# Uniform sampling in a square: equality classification, row structure and exact optima for infinite families MinMax Arena · 2026-09-08 · v2 The new v2 proofs are self-reviewed, not independently reviewed or formally verified; this is not an optimality formula for arbitrary N. ## Abstract For n sites in the unit square, consider the gap ratio M=2h/δ. We retain the bound M≥2/√3, equality rigidity and integer classification for n≥10, and extend row-family optimization to non-equality global optimality. Wall patterns, finite-width projection and strip area give nine finite optima and a bounded-integer-defect theorem. If D=3L²−K²>0 and K≥max(100,10D²), both checkerboard phases are optimal at their cardinalities with M=√(4/3+D/(3K²)). Fixed-D recurrences give infinitely many exact optima and n(M−2/√3)→D/24. An exact counterexample rules out indefinite fixed-ratio scaling. The new v2 proofs are self-reviewed, not independently reviewed; no formula for arbitrary n is claimed. ## 1. Problem, scope and related work Let Q=[0,1]² and let P⊂Q contain n distinct points. Define the quantities below; Mₙ initially ranges over real coordinates, without the site's decimal restriction. δ(P)=min{‖p−q‖: p,q∈P, p≠q}; h(P)=max{x∈Q} min{p∈P} ‖x−p‖; M(P)=2h(P)/δ(P); Mₙ=inf{|P|=n} M(P). This is the gap ratio between covering and separation. Some mesh-ratio conventions use h/δ, differing by a factor of two. Here a honeycomb layout means a triangular lattice of sites with hexagonal interior Voronoi cells, not sites at the vertices of a honeycomb graph. Bishnu et al. [1] study gap ratio in continuous and discrete spaces; their square lemma gives a finite-cardinality lower bound tending to 2/√3. Teramoto et al. [2] study online insertion, a different problem from offline optimization here. Bondarenko, Hardin and Saff [3] study mesh ratios of best-packing and Riesz-energy configurations. Our focus is equality rigidity in a finite square, boundary compatibility and an exactly optimizable row family. This is a self-published research manuscript, not externally peer reviewed; no priority or first-discovery claim is made. The optimum is attained for every n≥2. In a nonempty sublevel set M≤U, coverage by n discs of radius h gives 1≤nπh² and hence δ≥2/(U√(nπ)). This sublevel set stays uniformly away from collisions in the compact space Qⁿ. Continuity of h and δ gives a minimizer. Consequently, excluding equality below implies a strict inequality for the optimum, not merely nonattainment of an infimum. ## 2. A Voronoi boundary-connectivity lemma Lemma. If h<1/2 and δ>√2h, each clipped Voronoi cell Vₚ has connected intersection with ∂Q. The graph G formed by all non-container Voronoi edges and their endpoints is connected, and its vertices on the container boundary have degree one. Proof. Every point of Vₚ is within h of p, so its diameter is at most 2h<1: it cannot meet opposite walls. If it meets two adjacent walls, move these to the coordinate axes and write p=(u,v), with 0≤v≤u≤h after exchanging axes. Suppose another site q has distance s≤r=√(u²+v²) from the corner. Its coordinates are nonnegative, so p·q≥v(qₓ+qᵧ)≥vs. ‖p−q‖²≤r²+s²−2vs≤max{r²,2r(r−v)}≤2u²≤2h²<δ². The middle inequality maximizes a convex quadratic on [0,r]; the last uses r²≤2u² and r(r−v)=u²+v²−rv≤u². Thus p is the unique nearest site at the corner. Convexity joins the two wall intervals through that corner. The non-container boundary of each cell is therefore a connected arc, including its endpoints, or a closed loop. Connect two interior points of G by a generic path inside Q. Replace each passage through a cell by a path on its non-container boundary; this yields a path in G. Boundary endpoints connect to the interior along their incident edges. Three nearest sites at a wall point lie in one semicircle, forcing a pair with central angle at most 90° and δ≤√2h. At a corner even two nearest sites are impossible. Boundary graph vertices consequently have degree one. We use the geometric Voronoi graph, without artificial vertices subdividing straight edges. ∎ ## 3. Finite-cardinality lower bound and equality rigidity Theorem 1. For every n≥5, Mₙ≥2/√3. If equality holds, there is a triangular lattice Λ of nearest-neighbour spacing δ such that P=Λ∩Q. Proof. It suffices to consider M≤2/√3. Partition Q into four smaller squares, assigning shared-boundary points to one part. Pigeonhole gives δ≤1/√2. Hence h≤1/√6<1/2 and δ>√2h, so the lemma applies. G must have an interior vertex: otherwise its components are straight boundary-to-boundary chords; connectivity permits only one chord and two cells, contradicting n≥5. At an interior vertex at least three nearest sites lie on a circle of radius R≤h. One central gap is at most 120°, so δ≤√3R≤√3h, proving the bound. At equality there are exactly three sites, all central gaps are 120°, and R=h: they form an equilateral triangle of side δ. Four sites would force δ≤√2h and are impossible. Connectivity and degree-one boundary endpoints connect all interior vertices by interior edges. Every edge has an interior endpoint, or it would be a separate chord component. Every cell has a non-container edge, so every site belongs to an interior triangle. Adjacent interior vertices give equilateral triangles sharing a δ-edge, with third vertices on opposite sides. A single triangular lattice propagates throughout G to all sites. A missing site q∈(Λ∩Q)∖P would have distance at least δ>h from P, contradicting coverage. Thus P=Λ∩Q. ∎ ## 4. Exact formula for staggered rectangular arrays Take integers K≥L≥1, K≥2, a phase e∈{0,1}, and 1≤t≤K/L. The following array has symmetric top and bottom margins; t controls its aspect ratio. a=1/K, b=t/K, m=(1−Lb)/2; P(K,L,e,t)={(ia,m+jb): 0≤i≤K, 0≤j≤L, i+j≡e (mod 2)}. N(K,L,e)=((K+1)(L+1)+(−1)ᵉ)/2 if K,L are even; otherwise N(K,L,e)=(K+1)(L+1)/2. Theorem 2. Its separation, covering radius and objective are as follows. The objective is independent of phase, although the cardinality may not be. δ=a min{2,√(1+t²)}; h=a max{(1+t²)/(2t),√(1+(K−Lt)²/4)}; M=2 max{(1+t²)/(2t),√(1+(K−Lt)²/4)}/min{2,√(1+t²)}. Proof. Candidate shortest vectors have lengths 2a, √(a²+b²), and 2b. Since b≥a and the finite array contains pairs of the first two kinds, this gives δ. The infinite array has elementary triangles of base 2a and height b, with circumradius R=(a²+b²)/(2b). For b≥a these form a Delaunay triangulation (with cocircular degeneracy at b=a): the angle opposite a horizontal edge is 2 arctan(a/b)≤90°, and that opposite a sloping edge is arctan(b/a)<90°, so adjacent opposite angles sum to at most 180°. Circumcentres lie inside or on these triangles, and the empty-circle radius is R. Reflection across each side of B=[0,1]×[m,1−m] preserves the infinite array. Folding a nearest lattice site into B does not increase its distance to a given point of B, so B is covered with radius R. Each phase contains a full elementary triangle in B, attaining R. The outer bands are within C=√(a²+m²) of the extreme row. At a bottom-wall column missing from that row, the horizontal neighbours are exactly C away, while subsequent rows are at distance at least m+b≥C. Thus C is attained as well, and h=max{R,C}. ∎ ## 5. Three family-optimal regimes Corollary 2.1. Fix K,L,e and optimize t only within this family. A: if K/L≤√3, the array is height-limited and t*=K/L, M*=√(1+(L/K)²). B: if √3L≤K≤√3L+2/√3, take t*=√3 and M*=2/√3. C: if K>√3L+2/√3, t* is the unique root in (√3,K/L) of the following equation. (t+1/t)/2=√(1+(K−Lt)²/4); (L²−1)t⁴−2KLt³+(K²+2)t²−1=0; M*=(t*+1/t*)/2. Proof. For t≤√3, the interior term divided by separation is √(1+t²)/(2t), strictly decreasing. The boundary term has decreasing numerator and increasing denominator, so also decreases. For t≥√3, separation stays at 2a; the interior term increases and the boundary term decreases. The minimum is therefore at the allowed endpoint, at √3, or at their unique crossing, giving A, B and C. ∎ For example, (K,L,e)=(9,5,0) attains the global optimum at n=30; (10,5,0) gives the family value (7√31−25)/12≈1.164529211651 at n=33; and (9,6,0) gives √13/3≈1.201850425155 at n=35. The latter two are family optima, not lower bounds against arbitrary deformations of the sites. ## 6. Equality lattices must align when n≥10 Lemma. For n≥10, an equality lattice has a family of lines parallel to a square side. The proof excludes nonzero angles analytically, without a numerical scan. Combining 90° square symmetry, 60° lattice symmetry and reflection reduces the angle to 0≤θ≤15°. Suppose θ>0, normalize δ=1, and put r=1/√3. Extend the finite window to all lattice sites in x≥0; this only improves left-wall coverage. a⃗=(cos θ,sin θ), b⃗=(cos(60°+θ),sin(60°+θ)), A=cos θ, c=cos(60°+θ), p=√3/2. The j-th line parallel to a⃗ meets the wall at height Yⱼ, with successive intercepts p/A apart. Its first site in the half-plane has 0≤xⱼr, only this site can serve the wall. Its interval is Iⱼ=[yⱼ−√(r²−xⱼ²),yⱼ+√(r²−xⱼ²)], empty when xⱼ>r. In a non-wrapping step the centres differ by b⃗ and are one unit apart. Their radius-r lens has leftmost coordinate xⱼ+c/2−sin(60°+θ)/(2√3)=xⱼ−sinθ/√3. Their wall intervals overlap exactly when xⱼ≤sinθ/√3. The leftmost lens point really is a circle intersection: a disc's left horizontal extremum is outside the other disc, since cos(60°+θ)<√3/2; the rightmost lens coordinate is positive. In a wrapping step put x′=xⱼ₊₁≥0. The corresponding lens begins at x′+sinθ/√3>0, so the wall intervals cannot overlap. Two consecutive overlapping non-wrapping steps are also impossible: they would require xⱼ+c≤sinθ/√3, whereas c−sinθ/√3=(√3cosθ−5sinθ)/(2√3)>0 because tanθ≤2−√3<√3/5 (the final comparison reduces to 100<108). Centres at least two rows apart have vertical separation at least √3/cosθ−sinθ≥√3−1/2>2/√3=2r, so nonadjacent rows cannot connect. The interval family is locally finite; each connected covered component contains at most two discs and has length at most sin(60°+θ)+r+√(r²−c²). Since cos75°>1/4, this is strictly below 1+1/√3+√39/12<101/48; the rational estimates follow from 48<49 and 16·39<625. Single-disc components satisfy the same bound. Restoring scale, each covered component has length below 101δ/48. With n≥10, a 3×3 partition gives δ≤√2/3, so this length is below 101√2/144<1, since 20402<20736. A covered unit wall must lie in a single connected component, a contradiction. Hence θ=0. ∎ ## 7. Complete integer classification of equality cardinalities (n≥10) Theorem 3. For n≥10, Mₙ=2/√3 if and only if integers K≥2, L≥1 and e∈{0,1} satisfy the following inequalities and n=N(K,L,e). 3L²≤K², 3K²≤(3L+2)²; equivalently √3L≤K≤√3L+2/√3. Necessity. Theorem 1 and the preceding lemma give an aligned complete lattice window. At θ=0 the wall calculation requires the nearer row to touch the wall for adjacent-row intervals to overlap. A single disc covers less than one unit and nonadjacent rows cannot connect, so both vertical walls contain sites. Thus 1=Kδ/2. Successive horizontal rows are √3/K apart. With L spacings between extreme rows and margins tᵦ,tₜ, one has tᵦ+tₜ=1−L√3/K. Only the extreme row covers its corresponding horizontal wall, since the next row is at distance at least √3δ/2>h. Each row contains at least two sites spaced δ apart, and its worst horizontal distance is exactly δ/2. Consequently 0≤tᵦ,tₜ≤δ/(2√3), giving precisely the stated inequalities. Completeness of the window gives the alternating row counts in N. Conversely, choose symmetric margins m=(1−L√3/K)/2 and t=√3. Theorem 2 attains 2/√3. ∎ The margins need not be equal: all splits satisfying those bounds are allowed. For 10≤n≤40 the equality cardinalities are exactly 14,20,22,23,30. The two phases at K=4,L=2, together with Theorem 1, also settle n=7,8. The theorem does not completely classify n=5,6,9. Equality cardinalities have natural density zero. For each L the K interval has length 2/√3<2, permitting at most two integers, each giving at most two cardinalities. Since n≥((L+1)²−1)/2, there are only O(√X) equality cardinalities up to X. In particular M₃₃,M₃₅>2/√3, but neither their exact optimum nor an explicit gap is obtained. Sparsity does not imply a uniform positive gap for other n and does not exclude Mₙ→2/√3. ## 8. Exact constructions, decimal certificates and competition settlement Each row of the construction table gives an exact real-coordinate witness through Section 4 with t=√3. Theorem 1 supplies its matching lower bound. Figures are illustrations only; the proofs do not depend on pixels or floating-point sampling. Finite-decimal coordinates cannot attain this bound. Equality in Theorem 1 contains a nondegenerate equilateral triangle. If its vertex coordinates were all rational, the determinant formula would give rational area, whereas the area is √3δ²/4 with δ² positive rational, a contradiction. The P56 verifier computes M² exactly from nine-decimal coordinates and rounds M² upward at 10⁻¹⁵; the page separately displays an approximation to M. The internal target 4/3 and the public quantity 2/√3 must not be confused, and tolerances must not be widened to label an approximate record exactly optimal. The site marks n=7,8,14,20,22,23,30 as “Proven optimum” and closes competition because their continuous problems are solved. Historical records, authorship and existing contribution points remain intact; the current certificates are not asserted optimal on the nine-decimal grid. Other cases remain open, particularly n=33,35: numerical agreement with a regular construction is not grounds for closure. | n | K | L | e | |---|---|---|---| | 7 | 4 | 2 | 1 | | 8 | 4 | 2 | 0 | | 14 | 6 | 3 | 0 | | 20 | 7 | 4 | 0 | | 22 | 8 | 4 | 1 | | 23 | 8 | 4 | 0 | | 30 | 9 | 5 | 0 | ## 9. Near-equality geometry and wall patterns Sections 9–14 are new research proofs in v2, self-reviewed but not independently reviewed or formally verified. We develop the topology, finite criterion and parameter theorem in dependency order; optimality is not inferred from visual resemblance or numerical fitting. Detailed derivation notes, rigorous rational constant checks and unsuccessful cases are linked in the reproducibility materials. Put M₀=2/√3 and consider competitors with M≤U, where M₀h. Two same-sign large turns exceed the wall direction cone. Hence each wall is either entirely j=3 (S), or alternating j=2,4 (Z). S means combinatorially straight, not physically collinear. Adjacent walls cannot both be S: the corner forces three steps on one wall with same-sign normal displacement at least δ[sin(π/12−2β)+sin(π/12−8β)+sin(π/12−14β)]>h. Three steps require only four wall sites. Z steps are near ±π/6 with error ≤π/30+8β. If adjacent walls were both Z, their corner rays would meet near π/6, π/2 or 5π/6 with error ≤π/15+16β, whereas corner fans are near π/3 or 2π/3 with error ≤4β. These possibilities remain separated by π/10−20β>0. Rotate the square so that left/right are Z and top/bottom are S. S endpoint steps have direction bound A=π/30+12β and internal row turns have bound κ=6β. All strict inequalities hold uniformly at the stated caps on E,U; the rational audit uses a cubic lower bound for sine, not angle sampling. ## 10. Global row structure and strip area Replace every triangle of T by a unit equilateral triangle and develop across edges. Interior degree six makes continuation around each interior vertex trivial; these loops generate continuation loops in the disc. Thus developed coordinates are well-defined, without requiring global injectivity. They assign three unoriented edge colours. S edges have one colour and Z edges alternate the other two: if an adjacent S colour occurred on Z, continuation across at most six triangles would make same-colour physical directions differ by ≤12β, contradicting the π/10 separation of adjacent wall cones. The top and bottom S walls consequently have the same colour; select it as the row colour. The row-colour graph contains every site: interior and non-corner S sites have degree two, while Z sites and corners have degree one. An inward Z row edge points inward with direction error ≤π/30+2β≤A; at a corner it follows S. A cycle is impossible: along an oriented row each developed step adds the same unit vector v, so an r-step cycle would require Φ(p)=Φ(p)+rv. All components are therefore paths, but left-to-right spanning has not yet been assumed. If d₀<δ and a positive integer t satisfies the following projection barrier, no path can have t steps: its first t steps from a left endpoint all point right and already span more than one unit. All remaining steps also point right. A second left endpoint would have an inward edge pointing right, so the last incoming step would point left, a contradiction. Monotonicity of x alone would not suffice, since wall sites need not have x=0. The right-wall argument is symmetric. Thus every row spans left to right and rows do not cross. A+(t−1)κ<π/2; d₀ Σᵢ₌₀ᵗ⁻¹ cos(A+iκ)>1. The ordered rows cut the disc into strips. Each triangle has exactly one row-colour base; its third vertex is not on that row (its developed height differs), and cannot cross another row. Hence it belongs to a strip between consecutive rows. The strip has no interior vertices and its triangle dual is a path. Bases must alternate between the two rows: consecutive bases on one row would leave only two triangles on that side of their common vertex instead of three. The one- or two-triangle side fans at Z endpoints alternate from row to row. Consequently row counts are either all m (type A), or alternating m,m−1 (type B). Each row has at least two sites. Use physical, not developed, coordinates for the exterior wall regions. The left chain is a y-monotone graph x=xₗ(y); the right is analogous and the wall strips are disjoint since 2h<1. After its first step the bottom chain has x>h. If that first step rises, its dx/dy≥√(δ²−h²)/h>1, whereas the left chain has |dx/dy|≤h/√(δ²−h²)<1. The bottom step changes height by at most h while the first left step rises by more than h. Thus the bottom chain cannot enter the region between left chain and wall. The other corners are symmetric: the subtracted trapezoids are outside T and do not overlap. Put s=√(M²−1). For two left endpoints let X be the sum of their x-coordinates, dy their vertical difference, b their distance, and ρ≤h the common wall distance. Then 4ρ²/b²=1+X²/dy², giving exterior trapezoid area ≥X²/(2s); the right-wall distance sum Y behaves identically. In a good triangle with base b, height q and other sides c₁,c₂≥δ, one has M≥c₁c₂/(qδ)≥max(c₁,c₂)/q≥√(1+b²/(4q²)), hence area ≥b²/(4s). For adjacent rows of a,b sites and horizontal spans σ,τ, one has X+Y+σ+τ=2. The sum of row-edge lengths is at least the horizontal span. Two applications of Cauchy give the following lower bound for the strip and its exterior trapezoids. Summing over disjoint strips within unit area proves the area lemma without assuming straight rows. s·area ≥ X²/2+Y²/2+σ²/[4(a−1)]+τ²/[4(b−1)] ≥ 1/(a+b−1); s ≥ Σⱼ₌₁ᵏ⁻¹ 1/(mⱼ+mⱼ₊₁−1). ## 11. A finite criterion and nine non-equality optima This is a sufficient criterion: failure does not prove a construction suboptimal. For a compressed-window candidate U from Section 5, disc coverage and separated-disc packing give strict outward bounds d₀<δ 1/(√(nπ)/2−1); A=π/30+12β; α=2π/15−8β; S=√(U²−1). After spanning is established, step i of an r-step row is controlled by both endpoints: |θᵢ|≤A+κ min(i,r−1−i). A corner site's distance to its S wall is at most δ(S+A)/2: wall equidistance gives z≤√(h²−b²/4)cosθ+(b/2)|sinθ|, decreasing in b≥δ when U|sinθ|≤1; use |θ|≤A and UA<1. A Z step has vertical component ≤Bδ cosα. Thus the following horizontal and vertical span bounds must both hold. Hᵣ=Σᵢ₌₀ʳ⁻¹ cos(A+κ min(i,r−1−i)); Cₖ=S+A+(k−1)B cosα; δHᵣ≤1≤δCₖ. Exhaust all A/B integer representations with k≥w, mⱼ≥2 and at least w sites in the first and last rows. Put r=max mⱼ−1. Every representation must either be excluded by r≥t, Hᵣ>Cₖ, d₀Hᵣ>1 or d₁Cₖ<1, or already have area bound 1+(Σ1/(mⱼ+mⱼ₊₁−1))²≥U². Only when all representations pass do we conclude Mₙ=U. Trigonometric functions, roots and π are compared with rational enclosures; the final area comparison uses exact fractions. The criterion yields the nine cases in the table, all witnessed by {(i/K,j/L):i+j≡e mod 2}. For example, the original n=52 wall count leaves five alternating 10/11 rows, seven alternating 7/8 rows, and eight alternating 6/7 rows. Span bounds exclude the first two; the last gives s≥7/12. For n=99, the remaining types are 9×11 and 11×9. The former has H₁₀−C₉>1.11886; the latter gives s≥10/17. These arguments address continuous configurations, not just optimization inside a row family. The screen for 19≤n≤400 has seven criterion successes and 26 universal-bound equality constructions. Cases 367,368 fail at row projection; 339 lie outside the local angle range and eight have no enumerated compressed candidate. Cases 621,1512 are checked separately to complete the sequence prefixes below. Failed and out-of-scope cases remain inconclusive, not 'necessarily tilted'. Observed improvements by relaxing tilted rows are not structural necessity theorems either. | n | K | L | e | Mₙ | |---|---|---|---|---| | 52 | 12 | 7 | 0 | √193/12 | | 99 | 17 | 10 | 0 | √389/17 | | 120 | 19 | 11 | 0 | √482/19 | | 187 | 24 | 14 | 1 | √193/12 | | 188 | 24 | 14 | 0 | √193/12 | | 270 | 29 | 17 | 0 | √1130/29 | | 304 | 31 | 18 | 0 | √1285/31 | | 621 | 45 | 26 | 0 | √2701/45 | | 1512 | 71 | 41 | 0 | √6722/71 | ## 12. A bounded-integer-defect optimality theorem Theorem 4 (v2, self-reviewed proof). Let K,L be positive integers, D=3L²−K²>0, and K≥max(100,10D²). For e=0,1, each checkerboard window Pₑ is continuously globally optimal at its own cardinality nₑ. If (K+1)(L+1) is even the phases have the same cardinality; otherwise their cardinalities are the product plus or minus one, divided by two. Uniqueness of the optimizer is not asserted. Pₑ={(i/K,j/L): 0≤i≤K, 0≤j≤L, i+j≡e (mod 2)}; Mₙₑ=U=√(1+(L/K)²)=√(4/3+D/(3K²)). First sharpen the wall direction bound so its error vanishes with E. Adjacent left-wall sites with coordinate sum X, vertical difference dy and distance b satisfy the wall identity of Section 10. Since |dx|≤X, their inclination is |θ|≤arctan√(M²−1)≤π/6+β, using derivative ≤3/2. Each Z step adjoins a turn near π/3, giving |θ|≥π/6−9β. Corner fans are near π/3 or 2π/3, so S endpoint directions lie near 0 or ±π/3 with error ≤13β; 14β<π/6 excludes the last two branches. At non-corner Z sites the inward row edge is bounded by both side fans, with error ≤3β. Hence use the sharpened A,α below. E=D/(6K²); β=D/(4K²); A=13D/(4K²); κ=3D/(2K²); α=π/6−9β. Assume M≤U. The threshold gives 1/K≤1/100, D/K<1/30 and D²/K≤1/10. The count formula gives K²/(2√3)3 and √3<7/4 yield δ<1/(√(nπ)/2−1), hence Uδ<3/K and h<3/(2K)<1/4. Coverage area with π<22/7 yields δ>8/(5K). Also U−M₀=D/[3K²(U+M₀)]1. Thus every row has fewer than K steps, and the Section 10 barrier proves spanning and A/B counts. Only now use both endpoints: in any rK, necessarily L′L and K/L<√3 place the candidate in the compressed regime of Section 5, attaining U exactly. The lower and upper bounds agree. ∎ Hᵣ−C > 1/(2√3)−[8D²/25+5D/4]/K−4D/K² > 1/4−4/125−1/8−1/250 = 89/1000 > 0. ## 13. Infinite sequences and exact asymptotics The recurrence (K,L)↦(2K+3L,K+2L) preserves D=3L²−K² and strictly increases K. Thus whenever a fixed positive D has a positive integer solution, its orbit eventually satisfies Theorem 4, producing infinitely many non-equality optima. This does not require one seed to generate every integer solution of that equation. D=2: (19,11)→(71,41)→(265,153)→(989,571)→⋯; n=120,1512,20482,283140,3935672,⋯. D=3: (12,7)→(45,26)→(168,97)→(627,362)→⋯; n=52,621,8281,113982,1582516,⋯. The first two terms of each sequence are covered by Section 11. From the third term, K≥265 for D=2 and K≥168 for D=3 meet the threshold. Thus these entire specified sequences are proved, not just their infinite tails. Defects such as 11 and 12 also produce infinite tails, but unchecked finite prefixes must not be counted as solved. Along any proved fixed-D recurrence tail, n/K²→1/(2√3). Rationalizing the score difference gives the exact asymptotic constant D/24, namely 1/12 and 1/8 for D=2,3. This concerns optimal values, but only along these sequences, not a convergence rate for all n. K²(Mₙ−2/√3)=D/[3(Mₙ+2/√3)]→D/(4√3); n(Mₙ−2/√3)→D/24. ## 14. A counterexample to fixed-ratio extrapolation and scope limits Although 52,187,188 share optimum √193/12, one cannot extrapolate that (K,L)=(12t,7t) is optimal at every scale. At t=545, windows (6540,3815) and (6571,3797) both contain 12,480,228 sites. The second remains in the compressed regime and has strictly smaller score, certified entirely by the following integer and rational identities. No array of over twelve million points is needed, and optimality of the improved construction is not claimed. 6541·3816=6572·3798; 6571²<3·3797²; 193/144−57595250/43178041=39645913/6217637904>0. This explains why the useful parameter is integer defect D, not merely the ratio K/L: fixed-ratio scaling makes D grow quadratically with scale, whereas fixed D controls near-equilateral error. The theorem's threshold is conservative and sufficient, not a necessary boundary of validity. Tilted relaxed configurations at n=13,19, the non-equality global optima at 33,35, and the failed finite-criterion cases 367,368 remain unresolved by this work. The new coordinates i/K,j/L are rational: the rational-coordinate impossibility argument in Section 8 for equilateral equality constructions does not apply to them. Rational numbers nevertheless need not be exactly representable with nine decimals; rounding requires verification by the original verifier. This revision updates the paper only: it neither expands the catalogue, changes settlement beyond the existing seven cases, nor alters contribution points. ## 15. Conclusions, revision history and reproducibility Version 1 (2026-09-07) gave the universal bound, equality rigidity, equality-cardinality classification for n≥10 and optimization inside staggered rows. Version 2 (2026-09-08) adds near-equality global row structure, a finite sufficient criterion, nine non-equality optima, the bounded-integer-defect theorem, two complete infinite sequences, the D/24 asymptotic constant and a fixed-ratio counterexample. These turn some numerical patterns into structural results under explicit assumptions, not a formula for every Mₙ. Enumeration of the classification needs only integer arithmetic; all coordinates and objective formulas are given above. Accompanying tests check cardinalities, integer conditions, the list of proved cases, and that decimalized witnesses pass the existing exact verifier without attaining 4/3. These tests check implementation consistency, not replace the analytic proofs. The new audit materials use Python's standard-library Fraction, integer square roots, Machin's identity and trigonometric Taylor bounds with controlled remainder sign. Run tools/research-p56-row-criterion-2026-09-08.py followed by tools/research-p56-defect-families-2026-09-08.py to regenerate the finite enumeration and parameter-constant certificates. Finite enumeration is not the proof of the infinite theorem, which rests on the uniform inequalities of Section 12. Independent review of the geometric topology remains a higher priority than extending numerical tables. Authorship remains MinMax Arena. Claude assisted the wall-pattern, row-count and area approach; Codex assisted global row organization, scope corrections, parameter estimates, self-review and exposition in this revision. Both are tools, not independent peer reviewers. The new proofs do not rely on unverified Landau estimates from earlier drafts. Proof gaps, prior literature and stronger results are welcome in the P56 discussion board. Version: 2026-09-08, v2. ## References [[1] Bishnu, Desai, Ghosh, Goswami and Paul — Uniformity of point samples in metric spaces using gap ratio (2015)](https://arxiv.org/abs/1411.7819) [[2] Teramoto, Asano, Katoh and Doerr — Inserting Points Uniformly at Every Instance (2006)](https://globals.ieice.org/en_transactions/information/10.1093/ietisy/e89-d.8.2348/_p) [[3] Bondarenko, Hardin and Saff — Mesh ratios for best-packing and limits of minimal energy configurations](https://arxiv.org/abs/1212.6211) ## Derivations and reproducibility materials [Detailed row-structure and exterior-region proof](https://minmaxarena.com/papers/p56-v2/docs/P56-N52-ROW-STRUCTURE-2026-09-08.md) [Finite criterion and inconclusive cases](https://minmaxarena.com/papers/p56-v2/docs/P56-ROW-CRITERION-2026-09-08.md) [Full parameter-theorem derivation](https://minmaxarena.com/papers/p56-v2/docs/P56-BOUNDED-DEFECT-THEOREM-2026-09-08.md) [Author self-review record](https://minmaxarena.com/papers/p56-v2/docs/P56-N52-SELF-REVIEW-2026-09-08.md) [Finite-criterion reproduction (Python standard library)](https://minmaxarena.com/papers/p56-v2/tools/research-p56-row-criterion-2026-09-08.py) [Parameter-theorem and sequence reproduction](https://minmaxarena.com/papers/p56-v2/tools/research-p56-defect-families-2026-09-08.py) [Finite enumeration certificate](https://minmaxarena.com/papers/p56-v2/docs/p56-row-criterion-to-400-2026-09-08.json) [Parameter constants, sequence prefixes and counterexample certificate](https://minmaxarena.com/papers/p56-v2/docs/p56-defect-families-audit-2026-09-08.json)