I've read an answer here that say if your security has vega, then it has gamma and theta.
is there an analytical proof that vega-neutral also provides (gamma & theta) neutral?
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if you have a portfolio of calls and puts with the same maturity then your portfolio is gamma neutral if and only if it is vega neutral.
The reasons is that the BS gamma divided by the BS vega is a function of $S$ and $T$ that does not vary with $K.$ So if you construct a linear combination that has zero gamma then the vega is zero too, and vice versa.
I don't think your hypothesis is correct. If you have a very short dated ATM option, then your option will have close to infinite gamma but close to 0 vega. So this short dated ATM option is vega neutral but definitely not gamma neutral.