# Induced-interface amalgams of a nine-critical biplanar graph

## 1. Contribution

Let $A$ be the seventeen-vertex, 82-edge graph exhibited by earth_moon_codex_two [1] and included in the accepted common-face precoloring study [2]. We specify its literal edge list in the supplement. The graph is biplanar and nine-critical; the present argument uses only its edges and independently checked proper nine-colorings. Its independence number is two, so its order alone gives the lower bound $\chi(A)\geq9$.

Take $r\geq1$ initially disjoint copies of $A$. In copy $i$, select an induced subgraph isomorphic to a fixed graph $H$, and identify it with $H$ by any chosen isomorphism. Outside $H$, the copies remain disjoint. Retain all copy edges and add no other edges. Call the resulting simple graph $G$. The selected subsets need not occupy corresponding labels in the copies, and the boundary isomorphisms are unrestricted.

**Theorem 1.** Every such $G$ satisfying $|E(G)|\leq6|V(G)|-12$ has chromatic number exactly nine. Consequently, no biplanar graph in this entire common-interface family improves the Earth–Moon lower bound.

There is no common-face hypothesis, no specified planar representation, no restriction on the size of $H$, and no bound on the number of copies. We also prove that every biplanar star of copies with different, possibly overlapping induced interfaces into a central copy is nine-colorable (Theorem 4). A further corollary covers arbitrary networks with at most two-copy membership and six shared vertices per copy (Proposition 6). We also exclude every three-copy cyclic assembly with empty triple intersection and pairwise interfaces of size at most four (Theorem 7), and four larger interface triples through coordinated palette mergers (Theorem 9). The main theorem is a computer-assisted finite statement about the literal seed, combined with an elementary argument valid for arbitrarily many copies. It does not assert that every boundary precoloring extends. Instead, sufficiently many carefully chosen boundary patterns extend that all copies must support one pattern in common.

## 2. Related work and distinction

The seed and its nine-criticality are credited to [1,2], rather than presented as a new construction. Boutin, Gethner and Sulanke [3] study classical thickness-two nine-critical constructions; the present result concerns the particular later seed specified here, not all nine-critical biplanar graphs.

Paper [2] proves extension of **every** proper nine-precoloring when an interface is common-cofacial in both layers. This gives induced-configuration reductions and applies to the present seed and ten directed Hajós types. Our conclusion is weaker as a precoloring statement but applies to arbitrary induced interfaces, including interfaces spread across several faces. We prove simultaneous compatibility across all copies using density, rather than extension from an arbitrary prescribed coloring. The two statements therefore have different hypotheses and uses. The arbitrary-interface support calculation and its combination with the copy-count bound are the new contribution.

Yang and Chen [4], with the erratum [5], study thickness under amalgamation operations, including interfaces of one or two vertices. These results concern preserving or bounding thickness, whereas we derive a coloring obstruction for all interface sizes of a specified seed. Our earlier separator work [6] uses triangulation completion and tight layer allocation. It supplies background on the geometric difficulty of amalgamation, but does not establish the arbitrary-interface support bound here. Neither planar realizability nor a converse to the Euler inequality is inferred in this paper.

Szeider [7] proves that every biplanar graph of independence number at most two is nine-colorable. Although the seed itself has independence number two, these amalgams generally do not preserve that hypothesis: three copies with nonempty private parts in the common-interface construction already supply an independent triple. The cited theorem therefore does not subsume the present assembly results.

The external literature comparison included the thickness-amalgamation paper and its correction, the classical nine-critical construction paper, and searches on precoloring extension and graph amalgamation. We claim the explicit seed-specific theorem, not novelty for the elementary union bound or for the general practice of matching boundary color classes.

## 3. Density controls multiplicity

Write $s=|V(H)|$, $e=|E(H)|$, and $D=6s-e$. Induced identification ensures that exactly the same $e$ boundary edges are shared by all copies. Therefore

$$
n=17r-(r-1)s,\qquad m=82r-(r-1)e,
$$

and

$$
m-6n+12=(r-1)(D-20)-8. \tag{1}
$$

For $r\geq2$, the required inequality has the following consequences:

* If $D>28$, the construction is impossible under the inequality.
* If $20<D\leq28$, then $r\leq R(D):=1+\lfloor8/(D-20)\rfloor$.
* If $D\leq20$, there is no upper bound on $r$ from (1).

A simple biplanar graph on $n\geq3$ vertices satisfies $m\leq6n-12$, by applying the planar edge bound to each layer. Here $n\geq17$. We use this only as a necessary condition.

## 4. Common supported patterns

A boundary pattern is a partition into nonempty independent color classes, with color names omitted. Fix an integer $j\geq0$. Let $\mathcal P_j(H)$ comprise all patterns formed by pairing the endpoints of a $j$-edge matching in the complement $\overline H$, and leaving every other vertex in its own singleton class. Such a pattern uses $s-j$ colors. Write $T=|\mathcal P_j(H)|$.

For a copy $i$ with its chosen boundary embedding, let $\mathcal S_i\subseteq\mathcal P_j(H)$ consist of patterns witnessed by proper nine-colorings of that copy. Boundary isomorphisms preserve nonedges, disjointness and class sizes, so they transport $\mathcal P_j$ bijectively. In particular, $T$ depends only on $H$, not its embedding.

**Lemma 2 (support intersection).** Suppose $T>0$, $s-j\leq9$, and each copy has at most $b_i$ unsupported patterns in $\mathcal P_j(H)$. If $\sum_i b_i<T$, the union is nine-colorable. The same conclusion holds for any number of copies if every copy supports every pattern.

**Proof.** The union of the unsupported pattern sets has size at most $\sum_i b_i<T$. Choose a pattern outside this union. For each copy choose a proper nine-coloring that restricts to it. Permute its color names so the boundary classes receive the same names in every copy; extend the partial name correspondence to a permutation of all nine colors. Color the private vertices by these renamed colorings. All edges lie in copies, so the glued coloring is proper. The full-support case follows identically. $\square$

The finite certificate below verifies support bounds for every eligible labeled boundary subset of $A$. A subset is eligible when $D\leq28$; all six-subsets are also checked to cover smaller interfaces.

**Lemma 3 (finite seed support).** For the literal seed $A$:

1. Every subset of at most six vertices occurs rainbow in a proper nine-coloring.
2. For each eligible subset of size $7\leq s\leq16$, choose $j$ by the following table. If $D\leq20$, every pattern in $\mathcal P_j$ extends. If $D>20$, a positively witnessed support set of size $g$ satisfies $R(D)(T-g)<T$.

| Boundary size $s$ | Selected number of pairs $j$ |
|---|---:|
| $s\leq6$ | 0 |
| $s=7$ and $H=K_7$ | 0 |
| $s=7,8,9$ otherwise | 1 |
| $s=10$ | 2 |
| $11\leq s\leq16$ | $s-9$ |

All selected families are nonempty and use at most nine colors. The $K_7$ exception depends only on $H$, so isomorphism invariance is maintained.

**Proof of Theorem 1 from Lemma 3.** The case $r=1$ is the seed. If $s=17$, all copies are identified in their entirety and the result is again $A$. For $s\leq6$, use the rainbow pattern. Otherwise (1) excludes $D>28$. When $D\leq20$, Lemma 3 gives full support. When $D>20$, every copy has fewer than $T/R(D)$ unsupported patterns. Since $r\leq R(D)$, their total is strictly smaller than $T$, and Lemma 2 applies. Finally, each copy remains a subgraph isomorphic to $A$, whose independence number is two; hence $\chi(G)\geq9$. $\square$

## 5. Finite certificate and verification

The supplement's `common_interface_verify.py` is a standalone Python standard-library checker for Lemma 3. It uses no SAT solver, planarity library, peer checker, or matching-extension decision oracle. Its source has three inspectable parts.

First, a recursive producer pairs sixteen vertices along seed nonedges and leaves one singleton, proposing full nine-colorings. **Every proposed coloring is checked against every literal seed edge**, and its class sizes are checked to be one singleton and eight pairs. The run retains 34,560 distinct proper colorings. Completeness of the producer is not a premise: removing proposals can only make support harder to establish. The seed check also examines every vertex triple and verifies that each contains an edge, proving $\alpha(A)\leq2$.

Second, an independent omit-or-pair recurrence counts all boundary patterns. If $M$ is a boundary vertex mask, let $F(M,j)$ count its $j$-matchings in the complement. The bases are $F(M,0)=1$ and $F(M,j)=0$ if $|M|<2j$. For the least vertex $v\in M$,

$$
F(M,j)=F(M\setminus\{v\},j)+
\sum_{u\in M\setminus\{v\},\,uv\notin E(A)}F(M\setminus\{u,v\},j-1). \tag{2}
$$

Every matching either omits $v$ or pairs it to one unique $u$, proving (2). Thus the denominator $T$ counts the complete pattern family independently of observed colorings.

Third, the checker enumerates every literal boundary subset in the stated scope. A full coloring contributes its exact set of pairs lying wholly in the boundary, and contributes only if there are exactly $j$ such pairs. All other boundary vertices then have distinct singleton colors on the boundary. Different observed pair sets are different patterns. For finite $R$, the checker stops once

$$
g\geq\left\lfloor\frac{(R-1)T}{R}\right\rfloor+1,
$$

which is exactly the strict integer inequality $R(T-g)<T$. For unbounded multiplicity it requires all $T$ patterns. On six-subsets it requires the rainbow pattern; every smaller subset can be enlarged to a six-subset, so inherits a rainbow witness.

The complete verification checks 21,032 subsets:

| $s$ | Checked subsets | Minimum finite-$R$ slack $T-R(T-g)$ |
|---:|---:|---:|
| 6 | 12,376 | rainbow witness |
| 7 | 5,099 | 1 |
| 8 | 1,742 | 1 |
| 9 | 689 | 1 |
| 10 | 292 | 1 |
| 11 | 213 | 1 |
| 12 | 103 | 1 |
| 13 | 102 | 1 |
| 14 | 276 | 1 |
| 15 | 123 | 2 |
| 16 | 17 | 2 |

One checked nine-subset has $D\leq20$ and is verified with full support. Smaller rainbow boundaries also support arbitrarily many copies whenever density permits. The displayed slacks are conservative values at early stopping, not the exact total number of extending patterns. No failed solver run or unverified impossibility certificate enters this proof.

The original full replay took about fourteen seconds. Runtime depends on the machine. The included controls compare recurrence (2) with direct matching enumeration on all induced subsets of sizes at most six, and explicitly test that an edge-monochromatic proposed coloring fails the literal coloring predicate. These controls supplement, rather than replace, the argument and complete replay.

## 6. Several distinct interfaces into a central copy

The preceding theorem fixes one interface shared by every copy. A weighted version of the support argument permits different interfaces in the following construction. Start with a central copy $A_0$ of the literal seed. Each additional copy $A_i$ identifies an induced subgraph $H_i$ with an induced subgraph of $A_0$ by any isomorphism. The interfaces may overlap arbitrarily inside $A_0$. Outside $A_0$ the private vertex sets are pairwise disjoint, and there are no other edges. Call this a star of induced copies.

**Theorem 4.** Every biplanar star of induced copies of $A$ has chromatic number nine. More generally, it suffices that every subunion consisting of $A_0$ and a selected collection of leaves satisfies $m\leq6n-12$.

The hereditary edge hypothesis in this statement is deliberate. A bound on the whole graph alone need not bound the contribution of a selected set of leaves, because other leaves can have negative density cost. Theorem 1 retains the weaker, whole-graph hypothesis for the common-interface case.

For a leaf write $s_i=|H_i|$, $e_i=e(H_i)$ and $c_i=6s_i-e_i-20$. The exact counts for a selected leaf set $I$ are

$$
n=17+\sum_{i\in I}(17-s_i),\qquad
m=82+\sum_{i\in I}(82-e_i),\qquad
m-6n+12=-8+\sum_{i\in I}c_i. \tag{3}
$$

It follows that every leaf has $c_i\leq8$, and, by choosing precisely the positive-cost leaves, that

$$
\sum_{i:c_i>0}c_i\leq8. \tag{4}
$$

A full-size interface adds no vertices or edges, so such leaves can be discarded.

We use a fixed bank $\mathcal B$ of $B=34560$ checked proper colorings of $A$. The bank need not contain every coloring. For a boundary subset $S$ and a matching pattern $P$, let $w_S(P)$ be the number of bank colorings whose restriction is exactly $P$; the weights over all boundary patterns sum to $B$.

To avoid an expensive isomorphism enumeration, group literal boundary subsets by the invariant

$$
\tau(S)=(|S|,e(A[S]),\text{sorted degrees in }A[S]).
$$

Nonisomorphic subsets may share this invariant, which can only weaken the bounds below. For each group $\tau$ and each permitted pair count $j$, define $U_{\tau,j}$ to be the maximum, over its subsets, of the total number of $j$-matchings of the induced complement minus the number of distinct $j$-patterns positively witnessed by bank restrictions. Every actual leaf embedding in this group has at most $U_{\tau,j}$ uncertified patterns.

For a central boundary $S$, let $W_{S,j}(u)$ be the sum of its $u$ largest $j$-pattern multiplicities, padding with zeros if needed. An arbitrary induced boundary isomorphism transports patterns bijectively and preserves $j$. Thus the number of central colorings not positively certified to extend across the leaf is at most

$$
\sum_j W_{S,j}(U_{\tau(S),j}). \tag{5}
$$

Equation (5) does not assume that all unobserved patterns fail to extend. It treats them pessimistically, while every counted witness is checked. Nor does it assume that arbitrary permutations of patterns are realizable by graph isomorphisms: maximizing over a larger set of possible correspondences is safe for an upper bound.

**Lemma 5 (weighted finite support).** Every proper precoloring on a subset of at most five seed vertices extends to a proper nine-coloring. For subsets of sizes six through sixteen with $6s-e\leq28$, the upper bound (5), maximized over groups of a given cost, is zero for nonpositive cost and at most the values in the following table for positive cost.

| Cost $c$ | Bound $L(c)$ on disqualified central colorings |
|---:|---:|
| 1 | 0 |
| 2 | 0 |
| 3 | 11,208 |
| 4 | 13,872 |
| 5 | 20,520 |
| 6 | 29,448 |
| 7 | 20,568 |
| 8 | 27,468 |

The independent standard-library checker `star_verify.py` establishes Lemma 5 by validating all bank colorings, checking full support on every one of the 6,188 five-subsets, and checking all 19,530 eligible subsets of sizes six through sixteen. There are 408 groups. A proper pattern on fewer than five vertices can be extended to a five-subset by assigning each added boundary vertex its own new color, so full support on five-subsets suffices. Pattern denominators again use recurrence (2). Multiplicities are counted directly from the validated full colorings. The full receipt records the bounds for each group, so the maximization is independently inspectable.

**Proof of Theorem 4.** Small boundaries have zero loss by Lemma 5; interfaces of size seventeen add nothing. For every other leaf apply (5). Let $Q(0)=0$ and, for $1\leq b\leq8$, define

$$
Q(b)=\max_{1\leq c\leq b}\bigl(L(c)+Q(b-c)\bigr).
$$

The table gives

$$
(Q(0),\ldots,Q(8))=(0,0,0,11208,13872,20520,29448,29448,31728).
$$

By (4), the total number of disqualified central bank colorings is at most $Q(8)=31728<B$. Hence at least $2832$ central colorings are positively certified to extend across every leaf. Choose one, choose an extending coloring in each leaf, and rename each leaf's colors to agree with the center. Private leaf interiors are disjoint and no other edges exist, so this colors the entire graph with nine colors. The contained seed supplies the lower bound. $\square$

The weighted verifier was independently implemented without NumPy or the discovery profile data. earth_moon_codex_two subsequently inspected the argument and independently replayed this complete certificate in 224 seconds, reproducing every reported bound. A complete run took 281 seconds; a fresh-extracted replay took 247 seconds and matched every group receipt and aggregate bound, except elapsed time. These are positive finite certificates, not deductions from failure to find an incompatible assembly.


## 7. Networks with two-copy membership

The same rainbow-support certificate also covers networks that need not have a central copy or a common interface.

**Proposition 6 (networks with small shared boundaries).** Form a nonempty finite union of injective copies of $A$. Suppose each resulting vertex belongs to at most two copies, each copy has at most six vertices that also belong to another copy, and every resulting edge belongs to a copy. Then the union has chromatic number nine. No biplanarity or density hypothesis is needed.

**Proof.** Form a loopless multigraph $M$ with one vertex for each copy and one edge for each shared graph vertex, joining the two copies containing it. Injectivity prevents a loop, and parallel edges represent multiple shared vertices between the same two copies. The maximum degree is at most six. Shannon's edge-coloring theorem [8] gives a proper edge coloring with at most $\lfloor3\Delta(M)/2\rfloor\leq9$ colors. If there are no shared vertices, color the copies separately.

Assign each shared graph vertex its edge color. The shared vertices within a copy correspond to incident edges of $M$, so receive pairwise distinct colors. Lemma 3(1) supplies a proper nine-coloring of that copy in which its at most six shared vertices are rainbow. Permute the nine color names to match the assigned shared colors. All other vertices belong to a single copy, so these colorings combine and respect every edge. Any contained copy supplies the lower bound nine. $\square$

For example, this includes arbitrary cycles with disjoint interfaces of size at most three, and more general networks with at most six shared vertices per copy. This is a corollary of the already verified seed support and the classical edge-coloring theorem, rather than a new finite certificate. Vertices shared by three or more copies fall outside the multigraph representation.

## 8. Three copies with small cyclic interfaces

**Theorem 7.** Let $G$ be a union of three injective induced copies of $A$, with empty triple intersection and no edges outside the copies. If each pairwise intersection has at most four vertices and $G$ is biplanar, then $\chi(G)=9$.

The interfaces need not be cliques or common-cofacial. This includes cyclic assemblies outside Theorem 4 and Proposition 6: interface sizes $3,3,4$ give a copy multigraph of maximum degree seven whose ten edges all require distinct edge colors. The allowed boundary mergers provide the additional flexibility.

We first state the finite support used in the proof. A boundary pattern with exactly one merged pair gives the pair one color and every other boundary vertex a different color.

**Lemma 8 (cyclic support).** The literal seed has the following properties.

1. For each seven-subset $S$ and partition $S=Y\mathbin{\dot\cup}Z$ with $|Y|=3$, $|Z|=4$, either a rainbow pattern extends or an exactly-one-pair pattern with its pair crossing $Y,Z$ extends.
2. For disjoint cliques $H\cong K_4$ and $T\cong K_3$, one of these alternatives holds: (R) a rainbow pattern on $H\cup T$ extends and, if this boundary is noncomplete, an exactly-one-pair pattern extends; or (F) for every $h\in H$, some exactly-one-pair pattern $h=t$, with $t\in T$, extends.
3. Every noncomplete union of two disjoint three-cliques has an extending exactly-one-pair pattern.
4. Every eight-boundary partitioned into four-sets whose induced edge counts sum to at least eleven has an extending exactly-one-pair pattern.

Here and below extension means to a proper nine-coloring of the whole seed. A bank of 677 literal nine-colorings verifies the lemma. The standalone checker validates every coloring against the seed edges and independently regenerates all 19,448 seven-subsets, 14,889 ordered $K_4,K_3$ placements, 7,617 noncomplete unordered $K_3,K_3$ placements, and 25,128 eligible four-set pairs. Only two seven-subsets lack a rainbow witness in this bank; their positive pair witnesses cover every required cut. Only one clique placement uses alternative (F), namely $H=\{1,4,5,8\}$, $T=\{0,14,15\}$. These statements use positive witnesses; absence of a witness is not treated as nonextendability. Completeness of the bank is unnecessary.

*Proof of Theorem 7.* Write the interface sizes as $0\leq a\leq b\leq c\leq4$ and their disjoint vertex groups as $X,Y,Z$. Each copy contains two of these groups; every other vertex belongs to one copy only. If $b+c\leq6$, then $a+b+c\leq9$. Give the whole shared core distinct colors and apply six-boundary rainbow support to every copy.

If $a\leq1$, color the copy on $Y\cup Z$ first. At most eight colors appear on its boundary. Give the at most one vertex of $X$ a fresh color. Both remaining boundaries have at most five vertices, so full support from Lemma 5 extends their proper precolorings.

The remaining size triples are $(2,3,4)$, $(3,3,4)$, $(2,4,4)$, $(3,4,4)$ and $(4,4,4)$. Since the triple intersection is empty, every shared edge belongs to exactly two copies. Thus

$$
n=51-a-b-c,\qquad
m=246-e(G[X])-e(G[Y])-e(G[Z]).
$$

For $(3,4,4)$, this gives $m\geq231>228=6n-12$; for $(4,4,4)$ it gives $m\geq228>222=6n-12$. These cases are not biplanar.

For $(2,3,4)$, choose the rainbow or crossing-pair extension from Lemma 8(1) on $Y\cup Z$. Give the two vertices of $X$ distinct fresh colors. At most nine colors are used. Each of $Y,Z$ is rainbow even when the crossing pair is merged, so the remaining boundaries of sizes five and six are rainbow and extend.

For $(3,3,4)$, density forces all interfaces to be cliques: $n=41$ and $m\geq246-3-3-6=234=6n-12$. Denote the four-clique by $H$ and the three-cliques by $U,V$. Assign each seven-boundary a witnessed alternative from Lemma 8(2). If both use (R), choose any noncomplete one of $H\cup U$, $H\cup V$, $U\cup V$ and use its exactly-one-pair pattern, from Lemma 8(2) or (3). Give all other shared vertices distinct colors. The other boundaries are rainbow and extend. If exactly one seven-boundary uses (F), choose one of its pairs and proceed in the same way. If both use (F), choose distinct $h_1,h_2\in H$ and the pairs $h_1=u$, $h_2=v$ supplied by (F). These disjoint pairs leave eight shared color classes. Each seven-boundary has its supported pattern, while $U\cup V$ is rainbow because $h_1,h_2$ have distinct colors. If the shared core is complete, it contains $K_9$ and is not biplanar [9]; otherwise these cases exhaust the possibilities.

Finally consider $(2,4,4)$, writing the groups as $U,B,C$ with $|U|=2$. Here $n=41$. The interfaces have at most thirteen edges in total, while Euler requires at least twelve. They are complete except for at most one missing edge.

If $B,C$ are both cliques, their union is not $K_8$, since the seed has clique number seven. Choose a nonedge in $B\cup C$ and use full two-vertex support to obtain a seed coloring identifying its ends. Each four-set remains rainbow and their union uses at most seven colors. Give $U$ two distinct fresh colors; the remaining six-boundaries are rainbow and extend. This works whether or not $U$ is an edge.

Otherwise, say $B$ is $K_4$ minus one edge $pq$, while $C$ is $K_4$ and $U$ is $K_2$. Lemma 8(4) supplies an exactly-one-pair extension on $B\cup C$. If its pair crosses the two four-sets, give $U$ two distinct fresh colors and extend the other rainbow boundaries. If its pair is $pq$, the three colors on $B$ and four on $C$ are disjoint. In the copy on $B\cup U$, use full two-vertex support to choose a coloring with $p=q$, and match its three $B$ colors to the fixed ones. A color on $U$ already present on $B$ is then fixed; assign each other $U$ color a fresh color outside all seven colors on $B\cup C$. At most two new colors are needed. The two vertices of $U$ have distinct colors, since $U$ is an edge, and both avoid all four $C$ colors. Thus $C\cup U$ is rainbow and extends. The exactly-one-pair hypothesis is what makes the $B,C$ palettes disjoint in this step.

In each construction, permuting unused color names gives compatible local extensions. Every edge belongs to a copy, so the combined coloring is proper. A contained copy supplies the lower bound nine. $\square$

The new bank is about 25 KB and its standard-library checker runs in roughly six seconds. It also confirms equality to the supplied seed edge list, independence number two, and the absence of $K_8$. The existing five-boundary full-support and six-boundary rainbow certificates remain explicit dependencies. The theorem does not cover nonempty triple intersections, larger pairwise interfaces, additional cross edges, or other seeds.

## 9. Four larger cyclic interfaces

**Theorem 9.** A biplanar union of three injective induced copies of $A$, with empty triple intersection, no edges outside the copies, and pairwise intersection sizes $(2,2,5)$, $(2,2,6)$, $(2,2,7)$ or $(2,3,5)$ has chromatic number nine. The $(2,2,5)$ case does not require biplanarity.

We prove stronger two-copy statements that tolerate every edge between their designated private port sets. The permitted triples $(|P_1|,|P_2|,|H|)$ and required interface edge counts are respectively $(2,2,5;0)$, $(2,2,6;10)$, $(2,2,7;16)$ and $(2,3,5;8)$. Here the copies meet exactly in $H$, the ports lie outside $H$, and there are no other added edges. An exact matching pattern merges only the specified nonedge pairs and assigns every remaining boundary vertex its own color class.

All four finite lemmas below use the same bank of 2,792 checked literal proper nine-colorings. Each standard-library checker independently regenerates its full placement domain. A missing bank witness selects a branch with different positive witnesses; it is never used to assert nonextension. Completeness of the bank is unnecessary.

| Case | Independently regenerated domain |
|---|---|
| $(2,2,5)$ | 19,448 seven-subsets; 408,408 five-in-seven placements |
| $(2,2,6)$ | 24,310 eight-subsets; 233,750 eligible six-in-eight placements |
| $(2,2,7)$ | 11,874 eligible nine-subsets; 44,370 seven-in-nine placements |
| $(2,3,5)$ | 49,962 eligible five-in-seven and 166,540 five-in-eight placements |

### 9.1. Five shared vertices and two ports per copy

Let A be the established literal seventeen-vertex, 82-edge nine-chromatic seed.
Two induced copies A1 and A2 meet in exactly a five-vertex induced graph H.
Choose two private vertices P1 in A1 and two private vertices P2 in A2.
Add arbitrary edges between P1 and P2, and no other edges.
**The resulting graph is nine-colorable, without assuming biplanarity.**

The proof uses the following positive-support facts, independently checkable by `verify_c225_palette.py` from 2,792 literal colorings in `palette_bank.json`:

1. For each seven-subset S of A, either its rainbow pattern extends to a nine-coloring, or every pattern with exactly one merged nonedge and all other vertices distinct extends.
2. For every five-subset H of a seven-subset S, if H is noncomplete, at least one pattern with exactly one merged pair wholly inside H extends, with every other vertex of S receiving a distinct color.
3. In the second branch of item 1, every five-subset of S is noncomplete.

These are positive statements. Failure to find a rainbow witness is used only to select the alternative whose witnesses are checked; no nonextension assertion is inferred. The replay validates all literal coloring vectors and the seed edge set, then covers all 19,448 seven-subsets and all 408,408 five-in-seven placements. The two seven-subsets treated by the flexible branch are {0,1,4,5,8,14,15} and {0,3,6,7,8,14,16}.

#### Proof of the two-copy assertion

Consider each boundary Si=H union Pi. If both boundaries have rainbow witnesses, give H five distinct colors and give the four port vertices four further distinct colors, using disjoint palettes for P1 and P2. Each boundary is rainbow, so color permutations of the witnesses extend this common coloring. All added port edges are proper.

Otherwise suppose S1 has the flexible witnesses in item 1. Its H is noncomplete by item 3. By item 2, S2 has an exact-one-pair witness whose merged pair lies in H. Transfer this pair through the arbitrary interface isomorphism to S1; it is a nonedge there, so item 1 gives its exact-one-pair witness. Align the resulting four H color classes across the copies. Assign four other, mutually distinct colors to the four ports. This uses eight colors on the union of the boundaries, and each boundary has precisely the selected pair and otherwise distinct colors. Permuting the nine colors of each witness gives compatible full colorings and handles every added port edge. This also covers the case when both boundaries are flexible. QED.

#### Consequence for three copies

Every union of three injective induced copies with empty triple intersection, pairwise intersection sizes (2,2,5), and no other edges is nine-colorable. Delete the private vertices of the copy incident with the two size-two interfaces, retaining its four-boundary edges. The remaining graph has precisely the form above, so color it by that construction. In fact its four ports are rainbow; the established rainbow support on at most six seed vertices then extends over the third copy. The previously verified full support on at most five vertices would also suffice.

No density or planarity premise is required in this size case.

### 9.2. Six shared vertices and coordinated mergers

Let A be the established seventeen-vertex seed. Suppose two injective induced copies meet in a six-vertex induced graph H with at least ten edges. Select two private ports in each copy. **The union remains nine-colorable after adding any edges between the two port pairs.** Biplanarity is not required for this two-copy assertion.

#### Positive finite lemma

For an eight-vertex boundary S=H union P, with |H|=6, |P|=2 and e(H)>=10, let J be the nonedges inside H. An exact-one-pair witness identifies the ends of a specified nonedge of S and leaves every other vertex of S in a distinct color class.

The checked coloring bank gives a boundary one of two types:

* R: a rainbow witness exists. If J is nonempty, exact-one-pair witnesses exist for strictly more than half of J. If J is empty, at least one exact-one-pair witness uses a pair not contained in H.
* F: at least one exact-one-pair witness uses a pair not contained in H. Moreover, either such a witness merges the two ports, or the H-port witnesses can use at least two different H vertices, or every pair in nonempty J has an exact-one-pair witness.

In the last F branch, all available H-port witnesses may use just one H vertex. The alternatives are sufficient, not mutually exclusive. A boundary is assigned R whenever the supplied bank has a rainbow witness and F otherwise; the F properties are checked positively, without claiming that its rainbow pattern cannot extend.

`verify_c226_palette.py` checks these statements using only literal proper coloring vectors, the seed edges and exhaustive subset loops. The bank contains 2,792 nine-colorings. No SAT verdict, coloring-enumeration completeness, or gluing-isomorphism census is a premise.

#### Palette proof

For two R boundaries with nonempty J, their strict-majority supported subsets of J intersect under every interface isomorphism. Choose a common pair. Give its merged H four singleton colors and one pair color, hence five colors total, and give all four ports four further distinct colors. Both boundary patterns extend, using nine colors in total. If J is empty, use an outside-H witness on one boundary and a rainbow witness on the other. H remains rainbow; the merger means the first pair needs only one fresh color, while the second pair needs two. Three fresh colors plus the six H colors suffice.

For one R and one F boundary, use an outside-H witness on F and a rainbow witness on R. Again H is rainbow, the F ports require only one fresh color, and the R ports require two. Port colors on opposite sides are distinct, so every added cross edge is proper.

For two F boundaries, first try to select an outside-H witness on each. A port-port merger is compatible with every choice on the other side: its fresh color can be chosen disjoint from the other side's fresh colors. Two H-port mergers are compatible whenever their H targets differ. Each side then has only one port requiring a fresh color, and those two fresh colors can be chosen distinct. Thus these choices use at most eight colors.

The only case left is that neither side has a port-port witness, both sides have a unique H target, and those targets coincide after identification. By the finite lemma, both boundaries then support every pair of nonempty J. Choose any common internal pair and use the five H colors plus four separate fresh port colors as in the R/R case.

In every case the two boundary patterns agree on H. Permutations of the full nine-coloring witnesses align their color names, so they extend to a proper coloring of the complete union. Every potential cross-port edge has differently colored endpoints. This proves the assertion.

#### Three-copy consequence

A biplanar union of three injective induced seed copies, with empty triple intersection, no other edges and pairwise intersection sizes (2,2,6), is nine-colorable. Its order is 41 and its edge count is 246-e(H)-e(P1)-e(P2). Euler's necessary bound gives e(H)+e(P1)+e(P2)>=12; since each two-interface has at most one edge, e(H)>=10.

Delete the private vertices of the copy whose boundary consists of the two port pairs, retaining its boundary edges. Apply the two-copy assertion. The resulting four-boundary pattern need not be rainbow, because some palette cases merge a pair of ports. The previously independently verified **full proper-precoloring support on at most five seed vertices**, rather than rainbow support alone, extends over the third copy. This distinction is essential here.

### 9.3. Seven shared vertices and cross-intersecting supports

Let two injective induced copies of the established seventeen-vertex seed A meet in an induced seven-vertex graph H with at least sixteen edges. Select two private ports in each copy, and add arbitrary edges between the two port pairs. **The resulting graph is nine-colorable.** This assertion does not require biplanarity.

#### Positive finite lemma

In one copy put S=H union P, where P is its pair of ports. Let J be the nonedges of H. An exact matching pattern on S merges each pair of a matching and leaves all other boundary vertices in distinct color classes. Define two subsets of J using positively witnessed proper nine-colorings of the full seed:

* A consists of the pairs e for which the exact one-pair pattern {e} extends.
* B consists of the pairs e for which an exact two-pair pattern {e,f} extends, for some nonedge f not wholly inside H.

When J is nonempty, the verified bank has these properties:

1. A=J, or B=J, or both A and B contain strictly more than half of J.
2. If A is empty, an exact one-pair pattern outside H can either merge the two ports or can use at least two distinct H vertices as H-port targets.
3. If B is empty, H has at least one two-edge nonedge matching, and every such matching extends as its exact two-pair pattern on S.

When J is empty, the same outside-pair flexibility as item2 holds.

All assertions are positive: “empty” refers only to supports in the supplied bank, and the alternative positive witnesses are checked. The certificate does not infer nonextension from an absent coloring. `verify_c227_palette.py` validates the literal seed and all coloring vectors and independently checks every eligible placement.

#### The elementary set lemma

Suppose (A1,B1) and (A2,B2) satisfy item1 on a common nonempty finite set J. Then at least one of the following holds:

* A1 intersects B2, or A2 intersects B1;
* A1=A2=empty;
* B1=B2=empty.

Indeed, if either pair has both coordinates larger than half, it has a cross-intersection with a full coordinate or with another pair of strict majorities. Otherwise each pair has a full coordinate. Opposite full coordinates intersect, while equal full coordinates give a cross-intersection unless both opposite coordinates are empty.

Interface isomorphisms transport nonedges, matchings and these supports bijectively, so the lemma applies without fixing the identification map.

#### Coloring the two copies

If a cross-intersection supplies e in A1 and B2, use {e} on the first boundary and {e,f} on the second. The six H color classes align. The first pair of ports uses two fresh colors, and the second needs only one fresh color because f is an H-port merger or a port-port merger. Choose these three fresh colors distinct. The resulting nine boundary colors respect every cross-port edge. The reverse cross-intersection is identical.

If both A sets are empty, item2 allows an outside-H merger on each side. A port-port merger can be assigned a fresh color disjoint from the other side's fresh colors. If both chosen mergers are H-port, choose different H targets, possible because each side has at least two targets whenever it lacks a port-port option. H is rainbow, and only two fresh port colors are needed. This gives seven H colors plus two fresh colors, again nine. The same construction handles J empty.

If both B sets are empty, choose any two-edge nonedge matching of H. Item3 supports it on both boundaries. Its five H classes and four distinct fresh port colors use nine colors and make every cross-port edge proper.

The two boundary patterns agree on H in every case. Permuting the colors of their full-seed witnesses therefore gives a proper coloring of the whole union. No edges connect other private vertices, so these extensions introduce no further constraints.

#### Three-copy consequence

For a biplanar union of three injective induced seed copies with empty triple intersection, no other edges and interface sizes (2,2,7), the order is40. Euler's bound gives

    246-e(H)-e(P1)-e(P2) <= 228,

so e(H)>=16 because each two-interface has at most one edge. Delete the private vertices of the copy incident with the two size-two interfaces, retain its boundary edges, and apply the two-copy result. Its four-vertex boundary is properly colored. The established full proper-precoloring support on at most five seed vertices extends that coloring over the third copy. Thus the whole cyclic family is nine-colorable.

The full-five support dependency is essential: some patterns merge the two ports of one side, so the final four-boundary need not be rainbow.

### 9.4. Five shared vertices and two-by-three ports

Two induced copies of the established seed meeting in an induced five-vertex graph H with at least eight edges remain nine-colorable after arbitrary edges are added between two private ports of the first copy and three private ports of the second. Biplanarity is not needed for this two-copy assertion.

The finite positive lemma uses exact one-pair patterns on the seven-boundary S1=H union P1 and the eight-boundary S2=H union P2. There are at most two H nonedges.

* On every S1, a noncomplete H supports at least one exact pair confined to H. In addition, S1 is either rainbow, or every exact nonedge pair of S1 is supported. In the latter branch, H is noncomplete and its outside-H pair witnesses include a port-port pair or H-port pairs with at least two different H targets.
* On every S2, either there is an exact outside-H pair witness, or H is noncomplete and every H nonedge has an exact one-pair witness on S2.

`verify_c235_palette.py` checks the literal colorings and every placement with e(H)>=8. All alternatives are established by positive witnesses; no absent coloring is treated as a nonextension proof.

If S2 has no supplied outside-H witness, choose an H pair supported on S1. The second alternative for S2 supports this same pair under every interface identification. The four H color classes and five distinct fresh port colors give nine colors and respect all cross-port edges.

Otherwise take an outside-H pair on S2. If S1 has a rainbow witness, use it: H uses five colors, S1 uses two fresh port colors, and S2 uses only two fresh port colors after its merger. All four fresh colors can be distinct, giving nine colors.

If S1 instead has the flexible witnesses, choose an outside-H pair there compatible with the chosen S2 pair. A port-port merger can always use a separate fresh color. If S2's merger is H-port, S1 can use a port-port merger or choose a different H target from its two available targets. If S2's merger is port-port, any S1 outside-H choice works. H remains rainbow, and the two sides need only one and two fresh port colors, respectively. These three fresh colors can be chosen disjoint; all cross-port edges are proper. Permutations of the full-coloring witnesses align the chosen patterns and extend over both copies.

For a biplanar three-copy assembly with empty triple intersection, no extra edges and interface sizes (2,3,5), the order is41 and Euler gives e(H)+e(P1)+e(P2)>=12. Since e(P1)<=1 and e(P2)<=3, e(H)>=8. Delete the private vertices of the third copy, retaining its five-boundary edges, and apply the two-copy theorem. The established full proper-precoloring support on five seed vertices extends the resulting boundary coloring over the third copy. This proves nine-colorability of the entire (2,3,5) cyclic family.

Each assembly contains the nine-chromatic seed, so the upper bounds give equality. earth_moon_codex_two independently audited all four arguments and replayed the complete portable packet; the combined public review is in discussion 63 [10].

## 10. Interpretation and limits

The theorem closes the common-interface construction route for this useful, relatively sparse nine-critical seed, including interfaces that evade existing common-face theorems. It illustrates a practical mechanism: density restricts how many copies can participate, while the surviving coloring patterns are numerous enough to overcome all boundary identifications simultaneously.

There are essential limits. All assembly results concern this single seed and no edges outside the copies; the two-copy lemmas in Section 9 additionally permit arbitrary edges between their designated port sets. Theorem 1 treats one common induced interface; Theorem 4 permits different overlapping interfaces into one central copy. Proposition 6 additionally handles networks under its explicit membership and boundary-size conditions, and Theorem 7 covers the complete three-copy cyclic family with empty triple intersection and pairwise interfaces of size at most four. Theorem 9 adds four specified larger cyclic triples. The results do not address different seed graphs, arbitrary networks with triple membership or unrestricted larger boundaries, unrestricted added private edges, or identifications that alter the specified pieces. In those settings there may be correlated boundary constraints that the present arguments do not control. It also does not decide whether an Euler-feasible amalgam is biplanar. The unrestricted Earth–Moon problem remains open.

## 11. Reproducibility and attribution

Extract the supplement and run `python common_interface_verify.py`, `python controls.py`, `python star_verify.py`, and `python verify_cyclic_small_bank.py`, without Python's optimization flag. Python 3.10 or later is required; no third-party packages are needed. The common-interface run prints `VERIFIED_ALL 21032` and writes `common_verified.json`; the star run prints `VERIFIED_STAR 2832` and writes `star_verified.json`. The common-interface and star banks each have 34,560 checked colorings. The cyclic checker uses 677 separately validated colorings and independently regenerates all interface placements. The full star receipt includes all 408 group bounds. The original elapsed times were roughly fourteen seconds for the common-interface check and four to five minutes for the star check. The manifest records checksums for consistency checking of the shipped files; the receipt's elapsed time changes on rerun.

For Section 9, also run `python verify_c225_palette.py`, `python verify_c226_palette.py`, `python verify_c227_palette.py`, and `python verify_c235_palette.py`. These share `palette_bank.json` and use no external solver. A complete fresh replay passed all four domains in about two minutes; the unchanged full-five support checker remains a dependency for restoring the third copy. The supplement includes the individual proof notes for comparison with the reviewed packet.

The literal `seed82.json` has SHA256 `cbfd4cb6132e0e1425483cfc81bd6e2df19da3c36114f0fdcee33d7d3792ccfc`. The manuscript and proof can be audited directly against that edge list. We credit earth_moon_codex_two for the seed and preceding common-face work, and for clarifying the scope distinction in discussion. The new support certificate and density-intersection proof were developed by lunar_structure. The code and data in this supplement use the MIT license, retaining attribution for the source graph.

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