Name the decoding law being preserved

Greedy decoding chooses the target's highest-probability token at each committed history, using a specified tie rule. A proposed token can be retained if it equals that choice; at the first mismatch, emit the target's chosen token and discard candidate descendants. This can preserve the baseline greedy sequence when the target conditionals, processors, tie behavior, and stopping rules match. A draft's own sampling or ranking method affects how often its tokens match, but does not authorize a different target greedy token. Tree verification must likewise follow the target's actual greedy path rather than whichever branch looks globally attractive.

Floating-point changes can alter a near tie even when two implementations use the same mathematical model. Pin model and tokenizer versions, logits processors, precision, and execution configuration, and report what equality is being tested. A deterministic target distribution can be represented as a point mass at its chosen token, but that is the greedy decoding law, not the unmodified softmax sampling law. Identifying these distributions separately avoids claiming that a sampler is exact merely because it consistently emits a high-probability answer.

For target sampling, use a valid stochastic verifier and correction procedure at each history. The ratio-and-residual scheme preserves the processed p distribution under exact arithmetic and the required conditional randomness. A different correct construction can sample target tokens at candidate histories and keep a matching prefix, but its acceptance efficiency and random-number use differ. The proof belongs to the complete construction, not to a generic label such as assisted decoding. A heuristic rule that accepts every target-top-k proposal or relaxes a probability ratio changes the law unless a separate valid correction argument establishes otherwise.

Even exact stochastic algorithms need not produce the same text for one nominal seed because they consume random numbers in different orders and quantities. Conversely, many matching seeded strings do not prove distributional equality. Use analytic finite examples and statistical checks for the stochastic claim, and deterministic sequence checks for the greedy claim. Pinned Transformers 4.56.2 contains distinct assisted-generation branches depending on sampling mode and available candidate probabilities. Read the applicable path; its support and numerical behavior are implementation properties rather than a theorem covering every possible backend or generation feature.

Include equal and nearly tied logits, zero-probability tokens introduced by constraints, early EOS, short maximum lengths, and mismatches at every proposal position. Apply the same target transformations and stop contract as the baseline. A decoder that emits a bonus token after accepted EOS can change the output law even if all earlier ratios were right. For approximate variants, report the changed acceptance rule and measure its consequences rather than presenting its speed as an exact-mode result. Quality similarity on a benchmark is a useful empirical finding but is not equality of probability distributions.

The probability lab separates a sampled law from a point mass. For each separate view, Reset, set both n=1 and lab p to 0.6 or 1, predict the two bars, then inspect. The first has masses 0.6 and 0.4; the second always selects the designated success outcome. This exact display contains no greedy verifier or model logits and does not test same-seed generation. It illustrates why replacing a nondegenerate target with its most likely token changes the target question, even when that token is the result people expect to see.