Encode branch histories in masks and positions

Let a committed prefix have length L and a candidate node v have depth d(v), where the first proposed tokens have depth one. Its representation can attend to the shared prefix and its own ancestor path including itself, but not sibling branches. For a decoder-only model, that representation supplies next-token logits after the path through v; the probability of v's token is obtained from its parent history. Preserve that prediction offset when gathering verification scores. Otherwise even a correctly shaped attention tensor can associate a token with the wrong conditional distribution.

A flattened storage order is an implementation choice. Siblings at the same logical depth receive the same sequence-position index for their respective alternative paths, even if their tensor offsets differ. Rotary or other positional features must follow the model's supported position semantics rather than simply count every flattened node as a new chronological token. Store parent and depth mappings and compare small tree outputs against separate ordinary target runs on each root-to-node path. Shape equality alone does not show that the target saw the same histories.

SpecInfer and Medusa describe tree-based verification mechanisms with particular proposal and selection designs. Their tree attention shares prefix work and evaluates branch histories, but the attention mask itself is not a rejection-sampling proof. Greedy verification can follow the target's chosen token from the current history if that child exists, then emit the target choice when no matching child remains. Stochastic verification must account for how candidates were proposed, their probabilities, the order of tests, and any residual updates. Taking the highest-scoring available branch generally defines a search policy rather than sampling the original target law.

More branches increase candidate coverage but also target arithmetic, mask metadata, logits, and temporary KV storage. A tree with many nodes can be expensive even if its maximum depth is small. Duplicate tokens at different histories remain different nodes because their context differs. If an implementation merges nodes or reuses state, check that the represented histories and required probabilities remain equivalent. Approximate acceptance modes can be useful empirical tradeoffs, but they should be named separately from exact stochastic verification. Report the actual mode and its evidence rather than infer exactness from a high branch acceptance count.

Construct a tiny tree with unequal branch lengths and repeated token labels, then compare every node representation's next-token distribution with an independent forward pass on its exact path. Check positions, parent-to-child probability gathering, cache selection, EOS, and branch pruning. Validate numerical tolerances under the actual attention kernel. Dense and specialized tree kernels can have different work and storage behavior, so count actual candidates and profile the supported implementation. A logical mask with many forbidden pairs does not guarantee that a dense kernel skips all corresponding arithmetic.

The function lab models only the quadratic size of a dense candidate-to-candidate mask block. For each separate view, Reset, set both a=1 and dimensionless size ratio x to 1 or 2, predict y=ax², then inspect. Doubling the size ratio changes the value from one to four. This is not a tree, attention kernel, or mask validator, and it omits the shared-prefix block and sparse representations. Use it for the specific N² dense-storage scaling, while real verification correctness still comes from path-level comparisons and the actual sampling procedure.