Clarify Pollard-Rho parity guard and seed CPU batches

PerfectNumbers.Core/Cpu/MersenneCpuDivisorScanSession.cs

@@ -43,16 +43,13 @@ internal sealed class MersenneCpuDivisorScanSession : IMersenneNumberDivisorByDi
         // 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)
-        {
-            throw new InvalidOperationException($"Invalid modulus {cachedData.Modulus} for divisor scans.");
-        }
 
         var cycleStepper = new CycleRemainderStepper(divisorCycle);
 
+        bool initialUnity = exponentStepper.InitializeCpuIsUnity(primes[0]);
         ulong remainder = cycleStepper.Initialize(primes[0]);
         hits[0] = remainder == 0UL
-            ? (exponentStepper.ComputeNextIsUnity(primes[0]) ? (byte)1 : (byte)0)
+            ? (initialUnity ? (byte)1 : (byte)0)
             : (byte)0;
 
         for (int i = 1; i < length; i++)

PerfectNumbers.Core/ExponentRemainderStepper.cs

@@ -1,5 +1,9 @@
 namespace PerfectNumbers.Core;
 
+/// <summary>
+/// Tracks successive Montgomery residues for a single divisor while scanning exponent candidates.
+/// Callers are responsible for seeding the stepper once per divisor and then advancing exponents in ascending order.
+/// </summary>
 internal struct ExponentRemainderStepper
 {
     private readonly MontgomeryDivisorData _divisor;
@@ -8,7 +12,6 @@ internal struct ExponentRemainderStepper
     private readonly ulong _montgomeryOne;
     private ulong _previousExponent;
     private ulong _currentMontgomery;
-    private bool _hasState;
 
     public ExponentRemainderStepper(in MontgomeryDivisorData divisor)
     {
@@ -18,13 +21,8 @@ internal struct ExponentRemainderStepper
         _montgomeryOne = divisor.MontgomeryOne;
         _previousExponent = 0UL;
         _currentMontgomery = divisor.MontgomeryOne;
-        _hasState = false;
     }
 
-    public bool IsValidModulus => _modulus > 1UL && (_modulus & 1UL) != 0UL;
-
-    public bool HasState => _hasState;
-
     public ulong PreviousExponent => _previousExponent;
 
     public bool MatchesDivisor(in MontgomeryDivisorData divisor)
@@ -32,29 +30,47 @@ internal struct ExponentRemainderStepper
         return _modulus == divisor.Modulus && _nPrime == divisor.NPrime && _montgomeryOne == divisor.MontgomeryOne;
     }
 
+    /// <summary>
+    /// Clears the cached Montgomery residue so the stepper can be reused for another divisor.
+    /// Callers must reinitialize the stepper before consuming any residues after a reset.
+    /// </summary>
     public void Reset()
     {
         _previousExponent = 0UL;
         _currentMontgomery = _montgomeryOne;
-        _hasState = false;
     }
 
-    public ulong ComputeNext(ulong exponent)
+    /// <summary>
+    /// Initializes the stepper for CPU calculations using the first exponent in a strictly ascending sequence.
+    /// The returned value is the canonical (non-Montgomery) residue corresponding to the provided exponent.
+    /// </summary>
+    /// <param name="exponent">First exponent evaluated for the current divisor. The caller must reuse the same divisor metadata.</param>
+    /// <returns>The canonical residue produced by <c>2^exponent mod divisor</c>.</returns>
+    public ulong InitializeCpu(ulong exponent)
     {
-        if (!IsValidModulus)
-        {
-            _hasState = false;
-            return 0UL;
-        }
+        InitializeCpuState(exponent);
+        return ReduceCurrent();
+    }
 
-        if (!_hasState || exponent <= _previousExponent)
-        {
-            _currentMontgomery = exponent.Pow2MontgomeryModWindowedCpu(_divisor, keepMontgomery: true);
-            _previousExponent = exponent;
-            _hasState = true;
-            return ReduceCurrent();
-        }
+    /// <summary>
+    /// Initializes the CPU stepping state and reports whether the seed exponent already yields unity.
+    /// </summary>
+    /// <param name="exponent">First exponent evaluated for the current divisor. Must match the divisor passed to the constructor.</param>
+    /// <returns><see langword="true"/> when the seed exponent produces a Montgomery residue equal to one.</returns>
+    public bool InitializeCpuIsUnity(ulong exponent)
+    {
+        InitializeCpuState(exponent);
+        return _currentMontgomery == _montgomeryOne;
+    }
 
+    /// <summary>
+    /// Advances the CPU stepping state to the next exponent in a strictly ascending sequence that shares the same divisor.
+    /// Callers must invoke <see cref="InitializeCpu(ulong)"/> beforehand.
+    /// </summary>
+    /// <param name="exponent">Next exponent in the ascending sequence. Callers must never move backwards or reuse an older exponent.</param>
+    /// <returns>The canonical residue produced by <c>2^exponent mod divisor</c>.</returns>
+    public ulong ComputeNextCpu(ulong exponent)
+    {
         ulong delta = exponent - _previousExponent;
         // TODO: Once divisor cycle lengths are mandatory, pull the delta multiplier from the
         // single-block divisor-cycle snapshot so we can skip the powmod entirely and reuse the
@@ -65,23 +81,37 @@ internal struct ExponentRemainderStepper
         return ReduceCurrent();
     }
 
+    /// <summary>
+    /// Initializes the stepper for GPU calculations using the first exponent in a strictly ascending sequence.
+    /// </summary>
+    /// <param name="exponent">First exponent evaluated on the GPU path. The same divisor metadata must be reused for future calls.</param>
+    /// <returns>The canonical residue produced by <c>2^exponent mod divisor</c>.</returns>
+    public ulong InitializeGpu(ulong exponent)
+    {
+        InitializeGpuState(exponent);
+        return ReduceCurrent();
+    }
+
+    /// <summary>
+    /// Initializes the GPU stepping state and reports whether the seed exponent produces unity.
+    /// </summary>
+    /// <param name="exponent">First exponent evaluated on the GPU path. Must match the divisor supplied during construction.</param>
+    /// <returns><see langword="true"/> when the seed exponent produces a Montgomery residue equal to one.</returns>
+    public bool InitializeGpuIsUnity(ulong exponent)
+    {
+        InitializeGpuState(exponent);
+        return _currentMontgomery == _montgomeryOne;
+    }
+
+    /// <summary>
+    /// Advances the CPU stepping state to the next exponent and reports whether it evaluates to unity.
+    /// Callers must seed the stepper through <see cref="InitializeCpu(ulong)"/> or <see cref="InitializeCpuIsUnity(ulong)"/> and then
+    /// pass exponents in strictly ascending order.
+    /// </summary>
+    /// <param name="exponent">Next exponent in the ascending sequence.</param>
+    /// <returns><see langword="true"/> when the canonical residue equals one.</returns>
     public bool ComputeNextIsUnity(ulong exponent)
     {
-        if (!IsValidModulus)
-        {
-			throw new InvalidOperationException($"Modulus is invalid for exponent {exponent}");
-            // _hasState = false;
-            // return false;
-        }
-
-        if (!_hasState || exponent <= _previousExponent)
-        {
-            _currentMontgomery = exponent.Pow2MontgomeryModWindowedCpu(_divisor, keepMontgomery: true);
-            _previousExponent = exponent;
-            _hasState = true;
-            return _currentMontgomery == _montgomeryOne;
-        }
-
         ulong delta = exponent - _previousExponent;
         // TODO: Reuse the divisor-cycle derived Montgomery delta once the cache exposes single-cycle
         // lookups so this branch also avoids recomputing powmods when the snapshot lacks the divisor.
@@ -91,40 +121,32 @@ internal struct ExponentRemainderStepper
         return _currentMontgomery == _montgomeryOne;
     }
 
+    /// <summary>
+    /// Seeds the stepper with a precomputed Montgomery residue produced by the caller.
+    /// The provided exponent becomes the new baseline and all subsequent calls must continue with greater exponents.
+    /// </summary>
+    /// <param name="exponent">Exponent associated with <paramref name="montgomeryResult"/>.</param>
+    /// <param name="montgomeryResult">Residue in Montgomery form produced externally for the configured divisor.</param>
+    /// <param name="isUnity">Outputs whether the supplied residue equals Montgomery one.</param>
+    /// <returns><see langword="true"/> when the residue is accepted and cached.</returns>
     public bool TryInitializeFromMontgomeryResult(ulong exponent, ulong montgomeryResult, out bool isUnity)
     {
-        if (!IsValidModulus)
-        {
-            _hasState = false;
-            isUnity = false;
-            return false;
-        }
-
         _currentMontgomery = montgomeryResult;
         _previousExponent = exponent;
-        _hasState = true;
         isUnity = _currentMontgomery == _montgomeryOne;
         return true;
     }
 
+    /// <summary>
+    /// Advances the stepper using a caller-supplied Montgomery delta that represents the gap to the next exponent.
+    /// Callers must seed the stepper first, reuse the same divisor metadata, and continue feeding exponents in ascending order so the delta remains valid.
+    /// </summary>
+    /// <param name="exponent">Next exponent in the ascending sequence.</param>
+    /// <param name="montgomeryDelta">Montgomery residue delta corresponding to the exponent gap.</param>
+    /// <param name="isUnity">Outputs whether the resulting residue equals Montgomery one.</param>
+    /// <returns><see langword="true"/> when the advance succeeds.</returns>
     public bool TryAdvanceWithMontgomeryDelta(ulong exponent, ulong montgomeryDelta, out bool isUnity)
     {
-        if (!IsValidModulus)
-        {
-            _hasState = false;
-            isUnity = false;
-            return false;
-        }
-
-        if (!_hasState || exponent <= _previousExponent)
-        {
-            _currentMontgomery = exponent.Pow2MontgomeryModWindowedCpu(_divisor, keepMontgomery: true);
-            _previousExponent = exponent;
-            _hasState = true;
-            isUnity = _currentMontgomery == _montgomeryOne;
-            return true;
-        }
-
         // TODO: Once the divisor-cycle cache exposes a direct Montgomery delta, multiply it here instead
         // of relying on the caller-provided delta so incremental scans remain in sync with the snapshot
         // without computing additional cycles or mutating cache state.
@@ -134,5 +156,17 @@ internal struct ExponentRemainderStepper
         return true;
     }
 
+    private void InitializeCpuState(ulong exponent)
+    {
+        _currentMontgomery = exponent.Pow2MontgomeryModWindowedCpu(_divisor, keepMontgomery: true);
+        _previousExponent = exponent;
+    }
+
+    private void InitializeGpuState(ulong exponent)
+    {
+        _currentMontgomery = exponent.Pow2MontgomeryModWindowedCpu(_divisor, keepMontgomery: true);
+        _previousExponent = exponent;
+    }
+
     private ulong ReduceCurrent() => _currentMontgomery.MontgomeryMultiply(1UL, _modulus, _nPrime);
 }

PerfectNumbers.Core/MersenneDivisorCycles.cs

@@ -564,7 +564,8 @@ public class MersenneDivisorCycles
 
         stepper.Reset();
 
-        for (int j = 0; j < actual; j++)
+        evaluations[0] = stepper.InitializeCpuIsUnity(candidates[0]);
+        for (int j = 1; j < actual; j++)
         {
             evaluations[j] = stepper.ComputeNextIsUnity(candidates[j]);
         }
@@ -582,10 +583,13 @@ public class MersenneDivisorCycles
 
     private static ulong PollardRho64(ulong n)
     {
-        if ((n & 1UL) == 0UL)
-        {
-            return 2UL;
-        }
+        // The EvenPerfectBitScanner path removes all factors of two before invoking Pollard-Rho, so this branch never executes
+        // outside synthetic tests and benchmarks. Keep the guard documented to prevent future regressions without paying the hot-
+        // path cost.
+        // if ((n & 1UL) == 0UL)
+        // {
+        //     return 2UL;
+        // }
 
         while (true)
         {

PerfectNumbers.Core/PrimeOrderCalculator.Cpu.cs

@@ -165,7 +165,14 @@ internal static partial class PrimeOrderCalculator
 
 		ExponentRemainderStepper stepper = ThreadStaticPools.RentExponentStepper(divisorData);
 		int candidateCount = candidates.Length;
-		for (int i = 0; i < candidateCount; i++)
+
+		if (stepper.InitializeCpuIsUnity(candidates[0]))
+		{
+			ThreadStaticPools.ReturnExponentStepper(stepper);
+			return false;
+		}
+
+		for (int i = 1; i < candidateCount; i++)
 		{
 			if (stepper.ComputeNextIsUnity(candidates[i]))
 			{
@@ -300,7 +307,8 @@ internal static partial class PrimeOrderCalculator
 
 		stepper.Reset();
 
-		for (int j = 0; j < actual; j++)
+		evaluations[0] = stepper.InitializeCpuIsUnity(candidates[0]);
+		for (int j = 1; j < actual; j++)
 		{
 			evaluations[j] = stepper.ComputeNextIsUnity(candidates[j]);
 		}
@@ -459,7 +467,12 @@ internal static partial class PrimeOrderCalculator
 
 		stepper.Reset();
 
-		for (int j = 0; j < actual; j++)
+		if (stepper.InitializeCpuIsUnity(candidates[0]))
+		{
+			return true;
+		}
+
+		for (int j = 1; j < actual; j++)
 		{
 			if (stepper.ComputeNextIsUnity(candidates[j]))
 			{
@@ -535,6 +548,7 @@ internal static partial class PrimeOrderCalculator
 			bool allowGpuBatch = true;
 			Span<ulong> candidateSpan = CollectionsMarshal.AsSpan(candidates);
 			ExponentRemainderStepper powStepper = ThreadStaticPools.RentExponentStepper(divisorData);
+			bool powStepperInitialized = false;
 
 			// DebugLog(() => $"Checking candidates ({candidateCount} candidates, {powBudget} pow budget)");
 			int index = 0;
@@ -594,7 +608,33 @@ internal static partial class PrimeOrderCalculator
 					}
 				}
 
-				for (int i = 0; i < batchSize && powUsed < powBudget; i++)
+				int startIndex = 0;
+
+				if (!gpuSuccess && !powStepperInitialized)
+				{
+					ulong candidate = batch[0];
+					powUsed++;
+
+					bool equalsOne = powStepper.InitializeCpuIsUnity(candidate);
+					powStepperInitialized = true;
+					if (equalsOne && TryConfirmCandidateCpu(prime, candidate, divisorData, config, ref powUsed, powBudget))
+					{
+						if (gpuPool is not null)
+						{
+							pool.Return(gpuPool, clearArray: false);
+						}
+
+						candidates.Clear();
+						ThreadStaticPools.ReturnUlongList(candidates);
+						ThreadStaticPools.ReturnExponentStepper(powStepper);
+						result = candidate;
+						return true;
+					}
+
+					startIndex = 1;
+				}
+
+				for (int i = startIndex; i < batchSize && powUsed < powBudget; i++)
 				{
 					ulong candidate = batch[i];
 					powUsed++;
@@ -936,7 +976,18 @@ internal static partial class PrimeOrderCalculator
 
 		stepper.Reset();
 
-		for (int j = 0; j < actual; j++)
+		if (powUsed >= powBudget && powBudget > 0)
+		{
+			return true;
+		}
+
+		powUsed++;
+		if (stepper.InitializeCpuIsUnity(candidates[0]))
+		{
+			return true;
+		}
+
+		for (int j = 1; j < actual; j++)
 		{
 			if (powUsed >= powBudget && powBudget > 0)
 			{