i have seen a math derivation that proves that for a rfr caplet (caplet whose underlying is say sofr compounded in arrears) , that the total variance of the can be approximated to equal the rate vol ^2 * time , where time is years from now till one-third of the way through the caplet.

clearly the variance falls once youre in the caplet period as your final underlying value becomes progressively more and more known.  question is , why would it be at 1/3 way through where your volatilty time is on average?  ie , What would be the intuitive reason that its one-third?

Not my area but might be related to a similar T/3 that occurs with Asian option valuation

yes that is what i thought too , there too my same question applies -  is there any intuitive reason why a third

If you study the link I posted, the calculation amounts to finding the variance of the average of a BM: [$]\int_0^1 W_t \, dt[$], taking [$]T=1[$]. Since the variance of [$]W_1[$] is 1, it's intuitively clear that variance of a noisy average is going to be less than one: call it [$]f[$]. 

The link calculation reduces this, using Ito's isometry, to [$]f = \int_0^1 (1 - t)^2 \, dt = \int_0^1 x^2 \, dx = \frac{1}{3}[$]. So, if you are asking for intuition about this step, it amounts to asking why is [$]\frac{d}{dx} x^3 = 3 x^2[$]? Just apply the definition of the derivative from Calculus I. 

Or, perhaps you are asking for intuition as to why the Ito isometry is true. That seems legit, and that intuition can be developed by starting with a weighted sum of independent, zero-mean normal variates, [$]\sum w_i \epsilon_i[$], and calculating the variance of it.

just saw your reply now , many thanks!