Local repairs and reconfiguration barriers in no-three-in-line sets

Athanor lab (Aleksei Kudriashov with AI research agents)

Contributions. Aleksei Kudriashov initiated and directed the research programme. The public submission account is athanor_lab. Astra (gpt-6-astra) developed the repair formulation and prepared the mathematical exposition. Guide (Claude Code) coordinated the research records and integrated sources. Four independent research-core roles, each using gpt-6.1-sol, are credited: the $G_6$ enumerator; the representation-note outside reader; the matching/carrier auditor and $G_7,G_8$ certificate searcher; and the final external reader, which independently replayed $n\le6$ and audited proofs, literature, and attribution. These named AI contributions do not replace the mathematical evidence and certificates described below.

1. Introduction

The no-three-in-line problem is usually posed as an extremal question: how large can a subset of a carrier be if it contains no three collinear points? Here the object is the landscape below a maximum. A safe set can be maximal without being maximum. How much must it shrink before it can grow into a maximum configuration?

This question has a local and a global part. At a specified empty site, selected pairs on lines through that site record exactly how many deletions make the insertion possible. But that insertion may enter a component that cannot reach any maximum at the same cardinality floor. The first full-square separation of these two notions among the boards exhaustively enumerated here occurs in the complete $6\times6$ census: some 11-point maximal sets have a one-deletion insertion, yet every path to a 12-point maximum falls to nine points.

The paper develops the language needed to state that phenomenon without overclaiming it.

1. Section 3 gives the matching and vertex-cover repair calculus.
2. Sections 4–5 turn a path-floor question into a threshold-connectivity question and give a compact matching certificate for a one-extra-loss barrier.
3. Sections 6–7 record the complete $G_6$ census and its four exceptional component types.
4. Sections 8–9 separate an unbounded restricted-carrier theorem from the still open complete-square problem, with certified exclusions and specified-layer path bounds on $G_7,G_8$.

The local matching observation and the add/remove versus swap translation are elementary or standard. The prospective contribution is the exact Euclidean-grid barrier classification, with reproducible certificates, not a claim of a new general reconfiguration formalism. In particular, the six-state C4 component was publicly exhibited earlier by u/cedar_conics [10]. Section 7 identifies its representative under the reflection $(x,y)\mapsto(5-x,y)$; the contribution here is the exhaustive classification of all 124 labelled strict examples into four component types, together with the accompanying finite certificates.

2. Definitions and move models

Let $X\subset\mathbb R^2$ be finite. A subset $S\subseteq X$ is safe if no three distinct points of $S$ are collinear. Write $\mathcal I(X)$ for the hereditary family of safe sets and

$$ \alpha(X)=\max\{|S|:S\in\mathcal I(X)\}. $$

A safe set is maximal if it has no safe one-point extension, and maximum if it has size $\alpha(X)$. The main carrier is the ordinary integer square

$$ G_n=\{0,\ldots,n-1\}^2, $$

with all Euclidean slopes allowed. A later construction uses a finite restricted carrier $X\subset\mathbb Z^2$. Such a carrier remains restricted even if it is embedded in a larger square.

For safe $S$ and $x\in X\setminus S$, define the blocker matching

$$ M_x(S)=\bigl\{\{p,q\}\subseteq S:p,q,x\text{ are collinear}\bigr\}, \qquad b_S(x)=|M_x(S)|. \tag{2.1} $$

Whenever $X\setminus S\ne\varnothing$, define

$$ \beta(S)=\min_{x\in X\setminus S}b_S(x). \tag{2.2} $$

We use comparisons involving $\beta$ only for submaximum sets. A maximum can have exterior points and therefore can have a defined $\beta$, but it has path deficit zero and is not a repair-barrier candidate.

For a submaximum safe $k$-set, its direct improvement depth is

$$ d(S)=\min_{\substack{T\in\mathcal I(X)\\ |T|=k+1}}|S\setminus T|. \tag{2.3} $$

For every safe $S$, its escape floor is

$$ F(S)=\max_{\gamma:S\leadsto Q,\ |Q|=\alpha(X)} \ \min_{U\in\gamma}|U|, \tag{2.4} $$

where $\gamma$ uses one insertion or deletion per step and remains safe. The path deficit is

$$ \delta(S)=|S|-F(S). \tag{2.5} $$

Thus a maximum has $\delta(S)=0$. An atomic neutral swap abbreviates a deletion followed by an insertion. It therefore visits the next lower cardinality level and is not a zero-cost operation in the definition of $F$.

3. Exact local repair calculus

3.1 One prescribed insertion

Proposition 3.1. For safe $S$ and $x\notin S$, $M_x(S)$ is a matching. The point $x$ is insertable exactly when $b_S(x)=0$, and the minimum number of selected points that must be deleted before inserting this specified $x$ is $b_S(x)$.

Proof. Two edges of $M_x(S)$ cannot share an endpoint: their three selected endpoints would lie on the same line through $x$, contradicting safety. Thus the edges are disjoint. Inserting $x$ creates exactly the triples indexed by these edges. Every deletion set has to meet each edge, and one endpoint per edge suffices. $\square$

Consequently, $S$ is maximal exactly when every empty site has positive blocker number. Two useful audit identities are

$$ \sum_{x\in X\setminus S}b_S(x) =\sum_{\{p,q\}\subseteq S}(|X\cap\ell(p,q)|-2), \tag{3.1} $$

and, if $E(U)$ counts collinear triples in $U$,

$$ E(S\cup\{x\})-E(S)=b_S(x). \tag{3.2} $$

The first double-counts a selected pair and an empty third carrier point on its full line. These identities do not imply that different empty sites act independently.

3.2 Several prescribed insertions

Let $A\subseteq X\setminus S$ itself be safe. Define

$$ D_A=\{s\in S:s\text{ is collinear with two distinct points of }A\} $$

and

$$ G_A=\bigcup_{a\in A}M_a(S)\big|_{S\setminus D_A}. \tag{3.3} $$

The restriction in (3.3) deletes every vertex of $D_A$, and $\tau$ denotes the minimum vertex-cover number.

Proposition 3.2. The minimum number of points of $S$ that must be deleted before inserting all points of $A$ is

$$ r_S(A)=|D_A|+\tau(G_A). \tag{3.4} $$

Proof. A triple comprising two new points and one old point forces the old point into $D_A$. Once these forced deletions are made, every remaining mixed triple consists of one new point and one edge of $G_A$. Thus the additional deleted old points must form a vertex cover. The conditions are sufficient because both $S$ and $A$ are safe. $\square$

For two inserted points, $G_A$ is a union of two matchings, so each component is a path or an even cycle. This makes $\tau(G_A)$ immediate. For three inserted points, odd cycles can occur. For example, with

$$ S=\{(0,0),(2,0),(0,2)\},\qquad A=\{(1,0),(1,1),(0,1)\}, $$

the graph $G_A$ is a triangle and has vertex-cover number two.

3.3 Prior-work boundary

Maximal general-position subsets of grids are studied as independent geometric dominating sets by Aichholzer, Eppstein, and Hainzl [1]. Their historical discussion credits Adena, Holton and Kelly (1974), and Martin Gardner (1976), for the minimum-maximal problem. Nagy, Nagy and Woodroofe use the term “saturated” for this maximality condition [7]. Wilhelmina's public blocker/exchange computation [11] and Kudriashov's certified work on nearby blocker and rigidity questions [12] are related local antecedents, but do not establish the planar path-floor statements here. Multiple saturation by secants in finite projective planes is another analogy [8], not a transfer to integer squares.

Propositions 3.1–3.2 are elementary. This manuscript makes no novelty claim for these identities, for vertex-cover language, or for local search.

4. Threshold components and local gates

For $h\ge0$, let $H_h$ be the add/delete graph induced by safe sets of size at least $h$. Let $\Sigma_h$ be the union of components of $H_h$ that meet a maximum set. By definition,

$$ F(S)=\max\{h:S\in\Sigma_h\}. \tag{4.1} $$

The next statement says precisely what is absent from a scalar local repair cost.

Theorem 4.1 (gate criterion). Let $S$ be a maximal submaximum $k$-set and put $b=\beta(S)$. For every empty $x$ with $b_S(x)=b$, let $C$ range over the minimum vertex covers of $M_x(S)$, and put

$$ P_{x,C}=S\setminus C,\qquad U_{x,C}=P_{x,C}\cup\{x\}. $$

Then

$$ \delta(S)=b \quad\Longleftrightarrow\quad P_{x,C}\in\Sigma_{k-b}\text{ for some }x,C. \tag{4.2} $$

Proof. Suppose a path from $S$ to a maximum stays at size at least $k-b$, and consider its first inserted point $x\notin S$. Immediately before that move the path is at a subset $R\subseteq S$. The deletion set $S\setminus R$ meets every edge of $M_x(S)$, so it has at least $b_S(x)$ points. The threshold bound gives at most $b$. Both inequalities are equalities: $b_S(x)=b$, $R=P_{x,C}$, and $C$ is a minimum cover. The converse follows by deleting $C$, inserting $x$, and following a path within $\Sigma_{k-b}$. $\square$

Hence $\delta(S)\ge\beta(S)$ for every submaximum maximal state. Equality fails exactly when every minimum local gate lies in a maximum-free threshold component. This is an exact location of the missing global information, not an efficient replacement for a component calculation: all gates remain connected through $S$ at the same threshold.

5. Swap layers and a matching certificate for a second loss

Let $J_h$ be the graph on safe $h$-sets in which two vertices are adjacent if they have a common safe $(h-1)$-subset. Thus a $J_h$-edge is an atomic neutral swap. Every pair of $h$-subsets of a fixed safe set of size at least $h$ lie in one $J_h$-component; call it $j_h(U)$.

Theorem 5.1 (threshold/TJ lemma). Let $h\ge1$. If $S,Q$ are safe and have size at least $h$, then

$$ S\text{ and }Q\text{ are connected in }H_{h-1} \quad\Longleftrightarrow\quad j_h(S)=j_h(Q). \tag{5.1} $$

Proof. A path at threshold $h-1$ either moves above the $h$-layer, which preserves the component of its $h$-subsets, or passes through a configuration $A\supset B\subset A'$ with $|B|=h-1$, which is one $J_h$-edge. Conversely, shrink the endpoints to $h$-subsets, implement each $J_h$-edge as delete then add through its common $(h-1)$-subset, and regrow. $\square$

This is the hereditary-family form of the familiar token addition/removal–token jumping relation. For graph independent sets, see Kamiński, Medvedev, and Milanič [2], and the threshold/buffer context of de Berg, Jansen and Mukherjee [3]. The proof is included because no-three-in-line safety is independence in a 3-uniform hypergraph.

Define the maximum-target label set

$$ T_h=\{R:|R|=h,\ R\subseteq Q\text{ for some maximum }Q\}. \tag{5.2} $$

For $1\le d\le |S|$, Theorem 5.1 gives

$$ \delta(S)\le d \quad\Longleftrightarrow\quad j_{|S|-d+1}(S)\cap T_{|S|-d+1}\ne\varnothing. \tag{5.3} $$

This is a connectivity characterization, not a claimed fast algorithm.

Comparison with completable-subgraph connectivity. Paper [17] studies, for $G_9$, fixed-cardinality exchange graphs restricted to sets contained in an 18-point maximum. In the notation above, these are the induced graphs $J_h[T_h]$. Our graph $J_h$ also contains safe sets with no maximum completion. Consequently two components of $J_h[T_h]$ may be joined by a path in $J_h$ through vertices omitted from the induced graph; conversely, a component of $J_h$ disjoint from $T_h$ is entirely invisible in that restricted graph. The move convention also matters: a $J_h$-edge becomes a deletion and insertion through size $h-1$, so its threshold counterpart is $H_{h-1}$, not $H_h$. Thus the completable-future islands in [17] and the maximum-free threshold components classified here answer different questions, despite sharing enumeration background and exchange moves.

We now record a local closure certificate for the case $\beta=1$. For safe $U$ and any carrier point $y$, selected or empty, let

$$ L_y(U)=\{\{a,b\}\subseteq U\setminus\{y\}:a,b,y \text{ are collinear}\}. \tag{5.4} $$

Suppose $x\notin U$, $L_x(U)=\{\{p,q\}\}$, $r\in\{p,q\}$, and $U'=U-r+x$. For $y\notin U'$,

$$ L_y(U')=L_y(U)|_{U\setminus\{r\}} \cup E_y(x;U\setminus\{r\}), \tag{5.5} $$

where

$$ E_y(x;W)= \begin{cases} \{\{x,s\}\},&\text{if a unique }s\in W\text{ is collinear with }x,y,\\ \varnothing,&\text{otherwise.} \end{cases} \tag{5.6} $$

There is at most one such $s$, and this new edge is disjoint from all retained edges, because $U'$ is safe. Equations (5.5)–(5.6) are therefore an exact labelled-matching update. Blocker numbers alone omit the old–new–new incidence that decides whether the new edge is created.

Theorem 5.2 (matching-closure certificate). Let $S$ be a submaximum maximal safe $k$-set with $\beta(S)=1$. Let $C$ be its component in the labelled neutral-swap graph generated by singleton blocker matchings, and suppose every $U\in C$ is maximal. Define

$$ D(C)=\{P:|P|=k-1,\ P\subset U\text{ for some }U\in C\}. $$

Then the component of $S$ in $H_{k-1}$ is exactly

$$ C\cup D(C). \tag{5.7} $$

In particular, $\delta(S)\ge2$.

Proof. No $U\in C$ has a size-$k+1$ coface. Take $P=U\setminus\{r\}\in D(C)$, and let $U'=P\cup\{x\}$ be another safe $k$-coface. Maximality of $U$ makes $M_x(U)$ nonempty. Safety of $U'$ forces every edge of this matching to contain $r$. As it is a matching, it has one edge, so $U'$ is a declared singleton-blocker swap and lies in $C$. Thus every threshold path alternates only between $C$ and $D(C)$. Conversely, $C$ is connected by its definition, each neutral edge is realized through a member of $D(C)$, and each member of $D(C)$ is adjacent to a member of $C$. Thus their union is connected and is the entire threshold component. $\square$

Connectedness matters: for an arbitrary swap-closed family of maximal $k$-sets, the same boundary argument gives a closed union of threshold components. It gives one component only if the family is connected. The theorem explicitly takes $C$ to be a component.

6. Exact $G_6$ census

This section records exactly checked finite computation. The results are attributed to the complete-grid enumeration core and its independent verifier. They are reproducible mathematical evidence, not proof-assistant formalizations.

Census algorithm. The archived fresh_grid.cpp first constructs the maximal grid lines containing at least three sites by exact integer determinants. It enumerates safe sets by depth-first insertion in strictly increasing site order, maintaining line occupancies. A site is available exactly when it is unselected and belongs to no line already containing two selected sites. Every safe set appears once: its increasing insertion order is unique, and all its prefixes are safe. An empty available-site list identifies an inclusion-maximal set. The largest occupied cardinality layer identifies all maxima.

For the escape floors, the program stores the labelled masks and processes cardinality thresholds $h$ in decreasing order. Every safe $h$-set is represented, initially as a singleton if it has no larger coface. At threshold $h$, the program adds every deletion edge from an $(h+1)$-set to each of its $h$-subsets. A disjoint-set structure therefore represents precisely the components of $H_h$; each component records whether it contains a maximum. The first threshold at which a state joins a marked component is its escape floor $F(S)$, because all higher thresholds have already been tested. This gives $\delta(S)=|S|-F(S)$. Blocker counts and direct improvement depths are computed separately from the resulting state list. No symmetry quotient is used in enumeration or threshold connectivity. Dihedral symmetry is applied only afterward to classify the complete labelled exception list and its components. The separate motif checker reconstructs all one-deletion cofaces and verifies the saved escape paths.

| $n$ | safe subsets | maximal subsets | $\alpha(G_n)$ | maximum subsets | maximal $\delta=1$ | maximal $\delta=2$ | |---:|---:|---:|---:|---:|---:|---:| | 1 | 2 | 1 | 1 | 1 | 0 | 0 | | 2 | 16 | 1 | 4 | 1 | 0 | 0 | | 3 | 230 | 23 | 6 | 2 | 21 | 0 | | 4 | 4,812 | 347 | 8 | 11 | 336 | 0 | | 5 | 109,536 | 5,646 | 10 | 32 | 5,586 | 28 | | 6 | 3,599,697 | 116,411 | 12 | 50 | 115,876 | 485 |

The complete $n\le5$ records have $\beta=\delta$ for every submaximum maximal state. In $G_6$, 124 labelled maximal 11-sets, in 16 dihedral state orbits, satisfy

$$ \beta=1,\qquad \delta=2. \tag{6.1} $$

They occupy 28 labelled nontrivial $J_{11}$-components in four geometric component orbits. The other 361 maximal $G_6$ states with $\delta=2$ have $\beta=2$ and are neutral-swap isolated: one has size eight, four have size nine, 28 size ten, and 328 size eleven.

The direct-depth distribution among submaximum maximal $G_6$ states is:

| $d$ | labelled states | |---:|---:| | 1 | 104,050 | | 2 | 9,173 | | 3 | 2,014 | | 4 | 1,072 | | 5 | 52 |

Forty-eight of the 52 depth-five states have $\delta=1$. This demonstrates that direct replacement difficulty and path-floor difficulty are distinct.

The census also finds $\delta\le2$ for every maximal $G_6$ state. The underlying layer record is: $J_h$ is connected for $h\le7$; at $h=8,9,10$, there are respectively $1,4,28$ maximum-free components, all singleton maximal states; and every nonmaximal 11-set has a 12-point coface. Thus every nonmaximal state has deficit at most one. The following general identity explains the asserted maximal-state bound.

Lemma 6.2 (floor identity). For safe $U\subseteq S$,

$$ F(U)=\min\{|U|,F(S)\},\qquad \delta(U)=\max\{\delta(S)-(|S|-|U|),0\}. \tag{6.2} $$

Proof. Grow $U$ to $S$ within $S$, then use an optimal path from $S$, for one inequality. In the other direction, delete from $S$ to $U$, then use an optimal path from $U$; also $F(U)\le|U|$. $\square$

It follows on any finite carrier that the assertion “every maximal state has $\delta\le2$” is equivalent to “every nonmaximal state has $\delta\le1$.” The equivalence is proved; its $G_6$ hypothesis is a finite checked fact, not an all-$n$ theorem.

7. The four second-barrier motifs in $G_6$

Coordinates are zero based. In each row, the listed core is common to every state in a representative neutral component. The residual description, component count, and full threshold boundary are exactly checked in the archived complete-grid certificate. The visible local geometry does not, by itself, exclude an escape through an unlisted grid site; that exclusion comes from the complete labelled-matching boundary check.

| type | common core | variable sites | residual state description | components / states | $H_{10}$ size | |:--|:--|:--|:--|--:|--:| | C1 | $(0,0),(0,4),(1,2),(2,1),(2,2),(3,5),(4,1),(4,3),(5,0)$ | $(1,5),(3,4),(5,3)$ | choose any two collinear variables | 8 / 24 | 33 | | C2 | $(0,2),(0,5),(1,2),(2,0),(2,1),(3,4),(4,3),(4,4),(5,0)$ | $(1,5),(3,3),(5,1)$ | choose any two collinear variables | 4 / 12 | 33 | | C3 | $(0,2),(1,4),(2,5),(3,3),(4,3),(4,5),(5,1)$ | $(0,4),(2,2),(3,1);(1,0),(2,0),(5,0)$ | one vacancy per triple, with one cross-pair forbidden | 8 / 40 | 54 | | C4 | $(0,0),(0,4),(1,5),(2,1),(3,5),(4,0),(4,1)$ | $(1,2),(2,3),(3,3),(5,2),(5,3),(5,4)$ | choose four; vacancies hit all three lines | 8 / 48 | 64 |

Neutral exchange graphs of the four component types: two triangles, a bowtie, and a 3-sun.

Figure 1. Each vertex represents a maximal 11-point set; each edge is one safe neutral exchange. C1 and C2 have the same abstract graph but are geometrically inequivalent under the square's dihedral group. The layouts are schematic. Vertex-to-state masks and exact adjacency checks are archived in figures/motif-data.json in the supplement.

In C1 and C2, the three variable points are collinear. The missing variable has the other two as its singleton blocker edge. A neutral move therefore rotates the vacancy around a triangle of maximal sets.

For C3, the variable triples are

$$ L_1=\{(0,4),(2,2),(3,1)\},\qquad L_2=\{(1,0),(2,0),(5,0)\}. $$

The core point $(2,5)$ forbids the cross-pair $\{(2,2),(2,0)\}$. Of the nine nominal vacancy pairs, five remain. Their swap graph is two triangles sharing one vertex.

For C4, set

$$ A=(1,2),\ B=(2,3),\ C=(3,3),\ D=(5,2),\ E=(5,3),\ F=(5,4). $$

The collinear variable triples are $\{B,C,E\}$, $\{A,C,F\}$, and $\{D,E,F\}$. A four-variable selection is safe exactly when its two vacancies meet all three triples. The six states form a 3-sun: a central triangle with one further vertex attached to each edge.

This C4 component has a public antecedent: u/cedar_conics posted its six-state family on 6 October 2026 [10]. The reflection $(x,y)\mapsto(5-x,y)$ identifies that family exactly with the C4 representative above. The public example has 58 one-deletion states and a closed 64-state component at height ten; every top state has $\beta=1$ and $\delta=2$. The complete four-type classification in this section is a separate finite result.

Every listed 11-point component consists of maximal states and has a closed one-deletion layer. Theorem 5.2 therefore supplies $\delta\ge2$; saved paths to a 12-point state at floor nine give equality.

A compact C1 witness is its core together with $(1,5)$ and $(3,4)$. Its three 11-point states and thirty 10-point deletion states form a closed 33-state component of $H_{10}$. It certifies

$$ \beta=1<2=\delta. $$

A representative C1 set on the 6 by 6 grid, with nine core points, two selected variable sites, and the third variable site empty.

Figure 2. The C1 core and the selected variables $(1,5),(3,4)$ form one maximal 11-set; $(5,3)$ is the empty variable site. The three variable sites lie on one line. Rotating the vacancy gives the triangle in Figure 1. Closure against all other grid sites is supplied by the boundary check, not by the drawing alone.

The classification is exactly checked finite computation. A determinant-based reconstruction independently recovered the exchanges and one-deletion boundaries, including the 28 labelled components and the threshold-component sizes 33, 33, 54 and 64. A separate fresh enumeration recovered the complete $G_6$ census and its exhaustiveness claim; both receipts are in the supplement.

8. Arbitrarily high barriers on restricted carriers

The next theorem is intentionally not a full-square result.

Theorem 8.1 (restricted carriers). For every $m\ge1$, there is a finite carrier $X_m\subset\mathbb Z^2$ and an inclusion-maximal submaximum safe set $S_m$ satisfying

$$ \beta_{X_m}(S_m)=\delta_{X_m}(S_m)=m. \tag{8.1} $$

Proof. The rational realization used below is the edge-blocker construction of Froese, Kanj, Nichterlein and Niedermeier [5, Proposition 1], applied to $K_{m,m+1}$. Their construction credits the underlying point-visibility transformation to Ghosh and Roy [6]. We include the specialized argument because the subsequent barrier calculation uses its exact incidence pattern. Choose rational points

$$ B=\{b_1,\ldots,b_m\},\qquad W=\{w_1,\ldots,w_{m+1}\} $$

in general position, fixing all of $B\cup W$ before any anchor. For each pair $(i,j)$, choose a rational anchor $a_{ij}$ on the line $b_iw_j$, avoiding all existing points and the finitely many intersections that would create a collinear triple other than the prescribed one. This finite-avoidance construction works because every rational line has infinitely many rational points. Distinct prescribed $B$–$W$ lines are distinct by general position. Every future prescribed line already joins an existing base pair, so each earlier anchor avoids it unless it is that anchor's own prescribed line. Hence the designated pair is the only old pair on the line chosen for a new anchor. It produces

$$ X=B\cup W\cup A,\qquad A=\{a_{ij}:1\le i\le m,\ 1\le j\le m+1\}, $$

whose only collinear triples are

$$ \{b_i,w_j,a_{ij}\}. \tag{8.2} $$

Common rational scaling and translation put $X$ in $\mathbb Z^2$.

Let $S=A\cup B$, $Q=A\cup W$, and $E=|A|=m(m+1)$. At each absent $w_j$, the $m$ disjoint pairs $\{b_i,a_{ij}\}$ are its blockers. Thus $S$ is maximal and $\beta(S)=m$.

To show that $Q$ is the unique maximum, let $T$ be safe, and put $r=|T\cap B|$, $s=|T\cap W|$. Every selected pair $b_i,w_j$ excludes the distinct anchor $a_{ij}$, giving

$$ |T|\le E+r+s-rs. \tag{8.3} $$

If $r,s>0$, the right side is at most $E+1$. If $r=0$, it is at most $E+m+1$, with equality only when $T=Q$; if $s=0$, it is at most $E+m$. So $Q$ is the unique maximum.

Before the first $w_j$ can be inserted on a path from $S$ to $Q$, one endpoint of every pair $\{b_i,a_{ij}\}$ must be deleted. The path reaches size at most $E$. Deleting all $b_i$ and then inserting all $w_j$ attains that floor. Hence $\delta(S)=m$. $\square$

Comparison with complete-bipartite reconfiguration. If all anchors $A$ are held selected, (8.2) identifies the safe states $A\cup I$, $I\subseteq B\cup W$, with the independent sets of $K_{m,m+1}$. Reaching the $W$ side in this subfamily requires deleting all $m$ vertices of $B$ before inserting the first $w_j$. This is the forced-clearance mechanism in de Berg, Jansen and Mukherjee's $K_{n,n}$ example [3, Section 5], where reconfiguration between the two sides must pass through the empty independent set. Their example concerns two specified equal-size maximum independent sets. Here $B$ is maximal but submaximum, $W$ is the unique maximum, and the geometric endpoints have different sizes. The first-insertion argument and (8.3) show that even allowing anchors to be deleted neither reduces the required loss nor creates another maximum. The realization and the complete-bipartite barrier mechanism are prior ingredients; the escape-deficit calculation states their application to this restricted geometric carrier. It supplies no transfer to the full square.

For $m\ge2$, adjoin an independent collinear triple $\{u,v,w\}$, with no mixed collinear triples, and start from $S'=S\cup\{u,v\}$. The missing $w$ has one blocker pair while every $w_j$ retains $m$ blockers. Thus $\beta(S')=1$. The neutral component on the added line is a triangle. Every maximum is $Q$ plus two points of this line; projecting a path onto the original carrier forces a projected state of size at most $E$. At most two points can remain on the added line, so the full path has a state of size at most $E+2$. Keeping the two added-line points while using the original deletion/insertion path attains this floor. Thus

$$ \delta(S')=m. \tag{8.4} $$

So a three-state closed neutral component, by itself, does not bound how far below it eventual escape occurs. Exact exhaustive checks of supplied 19-site and 22-site carriers give deficit three, with 164,305 and 1,150,135 safe subsets respectively; these finite checks supplement the all-$m$ proof.

9. The full-square transfer gap

9.1 A conditional transfer theorem

Embedding the carrier of Theorem 8.1 in a large $G_N$ does not preserve its proof. The extra lattice sites need not be blocked, so the old start need not be maximal. Greedily extending it can create low-cost gates and threshold paths around the intended lock. Square-grid maxima need not resemble $Q$, and the first point inserted on an escape path need not be one of the $w_j$. Thus induced-subcarrier reasoning is invalid.

There is, however, a precise sufficient condition for transfer. It tests the whole component above a proposed cardinality floor, rather than only the start.

Proposition 9.1. Let $X\subseteq G_n$, let $S\subseteq X$ be safe, fix $0\le h\le |S|$, and let $C$ be its component in $H_h(X)$. Suppose

$$ M_z(U)\ne\varnothing \quad\text{for every }U\in C\text{ and }z\in G_n\setminus X. \tag{9.1} $$

Then $C$ is also the entire component of $S$ in $H_h(G_n)$. If $|U|\le |S|=k$ for all $U\in C$ and a safe $(k+1)$-set exists in $G_n$, then

$$ \delta_{G_n}(S)\ge k-h+1. \tag{9.2} $$

Proof. An internal move stays in $C$ by its definition. A first insertion outside $X$ is forbidden by (9.1). Thus no threshold path leaves $C$, whereas all its original paths remain available. Every maximum of $G_n$ is larger than every state of $C$, proving (9.2). $\square$

For the carrier in Theorem 8.1, the component of $S=A\cup B$ at height $E+1$ is exactly

$$ \{S\setminus D:D\subseteq S,\ |D|\le m-1\}. \tag{9.3} $$

Indeed, no $w_j$ can be the first inserted new point until at least $m$ deletions have occurred, and every displayed deletion state is reachable. For an outside site $z$, blocking at every state in (9.3) is equivalent to $b_S(z)\ge m$: at most $m-1$ deletions cannot hit $m$ disjoint matching edges, while fewer than $m$ edges can all be hit by an allowed deletion set. Therefore a full-square embedding with this uniform outside guard would preserve $\delta\ge m$, witnessed by the old larger set $A\cup W$. No such embedding is asserted here.

For the $\beta=1$ extension with $m\ge2$, the relevant threshold is

$$ h=E+3=|S'|-m+1. $$

At that height the component is the union of the deletion families with at most $m-1$ deletions from each of the three neutral top states. The uniform guard is $b_T(z)\ge m$ for every outside site $z$ and each top state $T$. The preceding base height $E+1$ is not the extension threshold: at that lower height the extension can escape. This specifies the component to which Proposition 9.1 applies.

An elementary obstruction already excludes naive dilation. Every pair of a safe $k$-set blocks at most $n-2$ additional grid sites, so maximality requires

$$ n^2-k\le {k\choose2}(n-2). \tag{9.4} $$

The checked 17-point start on the 22-site restricted carrier fits inside $G_{147}$, but (9.4) leaves at least $147^2-17-{17\choose2}145=1872$ immediately free sites. This proves failure of maximality for that embedding, not a value of its full-square path deficit.

9.2 A full-square obstruction pattern

A full-square certificate for one example with $\delta\ge3$ would require:

1. a safe maximal $S\subseteq G_n$ with blockers recorded at every empty grid site;
2. a threshold-component lower-bound certificate, not only a large blocker count at selected named sites;
3. a larger safe set, or a sharp maximum proof and an escape path.

An unbounded full-square family would additionally need a scaling mechanism that keeps the extra lattice sites saturated and excludes alternative routes.

One decisive sufficient certificate would be a submaximum safe $S\subseteq G_n$ with $b_S(x)\ge3$ for every empty $x$, together with any larger safe set. The first-insertion bound then gives $\delta(S)\ge3$. This is a sufficient condition, not an existence assertion; Section 9.3 excludes it on $G_7,G_8$.

There is a useful reduction at size $2n-1$. If $S\subseteq G_n$ is safe with $|S|=2n-1$, each row and column contains at most two selected points. Thus exactly one row $r_0$ and one column $c_0$ contain one selected point; every other row and column contains two. For an empty point $x$,

$$ b_S(x)=\mathbf 1[x\text{ is on a full row}] +\mathbf 1[x\text{ is on a full column}] +b_S^{\mathrm{obl}}(x), \tag{9.5} $$

where $b_S^{\mathrm{obl}}(x)$ counts non-axis-parallel blocker secants. Therefore a size-$2n-1$ candidate with $\beta\ge3$ needs at least one oblique blocker away from the deficient row and column, at least two on either deficient axis away from their intersection, and, if their intersection is empty, at least three there. Whenever a $2n$-point safe comparison configuration is available, it is a larger witness. These are full-square-specific requirements for a decisive search; they do not assert that such a set exists.

Independent copies of small barriers do not add deficits. If there are no mixed collinear triples between the carriers, repair each part while holding the others fixed. Earlier repairs only increase the total cardinality, so the combined deficit is at most the largest individual deficit. Thus copies of a fixed independent gadget alone cannot produce an unbounded family.

More generally, extra selected points can compensate for a projected loss on the old carrier. None of the preceding conditions gives a universal bound for complete squares.

9.3 Certified exclusions and path bounds on $G_7,G_8$

The following exactly checked results use the certificate package described in Section 11. It distinguishes complete certificates from bounded searches.

Result 9.2 (high-blocker exclusion). Every submaximum safe set on $G_7$ or $G_8$ has $\beta\le2$.

Encoding and certificate. Introduce occupation variables $x_p$ and pair variables $e_{pq}\leftrightarrow(x_p\wedge x_q)$. Impose a negative clause for every collinear triple and, for each site $y$,

$$ \sum_{\substack{\{p,q\}\subseteq G_n\setminus\{y\}\\p,q,y\text{ collinear}}} e_{pq}\ \ge\ 3(1-x_y). \tag{9.6} $$

At an empty site the sum is precisely $b_S(y)$; at an occupied site the inequality is vacuous. Exact unary threshold equivalences encode these sums and $|S|\le2n-1$. For successive input literals $\ell_i$, the recurrence is $z_{i,t}\leftrightarrow z_{i-1,t}\vee(\ell_i\wedge z_{i-1,t-1})$, with $z_{i,0}=1$, $z_{0,t}=0$ for $t>0$, and explicit boundary cases. Induction gives the intended threshold semantics. Thus satisfiability is equivalent to the claimed obstruction; no symmetry restriction is used. Independently checked 14- and 16-point witnesses, together with the row bound, establish $\alpha(G_7)=14$, $\alpha(G_8)=16$.

| Board | Cardinality scope | Collinear triples | CNF variables | CNF clauses | DRAT-trim result | |---|---|---:|---:|---:|---| | $G_7$ | $0\le |S|\le13$ | 824 | 9,089 | 33,176 | VERIFIED UNSAT | | $G_8$ | $0\le |S|\le15$ | 1,544 | 16,688 | 61,185 | VERIFIED UNSAT |

The supplement's proof receipt records hashes of the historical instances, proofs, checker source and logs. Section 11 distinguishes these receipts from the archived source and the regeneration procedure. This result excludes a local first-insertion barrier of three on these boards. It does not exclude $\delta\ge3$ when $\beta=1$ or 2.

Result 9.3 (complete specified layers). Every maximal safe set in each of the following classes has $\delta\le2$.

| Board | Size | All safe sets of that size | All labelled maximal sets | D4 representatives with paths | |---|---:|---:|---:|---:| | $G_7$ | 12 | 471,860 | 305,960 | 38,493 | | $G_7$ | 13 | 15,532 | 13,684 | 1,722 | | $G_8$ | 15 | 79,968 | 73,888 | 9,251 |

For a maximal $k$-set, the search traverses $J_{k-1}$, testing whether a state extends all the way to a maximum. Each route is expanded into ordinary insertions and deletions at floor at least $k-2$. Separate determinant code checks every move. A separate row-recursion verifier considers every zero-, one- or two-point row choice, tests safety and maximality, and checks each maximal leaf against the symmetry-expanded path starts. This establishes coverage, rather than inferring it from a search completion flag. The three coverage receipts are included in the supplement.

In the first row the target test requires three compatible insertions from an 11-set to reach size fourteen; extension merely to thirteen would not certify a maximum route. Nonmaximal sets of sizes 13 on $G_7$ and 15 on $G_8$ insert directly to maxima. A nonmaximal 12-set on $G_7$ extends to the covered 13-layer and has $\delta\le1$ by Lemma 6.2.

Additional bounded row searches, nonuniform random greedy samples and targeted $\beta=2$ candidates yielded no third-barrier witness. An independent path audit verified 251,579 path records from 244,443 distinct tested starts, all reaching a maximum at floor at least their starting size minus two. Its compact scope receipt is included in the supplement; the large path list is omitted as described in Section 11. The sample lists overlap; the complete-class counts above are separate coverage statements. No whole-class conclusion is inferred for the remaining sizes. These data settle neither $\delta\le2$ for every maximal state of $G_7,G_8$ nor a bound for larger boards.

10. Open questions

1. Is $\delta\le2$ for every maximal safe subset of every $G_n$, or does a full-square example with $\delta\ge3$ exist?
2. Does a submaximum $S\subseteq G_n$ exist with $b_S(x)\ge3$ at every empty $x$? The complete $n\le6$ census and Result 9.2 exclude this on all boards through eight; the next possible board is $G_9$. At size $2n-1$, equation (9.5) supplies a particularly sharp necessary condition whenever a $2n$-point comparison set is available.
3. Can the four $G_6$ motifs be coupled into serial threshold locks rather than placed as independent gadgets?
4. What statistics of the labelled matching field predict which threshold component contains a maximal set?
5. Is there prior work already containing a Euclidean-grid reconfiguration census, these component types, or the restricted-carrier construction?

11. Reproducibility, scope, and software

The accompanying supplement contains the source needed for deterministic reconstruction of the finite enumerations, matching-component data, SAT encodings, and coverage checks. It also contains compact result receipts and SHA-256 hashes. Before submission, a separate manuscript reader wrote a fresh enumerator for the $G_n$ state spaces, $1\le n\le6$, without importing the producer's state list. This recovered the $G_6$ counts and the C1–C4 classification; separate checks verified the displayed path and component certificates. That implementation is archived as fresh_grid.cpp. The finite claims in Sections 6--7 are therefore exactly checked; they are not proof-assistant formalizations.

The $G_7,G_8$ high-blocker exclusions are certified by DRAT traces whose raw compressed files exceed the venue's supplement limit. The archive therefore includes the exact encoding generator, semantic checker, source and result hashes, and independently checked verification receipts, but not the raw traces or the large path lists. Those receipts are evidence of a verification run, not substitutes for the omitted traces. The supplement supplies commands for regenerating the CNFs, new DRAT proofs, and the three complete path classes, with independent checks. The additional sampled-path aggregate is documented by receipts and original hashes only; its entire historical sample is not archived in this package. The three complete path classes in Result 9.3 are precisely the three rows in its table. Across the combined, overlapping runs, 244,443 distinct tested starts were checked; the bounded or sampled populations support no unlisted whole-layer conclusion.

The certified SAT work used PySAT [13] with Glucose [14]; MiniSat [15] is credited as an acknowledged basis of that solver lineage. DRAT validation used DRAT-trim [16]. The supplied C++ and Python programs require a standards-conforming C++ compiler and Python 3; exact package versions and commands are listed in the supplement.

The acceptance reviewer of this paper's version 1 [18] reran the supplied $G_6$ census, motif reconstruction and two restricted-carrier enumerations. These were replays of inspected submitted implementations, not another newly written independent census. The reviewer also regenerated the $G_7,G_8$ CNFs, checked their gate semantics and checked the comparison maxima, but did not repeat the SAT/DRAT runs or the three complete path-class searches. The present revision changes exposition, comparisons and figures; it adds no new census or larger-board certificate.

The literature review found a public prior C4 occurrence and established realization/reconfiguration antecedents cited above. No claim of historical priority is made. In particular, the results do not prove a full-square unbounded-deficit theorem or a universal upper bound on $\delta$.

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