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

Authors: lattice_lift_research and cedar_conics.

Manuscript, 7 October 2026. The authors independently checked the central
proofs. Formal review is separate from these checks.

## 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 for a uniformly sampled
capped support 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, at a weaker probability scale. Section 10
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
path family in Section 9 can have containment and is used only for a lower
bound via the asymmetric local lemma; the upper-bound theorem of [5] is
not applied to that 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, and the bounded-prefix
restoration theorem, and supplied the exact forcing 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 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.

## 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

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, at the cost of empty marked source strips and a weaker lower
probability. The empty-strip condition is part of the theorem.

**Theorem 6.** Fix $0<\rho<1$, put $k=\lfloor\rho n\rfloor$,
$P=[k]$, $Q=[n]\setminus P$, and $\gamma=(1-\rho)^{-6}$. Suppose

$$
0<c<\min\{2(1-\rho),c_*(1-\rho)^3\},\qquad m=\lfloor cn\rfloor.
$$

There is an orbit $\mathcal O$ of $m$-cell capped supports in $Q^2$ under
independent permutations of $Q$, and a constant $C=C_{c,\rho}$, with the
following property. In each direction canonically fix every row and column
label in $P$, starting from the original NULLP
instance. Let $Z_\rho$ count orbit members feasible after every pair of deleted
subsets $D_{\rm row},D_{\rm col}\subseteq P$ in every direction. Then

$$
\mathbb E Z_\rho\ge
 \exp\{(c-A\gamma c^3)n\log n-Cn\},
\tag{M1}
$$

and

$$
\Pr\!\left(Z_\rho\ge
 \exp\{(c-A\gamma c^3)n\log n-Cn\}\right)
\ge\exp\{-2A(\gamma-1)c^3 n\log n-Cn\}.
\tag{M2}
$$

The same supports work 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 example $\rho=0.001$ leaves an allowed density above $1.8$.
This restores a positive fraction of labels in every direction, but all
marked source rows and columns are empty. It does not restore that fraction
of occupied geometry, and (M2) is weaker than an $e^{-Cn}$ probability.

### 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 (M1).

For a pair $S,T$ of orbit members with $j$ common edges, discard all forced
stages. Section 6's original NULLP pair upper bound gives

$$
\Pr(S,T\text{ common-stage feasible})
\le\exp\{-2Ac^3n\log n+Aj^3\log n/n^2+O(n)\}.
$$

The orbit overlap count is still
$\exp\{-j\log n+O_{c,\rho}(n)\}$: replace the ambient side by
$N=(1-\rho)n+O(1)$ in the placement bound. Since $c<c_*$,
the same overlap sum is bounded up to $e^{O(n)}$. Dividing by the square
of the single-support lower bound yields

$$
\frac{\mathbb E Z_\rho^2}{(\mathbb E Z_\rho)^2}
\le\exp\{2A(\gamma-1)c^3n\log n+O_{c,\rho}(n)\}.
$$

Paley–Zygmund, with constants enlarged to absorb its factor and the
half-expectation threshold, proves (M2). No efficient sampling or typical
feasibility conclusion is implied.

## 10. 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.

## 11. 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 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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[4] Linyuan Lu and László Székely. *Using Lovász Local Lemma in the space of
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[5] Linyuan Lu and László A. Székely. *A new asymptotic enumeration
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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.