Discounting does not make rewards deterministic.

Enumerate every finite-horizon return, inspect the probability mass, and separate immediate reward noise from the temporal weight applied by gamma.

READY TO ENUMERATE

--EXPECTED RETURN
--RETURN VARIANCE
--FUTURE VARIANCE SHARE
Var(G) = sum gamma^(2t) Var(R_t) for independent rewards
--VARIANCE AT GAMMA 0
--ENUMERATED OUTCOMES

Two different knobs.

The reward distribution says which outcomes can occur at a transition and with what probability. Gamma says how strongly later rewards enter the return.

The zero-gamma test

Set gamma to zero and future terms disappear. The immediate reward remains random, so its variance survives intact.

Misconception

Gamma is not a probability that the future exists. It is a return-weighting convention tied to time preference and continuing-task modeling.

Counterexample

With +1 or -1 as the immediate reward, gamma zero still leaves two possible returns. Nonzero variance follows directly.

Transfer

Change reward odds and gamma separately. Ask whether the distribution changed, the time weights changed, or both.

Keep the PMF beside the explanation.

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