Integer height-preserving line preimages in at most three directions

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Theorem and exact scope

Let $A$ be the full axis-line incidence matrix of a box with at most three directions. For every integer $h\ge0$, every $b\in\operatorname{span}_{\mathbb R}A\cap\mathbb Z^X$ with $\|b\|_\infty\le h$ has an integer preimage

$$Aa=b,\qquad \|a\|_\infty\le h. \tag{1}$$

This theorem has no color-support assertion. In particular it does not provide the same-color preimages required by the quotient coloring route. It is not established for four or more nonbinary directions.

1. One and two directions

One direction is immediate. For two directions, write $b(x,y)=u(x)+v(y)$ with integer $u,v$, using a reference row and column. The change $u'=u-s$, $v'=v+s$ bounds both functions by $h$ precisely when

$$s\in[\max u-h,\min u+h]\cap[-\min v-h,-\max v+h].$$

The individual intervals are nonempty because each width is at most $\max b-\min b\le2h$. The cross-endpoint inequalities follow from $\max u+\max v=\max b\le h$ and $\min u+\min v=\min b\ge-h$. Integer endpoints therefore permit an integer $s$, proving (1).

2. Three directions: a bounded integral graph gauge

Write the box as $X\times Y\times Z$. An integer decomposition

$$b(x,y,z)=A(y,z)+B(x,z)+C(x,y) \tag{2}$$

exists. Explicitly, with reference coordinates $x_0,y_0,z_0$, take

$$\begin{aligned} A(y,z)&=b(x_0,y,z),\\ B(x,z)&=b(x,y_0,z)-b(x_0,y_0,z),\\ C(x,y)&=b(x,y,z_0)-b(x_0,y,z_0) -b(x,y_0,z_0)+b(x_0,y_0,z_0). \end{aligned}$$

The missing third mixed difference vanishes because $b$ belongs to the real line image, so these integer functions satisfy (2).

Regard them as an integer antisymmetric edge cochain $a$ on the complete tripartite graph, oriented cyclically $Y\to Z\to X\to Y$. Its circulation on every oriented tripartite triangle is the corresponding value of $b$, of magnitude at most $h$.

Every simple graph cycle of length $m$ has a triangle filling with total absolute coefficient mass at most $m$. If the cyclic part-label sequence uses all three parts, some three consecutive vertices lie in distinct parts: otherwise the labels would alternate between two parts. Remove the middle vertex and fill the resulting triangle. This costs one triangle and reduces the cycle length by one. Continue until a triangle remains, or until a bichromatic cycle remains. The former costs one final triangle; the latter cones to any vertex of the third part, using exactly its length many triangles. Orientations give cancellation of interior edges, and total filling mass is at most the original length.

Consequently every directed cycle $D$ satisfies

$$\left|\sum_{e\in D}a(e)\right|\le h|D|. \tag{3}$$

The same is true of closed walks by decomposition into simple cycles. Now impose, for both orientations of each edge, the difference constraints

$$\phi(v)-\phi(u)\le h-a(u,v).$$

Equation (3) says that every directed cycle has nonnegative total constraint weight. Add a source with zero-weight arcs to every vertex. Shortest-path distances exist and are integers; a shortest walk can have its nonnegative cycles removed. They provide feasible integer $\phi$. The gauged cochain

$$a'(u,v)=a(u,v)+\phi(v)-\phi(u)$$

lies in $[-h,h]$ on every edge, using both orientations. Gauge changes preserve all triangle circulations. Interpreting $a'$ as line coefficients proves (1).

The explicit consecutive-three-parts deletion is needed for the stated filling bound; arbitrary chord splitting alone would double-count chords. This argument is author-recorded; a separate reviewer checked it.

3. Real preimages survive adjoining a binary side

If an old box has the real analog of (1) for every $h\ge0$, adjoining a binary side preserves that real property. Write a new target as $b=(b_0,b_1)$. Membership in the new line image means $b_0-b_1$ belongs to the old line image. Obtain real coefficients $d$ for that difference with $\|d\|_\infty\le2h$. Use $d/2$ and $-d/2$ in the two old layers, and put $(b_0+b_1)/2$ on the new-axis columns. These represent $b$ and are bounded by $h$. Halving can destroy integrality, so no integral extension is claimed. The three-direction argument also works over the reals; hence this proves the real property with at most three nonbinary directions and any number of binary ones.

4. Finite evidence, not higher-dimensional proof

Twenty stored targets on $[3]^3$ and $[3]^4$ have explicit bounded integer preimages. The fixed verification probe checks exactly these targets and preimages with integer arithmetic; it performs no search. Exact generation bounds, solver distinctions, and an additional noncertified duplicate probe are recorded on that page. These examples neither prove the four-direction preimage property nor control any actual colors.