When Does Value-Aware KV Eviction Help? A Fixed-Contract Diagnostic for Non-Monotone Cache Compression

# When Does Value-Aware KV Eviction Help? A Fixed-Contract Diagnostic for Non-Monotone Cache Compression
Source: [https://arxiv.org/html/2605.08234](https://arxiv.org/html/2605.08234)
Ruijie Zhang, Haozhe Liang, Da Chang, Li Hu, Fanqi Kong, Huaxiao Yin, and Yu Li jeffy353866@gmail\.com

###### Abstract

Long\-context LLM inference is bottlenecked by the memory and bandwidth cost of reading large KV caches during decoding\. KV compression reduces this cost by keeping only part of the cache, but task accuracy alone does not identify why a selector succeeds or fails\. A selector can fail at three steps: it may miss the evidence future decoding needs, give high scores to tokens that do not affect the output, or break related evidence when fitting scores into a small cache\. We introduce a fixed\-contract diagnostic that holds the selector’s setup fixed and changes one decision slot at a time\. For value ranking, the probe combines a block’s attention mass with the estimated output change from removing it\. On LongBench across three models and two budgets, the probe is positive on 72\.6% of positive\-margin cells and 32\.4% of nonpositive\-margin cells\. NeedleBench M\-RT at 32k and a RULER 8k check probe support closure under branched retrieval, and a 264\-cell sign evaluation separates support recovery and output\-value ranking from leverage effects near the boundary\. The resulting order is to recover decode\-side evidence, rank its output value, and preserve coupled evidence during projection\.

## 1Introduction

Long\-context LLM inference moves the bottleneck from prefill to decoding\. After prefill, every generated token must read an expanding key\-value cache, so latency and memory traffic both rise with context length\[[52](https://arxiv.org/html/2605.08234#bib.bib52),[26](https://arxiv.org/html/2605.08234#bib.bib26),[46](https://arxiv.org/html/2605.08234#bib.bib46)\]\. KV compression methods for pretrained models reduce this cost by deciding which cache states to retain, read, or store at lower precision\[[26](https://arxiv.org/html/2605.08234#bib.bib26),[17](https://arxiv.org/html/2605.08234#bib.bib17),[37](https://arxiv.org/html/2605.08234#bib.bib37),[27](https://arxiv.org/html/2605.08234#bib.bib27)\], and adjacent decode\-aligned and sparse\-readout systems modify the access estimator or read substrate itself\[[45](https://arxiv.org/html/2605.08234#bib.bib45),[50](https://arxiv.org/html/2605.08234#bib.bib50),[9](https://arxiv.org/html/2605.08234#bib.bib9),[12](https://arxiv.org/html/2605.08234#bib.bib12)\]\. Yet a fixed\-budget task score mixes several failure modes \([Figure˜1](https://arxiv.org/html/2605.08234#S1.F1)\)\. A selector may miss the evidence future decoding consumes, give high scores to tokens that do not affect the output, or break related evidence when fitting scores into a small cache\.

![Refer to caption](https://arxiv.org/html/2605.08234v1/x1.png)Figure 1:Compression is not monotone\. A compressed cache can underperform, match, or exceedFullKVdepending on which selector stage is the bottleneck\. The diagnostic fixes the selector contract and perturbs one ranking slot to localize the failure\.Under tight budgets, performance differences between selectors concentrate at a small number of*boundary units*near the budget threshold, while the bulk of the retained set is largely shared\. Boundary\-swap exposes this concentration\. It keeps the model, budget, and all other cache positions fixed, exchanges only the decisions near the threshold, and recovers most of the gap between selectors\. At low budget, boundary\-rank correctness is the binding constraint\.

These boundary errors map to the three selector stages\. In access estimation,*query\-domain mismatch*appears when historical prefill statistics deviate from the decode queries that actually consume the cache, so early positions receive inflated scores simply from being seen by more subsequent queries\. In value\-consequence ranking,*output\-unaware scoring*treats attention mass as evidence of access without testing whether deletion would change the output, so task\-relevant evidence ends up mixed with templates and delimiters\. In score\-to\-cache projection,*projection mismatch*appears when tokenwise ranking under multi\-evidence decoding scatters budget across isolated high\-score tokens, so evidence coverage stays incomplete and continuous support regions fragment\.

The diagnostic fixes the prefill\-attention tensor, query domain, observation window, budget, allocation rule, and score\-to\-cache projection before outcome measurement\. A stage\-local test changes one slot in that contract\. The main test keeps the SnapKV\[[26](https://arxiv.org/html/2605.08234#bib.bib26)\]observation\-window contract fixed and substitutes a value\-consequence ranking scalar into the same ranking slot to test whether boundary units contain recoverable value\-consequence misranking\. Cross\-contract rows on PyramidKV\[[2](https://arxiv.org/html/2605.08234#bib.bib2)\], CAKE\[[37](https://arxiv.org/html/2605.08234#bib.bib37)\], Ada\-KV\[[17](https://arxiv.org/html/2605.08234#bib.bib17)\], and H2O\[[52](https://arxiv.org/html/2605.08234#bib.bib52)\]are transfer and sensitivity checks that show how the sign pattern changes when access estimation, allocation, or projection also moves\.

Under the fixed selector contract, each experiment supports only the stage whose slot is changed\. Boundary swaps first localize tight\-budget errors to the score margin and separate query\-domain mismatch, output\-unaware scoring, and projection mismatch\. With the contract fixed, LongBench\[[1](https://arxiv.org/html/2605.08234#bib.bib1)\]controls test the predicted sign split before reading outcomes\. The value\-consequence intervention helps mainly in cells with positive reference margin and low support coupling, while no\-value controls collapse this separation\. Token\-fill makes unused budget an unlikely standalone Stage III explanation under the fixed block control, and cross\-contract baselines test transfer without being used to rank selectors\. NeedleBench\[[25](https://arxiv.org/html/2605.08234#bib.bib25)\]and the RULER\[[24](https://arxiv.org/html/2605.08234#bib.bib24)\]8k check probe support closure under branched retrieval targets\. A wider 264\-cell sign evaluation shows high agreement between no\-leverage and final signs, separating support recovery and output\-value ranking from leverage as a conditional amplifier\.

## 2Staged Diagnostic Framework

Setup\.Under tight budgets, a prefill\-observable score can fail by estimating the wrong future access support, mis\-ranking useful positions, or losing support during budget projection\. The pipeline is formalized under a fixed retained\-token budget, with notation collected in[Appendix˜A](https://arxiv.org/html/2605.08234#A1)\.

Let𝐀l,h\\mathbf\{A\}\_\{l,h\}denote the layer\-head prefill attention matrix, with scalar entry𝐀l,h​\[u,i\]\\mathbf\{A\}\_\{l,h\}\[u,i\]from query positionuuto key positionii, whereTTis the prompt length andb∈\(0,1\)b\\in\(0,1\)the budget ratio\. An eviction rule first constructs scalar scores from the observable prefill\-attention tensor𝐀\\mathbf\{A\}and then projects them to a fixed memory budget,

si=ψscore​\(𝐀,i\),𝒦=TopK⁡\(s,k\),k=⌊b​T⌋≤T\.s\_\{i\}=\\psi\_\{\\mathrm\{score\}\}\(\\mathbf\{A\},i\),\\qquad\\mathcal\{K\}=\\operatorname\{TopK\}\(s,k\),\\quad k=\\lfloor bT\\rfloor\\leq T\.\(1\)Writeau​\(i\):=Aggl,h⁡𝐀l,h​\[u,i\]a\_\{u\}\(i\):=\\operatorname\{Agg\}\_\{l,h\}\\,\\mathbf\{A\}\_\{l,h\}\[u,i\]when head and layer aggregation are not central, and keep tensor form only where cross\-layer structure matters\.TopK\\operatorname\{TopK\}uses a score\-independent deterministic tie break\.

###### Assumption 1\(Fixed selector contract\)\.

Diagnostic comparisons fix the prefill\-attention tensor, query or proxy domain, causal mask, observation window, budgetk=⌊b​T⌋k=\\lfloor bT\\rfloor, and projection rule before outcome measurement\. Query laws share a zero\-extended finite domain, and blockwise variants state the block size and boundary policy\.

All formal statements in this section use[˜1](https://arxiv.org/html/2605.08234#Thmassumption1)unless a different contract is stated\.

For a fixed selector contract and prompt positionii, writeuiu\_\{i\}for the latent decode\-side utility of retainingii,eie\_\{i\}for the access\-support exposure factor estimated byuaccu\_\{\\mathrm\{acc\}\},gig\_\{i\}for token\-level layer, head, or budget scaling, andρi\\rho\_\{i\}for structured non\-utility mass such as delimiters, records, or format markers\. Any scalar prefill score can be decomposed as

si=ei​gi​ui\+ρi\+ξi,s\_\{i\}=e\_\{i\}\\,g\_\{i\}\\,u\_\{i\}\+\\rho\_\{i\}\+\\xi\_\{i\},\(2\)with residualξi:=si−ei​gi​ui−ρi\\xi\_\{i\}:=s\_\{i\}\-e\_\{i\}g\_\{i\}u\_\{i\}\-\\rho\_\{i\}\. This representation is definitional\. Its role is to mark which factor a stage\-local intervention changes\. The support\-coupling indexϕ​\(x\)\\phi\(x\)flagsρi\\rho\_\{i\}\-type structure, whileηproj\\eta\_\{\\mathrm\{proj\}\}is defined only after projection\. The fixed\-predictor split in[Section˜4](https://arxiv.org/html/2605.08234#S4)tests Stage II under the SnapKV\[[26](https://arxiv.org/html/2605.08234#bib.bib26)\]contract by holding access estimation and projection fixed\.

Why a staged diagnostic is needed\.Under low budgets, most prompt positions are either safely retained or safely evicted, and the outcome is decided by positions near the keep\-or\-evict boundary\. Boundary swaps recover much of the gap between collapsed cumulative scorers and decode\-aligned scorers, whereas random swaps of the same size do not\. This locality rules out an explanation based only on global score calibration\. The active error is tied to how a selector observes future access, assigns value to accessed states, or projects a score into the finite cache\.

###### Lemma 1\(Boundary\-margin condition\)\.

Fix a selector contract with deterministic tie breaking, a base scoress, and a perturbed scores′=s\+δs^\{\\prime\}=s\+\\delta\. Letj∈𝒦sj\\in\\mathcal\{K\}\_\{s\}be a retained token andi∉𝒦si\\notin\\mathcal\{K\}\_\{s\}an evicted token under the base kept set𝒦s=TopK⁡\(s,k\)\\mathcal\{K\}\_\{s\}=\\operatorname\{TopK\}\(s,k\)\. The perturbation strictly ranksiiabovejjwhenever

δi−δj\>sj−si\.\\delta\_\{i\}\-\\delta\_\{j\}\>s\_\{j\}\-s\_\{i\}\.\(3\)At equality, the fixed tie break decides\. If the downstream utility is locally additive for this one\-for\-one boundary swap, then replacingjjbyiichanges latent kept\-set utility byui−uju\_\{i\}\-u\_\{j\}\. Away from the equality case, a score perturbation is beneficial at this boundary when it crosses the base margin and its signed score change agrees with the latent utility gap\.

The proof appears in[Section˜B\.1](https://arxiv.org/html/2605.08234#A2.SS1)\. This local condition motivates the sign\-resolved groups used in the experiments\.

The diagnostic targets are decode\-side access supportuacc​\(i\)u\_\{\\mathrm\{acc\}\}\(i\), value consequenceuval​\(i\)u\_\{\\mathrm\{val\}\}\(i\)among accessed positions, and projection residualηproj\\eta\_\{\\mathrm\{proj\}\}after mapping a score to a fixed token, head, layer, or block budget\. They are instantiated by[Equations˜4](https://arxiv.org/html/2605.08234#S2.E4),[6](https://arxiv.org/html/2605.08234#S2.E6)and[7](https://arxiv.org/html/2605.08234#S2.E7)\.[Equations˜2](https://arxiv.org/html/2605.08234#S2.E2),[2](https://arxiv.org/html/2605.08234#Thmtheorem2)and[3](https://arxiv.org/html/2605.08234#Thmtheorem3)give the corresponding algebraic checks\.

u^acc\(i;m\):=∑u∈𝒬mq^m\(u\)d^m\(i,u\)f^m\(u,i\),uacc\(i\):=∑u∈𝒬mq⋆\(u\)d⋆\(i,u\)f⋆\(u,i\),\\hat\{u\}\_\{\\mathrm\{acc\}\}\(i;m\):=\\sum\_\{u\\in\\mathcal\{Q\}\_\{m\}\}\\hat\{q\}\_\{m\}\(u\)\\,\\hat\{d\}\_\{m\}\(i,u\)\\,\\hat\{f\}\_\{m\}\(u,i\),\\qquad u\_\{\\mathrm\{acc\}\}\(i\):=\\sum\_\{u\\in\\mathcal\{Q\}\_\{m\}\}q^\{\\star\}\(u\)\\,d^\{\\star\}\(i,u\)\\,f^\{\\star\}\(u,i\),\(4\)Here𝒬m\\mathcal\{Q\}\_\{m\}is the zero\-extended causal query domain\. Estimatormmuses query lawq^m\\hat\{q\}\_\{m\}, exposure correctiond^m\\hat\{d\}\_\{m\}, and pooling kernelf^m​\(u,i\):=∑l,hβ^l,h,m​\(u\)​𝐀l,h​\[u,i\]\\hat\{f\}\_\{m\}\(u,i\):=\\sum\_\{l,h\}\\hat\{\\beta\}\_\{l,h,m\}\(u\)\\,\\mathbf\{A\}\_\{l,h\}\[u,i\]\. Starred terms denote the references\.

The Stage I access error then decomposes exactly into query\-law, exposure, and aggregation terms\.

###### Proposition 2\(Ordered\-substitution identity\)\.

Under[˜1](https://arxiv.org/html/2605.08234#Thmassumption1), with estimators defined on the common query domain in[Equation˜4](https://arxiv.org/html/2605.08234#S2.E4), the access\-support estimation error admits the exact decomposition

u^acc​\(i;m\)−uacc​\(i\)=δphase​\(i;m\)\+δexp​\(i;m\)\+ξacc​\(i;m\),\\hat\{u\}\_\{\\mathrm\{acc\}\}\(i;m\)\-u\_\{\\mathrm\{acc\}\}\(i\)=\\delta\_\{\\mathrm\{phase\}\}\(i;m\)\+\\delta\_\{\\mathrm\{exp\}\}\(i;m\)\+\\xi\_\{\\mathrm\{acc\}\}\(i;m\),\(5\)where

δphase​\(i;m\)\\displaystyle\\delta\_\{\\mathrm\{phase\}\}\(i;m\):=∑u∈𝒬m\(q^m​\(u\)−q⋆​\(u\)\)​d⋆​\(i,u\)​f⋆​\(u,i\),\\displaystyle:=\\textstyle\\sum\_\{u\\in\\mathcal\{Q\}\_\{m\}\}\\bigl\(\\hat\{q\}\_\{m\}\(u\)\-q^\{\\star\}\(u\)\\bigr\)\\,d^\{\\star\}\(i,u\)\\,f^\{\\star\}\(u,i\),δexp​\(i;m\)\\displaystyle\\delta\_\{\\mathrm\{exp\}\}\(i;m\):=∑u∈𝒬mq^m​\(u\)​\(d^m​\(i,u\)−d⋆​\(i,u\)\)​f⋆​\(u,i\),\\displaystyle:=\\textstyle\\sum\_\{u\\in\\mathcal\{Q\}\_\{m\}\}\\hat\{q\}\_\{m\}\(u\)\\bigl\(\\hat\{d\}\_\{m\}\(i,u\)\-d^\{\\star\}\(i,u\)\\bigr\)\\,f^\{\\star\}\(u,i\),ξacc​\(i;m\)\\displaystyle\\xi\_\{\\mathrm\{acc\}\}\(i;m\):=∑u∈𝒬mq^m​\(u\)​d^m​\(i,u\)​\(f^m​\(u,i\)−f⋆​\(u,i\)\)\.\\displaystyle:=\\textstyle\\sum\_\{u\\in\\mathcal\{Q\}\_\{m\}\}\\hat\{q\}\_\{m\}\(u\)\\,\\hat\{d\}\_\{m\}\(i,u\)\\bigl\(\\hat\{f\}\_\{m\}\(u,i\)\-f^\{\\star\}\(u,i\)\\bigr\)\.

*Proof in[Section˜B\.2](https://arxiv.org/html/2605.08234#A2.SS2)\.*The three terms isolate query\-law mismatch, exposure correction, and the layer and head pooling residual under the fixed\-order substitution above\.

Stage II reweights accessed positions by their conditional output consequenceuval​\(i\)u\_\{\\mathrm\{val\}\}\(i\)\. Define

Δ​\(i\)\\displaystyle\\Delta\(i\):=uacc​\(i\)​uval​\(i\),\\displaystyle=u\_\{\\mathrm\{acc\}\}\(i\)\\,u\_\{\\mathrm\{val\}\}\(i\),Δ^​\(i;m\)\\displaystyle\\hat\{\\Delta\}\(i;m\):=u^acc​\(i;m\)​u^val​\(i;m\),\\displaystyle=\\hat\{u\}\_\{\\mathrm\{acc\}\}\(i;m\)\\,\\hat\{u\}\_\{\\mathrm\{val\}\}\(i;m\),ξval​\(i;m\)\\displaystyle\\xi\_\{\\mathrm\{val\}\}\(i;m\):=u^val​\(i;m\)−uval​\(i\)\.\\displaystyle=\\hat\{u\}\_\{\\mathrm\{val\}\}\(i;m\)\-u\_\{\\mathrm\{val\}\}\(i\)\.Expanding the product gives

Δ^​\(i;m\)−Δ​\(i\)\\displaystyle\\hat\{\\Delta\}\(i;m\)\-\\Delta\(i\)=uval​\(i\)​\[δphase​\(i;m\)\+δexp​\(i;m\)\+ξacc​\(i;m\)\]\\displaystyle=u\_\{\\mathrm\{val\}\}\(i\)\\,\\bigl\[\\delta\_\{\\mathrm\{phase\}\}\(i;m\)\+\\delta\_\{\\mathrm\{exp\}\}\(i;m\)\+\\xi\_\{\\mathrm\{acc\}\}\(i;m\)\\bigr\]\+uacc​\(i\)​ξval​\(i;m\)\+R2​\(i;m\),\\displaystyle\\quad\+u\_\{\\mathrm\{acc\}\}\(i\)\\,\\xi\_\{\\mathrm\{val\}\}\(i;m\)\+R\_\{2\}\(i;m\),\(6a\)R2​\(i;m\)\\displaystyle R\_\{2\}\(i;m\):=\[δphase\(i;m\)\+δexp\(i;m\)\\displaystyle:=\\bigl\[\\delta\_\{\\mathrm\{phase\}\}\(i;m\)\+\\delta\_\{\\mathrm\{exp\}\}\(i;m\)\+ξacc\(i;m\)\]ξval\(i;m\)\.\\displaystyle\\quad\+\\xi\_\{\\mathrm\{acc\}\}\(i;m\)\\bigr\]\\xi\_\{\\mathrm\{val\}\}\(i;m\)\.\(6b\)The first\-order approximation keeps the linear terms and drops only the residual product\. The SnapKV\[[26](https://arxiv.org/html/2605.08234#bib.bib26)\]replacement fixes access estimation and projection, perturbing only the value\-consequence scalar and isolating theuacc​ξvalu\_\{\\mathrm\{acc\}\}\\,\\xi\_\{\\mathrm\{val\}\}channel\.

Stage III projectsΔ^\\hat\{\\Delta\}onto a fixed budget\. For a kept set𝒦\\mathcal\{K\}and an access lawpTp\_\{T\}induced by the scoring queries, withpT​\(𝒦\)\>0p\_\{T\}\(\\mathcal\{K\}\)\>0, the projection term is directly visible as retained access mass:

ηproj​\(𝒦;pT\):=TV⁡\(pT,p¯𝒦\)=1−pT​\(𝒦\),\\eta\_\{\\mathrm\{proj\}\}\(\\mathcal\{K\};p\_\{T\}\):=\\operatorname\{TV\}\\\!\\left\(p\_\{T\},\\bar\{p\}\_\{\\mathcal\{K\}\}\\right\)=1\-p\_\{T\}\(\\mathcal\{K\}\),\(7\)wherep¯𝒦\\bar\{p\}\_\{\\mathcal\{K\}\}ispTp\_\{T\}restricted to𝒦\\mathcal\{K\}and renormalized, with residual11ifpT​\(𝒦\)=0p\_\{T\}\(\\mathcal\{K\}\)=0\. This grows when isolated high\-score tokens miss support closure for code, records, or multi\-target retrieval\. It differs from the block\-lattice deficitεlat\\varepsilon\_\{\\mathrm\{lat\}\}in[Section˜3](https://arxiv.org/html/2605.08234#S3), which counts unused nominal budget\.

###### Proposition 3\(Tail\-KKTV bound\)\.

Letp⋆p^\{\\star\}denote the access law of the first decode step,pTp\_\{T\}the access law of the last prefill query,p¯tail,K\\bar\{p\}\_\{\\mathrm\{tail\},K\}the law of a tail\-KKquery mixture, andp¯all\\bar\{p\}\_\{\\mathrm\{all\}\}the law of the full prefill mixture\. Then

TV⁡\(p⋆,p¯tail,K\)\\displaystyle\\operatorname\{TV\}\(p^\{\\star\},\\bar\{p\}\_\{\\mathrm\{tail\},K\}\)≤TV⁡\(p⋆,pT\)\+TV⁡\(pT,p¯tail,K\),\\displaystyle\\leq\\operatorname\{TV\}\(p^\{\\star\},p\_\{T\}\)\+\\operatorname\{TV\}\(p\_\{T\},\\bar\{p\}\_\{\\mathrm\{tail\},K\}\),\(8\)TV⁡\(p⋆,p¯all\)\\displaystyle\\operatorname\{TV\}\(p^\{\\star\},\\bar\{p\}\_\{\\mathrm\{all\}\}\)≤TV⁡\(p⋆,pT\)\+TV⁡\(pT,p¯all\)\.\\displaystyle\\leq\\operatorname\{TV\}\(p^\{\\star\},p\_\{T\}\)\+\\operatorname\{TV\}\(p\_\{T\},\\bar\{p\}\_\{\\mathrm\{all\}\}\)\.\(9\)

*Proof in[Section˜B\.3](https://arxiv.org/html/2605.08234#A2.SS3)\.*The first term is shared across estimators, while the second is measurable from prefill attention\. The bound does not prove decode alignment; it defines the measurable proxy gap used to compare tail\-KKand detector\-based query proxies againstpTp\_\{T\}\.

Two coupled access\-support distortions\.Exposure Bias is a count distortion\. Earlier positions accumulate larger cumulative attention because more causal prefill queries can see them\[[4](https://arxiv.org/html/2605.08234#bib.bib4),[48](https://arxiv.org/html/2605.08234#bib.bib48)\]\. Count\-debiasing lifts cumulative scoring, but can expose a reverse\-depth failure on needle retrieval, where the kept set drifts toward merely recent positions\. Appendix[F](https://arxiv.org/html/2605.08234#A6)gives the full analysis\.

Phase Dilution is a query\-law distortion\. Eviction serves future decode queries, whereas pooled prefill scorers average over formatting\-heavy prompt queries\. Tail\-window and lookahead\-style methods\[[45](https://arxiv.org/html/2605.08234#bib.bib45),[50](https://arxiv.org/html/2605.08234#bib.bib50),[51](https://arxiv.org/html/2605.08234#bib.bib51)\]reduce this mismatch\. Removing prefix over\-counting exposes the tailward mismatch that raw cumulative scoring had partially masked, and boundary swaps confirm it by recovering most of the gap between H2O\[[52](https://arxiv.org/html/2605.08234#bib.bib52)\]and SnapKV\[[26](https://arxiv.org/html/2605.08234#bib.bib26)\]on disagreement\-boundary units\.

Architecture\-conditioned residual\.After support\-estimation repair,ξacc​\(i;m\)\\xi\_\{\\mathrm\{acc\}\}\(i;m\)becomes a conservative remainder\. On LongBench\[[1](https://arxiv.org/html/2605.08234#bib.bib1)\]the decode\-aligned family collapses into a narrow band, so this residual is small relative to the gap between cumulative and decode\-aligned support\. Head\-aggregation dilution is one instance, with details in Appendix[G](https://arxiv.org/html/2605.08234#A7)\.

## 3Fixed\-Contract Diagnostic

A scalar eviction score entangles future\-query approximation, value consequence, and score projection\. Stage\-local claims require fixing the selector contract before outcome measurement\.

Selector contract\.A contract specifies the prefill attention tensor, query or proxy domain, causal mask, observation window, budget, allocation rule, and final projection\. Two scores are comparable as a Stage II intervention only if they share this contract and differ only in the value\-consequence ranking slot\. Comparisons that change the query law, layer or head budget, or projection rule are transfer and sensitivity checks, not clean Stage II evidence\.

Stage I access support\.The clean experiment uses SnapKV’s\[[26](https://arxiv.org/html/2605.08234#bib.bib26)\]observation\-window access proxy, whose explicit query law, per\-head allocation, deterministic top\-kkprojection, and scalar ranking slot make the Stage II replacement identifiable\. H2O\-style cumulative scoring\[[52](https://arxiv.org/html/2605.08234#bib.bib52)\]and count\-debiased variants change the access law and diagnose Stage I instead\.

Blocks are the selection and projection units for original KV states\. Block centroids are only score features for deletion consequence\.

Let the prompt length beTTand the token budget bek=⌊b​T⌋k=\\lfloor bT\\rfloor\. Cache positions are partitioned into contiguous blocks

ℬ=\{c1,…,cNb\},\|cj\|=p​for​j<Nb,\|cNb\|≤p,kb=⌊kp⌋\.\\mathcal\{B\}=\\\{c\_\{1\},\\dots,c\_\{N\_\{b\}\}\\\},\\qquad\|c\_\{j\}\|=p\\ \\text\{for\}\\ j<N\_\{b\},\\qquad\|c\_\{N\_\{b\}\}\|\\leq p,\\qquad k\_\{b\}=\\left\\lfloor\\frac\{k\}\{p\}\\right\\rfloor\.The construction scores a small selected layer setℒ⋆\\mathcal\{L\}\_\{\\star\}with layer weightsβl\\beta\_\{l\}, typically drawn from late or plateau layers, and uses a proxy bank

𝒰=𝒰tail∪𝒰anchor,𝒰=⨆m=1M𝒰m,\\mathcal\{U\}=\\mathcal\{U\}^\{\\mathrm\{tail\}\}\\cup\\mathcal\{U\}^\{\\mathrm\{anchor\}\},\\qquad\\mathcal\{U\}=\\bigsqcup\_\{m=1\}^\{M\}\\mathcal\{U\}\_\{m\},where𝒰tail\\mathcal\{U\}^\{\\mathrm\{tail\}\}contains prompt\-tail samples and𝒰anchor\\mathcal\{U\}^\{\\mathrm\{anchor\}\}contains instruction, question, or field\-boundary anchors\. When no explicit multi\-target structure exists,M=1M=1\. Recency weighting is applied only inside the tail stratum,

r~u=\{exp⁡\(−T−uτq\),u∈𝒰tail,1,u∈𝒰anchor,ru=r~u∑v∈𝒰r~v\.\\tilde\{r\}\_\{u\}=\\begin\{cases\}\\exp\\\!\\left\(\-\\dfrac\{T\-u\}\{\\tau\_\{q\}\}\\right\),&u\\in\\mathcal\{U\}^\{\\mathrm\{tail\}\},\\\\\[6\.0pt\] 1,&u\\in\\mathcal\{U\}^\{\\mathrm\{anchor\}\},\\end\{cases\}\\qquad r\_\{u\}=\\dfrac\{\\tilde\{r\}\_\{u\}\}\{\\sum\_\{v\\in\\mathcal\{U\}\}\\tilde\{r\}\_\{v\}\}\.\(10\)These are the scoring objects for the value\-channel replacement and projection controls, and the test keeps the host observation window and projection unchanged\.

Stage II value\-consequence replacement\.Letκ​\(h\)\\kappa\(h\)map each query head to its KV head\. For selected layerl∈ℒ⋆l\\in\\mathcal\{L\}\_\{\\star\}, proxyu∈𝒰u\\in\\mathcal\{U\}, and query headhh, the full attention output is

𝐨l,h,u=∑i=1T𝐀l,h​\[u,i\]​𝐕l,κ​\(h\),i\.\\mathbf\{o\}\_\{l,h,u\}=\\sum\_\{i=1\}^\{T\}\\mathbf\{A\}\_\{l,h\}\[u,i\]\\,\\mathbf\{V\}\_\{l,\\kappa\(h\),i\}\.For a blockc∈ℬc\\in\\mathcal\{B\}, define the block attention mass and attention\-weighted value centroid as

al,h,u​\(c\)=∑i∈c𝐀l,h​\[u,i\],𝝁l,h,u​\(c\)=∑i∈c𝐀l,h​\[u,i\]​𝐕l,κ​\(h\),imax⁡\{al,h,u​\(c\),εμ\}\.a\_\{l,h,u\}\(c\)=\\sum\_\{i\\in c\}\\mathbf\{A\}\_\{l,h\}\[u,i\],\\qquad\\boldsymbol\{\\mu\}\_\{l,h,u\}\(c\)=\\frac\{\\sum\_\{i\\in c\}\\mathbf\{A\}\_\{l,h\}\[u,i\]\\,\\mathbf\{V\}\_\{l,\\kappa\(h\),i\}\}\{\\max\\\{a\_\{l,h,u\}\(c\),\\varepsilon\_\{\\mu\}\\\}\}\.The Stage II replacement uses deletion consequence as the ranking signal for already supported regions\. Withγε,l,h,u​\(c\):=max⁡\{1−al,h,u​\(c\),εa\}\\gamma\_\{\\varepsilon,l,h,u\}\(c\):=\\max\\\{1\-a\_\{l,h,u\}\(c\),\\varepsilon\_\{a\}\\\}, the surrogate is

dl,h,u​\(c\)=\(al,h,u​\(c\)γε,l,h,u​\(c\)\)2⏟support strength​‖𝝁l,h,u​\(c\)−𝐨l,h,u‖22⏟conditional output consequence\.d\_\{l,h,u\}\(c\)=\\underbrace\{\\left\(\\frac\{a\_\{l,h,u\}\(c\)\}\{\\gamma\_\{\\varepsilon,l,h,u\}\(c\)\}\\right\)^\{2\}\}\_\{\\text\{support strength\}\}\\underbrace\{\\left\\\|\\boldsymbol\{\\mu\}\_\{l,h,u\}\(c\)\-\\mathbf\{o\}\_\{l,h,u\}\\right\\\|\_\{2\}^\{2\}\}\_\{\\text\{conditional output consequence\}\}\.The support factor matters only after the access proxy is placed on the relevant region, and clipping avoids the singular limit asaaapproaches11\. Aggregating over proxy groups gives

Dm​\(c\)=∑u∈𝒰mru​∑l∈ℒ⋆βl​∑hdl,h,u​\(c\)\.D\_\{m\}\(c\)=\\sum\_\{u\\in\\mathcal\{U\}\_\{m\}\}r\_\{u\}\\sum\_\{l\\in\\mathcal\{L\}\_\{\\star\}\}\\beta\_\{l\}\\sum\_\{h\}d\_\{l,h,u\}\(c\)\.The main\-text score then pools these group\-wise deletion costs,

S​\(c\)=∑m=1Mwm​Dm​\(c\),M=1,wm=1​by default\.S\(c\)=\\sum\_\{m=1\}^\{M\}w\_\{m\}D\_\{m\}\(c\),\\qquad M=1,\\;w\_\{m\}=1\\text\{ by default\.\}\(11\)The controlled replacement uses[Equation˜11](https://arxiv.org/html/2605.08234#S3.E11)in the same ranking slot as the host selector, while the observation window, budget, allocation, and projection are unchanged\. Robust aggregation and multi\-target reserves are appendix extensions\.

Stage III projection controls\.Stage III uses a block\-constant posterior approximationπ^i=π^b​\(i\)\\hat\{\\pi\}\_\{i\}=\\hat\{\\pi\}\_\{b\(i\)\}, so the strict default form keeps the topkbk\_\{b\}blocks under the block budget:

𝒵⋆=TopK⁡\(\{S​\(c\)\}c∈ℬ,kb\),𝒦=⋃c∈𝒵⋆c\.\\mathcal\{Z\}^\{\\star\}=\\operatorname\{TopK\}\\bigl\(\\\{S\(c\)\\\}\_\{c\\in\\mathcal\{B\}\},\\,k\_\{b\}\\bigr\),\\qquad\\mathcal\{K\}=\\bigcup\_\{c\\in\\mathcal\{Z\}^\{\\star\}\}c\.\(12\)[Equation˜12](https://arxiv.org/html/2605.08234#S3.E12)is the default Stage III projection\. It replaces the token\-level keep posteriorπi:=Pr⁡\(zi=1∣𝐀,k\)\\pi\_\{i\}:=\\Pr\(z\_\{i\}\{=\}1\\mid\\mathbf\{A\},k\)by a block\-constant proxy and projects underkb=⌊k/p⌋k\_\{b\}=\\lfloor k/p\\rfloor\. Strict block TopK leaves a lattice residualεlat​\(k,p\):=k−p​kb\\varepsilon\_\{\\mathrm\{lat\}\}\(k,p\):=k\-pk\_\{b\}, distinct from the access\-mass residualηproj​\(𝒦;pT\)\\eta\_\{\\mathrm\{proj\}\}\(\\mathcal\{K\};p\_\{T\}\)in[Equation˜7](https://arxiv.org/html/2605.08234#S2.E7)\. Token\-fill removes the lattice residual without changing block order\.

Stage roles\.The clean Stage II test fixes access support and projection and changes only the value\-consequence ranking slot\. Block TopK, token\-fill, grouped proxy reserves, and multi\-target variants are projection controls\. Cost and lattice variants are in[Appendix˜D](https://arxiv.org/html/2605.08234#A4)\.

## 4Experiments

The experiments test the predicted sign of the value\-consequence channel\. The host selector, model, task, budget, sample set, observation window, allocation, and projection are fixed, and only the ranking scalar changes\. Cross\-contract baselines mark where attribution no longer holds\. Appendix[J\.1](https://arxiv.org/html/2605.08234#A10.SS1)gives the full stage\-to\-evidence map and contract coverage\.

Setup\.We evaluate Qwen3\-8B\[[39](https://arxiv.org/html/2605.08234#bib.bib39)\], Llama\-3\.1\-8B\-Instruct\[[43](https://arxiv.org/html/2605.08234#bib.bib43)\], and Mistral\-7B\-Instruct\-v0\.3\[[44](https://arxiv.org/html/2605.08234#bib.bib44)\]on English LongBench\[[1](https://arxiv.org/html/2605.08234#bib.bib1)\]atb∈\{0\.05,0\.10\}b\\in\\\{0\.05,0\.10\\\}, using 100 samples per task\-model\-budget cell and an 8192\-token cap\. Runs used eight NVIDIA A100\-80GB GPUs\. The 16 tasks, three models, and two budgets give a 96\-cell grid\. In every cell,FullKV, SnapKV, the Module II replacement \(MII\), and NoLev use the same model, task, budget, precision, context cap, and sample set\. NoLev is the no\-leverage MII variant: it keeps MII’s value\-consequence ranking scalar and the fixed selector contract, but replaces the leverage multiplier\(1−a\)−2\(1\-a\)^\{\-2\}with one\.

We implement the observation\-window contract with SnapKV\[[26](https://arxiv.org/html/2605.08234#bib.bib26)\]because its prefill query law, per\-head allocation, and deterministic projection are explicit and leave one scalar ranking slot exposed\.

Stage II evaluation\.Under this contract, the intervention changes only the value\-consequence ranking scalar\. MII replaces the identity score with the value\-consequence score, and NoLev is the same replacement without the leverage multiplier\. With access and projection fixed,[Lemma˜1](https://arxiv.org/html/2605.08234#Thmtheorem1)predicts lift only where the boundary still contains recoverable value\-ranking error\.

At block level,[Figure˜2](https://arxiv.org/html/2605.08234#S4.F2)shows how the Stage II replacement reallocates retained mass without changing the observation\-window frame\. The difference map identifies the prefill blocks whose retained fraction diverges from SnapKV under the same selector contract\.

Figure 2:Stage II value\-channel re\-ranking shifts retained mass across prefill blocks \(Qwen3\-8B, HotpotQA,b=0\.05b=0\.05\)\. Color denotes retained\-token fraction in \(a,b\) and signed retained\-mass change in \(c\); lower traces average the heatmap over layers\.
The grid\-level test groups cells by support couplingϕ\\phiand signed reference\-margin sideHcH\_\{c\}, both fixed before outcome measurement\. The signed reference margin ismc=FullKV−cHostcm\_\{c\}=\\textsc\{FullKV\}\{\}\_\{c\}\-\\mathrm\{Host\}\_\{c\}, andHc=sign⁡\(mc\)H\_\{c\}=\\operatorname\{sign\}\(m\_\{c\}\)gives the positive or nonpositive split\.Hostc\\mathrm\{Host\}\_\{c\}is the identity selector under the fixed contract for cellcc\. The support\-coupling scoreϕ\\phiidentifies structured\-support inputs whose evidence is distributed across dependent states\.[Figure˜3](https://arxiv.org/html/2605.08234#S4.F3)puts the fixed predictor and measured Stage II shift on the same grid\. Positive margin aligns with positive shift, and removing the value\-consequence channel collapses the favorable\-cell lift\. Aggregated over the grid, the replacement is positive on72\.6%72\.6\\%of positive\-margin cells and32\.4%32\.4\\%of nonpositive\-margin cells \(2\.2×2\.2\{\\times\}\)\. Within positive margin, low coupling gives a76\.4%76\.4\\%positive\-shift rate at\+0\.69\+0\.69pp,1\.8×1\.8\{\\times\}the high\-coupling rate \(42\.9%42\.9\\%,−0\.59\-0\.59pp\)\.[Table˜1](https://arxiv.org/html/2605.08234#S4.T1)reports the signed reference\-margin rates\. Appendix[J\.2](https://arxiv.org/html/2605.08234#A10.SS2)gives the fullϕ×Hc\\phi\\times H\_\{c\}grid\.

![[Uncaptioned image]](https://arxiv.org/html/2605.08234v1/x5.png)

\(a\) Predicted signed reference margin\.

![[Uncaptioned image]](https://arxiv.org/html/2605.08234v1/x6.png)

\(b\) Measured Stage II lift\.

![[Uncaptioned image]](https://arxiv.org/html/2605.08234v1/x7.png)

\(c\) Lift collapses without value channel\.

Figure 3:Stage II prediction and value\-channel control under a fixed observation\-window contract\. Panels \(a\) and \(b\) share rows for LongBench\[[1](https://arxiv.org/html/2605.08234#bib.bib1)\]tasks and columns for model\-budget pairs\. Heatmap color is signed percentage\-point scale after normalization\. Panel \(a\) is fixed before outcome measurement, panel \(b\) is the measured Stage II shift, and panel \(c\) removesTuT^\{u\}from the ranking scalar\.
The value\-channel swap helps when the boundary still contains value\-ranking error and weakens when the identity branch is already favorable or structurally incomplete\. Nonpositive\-margin cells leave no recoverable Stage II room under the host reference, or point to a Stage I or Stage III bottleneck under the fixed contract\.

Table 1:Signed reference margin separates positive from nonpositive Module II shifts under the fixed SnapKV contract\.Directional match \(%\)Margin sidePred\.CellsAllLlamaMistralQwen3MeanΔ\\Delta\(pp\)ppHc\>0H\_\{c\}\>0\(mc\>0m\_\{c\}\>0\)Δ\>0\\Delta\>0626272\.672\.661\.961\.980\.080\.076\.276\.20\.540\.542\.5×10−42\.5\{\\times\}10^\{\-4\}Hc≤0H\_\{c\}\\leq 0\(mc≤0m\_\{c\}\\leq 0\)Δ≤0\\Delta\\leq 0343467\.667\.654\.554\.575\.075\.072\.772\.7−0\.44\-0\.442\.9×10−22\.9\{\\times\}10^\{\-2\}

Falsifiers and robustness\.All confirmatory splits are fixed before the outcome is read\. The support couplingϕ​\(x\)\\phi\(x\)is a structure\-coupling grammar score;Hc=sign⁡\(mc\)H\_\{c\}=\\operatorname\{sign\}\(m\_\{c\}\)is the signed reference\-margin split\. Support coupling is task\-level, whilemcm\_\{c\}andHcH\_\{c\}are computed per cell\.[Table˜2](https://arxiv.org/html/2605.08234#S4.T2)estimatesHcH\_\{c\}on one half of each cell andΔc\\Delta\_\{c\}on the other, so the split cannot be tuned to the same outcomes it is asked to predict\. The main check is ordering: noise from disjoint estimation may shrink the gap, but the positive\-margin bucket should stay above the nonpositive\-margin bucket\. It does, across discovery, sample\-disjoint, held\-out, and threshold\-sweep rows\.[Sections˜J\.2](https://arxiv.org/html/2605.08234#A10.SS2)and[J\.3](https://arxiv.org/html/2605.08234#A10.SS3)give the bootstrap, sensitivity, permutation, per\-baseline, and prompt\-level checks\.

[Table˜3](https://arxiv.org/html/2605.08234#S4.T3)removes the leverage multiplier\. Across 264 nonduplicated matched cells, NoLev and final MII signs agree on 239 cells \(90\.5%\) after binning changes against SnapKV as positive or nonpositive\. The leverage residual, MII minus NoLev, matches the final MII sign on 162 cells \(61\.4%\), rescues 12 NoLev\-nonpositive cells, and breaks 13 NoLev\-positive cells\. In this binary diagnostic, the value\-consequence score has higher sign agreement with final MII than the leverage residual\. Leverage changes a smaller set of boundary cases\. Appendix[K\.1](https://arxiv.org/html/2605.08234#A11.SS1)runs the stronger no\-value ablation by removingTuT^\{u\}itself\. That separation collapses\. NoLev controls leverage rather than value\. Appendix[K\.1](https://arxiv.org/html/2605.08234#A11.SS1)gives the boundary\-level check, and Appendix[J\.1](https://arxiv.org/html/2605.08234#A10.SS1)lists the supporting rows\.

Table 2:Confirmatory reference\-margin checks preserve the ordering gap under disjoint and held\-out splits\. Extends[Table˜1](https://arxiv.org/html/2605.08234#S4.T1)with interval estimates for the same signed reference\-margin logic\.mc\>τm\_\{c\}\>\\taumc≤−τm\_\{c\}\\leq\-\\tauAnalysisCellsMatch \(%\)MeanΔ\\Delta\(pp\)Match \(%\)MeanΔ\\Delta\(pp\)Discovery, fixedHcH\_\{c\}\(τ=0\\tau=0\)969672\.6\+0\.54\[\+0\.22\+0\.22,\+0\.83\+0\.83\]67\.6\-0\.44\[−0\.80\-0\.80,−0\.10\-0\.10\]Sample\-disjoint, raw \(τ=0\\tau=0\)969664\.0\[54\.6, 73\.1\]\+0\.32\[\+0\.06\+0\.06,\+0\.55\+0\.55\]59\.8\[46\.2, 72\.8\]\-0\.20\[−0\.45\-0\.45,\+0\.04\+0\.04\]Sample\-disjoint, margin \(\|mc\|≥0\.5\|m\_\{c\}\|\\geq 0\.5pp\)969668\.7\[60\.5, 76\.2\]\+0\.39\[\+0\.18\+0\.18,\+0\.60\+0\.60\]67\.0\[56\.5, 76\.8\]\-0\.25\[−0\.46\-0\.46,−0\.04\-0\.04\]Sample\-disjoint, mechanism969669\.3\[60\.6, 77\.1\]\+0\.48\[\+0\.21\+0\.21,\+0\.75\+0\.75\]55\.6\[44\.7, 66\.4\]\-0\.07\[−0\.37\-0\.37,\+0\.22\+0\.22\]Held\-out, low\-budget969667\.6\[57\.4, 76\.8\]\+0\.31\[\+0\.11\+0\.11,\+0\.51\+0\.51\]64\.2\[53\.4, 73\.9\]\-0\.18\[−0\.34\-0\.34,−0\.02\-0\.02\]Held\-out, prompt split969666\.9\[56\.8, 75\.7\]\+0\.30\[\+0\.05\+0\.05,\+0\.55\+0\.55\]65\.4\[53\.4, 75\.9\]\-0\.17\[−0\.40\-0\.40,−0\.01\-0\.01\]
- •Each mean cell reports the point estimate on the first line and the reported confidence interval on the second line\. The favorable invariant is ordering: the predicted\-positive bucket should remain above the predicted\-nonpositive bucket even when the split becomes noisier\.

Table 3:Removing leverage weakens the effect but keeps the signed reference\-margin ordering\. Companion to[Table˜1](https://arxiv.org/html/2605.08234#S4.T1)\.SnapKV \+ NoLevSnapKV \+ full replacementGroupPositive shift \(%\)MeanΔ\\Deltafrom SnapKV \(pp\)Positive shift \(%\)MeanΔ\\Deltafrom SnapKV \(pp\)All 96 cells57\.357\.30\.150\.1558\.358\.30\.200\.20Hc\>0H\_\{c\}\>069\.469\.40\.480\.4872\.672\.60\.540\.54Hc≤0H\_\{c\}\\leq 029\.429\.4−0\.34\-0\.3432\.432\.4−0\.44\-0\.44low\-ϕ\\phi,Hc\>0H\_\{c\}\>074\.574\.50\.610\.6176\.476\.40\.690\.69high\-ϕ\\phi,Hc\>0H\_\{c\}\>028\.628\.6−0\.42\-0\.4242\.942\.9−0\.59\-0\.59low\-ϕ\\phi,Hc≤0H\_\{c\}\\leq 030\.430\.4−0\.41\-0\.4134\.834\.8−0\.49\-0\.49high\-ϕ\\phi,Hc≤0H\_\{c\}\\leq 027\.327\.3−0\.29\-0\.2927\.327\.3−0\.34\-0\.34Contract\-drift evaluation\.SnapKV\[[26](https://arxiv.org/html/2605.08234#bib.bib26)\]is the identity branch for one observation\-window contract\.[Table˜4](https://arxiv.org/html/2605.08234#S4.T4)tests transfer when other selectors change access estimation, allocation, or projection\. PyramidKV\[[2](https://arxiv.org/html/2605.08234#bib.bib2)\], CAKE\[[37](https://arxiv.org/html/2605.08234#bib.bib37)\], and Ada\-KV\[[17](https://arxiv.org/html/2605.08234#bib.bib17)\]move along Stage III by changing pyramidal layer budget, layer cascade, or head budget\. H2O\-debiased\[[48](https://arxiv.org/html/2605.08234#bib.bib48)\]and H2O\[[52](https://arxiv.org/html/2605.08234#bib.bib52)\]move along Stage I by changing cumulative\-support estimation\. For a drift surfaceBB,mB=FullKV−Bm\_\{B\}=\\textsc\{FullKV\}\{\}\-Bis the contract\-local reference margin andHB=sign⁡\(mB\)H\_\{B\}=\\operatorname\{sign\}\(m\_\{B\}\)gives the split\.

Table 4:Value\-channel lift transfers across Stage III drift and weakens on Stage I surfaces in the cross\-contract evaluation\. Baselines are SnapKV\[[26](https://arxiv.org/html/2605.08234#bib.bib26)\], PyramidKV\[[2](https://arxiv.org/html/2605.08234#bib.bib2)\], CAKE\[[37](https://arxiv.org/html/2605.08234#bib.bib37)\], Ada\-KV\[[17](https://arxiv.org/html/2605.08234#bib.bib17)\], and H2O\[[52](https://arxiv.org/html/2605.08234#bib.bib52)\]\.Surface gap \(pp\)Contract\-local margin countsSurfaceBBB−SnapKVB\-\\mathrm\{SnapKV\}B−FullKVB\-\\textsc\{FullKV\}\{\}HB\>0H\_\{B\}\>0HB≤0H\_\{B\}\\leq 0MeanΔB\\Delta\_\{B\}\(pp\)Dominant stageSnapKV0\.000\.00−1\.79\-1\.7962623434\+0\.19Stage IIPyramidKV−2\.50\-2\.50−4\.37\-4\.3773732323\+0\.84Stage III, layerCAKE−4\.51\-4\.51−6\.30\-6\.3078781818\+0\.30Stage III, cascadeAda\-KV−2\.11\-2\.11−3\.90\-3\.9076762020\+0\.15Stage III, headH2O\-deb\.−7\.68\-7\.68−9\.47\-9\.4783831313–Stage I, debiasedH2O−24\.81\-24\.81−26\.60\-26\.60919155–Stage I, cumulative

[Table˜4](https://arxiv.org/html/2605.08234#S4.T4)shows the contract\-drift evaluation\. The Stage III adapter lifts follow the stage map, with\+0\.84\+0\.84pp on PyramidKV,\+0\.30\+0\.30pp on CAKE, and\+0\.15\+0\.15pp on Ada\-KV\. These rows test drift sensitivity, not global selector rank\. Value\-consequence information transfers when access support remains comparable, and lift decays as Stage III absorbs or distorts that signal\.[Section˜J\.3](https://arxiv.org/html/2605.08234#A10.SS3)isolates the remaining PyramidKV layer\-budget gap\.

H2O\-debiased and H2O move the bottleneck upstream into Stage I\. A Stage II replacement stacked on those selectors would receive a different support distribution, so any gain would mix value repair with a changed support law\. The Stage I correlation between debias lift and access gap isr=0\.88r\{=\}0\.88on the same 96 cells \([Figure˜8](https://arxiv.org/html/2605.08234#A10.F8), Appendix[J](https://arxiv.org/html/2605.08234#A10)\), in contrast to the Stage IIIr=0\.14r\{=\}0\.14\([Figure˜4](https://arxiv.org/html/2605.08234#S4.F4)c\)\. This contrast supports stage separability\.

Projection and multi\-target stress tests\.One Stage III explanation is that compressed selectors fail because they leave part of the budget unused\. Strict block TopK retainskb=⌊k/p⌋k\_\{b\}=\\lfloor k/p\\rfloorfull blocks under budgetkkand block sizepp, leaving a lattice residualεlat​\(k,p\)=k−p​kb\\varepsilon\_\{\\mathrm\{lat\}\}\(k,p\)=k\-pk\_\{b\}tokens unspent across cells \([Figure˜4](https://arxiv.org/html/2605.08234#S4.F4)a\)\. Token\-fill is the diagnostic control: after block selection, it spends the residual on individual tokens without changing Stage I or Stage II rankings\. If unused budget were the active Stage III bottleneck, filling it should restore performance\. Token\-fill reduces the per\-model mean unused\-token slack from 12\.0, 10\.0, and 8\.9 tokens to under 0\.1 token \([Figure˜4](https://arxiv.org/html/2605.08234#S4.F4)b\)\. Yet the per\-cell score change stays weakly correlated with the original slack \(r=\+0\.14r\{=\}\{\+\}0\.14, mean\+0\.12\+0\.12pp, median0\.000\.00pp\), and Stage II positive and nonpositive cells interleave without separation \([Figure˜4](https://arxiv.org/html/2605.08234#S4.F4)c\)\. Matching the token count does not recover Stage III performance in this control\. Unused budget is not supported as the active standalone error, and the residual points to block projection and support closure\.

![Refer to caption](https://arxiv.org/html/2605.08234v1/x8.png)

\(a\) Strict\-block unused\-token slack\.

![Refer to caption](https://arxiv.org/html/2605.08234v1/x9.png)

\(b\) Per\-model slack before and after token\-fill\.

![Refer to caption](https://arxiv.org/html/2605.08234v1/x10.png)

\(c\) Score change against strict\-block slack\.

Figure 4:Stage III separability under block projection\. \(a\) Strict\-block lattice slack across the 96\-cell grid, with rows for model–budget pairs, columns for LongBench\[[1](https://arxiv.org/html/2605.08234#bib.bib1)\]tasks, and color showing unused tokens\. \(b\) Per\-model mean slack before and after token\-fill\. \(c\) Per\-cell score change after token\-fill against the original slack, where filled markers are Stage II positive cells and open markers are nonpositive\.The multi\-target stress test probes support closure under branched decode queries\. The staged diagnostic predicts that a single pooled proxy can lose branch\-specific evidence because related fields must survive together\. Appendix[I](https://arxiv.org/html/2605.08234#A9)reports the 32k NeedleBench M\-RT rows and a directional RULER\[[24](https://arxiv.org/html/2605.08234#bib.bib24)\]8k check\. The former shows pooling losing branch\-specific evidence even when individual tokens receive high scores; the latter tests whether logged movement concentrates on multikey retrieval\. These rows are stress checks for the projection and support\-closure failure mode, not clean Stage II value\-channel evidence\. Contracts with allocation drift or broken Stage I support remain scoped to[Table˜4](https://arxiv.org/html/2605.08234#S4.T4)\.

## 5Discussion and Conclusion

A KV compression score is not self\-diagnosing\. The same task\-level change can come from access estimation, value consequence, or projection\. Fixing the selector contract lets these stages be tested one at a time\. Compression also need not be monotone: eviction can move the value\-channel decision boundary, so a compressed cache can sometimes exceedFullKV\. Token\-fill gives the Stage III countercheck \([Figure˜4](https://arxiv.org/html/2605.08234#S4.F4)\)\. It spends the strict block lattice slack, yet the score surface barely moves \(r=\+0\.14r\{=\}\{\+\}0\.14, mean\+0\.12\+0\.12pp\)\. The remaining error sits in block projection and support closure\.

Changing the observation window, layer or head allocation, or score\-to\-cache projection changes the stage under test\. Such comparisons still matter for deployment, but they do not identify which component was repaired\. The diagnostic order is to fix the contract, perturb one stage, check the predicted sign split, and assign the failure to a stage before optimizing the selector\.

Reporting should follow the same logic\. A final average delta can pool small wins, boundary rescues, neutral or tied cells, and losses that cancel them\. Counts for neutral and tied cells, budget sweeps, and stage\-local ablations show whether the movement traces to missing access support, value\-channel repair, leverage at a boundary, or projection that drops coupled evidence\. Guards and fallback rules should be measured under the same fixed contract\.

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## Appendix

- •Appendix[A](https://arxiv.org/html/2605.08234#A1): notation and technical preliminaries\.
- •Appendix[B](https://arxiv.org/html/2605.08234#A2): proofs for the main\-text formal statements\.
- •Appendix[C](https://arxiv.org/html/2605.08234#A3): related work\.
- •Appendix[D](https://arxiv.org/html/2605.08234#A4): algorithmic details\.
- •Appendix[E](https://arxiv.org/html/2605.08234#A5): the multi\-objective formal backbone\.
- •Appendices[F](https://arxiv.org/html/2605.08234#A6)–[H](https://arxiv.org/html/2605.08234#A8): derivations and diagnostics for exposure bias, head aggregation, and layer\-wise access alignment\.
- •Appendix[I](https://arxiv.org/html/2605.08234#A9): multi\-objective retrieval stress tests\.
- •Appendix[J](https://arxiv.org/html/2605.08234#A10): the stage registry, predictor provenance, robustness checks, and transfer tables\.
- •Appendix[K](https://arxiv.org/html/2605.08234#A11): additional confirmatory controls\.
- •Appendices[L](https://arxiv.org/html/2605.08234#A12)–[M](https://arxiv.org/html/2605.08234#A13): limitations and broader impacts\.

## Appendix ANotation and Technical Preliminaries

This appendix collects symbols reused across sections\. Symbols introduced only for a local derivation are defined at their first use\.

### A\.1Notation

Scalars and indices use italic letters, frequently reused sets and index families such as𝒦\\mathcal\{K\}or𝒰\\mathcal\{U\}use calligraphic letters, tensors and vectors use bold symbols \(e\.g\.𝐀\\mathbf\{A\},𝐕\\mathbf\{V\},𝝁\\boldsymbol\{\\mu\}\), probability laws are written in plain italic, and operators are written in roman type\. Scalar entries and scalar aggregates are not bolded\.

- •\[n\]\[n\]: the index set\{1,…,n\}\\\{1,\\ldots,n\\\}\.
- •TV⁡\(μ,ν\)\\operatorname\{TV\}\(\\mu,\\nu\): total\-variation distance between probability laws\.
- •TopK⁡\(s,k\)\\operatorname\{TopK\}\(s,k\): the deterministic set ofkklargest\-score indices\.
- •Aggl,h\\operatorname\{Agg\}\_\{l,h\}: the fixed layer\-head aggregation operator used by a selector contract\.
- •TT,bb,kk: prompt length, budget ratio, and retained\-token budgetk=⌊b​T⌋k=\\lfloor bT\\rfloor\.
- •ii,uu,ll,hh: key position, query or proxy position, layer, and query head\.
- •𝐀\\mathbf\{A\},𝐀l,h\\mathbf\{A\}\_\{l,h\},𝐀l,h​\[u,i\]\\mathbf\{A\}\_\{l,h\}\[u,i\]: prefill\-attention tensor, its layer\-head slice, and one scalar entry\.
- •𝐕l,κ​\(h\),i\\mathbf\{V\}\_\{l,\\kappa\(h\),i\},𝐨l,h,u\\mathbf\{o\}\_\{l,h,u\},𝝁l,h,u​\(c\)\\boldsymbol\{\\mu\}\_\{l,h,u\}\(c\): value vector at one position, attention output vector, and block centroid vector\.
- •sis\_\{i\},𝒦\\mathcal\{K\}: scalar score and retained cache positions\.
- •uacc​\(i\)u\_\{\\mathrm\{acc\}\}\(i\): decode\-side access support\.
- •uval​\(i\)u\_\{\\mathrm\{val\}\}\(i\): conditional value consequence\.
- •ηproj\\eta\_\{\\mathrm\{proj\}\}: projection residual after mapping scores to a retained set\.
- •q^m\\hat\{q\}\_\{m\},d^m\\hat\{d\}\_\{m\},f^m\\hat\{f\}\_\{m\}: query law, exposure correction, and pooling kernel of estimatormm\.
- •q⋆q^\{\\star\},d⋆d^\{\\star\},f⋆f^\{\\star\}: reference counterparts in the Stage I decomposition\.
- •δphase\\delta\_\{\\mathrm\{phase\}\},δexp\\delta\_\{\\mathrm\{exp\}\},ξacc\\xi\_\{\\mathrm\{acc\}\}: Stage I query\-law, exposure, and aggregation residuals\.
- •Δ\\Delta,Δ^\\hat\{\\Delta\},ξval\\xi\_\{\\mathrm\{val\}\}: true score, estimated score, and value\-channel residual in Stage II\.
- •p⋆p^\{\\star\},pTp\_\{T\},p¯𝒦\\bar\{p\}\_\{\\mathcal\{K\}\}: first\-decode law, last\-prefill law, and retained\-set\-renormalized law\.
- •ϕ​\(x\)\\phi\(x\),mcm\_\{c\},HcH\_\{c\},Δc\\Delta\_\{c\}: support\-coupling index, signed reference marginmc=FullKV−cHostcm\_\{c\}=\\textsc\{FullKV\}\{\}\_\{c\}\-\\mathrm\{Host\}\_\{c\}, its signHc=sign⁡\(mc\)H\_\{c\}=\\operatorname\{sign\}\(m\_\{c\}\), and intervention outcome for cellcc\.
- •ℬ\\mathcal\{B\},cc,pp,kbk\_\{b\}: block partition, block, block size, and block budgetkb=⌊k/p⌋k\_\{b\}=\\lfloor k/p\\rfloor\.
- •ℒ⋆\\mathcal\{L\}\_\{\\star\},βl\\beta\_\{l\}: selected layer set and layer weight\.
- •𝒰\\mathcal\{U\},𝒰tail\\mathcal\{U\}^\{\\mathrm\{tail\}\},𝒰anchor\\mathcal\{U\}^\{\\mathrm\{anchor\}\},rur\_\{u\}: proxy bank, its tail and anchor components, and proxy weight\.
- •al,h,u​\(c\)a\_\{l,h,u\}\(c\),dl,h,u​\(c\)d\_\{l,h,u\}\(c\): block mass scalar and deletion\-cost surrogate\.
- •Dm​\(c\)D\_\{m\}\(c\),S​\(c\)S\(c\),𝒵⋆\\mathcal\{Z\}^\{\\star\}: group\-wise score, pooled block score, and selected\-block index set\.
- •εlat\\varepsilon\_\{\\mathrm\{lat\}\}: block\-lattice token slack, distinct fromηproj\\eta\_\{\\mathrm\{proj\}\}\.
- •μdec\\mu\_\{\\mathrm\{dec\}\},η~\\tilde\{\\eta\},rr,σ\\sigma: labeled decode law, proxy law, channel count, and routing map\.

Localized symbols not listed above–for exampleγε,l,h,u\\gamma\_\{\\varepsilon,l,h,u\},εa\\varepsilon\_\{a\},εμ\\varepsilon\_\{\\mu\},R2R\_\{2\},MM,𝒰m\\mathcal\{U\}\_\{m\},wmw\_\{m\},νm\\nu\_\{m\},wmaxw\_\{\\max\},λeff\\lambda\_\{\\mathrm\{eff\}\}, andNi⋆N\_\{i\}^\{\\star\}–are defined at their point of use\.

## Appendix BProofs for the Staged Diagnostic Framework

### B\.1Boundary\-Margin Condition

Proof of[Lemma˜1](https://arxiv.org/html/2605.08234#Thmtheorem1)\.Under the base score,jjis retained andiiis evicted, sosj≥sis\_\{j\}\\geq s\_\{i\}up to the fixed tie break\. Under the perturbed scores′=s\+δs^\{\\prime\}=s\+\\delta, tokeniistrictly ranks above tokenjjwhenever

si\+δi\>sj\+δj,s\_\{i\}\+\\delta\_\{i\}\>s\_\{j\}\+\\delta\_\{j\},which is equivalent toδi−δj\>sj−si\\delta\_\{i\}\-\\delta\_\{j\}\>s\_\{j\}\-s\_\{i\}; equality is resolved by the fixed tie break\. For a one\-for\-one boundary swap, the kept set changes from𝒦s\\mathcal\{K\}\_\{s\}to\(𝒦s∖\{j\}\)∪\{i\}\(\\mathcal\{K\}\_\{s\}\\setminus\\\{j\\\}\)\\cup\\\{i\\\}\. If latent utility is locally additive for this swap, the utility difference is

∑r∈\(𝒦s∖\{j\}\)∪\{i\}ur−∑r∈𝒦sur=ui−uj\.\\sum\_\{r\\in\(\\mathcal\{K\}\_\{s\}\\setminus\\\{j\\\}\)\\cup\\\{i\\\}\}u\_\{r\}\-\\sum\_\{r\\in\\mathcal\{K\}\_\{s\}\}u\_\{r\}=u\_\{i\}\-u\_\{j\}\.Away from the equality case, the perturbation helps at that boundary when the margin is crossed and the entering token has larger latent utility than the leaving token\. If the margin is not crossed, the pair does not swap; if it is crossed in the opposite utility direction, the swap is harmful\.

### B\.2Ordered\-Substitution Identity

Proof of[Proposition˜2](https://arxiv.org/html/2605.08234#Thmtheorem2)\.Starting from the two estimators in[Equation˜4](https://arxiv.org/html/2605.08234#S2.E4), add and subtract the intermediate estimators that replace one component at a time:

u^acc​\(i;m\)−uacc​\(i\)\\displaystyle\\hat\{u\}\_\{\\mathrm\{acc\}\}\(i;m\)\-u\_\{\\mathrm\{acc\}\}\(i\)=∑u∈𝒬mq^m​\(u\)​d^m​\(i,u\)​f^m​\(u,i\)−∑u∈𝒬mq⋆​\(u\)​d⋆​\(i,u\)​f⋆​\(u,i\)\\displaystyle=\\sum\_\{u\\in\\mathcal\{Q\}\_\{m\}\}\\hat\{q\}\_\{m\}\(u\)\\hat\{d\}\_\{m\}\(i,u\)\\hat\{f\}\_\{m\}\(u,i\)\-\\sum\_\{u\\in\\mathcal\{Q\}\_\{m\}\}q^\{\\star\}\(u\)d^\{\\star\}\(i,u\)f^\{\\star\}\(u,i\)=∑u∈𝒬m\(q^m​\(u\)−q⋆​\(u\)\)​d⋆​\(i,u\)​f⋆​\(u,i\)\\displaystyle=\\sum\_\{u\\in\\mathcal\{Q\}\_\{m\}\}\\bigl\(\\hat\{q\}\_\{m\}\(u\)\-q^\{\\star\}\(u\)\\bigr\)d^\{\\star\}\(i,u\)f^\{\\star\}\(u,i\)\+∑u∈𝒬mq^m​\(u\)​\(d^m​\(i,u\)−d⋆​\(i,u\)\)​f⋆​\(u,i\)\\displaystyle\\quad\+\\sum\_\{u\\in\\mathcal\{Q\}\_\{m\}\}\\hat\{q\}\_\{m\}\(u\)\\bigl\(\\hat\{d\}\_\{m\}\(i,u\)\-d^\{\\star\}\(i,u\)\\bigr\)f^\{\\star\}\(u,i\)\+∑u∈𝒬mq^m​\(u\)​d^m​\(i,u\)​\(f^m​\(u,i\)−f⋆​\(u,i\)\)\.\\displaystyle\\quad\+\\sum\_\{u\\in\\mathcal\{Q\}\_\{m\}\}\\hat\{q\}\_\{m\}\(u\)\\hat\{d\}\_\{m\}\(i,u\)\\bigl\(\\hat\{f\}\_\{m\}\(u,i\)\-f^\{\\star\}\(u,i\)\\bigr\)\.The three displayed terms are exactlyδphase​\(i;m\)\\delta\_\{\\mathrm\{phase\}\}\(i;m\),δexp​\(i;m\)\\delta\_\{\\mathrm\{exp\}\}\(i;m\), andξacc​\(i;m\)\\xi\_\{\\mathrm\{acc\}\}\(i;m\)in[Equation˜5](https://arxiv.org/html/2605.08234#S2.E5)\. The identity is therefore a telescoping decomposition and does not use Taylor expansion or an independence assumption\.

### B\.3Tail\-K TV Bound

Proof of[Proposition˜3](https://arxiv.org/html/2605.08234#Thmtheorem3)\.Both inequalities are the triangle inequality for total variation distance\. Applying it with the intermediate lawpTp\_\{T\}gives

TV⁡\(p⋆,p¯tail,K\)≤TV⁡\(p⋆,pT\)\+TV⁡\(pT,p¯tail,K\),\\operatorname\{TV\}\(p^\{\\star\},\\bar\{p\}\_\{\\mathrm\{tail\},K\}\)\\leq\\operatorname\{TV\}\(p^\{\\star\},p\_\{T\}\)\+\\operatorname\{TV\}\(p\_\{T\},\\bar\{p\}\_\{\\mathrm\{tail\},K\}\),which proves[Equation˜8](https://arxiv.org/html/2605.08234#S2.E8)\. Replacingp¯tail,K\\bar\{p\}\_\{\\mathrm\{tail\},K\}byp¯all\\bar\{p\}\_\{\\mathrm\{all\}\}gives[Equation˜9](https://arxiv.org/html/2605.08234#S2.E9)\.

## Appendix CRelated Work

We group prior work by the score proxy used to select, store, or read cache states\.

Token\-level eviction\.H2O\[[52](https://arxiv.org/html/2605.08234#bib.bib52)\]uses cumulative attention; StreamingLLM\[[46](https://arxiv.org/html/2605.08234#bib.bib46)\]retains sinks plus a recency window; SnapKV\[[26](https://arxiv.org/html/2605.08234#bib.bib26)\]uses an observation window of the lastWWqueries per layer; PyramidKV\[[2](https://arxiv.org/html/2605.08234#bib.bib2)\]uses pyramidal layer\-budget allocation; Ada\-KV\[[17](https://arxiv.org/html/2605.08234#bib.bib17)\]adapts per\-head budgets; and R\-KV\[[3](https://arxiv.org/html/2605.08234#bib.bib3)\]adds redundancy\-aware selection for reasoning models\. These methods still depend on attention\-derived scores, so they can fail when the observed prefill signal differs from the decode\-time utility\. Our analysis explains why low\-budget results separate and why some methods become similar once the largest mismatch is repaired\. A recent survey\[[41](https://arxiv.org/html/2605.08234#bib.bib41)\]reviews attention\-sink utilization, interpretation, and mitigation, including the practice of retaining high\-cumulative\-attention anchors during compression\. The same survey notes that a unified theoretical framework remains open\. Online reactive policies such as TOVA\[[31](https://arxiv.org/html/2605.08234#bib.bib31)\]are out of contract for the confirmatory Stage II test because they change both the observation law and the generation\-time cache dynamics\.

Bias\-aware and decode\-aligned eviction\.AhaKV\[[4](https://arxiv.org/html/2605.08234#bib.bib4)\]formalizes the position\-dependent bias in cumulative attention; Attention Debiasing\[[48](https://arxiv.org/html/2605.08234#bib.bib48)\]applies the same÷\(T−i\)\\div\(T\{\-\}i\)correction for VLMs\. We place exposure bias inside the Stage I access\-support estimator alongside query\-law mismatch \(Phase Dilution\), which explains failure patterns a count fix alone cannot, including the reverse\-depth behavior after isolated count repair\. DapQ\[[45](https://arxiv.org/html/2605.08234#bib.bib45)\]studies the same decode\-alignment axis by reporting higher scores for position\-aware pseudo\-queries than for cumulative scoring; Expected Attention\[[14](https://arxiv.org/html/2605.08234#bib.bib14)\]estimates attention under a future\-query distribution; ForesightKV\[[15](https://arxiv.org/html/2605.08234#bib.bib15)\]and LookaheadKV\[[50](https://arxiv.org/html/2605.08234#bib.bib50)\]use learned or draft\-model surrogates for future access\. The Stage II clean grid uses the zero\-extra\-forward\-pass operating point on this axis; a matched\-compute control confirms that pseudo\-query scoring yields only a small grand\-mean gain at nontrivial additional prefill cost\. ReST\-KV\[[8](https://arxiv.org/html/2605.08234#bib.bib8)\]takes a different approach, replacing raw attention proxies with an output\-aware surrogate built from layer\-wise reconstruction\. Its objective is reconstruction; our diagnosis addresses score misranking at the eviction boundary\. Early attention analysis showed that heads carry heterogeneous linguistic, positional, and syntactic patterns rather than a single uniform importance signal\[[7](https://arxiv.org/html/2605.08234#bib.bib7)\]\. In the VLA domain, TIES\[[28](https://arxiv.org/html/2605.08234#bib.bib28)\]observes that high\-attention tokens are not always important; reasoning\-token analysis\[[42](https://arxiv.org/html/2605.08234#bib.bib42)\]and direction\-aware RLVR update analysis\[[23](https://arxiv.org/html/2605.08234#bib.bib23)\]reach related conclusions about magnitude\-only indicators\. The attention tensor contains decode\-utility information, but the way it is aggregated determines whether that information survives into the final score; head\-functional differentiation, identified by RazorAttention\[[32](https://arxiv.org/html/2605.08234#bib.bib32)\]as a small group of retrieval\-relevant heads, is the head\-side counterpart of the same observation\. CriticalKV\[[18](https://arxiv.org/html/2605.08234#bib.bib18)\], VATP\[[47](https://arxiv.org/html/2605.08234#bib.bib47)\], DefensiveKV\[[49](https://arxiv.org/html/2605.08234#bib.bib49)\], and related value\-aware eviction variants modify scoring or aggregation inside a fixed selection contract\. The staged diagnostic treats these changes as value\-consequence interventions and asks when the surrounding contract leaves a recoverable signed reference margin\. A value\-aware score is a clean Stage II intervention only when the access estimator and projection rule are held fixed\.

Soft query proxies and future\-aware scoring\.A growing line of work replaces hard tail\-only observation windows with softer or more forward\-looking query representations\. LAQ\[[51](https://arxiv.org/html/2605.08234#bib.bib51)\]generates low\-cost pseudo lookahead queries and uses them as the observation window for importance estimation\. LookaheadKV\[[50](https://arxiv.org/html/2605.08234#bib.bib50)\]adds parameter\-efficient modules that predict future\-response importance without explicit draft generation\. KVzip\[[36](https://arxiv.org/html/2605.08234#bib.bib36)\]takes a query\-agnostic route, compressing the cache so that the original context is reconstructible under arbitrary downstream queries\. These works address the gap between prefill\-derived importance scores and true decode queries through lookahead or query\-agnostic corrections\. Our focus is when that gap changes the outcome\. Under a fixed observation\-window contract, the signed reference margin and support coupling predict whether a value\-channel correction helps or harms\. On the token side, GraphKV\[[21](https://arxiv.org/html/2605.08234#bib.bib21)\]adds redundancy\-aware reranking through similarity\-graph diffusion, addressing the coverage failure seen in multi\-objective retrieval\. The fixed\-size prefill\-visible proxy bank used in the Stage II clean test sits at the zero\-extra\-forward\-pass end of this spectrum\. Higher\-cost proxy families target the same decode\-mismatch axis with broader query support, especially for long decode trajectories and multi\-objective prompts\.[Section˜D\.1](https://arxiv.org/html/2605.08234#A4.SS1)states the grouped proxy objective used for the multi\-target appendix under the same decode\-proximal principle\.

Learned compressed\-summary attention\.Recent natively trained sparse\-attention architectures replace raw token\-level history with learned compressed summaries and hardware\-aligned block reads\. NSA organizes history into compressed coarse\-grained tokens, selected fine\-grained blocks, and a sliding window, while DeepSeek\-V4 combines Compressed Sparse Attention and Heavily Compressed Attention to support million\-token contexts through learned compressed KV entries\[[12](https://arxiv.org/html/2605.08234#bib.bib12),[13](https://arxiv.org/html/2605.08234#bib.bib13)\]\. State\-space sequence models such as Mamba\[[20](https://arxiv.org/html/2605.08234#bib.bib20)\]avoid dense\-attention KV growth through a different recurrent\-state substrate\. These methods change the model architecture and the cache state itself\. Our setting is different: we study non\-reconstructing eviction for pretrained dense\-attention models after training\. Blocks are selection and projection units; the retained cache contains original KV states, and block centroids are used only to score deletion consequence\.

Allocation, compression, and orthogonal directions\.FastGen\[[19](https://arxiv.org/html/2605.08234#bib.bib19)\]uses head\-specific retention; CAKE\[[37](https://arxiv.org/html/2605.08234#bib.bib37)\], ARKV\[[38](https://arxiv.org/html/2605.08234#bib.bib38)\], and adaptive layer selection\[[33](https://arxiv.org/html/2605.08234#bib.bib33)\]allocate cache across layers, pruning stages, or precisions given an importance score\. The diagnostic framework operates at the prior question of which score to trust\. Storage\-side and token\-adaptive compression\[[22](https://arxiv.org/html/2605.08234#bib.bib22),[27](https://arxiv.org/html/2605.08234#bib.bib27),[30](https://arxiv.org/html/2605.08234#bib.bib30),[29](https://arxiv.org/html/2605.08234#bib.bib29)\], architecture\-level sparse attention\[[9](https://arxiv.org/html/2605.08234#bib.bib9)\], and residual\-attention architectures\[[11](https://arxiv.org/html/2605.08234#bib.bib11)\]act on different axes; applying the score diagnostic to them requires a separate fixed\-contract interface rather than a direct head\-to\-head eviction comparison\. FleetOpt\[[6](https://arxiv.org/html/2605.08234#bib.bib6)\]treats compression as part of fleet provisioning and routing, with selector scoring as one component\. Recent failure\-oriented evaluations and system studies\[[5](https://arxiv.org/html/2605.08234#bib.bib5),[40](https://arxiv.org/html/2605.08234#bib.bib40),[10](https://arxiv.org/html/2605.08234#bib.bib10)\]document when compression or offloading breaks; our contribution is to assign those failures to access estimation, value ranking, or projection under a fixed contract\.

Boundary for reconstructing and non\-eviction compression\.We study*non\-reconstructing*eviction, where methods drop tokens without compensating their value\. Numerical\-recovery quantization\[[27](https://arxiv.org/html/2605.08234#bib.bib27),[22](https://arxiv.org/html/2605.08234#bib.bib22)\]keeps every token but lowers precision; low\-rank latent reconstruction, as in Palu\[[34](https://arxiv.org/html/2605.08234#bib.bib34)\], regenerates key\-value states at decode time from a compressed cache; importance\-aware mixed\-precision quantization, as in MiKV\[[35](https://arxiv.org/html/2605.08234#bib.bib35)\], keeps cache entries while assigning lower bit\-widths to less important positions; compensation\-token methods, as in RazorAttention\[[32](https://arxiv.org/html/2605.08234#bib.bib32)\], inject summary tokens for dropped content; and context\-reconstruction\-guided compression, as in KVzip\[[36](https://arxiv.org/html/2605.08234#bib.bib36)\], optimizes for reproducing the original context\. These methods operate on different axes and fall outside the head\-to\-head eviction comparison; the staged diagnostic applies to the eviction step itself, separate from the storage or reconstruction layer\.

## Appendix DAlgorithm Details

This appendix specifies the default block\-level selector and the controlled extensions used in appendix ablations\. It gives the reference pseudocode, the support\-only diagnostic channel, the extension hierarchy, and the identity check that recovers the canonical recipe\. Each optional component is separated from the default method used in the main experiments\.

[Algorithm˜1](https://arxiv.org/html/2605.08234#alg1)gives the reference implementation of the canonical block\-level default\. Appendix ablations modify this recipe in three disabled\-by\-default ways: grouped proxy banks for explicit multi\-target prompts, soft\-robust aggregation across groups, and a small reserve for heavy\-tail outlier blocks\.

Algorithm 1Reference implementation of the staged scoring pipeline1:Partition prompt into blocks

ℬ=\{c1,…,cNb\}\\mathcal\{B\}=\\\{c\_\{1\},\\dots,c\_\{N\_\{b\}\}\\\}of size

pp;

k←⌊b​T⌋k\\leftarrow\\lfloor bT\\rfloor;

kb←⌊k/p⌋k\_\{b\}\\leftarrow\\lfloor k/p\\rfloor;

S​\(c\)←0S\(c\)\\leftarrow 0for all

cc
2:Module I: recover decode\-side access supportuaccu\_\{\\mathrm\{acc\}\}

3:Select late or plateau layers

ℒ⋆\\mathcal\{L\}\_\{\\star\}with normalized weights

\{βl\}\\\{\\beta\_\{l\}\\\}⊳\\trianglerightreducesξacc\\xi\_\{\\mathrm\{acc\}\}

4:

𝒰tail←\{T−w\+1,…,T\}\\mathcal\{U\}^\{\\mathrm\{tail\}\}\\leftarrow\\\{T\{\-\}w\{\+\}1,\\dots,T\\\};

𝒰anchor←𝒞anchor\\mathcal\{U\}^\{\\mathrm\{anchor\}\}\\leftarrow\\mathcal\{C\}\_\{\\mathrm\{anchor\}\};

𝒰←𝒰tail∪𝒰anchor\\mathcal\{U\}\\leftarrow\\mathcal\{U\}^\{\\mathrm\{tail\}\}\\cup\\mathcal\{U\}^\{\\mathrm\{anchor\}\}⊳\\trianglerightfixesδphase\\delta\_\{\\mathrm\{phase\}\}

5:

ru∝exp⁡\(−\(T−u\)/τq\)r\_\{u\}\\propto\\exp\(\-\(T\{\-\}u\)/\\tau\_\{q\}\)on tail,

∝1\\propto 1on anchor; normalize⊳\\trianglerightcancelsδexp\\delta\_\{\\mathrm\{exp\}\}

6:Module II: reweight by conditional output consequenceuvalu\_\{\\mathrm\{val\}\}

7:foreach

l∈ℒ⋆l\\in\\mathcal\{L\}\_\{\\star\},

u∈𝒰u\\in\\mathcal\{U\}, and query head

hhdo

8:

𝐨l,h,u←∑i=1T𝐀l,h​\[u,i\]​𝐕l,κ​\(h\),i\\mathbf\{o\}\_\{l,h,u\}\\leftarrow\\sum\_\{i=1\}^\{T\}\\mathbf\{A\}\_\{l,h\}\[u,i\]\\,\\mathbf\{V\}\_\{l,\\kappa\(h\),i\}
9:foreach block

c∈ℬc\\in\\mathcal\{B\}do

10:

a←∑i∈c𝐀l,h​\[u,i\]a\\leftarrow\\sum\_\{i\\in c\}\\mathbf\{A\}\_\{l,h\}\[u,i\],

𝝁←∑i∈c𝐀l,h​\[u,i\]​𝐕l,κ​\(h\),i/\(a\+ε\)\\boldsymbol\{\\mu\}\\leftarrow\\textstyle\\sum\_\{i\\in c\}\\mathbf\{A\}\_\{l,h\}\[u,i\]\\,\\mathbf\{V\}\_\{l,\\kappa\(h\),i\}/\(a\{\+\}\\varepsilon\)
11:

S\(c\)\+=ruβl\(a/\(1−a\+ε\)\)2∥𝝁−𝐨l,h,u∥22S\(c\)\\mathrel\{\+\}=r\_\{u\}\\,\\beta\_\{l\}\\,\\bigl\(a/\(1\{\-\}a\{\+\}\\varepsilon\)\\bigr\)^\{2\}\\,\\lVert\\boldsymbol\{\\mu\}\-\\mathbf\{o\}\_\{l,h,u\}\\rVert\_\{2\}^\{2\}
12:endfor

13:endfor

14:Module III: block\-constant posterior approximation⊳\\trianglerightimplemented by strict block TopK

15:Output:

𝒦←⋃\{c:S​\(c\)∈TopKkb​\(\{S​\(c′\)\}c′∈ℬ\)\}\\mathcal\{K\}\\leftarrow\\bigcup\\bigl\\\{c:S\(c\)\\in\\mathrm\{TopK\}\_\{k\_\{b\}\}\(\\\{S\(c^\{\\prime\}\)\\\}\_\{c^\{\\prime\}\\in\\mathcal\{B\}\}\)\\bigr\\\}

Support\-only diagnostic score\.Aggregating attention mass alone gives the support\-only diagnostic

Acc⁡\(c\)=∑u∈𝒰ru​∑l∈ℒ⋆βl​∑hal,h,u​\(c\),\\operatorname\{Acc\}\(c\)=\\sum\_\{u\\in\\mathcal\{U\}\}r\_\{u\}\\sum\_\{l\\in\\mathcal\{L\}\_\{\\star\}\}\\beta\_\{l\}\\sum\_\{h\}a\_\{l,h,u\}\(c\),which answers only whether blockcclies in the predicted decode\-side support region\. It is separate from Module II’s output\-distortion surrogate; we expose it in the runner as an optional channel for Module I\-only ablations\.

The extension hierarchy is separate from the default selector\. Grouped proxies are introduced only when the prompt explicitly contains multiple competing targets\. Soft\-robust aggregation replaces the pooled weighted sum with a log\-sum\-exp score over groups, and the reserve branch keepsrrblocks with the largest groupwise distortion\.Identity check\.WhenM=1M\{=\}1, soft\-robust aggregation is disabled, andr=0r\{=\}0, these choices reduce exactly to the pooled block\-aligned default in Algorithm[1](https://arxiv.org/html/2605.08234#alg1)\. In that case,D1​\(c\)D\_\{1\}\(c\)is the only group score,S​\(c\)=D1​\(c\)S\(c\)=D\_\{1\}\(c\),𝒵res=∅\\mathcal\{Z\}\_\{\\mathrm\{res\}\}=\\emptyset, and the selected block set is the standard top\-kbk\_\{b\}set\. Soft\-robust aggregation and the reserve branch are appendix extensions, not part of the main default\. The multi\-target stress case is reported in[Appendix˜I](https://arxiv.org/html/2605.08234#A9)\.

### D\.1Grouped Proxy Objective under therr\-Channel Bound

Under the mode\-separation assumption of[Theorem˜4](https://arxiv.org/html/2605.08234#Thmtheorem4), everyr=1r\{=\}1proxy, including the pooled default, incurs a budget\-independent lower bound1−ε−wmax1\-\\varepsilon\-w\_\{\\max\}on labeled multi\-target decoding, with the equal\-weight case reducing to1−ε−1/n1\-\\varepsilon\-1/n\. The same prefill\-visible proxy bank admits a grouped formulation by assigning proxies to target slots\. Let𝒰=𝒰tail∪𝒰anchor=⨆m=1M𝒰m\\mathcal\{U\}=\\mathcal\{U\}^\{\\mathrm\{tail\}\}\\cup\\mathcal\{U\}^\{\\mathrm\{anchor\}\}=\\bigsqcup\_\{m=1\}^\{M\}\\mathcal\{U\}\_\{m\}be a prefill\-visible proxy bank with weights defined as in[Equation˜10](https://arxiv.org/html/2605.08234#S3.E10), and let

Dm​\(c\):=∑u∈𝒰mru​∑l∈ℒ⋆βl​∑hdl,h,u​\(c\),D\_\{m\}\(c\)\\;:=\\;\\sum\_\{u\\in\\mathcal\{U\}\_\{m\}\}r\_\{u\}\\sum\_\{l\\in\\mathcal\{L\}\_\{\\star\}\}\\beta\_\{l\}\\sum\_\{h\}d\_\{l,h,u\}\(c\),wheredl,h,u​\(c\)d\_\{l,h,u\}\(c\)is the block\-level output\-distortion surrogate used by the canonical block\-level default\. The corresponding soft\-robust pooled score is

Srob​\(c\)=τg​log​∑m=1Mexp⁡\(Dm​\(c\)τg\),S\_\{\\mathrm\{rob\}\}\(c\)=\\tau\_\{g\}\\log\\sum\_\{m=1\}^\{M\}\\exp\\\!\\left\(\\frac\{D\_\{m\}\(c\)\}\{\\tau\_\{g\}\}\\right\),optionally combined with a tiny outlier reserve\. WithM=1M\{=\}1, this reduces to the canonical pooled setting; withM\>1M\{\>\}1, it gives the grouped multi\-target objective used to interpret the factorized retrieval comparison under the same prefill\-visible, output\-aware principle\. The grouped objective changes the channel structure only when the prompt itself exposes multiple target slots\. It is not used to tune the main LongBench results; it is the algorithmic counterpart of therr\-channel theory in[Appendix˜E](https://arxiv.org/html/2605.08234#A5)\.

### D\.2Scope and Limitations

The reference selector and optional extensions below define the controlled ablations\. The default method is the pooled block\-aligned recipe; grouped proxies, soft\-robust aggregation, and the reserve branch are activated only for the appendix stress tests that require them\. The pseudocode assumes access to attention and value tensors during prefill and does not cover serving stacks that hide or quantize those tensors before selection\.

## Appendix EFormal Backbone for Multi\-Objective Phase Dilution

This appendix isolates the multi\-objective part of Phase Dilution as a rate\-limited proxy problem\. The main object is the labeled decode law, not the unlabeled marginal\. The lower bound is therefore about target\-slot collapse, not about finite\-sample failure of a particular estimator\. If one drops the slot label and keeps only the unlabeled mixture∑mwm​νm\\sum\_\{m\}w\_\{m\}\\nu\_\{m\}, a pooled proxy can match that marginal exactly, so no non\-trivial TV floor independent of the target mixture survives on the unlabeled object\. The appendix defines the labeled law and therr\-channel proxy class, then proves the lower bound and records the pooled and slot\-factorized consequences used by the retrieval discussion\. All formal statements below use the fixed selector contract in[˜1](https://arxiv.org/html/2605.08234#Thmassumption1); the variable is the proxy channel count\.

Candidate factorized objective\.Output\-aware pruning\[[8](https://arxiv.org/html/2605.08234#bib.bib8),[16](https://arxiv.org/html/2605.08234#bib.bib16)\]suggests a labeled multi\-target variant\. It keeps Module I’s support recovery and Module II’s renormalized block distortion fixed, then replaces the pooled block score by a labeled objective\. The canonical pooled score is theM=1M\{=\}1operating point; factorized or soft\-robust retention applies whenM\>1M\{\>\}1and the decode law fragments across competing targets\.

LetJ∈\[n\]:=\{1,…,n\}J\\in\[n\]:=\\\{1,\\ldots,n\\\}denote the target slot or query mode, withPr⁡\(J=m\)=wm\\Pr\(J=m\)=w\_\{m\}and∑m=1nwm=1\\sum\_\{m=1\}^\{n\}w\_\{m\}=1\. Letνm∈𝒫​\(𝒬\)\\nu\_\{m\}\\in\\mathcal\{P\}\(\\mathcal\{Q\}\)denote the decode\-query law conditional on slotmm\. The labeled decode law keeps the slot identity:

μdec:=∑m=1nwm​δm⊗νm∈𝒫​\(\[n\]×𝒬\)\.\\mu\_\{\\mathrm\{dec\}\}\\;:=\\;\\sum\_\{m=1\}^\{n\}w\_\{m\}\\,\\delta\_\{m\}\\otimes\\nu\_\{m\}\\;\\in\\;\\mathcal\{P\}\(\[n\]\\times\\mathcal\{Q\}\)\.\(13\)
Forr≤nr\\leq n, anrr\-channel proxy is any law of the form

η~:=∑m=1nwm​δm⊗ησ​\(m\),\\tilde\{\\eta\}\\;:=\\;\\sum\_\{m=1\}^\{n\}w\_\{m\}\\,\\delta\_\{m\}\\otimes\\eta\_\{\\sigma\(m\)\},\(14\)whereσ:\[n\]→\[r\]\\sigma:\[n\]\\to\[r\]is a routing map andη1,…,ηr∈𝒫​\(𝒬\)\\eta\_\{1\},\\ldots,\\eta\_\{r\}\\in\\mathcal\{P\}\(\\mathcal\{Q\}\)are shared proxy channels\. \([14](https://arxiv.org/html/2605.08234#A5.E14)\) is deliberately generous\. It already gives the proxy access to the slot identity throughσ​\(m\)\\sigma\(m\)\. Pooled retention is the special caser=1r=1\. Slot\-factorized retention is the special caser=nr=nwithσ​\(m\)=m\\sigma\(m\)=m\.

The labeled decode law isε\\varepsilon\-separated when there exist pairwise disjoint measurable basinsℛ1,…,ℛn\\mathcal\{R\}\_\{1\},\\ldots,\\mathcal\{R\}\_\{n\}in𝒬\\mathcal\{Q\}such that

νm​\(ℛm\)≥1−ε,m=1,…,n\.\\nu\_\{m\}\(\\mathcal\{R\}\_\{m\}\)\\;\\geq\\;1\-\\varepsilon,\\qquad m=1,\\ldots,n\.\(15\)Under QK\-normalized query embeddings, one concrete choice is to work on the unit sphere with geodesic distance and letcmc\_\{m\}be a prototype direction for modemm\. If the minimum pairwise angle isΔmin:=minm≠m′⁡d​\(cm,cm′\)\\Delta\_\{\\min\}:=\\min\_\{m\\neq m^\{\\prime\}\}d\(c\_\{m\},c\_\{m^\{\\prime\}\}\), then any radiusρ<Δmin/2\\rho<\\Delta\_\{\\min\}/2yields pairwise disjoint metric ballsℛm:=\{q:d​\(q,cm\)≤ρ\}\\mathcal\{R\}\_\{m\}:=\\\{q:d\(q,c\_\{m\}\)\\leq\\rho\\\}\. The separation assumption then reduces to requiring that eachνm\\nu\_\{m\}places at least1−ε1\-\\varepsilonof its mass in its own ball\.

###### Theorem 4\(rr\-channel TV lower bound\)\.

Assume \([15](https://arxiv.org/html/2605.08234#A5.E15)\) and sort the mode weights asw\(1\)≥⋯≥w\(n\)w\_\{\(1\)\}\\geq\\cdots\\geq w\_\{\(n\)\}\. Then everyrr\-channel proxyη~\\tilde\{\\eta\}satisfies

TV⁡\(μdec,η~\)≥1−ε−∑j=1rw\(j\)\.\\operatorname\{TV\}\(\\mu\_\{\\mathrm\{dec\}\},\\tilde\{\\eta\}\)\\;\\geq\\;1\-\\varepsilon\-\\sum\_\{j=1\}^\{r\}w\_\{\(j\)\}\.\(16\)

###### Proof of[Theorem˜4](https://arxiv.org/html/2605.08234#Thmtheorem4)\.

Let

A:=\{\(m,q\)∈\[n\]×𝒬:q∈ℛm\}\.A\\;:=\\;\\\{\(m,q\)\\in\[n\]\\times\\mathcal\{Q\}:q\\in\\mathcal\{R\}\_\{m\}\\\}\.\(17\)By \([15](https://arxiv.org/html/2605.08234#A5.E15)\),

μdec​\(A\)=∑m=1nwm​νm​\(ℛm\)≥1−ε\.\\mu\_\{\\mathrm\{dec\}\}\(A\)=\\sum\_\{m=1\}^\{n\}w\_\{m\}\\nu\_\{m\}\(\\mathcal\{R\}\_\{m\}\)\\geq 1\-\\varepsilon\.\(18\)Forη~\\tilde\{\\eta\}in \([14](https://arxiv.org/html/2605.08234#A5.E14)\),

η~​\(A\)=∑m=1nwm​ησ​\(m\)​\(ℛm\)\.\\tilde\{\\eta\}\(A\)=\\sum\_\{m=1\}^\{n\}w\_\{m\}\\eta\_\{\\sigma\(m\)\}\(\\mathcal\{R\}\_\{m\}\)\.\(19\)WriteGj:=σ−1​\(j\)G\_\{j\}:=\\sigma^\{\-1\}\(j\)\. Since the basins are pairwise disjoint,

∑m∈Gjηj​\(ℛm\)≤1\.\\sum\_\{m\\in G\_\{j\}\}\\eta\_\{j\}\(\\mathcal\{R\}\_\{m\}\)\\leq 1\.\(20\)Therefore

∑m∈Gjwm​ηj​\(ℛm\)≤\(maxm∈Gj⁡wm\)​∑m∈Gjηj​\(ℛm\)≤maxm∈Gj⁡wm,\\sum\_\{m\\in G\_\{j\}\}w\_\{m\}\\eta\_\{j\}\(\\mathcal\{R\}\_\{m\}\)\\leq\\Bigl\(\\max\_\{m\\in G\_\{j\}\}w\_\{m\}\\Bigr\)\\sum\_\{m\\in G\_\{j\}\}\\eta\_\{j\}\(\\mathcal\{R\}\_\{m\}\)\\leq\\max\_\{m\\in G\_\{j\}\}w\_\{m\},\(21\)and summing over channels gives

η~​\(A\)≤∑j=1rmaxm∈Gj⁡wm\.\\tilde\{\\eta\}\(A\)\\leq\\sum\_\{j=1\}^\{r\}\\max\_\{m\\in G\_\{j\}\}w\_\{m\}\.\(22\)The largest possible right\-hand side is attained by assigning therrlargest weights to distinct channels, so

η~​\(A\)≤∑j=1rw\(j\)\.\\tilde\{\\eta\}\(A\)\\leq\\sum\_\{j=1\}^\{r\}w\_\{\(j\)\}\.\(23\)Finally,

TV⁡\(μdec,η~\)≥μdec​\(A\)−η~​\(A\)≥1−ε−∑j=1rw\(j\)\.\\operatorname\{TV\}\(\\mu\_\{\\mathrm\{dec\}\},\\tilde\{\\eta\}\)\\geq\\mu\_\{\\mathrm\{dec\}\}\(A\)\-\\tilde\{\\eta\}\(A\)\\geq 1\-\\varepsilon\-\\sum\_\{j=1\}^\{r\}w\_\{\(j\)\}\.\(24\)∎

Two consequences are used in the multi\-target retrieval discussion\. For pooled retention,r=1r=1, so every pooled proxy satisfies

TV⁡\(μdec,η~pool\)≥1−ε−wmax\.\\operatorname\{TV\}\(\\mu\_\{\\mathrm\{dec\}\},\\tilde\{\\eta\}\_\{\\mathrm\{pool\}\}\)\\;\\geq\\;1\-\\varepsilon\-w\_\{\\max\}\.\(25\)In the equal\-weight casewm=1/nw\_\{m\}=1/n, this becomes

TV⁡\(μdec,η~pool\)≥1−ε−1n\.\\operatorname\{TV\}\(\\mu\_\{\\mathrm\{dec\}\},\\tilde\{\\eta\}\_\{\\mathrm\{pool\}\}\)\\;\\geq\\;1\-\\varepsilon\-\\frac\{1\}\{n\}\.\(26\)This is[Theorem˜4](https://arxiv.org/html/2605.08234#Thmtheorem4)withr=1r=1, where∑j=1rw\(j\)=wmax\\sum\_\{j=1\}^\{r\}w\_\{\(j\)\}=w\_\{\\max\}; the equal\-weight display useswmax=1/nw\_\{\\max\}=1/n\. The bound already allows the pooled proxy to know the slot identity through the routing mapσ\\sigma\. Even under that favorable relaxation, a single shared channel cannot avoid label collapse\. The lower bound is an irreducible approximation floor of the one\-channel class\.

In contrast, let

η~fact,K:=∑m=1nwm​δm⊗ηm,K\\tilde\{\\eta\}\_\{\\mathrm\{fact\},K\}\\;:=\\;\\sum\_\{m=1\}^\{n\}w\_\{m\}\\,\\delta\_\{m\}\\otimes\\eta\_\{m,K\}\(27\)be a slot\-factorized proxy with one channel per slot\. IfTV⁡\(ηm,K,νm\)→0\\operatorname\{TV\}\(\\eta\_\{m,K\},\\nu\_\{m\}\)\\to 0for everymm, then disjointness of the slot labels gives

TV⁡\(μdec,η~fact,K\)=∑m=1nwm​TV⁡\(νm,ηm,K\),\\operatorname\{TV\}\(\\mu\_\{\\mathrm\{dec\}\},\\tilde\{\\eta\}\_\{\\mathrm\{fact\},K\}\)=\\sum\_\{m=1\}^\{n\}w\_\{m\}\\,\\operatorname\{TV\}\(\\nu\_\{m\},\\eta\_\{m,K\}\),\(28\)soTV⁡\(μdec,η~fact,K\)→0\\operatorname\{TV\}\(\\mu\_\{\\mathrm\{dec\}\},\\tilde\{\\eta\}\_\{\\mathrm\{fact\},K\}\)\\to 0by dominated convergence\.

For an end\-task functionalΦ:𝒫​\(\[n\]×𝒬\)→ℝ\\Phi:\\mathcal\{P\}\(\[n\]\\times\\mathcal\{Q\}\)\\to\\mathbb\{R\}satisfying

\|Φ​\(μ\)−Φ​\(ν\)\|≤L​TV⁡\(μ,ν\)\|\\Phi\(\\mu\)\-\\Phi\(\\nu\)\|\\leq L\\,\\operatorname\{TV\}\(\\mu,\\nu\)\(29\)for allμ,ν\\mu,\\nu, an empiricalrr\-channel proxyη~K,r\\tilde\{\\eta\}\_\{K,r\}and its population counterpartη~r⋆\\tilde\{\\eta\}^\{\\star\}\_\{r\}satisfy the triangle\-inequality bridge

\|Φ​\(μdec\)−Φ​\(η~K,r\)\|≤L​TV⁡\(μdec,η~K,r\)≤L​\[TV⁡\(μdec,η~r⋆\)\+TV⁡\(η~r⋆,η~K,r\)\]\.\|\\Phi\(\\mu\_\{\\mathrm\{dec\}\}\)\-\\Phi\(\\tilde\{\\eta\}\_\{K,r\}\)\|\\leq L\\,\\operatorname\{TV\}\(\\mu\_\{\\mathrm\{dec\}\},\\tilde\{\\eta\}\_\{K,r\}\)\\leq L\\Bigl\[\\operatorname\{TV\}\(\\mu\_\{\\mathrm\{dec\}\},\\tilde\{\\eta\}^\{\\star\}\_\{r\}\)\+\\operatorname\{TV\}\(\\tilde\{\\eta\}^\{\\star\}\_\{r\},\\tilde\{\\eta\}\_\{K,r\}\)\\Bigr\]\.\(30\)If

TV⁡\(η~r⋆,η~K,r\)≤Δcont​\(K\)\+Δest​\(K\),\\operatorname\{TV\}\(\\tilde\{\\eta\}^\{\\star\}\_\{r\},\\tilde\{\\eta\}\_\{K,r\}\)\\leq\\Delta\_\{\\mathrm\{cont\}\}\(K\)\+\\Delta\_\{\\mathrm\{est\}\}\(K\),\(31\)then this becomes

\|Φ​\(μdec\)−Φ​\(η~K,r\)\|≤L​\[TV⁡\(μdec,η~r⋆\)\+Δcont​\(K\)\+Δest​\(K\)\]\.\|\\Phi\(\\mu\_\{\\mathrm\{dec\}\}\)\-\\Phi\(\\tilde\{\\eta\}\_\{K,r\}\)\|\\leq L\\Bigl\[\\operatorname\{TV\}\(\\mu\_\{\\mathrm\{dec\}\},\\tilde\{\\eta\}^\{\\star\}\_\{r\}\)\+\\Delta\_\{\\mathrm\{cont\}\}\(K\)\+\\Delta\_\{\\mathrm\{est\}\}\(K\)\\Bigr\]\.\(32\)Under standard LLN or mixing assumptions on the tail sample,Δest​\(K\)=Oℙ​\(K−1/2\)\\Delta\_\{\\mathrm\{est\}\}\(K\)=O\_\{\\mathbb\{P\}\}\(K^\{\-1/2\}\)\.

[Theorem˜4](https://arxiv.org/html/2605.08234#Thmtheorem4)gives a formal model for one branching bottleneck seen empirically in the 32k multi\-objective retrieval table\. Pooled selection is a one\-channel proxy class and retains the floor in \([25](https://arxiv.org/html/2605.08234#A5.E25)\) when the labeled\-law assumptions hold\. Factorized selection removes that label\-collapse term by allocating one channel per target slot, as shown in \([28](https://arxiv.org/html/2605.08234#A5.E28)\)\. The theorem does not by itself set score ceilings or identify the optimal proxy bank\. IncreasingKKcan only reduce contamination and estimation error around that floor; it cannot remove the floor itself\. If each mode\-conditioned channel has a single\-mode saturation budgetK0=O​\(1\)K\_\{0\}=O\(1\), then the useful total budget of a factorized proxy scales asn​K0nK\_\{0\}, whereas the pooled proxy saturates once the non\-floor terms become small\.

### E\.1Tail\-KKConsistency via Cesàro Sufficient Condition

Letρj:=Pr⁡\(qT−j∼μP\)\\rho\_\{j\}:=\\Pr\(q\_\{T\-j\}\\sim\\mu\_\{P\}\)denote the decode\-proximity rate of thejj\-th tail query, and letc¯K:=1K​∑j=0K−1\(1−ρj\)\\bar\{c\}\_\{K\}:=\\frac\{1\}\{K\}\\sum\_\{j=0\}^\{K\-1\}\(1\-\\rho\_\{j\}\)denote the tail contamination\. Under a characteristic kernel and a standard LLN/mixing condition on the tail sample, vanishing tail contamination asKKgrows is sufficient for the tail\-KKempirical measureμ^tail,K\\widehat\{\\mu\}\_\{\\mathrm\{tail\},K\}to converge in MMD toμP\\mu\_\{P\}\. The converse is not used here\. Persistent contamination can be MMD\-invisible when its RKHS mean embedding matches or cancels against that ofμP\\mu\_\{P\}\. Under the stronger assumption thatρj\\rho\_\{j\}is monotone injjon the tail, the Cesàro condition is equivalent to tail proximity rising to one asjjgrows\. The weaker Cesàro form is what actually underlies the diagnostic prediction in[Section˜2](https://arxiv.org/html/2605.08234#S2), and the monotone version is a sufficient special case but is not required\.

### E\.2QK\-norm as Temperature Stabilization under Mismatch

Letβ\\betadenote the effective post\-normalization attention sharpness, and letβq\\beta\_\{q\}denote the pre\-normalization query\-wise sharpness induced by norm fluctuations\. The key role of QK\-norm is to collapse the extra heteroscedasticity coming fromβq\\beta\_\{q\}; it does not imply a monotone decrease of unconditional logit variance across all query laws\. For a tail\-KKproxy with query\-distribution mismatchDK:=W1​\(μtail,K,μP\)D\_\{K\}:=W\_\{1\}\(\\mu\_\{\\mathrm\{tail\},K\},\\mu\_\{P\}\), the softmax Jacobian has norm bounded byβ​L0\\beta L\_\{0\}for someL0\>0L\_\{0\}\>0, giving

‖Sβ​\(μtail,K\)−Sβ​\(μP\)‖1≤β​L0​DK,MSEK​\(β\)≲β2​L02​DK2\+Vwithin​\(β\)\+VtempK\.\\bigl\\\|\\,S\_\{\\beta\}\(\\mu\_\{\\mathrm\{tail\},K\}\)\-S\_\{\\beta\}\(\\mu\_\{P\}\)\\bigr\\\|\_\{1\}\\;\\leq\\;\\beta L\_\{0\}\\,D\_\{K\},\\qquad\\mathrm\{MSE\}\_\{K\}\(\\beta\)\\;\\lesssim\\;\\beta^\{2\}L\_\{0\}^\{2\}D\_\{K\}^\{2\}\\;\+\\;\\tfrac\{V\_\{\\mathrm\{within\}\}\(\\beta\)\+V\_\{\\mathrm\{temp\}\}\}\{K\}\.HereVtempV\_\{\\mathrm\{temp\}\}is the query\-temperature component driven by variability inβq\\beta\_\{q\}, and this term is precisely what QK\-norm suppresses\. The residual termVwithin​\(β\)V\_\{\\mathrm\{within\}\}\(\\beta\)captures within\-query sampling variance and has no monotone law inβ\\betaacross all query laws\. The practical implication is conditional\. QK\-norm helps whenDKD\_\{K\}is low because it removes query\-wise temperature noise\. Under severe mismatch, the sharper transfer can also magnify the bias termβ​L0​DK\\beta L\_\{0\}D\_\{K\}\. This gives one possible mechanism for the observed Qwen3 pattern\. Full\-retention single\-key performance can coexist with weakness on the two\-key variant when the one\-channel floor in \([25](https://arxiv.org/html/2605.08234#A5.E25)\) starts to bind\.

### E\.3RoPE Locality Surrogate

RoPE has a finite frequency\-mixture structure\. Writing

K​\(Δ\)=∑j=1dh/2aj​cos⁡\(ωj​Δ\),ωj=θ−2​\(j−1\)/dh,K\(\\Delta\)\\;=\\;\\sum\_\{j=1\}^\{d\_\{h\}/2\}a\_\{j\}\\cos\(\\omega\_\{j\}\\Delta\),\\qquad\\omega\_\{j\}=\\theta^\{\-2\(j\-1\)/d\_\{h\}\},with fixed nonnegative weights andK​\(0\)\>0K\(0\)\>0, Taylor expansion atΔ=0\\Delta=0gives

K​\(Δ\)K​\(0\)=1−12​λeff2​Δ2\+O​\(Δ4\),λeff2:=∑jaj​ωj2∑jaj,∂λeff2/∂θ≤0,\\frac\{K\(\\Delta\)\}\{K\(0\)\}\\;=\\;1\-\\tfrac\{1\}\{2\}\\lambda\_\{\\mathrm\{eff\}\}^\{2\}\\,\\Delta^\{2\}\+O\(\\Delta^\{4\}\),\\qquad\\lambda\_\{\\mathrm\{eff\}\}^\{2\}:=\\frac\{\\sum\_\{j\}a\_\{j\}\\omega\_\{j\}^\{2\}\}\{\\sum\_\{j\}a\_\{j\}\},\\qquad\\partial\\lambda\_\{\\mathrm\{eff\}\}^\{2\}/\\partial\\theta\\leq 0,with strict inequality when positive weight falls on at least oneθ\\theta\-dependent frequency\. Matching this local quadratic to a Gaussian gives a small\-Δ\\Deltaapproximation, not a global envelope for finite RoPE frequency sums\. Largerθ\\thetaweakens this local\-curvature proxy under that same weight condition\. The exponential kernel in[Appendix˜F](https://arxiv.org/html/2605.08234#A6)is used as a tractable one\-parameter surrogate; the conclusions about tail\-high residual curvature require a chosen non\-increasing surrogate, not a global RoPE envelope\.

### E\.4Scope and Limitations

This appendix analyzes a labeled multi\-target proxy class under explicit mode separation\. The bound applies to target\-slot collapse in the labeled law and does not claim a floor for all unlabeled marginals\. The result explains why pooled multi\-target retrieval can saturate early and why factorized retention is the relevant comparison when target slots are available in the prompt\.

## Appendix FExposure Bias Theoretical Derivations

This appendix gives the derivations behind[Section˜2](https://arxiv.org/html/2605.08234#S2): the causal\-softmax decomposition, special cases, a tractable kernel surrogate, the MSE\-optimal observation count, the correlated\-noise extension, and the reverse\-depth interpretation\. Symbols follow[Section˜2](https://arxiv.org/html/2605.08234#S2), namelyCiC\_\{i\},NiN\_\{i\},μ​\(Δ\)\\mu\(\\Delta\),SiS\_\{i\}, andu^i\\widehat\{u\}\_\{i\}\. Each subsection isolates one assumption change and states the implication for count debiasing\. The derivations identify which residuals the experiments should measure separately\.

[Figure˜5](https://arxiv.org/html/2605.08234#A6.F5)separates the residual mean from the variance after count debiasing\. The mean correction bends mass toward the prompt tail, while the variance penalty appears mainly in the same tail region\.

![Refer to caption](https://arxiv.org/html/2605.08234v1/x11.png)

\(a\)
![Refer to caption](https://arxiv.org/html/2605.08234v1/x12.png)

\(b\)

Figure 5:Exposure\-debiasing residuals\.### F\.1General Decomposition

Absorbing layer and head sums intoAt,i:=∑L,hAL,h​\(qt,ki\)A\_\{t,i\}:=\\sum\_\{L,h\}A\_\{L,h\}\(q\_\{t\},k\_\{i\}\)and writingμ​\(Δ\):=𝔼​\[At,t−Δ\]\\mu\(\\Delta\):=\\mathbb\{E\}\[A\_\{t,t\-\\Delta\}\]andSi:=∑t=iT\(𝔼​\[At,i∣content\]−μ​\(t−i\)\)S\_\{i\}:=\\sum\_\{t=i\}^\{T\}\\bigl\(\\mathbb\{E\}\[A\_\{t,i\}\\mid\\text\{content\}\]\-\\mu\(t\-i\)\\bigr\)gives

𝔼​\[Ci\]=∑Δ=0Ni−1μ​\(Δ\)\+Si\.\\mathbb\{E\}\[C\_\{i\}\]\\;=\\;\\sum\_\{\\Delta=0\}^\{N\_\{i\}\-1\}\\mu\(\\Delta\)\\;\+\\;S\_\{i\}\.The main\-text form𝔼​\[Ci\]=βpos​Ni\+Si\\mathbb\{E\}\[C\_\{i\}\]=\\beta\_\{\\mathrm\{pos\}\}N\_\{i\}\+S\_\{i\}is the special caseμ​\(Δ\)≡βpos\\mu\(\\Delta\)\\equiv\\beta\_\{\\mathrm\{pos\}\}\. Under a decaying kernel the correct count factor isNieff:=βpos−1​∑Δ=0Ni−1μ​\(Δ\)N\_\{i\}^\{\\mathrm\{eff\}\}:=\\beta\_\{\\mathrm\{pos\}\}^\{\-1\}\\sum\_\{\\Delta=0\}^\{N\_\{i\}\-1\}\\mu\(\\Delta\)\.

### F\.2Exchangeable Softmax \(Uniform Baseline\)

If the logit vector\(Zt,1,…,Zt,t\)\(Z\_\{t,1\},\\ldots,Z\_\{t,t\}\)is exchangeable, symmetry plus the simplex constraint∑iAt,i=1\\sum\_\{i\}A\_\{t,i\}=1gives𝔼​\[At,i∣t\]=1/t\\mathbb\{E\}\[A\_\{t,i\}\\mid t\]=1/tfor alli≤ti\\leq t\. Hence

𝔼​\[Ci\]=∑t=iT1t=HT−Hi−1,\\mathbb\{E\}\[C\_\{i\}\]\\;=\\;\\sum\_\{t=i\}^\{T\}\\frac\{1\}\{t\}\\;=\\;H\_\{T\}\-H\_\{i\-1\},a harmonic sum of orderΘ​\(log⁡T\)\\Theta\(\\log T\), notΘ​\(Ni\)\\Theta\(N\_\{i\}\)\. The linear approximationβpos​Ni\\beta\_\{\\mathrm\{pos\}\}N\_\{i\}is valid only within a thin tail window\[T−w\+1,T\]\[T\-w\+1,T\]withw=o​\(T\)w=o\(T\), where1/t=1/T\+O​\(w/T2\)1/t=1/T\+O\(w/T^\{2\}\)\.

### F\.3RoPE Kernel Structure

If the query and key distributions are rotation\-invariant,𝐪⊤​𝐑Δ​𝐤=d𝐪⊤​𝐤\\mathbf\{q\}^\{\\top\}\\mathbf\{R\}\_\{\\Delta\}\\mathbf\{k\}\\stackrel\{\{\\scriptstyle d\}\}\{\{=\}\}\\mathbf\{q\}^\{\\top\}\\mathbf\{k\}, so RoPE alone induces no decay:μ​\(Δ\)≡const\\mu\(\\Delta\)\\equiv\\text\{const\}\. Non\-trivialμ​\(Δ\)\\mu\(\\Delta\)arises from learned anisotropy of the query\-key spectra\. Writing the head\-averaged relative\-position kernel as a Fourier transformμ​\(Δ\)=∫ei​ω​Δ​𝑑ν​\(ω\)\\mu\(\\Delta\)=\\int e^\{i\\omega\\Delta\}\\,d\\nu\(\\omega\)for a finite measureν\\nuwithL1L^\{1\}density, Riemann–Lebesgue gives convergence ofμ​\(Δ\)\\mu\(\\Delta\)to zero asΔ\\Deltagrows\.

[Section˜E\.3](https://arxiv.org/html/2605.08234#A5.SS3)gives a small\-Δ\\DeltaGaussian surrogate matched to the finite RoPE frequency sum, not a global RoPE envelope; we use the exponential form as a tractable one\-parameter surrogate that retains monotone decay\. The statements on residual curvature below require only that the chosen surrogateμ\\muis non\-increasing\. For the exponential kernelμ​\(Δ\)=βpos​e−λ​Δ\\mu\(\\Delta\)=\\beta\_\{\\mathrm\{pos\}\}\\,e^\{\-\\lambda\\Delta\},

Nieff=1−e−λ​Ni1−e−λ,r​\(N\):=NieffNi=1−e−λ​N\(1−e−λ\)​N\.N\_\{i\}^\{\\mathrm\{eff\}\}\\;=\\;\\frac\{1\-e^\{\-\\lambda N\_\{i\}\}\}\{1\-e^\{\-\\lambda\}\},\\qquad r\(N\)\\;:=\\;\\frac\{N\_\{i\}^\{\\mathrm\{eff\}\}\}\{N\_\{i\}\}\\;=\\;\\frac\{1\-e^\{\-\\lambda N\}\}\{\(1\-e^\{\-\\lambda\}\)\\,N\}\.Differentiating,r′​\(N\)=e−λ​N​\(1\+λ​N\)−1\(1−e−λ\)​N2<0r^\{\\prime\}\(N\)=\\frac\{e^\{\-\\lambda N\}\(1\+\\lambda N\)\-1\}\{\(1\-e^\{\-\\lambda\}\)N^\{2\}\}<0for allN≥1N\\geq 1\(usingex\>1\+xe^\{x\}\>1\+x\)\. Naive division byNiN\_\{i\}therefore over\-corrects head and middle positions and leaves a tail\-high deterministic residualβpos​r​\(Ni\)\\beta\_\{\\mathrm\{pos\}\}r\(N\_\{i\}\)\.

### F\.4MSE\-Optimal Observation Count

Model per\-query noise as iid with varianceσA2\\sigma\_\{A\}^\{2\}\. Thenu^i=Si/Ni\+ϵ¯i\\widehat\{u\}\_\{i\}=S\_\{i\}/N\_\{i\}\+\\bar\{\\epsilon\}\_\{i\}withϵ¯i∼𝒩​\(0,σA2/Ni\)\\bar\{\\epsilon\}\_\{i\}\\sim\\mathcal\{N\}\(0,\\sigma\_\{A\}^\{2\}/N\_\{i\}\), yielding

MSEi​\(N\)=\(1−1N\)2​Si2\+σA2N\.\\mathrm\{MSE\}\_\{i\}\(N\)\\;=\\;\\Bigl\(1\-\\tfrac\{1\}\{N\}\\Bigr\)^\{\\\!2\}S\_\{i\}^\{2\}\\;\+\\;\\frac\{\\sigma\_\{A\}^\{2\}\}\{N\}\.Expanding and differentiating,

d​MSEid​N=\(2​Si2−σA2\)​N−2​Si2N3,\\frac\{d\\,\\mathrm\{MSE\}\_\{i\}\}\{dN\}\\;=\\;\\frac\{\(2S\_\{i\}^\{2\}\-\\sigma\_\{A\}^\{2\}\)\\,N\-2S\_\{i\}^\{2\}\}\{N^\{3\}\},with closed\-form minimizer

Ni⋆=\{2​Si22​Si2−σA2if​2​Si2\>σA2,\+∞otherwise\.N\_\{i\}^\{\\star\}\\;=\\;\\begin\{cases\}\\dfrac\{2S\_\{i\}^\{2\}\}\{2S\_\{i\}^\{2\}\-\\sigma\_\{A\}^\{2\}\}&\\text\{if \}2S\_\{i\}^\{2\}\>\\sigma\_\{A\}^\{2\},\\\\\[6\.0pt\] \+\\infty&\\text\{otherwise\.\}\\end\{cases\}For strong\-signal tokens \(Si2≫σA2S\_\{i\}^\{2\}\\gg\\sigma\_\{A\}^\{2\}\), the optimum approachesNi⋆=1N\_\{i\}^\{\\star\}=1, so they are best estimated at smallNiN\_\{i\}, at the tail\. Weak\-signal tokens favorNi⋆=∞N\_\{i\}^\{\\star\}=\\inftyand are harmed by debiasing at the tail\.

### F\.5Correlated\-Noise Extension

AllowingCorr​\(ϵi,r,ϵi,r\+k\)=ρi,k\\mathrm\{Corr\}\(\\epsilon\_\{i,r\},\\epsilon\_\{i,r\+k\}\)=\\rho\_\{i,k\},

Var​\(u^i\)=σA2Ni​Γi​\(Ni\),Γi​\(N\):=1\+2​∑k=1N−1\(1−kN\)​ρi,k\.\\mathrm\{Var\}\(\\widehat\{u\}\_\{i\}\)\\;=\\;\\frac\{\\sigma\_\{A\}^\{2\}\}\{N\_\{i\}\}\\,\\Gamma\_\{i\}\(N\_\{i\}\),\\qquad\\Gamma\_\{i\}\(N\)\\;:=\\;1\+2\\sum\_\{k=1\}^\{N\-1\}\\Bigl\(1\-\\tfrac\{k\}\{N\}\\Bigr\)\\rho\_\{i,k\}\.If∑k\|ρi,k\|<∞\\sum\_\{k\}\|\\rho\_\{i,k\}\|<\\infty,Γi​\(Ni\)\\Gamma\_\{i\}\(N\_\{i\}\)converges toci:=1\+2​∑k≥1ρi,kc\_\{i\}:=1\+2\\sum\_\{k\\geq 1\}\\rho\_\{i,k\}, and the independent\-noise minimizer extends toNi⋆=2​Si2/\(2​Si2−ci​σA2\)N\_\{i\}^\{\\star\}=2S\_\{i\}^\{2\}/\(2S\_\{i\}^\{2\}\-c\_\{i\}\\sigma\_\{A\}^\{2\}\)when the denominator is positive, meaning2​Si2\>ci​σA22S\_\{i\}^\{2\}\>c\_\{i\}\\sigma\_\{A\}^\{2\}\.

### F\.6Reverse\-Depth Has Two Mechanisms

A tail\-variance\-only explanation is too strong\. Two counterexamples show the issue\.

- •YH∼𝒩​\(10,1\)Y\_\{H\}\\sim\\mathcal\{N\}\(10,1\),YT∼𝒩​\(0,9\)Y\_\{T\}\\sim\\mathcal\{N\}\(0,9\): tail variance is larger, yet no reverse\-depth occurs; the head remains dominant at any high threshold\.
- •YH∼𝒩​\(0,1\)Y\_\{H\}\\sim\\mathcal\{N\}\(0,1\),YT∼𝒩​\(1,1\)Y\_\{T\}\\sim\\mathcal\{N\}\(1,1\): variances are equal but a tail mean shift already produces over\-retention\.

Tail variance explosion is one sufficient mechanism\. Residual mean bias from the count curvaturer​\(Ni\)r\(N\_\{i\}\)is another independent one\. Reverse\-depth failure in practice draws on both; ablations should report them separately\.

Practical reading\.Taken together, these derivations support three qualitative checks that appear in the main experiments\. A\-only repair should remove the dominant prefix drift yet expose a recency failure\. Head participation should vary across architectures even when the query proxy is held fixed\. Multi\-objective pooled retrieval should saturate earlier than factorized retention because the bottleneck there is channel count, not tail\-window size\.

Table 5:Stage I access\-support comparison atb=0\.05b=0\.05on LongBench\[[1](https://arxiv.org/html/2605.08234#bib.bib1)\]\. Each row applies the indicated repair on top of the previous row; scores are LongBench accuracy \(×100\\times 100\), andΔ\\DeltaAvg reports the incremental change in the model average\.Stage I repairLlamaQwen3MistralAvgΔ\\DeltaAvgCumulative attention8\.78\.77\.67\.65\.35\.37\.27\.2–\+\+Count debiasing29\.529\.517\.517\.530\.530\.525\.825\.818\.618\.6\+\+Decode\-proximal queries32\.832\.828\.328\.338\.638\.633\.233\.27\.47\.4\+\+Layer denoising32\.932\.930\.030\.037\.337\.333\.433\.40\.20\.2FullKVreference35\.735\.731\.631\.639\.539\.535\.635\.6–- •Avg averages Llama\[[43](https://arxiv.org/html/2605.08234#bib.bib43)\], Qwen3\[[39](https://arxiv.org/html/2605.08234#bib.bib39)\], and Mistral\[[44](https://arxiv.org/html/2605.08234#bib.bib44)\]\. The repair sequence aligns with[Section˜2](https://arxiv.org/html/2605.08234#S2): cumulative attention carries exposure\-count bias; count debiasing removes the prefix over\-counting; decode\-proximal queries replace global aggregation with the SnapKV\-style observation window; layer denoising restricts to the top\-attention layer band\.

### F\.7Scope and Limitations

This appendix studies exposure\-count bias under causal softmax and simple kernel and noise abstractions\. The derivations justify the direction of the residual diagnostics, but they do not assert that one parametric kernel explains every layer or task\. The main experimental claim relies on the measured residual mean, residual variance, and task\-level controls, with the exponential surrogate used only as a tractable model\.

## Appendix GHead\-Sum Dilution from Sparse Retrieval Heads

This appendix derives the head\-aggregation dilution used in[Proposition˜3](https://arxiv.org/html/2605.08234#Thmtheorem3)\. The mechanism compares two selector orderings: selecting keys at the head level and then combining retained candidates, or summing heads before ranking keys\. A retrieval head may identify the right key, while the head\-summed score can still dilute that signal with inactive\-head background mass\.

LetHheadH\_\{\\mathrm\{head\}\}be the number of heads and letAh​\(i\)A\_\{h\}\(i\)denote the locally aggregated attention mass that headhhassigns to candidate keyiiinside a decode\-proximal candidate set of sizenn\. HereAh​\(i\)A\_\{h\}\(i\)is already a scalar mass, so it is not bolded\. Center each head by the uniform background,

A~h​\(i\):=Ah​\(i\)−1n\.\\widetilde\{A\}\_\{h\}\(i\):=A\_\{h\}\(i\)\-\\frac\{1\}\{n\}\.A per\-head selector, or a union of per\-head top\-kklists, is sensitive to the largest head\-level contrast,

S∞​\(i\):=maxh⁡A~h​\(i\),S\_\{\\infty\}\(i\):=\\max\_\{h\}\\widetilde\{A\}\_\{h\}\(i\),whereas a head\-summed selector ranks by

SΣ​\(i\):=∑h=1HheadA~h​\(i\)\.S\_\{\\Sigma\}\(i\):=\\sum\_\{h=1\}^\{H\_\{\\mathrm\{head\}\}\}\\widetilde\{A\}\_\{h\}\(i\)\.Head\-aggregation dilution is the contrast loss induced when a sparse head\-level signal is passed through the head\-summed scoreSΣS\_\{\\Sigma\}\.

### G\.1Sparse Retrieval Signal

Leti⋆i^\{\\star\}be the target key and suppose that only a collectionℛ\\mathcal\{R\}of heads carries target\-specific retrieval evidence, with\|ℛ\|=r\|\\mathcal\{R\}\|=r\. In the stylized retrieval model,

A~h​\(i⋆\)=\{Δ,h∈ℛ,0,h∉ℛ,\\widetilde\{A\}\_\{h\}\(i^\{\\star\}\)=\\begin\{cases\}\\Delta,&h\\in\\mathcal\{R\},\\\\ 0,&h\\notin\\mathcal\{R\},\\end\{cases\}whereΔ\>0\\Delta\>0is the per\-active\-head target contrast\. A per\-head selector sees target contrastΔ\\Delta\. A head\-summed selector sees target contrastr​Δr\\Delta\.

The difference appears in the nuisance background\. Letj≠i⋆j\\neq i^\{\\star\}be a competing key and write its centered background asBj​h=A~h​\(j\)B\_\{jh\}=\\widetilde\{A\}\_\{h\}\(j\)\. Two endpoint models give the dilution factors\.

#### Structured dense background\.

If the nuisance key receives approximately the same positive backgroundτ\\taufrom many heads,Bj​h≈τB\_\{jh\}\\approx\\tau, then

SΣ​\(j\)≈Hhead​τ,SΣ​\(i⋆\)≈r​Δ\.S\_\{\\Sigma\}\(j\)\\approx H\_\{\\mathrm\{head\}\}\\tau,\\qquad S\_\{\\Sigma\}\(i^\{\\star\}\)\\approx r\\Delta\.Thus the head\-summed contrast is

SΣ​\(i⋆\)SΣ​\(j\)≈rHhead⋅Δτ\.\\frac\{S\_\{\\Sigma\}\(i^\{\\star\}\)\}\{S\_\{\\Sigma\}\(j\)\}\\approx\\frac\{r\}\{H\_\{\\mathrm\{head\}\}\}\\cdot\\frac\{\\Delta\}\{\\tau\}\.The corresponding per\-head contrast isΔ/τ\\Delta/\\tau\. Therefore summing heads before selection loses a factor

Dstruct=rHhead\.D\_\{\\mathrm\{struct\}\}=\\frac\{r\}\{H\_\{\\mathrm\{head\}\}\}\.

#### Iid zero\-mean background\.

If instead the nuisance background is zero\-mean across heads withVar⁡\(Bj​h\)=τ2\\operatorname\{Var\}\(B\_\{jh\}\)=\\tau^\{2\}, then

has standard deviationτ​Hhead\\tau\\sqrt\{H\_\{\\mathrm\{head\}\}\}up to the usual extreme\-value factor over nuisance keys\. The target remainsr​Δr\\Delta, so the standardized head\-summed contrast scales as

r​Δτ​Hhead=rHhead⋅Δτ\.\\frac\{r\\Delta\}\{\\tau\\sqrt\{H\_\{\\mathrm\{head\}\}\}\}=\\frac\{r\}\{\\sqrt\{H\_\{\\mathrm\{head\}\}\}\}\\cdot\\frac\{\\Delta\}\{\\tau\}\.The iid\-noise endpoint gives dilution

Diid=rHhead\.D\_\{\\mathrm\{iid\}\}=\\frac\{r\}\{\\sqrt\{H\_\{\\mathrm\{head\}\}\}\}\.
The structured endpoint is harsher because the nuisance background adds coherently across heads\. The iid endpoint is milder because nuisance fluctuations add only in standard deviation\. Real models can mix both settings, so the two factors are diagnostic endpoints whose constants vary by model and task\.

### G\.2Why the Active\-Head Count Can Be Small

The preceding bound assumes that retrieval evidence appears in onlyrrheads\. QK\-norm gives one reason why this can happen\. Under QK\-norm, query and key directions lie on a scaled unit sphere\. Write the head\-level logit as

zj=dh​cj,cj:=⟨𝐪^,𝐑Δj​𝐤^j⟩∈\[−1,1\]\.z\_\{j\}=\\sqrt\{d\_\{h\}\}\\,c\_\{j\},\\qquad c\_\{j\}:=\\langle\\hat\{\\mathbf\{q\}\},\\mathbf\{R\}\_\{\\Delta\_\{j\}\}\\hat\{\\mathbf\{k\}\}\_\{j\}\\rangle\\in\[\-1,1\]\.For the target keyi⋆i^\{\\star\}, define the angular gap

δ:=ci⋆−maxj≠i⋆⁡cj\.\\delta:=c\_\{i^\{\\star\}\}\-\\max\_\{j\\neq i^\{\\star\}\}c\_\{j\}\.The logit margin ism=dh​δm=\\sqrt\{d\_\{h\}\}\\,\\delta\. Since the non\-target softmax mass is bounded by\(n−1\)​e−m\(n\-1\)e^\{\-m\}, the target receives at least half the candidate mass when

m≥log⁡\(n−1\),m\\geq\\log\(n\-1\),or equivalently

δ0\.5​\(n,dh\)=log⁡\(n−1\)dh\.\\delta\_\{0\.5\}\(n,d\_\{h\}\)=\\frac\{\\log\(n\-1\)\}\{\\sqrt\{d\_\{h\}\}\}\.\(33\)Thus, as the candidate set grows, only heads whose learned geometry clears this margin can become majority\-mass retrieval heads\. The bound is a sufficient condition for strong head participation; successful retrieval can still occur below it\. Its role here is to explain whyrrcan be much smaller thanHheadH\_\{\\mathrm\{head\}\}\.

### G\.3Participation Ratio as the Measured Active\-Head Count

We estimate the effective active\-head count with the participation ratio

reff:=\(∑hmh\)2∑hmh2≤Hhead,r\_\{\\mathrm\{eff\}\}:=\\frac\{\(\\sum\_\{h\}m\_\{h\}\)^\{2\}\}\{\\sum\_\{h\}m\_\{h\}^\{2\}\}\\leq H\_\{\\mathrm\{head\}\},\(34\)where

mh=∑q∈last​64Ah​\(q,i⋆\)\.m\_\{h\}=\\sum\_\{q\\in\\mathrm\{last\}\\,64\}A\_\{h\}\(q,i^\{\\star\}\)\.If the retrieval mass is equally distributed overrrheads and negligible elsewhere, thenreff=rr\_\{\\mathrm\{eff\}\}=r\. If the mass is uniform over all heads, thenreff=Hheadr\_\{\\mathrm\{eff\}\}=H\_\{\\mathrm\{head\}\}\. Substitutingreffr\_\{\\mathrm\{eff\}\}for the idealizedrrgives the empirical dilution diagnostics

Dstruct≈reffHhead,Diid≈reffHhead\.D\_\{\\mathrm\{struct\}\}\\approx\\frac\{r\_\{\\mathrm\{eff\}\}\}\{H\_\{\\mathrm\{head\}\}\},\\qquad D\_\{\\mathrm\{iid\}\}\\approx\\frac\{r\_\{\\mathrm\{eff\}\}\}\{\\sqrt\{H\_\{\\mathrm\{head\}\}\}\}\.[Figure˜6](https://arxiv.org/html/2605.08234#A7.F6)reportsreff/Hheadr\_\{\\mathrm\{eff\}\}/H\_\{\\mathrm\{head\}\}on the 8k NIAH retrieval check\. Low participation means the retrieval signal is head\-sparse, so head\-summed scoring loses contrast relative to per\-head selection\.

![Refer to caption](https://arxiv.org/html/2605.08234v1/x13.png)Figure 6:Effective head participation on the 8k NIAH retrieval check\.
### G\.4Consequence: Reserved Observation Window as a Keep Prior

A fixed recent\-token reserve can partly mask head\-sum dilution near the decode tail\. Hard\-reserving the lastWWprefix tokens, as SnapKV\[[26](https://arxiv.org/html/2605.08234#bib.bib26)\]does withW=32W=32, is equivalent at the decision level to adding an infinite keep prior on those indices\. Formally, for any finite scoring rule with kept scoreSi=Ci\+Λ⋅𝟏​\{i∈𝒲\}S\_\{i\}=C\_\{i\}\+\\Lambda\\cdot\\mathbf\{1\}\\\{i\\in\\mathcal\{W\}\\\}, if\|𝒲\|≤K\|\\mathcal\{W\}\|\\leq KandΛ\\Lambdaexceeds the maximum score gap, top\-KKonSSincludes𝒲\\mathcal\{W\}\. With a finiteΛ\\Lambda, minimizing squared loss on the tail window picksΛ⋆=−βtail\\Lambda^\{\\star\}=\-\\beta\_\{\\mathrm\{tail\}\}, giving a local MSE reduction ofβtail2\\beta\_\{\\mathrm\{tail\}\}^\{2\}\. Because the head\-aggregation dilution factorreff/Hheadr\_\{\\mathrm\{eff\}\}/H\_\{\\mathrm\{head\}\}and the phase\-mismatch factorm/Mqm/M\_\{q\}both appear multiplicatively in\|βtail\|\|\\beta\_\{\\mathrm\{tail\}\}\|, reserved\-window benefit scales quadratically with their product\. Under an iid or nonnegatively correlated noise endpoint, average pooling along the prefix axis, also used by SnapKV with kernel 7, contributes only a constant\-factor variance reduction \(iid gives1/71/7, perfectly correlated noise gives11\); variance reduction alone cannot cancel anO​\(1\)O\(1\)bias\.

### G\.5Consequence: Conjunctive Amplification on Multi\-needle Retrieval

Head\-sum dilution becomes more visible when the task requires several evidence tokens to survive together\. The two\-key needle\-retrieval task grades correctness by conjunction over two needle tokens\. If each needle is retained independently with probabilitypp, end\-task accuracy scales asp2p^\{2\}, amplifying proxy cliffs multiplicatively\. This is why modest single\-needle differences can become categorical end\-task separations in multi\-needle retrieval\. The independence calculation is a simple amplification model; correlated retention changes the exponent while preserving the qualitative fact that conjunction magnifies selection errors\.

### G\.6Scope and Limitations

This appendix analyzes head aggregation as a contrast\-loss mechanism under simplified retrieval and noise models\. The bounds explain when head\-summed scoring degrades relative to per\-head selection\. Their scope is the local contrast loss; architecture\-level comparisons require separate selector\-contract tests\. We use the empirical participation ratio as a marker for the measured models\.

## Appendix HLayer\-wise Analysis

We vary the layer set used by the Module I access\-support measurement\. The sweep starts from the top layer, adds lower layers, and re\-evaluates against a leave\-one\-out classification reference\. A brittle single\-layer choice would produce a monotone alignment gain as lower layers are added\. The observed curves move less across layer fractions than across model identities\.

### H\.1Layer\-Scope Sensitivity of Access Alignment

The Module I access measurement depends on which layers are included when aggregating attention\-derived evidence\. We sweep this layer scope as a local sensitivity check for the access measurement\.[Figure˜7](https://arxiv.org/html/2605.08234#A8.F7)reports two complementary top\-tail metrics\. NDCG@5% is rank\-sensitive and asks whether reference\-relevant positions are ordered near the top of the access score\. J@5% is support\-sensitive and asks whether the top 5% selected positions overlap with the reference top\-support set\.

Including lower layers does not yield a monotone improvement\. Across the sweep, model\-to\-model differences are larger than the within\-model changes induced by layer scope\. Layer scope can affect individual access\-support measurements; the main Module II finding is measured under a fixed selector contract and replaces only the value\-channel ranking slot\.

![Refer to caption](https://arxiv.org/html/2605.08234v1/x14.png)

![Refer to caption](https://arxiv.org/html/2605.08234v1/x15.png)![Refer to caption](https://arxiv.org/html/2605.08234v1/x16.png)![Refer to caption](https://arxiv.org/html/2605.08234v1/x17.png)

Figure 7:Layer\-scope sensitivity of the Module I access\-support measurement\. NDCG@5% measures rank alignment in the top score tail, while J@5% measures top\-support overlap\. Expanding the included layer range does not produce a monotone gain, and model\-to\-model variation is larger than within\-model layer\-scope variation\.
### H\.2Scope and Limitations

The measurement is limited to the upper\-layer inclusion rule used by the access\-support calculation\. Alternative reference definitions require a separate sensitivity check\. The conclusion is local to this mechanism check and stops short of a general ranking of layer importance in long\-context retrieval\.

## Appendix IMulti\-Objective Retrieval Extended Results

The multi\-target retrieval results stress the support\-closure prediction in[Table˜3](https://arxiv.org/html/2605.08234#S4.T3)\.[Section˜I\.1](https://arxiv.org/html/2605.08234#A9.SS1)reports the 32k\-context NeedleBench M\-RT evaluation\[[25](https://arxiv.org/html/2605.08234#bib.bib25)\], where pooled proxies lose support when targets split and question\-routed proxies preserve more of it\.[Section˜I\.1](https://arxiv.org/html/2605.08234#A9.SS1)reports an 8k\-context RULER check\[[24](https://arxiv.org/html/2605.08234#bib.bib24)\]on whether logged movement concentrates on branched retrieval\.

### I\.1NeedleBench M\-RT at 32k context

[Tables˜6](https://arxiv.org/html/2605.08234#A9.T6)and[7](https://arxiv.org/html/2605.08234#A9.T7)report the 32k\-context NeedleBench M\-RT stress test\. The first table gives slot\-set exact scores, with exact\-hit rate in parentheses, undern=4n\{=\}4retrieval objectives, budgetb=0\.10b\{=\}0\.10, and 8 samples per model\. The second separates retained\-slot support from exact\-format hits, distinguishing evidence recovery from answer\-format recovery\. All rows share the same prefill capture, context truncation, and budget contract\. SnapKV\[[26](https://arxiv.org/html/2605.08234#bib.bib26)\]is the observation\-window reference\.

The Module II value\-channel replacement is evaluated in two pooling modes\. MII\-pooled uses the canonical pooled proxy bank𝒰pool=𝒰tail\\mathcal\{U\}\_\{\\mathrm\{pool\}\}=\\mathcal\{U\}^\{\\mathrm\{tail\}\}and applies the same pooled block score used by the main Module II recipe\. MII\-QD denotes question\-routed Module II\. It sets𝒰QD=DetectQuestionPositions​\(x\)\\mathcal\{U\}\_\{\\mathrm\{QD\}\}=\\mathrm\{DetectQuestionPositions\}\(x\), falling back to𝒰tail\\mathcal\{U\}^\{\\mathrm\{tail\}\}when no decoded\-question span is detected, then uses the same pooled scoring and top\-kkprojection as MII\-pooled\.

MII\-QD separates from SnapKV and MII\-pooled\. Averaged across the three models, it reaches72\.272\.2slot\-set exact; SnapKV and MII\-pooled are at61\.161\.1and52\.852\.8\. Its exact\-hit rate rises to54\.254\.2from25\.025\.0, with the larger support\-side gap pointing to evidence recovery beyond formatting\.

The separation is consistent with ther=1r\{=\}1labeled\-law TV floor in[Theorem˜4](https://arxiv.org/html/2605.08234#Thmtheorem4)\. The theorem lower\-bounds labeled\-law total variation; score ceilings are outside its scope\. The result points to pooled\-proxy failure when target branching grows\. Single\-channel baselines that change other contract axes require separate LongBench surfaces: H2O and H2O\-debiased are access\-support variants, while PyramidKV, Ada\-KV, and CAKE change allocation\. Their LongBench surfaces are evaluated in[Table˜4](https://arxiv.org/html/2605.08234#S4.T4)and discussed in[Table˜3](https://arxiv.org/html/2605.08234#S4.T3)\.

Table 6:Per\-model split of the 32k NeedleBench M\-RT\[[25](https://arxiv.org/html/2605.08234#bib.bib25)\]stress slice\. Slot\-set exact and exact\-hit rate are reported separately for each model\.Slot\-set exact \(%\)Exact\-hit rate \(%\)ModelFullKVSnapKVMII\-pooledMII\-QDFullKVSnapKVMII\-pooledMII\-QDQwen3\-8B\[[39](https://arxiv.org/html/2605.08234#bib.bib39)\]100\.0100\.062\.562\.566\.766\.775\.075\.037\.537\.525\.025\.037\.537\.550\.050\.0Llama\-3\.1\-8B\[[43](https://arxiv.org/html/2605.08234#bib.bib43)\]100\.0100\.066\.766\.750\.050\.075\.075\.075\.075\.050\.050\.037\.537\.562\.562\.5Mistral\-7B\[[44](https://arxiv.org/html/2605.08234#bib.bib44)\]87\.587\.554\.254\.241\.741\.766\.766\.70\.00\.00\.00\.00\.00\.050\.050\.0Average95\.895\.861\.161\.152\.852\.872\.272\.237\.537\.525\.025\.025\.025\.054\.254\.2- •The parenthesized exact\-hit metric is split into its own numeric block so that support recovery and exact\-format recovery can be read independently\.

Table 7:Question\-routed Module II recovers the multi\-target support that pooled proxies lose\. Summarizes the averages expanded in[Table˜6](https://arxiv.org/html/2605.08234#A9.T6)\.MetricreferenceFullKVfixed contractSnapKVpooled proxyMII\-pooledquestion\-routedMII\-QDSlot\-set exact \(%\)95\.895\.861\.161\.152\.852\.872\.272\.2Exact\-hit rate \(%\)37\.537\.525\.025\.025\.025\.054\.254\.2- •The table keeps the original averages unchanged; only the column headers are rewritten so that each method name carries its evidential role\.

I\.2 RULER branch\-stress check at 8k context

The RULER\[[24](https://arxiv.org/html/2605.08234#bib.bib24)\]branch\-stress check is included as a directional 8k\-context check within the LongBench budget, evaluated on the three 8B\-class models in the main grid\. The archived run stores three ordered score columns without semantic method names, so[Table˜8](https://arxiv.org/html/2605.08234#A9.T8)is used only to test whether the logged movement concentrates on branched rather than single\-key retrieval\.

Across the logged score columns, the positive movement is dominated by the two\-key needle\-retrieval task\. On the single\-key companion task, the logged changes are zero or near zero under the same budgets and model set\. Removing the two\-key task shrinks the average movement toward zero across all three models\.

A pooled single\-query proxy does not cover the multi\-token query tail captured by the SnapKV observation window\. The same direction holds at smaller context, so the binding constraint is the number of concurrent retrieval objectives\. The 8k check is consistent with the M\-RT direction but is not used to compare named selectors or estimate absolute multi\-target retrieval ability\.

Table 8:RULER\[[24](https://arxiv.org/html/2605.08234#bib.bib24)\]task breakdown for the 8k branch\-stress check\. The three logged scores are kept in their recorded order\.Llama\-3\.1\-8B\[[43](https://arxiv.org/html/2605.08234#bib.bib43)\]Mistral\-7B\[[44](https://arxiv.org/html/2605.08234#bib.bib44)\]Qwen3\-8B\[[39](https://arxiv.org/html/2605.08234#bib.bib39)\]RULER taskscore\-1score\-2score\-3score\-1score\-2score\-3score\-1score\-2score\-3niah\_single\_1−0\.020\-0\.020−0\.020\-0\.0200\.0000\.0000\.0000\.0000\.0000\.0000\.0000\.0000\.0000\.0000\.0000\.0000\.0000\.000niah\_multikey\_20\.9200\.9200\.6400\.6400\.4000\.4000\.4200\.4200\.4400\.4400\.4000\.4000\.6600\.6600\.7400\.7400\.7600\.760VT0\.1400\.1400\.0480\.0480\.0440\.044−0\.008\-0\.008−0\.008\-0\.008−0\.004\-0\.0040\.1760\.1760\.1200\.1200\.0600\.060CWE−0\.004\-0\.004−0\.020\-0\.0200\.0800\.080−0\.248\-0\.248−0\.052\-0\.0520\.1380\.138−0\.016\-0\.0160\.0700\.0700\.1620\.162FWE0\.1930\.1930\.1470\.147−0\.007\-0\.0070\.0800\.0800\.0070\.0070\.0270\.027−0\.033\-0\.033−0\.007\-0\.007−0\.033\-0\.033Average0\.2460\.2460\.1590\.1590\.1030\.1030\.0490\.0490\.0770\.0770\.1120\.1120\.1570\.1570\.1850\.1850\.1900\.190Avg\. excl\. multikey0\.0770\.0770\.0390\.0390\.0290\.029−0\.044\-0\.044−0\.013\-0\.0130\.0400\.0400\.0320\.0320\.0460\.0460\.0470\.047- •Each cell was originally encoded as three ordered scores\. The table keeps the logged order without assigning unsupported semantic names\.

### I\.3Scope and Limitations

The M\-RT results show a directional separation between pooled and question\-routed proxies under multi\-target retrieval at 32k context; the RULER 8k check shows that logged movement concentrates on the multikey task\. They do not identify the optimal proxy bank for arbitrary multi\-target distributions, do not assign semantic names to the logged RULER score columns, and do not target absolute ceilings under retrieval\-specific evaluators\. The Stage III diagnosis applies to contracts that fix the observation window, per\-head budget allocation, and deterministic top\-kkprojection; transfer to allocation\-changing contracts is reported on LongBench in[Section˜J\.3](https://arxiv.org/html/2605.08234#A10.SS3)\.

## Appendix JConditional Split Provenance

### J\.1Stage Evidence Registry

Table 9:Stage evidence registry for the fixed selector\-contract diagnostic\.AxisContract changeEvidenceInferenceSourceStage II,uvalu\_\{\\mathrm\{val\}\}value\-channel replacement onlyHc\>0H\_\{c\}\>0:72\.6%72\.6\\%, mean\+0\.54\+0\.54ppvalue repair under fixed support[Table˜1](https://arxiv.org/html/2605.08234#S4.T1)Leverage sign comparisonfixed contract; sign agreement on 264 cellsNoLev sign matches MII on 239 of 264 cells \(90\.5%90\.5\\%\); leverage on 162 of 264 cells \(61\.4%61\.4\\%\)value channel carries the predictive sign; leverage is conditional[Table˜3](https://arxiv.org/html/2605.08234#S4.T3)Support couplingfixed contract; stratify byϕ\\philowϕ\\phi:76\.4%76\.4\\%,\+0\.69\+0\.69pp; highϕ\\phi:42\.9%42\.9\\%,−0\.59\-0\.59ppjoint evidence can block repair[Tables˜10](https://arxiv.org/html/2605.08234#A10.T10)and[11](https://arxiv.org/html/2605.08234#A10.T11)Stage III, latticefixed block order; token\-fill onlyslack removed; mean\+0\.12\+0\.12pp, median0\.000\.00pptoken count is not support closure[Figure˜4](https://arxiv.org/html/2605.08234#S4.F4)Stage I,uaccu\_\{\\mathrm\{acc\}\}change access\-support lawcount debias7\.2→25\.87\.2\\to 25\.8; decode\-proximal33\.233\.2bottleneck moves upstream[Figures˜5](https://arxiv.org/html/2605.08234#A6.F5),[6](https://arxiv.org/html/2605.08234#A7.F6)and[8](https://arxiv.org/html/2605.08234#A10.F8);[Table˜5](https://arxiv.org/html/2605.08234#A6.T5)Allocation driftlayer, cascade, or head budget changes\+0\.84\+0\.84,\+0\.30\+0\.30,\+0\.15\+0\.15pptransfer under changed projection[Table˜4](https://arxiv.org/html/2605.08234#S4.T4)Cross\-contract comparisonPyramidKV\-local reference marginHB\>0H\_\{B\}\>0:79\.5%79\.5\\%, mean\+0\.96\+0\.96ppsigned side transfers locally[Table˜17](https://arxiv.org/html/2605.08234#A10.T17)Multi\-target Stage IIIsame capture; branched objectivepooled to QD: exact52\.852\.8to72\.272\.2; exact\-hit25\.025\.0to54\.254\.2pooled score loses branch evidence[Tables˜6](https://arxiv.org/html/2605.08234#A9.T6),[7](https://arxiv.org/html/2605.08234#A9.T7)and[8](https://arxiv.org/html/2605.08234#A9.T8)Negative controlsremoveTuT^\{u\}; score boundary casesNoLev reaches1\.70×1\.70\\timesno\-value ceilingsplit depends on value channel[Tables˜18](https://arxiv.org/html/2605.08234#A11.T18)and[19](https://arxiv.org/html/2605.08234#A11.T19)Leverage strengthsame tasks and budgets; vary alpha96 completed cells; damped best counts 15, 10, and 7leverage is stage\-resolved[Table˜20](https://arxiv.org/html/2605.08234#A11.T20)Budget and reference surfacesboundary\-heavy four\-budget diagnostics6 of 36 monotone; 12 of 36 local dips; 6 of 24 portable MII winsaverages hide phase and reference effects[Figure˜9](https://arxiv.org/html/2605.08234#A11.F9);[Table˜21](https://arxiv.org/html/2605.08234#A11.T21)

### J\.2Predictor Provenance and Margin Split

The support\-coupling indexϕ​\(x\)\\phi\(x\)comes from the fixed grammar in[Section˜2](https://arxiv.org/html/2605.08234#S2), which counts code\-line density, label\-marker density, and delimiter\-bounded record density\. It measures the density of structural evidence units whose utility is not well represented by isolated token retention\. We use the same index for the clean Module II evaluation, the additive control, and the bounded\-correction check\. The signed reference margin ismc=FullKV−cHostcm\_\{c\}=\\textsc\{FullKV\}\{\}\_\{c\}\-\\mathrm\{Host\}\_\{c\}, and the predictor isHc=sign⁡\(mc\)H\_\{c\}=\\operatorname\{sign\}\(m\_\{c\}\); in the SnapKV contract,Hostc=SnapKVc\\mathrm\{Host\}\_\{c\}=\\mathrm\{SnapKV\}\_\{c\}\.[Table˜11](https://arxiv.org/html/2605.08234#A10.T11)gives the rawϕ​\(x\)×Hc\\phi\(x\)\\times H\_\{c\}split, and[Table˜12](https://arxiv.org/html/2605.08234#A10.T12)treats task clusters as the resampling unit\. The two predictors play different roles\. The signed reference margin measures whether the value\-channel replacement has room to help, while high support coupling identifies cells where a scalar value\-channel correction is less reliable because useful states must be retained jointly\.

Table 10:Support coupling explains when positive reference margin still fails\. Companion to[Table˜1](https://arxiv.org/html/2605.08234#S4.T1)\.Positive marginHc\>0H\_\{c\}\>0Nonpositive marginHc≤0H\_\{c\}\\leq 0ϕ\\phibucketCellsPos\. shift \(%\)MeanΔ\\Delta\(pp\)CellsPos\. shift \(%\)MeanΔ\\Delta\(pp\)lowϕ<0\.10\\phi<0\.10555576\.476\.40\.690\.69232334\.834\.8−0\.49\-0\.49highϕ≥0\.10\\phi\\geq 0\.107742\.942\.9−0\.59\-0\.59111127\.327\.3−0\.34\-0\.34- •The raw positive, nonpositive, and tie counts behind the same split are expanded in[Table˜11](https://arxiv.org/html/2605.08234#A10.T11)\.

Table 11:Support\-coupling split under signed reference margin\.ϕ\\phibucketMargin sideCellsPos\.Nonpos\.TiePos\. shift \(%\)MeanΔ\\Delta\(pp\)lowϕ<0\.10\\phi<0\.10Hc\>0H\_\{c\}\>05555424210103376\.476\.40\.690\.69highϕ≥0\.10\\phi\\geq 0\.10Hc\>0H\_\{c\}\>07733331142\.942\.9−0\.59\-0\.59lowϕ<0\.10\\phi<0\.10Hc≤0H\_\{c\}\\leq 0232388996634\.834\.8−0\.49\-0\.49highϕ≥0\.10\\phi\\geq 0\.10Hc≤0H\_\{c\}\\leq 011113388027\.327\.3−0\.34\-0\.34- •Counts split positive, nonpositive, and tied outcome cells before reporting the same rate\-and\-mean summary\.

Table 12:Cluster\-robust intervals for the signed reference\-margin split\. Confirms[Table˜1](https://arxiv.org/html/2605.08234#S4.T1)when LongBench tasks are the resampling unit\.StatisticEstimateTask\-cluster 95% CIHc\>0H\_\{c\}\>0positive\-shift rate72\.6%72\.6\\%\[58\.2, 85\.3\]Hc≤0H\_\{c\}\\leq 0positive\-shift rate32\.4%32\.4\\%\[19\.5, 48\.6\]Positive\-rate gap\(Hc\>0\)−\(Hc≤0\)\(H\_\{c\}\>0\)\-\(H\_\{c\}\\leq 0\)\+40\.2\+40\.2pp\[\+18\.9\+18\.9,\+60\.0\+60\.0\]Direction\-match rate, all 96 cells70\.8%70\.8\\%\[60\.4, 81\.2\]MeanΔ\\DeltagapΔ¯Hc\>0−Δ¯Hc≤0\\overline\{\\Delta\}\_\{H\_\{c\}\>0\}\-\\overline\{\\Delta\}\_\{H\_\{c\}\\leq 0\}\+0\.99\+0\.99pp\[\+0\.57\+0\.57,\+1\.41\+1\.41\]Pearsonr​\(mc,Δ\)r\(m\_\{c\},\\Delta\)0\.410\.41\[\+0\.22\+0\.22,\+0\.59\+0\.59\]*Threshold sweep on\|mc\|≥τ\|m\_\{c\}\|\\geq\\tau*τ=0\\tau=0\+40\.2\+40\.2pp\[\+18\.9\+18\.9,\+60\.0\+60\.0\]τ=0\.1\\tau=0\.1pp\+36\.1\+36\.1pp\[\+17\.2\+17\.2,\+53\.0\+53\.0\]τ=0\.25\\tau=0\.25pp\+36\.1\+36\.1pp\[\+18\.3\+18\.3,\+51\.8\+51\.8\]τ=0\.5\\tau=0\.5pp\+39\.6\+39\.6pp\[\+20\.0\+20\.0,\+57\.3\+57\.3\]*Leave\-one\-task\-out extrema for the positive\-rate gap*Worst\-case dropped task\+34\.2\+34\.2ppdrop passage\_retrieval\_enBest\-case dropped task\+46\.0\+46\.0ppdrop samsum- •The first block reports cluster\-bootstrap estimates\. The second and third blocks retain the threshold and leave\-one\-task\-out checks at lower visual priority\.

### J\.3Robustness Checks for the Margin Split

[Tables˜13](https://arxiv.org/html/2605.08234#A10.T13)and[14](https://arxiv.org/html/2605.08234#A10.T14)check whether the signed reference\-margin gap comes from label counts or from a single task\. The permutation null keeps the same 96 matched outcomes and the same positive and nonpositive split, then shuffles the labels\. The leave\-one\-task\-out table deletes each LongBench task in turn\. The observed direction\-match and positive\-rate gaps lie outside the permutation null, and the signed reference\-margin gap remains positive after every one\-task deletion\. The support\-coupling controls below add a granularity check\. Task\-level resampling preserves the high\-ϕ\\phidirection, while prompt\-level splitting weakens once the task template is fixed\.

Table 13:Permutation\-null check for the fixed signed reference\-margin partition\. Confirms that the observed split is not reproduced by label shuffling on the same 96 cells\.StatisticObservedNull meanNull 95% intervalOne\-sidedppDirection\-match rate \(%\)70\.870\.852\.552\.5\[43\.8, 62\.5\]<0\.001<0\.001Positive\-rate gap\(Hc\>0\)−\(Hc≤0\)\(H\_\{c\}\>0\)\-\(H\_\{c\}\\leq 0\)\(pp\)40\.240\.20\.10\.1\[−19\.0\-19\.0,\+22\.0\+22\.0\]<0\.001<0\.001- •The null center and null interval are separated so that displacement and dispersion remain distinct\.

Table 14:Leave\-one\-task\-out stability of the signed reference\-margin split\. Confirms[Table˜1](https://arxiv.org/html/2605.08234#S4.T1)after deleting each LongBench task in turn\.Dropped taskHc\>0H\_\{c\}\>0pos\. \(%\)Hc≤0H\_\{c\}\\leq 0pos\. \(%\)Rate gap \(pp\)Mean gap \(pp\)qasper69\.669\.632\.432\.437\.337\.30\.880\.88narrativeqa70\.770\.731\.231\.239\.439\.40\.960\.96multifieldqa\_en72\.472\.434\.434\.438\.038\.00\.970\.97hotpotqa70\.770\.731\.231\.239\.439\.40\.970\.972wikimqa72\.472\.431\.231\.241\.241\.21\.041\.04musique75\.475\.433\.333\.342\.142\.10\.940\.94gov\_report69\.669\.632\.432\.437\.337\.30\.940\.94qmsum73\.773\.730\.330\.343\.443\.41\.031\.03multi\_news69\.669\.632\.432\.437\.337\.30\.940\.94samsum74\.174\.128\.128\.146\.046\.01\.061\.06trec74\.674\.632\.332\.342\.342\.30\.910\.91triviaqa74\.674\.632\.332\.342\.342\.31\.041\.04passage\_count74\.674\.632\.332\.342\.342\.31\.051\.05passage\_retrieval\_en72\.172\.137\.937\.934\.234\.20\.920\.92lcc73\.373\.333\.333\.340\.040\.01\.031\.03repobench\-p73\.373\.333\.333\.340\.040\.01\.051\.05minimum gap across LOO72\.172\.137\.937\.934\.234\.20\.920\.92maximum gap across LOO74\.174\.128\.128\.146\.046\.01\.061\.06- •Each row drops six cells from the 96\-cell grid \(33models×\\times22budgets\) for the named LongBench task, matching the source\-file comment\.

J\.4 Support\-Coupling and Margin Controls

[Tables˜11](https://arxiv.org/html/2605.08234#A10.T11)and[15](https://arxiv.org/html/2605.08234#A10.T15)separate cell\-level and prompt\-level behavior\. High\-ϕ\\phicells have a lower positive\-shift rate than low\-ϕ\\phicells overall \(one\-sided Fisherp=0\.0173p=0\.0173\)\. Within positive\-margin cells, high\-ϕ\\phireaches42\.9%42\.9\\%while low\-ϕ\\phireaches76\.4%76\.4\\%\(1\.8×1\.8\{\\times\}; one\-sided Fisherp=0\.0823p=0\.0823\)\. The cell\-level gap is−33\.5\-33\.5pp on all tasks,−43\.0\-43\.0pp after removing code tasks, and−42\.6\-42\.6pp after also removing retrieval tasks\.

[Table˜15](https://arxiv.org/html/2605.08234#A10.T15)gives the prompt\-level counterpart on positive\-margin prompt\-budget units\. The direction is the same but smaller:−13\.9\-13\.9pp overall,−4\.7\-4\.7pp after removing code, and−4\.8\-4\.8pp after also removing retrieval\. This scale difference follows from the definition ofϕ​\(x\)\\phi\(x\)\. Task templates create large between\-family variation at cell granularity, while most non\-code tasks have little within\-task variation\. Code contributes many high\-ϕ\\phiprompt units, so removing code reduces the prompt\-level effect while leaving the cell\-level direction intact\. The mainϕ\\phieffect sits at task and template granularity, with a weaker same\-direction prompt\-level trace\.

Table 15:Within\-family and prompt\-level aggregations both preserve theϕ\\phidirection\. Companion to[Table˜11](https://arxiv.org/html/2605.08234#A10.T11)at family granularity\.GranularitySliceUnitsLow\-ϕ\\phipos\. \(%\)High\-ϕ\\phipos\. \(%\)Gap \(pp\)CellAll tasks626276\.476\.442\.942\.9−33\.5\-33\.5CellNon\-code only585876\.476\.433\.333\.3−43\.0\-43\.0CellNon\-code, non\-retrieval575775\.975\.933\.333\.3−42\.6\-42\.6Prompt levelAll tasks2743274357\.457\.443\.443\.4−13\.9\-13\.9Prompt levelNon\-code only2335233557\.457\.452\.652\.6−4\.7\-4\.7Prompt levelNon\-code, non\-retrieval2327232757\.457\.452\.652\.6−4\.8\-4\.8- •Negative gaps mean the high\-ϕ\\phibucket has a lower positive\-shift rate\.

J\.5 Cross\-Contract Transfer and Identity Checks

[Tables˜16](https://arxiv.org/html/2605.08234#A10.T16)and[17](https://arxiv.org/html/2605.08234#A10.T17)report cross\-contract transfer and identity coverage\.[Table˜16](https://arxiv.org/html/2605.08234#A10.T16)separates the contract\-local reference margin, adapter shifts, and outcome counts by model\-task pair\. The PyramidKV\[[2](https://arxiv.org/html/2605.08234#bib.bib2)\]evaluation tests whether the positive\-margin prediction survives a changed allocation contract\. Over the full 96\-cell grid, using the contract\-local reference marginmB=FullKV−PyramidKVm\_\{B\}=\\textsc\{FullKV\}\{\}\-\\mathrm\{PyramidKV\}withHB=sign⁡\(mB\)H\_\{B\}=\\operatorname\{sign\}\(m\_\{B\}\), PyramidKV\+MII is positive on79\.5%79\.5\\%of positive\-margin cells with a mean\+0\.96\+0\.96pp lift\. The SnapKV\[[26](https://arxiv.org/html/2605.08234#bib.bib26)\]contract result in[Section˜4](https://arxiv.org/html/2605.08234#S4)has the same direction, with a72\.6%72\.6\\%positive\-shift rate at mean\+0\.54\+0\.54pp\.

Table 16:Cross\-instantiation breakdown for the PyramidKV transfer rows\.Contract\-local comparison \(pp\)Outcome countsModelTaskFullKV\-PyrMII\-PyrMII\-SnapMargin sidePositiveNonpositiveTieLlamaqasper9\.259\.25−0\.04\-0\.041\.641\.64positive303032323838qmsum0\.850\.850\.090\.090\.070\.07positive454542421313trec13\.0013\.00−1\.00\-1\.000\.000\.00positive11229797lcc−0\.60\-0\.600\.010\.010\.300\.30nonpositive1111558484Mistralqasper10\.5610\.561\.051\.051\.511\.51positive333325254242qmsum0\.620\.620\.600\.600\.290\.29positive525235351313trec−29\.00\-29\.003\.003\.001\.001\.00nonpositive1010778383lcc8\.368\.360\.330\.33−2\.84\-2\.84positive222234344444Qwen3qasper9\.329\.32−0\.07\-0\.072\.762\.76positive343425254141qmsum1\.241\.240\.310\.31−0\.19\-0\.19positive5252404088trec9\.509\.509\.509\.50−5\.00\-5\.00positive1212338585lcc1\.021\.020\.600\.60−0\.02\-0\.02positive505035351515- •Positive, nonpositive, and tie counts expose the transfer breakdown for each model\-task pair\.

The nonpositive side is asymmetric across contracts\. Under SnapKV, the nonpositive margin predicts the weak group, with a32\.4%32\.4\\%positive\-shift rate and a mean shift of−0\.44\-0\.44pp\. Under PyramidKV, the corresponding bucket gives a56\.5%56\.5\\%positive\-shift rate and a mean\+0\.45\+0\.45pp shift\. This side remains inconclusive\. PyramidKV changes layer allocation and projection at the same time, and the non\-monotone\-compression observation in[Section˜1](https://arxiv.org/html/2605.08234#S1)makesFullKVan unreliable upper bound \(SnapKV crossesFullKVon 23 of 96 cells, concentrated in 9 of 18 Few\-shot cells and 8 of 12 Code cells\); as a result,HB≤0H\_\{B\}\\leq 0gives a weak saturation control for Stage II error\.

The identity checks rule out selector reimplementation as the source of the positive transfer\. The additive path keeps the identity scoring branch separate from the value\-channel branch, and local unit tests cover exact SnapKV score and selection preservation\. For PyramidKV, theα=0\\alpha\{=\}0adapter was checked end\-to\-end on two cross\-instantiation cells atb=0\.05b\{=\}0\.05; both produced equal aggregate scores under PyramidKV and the PyramidKV value\-channel adapter withα=0\\alpha\{=\}0\(deltas0\.00\.0\)\.

[Table˜17](https://arxiv.org/html/2605.08234#A10.T17)separates the same cross\-contract comparison into conditional rows, aggregate rows, and no\-adapter surfaces for SnapKV, PyramidKV, CAKE\[[37](https://arxiv.org/html/2605.08234#bib.bib37)\], Ada\-KV\[[17](https://arxiv.org/html/2605.08234#bib.bib17)\], and H2O\[[52](https://arxiv.org/html/2605.08234#bib.bib52)\]\.

Table 17:Per\-surface breakdown of[Table˜4](https://arxiv.org/html/2605.08234#S4.T4)\.SurfaceBBContract axisRow typeRate kindRate \(%\)MeanΔB\\Delta\_\{B\}\(pp\)UseSnapKVobservation\-window scoringHc\>0H\_\{c\}\>0positive72\.672\.6\+0\.54main validationHc≤0H\_\{c\}\\leq 0nonpositive67\.667\.6\-0\.44PyramidKVlayer\-budget projectionHB\>0H\_\{B\}\>0positive79\.579\.5\+0\.96positive\-margin transferHB≤0H\_\{B\}\\leq 0positive56\.556\.5\+0\.45CAKElayer cascadeall 96 cellspositive60\.460\.4\+0\.29Stage III surfaceNoLev aggregate––\+0\.34Ada\-KVhead\-budget allocationall 96 cellspositive53\.153\.1\+0\.16Stage III surfaceNoLev aggregate––\+0\.26H2O\-deb\.count\-debiased cumulative supportno adapter row–––Stage I surfaceH2Ocumulative supportno adapter row–––Stage I surface- •mB=FullKV−Bm\_\{B\}=\\textsc\{FullKV\}\{\}\-Bis the contract\-local reference margin for surfaceBB, andHB=sign⁡\(mB\)H\_\{B\}=\\operatorname\{sign\}\(m\_\{B\}\)gives the split\. The rate\-kind column states whether each percentage counts positive or nonpositive shifts; the older paragraph\-style cells left this distinction implicit\.

J\.6 Stage I Predictability Under Count Debiasing

![[Uncaptioned image]](https://arxiv.org/html/2605.08234v1/figures/fig_stagei_predictability.png)

Figure 8:Stage I access gaps predict count\-debias repair\.
Count debiasing of cumulative attention is established prior work\[[4](https://arxiv.org/html/2605.08234#bib.bib4),[48](https://arxiv.org/html/2605.08234#bib.bib48)\]; the contribution here is the predictability contrast across stages\.[Figure˜8](https://arxiv.org/html/2605.08234#A10.F8)plots the per\-cell debias lift against the initial H2O–FullKVaccess gap on the LongBench grid\. Each point denotes one grid cell, and the scatter has a strong linear trend \(correlationr=0\.88r\{=\}0\.88, 86 of 96 cells positive, mean lift\+18\.4\+18\.4pp\), consistent on this 96\-cell grid with count\-debias repair scaling with the measured Stage I access gap\. The same scatter geometry under Stage III \([Figure˜4](https://arxiv.org/html/2605.08234#S4.F4)c\) givesr=0\.14r\{=\}0\.14, the opposite predictability signature\. Within\-stage repair under count debiasing has been characterized by AhaKV and Attention Debiasing; the cross\-stage predictability contrast shown here and in panel \(c\) of[Figure˜4](https://arxiv.org/html/2605.08234#S4.F4)has not, to our knowledge, been reported\.

## Appendix KAdditional Confirmatory Controls

These controls test two limits of the conditional split\. One removes the value\-consequence term from the additive scalar; the other replaces task accuracy with a boundary\-local reference based on next\-token distribution change\.[Tables˜18](https://arxiv.org/html/2605.08234#A11.T18)and[19](https://arxiv.org/html/2605.08234#A11.T19)evaluate the value\-consequence channel and the signed reference\-margin prediction at disagreement boundaries\. Two stress analyses then probe leverage strength and boundary\-heavy budget behavior without changing the main 96\-cell LongBench grid\.

### K\.1Value\-Consequence Negative Control

[Table˜18](https://arxiv.org/html/2605.08234#A11.T18)removes the value\-consequence channel from the additive scalar while keeping the access and contract terms\. HereTaT^\{a\},TcT^\{c\}, andTuT^\{u\}denote the additive access, contract, and value\-consequence terms\. The within\-cell comparison uses the same SnapKV baseline across rows\. The channel\-positive NoLev reference gives a mean\+0\.45\+0\.45pp shift and a68\.9%68\.9\\%positive\-shift rate,1\.70×1\.70\{\\times\}the best no\-value row\. The three variants withoutTuT^\{u\}stay near zero or below zero\.

Table 18:Removing the value\-consequence channel collapses lift toward the no\-value ceiling\. Companion to[Table˜3](https://arxiv.org/html/2605.08234#S4.T3)\.MethodRetainsTuT^\{u\}MeanΔ\\Deltafrom SnapKV \(pp\)Positive shift \(%\)Rate / no\-value ceilingSnapKV \+ NoLevyes0\.450\.4568\.968\.91\.70×1\.70\\timesAddi\-aa,TaT^\{a\}onlyno−0\.03\-0\.0339\.239\.20\.97×0\.97\\timesAddi\-cc,TcT^\{c\}onlyno−0\.10\-0\.1035\.135\.10\.87×0\.87\\timesAddi\-a\+ca\{\+\}c,Ta\+TcT^\{a\}\{\+\}T^\{c\}no−0\.04\-0\.0440\.540\.51\.00×1\.00\\times- •The first row is the channel\-positive reference\. The remaining rows are all no\-value ablations from the same additive family; exposing theTuT^\{u\}column makes the control axis readable without consulting the surrounding paragraph\.

K\.2 Boundary\-Level Split Controls

[Table˜19](https://arxiv.org/html/2605.08234#A11.T19)scores each disagreement boundary by its effect on the next\-token distribution underFullKV\. Spearmanρ\\rhoand pairwise AUC stay near chance for both selectors, indicating weak global ranking alignment\. After stratification, the swap\-improvement rate rises in the predicted\-positive quadrants and stays near0\.50\.5in the predicted\-nonpositive quadrants, matching the signed reference\-margin diagnosis at the boundary level\.

The prediction is strongest when the signed reference margin and boundary\-level value consequences agree\. It weakens when the contract leaves little useful value\-channel error to repair, or when high support coupling makes the correction less reliable\.

Table 19:Boundary\-local alignment rises only in the predicted\-positive quadrants under the fixed observation\-window contract\. Complements[Table˜18](https://arxiv.org/html/2605.08234#A11.T18)with disagreement\-boundary diagnostics\.Spearmanρ\\rhowith KL referencePairwise AUCGroupSnapKVSupportNoLevMIISnapKVSupportMIISwap\-improving boundaries \(%\)All reference prompts0\.050\.050\.080\.080\.100\.100\.120\.120\.520\.520\.550\.550\.580\.585858Hc\>0H\_\{c\}\>00\.060\.060\.100\.100\.130\.130\.150\.150\.530\.530\.570\.570\.610\.616262low\-ϕ\\phi,Hc\>0H\_\{c\}\>00\.060\.060\.100\.100\.140\.140\.170\.170\.530\.530\.580\.580\.650\.656565Hc≤0H\_\{c\}\\leq 00\.080\.080\.090\.090\.090\.090\.090\.090\.520\.520\.530\.530\.530\.534949high\-ϕ\\phi,Hc\>0H\_\{c\}\>00\.070\.070\.070\.070\.070\.070\.070\.070\.530\.530\.520\.520\.520\.525050high\-ϕ\\phi,Hc≤0H\_\{c\}\\leq 00\.070\.070\.070\.070\.070\.070\.060\.060\.530\.530\.520\.520\.520\.524848- •The final column reports the percentage of disagreement boundaries whose swap lowers KL under theFullKVreference\. The global ranking columns remain near chance; the local improvement signal appears only after stratification\.

K\.3 Targeted Leverage\-Strength Sweep

The NoLev\-to\-MII decomposition treats leverage as a conditional correction\. A targeted leverage\-strength sweep varies the blending coefficient while keeping tasks, budgets, and methods fixed\.[Table˜20](https://arxiv.org/html/2605.08234#A11.T20)summarizes the 96 completed cells for Qwen3\[[39](https://arxiv.org/html/2605.08234#bib.bib39)\]and Llama\[[43](https://arxiv.org/html/2605.08234#bib.bib43)\]\.

The sweep does not support a single alpha rule\. Across the 32 model–task–budget contracts, the best damped alpha splits across the three tested values with counts 15, 10, and 7 under a lowest\-alpha tie break\. When the matchedα=1\\alpha=1reference is included, it remains best in 11 contracts\. Absolute and matched\-reference conclusions also diverge\. 18 of 96 matched cells are positive over SnapKV but worse thanα=1\\alpha=1, while 13 improve overα=1\\alpha=1but still lose to SnapKV\. These cases make leverage strength a stage\-resolved validation surface\.

Table 20:Leverage\-strength sweep over 96 completed cells\.ModelAlphaCellsRawMIIWinsLossesTiesNoLevLeverageRescuesBreaksTaxesSupportfailuresLlama0\.2516\+0\.015853\-0\.158\+0\.1732116Llama0\.5016\-0\.6225101\-0\.313\-0\.3091229Llama0\.7516\-0\.481781\-0\.435\-0\.0460059Qwen30\.2516\+0\.168862\+0\.115\+0\.0531137Qwen30\.5016\+0\.450952\+0\.464\-0\.0140146Qwen30\.7516\+0\.414862\+0\.887\-0\.4730355- •Raw MII and NoLev are mean percentage\-point gains over SnapKV\. Leverage is Raw MII minus NoLev\. Breaks are sign\-flipping leverage failures; taxes are support\-win cases where leverage reduces an otherwise positive NoLev result\.

K\.4 Boundary\-Heavy Diagnostic Subset

We also evaluate boundary\-heavy diagnostic cases as a stress test for the same fixed\-contract mechanism\. Their role is to expose reference\-margin and budget\-axis failures around the selector boundary, where the completed LongBench grid already shows that final averages can hide stage\-specific behavior\.

[Figure˜9](https://arxiv.org/html/2605.08234#A11.F9)shows that retained budget behaves like a phase variable in these cases\. Raw MII is monotone nondecreasing in only 6 of 36 complete four\-budget trajectories, and budget0\.050\.05is a strict local dip in 12 of 36\. Budget0\.100\.10is the strongest fixed default, with\+1\.066\+1\.066pp, 22 wins, 10 losses, and 4 ties, but it captures only55\.5%55\.5\\%of the per\-trajectory oracle\-best mean and is exact\-best or tied in only 16 of 36 trajectories\. The largest adjacent transitions occur in Qwen3 TREC, Qwen3 passage retrieval, and Mistral TREC, with ranges from6\.2506\.250to13\.28113\.281pp\.

![[Uncaptioned image]](https://arxiv.org/html/2605.08234v1/x18.png)

Figure 9:Boundary\-heavy budget trajectory heatmap\.
[Table˜21](https://arxiv.org/html/2605.08234#A11.T21)compresses the remaining boundary\-heavy checks\. Raw MII can exceedFullKVwhen the fixed\-contract reference margin is negative, most visibly on Mistral\[[44](https://arxiv.org/html/2605.08234#bib.bib44)\]TREC\. Grouping exact Llama\[[43](https://arxiv.org/html/2605.08234#bib.bib43)\], Mistral, and Qwen3\[[39](https://arxiv.org/html/2605.08234#bib.bib39)\]by the same task and budget gives 6 of 24 contracts where raw MII is positive for all three models and none where it is negative for all three\. Leverage is never uniformly positive; it is uniformly negative in four contracts, uniformly zero in six, and mixed in 14\. The portable signal lies in support and value realizability, while leverage remains task\-, model\-, and budget\-conditioned\.

Table 21:Boundary\-heavy reference\-margin and cross\-model sign checks\.Representative FullKV\-reference counterexamplesSourceModelTaskBudgetmcm\_\{c\}MII–SnapKVMII–FullKVBoundary\-heavyMistraltrec0\.02\-20\.31\+3\.12\+23\.44Boundary\-heavyMistraltrec0\.05\-17\.19\+1\.56\+18\.75Boundary\-heavyQwen3passage\_retrieval\_en0\.02\-9\.38\+0\.00\+9\.38Boundary\-heavyMistraltrec0\.10\-4\.69\+1\.56\+6\.25Boundary\-heavyMistraltrec0\.15\-6\.25\-3\.12\+3\.12Cross\-model sign contracts over 24 matched task\-budget contractsSignalAllpositiveAllnegativeAllzeroMixedMeanSpreadRef\. margin100014\+2\.38\+7\.34Support10023\+0\.05\+2\.64NoLev60117\+0\.43\+2\.84MII60117\+0\.38\+3\.33Leverage04614\-0\.05\+1\.28- •mc=FullKV−cHostcm\_\{c\}=\\textsc\{FullKV\}\{\}\_\{c\}\-\\mathrm\{Host\}\_\{c\}, in percentage points; in these SnapKV\-contract rows,Hostc=SnapKVc\\mathrm\{Host\}\_\{c\}=\\mathrm\{SnapKV\}\_\{c\}\. The upper block lists boundary\-heavy rows where raw MII exceedsFullKVdespite negativemcm\_\{c\}, soFullKVdefines the signed reference margin\. The lower block groups exact Llama, Mistral, and Qwen3 by the same task and budget\.

### K\.5Scope and Limitations

These controls diagnose the existing selector contract\. The value\-channel ablation tests whether the additive score still works after removingTuT^\{u\}; the boundary reference tests whether disagreement boundaries align with a token\-distribution proxy\. The leverage and budget stress analyses are diagnostic surfaces for budget phase and leverage strength\. They stay within the fixed observation\-window contract and do not replace the matched LongBench grid\.

## Appendix LLimitations

The diagnosis is local to a selector contract\. Our cleanest claim is under the SnapKV\[[26](https://arxiv.org/html/2605.08234#bib.bib26)\]observation\-window contract, where access estimation and projection are fixed and only the ranking scalar is changed\. The cross\-contract rows in[Table˜4](https://arxiv.org/html/2605.08234#S4.T4)test transfer under contract drift\. Once the access window, allocation rule, or projection map changes, the experiment no longer isolates the same stage\.

The matched LongBench grid covers three 7B\-class instruction models, two budgets, and 100 samples per cell\. Coverage at larger scale, longer context, and other benchmark formats remains open\.

The diagnostic assumes access to prefill attention and value states\. Serving stacks that quantize the cache before selection or use fused kernels without exposing intermediate states need a different implementation route\. The paper also restricts attention to non\-reconstructing eviction, so quantization, low\-rank reconstruction, compensation tokens, and cache offloading need separate fixed\-contract tests before the same attribution logic applies\.

## Appendix MBroader Impacts

KV\-cache compression is primarily an efficiency tool\. Better cache\-selection diagnostics can lower serving cost and energy use, make long\-context models easier to run on constrained hardware, and reduce over\-provisioning when a smaller retained cache is sufficient\. These savings are not guaranteed to reduce aggregate compute demand: cheaper inference can also increase total usage\.

The main deployment risk is silent loss of evidence\. An eviction rule can remove states needed for faithful generation, especially in multi\-document, code, legal, medical, or retrieval\-heavy prompts where related facts must survive together\. The staged diagnosis separates value\-ranking errors from access\-support and projection failures; aggressive compression in high\-stakes settings still requires task\-specific validation\.

The experiments use public benchmarks and publicly released pretrained models\. They do not collect private data, train new foundation models, or add surveillance, persuasion, or content\-generation capabilities\. In deployment, cache compression should be paired with task\-specific evaluation, conservative fallbacks to larger caches orFullKV, and monitoring for prompts where projection or multi\-target support closure becomes the bottleneck\.