed evaluator

EE, islands

ℐ\\mathcal\{I\}, number of mutations

TT, target fitness

11
2:evaluate

p0p\_\{0\}with

EE; insert its source, metrics, and features into the database

3:for

t=1,…,Tt=1,\\ldots,Tdo

4:choose the active island and sample a parent from its population/archive

5:assemble system and user messages from the fixed task prompt, parent source and metrics, execution artifacts, lineage, and selected top/diverse programs

6:request an exactSEARCH/REPLACEdiff fromgpt\-5\.6\-sol

7:apply valid edits only inside the mutableEVOLVE\-BLOCK; reject a full rewrite

8:execute the child with a fresh RNG seed

4242and a shared allowance of

1,0001\{,\}000internalfind\_rpscalls

9:

EEindependently validates and scores at most

3232returned candidates

10:store source, parent, prompt, response, metrics, features, and candidate artifact

11:update the island and global archive; migrate after every ten island generations

12:checkpoint;stopifcombined\_score=1=1

13:endfor

Table 7:AlphaEvolve\-style program\-evolution settings\.
#### Outcomes\.

Each AlphaEvolve\-style program\-evolution experiment completes 20 mutations\.

Every child in both runs returns 32 convex target configurations, uses exactly 1,000 internal oracle calls, and has evaluator error flag zero\. Each lineage therefore returns20×32=64020\\times 32=640candidates\. Neither returns a Radon\-free target\. The best 3D program \(first seen at mutation 7\) and best 4D program \(first seen at mutation 11\) each attain one Radon point and fitness0\.7450\.745\.

### C\.6AlphaZero\-Radon experiment configuration

The reported runs use the Plan\-D search and training implementation\. The 3D instance hasL=\(20,20,5\)L=\(20,20,5\),N=11N=11, and2,0002\{,\}000actions\. The 4D instance hasL=\(6,6,6,6\)L=\(6,6,6,6\),N=21N=21, and1,2961\{,\}296actions\. Both complete 2,000 iterations, comprising 200,000 generated trajectories per dimension; all completed iterations enter the reported outcomes\. The implementation calls this data\-generation phase “self\-play,” but the adapted task is single\-agent and has no opponent\.

#### States, actions, and terminal outcomes\.

A state is the unordered set of placed lattice points, serialized internally by mixed\-radix tokens

a⁡\(x\)=∑j=1dxj​∏k=1j−1Lk,a\(x\)=\\sum\_\{j=1\}^\{d\}x\_\{j\}\\prod\_\{k=1\}^\{j\-1\}L\_\{k\},so the first axis varies fastest\. Every game starts from the fixed origin\(0,…,0\)\(0,\\ldots,0\)and the agent selects the remainingN−1N\-1points\. The origin is an explicit finite\-box anchor, not a without\-loss\-of\-generality reduction; it contributes neither a policy target nor an edge reward\. At an expanded node, occupied actions are zeroed before candidate ranking\. Applying an action terminates the branch when the new set is non\-convex, has positivenrpn\_\{\\mathrm\{rp\}\}, or reachesNN\. Rewards follow[Equation2](https://arxiv.org/html/2610.33259#S3.E2)and are attached to the edge entering the successor\. The value head estimates the undiscounted future reward\-to\-go\.

#### Plan\-D top\-KKPUCT\.

LetPθ\(⋅∣s\)P\_\{\\theta\}\(\\cdot\\mid s\)be the full\-vocabulary softmax policy\. Every newly expanded node zeros occupied actions, retains theK=64K=64largest remaining probabilities, and renormalizes them\. Writing the retained support asC⁡\(s\)C\(s\), the tree prior is

P^​\(a∣s\)=Pθ\(a∣s\)𝟏\{a∈C\(s\)\}∑b∈C⁡\(s\)Pθ​\(b∣s\)\.\\widehat\{P\}\(a\\mid s\)=\\frac\{P\_\{\\theta\}\(a\\mid s\)\\mathbf\{1\}\\\{a\\in C\(s\)\\\}\}\{\\sum\_\{b\\in C\(s\)\}P\_\{\\theta\}\(b\\mid s\)\}\.At the root only, self\-play replaces this prior by0\.75​P^\+0\.25​η0\.75\\widehat\{P\}\+0\.25\\eta, whereη∼Dir⁡\(0\.1​𝟏\)\\eta\\sim\\operatorname\{Dir\}\(0\.1\\mathbf\{1\}\)\. There is no eager root probe: a transition is checked only when tree selection reaches it\. Once a child is found terminal, Plan D marks that edge and excludes it from later tree selections\. Terminal means any non\-convex or Radon\-positive set as well as a target success, so a scored terminal receives one backup before it is masked\. Search does not stop early upon discovering a success\.

Algorithm 6AlphaZero\-Radon single\-agent trajectory\-generation and training loop\.1:lattice sides

LL, target

NN, iterations

II, games

GG, reads

MM, candidate count

KK,

cpuctc\_\{\\mathrm\{puct\}\}, Dirichlet parameter

α\\alpha
2:initialize policy–value parameters

θ\\theta, Adam state, and FIFO buffer

ℬ\\mathcal\{B\}
3:for

i=1,…,Ii=1,\\ldots,Ido

4:

𝒟i←∅\\mathcal\{D\}\_\{i\}\\leftarrow\\emptyset
5:for

g=1,…,Gg=1,\\ldots,Gdo

6:initialize

s0=\{\(0,…,0\)\}s\_\{0\}=\\\{\(0,\\ldots,0\)\\\}and empty state, policy, and reward lists

7:while

sts\_\{t\}is not terminaldo

8:

πt←PUCTSearch​\(st,θ,M,K,cpuct,α\)\\pi\_\{t\}\\leftarrow\\textsc\{PUCTSearch\}\(s\_\{t\},\\theta,M,K,c\_\{\\mathrm\{puct\}\},\\alpha\)
9:store

\(st,πt\)\(s\_\{t\},\\pi\_\{t\}\); sample

at∼πta\_\{t\}\\sim\\pi\_\{t\}; apply

ata\_\{t\}
10:append the realized edge reward and continue unless terminal

11:endwhile

12:compute

Gt=∑j=tTend−1rjG\_\{t\}=\\sum\_\{j=t\}^\{T\_\{\\rm end\}\-1\}r\_\{j\}backwards

13:add all

\(st,πt,Gt,nrpfinal\)\(s\_\{t\},\\pi\_\{t\},G\_\{t\},n\_\{\\rm rp\}^\{\\rm final\}\)to

𝒟i\\mathcal\{D\}\_\{i\}
14:endfor

15:append

𝒟i\\mathcal\{D\}\_\{i\}to

ℬ\\mathcal\{B\}; retain its latest

4040iterations

16:for

h=1,…,10h=1,\\ldots,10do

17:sample a set\-length bucket proportional to its buffered count

18:draw at most

128128examples from that bucket without replacement

19:update

θ\\thetawith Adam on[Equation6](https://arxiv.org/html/2610.33259#A3.E6)

20:endfor

21:every 10 iterations, save the model weights

22:endfor

Algorithm 7Plan\-D fixed\-candidate PUCT search used by the reported runs\.1:state

ss, network

θ\\theta, reads

MM, candidate count

KK,

cpuctc\_\{\\mathrm\{puct\}\}, Dirichlet parameter

α\\alpha
2:functionPUCTSearch\(

s,θ,M,K,cpuct,αs,\\theta,M,K,c\_\{\\mathrm\{puct\}\},\\alpha\)

3:create a fresh root; retain and renormalize its occupancy\-legal policy top\-

KK
4:mix Dirichlet noise over those root candidates

5:for

m=1,…,Mm=1,\\ldots,Mdo

6:descend over unmarked retained edges by maximizing

Q⁡\(u,a\)\+cpuct​U​\(u,a\)Q\(u,a\)\+c\_\{\\mathrm\{puct\}\}U\(u,a\)
7:evaluate the selected successor withStepOutcome

8:ifterminalthen

9:mark its parent edge terminal; back up its edge reward with leaf value

00
10:else

11:retain the policy top\-

KKcandidate set at the leaf; set leaf value to

VθV\_\{\\theta\}
12:back up intervening edge rewards plus the leaf value

13:endif

14:endfor

15:returnthe normalized full\-vocabulary root visit counts

π\\pi
16:endfunction

Table 8:AlphaZero\-Radon hyperparameters shared across dimensions\.
#### PUCT and training targets\.

PUCT uses

Q⁡\(s,a\)=W⁡\(s,a\)1\+N⁡\(s,a\),U⁡\(s,a\)=P^​\(s,a\)​N⁡\(s\)1\+N⁡\(s,a\)\.Q\(s,a\)=\\frac\{W\(s,a\)\}\{1\+N\(s,a\)\},\\qquad U\(s,a\)=\\widehat\{P\}\(s,a\)\\frac\{\\sqrt\{N\(s\)\}\}\{1\+N\(s,a\)\}\.The root visit count starts at zero\. Each simulation accumulates normalized per\-edge rewards between the leaf and every ancestor and addsVθV\_\{\\theta\}only at a nonterminal leaf\. After 500 simulations, the root visit counts are normalized over the full vocabulary and the played action is sampled from them withT=1T=1\.

For a minibatch of sizeBB, define the full\-vocabulary policy by

pθ​\(a∣sb\)=exp⁡ℓθ​\(sb,a\)∑c∈Λ⁡\(L\)exp⁡ℓθ​\(sb,c\)\.p\_\{\\theta\}\(a\\mid s\_\{b\}\)=\\frac\{\\exp\\ell\_\{\\theta\}\(s\_\{b\},a\)\}\{\\sum\_\{c\\in\\Lambda\(L\)\}\\exp\\ell\_\{\\theta\}\(s\_\{b\},c\)\}\.The loss is

ℒ=−1B∑b=1B∑a∈Λ⁡\(L\)πb\(a\)logpθ\(a∣sb\)\+λV1B∑b=1B\(Vθ\(sb\)−Gb\)2\.\\mathcal\{L\}=\-\\frac\{1\}\{B\}\\sum\_\{b=1\}^\{B\}\\sum\_\{a\\in\\Lambda\(L\)\}\\pi\_\{b\}\(a\)\\log p\_\{\\theta\}\(a\\mid s\_\{b\}\)\+\\lambda\_\{V\}\\frac\{1\}\{B\}\\sum\_\{b=1\}^\{B\}\\bigl\(V\_\{\\theta\}\(s\_\{b\}\)\-G\_\{b\}\\bigr\)^\{2\}\.\(6\)HereλV=20\\lambda\_\{V\}=20,πb\\pi\_\{b\}is the normalized MCTS visit vector, andGbG\_\{b\}is the undiscounted reward\-to\-go\. Althoughπb\\pi\_\{b\}is sparse because search expands top\-KKactions, the softmax denominator is the full action vocabulary\.

#### Network\.

Each coordinate is normalized asuj=xj/\(Lj−1\)u\_\{j\}=x\_\{j\}/\(L\_\{j\}\-1\)and mapped to the Fourier feature vector

\[sin⁡\(2k​π​uj\),cos⁡\(2k​π​uj\)\]k=0Bd−1,\[\\sin\(2^\{k\}\\pi u\_\{j\}\),\\cos\(2^\{k\}\\pi u\_\{j\}\)\]\_\{k=0\}^\{B\_\{d\}\-1\},where the Nyquist\-safeBd=3B\_\{d\}=3in both runs, followed by a learned linear projection to width256256\. Four layer\-normalized set\-attention blocks with four heads encode the placed\-point set\. Each multihead\-attention block uses scaled dot\-product attention, a query residual, a ReLU output projection with a second residual, and LayerNorm after each residual\. Every candidate cell uses the same coordinate encoder to form a cross\-attention query over the set latents; a256→256→1256\\rightarrow 256\\rightarrow 1GELU MLP produces its policy logit\. The value head mean\-pools the set latents and applies an independent256→256→1256\\rightarrow 256\\rightarrow 1GELU MLP, whose scalar output is used directly\. All linear layers use PyTorch defaults, and the dropout rate is zero\.

#### Replay and optimization\.

One randomly initialized network and one Adam optimizer persist across all iterations\. At each iteration, every self\-play game receives the latest evaluation\-mode weights, and every move constructs a fresh tree\. Deterministic verifier outcomes may be memoized within a trajectory; this does not alter transitions, terminal labels, rewards, or the search budget\.

For each decision after the origin, self\-play stores the pre\-move token set, the normalized visit policy, the final Radon count, and the reward\-to\-go\.

The learner appends the current iteration before training\. For each of ten updates, it selects a set\-length bucket with probability proportional to its buffered sample count and drawsmin⁡\(128,\|bucket\|\)\\min\(128,\\lvert\\text\{bucket\}\\rvert\)examples uniformly without replacement\. Thus each iteration makes at most1,2801\{,\}280replay draws, independent of the current FIFO population; samples may reappear in later updates\. Adam uses the fixed learning rate10−410^\{\-4\}, PyTorch defaults for\(β1,β2,ϵ\)\(\\beta\_\{1\},\\beta\_\{2\},\\epsilon\), and coupledℓ2\\ell\_\{2\}weight decay10−410^\{\-4\}\.

Both jobs run uninterrupted from a clean initialization to iteration 2,000\. The global initialization seed is4242; per\-game and replay\-sampling seeds are deterministic functions of the iteration and game indices\. End\-to\-end wall times are approximately85\.985\.9hours in 3D and561\.4561\.4hours in 4D\.

Figure 5:Training diagnostics for the complete 2,000\-iteration AlphaZero\-Radon runs\. The top and bottom rows show the 3D \(N=11N=11\) and 4D \(N=21N=21\) instances, respectively\. Curves are 50\-iteration moving averages\. The trajectory panels show only the mean trajectory length; the dashed line is the full horizon of 10 or 20 policy\-selected placements after the fixed origin\.
#### Aggregation over the complete runs\.

In 3D,157,485157\{,\}485of 200,000 generated trajectories place an eleventh point\. Of those terminals,8,6518\{,\}651are convex and receive a complete Radon count; the minimum is22\. In 4D,120,973120\{,\}973games place a twenty\-first point,2,3022\{,\}302of those terminals are convex, and the minimum Radon count is22; a trajectory with terminal score two first appears at iteration 702\. No target terminal has score zero\. The remaining target\-cardinality terminals are non\-convex and are not included in the “target candidates” column\. The largest fully verified prefixes have sizes 10 and 20, respectively\.[Figure5](https://arxiv.org/html/2610.33259#A3.F5)reports the corresponding optimization and trajectory\-length diagnostics over the same complete runs\.