diff --git a/src/core/LinearLayout.test.ts b/src/core/LinearLayout.test.ts
index 77a3341..b2788c0 100644
--- a/src/core/LinearLayout.test.ts
+++ b/src/core/LinearLayout.test.ts
@@ -249,10 +249,54 @@ describe('LinearLayout', () => {
       expect(() => nonSquare.invert()).toThrowError()
     })
 
-    it('should reject layouts where output bits are zero but inputs are not', () => {
+    it('should reject rank-deficient rectangular layouts', () => {
+      const matrix = [
+        [1, 0, 1],
+        [1, 0, 1],
+      ]
+      const layout = LinearLayout.fromBitMatrix(
+        matrix,
+        [{ name: 'in', size: 8 }],
+        [
+          { name: 'out0', size: 2 },
+          { name: 'out1', size: 2 },
+        ]
+      )
+
+      expect(layout.isInvertible()).toBe(false)
+      expect(() => layout.invert()).toThrowError(
+        'Layout matrix is not invertible (matrix is not full row rank)'
+      )
+    })
+
+    it('should handle layouts where output bits are zero but inputs are not', () => {
       const broadcast = LinearLayout.zeros1D(8, 'reg', 'zero')
-      expect(broadcast.isInvertible()).toBe(false)
-      expect(() => broadcast.invert()).toThrowError(/square and surjective/i)
+      expect(broadcast.isInvertible()).toBe(true)
+      const inverted = broadcast.invert()
+      const recovered = inverted.apply({ zero: 0 })
+      expect(recovered.reg).toBe(0)
+      const roundTrip = broadcast.apply(recovered)
+      expect(roundTrip.zero).toBe(0)
+    })
+
+    it('should compute a right inverse for rectangular surjective layouts', () => {
+      const layout = new LinearLayout(
+        [
+          ['in', [[1, 0], [0, 1], [1, 1]]],
+        ],
+        ['out0', 'out1']
+      )
+      expect(layout.isInvertible()).toBe(true)
+      const inverse = layout.invert()
+      const out0Size = layout.getOutDimSize('out0')
+      const out1Size = layout.getOutDimSize('out1')
+      for (let out0 = 0; out0 < out0Size; out0++) {
+        for (let out1 = 0; out1 < out1Size; out1++) {
+          const recoveredInput = inverse.apply({ out0, out1 })
+          const roundTrip = layout.apply(recoveredInput)
+          expect(roundTrip).toEqual({ out0, out1 })
+        }
+      }
     })
 
     it('should invert layouts whose input/output spaces collapse to a single value', () => {
diff --git a/src/core/LinearLayout.ts b/src/core/LinearLayout.ts
index 8151946..35a793f 100644
--- a/src/core/LinearLayout.ts
+++ b/src/core/LinearLayout.ts
@@ -272,48 +272,71 @@ function basesToMatrix(
 }
 
 function invertBinaryMatrix(matrix: number[][]): number[][] {
-  const n = matrix.length
-  if (n === 0) {
+  const rowCount = matrix.length
+  if (rowCount === 0) {
     return []
   }
-  const width = matrix[0]?.length ?? 0
-  if (width !== n) {
-    throw new Error('Layout matrix must be square to invert')
+  const colCount = matrix[0]?.length ?? 0
+  if (colCount < rowCount) {
+    throw new Error('Layout matrix must have at least as many columns as rows to be invertible')
   }
 
   const augmented: number[][] = matrix.map((row, i) => {
-    if (row.length !== width) {
+    if (row.length !== colCount) {
       throw new Error('Layout matrix rows must have consistent width')
     }
     const left = row.map((value) => (value & 1 ? 1 : 0))
-    const right = new Array(n).fill(0)
+    const right = new Array(rowCount).fill(0)
     right[i] = 1
     return [...left, ...right]
   })
 
-  for (let col = 0; col < n; col++) {
-    let pivot = col
-    while (pivot < n && augmented[pivot]?.[col] !== 1) {
+  const totalCols = colCount + rowCount
+  let pivotRow = 0
+  const pivotColumns: number[] = []
+
+  for (let col = 0; col < colCount && pivotRow < rowCount; col++) {
+    let pivot = pivotRow
+    while (pivot < rowCount && augmented[pivot]?.[col] !== 1) {
       pivot++
     }
-    if (pivot === n) {
-      throw new Error('Layout matrix is not invertible')
+    if (pivot === rowCount) {
+      continue
     }
-    if (pivot !== col) {
-      const temp = augmented[col]!
-      augmented[col] = augmented[pivot]!
+    if (pivot !== pivotRow) {
+      const temp = augmented[pivotRow]!
+      augmented[pivotRow] = augmented[pivot]!
       augmented[pivot] = temp
     }
-    for (let row = 0; row < n; row++) {
-      if (row !== col && augmented[row]?.[col] === 1) {
-        for (let k = col; k < 2 * n; k++) {
-          augmented[row]![k]! ^= augmented[col]![k]!
+    for (let row = 0; row < rowCount; row++) {
+      if (row !== pivotRow && augmented[row]?.[col] === 1) {
+        for (let k = col; k < totalCols; k++) {
+          augmented[row]![k]! ^= augmented[pivotRow]![k]!
         }
       }
     }
+    pivotColumns[pivotRow] = col
+    pivotRow++
+  }
+
+  if (pivotRow !== rowCount) {
+    throw new Error('Layout matrix is not invertible (matrix is not full row rank)')
+  }
+
+  const rightInverse = Array.from({ length: colCount }, () => new Array(rowCount).fill(0))
+  for (let row = 0; row < rowCount; row++) {
+    const pivotCol = pivotColumns[row]
+    if (pivotCol === undefined) continue
+    const augRow = augmented[row]
+    if (!augRow) continue
+    const targetRow = rightInverse[pivotCol]
+    if (!targetRow) continue
+    for (let j = 0; j < rowCount; j++) {
+      targetRow[j] = augRow[colCount + j] ?? 0
+    }
   }
 
-  return augmented.map((row) => row.slice(n))
+  return rightInverse
 }
 
 /**
@@ -584,13 +607,13 @@ export class LinearLayout {
 
     const inputBitCount = totalBitCount(inputSpecs)
     const outputBitCount = totalBitCount(outputSpecs)
-    if (inputBitCount !== outputBitCount) {
-      throw new Error('Cannot invert layout: layout must be square and surjective (input/output bit counts differ)')
+    if (inputBitCount < outputBitCount) {
+      throw new Error('Cannot invert layout: layout must be surjective (input bit count is smaller than output bit count)')
     }
 
     let inverseMatrix: number[][]
     if (outputBitCount === 0) {
-      inverseMatrix = []
+      inverseMatrix = Array.from({ length: inputBitCount }, () => [])
     } else {
       const matrix = basesToMatrix(this.bases, inputSpecs, outputSpecs)
       inverseMatrix = invertBinaryMatrix(matrix)
@@ -610,10 +633,10 @@ export class LinearLayout {
       }
       const inputBitCount = totalBitCount(inputSpecs)
       const outputBitCount = totalBitCount(outputSpecs)
-      if (inputBitCount !== outputBitCount) {
+      if (inputBitCount < outputBitCount) {
         return false
       }
-      if (inputBitCount === 0) {
+      if (outputBitCount === 0) {
         return true
       }
       const matrix = basesToMatrix(this.bases, inputSpecs, outputSpecs)