Certified 178-XOR implementation of the released 4×4×4 rank-47 F₂ factorization

Hello,

I reproduced the released rank-47 factorization for 4×4 matrix multiplication over F₂ and synthesized an exact binary linear circuit for its two encoders and decoder.

### Result

Using the factorization contained in the public AlphaTensor data:

- left encoder `U`: **46 XORs**
- right encoder `V`: **46 XORs**
- decoder `Wᵀ`: **86 XORs**
- bilinear products: **47 ANDs**
- total linear cost: **178 XORs**

The verifier checks coefficient-by-coefficient that the resulting circuit computes the full 4×4×4 matrix-multiplication tensor over F₂.

### Additional structural check

For the fixed 47-dimensional span of bilinear encoders, I enumerated all rank-one points in that span:

- the span contains exactly **48** rank-one points;
- the 47 released points plus one additional global point;
- none of the `C(48,46) = 1128` subsets of size 46 spans the target-slice space.

Therefore, a reduction from 47 to 46 products cannot be obtained merely by replacing products inside the same bilinear encoder span and redesigning the decoder. This does **not** rule out a completely different rank-46 factorization.

### Questions

Could you please confirm:

1. whether the 178-XOR count matches the cost model used in your benchmarking pipeline;
2. whether an equal or lower XOR circuit for this released factorization is already known internally or publicly;
3. whether you would be interested in a pull request adding the circuit and verifier to the repository?

I am not claiming that 178 is globally optimal among all implementations or all rank-47 factorizations. The claim is only that this is an exact, reproducible implementation of the released factorization.

### Reproducibility package

Please attach:

[alphatensor_rank47_add178_submission.zip](https://github.com/user-attachments/files/30410086/alphatensor_rank47_add178_submission.zip)

The package contains:

- the released factorization in JSON form;
- the 178-XOR circuit;
- an exact tensor verifier;
- the fixed-span rank-one enumeration certificate;
- instructions for reproducing both checks.

Thank you for making the AlphaTensor factorizations publicly available.

Start with the attached reproducibility package and its instructions, then inspect the released factorization JSON, 178-XOR circuit, exact tensor verifier, and fixed-span rank-one certificate. Run both checks to understand the reported results; completion would require agreement on the benchmarking cost model and a defined repository change for adding the circuit and verifier.

Written by the indexing model from the issue text.