Post-Boundary Bridge: Must Local Attention Go Global Between Global Layers?

Sparse attention is widely deployed in long-context serving stacks, yet no framework audits how discarding blocks changes the influence of specific content on model output. We first establish that the phenomenon is real and causal: Block Sparse Flash Attention (BSFA) route replay across four architectures changes output decisions in 13 of 16 cells, with zero identity-replay label flips. We then introduce a dense-calibrated counterfactual audit using matched probe cards---Gold (carrying the correct answer label), Poison (carrying a target wrong label), and Benign (filler only)---under six-layout position symmetry, isolating the sparsification-specific effect. Two patterns compete. Signal concentration: the selector preserves Gold and Poison blocks far above filler-matched Benign blocks (G$\approx$P$\gg$B across all model--task pairs). Integration loss: discarding blocks severs cross-block attention---confirmed by an ablation where isolating the probe block collapses its influence from 4.48 logits to zero. Compression ratio governs the balance: a full sweep from mild ($c=0.25$) to aggressive ($c=0.75$) compression across four model--task pairs reveals that three of four cells move toward stronger sparse amplification at higher compression, with two exhibiting sign reversals. Three independent arms---BSFA route replay, controlled block-top-$k$, and KV-cache eviction---converge: sparsification changes content influence in ways aggregate accuracy cannot detect. We provide an open measurement framework deployable on any model exposing block identities.

When a language model processes a hallucinated response, its attention routing tends to fail in one of two shapes: over-concentrating on a narrow set of positions, or spreading so diffusely that relevance is diluted, and the shape of the failure carries diagnostic signal. We study these shapes as a diagnostic characterization, computed from attention matrices under \emph{forced scoring} of benchmark-labeled responses rather than during live generation. A widely used family of spectral methods analyzes the symmetric component of the degree-normalized attention operator, which governs transport \emph{capacity}; we prove that every transpose-invariant spectral diagnostic of this operator is structurally \emph{orientation-blind} (it cannot distinguish an operator from its transpose, and therefore cannot detect information-flow direction), with a converse to the blindness theorem bounding any Lipschitz diagnostic's transpose sensitivity by the asymmetry coefficient $G$. Pairing this with a closed-form bipartite-Cheeger landscape for canonical causal architectures, we show that uniform causal attention satisfies an $n$-independent floor $φ\ge 1/5$, while window attention pierces the floor as $O(w/n)$; failure modes are shape-different, not just value-different. This floor is an idealized-architecture benchmark, not an empirical attractor: the fraction of real attention heads that pierce it is itself an architectural signature. The resulting two-axis diagnostic ($φ$ for capacity, $G$ for direction) yields a falsifiable polarity prediction: bottleneck- and diffuse-dominated benchmarks should exhibit opposite polarity. Under length-controlled evaluation, transport features retain interpretable signal (0.62-0.84 LC-AUROC) across the tested decoder-only, encoder-only, and encoder-decoder models, with polarity reversing as predicted between HaluEval and MedHallu.

Token-level sparse attention mechanisms, exemplified by DeepSeek Sparse Attention (DSA), achieve fine-grained key selection by scoring every historical key for each query through a lightweight indexer, then computing attention only on the selected subset. While the downstream sparse attention itself scales favorably, the indexer must still scan the entire prefix for every query, introducing an per-layer bottleneck that grows prohibitively with context length. We propose HISA (Hierarchical Indexed Sparse Attention), a plug-and-play replacement for the indexer that rewrites the search path from a flat token scan into a two-stage hierarchical procedure: (1) a block-level coarse filtering stage that scores pooled block representations to discard irrelevant regions, followed by (2) a token-level refinement stage that applies the original indexer exclusively within the retained candidate blocks. HISA preserves the identical token-level top-sparse pattern consumed by the downstream Sparse MLA operator and requires no additional training. On kernel-level benchmarks, HISA achieves up to speedup at 64K context. On Needle-in-a-Haystack and LongBench, we directly replace the indexer in DeepSeek-V3.2 and GLM-5 with our HISA indexer, without any finetuning. HISA closely matches the original DSA in quality, while substantially outperforming block-sparse baselines.