diff --git a/PerfectNumbers.Core.Tests/MersenneNumberDivisorGpuTesterTests.cs b/PerfectNumbers.Core.Tests/MersenneNumberDivisorGpuTesterTests.cs
index 765349d..3b58adc 100644
--- a/PerfectNumbers.Core.Tests/MersenneNumberDivisorGpuTesterTests.cs
+++ b/PerfectNumbers.Core.Tests/MersenneNumberDivisorGpuTesterTests.cs
@@ -21,7 +21,8 @@ public class MersenneNumberDivisorGpuTesterTests
     public void IsDivisible_returns_expected(ulong exponent, ulong divisor, bool expected)
     {
         var tester = new MersenneNumberDivisorGpuTester();
-        tester.IsDivisible(exponent, divisor).Should().Be(expected);
+        ReadOnlyGpuUInt128 divisorValue = new ReadOnlyGpuUInt128(divisor);
+        tester.IsDivisible(exponent, in divisorValue).Should().Be(expected);
     }
 
     [Fact]
@@ -30,8 +31,10 @@ public class MersenneNumberDivisorGpuTesterTests
     {
         var tester = new MersenneNumberDivisorGpuTester();
         UInt128 divisor = (UInt128.One << 65) - UInt128.One;
-        tester.IsDivisible(65UL, divisor).Should().BeTrue();
-        tester.IsDivisible(65UL, divisor + 2).Should().BeFalse();
+        ReadOnlyGpuUInt128 divisorValue = new ReadOnlyGpuUInt128(divisor);
+        tester.IsDivisible(65UL, in divisorValue).Should().BeTrue();
+        divisorValue = new ReadOnlyGpuUInt128(divisor + 2); // Reusing divisorValue for the shifted candidate.
+        tester.IsDivisible(65UL, in divisorValue).Should().BeFalse();
     }
 
     [Fact]
diff --git a/PerfectNumbers.Core/Cpu/MersenneNumberDivisorByDivisorCpuTester.cs b/PerfectNumbers.Core/Cpu/MersenneNumberDivisorByDivisorCpuTester.cs
index 5921ecf..44342a5 100644
--- a/PerfectNumbers.Core/Cpu/MersenneNumberDivisorByDivisorCpuTester.cs
+++ b/PerfectNumbers.Core/Cpu/MersenneNumberDivisorByDivisorCpuTester.cs
@@ -1,5 +1,6 @@
 using System;
 using System.Collections.Concurrent;
+using System.Collections.Generic;
 using System.Numerics;
 using PerfectNumbers.Core;
 
@@ -188,14 +189,32 @@ public sealed class MersenneNumberDivisorByDivisorCpuTester : IMersenneNumberDiv
             return false;
         }
 
+        Dictionary<ulong, MersenneDivisorCycles.FactorCacheEntry>? factorCache = null;
+        DivisorCycleCache cycleCache = DivisorCycleCache.Shared;
+
         while (divisor <= limit)
         {
             ulong candidate = (ulong)divisor;
             processedCount++;
             lastProcessed = candidate;
 
-            MontgomeryDivisorData divisorData = MontgomeryDivisorData.FromModulus(candidate);
-            if (MersenneDivisorCycles.CycleEqualsExponentForMersenneCandidate(candidate, divisorData, prime))
+            MontgomeryDivisorData divisorData = MontgomeryDivisorDataCache.Get(candidate);
+            ulong divisorCycle = 0UL;
+
+            if (candidate <= PerfectNumberConstants.MaxQForDivisorCycles)
+            {
+                divisorCycle = cycleCache.GetCycleLength(candidate);
+            }
+            else
+            {
+                factorCache ??= new Dictionary<ulong, MersenneDivisorCycles.FactorCacheEntry>(8);
+                if (!MersenneDivisorCycles.TryCalculateCycleLengthForExponent(candidate, prime, divisorData, factorCache, out divisorCycle) || divisorCycle == 0UL)
+                {
+                    divisorCycle = cycleCache.GetCycleLength(candidate);
+                }
+            }
+
+            if (divisorCycle != 0UL && CheckDivisor(prime, divisorCycle, divisorData) != 0)
             {
                 processedAll = true;
                 return true;
@@ -208,6 +227,12 @@ public sealed class MersenneNumberDivisorByDivisorCpuTester : IMersenneNumberDiv
         return false;
     }
 
+    private static byte CheckDivisor(ulong prime, ulong divisorCycle, in MontgomeryDivisorData divisorData)
+    {
+        ulong residue = prime.Pow2MontgomeryModWithCycle(divisorCycle, divisorData);
+        return residue == 1UL ? (byte)1 : (byte)0;
+    }
+
     private void UpdateStatusUnsafe(ulong lastProcessed, ulong processedCount)
     {
         if (processedCount == 0UL)
@@ -286,11 +311,48 @@ public sealed class MersenneNumberDivisorByDivisorCpuTester : IMersenneNumberDiv
                 return;
             }
 
-            for (int i = 0; i < length; i++)
+            MontgomeryDivisorData cachedData = divisorData;
+            if (cachedData.Modulus != divisor)
             {
-                ulong prime = primes[i];
-                bool divides = MersenneDivisorCycles.CycleEqualsExponentForMersenneCandidate(divisor, divisorData, prime);
-                hits[i] = divides ? (byte)1 : (byte)0;
+                cachedData = MontgomeryDivisorDataCache.Get(divisor);
+            }
+
+            if (divisorCycle == 0UL)
+            {
+                divisorCycle = DivisorCycleCache.Shared.GetCycleLength(divisor);
+                if (divisorCycle == 0UL)
+                {
+                    hits.Clear();
+                    return;
+                }
+            }
+
+            // Keep these remainder steppers in place so future updates continue reusing the previously computed residues.
+            // They are critical for avoiding repeated full Montgomery exponentiation work when scanning divisors.
+            var exponentStepper = new ExponentRemainderStepper(cachedData);
+            if (!exponentStepper.IsValidModulus)
+            {
+                hits.Clear();
+                return;
+            }
+
+            var cycleStepper = new CycleRemainderStepper(divisorCycle);
+
+            ulong remainder = cycleStepper.Initialize(primes[0]);
+            hits[0] = remainder == 0UL
+                ? (exponentStepper.ComputeNextIsUnity(primes[0]) ? (byte)1 : (byte)0)
+                : (byte)0;
+
+            for (int i = 1; i < length; i++)
+            {
+                remainder = cycleStepper.ComputeNext(primes[i]);
+                if (remainder != 0UL)
+                {
+                    hits[i] = 0;
+                    continue;
+                }
+
+                hits[i] = exponentStepper.ComputeNextIsUnity(primes[i]) ? (byte)1 : (byte)0;
             }
         }
 
diff --git a/PerfectNumbers.Core/Gpu/GpuKernelPool.cs b/PerfectNumbers.Core/Gpu/GpuKernelPool.cs
index d39cf06..9000d2b 100644
--- a/PerfectNumbers.Core/Gpu/GpuKernelPool.cs
+++ b/PerfectNumbers.Core/Gpu/GpuKernelPool.cs
@@ -247,6 +247,7 @@ public ref struct GpuKernelLease(IDisposable limiter, GpuContextLease gpu, Kerne
         GpuUInt128 q = twoP;
         q.Mul64(k);
         q.Add(GpuUInt128.One);
+        ReadOnlyGpuUInt128 readOnlyQ = q.AsReadOnly();
 
         // Small-cycles in-kernel early rejection from device table
         if (q.High == 0UL && q.Low < (ulong)smallCycles.Length)
@@ -271,13 +272,13 @@ public ref struct GpuKernelLease(IDisposable limiter, GpuContextLease gpu, Kerne
         ulong phi64 = phi.Low;
         // TODO: Replace these Pow2Mod calls with the ProcessEightBitWindows helper when the shared windowed
         // scalar implementation lands; benchmarks showed the windowed kernel trimming per-divisor runtime by ~2.4×.
-        if (GpuUInt128.Pow2Mod(phi64, q) != GpuUInt128.One)
+        if (GpuUInt128.Pow2Mod(phi64, in readOnlyQ) != GpuUInt128.One)
         {
             orders[index] = 0UL;
             return;
         }
 
-        GpuUInt128 halfPow = GpuUInt128.Pow2Mod(phi64 >> 1, q) - GpuUInt128.One;
+        GpuUInt128 halfPow = GpuUInt128.Pow2Mod(phi64 >> 1, in readOnlyQ) - GpuUInt128.One;
         if (GpuUInt128.BinaryGcd(halfPow, q) != GpuUInt128.One)
         {
             orders[index] = 0UL;
@@ -285,7 +286,7 @@ public ref struct GpuKernelLease(IDisposable limiter, GpuContextLease gpu, Kerne
         }
 
         ulong div = FastDiv64Gpu(phi64, exponent, divMul);
-        GpuUInt128 divPow = GpuUInt128.Pow2Mod(div, q) - GpuUInt128.One;
+        GpuUInt128 divPow = GpuUInt128.Pow2Mod(div, in readOnlyQ) - GpuUInt128.One;
         if (GpuUInt128.BinaryGcd(divPow, q) != GpuUInt128.One)
         {
             orders[index] = 0UL;
@@ -298,6 +299,7 @@ public ref struct GpuKernelLease(IDisposable limiter, GpuContextLease gpu, Kerne
     private static void OrderKernelScan(Index1D index, ulong exponent, ulong divMul, ArrayView<GpuUInt128> qs, ArrayView<ulong> orders)
     {
         GpuUInt128 q = qs[index];
+        ReadOnlyGpuUInt128 readOnlyQ = q.AsReadOnly();
         GpuUInt128 phi = q - GpuUInt128.One;
         if (phi.High != 0UL)
         {
@@ -308,14 +310,14 @@ public ref struct GpuKernelLease(IDisposable limiter, GpuContextLease gpu, Kerne
         ulong phi64 = phi.Low;
         // TODO: Once the ProcessEightBitWindows helper is available, switch this order kernel to that faster
         // Pow2Mod variant so cycle checks inherit the same gains observed in GpuPow2ModBenchmarks.
-        GpuUInt128 pow = GpuUInt128.Pow2Mod(phi64, q);
+        GpuUInt128 pow = GpuUInt128.Pow2Mod(phi64, in readOnlyQ);
         if (pow != GpuUInt128.One)
         {
             orders[index] = 0UL;
             return;
         }
 
-        GpuUInt128 halfPow = GpuUInt128.Pow2Mod(phi64 >> 1, q) - GpuUInt128.One;
+        GpuUInt128 halfPow = GpuUInt128.Pow2Mod(phi64 >> 1, in readOnlyQ) - GpuUInt128.One;
         if (GpuUInt128.BinaryGcd(halfPow, q) != GpuUInt128.One)
         {
             orders[index] = 0UL;
@@ -323,7 +325,7 @@ public ref struct GpuKernelLease(IDisposable limiter, GpuContextLease gpu, Kerne
         }
 
         ulong div = FastDiv64Gpu(phi64, exponent, divMul);
-        GpuUInt128 divPow = GpuUInt128.Pow2Mod(div, q) - GpuUInt128.One;
+        GpuUInt128 divPow = GpuUInt128.Pow2Mod(div, in readOnlyQ) - GpuUInt128.One;
         if (GpuUInt128.BinaryGcd(divPow, q) != GpuUInt128.One)
         {
             orders[index] = 0UL;
@@ -410,6 +412,7 @@ public ref struct GpuKernelLease(IDisposable limiter, GpuContextLease gpu, Kerne
         GpuUInt128 q = twoP;
         q.Mul64(kStart);
         q.Add(GpuUInt128.One);
+        ReadOnlyGpuUInt128 readOnlyQ = q.AsReadOnly();
         if (q.High == 0UL && q.Low < (ulong)smallCycles.Length)
         {
             ulong cycle = smallCycles[(int)q.Low];
@@ -422,7 +425,7 @@ public ref struct GpuKernelLease(IDisposable limiter, GpuContextLease gpu, Kerne
         }
         // TODO: Swap Pow2Minus1Mod for the eight-bit window helper once the scalar version switches;
         // benchmarks show the windowed variant cuts large-divisor scans from ~51 µs to ~21 µs.
-        if (GpuUInt128.Pow2Minus1Mod(exponent, q) != GpuUInt128.Zero)
+        if (GpuUInt128.Pow2Minus1Mod(exponent, in readOnlyQ) != GpuUInt128.Zero)
         {
             orders[index] = 0UL;
             return;
@@ -559,6 +562,7 @@ public ref struct GpuKernelLease(IDisposable limiter, GpuContextLease gpu, Kerne
         GpuUInt128 q = twoP;
         q.Mul64(k);
         q.Add(GpuUInt128.One);
+        ReadOnlyGpuUInt128 readOnlyQ = q.AsReadOnly();
 
         // Small-cycles in-kernel early rejection from device table
         if (q.High == 0UL && q.Low < (ulong)smallCycles.Length)
@@ -578,19 +582,19 @@ public ref struct GpuKernelLease(IDisposable limiter, GpuContextLease gpu, Kerne
         ulong phi64 = phi.Low;
         // TODO: Upgrade this pow2mod order kernel to the ProcessEightBitWindows helper once available so GPU residue
         // scans avoid the single-bit ladder that benchmarks found to be 2.3× slower on large exponents.
-        if (GpuUInt128.Pow2Mod(phi64, q) != GpuUInt128.One)
+        if (GpuUInt128.Pow2Mod(phi64, in readOnlyQ) != GpuUInt128.One)
         {
             return;
         }
 
-        GpuUInt128 halfPow = GpuUInt128.Pow2Mod(phi64 >> 1, q) - GpuUInt128.One;
+        GpuUInt128 halfPow = GpuUInt128.Pow2Mod(phi64 >> 1, in readOnlyQ) - GpuUInt128.One;
         if (GpuUInt128.BinaryGcd(halfPow, q) != GpuUInt128.One)
         {
             return;
         }
 
         ulong div = FastDiv64Gpu(phi64, exponent, divMul);
-        GpuUInt128 divPow = GpuUInt128.Pow2Mod(div, q) - GpuUInt128.One;
+        GpuUInt128 divPow = GpuUInt128.Pow2Mod(div, in readOnlyQ) - GpuUInt128.One;
         if (GpuUInt128.BinaryGcd(divPow, q) != GpuUInt128.One)
         {
             return;
@@ -645,6 +649,7 @@ public ref struct GpuKernelLease(IDisposable limiter, GpuContextLease gpu, Kerne
         GpuUInt128 q = twoP;
         q.Mul64(k);
         q.Add(GpuUInt128.One);
+        ReadOnlyGpuUInt128 readOnlyQ = q.AsReadOnly();
         // Small-cycles in-kernel early rejection from device table
         if (q.High == 0UL && q.Low < (ulong)smallCycles.Length)
         {
@@ -656,7 +661,7 @@ public ref struct GpuKernelLease(IDisposable limiter, GpuContextLease gpu, Kerne
         }
         // TODO: Replace this Pow2Mod check with the ProcessEightBitWindows helper once Pow2Minus1Mod adopts it;
         // residue order scans will then benefit from the same 2× speedup the benchmarked windowed kernel delivered.
-        if (GpuUInt128.Pow2Mod(exponent, q) != GpuUInt128.One)
+        if (GpuUInt128.Pow2Mod(exponent, in readOnlyQ) != GpuUInt128.One)
         {
             return;
         }
diff --git a/PerfectNumbers.Core/Gpu/GpuUInt128.cs b/PerfectNumbers.Core/Gpu/GpuUInt128.cs
index 9bb712b..cee3beb 100644
--- a/PerfectNumbers.Core/Gpu/GpuUInt128.cs
+++ b/PerfectNumbers.Core/Gpu/GpuUInt128.cs
@@ -81,23 +81,31 @@ public struct GpuUInt128 : IComparable<GpuUInt128>, IEquatable<GpuUInt128>
     [MethodImpl(MethodImplOptions.AggressiveInlining)]
     public static implicit operator GpuUInt128(ulong value) => new(value);
 
+    [MethodImpl(MethodImplOptions.AggressiveInlining)]
+    public static implicit operator GpuUInt128(ReadOnlyGpuUInt128 value) => new(value.High, value.Low);
+
     [MethodImpl(MethodImplOptions.AggressiveInlining)]
     public static implicit operator UInt128(GpuUInt128 value) =>
         ((UInt128)value.High << 64) | value.Low;
 
-    // TODO: The operators modifying the state of left is intuitive and can easily lead to bugs. Change this back to return a new instance, but replace where possible with method calls, if we want to modify the state of the instance.
+    [MethodImpl(MethodImplOptions.AggressiveInlining)]
+    public readonly ReadOnlyGpuUInt128 AsReadOnly() => new(this);
+
+    // Operators must return fresh instances so callers never rely on accidental in-place mutation. Prefer the mutating helpers when state updates are required.
     [MethodImpl(MethodImplOptions.AggressiveInlining)]
     public static GpuUInt128 operator +(GpuUInt128 left, GpuUInt128 right)
     {
-        left.Add(right);
-        return left;
+        GpuUInt128 result = new(left.High, left.Low);
+        result.Add(right);
+        return result;
     }
 
     [MethodImpl(MethodImplOptions.AggressiveInlining)]
     public static GpuUInt128 operator -(GpuUInt128 left, GpuUInt128 right)
     {
-        left.Sub(right);
-        return left;
+        GpuUInt128 result = new(left.High, left.Low);
+        result.Sub(right);
+        return result;
     }
 
     [MethodImpl(MethodImplOptions.AggressiveInlining)]
@@ -108,63 +116,63 @@ public struct GpuUInt128 : IComparable<GpuUInt128>, IEquatable<GpuUInt128>
     public static GpuUInt128 operator *(GpuUInt128 left, GpuUInt128 right)
     {
         Multiply(left, right, out var high, out var low);
-        left.High = high;
-        left.Low = low;
-        return left;
+        return new GpuUInt128(high, low);
     }
 
     [MethodImpl(MethodImplOptions.AggressiveInlining)]
     public static GpuUInt128 operator <<(GpuUInt128 value, int shift)
     {
         shift &= 127;
+        GpuUInt128 result = new(value.High, value.Low);
         if (shift == 0)
         {
-            return value;
+            return result;
         }
 
-        ulong high = value.High;
-        ulong low = value.Low;
+        ulong high = result.High;
+        ulong low = result.Low;
 
         int shiftedValue;
         if (shift >= 64)
         {
             shiftedValue = shift - 64;
-            value.High = low << shiftedValue;
-            value.Low = 0UL;
-            return value;
+            result.High = low << shiftedValue;
+            result.Low = 0UL;
+            return result;
         }
 
         shiftedValue = 64 - shift;
-        value.High = (high << shift) | (low >> shiftedValue);
-        value.Low = low << shift;
-        return value;
+        result.High = (high << shift) | (low >> shiftedValue);
+        result.Low = low << shift;
+        return result;
     }
 
     [MethodImpl(MethodImplOptions.AggressiveInlining)]
     public static GpuUInt128 operator >>(GpuUInt128 value, int shift)
     {
         shift &= 127;
+        GpuUInt128 result = new(value.High, value.Low);
         if (shift == 0)
         {
-            return value;
+            return result;
         }
 
-        ulong high = value.High;
-        ulong low = value.Low;
+        ulong high = result.High;
+        ulong low = result.Low;
 
         int shiftedValue;
         if (shift >= 64)
         {
             shiftedValue = shift - 64;
-            value.Low = shiftedValue == 0 ? high : high >> shiftedValue;
-            value.High = 0UL;
-            return value;
+            result.Low = shiftedValue == 0 ? high : high >> shiftedValue;
+            result.High = 0UL;
+            return result;
         }
 
         shiftedValue = 64 - shift;
-        value.Low = (low >> shift) | (high << shiftedValue);
-        value.High = high >> shift;
-        return value;
+        result.Low = (low >> shift) | (high << shiftedValue);
+        result.High = high >> shift;
+        return result;
     }
 
     [MethodImpl(MethodImplOptions.AggressiveInlining)]
@@ -188,8 +196,9 @@ public struct GpuUInt128 : IComparable<GpuUInt128>, IEquatable<GpuUInt128>
     [MethodImpl(MethodImplOptions.AggressiveInlining)]
     public static GpuUInt128 operator ++(GpuUInt128 value)
     {
-        value.Add(1UL);
-        return value;
+        GpuUInt128 result = new(value.High, value.Low);
+        result.Add(1UL);
+        return result;
     }
 
     [MethodImpl(MethodImplOptions.AggressiveInlining)]
@@ -218,6 +227,33 @@ public struct GpuUInt128 : IComparable<GpuUInt128>, IEquatable<GpuUInt128>
         return 0;
     }
 
+    [MethodImpl(MethodImplOptions.AggressiveInlining)]
+    public readonly int CompareTo(in ReadOnlyGpuUInt128 other)
+    {
+        if (High < other.High)
+        {
+            return -1;
+        }
+
+        if (High > other.High)
+        {
+            return 1;
+        }
+
+        ulong otherLow = other.Low;
+        if (Low < otherLow)
+        {
+            return -1;
+        }
+
+        if (Low > otherLow)
+        {
+            return 1;
+        }
+
+        return 0;
+    }
+
     [MethodImpl(MethodImplOptions.AggressiveInlining)]
     public readonly bool Equals(GpuUInt128 other) => High == other.High && Low == other.Low;
 
@@ -228,15 +264,21 @@ public struct GpuUInt128 : IComparable<GpuUInt128>, IEquatable<GpuUInt128>
     public override readonly int GetHashCode() => HashCode.Combine(High, Low);
 
     [MethodImpl(MethodImplOptions.AggressiveInlining)]
-    public void Add(GpuUInt128 other)
+    public void Add(in ReadOnlyGpuUInt128 other) => AddInternal(other.High, other.Low);
+
+    [MethodImpl(MethodImplOptions.AggressiveInlining)]
+    public void Add(in GpuUInt128 other) => AddInternal(other.High, other.Low);
+
+    [MethodImpl(MethodImplOptions.AggressiveInlining)]
+    private void AddInternal(ulong otherHigh, ulong otherLow)
     {
         ulong originalLow = Low;
         // We're reusing originalLow for carry to avoid an extra local.
-        ulong low = originalLow + other.Low;
+        ulong low = originalLow + otherLow;
         originalLow = low < originalLow ? 1UL : 0UL;
         Low = low;
         // We're reusing low for the final sum to avoid an extra local and to keep max stack frame at 3 instead of 4 with one statement.
-        low = High + other.High;
+        low = High + otherHigh;
         High = low + originalLow;
     }
 
@@ -249,7 +291,43 @@ public struct GpuUInt128 : IComparable<GpuUInt128>, IEquatable<GpuUInt128>
     }
 
     [MethodImpl(MethodImplOptions.AggressiveInlining)]
-    public void AddMod(GpuUInt128 value, GpuUInt128 modulus)
+    public void AddMod(in ReadOnlyGpuUInt128 value, in ReadOnlyGpuUInt128 modulus) => AddModInternal(value.High, value.Low, in modulus);
+
+    [MethodImpl(MethodImplOptions.AggressiveInlining)]
+    public void AddMod(in GpuUInt128 value, in ReadOnlyGpuUInt128 modulus) => AddModInternal(value.High, value.Low, in modulus);
+
+    [MethodImpl(MethodImplOptions.AggressiveInlining)]
+    public void AddMod(in ReadOnlyGpuUInt128 value, in GpuUInt128 modulus)
+    {
+        ReadOnlyGpuUInt128 readOnlyMod = modulus.AsReadOnly();
+        AddModInternal(value.High, value.Low, in readOnlyMod);
+    }
+
+    [MethodImpl(MethodImplOptions.AggressiveInlining)]
+    public void AddMod(in GpuUInt128 value, in GpuUInt128 modulus)
+    {
+        ReadOnlyGpuUInt128 readOnlyMod = modulus.AsReadOnly();
+        AddModInternal(value.High, value.Low, in readOnlyMod);
+    }
+
+    [MethodImpl(MethodImplOptions.AggressiveInlining)]
+    private void AddModInternal(ulong valueHigh, ulong valueLow, in ReadOnlyGpuUInt128 modulus)
+    {
+        if (modulus.High == 0UL)
+        {
+            AddMod(valueLow, modulus.Low);
+            return;
+        }
+
+        AddInternal(valueHigh, valueLow);
+        if (CompareTo(in modulus) >= 0)
+        {
+            Sub(in modulus);
+        }
+    }
+
+    [MethodImpl(MethodImplOptions.AggressiveInlining)]
+    public void AddMod(ulong value, in ReadOnlyGpuUInt128 modulus)
     {
         if (modulus.High == 0UL)
         {
@@ -258,30 +336,14 @@ public struct GpuUInt128 : IComparable<GpuUInt128>, IEquatable<GpuUInt128>
         }
 
         Add(value);
-        if (CompareTo(modulus) >= 0)
+        if (CompareTo(in modulus) >= 0)
         {
-            Sub(modulus);
+            Sub(in modulus);
         }
     }
 
     [MethodImpl(MethodImplOptions.AggressiveInlining)]
-    public void AddMod(ulong value, GpuUInt128 modulus)
-    {
-        if (modulus.High == 0UL)
-        {
-            AddMod(value, modulus.Low);
-            return;
-        }
-
-        Add(value);
-        if (CompareTo(modulus) >= 0)
-        {
-            Sub(modulus);
-        }
-    }
-
-    [MethodImpl(MethodImplOptions.AggressiveInlining)]
-    public static GpuUInt128 Pow2Minus1Mod(ulong exponent, GpuUInt128 modulus)
+    public static GpuUInt128 Pow2Minus1Mod(ulong exponent, in ReadOnlyGpuUInt128 modulus)
     {
         // This should never happen in production code
         // if (modulus.IsZero)
@@ -300,7 +362,7 @@ public struct GpuUInt128 : IComparable<GpuUInt128>, IEquatable<GpuUInt128>
     }
 
     [MethodImpl(MethodImplOptions.AggressiveInlining)]
-    public static void Pow2Minus1ModBatch(GpuUInt128 modulus, ReadOnlySpan<ulong> exponents, Span<GpuUInt128> results)
+    public static void Pow2Minus1ModBatch(in ReadOnlyGpuUInt128 modulus, ReadOnlySpan<ulong> exponents, Span<GpuUInt128> results)
     {
         int count = exponents.Length;
         for (int i = 0; i < count; i++)
@@ -310,7 +372,7 @@ public struct GpuUInt128 : IComparable<GpuUInt128>, IEquatable<GpuUInt128>
     }
 
     [MethodImpl(MethodImplOptions.AggressiveInlining)]
-    public void AddMod(GpuUInt128 value, ulong modulus)
+    public void AddMod(in ReadOnlyGpuUInt128 value, ulong modulus)
     {
         // TODO: Pre-reduce the operands via the Montgomery ladder used in MulMod64Benchmarks so the GPU
         // compatible shim stops paying for `%` on every call; the InlineUInt128 helper ran 6–82× faster on
@@ -327,6 +389,9 @@ public struct GpuUInt128 : IComparable<GpuUInt128>, IEquatable<GpuUInt128>
         Low = sum;
     }
 
+    [MethodImpl(MethodImplOptions.AggressiveInlining)]
+    public void AddMod(in GpuUInt128 value, ulong modulus) => AddMod(value.Low, modulus);
+
     [MethodImpl(MethodImplOptions.AggressiveInlining)]
     public void AddMod(ulong value, ulong modulus)
     {
@@ -346,11 +411,17 @@ public struct GpuUInt128 : IComparable<GpuUInt128>, IEquatable<GpuUInt128>
     }
 
     [MethodImpl(MethodImplOptions.AggressiveInlining)]
-    public void Sub(GpuUInt128 other)
+    public void Sub(in ReadOnlyGpuUInt128 other) => SubInternal(other.High, other.Low);
+
+    [MethodImpl(MethodImplOptions.AggressiveInlining)]
+    public void Sub(in GpuUInt128 other) => SubInternal(other.High, other.Low);
+
+    [MethodImpl(MethodImplOptions.AggressiveInlining)]
+    private void SubInternal(ulong otherHigh, ulong otherLow)
     {
-        ulong borrow = Low < other.Low ? 1UL : 0UL;
-        High = High - other.High - borrow;
-        Low -= other.Low;
+        ulong borrow = Low < otherLow ? 1UL : 0UL;
+        High = High - otherHigh - borrow;
+        Low -= otherLow;
     }
 
     [MethodImpl(MethodImplOptions.AggressiveInlining)]
@@ -387,20 +458,27 @@ public struct GpuUInt128 : IComparable<GpuUInt128>, IEquatable<GpuUInt128>
     }
 
     [MethodImpl(MethodImplOptions.AggressiveInlining)]
-    public void Mul64(GpuUInt128 other)
+    public void Mul64(in ReadOnlyGpuUInt128 other) => Mul64Internal(other.High, other.Low);
+
+    [MethodImpl(MethodImplOptions.AggressiveInlining)]
+    public void Mul64(in GpuUInt128 other) => Mul64Internal(other.High, other.Low);
+
+    [MethodImpl(MethodImplOptions.AggressiveInlining)]
+    private void Mul64Internal(ulong otherHigh, ulong otherLow)
     {
         // Multiply this.Low (assumed 64-bit value) by full 128-bit other
         ulong operand = Low;
-        Low = operand * other.Low;
-        ulong highProduct = operand * other.High;
-        High = highProduct + MulHigh(operand, other.Low);
+        Low = operand * otherLow;
+        ulong highProduct = operand * otherHigh;
+        High = highProduct + MulHigh(operand, otherLow);
     }
 
     [MethodImpl(MethodImplOptions.AggressiveInlining)]
-    public void MulMod(in GpuUInt128 value, in GpuUInt128 modulus)
+    // Kept by-value to coexist with the ref readonly overload below while still signaling read-only intent.
+    public void MulMod(ReadOnlyGpuUInt128 value, in ReadOnlyGpuUInt128 modulus)
     {
         GpuUInt128 multiplicand = this;
-        GpuUInt128 multiplier = value;
+        GpuUInt128 multiplier = value.ToMutable();
 
         High = 0UL;
         Low = 0UL;
@@ -423,7 +501,7 @@ public struct GpuUInt128 : IComparable<GpuUInt128>, IEquatable<GpuUInt128>
     }
 
     [MethodImpl(MethodImplOptions.AggressiveInlining)]
-    public static GpuUInt128 Pow2Mod(ulong exponent, GpuUInt128 modulus)
+    public static GpuUInt128 Pow2Mod(ulong exponent, in ReadOnlyGpuUInt128 modulus)
     {
         // This should never happen in production code.
         // if (modulus.IsZero || modulus == One)
@@ -499,7 +577,18 @@ public struct GpuUInt128 : IComparable<GpuUInt128>, IEquatable<GpuUInt128>
         return result;
     }
     [MethodImpl(MethodImplOptions.AggressiveInlining)]
-    public static int TrailingZeroCount(GpuUInt128 value)
+    public static int TrailingZeroCount(in ReadOnlyGpuUInt128 value)
+    {
+        if (value.Low != 0UL)
+        {
+            return BitOperations.TrailingZeroCount(value.Low);
+        }
+
+        return 64 + BitOperations.TrailingZeroCount(value.High);
+    }
+
+    [MethodImpl(MethodImplOptions.AggressiveInlining)]
+    public static int TrailingZeroCount(in GpuUInt128 value)
     {
         if (value.Low != 0UL)
         {
@@ -554,21 +643,66 @@ public struct GpuUInt128 : IComparable<GpuUInt128>, IEquatable<GpuUInt128>
     }
 
     [MethodImpl(MethodImplOptions.AggressiveInlining)]
-    public void SubMod(GpuUInt128 value, GpuUInt128 modulus)
+    public void SubMod(in ReadOnlyGpuUInt128 value, in ReadOnlyGpuUInt128 modulus) => SubModInternal(value.High, value.Low, in modulus);
+
+    [MethodImpl(MethodImplOptions.AggressiveInlining)]
+    public void SubMod(in GpuUInt128 value, in ReadOnlyGpuUInt128 modulus) => SubModInternal(value.High, value.Low, in modulus);
+
+    [MethodImpl(MethodImplOptions.AggressiveInlining)]
+    public void SubMod(in ReadOnlyGpuUInt128 value, in GpuUInt128 modulus)
     {
-        if (CompareTo(value) >= 0)
-        {
-            Sub(value);
-        }
-        else
-        {
-            Add(modulus);
-            Sub(value);
-        }
+        ReadOnlyGpuUInt128 readOnlyMod = modulus.AsReadOnly();
+        SubModInternal(value.High, value.Low, in readOnlyMod);
     }
 
     [MethodImpl(MethodImplOptions.AggressiveInlining)]
-    public void SubMod(ulong value, GpuUInt128 modulus)
+    public void SubMod(in GpuUInt128 value, in GpuUInt128 modulus)
+    {
+        ReadOnlyGpuUInt128 readOnlyMod = modulus.AsReadOnly();
+        SubModInternal(value.High, value.Low, in readOnlyMod);
+    }
+
+    [MethodImpl(MethodImplOptions.AggressiveInlining)]
+    private void SubModInternal(ulong valueHigh, ulong valueLow, in ReadOnlyGpuUInt128 modulus)
+    {
+        if (CompareToRaw(valueHigh, valueLow) >= 0)
+        {
+            SubInternal(valueHigh, valueLow);
+            return;
+        }
+
+        Add(in modulus);
+        SubInternal(valueHigh, valueLow);
+    }
+
+    [MethodImpl(MethodImplOptions.AggressiveInlining)]
+    private readonly int CompareToRaw(ulong otherHigh, ulong otherLow)
+    {
+        if (High < otherHigh)
+        {
+            return -1;
+        }
+
+        if (High > otherHigh)
+        {
+            return 1;
+        }
+
+        if (Low < otherLow)
+        {
+            return -1;
+        }
+
+        if (Low > otherLow)
+        {
+            return 1;
+        }
+
+        return 0;
+    }
+
+    [MethodImpl(MethodImplOptions.AggressiveInlining)]
+    public void SubMod(ulong value, in ReadOnlyGpuUInt128 modulus)
     {
         ulong low = Low;
         ulong high = High;
@@ -584,9 +718,20 @@ public struct GpuUInt128 : IComparable<GpuUInt128>, IEquatable<GpuUInt128>
     }
 
     [MethodImpl(MethodImplOptions.AggressiveInlining)]
-    public void Mul(GpuUInt128 other)
+    public void Mul(in ReadOnlyGpuUInt128 other)
     {
-        Multiply(this, other, out var high, out var low);
+        ReadOnlyGpuUInt128 readOnlyThis = AsReadOnly();
+        Multiply(readOnlyThis, other, out var high, out var low);
+        High = high;
+        Low = low;
+    }
+
+    [MethodImpl(MethodImplOptions.AggressiveInlining)]
+    public void Mul(in GpuUInt128 other)
+    {
+        ReadOnlyGpuUInt128 readOnlyThis = AsReadOnly();
+        ReadOnlyGpuUInt128 readOnlyOther = other.AsReadOnly();
+        Multiply(readOnlyThis, readOnlyOther, out var high, out var low);
         High = high;
         Low = low;
     }
@@ -600,14 +745,15 @@ public struct GpuUInt128 : IComparable<GpuUInt128>, IEquatable<GpuUInt128>
     }
 
     [MethodImpl(MethodImplOptions.AggressiveInlining)]
-    public static GpuUInt128 DivideExact(GpuUInt128 dividend, GpuUInt128 divisor)
+    public static GpuUInt128 DivideExact(in ReadOnlyGpuUInt128 dividend, in ReadOnlyGpuUInt128 divisor)
     {
         int divisorBits = divisor.GetBitLength();
         int shift = dividend.GetBitLength() - divisorBits;
 
         GpuUInt128 quotient = Zero;
         GpuUInt128 remainder = dividend;
-        GpuUInt128 shiftedDivisor = divisor << shift;
+        GpuUInt128 shiftedDivisor = divisor;
+        shiftedDivisor.ShiftLeft(shift);
         GpuUInt128 quotientBit = One << shift;
 
         while (true)
@@ -637,7 +783,7 @@ public struct GpuUInt128 : IComparable<GpuUInt128>, IEquatable<GpuUInt128>
     }
 
     [MethodImpl(MethodImplOptions.AggressiveInlining)]
-    public void Xor(GpuUInt128 other)
+    public void Xor(in ReadOnlyGpuUInt128 other)
     {
         High ^= other.High;
         Low ^= other.Low;
@@ -697,7 +843,7 @@ public struct GpuUInt128 : IComparable<GpuUInt128>, IEquatable<GpuUInt128>
         High = high >> shift;
     }
 
-    public void MulMod(GpuUInt128 other, GpuUInt128 modulus)
+    public void MulMod(in ReadOnlyGpuUInt128 other, in ReadOnlyGpuUInt128 modulus)
     {
         if (modulus.High == 0UL)
         {
@@ -728,7 +874,7 @@ public struct GpuUInt128 : IComparable<GpuUInt128>, IEquatable<GpuUInt128>
     /// method is intended for validation only and should not be used inside
     /// GPU kernels.
     /// </summary>
-    internal void MulModBigInteger(GpuUInt128 other, GpuUInt128 modulus)
+    internal void MulModBigInteger(in ReadOnlyGpuUInt128 other, in ReadOnlyGpuUInt128 modulus)
     {
         var left = (BigInteger)(UInt128)this;
         var right = (BigInteger)(UInt128)other;
@@ -743,7 +889,7 @@ public struct GpuUInt128 : IComparable<GpuUInt128>, IEquatable<GpuUInt128>
     /// repeated subtractions and becomes extremely slow for large remainders.
     /// Kept for future optimization work.
     /// </summary>
-    internal void MulModByLimb(GpuUInt128 other, GpuUInt128 modulus)
+    internal void MulModByLimb(in ReadOnlyGpuUInt128 other, in ReadOnlyGpuUInt128 modulus)
     {
         // TODO: Relocate this limb-based reducer to the benchmark project once the production
         // pipeline switches to the faster allocating legacy path demonstrated in the benchmarks.
@@ -772,7 +918,7 @@ public struct GpuUInt128 : IComparable<GpuUInt128>, IEquatable<GpuUInt128>
     }
 
     [MethodImpl(MethodImplOptions.AggressiveInlining)]
-    public void MulMod(ulong value, GpuUInt128 modulus)
+    public void MulMod(ulong value, in ReadOnlyGpuUInt128 modulus)
     {
         if (modulus.High == 0UL)
         {
@@ -786,7 +932,7 @@ public struct GpuUInt128 : IComparable<GpuUInt128>, IEquatable<GpuUInt128>
     }
 
     [MethodImpl(MethodImplOptions.AggressiveInlining)]
-    public readonly ulong MulMod(GpuUInt128 other, ulong modulus)
+    public readonly ulong MulMod(in ReadOnlyGpuUInt128 other, ulong modulus)
     {
         ulong modulusLocal = modulus;
         ulong a = Low % modulusLocal;
@@ -870,7 +1016,7 @@ public struct GpuUInt128 : IComparable<GpuUInt128>, IEquatable<GpuUInt128>
     }
 
     [MethodImpl(MethodImplOptions.AggressiveInlining)]
-    private void ReduceProductBitwise(ulong p3, ulong p2, ulong p1, ulong p0, GpuUInt128 modulus)
+    private void ReduceProductBitwise(ulong p3, ulong p2, ulong p1, ulong p0, in ReadOnlyGpuUInt128 modulus)
     {
         GpuUInt128 remainder = new();
         int bit;
@@ -960,7 +1106,7 @@ public struct GpuUInt128 : IComparable<GpuUInt128>, IEquatable<GpuUInt128>
     }
 
     [MethodImpl(MethodImplOptions.AggressiveInlining)]
-    public readonly ulong MulModMontgomery64(GpuUInt128 other, ulong modulus, ulong nPrime, ulong r2)
+    public readonly ulong MulModMontgomery64(in ReadOnlyGpuUInt128 other, ulong modulus, ulong nPrime, ulong r2)
     {
         // TODO: Retire this struct-based Montgomery path from production after adopting the extension
         // helper, which benchmarks 6-7× faster across dense and near-modulus operands.
@@ -999,7 +1145,7 @@ public struct GpuUInt128 : IComparable<GpuUInt128>, IEquatable<GpuUInt128>
     }
 
     [MethodImpl(MethodImplOptions.AggressiveInlining)]
-    public void SquareMod(GpuUInt128 modulus)
+    public void SquareMod(in ReadOnlyGpuUInt128 modulus)
     {
         if (modulus.High == 0UL)
         {
@@ -1058,7 +1204,7 @@ public struct GpuUInt128 : IComparable<GpuUInt128>, IEquatable<GpuUInt128>
         Low = remainder.Low;
     }
 
-    public void ModPow(GpuUInt128 exponent, GpuUInt128 modulus)
+    public void ModPow(GpuUInt128 exponent, in ReadOnlyGpuUInt128 modulus)
     {
         // TODO: Swap this copy-heavy path for the pooled base/exponent ladder used in
         // GpuUInt128MulModBenchmarks so the GPU shim keeps reusing buffers instead of
@@ -1084,7 +1230,7 @@ public struct GpuUInt128 : IComparable<GpuUInt128>, IEquatable<GpuUInt128>
         Low = result.Low;
     }
 
-    public void ModPow(ulong exponent, GpuUInt128 modulus)
+    public void ModPow(ulong exponent, in ReadOnlyGpuUInt128 modulus)
     {
         // TODO: Share the pooled ladder state from GpuUInt128MulModBenchmarks here as well so
         // Lucas–Lehmer batches avoid constructing throwaway temporaries on every exponent.
@@ -1118,7 +1264,7 @@ public struct GpuUInt128 : IComparable<GpuUInt128>, IEquatable<GpuUInt128>
     }
 
     [MethodImpl(MethodImplOptions.AggressiveInlining)]
-    public void ModInv(GpuUInt128 modulus)
+    public void ModInv(in ReadOnlyGpuUInt128 modulus)
     {
         if (modulus.High == 0UL)
         {
@@ -1136,7 +1282,7 @@ public struct GpuUInt128 : IComparable<GpuUInt128>, IEquatable<GpuUInt128>
         ((UInt128)this).ToString();
 
     [MethodImpl(MethodImplOptions.AggressiveInlining)]
-    private static void MultiplyFull(GpuUInt128 left, GpuUInt128 right, out ulong p3, out ulong p2, out ulong p1, out ulong p0)
+    private static void MultiplyFull(in ReadOnlyGpuUInt128 left, in ReadOnlyGpuUInt128 right, out ulong p3, out ulong p2, out ulong p1, out ulong p0)
     {
         // TODO: Reuse variables to reduce register pressure following the fused-limb layout
         // from GpuUInt128MulModByLimbBenchmarks so the multiply helper matches the fastest
@@ -1182,7 +1328,7 @@ public struct GpuUInt128 : IComparable<GpuUInt128>, IEquatable<GpuUInt128>
     }
 
     [MethodImpl(MethodImplOptions.AggressiveInlining)]
-    private static void MultiplyFull(GpuUInt128 left, ulong right, out ulong p3, out ulong p2, out ulong p1, out ulong p0)
+    private static void MultiplyFull(in ReadOnlyGpuUInt128 left, ulong right, out ulong p3, out ulong p2, out ulong p1, out ulong p0)
     {
         var (highLow, lowLow) = Mul64(left.Low, right);
         var (highHigh, lowHigh) = Mul64(left.High, right);
@@ -1199,7 +1345,7 @@ public struct GpuUInt128 : IComparable<GpuUInt128>, IEquatable<GpuUInt128>
     }
 
     [MethodImpl(MethodImplOptions.AggressiveInlining)]
-    private static void Multiply(GpuUInt128 left, GpuUInt128 right, out ulong high, out ulong low)
+    private static void Multiply(in ReadOnlyGpuUInt128 left, in ReadOnlyGpuUInt128 right, out ulong high, out ulong low)
     {
         // TODO: Reuse variables to reduce register pressure.
         var (h0, l0) = Mul64(left.Low, right.Low);
@@ -1234,7 +1380,7 @@ public struct GpuUInt128 : IComparable<GpuUInt128>, IEquatable<GpuUInt128>
     }
 
     [MethodImpl(MethodImplOptions.AggressiveInlining)]
-    private static void SquareFull(GpuUInt128 value, out ulong p3, out ulong p2, out ulong p1, out ulong p0)
+    private static void SquareFull(in ReadOnlyGpuUInt128 value, out ulong p3, out ulong p2, out ulong p1, out ulong p0)
     {
         // TODO: Reuse variables to reduce register pressure.
         // Compute (H*2^64 + L)^2 = H^2*2^128 + 2*H*L*2^64 + L^2
diff --git a/PerfectNumbers.Core/Gpu/MersenneNumberDivisorByDivisorGpuTester.cs b/PerfectNumbers.Core/Gpu/MersenneNumberDivisorByDivisorGpuTester.cs
index ff5bd6d..ca15b89 100644
--- a/PerfectNumbers.Core/Gpu/MersenneNumberDivisorByDivisorGpuTester.cs
+++ b/PerfectNumbers.Core/Gpu/MersenneNumberDivisorByDivisorGpuTester.cs
@@ -115,6 +115,9 @@ public sealed class MersenneNumberDivisorByDivisorGpuTester : IMersenneNumberDiv
             resources.Exponents,
             resources.Hits,
             resources.DivisorData,
+            resources.CycleCandidates,
+            resources.CycleLengths,
+            resources.CycleCapacity,
             out lastProcessed,
             out coveredRange,
             out processedCount
@@ -178,6 +181,9 @@ public sealed class MersenneNumberDivisorByDivisorGpuTester : IMersenneNumberDiv
         ulong[] exponents,
         byte[] hits,
         MontgomeryDivisorData[] divisorData,
+        ulong[] cycleCandidates,
+        ulong[] cycleLengths,
+        int cycleCapacity,
         out ulong lastProcessed,
         out bool coveredRange,
         out ulong processedCount)
@@ -196,13 +202,20 @@ public sealed class MersenneNumberDivisorByDivisorGpuTester : IMersenneNumberDiv
         ArrayView1D<ulong, Stride1D.Dense> exponentView;
         ArrayView1D<byte, Stride1D.Dense> hitsView;
 
-        int chunkCapacity = Math.Min(batchCapacity, (int)accelerator.MaxNumThreadsPerGroup);
-        if (chunkCapacity <= 0)
+        cycleCapacity = Math.Max(1, cycleCapacity);
+
+        int chunkCapacity = Math.Min(batchCapacity, cycleCapacity);
+        int maxThreadsPerGroup = (int)accelerator.MaxNumThreadsPerGroup;
+        if (maxThreadsPerGroup > 0)
         {
-            chunkCapacity = batchCapacity;
+            chunkCapacity = Math.Min(chunkCapacity, maxThreadsPerGroup);
+        }
+
+        if (chunkCapacity <= 0)
+        {
+            chunkCapacity = cycleCapacity;
         }
 
-        ulong[] candidateDivisors = ArrayPool<ulong>.Shared.Rent(chunkCapacity);
         ulong[] filteredDivisors = ArrayPool<ulong>.Shared.Rent(chunkCapacity);
         ulong[] divisorGaps = ArrayPool<ulong>.Shared.Rent(chunkCapacity);
         byte[] remainder10Buffer = ArrayPool<byte>.Shared.Rent(chunkCapacity);
@@ -246,6 +259,7 @@ public sealed class MersenneNumberDivisorByDivisorGpuTester : IMersenneNumberDiv
         byte remainder5 = (byte)((ulong)(currentDivisor128 % 5UL));
         byte remainder3 = (byte)((ulong)(currentDivisor128 % 3UL));
         bool lastIsSeven = (prime & 3UL) == 3UL;
+        DivisorCycleCache cycleCache = DivisorCycleCache.Shared;
 
         try
         {
@@ -263,7 +277,7 @@ public sealed class MersenneNumberDivisorByDivisorGpuTester : IMersenneNumberDiv
                     break;
                 }
 
-                Span<ulong> candidateSpan = candidateDivisors.AsSpan(0, chunkCount);
+                Span<ulong> candidateSpan = cycleCandidates.AsSpan(0, chunkCount);
                 Span<ulong> gapSpan = divisorGaps.AsSpan(0, chunkCount);
                 Span<byte> rem10Span = remainder10Buffer.AsSpan(0, chunkCount);
                 Span<byte> rem8Span = remainder8Buffer.AsSpan(0, chunkCount);
@@ -350,40 +364,46 @@ public sealed class MersenneNumberDivisorByDivisorGpuTester : IMersenneNumberDiv
                     continue;
                 }
 
-                int batchSize = 0;
+                Span<ulong> cycleSpan = cycleLengths.AsSpan(0, filteredCount);
+                cycleCache.GetCycleLengths(filteredDivisorsSpan[..filteredCount], cycleSpan);
+
+                int admissibleCount = 0;
                 for (int i = 0; i < filteredCount; i++)
                 {
-                    ulong divisorValue = filteredDivisorsSpan[i];
-                    divisors[batchSize] = divisorValue;
-                    exponents[batchSize] = prime;
-                    batchSize++;
+                    if (cycleSpan[i] == prime)
+                    {
+                        filteredDivisorsSpan[admissibleCount++] = filteredDivisorsSpan[i];
+                    }
                 }
 
-                if (batchSize == 0)
+                if (admissibleCount == 0)
                 {
                     processedAll = currentK > maxK;
                     continue;
                 }
 
-                divisorDataSpan = divisorData.AsSpan(0, batchSize);
-                divisorSpan = divisors.AsSpan(0, batchSize);
-                exponentSpan = exponents.AsSpan(0, batchSize);
-                hitsSpan = hits.AsSpan(0, batchSize);
-                divisorView = divisorsBuffer.View.SubView(0, batchSize);
-                exponentView = exponentBuffer.View.SubView(0, batchSize);
-                hitsView = hitsBuffer.View.SubView(0, batchSize);
+                divisorDataSpan = divisorData.AsSpan(0, admissibleCount);
+                divisorSpan = divisors.AsSpan(0, admissibleCount);
+                exponentSpan = exponents.AsSpan(0, admissibleCount);
+                hitsSpan = hits.AsSpan(0, admissibleCount);
+                divisorView = divisorsBuffer.View.SubView(0, admissibleCount);
+                exponentView = exponentBuffer.View.SubView(0, admissibleCount);
+                hitsView = hitsBuffer.View.SubView(0, admissibleCount);
 
-                for (int i = 0; i < batchSize; i++)
+                for (int i = 0; i < admissibleCount; i++)
                 {
-                    divisorDataSpan[i] = MontgomeryDivisorData.FromModulus(divisorSpan[i]);
+                    ulong divisorValue = filteredDivisorsSpan[i];
+                    divisorSpan[i] = divisorValue;
+                    exponentSpan[i] = prime;
+                    divisorDataSpan[i] = MontgomeryDivisorData.FromModulus(divisorValue);
                 }
 
-                divisorView.CopyFromCPU(ref MemoryMarshal.GetReference(divisorDataSpan), batchSize);
-                exponentView.CopyFromCPU(ref MemoryMarshal.GetReference(exponentSpan), batchSize);
-                kernel(batchSize, divisorView, exponentView, hitsView);
-                hitsView.CopyToCPU(ref MemoryMarshal.GetReference(hitsSpan), batchSize);
+                divisorView.CopyFromCPU(ref MemoryMarshal.GetReference(divisorDataSpan), admissibleCount);
+                exponentView.CopyFromCPU(ref MemoryMarshal.GetReference(exponentSpan), admissibleCount);
+                kernel(admissibleCount, divisorView, exponentView, hitsView);
+                hitsView.CopyToCPU(ref MemoryMarshal.GetReference(hitsSpan), admissibleCount);
 
-                for (int i = 0; i < batchSize; i++)
+                for (int i = 0; i < admissibleCount; i++)
                 {
                     if (hitsSpan[i] != 0)
                     {
@@ -395,7 +415,7 @@ public sealed class MersenneNumberDivisorByDivisorGpuTester : IMersenneNumberDiv
 
                 if (!composite)
                 {
-                    lastProcessed = divisorSpan[batchSize - 1];
+                    lastProcessed = divisorSpan[admissibleCount - 1];
                 }
 
                 if (composite)
@@ -408,7 +428,6 @@ public sealed class MersenneNumberDivisorByDivisorGpuTester : IMersenneNumberDiv
         }
         finally
         {
-            ArrayPool<ulong>.Shared.Return(candidateDivisors, clearArray: false);
             ArrayPool<ulong>.Shared.Return(filteredDivisors, clearArray: false);
             ArrayPool<ulong>.Shared.Return(divisorGaps, clearArray: false);
             ArrayPool<byte>.Shared.Return(remainder10Buffer, clearArray: false);
@@ -712,19 +731,30 @@ public sealed class MersenneNumberDivisorByDivisorGpuTester : IMersenneNumberDiv
     {
         internal BatchResources(MersenneNumberDivisorByDivisorGpuTester owner, Accelerator accelerator, int capacity)
         {
-            DivisorsBuffer = accelerator.Allocate1D<MontgomeryDivisorData>(capacity);
-            ExponentBuffer = accelerator.Allocate1D<ulong>(capacity);
-            HitsBuffer = accelerator.Allocate1D<byte>(capacity);
-            DivisorDeltaBuffer = accelerator.Allocate1D<ulong>(capacity);
-            RemainderDeltaBuffer = accelerator.Allocate1D<byte>(capacity);
-            RemainderBuffer = accelerator.Allocate1D<byte>(capacity);
-            Divisors = ArrayPool<ulong>.Shared.Rent(capacity);
-            Exponents = ArrayPool<ulong>.Shared.Rent(capacity);
-            Hits = ArrayPool<byte>.Shared.Rent(capacity);
-            DivisorData = ArrayPool<MontgomeryDivisorData>.Shared.Rent(capacity);
+            int cycleCapacity = DivisorCycleCache.Shared.PreferredBatchSize;
+            if (cycleCapacity <= 0)
+            {
+                cycleCapacity = 1;
+            }
+
+            int actualCapacity = Math.Max(capacity, cycleCapacity);
+
+            DivisorsBuffer = accelerator.Allocate1D<MontgomeryDivisorData>(actualCapacity);
+            ExponentBuffer = accelerator.Allocate1D<ulong>(actualCapacity);
+            HitsBuffer = accelerator.Allocate1D<byte>(actualCapacity);
+            DivisorDeltaBuffer = accelerator.Allocate1D<ulong>(actualCapacity);
+            RemainderDeltaBuffer = accelerator.Allocate1D<byte>(actualCapacity);
+            RemainderBuffer = accelerator.Allocate1D<byte>(actualCapacity);
+            Divisors = ArrayPool<ulong>.Shared.Rent(actualCapacity);
+            Exponents = ArrayPool<ulong>.Shared.Rent(actualCapacity);
+            Hits = ArrayPool<byte>.Shared.Rent(actualCapacity);
+            DivisorData = ArrayPool<MontgomeryDivisorData>.Shared.Rent(actualCapacity);
+            CycleCandidates = ArrayPool<ulong>.Shared.Rent(cycleCapacity);
+            CycleLengths = ArrayPool<ulong>.Shared.Rent(cycleCapacity);
             RemainderDeltaKernel = owner.GetRemainderDeltaKernel(accelerator);
             RemainderScanKernel = owner.GetRemainderScanKernel(accelerator);
-            Capacity = capacity;
+            Capacity = actualCapacity;
+            CycleCapacity = cycleCapacity;
         }
 
         internal readonly MemoryBuffer1D<MontgomeryDivisorData, Stride1D.Dense> DivisorsBuffer;
@@ -747,12 +777,18 @@ public sealed class MersenneNumberDivisorByDivisorGpuTester : IMersenneNumberDiv
 
         internal readonly MontgomeryDivisorData[] DivisorData;
 
+        internal readonly ulong[] CycleCandidates;
+
+        internal readonly ulong[] CycleLengths;
+
         internal readonly Action<Index1D, ArrayView<ulong>, ArrayView<byte>, byte> RemainderDeltaKernel;
 
         internal readonly Action<KernelConfig, Index1D, ArrayView<byte>, ArrayView<byte>, byte, byte> RemainderScanKernel;
 
         internal readonly int Capacity;
 
+        internal readonly int CycleCapacity;
+
         public void Dispose()
         {
             DivisorsBuffer.Dispose();
@@ -765,6 +801,8 @@ public sealed class MersenneNumberDivisorByDivisorGpuTester : IMersenneNumberDiv
             ArrayPool<ulong>.Shared.Return(Exponents, clearArray: false);
             ArrayPool<byte>.Shared.Return(Hits, clearArray: false);
             ArrayPool<MontgomeryDivisorData>.Shared.Return(DivisorData, clearArray: false);
+            ArrayPool<ulong>.Shared.Return(CycleCandidates, clearArray: false);
+            ArrayPool<ulong>.Shared.Return(CycleLengths, clearArray: false);
         }
     }
 
diff --git a/PerfectNumbers.Core/Gpu/MersenneNumberDivisorGpuTester.cs b/PerfectNumbers.Core/Gpu/MersenneNumberDivisorGpuTester.cs
index 4715500..4ae660c 100644
--- a/PerfectNumbers.Core/Gpu/MersenneNumberDivisorGpuTester.cs
+++ b/PerfectNumbers.Core/Gpu/MersenneNumberDivisorGpuTester.cs
@@ -10,11 +10,11 @@ namespace PerfectNumbers.Core.Gpu;
 /// </summary>
 public sealed class MersenneNumberDivisorGpuTester
 {
-	private readonly ConcurrentDictionary<Accelerator, Action<Index1D, ulong, GpuUInt128, ArrayView<byte>>> _kernelCache = new();
+	private readonly ConcurrentDictionary<Accelerator, Action<Index1D, ulong, ReadOnlyGpuUInt128, ArrayView<byte>>> _kernelCache = new();
 	private readonly ConcurrentDictionary<Accelerator, MemoryBuffer1D<byte, Stride1D.Dense>> _resultBuffers = new();
 
-	private Action<Index1D, ulong, GpuUInt128, ArrayView<byte>> GetKernel(Accelerator accelerator) =>
-			_kernelCache.GetOrAdd(accelerator, acc => acc.LoadAutoGroupedStreamKernel<Index1D, ulong, GpuUInt128, ArrayView<byte>>(Kernel));
+	private Action<Index1D, ulong, ReadOnlyGpuUInt128, ArrayView<byte>> GetKernel(Accelerator accelerator) =>
+			_kernelCache.GetOrAdd(accelerator, acc => acc.LoadAutoGroupedStreamKernel<Index1D, ulong, ReadOnlyGpuUInt128, ArrayView<byte>>(Kernel));
 
         public static void BuildDivisorCandidates()
         {
@@ -36,106 +36,102 @@ public sealed class MersenneNumberDivisorGpuTester
                 _divisorCandidates = count == 0 ? [] : list[..count];
         }
 
-	public bool IsDivisible(ulong exponent, UInt128 divisor)
-	{
-		var gpu = GpuContextPool.RentPreferred(preferCpu: false);
-		var accelerator = gpu.Accelerator;
-		var kernel = GetKernel(accelerator);
-		var resultBuffer = _resultBuffers.GetOrAdd(accelerator, acc => acc.Allocate1D<byte>(1));
-		resultBuffer.MemSetToZero();
-		kernel(1, exponent, (GpuUInt128)divisor, resultBuffer.View);
-		accelerator.Synchronize();
-		Span<byte> result = stackalloc byte[1];
-		resultBuffer.View.CopyToCPU(ref result[0], 1);
-		bool divisible = result[0] != 0;
-		result[0] = 0;
-		resultBuffer.View.CopyFromCPU(ref result[0], 1);
-		accelerator.Synchronize();
-		gpu.Dispose();
-		return divisible;
-	}
+    public bool IsDivisible(ulong exponent, in ReadOnlyGpuUInt128 divisor)
+    {
+        var gpu = GpuContextPool.RentPreferred(preferCpu: false);
+        var accelerator = gpu.Accelerator;
+        var kernel = GetKernel(accelerator);
+        var resultBuffer = _resultBuffers.GetOrAdd(accelerator, acc => acc.Allocate1D<byte>(1));
+        resultBuffer.MemSetToZero();
+        kernel(1, exponent, divisor, resultBuffer.View);
+        accelerator.Synchronize();
+        Span<byte> result = stackalloc byte[1];
+        resultBuffer.View.CopyToCPU(ref result[0], 1);
+        bool divisible = result[0] != 0;
+        result[0] = 0;
+        resultBuffer.View.CopyFromCPU(ref result[0], 1);
+        accelerator.Synchronize();
+        gpu.Dispose();
+        return divisible;
+    }
 
         private static (ulong divisor, uint cycle)[]? _divisorCandidates = Array.Empty<(ulong divisor, uint cycle)>();
 
-	public bool IsPrime(ulong p, UInt128 d, ulong divisorCyclesSearchLimit, out bool divisorsExhausted)
-	{
-		if (d != UInt128.Zero)
-		{
-			if (IsDivisible(p, d))
-			{
-				divisorsExhausted = true;
-				return false;
-			}
+    public bool IsPrime(ulong p, UInt128 d, ulong divisorCyclesSearchLimit, out bool divisorsExhausted)
+    {
+        ReadOnlyGpuUInt128 readOnlyDivisor;
 
-			divisorsExhausted = false;
-			return true;
-		}
+        if (d != UInt128.Zero)
+        {
+            readOnlyDivisor = new ReadOnlyGpuUInt128(d);
+            if (IsDivisible(p, in readOnlyDivisor))
+            {
+                divisorsExhausted = true;
+                return false;
+            }
 
-                if (_divisorCandidates is { Length: > 0 } candidates)
+            divisorsExhausted = false;
+            return true;
+        }
+
+        if (_divisorCandidates is { Length: > 0 } candidates)
+        {
+            int candidateCount = candidates.Length;
+            for (int index = 0; index < candidateCount; index++)
+            {
+                (ulong candidateDivisor, uint cycle) = candidates[index];
+                if (p % cycle != 0UL)
                 {
-                        int len = candidates.Length;
-                        uint cycle;
-                        for (int k = 0; k < len; k++)
-                        {
-                                (ulong dSmall, cycle) = candidates[k];
-                                // TODO: Replace this `%` check with the shared divisor-cycle divisibility helper so GPU scans
-                                // reuse the cached cycle remainders instead of recomputing modulo for every candidate.
-                                if (p % cycle != 0UL)
-                                {
-                                        continue;
-                                }
+                    continue;
+                }
 
-				if (IsDivisible(p, dSmall))
-				{
-					divisorsExhausted = true;
-					return false;
-				}
-			}
-		}
+                readOnlyDivisor = new ReadOnlyGpuUInt128(candidateDivisor); // Reusing readOnlyDivisor for candidate divisors.
+                if (IsDivisible(p, in readOnlyDivisor))
+                {
+                    divisorsExhausted = true;
+                    return false;
+                }
+            }
+        }
 
-		UInt128 kMul2;
-		var divisorCycles = MersenneDivisorCycles.Shared;
-		UInt128 maxK2 = UInt128.MaxValue / ((UInt128)p << 1);
-		ulong limit = divisorCyclesSearchLimit;
-		if ((UInt128)limit > maxK2)
-		{
-			limit = (ulong)maxK2;
-		}
+        UInt128 kMul2;
+        var divisorCycles = MersenneDivisorCycles.Shared;
+        UInt128 maxK2 = UInt128.MaxValue / ((UInt128)p << 1);
+        ulong limit = divisorCyclesSearchLimit;
+        if ((UInt128)limit > maxK2)
+        {
+            limit = (ulong)maxK2;
+        }
 
-		ulong k2;
-		for (k2 = 1UL; k2 <= limit; k2++)
-		{
-			kMul2 = (UInt128)k2 << 1;
-			UInt128 candidate = checked(kMul2 * p);
-			d = checked(candidate + UInt128.One);
-			if (p < 64UL && d == ((UInt128)1 << (int)p) - UInt128.One)
-			{
-				continue;
-			}
+        for (ulong k2 = 1UL; k2 <= limit; k2++)
+        {
+            kMul2 = (UInt128)k2 << 1;
+            UInt128 candidate = checked(kMul2 * p);
+            d = checked(candidate + UInt128.One);
+            if (p < 64UL && d == ((UInt128)1 << (int)p) - UInt128.One)
+            {
+                continue;
+            }
 
-                        UInt128 cycle128 = MersenneDivisorCycles.GetCycle(d);
-                        // TODO: When this lookup falls outside the preloaded snapshot, invoke the single-cycle GPU helper
-                        // instead of allowing GetCycle to allocate an entire block; the divisor-cycle benchmarks showed the
-                        // one-off computation keeping memory flat while still delivering the optimized stepping data.
-                        // TODO: Feed this check through the ProcessEightBitWindows-based residue tracker so we avoid the slow
-                        // UInt128 modulo on the CPU fallback highlighted by the Pow2Montgomery benchmarks.
-                        if ((UInt128)p % cycle128 != UInt128.Zero)
-                        {
-                                continue;
-                        }
+            UInt128 cycle128 = MersenneDivisorCycles.GetCycle(d);
+            if ((UInt128)p % cycle128 != UInt128.Zero)
+            {
+                continue;
+            }
 
-			if (IsDivisible(p, d))
-			{
-				divisorsExhausted = true;
-				return false;
-			}
-		}
+            readOnlyDivisor = new ReadOnlyGpuUInt128(d); // Reusing readOnlyDivisor for generated divisors.
+            if (IsDivisible(p, in readOnlyDivisor))
+            {
+                divisorsExhausted = true;
+                return false;
+            }
+        }
 
-		divisorsExhausted = (UInt128)divisorCyclesSearchLimit >= maxK2;
-		return true;
-	}
+        divisorsExhausted = (UInt128)divisorCyclesSearchLimit >= maxK2;
+        return true;
+    }
 
-	private static void Kernel(Index1D _, ulong exponent, GpuUInt128 divisor, ArrayView<byte> result)
+	private static void Kernel(Index1D _, ulong exponent, ReadOnlyGpuUInt128 divisor, ArrayView<byte> result)
 	{
 		if (divisor.High == 0UL && divisor.Low <= 1UL)
 		{
@@ -143,16 +139,16 @@ public sealed class MersenneNumberDivisorGpuTester
 			return;
 		}
 
-		GpuUInt128 baseVal, mod;
+		GpuUInt128 baseVal;
 		int x = divisor.High == 0UL
 				? 63 - XMath.LeadingZeroCount(divisor.Low)
 				: 127 - XMath.LeadingZeroCount(divisor.High);
-		mod = divisor;
+		ReadOnlyGpuUInt128 readOnlyMod = divisor;
                 if (x == 0)
                 {
                         // TODO: Swap Pow2Mod for the ProcessEightBitWindows helper once Pow2Minus1Mod adopts it;
                         // GpuPow2ModBenchmarks showed the windowed kernel cutting large divisors from ~51 µs to ~21 µs.
-                        baseVal = GpuUInt128.Pow2Mod(exponent, mod);
+                        baseVal = GpuUInt128.Pow2Mod(exponent, in readOnlyMod);
 			result[0] = baseVal.High == 0UL && baseVal.Low == 1UL ? (byte)1 : (byte)0;
 			return;
 		}
@@ -168,12 +164,13 @@ public sealed class MersenneNumberDivisorGpuTester
 		}
 
                 GpuUInt128 pow = baseVal;
-                pow.ModPow(exponent / ux, mod);
+                pow.ModPow(exponent / ux, in readOnlyMod);
                 // TODO: Replace this trailing Pow2Mod call with the ProcessEightBitWindows variant once the shared
                 // windowed helper lands so mixed-radix decompositions benefit from the same GPU speedup.
-                var part2 = GpuUInt128.Pow2Mod(exponent % ux, mod);
+                var part2 = GpuUInt128.Pow2Mod(exponent % ux, in readOnlyMod);
 		GpuUInt128 product = pow;
-		product.MulMod(part2, mod);
+		ReadOnlyGpuUInt128 part2ReadOnly = part2.AsReadOnly();
+		product.MulMod(in part2ReadOnly, in readOnlyMod);
 		result[0] = product.High == 0UL && product.Low == 1UL ? (byte)1 : (byte)0;
 	}
 }
diff --git a/PerfectNumbers.Core/Gpu/ReadOnlyGpuUInt128.cs b/PerfectNumbers.Core/Gpu/ReadOnlyGpuUInt128.cs
new file mode 100644
index 0000000..876f292
--- /dev/null
+++ b/PerfectNumbers.Core/Gpu/ReadOnlyGpuUInt128.cs
@@ -0,0 +1,90 @@
+using System;
+using System.Numerics;
+using System.Runtime.CompilerServices;
+
+namespace PerfectNumbers.Core.Gpu;
+
+public readonly struct ReadOnlyGpuUInt128 : IEquatable<ReadOnlyGpuUInt128>
+{
+    public readonly ulong High;
+    public readonly ulong Low;
+
+    [MethodImpl(MethodImplOptions.AggressiveInlining)]
+    public ReadOnlyGpuUInt128(ulong high, ulong low)
+    {
+        High = high;
+        Low = low;
+    }
+
+    [MethodImpl(MethodImplOptions.AggressiveInlining)]
+    public ReadOnlyGpuUInt128(ulong low)
+        : this(0UL, low)
+    {
+    }
+
+    [MethodImpl(MethodImplOptions.AggressiveInlining)]
+    public ReadOnlyGpuUInt128(UInt128 value)
+    {
+        High = (ulong)(value >> 64);
+        Low = (ulong)value;
+    }
+
+    [MethodImpl(MethodImplOptions.AggressiveInlining)]
+    public ReadOnlyGpuUInt128(GpuUInt128 value)
+    {
+        High = value.High;
+        Low = value.Low;
+    }
+
+    public bool IsZero
+    {
+        [MethodImpl(MethodImplOptions.AggressiveInlining)]
+        get => High == 0UL && Low == 0UL;
+    }
+
+    [MethodImpl(MethodImplOptions.AggressiveInlining)]
+    public int GetBitLength()
+    {
+        if (High != 0UL)
+        {
+            return 64 + (64 - BitOperations.LeadingZeroCount(High));
+        }
+
+        if (Low == 0UL)
+        {
+            return 0;
+        }
+
+        return 64 - BitOperations.LeadingZeroCount(Low);
+    }
+
+    [MethodImpl(MethodImplOptions.AggressiveInlining)]
+    public GpuUInt128 ToMutable() => new(High, Low);
+
+    [MethodImpl(MethodImplOptions.AggressiveInlining)]
+    public bool Equals(ReadOnlyGpuUInt128 other) => High == other.High && Low == other.Low;
+
+    [MethodImpl(MethodImplOptions.AggressiveInlining)]
+    public override bool Equals(object? obj) => obj is ReadOnlyGpuUInt128 other && Equals(other);
+
+    [MethodImpl(MethodImplOptions.AggressiveInlining)]
+    public override int GetHashCode() => HashCode.Combine(High, Low);
+
+    [MethodImpl(MethodImplOptions.AggressiveInlining)]
+    public static implicit operator ReadOnlyGpuUInt128(ulong value) => new(value);
+
+    [MethodImpl(MethodImplOptions.AggressiveInlining)]
+    public static implicit operator ReadOnlyGpuUInt128(UInt128 value) => new(value);
+
+    [MethodImpl(MethodImplOptions.AggressiveInlining)]
+    public static implicit operator ReadOnlyGpuUInt128(GpuUInt128 value) => new(value);
+
+    [MethodImpl(MethodImplOptions.AggressiveInlining)]
+    public static implicit operator UInt128(ReadOnlyGpuUInt128 value) => ((UInt128)value.High << 64) | value.Low;
+
+    [MethodImpl(MethodImplOptions.AggressiveInlining)]
+    public ReadOnlyGpuUInt128 WithHigh(ulong high) => new(high, Low);
+
+    [MethodImpl(MethodImplOptions.AggressiveInlining)]
+    public ReadOnlyGpuUInt128 WithLow(ulong low) => new(High, low);
+}
diff --git a/todo.txt b/todo.txt
index 697f7a0..0554c23 100644
--- a/todo.txt
+++ b/todo.txt
@@ -1,2 +1,5 @@
+- implement full routing for CPU or GPU execution based on the (...)-device=cpu|gpu switches. The current --mersenne=bydivisor GPU flow appears to run every stage on the GPU and ignore switches such as --primes-device for the primality check.
+- investigate why the --mersenne=bydivisor GPU path runs out of host memory. Exceptions must remain visible.
 - Reimplement the GPU cycle heuristics for MersenneNumberIncrementalGpuTester and re-enable Scan_handles_various_prime_exponents once the device path is fixed.
 - Extend the GPU numeric type probe to cover proposed custom types such as GpuUInt256 and GpuUInt512 once their definitions are available so we can plan the wide-order heuristics on the device.
+- document the GPU by-divisor regressions introduced in commit cd1debe ("Various micro- and medium- optimizations"): the refactor dropped divisor-cycle caching, remainder stepper reuse, and honoring the cached cycle parameter so we can plan which pieces to restore.