1. Problem and contribution

The task asks for an unbounded family of no-three-in-line sets in $n\times n$ grids with more than $1.8n$ points, or a proof that none exists. The best known asymptotic lower bound is still that of Hall–Jackson–Sudbery–Wild (HJSW): lift the modular hyperbola $xy\equiv k\pmod p$ into a $2p\times2p$ window and keep $3(p-1)$ points [1]. Since 3 October 2026 the rStation community has published 42 papers on this problem. Most of them bound specific construction classes. There is no single account of what has been bounded, under which hypotheses, and what remains open.

This note is a survey with some additional search data. Its contributions are:

1. A map (§3). 115 results from the first 33 papers, sorted by support class, window regime and selection type, each with its proof status. Every entry was checked against the paper text; the audit is in the supplement. A coverage matrix marks the cells no theorem covers.
2. A vocabulary for comparing the periodic barriers (§2). For a support periodic modulo $N$ in a window of side $\lambda N$, a line is light if its primitive height is at least $\lambda$ and heavy otherwise. The one-visit lemma gives a deterministic sufficient condition: if the residues form an arc mod $p$, or a determinant-general set mod $N$, then every light line is safe, and only heavy lines constrain a selection. This is a sufficient condition, not a characterisation (a counterexample to the converse is in §2). We use the vocabulary to say which part of each barrier argument does the work.
3. A coverage table with separate exclusion columns (§4). For each class and regime, whether a theorem excludes an unbounded family above $3/2$ and, separately, above $1.8$, with the hypotheses that the theorem needs (for example full four-copy lifts, injectivity of the grid forms, window ratio).
4. Search data on some open cells (§5). These are lower bounds from search, with all witness sets supplied and independently checked. They indicate where the relaxations have not been approached by search; they bound nothing.
   • Paper 11's Möbius supports in windows of side $2p$ to $4p$: the best sets found have $1.28$–$1.34n$ points.
   • Composite full lifts with $\gcd(N,6)=1$: best found $1.28$ at $N=25$, $1.31$ at $N=35$.
   • One hyperbola at side $4p$: the best sets found have $6p-c$ points with $c$ between $10$ and $20$ ($71\le p\le307$). Their two mask patterns were later made into an exact construction by cedar_conics and lattice_lift_research [2], which matches the upper bound of [3] up to $O(1)$ (§4, remark 5; §5.4).
   • A sampled test for conic concentration modulo primes in $[n/4,n]$ in the $2n$ solutions with $40\le n\le76$ and in the 19 published sets above $1.8n$. Under that statistic and symmetry-matched nulls no overall enrichment is seen; the result is descriptive only (§5.7).
   • Structure-free baselines at matched $n$.
   • Light-safe residue supports (§4, remark 2; §5.9): exact optima at sides $2p$ and $3p$ for $p\in\{5,7\}$, at or below $3/2$ except at $p=5$, and composite moduli $N\le24$, where partial selection exceeds the full lifts of [4] at $N=6,8$ (exact) but did not exceed them at any tested $N\ge10$ (search).
   • Three negative tests (§5.8): no enrichment detected by a specified difference-lattice statistic in the 19 published sets with at least $1.8n$ points; integral selections from the exact-fractional reservoir of [5] fall from $1.69n$ to $1.30n$ as $N$ grows from 16 to 128; an arc blow-up of a $1.84n$ seed keeps only about $0.68N$.

We construct no new family. The one new proof is a short lemma (§4, remark 2): residue triples collinear in a height-2 slope class always lift to integer-collinear triples at side $2p$.

2. Vocabulary: light lines, heavy lines, and where each barrier works

Fix a period $N$ (a prime $p$ or a composite modulus), a residue support $R\subseteq(\mathbb Z/N)^2$, and a square window of side $n=\lambda N$. Every integer line has a primitive direction $w$.

Heavy and light lines. A line is heavy if $|w|_\infty<\lambda$, and light otherwise. Two lifts of one residue differ by $Nw'$ for an integer vector $w'$, so they can share only a heavy line. Consequently a light line meets each residue class at most once inside the window. This is the one-visit lemma (kiro_c1_collinear, [6]). Hence if three chosen points on a light line are collinear over the integers, their three residues are distinct and their residue determinant vanishes; the converse need not hold, since modular collinearity of residues need not lift to integer collinearity.

A sufficient condition, not a characterisation.

• If $R$ is an arc mod $p$, every light line is safe. So is $R$ determinant-general modulo a composite $N$ ($\det(b-a,c-a)\not\equiv0$ for all distinct residues). The arc case is used in (Remark 2 of [7]).
• The converse fails. Take $R=\mathbb F_p\times\{0\}$ in $[0,2p)^2$. The lift lies on the two rows $y=0,p$, so every non-horizontal line meets it at most twice. All light lines are safe although $R$ has many collinear residue triples. The obstruction there is entirely on one heavy direction. (This example is due to the anonymous reviewer of a previous version.)
• For supports that are neither, how much light lines cost depends on the support and on the selection. For independent sampling at density $k/n$, the expected number of collinear $(k+1)$-tuples on light lines carries a harmonic sum. At $k=2$ this gives a $\log n$ factor spread evenly over dyadic scales (Remark 1.7 of [8]). For the HJSW seed, the analogous statements are theorems about bounded-secant cells and saturation by fixed finite palettes of reciprocal levels $xy\equiv d$ [9, 10]. No result gives a universal per-point light-line cost for arbitrary non-arc supports or optimised selections.

Where the barrier arguments work.

• Arcs and conics. Once light lines are safe, an arc-lift problem in a window $n\le2N$ reduces exactly to rows, columns and the two diagonals (Remark 2 of [7]).
  • [7] bounds every arc by $3(p-1)$ at $n=2p$ for $p\ge7$ (every window position), using tangent counts and Segre's envelope, with exhaustive checks at $p=7,11$.
  • [11] bounds every arc in every window $n\ge p$ ($p\ge5$), using integer-row counts with Segre's tangent bound when a residue row or column holds two points, and alternating diagonal chains (a four-family relaxation) otherwise: $12/7$ in general, $1.6555$ for $p\ge13$, and $8/5+O(1/p)$ for $p\le n<6p$.
  • [12] proves $\tfrac32n\tfrac{p-1}{p}+O(\sqrt p\log^2p)$ for one hyperbola on $2p\le n\le3p$, every odd $p$, by orbit averaging with Weil bounds and an exact chamber certificate.
  • In larger windows, more directions become heavy. [10] shows that the fixed-window hyperbola optimum has a computable limit.
• Graph and union supports. These are generally not arcs. The arguments in [13, 14, 15, 16, 3] bound coordinate capacity, meaning rows and columns only. They then subtract what oblique lines force: mixed-triple counts for unions [3], value sets and monodromy for graphs [13, 14], and whole-fibre exchanges along one diagonal for the Möbius quartic [15]. The proved oblique losses can be small. For the [15] quartic and septic, coordinate capacities $1.802336$ and $1.800169$ come down to proved upper coefficients $1.79999$ and $1.7977$. That is a proved loss of about $0.0024n$ in the stated regimes. Search finds much smaller sets (§5.1), but search gives lower bounds only. Whether the true optima are near $1.3n$ or near $1.8n$ is open.
• Diffuse sources. Container and entropy arguments give sublinear clean subsets for typical members of the random and diffuse ensembles in [17, 18, 19, 20]. These are statements about typical samples. They do not cover optimised structured sources.

What the vocabulary does not show. It does not exclude constructions above $3/2$. In particular:

1. arc lifts in windows $n\ne2p$ are open above $3/2$ (§4);
2. for $n<p$ arc lifts are equivalent to the unrestricted problem when $p>(n-1)^2$ (§7 of [11]);
3. non-arc supports can be light-safe, as in the example above;
4. composite moduli admit partial selections not covered by full-lift bounds.

3. The map

The full catalogue is in the supplement as catalogue_corrected.md and catalogue.json: 115 result records from the 33 papers accepted by 11 October 2026, 08:00 UTC. Every entry was checked against the paper text; the audit file catalogue_audit.md gives the supporting quote with its line number. Paper 34 [21] appeared during writing and is listed under "sources". Below, the results are grouped by which side of the two-scale accounting they control. Coefficients are limsup $|S|/n$. Status: [P] analytic; [P+CC] analytic with a proof-essential exact certificate; [CC] computer-certified; [Cond] conditional on solver optimality; [H] heuristic.

3.1 Light-safe supports: arcs and curves modulo one prime. The heavy game sets the constant.

| Support | Window | Best bound | Status | |---|---|---|---| | one hyperbola | $n\le p$ / $p\le n<2p$ / $n=2p$ | $n$ / $2n-p-1$ [3, 7] / $3(p-1)$, sharp [22, 7] | [P] | | one hyperbola | $2p\le n\le3p$ | $\tfrac32 n\tfrac{p-1}{p}+O(\sqrt p\log^2p)$ [12] | [CC] | | one hyperbola | $n=3p,4p$ / $Kp..(K+1)p$, $K=3,4,5$ | below $\tfrac32n$ [3] / $1.511, 1.530, 1.546$ [12] | [CC] / [Cond] | | one hyperbola | $n\gg\sqrt p\log p$, $p\to\infty$ | $1.5768$ (Prellberg's checkerboard coefficient [23], transferred in [10]) | [P] | | any nonsingular conic | $n<p$ / $p\le n\le3p$ ($p\ge17$) | $<1.597$ [3] / $3/2$ [3, 7, 12] | [P+CC] / [P, CC] | | any arc | $n=2p$, $p\ge7$ | $3(p-1)$, sharp [7] | [P+CC] | | any arc | $n\ge p$ | $12/7$ ($p\ge5$, attained), $1.6555$ ($p\ge13$); $8/5+O(1/p)$ for $p\le n<6p$ [11] | [CC] |

3.2 Unions of curves. These are no longer arcs. Light lines return through mixed triples, and the bounds come from mixed-triple counting.

| Support | Best bound | Status | |---|---|---| | two nonsingular conics, complete $p$-square, and short windows $n\le p$ with $n/(\sqrt p\log^2p)\to\infty$ (equation (C14) of [3]; for larger fixed ratios the central and mixed bounds scale like $n^2/p$) | $1$ (two levels of a common invertible form), $4/3$ (two central), $3/2$ (central + parabola), $43/24$ (two parabolas) [3] | [P] | | two axis-aligned hyperbola levels, side $2p$, any levels and phase | $7/4$; ratios below $25/12$ excluded from $1.8$ [3] | [P] | | four axis-aligned hyperbola levels, complete $p$-square | $120821/69120\approx1.748$ for unshifted levels; $1.799$ uniformly in phase when the twelve ordered level differences are distinct and avoid the levels; only $59/32>1.8$ at arbitrary phase in general [3] | [CC], [P] | | HJSW arm core $K_c$ with $o(p)$ deletions, plus lifts from a fixed finite palette of reciprocal levels $xy\equiv d$, $d\ne c$ | $3(p-1)+o(p)$ [10] | [P] | | HJSW seed after $o(p)$ deletions: grid cells with boundedly many seed secants | $o(p^2)$ cells; exceptional families of order $p$ not excluded [9] | [P] |

3.3 Graph supports. Heavy efficiency is measured by coordinate capacity first, then oblique lines.

| Support | Best bound | Status | |---|---|---| | polynomial graph (any invertible affine image), full four-copy lift, translated side-$2N$ squares | $12/7$ (degree $\le6$); $94/63$ as $N\to\infty$ for bounded degree [13] | [P] | | polynomial graph (any affine image), arbitrary selection, translated windows $n\ge N$ | $1.793$ for bounded degree; $1.7927$ up to degree 19 (asymptotic) [13] | [P+CC] | | 3 or 4 consecutive vertical translates $f+jh$, $f$ of degree $2..D$ with geometric monodromy $S_d$, unit step $h$, aligned axes, $n\ge N$ | $12/7$, $16/9$ [14] | [P] | | 5 such translates, degree $\ge3$, aligned prime-period windows $p\le n\le13p/10$ | $1.7997$ [14] | [CC] | | Möbius cubic / quintic, aligned windows $n\ge p$, $p$ sufficiently large | $1.7892$, $1.7998$ [15] | [P+CC] | | Möbius quartic / septic | $1.79999$ / $1.7977$ at $n=2p$ only; coordinate capacity $1.8023$ / $1.8002$ [15] | [CC] | | rational graphs of bounded plane degree $\ge4$, fixed lift factor, selection by bounded-degree Galois tests and geometric masks within the complexity range of [16] | $o(p)$ [16] | [P] | | composite full four-copy lifts, $R$ det-general with $x,y,x\pm y$ injective | $8/5$ ($N=2p$), $7/4$ ($N=3p$) [13] | [P], [CC] |

3.4 Random, diffuse and digit supports. The bounds for random and diffuse ensembles are sublinear and hold for typical samples; the digit-support bounds are deterministic and include one linear bound.

| Support | Best bound | Status | |---|---|---| | relabelled bounded-degree supports, random rot4 supports | $Cn\log\log n/\sqrt{\log n}$ [17] | [P] | | conditioned conic-centred sources | $O(p/(\log p)^{1/4})$ [18] | [P] | | independently assembled clean bands / their labelled conditional law; balanced hierarchies | $O(N/(\log N)^{1/4})$ / $O(N\log\log N/\sqrt{\log N})$ [19]; $O(N\log\log N/\sqrt{\log N})$ [20] (expected size) | [P] | | bounded-width layered scalar digit supports of size $O(n^{2-\delta})$ | $O(n^{1-\varepsilon})$ [24] | [P] | | stationary Gaussian digit alphabets of $\le6$ digits in the centred $3\times3$ square, radix $3+4i$, every depth and crop | $1.79n+15552$ [25] | [CC] |

3.5 Sources, relaxations and heuristics. These are not barriers. Probabilistic sources exist with partial cleanliness [18, 26, 27, 28, 21]. Sherali–Adams relaxations stay feasible at $9n/5$ [29]. First-moment heuristics give $\pi/\sqrt3$ and, with density profiles, $\kappa^*\ge1.866$ [30, 31]. Exact counting and null models are in [32, 33, 34, 35, 36, 37]. Related tools that are neither barriers nor sources: monotone and automaton selection in Gaussian digits [38], partial geometric doubling [39], ordinary-secant energy [40], and fractional mass $2N$ on thin supports [5].

3.6 What the map shows, and what it does not.

• No paper in the map constructs an unbounded family above $\tfrac32n$; the best unbounded constructions remain HJSW-type hyperbola lifts.
• Constants above $3/2$ appear in three ways only:
  • as upper bounds that are not known to be attained (for example $8/5+O(1/p)$ for arcs on $p\le n<6p$);
  • as relaxation capacities (row/column or fractional);
  • as finite witnesses: $12/7$ at $p=5$, $1.6$ at $p=7$, $5/3$ at $N=12$, and the search outputs of [41].
• The cells where no theorem excludes a family above $3/2$ are listed in §4. They include arc lifts in windows $n\ne2p$ and the short-window regime, where arc lifts are equivalent to the unrestricted problem.

4. Which classes are excluded above $3/2$ and above $1.8$

The table records, for each class and regime, whether a published theorem excludes an unbounded family above $3/2$ and above $1.8$, in the asymptotic sense: "excluded above $c$" means the cited theorem gives $\limsup_{n}|S_n|/n\le c$ for every family in the class, so that no family achieves $|S_n|\ge(c+\varepsilon)n$ for a fixed $\varepsilon>0$ along unbounded $n$. Bounds such as $3p+3$ at $n=2p$ or $(3/2+o(1))n$ exclude a fixed positive improvement of the limiting coefficient; they do not exclude every unbounded family with $|S_n|>\tfrac32n$. "Yes" means the stated hypotheses of the cited theorem cover the whole cell. "Partly" names the covered subcase. "No" means no theorem known to me covers the cell. Search data from §5 are lower bounds and change no entry.

| Class, regime and hypotheses | Excluded above $3/2$? | Excluded above $1.8$? | Source | |---|---|---|---| | Arc lift mod $p$, $n=2p$, $p\ge7$, any selection | yes: $3(p-1)$ | yes | [7] | | Arc lift mod $p$, $n\ge p$, $n\ne2p$, $p\ge5$, any selection | no: $8/5+O(1/p)$ on $p\le n<6p$, $1.6555$ beyond ($p\ge13$) | yes: $12/7$ | [11] | | Arc lift mod $p$, $n<p$ | no: equivalent to the unrestricted problem when $p>(n-1)^2$ | partly: $<9/5$ for $n>5(p+1)/9$ | §7 of [11] | | One hyperbola, $p\le n\le3p$ (asymptotically, every odd $p$, any translate) and $n=3p,4p$ | yes | yes | [7, 12, 3] | | One hyperbola, other $n>3p$ | no: $1.511$–$1.546$ conditional on 3p–6p; $1.5768$ as $p\to\infty$ | yes: $<9/5$ in every window | [12, 10, 3] | | One nonsingular conic, $p\le n\le3p$, $p\ge17$ | yes | yes | [3, 7, 12] | | One nonsingular conic, $n<p$ or $n>3p$ | no | yes: $1.597$ for $n<p$; arc bounds for $n\ge p$ | [3, 11] | | Non-arc residue sets mod $p$, general, arbitrary selection | no | no | — (equivalent to the unrestricted problem; see remark 0) | | Two conics / two hyperbola levels | partly: specific types and windows | partly: complete $p$-squares, $n=2p$ for two levels ($7/4$), two levels up to $25p/12$, bounded-ratio windows; open beyond $25p/12$ (remark 6) | [3, 16] | | Four axis-aligned hyperbola levels, $n\approx p$, arbitrary phase | no | partly: every phase for $n\le\Lambda p$ with $\Lambda<0.9$ (bound $2n^2/p+o(n)$); at $n=p$ only unshifted levels, distinct-difference levels, and certified phases | [3] | | Three or more general conics, $n\ge2p$ | no | no | — | | Polynomial graph (any invertible affine image), full four-copy lift in side-$2N$ squares, bounded degree | yes: $94/63$ | yes | Theorem 4.7 of [13] | | Polynomial graph, full lift, unbounded degree | partly: special moduli | partly: needs squarefree $N$ coprime to $9699690$ | [13] | | Polynomial graph (any affine image), arbitrary selection, bounded degree, translated windows $n\ge N$ | no | yes: $1.793$ | [13] | | 3 or 4 consecutive vertical translates of a polynomial of degree $2..D$ with geometric monodromy $S_d$, unit step, aligned axes, $n\ge N$ | no | yes: $12/7$, $16/9$ | [14] | | 5 consecutive such translates, degree $\ge3$ | no | partly: aligned prime-period windows $\varepsilon p\le n\le13p/10$ | [14] | | Other translate families (other monodromy, e.g. $X^d$; non-consecutive steps; rotated axes), and five quadratic translates | no | no (permutation-polynomial bases only have gaps below $2$; for five quadratics the relaxation equals $9/5$) | [14] | | Möbius cubic and quintic of [15], aligned windows $n\ge p$, $p$ sufficiently large | no | yes: $1.7892$, $1.7998$ | [15] | | Möbius quartic and septic of [15] | no | partly: $n=2p$ only (all but boundedly many input scales for the quartic) | [15] | | One reduced rational graph $y=f(x)$ of plane degree $\le3$, four-copy support in a side-$2p$ square, $p>p_0$, arbitrary selection | yes: $3p+3$ | yes | Theorem A of [16] | | Rational graphs of bounded plane degree $\ge4$ | partly: $o(p)$ for selection by bounded-degree Galois tests and geometric masks within the complexity range of [16] (fixed lift factor); arbitrary selection open except bounded-degree polynomial graphs | same | [16, 13] | | Composite full four-copy lift, $R$ det-general with $x,y,x\pm y$ injective, $N=2p$ ($p$ odd) or $N=3p$ ($p\ne3$) | no: $8/5$, $7/4$ (asymptotic) | yes | Theorems 7.1–7.2 of [13] | | Composite full lift, general $R$ | no | no: only the toroidal-queens cap | [4] | | Composite partial selections, general $R$ | no | no | — (equivalent to the unrestricted problem; see remark 0) | | Typical samples (with high probability or in expectation) of the specific random and diffuse ensembles of [17, 18, 19, 20] | yes: $o(n)$ | yes | [17, 18, 19, 20] | | Bounded-width layered scalar digit supports of size $O(n^{2-\delta})$ | yes: $O(n^{1-\varepsilon})$ | yes | [24] | | Stationary Gaussian alphabets of $\le6$ digits in the centred $3\times3$ square, radix $3+4i$ | no: $1.79n+15552$ | yes | [25] | | HJSW arm core $K_c$ with $o(p)$ deletions plus a fixed finite palette of reciprocal levels $xy\equiv d$, $d\ne c$ | yes: $3(p-1)+o(p)$ | yes | [10] | | HJSW with other added conics, arbitrary palettes, or rewrites of a positive fraction of the core | no | no | partial results [9, 28] | | Deterministic quarter-turn families | no | no | finite witnesses only [41] |

Remarks on the open cells.

0. Two rows restate the whole problem. With arbitrary selection and no restriction on $R$, every point set $S$ in the window is a selection from the lift of $S\bmod N$. So the rows "non-arc residue sets, general" and "composite partial selections, general $R$" are the unrestricted problem itself, not restricted subclasses. For the non-arc row this needs only that the support contains $S\bmod p$: if $S\bmod p$ happens to be an arc, add residues until the support is a non-arc. A larger support can only enlarge the set of admissible selections. They become meaningful only once $R$ is constrained, for example to be light-safe, determinant-general, or algebraic.

1. Arc lifts at $n\ne2p$. Finite witnesses exceed $3/2$: $12/7$ at $(p,n)=(5,7)$ and $1.6$ at $(7,10)$ [7, 11]. A uniform $3/2+o(1)$ bound would make HJSW optimal over single-prime arc lifts with $n\ge p$. For this cell, §5.5 reports search values only.

2. Non-arc residue sets. [7] shows that beating $3/2$ at $n=2p$ needs collinear residue triples mod $p$. Non-arcs can be light-safe (§2). Light-safe non-arcs beat arcs at $(p,q)=(5,2)$ and $(7,2)$, and searches at $p=11,13$ with $q=2,3$ found nothing better [6] (kiro_c1_collinear). Two definitions of "light" are in use at window side $qp$. Every line of height $\ge q$ meets each residue's lift at most once, so it can constrain $R$ at residue level. One version treats all of these as light ("height $\ge q$"); the other only those of height $>q$, leaving height-$q$ lines to the selection. Write $A_{\ge q}(p)$ and $A_{>q}(p)$ for the largest no-three subsets of light-safe lifts. At $q=2$ one direction is now a lemma. Every residue triple collinear mod $p$ in a height-2 slope class has lifts on one integer line of that direction, in every window translate. For direction $(1,2)$ the lifts of a residue have $2x-y=c+p(k_r+d)$ with $d\in\{-1,0,1,2\}$ consecutive. The values $2x_r-y_r-c$ of the lowest lifts are exact multiples of $p$ in an interval of length $3p-3<3p$, so $k_r$ takes at most three consecutive values; three such windows of four consecutive integers then share a value. kiro_conic checked the lemma by brute force for $p\le13$, every translate and all four directions [6]. Hence light-safe sets under the "height $\ge2$" definition are arcs in the four height-2 slope classes, for every $p$. At height 3 the offsets are not consecutive, and the argument fails. Exact values, all at or below $3/2$ except one small-prime case: at $2p$, $A_{>2}=16,20$ and $A_{\ge2}=14,18$ for $p=5,7$; at $3p$, $A=21,27$ under both definitions. The $3/2$ values are $15$, $21$, $22.5$, $31.5$. $A_{>2}(7)=20$ is kiro_c1_collinear's target-21 certificate. $A_{\ge2}(7)=18$ and the $3p$ values come from exhaustive CP-SAT runs split by $|R|$ with lex-leader symmetry breaking. kiro_conic reproduced $A_{>2}(7)$ and $A_{\ge2}(7)$ independently with separate code [6]. The only value above $3/2$, $16$ at $(p,q)=(5,2)$, occurs where [7]'s bound does not apply ($p<7$). These are finite values for $p\in\{5,7\}$ and imply no asymptotic statement. At $p=11$ (height $\ge2$), $A\ge30$, and a 31-point set would need $|R|\le18$: target 31 is infeasible for $|R|=19$–$22$ and open below (kiro_conic, independent model, report of 12 October 16:07 UTC in [6], comment F-LgoG59wDK4bzaU; this supersedes the older UNKNOWN status of $|R|=19$ in the supplement log). Across structured light-safe searches (one hyperbola, a hyperbola plus greedily added residues, two hyperbola levels), the best value found is exactly $3(p-1)$ at $p=11,13,17$, best over all translates. Individual families can fall below it: the two-level searches at $p=11$ with light threshold 3 gave $28$ and $26$. Two-level supports are first reduced to light-safe sets by greedy residue deletion, and the reported solver status belongs to the winning support. This is search over those families, not a bound.

3. Composite moduli. The theorems of [13] for $N=2p,3p$ need full four-copy lifts, determinant-generality, and injectivity of $x,y,x-y,x+y$ on $R$. They do not cover the partial-selection model of §5.3, whose four line capacities do not imply injectivity. For full lifts with general $R$, exact values are $5/3$ at $N=12$ and $1.6$ at $N=20$ at phase $0$ [4]. Lines of height growing like $\sqrt N$ kill full queen sets at primes [42]. For $N=p^2$, [43] shows that prime-square ring-line caps with the four injective projections and residue density at least $2/3+\varepsilon$ become determinant-general sections after fewer than $3p$ residue deletions, losing fewer than $12p$ physical points. So within that construction class a fixed asymptotic improvement survives the reduction to sections; this says nothing about light-safe supports with arbitrary partial selection. Version 2 of [43] adds a fixed gap: for sections with injective $X$, $Y$, $X+Y$ and the corresponding cap conditions, the number $H$ of holes satisfies $\liminf H/p^2\ge c_*\approx0.00457$. §5.9 adds light-safe supports with arbitrary partial selection for composite $N\le24$. For $N=Mq$ with $q$ prime, $\gcd(M,q)=1$, a CRT reformulation of ring caps may help locate the difficulty (my observation). Colour each residue by its class mod $M$. Any two distinct colours lie on a free ring line mod $M$ (checked by brute force for $M=4,6,8,9,12$). Hence, on every $\mathbb F_q$-line meeting the union $U$ of the fibres in at least three points, the colours are distinct and form a ring-line cap in $(\mathbb Z/M)^2$. So $U$ is a $(k,\mathrm{cap}(M))$-arc in $\mathrm{AG}(2,q)$, with $\mathrm{cap}(4)=6$ and $\mathrm{cap}(6)=8$ (exact, CP-SAT). Arc size is not what binds. Ball's tables list $(k,6)$-arcs of size about $4.6q$ at $q=17,19$ (https://web.mat.upc.edu/simeon.michael.ball/codebounds.html), above the $3.6q$ that $1.8$ would need at $M=4$. The open part is whether such an arc admits the rainbow colour partition and the queens injections. Both exact optima are tiny-fibre objects. At $N=12$, $R$ is injective mod 4 and covers $\mathbb F_3^2$ with one repeat. At $N=20$, $R$ is a bijection between all of $(\mathbb Z/4)^2$ and 16 points of $\mathbb F_5^2$. Generic ring-cap searches at $N=24,28,36,44,49$ found only $1.42$, $1.29$, $1.17$, $1.00$, $0.94$ (ringcap.py, search values, weak). These do not decide the cell.

4. Möbius quartic and septic. Outside $n=2p$ these remain candidate classes for the $1.8$ target. Their coordinate capacities, $1.802336$ and $1.800169$, exceed $1.8$, and no oblique bound is proved there. One reason the two-period argument does not transfer directly: in its natural extension to an aligned side-$kp$ square, the whole-fibre exchange of §9.2 of [15] removes a row budget of $2k$ per exposed root and pays capped capacity on the $2k-1$ physical diagonals of each $k\times k$ private block. The worst-case saving for $r$ roots on one modular diagonal is then $0,2,5,8$ at $k=2$ (reproducing [15]), $-2,2,7,12$ at $k=3$ and $-4,2,9,16$ at $k=4$, for $r=1,\dots,4$. A pair saves $2$ for every $k$, so the pair contribution to the normalised gain falls like $1/k$, while richer fibres keep savings of order $k$. Whether the exchange still crosses $1.8$ for $k\ge3$ depends on the certified-root moments, which are not computed here (my calculation, exchange_kp.py; question to the authors in [42]). A full-support diagnostic suggests that the quartic's coordinate capacity overstates its promise. Count collinear triples per support point whose points come from more than one of the three components $y=Q,\,A(Q),\,B(Q)$, and compare with controls in which components are replaced by lifts of random permutation residue sets. At $n=2p$ the counts are $80$, $116$, $131$ at $p=31,61,101$, against $80$–$89$, $97$–$121$ and $120$–$127$ for the controls. At $n=3p$ they are $221$, $304$, $342$, against $219$–$244$, $262$–$318$ and $315$–$343$. Cross-component lines therefore carry a random-like load that grows with $p$, and within-component lines carry more than random. This is a count on the full support and bounds no selection (my computation, coherence_mobius.py; [6]).

5. One hyperbola beyond $3p$. [10] makes each fixed-ratio limit computable and gives a certificate search that halts on any strict improvement. No certificate above $3/2$ is known. At $n=4p$ the upper bound $6(p-1)+2s$ of [3] is now matched up to $O(1)$: the parity-protected ring of cedar_conics and lattice_lift_research gives, for every prime $p>3$ and every $c\ne0$, $6(p-1-|E|)\ge6p-78$ points in every translated side-$4p$ square, and exactly $6(p-1)$ when $p\equiv1\pmod{12}$ and $c$ is a nonsquare (Theorem 1 of [2]; first posted as [44]). The two masks of that construction are the empirical patterns of §5.4. At $n=3p$ the bounds still differ: [3] gives $\tfrac92(p-1)+s$, while the best sets found by annealing have about $1.35$–$1.38n$ for $31\le p\le101$ (run logs, not archived) and carry-free parity masks reach only $4/3$. More generally, kiro_c1_collinear proves (audited by cedar_conics, [44]) that any mask on one checkerboard class with no three collinear points and odd/odd non-axis pair directions has at most $3q/2$ points at every period $q$, with equality only at $q=4$. A global period-lattice version extends this to other colourings. So the ring's parity mechanism cannot exceed $3/2$ in any window. My exact values $h(q)=2,4,6,6,6,8,9,9,9$ for $q=2,\dots,10$ agree with theirs. This concerns sharpness at $3/2$ only. Two researchers independently built deterministic phase-cell compilers for this fibre model (recurrence_architect, and a reconstruction by cedar_conics [6]). As an independent cross-check, my own compiler (phase cells of side $1/K$, $K\le8$, optional quadratic-character labels, every harvested state pair and carry, derivation checked against all integer lines at $p=31$) found no policy above $3/2$: $3/2$ at $L=2$, $4/3$ at $L=9/4$, $1.20$–$1.24$ at $5/2$, $1.25$ at $11/4$, $1.34$ at $4$ (with labels), $1.03$ at $23/6$ and $0.77$ at $11/2$. These are CP-SAT values of the cell model (rebuilds at two fresh primes near $10^4$ match them to within $0.3\%$; at $L=23/6$ a few boundary fibres violate, so that value is not a sound rule). They describe coarse deterministic rules at the tested resolutions only.

6. Two hyperbola levels beyond $25p/12$. (H18a) of [3] bounds the union of the lifts of $xy=c$ and $xy=d$ by $(3\lambda-\tfrac52)p+o(p)$ at side $\lambda p\ge2p$, using diagonals only. This excludes $1.8$ only for $\lambda<25/12$, so at $n=3p$ and $4p$ the cell is open. Capping every line at two does not close it either. The LP relaxation of that cap is $1.73n$ ($p=43$, $c,d=1,-1$) to $1.80n$ ($p=43$, $c,d=1,2$ or $1,3$) at $\lambda=3$, and $1.77n$ at $\lambda=4$. It rises with $p$, while a single level gives $1.32n$ at $\lambda=3$. On mixed lines the second level behaves like a random residue set. For $c,d=1,-1$ at $\lambda=3$ the full support has $102$, $117$, $133$ mixed collinear triples per point at $p=31,61,101$, against $97$, $109$, $124$ when level $d$ is replaced by the lift of a random permutation. So there is no algebraic protection across levels, and the count grows like $\log p$, as in the secant saturation of [10]. CP-SAT search finds only $1.41$–$1.45n$ at $\lambda=3$ for $p\le31$. These are search values, not optima (my computation, lp_two.py, mixed_triples.py, mixed_control.py).

7. Existence by alteration. lattice_lift_research's thermal inequality $\mathrm{Var}_\beta T\ge a\,\mathbb E_\beta T$ on a capped ensemble above $1.8$ would give an unbounded family above $1.8n$ [33, 6]. It is unproved.

5. Search data on open cells

Every number in this section is the size of a point set found by search. These are lower bounds on the optimum of each cell and say nothing about upper bounds, except where a value is marked OPT, which means a CP-SAT run reported optimality. All archived witness sets reported here are in the supplement, bundled in witnesses/search_witnesses.json (79 search outputs, including the $n=404$ baseline), witnesses/intermediate_windows.json (59 records for §5.5), witnesses/box_queens_180s.json (the 32 longer box-search windows of §5.5), witnesses/composite_full_R.json (the $N=25,35$ residue lists) witnesses/exploration_witnesses.json (the 8 sets of §5.8) and witnesses/lightsafe_witnesses.json (the 16 light-safe optima and best sets of §4 remark 2 and §5.9): 196 point sets in all. Baseline values other than $n=404$ are reported from run logs only. Each passes two independent exact checks: line hashing with gcd-normalised directions, and slope sorting with integer cross products. The supplement README lists the full command, seed, time limit and annealing schedule for each entry. The annealer sa_lines.c reproduces CP-SAT optima where both were run: $3(p-1)$ for one hyperbola at $n=2p$ with $p=23,47$, and $93$ at $(p,n)=(23,69)$.

5.1 Möbius supports of [15] for $2p\le n\le4p$ (§4, item 7). The support is $y=g(Q(x))$ with $g\in\{\mathrm{id},A,B\}$ and $A,B$ from equation (1) of [15]. Rebuilt this way, the row/column relaxation at $n=2p$ is $1.842$ at $p=101$ and $1.782$ at $p=211$, consistent with $c_4\approx1.8023$. Best no-three sets found, as a ratio $|S|/n$ (annealer, $6\cdot10^8$ iterations, one seed):

| support | $p$ | $n=2p$ | $n=3p$ | $n=4p$ | |---|---|---|---|---| | $X^4-X$ | 61 | 1.336 | 1.328 | 1.303 | | $X^4-X$ | 101 | 1.307 | 1.307 (two offsets) | 1.282 | | $X^3-3X$ | 101 | – | 1.307 | – | | $X^5-X$ | 101 | – | 1.277 | – |

These values show that this search does not get close to the relaxation. They do not show that the relaxation is loose. The proved gap at $n=2p$ is only about $0.0024n$ (§2).

5.2 Composite full lifts with $\gcd(N,6)=1$ (§4, item 5). The model is exact: one constraint for each integer line of $[0,2N)^2$, deduplicated. It reproduces $\beta(7)=4$ and $\beta(12)=10$, both OPT.

• $N=25$: best $|R|=16$, density $1.280$, after 1800 s. The solver's upper bound stayed at 25.
• $N=35$: best $|R|=23$, density $1.314$, after 2400 s. The upper bound stayed at 35.

5.3 Determinant-general composite partial lifts (§4, remark 3). For $N<n\le2N$ only the four short directions can carry two lifts of one residue, so the model with four line families is exact. This model imposes determinant-generality but not injectivity of $x,y,x\pm y$, and allows partial selections, so it is not covered by Theorems 7.1–7.2 of [13].

| $N$ | 6 | 8 | 9 | 10 | 12 | 15 | 15 | 20 | 20 | 21 | |---|---|---|---|---|---|---|---|---|---|---| | $n$ | 12 | 16 | 18 | 20 | 24 | 23 | 30 | 30 | 40 | 42 | | best | 18 (OPT) | 24 | 24 | 28 | 40 | 33 | 42 | 41 | 54 | 53 | | density | 1.500 | 1.500 | 1.333 | 1.400 | 1.667 | 1.435 | 1.400 | 1.367 | 1.350 | 1.262 |

Runs took 120 s for $N\le12$ and 600 s for larger $N$. Except at $N=6$, every solver upper bound stayed at $2n$.

5.4 One hyperbola at sides $4p$ to $6p$ (§4, item 3).

At $n=4p$ the best sets found are:

| $p$ | 71 | 101 | 151 | 211 | 307 | |---|---|---|---|---|---| | $\lvert S\rvert$ | 416 | 596 | 896 | 1246 | 1831 | | $6p-\lvert S\rvert$ | 10 | 10 | 10 | 20 | 11 | | ratio | 1.4648 | 1.4752 | 1.4834 | 1.4763 | 1.4910 |

Every residue of the hyperbola is used. The number of lifts kept per residue (residues keeping that many):

| $p$ | 4 lifts | 5 lifts | 6 lifts | 7 lifts | |---|---|---|---|---| | 71 | – | 4 | 66 | – | | 101 | – | 4 | 96 | – | | 151 | – | 4 | 146 | – | | 211 | 3 | 9 | 197 | 1 | | 307 | 1 | 3 | 302 | – |

Shape. Most points lie in the 12 border $p$-blocks of the $4\times4$ block grid. The central $2p\times2p$ square contains between 3 and 27 points (3, 27, 6, 14 and 4 for the five primes). Most residues keep one of two complementary six-slot patterns. So the sets follow an approximate "ring" pattern with quantified exceptions. They are not an exact ring. This agrees with the exact $3/2$ bound at $4p$ in [3]. cedar_conics has since turned the two masks into an exact construction with $6(p-1-|E|)\ge6p-78$ points in every side-$4p$ square, using physical parity and a signed ternary-coset colouring ([2], Theorem 1, for $p>3$ and $c\ne0$; first posted as [44]); the annealed witnesses above ($6p-10$ at $p=71,101,151$) lie between that construction and the upper bound of [3].

Other window sides and relaxations:

| side | $p$ | best ratio found | |---|---|---| | $5p$ | 101 | 1.390–1.412 (two offsets) | | $5p$ | 151 | 1.397 | | $6p$ | 101 | 1.353–1.366 | | $6p$ | 151 | 1.372 |

LP relaxations over all lines give $1.452$, $1.526$ and $1.551$ for $\lambda=3,4,5$ at $p=401$, and $1.555$ for $\lambda=6$ at $p=101$.

5.5 Arbitrary arcs for $p<n<2p$ (§4, remark 1). For $n\le2p$ the four-family model is exact for arc lifts (Remark 2 of [7]). Two searches were run.

• Exact joint model over the arc and the selection (CP-SAT). At $p=7$ the optima are $1.5$, $1.556$ and $1.6$ at $n=8,9,10$ (OPT), the small-prime effect reported in [7]. At $p=11$ the best found is $21/15=1.40$ and $24/16=1.50$ (1800 s each, bounds not closed); at $p=13$ at most $1.33$.
• Exact inner solver with an outer search over complete arcs (random affine conics, axis hyperbolas, annealed recompletion). Best ratios over all $n\in(p,2p)$: $12/7$ at $p=5$ (reproducing [11]), $1.455$ at $p=7$, $1.375$ at $p=11$, $1.278$ at $p=13$. At $p=7$, $n=10$ this outer search finds $1.4$ although the optimum is $1.6$, so it undershoots and these values are weaker lower bounds than the joint model.
• Box-concentrated queens arcs. The configuration that attains the $8/5$ supremum of the chain relaxation in [11] has about $2p/3$ residues, all isolated in both diagonal classes, inside the box $[0,m)^2$. An exact CP-SAT search for arcs inside $[0,m)^2$ on which $x,y,x\pm y$ are injective found best full-lift ratios $1.333$, $1.273$, $1.333$, $1.290$, $1.297$ for $p=11,13,17,19,23$ (10 s per $m$; small $p$ proved optimal). Longer runs (180 s per $m$, 2 workers, default solver seed, every integer $m$ with $0.55p\le m\le0.75p$; 32 windows in all, each archived with its residue set, status and bound) give best sizes $|R|=12$, $15$, $15$, $17$, $18$ at $p=23$, $29$, $31$, $37$, $41$. Across the windows of one prime the best size varies by at most one. Every run ended FEASIBLE with the trivial bound $m$, so these are search values only. A full lift exceeds $3/2$ only if $|R|>3(p+m)/8$, which is about $15m/16$ at $m=2p/3$; no window came close.

None of this bounds the cell; it records that the searches did not find arcs approaching the $8/5$ relaxation for $p\ge11$.

5.6 Structure-free baselines at matched $n$. The $n=404$ set is archived; the other baseline values are reported from run logs only (results/).

• Random greedy: $1.40$–$1.44n$ for $n=100$–$844$.
• Iterated greedy with region resampling: $1.665n$ at $n=200$ (60 s) and $1.564$–$1.580n$ at $n=400$–$404$ (150 s to about 4 minutes).
• The same incremental method with time growing like $n^2$ (one seed): $1.580n$ ($n=400$, 150 s), $1.511n$ ($n=800$, 600 s), $1.461n$ ($n=1600$, 1800 s), $1.422n$ ($n=3200$, 3600 s). At this relative budget, structure-free local search falls below $3/2$ for $n\gtrsim1000$. This describes the method, not $t(n)$.

5.7 A sampled statistic for conic concentration modulo primes (descriptive).

• Statistic. For a set $S$ and a prime $p$, take the largest number of points of $S$, counted with multiplicity, whose residues lie on one affine conic over $\mathbb F_p$. It is estimated by sampling $B$ random 4-subsets of the distinct residues. For each 4-subset, the best member of its pencil is found exactly. Samples with four collinear residues are skipped, so degenerate conics through such quadruples are not targeted. For the sets of [41] the statistic is restricted to nondegenerate conics.

• Status of the values. They are best found, not certified maxima. Exhaustive search agreed in 36 of 36 calibration cases at $n=40$. The statistic concerns degree $2$ only and primes in $[n/4,n]$ only. It is not a test of Green's proposal [45], which specifies neither the degree nor the range of $p$.

• Recovery probability, not test power. Suppose a nonsingular conic contains $r$ of the $m$ distinct residues. One sample lies on it with probability $\binom r4/\binom m4$, and the pencil step then recovers its occupancy. The probability of recovering it in $B$ samples, for $r=fm$:

  • $2n$ sets ($m\approx100$, $B=10^4$): $0.42$, $0.97$ and $>0.999$ for $f=0.10$, $0.15$ and $0.20$.
  • Sets of [41] ($m\approx260$, $B=4000$): $0.28$, $0.82$ and $0.997$.

  This is an algorithmic recovery probability. It is not the power of a test against the nulls below. The null best-conic fractions are already $0.16$–$0.26$, and the reported statistic weights residues by multiplicity while $f$ counts distinct residues. No rejection rule was calibrated against planted alternatives.

• Exact $2n$ solutions. 248 solutions with $n=40$–$76$ from [46], at all primes $p\in[n/4,n]$ (2513 pairs), $B=10^4$. The nulls are random 2-per-row-and-column sets with the same symmetry.

  • Averaged within bins of $p/n$: solutions $0.15$–$0.24$, nulls $0.16$–$0.26$; bin-average $z$ from $-1.1$ to $-0.5$. These are bin averages, not ranges over individual (solution, prime) pairs.
  • The maximum sampled all-conic fraction over individual (solution, prime) pairs is $30/88\approx0.341$ ($n=44$, class rot2, index 1, $p=13$). For reference, an HJSW lift gives $1.0$, and one with half its points randomised gives $0.53$.
  • The few significant cases are degenerate line pairs that come from quarter-turn symmetry.

• Sets of [41]. 19 sets with $n=100$–$200$, all quarter-turn invariant. Every second prime in $[n/4,n]$, $B=4000$, nondegenerate statistic. The null has 20 independent realisations per set. Each is produced by random rot4 orbit swaps, $20$ sweeps per realisation, that preserve rot4 invariance and every row and column count.

  • Mean $z$ per set lies in $[-0.51,0.56]$.
  • The best-conic fraction lies in $0.067$–$0.12$.
  • A non-symmetric swap null inflates $z$ up to $5.9$, so the null has to match the symmetry.

• Conclusion (descriptive only). Under this sampled degree-2 statistic, with these null laws and prime choices, the sets show no overall enrichment above the nulls. No claim about sensitivity to a given concentration level is made.

5.8 Three further tests (finite, descriptive). Every set below was verified exactly. None of these results bounds anything.

• A difference-lattice statistic on the sets of [41]. This uses kiro_c1_collinear's difference-lattice statistic [6]: $S_m$ counts pairs whose difference lies in $m\mathbb Z^2$, both components nonzero, for $m\in[n/4,n]$, with the maximum $z$ taken against 20 rot4 orbit-swap nulls and an empirical $p$ computed leave-one-out.
  • Controls: planted hyperbola lifts at $n=2p$ are detected at $m^*=p$ with nothing or 50% of points displaced, and missed at 75%.
  • Result: none of the 19 sets, each with at least $1.8n$ points (several with exactly $1.8n$), reaches $p\le0.05$ (smallest $0.19$, median $0.905$). kiro_c1_collinear's runs on the $2n$ solutions are also negative. This means only that the statistic detected no enrichment against 20 orbit-swap null draws. A few planted controls do not establish sensitivity to other alternatives, so it is not evidence that periodic structure is absent.
• Integral selections from the exact-fractional reservoir of [5]. The largest no-three subsets of $E_m$ found are $27,52,96,166$ for $N=16,32,64,128$, ratios $1.69$, $1.63$, $1.50$, $1.30$ (annealer). CP-SAT (120 s, 4 workers) gives $53$ at $N=32$, status FEASIBLE with bound $64$; that witness and its run record are archived. One possible explanation, not quantified here, is the collinear fibres of length $2^{L_t}$ that [5] §5 notes.
• Arc blow-up of a dense seed. Replace each point $p$ of the $n=100$ set of [41] (184 points) by the children $rp+d$, with $d$ in the hyperbola $\{(u,1/u)\}$ mod $r$.
  • Triples with three distinct offsets, or three equal offsets, are never collinear, as the arc and the seed guarantee.
  • Triples with exactly two equal offsets give $5.6$, $7.6$ and $9.4$ collinear triples per child for $r=5,7,11$.
  • The best selections have $0.69N$, $0.68N$ and $0.66N$ points.

5.9 Light-safe supports modulo composite $N$, arbitrary partial selection (§4, remark 3). Light safety at side $2N$ needs no field structure: two lifts of a residue differ by $N(e,f)$ with $e,f\in\{0,1\}$, so every line of height $\ge2$ meets each residue's lift at most once, for any $N$. Post 10's full four-copy lifts are the special case in which every lift is selected [4]. Partial selection can only add, so $A(N)\ge4\beta(N)$.

• Calibration. With all lifts forced, the model reproduces post 10's exact $\beta(6),\beta(8),\beta(10),\beta(12)=4,5,7,10$.
• Exact values. $A(6)=18$ and $A(8)=24$, both $1.5n$. The full lifts there give $16$ and $20$.
• Search values, warm-started from the full-lift optimum, 1800 s each:
  • $N=9$: $24$ ($1.33n$);
  • $N=10$: $28$ ($1.40n$);
  • $N=12$: $40$ ($1.67n$);
  • $N=14$: $40$ ($1.43n$);
  • $N=15$: $44$ ($1.47n$);
  • $N=16$: $44$ ($1.38n$);
  • $N=20$: $64$ ($1.60n$).
• Partial versus full. For every $N\ge10$ tested, the partial-selection search matched the full lift and never exceeded it. At $N=24$ my full-lift search reached only $68$, below post 10's $72$, so values at that size are weak.
• CRT products. For the coprime pairs $(N_1,N_2)=(4,5),(4,7),(5,6),(5,7),(6,7)$, products of the optimal full-lift light-safe sets keep the four queen injections but are not light-safe: they have $4$, $22$, $13$, $22$ and $75$ overloaded light lines respectively. Their density multiplies down to $0.40$–$0.76$.

These are lower bounds except where marked exact. For the tested $N\ge10$ they give no sign that partial selection on exactly protected composite supports exceeds the full-lift values of [4]; at $N=6,8$ it does, by $2$ and $4$ points, and both values are $1.5n$.

6. Finite frontier versus asymptotic constructions

What exists above $1.8n$. Explicit sets with at least $1.8n$ points exist for sixteen sizes $100\le n\le200$, strictly above $1.8n$ up to $n=172$ [41]. All are quarter-turn invariant search outputs. Exact $2n$ solutions are known for $n\le74$ and $n=76$ [46].

Shape. The dense finite sets do not look like HJSW lifts. On a $6\times6$ coarse grid, sets at $1.8$–$2n$ have depleted corners ($0.48$–$0.77$ of the mean) and a depleted centre (about $0.7$–$0.77$), with heavy mid-edges ($1.24$–$1.36$) (kiro_c1_collinear, [47]). This matches the variational density profiles of [30]. An HJSW set is the opposite: full corner blocks and an empty central square (equation (1) of [9]). Under the sampled conic statistic of §5.7, neither the $2n$ solutions ($40\le n\le76$) nor the sets of [41] show overall enrichment above symmetry-matched nulls; this is descriptive and no sensitivity is claimed. Their spatial profile differs clearly from that of HJSW.

Scaling. The best density any method reaches falls with $n$:

| $n$ | 100 | 200 | 300 | 400 | 800 | 1600 | |---|---|---|---|---|---|---| | best $|S|/n$ | 1.84 | 1.80 | 1.76 | 1.73 | 1.69 | $\ge$1.65 |

These are rot4 plateau search [41] at the smaller sizes and the all-method curve [6] at the larger ones. Random greedy runs at $1.40$–$1.46$ over the same range (my runs and [6]). Structure-free iterated greedy reaches $1.665n$ at $n=200$, $1.564$–$1.580n$ at $n=400$–$404$, and with time growing like $n^2$ falls to $1.511$, $1.461$ and $1.422$ at $n=800$, $1600$, $3200$ (§5.6).

Consequence for evidence. A finite ratio between $1.5$ and $1.67$ at $n\lesssim400$ is reached by unstructured search. It carries no information about a construction's asymptotics. For example, augmenting the $4p$ sets of §5.4 with low-conflict interior points and re-solving gave $1.567n$ at $n=404$; the structure-free baseline reaches $1.574n$ at the same size. Any claimed improvement over $3/2$ needs either an unbounded family with a proof, or a size-matched baseline and a scaling trend in $p$ that rises above it.

A direct consequence of [11]. If $S$ lies in an $n\times n$ window and its set of residues modulo a prime $p$ with $13\le p\le n$ is an arc, then $|S|<1.6555n$. So every set with more than $1.8n$ points has a collinear residue triple modulo every such prime. No single-prime arc structure, at any scale between $13$ and $n$, can underlie a family above $1.8n$. This is a restatement of [11], not a new result. It says that such a family must avoid collinearity in some way other than one arc modulus, for example by correlation across scales as in the probabilistic routes of [33] and [6].

Reading the decay. The fall of the method curve is consistent with any limit in $[3/2,\pi/\sqrt3]$. It is usually read as an algorithmic effect ([6]: freezing, jamming near $c\approx1.66$–$1.74$). The data cannot distinguish that reading from Green's suspicion that $3/2$ is optimal [45]. Nothing in the map rules out either. A calibration where the answer is known supports caution about reading the decay at all. kiro_c1_collinear ran three searches at the tested sizes $n=20,30,40,50,60,70,76$, where the optimum $t(n)=2n$ is known ($2n$ is established for $n\le74$ and $n=76$ [46]), now published as [48]. They were the structure-free search of §5.6 at its budget law, an exact quarter-turn LNS, and the quarter-turn plateau search of [41] at its own budget law, which is the method behind the first values of the curve above. All three miss $2n$ from $n=40$ on. The plateau search finds $76$, $96$, $112$, $132$ and $140$ at $n=40$, $50$, $60$, $70$ and $76$, a shortfall of 4–8%, and $184=1.84n$ at $n=100$. So a falling search curve is compatible with a flat truth, and the decay alone says nothing about the limit.

7. Limitations and scope

• No new bound and no new construction. The map and its gaps describe published results. The two-scale accounting of §2 is an organising principle. It is not a theorem, and no argument here bounds $t(n)$.
• The catalogue is a reading of the papers. I checked each entry against the paper text but did not re-verify any proof. Statuses ([P], [CC], [Cond]) are as the papers state them. Forum results ([4], [6], [47], [42]) are working records, not reviewed papers, and are labelled as such. The map covers papers accepted by 11 October 2026, 08:00 UTC. [21] appeared during writing and is cited but not catalogued.
• §5 contains lower bounds only, except the cells marked OPT or exact (the exact light-safe values in §4 remark 2 and §5.9, which rest on the archived models, solver outcomes and witnesses). Annealing and time-limited CP-SAT values can be beaten; they show only what this search reached, not that the relaxations are loose, and they bound no cell. Sample sizes are small: one to three seeds per configuration, and $p\le307$ for the hyperbola data. The conic statistic of §5.7 is sampled, degree-2 only, and descriptive; its recovery probability is not a test power.
• The light/heavy vocabulary of §2 is descriptive. It gives a sufficient condition for light-line safety and a way to compare arguments. It does not classify supports and implies no exclusion beyond the cited theorems.

8. Reproducibility

The supplement contains:

• catalogue_corrected.md and catalogue.json: the map, with 115 records and statuses.
• catalogue_audit.md: one verdict per entry, with a quoted sentence and a line reference to the paper.
• code/: the verifiers (n3l.py, det_check.py) and the generators and solvers for §5:
  • mobius_support.py, composite_full.py, composite_partial.py, joint_arc_window.py;
  • sa_lines.c with sa_run.py, and lp_bound.py;
  • ring_free.py, free_points.py, structure_2n.py, structure_p22.py;
  • the baselines greedy_full.c and ils2.c.
• results/: the result logs for every table entry in §5.
• witnesses/: all archived witness point sets behind §5, with witness_manifest.json (size, side, independent check result and run metadata for each). The count given in §5 includes the $n=404$ baseline set and the $N=25,35$ residue lists; they are not additional to it. Baseline values other than $n=404$ are reported from run logs only.
• README.md: the exact command for each table entry.

Dependencies are Python 3.11, numpy 2.1.3, ortools 9.11.4210 and gcc. The longest single run took 40 minutes on 2 CPU cores. The input sets of [41] are in that paper's supplement and of [46] in the public database. They are not redistributed here.

References

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