Accept each proposed token with its actual probability ratio

Let p and q be normalized nonnegative distributions on one finite vocabulary at a fixed prefix. Draw X from q and an independent uniform random variable U for the test. For every possible proposal with q(X)>0, accept when U is below min(1, p(X)/q(X)), using a consistent endpoint convention. The continuous uniform ideal makes endpoint equality probability zero. If p(X)=0 while q(X)>0, acceptance is zero. If p(X) is at least q(X), acceptance is one. The proposal denominator is the probability actually used to draw X, not a raw logit or another temperature's softmax.

Tokens with q(x)=0 cannot be drawn by the ideal proposal sampler, so no division by zero is needed for them. They can still have positive target probability and must remain reachable through the full correction procedure. The algorithm does not require draft support to contain all target support when the residual is implemented correctly. If an implementation nevertheless presents a zero-q token as a sampled proposal, investigate inconsistent distributions, masking, or numerical behavior. Assigning an arbitrary acceptance value to a 0/0 expression is not part of the proof and can conceal that inconsistency.

The chance that token x is both proposed and accepted is q(x)a(x)=min(p(x), q(x)) when q(x)>0, and is zero for q(x)=0. Summing gives the overall one-position acceptance probability α. It equals one minus the total variation distance between the two finite distributions. This is distributional overlap, not the fraction of tokens whose argmax labels agree, a perplexity guarantee, or the number of semantically acceptable answers. A smaller draft can have useful overlap without predicting the target's most likely token at every history.

The overlap calculation must use the same conditioning state and declared decoding transforms on which the test relies. Sampling q after top-k truncation but dividing by an untruncated q changes the proposed-and-retained mass. Likewise, accepting any token whose target probability exceeds a fixed threshold is generally another law. Such heuristics can be deliberate approximate decoding choices, but they need their own quality and distributional evaluation and must not borrow the exact rule's proof. Keep proposal generation, probability collection, verification, and fallback paths versioned together.

A practical sampler uses finite precision and a pseudorandom generator, while the algebra describes exact probabilities and conditionally independent random draws. Compare log probabilities or otherwise use numerically appropriate arithmetic without silently modifying the mathematical rule. Underflow, inconsistent normalization, or a wrong mask can turn a positive probability into an apparent zero. Pin the target, draft, processors, dtype, and backend when validating behavior. Transformers 4.56.2 implements proposal-ratio checks in its speculative sampling path, but that concrete floating-point implementation still needs tests of its supported models and decoding configuration.

The probability lab shows one conditional acceptance event. Reset, set both the lab's success probability p=0.6 and n=1, predict the acceptance and rejection bars, then inspect. They carry masses 0.6 and 0.4 for a hypothetical already-proposed token whose target/proposal ratio is 0.6. This is not the entire vocabulary calculation and no random draws or language-model verification occur. Use the display to separate a conditional token acceptance probability from the total acceptance α, which additionally averages over which token q proposes.