Pseudo random number generation on the gpu

Using a C# script in the Unity3D game engine to control a HLSL compute shader, I'm trying to generate pseudo random numbers on the GPU and store them in a Texture2D. Following along with GPU Gems 3 Hybrid Tausworthe method and another thread Pseudo Random Number Generation on the GPU, I've come across an issue.

The problem:

the resulting texture appears to be one solid color. If I run the shader multiple times, I get a different solid color texture result every time, but the entire texture is the one color.

Compute shader code

#pragma kernel CSMain

RWTexture2D<float4> result; // 256 resolution texture to write to
uint4 seed;  //four uniform random numbers generated on the CPU in a C# script

struct RandomResult
{
    uint4 state;
    float value;
};

uint TausStep(uint z, int S1, int S2, int S3, uint M)
{
    uint b = (((z << S1) ^ z) >> S2);
    return ((z & M) << S3) ^ b;
}
uint LCGStep(uint z, uint A, uint C)
{
    return A * z + C;
}
RandomResult HybridTaus(uint4 state)
{
    state.x = TausStep(state.x, 13, 19, 12, 4294967294);
    state.y = TausStep(state.y, 2, 25, 4, 4294967288);
    state.z = TausStep(state.z, 3, 11, 17, 4294967280);
    state.w = LCGStep(state.w, 1664525, 1013904223);

    RandomResult rand;
    rand.state = state;
    rand.value = 2.3283064365387e-10 * (state.x ^ state.y ^ state.z ^ state.w);

    return rand;
}

[numthreads(8, 8, 1)]
void CSMain(uint3 id)
{
    result[id.xy] = HybridTaus(seed).value;
}

Do I need to save the state on the gpu? If so, how would I do that? Do I need to deallocate the memory afterwards?

I tried to assign the result of the HybridTaus() function to seed in hopes that it would use the new value in the following HybridTaus(seed) call to see if that would make a difference. I also tried to add unique arbitrary numbers based on the thread id, which is the id parameter. This gave some improved results, but I suspect the randomness is only as good as I can make it, coming from maths performed on the thread ids and not effectively from the random number generator.

[numthreads(8, 8, 1)]
void CSMain(uint3 id)
{
    //first thing I tried
    //RandomResult rand = HybridTaus(seed);
    //seed = rand.state;  // re-assign seed with the new state

    //result[id.xy] = rand.value;

    //second thing I tried
    RandResult rand = HybridTaus(seed * uint4(id.x*id.y*id.x*id.y,
                                              id.x*id.y/id.x*id.y, 
                                              id.x*id.y+id.x*id.y,
                                              id.x*id.y-id.x*id.y));
    result[id.xy] = rand.value;
}