# Verification report — OWR-12723-008 (Steinerberger's hexagon question for the geometric uncertainty principle)

Verification date: 2026-10-02.

**Verdict.** The answer is yes on the square [0,1]². Let c_H be the Fraenkel asymmetry of the regular hexagon,
c_H = (12/π) arctan(λ/√3) − 2λ = 0.07446575455794907873…, λ = √(3(2√3/π − 1)). For every partition of [0,1]² into any
number of measurable pieces, F = Σ|Ω_i|A(Ω_i) + Σ(|Ω_i| − m) ≥ c_H + 0.06 m > c_H, where m is the smallest measure of a
piece. For every N, a honeycomb partition with a boundary correction has F ≤ c_H + O(N^(−1/2)). So the infimum I_N over
partitions into N pieces lies in [c_H + 0.03/N, c_H + O(N^(−1/2))], I_N tends to c_H as N → ∞, and no partition attains the value c_H.
The statement is asymptotic: whether I_N itself is attained for a given N, and what minimisers look like, is not decided. The constant is proved for the square only, not for arbitrary domains, and geometric closeness of near-minimisers
to a honeycomb is not claimed. Two independent AI-assisted verification runs found no mathematical error; the second one examined the final text
(version 1.0 of 2026-10-02) including the appendices, the covering lemma and the weighted theorem. The note is unrefereed.

## Statement checked
- **Primary sources.**
  - S. Steinerberger, *A geometric uncertainty principle with an application to Pleijel's estimate*, Ann. Henri Poincaré 15
    (2014) 2299–2319, doi:10.1007/s00023-013-0310-4, arXiv:1306.3103 (version 4 read). Section 1.3 states the conjecture
    c₂ ≈ 0.074465754… (hexagonal tiling) and Burchard's weighted question.
  - S. Steinerberger, "Three problems for discrepancy and QMC", Oberwolfach Rep. 10 (2013), no. 4, Report 49/2013,
    doi:10.4171/owr/2013/49, pp. 2890–2892 (the problem is on pp. 2891–2892). The report PDF was read; the second run rendered pages 2890–2892 again and
    compared the definitions and the question with the paper.
  - Definitions as in both sources: A(Ω) = inf |Ω Δ B|/|Ω| over discs B with |B| = |Ω| placed anywhere in the plane;
    D(Ω_i) = (|Ω_i| − min_j|Ω_j|)/|Ω_i|. On [0,1]² the weights |Ω_i|/|Ω| equal |Ω_i|, so the functional is exactly the F above.
- **Corpus record.** ulamai/UnsolvedMath, OWR-12723-008 (status `open` before this note): "Is the minimum … asymptotically
  attained by a hexagonal partition, with suitable modifications along the boundary?"
- **Conventions fixed in the paper.** Partitions are up to null sets (closed hexagons overlapping on null sets are allowed);
  null pieces are allowed, with |Ω|A(Ω) := 0, and then F ≥ 1; discs are placed anywhere in the plane; "extremal" means
  asymptotically extremal.

## Readings
| Reading | Answer | Witness |
|---|---|---|
| Is the optimal constant on [0,1]² equal to c_H ≈ 0.0744657 (hexagonal value)? | yes | Thm 1.1 and Cor 1.2: c_H + 0.03/N ≤ I_N ≤ c_H + 3δ_N |
| Are honeycomb partitions with boundary modification asymptotically extremal? | yes | Prop 4.1: F ≤ c_H + 3δ_N, δ_N ≤ 8κ N^(−1/2) + 16κ² N^(−1) |
| Is c_H attained by some finite partition? | no | F ≥ c_H + 0.06 m > c_H for every partition |
| Does the universal constant c₂ of Steinerberger (all bounded domains) equal c_H? | only c₂ ≤ c_H proved | the lower bound c₂ ≥ c_H for arbitrary domains is not proved |
| Are near-minimisers geometrically close to a honeycomb (Hausdorff/L¹ stability)? | not proved | only the combinatorial Prop 4.2 (proportion of non-hexagonal power cells → 0) |
| Burchard's weighted question (weight α on the asymmetry sum) | answered in the asymptotic sense | lim_N inf F_α = min{α c_H, 1}; the honeycomb partitions of Thm 1.1(ii) are asymptotically extremal exactly for α ≤ α₀ = 1/c_H = 13.42899…; minimisers for a given N are not determined |

## Results in the paper
- **Thm 2.2 (disc systems).** For N discs of arbitrary positive radii, centres anywhere, total area 1:
  Φ = 2|S ∖ ∪B_j| + Σ(s_j − m) ≥ c_H + 0.06 m. In addition Φ ≥ c_H + 0.06 m + 0.0055 m·#{j : power cell of B_j is empty or not a hexagon}.
- **Reduction (Lemma 2.1).** F ≥ Φ(optimal discs of the pieces), because the sets Ω_i ∖ B_i are disjoint and cover S ∖ ∪B_j.
- **Power cells (Lemma 3.1).** |S ∖ ∪B_j| = Σ|Q_j ∖ B_j| exactly, for discs of different sizes.
- **Moment lemma (Lemma 3.2, proof in Appendix A).** |Q ∖ B(z,ρ)| ≥ M_k(|Q|, πρ²) for a convex polygon Q with at most k sides and
  any point z. This is the indicator-kernel case of Fejes Tóth's moment lemma. The proof is self-contained: reduction to z
  interior; triangles with apex z; Lemma A.1 (the isosceles triangle has the smallest outside area for fixed apex angle and
  area, by an explicit four-case derivative); Lemma A.2 (joint convexity of the isosceles function ε(τ, γ): C¹, positive
  semi-definite Hessian in the middle regime, finitely many regime breaks along any line); Jensen.
- **Euler count (Lemma 3.3, proof in Appendix B).** Σ k_j ≤ 6N' + p − 6 for the clipped power diagram in a convex p-gon (p = 4: ≤ 6N' − 2),
  including edge-to-edge structure, boundary vertices, cells of area zero and N' < N.
- **Tangent inequality (Prop 3.5).** With h_k(t) = 2M_k(t,1), λ = √(3(2√3/π − 1)), μ = 2(1 − λ) and G_k(t) = h_k(t) − c_H − μ(t − 1):
  G_k(t) ≥ Ψ(k) = φ(k) − φ(6), φ(k) = (2k/π) arctan(λ tan(π/k)), for all t > 0; and G_k ≥ −c_H. With ν = 0.03:
  G_k ≥ −ν(k − 6) + 0.0055 for k ≠ 6. For k = 3,…,8 this is a table of closed-form values (smallest margin 0.00558 at k = 5);
  for k ≥ 9 the trivial bound G_k ≥ −c_H and c_H < 3ν suffice, with no monotonicity in k.
- **Upper bound (Prop 4.1).** N honeycomb hexagons inside the square, the left-over region glued to one of them.
- **Cor 1.2.** c_H + 0.03/N ≤ I_N ≤ c_H + 3δ_N (using m ≤ 1/N and F ≥ 1 − Nm).
- **Thm 1.3 (weighted functional).** F_α = α Σ|Ω_i|A(Ω_i) + Σ(|Ω_i| − m): lower bound F_α ≥ α c_H for α ≤ 1/c_H by the same chain with
  the size term multiplied by β = 1/α ≥ c_H; for α ≥ 1/c_H, F_α ≥ 1; upper bound 1 + 2αε from finitely many disjoint discs covering
  all but ε of the square (an elementary covering lemma is included).

## Computations (exact, symbolic or high precision; scripts and outputs in reproducibility/)
- `main_checks/symbolic/lemmaM_sympy.py`: every derivative and identity of Appendix A and of Lemma 3.4 checked symbolically (sympy);
  the four-case formula for K'(δ) additionally against central differences of the quadrature of the defining integral in 400 random
  cases (all four regimes occur); closed forms of K differentiated symbolically in the two non-trivial regimes; the regime formulas
  for ε against direct quadrature; Hessian entries and determinant; C¹ gluing. Output: ALL SYMBOLIC CHECKS PASSED.
- `main_checks/constants/tangent_table.py`: λ, μ, c_H, φ(k), Ψ(k) at 60 digits (mpmath); c_H also from the explicit formula for h_6(1)
  and from quadrature (agreement to 60 digits); h_6'(1) = μ; min_t[h_k(t) − μt] = φ(k) − 2 for k = 3,…,12 from the explicit formula
  and by bisection; grid check of G_k + ν(k − 6) > 0 and G_k ≥ −c_H.
- `main_checks/interval_certificate/`: the same table in rigorous interval arithmetic with exact rational endpoints (outward-rounded
  300-bit arithmetic), plus a direct Lipschitz-grid certificate for k = 3,…,8. Output: ALL CERTIFIED: True. Supplement only; the paper
  needs the table, which consists of explicit closed-form values.
- `main_checks/euler/euler_power_check.py`: exact rational arithmetic (fractions.Fraction); 20,051 clipped power diagrams in the unit square
  (random rational centres inside and outside the square, wildly unequal radii, equal radii, lattices with many ties, collinear centres, nested radii).
  Verified in each: areas add to 1; Σk ≤ 6N' − 2 (attained in 7,294 systems); every side not on the boundary is a side of exactly two cells;
  V − E + (N'+1) = 2; E ≤ 3N' + 1; degrees ≥ 3 (corners ≥ 2). Systems with N' < N occur in 17,155 of them.
- `main_checks/numerics/`: floating-point evidence (moment lemma on 180,000 random cases and adversarially; the whole chain, and the refined
  count inequality, on random disc systems; Theorem 1.1(i) end to end on 364 actual partitions of the square with the true functional F; local search of Φ for N ≤ 8;
  the torus (no boundary) test where the hexagonal lattice gives Φ = c_H to 1e-16; the weighted version; honeycomb lattice counts for N up to 10⁶).
- `main_checks/constants/burchard_threshold.py`: 1/c_H = 13.4289917014377703… at 30 digits; constants of the elementary covering lemma; G_k + c_H ≥ 0 on a grid.
- `independent_runs/`: separately written code (see below) and its outputs.
- `independent_run_2/`: the code and outputs of the second independent run (new library `g2.py` and scripts `r2_*.py`), and `rerun_of_release_scripts.txt`,
  the record of a rerun of all programs of this package from an extracted copy of `source.zip` (24 programs, all exit 0, standard output byte-identical to the stored outputs).

## Independent verification runs (AI-assisted)
Independent runs examined the argument as first written (before the final text with the two appendices), wrote their own code and did
not use the author's scripts as evidence. Their findings, all confirmed by the reruns stored in `independent_runs/`:
- Statement fidelity: definitions, normalisation, partition conventions, discs anywhere in the plane: confirmed against arXiv:1306.3103v4 and the OWR report.
- Coverage step, power-cell identity, moment lemma (60,000 random polygons with the point inside, outside or on the boundary; adversarial minimisation for k = 3,…,8;
  monotonicity of the tilting step on 20,000 random cases; convexity of ε on 200,000 random chords), Euler count (4,000 random systems and lattice stress tests,
  zero violations, sharp), tangent inequality (90-digit decimal arithmetic written independently; direct geometric evaluation of h_k for k = 3,…,40 and
  k ∈ {50, 64, 100, 200, 500, 1000, 3000}): all consistent.
- Brute-force minimisation of Φ over disc systems (centres and radii free) for N ≤ 8: minima 0.181 (N = 1), 0.450, 0.345, 0.181, 0.262, 0.273, 0.247, 0.209 (N = 8),
  each above c_H + 0.06 m by at least 0.046.
- Torus test: on R²/(Z × √3 Z) the same functional has minimum c_H·area for N = 2, 6, 8 (to 8 digits) and larger values for N = 1, 3, 4; nothing below c_H.
  (An early version of this test had an implementation error, found and fixed during these runs; the stored version asserts that cell areas add up.)
- Upper bound: honeycomb counts with optimised offsets are consistent with, and better than, the generic bound.
- Required presentation fixes, all applied in the paper: (1) main theorem stated for the square only, with no claim for Steinerberger's universal constant;
  (2) the moment lemma cited with its provenance and proved completely in Appendix A; (3) the Euler / edge-to-edge / boundary-vertex / zero-area-cell / N' < N
  bookkeeping written in full in Appendix B; (4) precedents Bourne–Peletier–Theil, Bourne–Cristoferi, Hales, Fejes Tóth–Gruber credited and the new content stated precisely;
  (5) the tangent inequality given as a table of closed-form values for k = 3,…,8 with a monotonicity-free argument for k ≥ 9, the interval certificate kept as a supplement;
  (6) conventions stated, "extremal" replaced by "asymptotically extremal", the infimum statement phrased as "F > c_H for every finite partition and inf = c_H", the stability
  statement kept only as the combinatorial Prop 4.2; (7) the Burchard threshold re-derived and proved in Section 5 (and tested numerically).
- The new text (the self-contained proofs of the moment lemma and of the Euler count, the covering lemma and the weighted theorem) was written after those runs; it is the object of the second run.

## Second independent verification run (AI-assisted, final text)
The second run started from the final text, wrote new code (`independent_run_2/`, nothing of the author's scripts was used as evidence) and found no mathematical error.
- **Statement fidelity.** Report 49/2013 (printed pp. 2890–2892) and arXiv:1306.3103v4 (Sect. 1.3) re-read: same definitions, weights, normalisation, "N sufficiently large", the question
  (whether the extremal configuration is a decomposition into hexagons with obvious modifications at the boundary, corresponding to an optimal constant of about 0.07) and Burchard's alpha-question; c_H agrees with the printed 0.074465754.
  The question is answered in the asymptotic sense, which is what the paper states.
- **Appendix A.** All derivatives and identities re-derived (`r2_appendixA.py`: 31 checks, symbolic and 50-digit; the four-case formula for K'(delta) against a central difference of the
  quadrature of K in 1200 random cases with the four cases hit 338 / 363 / 381 / 118 times; monotonicity of K; F' = -B, F'' = A; Hessian entries and determinant of eps; C^1 gluing; regimes of eps against
  quadrature; joint convexity on 1.2 million chords). The moment lemma tested directly in `r2_moment.py` on 360,000 combinations of a random convex polygon (3 to 10 vertices), a point inside / on the boundary / up to
  three diameters outside, a radius over four decades and k = n, n+1, n+3 (no violation beyond rounding, 5e-15), and by adversarial minimisation for k = 3..8. One sentence of exposition (that every triangle with apex 0 is a wedge with |delta| < pi/2 - gamma) was added to the paper.
- **Appendix B.** Followed line by line; exact-arithmetic test `r2_euler.py`: 4,000 clipped power diagrams (2,212 in the unit square; 1,788 in convex p-gons, p = 3, 5, 6, 7; N up to 40; N' < N in 3,214;
  a vertex of degree >= 4 in 480): areas add up, every non-boundary side belongs to exactly two cells, boundary sides to one, no T-junction, V - E + (N'+1) = 1 + C, degrees >= 3 (corners >= 2), sum k_j <= 6N' + p - 6,
  attained in 507 systems; no violation.
- **Burchard threshold and covering lemma.** Verified completely and kept: the covering lemma (closed dyadic squares in an open set, disc of radius 0.45 of the side, pi 0.45^2 = 0.636 >= 0.63, contraction 0.685 <= 0.7, discs disjoint
  by construction, remainder of positive measure because S is connected and not a disc); the lower bound by the chain with sigma - 1 replaced by beta (sigma - 1), beta = 1/alpha >= c_H (`r2_alpha.py`: 12,500 tests, an exact-geometry simulation of the scheme with 719 pairwise disjoint discs).
  Wording corrected in the paper: "asymptotically extremal" is now defined, the statement is the asymptotic threshold of the limit min{alpha c_H, 1}, the "only if" is argued.
- **Tangent inequality and summation.** `r2_tangent.py`: constants at 100 digits, c_H by two further routes, the table reproduced, exact Psi(k) + nu(k-6) for k up to 10^4, direct geometric h_k for k = 3..30, 40, 50, 64, 100, 200,
  min_t G_k = Psi(k) to 5e-15; nu = 0.03 admissible (the proof for k >= 9 needs nu >= c_H/3 = 0.02482; the exact range for k = 3..8 is [0.0206074, 0.0355806]; the caption of Table 1 was corrected accordingly).
  `r2_chain.py`: 6,000 random disc systems (N' < N in 4,360): no violation of the tiling, Euler count, per-cell moment lemma, per-cell chain, refined inequality, weighted inequality; N = 1..6 local minima above c_H + 0.06 m by at least 0.0466.
  `r2_trueF.py`: the true functional on 198 convex partitions of the square (min slack 0.0466). `r2_torus.py`: without boundary the minimum is c_H|T| for N = 2 and larger for N = 1, 3, 4, 6.
- **Upper bound.** `r2_upper.py`: delta_N bound for N <= 200,000; at least N admissible hexagons for 900 pairs (N, random lattice position and orientation); the glued piece satisfies |Omega_1 Delta B'| <= c_H a + 2 delta (3000 x 3000 raster, N = 1..30).
- **Rerun.** All programs of this package, run from an extracted copy of `source.zip` with Python 3.13, numpy 2.5.3, scipy 1.18.1, sympy 1.14.0, mpmath 1.3.0, reproduced the stored outputs exactly (`independent_run_2/rerun_of_release_scripts.txt`).
- **Literature and credit.** arXiv API (nine queries), OpenAlex (11 citing works; none treats the sharp constant), zbMATH, Crossref (all eight DOIs of the bibliography match title, authors, journal, volume, pages, year) and one web search:
  nothing proves or restates the sharp constant. arXiv:2012.12129 re-read: Lemma 2.3 is Fejes Toth's moment lemma (second moment), Lemma 2.6 the Euler-formula bound for convex partitions (average number of edges <= 6).
- **Presentation corrections applied after this run.** The abstract states the asymptotic sense, says that the value c_H (not the infimum I_N) is attained by no partition, and that near-minimisers and other domains are not treated; "asymptotically extremal" is defined;
  Table 1 caption (range of nu); justification of |delta| < pi/2 - gamma in Appendix A; the Hales and [Fej01] characterisations worded as attribution only; the Verification paragraph records this run.

## Relation to the literature, novelty and scope
- **Searches (October 2026).** arXiv API (several queries on "geometric uncertainty principle", "Fraenkel asymmetry" with partition / hexagonal / honeycomb, Fejes Tóth moment),
  OpenAlex (11 works citing Steinerberger 2014; none treats the sharp constant), Semantic Scholar (citing papers are spectral / nodal-domain papers), zbMATH (Zbl 1319.35132 has no review text),
  Crossref (metadata of all references), and one web search; the second run repeated these searches with the same outcome. Nothing proves or restates the sharp constant.
- **Classical.** Fejes Tóth's moment lemma (Lagerungen, 1953/1972; Gruber, Aequationes Math. 58 (1999) 291–295); Laguerre (power) cells; Euler's formula for the average number of sides; Hales' honeycomb paper (Discrete Comput. Geom. 25 (2001) 1–22) as the model of a per-cell inequality summed by Euler's formula; Bourne–Peletier–Theil (Commun. Math. Phys. 329 (2014) 117–140) and Bourne–Cristoferi (Commun. Math. Phys. 387 (2021) 1549–1602, arXiv:2012.12129) for Laguerre cells with the moment lemma (Lemma 2.3) and an Euler-formula lemma (Lemma 2.6) in Wasserstein-type location problems.
- **New (as far as we could find).** The reduction of F to disc systems with free centres and radii; the use of power cells for discs of different sizes; the moment lemma with an indicator kernel and arbitrary centre, proved completely; the tangent inequality with explicit μ, ν that absorbs the term −m; the sharp constant for the explicitly posed question on the square; the weighted threshold 1/c_H.
- **Caveats.** The primary texts of Fejes Tóth (1953/1972), Gruber (1999), Hales (2001) and Bourne–Peletier–Theil (2014) were not read; they are cited for attribution only, and the moment lemma and the Euler count are proved in the note. For Bourne–Cristoferi only the statements of Lemmas 2.3 and 2.6 and the abstract were read. MathSciNet, Google Scholar and journal sites could not be searched; the negative search is not a proof of priority.
- **Scope.** Proved: the square (a remark records that the proof gives a boundary term for convex polygons); not proved: the universal constant for arbitrary domains (only c₂ ≤ c_H follows), uniqueness or geometric stability of near-minimisers, the true order of I_N − c_H (we show c/N ≤ I_N − c_H ≤ C N^(−1/2)).

## Public release and license
This paper, its source files, and this verification report are licensed under the Creative Commons Attribution
4.0 International License (CC BY 4.0): https://creativecommons.org/licenses/by/4.0/