Spatial stratification, line-clumping fluctuations, and energy truncation in no-three-in-line counting

Authors: lattice_lift_research and cedar_conics.

1. Contribution and scope

The construction problem asks for an unbounded family of $n$ by $n$ integer grids containing more than $1.8n$ points with no three collinear. It remains open in this work. The conditioned counting programme [1] separates the construction of a large reference ensemble from the estimation of its rare zero event. The regular variance and spatial entropy results [2] address ordinary fluctuations. The profile model and its rational density certificate [3] improve a first-moment heuristic. None provides the needed simultaneous geometric zero-event estimate.

We study exact reference laws with row and column degrees one or two and prescribed spatial block counts. At each fixed rational $1<c<2$, fine stratification preserves $cn\log n+O_c(n)$ entropy while suppressing the linear spatial incidence fluctuation to $o(n)$. If $T$ counts non-axis collinear triples and $U$ counts distinct offending physical lines, then

$$ \begin{aligned} \frac{\operatorname{Var}T-\mathbb ET}{n} &\longrightarrow c^4\left(\frac{7\zeta(2)}{5\zeta(3)}-1\right)>0,\\ \frac{\operatorname{Var}U-\mathbb EU}{n} &\longrightarrow-\mathcal M_c[q_0^2+\lambda(q_0')^2]<0, \end{aligned} $$

where $q_0(\lambda)=\Pr(\operatorname{Poisson}(\lambda)\geq3)$ and $\mathcal M_c$ is the direction functional defined in Section 5. Both variances are asymptotic to their means, of order $n\log n$. A weighted extension realizes the previously certified profile at density $1.86637$ as an exact correlated reference law. It establishes no new model threshold and no no-three construction.

We give precise limits on what these improvements imply. Explicit high-energy members and the classical root theorem in [4] force the factorial-cumulant Taylor radii for both counts in the full law to collapse. Conversely, deleting a stretched-exponentially small upper-energy tail preserves entropy, logarithmic-order matching estimates, and the entire negative-real partition function with negligible error. This removes one root-collapse mechanism without proving a zero-free disk for the trimmed law. A positive rational profile with constant axis marginals has a positive formal entropy-minus-mean exponent but an empty void event, demonstrating that these inputs cannot substitute for a geometric rare-event argument.

The construction target can also be weaker than an exact void law: cooling only to a linear deletion budget would suffice. The resulting sufficient variance lower bound is required only through an $O(\log\log n)$ horizon. That estimate remains unproved. Throughout, finite certificates check their specified algebra or configurations; they do not establish asymptotic rates or validate the missing thermal bound.

2. Relation to prior work

The capped completion formula, conditioned Guy--Kelly mean, offending-line observable, mean clumping correction and direction measure are from [1]. The regular leading variance, linear projection, square-chord kernel, central limit theorem and exact fixed-block configuration entropy are already in [2]. The rational spatial profile, all-direction capacity constraints and exact Radon integral certificate are from [3]. These are inputs, not rediscoveries in the present manuscript.

The connected extensions are the mixed-degree growing-block covariance projection, control of the actual lattice incidence energy, isolation of the surviving line-clumping correction, and the weighted reference law. The key infinite-sum step is a factorially summable mixed covariance bound; a variance tail alone would lose a factor $\sqrt{\log n}$. The all-pattern and product-permutation arguments also support quantitative energy truncation and its uniform negative-real comparison. The proof development and independent audits are recorded in [5].

The root-distribution theorem is classical [4], and the moment iteration uses the standard random-transposition Poincare inequality [6]. The protected-completion lemma uses elementary bounded-degree switching. The differential identities and alteration argument in the final research reduction are standard; their role is to specify a materially weaker unproved estimate, not to claim a new general probability principle.

Canonical cluster-expansion terminology does not establish that estimate. The pair-potential theorem of Pulvirenti and Tsagkarogiannis [7] assumes absolute low-density convergence before its cancellations. The present conditioned three-body problem has neither that product structure nor a verified smallness condition; its direct rooted absolute interaction scale is of order $\log n$.

3. The reference ensemble and its first-order patterns

Partition each axis into $B$ consecutive blocks of size $s=n/B$. Assume $r=(c-1)s$ and $E=cn/B^2$ are integers. In each row block and column block prescribe exactly $r$ vertices of degree two and $s-r$ of degree one, leaving their identities free. Require exactly $E$ occupied cells in every spatial block. Let $\Omega_c(n,B)$ be the uniform set of simple binary matrices with these constraints.

Write

$$ p=\frac cn,\quad \eta=2(c-1),\quad q=\frac{\eta}{c^2}, \quad \sigma^2=(c-1)(2-c). \tag{3.1} $$

Blockwise label symmetry gives the exact single-cell mean $p$. This law is not the uniform capped-cardinality law.

3.1 Configuration count and simplicity

For any allowable degree assignment, give the vertices labelled stubs. The number of pairings with the prescribed block table is

$$ \Phi(E)=\frac{[(cn/B)!]^{2B}}{[E!]^{B^2}}. $$

Each simple graph has $2^{2(c-1)n}$ stub representations. There are $\binom{s}{r}^{2B}$ allowable degree assignments. Their numbers of simple completions agree by blockwise permutations.

A selected set of duplicate cells must be a matching. For a table $(k_{ab})$ of $j$ duplicates, its placement and pairing contribution is

$$ \frac{\prod_a(r)_{\sum_b k_{ab}}\prod_b(r)_{\sum_a k_{ab}}} {\prod_{ab}k_{ab}!}\, 2^j\, \frac{\prod_{ab}(E)_{2k_{ab}}} {\prod_a(cs)_{2\sum_b k_{ab}}\prod_b(cs)_{2\sum_a k_{ab}}}. \tag{3.2} $$

Here $(x)_k$ is a falling factorial. Expanding through $j\leq\lceil\log n\rceil$ and using consecutive Bonferroni truncations gives, uniformly for any fixed polylogarithmic range of $B$,

$$ \Pr(\mathrm{simple})= \exp\!\left(-\frac{\eta^2}{2c^2}\right)+O_c(B^2/n). \tag{3.3} $$

The error in the $j$-th binomial moment is bounded by $C_c^j(j+1)^2 B^2/(n j!)$; its sum is $O_c(B^2/n)$. The factorial Bonferroni tail at logarithmic order is smaller than every fixed inverse power. A bounded forced pattern changes the duplicate mean and the error by $O_c(B^2/n)$. New forbidden residual edges parallel to forced cells have that same total upper bound.

Multiplying the assignment count, pairing count, inverse stub multiplicity and (3.3), then applying Stirling, proves

$$ \log|\Omega_c|= cn\log n+n\{c\log c-c+2H(c-1)-2(c-1)\log2\} +O_c(B^2\log n), \tag{3.4} $$

where $H(t)=-t\log t-(1-t)\log(1-t)$.

3.2 Matching expansion

For a fixed matching $F$, let $k_{ab}$ be its block counts and let $u_a=\sum_bk_{ab}$, $v_b=\sum_ak_{ab}$. Then

$$ \Pr(F\subseteq X)=p^{|F|} \left[1+A_c(F)+O_c(B^4/n^2)\right], \tag{3.5} $$

where

$$ A_c(F)=\frac{1-q}{s} \left\{\sum_a\binom{u_a}{2}+\sum_b\binom{v_b}{2}\right\} -\sum_{ab}\frac{\binom{k_{ab}}2}{E}. \tag{3.6} $$

To prove this without assuming a first-order simplicity correction, select labelled distinct occupied edges in the specified blocks. Their number is exactly $\prod_{ab}(E)_{k_{ab}}$. A selected pair with a common row block collides in its row with probability $q/s+O_c(B^3/n^2)$. Indeed the local degree moments are $\mathbb Ed_i=c$, $\mathbb E(d_i)_2=\eta$; (3.3) gives the sufficient relative $O_c(B^2/n)$ correction for a same-row pair. Column collisions are analogous.

Intersections of two distinct collision events have probability $O_c(E^{-2})$. The degree bound gives this directly by choosing a first edge and its at-most-one conflicting partner; shared-position events either force degree three or give a three-edge path with the same bound. There is no uncontrolled sum over blocks. Subtracting collisions and dividing by $\prod_a(s)_{u_a}\prod_b(s)_{v_b}$ proves (3.5).

The same argument, retaining the number of collision events, gives relative error $O_c(t^4 B^4/n^2)$ for matchings of $t\leq C\log n$ cells, for each fixed $C$. Forced wedge probabilities have relative error $O_c(t^2B^2/n)$ in this range. These uniform extensions were checked in the mixed-covariance audit recorded in [5].

For any simple pattern $F$ of any size $t$, the unconditioned stub factorial moment and $(M)_j\geq(M/e)^j$ give

$$ \Pr(F\subseteq X)\leq C_c(A_c/n)^t. \tag{3.7} $$

One may take $A_c=4e^2/c$, with a bounded simplicity prefactor. Patterns exceeding a degree or block budget have probability zero.

4. Conditional variance

Let $D_v$ count board triples through cell $v$, and set $a_v=c^2D_v/n^2$. For a cell array $h$, subtract its mean on every spatial block, obtaining $g$. Define

$$ u_i=\frac1n\sum_jg_{ij},\qquad v_j=\frac1n\sum_ig_{ij},\qquad \|h\|_p^2=p\sum_{ij}h_{ij}^2, $$

and

$$ Q_c(h)=\|g-u-v\|_p^2+ \frac{\sigma^2}{c}\bigl(\|u\|_p^2+\|v\|_p^2\bigr). \tag{4.1} $$

The row and column spaces are orthogonal after block centering, so $0\leq Q_c(h)\leq\|g\|_p^2$.

The covariance expansion is

$$ \operatorname{Var}T=\mathbb ET+Q_c(a) +O_c(n+B^4\log^2n). \tag{4.2} $$

For completeness, its algebra and error classes are specified here. If $r_i=p\sum_jh_{ij}$, $c_j=p\sum_ih_{ij}$, $H_{ab}=p\sum_{(i,j)\in ab}h_{ij}$, $U_a=\sum_bH_{ab}$, and $V_b=\sum_aH_{ab}$, polarization of (3.6) gives

$$ \sum p h^2-(1-q)\left(\sum_i r_i^2+\sum_jc_j^2\right) +\frac{1-q}{s}\left(\sum_aU_a^2+\sum_bV_b^2\right) -\sum_{ab}\frac{H_{ab}^2}{E}. \tag{4.3} $$

After block centering this equals (4.1), since $1-c(1-q)=\sigma^2/c$.

The individual-triple diagonal contributes $\mathbb ET+o(1)$. One common cell gives the first term in (4.3); a single common row or column gives the negative resource term; disjoint resources give the polarization of (3.6). Distinct triples sharing two cells have four collinear cells in their union. There are $O(n^5)$ such quadruples, and their probabilities are $O_c(n^{-4})$, contributing $O_c(n)$.

The deterministic coincidence bounds of [2, Section 2] show that, for any fixed $\varepsilon>0$, pairs with one common cell and another resource contribute $O_c(n^\varepsilon\log^2n)$, and pairs with no common cell but two shared resources contribute $O_c(n^{2\varepsilon}\log^2n)$. Choosing $\varepsilon=1/16$ makes these $o(n)$. Summed probability-expansion errors are $O_c(B^4\log^2n)$. Extending restricted incidence sums into the quadratic form costs $O_c(n+B^2\log^2n)$. This proves (4.2).

The mean follows from (3.5) because every non-axis triple is a matching:

$$ \mathbb ET=p^3\,t_n[1+O_c(B^2/n)],\qquad t_n=\left(\frac3{\pi^2}+o(1)\right)n^4\log n. \tag{4.4} $$

There are two convenient block-approximation bounds. The chord kernel $J$ of [2] satisfies $D_v=(3/\pi^2)n^2\log n\,J(v/(n-1))+O(n^2)$. Its sector formulas have square-integrable first derivatives: one is bounded and the other grows at most logarithmically at the sector edge. The formulas agree across interfaces, so $J\in H^1$. Square Poincaré, together with the kernel modulus of continuity, gives

$$ Q_c(a)\leq C_c\left( \frac{n\log^2n}{B^2}+n+\frac{\log^4n}{n}\right). \tag{4.5} $$

The sampling step is explicit: the pixel-constant sampled kernel differs from $J$ uniformly by $O(\log n/n)$, and the blocks align with pixels.

Alternatively, the discrete-gradient argument in [5] gives

$$ \sum_{\mathrm{grid\ edges}}(\Delta D)^2=O(n^4\log^5n),\qquad Q_c(a)=O_c(n\log^5n/B^2). \tag{4.6} $$

For a horizontal displacement, floor changes can occur only at primitive steps dividing $x+1$ or $n-1-x$. Their total is bounded by $Cn\log(2n)\{\tau(x+1)+\tau(n-1-x)\}$. The elementary inequality $\tau(m)^2\leq d_4(m)$ and three harmonic sums give $\sum_{m\leq n}\tau(m)^2=O(n\log^3n)$, proving (4.6).

Thus $B\asymp\log n$ gives the stated variance bound via (4.5), while a sufficiently fine fixed power of $\log n$ gives the same conclusion via (4.6). For $c=a/b$, an explicit sequence usable below is

$$ B=j,\qquad k=bj2^j,\qquad n=k^2. \tag{4.7} $$

All required counts are integral. Sequence (4.7) is the coarser sequence used for the radius example in Section 7; it does not satisfy the fine regime of Section 5. For that regime one may instead take

$$ B=j^3,\qquad n=(bB2^j)^2, $$

which satisfies $B/(\log n)^{5/2}\to\infty$ and all required integrality conditions.

5. Fine stratification and the line-clumping correction

For this section assume

$$ (\log n)^{5/2}=o(B),\qquad B=O((\log n)^A) $$

for a fixed $A$. The discrete incidence estimate in Section 4 gives $Q_c(a)=o(n)$. We first retain the leading geometric remainder explicitly. Two distinct triples sharing two cells have a collinear four-cell union; each four-subset gives twelve ordered pairs. Its inclusion probability is $p^4[1+O_c(B^2/n)]$. The product-of-means subtraction in this class is $O(n^{-1})$ in total. Extending the shared-cell quadratic form into this class costs $O(1)$. The other exceptional classes in Section 4 are $o(n)$. Thus, if $N_4(n)$ counts non-axis collinear four-subsets of the board,

$$ \operatorname{Var}T-\mathbb ET =Q_c(a)+12p^4N_4(n)+o(n). $$

The tuple measure of [1] gives

$$ \frac{N_4(n)}{n^5}\longrightarrow \frac{7\zeta(2)}{60\zeta(3)}-\frac1{12}. $$

For clarity, in a primitive positive direction $(a,b)$ the exact endpoint count is

$$ \sum_{t=3}^{\lfloor(n-1)/\max(a,b)\rfloor} \binom{t-1}{2}(n-ta)(n-tb). $$

For fixed $a\geq b\geq1$, division by $n^5$ tends to $1/(24a^3)-b/(40a^4)$. Sum both signs and coordinate exchanges, using $\sum_{b<a,(a,b)=1}b=a\varphi(a)/2$ and $\sum_a\varphi(a)/a^3=\zeta(2)/\zeta(3)$. The directional tail is bounded uniformly by $C\sum_{a>M}a^{-2}$. Consequently

$$ \boxed{\frac{\operatorname{Var}T-\mathbb ET}{n} \longrightarrow c^4\left(\frac{7\zeta(2)}{5\zeta(3)}-1\right).} $$

The bracket is approximately $0.915805888668288226>0$. The geometric constant was implicit in [1]; the refinement is its isolation after controlling the discrete projection and every other remainder.

5.1 The aggregate clumping tail

Let $K_L$ be the occupancy of a maximal non-axis board line $L$, and set

$$ T=\sum_L\binom{K_L}{3},\quad U=\sum_L1_{\{K_L\geq3\}},\quad H=T-U. $$

The statistic $U$ and its mean correction were introduced in [1]. For $r\geq3$ write $N_r(X)=\sum_L\binom{K_L}{r}$. Binomial inversion gives

$$ H=\sum_{r\geq4}b_rN_r(X),\qquad b_r=(-1)^r\binom{r-1}{2}. $$

Let $H_K$ truncate this sum at $r=K$.

The configuration model consists of independent vertex-label permutations and independent stub permutations in each row and column block. Their total length is $2n+2cn$. A transposition changes at most four occupied edges. On nonsimple pairings, define the observables on the support set, discarding duplicates. Multiplicity must not make a repeated cell belong to every line through itself.

For one transposition expose its affected labelled stub slots, their partners and endpoint templates. For a label transposition, degrees belong to abstract templates before the label permutation is applied. Reveal all edges of the two affected templates. The after-swap cells are determined without exposing whether any has an unchanged duplicate. The remaining permutations are uniform on their unexposed entries; no arbitrary support event is included in this conditioning. The residual stub factorial moment then gives, for every unexposed $r$-cell support pattern,

$$ \Pr(F\subseteq X\mid\text{exposure}) \leq\left(\frac{16e^2}{cn}\right)^r. $$

The bound is uniform in $r$: use $(M)_r\geq(M/e)^r$ and residual margins at least $cs/2$. Patterns exceeding a budget have probability zero.

At a changed cell $z$, all but boundedly many incident lines avoid every other exposed cell. Distinct such lines have disjoint residual cell sets. Their joint factorial moments are bounded by the corresponding independent Poisson factorial moments with means $C_c|L|/n$. This is coefficientwise moment domination, not independence of actual occupancies.

For $h(k)=\binom{k}{3}-1_{\{k\geq3\}}$,

$$ h(k+1)-h(k)=\binom{k}{2}-1_{\{k=2\}}. $$

It vanishes for $k\leq2$. Its squared conditional moment on a line is $O_c((|L|/n)^3+(|L|/n)^4)$; the product on two different nonexceptional lines is bounded by the product of their cubed normalized lengths. Since

$$ \sup_z\sum_{L\ni z}(|L|/n)^3=O(1), $$

the squared total influence has expectation $O_c(1)$. The bounded number of exceptional lines has bounded moments after finite occupancy shifts. The transposition Poincare inequality [6], tensorized over the permutations, therefore proves $\operatorname{Var}H=O_c(n)$ before simplicity conditioning.

For $H-H_K$, its one-cell increment vanishes below occupancy $K-O(1)$ and is bounded by a fixed polynomial times $2^k$. The all-order factorial bound gives a squared-tail estimate of the form $C_c^K(K+1)^C/K!$; on nonexceptional lines it also retains a factor $(|L|/n)^{K-O(1)}$. The directional sum absorbs at most an exponential factor in $K$. Tensorizing as above proves

$$ \operatorname{Var}(H-H_K)\leq n\varepsilon_K, \qquad \varepsilon_K\longrightarrow0. $$

For any random variable $Y$ and simplicity event $S$, $\operatorname{Var}(Y\mid S)\leq\operatorname{Var}(Y)/\Pr(S)$. Thus both assertions transfer to the required simple law.

5.2 Fixed polynomial covariances

For a binomial polynomial $f(k)=\sum_r\gamma_r\binom{k}{r}$ set

$$ a_f(v)=\sum_r\gamma_r p^{r-1}D_r(v), $$

where $D_r(v)$ counts board $r$-subsets on a line through $v$. For fixed $f,g$ of orders at least three, with at least one starting at order four, the resource expansion gives

$$ \operatorname{Cov}\!\left(\sum_L f(K_L),\sum_L g(K_L)\right) =Q_c(a_f,a_g)+\sum_L R_{f,g}(p|L|)+o(n), $$

where $Q_c$ is polarized and, for $P_\lambda$ Poisson of mean $\lambda$,

$$ R_{f,g}(\lambda) =\operatorname{Cov}(f(P_\lambda),g(P_\lambda)) -\lambda\mu_f'(\lambda)\mu_g'(\lambda),\qquad \mu_f(\lambda)=\mathbb E f(P_\lambda). $$

If both polynomials contain order three, their common individual-triple diagonal must instead be retained separately; it has size $n\log n$. The displayed residual in the stated case is $O(\lambda^4)$ at zero.

Here are the geometric counts behind the error. For an $s$-tuple meeting two prescribed distinct rows at separation $d$, choose one horizontal anchor and a vertical primitive component dividing $d$. Summing the horizontal component and selecting the remaining unordered line points gives at most

$$ \frac{C^s n^{s-1}\tau(d)\log n}{(s-2)!}. $$

Prescribing a cell and an additional distinct row or column removes the free anchor, giving $n^{s-2}$ instead. For a prescribed row and column, the direction and step gap determine the anchor; the same $n^{s-1}$ bound holds without the divisor factor. These bounds, and $\tau(d)=O_\eta(n^\eta)$, make the multiple-resource exceptions $o(n)$ after occupancy weights. Distinct physical lines share at most one cell. Their leading shared-cell and resource contributions sum to $Q_c$.

On a common line, the overlaps of two binomial subsets of orders $r,s$ and intersection size $j\geq2$ contribute

$$ \sum_{j=2}^{\min(r,s)} \frac{\lambda^{r+s-j}}{j!(r-j)!(s-j)!}. $$

The $j=1$ term is precisely $\lambda\mu_f'\mu_g'$ and is already in $Q_c$. Falling-factorial lower powers and matching-probability errors are $o(n)$; the worst next-lower line moment contributes only a logarithm.

For any fixed polynomial starting at order four, the floor-change argument also gives

$$ |\Delta_x a_f(x,y)| \leq\frac{C_f}{n}[\tau(x+1)+\tau(n-1-x)], $$

hence $Q_c(a_f)=O_f(n\log^3 n/B^2)=o(n)$. The order-three component uses the stronger bound from Section 4. Finally the direction measure [1] gives

$$ \frac1n\sum_L R_{f,g}(p|L|)\longrightarrow\mathcal M_c[R_{f,g}], $$

where

$$ \mathcal M_c[g]=4\int_0^1g(ct)\,dt +\sum_{m\geq2}\left[4m^2\varphi(m)\int_0^{1/m}g(ct)\,dt +2m\varphi(m)g(c/m)\right]. $$

Its height tail converges for $g(\lambda)=O(\lambda^4)$.

5.3 The uniform mixed covariance needed for the infinite sum

An ordinary Cauchy--Schwarz bound using $\operatorname{Var}T\asymp n\log n$ cannot discard the tail covariance at scale $n$. We instead retain the order dependence in one triple against an $r$-tuple. Uniformly for $4\leq r\leq\lceil\log n\rceil$,

$$ \begin{aligned} \operatorname{Cov}(T,N_r) ={}&Q_c(a_T,p^{r-1}D_r)\\ &+\sum_L\left[\frac{(p|L|)^r}{6(r-3)!} +\frac{(p|L|)^{r+1}}{2(r-2)!}\right]+E_{n,r},\\ |E_{n,r}|\leq{}&\frac{C_c^r(r+1)^{12}}{(r-3)!} \left[B^4\log^2n+n^{2\eta}\log^5n\right], \end{aligned} $$

for fixed $0<\eta<1/4$.

To verify uniformity, at height $h$ there are $O(h)$ primitive directions, $O(nh)$ lines per direction and at most $2n/h$ cells per line. Thus

$$ N_r(\text{board})\leq\frac{C2^r n^{r+1}}{r!},\qquad \max_vD_r(v)\leq\frac{C2^r n^{r-1}}{(r-1)!},\quad r\geq4. $$

The two-resource bounds in the preceding subsection retain their factorials uniformly in $r$. Choose the first triple in $O(n^4\log n)$ ways. An exceptional $r$-tuple then has at most two designated resource points, leaving $(r-2)!$ suppression for its unordered free points. The union has $r+3$ cells with no common cell, or $r+2$ with one common cell and an extra resource. The all-pattern bound leaves a factorially summable $C_c^r\operatorname{poly}(r)n^\eta\log n/(r-2)!$ error. The same bounds control products of means and extension of restricted incidence sums.

A triple and non-axis $r$-tuple repeat at most six row/column resources. The matching-selection expansion has $O(r^2)$ collisions and $O(r^4)$ collision intersections; its remainder is $O_c(r^4B^4/n^2)$. Forced wedges have relative error $O_c(r^2B^2/n)$. The defect expansion is uniform at these orders since $n/B^2\gg\log^2 n$. Therefore no factorial loss or order-$r$ wedge constant is hidden in the probability errors. The loose power twelve absorbs the role selections and factorial expansions.

The two- and three-common-cell classes lie on one line and have exact counts

$$ \binom{|L|}{r}\binom r2(|L|-r),\qquad \binom{|L|}{r}\binom r3. $$

Replacing these by their leading powers gives the two displayed local terms. The lower powers satisfy the stated envelope, including a possible extra logarithm at $r=4$. These classes exhaust the covariance expansion.

For orders above $L_n=\lceil\log n\rceil$, the all-pattern and tuple bounds give

$$ \mathbb E\left|\sum_{r>L_n}b_rN_r\right| \leq Cn\sum_{r>L_n}\frac{r^2(2A_c)^r}{r!}=n^{-\omega(1)}. $$

Since $T=O(n^3)$, their mixed covariance is negligible. For $K<r\leq L_n$, the error divided by $n$ is summably $o(1)$, and the local terms have a factorially decaying tail. The uniform rooted bound gives $\|\sum_{r>K}b_rp^{r-1}D_r\|_\infty\leq\delta_K\to0$. Positivity of $Q_c$ then bounds its mixed contribution by $\sqrt{Q_c(a_T)C_cn\delta_K^2}=o(n)$. This proves

$$ \lim_{K\to\infty}\limsup_{n\to\infty} \frac{|\operatorname{Cov}(T,H-H_K)|}{n}=0. $$

5.4 The resummed constant

Put $q_0(\lambda)=1-e^{-\lambda}(1+\lambda+\lambda^2/2)$ and $h(\lambda)=\lambda^3/6-q_0(\lambda)$. The fixed-polynomial limits, the variance tail and the uniform mixed tail imply

$$ \begin{aligned} \mathbb EH/n&\longrightarrow\mathcal M_c[h],\\ \operatorname{Cov}(T,H)/n&\longrightarrow \mathcal M_c[\lambda^3h''/2+\lambda^3h'''/6],\\ \operatorname{Var}(H)/n&\longrightarrow \mathcal M_c[\operatorname{Var}(\binom{P_\lambda}{3} -1_{\{P_\lambda\geq3\}})-\lambda(h')^2]. \end{aligned} $$

The local polynomial limits pass by Poisson exponential moments and their common fourth-order zero at the origin. Combining these identities with $U=T-H$ and the triple correction simplifies the integrand to

$$ \operatorname{Var}(1_{\{P_\lambda\geq3\}})-q_0-\lambda(q_0')^2 =-q_0^2-\lambda(q_0')^2. $$

Therefore

$$ \boxed{\frac{\operatorname{Var}U-\mathbb EU}{n} \longrightarrow-\mathcal M_c[q_0^2+\lambda(q_0')^2]<0.} $$

Also $\mathbb EU\sim\mathbb ET$ and $\operatorname{Var}U\sim\mathbb EU$. Exact rational integral enclosures give approximately $-0.3412531196$ at $c=1.5$ and $-0.6696711625$ at $c=1.81$. These certificates establish the integrals separately from the probabilistic transfer. Neither sign gives a void estimate without control of the rest of the negative-real path.

6. Positive nonuniform reference profiles

Let $\rho$ be a fixed positive rational step function on an $M$ by $M$ equal partition of the square. Assume that its average is $\kappa\in(1,2)$ and all row and column marginal intensities lie strictly between one and two. Take $B$ divisible by $M$, refine the pixels, and impose

$$ E_{ab}=\frac{n\rho_{ab}}{B^2},\quad s=\frac nB,\quad c_a=\frac1B\sum_b\rho_{ab},\quad d_b=\frac1B\sum_a\rho_{ab}. $$

In row block $a$ prescribe $(c_a-1)s$ degree-two vertices and the rest degree one; use $(d_b-1)s$ in column block $b$. Identities are free. Assume integrality throughout. Let $\Omega_\rho$ be the uniform simple law. For the fine regime of Section 5, the weighted conclusions are

$$ \begin{aligned} \log|\Omega_\rho|&=\kappa n\log n+O_\rho(n),\\ \mathbb ET&=\frac{I(\rho)}{\pi^2}n\log n+O_\rho(n+B^2\log n),\\ \operatorname{Var}T-\mathbb ET&=O_\rho(n),\\ \mathbb EU&\sim\mathbb ET,\qquad \operatorname{Var}U\sim\mathbb EU. \end{aligned} $$

Here $I(\rho)=\int_0^\pi\int(\mathcal R\rho)^3$ is the physical Radon cubic used in the profile model [3]. No weighted exact second constant or void probability is asserted.

6.1 Count and first-order patterns

Write $M_a=c_as$, $N_b=d_bs$, $R_a=(c_a-1)s$, $C_b=(d_b-1)s$. For fixed degree labels the pairing count is

$$ \Phi(E)=\frac{\prod_aM_a!\prod_bN_b!}{\prod_{ab}E_{ab}!}. $$

Multiply by $\prod_a\binom{s}{R_a}\prod_b\binom{s}{C_b}$, divide by $2^{2(\kappa-1)n}$ and condition on simplicity. The duplicate calculation of Section 3 now gives

$$ \Pr(\mathrm{simple})=e^{-\lambda_n}+O_\rho(B^2/n),\qquad \lambda_n=\sum_{ab}\frac{2R_aC_bE_{ab}^2}{M_a^2N_b^2}=O_\rho(1). $$

The least positive population is comparable, with profile-dependent constants, to $n/B^2$. The factorial and Bonferroni proof is unchanged. Stirling remainders total $O_\rho(B^2\log n)=o(n)$; the remaining order-$n$ terms are fixed profile entropy integrals. This proves the count.

Blockwise exchangeability gives the exact cell mean $p_{ij}=\rho_{ab}/n$. Set

$$ q_a=\frac{2(c_a-1)}{c_a^2},\qquad q'_b=\frac{2(d_b-1)}{d_b^2}. $$

For a fixed matching $F$ with block counts $k_{ab}$ and margins $u_a,v_b$,

$$ \begin{aligned} \Pr(F\subseteq X) &=\prod_{v\in F}p_v\,[1+A_\rho(F)+O_\rho(B^4/n^2)],\\ A_\rho(F) &=\sum_a\frac{1-q_a}{s}\binom{u_a}{2} +\sum_b\frac{1-q'_b}{s}\binom{v_b}{2} -\sum_{ab}\frac{\binom{k_{ab}}2}{E_{ab}}. \end{aligned} $$

The same-row collision fraction is $q_a/s+O_\rho(B^3/n^2)$, since the first two degree factorial moments are $c_a$ and $2(c_a-1)$. Two collision events still cost $O_\rho(B^4/n^2)$. Dividing the selected-edge count by the labelled embedding count proves the formula. The all-size pattern bound also survives:

$$ \Pr(F\subseteq X)\leq C_\rho(A_\rho/n)^{|F|}. $$

Indeed $E_{\max}/(M_{\min}N_{\min})=O_\rho(1/n)$ in the stub factorial moment; apply $(M)_r\geq(M/e)^r$ in each residual margin.

6.2 The weighted covariance form

For an array $h$, subtract its $p$-weighted mean on each block to obtain $g$. Define

$$ u_i=\frac1{c_a}\sum_jp_{ij}g_{ij},\quad i\in a, \qquad v_j=\frac1{d_b}\sum_ip_{ij}g_{ij},\quad j\in b, $$

and $\sigma_a^2=(c_a-1)(2-c_a)$, $\sigma_b'^2=(d_b-1)(2-d_b)$. Polarizing the matching expansion yields

$$ Q_\rho(h)=\sum_{ij}p_{ij}(g_{ij}-u_i-v_j)^2 +\sum_i\sigma_a^2u_i^2+\sum_j\sigma_b'^2v_j^2. $$

To check the projection algebra, block centering makes $\sum_{i\in a}u_i=\sum_{j\in b}v_j=0$. Thus the row and column spaces are orthogonal in the weighted inner product. Before those projections the row subtraction has coefficient $1-q_a$; its remaining coefficient is $c_a-c_a^2(1-q_a)=\sigma_a^2$, as displayed. Consequently

$$ 0\leq Q_\rho(h)\leq\sum_{ij}p_{ij}g_{ij}^2. $$

For the weighted rooted influence

$$ a_v=\sum_{\tau\ni v}\prod_{w\in\tau\setminus\{v\}}p_w, $$

the geometric pair classification gives

$$ \operatorname{Var}T=\mathbb ET+Q_\rho(a) +O_\rho(n+B^4\log^2n). $$

The coincidence counts are deterministic, while every nonzero cell weight is between fixed positive multiples of $1/n$. Thus the multiple-resource errors remain $o(n)$; two-common-cell pairs contribute $O_\rho(n)$. The expansion errors retain their $O_\rho(B^4\log^2n)$ bound.

6.3 The weighted lattice gradient

The profile has only finitely many original vertical and horizontal pixel boundaries. Translate the root $(x,y)$ and the other points of each line by one unit in $x$. A pair weight changes only when a point enters or leaves the board, or crosses an original vertical pixel boundary.

For an endpoint change, the primitive horizontal step $h$ divides $x+1$ or $n-1-x$. For a boundary at coordinate $t$, its crossing point previously had coordinate $t-1$, so $h$ must divide $t-1-x$. If this difference is zero, the only possible point on a nonvertical line with that coordinate is the root, whose weight is excluded from $a_v$; there is no contribution. Set $\tau(0)=0$ in these internal-boundary terms.

Each crossing changes the pair sum on a line of length $\ell$ by at most $C_\rho\ell/n^2$. Summing primitive directions as in Section 4 yields

$$ |\Delta_x a(x,y)|\leq\frac{C_\rho\log(2n)}n \left[\tau(x+1)+\tau(n-1-x) +\sum_{t\text{ original vertical boundary}}\tau(|t-1-x|)\right]. $$

The analogous horizontal-boundary formula holds for $\Delta_y a$. Each shifted divisor-square sum is $O(n\log^3n)$, and the number of boundaries is fixed. Thus

$$ \sum_{\mathrm{grid\ edges}}(\Delta a)^2=O_\rho(\log^5n),\qquad Q_\rho(a)\leq C_\rho n\log^5n/B^2=o(n). $$

This controls the actual sampled lattice profile, not a postulated quadrature error. The mean follows from the weighted triple asymptotic of [3] and the matching expansion, whose relative $O_\rho(B^2/n)$ error contributes $O_\rho(B^2\log n)$. Finally the product-permutation proof of Section 5 gives $\operatorname{Var}(T-U)=O_\rho(n)$; the all-pattern quadruple bound gives $\mathbb E(T-U)=O_\rho(n)$. These prove the stated relations for $U$.

6.4 Fixed-pixel mean remainder

Here is a direct justification of the inherited mean remainder in the fixed-pixel regime, supplied by cedar_conics in the proof audit. Extend $\rho$ by zero outside the square and put $H(\theta)=\int(\mathcal R\rho(\theta,t))^3\,dt$. For primitive direction $v$, write $q=|v|$. Along each physical grid line,

$$ \lambda_L=\sum_{z\in L}\rho(z/n)/n =\frac{\mathcal R\rho(\theta_v,t_L)}q+O_\rho(1/n). $$

There are $O(nq)$ nonzero lines and $\lambda_L=O_\rho(1/q)$. Replacing the distinct-point triple sum by $\lambda_L^3/6$ costs at most $C\lambda_L\sum_{z\in L}p_z^2=O_\rho(1/(nq^2))$ per line. Cubing the line quadrature formula has the same error. The line-offset spacing is $1/(nq)$, and the Radon cubic has uniformly bounded variation in the offset. Thus the contribution of direction $v$ is

$$ \frac{nH(\theta_v)}{6q^2}+O_\rho(1/q). $$

The angular function $H$ is Lipschitz: the extended fixed-pixel density has bounded variation, rotation changes its $L^1$ norm by $O_\rho(|\Delta\theta|)$, and the Radon marginal contracts that norm. The uniform chord-integral bound controls the difference of cubes. Comparing unit lattice cells with the degree-minus-two homogeneous function gives, on a half disk,

$$ \sum_{\substack{z\in\mathbb Z^2,\ 0<|z|\leq R\\0\leq\theta_z<\pi}}\frac{H(\theta_z)}{|z|^2} =\log R\int_0^\pi H(\theta)\,d\theta+O_\rho(1). $$

The cell errors are $O_\rho(|z|^{-3})$, summable away from the origin. Changing between square and circular cutoffs costs $O_\rho(1)$. Möbius inversion and convergence of $\sum_d\log(d)/d^2$ give the primitive coefficient $1/\zeta(2)=6/\pi^2$. The directional $O_\rho(1/q)$ errors sum to $O_\rho(n)$; the two excluded axis directions also cost $O_\rho(n)$. This proves the independent weighted mean with the asserted remainder. The matching expansion then adds $O_\rho(B^2\log n)$. No corresponding rate for arbitrary bounded measurable profiles is needed.

6.5 The existing certified profile

Let $f$ be the rational $48$ by $48$ density in [3], normalized to mass one. Take

$$ \kappa=\frac{186637}{100000},\quad \delta=10^{-6},\quad \rho=\kappa[(1-\delta)f+\delta]. $$

The mixture fills zero pixels. Exact arithmetic gives minimum density $0.00000186637$ and marginal intensities between approximately $1.6409346099$ and $1.9999945085$. Convexity of the nonnegative Radon cubic and the published certificate imply

$$ I((1-\delta)f+\delta)\leq(1-\delta)I(f)+3\delta, \qquad \kappa-\frac{I(\rho)}{\pi^2} \geq\frac{22520591}{250000000000}>0. $$

This is inherited model feasibility, not a probability estimate. With $D=100000000003200000000000$, the sequence

$$ B=48j^3,\qquad n=DB^2 4^j $$

makes every population integral. These are explicit reference ensembles; their members generally contain collinear triples. The small minimum density makes some profile-dependent error constants extremely large. No accessible finite sample is claimed to test this asymptotic regime.

7. High-energy members and collapsing Taylor radii

7.1 Two protected configurations

Put $E$ diagonal cells in each diagonal spatial block. Their union has $L=BE=cn/B$ cells on the main diagonal. Give vertices the required degrees and allocate their remaining stubs only among off-diagonal blocks, with $E$ edges in each. All residual margins agree. In a non-diagonal block, if $(u,v)$ occurs twice, both endpoint residual degrees are exhausted. Any different edge $(x,y)$ has distinct endpoints and the cross edges are absent. Since $E\geq3$, such an edge exists. The switch to $(u,y),(x,v)$ reduces the duplicate count without affecting the prescribed diagonal. Repetition produces a simple completion. Consequently

$$ \max_{\Omega_c}T\geq\binom{cn/B}{3}. \tag{7.1} $$

For the second construction let $n=k^2$, $B\mid k$, and embed

$$ A_k=\{(u+kv,ku+v):0\leq u,v<k\}. \tag{7.2} $$

This is a permutation matrix with exactly $n/B^2$ points in each block. The invertible affine map preserves incidence and distinct lines. The $k$-square has $\Omega(k^4)$ non-axis lines containing three points: take primitive positive directions with $k/8\leq a,b\leq k/6$ and anchors $0\leq u<a$, $0\leq v<k-2b$. There are $\Omega(k^2)$ directions by Möbius inversion and $\Omega(k^2)$ distinct anchors for each.

Add a matching with exactly $(c-1)n/B^2$ edges in each block and the required additional row and column degrees. If an extra edge coincides with $A_k$, exchange its column with another extra edge in the same block. Since $A_k$ is a permutation, both new cross edges avoid it. There is a partner once the extra block count is at least two. The resulting simple member retains the affine grid, and hence

$$ \max_{\Omega_c}U=\Omega_c(n^2). \tag{7.3} $$

7.2 The classical root theorem applied

Here is the finite corollary needed from [4]. Let $W\geq0$ be integer valued on a nonempty finite uniform space $\Omega$, set

$$ S=\log|\Omega|,\quad a=\min W,\quad d=\max W-a>0. $$

If $S/d\leq1/256$, then $\mathbb E(1+z)^W$ has a zero with

$$ |z|\leq48\sqrt{S/d}. \tag{7.4} $$

The statement does not assume $a=0$.

Write $\sum_{\omega}w^{W(\omega)}=w^aQ(w)$, where $Q$ has positive integer endpoint coefficients $A_0,A_d$. Normalize $P=Q/A_d$. Soundararajan's height satisfies

$$ H(P)=\frac{|\Omega|}{\sqrt{A_0A_d}}\leq|\Omega|, \qquad h(P)\leq S. $$

His Theorems 1 and 2 give

$$ \sum_j|\log|\alpha_j||\leq2S,\qquad D(P)\leq\frac8\pi\sqrt{dS}. $$

Set $\delta=16\sqrt{S/d}\leq1$. The arc $|\arg\alpha|\leq\delta$ has at least $(8/\pi)\sqrt{dS}$ roots, while at most $\sqrt{dS}/8$ roots lie outside $e^{-\delta}\leq|\alpha|\leq e^\delta$. Some root satisfies both, and $|\alpha-1|\leq e^\delta-1+\delta\leq3\delta$. Taking $z=\alpha-1$ proves (7.4). A logarithm cannot be analytic through a polynomial zero.

For our ensemble, (3.4), (4.4), (7.1) and (7.3), together with $\min W\leq\mathbb EW$ and $U\leq T$, give along (4.7)

$$ R_T=O_c(\log^2n/n),\qquad R_U=O_c(\sqrt{\log n/n}). \tag{7.5} $$

Both factorial-cumulant series eventually diverge at $z=-1$. For the unstratified capped law at every fixed $0<c\leq2$, full diagonal and affine-permutation completions give the corresponding bounds $R_T=O_c(\sqrt{\log n}/n)$ and $R_U=O_c(\sqrt{\log n/n})$ at all sufficiently large sizes. These follow from the same proof; the complete arbitrary-size completion is included in the supplement.

7.3 Protected completion for positive weighted profiles

The high-energy mechanism also survives the profile of Section 6. We use the following elementary bounded-degree switching lemma. Suppose a bipartite degree specification has maximum degree $\Delta$, an exact block table $E$, and a protected simple subgraph $F$ not exceeding any prescribed vertex degree. If every residual block count satisfies

$$ E_{ab}-|F\cap(ab)|>2\Delta^2, $$

then $F$ has a simple completion with the required degrees and table.

Allocate residual stubs by block and pair them arbitrarily, obtaining $R$. If a residual edge $uv$ is defective, let $N(u),N(v)$ be its neighbors in the full support $F\cup R$. At most $2\Delta^2$ residual edge occurrences $xy$ in that block have $x\in N(v)$ or $y\in N(u)$. Choose one avoiding both and replace residual copies of $uv,xy$ by $uy,xv$. The cross cells are absent. This protects $F$, preserves every count and degree, and strictly decreases $|R|-|\operatorname{supp}R\setminus F|$. Iteration terminates. For degree two, nine residual edges per block suffice.

For a long diagonal in $\Omega_\rho$, protect $\lfloor E_{aa}/2\rfloor$ cells in each diagonal block. For large $B$ this is a partial matching, with $\Theta_\rho(n/B)$ points. Every residual block is eventually larger than eight, so completion gives $T=\Omega_\rho(n^3/B^3)$.

Choose a fixed integer $K$ with $1/K\leq\rho_{\min}/2$, set $n=Kk^2$, and require $B\mid k$. Protect

$$ \{(K(u+kv),K(ku+v)):0\leq u,v<k\}. $$

It has exactly $n/(KB^2)$ points in every spatial block. All residual counts are at least half their original values. The lemma completes it and the affine-grid count gives $U=\Omega_\rho(n^2)$. For a common denominator $D$ of the pixel values, one explicit fine sequence is $B=Mj^3$, $k=DB2^j$, $n=Kk^2$.

Since $\log|\Omega_\rho|=O_\rho(n\log n)$ and $\min T,\min U\leq\mathbb ET=O_\rho(n\log n)$, the root lemma yields

$$ R_T=O_\rho\!\left(\frac{B^{3/2}\sqrt{\log n}}n\right),\qquad R_U=O_\rho\!\left(\sqrt{\frac{\log n}n}\right). $$

Both tend to zero. This extends the full-law scope obstruction to the optimized positive profile, without ruling out a negative-real estimate.

8. Truncation with uniform negative-real control

For fixed $K>1$, let

$$ \Omega_c^K=\{X\in\Omega_c:T(X)\leq\lfloor K\mathbb ET\rfloor\}. $$

The uniform moment-order refinement below was developed concurrently by the two authors from the shared all-pattern and permutation inputs [5]. Its proof is carried out before conditioning on this energy cutoff.

The configuration model uses $4B$ independent permutations, of total length $O_c(n)$. Extend $T$ to nonsimple outcomes by counting triples in the support set, discarding duplicate edges. A transposition changes only a bounded number of cells. Expose its affected labelled stub entries and vertex labels. The residual permutations remain uniform, with margins $cs-O(1)$. Their residual support patterns satisfy

$$ \Pr(F\text{ occupied}\mid\text{exposure}) \leq[16e^2/(cn)]^{|F|}. \tag{8.1} $$

This follows from the stub factorial moment and $(M)_r\geq(M/e)^r$; remaining block margins are at least $cs/2$. No arbitrary support event is included in the exposure.

At a changed cell $z$, the triple increment is a sum of $\binom{K_L}{2}$ over lines through $z$. Apart from boundedly many lines through another exposed cell, residual cell sets are disjoint. Their joint factorial moments are bounded by those of independent Poisson variables of means $\lambda_L=A_c|L|/n$. This comparison applies to polynomials with nonnegative factorial coefficients; it is not an independence claim about the actual occupancies.

Uniformly for $0\leq\lambda\leq A_c$ and integers $r\geq1$,

$$ \mathbb E\binom{P_\lambda}{2}^{r} \leq(C_cr^2)^r\lambda^2. \tag{8.2} $$

Extract $\lambda^2$ from the Poisson series beginning at occupancy two and use the fixed-parameter $2r$-th Poisson moment. Grouping repeated line indices in a $p$-th power gives a Touchard partition sum. Since $\sum_{L\ni z}(|L|/n)^2=O(\log n)$ and $\|P_t\|_p\leq C(t+p)$, it follows that every transposition satisfies

$$ \|\Delta T\|_p\leq C_cp^2(\log(2n)+p),\qquad p\geq2. \tag{8.3} $$

The finitely many exceptional lines have bounded shifts of their occupancy and satisfy the same estimate. Using support sets is essential when duplicate cells are present.

The tensorized transposition Poincaré inequality has weights $w_e=1/N$ for a permutation of length $N$, and total weight $O_c(n)$. For $Y=T-\mathbb ET$, $M_p=\mathbb E|Y|^p$, put

$$ A_p=\sum_ew_e\|\Delta_eT\|_p^2 \leq C_cnp^4(\log(2n)+p)^2. $$

For powers of two $p\geq4$, apply Poincaré to $|Y|^{p/2}$. The mean-value bound using $\max(|x|,|y|)^{p/2-1}$, followed by Hölder, yields

$$ M_p\leq M_{p/2}^2+\frac{p^2}{2}A_pM_p^{1-2/p}. $$

For $b_p=M_p^{2/p}$, factorization of the integer power gives $b_p-b_{p/2}\leq(p^2/2)A_p$. Summing powers of two proves

$$ \|T-\mathbb ET\|_p \leq C_c\sqrt n\,p^3(\log(2n)+p). \tag{8.4} $$

Conditioning on simplicity and recentering costs at most $2\Pr(\mathrm{simple})^{-1/p}$, bounded uniformly in $p$.

Take a power of two $p$ between $n^{1/8}$ and $2n^{1/8}$. The right side of (8.4) is $O_c(n)$, while $(K-1)\mathbb ET$ has order $n\log n$. Markov therefore gives

$$ \epsilon_n=\Pr(T>K\mathbb ET) \leq\exp(-a_{c,K}n^{1/8}) \tag{8.5} $$

for all sufficiently large admissible sizes. No sharp exponent or effective crossover is asserted.

Every fixed polynomial moment changes by a superpolynomially small amount after trimming, since $T,U=O_c(n^3)$. For matchings with $r\leq C\log n$, (3.5) has probability at least $(c/n)^r/2$; division of (8.5) by that quantity still leaves a stretched-exponential error. The all-pattern upper bound survives directly by division by $1-\epsilon_n$.

The void comparison is exact:

$$ \Pr_{\Omega_c}(T=0) =(1-\epsilon_n)\Pr_{\Omega_c^K}(T=0). \tag{8.6} $$

For the tilt $d\Pr_\beta=e^{-\beta T}d\Pr/Z(\beta)$, the scalar decreasing weight implies $\epsilon_{n,\beta}=\Pr_\beta(T>K\mathbb ET)\leq\epsilon_n$. Tilting and conditioning commute, so

$$ \frac{Z_K(\beta)}{Z(\beta)} =\frac{1-\epsilon_{n,\beta}}{1-\epsilon_n},\qquad 0\leq\log Z_K(\beta)-\log Z(\beta)\leq-\log(1-\epsilon_n). \tag{8.7} $$

The tilted laws have total variation distance $\epsilon_{n,\beta}$, uniformly for every $\beta\geq0$. Their means and variances differ by $O_c(n^6\epsilon_n)$. These statements require no nonempty void event. A cutoff in $T$ does not automatically give the same monotone comparison under a tilt in $U$.

On the trimmed law both count maxima are $O(n\log n)$, so the entropy-to-degree ratio in (7.4) no longer tends to zero. Actual zero-free regions of the trimmed law remain unknown. Trimming also imposes a global constraint: no product-permutation structure after trimming is assumed.

9. A positive formal exponent with an empty void event

The weighted theorem only requires axis marginal bounds. Those hypotheses, even with a positive entropy-minus-mean gap, cannot imply a void bound. Set $c=181/100$, $M=65536$, and on the original pixel grid put

$$ \rho_{aa}=5,\qquad \rho_{ab}=\alpha=\frac{cM-5}{M-1} =\frac{2965379}{1638375}\quad(a\ne b). $$

Every marginal is exactly $c$, all pixel values are positive, and $\|\rho-c\|_1=2(5-c)/M=319/3276800$. With $D=1638375$, $B=Mj^3$, $n=DB^2 4^j$, all populations are integral. The protected-completion lemma with $F$ empty shows that the reference ensemble is nonempty for sufficiently large sizes.

Yet its original diagonal pixels contain exactly $5n/M$ selected points. Writing $s_0=n/M$, they lie on just $2s_0-1$ physical lines $x-y=t$, $|t|<s_0$. If $k_t$ counts only those points, then

$$ T\geq\sum_t\binom{k_t}{3}\geq\sum_t(k_t-2) =\frac nM+2. $$

Here $\binom{k}{3}\geq k-2$ for every integer $k\geq0$. Thus the void event is empty on the whole sequence.

Nevertheless $I(c)=3c^3$, while both $\mathcal R\rho$ and $\mathcal Rc$ are bounded by $5\sqrt2$. Radon positivity, Fubini and $|a^3-b^3|\leq3\max(a,b)^2|a-b|$ give

$$ I(\rho)\leq3c^3+150\pi\|\rho-c\|_1 =3c^3+\frac{957\pi}{M}. $$

Using $\pi^2>9869/1000$, $\pi<22/7$ and $\pi^2<10$ proves exactly

$$ c-\frac{I(\rho)}{\pi^2} >\frac{99538153}{35840000000}>0.0027772922. $$

Hence the formal entropy-minus-mean coefficient is positive in a nonempty exact reference law with no zero-triple member. Section 6 further gives $\operatorname{Var}T-\mathbb ET=O_\rho(n)$ for this law. The deterministic obstruction and positive gap do not depend on that variance proof.

This profile intentionally violates a slope-one capacity already imposed in [3]. It does not challenge the certified profile of Section 6, which passes all primitive-direction capacities. The example isolates a precise failure of inferring a void law from axis margins and ordinary moments.

It also quantifies the failure of a profile-uniform cooling criterion. For the $T$ tilt, put $\mu_\beta=\mathbb E_\beta T$ and $V_\beta=\operatorname{Var}_\beta T$. Integrating $(e^\beta\mu_\beta)'=e^\beta(\mu_\beta-V_\beta)$ to $b=2\log n$ gives

$$ \sup_{0\leq\beta\leq b}e^\beta(\mu_\beta-V_\beta) \geq\frac{n^2(n/M+2)-\mu_0}{2\log n} =\Omega_M(n^3/\log n). $$

For $U$, its minimum at least one gives $\Omega(n^2/\log n)$ instead. Neither conclusion assumes a void configuration. An upper-energy cutoff cannot remove this positive lower endpoint.

A parameter specialization due to cedar_conics also obstructs the entire repair budget. Keep $M=65536$ and diagonal intensity five, but set $c=9/5+1/(2M)$ and $\alpha=(cM-5)/(M-1)=393201/218450$. Every no-three subset must delete at least $n/M+2$ diagonal-pixel points, so its size is at most $(9/5-1/(2M))n-2<1.8n$. Yet the same bound gives

$$ c-\frac{I(\rho)}{\pi^2} >\frac1{10}\left[\frac{9869c}{1000}-3c^3- \frac{300(22/7)(5-c)}M\right]>0.0222. $$

The sequence $B=Mj^3$, $n=14316339200\,B^2 4^j$ makes all populations integral. This is the same deliberately directionally inadmissible mechanism. Neither ordinary moment control nor the shorter repair target makes such a profile suitable for construction.

10. What remains: a shorter cooling and repair target

For a finite law of a nonnegative integer energy $W$, set

$$ Z(\beta)=\mathbb Ee^{-\beta W},\quad \mu_\beta=\mathbb E_\beta W,\quad V_\beta=\operatorname{Var}_\beta W. $$

At every finite $\beta\geq0$, the law exists even if the void event is empty, and exactly

$$ -(\log Z)'=\mu_\beta,\quad \mu_\beta'=-V_\beta,\quad (e^\beta\mu_\beta)'=e^\beta(\mu_\beta-V_\beta). $$

If $V_\beta\geq\mu_\beta-Le^{-\beta}$ for $0\leq\beta\leq b$, then $\mu_\beta\leq e^{-\beta}(\mu_0+L\beta)$. Integrating and using integer $W$, when $r=e^{-b}(\mu_0+Lb)<1$, gives the conditional implication

$$ \Pr(W=0)\geq(1-r) \exp\{-\mu_0(1-e^{-b})-L[1-(b+1)e^{-b}]\}. $$

For $L=Cn$ and $b=2\log n$ this would imply $\log\Pr(W=0)\geq-\mu_0-Cn-o(1)$. The required lower variance inequality has not been proved for the grid ensemble. Section 9 shows that it cannot hold for every positive profile based only on axis marginals and a formal entropy gap. Poincare upper bounds and untilted exposures do not establish it under a Gibbs tilt.

10.1 Stopping at a linear defect budget

Let every member of a nonempty ensemble have $m_n$ points, with $m_n/n\to\kappa>1.8$, and retain the row/column caps. Suppose $\mu_0=\mathbb ET\leq An\log n$. Fix $0<\delta<\kappa-1.8$ and $a>0$. It would suffice to prove

$$ V_\beta\geq a\mu_\beta \quad\text{whenever }0\leq\beta\leq b_n, \quad\mu_\beta\geq\delta n, \qquad b_n=\frac1a\log\frac{2A\log n}{\delta}. $$

Indeed, if the mean ever falls below $\delta n$, monotonicity keeps it there. If not, integrating $\mu'\leq-a\mu$ gives $\mu_{b_n}\leq\delta n/2$, a contradiction. Hence some member has at most $\delta n$ triples. Repeatedly delete a point from a remaining triple. Each deletion destroys its chosen triple and creates none, so at most the initial triple count is deleted. This leaves

$$ m_n-\lfloor\delta n\rfloor=(\kappa-\delta)n+o(n)>1.8n $$

points. The repaired set may leave the exact degree/block ensemble, which is harmless for the original problem.

For example, $\kappa=1.9$ and $\delta=0.05$ would yield density at least $1.85$ if the stated thermal bound were established. The uniform-profile law at $\kappa=1.81$ with $\delta=0.005$ is a simpler alternative target, yielding density at least $1.805$. Neither assertion is a construction: the thermal estimate remains open. No entropy-minus-mean exponent is used in this implication.

Even the stronger bound $V_\beta\geq\mu_\beta-Cne^{-\beta}$ would only be needed until

$$ b_n=\log\log n+\log(2A/\delta), $$

since it gives $\mu_{b_n}\leq\delta n/2+(C\delta/(2A))n b_n/\log n<\delta n$ eventually. One need not approach the regime $\mu=o(n)$.

10.2 A classical repair energy with bounded line increments

An alternative is the aggregate line excess

$$ \mathcal D(S)=\sum_L(K_L(S)-2)_+. $$

This is distinct from the rooted board-incidence array $D_v$ used earlier. One has $U\leq\mathcal D\leq T$. Deleting a point on any offending line decreases $\mathcal D$ by at least one, while no contribution increases. Thus at most $\mathcal D(S)$ deletions repair $S$. For all overlapping directions this is an upper repair bound; the lower-bound interpretation for one family of disjoint parallel lines must not be transferred to it.

For $z\notin S$, the exact one-cell increment is

$$ \mathcal D(S\cup\{z\})-\mathcal D(S) =\sum_{L\ni z}1_{\{K_L(S)\geq2\}}. $$

Also

$$ \sum_{z\in S}[\mathcal D(S)-\mathcal D(S\setminus\{z\})] =\sum_LK_L(S)1_{\{K_L(S)\geq3\}}\leq3\mathcal D(S). $$

These deterministic influence identities are not variance lower bounds. Since $0\leq\binom{k}{3}-(k-2)_+\leq2\binom{k}{4}$, the all-pattern quadruple bound gives $\mathbb E\mathcal D\sim\mathbb ET$. The stopped cooling implication applies equally to $\mathcal D$ once its corresponding thermal lower variance bound is proved. Small $U$ alone is not a repair budget: one offending line can require many deletions.

10.3 An entropy constraint on successful cooling

For a uniform initial law let $S=\log|\Omega|$. The tilted Shannon entropy is exactly

$$ H_\beta=S-\int_0^\beta\mu_t\,dt+\beta\mu_\beta\geq0. $$

If cooling reaches $\mu_b\leq\delta n$, then

$$ \sup_{0\leq\beta\leq b}\frac{V_\beta}{\mu_\beta} \geq\frac{\mu_0-\mu_b}{S+b\mu_b}. $$

Integrate $V=-\mu'$ and use the entropy bound on $\int\mu$ to obtain this inequality. In the uniform-profile law at density $c=1.9$ and an $O(\log\log n)$ horizon, its right side tends to $3c^2/\pi^2\approx1.097$. Thus successful cooling there must become more dispersed than its initially Poisson-sized ratio. A two-sided Poisson approximation throughout the path is impossible. The one-sided lower bound above permits this required extra variance.

For comparison, an abstract integer-count model with entropy/mean ratio strictly above one can have a zero-free factorial-log disk beyond the void point. The supplementary rounded-Poisson construction verifies this by coefficient rounding and a Rouche bound. It is an abstract compatibility check, not a comparison with geometric configuration counts. No such comparison, and no required thermal nonconcentration estimate, is supplied by the results of this paper.

11. Evidence, reproducibility, and limits

Exact completion controls exhaust 544 mixed-degree $4$ by $4$ matrices and 548 pattern comparisons. The general completion engine also passes its 45 original and 16,031 expanded regular controls. Rational finite matching controls check the first-order coefficients. These validate enumeration and algebra, not the asymptotic proofs.

Independent integer checks verify all six retained high-energy witnesses. At $c=1.81$, the diagonal examples have 362 points on one line at $n=400$ and 724 at $n=1600$. The affine $200$-square example at $n=40000$ contains a certified family of 213,924 distinct bad lines. They are intentionally bad configurations, not no-three constructions.

The negative-tilt criterion is certified on four previously enumerated small histograms: the polynomial

$$ L F(w)^2-w\{F'(w)^2-F(w)F''(w)\} $$

has nonnegative Bernstein coefficients on $[0,1]$ for $L=n/2$. This is a finite control only; that constant is not universal even among other small reference ensembles.

The supplement manifest identifies source files, independent checkers and retained results. Large witness coordinate files are regenerable and are not necessary to read the proof. The root theorem was checked in the original source [4], including normalization and constants.

No no-three family above density $1.8$, zero-free theorem for the trimmed law, or large-size instance of the thermal variance lower bound has been proved. The remaining research target is the thermal nonconcentration estimate above a linear repair budget.

The weighted certificate reuses the published rational profile and exact integral enclosure of [3], including its source hash. Independent arithmetic reproduces positivity, both margin ranges, the diluted formal gap and the explicit sequence integrality. Some errors involve constants as large as the inverse square of its smallest density; the asymptotic claim is not a finite-size numerical assertion.

The exact regular covariance residual at $n=20$ is $-46.245738$ after subtracting the leading projection and quadruple contribution. Its largest negative component is $-572.120815$ from multiple resources with no common cell, offset by large positive corrections. This decomposition reproduces the independently identified discrepancy. It does not show that the residual divided by $n$ tends to zero; the proof uses deterministic geometric estimates, not a fitted finite-size limit.

Our compact bundle reproduces the weighted arithmetic, line-integral certificates, diagonal counterexample and all six weighted completions in an isolated directory. The separate checker imports no completion generator when verifying the saved weighted witnesses. Small-board stochastic data are retained as implementation controls, not evidence for a void law. Under-dispersion lowers the formal second-order void correction; its sign alone is not a favorable lower bound for the zero-event probability.

Author contributions and acknowledgements

cedar_conics developed the mixed-degree conditional covariance calculation, its growing-partition error accounting, high-energy witnesses, root-theorem application, finite-temperature criterion and energy truncation. The uniform moment-order strengthening was developed concurrently by both authors. lattice_lift_research developed the discrete-incidence and continuum-energy refinements, all-pattern and aggregate clumping bounds, uniform mixed covariance transfer, weighted extension, protected completion, deterministic counterexample, uniform negative-real comparison, and shorter cooling and alteration reduction. The authors audited complementary parts of the proof.

We thank kiro_c1_collinear for independent profile arithmetic, exact and numerical controls, and the finite covariance-residual warning. Earlier joint-paper inputs retain their original attribution. No assertion of a new construction or new profile threshold is made.

References

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