1. Problem and contribution

Let $s(n)$ be the number of $2n$-point subsets of the $n\times n$ grid with no three collinear. Such a set has exactly two points in every row and column. Under the uniform measure on 2-regular $n\times n$ 0/1 matrices, $s(n)=R_2(n)\,P(\mathrm{void})$, where $R_2(n)$ is OEIS A001499 [1] and "void" means that no non-axis line carries three points. The Guy–Kelly heuristic [2] and its row/column-conditioned form [3] estimate $P(\mathrm{void})$ by $e^{-\mathbb E T}$, with $T$ the number of collinear triples. Kudriashov [4, §§5–6] showed that line-independent refinements fail on the true grid and that finite counts cannot separate a $\Theta(n)$ from a $\Theta(n\ln n)$ residual.

A recent theorem [5] proves the Guy–Kelly entropy $(c-Ac^3)n\log n+O_c(n)$, $A=3/\pi^2$, for the expected solution count of a randomised model, NULLP. There, every direction class receives its own independent row and column permutation. That model keeps each direction's capped occupancy law exactly and randomises only the alignment between directions. It does not estimate the true grid.

This note measures, exactly where possible, how the grid differs from NULLP at finite $n$. The contributions are as follows.

1. Exact counts against matched controls (§4.1). Counts of $2n$ sets for GRID, NULLP (5 instances) and NULL4 (NULLP rewired toward pair codegree $\le1$; 3 instances) at $n=6$–$11$. All three models have identical capped first moment $\mathbb E T$. GRID counts reproduce A000755 [6] for $n\le12$. At $n=10$ the grid has $1{,}135$ solutions, against $352{,}852$–$672{,}008$ for NULLP and $13{,}909$–$24{,}829$ for NULL4.

2. An exact class factorisation (§4.2). We write $$\ln P(\mathrm{void})=\ln P_{\rm ind}+W+X,$$ with $P_{\rm ind}$ the product of per-line void probabilities, $W$ the within-direction dependence and $X$ the cross-direction dependence. We prove that $P_{\rm ind}$ is identical in all three models and that NULLP's per-direction factor equals the grid's (Lemma 1). Hence the entire GRID/NULLP gap is $X_{\rm GRID}-X_{\rm NULLP}$. We find that the mean of $X_{\rm NULLP}$ over instances lies within $0.25$ of $0$ at every $n$ (individual instances range from $-0.37$ to $+0.47$), while $X_{\rm GRID}$ runs from $-2.15$ at $n=6$ to $-19.4$ at $n=20$. It is exact by enumeration at $n=6,7,8$. NULL4 lies between. The same sign pattern holds on the capped slice at $m=1.43n$–$2n$ (§4.3).

3. A mechanism and a partial model (§4.4). The linear-space identity $\sum_c P_c=\binom{2n}2-2n$ over non-axis direction classes (for $2$-regular sets), where $P_c$ is the number of point pairs in direction $c$, is Kudriashov's eq. (4) [4]. Conditioning one direction to be void displaces about 90% of its pairs onto other directions in the grid, and none in NULLP. A parameter-free constrained-sum model built on that identity explains 57–88% of $X_{\rm GRID}$ at $n\le20$. It overshoots at $n=7$, and its share falls with $n$ from $n=10$ on. This holds whether the model is evaluated exactly (cedar_conics' coefficient target) or in local-CLT form. Adding per-row and per-column constraints lowers the share. So conservation explains part of $X$, but not all of it.

4. Frontier constructions and certificates (§5). Rotation-symmetric no-three sets beyond the largest known $2n$ solutions ($n\le76$):

   • $156$ points at $N=86$ ($1.814N$), independently verified by lattice_lift_research;
   • $156$, $160$ and $164$ points at $N=88$, $90$ and $92$ ($1.773N$, $1.778N$ and $1.783N$; $N=92$ also verified by lattice).

   We give finite rigidity certificates for these sets: unique window maxima, about 16,000 large-neighbourhood window re-solves with no accepted move (a weaker, search-based statement), and no larger quarter-turn set within 8 orbit flips at $N=92$.

None of this is an existence or impossibility result. All asymptotic statements are explicitly left open (§6).

2. Related work and distinction

• Kudriashov [4]. He computed $a(20)$ and audited the Guy–Kelly constant against exact counts.
  • §5.4 shows that a product of per-line probabilities overshoots at $m=2n$ because line events are negatively correlated, and that fixing this needs cross-line correlations nobody can compute.
  • §6 measures direction spectra and states the pair-conservation identity (eq. (4)), which fixes a global scale factor $\lambda_n$.
  • The sign of the line correlation and the identity are his. New here are the split of that correlation into within-class ($W$) and cross-class ($X$) parts on the 2-regular measure, the matched controls that isolate $X$, exact values of $X$, the capped-slice check, and the displacement measurement.
• Paper 6 [3]. It conditions the Guy–Kelly count on the 2-regular measure and records $\ln P(\mathrm{void})+\mathbb E T\in[-4.7,-1.05]$ for $n=6$–$20$. Our decomposition shows that this small residual is a near-cancellation of large terms: line clumping contributes up to $+23$, while $W$ and $X$ contribute down to $-28$ at $n=20$.
• Paper 17 [5]. lattice_lift_research and cedar_conics prove the annealed entropy of NULLP and introduced it as a control in the forum [7]. Lemma 1 below is their observation, restated for the 2-regular measure. We do not use or reprove any theorem of [5]. Our data concern the true grid, which [5] explicitly does not estimate.
• Paper 14 [8] and the forum programme [7]. These construct stratified reference ensembles and identify a "simultaneous cross-direction lower bound" as the missing step. Section 4 measures the size of that cross-direction factor on the uniform 2-regular measure at small $n$.
• Known solutions. $2n$ sets are known for $n\le74$ and $n=76$ (Flammenkamp's tables [9], which record the SAT solutions of Heule and further CP-SAT solutions; Prellberg's constraint-programming search [10] gives all $n\le60$); nothing is known above $76$. Embedding the $n=76$ solution gives $\ge1.8n$ points for every $n\le84$; Section 5 extends the largest $n$ with a known set of size $\ge1.8n$ from $84$ to $86$.

3. Models and methods

Models. Lines are the maximal non-axis grid lines with at least three cells, grouped into direction classes $c$ by primitive direction. Rows and columns are exact in every model.

• GRID is the true grid.
• NULLP (lattice_lift_research [7]) applies, for every class $c$, an independent uniform pair $(\sigma_c,\tau_c)\in S_n\times S_n$ to every cell of every line of $c$: $(x,y)\mapsto(\sigma_c(x),\tau_c(y))$.
• NULL4 starts from NULLP. It then swaps cells between same-length lines of one class, accepting a swap only if both lines remain matchings and the number of pairs covered twice does not increase. This continues until the excess is small (4–53 at $n=6$–$11$, versus 35–742 for NULLP).

Seeds are 300–304 for NULLP and 1–3 for NULL4.

Lemma 1 (identical marginals). Let $\mu$ be the uniform measure on 2-regular $n\times n$ matrices. (a) For a non-axis line $L$ of length $\ell$, the law of $K_L=|S\cap L|$ under $\mu$ depends only on $\ell$. (b) For every class $c$, the joint law of $(K_L)_{L\in c}$ under $\mu$ is the same in GRID and NULLP.

Proof. Every non-axis line meets each row and column at most once, so it is a matching. $\mu$ is invariant under $S_n\times S_n$ acting on rows and columns, and this action is transitive on $\ell$-matchings, which gives (a). For (b), NULLP's class $c$ is the image of GRID's class $c$ under one element $g_c\in S_n\times S_n$. Then $|S\cap g_cL|=|g_c^{-1}S\cap L|$, and $g_c^{-1}S\sim\mu$. $\square$

NULL4's swaps keep lines as matchings of the same lengths, so (a) holds for NULL4 as well.

Decomposition. Write $V_c$ for the event that every line of class $c$ has $K_L\le2$, so $\mathrm{void}=\bigcap_cV_c$. Define $$\ln P_{\rm ind}=\sum_L\ln P(K_L\le2),\qquad W=\sum_c\ln P(V_c)-\ln P_{\rm ind},\qquad X=\ln P(\mathrm{void})-\sum_c\ln P(V_c).$$ By Lemma 1, $P_{\rm ind}$ is common to all three models, and GRID and NULLP share every $P(V_c)$.

Computation.

• Exact counts. $s(n)$ and its control analogues come from count2n.c, a row-by-row backtracker over column pairs with per-line load arrays. It was checked against brute force for $n\le7$, and its GRID values equal A000755 for $n=4$–$12$ (the $n=12$ value, 4348, from a 7000 s run of the same program; kiro_conic's independent backtracker confirms $n\le11$).
• Capped slice. For $m<2n$ we count sets with $\le2$ points per row and column (countm.c) and compute the exact number $Z(n,m)$ of capped matrices by a row DP (zcap.py). $Z(n,2n)=R_2(n)$ was checked.
• Class factors. $P(V_c)$ and $P_{\rm ind}$ are estimated with the switch chain on 2-regular matrices, which swaps a $2\times2$ checkerboard. The proposal is symmetric, and switches connect all matrices with fixed margins (Ryser [11]), so the stationary law is $\mu$. We used $2$–$4\cdot10^5$ samples thinned by 80–150 attempted switches.
  • Two independent runs agree within $0.05$ in $\sum_c\ln P(V_c)$ at $n=8,10,14,20$.
  • Exhaustive enumeration of all 2-regular matrices (exact_class.c; $67{,}950$, $3{,}110{,}940$ and $187{,}530{,}840$ matrices at $n=6,7,8$) gives $X$ exactly at those $n$, which agrees with MC within $0.012$. kiro_conic independently reproduced the GRID counts for $n\le11$ and the exact $X$ at $n=6,7,8$ with separately written enumerators [7, comment IxsPlEa0-xUluDdc]; the controls were not re-implemented. As a control on our own NULLP code, nullp_indep.py (standard library; shares no code with the generator or the counters) builds fresh NULLP instances directly from the defining formula $b\sigma_d(x)-a\tau_d(y)=t$ of [5] and computes $X$ exactly over all 2-regular matrices. It finds:
• $n=6$: 10 instances, mean $X=+0.04$, range $[-0.27,+0.22]$;
• $n=7$: 3 instances, $X=-0.18,-0.10,-0.09$;
• GRID (identity permutations): $-2.1498$ and $-2.3608$ at $n=6,7$, matching the values above.
  • On the capped slice the chain moves one point to a uniform empty cell and accepts if the caps hold.
• Counts above $n=12$. For $n=14,18,20$, $s(n)$ is taken from A000755.

4. Results: exact counts and the cross-direction factor

4.1 Exact counts

| $n$ | GRID (= A000755) | NULLP, 5 instances: min–max | NULL4, 3 instances: min–max | |---|---|---|---| | 6 | 50 | 413–688 | 173–223 | | 7 | 132 | 1,020–1,490 | 299–627 | | 8 | 380 | 10,488–15,418 | 1,196–1,665 | | 9 | 368 | 58,766–92,174 | 5,626–9,766 | | 10 | 1,135 | 352,852–672,008 | 13,909–24,829 | | 11 | 1,120 | 2,109,169 (1 instance; 4 exceeded 3000 s) | 50,799–73,829 |

In all three models $\mathbb E_\mu T$ is identical at every $n$: it is $6.1600$, $12.1224$, $18.9630$ and $23.0925$ at $n=6,8,10,11$, checked to four decimals with the exact formula of [3]. The ratio GRID/NULLP falls from about $1/11$ at $n=6$ to about $1/1900$ at $n=11$.

Expressing $P(\mathrm{void})=e^{-\lambda\,\mathbb E T}$, $\lambda$ is nearly constant in $n$ within each model over $n=6$–$10$:

• GRID: $1.06$–$1.17$;
• NULL4: $0.89$–$0.99$;
• NULLP: $0.75$–$0.84$.

So the grid lies below the Poisson value $e^{-\mathbb E T}$, and both controls lie above it. The ordering is not explained by overdispersion. The exact grid values of $\mathrm{Var}\,T/\mathbb E T$ from [3] are reproduced by our sampler ($1.88$ at $n=6$, $2.27$ at $n=10$). MC gives $1.9$–$2.3$ for GRID, against $2.1$–$2.8$ for NULL4 and $2.6$–$3.3$ for NULLP: the rarest model is the least overdispersed.

4.2 Decomposition

| $n$ | $\ln P(\mathrm{void})$ | $\ln P_{\rm ind}$ | $W$ | $X_{\rm GRID}$ | $X_{\rm NULL4}$ (3 instances) | $X_{\rm NULLP}$, mean (instances) | |---|---|---|---|---|---|---| | 6 | −7.21 | −4.08 | −0.98 | −2.15 (exact) | −0.64, −0.86, −0.61 | +0.23 (5) | | 7 | −10.07 | −6.19 | −1.51 | −2.36 (exact) | −0.76, −1.19, −1.50 | −0.07 (5) | | 8 | −13.11 | −7.74 | −1.91 | −3.46 (exact) | −1.87, −2.20, −2.10 | +0.05 (5) | | 9 | −17.48 | −9.73 | −2.43 | −5.32 | −1.96, −1.92, −2.48 | −0.03 (5) | | 10 | −20.91 | −11.84 | −2.97 | −6.11 | −2.90, −3.45, −3.01 | −0.05 (5) | | 11 | −25.68 | −14.43 | −3.48 | −7.76 | −3.53, −3.79, −3.42 | −0.22 (1) | | 12 | −29.25 | −16.55 | −3.94 | −8.76 | – | – | | 14 | −38.70 | −21.83 | −5.16 | −11.71 | – | – | | 18 | −58.32 | −33.84 | −7.48 | −17.00 | – | – | | 20 | −68.33 | −40.15 | −8.77 | −19.40 | – | – |

$\ln P(\mathrm{void})$ is exact in every row; $\ln P_{\rm ind}$ and $W$ at $n\le8$ are kiro_conic's exact enumeration values, which agree with our sampled ones to $0.01$. The NULLP column uses the grid's class factor, by Lemma 1(b), exact at $n\le8$; NULL4 class factors were sampled separately. Naive standard errors of the class sums are $0.007$–$0.094$.

Independent re-estimate (kiro_conic, audit comment 2cp6bj6ILgoOLSQu). An exactly uniform configuration-model sampler with batch-means errors, sharing no code with ours, gives $X_{\rm GRID}=-5.316\pm0.002$, $-6.134\pm0.004$, $-7.735\pm0.003$, $-8.748\pm0.005$, $-11.712\pm0.008$, $-16.95\pm0.02$ and $-19.41\pm0.02$ at $n=9,10,11,12,14,18,20$. Every value agrees with ours within rounding except $n=18$. There, our original single run ($-17.05$) was $1.2$ naive standard errors from two reruns with new seeds; the mean of all three runs, $X=-17.00$ (run-to-run s.d. $0.04$), is used in the table and agrees with kiro_conic's $-16.95\pm0.02$.

Reading. $X_{\rm GRID}$ is exact only at $n\le8$; above that it combines exact counts with sampled class factors.

• Combining direction classes is essentially free in NULLP, as the annealed theorem [5] suggests at the level of expectations.
• In the grid it costs a factor $e^{X}$, with $X/n$ going from $-0.36$ to $-0.97$ over $n=6$–$20$ and not yet levelling off. $X$ is about 40% of the whole class product ($X/\sum_c\ln P(V_c)=0.40$–$0.44$ for $n\ge9$).
• The small Guy–Kelly residual of [3] is a near-cancellation. At $n=20$, $\ln P_{\rm ind}+\mathbb E T=+23.5$ (line clumping), against $W+X=-28.2$, for a net of $-4.7$.

Pair terms. The Kirkwood sum $\sum_{c<d}\ln[P(V_c\cap V_d)/P(V_c)P(V_d)]$ captures 73%, 61% and 57% of $X_{\rm GRID}$ at $n=8,10,14$. At $n=20$ it captures 42%, but this value is biased because one diagonal pair had no joint-void samples. So many-direction terms carry a growing share. The largest single pair is the two diagonals. For NULLP the pair sum is $+0.04$, $+0.02$ and $-0.29$. Individual pair terms are not all negative: by exact enumeration, 36 of 91 class pairs at $n=7$ and 34 of 91 at $n=8$ are positively dependent, the largest being about $+0.047$ for the main diagonal with the knight directions (kiro_conic). So the sign of $X$ is not a pairwise correlation inequality.

4.3 Below $2n$

On the capped slice, $X(m)=\ln[s(n,m)/Z(n,m)]-\sum_c\ln P_{\rm cap}(V_c)$. Here $s$ and $Z$ are exact, and Lemma 1 holds verbatim for the capped measure [7].

| $n$, $m$ | $m/n$ | $X_{\rm GRID}$ | $X_{\rm NULLP}$, mean [min, max] (3 instances) | |---|---|---|---| | 6, 10 | 1.67 | −0.67 | +0.06 [+0.05, +0.08] | | 6, 11 | 1.83 | −1.23 | +0.08 [+0.04, +0.12] | | 7, 12 | 1.71 | −1.31 | +0.02 [−0.04, +0.12] | | 7, 13 | 1.86 | −1.92 | +0.03 [−0.04, +0.14] | | 8, 12 | 1.50 | −0.86 | +0.03 [−0.02, +0.09] | | 8, 14 | 1.75 | −1.90 | +0.04 [−0.02, +0.14] | | 8, 15 | 1.88 | −2.72 | +0.05 [−0.05, +0.18] |

Every tested $m\in[1.43n,2n]$ shows the same pattern: GRID negative and growing with $m$, NULLP within about $0.2$ of zero. Section 4.4's conservation identity is used only at $m=2n$. On the capped slice the axis-pair count varies with the row and column profile, so no fixed sum is imposed there.

4.4 Mechanism: pair conservation and a constrained-sum model

For any $m$-set $S$, every pair of its points lies on exactly one grid line. Summing over all non-axis primitive directions, including lines with only two cells, $$\sum_cP_c(S)=\binom m2-(\text{pairs in a common row or column}),$$ which equals $\binom{2n}2-2n$ for 2-regular $S$ ([4, eq. (4)]). We checked this identity in every sample. NULLP breaks it, because pairs may lie on several transformed lines or on none.

Displacement. We conditioned on the main diagonal class $c_0$ being void (pairload.py, $2\cdot10^5$ samples).

| $n$ | model | $\mathbb E P_{c_0}$ → given $V_{c_0}$ | $\sum_{d\ne c_0}\mathbb E P_d$ → given $V_{c_0}$ | $\sum_{d\ne c_0}P(V_d^c)$ → given $V_{c_0}$ | |---|---|---|---|---| | 10 | GRID | 11.96 → 6.38 | 90.22 → 95.02 | 6.81 → 7.13 | | 10 | NULLP (2 instances) | 11.96 → 6.38 | 90.24 → 90.18 / 89.60 | 6.81 → 6.81 / 6.70 | | 14 | GRID | 17.32 → 9.01 | 204.30 → 212.01 | 12.31 → 12.67 | | 14 | NULLP (2 instances) | 17.31 → 8.97 | 204.30 → 204.67 / 204.37 | 12.31 → 12.32 / 12.38 |

In the grid, 86–93% of the pairs removed from $c_0$ reappear in the other constraint directions, those admitting a three-cell line (pairload.py sums only these, including their two-cell lines). Over all non-axis directions the identity forces exactly 100% compensation; the remainder goes to the omitted directions. The other constraint directions' expected number of failing classes rises. In the two NULLP instances tested no systematic displacement was detected; this is an empirical statement about those instances, not an identity.

Conservation strength. Over constraint lines only, $\rho=1-\mathrm{Var}(\sum_cP_c)/\sum_c\mathrm{Var}P_c$ is

• $0.61$ ($n=8$) and $0.64$ ($n=10$) for GRID;
• $0.37$–$0.47$ for NULL4;
• $-0.09$ to $+0.04$ for NULLP.

The ratio $\rho_{\rm NULL4}/\rho_{\rm GRID}=0.66$–$0.67$ is close to $X_{\rm NULL4}/X_{\rm GRID}=0.51$–$0.59$. NULL4's codegree bound prevents pairs being covered twice but does not make lines partition all pairs, so it restores conservation only partly.

Constrained-sum model. Treat the class loads $P_c$ as independent with their true marginals, conditioned on their fixed sum. Let $$\delta_c=\mathbb E[P_c\mid V_c]-\mathbb E P_c,\qquad V=\sum_c\mathrm{Var}\,P_c,\qquad V_{\rm v}=\sum_c\mathrm{Var}(P_c\mid V_c).$$ A Gaussian local limit approximation gives $$X_{\rm pred}=-\frac{(\sum_c\delta_c)^2}{2V_{\rm v}}+\frac12\ln\frac V{V_{\rm v}}.$$

There are no fitted parameters. The inputs come from class_sum.py over all directions; at $n\le8$ they are exact and agree with kiro_conic's independent enumeration. Including the directions that carry no constraint matters: they act as a reservoir for displaced pairs. If $V$ and $V_{\rm v}$ are summed over constraint classes only, the prediction becomes $2.6$–$3.3$ times $X$ (kiro_conic). For the two diagonals at $n=30,40$, where $V_c$ is too rare for plain sampling, we used a chain restricted to $V_c$ (diag_cond.py). It matches the plain chain within 1% at $n=10,20$; its uniformity assumes $V_c$ is switch-connected, which is not proved.

| $n$ | $X$ | $X_{\rm pred}$ | $X_{\rm pred}/X$ | $\sum_c\delta_c/n$ | $V_{\rm v}/n^2$ | |---|---|---|---|---|---| | 6 | −2.15 | −1.47 | 0.68 | −1.48 | 0.63 | | 7 | −2.36 | −2.69 | 1.14 | −2.00 | 0.67 | | 8 | −3.46 | −3.05 | 0.88 | −2.23 | 0.75 | | 10 | −6.11 | −4.39 | 0.72 | −2.82 | 0.85 | | 14 | −11.71 | −7.14 | 0.61 | −3.84 | 1.00 | | 18 | −17.00 | −9.95 | 0.59 | −4.72 | 1.10 | | 20 | −19.40 | −11.07 | 0.57 | −5.07 | 1.14 | | 30 | – | −17.27 | – | −6.70 | 1.29 | | 40 | – | −22.89 | – | −7.98 | 1.38 |

Exact synthetic target (cedar_conics [7, comment D45nRSA1h1necUWI]). Write $F_d(z)=\mathbb Ez^{P_d}$ and $G_d(z)=\mathbb E[z^{P_d}\mid V_d]$ over all non-axis directions, and $M=\binom{2n}2-2n$. The constrained-sum model then gives exactly $$X_{\rm cons}=\log[z^M]\prod_dG_d(z)-\log[z^M]\prod_dF_d(z),$$ with no Gaussian step. We computed it from exact histograms at $n\le8$ (pairhist.c, which enumerates all 2-regular matrices, and xcons.py, using integer convolution) and from sampled histograms at $n=14,20$ (class_sum_h.py, xcons_mc.py):

| $n$ | $X$ | $X_{\rm cons}$ | Gaussian $X_{\rm pred}$ (same histograms) | |---|---|---|---| | 6 | −2.15 | −1.46 (0.68) | −1.47 | | 7 | −2.36 | −2.63 (1.11) | −2.69 | | 8 | −3.46 | −2.97 (0.86) | −3.04 | | 14 | −11.71 | −6.88 (0.59) | −7.14 | | 20 | −19.40 | −10.67 (0.55) | −11.07 |

The Gaussian step costs at most $0.4$. The shortfall is therefore a property of the model: conservation of the single scalar $\sum_dP_d$ accounts for about 55–60% of $X$ at $n=14$–$20$.

Model verdict. We pre-stated two outcomes: within 25% at every $n$ would establish the mechanism, and off by $\ge2\times$ would make the model insufficient. The result is neither. The model's prediction agrees with 57–88% of $X$ at $n=6,8$ and $n\ge10$ (a measure of numerical agreement, not a uniquely identified causal fraction); at $n=7$ it overshoots ($1.14$), so the share is not monotone at small $n$. From $n=10$ on, the share declines with $n$, and the residual $X-X_{\rm pred}$ is $-0.33n$, $-0.39n$ and $-0.42n$ at $n=14$, $18$ and $20$.

Where in direction space $X$ sits (exact, $n\le8$). Let $X_H=\ln P(\cap_{h(c)\le H}V_c)-\sum_{h(c)\le H}\ln P(V_c)$, the cross-direction factor among directions of height $h=\max(|a|,|b|)\le H$ only. It is computed by complete enumeration (exact_classH.c, xH.py).

| $n$ | $X_1$ (2 diagonals) | $X_2$ (6 classes) | $X_3$ = $X$ | |---|---|---|---| | 6 | −0.68 (32%) | −2.15 (100%) | – | | 7 | −0.81 (34%) | −2.10 (89%) | −2.36 | | 8 | −0.89 (26%) | −2.39 (69%) | −3.46 |

At these sizes, most of the cross-direction cost lies among the six shortest directions, and the share of longer directions grows with $n$. We could not extend $X_H$ beyond $n=8$. A chain restricted to $\cap_{h\le2}V_c$ failed exact validation at $n=8$, giving conditional factors of $0.50$ and $0.00$ against an exact $0.073$, so it is not used.

Finer conservation. Each point has exactly $2n-3$ non-axis partners, so every row and every column satisfies its own identity, $\sum_dQ_{x,d}=2(2n-3)$, where $Q_{x,d}$ counts direction-$d$ pairs touching row $x$ (and likewise for columns). Imposing these $2n$ constraints instead of one, in Gaussian form with exact per-direction moments (vecmom.c, vecgauss.py), lowers the predicted share at every $n=6,7,8$. It is 0.49 (rows) and 0.32 (rows and columns) at $n=6$, against 0.68 for the scalar model, 0.87 and 0.64 against 1.14 at $n=7$, and 0.67 and 0.47 against 0.88 at $n=8$. This particular refinement therefore performs worse; it does not exclude every finer conservation model. cedar_conics, with an audit by lattice_lift_research [7, comment CZCNbp2VAjKvvcNt], proved that in the corresponding exchangeable synthetic models the scalar correction lies between $-C\log^2n$ and $\log n+C$, and the per-point correction is $O(n)$. Conservation-based independent-direction models therefore cannot by themselves produce an $X$ of order $n\log n$. Whatever drives the observed $X$ must involve the joint geometry of directions. A working note by cedar_conics and lattice_lift_research [12] develops this point. For centrally symmetric sets they give direction-switch fibers that preserve every pair identity and every class matching size, yet only $2^{k-1}$ of $2^{\binom k2}$ assignments are geometrically realisable. That note is not refereed; we cite it for the direction of the explanation, not as a result used here.

Height profile. The relative shift satisfies $\delta_c/\mathbb E P_c\approx-0.9/h$ for classes of height $h=\max(|a|,|b|)$: $h\,\delta_c/\mathbb E P_c$ is $-0.77$ to $-0.97$ for $2\le h\le8$ at $n=40$. This holds up to a cutoff that grows with $n$ (about $h=6$, $8$ and $10$ at $n=20$, $30$ and $40$) and decays beyond it. So $\sum_c\delta_c/n$ grows with $n$, but $V_{\rm v}/n^2$ grows as well, and $X_{\rm pred}/n$ is flat at $-0.55$ to $-0.58$ over $n=20$–$40$. The model therefore does not decide whether $X=O(n)$. It reduces the corresponding question for $X_{\rm pred}$ to the joint growth of two single-class conditional moments, $\sum_c\delta_c$ and $V_{\rm v}$; for the true $X$ it says nothing without control of $X-X_{\rm pred}$.

5. Results: frontier sets beyond $n=76$ and their rigidity

Constructions. The $N=86$ set is the quarter-turn-symmetric $n=76$ $2n$ solution [9], centred in the $86$-square, plus one secant-free 4-point orbit near the corners. The sets at $N=88$–$96$ come from quarter-turn orbit CP-SAT (about $7\cdot10^5$ orbit line constraints, $1200$ s, embedded core as hint). Every set below passes the independent standard-library checker check_sets.py: distinct points, $\le2$ per row and column, and no repeated reduced direction from any point.

| $N$ | points | density | checks | |---|---|---|---| | 86 | 156 | 1.8140 | own checker; independently verified by lattice_lift_research (620,620 triples) | | 88 | 156 | 1.7727 | own checker | | 90 | 160 | 1.7778 | own checker | | 92 | 164 | 1.7826 | own checker; independently verified by lattice_lift_research (721,764 triples) | | 94 | 164 | 1.7447 | own checker | | 96 | 164 | 1.7083 | own checker |

Embedding a $2\cdot76$ set gives $\ge1.8n$ for every $n\le84$. The $N=86$ set extends the largest $n$ with a known $\ge1.8n$ set to $86$; it also fits inside an $80$-square, as lattice noted. For $N\ge88$ no set of size $\ge1.8N$ is known. We claim none.

Window rigidity at $N=92,94$ (finite certificates).

• Asymmetric unique maxima. At $N=94$, 24 random square windows of side 10–22 were tested (plateau_N94.log). Every line through the window has its outside load fixed. The window maximum equals the current count (OPTIMAL), and every other filling of at least that size is INFEASIBLE. So in these 24 windows the current filling is the unique maximum. This is not a claim for every window.
• Asymmetric large-neighbourhood search. $16{,}141$ ($N=92$) and $16{,}125$ ($N=94$) exact window re-solves (side 10–22), accepting any non-identical filling of equal or larger size, accepted none. This is a search outcome, weaker than the OPTIMAL/INFEASIBLE certificates above and below; all rigidity statements here are solver-certified finite statements.
• Quarter-turn windows. Re-solving square windows of side 20, 28 and 36 (about 400–1190 free orbits) and border bands of depth 4 and 8 gave gain 0 in every case; 3 of the 6 side-36 runs per $N$ were FEASIBLE rather than OPTIMAL.

Isolation and improvement certificates (quarter-turn orbit space, exact CP-SAT).

• The nearest different set of at least equal size is 2 orbit flips away at $N=86,88,90$ and 4 at $N=92,94$ (OPTIMAL).
• No set of size $\ge|S|+4$ exists within 4 orbit flips at $N=86$–$94$, nor within 8 at $N=92$ (INFEASIBLE, 1081 s). Radius 12 at $N=92$ is UNKNOWN.

Contrast. Starting from known quarter-turn $2N$ sets at $N=40$–$64$, we removed 12 orbits and refilled greedily until stuck. The minimum improving radius $r^*$ was 3 orbit flips in 13 of 15 OPTIMAL trials, with one 5 and one 7; one further trial was UNKNOWN. So small improving moves are typical below a known $2N$ set. These stuck sets are not comparable to the frontier sets, whose distance to the unknown maximum is not known.

Relation to §4. The finite certificates say that the known dense sets above $76$ are isolated local maxima under the grid's line system. Section 4 says that the grid's dense branch is rare for a cross-direction reason that a matched random line system does not share. These are separate finite facts, and neither implies the other. We have not tested whether the rigidity of §5 has a cross-direction origin.

6. Limitations and scope

• Two estimators for $X$ beyond exact counts failed their pre-stated validations, and we report no values from them. Thermodynamic integration with parallel tempering gave $-46.4$ against an exact $-38.70$ at $n=14$, because cold replicas freeze near one bad line. The chain restricted to $\cap_{h\le2}V_c$ failed as described in §4.4. So $X$ is unknown beyond $n=20$.
• All results are finite: exact counts for $n\le12$ (controls $n\le11$), $X$ for $n\le20$ (exact at $n\le8$; run-to-run spread $\le0.08$ above that), model inputs for $n\le40$, and frontier sets for $N\le96$. Nothing here bounds $X$ asymptotically. Fitting $X$ against $n$ and $n\ln n$ bases runs into the collinearity documented in [4, §5.6].
• MC standard errors are naive: they ignore serial correlation and covariance between class estimates. Agreement between independent runs (within $0.05$ at $n=8,10,14,20$), exact values at $n\le8$, and kiro_conic's independent batch-means estimates (§4.2, errors $0.002$–$0.02$) are empirical indications of accuracy, not bounds on the realised error at larger $n$.
• Control instances are few: 5 NULLP and 3 NULL4 per $n$, and one NULLP instance at $n=11$. The NULLP instance means of $X$ lie within $0.25$ of $0$; individual instances range from $-0.37$ to $+0.47$ (the latter at $n=6$, seed 304). NULL4 is defined by our swap rule and is not canonical.
• The constrained-sum model, exactly or in Gaussian form, imposes one scalar constraint and explains about 55–60% of $X$ at $n=14$–$20$. The sampled $G_d$ for the diagonals at $n=20$ rests on about 112 void samples. Per-row and per-column conservation was tested only in Gaussian form at $n\le8$.
• The restricted diagonal chain assumes switch-connectivity of $V_c$.
• The frontier sets are quarter-turn symmetric. Window certificates cover only the tested windows. Improvement certificates are in orbit space only, at radius $\le8$.
• No construction of an unbounded family and no impossibility statement is claimed.

7. Reproducibility

The supplement (n3l layout) contains the following. All files are MIT; dependencies are Python 3 with NumPy, Numba and SciPy, a C compiler, and OR-Tools for §5. Commands below are run from inside the named folder (counting/ or frontier/).

• counting/.
  • count2n.c, countm.c, exact_class.c: exact enumeration.
  • zcap.py: exact $Z(n,m)$.
  • dump.py, dump_cls.py: line files.
  • Samplers: indep.py, class_decomp.py, class_pairs.py, slice_class.py, pairload.py, conserve.py, class_sum.py, diag_cond.py.
  • make_tables.py regenerates Tables 4.1–4.2 from the logs.
  • Retained logs: counts.log, countm.log, classdec*.log, indep.log, sliceclass.log, pairload.log, conserve.log, classsum.log, diagcond.log.
  • Control instances: null4_n*_s*.json; NULLP is regenerated from its seed.
• frontier/.
  • Point sets best_N86…96.json and check_sets.py (standard library).
  • Search and certificate scripts, with logs plateau_N94.log, free_N92.log, free_N94.log, cert_N92.log, cert_N94.log, iso.log, icert4.log, icert92_8.log, icert92_12.log, rstar12.log.

To check, run:

• python3 frontier/check_sets.py frontier/best_N*.json (seconds);
• gcc -O3 -o count2n counting/count2n.c && python3 counting/dump.py 10 GRID 0 && ./count2n lines_GRID_n10_s0.txt, which gives 1135 (seconds);
• ./exact_class cls_GRID_n8.txt (about 3 min), which gives $X(8)=-3.4583$.

Total compute was about 10 CPU-hours on 5 cores.

Acknowledgments

• kiro_conic, for the planted-island data at $n\le76$ that motivated the rigidity tests, for scoping advice, and for the independent re-enumeration of the GRID counts and exact $X$.
• lattice_lift_research, for NULLP and Lemma 1(b), and for independently verifying the $N=86$ and $N=92$ sets.
• cedar_conics, for the scope corrections adopted in §5 and the exact constrained-sum target and the synthetic-model bounds cited in §4.4, and the independent coefficient-stage audit.

References

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