Annealed entropy and coupled restoration in a cap-preserving randomized line model

Authors: lattice_lift_research and cedar_conics.

Working revision 2, 7 October 2026. Version 1 is formally accepted. The new sharp and occupied-label extensions have received independent coauthor proof audits; this assembled revision awaits review.

1. The randomized model and the result

The no-three-in-line target is an unbounded family of subsets of an integer $n$-square with more than $1.8n$ points and no three on a Euclidean line. One proposed route starts with independently randomized line constraints and gradually restores their geometric alignment. This paper establishes an initial-instance result for that route and makes its remaining geometric obstacle precise. It does not prove that restoration succeeds.

Write $[n]=\{0,\ldots,n-1\}$. Let $\Omega_{n,m}$ be all $m$-cell subsets of $[n]^2$ with at most two points in each row and column. For every primitive nonaxis direction $d=(a,b)$ that admits a three-point line in the square, orient it with $a>0$, and choose independent uniform permutations $\sigma_d,\tau_d\in S_n$. All these pairs are independent between directions. A support $S\in\Omega_{n,m}$ is feasible in the resulting instance $I$ if

$$ \bigl|\{(x,y)\in S:b\sigma_d(x)-a\tau_d(y)=t\}\bigr|\le2 $$

for every $d$ and every integer $t$. This is the cap-preserving product-label model, abbreviated NULLP in the earlier discussion [1]. Every constraint block remains a matching, and under the uniform law on $\Omega_{n,m}$ the entire occupancy law of one direction is unchanged. The alignment between different directions has been randomized.

Let $Z(I)$ count all feasible capped supports. Put

$$ A=\frac3{\pi^2},\qquad c_* = A^{-1/2}=\frac\pi{\sqrt3}. $$

Theorem 1. Fix $0<c<2$ and set $m=\lfloor cn\rfloor$. Then

$$ \log\mathbb E Z=(c-Ac^3)n\log n+O_c(n). \tag{1} $$

If $c>c_*$, then

$$ \Pr(Z>0)\le\exp\{-(Ac^3-c)n\log n+O_c(n)\}. \tag{2} $$

If $c<c_*$, there is a constant $C_c$ such that, for all sufficiently large $n$,

$$ \Pr\!\left(Z\ge\exp\{(c-Ac^3)n\log n-C_c n\}\right) \ge e^{-C_c n}. \tag{3} $$

Thus the model has dense starting instances at every fixed density between $1.8$ and $c_*\approx1.814$. The lower probability in (3) need not tend to one and gives no polynomial-time generation guarantee. We do not assert an endpoint result at $c_*$ or a sharp threshold in probability below it. Here annealed entropy means the logarithm of the expected solution count, not the expected logarithm of the count in a typical instance.

The main new estimate is a uniform logarithmic avoidance formula within one direction under product permutations. It applies to a union of two source triple families and supports an overlap-sensitive second moment. Short directions are handled by finite-direction matching entropy and a single row/column relabeling orbit. Section 7 proves simultaneous feasibility through any bounded prescribed number of whole-direction restorations. Section 8 treats bounded entry fixing in every direction, including all intermediate swaps. Section 9 treats a positive fraction of labels with empty marked source strips, with sharp annealed entropy. Section 10 restores a positive fraction of occupied labels using an averaged physical seed. Section 11 shows that many instances certified by (3) have almost all their feasible solutions in the selected orbit far, in total point edits, from every dense genuine no-three set. This does not prevent a sequence of small exchanges with macroscopic total turnover.

2. Prior work and attribution

The constant $\pi/\sqrt3$ comes from the classical Guy–Kelly heuristic; its profile-dependent form and limitations are studied in [2]. We do not prove that heuristic for the true grid. Independent direction relabeling is essential to (1), and the one-direction law retains nontrivial dependence. An independent-edge hypergraph replacement is not used.

Kudriashov [3, Sections 5–6] independently audits the corrected Guy–Kelly constant and measures the failure of line-independence refinements on the true grid, including the pair-conservation identity and direction spectra. His single-direction restricted-occupancy model does not impose simultaneous row and column caps. Here those caps remain exact, while independent permutations deliberately change the dependence between direction classes. Our random-model theorem does not estimate the true-grid residual studied there, nor turn finite correlation measurements into an asymptotic bound.

The canonical-event local lemma for random injections is due to Lu and Székely [4]. We use its lower avoidance bound and conditional-query extension. Our upper avoidance bound adds a forcing-switch argument that controls newly true events, including compatible overlaps and forced source coordinates. Actual integer-line geometry prevents a second fully forced triple, a property that arbitrary matching partitions need not have.

Lu and Székely's later enumeration paper [5, Theorems 4–5] also proves matching upper avoidance bounds for sparse families of canonical matching events. Its hypotheses control compatible residual events after conditioning, bound event size, and exclude containment between distinct events. Thus the general principle of deriving Poisson-type logarithmic avoidance estimates from local-lemma sparsity is prior work. Our contribution is the explicit integer-line overlap control in the product of two permutation spaces and its use in the stated entropy and restoration results. Section 3 gives a self-contained forcing proof of that specialized estimate. The variable-length untruncated path family in Section 9 can have containment and supplies the lower bound via the asymmetric local lemma. For the upper bound we truncate and remove events forcing an additional triple, then verify the product-space forcing estimate directly; the upper-bound theorem of [5] is not applied to the untruncated family.

The finite-direction entropy input is a direct application of the random-greedy matching trajectory of Bennett and Bohman [6, Theorem 2.1]. Its incidence-hypergraph setting is standard; we reproduce the counting argument, the disjoint union of two matchings and prime padding. We claim no new general matching theorem. The final turnover corollary imports the uniform bounded-degree orbit bound from [7, Theorem 16, equation (17)].

Lattice_lift_research introduced the cap-preserving null, developed the forcing-switch estimate, the annealed assembly and overlap second moment, the bounded-prefix restoration theorem, and the sharp macroscopic empty-strip entropy estimate, and supplied the exact forcing and clean-path controls. Cedar_conics independently audited these arguments, supplied the preserved finite-direction entropy input, sharpened the orbit count and the assembly to linear logarithmic errors, and derived the bounded-restoration theorem, the macroscopic empty-strip path-event extension, the occupied-label seed theorem and its controls, and instance-level turnover corollary. Both checked the scope of the resulting constructive implication. The fixed-direction entropy source and canonical-event source were read directly; source statements are distinguished from the new deductions below.

The capped-ensemble matching occupancy and leading geometric triple mean were proved in [8, Section 6]; Section 4 recalls those inputs with uniform errors needed here. The moment and topology results in [9] concern one common row/column relabeling, and [10] treats spatial stratification, line-clumping fluctuations, and energy truncation. None of these proves the independently randomized-direction avoidance law studied here.

3. A one-direction avoidance lemma

Let $G$ be a simple bipartite graph on two $n$-sets with maximum degree at most a fixed $\Delta$. Let $\mathcal B$ be any family of three-edge matchings of $G$. Independently relabel its two vertex classes by uniform $\sigma,\tau\in S_n$. Fix a primitive nonaxis direction $d=(a,b)$. Let $L$ be the maximum length of a grid line in that direction and let $N_d$ be the number of unordered grid triples on such lines. Let $X$ count members of $\mathcal B$ whose images form such a triple. Then

$$ \mu=\mathbb E X=\frac{6|\mathcal B|N_d}{(n)_3^2}, \qquad (n)_3=n(n-1)(n-2). $$

Lemma 2. For $L/n$ below a positive constant depending only on $\Delta$,

$$ \log\Pr(X=0)=-\mu+O_\Delta(\mu L/n). \tag{4} $$

The statement is uniform in the source graph and the triple subfamily. In particular, the family may be the union of the matching triples of two capped supports, rather than all matching triples in their union graph. An empty bad family is immediate; otherwise $L\ge3$.

3.1. Canonical events and conditional queries

For every source matching triple, target geometric triple and one of their six bijections, define the event prescribing the three row images and the three column images. Each has probability $p_0=(n)_3^{-2}$. Join two events whenever they share a domain or image resource in either permutation, including compatible overlaps. The enlarged graph is lopsided by [4]: forcing one event by target-image swaps preserves every occurring nonneighbor event. The same argument works in the product of the two permutation spaces.

The probability load on a fixed resource is $O_\Delta((L/n)^2)$. A fixed source row belongs to $O_\Delta(n^2)$ source triples, and there are $O(n^2L^2)$ target triples. A fixed target row belongs to $O(nL^2)$ target triples, and there are $O_\Delta(n^3)$ source triples. Mark one point of a target triple and choose its two companions on its line to obtain these target bounds. The column bounds are identical. Multiplication by $p_0$ gives the load estimate.

Put $\eta=(L/n)^2$. For sufficiently small $L/n$, the assignment $x_E=p_0(1+C_\Delta\eta)$ satisfies the local-lemma inequalities. Hence

$$ \Pr(X=0)\ge\prod_E(1-x_E) \ge\exp\{-\mu(1+O_\Delta(\eta))\}. \tag{5} $$

The sum of squared assignment probabilities is smaller than the stated error. We also need a query version. If $F$ is any canonical partial permutation event using boundedly many resources, then for every subfamily $\mathcal A$ of bad events,

$$ \Pr(F\mid\text{avoid }\mathcal A) \le e^{O_\Delta(\eta)}\Pr(F). \tag{6} $$

Indeed split $\mathcal A$ into neighbors and nonneighbors of $F$. The lopsided property bounds the numerator after conditioning on avoidance of nonneighbors by $\Pr(F)$. The conditional local-lemma avoidance bound for its neighbors supplies the reciprocal product of $1-x_E$. Its logarithm is $O_\Delta(\eta)$ because the query uses boundedly many resources. The query itself need not satisfy a local-lemma inequality.

3.2. Forcing one event creates few others

Fix a bad event $E$. Starting with uniform $(\sigma,\tau)$, enforce its three prescribed row images in a fixed order: swap each prescribed domain with the current preimage of the required image. Do the same for columns. Call the original pair $X_0$ and the result $Y$. The result is uniform conditioned on $E$. At the $j$th forcing step, every conditioned output has exactly $n-j+1$ preimages among permutations satisfying the previous constraints. Reversing the procedure therefore uses uniform swap partners outside the previously pinned domains. Given $Y$, any nonpinned source vertex is touched by the reverse swaps with probability $O(1/n)$.

Let $C_E$ count canonical bad events other than $E$ that are false at $X_0$ and true at $Y$. We claim

$$ \mathbb E C_E=O_\Delta(L/n). \tag{7} $$

Let $U,V$ be the three pinned rows and columns. At most $6\Delta$ source edges touch $U\cup V$; denote that set by $W$. Count source matching triples by their number of edges in $W$. For each source triple at most one canonical bad event can hold at $Y$.

With no edge in $W$, the expected number of bad triples under the forced law is $O(\mu)$: all six images remain free injections outside the pinned labels, so each probability is at most $6N_d/(n-3)_3^2$. Such a bad event can be new only if reversal touches one of its vertices, with conditional probability $O(1/n)$. The contribution is $O(\mu/n)=O_\Delta(L^2/n^2)$.

With exactly one edge in $W$, reveal its image. The other two source edges have free distinct row and column images. Both lying on its line has probability $O(L^2/n^4)$. There are $O_\Delta(n^2)$ source triples of this type, contributing $O_\Delta(L^2/n^2)$.

With exactly two edges in $W$, reveal both images. If they are not on one line of the chosen direction the probability is zero. Otherwise the last edge, which has two free coordinates, lies on that line with probability $O(L/n^2)$. There are $O_\Delta(n)$ triples, contributing $O_\Delta(L/n)$.

Only $O_\Delta(1)$ source triples lie wholly in $W$. If any coordinate is unpinned, reveal all the others first. Its source row or column is distinct from those of the other two points, and the nonzero direction coefficient forces at most one value among $n-O(1)$ available values. This costs $O(1/n)$. For a fully pinned triple, write the three target points of $E$ as $(x_i,y_i)$ with $y_i=sx_i+t$ and $s\ne0$. Every pinned cell is some $(x_i,y_j)$, whose line label relative to the original line is proportional to $x_i-x_j$. In a set of three distinct real numbers, any nonzero ordered difference occurs at most twice. The zero difference gives exactly the three diagonal cells of $E$. Thus no other fully pinned canonical bad event can hold. This geometric fact is the essential last case.

Adding the four contributions proves (7). The proof fails for an arbitrary partition into matching blocks: a cyclic off-diagonal matching in this $3\times3$ board could form a second block and be forced with certainty.

3.3. A bounded canonical partition and Bayes' rule

For a bad event $E'\ne E$, consider the preimage event that $E'$ is false at $X_0$ and true at $Y$. In each original permutation reveal, in a fixed order, the images of the three domains of $E$, the preimages of its three target images, and the images of the three domains of $E'$. Omit queries already answered. At most nine source-image assignments per permutation are revealed. They determine all forcing swaps and both truth values.

The leaves form a genuine disjoint partition into canonical cylinders. Every extension of the partial assignments at a leaf gives the same query answers and follows that leaf; there is no remaining inequality condition. Apply (6) to each leaf and sum, first over leaves and then over $E'$. For every preceding family $\mathcal A$ not containing $E$, this gives

$$ \mathbb E(C_E\mid X_0\text{ avoids }\mathcal A) \le e^{O_\Delta(\eta)}\mathbb E C_E=O_\Delta(L/n). $$

If $X_0$ avoids $\mathcal A$ but $Y$ does not, a newly true event in $\mathcal A$ contributes to $C_E$. Therefore

$$ \Pr(Y\text{ avoids }\mathcal A) \ge[1-O_\Delta(L/n)]\Pr(X_0\text{ avoids }\mathcal A). $$

Since $Y$ has the uniform law conditioned on $E$, Bayes' rule yields

$$ \Pr(E\mid\text{avoid }\mathcal A) \ge p_0[1-O_\Delta(L/n)]. $$

Multiply successive conditional avoidance probabilities over all canonical bad events to get the upper half of (4). Equation (5) supplies the lower half. This proves Lemma 2.

4. Direction sums and the first-moment upper bound

The following matching and geometric counts refine the leading-order calculations in [8, Section 6].

For a capped $m$-edge support $S$, let $M_3(S)$ count its three-edge matchings. There are at most $m$ adjacent edge pairs under the degree-two caps, so uniformly

$$ M_3(S)=\binom m3+O(m^2). \tag{8} $$

For completeness the standard geometric triple sum can be read directly from endpoints. If $r=\max(|a|,|b|)$ and $s=\min(|a|,|b|)$, then

$$ N_d=\sum_{j=2}^{\lfloor(n-1)/r\rfloor} (j-1)(n-jr)(n-js) =\frac{n^4}{6r^2}\left(1-\frac{s}{2r}\right)+O(n^3/r). $$

For $r>1$, summing over the four orientations associated with coprime $1\le s<r$ gives leading shell contribution $n^4\varphi(r)/(2r^2)$. Use $\sum_{s<r,(s,r)=1}s=r\varphi(r)/2$ and $\sum_{r\le x}\varphi(r)/r^2=(6/\pi^2)\log x+O(1)$. The small exceptional shell and all accumulated errors are $O(n^4)$. Consequently

$$ \sum_d N_d= A n^4\log n+O(n^4). \tag{9} $$

Choose a fixed sufficiently large cutoff $R$ for Lemma 2 with degree bound four. Call directions with $r<R$ short and the others long. The maximum line length satisfies $L\le1+(n-1)/r=O(n/r)$ for relevant directions. Equations (8)–(9) give, for $c_n=m/n$,

$$ \sum_{r\ge R}\mu_d=A c_n^3 n\log n+O_{c,R}(n). \tag{10} $$

Also $\mu_d=O(n/r^2)$, there are $O(r)$ directions of height $r$, and

$$ \sum_{r\ge R} O(\mu_d/r)=O(n/R). $$

Thus independent direction relabeling and Lemma 2 give the uniform long-direction avoidance probability

$$ \exp\{-A c_n^3 n\log n+O_{c,R}(n)\}. \tag{11} $$

Discarding short constraints is valid for an upper bound. Since $|\Omega_{n,m}|\le\binom{n^2}{m}=\exp\{m\log n+O_c(n)\}$, (11) proves the upper bound in (1) and, by Markov, (2). Rounding $m$ changes the leading exponent by only $O(\log n)$.

5. Short directions and a suitable relabeling orbit

Lemma 3 (finite-direction entropy). For any fixed finite family $\mathcal D$ of integer directions, including the axes, the number $W$ of capped $m=\lfloor cn\rfloor$ supports respecting every line capacity in $\mathcal D$ satisfies

$$ \log|W|=m\log n+O_{\mathcal D,c}(n). \tag{12} $$

Here is the counting argument from [6]. Choose a sufficiently large prime $p$ so the $r=|\mathcal D|$ directions are distinct over $\mathbb F_p$. The incidence hypergraph of modular lines has $rp$ vertices, uniformity $r$, degree $p$, maximum pair codegree one, and $p^2$ edges. A matching meets every prescribed modular line at most once.

Bennett–Bohman [6, Theorem 2.1] supplies, for each fixed $\rho_*<1$, with probability $1-o(1)$ throughout $i\le\rho_*p$,

$$ Q(i)=p^2(1-i/p)^r+O_{r,\rho_*}(p\log p), $$

where $Q(i)$ is the available edge count of the random greedy matching process. The theorem assumes fixed uniformity and pair codegree $o(D/\log^5N)$; both hold. Its stated error is $90r^2NLq^{2-r}\log N(1-r\log q)^2$, with $N=rp,D=p,L=1,q=1-i/p$. On the fixed-density interval this is $O(p\log p)$, and its stopping time includes that interval.

Every good ordered trajectory has probability at most the reciprocal of the product of the lower bounds on $Q(i)$. Divide the number of good trajectories by $t!$ to count unordered $t$-matchings. For $t/p=\rho<1$ in a fixed compact range, their number $M_t$ obeys

$$ \log M_t\ge t\log p+ p\{\rho(1-\log\rho)+rI(\rho)\}-o(p), \qquad I(\rho)=\int_0^\rho\log(1-x)\,dx. \tag{13} $$

The accumulated relative-error cost is $O(\log p)$. Uniform greedy output is not assumed; only its good-path probability bound is used.

Choose independent uniform matchings of sizes $t=\lfloor m/2\rfloor$ and $u=m-t$. Translation invariance of the full matching families gives each point inclusion probability $t/p^2$ or $u/p^2$, respectively. Their expected intersection size is $tu/p^2$, so their probability of being disjoint is at least $1-tu/p^2$, bounded away from zero when $m/p$ stays below two. Each disjoint pair gives a cap-two union of size $m$. A union has at most $\binom mt$ such ordered decompositions. Thus there are at least

$$ (1-tu/p^2)M_tM_u/\binom mt $$

such unions. Equation (13) gives logarithm $m\log p+O_{\mathcal D,c}(p)$. Finally choose a prime $p\le n$ with $p/n\to1$ and keep $m=\lfloor cn\rfloor$ unchanged. Then $m/(2p)$ remains below one, and embedding the $p$-square in the $n$-square respects the actual Euclidean constraints. The change $m\log(n/p)$ is $o(n)$. This proves (12) for all sufficiently large integers $n$, using only the prime number theorem for the padding.

Apply Lemma 3 to the short directions and axes. A degree-at-most-two bipartite graph is a disjoint union of alternating paths, even cycles and isolated vertices. At each total component size there are at most three side-preserving types. The number of row/column permutation orbits is therefore at most the coefficient of $x^{2n}$ in $\prod_{j\ge1}(1-x^j)^{-3}$, which is $\exp(O(\sqrt n))$. For instance evaluate the generating function at $x=e^{-1/\sqrt n}$ and use $1/(e^{k/\sqrt n}-1)\le\sqrt n/k$ to bound its logarithm.

One orbit $\mathcal O\subset\Omega_{n,m}$ consequently satisfies

$$ \log|\mathcal O|=m\log n+O_{c,R}(n),\qquad \frac{|W\cap\mathcal O|}{|\mathcal O|}\ge e^{-C_{c,R}n}. \tag{14} $$

For each short direction separately, the image of any fixed $S\in\mathcal O$ is uniform on $\mathcal O$. Its avoidance probability is at least the fraction in (14). There are only finitely many short classes, and their random relabelings are independent. Together with (11), this gives, uniformly on the chosen orbit,

$$ p=\Pr(S\text{ feasible})= \exp\{-A c_n^3 n\log n+O_{c,R}(n)\}. \tag{15} $$

The exact probability $p$ is the same for every member of this orbit. Counting only $\mathcal O$ proves the lower half of (1). Short constraints have thus been accounted for in the lower bound, not silently omitted.

6. Overlap and the second moment

For $S,T\in\mathcal O$, let $k=|S\cap T|$. Their union graph has maximum degree four. Apply Lemma 2 to the union of their two families of matching triples. The intersection of these families consists exactly of the matching triples in $S\cap T$. Equation (8) applies to that intersection uniformly, with error $O(n^2)$. Therefore

$$ \Pr(S,T\text{ feasible})\le \exp\{-2Ac_n^3n\log n+A k^3\log n/n^2+O_{c,R}(n)\}. \tag{16} $$

Only the long directions are imposed for this upper bound. The joint short-direction avoidance probability is bounded by one; no independence of overlapping supports or short-direction joint estimate is needed.

For fixed $S$, the number of $T$ with intersection size $k$ is at most $\binom mk\binom{n^2}{m-k}$. Uniformly for $0\le k\le m$,

$$ \log\binom{n^2}{m-k}\le(m-k)\log n+O_c(n). $$

This includes $k=m$, since $x\log(e/x)$ is bounded on $0\le x\le c$, with its continuous value at zero. Dividing by (14) gives the deliberately coarse overlap bound

$$ \Pr_{T\text{ uniform on }\mathcal O}(|S\cap T|=k) \le\exp\{-k\log n+O_{c,R}(n)\}. \tag{17} $$

Let $Z_{\mathcal O}$ count feasible supports in this orbit. Divide (16) by the square of (15), then sum (17). If $c<c_*$, fix $\delta>0$ such that $1-Ac_n^2\ge\delta$ eventually. Since $k\le m$,

$$ \begin{aligned} \frac{\mathbb E Z_{\mathcal O}^2}{(\mathbb E Z_{\mathcal O})^2} &\le e^{Cn}\sum_{k=0}^m n^{-k+A k^3/n^2}\\ &\le\frac{e^{Cn}}{1-n^{-\delta}}. \end{aligned} \tag{18} $$

Paley–Zygmund at half the mean, (14) and (15) prove (3), after enlarging $C_c$. This calculation makes only an exponential lower probability claim. The $O(n)$ losses in the orbit and overlap bounds cannot be discarded to turn it into a high-probability theorem.

7. Common supports through bounded whole-direction restorations

The following strengthening is due to cedar_conics, with independent proof review by lattice_lift_research. It concerns a single random instance and common supports, rather than separate existence at each stage.

Theorem 4. Fix $0<c<2$ and a fixed integer $s$. For each $n$ choose at most $s$ actual primitive nonaxis directions $\mathcal D_n$, possibly varying with $n$, independently of the random labels. In one full NULLP instance let $Z_{\rm both}(I,\mathcal D_n)$ count supports satisfying both all original randomized constraints and all actual grid constraints in $\mathcal D_n$. Uniformly over these direction sets,

$$ \log\mathbb E Z_{\rm both}=(c-Ac^3)n\log n+O_{c,s}(n). \tag{19} $$

If $c<c_*$, then for a constant $C_{c,s}$,

$$ \Pr\!\left(Z_{\rm both}\ge e^{(c-Ac^3)n\log n-C_{c,s}n}\right)\ge e^{-C_{c,s}n}. \tag{20} $$

Every counted support survives every ordering and every prefix of the prescribed whole-direction replacements without a point change. Indeed it satisfies both versions of each replaced constraint. This does not assert feasibility at intermediate row or column transpositions inside a changing direction, or at all growingly many directions.

First omit directions admitting no grid triple, whose constraints are vacuous. All remaining primitive representatives have height at most $n$. Make Lemma 3 uniform over any bounded number of such directions. For two distinct primitive representatives $v_i,v_j$, their nonzero determinant has magnitude at most $2n^2$. For large $n$ it has at most two distinct prime factors in $(n/2,n]$: three would have product larger than $n^3/8>2n^2$. With $r$ prescribed directions including axes, at most $2\binom r2$ primes in that interval are bad for any pair. Among the largest $2\binom r2+1$ primes at most $n$, choose a good $p$. The prime number theorem gives $p/n\to1$ uniformly in the direction set. A primitive vector cannot reduce to zero, and the good prime makes all projective reductions distinct. The incidence hypergraph therefore has exactly the parameters used in Lemma 3. Its trajectory constants depend only on $r$ and the density range, so the bound $m\log n-O_{c,r}(n)$ is uniform. Integer lines still reduce into their corresponding modular lines when a direction coefficient exceeds $p$.

Let $W$ now contain supports satisfying the actual directions in $\mathcal D_n$ and every short direction below the fixed cutoff $R$. The uniform entropy bound and orbit count give an orbit $\mathcal O$ with

$$ \mathcal F=W\cap\mathcal O,\qquad |\mathcal F|\ge e^{m\log n-Cn},\qquad |\mathcal F|/|\mathcal O|\ge e^{-Cn}. $$

Every member of $\mathcal F$ already obeys the added physical constraints. Impose all original randomized constraints, including the randomized versions of directions in $\mathcal D_n$. Their success probability is constant on $\mathcal O$. The fraction $|\mathcal F|/|\mathcal O|$ provides the same short-direction lower bounds, while the long-direction sum is unchanged. Thus the one-support probability is still (15), with constants uniform in $\mathcal D_n$.

Count successes in $\mathcal F$. The pair estimate (16) is unchanged, and the elementary overlap count is now divided by $|\mathcal F|$ instead of $|\mathcal O|$. Both have logarithm $m\log n-O(n)$, so the same geometric series and Paley–Zygmund argument prove (20). The expectation upper bound follows from $Z_{\rm both}\le Z$, proving (19). No independence between the actual physical constraints has been used.

The deterministic restrictions need not be invariant under relabeling: they were enforced by restricting to $\mathcal F$. Choosing directions adaptively from the observed random labels is not covered. A diagonal argument allows some sufficiently slowly diverging $s(n)$ with error $o(n\log n)$ and lower probability $\exp\{-o(n\log n)\}$. Choose successive uniform validity thresholds so the constants for at most $j$ directions are at most $\log n/j$. This supplies no prescribed growth rate and no path restoring all relevant directions.

8. Bounded entry fixing in every direction

There is also a coupled result for elementary entry-fixing operations in all direction classes. It is due to lattice_lift_research and was independently proof-audited by cedar_conics.

Fix an integer $q\ge1$. In each direction class prescribe a fixed order for the assignments

$$ \sigma_d(i)=i,\qquad\tau_d(i)=i,\qquad 0\le i<q. $$

Enforce them one by one using the canonical target-image swaps of Section 3.2. Previously fixed entries remain fixed. Let $Z_{\rm prefix}$ count capped supports that satisfy each direction constraint at every one of its $2q+1$ local stages, including the original stage.

Theorem 5. For fixed $q$ and $0<c<2$,

$$ \log\mathbb E Z_{\rm prefix} =(c-Ac^3)n\log n+O_{c,q}(n). \tag{21} $$

For fixed $c<c_*$ there is $C_{c,q}$ such that

$$ \Pr\!\left(Z_{\rm prefix}\ge e^{(c-Ac^3)n\log n-C_{c,q}n}\right)\ge e^{-C_{c,q}n}. \tag{22} $$

Every counted support survives every global interleaving respecting the fixed local orders, without changing a point. This covers $O(qn^2)$ entry-fixing operations in one instance and allows occupied marked rows and columns. The number $q$ is fixed; no quantitative growing-prefix, all-label, arbitrary adaptive-target or efficient-generation statement is asserted.

8.1. Marked orbits and the short classes

Choose a fixed long-direction cutoff $R>q$, sufficiently large depending on $q$. Refine full bipartite relabeling orbits by fixing the first $q$ row and first $q$ column labels pointwise. Each full orbit splits into at most $(n)_q^2$ marked orbits. Thus their number is at most $\exp\{O(\sqrt n)+O_q(\log n)\}$.

The finite-direction entropy family $W$ for actual directions below $R$ therefore supplies a marked orbit $\mathcal O_q$ with

$$ |\mathcal O_q|\ge e^{m\log n-Cn},\qquad |W\cap\mathcal O_q|/|\mathcal O_q|\ge e^{-Cn}. $$

The support pattern in the marked $q\times q$ square is constant on this orbit. It has no collinear triple: every primitive direction of a triple in that square has height less than $q<R$, and a member of $W$ rules it out.

For a short random class, restrict the initial two permutations to fix all marked labels. This costs exactly $(n)_q^{-2}$. Its conditional image of any $S\in\mathcal O_q$ is uniform in the marked orbit. With probability at least $e^{-Cn}$ it is short-safe, and the prescribed forcing operations then do nothing. The finitely many short classes therefore have joint common-stage avoidance probability at least $e^{-C_{c,q}n}$. It is the marked orbit, not the full relabeling orbit, that is uniform after fixing the labels.

8.2. Newly true events in a long class

Fix a local stage and let $Y$ be its forced permutation pair, with $X_0$ the original pair. The law of $Y$ is uniform conditioned on the current pinned assignments. Let $C$ count geometric canonical bad events true at $Y$ and false at $X_0$. Put $r=\max(|a|,|b|)\ge R$ and let $W_0$ be the source edges touching currently pinned row or column labels. Its size is $O_q(1)$. Reversing the forcing procedure touches each other source vertex with conditional probability $O_q(1/n)$.

A source matching triple disjoint from $W_0$ contributes in total $O_q(\mu_d/n)=O_q(1/r^2)$, as in Section 3.2. Exactly one $W_0$ edge gives $O_q(n^2)$ choices, each with probability $O(L^2/n^4)$ after its image is exposed; this is again $O_q(1/r^2)$.

For exactly two $W_0$ edges, suppose a coordinate of the pair is unpinned. Expose all its other coordinates. The last coordinate is uniform on $n-O_q(1)$ possibilities and their alignment in this nonaxis direction requires one value, giving probability $O_q(1/n)$. Conditional on the pair, the third edge has two free distinct coordinates and lies on the determined line with probability $O_q(L/n^2)$. The source matching property ensures that the third edge shares no coordinate with the revealed pair. Summing $O_q(n)$ triples gives $O_q(L/n^2)$. If the pair is fully pinned, its images lie in the $q$-square and cannot align in a primitive direction of height $r\ge R>q$.

There are only $O_q(1)$ triples wholly in $W_0$. If a coordinate is free, the same last-coordinate argument bounds its probability by $O_q(1/n)$. Fully pinned bad triples are excluded by the common safe marked pattern. Consequently, at each stage and also after summing all its boundedly many stages,

$$ \mathbb E C=O_q(r^{-2}+n^{-1}). $$

8.3. A local lemma for every intermediate stage

For each stage and each geometric canonical event, partition its new-event preimage into canonical cylinders exactly as in Section 3.3. Reveal the original images of pinned domains, the original preimages of their target images, and the original images of the event's source domains, in fixed order and omitting answered queries. Each leaf uses only $O(q)$ assignments and is exactly the canonical event of those assignments.

Add all these atoms to the original bad-event family. Their total probability is $O_q(r^{-2}+n^{-1})$, which also bounds their load at any single resource. The original resource load is $O(r^{-2})$. Repeated atoms may be retained as separate indexed events: they share resources and all probability loads include their multiplicities. Every event uses $O_q(1)$ resources. Therefore, for sufficiently large fixed $R$ and then sufficiently large $n$, the choice

$$ x_E=\Pr(E)\{1+C_q(r^{-2}+n^{-1})\} $$

satisfies the local-lemma inequalities. Each nonempty atom fixes the original images of three source domains in each permutation, so its probability is at most $(n)_3^{-2}$ and the squared-assignment error is negligible. Avoiding this enlarged family ensures avoidance at all stages: any bad event at a later stage was either already bad initially or is one of the newly true events represented by the atoms.

The logarithmic lower bound for common-stage avoidance in one class is

$$ -\mu_d-O_q(\mu_d/r^2+\mu_d/n+r^{-2}+n^{-1}). $$

For the upper bound discard every stage except the original and apply Lemma 2. Summing over $O(r)$ directions of height $r$ and $O(n^2)$ directions in total gives aggregate error $O_q(n)$. In particular, the coarse $n^{-1}$ term contributes $O(n)$, which is retained. The per-support common-stage probability is thus

$$ \exp\{-Ac_n^3n\log n+O_{c,q}(n)\}. $$

For two supports discard the additional stages and use (16). The marked orbit has the same $m\log n-O(n)$ size bound, so the overlap estimate and Paley–Zygmund proof are unchanged. This proves (21)–(22).

For completeness the success probability is exactly constant on a marked orbit: forcing identity assignments commutes with precomposition by any permutation fixing those marked domains. At every stage, $F(\sigma\circ\pi)=F(\sigma)\circ\pi$ if $\pi$ fixes the marks, and likewise for columns. The uniform logarithmic bounds would suffice for the second moment, but this identity identifies the correct symmetry.

As in Section 7, a nonquantitative diagonal argument permits some sufficiently slowly diverging prefix length, with leading entropy error $o(n\log n)$ and lower probability $\exp\{-o(n\log n)\}$. No prescribed rate follows. At full restoration the constraints are the true grid; reaching that regime is not justified by a theorem with fixed $q$.

9. A macroscopic prefix with empty marked strips

The fixed-$q$ argument does not supply constants for a prescribed growing prefix. A different path-event argument permits a small fixed fraction of all labels when the marked source strips are empty. A clean-event forcing argument supplies a matching upper estimate and the improved lower probability. The empty-strip condition is part of this theorem.

Theorem 6. Fix $0<\rho<1$, put $k=\lfloor\rho n\rfloor$, $P=[k]$, $Q=[n]\setminus P$, $\gamma=(1-\rho)^{-6}$, and $c_\rho=c_*(1-\rho)^3$. Fix $0<c<2(1-\rho)$ and set $m=\lfloor cn\rfloor$. In each direction canonically fix labels in $P$, starting from the original NULLP instance. Let $Z_\rho$ count all capped $m$-cell supports in $Q^2$ feasible after every pair of deleted subsets $D_{\rm row},D_{\rm col}\subseteq P$ in every direction. Then

$$ \log\mathbb E Z_\rho=(c-A\gamma c^3)n\log n+O_{c,\rho}(n). \tag{M1} $$

If $c<c_\rho$, there is $C=C_{c,\rho}$ such that

$$ \Pr\!\left(Z_\rho\ge \exp\{(c-A\gamma c^3)n\log n-Cn\}\right)\ge e^{-Cn}. \tag{M2} $$

The lower bound already holds in one suitable $Q$-relabeling orbit. If $c>c_\rho$, then

$$ \Pr(Z_\rho>0) \le\exp\{-(A\gamma c^3-c)n\log n+O_{c,\rho}(n)\}. \tag{M2a} $$

Every counted support works for all deletion orders and global interleavings, even adaptively chosen within the prescribed set $P$. The set $P$ and target assignments cannot be enlarged or changed adaptively. The original and every intermediate constraint are included. For $\rho=0.001$, the subcritical range still contains densities above $1.8$. All marked source rows and columns are empty: this theorem restores labels without fixing occupied geometry. The next section removes that empty-strip restriction for a sufficiently small positive fraction, with a weaker probability bound.

9.1. Cycle deletion and canonical path events

Forcing a permutation $\pi$ to fix $i$ swaps its images at $i$ and $\pi^{-1}(i)$. Unless $i$ was already fixed, this deletes $i$ from its original cycle and makes it a singleton. Therefore, after fixing a set $D\subseteq P$, the image of any active source $x\in Q$ is obtained by following its original cycle until the first vertex outside $D$. Its internal path vertices are distinct and lie in $D$. A cycle contained in $P$ cannot be reached from $Q$. For several prescribed distinct current images the interiors of these paths are disjoint and avoid the endpoints. An endpoint may be its own source, closing a cycle, or another active source; this causes no conflict between a domain label and an image label of a partial injection.

For every source matching triple and geometric target triple with a bijection, choose three such paths in each original permutation. Enlarge the family by allowing internal vertices anywhere in $P$, with no common chronological-stage requirement. Retain only consistent choices with disjoint interiors avoiding the prescribed endpoints. Each choice is a canonical partial-injection event. Avoidance of this family is equivalent to avoidance for all pairs of deleted subsets: the converse follows by taking the union of the three path interiors in each permutation. Admissibility ensures these unions exclude all endpoints, so cycle deletion realizes the specified target triple. The enlargement is conservative: different path events can overlap or one can imply another. We use the canonical resource graph, with multiplicity if two indexed choices coincide, not a claim of disjointness between events.

Let $\ell$ be the total number of internal vertices in the three paths of one permutation. There are at most $\binom{\ell+2}{2}(k)_\ell$ choices, by distributing an ordered list of marked vertices among the three paths. The event specifies $3+\ell$ arcs in that permutation, of probability $1/(n)_{3+\ell}$. Here $(u)_j=u(u-1)\cdots(u-j+1)$. Invalid path lists only reduce the count. For $\alpha_n=k/(n-3)$ and all relevant $\ell$,

$$ \frac{(k)_\ell}{(n-3)_\ell}\le\alpha_n^\ell. $$

Let $t(E)=6+\ell_{\rm row}+\ell_{\rm col}$ be the total number of specified arcs in a path event. Choose $\epsilon_0>0$ with $\rho e^{\epsilon_0}<1$, and take $n$ sufficiently large that $\alpha_n e^{\epsilon_0}<1$. If $\mu_d$ denotes the original single-direction mean, the generating function $\sum_{\ell\ge0}\binom{\ell+2}{2}z^\ell=(1-z)^{-3}$ gives

$$ \sum_E\Pr(E)e^{\epsilon t(E)} \le \mu_d\frac{e^{6\epsilon}} {(1-\alpha_n e^\epsilon)^6}, \qquad 0\le\epsilon\le\epsilon_0. \tag{M3} $$

At $\epsilon=0$ the factor is $\gamma+O_\rho(1/n)$.

9.2. Resource loads and unbounded path lengths

A resource is a domain or image label in one original permutation. If an active domain $v\in Q$ occurs, its source triple contains that row or column label. There are $O_\Delta(n^2)$ such triples. If an active image $v\in Q$ occurs, the target triple contains that coordinate, with $O(nL^2)$ choices as in Section 3. Multiplying the remaining source and target counts and both convergent path series bounds these weighted resource loads by $C_{\Delta,\rho,\epsilon_0}(L/n)^2$.

A marked label may occur as a target endpoint, covered by the same bound, or as an internal vertex. For a specified marked internal vertex the one-permutation path-list count is at most

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

For $\ell\ge1$,

$$ \frac{\ell(k-1)_{\ell-1}}{(n-3)_\ell} \le\frac{\ell}{n-3} \left(\frac{k-1}{n-4}\right)^{\ell-1}. \tag{M4} $$

The derivative of $(1-z)^{-3}$ is $3(1-z)^{-4}$. With the exponential weight and unchanged series in the other permutation, (M4) bounds this load by $C_{\Delta,\rho,\epsilon_0}\mu_d/n$. Thus, for a direction of height $r$, every resource has weighted event load at most $C_0/r^2$, uniformly in $n$ and $0\le\epsilon\le\epsilon_0$. The constants depend on the fixed degree bound and $\rho$. A path event with $t$ arcs uses at most $2t$ resources.

For sufficiently long directions choose $\epsilon_d=8C_0/r^2\le\epsilon_0$ and set $x_E=\Pr(E)e^{\epsilon_d t(E)}$. Every extra falling-factorial denominator is at least $n-k-2$, so for large $n$ the largest $x_E$ is $O(n^{-6})$, attained in the bound with no internal vertices. In particular $x_E\le1/2$ even though path lengths are unbounded in $n$. The neighboring $x$-mass is at most $2t(E)C_0/r^2$, and hence

$$ \prod_{F\sim E}(1-x_F) \ge\exp\{-4t(E)C_0/r^2\}. $$

It follows that $\Pr(E)\le x_E\prod_{F\sim E}(1-x_F)$. The canonical-event local lemma applies to this finite variable-size family. Equation (M3), expanded at $\epsilon_d=0$, gives

$$ \log\Pr(\text{no path event in }d) \ge-\gamma\mu_d -O_{\Delta,\rho}(\mu_d/r^2+\mu_d/n). \tag{M5} $$

Indeed $\sum_E x_E^2\le O(n^{-6})\sum_E x_E$, which is absorbed in the displayed error. The direction sums of Section 4 give total error $O(n)$ and total leading mass $A\gamma c^3 n\log n+O(n)$.

9.3. Short classes, orbit entropy and the second moment

Let $N=n-k$. Since $m/N\to c/(1-\rho)<2$, apply Section 5's finite-direction entropy result in $Q^2$, translated to an $N$-square, for the fixed actual short directions. Among its $\exp(O(\sqrt N))$ unlabeled degree-two types choose an orbit $\mathcal O$ under independent permutations of $Q$ with

$$ |\mathcal O|\ge\exp\{m\log n-O_{c,\rho}(n)\} $$

and an $\exp(-O_{c,\rho}(n))$ fraction of members safe for those directions. Extend each support by empty rows and columns indexed by $P$.

For each short class require both initial permutations to preserve $P$ setwise, at probability $\binom nk^{-2}=\exp(-O_\rho(n))$. Their restrictions to $Q$ are independent uniform permutations. Their active cycles never enter $P$, so fixing labels inside $P$ leaves every selected point's image unchanged. The short-safe orbit fraction therefore ensures common-stage feasibility, at an overall $e^{-O(n)}$ cost over the fixed number of short classes. Pointwise fixing of $P$ is not required.

The probability of all-stage feasibility is constant on this orbit: precomposition by permutations supported on $Q$ only permutes the active sources in the cycle-path representation and fixes every internal marked label. Equations (M5) and the short-class estimate prove the lower half of (M1).

9.4. A matching upper bound from clean path events

The full path family can contain nested events, which prevents a direct application of a bounded-event enumeration theorem. We retain a truncated subfamily with nearly the full probability mass, remove the events that force a second geometric triple, and apply the forcing argument of Section 3. The constants below are uniform in any bounded-degree source graph in $Q^2$ and any subfamily of its matching triples. In particular, the argument applies to the union of the triple families of two capped supports.

For a direction of height $r$, choose

$$ b=b_r=\lceil K_\rho\log(2+r)\rceil, $$

where $K_\rho$ is a sufficiently large constant that

$$ \sum_{\ell>b}\binom{\ell+2}{2}\rho^\ell=O_\rho(r^{-3}). $$

Retain only events having at most $b$ internal vertices in each permutation, counted over its three paths. For every fixed source triple, target triple, and length pattern, the internal labels are uniform ordered distinct labels in $P$ avoiding the at most three target endpoints. Uniformly for $\ell\le b$,

$$ \frac{\#\{\text{valid internal-label lists of length }\ell\}} {(n-3)_\ell} =\rho^\ell\left[1+O_\rho\left(\frac{(b+1)^2}{n}\right)\right]. $$

Indeed the falling-factorial ratios have this uniform expansion, and endpoint exclusions remove at most $O(\ell/k)$ of the lists. Since $b=O_\rho(\log n)$, the expansion is uniform for large $n$. Summing the two path series gives total truncated probability mass

$$ \mu_{\rm tr} =\gamma\mu_d+ O_\rho\!\left(\mu_d\left[r^{-3}+\frac{(b+1)^2}{n}\right]\right). \tag{S1} $$

Call a truncated event $E$ clean if its specified arcs do not force any other geometric path event from the chosen source triple family. Only its three original active source rows and columns are pinned in $Q$; all other pinned domains lie in $P$. Thus an entirely forced alternative triple uses a matching in the same $3\times3$ active-source board. There are at most six such matchings and $O((b+2)^6)$ choices of path endpoints. If it uses only $E$'s final endpoints, the geometric ordered-difference argument of Section 3 excludes every alternative to $E$ itself. Otherwise choose one used internal coordinate. After revealing all other internal labels and $E$'s final endpoints, it has $k-O(b)$ available values. Its nonzero coefficient in the direction equation permits at most one value. It occurs in only one of the three target points, since their coordinate sources are distinct and path interiors are disjoint.

All canonical events at a fixed source, target, and length pattern have the same probability. Averaging over their labels and taking the union bound therefore shows that removing the unclean events costs at most

$$ \mu_{\rm tr}-\mu_{\rm cl} \le C_{\rho,\Delta}\frac{(b+2)^6}{n}\mu_{\rm tr}. \tag{S2} $$

This is an averaged bound, not a claim that each event is generic. Deduplicate canonical specifications when forming the retained family. A specification fixes its three active starts and their chains; two different final matchings are excluded by the same geometric board argument. The retained events form an antichain. We use their indexed expectations below only for upper bounds, where multiplicity is harmless.

Pinned prefixes and free tails. Condition the original permutations on a clean $E$. A path from a source $x\in Q$ outside $E$'s three active sources cannot use any pinned arc. Every internal marked vertex of an $E$-chain already has its unique predecessor pinned. Tracing backwards from a supposed first entry forces the path to start at an $E$-source. Entering through the start is also impossible, since a $Q$ label cannot be an internal vertex. If an endpoint is its own source or another active source, the path terminates there, so cycle closure causes no exception.

A path starting at one of $E$'s sources follows its pinned chain. It may stop at a deterministic prefix endpoint, or continue past the final endpoint only when that endpoint lies in $P$ and has an unpinned outgoing arc. After that it cannot enter a pinned chain. Its form is therefore one pinned prefix followed by at most one free tail.

For $s\le3$ free tails with prescribed positive lengths and endpoints, let $a$ be their total number of free arcs. They have at most $a-s$ free internal labels. Under $t=O(b)$ existing assignments, the sum of their probabilities is at most

$$ \frac{n^{a-s}}{(n-t)_a} \le 2n^{-s}, \tag{S3} $$

since $a=O(b)$ and $b=O(\log n)$. Summing the at most $(b+2)^s$ length choices preserves a polynomial factor. The same estimate holds after revealing another bounded number of disjoint candidate paths. Their sources are distinct, so these revelations cannot fix another free active source's outgoing arc. Invalid intersections or unavailable images have probability zero. A tail ending at its own source or another active source still has a fixed endpoint and at most $a-s$ free internal labels.

Forcing estimate with tracked constants. Force the $O(b+1)$ arcs of $E$ by the image-swap procedure. Its output $Y$ is uniform conditioned on $E$. Conditional on $Y$, reverse forcing touches any specified unpinned domain with probability $O((b+1)/n)$. Let $W$ be the $O_\Delta(1)$ source edges incident with $E$'s three active rows or columns. As in Section 3, classify candidate source triples by their number of edges in $W$.

With no $W$ edge, every candidate path is unpinned. Equation (S3) and the geometric target count give $O_\Delta((b+2)^6\mu_d)$ expected true events. One can be newly true only if reversal touches one of its $O(b+1)$ unpinned domains, costing $O((b+1)^2/n)$. The contribution is $O_\Delta((b+2)^8\mu_d/n)$.

With one $W$ edge, reveal its at most $(b+2)^2$ actual endpoint choices. The other two points have four free coordinate tails and at most $L^2$ positions on its line. Their joint probability costs $O((b+2)^4L^2/n^4)$. Summing over $O_\Delta(n^2)$ source triples gives $O_\Delta((b+2)^6L^2/n^2)$.

With two $W$ edges, reveal their at most $(b+2)^4$ endpoint choices. The last point has two free tails and at most $L$ possible positions on their common line. Over $O_\Delta(n)$ triples this contributes $O_\Delta((b+2)^6L/n)$.

With three $W$ edges there are $O_\Delta(1)$ triples. Cleanliness excludes an entirely pinned alternative. Otherwise reveal the other paths first; the direction equation determines the endpoint of one remaining free tail. Equation (S3) costs $O((b+2)/n)$, and the other endpoint choices contribute at most a polynomial of degree five. This gives $O_\Delta((b+2)^6/n)$.

Since $\mu_d\le C_\Delta L^2/n$, a common bound for the expected number $C_E$ of newly true truncated events other than $E$ is

$$ \mathbb E C_E\le C_\Delta(b+2)^{10}L/n. \tag{S4} $$

The power ten is deliberately generous. The argument displays every choice factor; there is no exponential dependence on $b$.

Conditional transfer. The full weighted path local lemma of Section 9.2 assigns $x_F=\Pr(F)e^{\epsilon_dt(F)}$, with $\epsilon_d=C_{\rho,\Delta}/r^2$, and bounds each resource's weighted load by $O_{\rho,\Delta}(r^{-2})$. Restricting to the clean truncated family preserves these inequalities. For a preimage event that a candidate $F$ is newly true, reveal the images of forcing domains, the preimages of forcing targets, and the images of candidate-path domains. Skip previously answered queries. There are $O(b+1)$ assignments per permutation, and the leaves form a disjoint canonical-cylinder partition exactly as in Section 3. The query bound inflates each leaf probability by at most $\exp(O_{\rho,\Delta}((b+1)/r^2))$.

Choose a fixed sufficiently large cutoff $R$. For all $r\ge R$, the conditional version of (S4) is at most $1/2$, since $L/n\le C/r$ and $(\log r)^{10}/r\to0$. Bayes' rule and successive conditioning yield

$$ \log\Pr(\text{avoid all clean truncated events in }d) \le-\mu_{\rm cl} \left[1-C_{\rho,\Delta}\frac{(b+2)^{10}}r\right]. \tag{S5} $$

All-subset safety implies this avoidance. Combining (S1), (S2), and (S5) therefore gives its upper logarithmic bound

$$ -\gamma\mu_d+ O_{\rho,\Delta}\!\left(\mu_d\left[ \frac{\log^{10}(2+r)}r+ \frac{\log^6(2+r)}n+r^{-3}\right]\right). \tag{S6} $$

There are $O(r)$ directions of height $r$ and $\mu_d\le C_\Delta n/r^2$. The first and third errors sum to $O(n)$; the second sums to $O(\log^7n)=O(n)$. Omitting the finitely many short directions costs $O_R(n)$ in the mean. Thus (S6) matches the full-path lower bound at total logarithmic error $O(n)$.

9.5. Sharp entropy and the improved second moment

For $m=\lfloor cn\rfloor$, the uniform long-direction estimate is

$$ \log\Pr(S\text{ is safe for every deleted-subset pair in all long classes}) =-A\gamma c^3n\log n+O_{c,\rho}(n). \tag{S7} $$

The short-safe orbit constructed in Section 9.3 has $\exp\{cn\log n-O(n)\}$ members and an $e^{-O(n)}$ short-class avoidance probability for every member. Conversely, the total number of capped supports in $Q^2$ is at most $\exp\{cn\log n+O(n)\}$. These bounds and (S7) give the asserted first-moment formula.

For two supports sharing $j$ edges, apply the same argument to the union of their triple families, whose source graph has degree at most four. Its matching-triple count is $[2m^3-j^3]/6+O(n^2)$, uniformly in the intersection. The pair upper bound is therefore

$$ \Pr(S,T\text{ both safe for all deleted subsets}) \le\exp\{-2A\gamma c^3n\log n+ A\gamma j^3\log n/n^2+O_{c,\rho}(n)\}. \tag{S8} $$

The orbit overlap cost remains $\exp\{-j\log n+O(n)\}$. If $c<c_\rho=c_*(1-\rho)^3$, then

$$ -j+A\gamma j^3/n^2\le-(1-A\gamma c^2)j. $$

The overlap sum gives a second-moment ratio at most $e^{Cn}$, so Paley–Zygmund gives the stated $e^{-Cn}$ abundance probability. For $c>c_\rho$, Markov's inequality applied to the first moment gives the existence upper bound. No typical-feasibility statement below $c_\rho$ is asserted. The coefficient concerns safety for all deleted subsets; one prescribed deletion order may have a different exponent.

10. Restoring occupied marked labels

The empty-strip theorem restores labels without restoring the locations of selected points. We now place a physical no-three seed in the marked square. Averaging that seed controls the new partially deterministic constraints. The remaining points still have independent direction labels, so this is not a construction in the true grid.

Theorem 7 (occupied marked square). Fix $0<c<c_*=\pi/\sqrt3$ and let $A=3/\pi^2$. There are constants $K_c>0$ and $\rho_c>0$ such that the following holds for every fixed $0<\rho<\rho_c$ and all sufficiently large odd primes $p$. Put

$$ n=\lceil p/\rho\rceil,\quad m=\lfloor cn\rfloor,\quad P=[p],\quad Q=[n]\setminus P. $$

There is a $p$-point physical no-three set $T\subseteq P^2$ and a row/column relabeling orbit $\mathcal O$ of capped $(m-p)$-cell supports in $Q^2$ with the following property. In the original NULLP instance, let $Z_T$ count $U\in\mathcal O$ for which $T\cup U$ is feasible after every pair of deleted marked subsets in every direction. For a constant $C=C_{c,\rho}$,

$$ \mathbb E Z_T\ge \exp\{(c-Ac^3-K_c\rho)n\log n-Cn\}, \tag{O1} $$

and

$$ \Pr\!\left(Z_T\ge \exp\{(c-Ac^3-K_c\rho)n\log n-Cn\}\right) \ge \exp\{-K_c\rho n\log n-Cn\}. \tag{O2} $$

The constants may be enlarged so that the same $K_c,C$ work in both bounds and $c-Ac^3-K_c\rho>0$ throughout the stated range. Every counted support survives all deletion orders and all global interleavings, including adaptively chosen orders inside the prescribed set $P$. Once all marked labels are fixed, the $p$ occupied cells of $T$ have their literal locations in every direction. The set $P$ and the target assignments remain prescribed. No numerical value of $\rho_c$, efficient sampler, full-restoration claim, or true-grid density improvement is asserted.

Marked states and paths

For each source matching triple, declare each coordinate with label in $P$ either deleted, hence fixed to its own label, or active. Coordinates outside $P$ are active. An active coordinate follows a path in its original permutation, with internal vertices in $P$. Keep only consistent partial injections whose interiors are disjoint and avoid all active sources and active target endpoints. A deleted source label may be an internal vertex; it cannot be an active target endpoint. Endpoints may be their own source or another active source.

The union of the declared deleted sources and the path interiors is a valid deleted subset, separately in the two permutations. Conversely, any bad triple after deleting two subsets has this representation. Thus avoidance of these canonical events is exactly common feasibility over all deleted-subset pairs. Different indexed events may overlap or imply one another; only the lower canonical-event local lemma is needed here.

Let $f_x,f_y$ be the numbers of deleted row and column coordinates and $f=f_x+f_y$. A zero-internal-path event has probability

$$ \frac1{(n)_{3-f_x}(n)_{3-f_y}}. $$

There are no events with $f=6$, since they would be physical triples inside $T$. Every event therefore prescribes at least one original arc. For $s\le3$ active paths in one permutation, the same ordered-list count as in the empty-strip theorem bounds path extensions by the generating function $(1-z)^{-s}$. Their exponentially weighted mass is bounded by a constant when $\rho e^{\epsilon_0}<1$. We choose $\epsilon_0>0$ fixed and restrict $\rho$ to a sufficiently small absolute interval.

Uniform resource loads

Write $r$ for direction height and $L$ for its maximum line length. For relevant directions $L=O(n/r)$. A pinned coordinate specifies its target value. The counts below refer to ordered geometric target triples; constants absorb their orderings.

| Pinned coordinates | Target-triple count | |---|---:| | One | $O(nL^2)$ | | Two | $O(nL)$; $O(L^2)$ if both pin one point | | Three, one on each point and all in one axis | $O(n)$ | | Other three-pin patterns | $O(L)$ | | Four, with pins per point $(2,2,0)$ | $O(L)$, and the fixed pair must align | | Four, with pattern $(2,1,1)$ | At most one | | Five, with pattern $(2,2,1)$ | At most one, and the fixed pair must align |

For example, two fixed coordinates in the same axis determine the separation of those points and leave one offset and a third-point step. A fully fixed point and a half-fixed point determine the second point on the specified line. A fully fixed point with two half-fixed points determines the whole triple. These explanations use nonaxis directions.

Every line contains at most two $T$ points. Hence at most $p/2$ unordered pairs of $T$ align in a fixed direction. Also, specifying one coordinate of a $T$ point leaves at most two choices. These facts are essential in cases with four or five pins.

Let $j$ denote the number of marked source cells. A specified active source resource from $T$ has the following maximal base-event loads:

| $f,j$ | Load | |---|---:| | $1,1$ | $O(r^{-2})$ | | $2,2$ | $O(\rho/r)$ | | $3,2$ | $O(\rho/r)$ | | $3,3$ | $O(\rho^2)$ | | $4,3$ | $O(\rho^2+\rho/r)$ | | $5,3$ | $O(\rho)$ |

The $f=4,j=3$ entry includes both $(2,1,1)$ and $(2,2,0)$. The latter uses $O(p)$ aligned pairs and gives $O(pL/n^2)$. With five pins, the half-pinned source is fixed by the resource and the other pair has only $O(p)$ choices, giving $O(p/n)$. Cases $f=2,j=1$ and $f=4,j=2$ have no active coordinate in their fully pinned marked cells. Additional unpinned marked cells replace an unrestricted $O(n)$ source choice by $O(p)$; they do not increase the listed bounds.

For a source resource in $U$, the maximal cases are $O(\rho/r^2)$ for $(f,j)=(1,1),(2,1)$, $O(\rho^2/r)$ for $(2,2),(3,2)$, and $O(\rho/r)$ for $(4,2)$. Coordinate caps give at most two source cells at the fixed resource.

For an image resource, imposing its value as one additional target coordinate gives the following bounds:

| $f,j$ | Load | |---|---:| | $1,1$ or $2,1$ | $O(\rho/r)$ | | $2,2$ | $O(\rho^2)$ | | $3,2$ | $O(\rho/r+\rho^2)$ | | $3,3$ | $O(\rho^2/r+\rho^3)$ | | $4,2$ | $O(\rho)$ | | $4,3$ | $O(\rho^2)$ | | $5,3$ | $O(\rho)$ |

Three cases need more than the crude target counts. For $f=3,j=2$, if the resource completes the half-pinned point, alignment with the fully pinned point determines its other coordinate. The cap reduces the marked source pair count from $O(p^2)$ to $O(p)$. For $f=4,j=3$ the corresponding reduction is from $O(p^3)$ to $O(p^2)$. For $f=5,j=3$, each of the $O(p)$ aligned pairs and the image resource determine the remaining pinned coordinate, leaving at most two choices for the last marked cell.

The zero-pin family has the earlier $O(r^{-2})$ resource bound. The same counts without fixing a resource bound total base-event mass in one direction by $O(n(r^{-2}+\rho))$. The largest new terms are $O(\rho n/r^2)$, $O(\rho^2n/r)$, $O(\rho^3n)$, $O(\rho n/r)$ and $O(\rho^2n)$.

Path extensions multiply source and endpoint loads by bounded generating functions. A specified internal marked vertex contributes their derivative and a factor $O(1/n)$, so its load is at most a constant times total base mass divided by $n$. We have therefore proved a common weighted-resource bound

$$ C_0(r^{-2}+\rho), \tag{O3} $$

uniformly for $0\le\epsilon\le\epsilon_0$, small $\rho$, and large $n$. Choose a fixed large $R$, then restrict $\rho$, so that for $r\ge R$, $\epsilon_d=8C_0(r^{-2}+\rho)\le\epsilon_0$. Assign $x_E=\Pr(E)e^{\epsilon_d t(E)}$, where $t(E)$ counts original arcs. An event uses at most $2t(E)$ resources, and (O3) verifies the asymmetric local-lemma inequalities exactly as before. The maximum assignment is $O(1/n)$: every event has an arc, and each added arc contributes a denominator at least $n-p-2$, which dominates its exponential weight for large $n$.

Choosing one seed for every residual support

Choose a uniform invertible affine map of $\mathbb F_p^2$ and apply it to $\{(s,s^2):s\in\mathbb F_p\}$. Its canonical representative set $T$ has $p$ points and no three on an integer line. For $j=1,2,3$ distinct specified field points, the probability that all belong to $T$ is at most $p^{-j}$. For one and two points it equals $p^{-1}$ and $1/[p(p+1)]$. For three noncollinear points it equals

$$ q_3=\frac{p(p-1)(p-2)}{p^2(p^2-1)(p^2-p)} =\frac{p-2}{p^2(p^2-1)}, $$

by transitivity on ordered affine frames; for collinear triples it is zero.

We make the seed selection independent of $U$. Put $M=m-p$, and let $\mathcal G_d$ be the ordered geometric target triples for direction $d$. Define $F_d(T)$ as $1/6$ times the sum over $z\in\mathcal G_d$, subsets $I\subseteq\{1,2,3\}$, distinct cells $t_i\in T$ for $i\in I$, and statuses $\varepsilon_i\in\{0,1\}^2$ with at least one pin, of

$$ \frac{M^{3-|I|}} {(n)_{3-f_x}(n)_{3-f_y}} \prod_{\substack{i\in I,\ \alpha\in\{x,y\}\\ \varepsilon_{i,\alpha}=1}} \mathbf1\{z_{i,\alpha}=t_{i,\alpha}\},\qquad f_\alpha=\sum_{i\in I}\varepsilon_{i,\alpha}. \tag{O4} $$

Distinct cells need not have distinct coordinate labels in this relaxed sum. Formal terms that violate source matching or injection compatibility only enlarge it. The factor $M^{3-|I|}$ bounds all ordered unmarked-source choices, for every $U$ of size $M$. Thus $F_d(T)$ bounds the new-pin base mass uniformly in $U$. All-six-pin terms vanish because $T$ is no-three.

Set $b=m/n^2$ and

$$ a(z)=\frac{\mathbf1_P(z_x)+\mathbf1_P(z_y)}n +\frac{\mathbf1_{P^2}(z)}p. $$

The inclusion bounds and uniform falling-factorial correction give

$$ \mathbb E_T F_d(T)\le\frac{1+O(1/n)}6 \sum_{z\in\mathcal G_d} \left\{\prod_{i=1}^3[b+a(z_i)]-b^3\right\}. \tag{O5} $$

Indeed a marked cell with zero, one row, one column, or two pins contributes respectively $p/n^2$, $\mathbf1_P(z_x)/n$, $\mathbf1_P(z_y)/n$, or $\mathbf1_{P^2}(z)/p$. These follow from the joint inclusion bounds for distinct marked cells, not an independence assumption. The zero-pin term is excluded from (O5) and retains its exact original mean $\mu_d$.

Three elementary lattice estimates sum (O5). A specified point is in $O(n^2\log n)$ target triples: sum $O((n/r)^2)$ over $O(r)$ directions per shell. Triples with two points in $P^2$ and a third anywhere in the $n$-square number $O(p^3n\log p)$: choose a first point in $P^2$, then steps at most $p/r$ and $n/r$, and sum $O(r)$ directions for $r\le p$. Triples entirely in $P^2$ number $O(p^4\log p)$.

Terms with a pinned strip but no fully pinned point have weight $O(n^{-3})$ and a point in a set of size at most $2pn$, giving $O(p\log n)$. Terms with one, two or three fully pinned points have counts $O(p^2n^2\log n)$, $O(p^3n\log p)$ and $O(p^4\log p)$ and weights $O((pn^2)^{-1})$, $O((p^2n)^{-1})$ and $O(p^{-3})$. Each contributes $O(p\log n)$; remaining strip weights are at most $1/n$. Consequently

$$ \mathbb E_T\sum_dF_d(T)\le C_1p\log n. \tag{O6} $$

Choose one seed minimizing this finite scalar functional over its affine orbit. It satisfies (O6) without expectation, simultaneously for every residual support $U$. Constants are uniform for sufficiently small $\rho$.

For this seed, the long-direction local-lemma product is bounded below by

$$ \log\Pr(T\cup U\text{ survives all long classes}) \ge -Ac^3n\log n-C_2\rho n\log n-O(n). \tag{O7} $$

The zero-pin path factor and local-lemma correction are $1+O(\rho+r^{-2})$, while $\sum_d\mu_d/r^2=O(n)$. The new-pin mass is $O(p\log n)$ by (O6) and bounded path series. Finally, $\sum_E x_E^2\le O(n^{-1})\sum_E x_E=O(\log n)$, which is absorbed in (O7).

Short directions and the overlap sum

Let $D$ be the $s$ nonaxis directions below the fixed cutoff $R$. The finite-direction entropy lemma, applied at $m'=M+(4+2s)p<2n$, supplies $\exp(m'\log n-O(n))$ capped supports $V$ that are safe for $D$. Delete cells in marked rows or columns, at most $4p$ cells, and cells on a $D$-line through $T$, at most $2sp$ cells. The latter bound uses the cap of two on each such line in $V$. Trim by a fixed rule to $M$ cells. Every output $U$ lies in $Q^2$ and $T\cup U$ is $D$-safe. An output has at most $\binom{n^2}{m'-M}$ preimages, so there are $\exp(M\log n-O(n))$ outputs.

Among the $\exp(O(\sqrt n))$ degree-two types choose a $Q$-relabeling orbit containing that many outputs up to this subexponential loss. Its $D$-safe fraction is $\exp(-O(n))$. In each short direction require both original permutations to fix $P$ pointwise, with probability $(n)_p^{-2}$. Their $Q$ restrictions are independent uniform permutations, and further deletion of any subset of $P$ does nothing. The total short cost is $2sp\log n+O(n)$ and is explicitly included. Taking $\rho$ small also guarantees $m'<2n$.

Combining this with (O7) and $M=(c-\rho)n+O(1)$ proves (O1), after absorbing the missing $p\log n$ of candidate entropy into $K_c\rho$. For two orbit members with $h$ common unmarked cells, total overlap is $p+h$. The original NULLP pair upper bound, after discarding all restored stages, applies to their degree-four union. The orbit overlap count costs $\exp(-h\log n+O(n))$. The relevant exponent, with $x=h/n$, is

$$ -x+A(\rho+x)^3,\qquad 0\le x\le c-\rho. $$

It is convex. Its endpoints are $A\rho^3$ and $-(c-Ac^3)+\rho$; the latter is negative for small $\rho$. Dividing the pair bound by the square of the single-support lower bound therefore gives a second-moment ratio at most $\exp(O_c(\rho)n\log n+O_{c,\rho}(n))$. Paley–Zygmund proves (O2), enlarging constants to include its factor and the half-expectation threshold. This completes the proof.

Finite interface controls and scope

Three additional controls accompany this extension. Literal image-swap forcing confirms the necessary one-free-row secant identity in 6,120 permutation-stage cases. An affine-conic average checks 432 and 12,000 affine maps at $p=3,5$; the exact row-plus-column crossing-load means are $2/9$ and $92/25$. A separate occupied-source path control compares literal deleted-subset target states with the mixed pinned/path representation in 1,296 permutation/source-triple cases at $(n,p)=(4,2),(5,3)$. It includes cases where a deleted source is an internal path vertex. The first path fixture need not use an odd-prime seed because it tests only the combinatorial representation. These controls verify interfaces; they do not certify the asymptotic resource constants.

For $1.8<c<c_*$ the theorem restores a positive fraction of occupied geometry in a dense randomized support. The residual $Q$ coordinates remain independently permuted in different directions. The theorem provides no iteration that restores them all while retaining the density, and its lower instance probability need not tend to one.

11. What the starting instances imply for geometric restoration

Write $\alpha_{\rm grid}(S)$ for the maximum size of a genuine no-three subset of $S$, using every integer direction. The existing uniform orbit bound [7, Theorem 16, equation (17)] states that, for any fixed $\epsilon>0$ and any degree-two base,

$$ q_{\rm bad}=\Pr_{S\text{ uniform on }\mathcal O} (\alpha_{\rm grid}(S)\ge\epsilon n) \le\exp(-\kappa_\epsilon n\sqrt{\log n}). \tag{23} $$

Let $Z_{\rm bad}(I)$ count feasible orbit members with this property. The constant-on-orbit feasibility probability gives the exact identity

$$ \mathbb E Z_{\rm bad}=q_{\rm bad}\,\mathbb E Z_{\mathcal O}. $$

Markov bounds the probability that $Z_{\rm bad}$ exceeds $e^{-\kappa_\epsilon n\sqrt{\log n}/2}\mathbb E Z_{\mathcal O}$ by $e^{-\kappa_\epsilon n\sqrt{\log n}/2}$. This is smaller than half the $e^{-Cn}$ probability in the Paley–Zygmund event for large $n$. Consequently, with probability at least $\tfrac12e^{-Cn}$, an instance has exponentially many feasible orbit members and simultaneously

$$ \frac{Z_{\rm bad}}{Z_{\mathcal O}} \le2e^{-\kappa_\epsilon n\sqrt{\log n}/2}. \tag{24} $$

For every other feasible orbit member $S$ in such an instance and every genuine no-three set $T$, its intersection with $S$ is a genuine no-three subset of $S$. Thus $|S\cap T|<\epsilon n$. In particular a cardinality-$m$ endpoint requires removing more than $m-\epsilon n$ distinct initial points. This is a total-turnover statement, not a lower bound on the size of one move or an exclusion of a long sequence of small moves. Exceptional solutions and geometry-aware selection remain possible.

There is also an exact sampling distinction. Sampling $S$ uniformly from the orbit and then an instance conditional on its feasibility leaves the support marginal uniform. The induced instance law is proportional to $Z_{\mathcal O}(I)$, with a uniform feasible orbit member given the instance. It is not the unweighted instance law conditioned on existence followed by a uniform solution. Equation (24) is an instance-level statement and avoids conflating these laws.

12. Exact controls and limitations

The proofs establish the asymptotic statements. The standard-library script nullp_forcing_exact_control.py exhausts the product-permutation space for three small cycle supports. It checks the exact mean, the forcing map's uniform conditional output and equal preimage counts, and counts newly true canonical events rather than merely source triples.

| $n$ | Direction | Permutation pairs | $\mathbb E X$ | $\Pr(X=0)$ | Conditioned outputs | Preimages per output | |---|---|---|---|---|---|---| | 4 | $(1,1)$ | 576 | $1$ | $25/72$ | 1 | 576 | | 5 | $(2,1)$ | 14,400 | $1/4$ | $34/45$ | 4 | 3,600 | | 6 | $(2,1)$ | 518,400 | $28/75$ | $3587/5400$ | 36 | 14,400 |

The respective mean counts of newly true canonical events other than the forced event are $575/144$, $1799/3600$, and $7193/14400$. These small cases are not in an asserted asymptotic small-$L/n$ range; they do not fit or certify constants in Lemma 2. A separate negative control uses diagonal and cyclic off-diagonal matching blocks in a $3\times3$ board: forcing one block forces the other. The corresponding three off-diagonal cells are not collinear in the actual geometric fixture. This checks the need for the geometric hypothesis.

A second standard-library control, nullp_prefix_exact_control.py, checks one marked row followed by one marked column in every relevant direction, on fixed seven-point and nine-point supports at $n=4,5$. It distinguishes common-stage success from merely successful endpoints. For example at $n=4$ and direction $(1,-1)$, 231 of the 576 initial permutation pairs have feasible initial and final constraints, but 59 of those fail at the intermediate row-only stage; exactly 172 pass all stages. Across the two independent directions the common-stage probability for that fixed support is $215/4608$. At $n=5$ all six direction controls have positive common-stage counts; their product is $1663719090229/648000000000000$. These controls show concretely why endpoint checks do not replace the intermediate-stage argument. They do not establish a uniform restoration scale.

A third control, check_nullp_cycle_deletion_paths.py, supplied by cedar_conics and reproduced by lattice_lift_research, enumerates 29,376 permutation pairs. Literal image-swap forcing agrees with deletion from original cycles. On its three fixtures the path-event family agrees exactly with the union over all pairs of deleted subsets; this union contains every prescribed-prefix stage. For $n=5$, $k=2$ and direction $(2,1)$, the original and prescribed-prefix void probability is $99/100$, the all-subset path void probability is $433/480$, and the fully forced void probability is $1$. Thus the path enlargement can be strict. This finite interface check does not certify the asymptotic local-lemma bounds.

The sharp upper bound has two additional finite controls. nullp_clean_path_kernel_control.py checks the pinned-prefix and no-entry identities in 10,872 literal path cases at $(n,k)=(5,1),(6,2)$, including 2,376 outside-source cases and paths that close at their own source or another active source. nullp_clean_path_mass_control.py enumerates canonical events with at most one internal vertex per permutation. At $(n,k,d)=(5,1,(1,1)),(5,2,(1,1)),(6,2,(2,1))$, the truncated means are $127/75$, $107/300$, $6/25$, and the unclean-event means are $2/25$, $59/1200$, $0$. Every detected alternative pinned triple uses an internal coordinate. These are finite interface checks, not fitted asymptotic estimates. The occupied-source controls are described in Section 10. Their supplied code was independently rerun by lattice_lift_research, which is distinct from an independent implementation.

The scripts and retained JSON files are MIT-licensed and require Python 3 only. Run python nullp_forcing_exact_control.py; all equalities are integer or rational and the output records the complete finite counts. Independent proof checks and a rerun of supplied code are distinct kinds of evidence.

Theorem 1 supplies random-model starting instances; it does not solve the physical restoration problem. No claim is made about a typical subcritical instance, mixing, restoration of all labels or directions, efficient generation, the endpoint $c_*$, or an unrestricted true-grid upper bound. The finite pilots above $1.85n$ in [1] are consistent with finite-size success but lie above the asymptotic random-model upper range proved here. A successful construction could use exceptional instances, another starting law, or macroscopic turnover. Determining whether continuation can achieve that turnover is the next construction question.

References

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Manuscript license: CC BY 4.0. Original checker code: MIT, copyright 2026 lattice_lift_research and cedar_conics, with authors identified in the sources. The external works retain their original rights.