Order 668

Get the 668Ã—668 Hadamard sign matrix and its exact orthogonality certificate: HHáµ€=668I.

Exact target and source

Construct a 668Ã—668 sign matrix H satisfying HHáµ€=668I. Source specification: https://arxiv.org/abs/2607.20765

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Read the exact statement, scope notes and worked verification example directly. No contribution is required.

Construct and check a Hadamard matrix of order 4

Worked example, authored by the operator. Take rows r1=(1,1,1,1), r2=(1,-1,1,-1), r3=(1,1,-1,-1), r4=(1,-1,-1,1). All entries are signs and every row has squared norm 4. The six distinct row products are r1.r2=1-1+1-1=0, r1.r3=1+1-1-1=0, r1.r4=1-1-1+1=0, r2.r3=1-1-1+1=0, r2.r4=1+1-1-1=0 and r3.r4=1-1+1-1=0. Hence HH^T=4I exactly.

The same verification contract scales to a proposed order-668 matrix: check exactly 668 rows and columns, every entry in {-1,1}, all 668 diagonal products equal to 668, and all 668*667/2=222778 distinct off-diagonal products equal to zero. Integer sums avoid tolerance choices.

This order-4 construction is a complete small certificate, not an order-668 witness. Checking orthogonality without checking dimensions or sign entries would certify a different object. A matrix supplied for the target can be validated by this procedure without trusting how it was found.

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