diff --git a/language_theory/__init__.py b/language_theory/__init__.py
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diff --git a/language_theory/witnesses/__init__.py b/language_theory/witnesses/__init__.py
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diff --git a/language_theory/witnesses/collatz/README.md b/language_theory/witnesses/collatz/README.md
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+# Witness: The Collatz Conjecture
+
+This directory contains a concrete witness for the concept of "decidable refactoring," as proposed in the main `language_theory/THEORY.md` document. It separates the undecidable Collatz conjecture into two components:
+
+1.  **`collatz_total.py`**: A **total function** that computes a single step of the Collatz sequence. It is guaranteed to terminate because it takes a "fuel" parameter that limits the number of steps.
+
+2.  **`control_program.py`**: The **unbounded control logic** that orchestrates the Collatz sequence. This program is not guaranteed to terminate and represents the undecidable part of the problem.
+
+This separation provides a practical example of how to isolate and manage undecidability in a formal system, which is a core concept of the theoretical model.
diff --git a/language_theory/witnesses/collatz/__init__.py b/language_theory/witnesses/collatz/__init__.py
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diff --git a/language_theory/witnesses/collatz/collatz_total.py b/language_theory/witnesses/collatz/collatz_total.py
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+# A total function that computes one step of the Collatz sequence.
+# This function is guaranteed to terminate.
+
+def collatz_step(n):
+    if n % 2 == 0:
+        return n // 2
+    else:
+        return 3 * n + 1
+
+def collatz_total(n, fuel):
+    """
+    Computes the Collatz sequence for a given n, with a 'fuel' limit.
+    This function is a total function, guaranteed to terminate.
+    """
+    if fuel <= 0:
+        return n
+
+    current_n = n
+    for _ in range(fuel):
+        if current_n == 1:
+            break
+        current_n = collatz_step(current_n)
+    return current_n
+
+if __name__ == "__main__":
+    import sys
+    if len(sys.argv) != 3:
+        print("Usage: python -m language_theory.witnesses.collatz.collatz_total <n> <fuel>")
+        sys.exit(1)
+    n = int(sys.argv[1])
+    fuel = int(sys.argv[2])
+    result = collatz_total(n, fuel)
+    print(f"Result: {result}")
diff --git a/language_theory/witnesses/collatz/control_program.py b/language_theory/witnesses/collatz/control_program.py
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+# The control program for the Collatz sequence.
+# This program is not guaranteed to terminate.
+
+from .collatz_total import collatz_total
+
+def control_program(n, initial_fuel=10):
+    """
+    This program orchestrates the Collatz sequence computation.
+    It is not guaranteed to terminate for all inputs.
+    """
+    current_n = n
+    fuel = initial_fuel
+
+    while current_n != 1:
+        current_n = collatz_total(current_n, fuel)
+        if current_n != 1:
+            # Decide whether to allocate more fuel or give up.
+            # For this example, we'll just double the fuel.
+            fuel *= 2
+
+    return current_n
+
+if __name__ == "__main__":
+    import sys
+    if len(sys.argv) != 2:
+        print("Usage: python -m language_theory.witnesses.collatz.control_program <n>")
+        sys.exit(1)
+    n = int(sys.argv[1])
+    result = control_program(n)
+    print(f"Result: {result}")