CARVE-Q: Quantum-Proposed, Classically Certified Interactive Driving Repair

# CARVE-Q: Quantum-Proposed, Classically Certified Interactive Driving Repair
Source: [https://arxiv.org/html/2606.06531](https://arxiv.org/html/2606.06531)
###### Abstract

The critical question after a correct driving veto is not only whether a maneuver is unsafe, but whether the blocked interaction admits a lawful, auditable, and responsibility\-bounded repair\. Prediction and game\-theoretic planners can suggest plausible cooperation, yet they do not return a proof that the repair respects hard rules, right\-of\-way, cost allocation, and ego fallback\. We introduceCARVE,*Certified Affordable Repair of Vetoed maneuvers via Envelopes*, a certificate architecture for prediction\-free interactive repair\. Given a vetoed maneuver,CARVEconstructs a finite repair lattice and emits a structured certificate recording the binding rule, selected joint repair, right\-of\-way\-scaled cooperation envelope, responsibility\-weighted cost split, and ego\-only fallback\. This certificate view reveals the algorithmic bottleneck: multi\-owner repair induces a product latticeM=∏j\|𝒜j\|M=\\prod\_\{j\}\|\\mathcal\{A\}\_\{j\}\|\. We therefore introduceCARVE\-Q, a verifier\-shielded quantum\-AI search layer that applies quantum minimum finding only to this black\-box lattice while leaving all safety authority classical\. In the conservative verifier\-oracle model, exact classical minimum finding requiresΘ​\(M\)\\Theta\(M\)queries in the worst case, whereas Durr–Hoyer/Grover minimum finding usesO​\(M\)O\(\\sqrt\{M\}\)oracle queries with high probability\. We prove verifier\-shielded certificate soundness, priority non\-elicitation, black\-box query separation, and finite\-precision reversible\-oracle constructibility\. We then demonstrate statevector minimum finding onCARVErepair oracles up to 65,536 assignments and validate certificate preservation on Lanelet2\-grounded INTERACTION replay with 100% right\-of\-way respect, 100% blame consistency, and zero priority false positives\. The result is a trust\-bounded quantum\-AI pattern for certified autonomy: quantum proposes;CARVEcertifies\.

## Introduction

The most important decision in interactive driving may begin after the correct veto\. Consider an ego vehicle entering a dense urban merge\. A hard\-rule gate rejects the maneuver because another vehicle is marginally inside the required gap\. The rejection is safe, but it is not yet an explanation of what should happen next\. A predictor may guess that the other driver will slow down; a game\-theoretic planner may search for a compatible response\. Neither object is the proof a safety\-critical system needs: a certificate stating why the maneuver was vetoed, what bounded repair is allowed, who is asked to accommodate, how the cost is assigned, and what ego can still do safely if cooperation does not arrive\.

This example exposes a disconnect\. Hard\-rule vetoes are essential for unrecoverable danger, yet they are behaviorally rigid when a conflict can be resolved through a small, bounded accommodation such as waiting, yielding, or decelerating\. Prediction and social driving models estimate likely interactions\(Sadighet al\.[2016](https://arxiv.org/html/2606.06531#bib.bib22); Schwartinget al\.[2019](https://arxiv.org/html/2606.06531#bib.bib23); Alahiet al\.[2016](https://arxiv.org/html/2606.06531#bib.bib25); Deo and Trivedi[2018](https://arxiv.org/html/2606.06531#bib.bib27); Salzmannet al\.[2020](https://arxiv.org/html/2606.06531#bib.bib26)\), while rulebooks, RSS, shielding, and formal\-safety methods specify constraints\(Shalev\-Shwartzet al\.[2017](https://arxiv.org/html/2606.06531#bib.bib18); Censiet al\.[2019](https://arxiv.org/html/2606.06531#bib.bib19); Alshiekhet al\.[2018](https://arxiv.org/html/2606.06531#bib.bib20); Garcia and Fernandez[2015](https://arxiv.org/html/2606.06531#bib.bib21)\)\. The missing object is neither another trajectory predictor nor another veto rule; it is a certificate\-bearing interactive repair\.

We introduceCARVE,*Certified Affordable Repair of Vetoed maneuvers via Envelopes*\.CARVEelevates a rejected maneuver from terminal failure into a finite repair problem whose output is an auditable liability record\. The certificate names the binding hard rule, selects a bounded joint repair, checks right\-of\-way\-scaled cooperation envelopes, records a responsibility\-weighted cost split, and preserves an ego\-only fallback\. A repairable interaction must therefore answer three questions at once: does the repair make every declared hard\-rule margin nonnegative, does every requested accommodation stay inside a normatively admissible envelope, and can ego still recover without assuming another driver’s compliance?

CARVEalso exposes a search bottleneck\. Ifnnrepair owners each have a finite action set𝒜j\\mathcal\{A\}\_\{j\}, exact joint repair searches a product lattice of sizeM=∏j\|𝒜j\|M=\\prod\_\{j\}\|\\mathcal\{A\}\_\{j\}\|\. Dense unprotected turns, narrow construction\-zone negotiations, and multi\-agent merge conflicts can require several vehicles to choose among wait, decelerate, yield, or no\-op edits\. The quantum component is not imposed on the problem; it appears only afterCARVEconverts interaction repair into this finite certificate lattice\.

The black\-box verifier model is the safety assumption that makes the quantum layer meaningful\. If the verifier exposes stable convex, separable, or low\-treewidth structure, white\-box classical solvers should exploit it\. In safety\-critical integration, however, rulebooks can change, predicates can be nonconvex or adversarially coupled, and future shields may include proprietary perception modules\. A trust\-bounded design therefore assumes no exploitable structure and analyzes the verifier as an opaque cost oracle\. Under this model, exact classical minimum finding has a worst\-caseΘ​\(M\)\\Theta\(M\)query requirement, whereas Grover/Durr–Hoyer minimum finding givesO​\(M\)O\(\\sqrt\{M\}\)oracle queries with high probability\(Grover[1996](https://arxiv.org/html/2606.06531#bib.bib1); Durr and Hoyer[1996](https://arxiv.org/html/2606.06531#bib.bib2); Bennettet al\.[1997](https://arxiv.org/html/2606.06531#bib.bib3); Boyeret al\.[1998](https://arxiv.org/html/2606.06531#bib.bib4); Brassardet al\.[2002](https://arxiv.org/html/2606.06531#bib.bib5)\)\. This is the precise sense in which quantum search is necessary inCARVE\-Q\.

CARVE\-Qfollows one trust boundary throughout: quantum proposes;CARVEcertifies\. The quantum module searches for a low\-cost joint repair, but the deterministicCARVEverifier recomputes every predicate before any certificate can be emitted\. We do not claim a new quantum search algorithm, present\-day hardware speedup, universal superiority over white\-box solvers, or quantum\-certified safety\. The claim is architectural and oracle\-theoretic:CARVEdefines the certified repair object,CARVE\-Qtargets its black\-box joint\-lattice bottleneck, andCARVEremains the final certificate authority\.

Our contributions are:

- •Certified interactive repair\.We formulate hard\-rule\-vetoed driving interaction as a finite repair\-certificate problem rather than a prediction, game response, or ego\-only replanning problem\.
- •CARVEcertificate semantics\.We define right\-of\-way\-scaled cooperation envelopesBj​\(s\)=β​\(πj\)​αjmax​\(s\)B\_\{j\}\(s\)=\\beta\(\\pi\_\{j\}\)\\alpha\_\{j\}^\{\\max\}\(s\), responsibility\-weighted costs, affordability predicates, and fallback validity\.
- •Verifier\-shielded quantum\-AI search\.We show that multi\-owner repair induces a product latticeM=∏j\|𝒜j\|M=\\prod\_\{j\}\|\\mathcal\{A\}\_\{j\}\|and introduceCARVE\-Q, which applies quantum minimum finding to the black\-box verifier\-cost oracle while preserving classical certificate authority\.
- •Guarantees and evidence\.We prove verifier\-shielded soundness, priority non\-elicitation, black\-box query separation, and reversible\-oracle constructibility; we validate them with statevector minimum finding, black\-box stress, resource accounting, replay diagnostics, and QAOA/local\-search audits\.

![Refer to caption](https://arxiv.org/html/2606.06531v1/figure1.png)Figure 1:CARVE\-Qarchitecture\.CARVEis the certified repair architecture: it elevates a hard\-rule veto into an auditable certificate, exposes a finite multi\-owner repair lattice, and lets quantum minimum finding search only that black\-box bottleneck\. The quantum module is a proposal layer only; the final certificate is emitted by the classicalCARVEverifier\.
## From Vetoed Maneuvers to CARVE Certificates

CARVEbegins from a different object than prediction\-based interaction planning: a repair certificate\. The input is a sceness, a candidate maneuvermm, and a hard\-rule prefixℋ\\mathcal\{H\}\. If all margins are nonnegative,CARVEreturns an empty satisfied certificate\. If a binding ruleh⋆h^\{\\star\}has negative margin,CARVEconstructs a finite repair lattice with ego\-owned edits and bounded agent\-owned accommodation requests\.

###### Definition 1\(CARVEcertificate\)

Given a sceness, a vetoed maneuvermm, and a binding hard ruleh⋆h^\{\\star\}, aCARVEcertificate is

𝒞=\(κ,h⋆,x⋆,ρego,\{ρj\}j=1n,𝒜fb\),\\mathcal\{C\}=\(\\kappa,h^\{\\star\},x^\{\\star\},\\rho\_\{\\mathrm\{ego\}\},\\\{\\rho\_\{j\}\\\}\_\{j=1\}^\{n\},\\mathcal\{A\}\_\{\\mathrm\{fb\}\}\),whereκ\\kappais the certificate category,x⋆x^\{\\star\}is the selected joint repair assignment,ρego\\rho\_\{\\mathrm\{ego\}\}andρj\\rho\_\{j\}are responsibility\-weighted cost allocations, and𝒜fb\\mathcal\{A\}\_\{\\mathrm\{fb\}\}is an ego\-only fallback action set executable without external cooperation\.

![Refer to caption](https://arxiv.org/html/2606.06531v1/figure2.png)Figure 2:Repair certificate anatomy\.CARVEreturns a structured proof object, not merely a veto label or a predicted trajectory\. A proposal may come from quantum search, classical search, or another heuristic; acceptance depends only on the classical verifier recomputingHH,BB,FF, andΦ\\Phi\.The design separates three questions that are often conflated\. First, hard\-rule feasibility asks whether applying a repair makes all declared safety margins nonnegative\. Second, affordability asks whether ego effort and every requested agent accommodation remain within declared budgets\. Third, fallback asks whether ego retains an executable contingency if requested cooperation is not observed\. The certificate is accepted only when these predicates pass\.

This distinction is what makesCARVE\-Qan AI architecture rather than only a quantum algorithm\. The quantum subroutine is useful becauseCARVEdefines a finite but exponentially growing decision object\. The verifier shield is useful because safety\-critical semantics remain interpretable and classically auditable\.

## Certified Interactive Repair Semantics

### Rules and owners\.

Letsscontain ego state, agent states, semantic map context, and priority rolesπj\\pi\_\{j\}\. A hard rulehℓ∈ℋh\_\{\\ell\}\\in\\mathcal\{H\}returns margingℓ​\(m,s\)g\_\{\\ell\}\(m,s\)\. A repair owner can be ego or an interacting agent\. Ownerjjhas a finite set𝒜j\\mathcal\{A\}\_\{j\}containing no\-op and tactical edits such as wait, decelerate, yield, or nudge\. A joint repair assignment is

x=\(a1,…,an\)∈𝒜1×⋯×𝒜n\.x=\(a\_\{1\},\\ldots,a\_\{n\}\)\\in\\mathcal\{A\}\_\{1\}\\times\\cdots\\times\\mathcal\{A\}\_\{n\}\.The lattice uses semantic repair prototypes rather than arbitrary real\-valued controls\. Continuous quantities such as delay or deceleration are discretized conservatively into resolution cells\. If rule margins are Lipschitz in these repair parameters, refinement bounds the margin degradation between a feasible continuous repair and its representative prototype; missed repairs are therefore controlled false negatives, not unsafe false positives, because every accepted assignment is rechecked by the classical verifier\.

### Right\-of\-way cooperation envelopes\.

CARVEdoes not authorize arbitrary cooperation requests\. For an interacting agentjj, the request magnitudeΔj\\Delta\_\{j\}must satisfy

0≤Δj≤Bj​\(s\),Bj​\(s\)=β​\(πj\)​αjmax​\(s\)\.0\\leq\\Delta\_\{j\}\\leq B\_\{j\}\(s\),\\qquad B\_\{j\}\(s\)=\\beta\(\\pi\_\{j\}\)\\alpha\_\{j\}^\{\\max\}\(s\)\.The termαjmax​\(s\)\\alpha\_\{j\}^\{\\max\}\(s\)is a conservative kinematic accommodation bound, such as the maximum safe yield or speed reduction available to agentjjunder current speed and road conditions\. The factorβ​\(πj\)∈\[0,1\]\\beta\(\\pi\_\{j\}\)\\in\[0,1\]scales that bound by semantic right\-of\-way\. Priority holders useβ​\(πj\)=0\\beta\(\\pi\_\{j\}\)=0, so no nonzero request to a priority agent can be certified\. This structurally separates physical reachability from normative admissibility\.

### Cost and fallback\.

The objectiveΦ​\(x,s\)\\Phi\(x,s\)is a responsibility\-weighted cost over selected ego and agent edits:

Φ​\(x,s\)=ρego​\(x,s\)\+∑jw​\(πj\)​ρj​\(x,s\)\.\\Phi\(x,s\)=\\rho\_\{\\mathrm\{ego\}\}\(x,s\)\+\\sum\_\{j\}w\(\\pi\_\{j\}\)\\rho\_\{j\}\(x,s\)\.Hereρj​\(x,s\)≥0\\rho\_\{j\}\(x,s\)\\geq 0is a kinematic or temporal penalty, such as delay or integrated deceleration, andw​\(πj\)\>0w\(\\pi\_\{j\}\)\>0is a normative multiplier that penalizes inappropriate burden shifting\. Fallback prevents the system from accepting a maneuver that consumes the last recovery option\. In an elicited or joint certificate,𝒜fb\\mathcal\{A\}\_\{\\mathrm\{fb\}\}is an ego\-only contingency, such as a safe stop or wait action, that remains executable without relying on another driver’s compliance\. If the requested accommodation is not observed, ego follows the fallback or recertifies before proceeding\. The certificate never asserts that another driver will comply\.

### Certified joint\-repair problem\.

LetH​\(x,s\)H\(x,s\),B​\(x,s\)B\(x,s\), andF​\(x,s\)F\(x,s\)denote hard feasibility, affordability/envelope validity, and fallback validity\. The finite repair problem is

x⋆\\displaystyle x^\{\\star\}∈arg⁡minx∈𝒳​\(s\)⁡Φ​\(x,s\)\\displaystyle\\in\\arg\\min\_\{x\\in\\mathcal\{X\}\(s\)\}\\Phi\(x,s\)s\.t\.H​\(x,s\)\\displaystyle\\text\{s\.t\.\}\\quad H\(x,s\)=1,B​\(x,s\)=1,F​\(x,s\)=1,\\displaystyle=1,\\quad B\(x,s\)=1,\\quad F\(x,s\)=1,0\\displaystyle 0≤Δj​\(x,s\)≤Bj​\(s\),∀j,\\displaystyle\\leq\\Delta\_\{j\}\(x,s\)\\leq B\_\{j\}\(s\),\\quad\\forall j,where𝒳​\(s\)=∏j𝒜j​\(s\)\\mathcal\{X\}\(s\)=\\prod\_\{j\}\\mathcal\{A\}\_\{j\}\(s\)andM​\(s\)=\|𝒳​\(s\)\|M\(s\)=\|\\mathcal\{X\}\(s\)\|\.CARVEcan solve this exactly over the finite lattice, greedily for online use, or through a verifier\-oracle search layer\. Only the verifier can emit𝒞\\mathcal\{C\}\.

Table 1:CARVE\-side capability gap\. Exact denotes exhaustive enumeration over the joint lattice;CARVE\-Qadds verifier\-shielded quantum search while retaining the same certificate properties\.

## Verifier\-Shielded Quantum\-AI Search

Given theCARVEverifier, define a finite\-precision verifier\-cost oracle\. LetΦmax\\Phi\_\{\\max\}be a saturation value larger than any feasible encoded repair cost\. Then

f~​\(x;s\)=\{Φ​\(x,s\),H​\(x,s\)∧B​\(x,s\)∧F​\(x,s\)=1,Φmax,otherwise\.\\tilde\{f\}\(x;s\)=\\begin\{cases\}\\Phi\(x,s\),&H\(x,s\)\\wedge B\(x,s\)\\wedge F\(x,s\)=1,\\\\ \\Phi\_\{\\max\},&\\text\{otherwise\}\.\\end\{cases\}Infeasible assignments therefore implement\+∞\+\\infty\-style semantics in a bounded arithmetic register\. To remove degeneracy, define the composite key

K​\(x;s\)=\(f~​\(x;s\),lex⁡\(x\)\),K\(x;s\)=\(\\tilde\{f\}\(x;s\),\\operatorname\{lex\}\(x\)\),wherelex⁡\(x\)\\operatorname\{lex\}\(x\)is the integer assignment encoding\. For threshold keyτ\\tau, the phase oracle marks lower\-cost feasible assignments:

Oτ​\|x⟩=\(−1\)𝟏​\[K​\(x;s\)<lexτ\]​\|x⟩\.O\_\{\\tau\}\|x\\rangle=\(\-1\)^\{\\mathbf\{1\}\[K\(x;s\)<\_\{\\mathrm\{lex\}\}\\tau\]\}\|x\\rangle\.Here𝟏​\[⋅\]\\mathbf\{1\}\[\\cdot\]is the 0/1 indicator\. One verifier\-oracle query is one application of this reversible threshold phase oracle, including predicate evaluation, cost computation, comparison, phase flip, and uncomputation\.CARVE\-Qruns a Durr–Hoyer\-style minimum\-finding loop over this oracle\(Durr and Hoyer[1996](https://arxiv.org/html/2606.06531#bib.bib2)\): initialize a feasible threshold, amplify assignments below the threshold, sample a candidate, update the threshold if the candidate is lower cost, and finally pass the best candidate to theCARVEverifier\. Minimum finding is the right primitive because certified repair is a finite verifier\-oracle optimization problem, not a policy\-learning problem\. Variational quantum methods and QAOA are useful context, but they do not provide the load\-bearing worst\-case guarantee used here\.

Algorithm 1CARVE\-QVerifier\-Shielded Repair Search1:Build the finiteCARVErepair lattice\.

2:Initialize threshold

τ\\taufrom any feasible repair or fallback\.

3:ifno finite feasible threshold is availablethen

4:returnrefusal certificate or ego fallback\.

5:endif

6:repeat

7:Reversibly construct

OτO\_\{\\tau\}, mapping infeasible states to

Φmax\\Phi\_\{\\max\}\.

8:Apply amplitude amplification over assignment states\.

9:Sample candidate

x^\\hat\{x\}; evaluate

K​\(x^;s\)K\(\\hat\{x\};s\)classically\.

10:if

K​\(x^;s\)<lexτK\(\\hat\{x\};s\)<\_\{\\mathrm\{lex\}\}\\tauthen

11:Store

x^\\hat\{x\}and set

τ←K​\(x^;s\)\\tau\\leftarrow K\(\\hat\{x\};s\)\.

12:endif

13:untilthreshold stops improving or the prescribed minimum\-finding schedule ends

14:Re\-run theCARVEverifier on the stored candidate\.

15:Emit

𝒞\\mathcal\{C\}only if

H=1H=1,

B=1B=1, and

F=1F=1; otherwise return fallback/refusal\.

## Theory and Oracle Construction

###### Theorem 1\(Verifier\-shielded certificate soundness\)

For any proposal generatorGG, including quantum search, classical search, random sampling, or a faulty heuristic, ifCARVE\-Qreturns an accepting certificate, then the selected repair satisfies hard\-rule feasibility, right\-of\-way envelope validity, affordability, responsibility accounting, and fallback validity\.

*Proof sketch\.*The quantum routine is not a certificate authority\. The final assignment is accepted only after the deterministicCARVEverifier re\-runs the entire predicate suite on the returnedx^\\hat\{x\}\. The certificate is emitted only if hard feasibility, envelope validity, affordability, and fallback validity all pass\. Thus soundness is independent of whether the proposal came from quantum search, classical search, random sampling, or a faulty heuristic\. Full proofs are in the supplementary material\.

###### Lemma 1\(Priority non\-elicitation\)

Ifβ​\(πj\)=0\\beta\(\\pi\_\{j\}\)=0for a priority holderjj, then no acceptingCARVEcertificate contains a positive requested accommodation fromjj\.

*Proof sketch\.*Priority impliesBj​\(s\)=0B\_\{j\}\(s\)=0\. Any positive request hasΔj\>0\\Delta\_\{j\}\>0, so it violates0≤Δj≤Bj​\(s\)0\\leq\\Delta\_\{j\}\\leq B\_\{j\}\(s\)and is rejected by the envelope predicate\.

###### Theorem 2\(Verifier\-oracle query separation\)

In the black\-box verifier\-cost oracle model with a finite lattice of sizeMM, finite\-precision costs, lexicographic tie\-breaking byK​\(x;s\)K\(x;s\), and oracle access only, any deterministic classical algorithm that exactly solves minimum finding requiresΘ​\(M\)\\Theta\(M\)queries in the worst case; bounded\-error randomized algorithms requireΩ​\(M\)\\Omega\(M\)\. Quantum minimum finding returns the distinguished minimum\-cost feasible assignment with high probability usingO​\(M\)O\(\\sqrt\{M\}\)oracle queries\.

This is a black\-box verifier\-oracle result\. It does not assert dominance over white\-box classical solvers such as branch\-and\-bound, CP\-SAT, MILP, or structure\-exploiting local search on structured instances; E2 isolates the black\-box regime empirically\. If no feasible assignment exists,f~​\(x;s\)=Φmax\\tilde\{f\}\(x;s\)=\\Phi\_\{\\max\}for allxx, and the verifier issues a refusal or fallback certificate\.*Proof sketch\.*The classical lower bound follows from the unstructured search/minimum\-finding lower bound: a black\-box table can hide its unique optimum at any unqueried location\(Bennettet al\.[1997](https://arxiv.org/html/2606.06531#bib.bib3); Boyeret al\.[1998](https://arxiv.org/html/2606.06531#bib.bib4)\)\. Lexicographic tie\-breaking converts cost\-degenerate optima into a distinguished minimum\. The quantum upper bound follows from amplitude amplification and minimum finding\.

For scale, a 10\-owner lattice with four choices per owner hasM=410=1,048,576M=4^\{10\}=1\{,\}048\{,\}576andM=1,024\\sqrt\{M\}=1\{,\}024, before accounting for constant factors and oracle\-construction cost\.

###### Theorem 3\(Polynomial reversible constructibility\)

With finite action encodings,pppredicate blocks, pairwise constraints, and fixed\-point bit\-widthbbfor cost and request magnitudes, theCARVEverifier\-cost threshold predicate can be embedded into a reversible phase oracle withpoly⁡\(n,p,b\)\\operatorname\{poly\}\(n,p,b\)logical gates and ancillas\.

*Proof sketch\.*The verifier is composed of finite equality decoders, controlled additions, comparators, Boolean conjunctions, a phase flip, and uncomputation\. Each block has polynomial size in the encoded scene description\. The exponential term is the number of assignmentsMM, not the cost of one oracle evaluation\. Conservative finite\-precision bounds can introduce false negatives that waste search effort, but not unsafe false positives, because the continuous classical verifier rechecks every returned assignment before certification\.

## Experiments

The experiments follow the paper’s trust boundary\. E1 demonstrates minimum finding onCARVEverifier\-cost oracles\. E2 isolates the black\-box regime by destroying semantic neighborhood structure\. E3 checks reversible\-oracle constructibility\. E4 validates certificate semantics on replay data\. E5 reports a QAOA/local\-search audit explaining why minimum finding, not variational optimization, is the load\-bearing quantum primitive\.

### E1: Statevector minimum finding\.

We execute phase\-oracle flips and diffusion steps onCARVErepair oracles\. Because the simulated cost table is known, the statevector experiment uses the number of marked states to select near\-optimal Grover rotation counts at each threshold\. This is a noise\-free oracle\-model demonstration; standard Durr–Hoyer schedules with unknown marked count use randomized rotation schedules and retain the sameO​\(M\)O\(\\sqrt\{M\}\)expected query order\. Calibration only improves constants for transparent query accounting\.

Table 2:Statevector minimum\-finding query scaling onCARVErepair oracles\. DH denotes Durr–Hoyer\-style measured phase\-oracle calls; values are mean±\\pmstd over eight instances per size\.At eight agents with four choices each, exact enumeration evaluates 65,536 assignments; the calibrated statevector routine uses 434\.38 phase\-oracle calls on average and all returned assignments pass classical verification\.

### E2: Black\-box relabeling stress\.

To isolate the theorem’s model, we apply a secret reversible relabeling to the sameCARVEverifier\-cost table\. This preserves the table but destroys semantic neighborhood structure\. In a white\-box lattice, a small deceleration edit may be adjacent to a medium deceleration edit; after relabeling, Hamming neighbors no longer track semantic similarity\. Same\-budget bit\-flip local search and random search then lose their neighborhood advantage, while minimum finding remains exact\.

Table 3:Black\-box relabeled oracle stress\. Entries are exact\-hit rates under matched oracle\-interaction budgets\. The verifier\-cost table is unchanged; only assignment labels are reversibly relabeled to remove semantic locality\.
### E3: Oracle construction counts\.

The largest counted setting has 10 agents, 20 assignment bits, 45 pair constraints, 22,081 Toffoli gates, 51,872 CNOT gates, and 1,104 logical qubits\. These counts support constructibility and reveal overhead; they do not claim current hardware wall\-clock advantage\. The construction targets a fault\-tolerant quantum\-computing setting; wall\-clock advantage depends on future hardware and error\-correction overhead\.

### E4: Lanelet2\-grounded replay\.

We use 589 INTERACTION replay episodes with Lanelet2 geometry\(Zhanet al\.[2019](https://arxiv.org/html/2606.06531#bib.bib16); Poggenhanset al\.[2018](https://arxiv.org/html/2606.06531#bib.bib17)\)\. Replay Human Alignment \(RHA\) is a behavioral diagnostic, not a safety metric\. False\-veto recovery rate \(FVRR\) measures accepted certificates over initially hard\-vetoed but human\-resolved episodes\. Blame\-consistency rate \(BCR\) measures whether the responsibility allocation respects distinct\-duty ordering; pairwise BCR is its pair\-level form\. The RHA improvement comes from theCARVEcertificate layer, not from the quantum backend:CARVE\-Qsearches for assignments that must satisfy the same predicates\. The verifier\-gated layer improves full replay RHA from 28\.23% to 41\.82% and held\-out RHA from 26\.97% to 40\.45%, while preserving 97\.88% FVRR, 100% right\-of\-way \(RoW\) respect, 100% BCR, 100% pairwise BCR, and zero priority false positives\. The RHA gain is not imitation accuracy: rule\-compliant certificates should diverge from aggressive or unlawful human behavior\. It indicates better recovery of lawful compromises without weakening certificate predicates\.

MetricBaseGatedRoleFull RHA28\.2341\.82diagnosticHeld\-out RHA26\.9740\.45diagnosticFVRR97\.8897\.88certificateRoW respect100\.00100\.00certificateBCR100\.00100\.00certificatePriority FP00certificateTable 4:Replay metrics\. RHA is diagnostic only; FVRR, right\-of\-way respect, BCR, and priority false positives are certificate metrics\.![Refer to caption](https://arxiv.org/html/2606.06531v1/figure3.png)Figure 3:Query evidence and black\-box scaling\. Panels A and B report measured statevector evidence onCARVEverifier\-cost oracles; Panel C is explicitly a larger\-scale formula/accounting extension, not a measured statevector or hardware\-runtime result\.
### E5: Design\-discipline audit\.

Small white\-box QUBO tasks favor multi\-start local search: local search hits exact optima in 96\.67% of tasks, whilep=1p=1QAOA hits 23\.33%\. This is not a weakness of the paper’s main claim; it is whyCARVE\-Quses theorem\-backed minimum finding rather than making QAOA the load\-bearing result\. Local search dominates only when it can exploit the white\-box neighborhood structure that the conservative black\-box model refuses to assume\.

Table 5:Claim boundary\. The paper’s evidence supports a verifier\-oracle quantum\-AI search role insideCARVE; every safety\-critical claim remains classically certified\.

## Related Work

Rule\-constrained and certified autonomy\.RSS, rulebooks, runtime shielding, and formal safety define interpretable constraints and interventions\(Shalev\-Shwartzet al\.[2017](https://arxiv.org/html/2606.06531#bib.bib18); Censiet al\.[2019](https://arxiv.org/html/2606.06531#bib.bib19); Alshiekhet al\.[2018](https://arxiv.org/html/2606.06531#bib.bib20); Garcia and Fernandez[2015](https://arxiv.org/html/2606.06531#bib.bib21); Kochenderfer and Chryssanthacopoulos[2012](https://arxiv.org/html/2606.06531#bib.bib39); Fraichard and Asama[2004](https://arxiv.org/html/2606.06531#bib.bib38)\)\. They sit within a broader motion\-planning tradition that includes sampling, optimal planning, and urban\-driving surveys\(LaValle[2006](https://arxiv.org/html/2606.06531#bib.bib29),[1998](https://arxiv.org/html/2606.06531#bib.bib31); Karaman and Frazzoli[2011](https://arxiv.org/html/2606.06531#bib.bib30); Padenet al\.[2016](https://arxiv.org/html/2606.06531#bib.bib28)\)\.CARVE\-Qdiffers by asking whether a vetoed maneuver has an attributable, affordable, multi\-owner repair certificate\. Ego\-only recovery and trajectory repair alter a geometric plan;CARVErepairs an interactive decision object and records who owns each bounded accommodation\.

Interactive prediction and game\-theoretic driving\.Prediction and interaction\-aware planning model how agents may respond\(Sadighet al\.[2016](https://arxiv.org/html/2606.06531#bib.bib22); Schwartinget al\.[2019](https://arxiv.org/html/2606.06531#bib.bib23); Kudereret al\.[2015](https://arxiv.org/html/2606.06531#bib.bib24); Alahiet al\.[2016](https://arxiv.org/html/2606.06531#bib.bib25); Deo and Trivedi[2018](https://arxiv.org/html/2606.06531#bib.bib27); Salzmannet al\.[2020](https://arxiv.org/html/2606.06531#bib.bib26)\)\. Prediction can be useful for candidate generation, but likelihood is not a certificate\.CARVEuses replay alignment as a diagnostic while keeping certification in rule predicates\.

Multi\-agent combinatorial search\.MAPF and coordination methods study joint decisions in difficult search spaces\(Silver[2005](https://arxiv.org/html/2606.06531#bib.bib34); Sharonet al\.[2015](https://arxiv.org/html/2606.06531#bib.bib33); Sternet al\.[2019](https://arxiv.org/html/2606.06531#bib.bib35); Maet al\.[2017](https://arxiv.org/html/2606.06531#bib.bib36); Okumura[2023](https://arxiv.org/html/2606.06531#bib.bib37); van den Berget al\.[2011](https://arxiv.org/html/2606.06531#bib.bib32)\)\.CARVEdiffers because each assignment must also satisfy driving\-specific hard rules, right\-of\-way envelopes, responsibility costs, and fallback validity\.

Quantum AI and quantum optimization\.Quantum machine learning explores feature maps, variational policies, and QAOA\(Biamonteet al\.[2017](https://arxiv.org/html/2606.06531#bib.bib11); Schuld and Killoran[2019](https://arxiv.org/html/2606.06531#bib.bib12); Havliceket al\.[2019](https://arxiv.org/html/2606.06531#bib.bib13); Broughtonet al\.[2020](https://arxiv.org/html/2606.06531#bib.bib14); Jerbiet al\.[2021](https://arxiv.org/html/2606.06531#bib.bib15); Farhiet al\.[2014](https://arxiv.org/html/2606.06531#bib.bib7); Preskill[2018](https://arxiv.org/html/2606.06531#bib.bib8); Cerezoet al\.[2021](https://arxiv.org/html/2606.06531#bib.bib9); McCleanet al\.[2018](https://arxiv.org/html/2606.06531#bib.bib10)\); standard quantum computation references provide the circuit and oracle model used by the minimum\-finding layer\(Nielsen and Chuang[2010](https://arxiv.org/html/2606.06531#bib.bib6)\)\.CARVE\-Qtakes a different quantum\-AI route: it embeds a provable quantum search primitive inside a symbolic safety architecture and leaves the certificate auditable and classically verified\.CARVE\-Qdoes not claim a new quantum search algorithm; the contribution is the reduction of certified repair to a verifier\-oracle minimum\-finding problem and the verifier\-shielded safety architecture around that reduction\.

## Discussion and Claim Boundary

CARVE\-Qcontributes to trustworthy quantum\-AI integration: it does not replace classical AI safety with a quantum device, but uses quantum computation to search a formally defined bottleneck\. This is why the negative QAOA result helps the story\. Variational heuristics may be useful in other settings, but the load\-bearing result here is verifier\-oracle minimum finding\.

The conservative black\-box design is central\. A white\-box solver may be faster when the verifier exposes stable structure, and our theorem does not deny that\. The safety motivation is different: when verifier internals are complex, changing, hidden, or adversarially coupled, the robust assumption is to make no structural assumption at all\. Quantum minimum finding is valuable precisely because it retains a square\-root query guarantee in that worst\-case oracle model\.

When isCARVE\-Qpreferable? If the verifier is convex, separable, or otherwise amenable to source\-aware optimization, branch\-and\-bound, CP\-SAT, MILP, or local search should be used\.CARVE\-Qtargets the least structured regime: changing rulebooks, proprietary shields, relabeled semantics, or adversarially coupled predicates where no stable neighborhood model should be assumed\. In that regime, the square\-root verifier\-query guarantee is a robustness guarantee rather than a near\-term hardware\-speedup claim\.

The problem template is broader than driving\. The pattern “hard\-rule veto→\\rightarrowfinite multi\-owner repair→\\rightarrowauditable certificate” also appears in robot collaboration, air\-traffic management, and smart\-factory scheduling\. We evaluate driving because it exposes concrete right\-of\-way and fallback semantics; the verifier\-shielded search idea applies whenever a domain supplies a finite repair lattice and a deterministic certificate verifier\.

Finally, “quantum proposes;CARVEcertifies” is a transferable safe integration paradigm\. A noisy, suboptimal, or even adversarial quantum module can only propose\. A deterministic classical verifier retains sole authority, so proposal errors cannot become safety certificates\. This separation addresses a core adoption concern for quantum\-AI in safety\-critical systems\.

The claim boundary is precise\. Supported claims are certified interactive repair semantics, verifier\-shielded certificate soundness, black\-box query\-complexity separation, reversible\-oracle constructibility, statevector oracle evidence, and replay certificate preservation\. Excluded claims are present\-day quantum hardware speedup, universal quantum superiority, QAOA dominance, quantum safety certification, and real\-world deployment readiness\. This boundary strengthens the paper because every safety\-critical property is checked byCARVE\.

## Conclusion

We presentedCARVE, a certified repair architecture that elevates a hard\-rule veto from terminal rejection into an auditable certificate, andCARVE\-Q, a verifier\-shielded quantum search layer for the resulting joint\-repair bottleneck\. The key shift is to separate what is certified from how it is found\.CARVEdefines and verifies the binding rule, repair assignment, responsibility split, right\-of\-way envelope, and fallback;CARVE\-Qsearches the black\-box lattice with theorem\-backed minimum finding but never becomes a safety authority\. The broader pattern is deliberately trust\-bounded: quantum proposes;CARVEcertifies\.

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