FIELD MAP 01 / OPENAI MATH
Explore selected papers from OpenAI’s math repository through MILO’s diagrams and explanations. Compare a consequence claimed by each paper with a further question proposed here. Curated from repository snapshot fd4aeeb2ee4f; sources checked 9–10 October 2026. This is a static selection, not a live repository feed. The sources and limits are beside each result.
TRY THIS Choose a result and move its visual control. Then read the scope of the claim.
Twelve records from OpenAI’s math repository, with their original problem IDs.
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A wave can have a strong peak at one instant. The claimed estimate limits how much concentration can persist when we measure it across space and time.
A sharp peak at one moment is different from concentration that persists across space and time.
No damping is depicted. Packet paths are schematic: their overlaps neither compute the spacetime norm nor verify the estimate.
The first card reports a consequence claimed in the paper. The second proposes research that would need its own evidence.
Claimed consequence
The paper derives maximal Bochner–Riesz bounds and almost-everywhere Fourier summation for 3 ≤ p < ∞ and δ > 1−3/p, then states restriction and Kakeya implications with their classical provenance.
Speculative next step
Investigate whether the packet-control ideas extend to selected variable-coefficient or curved geometries. Existing counterexamples limit blanket extensions; this map does not claim they work.
The manuscript claims ||exp(it|D|)f|| over spacetime R³ × [1,2] in L³ is bounded by Cε ||(1−Δ)^(ε/2) f|| in L³, for every ε > 0 and Schwartz f. Interpolation gives 2 < p < ∞ with Sobolev loss α > max(0, 1−3/p).
Status: Preprint claim • no Lean link in the checked catalogue. Analytic proof presented; sections reviewed for statement, lineage and implications, not a full proof audit. No formalization entry found for this result in the checked catalogue. Independent proof verification: Not established.
Proof strategy, as reported: The paper decomposes waves into packets, tracks their overlap and packet masses across scales, and uses geometric and entropy estimates to rule out excessive concentration.
Every introduction reviewed here names earlier work. The names below show what the papers build on; this review does not establish whether those citations are complete.
Source checks: 9–10 October 2026, commit fd4aeeb2ee4f. The repository’s 7 October history reports withdrawals and repairs. This map is not a peer review, novelty audit, or proof certification.
These records remain available even when this viewer cannot run the interactive map.
RESULT 079 · Preprint claim
A wave can have a strong peak at one instant. The claimed estimate limits how much concentration can persist when we measure it across space and time.
A sharp peak at one moment is different from concentration that persists across space and time.
No damping is depicted. Packet paths are schematic: their overlaps neither compute the spacetime norm nor verify the estimate.
Claimed consequence
The paper derives maximal Bochner–Riesz bounds and almost-everywhere Fourier summation for 3 ≤ p < ∞ and δ > 1−3/p, then states restriction and Kakeya implications with their classical provenance.
Speculative next step
Investigate whether the packet-control ideas extend to selected variable-coefficient or curved geometries. Existing counterexamples limit blanket extensions; this map does not claim they work.
The manuscript claims ||exp(it|D|)f|| over spacetime R³ × [1,2] in L³ is bounded by Cε ||(1−Δ)^(ε/2) f|| in L³, for every ε > 0 and Schwartz f. Interpolation gives 2 < p < ∞ with Sobolev loss α > max(0, 1−3/p).
Status: Preprint claim • no Lean link in the checked catalogue. Analytic proof presented; sections reviewed for statement, lineage and implications, not a full proof audit. No formalization entry found for this result in the checked catalogue. Independent proof verification: Not established.
Every positive ε is allowed; ε = 0 is not claimed. Three-dimensional Euclidean geometry is essential. This is not a practical acoustics or imaging algorithm.
Proof strategy, as reported: The paper decomposes waves into packets, tracks their overlap and packet masses across scales, and uses geometric and entropy estimates to rule out excessive concentration.
RESULT 080 · Preprint claim
How rough can the starting wave be while still recovering its original value at almost every point as we rewind toward time zero? The claimed advance includes the exact 1/3-derivative threshold in two dimensions.
The claimed endpoint asks for exactly one-third of a derivative, with no extra positive regularity.
This smooth, one-dimensional periodic toy only illustrates the question. It cannot test almost-everywhere convergence for arbitrary H^(1/3)(R²) data.
Claimed consequence
If valid, the planar convergence statement covers H^(1/3) data itself, rather than requiring any extra positive regularity above 1/3.
Speculative next step
The scale-uniform packet and summation tools could be investigated for other dispersive endpoint questions. Extension to nonlinear equations, noisy measurements, or physical recovery algorithms is not shown.
For f in H^(1/3)(R²), the manuscript claims a full-measure spatial set on which Gaussian-regularized free Schrödinger evolution has a limit for every 0 < t < 1 and converges to the Lebesgue representative of f as t ↓ 0. Remove Gaussian regularization first, then take t ↓ 0.
Status: Preprint claim • no Lean link in the checked catalogue. Analytic proof presented; source statement and proof outline reviewed. No formalization entry found for this result in the checked catalogue. Independent proof verification: Not established.
This map covers the planar paper. The higher-dimensional results are outside its scope. Gaussian regularization is removed first; then time tends to zero. A finite animation does not establish convergence for arbitrary rough Sobolev data.
Proof strategy, as reported: The paper combines transverse packet estimates, a smooth-array bilinear estimate, and square summation across frequency scales on a common time graph.
RESULT 130 · Scoped Lean artifacts
A Fourier transform reorganizes samples into frequencies. The result claims a tiny asymptotic improvement in how many idealized arithmetic steps that reorganization requires.
The claimed saving concerns idealized arithmetic work, not a measured speedup on a computer.
Exponent saving: 10⁻¹³. The paper gives enormous constants, no useful crossover estimate, and no floating-point stability guarantee.
Claimed consequence
The manuscript derives the same asymptotic saving for linear convolution and multiplication of complex polynomials in its exact-root, unrestricted-coefficient model.
Speculative next step
Search for finite tensor savings with manageable constants and numerical conditioning. That would be necessary research toward useful numerical implementations; practical gains are not demonstrated.
The manuscript gives a deterministic exact complex-arithmetic algorithm for every length n with operation count O(n(log n)^(1−10^−13)). It assumes a supplied specified root of unity, unrestricted complex coefficients, and unit-cost exact complex operations plus logarithmic-size indexing; scalar preparation and array organization are counted.
Status: Preprint + scoped Lean artifacts • not independently rerun. Analytic/algorithmic proof presented. Repository provides uniform transform and convolution declarations, proof modules and Comparator configuration; scope was inspected. Exact arithmetic model only. The synthesis appendix's extraction/search/printer uniformization is outside selected formal statements. Independent proof verification: No build or kernel-check reproduction performed here.
The exponent saving is 10^−13, not a large practical factor. Complex precision is unbounded and exact operations have unit cost. Formalization availability alone does not establish novelty or speed on hardware.
Proof strategy, as reported: A fixed finite tensor-network saving is amplified through batching; a local Fourier compiler and synchronized coprime-axis schedule feed a chirp-convolution reduction for arbitrary lengths.
RESULT 362 · Scoped Lean artifacts
Charged particles create electromagnetic fields, and those fields bend particle motion. The claimed result prevents this idealized feedback loop from developing a finite-time loss of smoothness for the specified data.
Particles drive fields; fields steer particles. The proof strategy keeps cancellation in the feedback loop.
This is a signed-sum schematic, not a continuum plasma solver. Global smoothness does not mean a uniform momentum bound for all infinite time.
Claimed consequence
If valid, the stated arbitrary admissible data can be evolved uniquely through every finite time, with no size or symmetry restriction in this class.
Speculative next step
Use a validated global-regularity result as a starting point for decay, scattering, approximation-error analysis, or extensions beyond compact particle support. None of these follows automatically from existence alone.
For the three-dimensional one-species relativistic Vlasov–Maxwell system, the manuscript claims a unique global classical solution for nonnegative smooth compactly supported phase-space particle density and smooth finite-energy fields with every spatial derivative bounded, satisfying the Gauss constraints. Smoothness and compact particle support hold on each finite time interval.
Status: Preprint + scoped Lean artifacts • not independently rerun. Analytic proof presented; equations, hypotheses, lineage and proof strategy reviewed. Repository provides global classical existence/uniqueness scope and solution declaration plus Comparator configuration. Independent proof verification: No build or kernel-check reproduction performed here.
One species, collisionless, relativistic, no background charge. Initial constraints and regularity conditions matter. Global existence is not a practical numerical method or a statement that all plasma models stay smooth.
Proof strategy, as reported: The proof estimates the signed impulse along particle trajectories, preserving cancellation before taking absolute values. It aims to show that repeated momentum doublings need a divergent total time, then applies a classical continuation criterion.
RESULT 073 · Lean artifacts listed · not rerun
Take every pair of points in a compact fractal and measure the gap. Above half the surrounding dimension, the manuscript claims that these distances have positive total length.
d = 2: dim_H(E) > 1. All-pairs positive measure; equality is unclaimed.
A set can occupy no volume and still produce distances with positive total length.
Finite samples have Hausdorff dimension zero. This schematic cannot test the theorem. Positive measure need not contain an interval, and no single fixed pin is claimed at this threshold.
Claimed consequence
If valid, every compact planar set with Hausdorff dimension greater than 1 has an unpinned distance set of positive measure. The corresponding threshold is d/2 in every integer dimension d ≥ 2.
Speculative next step
Seek positive-measure distances from a single pin at the sharp threshold for arbitrary compact sets. The manuscript does not assert this stronger statement.
For every integer d ≥ 2 and compact E ⊂ R^d, if dim_H(E) > d/2, then ℒ¹(Δ(E)) > 0, where Δ(E) = {|x−y| : x,y ∈ E}. No Ahlfors-regularity, product structure or Fourier-decay assumption is imposed. The distance set uses all pairs; it is unpinned.
Repository manuscript claim. Lean scope notes list all-dimensional and planar artifacts. The all-dimensional config is linked by docs/073; the catalogue text lists the planar declaration and labels review status unchecked. Scope, statements, configurations and solution source were inspected, not independently rerun or proof-audited. No correction or withdrawal recorded at the 10 October snapshot.
The threshold is strict: dim_H(E) > d/2. Equality, an interval of distances, and a positive-measure distance set from one fixed pin are not asserted.
Proof strategy, as reported: The paper retains a positive collection of pairs, prepares critical angular cap bounds, controls Fourier frequency shells using a two-depth profile potential, and reconstructs a nonzero absolutely continuous measure carried by the original distance set.
RESULT 074 · Preprint claim · no Lean listed
One paper claims a sharp density-sensitive bound for thin tubes in three dimensions. A companion claims that any set containing a unit line segment in every four-dimensional direction has Hausdorff dimension four.
3D maximal claim: ‖Kδf‖₃ ≤ Cε δ^(−ε) ‖f‖₃, for every ε > 0. Bright shading is fixed at λ = 1/2.
4D set claim: dim_H(K) = 4. Full dimension does not assert positive volume or an L⁴ maximal bound.
Full dimension can coexist with zero volume. A tube-average bound is a separate, quantitative claim.
Both panels are schematic projections. A finite family of tubes or boxes cannot establish a maximal inequality or a Hausdorff dimension. Full dimension does not imply positive volume.
Claimed consequence
The 4D manuscript states that projection gives every Kakeya set in R^d, d ≥ 4, Hausdorff dimension at least 4. This does not reach full dimension when d > 4.
Speculative next step
The L⁴(R⁴) to L⁴(S³) tube-average estimate with δ^(−ε) loss remains a separate target. Full Hausdorff dimension alone does not supply that bound.
3D: for every ε > 0, 0 < δ < 1 and f ∈ L³(R³), ‖Kδf‖L³(S²) ≤ Cε δ^(−ε) ‖f‖L³(R³). Kδ takes the supremum of averages of |f| over all translated unit cylinders of radius δ in a given direction; the constant is uniform in δ and f. 4D: every K ⊂ R⁴ containing a unit segment in each direction in S³ satisfies dim_H(K) = 4, without compactness or measurability assumptions. The 4D proof imports a weighted plank lemma from the 3D paper, not the full 3D maximal theorem.
Repository manuscript claims. No accompanying Lean formalization is listed for either selected Kakeya manuscript at this snapshot. Statements, context and selected argument sections were reviewed; no full proof audit or independent Lean verification was performed. Wang–Zahl already established the 3D set-dimension theorem. No correction or withdrawal recorded at the 10 October snapshot.
The 3D bound allows δ^(−ε) for every ε > 0; a δ-independent endpoint is not claimed. The 4D statement is a dimension theorem, not an L⁴ maximal estimate or positive-volume claim.
Proof strategy, as reported: The 3D paper tracks shaded tube mass over scales and combines geometric projection bounds with entropy estimates. The 4D paper uses weighted line configurations and nested quadratic fits; its imported input is the 3D paper’s weighted full-time plank lemma.
RESULT 077 · Preprint claim · no Lean listed
Build a wave by adding frequencies from a compact, smooth, positively curved surface. The manuscript claims a sharp open-range bound on how much their sum can concentrate in three-dimensional space.
p = 4: inside the claimed open range 3 < p < ∞. This packet sketch does not evaluate the norm.
Curvature points neighboring packets in different directions. The estimate controls their sum across all of space.
The cap and packet are schematic two-dimensional projections, not a norm calculation or proof. The claimed estimate needs positive curvature and p > 3; the strong p = 3 sphere endpoint fails.
Claimed consequence
The general-surface paper derives ‖e^{itΔ}f‖Lp(R² × [0,1]) ≤ Cp,s ‖f‖W^{s,p} for 3 < p < ∞ and s > 2−6/p. This is a spacetime norm estimate, distinct from result 080’s almost-everywhere return to initial data.
Speculative next step
Can the growth of the operator constant as p approaches 3 be made explicit? The manuscript does not claim a polynomial rate, and the strong endpoint itself is excluded.
Let Σ⊂R³ be a compact smooth positively curved surface, possibly with smooth boundary, and dσ its induced surface measure. For every 3<p<∞, EΣf(x)=∫Σ f(ξ)e^{2πix·ξ}dσ(ξ) satisfies ‖EΣf‖Lp(R³)≤CΣ,p‖f‖Lp(Σ) for every complex-valued f∈Lp(Σ). The sphere paper separately states ‖Eg‖Lp(R³)≤Cp‖g‖L∞(S²) for every real p>3 and bounded measurable complex g. Its sphere-specific factorization consequence gives the diagonal and strict mixed-norm ranges on S².
Repository manuscript claims. No accompanying Lean formalization is listed for either the sphere or general-surface companion at this snapshot. Primary statements, context and selected proof-assembly sections were reviewed, not the full proofs. No correction or withdrawal recorded at the 10 October snapshot.
The general-surface diagonal claim has 3 < p < ∞ and fixed positive curvature. The separate sphere manuscript starts with bounded data. It does not make p = 3 valid or give constants uniform as curvature degenerates.
Proof strategy, as reported: The diagonal companion combines the independent 3D Kakeya theorem, which controls positive tube density, with the sphere paper’s packet-propagation theorem, which controls oscillatory amplitudes. It assembles the estimates across scales and surface charts.
RESULT 078 · Preprint claim · no Lean listed
Cut a signal off at a spherical frequency boundary, with a little fading at the edge. The manuscript reports exactly which strict smoothing range keeps the Lp norm bounded in three dimensions.
p = 4 needs δ > 0.25. Selected point is in the strict claimed range. White boundary dot: classical p = 2, δ = 0.
A softer frequency edge can preserve an Lp norm. The amount of softening depends on how that norm measures size.
This is a multiplier profile and a parameter-range diagram, not a numerical test on arbitrary inputs. The dashed equality boundary is excluded; p = 2, δ = 0 is a separate classical exception.
Claimed consequence
For 1 ≤ p < ∞ in the strict allowed range, smoothed cutoffs Tδ,Rf converge to f in Lp(R³) as R → ∞. At 3/2 ≤ p ≤ 3, any fixed δ > 0 suffices. This does not assert pointwise convergence.
Speculative next step
How fast do the constants worsen as δ decreases to zero? The manuscript does not claim a uniform zero-order limit; positive-order boundary estimates would require separate work.
For δ>0, let Tδf=F⁻¹[(1−|ξ|²)₊^δ Ff] on Schwartz functions in R³. For every δ>0 there is a finite Cδ with ‖Tδf‖L³(R³)≤Cδ‖f‖L³(R³); hence Tδ extends uniquely to a bounded L³ operator. Interpolation and duality give boundedness on Lp(R³), 1≤p≤∞, whenever δ>max{3|1/p−1/2|−1/2,0}. Dilation gives the same bound uniformly over positive frequency-ball radii R for fixed p and δ. For finite p only, the manuscript also states Lp norm convergence as R → ∞.
Repository manuscript claim. No accompanying Lean formalization is listed at this snapshot. Primary statements, context and selected consequence sections were reviewed; the proof was not independently audited. No correction or withdrawal recorded at the 10 October snapshot.
δ > max{3|1/p−1/2|−1/2, 0} is strict. At δ = 0 the hard ball multiplier fails for p ≠ 2. Uniform fixed-radius bounds do not themselves give a maximal estimate or almost-everywhere convergence.
Proof strategy, as reported: Planar incidence estimates control the growth of positional information across scales. A pseudoconformal reduction transfers that control to oscillatory packets and then multiplier shells; a small loss ν < δ leaves a summable series.
RESULT 087 · Lean artifacts listed · not rerun
A wide body has a narrow polar in the same direction. The manuscripts claim the smallest possible volume products in every dimension, with different answers for symmetric bodies and general bodies.
Symmetric planar lower bound: 8. A rectangle and its diamond polar attain it. This determinant-one stretch preserves the product; these are classical 2D examples.
Symmetric planar lower bound: 8. Ellipses have product π², strictly above 8. This determinant-one stretch preserves the product; these are classical 2D examples.
Nonsymmetric planar lower bound: 27/4. The triangle is centered at its Santaló point (its centroid). This determinant-one stretch preserves the product; these are classical 2D examples.
Stretching can trade width between a body and its polar without changing their volume product.
These are exact classical planar examples. The symmetric and general lower bounds are different. The triangle is placed at its centroid, which is its Santaló point. A few shapes cannot verify a universal inequality or classify equality in higher dimensions.
Claimed consequence
Because the Santaló point minimizes polar volume, the general claimed lower bound also holds with every interior center z. Equality for the minimized product identifies simplices. Symmetric bodies retain their separate, stronger constant.
Speculative next step
Seek a quantitative bound relating a small excess in volume product to distance from the appropriate equality class. That stability problem needs its own hypotheses, metric and proof; the diagrams supply none.
For n ≥ 1 and every origin-symmetric convex body K ⊂ Rⁿ, the symmetric manuscript claims |K||K°| ≥ 4^n/n!, with equality precisely for invertible linear images of Hanner polytopes. For an arbitrary convex body K with Santaló point s(K), the general manuscript claims |K||(K−s(K))°| ≥ (n+1)^(n+1)/(n!)², with equality precisely for simplices.
Repository manuscript claims, dated 22 September 2026. The scope notes and comparator configurations list the symmetric inequality, Hanner equality, and general inequality with simplex equality. The scope document is internally inconsistent: an older sentence excludes general Mahler, while later text includes it. GeneralMahler has a target, configuration and implementation but is absent from the machine catalogue. These are documentation mismatches, not a finding that a proof fails. The artifacts were inspected, not independently rerun or proof-audited. Functional corollaries are outside these selected targets; the scalar certificate runner was not run. No correction or withdrawal was recorded at the checked snapshot.
Convex bodies are compact, convex and have nonempty interior. The symmetric statement is centered at the origin. The general statement uses the Santaló point, the translation minimizing polar volume. The selected record covers the two geometric manuscripts; functional corollaries and the separate symplectic companion have additional claims.
Proof strategy, as reported: The symmetric paper builds a conformal lens, derives a holomorphic mass estimate and uses Hansen–Lima metric-median classification for equality. The general paper uses Klartag’s cone/Laplace correspondence, Moreau nearest-point projections, Gaussian covariance and Hermite estimates, then scalar certificates. These are separate proof routes; the general lower bound alone does not give the sharper symmetric constant.
RESULT 088 · Lean artifacts listed · not rerun
Collect a convex body’s shadow volumes in every direction. They describe a second body, its projection body. The selected manuscript claims that ellipsoids minimize its volume at fixed original volume in every dimension at least four.
Shadow length 1.600 equals h(ΠK,n) in the perpendicular direction n. This classical planar construction does not test the manuscript’s n ≥ 4 minimization claim.
Shadow length 1.600 equals h(ΠK,n) in the perpendicular direction n. This classical planar construction does not test the manuscript’s n ≥ 4 minimization claim.
The projection body stores all directional shadow sizes in one convex shape.
This is the classical planar construction ΠK = J(K−K), where J is a quarter turn. Every origin-symmetric planar body has |ΠK|/|K| = 4, including both shapes shown. The ellipse is therefore not a unique planar minimizer; the new selected claim begins at n = 4.
Claimed consequence
If valid, the n ≥ 4 theorem gives the sharp lower benchmark for projection-body volume at every fixed body volume, with ellipsoids as the equality class. The separate companion’s 20D product exceeds the simplex benchmark for a maximum.
Speculative next step
Ask how accurately finitely many directional measurements can bound projection-body volume for a specified class of convex bodies. This needs sampling and stability estimates beyond the stated minimization theorem.
For every n ≥ 4 and convex body K ⊂ Rⁿ, the selected manuscript claims that R_n(K) = |ΠK| / |K|^(n−1) is at least κ_(n−1)^n / κ_n^(n−2), with equality precisely for ellipsoids. Here κ_j is unit-ball volume and h_(ΠK)(u) = |proj_(u⊥)K|_(n−1). A companion gives an exact 20-dimensional product-of-simplices witness above the simplex value.
Repository manuscript claims, dated 24 September 2026. The listed Lean scope covers the n ≥ 4 Petty inequality with ellipsoid equality, failure of the simplex upper bound in dimension 20, and the explicit 20-dimensional product witness. It does not cover the sufficiently-large-dimensional exponential extension or all other projection-family inequalities. Artifacts were inspected, not independently rerun or proof-audited. The companion credits earlier counterexamples by Feng–Hu–Liu–Xu in every dimension ≥ 9; this is not the first disproof. No correction or withdrawal was recorded at the checked snapshot.
The minimization statement concerns the projection body itself, not its polar. It is distinct from the classical Petty polar-projection inequality. The diagram treats n = 2 only. The companion product counterexample concerns a maximum and has its own prior-art history.
Proof strategy, as reported: The Petty paper uses Brunn–Minkowski first variation, cosine/Funk transforms and a variational estimate; Brouwer’s theorem supplies an affine normalization. Separately, the product paper writes projection bodies of polytopes as zonotopes, proves a product identity and compares exact integers for two ten-dimensional simplices. The product witness does not depend on the claimed Petty minimum.
RESULT 091 · Lean artifacts listed · not rerun
Mix the support lengths of two origin-symmetric bodies geometrically. The manuscript claims that the resulting body’s volume stays above the geometric mean of their volumes, in every dimension.
Logarithmic mixture: half-sides 2^(1−2λ) and 2^(2λ−1). Area is exactly 4 for every λ. These origin-symmetric boxes form an equality family; the general-body inequality is not tested.
Ordinary Minkowski mixture: add the half-side lengths with weights (1−λ, λ). Its area is 6.250. This comparison is not the logarithmic operation.
The rule used to mix two shapes changes the intermediate shape and its volume.
The displayed rectangles are an exact equality family for logarithmic mixing, already covered by the classical unconditional case. They do not sample arbitrary convex bodies. The ordinary mixture is a separate comparison. No equality classification for general bodies is inferred here.
Claimed consequence
Using the stated transfer and support-subspace reduction, the paper claims that μ(exp(t)K), as a function of t, is log-concave for every even log-concave Radon measure μ and origin-symmetric convex body K. Its additive L_p consequence concerns Lebesgue volume and 0 < p < 1.
Speculative next step
For a specified smoothness class, ask whether a nearly tight logarithmic inequality forces the bodies near a described equality configuration. A useful answer would need a precise distance and a quantitative estimate.
For full-dimensional origin-symmetric convex bodies K,L ⊂ Rⁿ and 0 ≤ λ ≤ 1, define K ⊕_(0,λ) L = {x : x·u ≤ h_K(u)^(1−λ)h_L(u)^λ for every unit u}. The manuscript claims |K ⊕_(0,λ) L| ≥ |K|^(1−λ)|L|^λ. It then derives the symmetric additive L_p Brunn–Minkowski inequality for 0 < p < 1 and, using Saroglou’s transfer theorem plus support-subspace reduction, the scalar-dilation B-conjecture for even log-concave Radon measures.
Repository manuscript claim, dated 23 September 2026. The listed Lean target covers the symmetric Lebesgue-volume logarithmic inequality in all dimensions and all λ ∈ [0,1]. It does not certify the additional L_p inequalities or the measure and B-conjecture corollaries. The statement, configuration and solution source were inspected, not independently rerun or proof-audited. No correction or withdrawal was recorded at the checked snapshot.
Origin symmetry is required. The logarithmic mixture is the intersection of support half-spaces with bounds h_K(u)^(1−λ)h_L(u)^λ; that product need not itself be a support function. The scalar-dilation B-conjecture is distinct from a stronger coordinatewise matrix-dilation assertion.
Proof strategy, as reported: The paper approximates symmetric bodies by supporting slabs, softens their constraints and proves a variance estimate using moment coordinates and tensor identities. This makes a log partition function concave. It then takes the hard-slab and direction-density limits. Saroglou’s transfer theorem and support-subspace reduction are additional inputs for the measure consequence.
RESULT 092 · Lean artifacts listed · not rerun
Some convex bodies require overlapping copies to cover space. The manuscripts claim the worst possible optimal density grows like n log n, even if centers may be placed freely; one lattice is enough for an upper bound of the same order.
Square disk lattice: uncovered. Density = π/(s/r)². The planar drawing cannot test the claimed worst-case Θ(n log n) order in high dimension.
Triangular disk lattice: covered (equality at the threshold is included). Density = 2π/(√3(s/r)²). The planar drawing cannot test the claimed worst-case Θ(n log n) order in high dimension.
Overlap can be necessary. Density measures how much material is used per unit of space, including repeated coverage.
These two infinite periodic disk arrangements are classical planar examples, shown through a cropped window. Their formulas neither optimize over all coverings nor test a high-dimensional worst-case bound. An uncovered arrangement still has a nominal density, but is not a covering.
Claimed consequence
If the lower construction is valid, no absolute C can bound the optimal translative covering density of every n-dimensional convex body by Cn. The claimed lattice upper bound matches the n log n worst-case order.
Speculative next step
Ask whether optimal high-dimensional coverings by Euclidean balls need order n or n log n density, or something between. The source explicitly leaves the ball-specific gap open; the planar disk illustration does not decide it.
For every n ≥ 2 and every convex body K ⊂ Rⁿ, the upper manuscript claims one full-rank lattice Λ with K+Λ = Rⁿ and |K|/det(Λ) ≤ C n log n, for an absolute C > 0. For every sufficiently large n, the lower manuscript constructs a centrally symmetric convex body K_n with unrestricted translative covering density at least c n log n, for an absolute c > 0. Together these give worst-case order Θ(n log n), with or without the lattice restriction.
Repository manuscript claims, dated 23 September 2026. Scope notes and comparator configurations cover the single-lattice upper bound and the common asymptotic order of four worst-case suprema (lattice/translative, all/symmetric bodies). CoveringDensity has a statement, configuration and solution but is absent from the machine catalogue; SingleLatticeCovering is listed there. Those artifacts were inspected, not independently rerun or proof-audited. This does not certify every auxiliary assertion in the papers. No correction or withdrawal was recorded at the checked snapshot.
Θ(n log n) is a worst-case order over convex bodies as dimension grows, with unspecified positive absolute constants. It is not the density of every body or an exact finite-dimensional formula. The lower construction does not establish an n log n lower bound for Euclidean balls.
Proof strategy, as reported: For the upper bound, the paper combines Gaussian marginals, a random-lattice mean-hole estimate and binary correction blocks within one sheared lattice. The lower paper constructs random symmetric slab bodies, excludes finite rounded center patterns with Poisson/Janson estimates, then uses periodization to transfer the lower-bound exclusion to all locally finite translative coverings.
Selected problem 079: Do wave peaks persist?. Result, representation, and possible next questions updated.