CertBind from Multimodal Connectivity to Certifiable Retrieval Decisions

# CertBind from Multimodal Connectivity to Certifiable Retrieval Decisions
Source: [https://arxiv.org/html/2608.06516](https://arxiv.org/html/2608.06516)
Shuheng Cao1,∗, Zhenhao Zhang5,7,∗, Ruiqi Chen2,∗, Renjie Cao3,†, Weijia Zhang4,†, Siyu Zhang1,†, Jiaxin Liu5,†, Xiangyu Zeng6,†, Haotian Geng7,†, Fan Gu8,‡

###### Abstract

Lightweight connectors make frozen multimodal encoders composable at the representation level\. Deployment exposes a second problem at the level of task decisions\. A connected route can expand cross\-modal reach while changing an established native retrieval capability\. We introduce CertBind, a multiscale theory of certifiable composition for frozen multimodal connector graphs\. At the node scale, native anchors establish the exact task identification boundary under the stated chart model\. At the edge scale, contract\-aware conformal ranks provide graph\-wide family\-wise error control\. At the path scale, an overlap\-aware budget and clean calibration yield a finite\-sample recovery radius under declared conditions\. At the query scale, this radius yields a covered top\-kkcandidate set that becomes a point certificate when its size equalskk\. CertBind therefore retains supported routes asDirect, sends only flagged routes to recovery, returnsCertifiedfor decisive recovery, and returnsAbstainfor unresolved queries\. The evaluated C\-MCR shared route reduced native CLIP R@1 from0\.5240\.524to0\.2900\.290\. The production fallback recovered0\.963±0\.0020\.963\\pm 0\.002of clean retrieval, while the passing branch recorded a no\-harm value of1\.0001\.000\. CertBind extends multimodal composability from connected representations to certifiable task decisions\.

## Introduction

Frozen multimodal encoders provide strong but fragmented representations across image, text, audio, and video\. Lightweight connector maps make these encoders composable without rebuilding a universal model\. Existing systems either synthesize a common space or attach new modalities to a preserved base space\(Wanget al\.[2023](https://arxiv.org/html/2608.06516#bib.bib1); Zhanget al\.[2024](https://arxiv.org/html/2608.06516#bib.bib2); Wanget al\.[2024](https://arxiv.org/html/2608.06516#bib.bib6),[2025](https://arxiv.org/html/2608.06516#bib.bib7)\)\. This progress establishes representation\-level composability across frozen spaces and creates multiple routes to downstream tasks\. More broadly, multimodal models are increasingly evaluated by decision\-level capabilities such as explicit affordance reasoning and action\-region grounding\(Wanget al\.[2026](https://arxiv.org/html/2608.06516#bib.bib31)\)\.

A released connector checkpoint makes the decision\-level deployment problem visible\. On the same native CLIP image\-to\-text workload, the evaluated C\-MCR shared route reduced R@1 from0\.5240\.524to0\.2900\.290\. A route can expand multimodal reach without becoming the right replacement for an established native route\. Connector composition must therefore support expansion and preservation as distinct deployment decisions\.

Decision\-level certifiability is multiscale because a retrieval output inherits ambiguity from the graph that produced it\. At the node scale, graph consistency leaves a residual target gauge relative to a fixed native gallery\. At the edge scale, heterogeneous connector contracts require graph\-wide calibration\. At the path scale, shared failure incidence determines the value of overlapping routes\. At the query scale, the observed boundary margin determines whether recovered scores support a point top\-kkoutput\. Each scale resolves a different ambiguity and supplies the evidence required by the next\.

CertBind develops a multiscale theory of certifiable composition from graph structure to retrieval decisions\. At the node scale, native anchors identify task\-relevant target coordinates\. Contract\-aware screening determines whether the registered direct edge remains supported\. A majority path\-transversal budget and clean calibration recover flagged queries through failure\-diverse paths\. A covered top\-kkcandidate set then certifies decisive outputs and retains set\-valued uncertainty otherwise\. The resulting preservation\-first policy maps the reached certificate level to one deployment action\. Supported routes returnDirect, flagged routes enter recovery, decisive recovery returnsCertified, and unresolved queries returnAbstain\. Figure[1](https://arxiv.org/html/2608.06516#Sx1.F1)follows this progression from graph structure to the final retrieval decision\.

This work makes three contributions to certifiable multimodal composition over frozen connector graphs\.

1. 1\.Certifiable composition from graph consistency to native task scores\.We separate representation connectivity from native task identification by exposing the residual target gauge\. We characterize task\-score identification, establish the exact universal anchor rank, and bound registration error under noisy anchors\.
2. 2\.A multiscale certificate chain from edges to queries\.Contract\-stratified conformal ranks and Holm control graph\-wide clean\-edge false flags\. A majority path\-transversal budget quantifies robustness under shared failures across overlapping routes\. Together, these results yield a covered top\-kkcandidate set and an end\-to\-end error decomposition\.
3. 3\.Preservation\-first action and graph design\.A three\-state rule maps supported routes, decisive recovery, and unresolved queries toDirect,Certified, andAbstain\. The exact path budget is a quota\-cover integer program, while edge\-disjoint paths yield a polynomial Menger certificate\. Released\-system cases quantify the operational stakes of preservation, expansion, and recovery\.

Full proofs, computational details, and additional deployment records appear in the supplement\.

![Refer to caption](https://arxiv.org/html/2608.06516v1/x1.png)Figure 1:CertBind certifies multimodal composition across node, edge, path, and query scales\. \(A\) At the node scale, native anchors fix the target gauge when they span the declared task subspace\. \(B\) At the edge scale, conformal ranks calibrate contract\-aware evidence, Holm controls graph\-wide decisions, and supported edges retain the registered direct route asDirect\. \(C\) At the path and query scales, flagged edges enter overlap\-aware recovery\. CertBind returnsCertifiedwhen the covered top\-kkcandidate set containskkitems andAbstainotherwise\.
## Related Work

### Unified spaces and connector graphs\.

ImageBind, LanguageBind, and UniBind construct broad shared representations\(Girdharet al\.[2023](https://arxiv.org/html/2608.06516#bib.bib3); Zhuet al\.[2024](https://arxiv.org/html/2608.06516#bib.bib4); Lyuet al\.[2024](https://arxiv.org/html/2608.06516#bib.bib5)\)\. C\-MCR and Ex\-MCR connect frozen contrastive spaces, while FreeBind and OmniBind fuse or route among multiple spaces\(Wanget al\.[2023](https://arxiv.org/html/2608.06516#bib.bib1); Zhanget al\.[2024](https://arxiv.org/html/2608.06516#bib.bib2); Wanget al\.[2024](https://arxiv.org/html/2608.06516#bib.bib6),[2025](https://arxiv.org/html/2608.06516#bib.bib7)\)\. Multi\-Way Alignment constructs a jointly consistent universe from matched samples\(Acharaet al\.[2026](https://arxiv.org/html/2608.06516#bib.bib8)\)\. These systems establish representation\-level composability and support broad downstream utility\. CertBind starts from the resulting frozen graph and studies the task decisions supported after deployment\.

### Representation identifiability and stitching\.

Linear identifiability, CKA, model stitching, relative representations, and modality\-gap analyses study when learned spaces share geometry or communicate\(Roederet al\.[2021](https://arxiv.org/html/2608.06516#bib.bib13); Kornblithet al\.[2019](https://arxiv.org/html/2608.06516#bib.bib14); Bansalet al\.[2021](https://arxiv.org/html/2608.06516#bib.bib15); Moschellaet al\.[2023](https://arxiv.org/html/2608.06516#bib.bib16); Lianget al\.[2022](https://arxiv.org/html/2608.06516#bib.bib17); Huhet al\.[2024](https://arxiv.org/html/2608.06516#bib.bib18)\)\. Together, these analyses establish geometric comparability between learned representation spaces\. CertBind specializes the remaining chart ambiguity to native\-gallery retrieval, where native anchors determine whether the declared task scores are identified\.

### Robust recovery and conformal inference\.

Cycle\-edge message passing recovers corrupted group\-valued relative measurements\(Lerman and Shi[2022](https://arxiv.org/html/2608.06516#bib.bib9)\)\. Noisy\-correspondence methods repair sample pairs during training\(Huanget al\.[2021](https://arxiv.org/html/2608.06516#bib.bib10)\)\. Coordinate medians provide classical clean\-majority robustness\(Huber and Ronchetti[2009](https://arxiv.org/html/2608.06516#bib.bib20)\)\. Conformal ranks provide finite\-sample calibration\(Vovket al\.[2005](https://arxiv.org/html/2608.06516#bib.bib19)\)\. Holm’s step\-down procedure controls family\-wise error under arbitrary dependence among valid marginal tests\(Holm[1979](https://arxiv.org/html/2608.06516#bib.bib28)\)\. CertBind carries robust recovery and calibrated uncertainty through shared path failures and observed task margins to a retrieval decision\.

Prior work has established multimodal connectivity, representation comparison, robust recovery, and calibrated inference\. CertBind develops this decision layer as a multiscale theory linking native task identification, graph\-wide screening, overlap\-aware recovery, and query\-level retrieval certification\.

## Setting, Threat Model, and Certificate Semantics

CertBind associates node, edge, path, and query scales with distinct observables, assumptions, and certificates\. Routes and calibration roles are fixed before any candidate outcomes are observed\.

LetG=\(V,E\)G=\(V,E\)be a directed connector graph\. Nodevvcontains a frozen encoder with space𝒳v⊆ℝdv\\mathcal\{X\}\_\{v\}\\subseteq\\mathbb\{R\}^\{d\_\{v\}\}\. Edgee=\(u,v\)e=\(u,v\)is a deployed mapTe:𝒳u→𝒳vT\_\{e\}:\\mathcal\{X\}\_\{u\}\\to\\mathcal\{X\}\_\{v\}\. The formulation admits nonlinear and noninvertible maps\. A declared path family𝒫u​v\\mathcal\{P\}\_\{uv\}provides via outputsyp​\(q\)=Tp​\(q\)y\_\{p\}\(q\)=T\_\{p\}\(q\)for queryqq\. The direct output isy0​\(q\)=Tu​v​\(q\)y\_\{0\}\(q\)=T\_\{uv\}\(q\)\.

Before screening, a predeclared anchor operator𝒜v\\mathcal\{A\}\_\{v\}constructs the graph\-side matrixB=𝒜v​\(A\)B=\\mathcal\{A\}\_\{v\}\(A\)from fixed routes and aggregation rules\. The operator and the native anchor matrixAAare frozen before candidate outcomes\. The identification analysis writesB=R∗​A\+EB=R\_\{\*\}A\+E\. ThusEEcontains the total graph\-side anchor discrepancy\.

The fitted target registration is also frozen before screening\. Henceforthyp​\(q\)y\_\{p\}\(q\)denotes the registered output of routepp\. An edge that passes screening returns its registered direct output without routing\-layer modification\. A flagged edge is replaced only at the routing layer\.

CertBind uses the standard coordinate median and averages the two central order statistics for even sample sizes in each coordinate\. The via fallback is

y^​\(q\)=cmed⁡\{yp​\(q\):p∈𝒫u​v\}\.\\widehat\{y\}\(q\)=\\operatorname\{cmed\}\\\{y\_\{p\}\(q\):p\\in\\mathcal\{P\}\_\{uv\}\\\}\.\(1\)Each edge receives a predeclared audit contract

ξe=\(fu,fv,du,dv,𝗆𝖺𝗉e,𝗇𝗈𝗋𝗆e,R~v,𝒫e,De,ne,νe\)\.\\xi\_\{e\}=\(f\_\{u\},f\_\{v\},d\_\{u\},d\_\{v\},\\mathsf\{map\}\_\{e\},\\mathsf\{norm\}\_\{e\},\\widetilde\{R\}\_\{v\},\\mathcal\{P\}\_\{e\},D\_\{e\},n\_\{e\},\\nu\_\{e\}\)\.\(2\)Its fields specify the encoder families, dimensions, connector class, normalization, frozen registration, route family, score functional, owner count, and owner law\. A bounded owner\-level score aggregates direct–via disagreement

De​\(Oi,ξe\)∈\[0,B\],Ae=1ne​∑i=1neDe​\(Oi,ξe\),D\_\{e\}\(O\_\{i\},\\xi\_\{e\}\)\\in\[0,B\],\\qquad A\_\{e\}=\\frac\{1\}\{n\_\{e\}\}\\sum\_\{i=1\}^\{n\_\{e\}\}D\_\{e\}\(O\_\{i\},\\xi\_\{e\}\),\(3\)whereOiO\_\{i\}is the statistical owner cluster\. Cycle residual and ranking agreement are optional diagnostics\.

### Four certificate scales\.

A structural observation is invariant along a gauge orbit of graph parameterizations\. Node\-scale*task identification*asks when every orbit member induces the same native\-gallery scores\. Edge\-scale*screening*controls false connector flags under a calibration law\. Path\-scale*recovery*combines path incidence and clean\-path calibration to bound the recovered target\. Query\-scale*retrieval certification*guarantees a task decision for a particular recovered query\.

### Split discipline\.

Anchor owners, screening\-control owners, path\-calibration owners, and each future query owner occupy separate roles\. Conditional on the frozen registered design, screening controls and candidate audit owners are independent of the path\-calibration sample and future query owner\. Path\-calibration owners and the future owner follow the exchangeable deployment law\. Strata, routes, and thresholds are fixed independently of candidate outcomes\.

LetQQandGvG\_\{v\}denote the spans of possible target\-space queries and native gallery vectors, and letW=Q\+GvW=Q\+G\_\{v\}\. For an unknown target gaugeRR, exact task\-score identification requires

ProjGv⁡\(R−I\)​ProjQ=0\.\\operatorname\{Proj\}\_\{G\_\{v\}\}\(R\-I\)\\operatorname\{Proj\}\_\{Q\}=0\.\(4\)The universal result below specializes to the common conservative choiceQ=Gv=WQ=G\_\{v\}=W\. All top\-kkoperators use one fixed deterministic rule for score ties\.

Table 1:Each graph scale resolves an ambiguity left by the preceding scale and produces the evidence required for the next deployment decision\.The threat model separates three adversarial surfaces across the certificate chain\.*Gauge ambiguity*changes target coordinates while preserving the graph’s structural observations\.*Edge corruption*can make every route touching a bad edge arbitrary\.*Ordinary approximation error*remains on routes containing no corrupted edge\. Under the stated rank conditions, native anchors resolve gauge ambiguity before redundancy is evaluated\. Screening power requires failure separation, and graph redundancy operates only after native anchors establish task semantics\.

## Node\-Scale Task Identification from Native Anchors

At the node scale, certifiable composition requires a target chart that preserves native task scores\. A graph may remain internally consistent while the chart rotates relative to a fixed native gallery\. Native anchors determine when this residual gauge becomes irrelevant to the declared task\.

Although CertBind accepts arbitrary maps, the impossibility result already arises within a restricted submodel\. Assume a perfectly consistent target node admits an orthogonal chart and structural observations determine its output only up toR∈𝖮​\(W\)R\\in\\mathsf\{O\}\(W\)on anrr\-dimensional task subspaceWW\. Let native anchors be columns ofA=\[a1,…,am\]A=\[a\_\{1\},\\ldots,a\_\{m\}\]and define

Stab⁡\(A\)=\{R∈𝖮​\(W\)∣R​ai=ai​∀i\}\.\\operatorname\{Stab\}\(A\)=\\\{R\\in\\mathsf\{O\}\(W\)\\mid Ra\_\{i\}=a\_\{i\}\\ \\forall i\\\}\.\(5\)
###### Theorem 1\(Task identification and anchor complexity\)\.

LetS=span⁡\(A\)S=\\operatorname\{span\}\(A\)andU=S⟂∩WU=S^\{\\perp\}\\cap W\. Under the chart model

1. 1\.native scores for query spanQQand gallery spanGvG\_\{v\}are identifiable if and only if Eq\. \([4](https://arxiv.org/html/2608.06516#Sx3.E4)\) holds for everyR∈Stab⁡\(A\)R\\in\\operatorname\{Stab\}\(A\)\.
2. 2\.Stab⁡\(A\)=IS⊕𝖮​\(U\)\\operatorname\{Stab\}\(A\)=I\_\{S\}\\oplus\\mathsf\{O\}\(U\)\. Consequently, task scores are identifiable if and only ifQ⊆SQ\\subseteq SorGv⊆SG\_\{v\}\\subseteq S\.
3. 3\.in the universal caseQ=Gv=WQ=G\_\{v\}=W, identifiability holds if and only ifrank⁡\(A\)=r\\operatorname\{rank\}\(A\)=r\. Whenrank⁡\(A\)<r\\operatorname\{rank\}\(A\)<r, a residual reflection preserves every structural observation and anchor but reverses a valid two\-item native ranking\.
4. 4\.supposem≥rm\\geq r,rank⁡\(A\)=r\\operatorname\{rank\}\(A\)=r, and‖E‖op≤ηA\\\|E\\\|\_\{\\mathrm\{op\}\}\\leq\\eta\_\{A\}\. ForR~=B​A†\\widetilde\{R\}=BA^\{\\dagger\} ∥R~−R∗∥op,W≤ηAσr​\(A\)=:δA\.\\\|\\widetilde\{R\}\-R\_\{\*\}\\\|\_\{\\mathrm\{op\},W\}\\leq\\frac\{\\eta\_\{A\}\}\{\\sigma\_\{r\}\(A\)\}=:\\delta\_\{A\}\.\(6\)For a unit query observed asR∗​qR\_\{\*\}q, the fixed registrationR~⊤​R∗​q\\widetilde\{R\}^\{\\top\}R\_\{\*\}qis withinδA\\delta\_\{A\}ofqq\.

### Proof idea\.

Score invariance is equivalent to

⟨\(R−I\)​q,g⟩=0for all​q∈Q,g∈Gv\.\\langle\(R\-I\)q,g\\rangle=0\\quad\\text\{for all \}q\\in Q,\\ g\\in G\_\{v\}\.\(7\)This is Eq\. \([4](https://arxiv.org/html/2608.06516#Sx3.E4)\)\. EveryR∈Stab⁡\(A\)R\\in\\operatorname\{Stab\}\(A\)fixesSSpointwise\. Orthogonality preservesUU, and every element of𝖮​\(U\)\\mathsf\{O\}\(U\)extends to a residual gauge\. If either task span lies inSS, the score change vanishes\. Conversely, suppose both spans have nonzero projections ontoUU\. Chooseq∈Qq\\in Qandg∈Gvg\\in G\_\{v\}with nonzero residual components\. An orthogonal action onUUcan change their inner product\. Its identity extension onSSchanges the task score while fixing every anchor\. This proves the finite task\-specific criterion\. The universal condition follows by takingQ=Gv=WQ=G\_\{v\}=W\.

For noisy anchors, full row rank givesA​A†=IWAA^\{\\dagger\}=I\_\{W\}\. Therefore

R~−R∗=\(R∗​A\+E\)​A†−R∗=E​A†\.\\widetilde\{R\}\-R\_\{\*\}=\(R\_\{\*\}A\+E\)A^\{\\dagger\}\-R\_\{\*\}=EA^\{\\dagger\}\.\(8\)Submultiplicativity and‖A†‖op=1/σr​\(A\)\\\|A^\{\\dagger\}\\\|\_\{\\mathrm\{op\}\}=1/\\sigma\_\{r\}\(A\)prove Eq\. \([6](https://arxiv.org/html/2608.06516#Sx4.E6)\)\. Multiplication by the unit vectorR∗​qR\_\{\*\}qgives the registered\-query bound\. The supplement gives the reflection construction and complete proofs\.

### Design consequence\.

The theorem gives the exact identification boundary\. Task identification depends on anchor coverage of the query or gallery span\. Universal identification requires full rank onWW, while noisy stability depends on anchor conditioning rather than anchor count alone\. Under rank deficiency,σr​\(A\)=0\\sigma\_\{r\}\(A\)=0and the universal finite bound is unavailable\. At full rank, the stability radius scales as1/σr​\(A\)1/\\sigma\_\{r\}\(A\)\. A redundant anchor cluster can therefore be less stable than a smaller, well\-conditioned spanning set\. An offline selector should maximize the smallest singular value on the declared task subspace subject to acquisition cost\. If task queries and galleries occupy smaller spans, Eq\. \([4](https://arxiv.org/html/2608.06516#Sx3.E4)\) can certify them without fixing irrelevant target directions\.

###### Proposition 1\(Optimal bounded\-norm anchor geometry\)\.

Supposem≥rm\\geq rand‖ai‖2≤1\\\|a\_\{i\}\\\|\_\{2\}\\leq 1\. Thenσr​\(A\)2≤m/r\\sigma\_\{r\}\(A\)^\{2\}\\leq m/r\. Equality holds exactly whenA​A⊤=\(m/r\)​IWAA^\{\\top\}=\(m/r\)I\_\{W\}and every anchor has unit norm\. For a fixed operator\-norm discrepancy budgetηA\\eta\_\{A\}, a unit\-norm tight frame minimizes the bound in Eq\. \([6](https://arxiv.org/html/2608.06516#Sx4.E6)\)\.

## Edge\-Scale Contract\-Aware Graph\-Wide Screening

With node\-scale task identification fixed, edge\-scale screening asks which registered direct connectors remain supported under their audit contracts\. Contract strata define the calibration populations, while graph\-wide multiplicity control protects clean registered routes\.

Connector edges are rarely exchangeable across all architectures\. The audit stratums​\(e\)s\(e\)is the predeclared equivalence class of the complete contractξe\\xi\_\{e\}\. Edges share controls only when every contract field matches\. Each controlAj\(s\)A\_\{j\}^\{\(s\)\}is a complete owner\-aggregated score generated under that contract\. Write𝒟s=\(A1\(s\),…,AMs\(s\)\)\\mathcal\{D\}\_\{s\}=\(A\_\{1\}^\{\(s\)\},\\ldots,A\_\{M\_\{s\}\}^\{\(s\)\}\)\. Define

ps​\(a\)=1\+∑j=1Ms𝟏​\{Aj\(s\)≥a\}Ms\+1,pe=ps​\(Ae\),p\_\{s\}\(a\)=\\frac\{1\+\\sum\_\{j=1\}^\{M\_\{s\}\}\\mathbf\{1\}\\\{A\_\{j\}^\{\(s\)\}\\geq a\\\}\}\{M\_\{s\}\+1\},\\qquad p\_\{e\}=p\_\{s\}\(A\_\{e\}\),\(9\)and, for a frozen batch ofLLedges,

τs=sup\{a∈\[0,B\]∣ps​\(a\)\>δs/L\}\.\\tau\_\{s\}=\\sup\\\{a\\in\[0,B\]\\mid p\_\{s\}\(a\)\>\\delta\_\{s\}/L\\\}\.\(10\)An under\-populated stratum returnsAbstain\.

###### Theorem 2\(Stratified conformal screening and graph\-wide control\)\.

Suppose a clean candidate score and the clean controls are exchangeable conditional on its predeclared stratum\.

1. 1\.pep\_\{e\}is conditionally super\-uniform withℙ​\(pe≤α∣s​\(e\)\)≤α\\mathbb\{P\}\(p\_\{e\}\\leq\\alpha\\mid s\(e\)\)\\leq\\alpha\.
2. 2\.Holm’s procedure at levelδs\\delta\_\{s\}controls the probability of at least one clean\-edge false flag byδs\\delta\_\{s\}\. Candidate\-edge dependence is unrestricted\.
3. 3\.The smallest attainablepp\-value is1/\(Ms\+1\)1/\(M\_\{s\}\+1\)\. Rejection at levelα\\alpharequiresMs≥⌈1/α⌉−1M\_\{s\}\\geq\\lceil 1/\\alpha\\rceil\-1\.
4. 4\.every edge that passes Holm satisfiespe\>δs/Lp\_\{e\}\>\\delta\_\{s\}/Land thereforeAe≤τsA\_\{e\}\\leq\\tau\_\{s\}\.
5. 5\.conditional on the clean controls, suppose thenen\_\{e\}bad\-edge owner scores are independent in\[0,B\]\[0,B\]and satisfy 1ne​∑i=1ne𝔼bad​\[De​\(Oi,ξe\)∣𝒟s\]≥τs\+Δ\.\\frac\{1\}\{n\_\{e\}\}\\sum\_\{i=1\}^\{n\_\{e\}\}\\mathbb\{E\}\_\{\\rm bad\}\[D\_\{e\}\(O\_\{i\},\\xi\_\{e\}\)\\mid\\mathcal\{D\}\_\{s\}\]\\geq\\tau\_\{s\}\+\\Delta\.\(11\)Then ℙbad​\(pass∣𝒟s\)≤exp⁡\(−2​ne​Δ2/B2\)\.\\mathbb\{P\}\_\{\\rm bad\}\(\\mathrm\{pass\}\\mid\\mathcal\{D\}\_\{s\}\)\\leq\\exp\(\-2n\_\{e\}\\Delta^\{2\}/B^\{2\}\)\.\(12\)

The first three claims follow from the candidate rank amongMs\+1M\_\{s\}\+1exchangeable scores and the standard Holm step\-down argument\. For the fourth claim, every unrejected orderedpp\-value exceeds the threshold at the first failed Holm step\. That threshold is at leastδs/L\\delta\_\{s\}/L\. Hence a pass impliespe\>δs/Lp\_\{e\}\>\\delta\_\{s\}/L, which givesAe≤τsA\_\{e\}\\leq\\tau\_\{s\}by Eq\. \([10](https://arxiv.org/html/2608.06516#Sx5.E10)\)\.

Conditional on𝒟s\\mathcal\{D\}\_\{s\}, define

μe,s:=1ne​∑i=1ne𝔼​\[De​\(Oi,ξe\)∣𝒟s\]\.\\mu\_\{e,s\}:=\\frac\{1\}\{n\_\{e\}\}\\sum\_\{i=1\}^\{n\_\{e\}\}\\mathbb\{E\}\[D\_\{e\}\(O\_\{i\},\\xi\_\{e\}\)\\mid\\mathcal\{D\}\_\{s\}\]\.Since\{pass\}⊆\{Ae≤τs\}\\\{\\mathrm\{pass\}\\\}\\subseteq\\\{A\_\{e\}\\leq\\tau\_\{s\}\\\}, the mean\-separation assumption and Hoeffding’s inequality give

ℙbad​\(pass∣𝒟s\)\\displaystyle\\mathbb\{P\}\_\{\\rm bad\}\(\\mathrm\{pass\}\\mid\\mathcal\{D\}\_\{s\}\)≤ℙbad​\(Ae−μe,s≤−Δ∣𝒟s\)\\displaystyle\\leq\\mathbb\{P\}\_\{\\rm bad\}\\bigl\(A\_\{e\}\-\\mu\_\{e,s\}\\leq\-\\Delta\\mid\\mathcal\{D\}\_\{s\}\\bigr\)\(13\)≤exp⁡\(−2​ne​Δ2B2\)\.\\displaystyle\\leq\\exp\\\!\\left\(\-\\frac\{2n\_\{e\}\\Delta^\{2\}\}\{B^\{2\}\}\\right\)\.Thus the screening power statement uses the same pass event as the deployed Holm procedure\.

The screening theorem supplies the edge\-scale evidence required by preservation\-first routing\. Under within\-contract exchangeability, it controls graph\-wide clean\-edge false flags for the frozen batch\. Detecting a bad edge additionally requires the stated conditional mean separation\. Equation \([10](https://arxiv.org/html/2608.06516#Sx5.E10)\) connects that separation directly to the implemented Holm decision\. At marginal level0\.050\.05, rejection requires at least1919clean controls\. The first Holm threshold forLLedges can require substantially more\.

Multiple captions, frames, or segments from one item are aggregated before the concentration claim\. Holm permits dependence across candidate edges because it needs only marginal super\-uniformity\. The family\-wise guarantee applies to one frozen deployment batch ofLLhypotheses\. Each later batch requires a new predeclared family with its own error budget\.

Screening resolves the edge action by returning supported routes asDirectand sending flagged routes to path\-scale recovery under a separate certificate\.

## Path\-Scale Recovery and Query\-Scale Retrieval Certification

At the path scale, a flagged edge activates alternative routes whose value depends on shared failure incidence rather than route count\. The certificate measures how many corrupted edges can contaminate a majority of declared paths\.

LetF⊆EF\\subseteq Ebe corrupted edges\. A declared path is contaminated when it intersectsFF\. Form=\|𝒫\|m=\|\\mathcal\{P\}\|, define

C𝒫​\(F\)\\displaystyle C\_\{\\mathcal\{P\}\}\(F\)=\|\{p∈𝒫:p∩F≠∅\}\|,\\displaystyle=\|\\\{p\\in\\mathcal\{P\}:p\\cap F\\neq\\varnothing\\\}\|,\(14\)κ1/2​\(𝒫\)\\displaystyle\\kappa\_\{1/2\}\(\\mathcal\{P\}\)=minF⁡\{\|F\|:C𝒫​\(F\)≥⌈m/2⌉\}\.\\displaystyle=\\min\_\{F\}\\\{\|F\|:C\_\{\\mathcal\{P\}\}\(F\)\\geq\\lceil m/2\\rceil\\\}\.\(15\)Overlapping paths earn robustness only when more bad edges are required to contaminate half the route family\. For example, ten routes sharing one external edge haveκ1/2=1\\kappa\_\{1/2\}=1, whereas five edge\-disjoint routes haveκ1/2=3\\kappa\_\{1/2\}=3\.

Thusκ1/2\\kappa\_\{1/2\}converts shared failure incidence into the strict clean\-majority condition required by coordinate\-median recovery\.

A separate clean calibration split contains native targetsyi∗y\_\{i\}^\{\*\}and the outputs of the same*fixed, anchor\-registered*path family used at deployment\. Define owner residuals

Zi=maxp∈𝒫⁡‖yi,p−yi∗‖∞,Z\_\{i\}=\\max\_\{p\\in\\mathcal\{P\}\}\\\|y\_\{i,p\}\-y\_\{i\}^\{\*\}\\\|\_\{\\infty\},\(16\)and letε^δp\\widehat\{\\varepsilon\}\_\{\\delta\_\{p\}\}be the split\-conformal upper order statistic at rank⌈\(N\+1\)​\(1−δp\)⌉\\lceil\(N\+1\)\(1\-\\delta\_\{p\}\)\\rceil\(or\+∞\+\\inftywhen that rank exceedsNN\)\.

The path budget establishes structural recoverability, while the clean split calibrates a finite\-sample radius\. The chain then enters the query scale by converting this path evidence into a covered top\-kkcandidate set\.

###### Theorem 3\(Calibrated sparse\-graph prediction set\)\.

Fix a declared via\-edge corruption budgetb<κ1/2​\(𝒫\)b<\\kappa\_\{1/2\}\(\\mathcal\{P\}\)\. Assume future clean\-path residuals are exchangeable with theNNcalibration owners\. Also assume the realized corrupted via\-edge set satisfies\|F\|≤b\|F\|\\leq b\. Define

ρ:=d​ε^δp\.\\rho:=\\sqrt\{d\}\\,\\widehat\{\\varepsilon\}\_\{\\delta\_\{p\}\}\.\(17\)Under the joint law of the path\-calibration sample and one future owner,

ℙ​\(‖y^−y∗‖2≤ρ\)≥1−δp\.\\mathbb\{P\}\(\\\|\\widehat\{y\}\-y^\{\*\}\\\|\_\{2\}\\leq\\rho\)\\geq 1\-\\delta\_\{p\}\.\(18\)Let the gallery containJJvectors with norm at most one and fix1≤k<J1\\leq k<J\. Writesj=⟨y^,gj⟩s\_\{j\}=\\langle\\widehat\{y\},g\_\{j\}\\rangle\. Defineℓj=sj−ρ\\ell\_\{j\}=s\_\{j\}\-\\rhoanduj=sj\+ρu\_\{j\}=s\_\{j\}\+\\rho\. Letτk\\tau\_\{k\}be thekkth largest lower endpoint and set

𝒞k=\{j∣uj≥τk\}\.\\mathcal\{C\}\_\{k\}=\\\{j\\mid u\_\{j\}\\geq\\tau\_\{k\}\\\}\.\(19\)For the clean native setSk∗=Topk⁡\(y∗\)S\_\{k\}^\{\*\}=\\operatorname\{Top\}\_\{k\}\(y^\{\*\}\),

ℙ​\(Sk∗⊆𝒞k\)≥1−δp\.\\mathbb\{P\}\(S\_\{k\}^\{\*\}\\subseteq\\mathcal\{C\}\_\{k\}\)\\geq 1\-\\delta\_\{p\}\.\(20\)LetS^k=Topk⁡\(y^\)\\widehat\{S\}\_\{k\}=\\operatorname\{Top\}\_\{k\}\(\\widehat\{y\}\)and letγ^k\\widehat\{\\gamma\}\_\{k\}be its observed boundary gap\. Then

\|𝒞k\|=k⟺γ^k\>2​ρ\.\\lvert\\mathcal\{C\}\_\{k\}\\rvert=k\\quad\\Longleftrightarrow\\quad\\widehat\{\\gamma\}\_\{k\}\>2\\rho\.\(21\)DefineCertifiedby this equivalent condition\. Then

ℙ​\(Certified∩\{S^k≠Sk∗\}\)≤δp\.\\mathbb\{P\}\\bigl\(\\text\{\{Certified\}\}\\cap\\\{\\widehat\{S\}\_\{k\}\\neq S\_\{k\}^\{\*\}\\\}\\bigr\)\\leq\\delta\_\{p\}\.\(22\)When\|𝒞k\|\>k\\lvert\\mathcal\{C\}\_\{k\}\\rvert\>k, the method returns𝒞k\\mathcal\{C\}\_\{k\}and abstains from a point top\-kkoutput\.

### Proof idea\.

The inequality\|F\|<κ1/2​\(𝒫\)\|F\|<\\kappa\_\{1/2\}\(\\mathcal\{P\}\)leaves a strict majority ofFF\-free paths\. On the conformal event, every such path lies in the coordinate box centered aty∗y^\{\*\}with radiusε^δp\\widehat\{\\varepsilon\}\_\{\\delta\_\{p\}\}\. Both central order statistics lie in that box in every coordinate\. Their coordinate median therefore lies within the radius in Eq\. \([17](https://arxiv.org/html/2608.06516#Sx6.E17)\)\.

For every gallery item, Cauchy–Schwarz gives

\|⟨y^−y∗,gj⟩\|≤ρ\.\|\\langle\\widehat\{y\}\-y^\{\*\},g\_\{j\}\\rangle\|\\leq\\rho\.\(23\)Hence its clean score lies in\[ℓj,uj\]\[\\ell\_\{j\},u\_\{j\}\]\. At leastkkitems have lower endpoint at leastτk\\tau\_\{k\}\. An item outside𝒞k\\mathcal\{C\}\_\{k\}has upper endpoint belowτk\\tau\_\{k\}and cannot enterSk∗S\_\{k\}^\{\*\}\. This proves Eq\. \([20](https://arxiv.org/html/2608.06516#Sx6.E20)\) on the conformal event\.

All intervals have the same radius, soτk=s\(k\)−ρ\\tau\_\{k\}=s\_\{\(k\)\}\-\\rhoand

𝒞k=\{j∣sj≥s\(k\)−2​ρ\}\.\\mathcal\{C\}\_\{k\}=\\\{j\\mid s\_\{j\}\\geq s\_\{\(k\)\}\-2\\rho\\\}\.\(24\)It has sizekkexactly whens\(k\+1\)<s\(k\)−2​ρs\_\{\(k\+1\)\}<s\_\{\(k\)\}\-2\\rho\. This is Eq\. \([21](https://arxiv.org/html/2608.06516#Sx6.E21)\)\. On the coverage event, a certified set containsSk∗S\_\{k\}^\{\*\}and has the same cardinality\. It must equalSk∗S\_\{k\}^\{\*\}\. Therefore a certified error can occur only when coverage fails, proving Eq\. \([22](https://arxiv.org/html/2608.06516#Sx6.E22)\)\. The supplement proves the interval minimality statement\.

At the path scale, the theorem maps the budget and calibration to a recovery radius\. At the query scale, it maps that radius to a covered top\-kkcandidate set\. The output becomes point\-valued when the observed margin reduces this set tokkitems\.

The statistical certificate chain leaves two routes to an incorrect non\-abstaining output\. A separated bad direct edge can pass screening, or a query certificate can fail when its covered radius event fails\.

###### Corollary 1\(End\-to\-end error decomposition\)\.

Conditional on the screening controls, consider one bad direct edge that satisfies the separation in Theorem[2](https://arxiv.org/html/2608.06516#Thmtheorem2)\. Let the path\-calibration sample and future query owner follow the joint law in Theorem[3](https://arxiv.org/html/2608.06516#Thmtheorem3), independently of the screening data\. Then

ℙ​\(wrong non\-abstaining output∣𝒟s\)≤e−2​ne​Δ2/B2\+δp\.\\mathbb\{P\}\(\\text\{wrong non\-abstaining output\}\\mid\\mathcal\{D\}\_\{s\}\)\\leq e^\{\-2n\_\{e\}\\Delta^\{2\}/B^\{2\}\}\+\\delta\_\{p\}\.\(25\)

A wrongDirectoutput requires a bad\-edge pass\. A wrongCertifiedoutput requires failure of the covered radius event\. The union bound gives Eq\. \([25](https://arxiv.org/html/2608.06516#Sx6.E25)\)\. Separately, Holm controls the graph\-wide probability of changing any clean direct route byδs\\delta\_\{s\}\. The corollary closes the statistical chain at the returned query output\.

At the path scale, the corruption budget is sharp in the path\-output model\. An adversary with budgetκ1/2\\kappa\_\{1/2\}can choose an edge set that contaminates at least half the paths and can drive the coordinate median unbounded\. For cosine retrieval, normalization adds the deterministic factor in the supplement\. Ifrrselected paths are edge\-disjoint,κ1/2=⌈r/2⌉\\kappa\_\{1/2\}=\\lceil r/2\\rceil\. Maximizingrrgives directed edge connectivity by Menger’s theorem\(Menger[1927](https://arxiv.org/html/2608.06516#bib.bib21)\)\. Failure diversity, rather than route count, therefore becomes the next graph\-design objective\.

## Computation, Route Planning, and Preservation\-First Decision Rule

The path budget defines a graph\-design objective, while the multiscale chain maps the reached certificate level to one of three actions\.

For fixed path\-edge incidenceHp​e=𝟏​\{e∈p\}H\_\{pe\}=\\mathbf\{1\}\\\{e\\in p\\\}and costsce≥0c\_\{e\}\\geq 0, the weighted exact budget is

minz,h\\displaystyle\\min\_\{z,h\}\\quad∑ece​ze\\displaystyle\\sum\_\{e\}c\_\{e\}z\_\{e\}\(26\)s\.t\.hp≤∑eHp​e​ze,∑php≥⌈m/2⌉,\\displaystyle h\_\{p\}\\leq\\sum\_\{e\}H\_\{pe\}z\_\{e\},\\quad\\sum\_\{p\}h\_\{p\}\\geq\\lceil m/2\\rceil,\(27\)ze,hp∈\{0,1\}\.\\displaystyle z\_\{e\},h\_\{p\}\\in\\\{0,1\\\}\.\(28\)The weighted explicit\-incidence problem is NP\-hard by reduction from set cover\(Karp[1972](https://arxiv.org/html/2608.06516#bib.bib29)\)\. Equation \([28](https://arxiv.org/html/2608.06516#Sx7.E28)\) gives an exact offline formulation of this optimization problem\. Edge\-disjoint routes provide a polynomial\-time certificate for deployment planning\. A deployment planner should maximizeκ1/2\\kappa\_\{1/2\}or its weighted analogue subject to hop, latency, and edge\-quality constraints\.

Table 2:CertBind maps the certificate level reached by the deployment chain to one of three actions\.The deployment log records the complete audit contract, anchor rank,σr​\(A\)\\sigma\_\{r\}\(A\), conformalpp\-value, Holm threshold, declared budget, path radius, covered top\-kkcandidate set, observed margin, and final state\. By construction, the fallback aggregation excludes the registered direct\-edge output\. Add\-only insertion freezes old encoders, maps, registrations, route sets, and thresholds\. This rule therefore gives exact zero drift on retained routing layers\.

The preservation\-first rule operationalizes this chain by retaining supported routes asDirectand sending flagged routes to recovery\. Recovery becomesCertifiedonly when the query\-scale candidate set contracts tokkitems\. Otherwise, the output isAbstain, with the covered top\-kkcandidate set when available\.

## Deployment Evidence for Certifiable Composition

Released checkpoints and recorded protocols expose three operational demands of certifiable composition\. Table[3](https://arxiv.org/html/2608.06516#Sx8.T3)organizes the corresponding evidence as preservation, expansion, and preserve\-or\-recover behavior\.

Table 3:Evidence for preservation, expansion, and preserve\-or\-recover routing under released checkpoints and recorded protocols\.### Native\-route preservation\.

On the native CLIP image\-to\-text workload, the evaluated C\-MCR shared route changed R@1 from0\.5240\.524to0\.2900\.290\. This comparison identifies native preservation as a separate deployment target\.

### Cross\-modal expansion\.

On the disjoint 245\-clip Clotho subset, the original maps were applied without edge refitting\. C\-MCR and Ex\-MCR reached text\-to\-audio R@1 values of0\.1680\.168and0\.1800\.180, respectively\. Towers alone reached0\.2420\.242, while adding Ex\-MCR as optional graph evidence reached0\.2670\.267\.

### Preserve\-or\-recover routing\.

Across three seeds, the production via\-only fallback recovered0\.963±0\.0020\.963\\pm 0\.002of clean retrieval, while the passing branch recorded no\-harm1\.0001\.000\. A separate five\-family blind\-median stress suite included the direct output and recovered0\.9830\.983–0\.9890\.989\. Together, these cases separate native preservation, cross\-modal expansion, and selective recovery\.

## From Multimodal Connectivity to Certifiable Composition

Multimodal composability extends beyond representation connectivity to the task scores induced by a composed graph\. Graph consistency describes relative structure, while native anchors determine whether that structure preserves fixed native\-gallery scores\. A connector can therefore remain internally coherent while changing an established retrieval route\. This distinction makes task identification the node\-scale prerequisite for native preservation\.

A related distinction between literal information preservation and downstream recoverability also appears in lossy text compression, where strategically deleted content may be reconstructed by an LLM\(Zouet al\.[2026](https://arxiv.org/html/2608.06516#bib.bib30)\)\. CertBind addresses a different problem: it does not reconstruct omitted content, but certifies whether connector\-induced representation uncertainty preserves a fixed native retrieval decision\.

Calibration is part of composition design because each statistical certificate is indexed by its deployment population\. Contract strata determine where edge ranks are valid, while owner units and route families determine path calibration\. Broader strata can violate exchangeability, whereas narrower strata reduce attainablepp\-value resolution\. Control collection and route design must therefore be planned together\.

For median recovery, the relevant unit of graph redundancy is shared failure incidence rather than route count\. Paths that share one edge repeat outputs without adding protection against that edge\. The majority path\-transversal budget converts this failure structure into both a recovery criterion and a route\-planning objective\.

In CertBind, a recovered representation becomes a certified retrieval output only when its uncertainty clears the query’s observed top\-kkboundary\. The calibrated radius induces a covered top\-kkcandidate set whose size depends on the observed margin\. Well\-separated queries support a point certificate, while boundary\-adjacent queries remain set\-valued and returnAbstain\. Certification is therefore a property of the query decision rather than aggregate fallback accuracy\.

Task\-relevant symmetries provide a route beyond orthogonal charts for certifiable composition\. The present identification result studies orthogonal gauges because inner\-product retrieval is invariant to a common orthogonal change\. Other connector families induce different task\-relevant symmetry groups, whose stabilizers determine the corresponding identification condition\. Noncompact and nonlinear actions require separate identification and estimation results\.

## Scope and Ethical Considerations

The guarantees are indexed by explicit deployment conditions at each certificate scale\. The declared anchor condition identifies native task scores under the orthogonal chart model\. Within\-contract exchangeability establishes clean\-edge validity, while independent owner clusters and conditional mean separation give the miss bound\. The path theorem permits arbitrary outputs on routes touching corrupted edges when the declared edge budget remains belowκ1/2\\kappa\_\{1/2\}\. Exact budget computation may require offline integer optimization, while edge\-disjoint paths provide a polynomial certificate\.

The certificate chain also determines the evidence recorded for each deployment decision\. Before automatic recovery, an operator declares the task subspace and native gallery\. The record identifies native anchors, their conditioning, connector contracts, route incidence, calibration populations, and owner\-level path residuals\. Any missing required field maps the affected query toAbstain\.

CertBind supports auditable deployment decisions and retains human\-governed escalation for unresolved queries\. Deployment records contain connector and calibration metadata and should receive the same access controls as model artifacts\. The reported empirical record covers released checkpoints under the stated protocols\.

## Conclusion

Lightweight connector graphs make frozen multimodal systems composable at the representation level\. CertBind extends this composability to the decision level through a multiscale certificate chain spanning native task identification, contract\-aware graph\-wide screening, overlap\-aware recovery, and finite\-sample top\-kkcertification\. An exact node\-scale identification boundary, graph\-wide edge control, sharp path robustness, and finite\-sample query coverage provide the formal basis for this chain\. Its preservation\-first policy retains supported routes, certifies decisive recovery, and records unresolved queries asAbstain\. Connectivity is a property of the graph\. Certifiability is a property of the task decision\.

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