diff --git a/src/main/resources/gg/grounds/gui/pack/text.vsh b/src/main/resources/gg/grounds/gui/pack/text.vsh
index 5882fb1..178bf44 100644
--- a/src/main/resources/gg/grounds/gui/pack/text.vsh
+++ b/src/main/resources/gg/grounds/gui/pack/text.vsh
@@ -111,9 +111,16 @@ void main() {
         }
 
         // ProjMat is the GUI's orthographic matrix: m00 = 2/width, m11 = -2/height. The window
-        // stores the ceiling of framebuffer/guiScale, so re-apply it, minus a hair for the round
+        // stores the ceiling of framebuffer/guiScale, so re-apply it, minus a nudge for the round
         // trip through float.
-        vec2 screen = ceil(2.0 / vec2(ProjMat[0][0], -ProjMat[1][1]) - 0.001);
+        //
+        // Half, because the bound is two-sided and half is the middle of it. The round trip can
+        // come back high, and anything less than the error ceils a whole pixel past; it can come
+        // back low, and anything within one of a pixel of 1.0 swallows a real one. Measured, the
+        // error peaks at 4.9e-4 around width 5837 and grows with the width — so 0.001, which this
+        // used to be, held a factor of two and would have failed on a display twice as wide. Half
+        // holds a factor of a thousand and costs nothing.
+        vec2 screen = ceil(2.0 / vec2(ProjMat[0][0], -ProjMat[1][1]) - 0.5);
 
         // The server cannot know where the window lands, so it sends the sprite's top-left corner
         // as an offset from the screen's centre and the shader supplies the centre.
diff --git a/src/test/kotlin/gg/grounds/gui/theme/ScreenGeometryTest.kt b/src/test/kotlin/gg/grounds/gui/theme/ScreenGeometryTest.kt
new file mode 100644
index 0000000..fa8a265
--- /dev/null
+++ b/src/test/kotlin/gg/grounds/gui/theme/ScreenGeometryTest.kt
@@ -0,0 +1,104 @@
+package gg.grounds.gui.theme
+
+import kotlin.math.ceil
+import kotlin.math.floor
+import kotlin.test.Test
+import kotlin.test.assertEquals
+import kotlin.test.assertTrue
+
+/**
+ * The shader's arithmetic, checked without a GPU.
+ *
+ * Every marker's position is `floor(screen * 0.5) + offset`, where `screen` is recovered from the
+ * projection matrix rather than known. If that recovery is wrong by one pixel at some GUI scale,
+ * every themed screen is wrong by one pixel at that scale — and nobody plays at every scale, so it
+ * would be found by a player rather than by us.
+ *
+ * The recovery is float arithmetic, which is the one part of a shader that is the same on the CPU:
+ * GLSL's `highp float` and the JVM's `Float` are both IEEE binary32. So the formula can be replayed
+ * here over every size the client can produce.
+ *
+ * The epsilon is read out of the shipped shader rather than repeated, so a change there cannot pass
+ * this test by being invisible to it.
+ */
+class ScreenGeometryTest {
+    private val shader: String =
+        checkNotNull(javaClass.getResourceAsStream("/gg/grounds/gui/pack/text.vsh"))
+            .bufferedReader()
+            .use { it.readText() }
+
+    /** The `- 0.001` in `ceil(2.0 / ... - 0.001)`, taken from the shader itself. */
+    private val epsilon: Float =
+        Regex("""ceil\(2\.0 / vec2\(ProjMat\[0]\[0], -ProjMat\[1]\[1]\) - ([0-9.]+)\)""")
+            .find(shader)
+            ?.groupValues
+            ?.get(1)
+            ?.toFloat()
+            ?: error("the shader no longer recovers the screen size the way this test replays")
+
+    /** What the client puts in the matrix: an orthographic projection over the scaled GUI. */
+    private fun projection(scaledWidth: Int): Float = 2.0f / scaledWidth.toFloat()
+
+    /** What the shader gets back out of it. */
+    private fun recovered(scaledWidth: Int): Int =
+        ceil(2.0f / projection(scaledWidth) - epsilon).toInt()
+
+    @Test
+    fun `the screen size survives the round trip through the matrix`() {
+        // Every width the client can hand out: guiScaledWidth is ceil(framebuffer / guiScale), and
+        // a framebuffer runs from a small window to 8K. One pixel of error here is one pixel of
+        // error on every marker at that scale.
+        val wrong = (1..8192).filter { recovered(it) != it }
+        assertEquals(emptyList(), wrong, "widths the shader would misread")
+    }
+
+    @Test
+    fun `the nudge sits inside a two-sided bound, with room on both sides`() {
+        // Both directions matter, and that is what makes the value non-obvious. A round trip that
+        // comes back high needs the nudge to exceed the error, or the ceiling jumps a whole pixel.
+        // One that comes back low needs the nudge to stay a pixel clear of 1.0, or a real width is
+        // rounded away. So the safe range is [error, 1 - error] and the middle of it is the value
+        // that survives the widest displays — the first version used 0.001, which was legal but sat
+        // against the floor of that range.
+        val worst =
+            (1..8192).maxOf { width -> Math.abs(2.0f / projection(width) - width.toFloat()) }
+        assertTrue(worst < 0.01f, "round-trip error grew unexpectedly: $worst")
+        assertTrue(epsilon > worst, "nudge $epsilon must exceed the $worst error it absorbs")
+        assertTrue(epsilon < 1f - worst, "nudge $epsilon must stay clear of swallowing a pixel")
+
+        // And it should not merely be legal. Sitting near either end means the first wider display
+        // breaks it, which is a bug nobody can reproduce without that display.
+        val headroom = minOf(epsilon - worst, 1f - worst - epsilon)
+        assertTrue(
+            headroom > 0.4f,
+            "only $headroom of margin; the nudge is not centred in its range",
+        )
+    }
+
+    @Test
+    fun `a marker lands where the client puts the window, at both parities`() {
+        // The client centres a container with `leftPos = (width - imageWidth) / 2`, truncating.
+        // The shader computes `floor(screen / 2) + x - imageWidth / 2`. Those agree for even and
+        // odd screen widths, and the odd case is the one nobody checks by hand.
+        listOf(1920, 1921, 854, 855, 320, 321).forEach { screen ->
+            listOf(CONTAINER_WIDTH, 276).forEach { imageWidth ->
+                val clientLeft = (screen - imageWidth) / 2
+                val shaderOrigin = floor(screen / 2.0).toInt() + (0 - imageWidth / 2)
+                assertEquals(clientLeft, shaderOrigin, "screen $screen, window $imageWidth")
+            }
+        }
+    }
+
+    @Test
+    fun `a window wider than the screen still agrees, rather than drifting apart`() {
+        // A merchant window is 276 wide and the smallest GUI the client allows is 320, so this is
+        // reachable at scale 1 on a small window with a wide screen type.
+        listOf(280 to 276, 300 to 276).forEach { (screen, imageWidth) ->
+            assertEquals(
+                (screen - imageWidth) / 2,
+                floor(screen / 2.0).toInt() - imageWidth / 2,
+                "screen $screen, window $imageWidth",
+            )
+        }
+    }
+}