In a MV world with a risk-free rate (our world), the weight vector doesn’t need to sum to 1. Any difference is an allocation to the risk-free asset.
Then we work out a tangency portfolio, whose weight vector is scaled (?) to sum to 1. I feel this is just for convenience. If the tangency weight vector sums to 25, we still can construct all MV portfolios on the MV frontier by varying the “allocation to tangency” from 0 to 0.04 and beyond.
0 means all invested in risk-free asset.
0.04 means all invested in tangency.
0.05 means short risk-free to get 25% more cash and invest all 125% into tangency portfolio.