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Not All Agreement Counts as Corroboration: Provenance-Conserving Multi-View Fusion for Typed Action Admission in Human-Robot Collaboration
Zekai Jin, Hanrong Zhang, Yihong Tang, Fei Hu, Zhen Dong, Yi Shao
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 91%
Last extracted: 9/3/2026, 2:59:07 AM
Summary
The paper introduces PACT (Provenance-Conserving Multi-View Fusion), a framework for human-robot collaboration that distinguishes between computational agreement and evidential corroboration. PACT uses a supplied provenance partition to define countable evidence units, applying a coordinatewise meet within units and summing across units to prevent double-counting repeated inferences. It employs a typed action admission system (hold, confirm, fallback) to authorize actions based on risk-coverage metrics, demonstrating improved performance over singleton aggregation in offline human-robot scenarios.
Entities (6)
Relation Signals (5)
PACT → uses → Provenance Partition
confidence 95% · A supplied provenance partition defines countable units. PACT retains coordinatewise support shared within each unit...
Qwen3-VL-32B → isusedin → PACT
confidence 92% · camera-grouped PACT admits 47 of 57 Qwen3-VL-32B reference-consistent candidates
PACT → achieves → ncsAURC 0.0861
confidence 90% · Across 31,200 evaluations in 48 scene clusters, PACT attains a common-support normalized risk-coverage area (ncsAURC) of 0.0861.
Coordinatewise Meet → ispartof → PACT
confidence 90% · PACT retains coordinatewise support shared within each unit... the coordinatewise meet is the greatest budget
PACT → reduces → ncsAURC
confidence 85% · provenance-partition aggregation reduces ncsAURC by 0.0557 relative to singleton aggregation
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Abstract
Abstract:For embodied systems, predictive agreement alone does not determine whether evidence warrants action; evidential origin matters. Repeated inference over one observation can multiply agreement without adding evidence, while source-local values do not reveal whether outputs have separately countable origins. PACT treats evidence countability as a relational variable for provenance-conserving fusion and typed action admission. A supplied provenance partition defines countable units. PACT retains coordinatewise support shared within each unit, accumulates only across units, and maps unmet release conditions to hold, confirm, or fallback. Under the stated assumptions, source-local values cannot identify countability; the coordinatewise meet is the greatest budget satisfying singleton fidelity and insertion non-amplification, with coarsening monotonicity and fixed-partition stability. Across 31,200 evaluations in 48 scene clusters, PACT attains a common-support normalized risk-coverage area (ncsAURC) of 0.0861. Excluding the constructed adversarial-consensus arm, provenance-partition aggregation reduces ncsAURC by 0.0557 relative to singleton aggregation, while the corroboration contrast vanishes. On complete-source records, native scores favor PACT, but a common posterior-peak score narrows its difference from nested Dirichlet and favors product fusion. Reassigning provenance over unchanged predictions moves evidence budgets as predicted. In offline human-robot collaboration, eightfold within-camera duplication leaves 720 typed responses per checkpoint unchanged; camera-grouped PACT admits 47 of 57 Qwen3-VL-32B reference-consistent candidates with no observed reference-inconsistent admission in 60 episodes. PACT separates computational from evidential multiplicity: agreement constitutes corroboration only when provenance permits separate accumulation.
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- Source: https://arxiv.org/abs/2609.01662v1
- Canonical: https://arxiv.org/abs/2609.01662v1
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Not All Agreement Counts as Corroboration: Provenance-Conserving Multi-View Fusion for Typed Action Admission in Human–Robot Collaboration Zekai Jin zekai.jin@mail.mcgill.ca Hanrong Zhang Yihong Tang Fei Hu Zhen Dong Yi Shao yi.shao@ubc.ca Address: Department of Civil Engineering, McGill University, 817 Sherbrooke Street West, Montreal, Quebec, Canada Address: Department of Computer Science, University of Illinois Chicago, 850 West Taylor Street, Chicago, Illinois 60607, United States Address: Mila – Quebec AI Institute, 6666 Saint-Urbain Street, Montreal, Quebec, Canada Address: Department of Structural Engineering, Norwegian University of Science and Technology, Richard Birkelands vei 1A, Trondheim, Norway Address: Department of Computer Science, University of California, Santa Barbara, 2104 Harold Frank Hall, Santa Barbara, California, United States Address: NVIDIA Corporation, 2788 San Tomas Expressway, Santa Clara, California, United States Address: Department of Civil Engineering, University of British Columbia, 6250 Applied Science Lane, Vancouver, British Columbia, Canada Abstract For embodied systems, predictive agreement alone does not determine whether evidence warrants action; evidential origin matters. Repeated inference over one observation can multiply agreement without adding evidence, while source-local values do not reveal whether outputs have separately countable origins. PACT treats evidence countability as a relational variable for provenance-conserving fusion and typed action admission. A supplied provenance partition defines countable units. PACT retains coordinatewise support shared within each unit, accumulates only across units, and maps unmet release conditions to hold, confirm, or fallback. Under the stated assumptions, source-local values cannot identify countability; the coordinatewise meet is the greatest budget satisfying singleton fidelity and insertion non-amplification, with coarsening monotonicity and fixed-partition stability. Across 31,200 evaluations in 48 scene clusters, PACT attains a common-support normalized risk–coverage area (ncsAURC) of 0.0861. After removing the constructed adversarial-consensus arm, provenance-partition aggregation reduces ncsAURC by 0.0557 relative to singleton aggregation, whereas the corroboration contrast vanishes. Fusion determines candidate evidence; corroboration and selection shape risk–coverage ordering. On complete-source records, native scores favor PACT, but a common posterior-peak score narrows its difference from nested Dirichlet and favors product fusion. Reassigning provenance over unchanged predictions moves evidence budgets as predicted. In offline human–robot collaboration, multiplicity-eight within-camera duplication leaves 720 typed responses per checkpoint unchanged; camera-grouped PACT admits 47 of 57 Qwen3-VL-32B reference-consistent candidates with no observed reference-inconsistent admission in 60 episodes. PACT separates computational from evidential multiplicity. Agreement constitutes corroboration only when provenance permits separate accumulation. keywords Low-quality multi-view fusion ,Provenance-aware fusion ,Evidence countability ,Multimodal foundation models ,Selective action admission ,Human–robot collaboration †titlenote: Project: github.com/ZekaiJ/PACT†corresponding: Corresponding author. 1 Introduction Modern artificial intelligence systems can multiply predictions without acquiring new observations. Re-prompting or resampling can produce multiple outputs from the same input [1]. Predictions from different models can remain correlated across architectures and providers [2]. Agreement may reflect repeated computation over one evidential origin or corroboration across separately countable origins; output values alone do not reveal which. This distinction motivates evidence countability, which specifies what constitutes one evidential unit and which units may contribute support separately. When predictions determine whether an embodied system may act, miscounting evidence becomes a decision problem. Human–robot collaboration (HRC) places robots in shared workspaces, where human-aware planning must manage close-proximity interaction [3]. A handover decision may combine a command, scene geometry, path risk, and multiple vision–language model predictions. It must establish which action the outputs favor, whether their support comes from separately countable origins, and whether the release conditions are satisfied. Action admission denotes this pre-execution authorization. A typed rule preserves the reason for withholding by returning hold, confirm, or fallback instead of collapsing all failures into rejection. Fig. 1: Why agreement need not constitute corroboration. (a) Heterogeneous evidence informs whether a candidate handover may proceed to execution in a shared workspace. (b) Repeated inference over one acquired observation can yield agreeing candidates without adding another evidential origin. Because the outputs remain within one provenance component, the illustrated admission policy requests confirmation. (c) Reassigning the same numerical outputs to different parents changes which support may accumulate separately. Admission still depends on the remaining release conditions. Evidence countability is relational because it describes how outputs share evidential origin within a decision instance. Derived outputs may inherit the same observation and failure modes even when confidence, reliability, and conflict summaries are available. Two source collections can present identical numerical attributes even though one repeats an existing derivation and the other adds a separately countable source. Any fusion map that observes only those attributes must treat the collections identically despite their different admissible evidence counts. Proposition 1 formalizes this identifiability limit. Existing research addresses neighboring parts of this problem. Low-quality multi-view fusion adapts source contributions using quality, uncertainty, and conflict [4, 5, 6]. Dependence-aware fusion limits the accumulation of common or repeated information under an assumed dependence regime [7, 8]. Selective prediction determines whether uncertain outputs or weak views should be retained [9, 10]; foundation-model-based embodied agents generate and verify action candidates to improve search [11, 12]. These mechanisms regulate contribution strength, assumed dependence, retention, or candidate quality, but they do not resolve which retained outputs may accumulate as separate evidential units. PACT represents this instance-specific counting relation explicitly rather than inferring countability from numerical agreement. PACT makes provenance an explicit argument of fusion (Fig. 1). A supplied provenance partition defines the units that may accumulate separately. Within each component, a coordinatewise meet retains the support common to all members. PACT adds the resulting evidence budgets across components. Exact copies within one component are therefore inert. The resulting evidence defines a posterior over action contracts, each pairing a candidate action with the constraint governing its release. Accumulated evidence mass ranks instances for selective retention. Typed admission then evaluates command consistency, source validity, risk support, and component corroboration. The first unmet release condition determines whether the response is hold, confirm, or fallback. The evaluation separates what the conservation law fixes from what remains empirical. Paired copy, near-copy, and partition interventions test the law on a controlled benchmark, and multi-view and multi-camera studies then reassign learned outputs across alternative counting structures, where evidence budgets move in the directions the supplied partition predicts. Excluding the constructed adversarial-consensus arm, provenance-partition aggregation lowers ncsAURC by 0.0557 relative to singleton aggregation while the corroboration contrast vanishes, placing aggregation and corroboration at different points of the decision path. Complete-source comparisons mark a boundary of that result because method ordering also depends on the selection score. In offline HRC, camera-grouped PACT remains invariant to within-camera copies and admits 47 of 57 Qwen3-VL-32B candidates consistent with the reference action, with no reference-inconsistent admission observed. The paper makes three contributions: 1. Evidence countability as a relational fusion variable. PACT formalizes whether multiple outputs may contribute support separately and proves that source-local numerical attributes cannot distinguish repeated derivations from equally valued, separately countable sources. A supplied provenance partition provides the required relational information. 2. A maximal provenance-conserving evidence budget. The coordinatewise meet is the pointwise greatest component rule satisfying singleton fidelity and insertion non-amplification. Summing these budgets across components yields coarsening monotonicity and explicit fixed-partition stability bounds. 3. Distinct fusion, selection, and admission layers. PACT separates provenance-conditioned support from score-based retention and release authorization. Typed admission preserves the reason for non-release through hold, confirm, and fallback. Paired interventions, learned predictions, and offline multi-camera HRC show that evidence budgets follow the supplied partition, whereas ordering, calibration, and retention vary with the dataset, model, score, missing-source treatment, and final admission policy. 2 Related Work Reliability estimation and evidence countability answer different questions. Reliability, uncertainty, and conflict regulate contribution strength; countability determines which retained outputs may accumulate as separate evidential units. Four adjacent research lines frame this distinction. Low-quality multi-view fusion models contribution strength, whereas dependence-aware and hierarchical fusion address repeated or shared information. Selective prediction governs retention and clarification. Embodied foundation models generate, evaluate, and constrain action candidates. 2.1 Low-quality multi-view fusion under uncertainty Evidence theory, evidential learning, and distributional models of aleatoric uncertainty provide explicit uncertainty representations [13, 14, 15]. Low-quality multi-view learning further addresses noise, missingness, imbalanced view quality, and sample-dependent degradation [16]. Under degraded or heterogeneous observations, multi-view and multimodal fusion methods use reliability-, uncertainty-, or conflict-based signals to adapt source contributions [4, 5, 6, 17]. Recent methods extend scalar trust weights by combining feature neighborhoods with inter-view conflict or modeling trusted pairs [18, 19]. They refine contribution strength but leave evidential countability unresolved. 2.2 Dependence-aware fusion and provenance structure Dependence-aware fusion addresses the risk of accumulating repeated or shared information as if it were distinct. Covariance intersection provides conservative estimation when cross-covariances are unknown, whereas data-incest methods use information-flow structure to prevent recursive reuse of the same information [20, 21]. Dependence and information reuse therefore belong in fusion semantics even when outputs agree numerically. At the categorical level, belief-function and opinion-fusion methods likewise change the combination rule with the dependence regime. Denœux’s cautious rule is commutative, associative, and idempotent for reliable but potentially overlapping bodies of evidence [7]. Subjective Logic distinguishes cumulative from averaging fusion for independent and dependent opinions, respectively [8]. These are substantive precedents for preventing unsupported accumulation under dependence. Hierarchical fusion makes grouping part of aggregation. In the Transferable Belief Model, schools of thought combine evidence cautiously within groups and disjunctively across them [22]. PACT instead uses a supplied partition to specify countable evidential units for each decision instance. A coordinatewise meet retains each component budget, and commensurate budgets add across components. The partition encodes countability, not statistical independence. PACT contributes an instance-specific counting relation, a greatest-budget characterization under singleton fidelity and insertion non-amplification, and a decision path that carries this relation into selection and typed admission. Provenance provides adjacent structural information. It has been studied across evidence combination in belief-function theory [23], while computational provenance uses algebraic annotations to track how inputs participate in derivations [24]. Such records describe how an output came about. Countability instead asks which outputs may accumulate separately. PACT uses the supplied provenance relation to define those units before fusion. 2.3 Selective prediction and action admission Selective prediction asks whether an available prediction should be retained. Risk–coverage methods couple prediction with a reject option, and SelectiveNet learns prediction and selection jointly [25, 9]. Adaptive Rejection uses uncertainty-derived p-values to reject low-quality views while controlling false rejection at a prespecified level, then carries that rejection decision into multi-view fusion [10]. Conformal prediction represents uncertainty through prediction sets with coverage guarantees under exchangeability assumptions [26]. KnowNo uses prediction-set size operationally. Singleton sets permit autonomous execution, whereas larger sets trigger human clarification [27]. These methods decide whether a prediction should be retained and when clarification is required, leaving open how much support the retained outputs jointly carry. PACT resolves countability before score-based selection, and typed admission then returns hold, confirm, or fallback according to the first unmet release condition. 2.4 Embodied foundation models: generation, verification, and safeguards Foundation-model-based embodied systems span action generation, structured planning, candidate evaluation, and execution safeguards. Vision–language–action (VLA) policies and human-centric variants map visual and linguistic inputs to task-specific actions, including robot control commands and grounded driving waypoints [28, 29]. Information-theoretic behavior graphs, affordance-weighted skill selection, and vision–language-model-grounded 3D value maps provide structured interfaces between foundation-model representations and embodied action [30, 31, 32]. Candidate-generation and verification methods make computational multiplicity explicit. TapSampling draws actions from a learned posterior and verifies them through task-progress outcome prediction; RoboMonkey perturbs VLA proposals and uses a vision–language model (VLM) verifier to select among them [12, 11]. Such multiplicity can improve search. Candidate and evaluator counts remain insufficient to determine evidential countability. Safeguard methods address policy constraints, action-conditioned hazards, sequential confidence, or runtime intervention. SafeVLA constrains policy learning, EMBGuard evaluates action-conditioned physical hazards, and TDQC calibrates task-success confidence along an episode [33, 34, 35]. UNISafe combines calibrated epistemic uncertainty with latent reachability analysis in a safety filter for known and unseen hazards [36]. These mechanisms constrain policy learning and execution, complementing the pre-execution evidence-accounting layer studied here. Positioning of PACT. Across these research lines, instance-specific evidence countability determines which outputs may contribute support as separate evidential units for a particular decision. PACT represents that relation with a supplied provenance partition, applies meet-within/sum-across evidence conservation, and carries the resulting budget into selection and typed admission. This evidence-accounting layer connects candidate evidence to action authorization; execution-time safeguards and controllers address the subsequent control layer [37, 38]. 3 PACT: Provenance-Conserving Fusion and Typed Action Admission 3.1 Method overview PACT separates evidence accounting from action authorization. Source-local attributes map to nonnegative evidence vectors, while supplied provenance determines which outputs form one evidential unit. Within each provenance component, PACT retains the greatest coordinatewise budget that cannot grow when another component member is inserted. Budgets from separately countable components then add. The cumulative budget defines an action-contract posterior. The posterior selects a candidate, non-vacuity orders instances for retention, and typed admission determines execution eligibility (Figs. 1 and 1). PACT fusion denotes provenance-conditioned evidence conservation, posterior projection, and scoring. PACT denotes the complete decision path, including contract selection and typed admission. Table 11 summarizes the symbols; Appendix A contains the proofs and perturbation bounds. Algorithm 1 PACT fusion and typed action admission 1: Decision instance x, expected sources (x)=(ξi(x))i=1JS(x)=( _i(x))_i=1^J, evidence adapters (Φi)i=1J( _i)_i=1^J, base-eligibility predicate ζbase _base, threshold τ, base rate a, prior strength W, and typed admission rule V 2: Final decision Y(x)Y(x) 3: Stage 1: Construct evidence and countability 4: Identify observed, structurally valid sources using Eq. 1 5: Construct ΠP(x) _P(x) from parent-set overlaps using Eq. 3 6: for each expected source i=1,…,Ji=1,…,J do 7: Map ξi(x) _i(x) to nonnegative evidence i←Φi(ξi(x))e_i← _i( _i(x)) 8: end for 9: Stage 2: Conserve and accumulate support 10: for each component C∈ΠP(x)C∈ _P(x) do 11: C←⋀i∈Cib_C← _i∈ Ce_i 12: end for 13: Π(x)←∑C∈ΠP(x)CE_ (x)← _C∈ _P(x)b_C 14: Stage 3: Select and admit 15: Compute (,s,k^)( π,s, k) using Eqs. 8 and 9 16: y(0)←τ(k^,s,ζbase(x))y^(0) _τ( k,s, _base(x)) 17: return Y(x)←(y(0),x)Y(x) (y^(0),x) 3.2 Evidence, provenance, and the identifiability requirement Let =k1,…,kKK=\k_1,…,k_K\, with K≥2K≥ 2, be a finite action-contract set equipped with a fixed tie-breaking order. Each contract denotes a candidate action together with the constraint governing its release. Admission determines whether the selected contract becomes eligible for execution. Compound labels such as hold–confirm name action contracts; they are distinct from the typed admission responses introduced below. For decision x, let (x)=(ξ1(x),…,ξJ(x))S(x)=( _1(x),…, _J(x)) denote the J≥1J≥ 1 expected sources. Source i carries a categorical opinion i∈ΔK−1∪p_i∈ ^K-1∪\0\, quality and conflict scores qi,di∈[0,1]q_i,d_i∈[0,1], unavailability and structural-validity indicators zi,vi∈0,1z_i,v_i∈\0,1\, and a supplied finite nonempty parent set Pi(x)P_i(x). Here zi=1z_i=1 denotes an unavailable source; vi=1v_i=1 indicates that the source provides the required numerical attributes and satisfies their validity constraints. Source-local attributes determine the evidence vector, while the parent set determines evidential countability. Source role may index the adapter Φi _i; timing and other action-context attributes enter admission. PACT fusion operates on nonnegative evidence vectors rather than on a particular evidence extractor. A source-specific adapter Φi _i maps the local source representation to i(x)=Φi(ξi(x))∈ℝ+K.e_i(x)= _i\! ( _i(x) ) _+^K. The simulation benchmark instantiates Φi _i with reliability-scaled categorical evidence; the external studies use corresponding evidence mappings. Let ℐ=1,…,JI=\1,…,J\ denote the expected source indices. The observed, well-formed subset is ℐobs(x):=i:zi=0,vi=1,i∈ΔK−1.I_obs(x):= \i:\ z_i=0,\ v_i=1,\ p_i∈ ^K-1 \. (1) For i∈ℐobs(x)i _obs(x), let hi(x)=argmaxk∈pikh_i(x)= _k p_ik, with ties resolved by the order of K. For candidate k, define the supporting-source set ℐk(x):=i∈ℐobs(x):hi(x)=k.I_k(x):= \i _obs(x):h_i(x)=k \. (2) This set is used by the admission policies in Section 4.2. A provenance parent represents an evidence-producing process whose reuse creates a countability link between outputs. Sources whose parent sets overlap are connected, and each connected component becomes one unit for within-component evidence conservation. When acquisition or derivation history is unavailable, a checkpoint- or model-family grouping may instead be supplied as a surrogate countability hypothesis. The guarantees are conditional on the supplied partition; the external studies evaluate such surrogate groupings empirically. Two sources are adjacent when their provenance parent sets intersect. These overlaps induce the provenance partition Gprov(x):=(ℐ,ℰP(x)),(i,j)∈ℰP(x),i≠j⟺Pi(x)∩Pj(x)≠∅,ΠP(x)=C(Gprov(x)).G_prov(x):= (I,E_P(x) ), (i,j) _P(x),\ i≠ j P_i(x)∩ P_j(x)≠ , _P(x)=C\! (G_prov(x) ). (3) Connected-component closure makes parent overlap transitive. The graph encodes countability under the supplied relation; component membership does not establish statistical dependence. The vertex set contains every expected source, including unavailable ones, so missingness cannot erase an expected role or its provenance connections. Parent granularity specifies the evidence-producing step represented by one evidential unit. When acquisition is that step, all derived outputs share the raw observation as their parent. Subsequent measurements receive distinct parents when their generating steps define separate evidential units. The meet–sum construction rests on two explicit assumptions. Assumption 1 (Commensurate evidence units). All evidence vectors combined within a decision instance are expressed in commensurate additive units. Assumption 2 (Separate-component additivity). For distinct provenance components represented in commensurate evidence units, their retained budgets combine additively. These assumptions specify evidential units without asserting statistical independence. The benchmark enforces commensurability by construction. External studies either analyze each dataset–model pair under a fixed evidence mapping or use a common explicit mapping when multiple checkpoints are combined. For the benchmark, the source reliability factor ρi _i discounts the categorical opinion and κi>0 _i>0 sets its evidence scale. The resulting evidence mapping is i:=ρiκii,zi=0,vi=1,i∈ΔK−1,,otherwise.e_i:= cases _i _ip_i,&z_i=0,\ v_i=1,\ p_i∈ ^K-1,\\[2.0pt] 0,&otherwise. cases (4) Appendix A gives the benchmark construction of ρi _i. A component C is complete for instance x when C⊆ℐobs(x)C _obs(x). Under the primary expected-source convention, an unavailable or invalid expected source remains in its component as a zero vector, so an incomplete component carries no conserved evidence. Separate analyses compare alternative missing-source treatments and their effects on conserved evidence. Proposition 1 (Value-only indistinguishability). Let A be a space of source-local numerical attributes that excludes provenance parent sets. Its evidence coordinate lies in ℝ+KR_+^K; it may also include reliability, conflict, availability, and validity. Let ℳ()M(A) denote the finite multisets over A, and let H:ℳ()→H:M(A) be any value-only fusion map, where Y is an arbitrary output space. Consider two inputs presenting the same numerical attribute multiset ,\ α, α\, where the evidence coordinate of α is a nonzero vector e. In one input, the second source is a duplicate carrying the parent set of the first. In the other, it is an identically valued source assigned to a separate component. No value-only map can distinguish these inputs because both present ,\ α, α\ to H and therefore receive the same output. The result exposes an identifiability limit. Local reliability, conflict, availability, and validity can regulate contribution but cannot recover an absent counting relation. Conditional accumulation therefore requires relational information. Data provenance, for example, records source identity and preserves contributions through derivations [24]; analogous information in fusion includes source identity, dependence estimates, or a provenance partition. PACT treats the partition as an explicit modeling variable. Many supporting outputs can then correspond to fewer supporting provenance groups, even under high local reliability and low conflict, because countability follows component incidence instead of output value. 3.3 Provenance-conserving fusion and guarantees PACT operationalizes the supplied countability relation through two conservation requirements. Within a declared provenance component, insertion non-amplification forbids a newly inserted member from increasing the retained component budget. Across distinct components, retained budgets may add under Assumptions 1 and 2. Under the supplied relation, separately countable derivations occupy distinct components; repeated derivations remain within the same provenance component. Equip ℝ+KR_+^K with the coordinatewise partial order ⪯r if and only if rk≤skr_k≤ s_k for every k∈k . Singleton fidelity and insertion non-amplification imply that an admissible component budget C∈ℝ+Kg_C _+^K must satisfy C⪯ig_C _i for every i∈Ci∈ C. For commensurate evidence, the coordinatewise meet is the largest budget satisfying these inequalities. Because the coordinates represent distinct action contracts, a bound on total mass would not identify their shared support. For each provenance component C∈ΠP(x)C∈ _P(x), define C:=⋀i∈Ci,(C)k:=mini∈Ceik.b_C:= _i∈ Ce_i, (b_C)_k:= _i∈ Ce_ik. (5) The meet defines an evidence budget without estimating latent shared evidence or reproducing any individual opinion. Proposition 2 (Within-component characterization). Let g map every finite nonempty multiset ℰE of vectors in ℝ+KR_+^K to ℝ+KR_+^K, and let ⊎ denote multiset insertion. Among all rules satisfying singleton fidelity, g()=g(\e\)=e, and insertion non-amplification, g(ℰ⊎)⪯g(ℰ)g(E \e\) g(E), the coordinatewise meet is the unique pointwise greatest rule. Exact-copy invariance admits multiple component rules because coordinatewise maximum also remains invariant to duplicate insertion. Insertion non-amplification supplies the distinguishing conservation constraint, while pointwise greatestness retains the largest budget permitted by that constraint. Proposition 3 (Greatest admissible provenance-aware evidence budget). Under Assumptions 1 and 2, consider cumulative budgets of the form ∑C∈ΠP(x)C _C∈ _P(x)g_C, where each C∈ℝ+Kg_C _+^K satisfies C⪯ig_C _i for every i∈Ci∈ C. The unique greatest budget in this admissible class is ∑C∈ΠP(x)C=:Π(x). _C∈ _P(x)b_C=:E_ (x). (6) Adding a source that joins one or more existing components cannot increase ΠE_ coordinatewise. An exact duplicate carrying the original provenance parent set leaves ΠE_ invariant. By contrast, a source forming a disconnected component D gives Π+=Π+D⪰ΠE_ ^+=E_ +b_D _ . Insertion non-amplification and exact-copy invariance together define provenance-aware evidence conservation. Appendix A gives the full quantified definition. Corollary 1 (Partition monotonicity). For a given source index set and evidence collection, let Π2 _2 be a coarsening of Π1 _1. Then Π2⪯Π1.E_ _2 _ _1. (7) Coarsening the provenance partition cannot increase cumulative evidence. Refinement may increase it. Partition monotonicity governs changes in provenance structure, and coarsening cannot create evidence. For a fixed partition, the within-component meet is nonexpansive, so bounded source perturbations induce bounded changes in the cumulative budget, posterior, and selection score. A near-copy insertion may lower a component envelope but cannot add support. Sufficient posterior and score margins preserve the predicted contract and score-selection decision. Appendix A proves these results and gives the explicit bounds used in the near-copy analysis. 3.4 Selection scores and typed admission Fusion quantifies support but does not grant permission to act. PACT first determines an action contract from the fused posterior, then applies score-based selection, and finally evaluates the ordered conditions for typed admission. Let BΠ(x):=∥Π(x)∥1B_ (x):= _ (x) _1, let ∈ΔK−1a∈ ^K-1 be a base-rate vector, and let W>0W>0 be the total prior strength. All experiments use the uniform base rate ak=1/Ka_k=1/K. PACT projects the cumulative evidence into the categorical distribution (x):=W+Π(x)W+BΠ(x). π(x):= Wa+E_ (x)W+B_ (x). (8) The predicted contract, vacuity, and PACT selection score are k^(x):=argmaxk∈πk(x),u(x):=W+BΠ(x),s(x)=1−u(x), k(x):= _k _k(x), u(x):= WW+B_ (x), s(x)=1-u(x), (9) with ties in k k resolved by the order of K. The component budget in Eq. 5, cumulative evidence in Eq. 6, and posterior and score in Eqs. 8 and 9 define the map FΠF_ . The map is invariant to a within-component exact copy. The score s=1−us=1-u ranks retained evidence mass through BΠB_ ; calibration and correctness ordering remain empirical properties. Two limiting partitions bracket the provenance-conditioned aggregation. An all-singleton partition gives Π=∑iE_ = _ie_i; a single all-source component gives Π=⋀iE_ = _ie_i. With W=KW=K, a uniform base rate, and an all-singleton partition, π and u reduce to the summed-evidence Dirichlet projection and vacuity [14]. PACT introduces the provenance partition as the additional argument between these extremes. Score-based selection applies a threshold τ∈[0,1]τ∈[0,1] together with the method-independent eligibility condition ζbase(x)∈0,1 _base(x)∈\0,1\. Candidate k k is selected exactly when s≥τs≥τ and ζbase=1 _base=1. Otherwise, it is withheld. Section 4.2 specifies ζbase _base, and every fusion rule uses this uniform eligibility condition. Appendix A gives the exact selection map τA_τ. Typed admission is an application-specified ordered predicate–response map applied after score-based selection. A candidate withheld by score selection remains withheld. In the experiments, a retained candidate is tested for command consistency, source validity, risk support, and component corroboration. The first failed condition returns hold, fallback, hold, or confirm, respectively; passing all four yields admission. The response identifies the first unmet release condition. Algorithm 1 gives the decision path, and Appendix A defines the indicator functions and the equivalent case map V. The decision map composes provenance-conditioned fusion, prediction of an action contract, score-based selection, and typed admission: ((x),s(x)) ( π(x),s(x)) =FΠP(x)((i(x))i=1J), =F_ _P(x)\! ((e_i(x))_i=1^J ), (10) k^(x) k(x) =argmaxk∈πk(x), = _k _k(x), y(0)(x) y^(0)(x) =τ(k^(x),s(x),ζbase(x)), =A_τ\! ( k(x),s(x), _base(x) ), Y(x) Y(x) =(y(0)(x),x). =V (y^(0)(x),x ). The resulting decision is invariant to a within-component exact copy that preserves the evidence vectors, parent assignments, eligibility, and admission conditions. 3.5 Complexity and provenance misspecification With bounded parent-set cardinality, naive parent-overlap discovery and evidence aggregation require O(J2+JK)O(J^2+JK) time. Once the provenance graph is available in sparse form, connected-component traversal and aggregation require O(J+|ℰP(x)|+JK)O(J+ _P(x) +JK) time and space. Appendix A gives the full bound and the corresponding representation assumptions. The provenance partition can misstate countability in four ways. False splitting places shared-provenance outputs in separate components, over-broad parent assignments can suppress complementary support, transitive bridges merge components intended to remain separate, and unrecorded shared provenance leaves related opinions disconnected. These errors concern the supplied counting relation. PACT’s structural guarantees hold conditionally on that relation and do not establish its empirical correctness. 4 Experimental Design The experiments separate structural claims from predictive and admission consequences. Matched copy and partition interventions test what the supplied provenance relation and meet–sum operator imply; risk–coverage and calibration analyses then trace those effects through selection and admission. Paired interventions test transfer to learned predictions and offline HRC. Table 1 summarizes this evidence chain. 4.1 Evaluation strategy and benchmark design Table 1: Experimental tests of PACT’s central hypotheses. Hypothesis Experimental test Evaluation criteria Value-only indistinguishability Reassign the same source-local values to shared or separate parents Equality of value-only fusion outputs Within-component conservation Insert exact or near copies within a component; false-split them as a counting contrast Evidence-budget change, posterior ℓ1 _1 drift, and predicted-contract change rate Partition structure Coarsen, falsely split, bridge, or regroup components without changing numerical evidence Evidence-budget ratio; selective ncsAURC analyzed separately Selective consequences Vary selection scores, transfer a baseline-fitted score across multiplicities, and compare pre- versus post-admission ordering ncsAURC, random-ordering reference, and calibration metrics Typed admission Vary one admission requirement or corroboration rule for the same candidate and score Typed response and wrong-admission count The benchmark comprises 48 NVIDIA Isaac Sim scenes [39], balanced across three handover geometries. Each decision combines language (L), geometry (G), and path-risk (R) opinions over five action contracts =N,S,H,T,UK=\N,S,H,T,U\. The provenance partition L,G,R\\L\,\G,R\\ assigns language to one component and groups geometry with risk through their shared scene-context parent. A pre-perturbation label specifies the evaluation reference and initializes the language opinion; source conditions then alter the observed evidence. The benchmark targets evidence accounting, selective retention, and admission; cue recognition lies outside its scope. Fig. 2 traces this progression, and Table 2 defines the contracts. Fig. 2: PACT architecture and paired interventions in the simulation benchmark. (a) Language, geometry, and path-risk outputs are mapped to classwise evidence; unavailable or invalid sources contribute zero evidence. (b) Parent-set overlap induces the provenance graph and connected components L,G,R\\L\,\G,R\\. (c) PACT retains the coordinatewise meet within G,R\G,R\, preserves the singleton budget L\L\, and sums across separately countable components. (d) The posterior determines the action contract, the score 1−u1-u orders instances for retention, and the ordered admission checks stop at the first failed requirement. The bottom row shows the paired intervention families. Admission denotes eligibility for execution. Table 2: Action contracts used in the simulation benchmark. Symbol Contract Execution constraint Benchmark label N Normal Execute the nominal transfer without an added constraint. Neutral S Slow-clearance Proceed at reduced speed with additional clearance. Cautious H Hold–confirm Remain stationary while requesting or awaiting confirmation. Uncertain T Retreat–fallback Withdraw from the transfer or invoke a fallback before reattempting. Corrective U Bounded–urgent Act within defined time, resource, and state bounds. Time-critical The benchmark varies source availability, validity, timing, quality, conflict, and urgency while preserving the label and reference topology; separate interventions alter only the counting structure. Ten deterministic perturbations populate each scene–label–condition cell. The constructed adversarial-consensus arm contributes 2,400 high-confidence but incorrect agreements as a stress test of unsupported release. The resulting 31,200 evaluations per fusion rule are nested within 48 scene clusters. Because the label initializes the language opinion, Section 5.2 reports language-only baselines with and without base eligibility. Appendix B gives the perturbation and comparator specifications. The interventions isolate countability from numerical evidence. Exact copies test within-component invariance; false splits change only whether a copy counts separately, and near copies test fixed-partition stability. Coarsening and bridges merge counting units, while false splits and incorrect distinct parents refine them. Equal-cardinality regrouping isolates component identity from count. Multiplicity transfer separates score shift from score adaptation; missing-source variants compare a retained zero, exclusion from the meet, removal, and isolation. Nested-Dirichlet concentration and score thresholds are selected only within each outer training fold. At target coverages 0.100.10, 0.130.13, and 0.150.15, concentration candidates are ranked under the 10:1 relative decision cost Cλ=λNwrong+NnonreleaseN,λ=10,C_λ= λ N_wrong+N_nonreleaseN, λ=10, (11) where NwrongN_wrong and NnonreleaseN_nonrelease count wrong admissions and non-release decisions among N evaluations. The 10:1 ratio is illustrative and is not an application-estimated utility. Each concentration candidate is evaluated from nested-Dirichlet errors admitted under its own score cutoff and the shared admission rule. The selected concentration is transferred unchanged to PACT, so PACT does not influence comparator tuning. A 256-point scrambled Sobol design [40] tests whether the PACT–nested ordering persists under changes to opinion shape, opinion-generation concentration, quality, conflict, and occlusion strength. Appendix B gives the candidate-ranking and tie-breaking rules. 4.2 Admission, comparators, and evaluation The primary estimands are each fusion–score system before admission and its counterpart under a common policy. Methods share base eligibility, threshold estimation, ordered conditions, and the supplied partition, but retain method-specific scores and fold thresholds. Table 3 begins after eligibility and score selection; the first failed condition determines the response. Its final condition is component corroboration. A complete component has all expected members observed and well formed, and contributes one incidence when any member supports the candidate. With ν=3ν=3, three agreeing outputs corroborate only when they occupy three complete components. Under L,G,R\\L\,\G,R\\, unanimity therefore requests confirmation. This unanimity trigger applies only when all three expected sources support the candidate; the ν=2ν=2 component-count variant tests corroboration from two supporting sources onward. That variant and the cross-role rule serve as policy diagnostics; none is tuned for deployment. All rules and thresholds precede held-out evaluation. The base-eligibility indicator is ζbase(x)=|ℐobs(x)|≥2 _base(x)=1\|I_obs(x)|≥ 2\. At admission, k∗=k^(x)k^*= k(x) is the selected candidate, and 3(k∗,x)U_3(k^*;x) indicates that all three expected sources are observed and support k∗k^*. For observed source i, let pipeak=maxk∈pikp_i^peak= _k p_ik and Δi=pi,hi(x)−maxj≠hi(x)pij _i=p_i,h_i(x)- _j≠ h_i(x)p_ij. When ℐk∗(x)I_k^*(x) is nonempty, pminpeak,obsp_ ^peak,obs, qminobsq_ ^obs, and dmaxobsd_ ^obs are the minimum peak probability, minimum quality, and maximum conflict among its supporting sources. The number of complete supporting components is ncompcomplete(k∗;x)=|C∈ΠP(x):C∩ℐk∗(x)≠∅,C⊆ℐobs(x)|.n_comp^complete(k^*;x)= | \C∈ _P(x):C _k^*(x)≠ ,\ C _obs(x) \ |. A complete component contributes one incidence when at least one member supports the candidate. This binary admission statistic and the meet-derived evidence magnitude are distinct by construction. Table 3: Typed action admission: ordered failure responses (Panel A) and alternative component-corroboration rules (Panel B). Panel A: ordered admission conditions Criterion Observable condition Requirement Response Command consistency Missing timing information for a present language source; the current command differs from that underlying the evidence; or a stale-language state carries over the previous command Command identity and source status Hold Source validity A present source has an invalid opinion vector, an unrecognized contract label, or a missing required opinion, quality, or conflict attribute Source type and language attributes Fallback Risk support zR=1z_R=1, zL=0z_L=0, pLpeak≥0.90p_L^peak≥ 0.90, qL≥0.90q_L≥ 0.90, and ΔL≥0.70 _L≥ 0.70 0.900.90, 0.900.90, 0.700.70 Hold Component corroboration 3(k∗,x)U_3(k^*;x), pminpeak,obs≥0.40p_ ^peak,obs≥ 0.40, qminobs≥0.30q_ ^obs≥ 0.30, dmaxobs≤0.15d_ ^obs≤ 0.15, and ncompcomplete(k∗,x)<νn_comp^complete(k^*;x)<ν 0.400.40, 0.300.30, 0.150.15, and ν=3ν=3 Confirm Panel B: outcomes of alternative component-corroboration rules Supporting sources Complete components Role pair Unanimity Component count Cross-role 11 11 Any – – – 22 11 Language + geometry/risk – Confirm Confirm 22 22 Language + geometry/risk – Confirm Admit 22 22 No language + geometry/risk pair – Confirm Confirm 33 11 Language + geometry/risk Confirm Confirm Confirm 33 22 Language + geometry/risk Confirm Confirm Admit 33 33 Language + geometry/risk Admit Admit Admit Note. Unanimity is invoked by high-confidence support from all three sources. Component count is invoked by at least two supporting sources but admits only at the complete-component threshold. Cross-role admits when language and geometry or path risk support the candidate in two complete components. A dash means that the rule is not invoked; selection and preceding conditions determine the response. Table 4: Comparator methods by fusion rule, evidence form, provenance use, missing-source treatment, and score. Fusion specification Source handling Ordering Method Evidence form Combination rule Uses provenance Missing source Selection score Quality-weighted Probability Quality-weighted mean ✗ Omitted Peak Product Probability Reliability-discounted product ✗ Omitted Peak Nested Dirichlet Dirichlet evidence Sequential composition ✗ Omitted 1−u1-u Global cautious Belief weights Global cautious minimum ✗ Omitted Peak Hierarchy-matched cautious Belief weights Component minimum, then conjunctive combination ✓ Vacuous mass 1−m(Θ)1-m( ) Pooling without provenance Log probability Weighted log pool ✗ Penalized Peak Provenance-discounted pooling Log probability Edge-discounted log pool ✓ Penalized Peak PACT fusion Classwise evidence Meet, then sum ✓ Zero evidence 1−u1-u Note. ✓ marks provenance-aware and ✗ value-only computation. Nested Dirichlet and global cautious are published; hierarchy-matched cautious is structure-matched. Pooling isolates provenance discounting. All methods share opinions, source attributes, folds, and evaluation instances. Table 4 summarizes the matched comparison: value-only averaging and products, published Dirichlet and cautious operators, structure-matched cautious fusion, and pooling with and without provenance discounting [4, 7, 41]. All receive the same opinions, source attributes, folds, and instances; PACT and nested Dirichlet also share the fold-selected evidence concentration. A coordinatewise-maximum ablation replaces the within-component meet, separating exact-copy invariance from insertion non-amplification. A prespecified representative uses the first source in each component under the fixed source order and recovers the PACT budget only when i⋆=Ce_i =b_C. Native scores, posterior summaries, and a training-fitted correctness score separate candidate formation from score-induced ordering. Joint holdout also evaluates a separately trained Set Transformer [42]; Appendix B describes the comparator definitions. Each method fits separate pre- and post-admission cutoffs on stage-eligible training instances and transfers them unchanged to held-out scenes; the two stages therefore define distinct estimands. Coverage uses all evaluation instances. The reported operating point targets 0.130.13 coverage; risk–coverage targets span 0.01–0.60, with cross-method integration restricted to shared support. Among N instances, let nretn_ret, nerrn_err, and ncorn_cor count retained decisions, retained errors, and correct retained decisions. Then γ=nret/Nγ=n_ret/N, Rall=nerr/NR_all=n_err/N, Rcond=nerr/nretR_cond=n_err/n_ret, and Call=ncor/N=γ−RallC_all=n_cor/N=γ-R_all. The principal metric, normalized common-support area under the risk–coverage curve (ncsAURC), averages conditional risk over shared coverage [25]: ncsAURC[γmin,γmax]=1γmax−γmin∫γminγmaxRcond(γ)γ.ncsAURC_[ _ , _ ]= 1 _ - _ _ _ _ R_cond(γ)\,dγ. (12) Lower ncsAURC indicates lower retained error averaged over shared coverage. The primary support [0.10,0.39][0.10,0.39] is the largest grid interval attained by every method without extrapolation. Held-out decisions are aggregated, linearly interpolated at 36 equally spaced points, and integrated by the trapezoidal rule. Repeated fold assignments, joint holdout, and source-evidence perturbations use the shared support [0.10,0.35][0.10,0.35]. For a fixed combination of fusion method and selection score b, let AbrandA_b^rand denote its empirical error rate among instances satisfying ζbase _base. Under score-independent random ordering, this quantity equals the expected ncsAURC over the same support and quadrature grid. It provides the random-ordering reference used in Section 5.2. Negative log-likelihood (NLL) and top-confidence expected calibration error (ECE) assess probabilistic confidence [43]; the evaluation also reports the multiclass Brier score and instantiates ECE with ten equal-width bins, separately from selective ordering. With kprefk_pref denoting the reference contract, the unnormalized multiclass Brier score is ∑k=1K(πk−k=kpref)2∈[0,2] _k=1^K( _k-1\k=k_pref\)^2∈[0,2]. ECE10 uses posterior peak as confidence and ten equal-width bins weighted by their empirical frequencies. Empty bins contribute zero. An exploratory matched 2×22× 2 analysis contrasts fusion aggregation and component corroboration. All arms share the 31,200 instances, evidence mapping, prior, score, folds, threshold procedure, and remaining admission criteria. Fusion compares singleton with provenance-partition aggregation; admission toggles corroboration while evaluating the provenance partition in both arms. Because adversarial consensus is constructed to activate corroboration, a sensitivity analysis removes its 2,400 cases and retains the other 28,800. The contrasts isolate decision-stage differences under this construction, not an additive causal decomposition of PACT’s performance. All model selection, threshold estimation, and uncertainty analyses preserve scene grouping. Fifty grouped fold assignments quantify sensitivity to fold allocation but are not independent replications. Paired bootstrap intervals resample the 48 held-out scene contributions with the estimates fitted within each assignment held fixed. The 31,200 evaluations comprise 19,200 with three valid sources, 9,600 with one unavailable source, and 2,400 with one invalid source. Confidence intervals are contrast-specific and unadjusted for multiplicity. Joint holdout excludes test scenes from all fitting and withholds the evaluated source condition from threshold and learned-comparator training. 4.3 Evaluation on learned predictions The learned-prediction studies preserve model outputs and vary only provenance grouping or admission evidence. TMC [4] and RCML [6] provide per-view outputs for paired reassignment of the counting structure. The multi-view analysis uses one provenance component per original feature view as the reference grouping and compares the same predictions under false splitting and all-view merging. HandWritten/Mfeat supports exhaustive analysis of all 203 view partitions and 856 single-merge relations for both TMC and RCML. Scene15–TMC adds accuracy and selective-ordering outcomes; PIE–RCML supports the evidence-budget analysis. The HABIT analysis comprises 1,128 paired events from 696 episodes and six tasks, observed through five synchronized cameras and four prompts per image [44]. Subtask indices and gripper trajectories define offline early and event-proximal windows because the dataset lacks readiness annotations; these are evaluation constructs, not a deployment-time signal. Prompt outputs from one image form one component. Distinct camera acquisitions count separately at the evaluated granularity without implying statistical independence. Comparisons use per-output counting, acquisition grouping, identical-vector merging, all-view merging, and one output per camera. A complementary analysis tests whether model-family labels capture error association across eight vision–language checkpoints [45, 46, 47, 48]. The HRC admission study uses 60 evaluation episodes disjoint from the 82 development episodes. Across two temporal windows, one episode target, and five counterfactual targets, it contains 720 offline target–time evaluations; the episode is the sampling unit. Candidate evidence and a learned event-proximity signal share camera acquisitions and enter admission as complementary requirements, not separate votes. The primary model measures transfer to unseen episodes within the same six tasks. A leave-one-task-out analysis withholds each evaluated task from fitting to assess task transfer. Separate evaluations of absolute thresholds and within-episode score changes distinguish representation transfer from a shifted score baseline. Joint fusion and admission use Qwen3-VL-8B and Qwen3-VL-32B predictions for the same 60 episode-level events and both temporal windows. PACT meets prompt-derived evidence within each camera acquisition and sums across cameras; comparators count every prompt, merge all outputs, or retain one prompt per camera. Prompt multiplicity, exact copies, and camera removal test whether the admission decision depends on repeated computation or additional observation. The primary outcomes are reference-inconsistent admissions across all 720 queries and recall among the 60 reference-ready episode targets. Temporal mechanism tests use the paired-event population, kept distinct from the 60-episode admission subset. The permutation analysis excludes the sole singleton exchangeability block. The tests compare pre-early, early, and event-proximal scores and ask whether the early-to-proximal change exceeds exchangeable temporal variation. Appendix B gives the evidence mapping, model specification, thresholds, resampling units, and permutation design. 5 Results: Conservation, Ordering, and Admission Provenance structure fixes the conserved budget, fusion and scoring rank candidates, and typed admission determines whether and why a retained candidate may proceed. Exact copies kept within a provenance component preserve the PACT budget and posterior. False splitting converts computational multiplicity into apparent support. Provenance misspecification changes evidence budgets directionally, but its consequences for selection depend on the score and missing-source semantics. The algebra determines only the budget response; posterior quality, selective ordering, and admission remain empirical questions. 5.1 Conservation under copy and provenance interventions Exact copies. For PACT and hierarchy-matched cautious fusion, component assignment determines whether exact-copy repetition changes the fused result. Both show zero drift when exact copies remain within the source component, whereas assigning the same copies to distinct components produces a 0.3228 flip rate and mean posterior ℓ1 _1 drift of 0.3961 across 93,600 comparisons, comprising 31,200 instances with each of the three source roles duplicated in turn at multiplicity eight. Product and nested Dirichlet do not use the provenance partition, so the two assignments coincide numerically; their flip rates relative to the unduplicated output are 0.3339 and 0.3404. Provenance-discounted pooling preserves the predicted contract but drifts by up to 0.2269. Posterior invariance, prediction invariance, and selective ordering are distinct properties. Provenance-discounted pooling preserves the predicted contract even as normalization changes its posterior with accumulating copies. Near-copy stability. Across 960 comparisons spanning 48 scenes, five benchmark labels, and four perturbation magnitudes (48×5×448× 5× 4) under the undegraded source condition, near-copy insertion changes no predicted contract. The maximum posterior ℓ1 _1 drift is 0.0109, and the fitted slope constrained through the origin is 0.2156 with respect to ϵε. A within-component near copy may lower the component envelope but cannot add support, consistent with directional non-amplification. The fixed-partition bounds apply to evidence-space perturbations; this sweep measures sensitivity in opinion space. The role of insertion non-amplification. The coordinatewise-maximum ablation shows why exact-copy invariance is insufficient. Maximum shares exact-copy invariance and partition monotonicity with the meet but violates insertion non-amplification. Inserting a member raises its component budget on 96.4% of 24,000 instances where geometry and risk are observed and valid. The increases occur in ten of thirteen source conditions; by contrast, the meet raises the component budget in 0% of these instances. Across all instances, the ablation increases cumulative budget BΠB_ by 35.4% and changes 16.2% of predicted contracts. It raises ncsAURC from 0.6294 to 0.7918 (difference 0.1623, 95% CI [0.1535,0.1709][0.1535,0.1709]), despite improving ECE10 from 0.2454 to 0.1927. This ablation empirically illustrates why insertion non-amplification, rather than exact-copy invariance alone, distinguishes the meet (Proposition 2). A single representative does not generally recover the component meet. Using one designated source per component yields ncsAURC 0.8639 before admission, versus 0.6294 for PACT fusion under the same outer-fold threshold procedure (difference 0.2345, 95% CI [0.2294,0.2402][0.2294,0.2402]). After admission, the corresponding values are 0.1668 and 0.0861. At target coverage 0.13, the representative yields 246 wrong admissions among 4,051; PACT has none among 4,043. The representative changes the selection score on 82.3% of instances; even when both rules predict the same contract, 89.9% of instances receive different scores. These percentages are not nested because 1−u1-u depends on total evidence budget, not only on the predicted contract. Exact recovery requires the representative evidence vector to equal the component meet, which heterogeneous components do not guarantee. Sensitivity to provenance errors. Provenance misspecification produces asymmetric effects. Four interventions falsely split shared-source outputs, merge separately countable sources, delete recoverable edges, or assign incorrect distinct parents, none of them altering the numerical evidence. The coarsening and refinement cases can be read against Corollary 1, and the remainder test broader partition misspecification. At 50% corruption, false splitting and incorrect distinct-parent assignments yield Rall=0.0492R_all=0.0492, compared with 0.0106 after merging or recoverable deletion and 0.0875 for a provenance-unaware singleton reference. At this corruption level, the former error rate is 43.8% below that reference; at 100%, the maximum tested level, it equals the reference. Merging and recoverable deletion remain below it throughout the grid. Incorrect distinct parents and false splitting produce identical point estimates and outputs here. Multiplicity exposes the cumulative effect of false splitting. Within-component copies and an all-source merge remain invariant through m=32m=32, although the two provenance assignments yield different decision metrics. At m=32m=32, singleton counting raises the aggregate budget to 10.73×10.73× its own m=1m=1 value. False splitting increases Δsplit _split from 0.0243 (95% CI [0.0221,0.0268][0.0221,0.0268]) at m=1m=1 to 0.3676 ([0.3534,0.3801][0.3534,0.3801]) at m=8m=8 and 0.4338 ([0.4211,0.4469][0.4211,0.4469]) at m=32m=32 over support [0.10,0.39][0.10,0.39]. All six pointwise intervals exclude zero (Fig. 3). Grouping composition changes selective ordering even at equal component count. Relative to the provenance partition, grouping language with geometry while leaving risk separate raises ncsAURC[0.10,0.39] by 0.1988 (95% CI [0.1861,0.2107][0.1861,0.2107]); grouping language with risk while leaving geometry separate raises it by 0.2013 ([0.1846,0.2188][0.1846,0.2188]). All 2,000 paired cluster-bootstrap differences over scenes are positive. Fig. 3: False splitting increasingly inflates the evidence budget and reverses the method ranking under a score transferred from m=1m=1. (a) Δsplit=ncsAURCfalsesplit−ncsAURCprovenance _split=ncsAURC_false\ split-ncsAURC_provenance. (b) Budget relative to m=1m=1. (c) ncsAURC under the score estimated at m=1m=1. The singleton grouping in panel (b) is the singleton-component PACT rule in panel (c). Bands in panels (a, c) are 95% paired scene-bootstrap intervals; panel (b) is deterministic. All panels precede component corroboration and isolate the fusion response to source multiplicity. A score estimated at the baseline multiplicity exposes the ordering shift. Nested Dirichlet leads at m=1m=1, but the ordering reverses at m=2m=2 and the PACT–nested difference reaches −0.2543-0.2543 at m=8m=8 (Fig. 3). Removing observed-source count from the score delays the crossover to m=4m=4 but does not eliminate it. Because both scores are estimated only at m=1m=1, the reversal reflects multiplicity-induced distribution shift rather than refitting at each multiplicity. 5.2 Selective risk before and after admission Fig. 4 contrasts pre-admission selective ordering with the corresponding performance after typed admission. Fig. 4: Typed admission changes attainable coverage; the complete PACT rule has the lowest ncsAURC over shared support. Panels (a, b) show conditional risk; panels (c, d) report ncsAURC[0.10,0.39] and share the method labels shown at left. Rows in panels (c, d) follow the ordering in panel (d). Dashed lines mark target coverage 0.13. Panel (c) reports PACT fusion before admission; panel (d) reports the complete typed-admission rule. Hierarchy-matched cautious appears only in panels (c, d) because its curve does not span the full range in panels (a, b). 5.2.1 Fusion methods and selection scores before admission Before admission, PACT fusion paired with its prespecified score 1−u1-u yields the lowest ncsAURC[0.10,0.39] among the native method–score combinations. This comparison concerns the method and its selection score, not the fusion operator alone. Table 5 separates posterior quality from ordering. Table 5: Posterior quality and selective ordering rank fusion methods differently before admission. Panel A reports aggregate posterior metrics; Panel B compares native, posterior-derived, and learned selection scores. Panel A: posterior quality over all instances Fusion operator Acc.↑ Macro-F1↑ NLL↓ Brier↓ ECE10↓_10 Product 0.30200.3020 0.29960.2996 2.71892.7189 0.99030.9903 0.27780.2778 Nested Dirichlet 0.30110.3011 0.29870.2987 1.81461.8146 0.87880.8788 0.0944 Global cautious 0.27870.2787 0.27670.2767 2.09742.0974 0.95740.9574 0.23910.2391 PACT fusion 0.3689 0.3684 1.6735 0.8436 0.24540.2454 Panel B: methods and selection scores Fusion operator Native score Posterior-derived scores Learned pooled correctnessa Peak Margin Inv. entropy Product 0.76350.7635 0.76350.7635 0.76750.7675 0.7628 0.55470.5547 Nested Dirichlet 0.77490.7749 0.7547 0.7649 0.79610.7961 0.37960.3796 Global cautious 0.78060.7806 0.78060.7806 0.77630.7763 0.79040.7904 0.49840.4984 Provenance-discounted pooling 0.81250.8125 0.81250.8125 0.80650.8065 0.82560.8256 0.3711 Hierarchy-matched cautious 0.85970.8597 0.78700.7870 0.78400.7840 0.79430.7943 0.46590.4659 PACT fusion 0.6294 0.91720.9172 0.92340.9234 0.91580.9158 0.64890.6489 Note. Panel A aggregates thirteen source conditions, twelve of which introduce degradation; its absolute values describe the stress mixture rather than undegraded performance. Peak, margin, and inverse entropy are maxkπk _k _k, the top-two difference, and 1−H()/logK1-H( π)/ K. Bold marks the best value per column; the empirical-prior reference in Section 5.3.2 has lower NLL and Brier. Panel B adds the provenance-aware comparators. Reported differences are computed from unrounded values. a The pooled correctness score is described in Section 4.2 and specified in Appendix B; it is transferred without refitting. Fig. 4 shows all eight operators under their respective selection scores. Quality-weighted fusion (0.7977) and pooling without provenance (0.8417), shown only in the figure, both have higher ncsAURC than product fusion (0.7635). PACT ranks first with its selection score 1−u1-u and last with posterior peak, top-two margin, inverse normalized entropy, and pooled correctness. These reversals identify score choice as an essential part of the leading PACT combination because candidate formation and method-specific ordering jointly produce the result. The score 1−u1-u is monotone in the retained evidence budget BΠB_ , so it orders instances directly by the quantity that PACT is designed to conserve; the alternative scores are not constrained to be monotone in BΠB_ . Across the eight operators in Fig. 4, product fusion attains the lowest ncsAURC of the non-PACT methods under their respective selection scores. PACT fusion lowers ncsAURC by 0.1340 relative to product fusion (95% CI [−0.1401,−0.1277][-0.1401,-0.1277] for the PACT-minus-product contrast), with a negative difference in all 48 scenes. Over common support [0.10,0.35][0.10,0.35], the contrast remains negative across all 50 repeated scene-grouped split assignments. Let AbA_b denote the observed ncsAURC of method and selection-score combination b. The product–PACT gap decomposes exactly as Aprod−APACT A_prod-A_PACT =(Aprodrand−APACTrand)+[(Aprod−Aprodrand)−(APACT−APACTrand)] = (A_prod^rand-A_PACT^rand )+ [(A_prod-A_prod^rand)-(A_PACT-A_PACT^rand) ] =0.0725+0.0615=0.1340. =0.0725+0.0615=0.1340. The margin divides almost evenly between candidate formation and ordering. Of the 0.1340 product–PACT gap, 0.0725 (54.1%) is already present under random ordering and reflects differences in the candidate distributions produced by the fusion rules. Relative to their own random-ordering references, PACT and product have excess ncsAURC values of 0.0125 (95% CI [0.0053,0.0188][0.0053,0.0188]) and 0.0740, respectively, and the difference between these excesses is the remaining 0.0615. In this benchmark, neither native score improves correctness ranking over its random-ordering reference; the decomposition is algebraic rather than causal. Language is the strongest single source on the training folds, yet its ncsAURC is 0.8741, worse than every compared fusion rule and unchanged under base eligibility. Language alone cannot explain the gap to PACT fusion (0.6294). At full coverage, PACT fusion has lower NLL and Brier scores than the other reported fusion operators, although the empirical-prior reference in Section 5.3.2 is lower on both; nested Dirichlet has the lowest ECE10 (Table 5). 5.2.2 Fusion methods and selection scores under a common admission policy All methods receive the same source inputs and admission policy. They differ in the fused candidate, native selection score, and resulting fold threshold (Table 6). PACT attains an ncsAURC of 0.0861 under the complete method–score–admission rules, compared with 0.1479 for nested Dirichlet (difference −0.0618-0.0618, 95% CI [−0.0672,−0.0561][-0.0672,-0.0561]). At the 0.13 coverage target, the two methods retain nearly the same number of correct cases and produce no observed wrong admissions (Table 6). Their separation therefore arises across the coverage interval and belongs to the complete method combinations, including candidate formation and method-specific ordering. Method ordering depends on the selection score. Restricting the comparison to the 19,200 instances with three valid sources removes missing-source conventions from the contrast. Native scores favor PACT over nested Dirichlet (0.0012 versus 0.0199), whereas a common posterior-peak score narrows their difference (0.0956 versus 0.0979) and favors product fusion at 0.0909. The PACT-minus-nested difference under the common score is −0.0023-0.0023 (95% CI [−0.0028,−0.0018][-0.0028,-0.0018]). The leading complete method combination therefore does not imply a score-independent operator ranking. Excluding the constructed adversarial-consensus arm leaves this picture unchanged. A common posterior-peak score ties PACT and nested Dirichlet at 0.0532 and favors product fusion at 0.0517 on the remaining 16,800 complete-source records. Table 6: Selective performance (Panel A) and matched partition–corroboration analysis (Panel B). Panel A: common admission policy Method ncsAURC↓ Coverage Rall↓R_all Call↑C_all Product 0.24860.2486 0.13010.1301 0.01060.0106 0.11950.1195 Nested Dirichlet 0.14790.1479 0.13020.1302 0.0000 0.1302 Hierarchy-matched cautious 0.20940.2094 0.13000.1300 0.00740.0074 0.12260.1226 Provenance-discounted pooling 0.31010.3101 0.12980.1298 0.02240.0224 0.10750.1075 PACT 0.0861 0.12960.1296 0.0000 0.12960.1296 Panel B: fusion partition × component corroboration Fusion partition Corroboration ncsAURC all instances ncsAURC all sources available Singleton components Off 0.41350.4135 0.52140.5214 Singleton components On 0.15210.1521 0.04260.0426 Provenance partition Off 0.38910.3891 0.51890.5189 Provenance partition On 0.0861 0.0151 Note. Panel A uses method-specific scores and outer-fold thresholds; Rall=0.0000R_all=0.0000 denotes no observed wrong admissions. Panel B ranks all cells by 1−u1-u over [0.10,0.39][0.10,0.39]. All sources available denotes 21,600 records (19,200 valid; 2,400 with one invalid source). PACT is the provenance-partition/corroboration cell. Reported differences are computed from unrounded values. The factorial contrast assigns distinct roles to partitioning and corroboration. Provenance partitioning lowers ncsAURC by −0.0244-0.0244 (95% CI [−0.0267,−0.0221][-0.0267,-0.0221]) without corroboration and by −0.0661-0.0661 ([−0.0720,−0.0602][-0.0720,-0.0602]) with it, producing an interaction of −0.0417-0.0417 ([−0.0453,−0.0379][-0.0453,-0.0379]). Corroboration has the larger matched effect because it targets the constructed adversarial-consensus arm. The provenance-aware operator instead conserves evidence under within-component multiplicity and propagates exact-copy invariance through typed admission when the remaining admission inputs do not change. Removing component corroboration admits all 2,400 wrong cases in the constructed adversarial-consensus arm (Rall=7.69%R_all=7.69\%). Enabling it maps these cases to confirmation and leaves no observed wrong admission after thresholding. The arm isolates the policy response to high-confidence agreement rather than its external prevalence. Among instances with all sources available, corroboration strengthens the partition contrast from −0.0025-0.0025 ([−0.0029,−0.0021][-0.0029,-0.0021]) to −0.0276-0.0276 ([−0.0298,−0.0254][-0.0298,-0.0254]). Over [0.10,0.35][0.10,0.35], the corresponding contrasts are −0.00017-0.00017 ([−0.00038,0.00005][-0.00038,0.00005]) and −0.0216-0.0216 ([−0.0234,−0.0198][-0.0234,-0.0198]). Excluding the 2,400 adversarial-consensus cases leaves 28,800 instances on which toggling corroboration does not change ncsAURC. Under the shared 1−u1-u score, provenance-partition aggregation lowers ncsAURC from 0.1250 to 0.0693 relative to singleton aggregation (difference −0.0557-0.0557, 95% CI [−0.0608,−0.0508][-0.0608,-0.0508]). On these instances, the matched contrast identifies the selective-ordering contribution of provenance-conditioned aggregation; corroboration supplies the targeted response to consensus without sufficient separately countable support. 5.2.3 Admission responses conditional on fusion outputs and scores The adversarial-consensus arm isolates the component-corroboration requirement by reassigning the same numerical opinions across components. Language, geometry, and risk favor the same incorrect contract while satisfying the confidence, quality, and conflict thresholds. Geometry and risk share a scene-context parent, so their three local votes occupy two complete components. The corroboration check finds ncompcomplete(k∗,x)=2<ν=3n_comp^complete(k^*;x)=2<ν=3 and requests confirmation. Reassigning the same opinions to three complete components admits the candidate. Removing individual checks changes admission in different ways. Holding the nested-Dirichlet candidates and scores constant, removing command consistency adds 3,256 wrong admissions, while removing source validity, risk support, or component corroboration adds 2,400, 0, or 2,400, respectively. Risk support adds none because its rejected candidates arise under provenance-discounted pooling rather than nested composition. On the same candidates, three supporting components admit all 2,400 constructed adversarial-consensus cases, whereas two map them to confirmation. Incomplete geometry maps these cases to fallback, and marking language stale maps all 4,062 otherwise admitted candidates to hold. Admission-rule sensitivity shows that the transition depends on how supporting components are counted. Under the unanimity rule, PACT and nested Dirichlet obtain 0.0861 and 0.1479. The component-count variant closely matches them at 0.0868 and 0.1490 while retaining 18 fewer correct candidates per method. The cross-role rule raises ncsAURC to 0.3891 and 0.4116 and yields Rall=7.69%R_all=7.69\% at near-matched coverage. For PACT, these values coincide with the provenance-partition arm without the incidence check in Table 6, although the rules map two-source cases differently. Under the cross-role rule, the 2,400 additional wrong admissions under PACT have three agreeing source outputs but only two complete provenance components. Under the illustrative decision cost in Eq. 11, an always-withhold policy has C10=1C_10=1. The unanimity rule attains 0.8704 (95% CI [0.8644,0.8764][0.8644,0.8764]), compared with 6.3106 ([6.3032,6.3176][6.3032,6.3176]) for always continuing with the PACT candidate. Reducing the component requirement to ν=2ν=2 raises the cost to 1.6389 ([1.6250,1.6527][1.6250,1.6527]), crossing the always-withhold boundary. These values compare decision policies at the utility ratio used for training-fold selection; they do not independently validate λ=10λ=10. A shared non-release cost ranks admission boundaries but not the relative costs of hold, confirmation, and fallback. 5.3 Source availability, score dependence, and calibration 5.3.1 Expected-source availability Missing-source semantics materially change selective performance. Under the primary retained-zero convention, every declared source remains in its provenance component, and an incomplete component contributes no conserved support. This convention gives ncsAURC 0.6294 before admission and 0.0861 afterward. Three alternatives exclude the zero, remove the unavailable source, or place that source in an isolated component. Each yields 0.7748 before admission and 0.1875 afterward; the latter is worse than the 0.1479 obtained by nested Dirichlet under its own missing-source convention. This one-factor sensitivity analysis varies only PACT’s missing-source semantics; Section 5.2.2 reports the complete-source comparison in which no missing-source convention applies. The PACT–nested ordering reverses on the degraded availability stratum in a diagnostic that re-ranks the same held-out native scores within each stratum, without refitting models or re-estimating outer-fold thresholds. Across the 19,200 instances with three valid sources, pre-admission ncsAURC is 0.6172 for PACT fusion, 0.7245 for nested Dirichlet, and 0.7473 for product fusion. Across the 12,000 instances with an unavailable or invalid source, the corresponding values are 0.9308, 0.8671, and 0.9524, so nested Dirichlet leads on the degraded stratum. Restricting selection to valid-source instances while retaining the full denominator yields 0.6158, 0.6991, and 0.6350, respectively. The reversal therefore follows the instance mixture rather than threshold refitting. A convention adopted for absent sources can outweigh a headline margin. Changing only the treatment of unavailable sources raises PACT fusion ncsAURC from 0.6294 to 0.7748, a shift larger than the 0.1340 margin over product fusion. The three alternative conventions coincide because unavailable sources bridge no otherwise disconnected observed components. Excluding the zero changes 1,670 predictions among the 9,600 instances with an unavailable source, at mean posterior drift 0.0969, and none among the 21,600 instances without source unavailability. On the 19,200 instances with three observed valid sources, where no missing-source convention applies, this stratified diagnostic preserves the ordering reported above. In the same three-valid-source stratified diagnostic, the method ordering changes with the selection score. The pooled correctness score gives pre-admission ncsAURC values of 0.8032 for PACT fusion, 0.4319 for nested Dirichlet, 0.4047 for product fusion, and 0.3838 for global cautious fusion. After admission, the corresponding values are 0.3672, 0.0475, 0.2316, and 0.1092. The pooled score favors nested Dirichlet and global cautious fusion. PACT fusion leads before admission with 1−u1-u. 5.3.2 Selective ordering and probability calibration Selective ordering and calibration provide different assessments. Under the native selection scores, PACT fusion has lower selective risk than nested Dirichlet. Its raw posterior also has lower NLL and Brier score; nested Dirichlet has lower ECE10. PACT fusion’s raw NLL and Brier score are 1.6735 and 0.8436. The empirical-prior reference, estimated from the training portions of the outer folds, reaches 1.6094 and 0.8000 and achieves lower values on both proper scores. The differences between PACT fusion and nested Dirichlet are −0.1411-0.1411 for NLL (95% CI [−0.1491,−0.1333][-0.1491,-0.1333]), −0.0352-0.0352 for Brier ([−0.0360,−0.0342][-0.0360,-0.0342]), and +0.1510+0.1510 for ECE10 ([0.1408,0.1575][0.1408,0.1575]). Proper scores and binwise calibration point in opposite directions. Equal source scales define the primary benchmark specification. Source-specific rescaling selected on the training folds is reported as a sensitivity; it preserves the PACT–nested ordering before and after admission, but equal source scales yield lower PACT ncsAURC in both comparisons. Appendix B reports the selected scaling and the resulting sensitivity values. 6 Sensitivity and Decision-Layer Latency The PACT–nested ordering persists when the source-condition mixture is retained and foldwise cutoffs are re-estimated within each setting. Joint holdout shows condition-specific reversals. 6.1 Joint holdout of scenes and source conditions Within each outer scene fold, joint holdout withholds each source condition from threshold and learned-comparator fitting and evaluates it only on held-out scenes; concentration values selected in the main study remain unchanged. Table 7 reports the aggregate comparison and the four condition-specific contrasts whose curves span the common interval. Table 7: Joint scene-and-condition holdout over common support [0.10,0.35][0.10,0.35]: aggregate fusion results, a separately trained comparator, and PACT–nested contrasts by source condition. Panel A: fusion methods under the common admission policy Method ncsAURC↓ Coverage Rall↓R_all Call↑C_all PACT 0.0617 0.15380.1538 0.0000 0.1538 Nested Dirichlet composition 0.11680.1168 0.15380.1538 0.0000 0.1538 Hierarchy-matched cautious 0.17750.1775 0.16090.1609 0.00710.0071 0.1538 Product fusion 0.22470.2247 0.16440.1644 0.01060.0106 0.15380.1538 Provenance-discounted pooling 0.28280.2828 0.17080.1708 0.02250.0225 0.14830.1483 Panel B: separately trained learned comparator Method ncsAURC↓ 95% CI Set Transformer 0.55790.5579 [0.5362,0.5799][0.5362,0.5799] Panel C: PACT versus nested Dirichlet, by condition Held-out source condition PACT −- nested 95% CI Lower ncsAURC Missing geometry under low-quality language −0.4108-0.4108 [−0.4575,−0.3734][-0.4575,-0.3734] PACT Low-quality partial disagreement −0.0554-0.0554 [−0.0600,−0.0511][-0.0600,-0.0511] PACT Missing language under noisy scene context 0.02520.0252 [0.0092,0.0411][0.0092,0.0411] Nested Dirichlet High-confidence contradiction 0.11600.1160 [0.0920,0.1423][0.0920,0.1423] Nested Dirichlet Note. Panel A aggregates the 13 held-condition test subsets; the paired PACT-minus-nested difference is −0.0551-0.0551 (95% CI [−0.0595,−0.0507][-0.0595,-0.0507]). Rall=0.0000R_all=0.0000 denotes no observed wrong admissions, and boldface follows unrounded values. The apparent four-decimal tie in CallC_all includes 4,800 correct retentions for PACT, nested Dirichlet, and hierarchy-matched cautious fusion, and 4,799 for product fusion. Panel B reports the separately trained Set Transformer comparator. Panel C reports the four held conditions for which both curves span the common support interval, with intervals from 2,000 paired scene bootstraps. The aggregate advantage coexists with condition-specific reversals. Applied to the shifted test subset, the transferred fold thresholds do not enforce the 13% target within that subset (Table 7). When trained separately for joint holdout, the Set Transformer records higher ncsAURC than each of the five analytic rules in Panel A. It serves as a fixed-capacity learned reference rather than a tuned upper bound on learned fusion. At the reported operating point, PACT and nested Dirichlet have the same listed outcomes, with no observed wrong admissions and identical correct retention; their separation emerges over the full coverage interval. Among the four condition-specific comparisons with common support, two favor PACT and two favor nested Dirichlet, so the aggregate ordering is not uniform among support-eligible held conditions. 6.2 Sensitivity to evidence construction The ordering is insensitive to the evidence-construction parameters tested here. Across the 256 Sobol settings, with foldwise cutoffs re-estimated within each setting from non-test scene folds, post-admission ncsAURC[0.10,0.35] spans 0.0660–0.0691 for PACT, 0.1067–0.2231 for nested Dirichlet, and 0.1796–0.2864 for provenance-discounted pooling. PACT remains lower than both comparators in every setting and yields no observed wrong admission at the resulting 0.13 operating target. Joint holdout shows reversals after excluding held conditions from threshold fitting; the mixture-preserving sensitivity analysis does not. Together, the analyses distinguish sensitivity to evidence construction from nonuniform performance under held-condition shift. Appendix B reports the parameter ranges. 6.3 Decision-layer latency On the evaluated three-source CPU benchmark, PACT’s decision layer has a median latency of 195.8μs195.8\, and a 95th percentile of 251.7μs251.7\, . Appendix B specifies the timing boundary and source-count scaling. 7 External Evaluation on Learned Predictions and HRC Admission The external studies test whether provenance grouping still controls the conserved evidence budget on learned predictions, whether predictive consequences remain model- and score-dependent, and whether within-component copy invariance persists through typed admission. Multi-view and HABIT analyses reassign the same model outputs across counting structures. The HRC study then evaluates complementary admission evidence and temporal transfer at the final decision stage. 7.1 Multi-view provenance interventions Provenance reassignment preserves the benchmark’s directional budget response on TMC and RCML. HandWritten and Scene15 reveal how counting errors affect accuracy and selective ordering; PIE isolates evidence-budget conservation. Across the HandWritten and PIE dataset–model pairs, PACT fusion remains invariant to within-component copies through multiplicity eight. False splitting expands the evidence budget, whereas all-view merging contracts it; every paired evidence-budget interval for these two assignment errors excludes zero. Within this cohort, the largest expansion occurs on PIE–RCML (3.21×3.21× the original-view budget), while the strongest contraction occurs on HandWritten–TMC (0.0016×0.0016×). Scene15 shows the same directional budget response on learned TMC predictions. False splitting expands the mean evidence budget to 3.32×3.32× its original-view value, while all-view merging retains only 0.053×0.053×; both assignment errors reduce accuracy and worsen ncsAURC relative to original-view grouping. Because HandWritten uses a different common-support interval, ncsAURC is interpreted within rather than across datasets. Appendix B.4 reports the replication accuracy, paired intervals, and normalization sensitivity. Table 8: Effects of provenance-assignment errors on HandWritten/Mfeat at multiplicity m=8m=8. False splitting inflates evidence and worsens ordering. All-view merging yields the larger accuracy loss. Assignment error Dataset–model pair Δ ncsAURC Δ posterior ℓ1 _1 Δ accuracy (p) False split HandWritten–TMC +0.0049+0.0049 [0.0004,0.0108][0.0004,0.0108] +0.246+0.246 [0.243,0.249][0.243,0.249] −0.37-0.37 [−0.83,+0.03][-0.83,+0.03] HandWritten–RCML +0.0024+0.0024 [0.0008,0.0044][0.0008,0.0044] +0.214+0.214 [0.209,0.220][0.209,0.220] −1.25-1.25 [−1.91,−0.72][-1.91,-0.72] All-view merge HandWritten–TMC +0.4894+0.4894 [0.4643,0.5155][0.4643,0.5155] +0.900+0.900 [0.876,0.924][0.876,0.924] −62.60-62.60 [−65.80,−59.50][-65.80,-59.50] HandWritten–RCML +0.0182+0.0182 [0.0091,0.0286][0.0091,0.0286] +1.137+1.137 [1.117,1.155][1.117,1.155] −7.05-7.05 [−9.35,−5.00][-9.35,-5.00] Note. Contrasts subtract the reference assignment. Two-sided 95% intervals use 2,000 paired class-stratified bootstrap draws. Positive Δ ncsAURC means worse ordering; accuracy changes are percentage points. False splitting is referenced to within-component copies and merging to original views. Within-component copies leave accuracy at 98.0% for TMC and 98.5% for RCML. False splitting shifts the posterior by 0.21–0.25 in ℓ1 _1 and worsens selective ordering for both models. The TMC accuracy-change interval includes zero. RCML loses 1.25 percentage points (95% CI [−1.91,−0.72][-1.91,-0.72], Table 8). An incorrect provenance assignment can nevertheless improve aggregate probabilistic metrics. In a minority of replicated-source choices, false splitting lowers macro-averaged NLL, Brier score, and ECE10 while worsening ncsAURC. Across the 12 HandWritten/Mfeat replicated-source choices, six each for TMC and RCML, both NLL and Brier reverse jointly in four choices. Better aggregate probability scores therefore do not imply correct evidence accounting. The two assignment errors have asymmetric consequences. On HandWritten, all-view merging reduces accuracy by 62.60 percentage points for TMC and 7.05 for RCML, with instance-level paired bootstrap intervals reported in Table 8. Across five Scene15–TMC realizations, all-view merging reduces accuracy by 8.49 percentage points and false splitting by 4.62. Posterior displacement alone does not determine predictive loss, which varies with the dataset–model pair. 7.1.1 Exhaustive coarsening of original feature views Fig. 5: Evidence-budget contraction does not determine selective ordering under exhaustive coarsening of six-view HandWritten/Mfeat outputs. Panels (a,b) show selection-score ncsAURC across 203 partitions by component count for TMC and RCML, respectively, using different vertical scales. Panel (c) reports the percentage of the 856 single-merge cover relations for which coarsening lowers ncsAURC under the selection score and posterior confidence; whiskers show 95% class-stratified record-bootstrap intervals. Budget monotonicity holds on all 856 cover edges in Fig. 5, yet it imposes no order on selective performance. Under the selection score, coarsening worsens ncsAURC on 73.2% of TMC edges and 73.6% of RCML edges. It improves ncsAURC on 17.1% and 18.7% and leaves the remaining edges tied. Under posterior confidence, coarsening lowers ncsAURC on 9.3% and 6.2% of the respective edges. 7.2 Camera-acquisition and model-family grouping On HABIT [44], camera identity defines the acquisition boundary. Checkpoint families provide a model-based grouping for comparison with empirical error association. The analysis covers 1,128 paired pre-release events from 696 episodes across six tasks. Section 4.3 defines the temporal windows and outcome labels. 7.2.1 Camera-acquisition grouping Prompt multiplicity separates acquisition grouping from per-output counting. Under the selection score, acquisition grouping improves ordering at two prompts for both checkpoints; at four prompts, the contrast remains clear for Qwen3-VL-8B (0.1202, 95% CI [0.0893,0.1495][0.0893,0.1495]) but includes zero for Qwen3-VL-32B. Posterior confidence does not reproduce the 8B effect. The selective consequence of counting therefore varies across the tested checkpoints and scores (Fig. 6). The accuracy effect also reverses across checkpoints. Acquisition grouping improves Qwen3-VL-32B by 3.16 percentage points but lowers Qwen3-VL-8B by 3.14 points relative to per-output counting. Relative to acquisition grouping, all-view merging produces the largest posterior drift and lowers accuracy for both checkpoints, yet selection-score ncsAURC changes only from 0.2257 to 0.2327 for 32B and from 0.2562 to 0.2570 for 8B (Table 9). Risk–coverage performance alone therefore need not expose a provenance-assignment error. Equal-cardinality shuffling tests whether component size alone explains the budget response. At four prompts, shuffled assignments retain less evidence than acquisition grouping for both checkpoints. Under this construction, cardinality alone cannot explain the retained budget. Posterior confidence and equal-cardinality shuffling remain post-hoc sensitivity analyses on score choice and grouping structure. Fig. 6: Provenance grouping controls retained evidence, while its selective consequences depend on checkpoint and score across 1,128 HABIT events. (a) Difference in ncsAURC between per-output counting and acquisition grouping as prompt multiplicity increases. (b) At four prompts, the same contrast under the selection score (filled) and posterior confidence (open). (c) Evidence budgets under alternative counting rules relative to one output per camera; hollow markers denote equal-cardinality shuffled groupings. Whiskers in panels (a,b) are 95% paired episode-bootstrap intervals; positive values favor acquisition grouping. Table 9: Per-camera provenance assignments under four prompt variants. Acquisition grouping reduces retained evidence relative to per-output counting, while predictive effects vary across checkpoints. Predictive and selective performance Structural response Condition Selection-score ncsAURC↓ Posterior-confidence ncsAURC↓ Accuracy (%)↑ Budget ratio Posterior drift Qwen3-VL-32B Acquisition grouped 0.22570.2257 0.17020.1702 78.3578.35 0.7830.783 0.1450.145 Per-output counting 0.24810.2481 0.18810.1881 75.1975.19 4.0004.000 0.1590.159 Identical-vector merge 0.26610.2661 0.18450.1845 75.5975.59 3.8393.839 0.1580.158 All-view merge 0.23270.2327 0.23800.2380 54.9354.93 0.0380.038 0.2980.298 One output per camera (reference) 0.25540.2554 0.20640.2064 74.4674.46 1.0001.000 0.0000.000 Qwen3-VL-8B Acquisition grouped 0.25620.2562 0.25600.2560 59.2259.22 0.4760.476 0.2130.213 Per-output counting 0.37640.3764 0.24040.2404 62.3662.36 4.0004.000 0.2750.275 Identical-vector merge 0.43150.4315 0.24260.2426 61.3061.30 3.6773.677 0.2760.276 All-view merge 0.25700.2570 0.25510.2551 57.7157.71 0.0280.028 0.4920.492 One output per camera (reference) 0.46750.4675 0.25240.2524 53.2553.25 1.0001.000 0.0000.000 Note. Selection-score and posterior-confidence ncsAURC use common support [0.10,0.90][0.10,0.90], taskwise means, and equal task weights. Budget ratio and posterior drift are relative to one output per camera. Ratios remain checkpoint-specific because the meet depends on checkpoint predictions. 7.2.2 Model-family grouping Under pooled event-window aggregation on common support [0.10,0.90][0.10,0.90], the checkpoint analysis separates algebraic conservation from empirical error association. Several checkpoints predict ready for all or nearly all events, and the binary evidence mapping fixes every checkpoint budget at two. PACT fusion under model-family grouping reaches 86.6% accuracy, close to 86.5% for nested Dirichlet and hierarchy-matched cautious fusion. Product has the lowest NLL (0.441 versus 0.596 for PACT fusion with model-family grouping). Across product, nested Dirichlet, hierarchy-matched cautious, PACT fusion with model-family grouping, and PACT fusion with checkpoint separation, ncsAURC spans 0.1642–0.1654. Nested Dirichlet attains a lower ncsAURC than PACT fusion with model-family grouping by 0.0007 (95% CI [0.0003,0.0011][0.0003,0.0011]). The operators are close in accuracy and ncsAURC. Their probability-quality and calibration metrics differ, and the accounting response to regrouping is larger still. Fig. 7: Checkpoint heterogeneity does not determine a counting partition. (a) Early-to-event-proximal change in mean ready probability across eight checkpoints and six HABIT tasks; n gives the number of paired events. (b) Evidence budget under an all-checkpoint merge and checkpoint separation, relative to model-family grouping. (c) Under pooled event-window weighting, model-family pairing ranks seventh among 105 pair partitions by within-minus-between error association; nine pairings are tied or higher. Regrouping changes conserved evidence, but error association does not uniquely identify model-family grouping. Model-family grouping is a counting choice to be tested rather than inferred from architecture. Fig. 7 links taskwise response heterogeneity, the algebraic budget response to regrouping, and empirical support for the model-family partition from pairwise error association. Budget and posterior are invariant to exact copies. Checkpoint separation yields a mean matched-row budget ratio of 2.306×2.306× relative to model-family grouping, with posterior drift 0.006 and no accuracy change. Merging all checkpoints retains 9.9% of the budget under model-family grouping, produces drift 0.095, and lowers accuracy from 86.6% to 76.3%. Under pooled event-window weighting, the model-family pairing ranks seventh among 105 pairings, with nine tied or higher (9/105=0.0869/105=0.086). Equal task weighting moves it to 22nd, with 24 tied or higher (24/105=0.22924/105=0.229). A post-hoc sensitivity excluding the two LLaVA checkpoints, which predict ready for nearly every event, changes the rank from 7 of 105 to 3 of 15; the association remains unconfirmed (one-sided tail fraction 3/15=0.203/15=0.20). The partition defines which outputs may accumulate. The error-association statistic summarizes observed pairwise association between checkpoint errors. These criteria rank the checkpoints differently. 7.3 HRC admission with complementary evidence This study evaluates typed admission after candidate generation. The preceding studies evaluate the meet-within/sum-across fusion operator. Episode images supply candidate target-match and event-proximity signals, while HABIT task labels identify the current target. These signals enter as complementary requirements, not countable votes. A candidate is reference-consistent when it agrees with the dataset reference action and reference-inconsistent otherwise. The model trained on development episodes from all six task categories is evaluated on new episodes from those categories. The leave-one-task-out analysis tests whether the within-episode change rule transfers to a task absent from model fitting. Table 10: Offline HRC admission on 60 development-disjoint episodes. Target identity removes off-target admissions, and event-proximity evidence reduces early-window admissions; checkpoint sensitivity and leave-one-task-out transfer are reported separately. Panel A: ablation of admission evidence for Qwen3-VL-8B Admission evidence Reference-inconsistent admissions Reference-consistent admissions Off-target On-target early-window Total Candidate evidence only 6666 2525 9191 5353 Target identity 00 2525 2525 5353 Event proximity 5353 11 5454 5151 Target + event proximity 00 11 11 5151 Panel B: combined admission rule across VLM checkpoints Checkpoint Candidate-only admissions Candidate-only ready recall (%) Combined-rule admissions Combined-rule coverage (%) Consistent Inconsistent Consistent Inconsistent Qwen3-VL-8B 5353 9191 88.388.3 5151 11 7.27.2 Qwen3-VL-32B 5757 8686 95.095.0 5454 11 7.67.6 InternVL3-8B 2727 7171 45.045.0 2424 11 3.53.5 Panel C: leave-one-task-out temporal transfer Temporal rule Consistent Inconsistent ready recall (%) Absolute score (0.5) 3737 1010 61.761.7 Dev balanced accuracy 3434 1010 56.756.7 Dev FPR ≤0.05≤ 0.05 3838 1010 63.363.3 Dev Platt calibration 3737 1010 61.761.7 Paired positive change 5353 00 88.388.3 Note. Panel A uses Qwen3-VL-8B; Panel B compares the three listed checkpoints. All panels cover the same six tasks. Panel B applies the admission rule to candidate evidence; the joint fusion-and-admission configuration is reported in the text. Panel C excludes each evaluated task from temporal-model fitting and threshold/calibration selection while retaining candidate and target evidence. FPR means false-positive rate; consistency is agreement with the dataset reference action, not task success. Candidate-only ready recall uses 60 cases, whereas combined-rule coverage uses all 720 queries. The value 53 denotes reference-consistent admissions under candidate evidence alone in Panel A and cases retained by the paired positive-change rule in Panel C. Target identity removes all 66 observed off-target admissions without reducing reference-consistent admissions. Event proximity reduces early-window admissions of on-target candidates from 25 to one while retaining 51 of the 53 reference-consistent admissions allowed by candidate evidence alone (Table 10). Together, the signals withhold 90 of the 91 reference-inconsistent admissions allowed by candidate evidence alone, reducing their rate over the 720 repeated evaluations from 12.64% to 0.14%. The episode-stratified, equal-task-weighted difference is −12.50-12.50 percentage points (95% CI [−13.75,−11.25][-13.75,-11.25]). These numbers estimate offline reference consistency. Candidate quality varies across checkpoints, with reference-ready recall ranging from 45.0% to 95.0%. The combined rule nevertheless retains most reference-consistent candidates (24 of 27 to 54 of 57) and permits one reference-inconsistent admission per checkpoint. For Qwen3-VL-32B and InternVL3-8B, it reduces reference-inconsistent admissions relative to candidate evidence alone by 11.81 and 9.72 percentage points, respectively; both confidence intervals exclude zero (Table 10). Within-camera duplication to total multiplicity eight leaves all 720 final typed responses per checkpoint unchanged. When fusion and admission are evaluated jointly, camera-acquisition grouping assigns all prompt variants from one image to the same provenance component. Under this grouping, PACT admits 47 of 57 reference-consistent candidates for Qwen3-VL-32B and 43 of 53 for Qwen3-VL-8B, with no reference-inconsistent admission observed for either checkpoint. Repeating every selected output leaves the PACT budget, posterior, and temporal comparison unchanged. Counting the same outputs separately raises an event-window evidence budget by up to 280 budget units under the binary evidence mapping, which assigns total mass two to each output, yet changes none of the event decisions produced by the positive-change rule at this operating point. The other counting rules also yield no observed reference-inconsistent admissions, and their comparative retention ordering changes across checkpoints. PACT preserves the admission decision under within-acquisition duplication; comparative retention remains checkpoint-dependent. This aggregate result conceals a task boundary. PACT admits none of the ten reference-ready Hotel Towel episodes at either checkpoint. Within-task discrimination remains substantial across six leave-one-task-out folds (equal-task-weighted AUROC 0.8632, 95% CI [0.8397,0.8878][0.8397,0.8878]), but task-dependent score origins make the tested absolute thresholds unreliable (Fig. 8). A fixed 0.5 threshold admits ten reference-inconsistent candidates, and neither development-only thresholds nor Platt scaling removes them. Under the common admission policy, the within-episode positive-change rule instead retains 53 consistent and no inconsistent candidates (Table 10). Threshold placement is therefore a transfer bottleneck for the tested absolute interfaces, although these controls do not identify it as the sole source of error. The balanced 60-episode subset does not establish pairing specificity (p=0.1106p=0.1106); its 53/0 outcome evaluates the common admission policy rather than the temporal mechanism. In the larger block-exchangeable analysis, after excluding one nonexchangeable episode, early-to-event-proximal pairing has higher concordance than the task- and event-profile-stratified permutation reference (0.9629 versus 0.9261; p=0.0001p=0.0001). A cross-task sensitivity gives the same directional result. Positive change alone is insufficient for sequential admission. After averaging equally across tasks, the rate of positive score change is 0.3333 higher from early to event-proximal than from pre-early to early (episode-bootstrap 95% CI [0.2333,0.4333][0.2333,0.4333]). The same sign rule would nevertheless admit 14 reference-inconsistent early-window queries in the offline evaluation while retaining 53 reference-consistent queries. Under this offline policy, relative change supports admission only after a temporal reference exists. Before that reference exists, the temporal condition is unmet; the ordered policy returns confirm for an otherwise selected target mismatch and hold otherwise. Fig. 8: Within-episode score changes are more stable across tasks than absolute score levels. Descriptive panels use all 1,128 events from 696 episodes. Task-specific score origins shift while within-episode changes remain mostly positive. (a) Early and event-proximal distributions across six HABIT tasks; white ticks mark medians and the dashed line marks 0.5. (b) Within-task AUROC versus balanced error at 0.5. (c) Calibration intercepts with episode-clustered 95% intervals. (d) Paired early-to-proximal changes; the dashed line marks zero. 8 Discussion 8.1 Provenance partition as an evidence-counting variable View-specific reliability characterizes individual reports; conflict characterizes relations between reports [4, 6, 16]. PACT adds the distinct relational variable of whether two outputs are licensed to contribute support separately. The component meet limits evidence magnitude within one provenance unit, and component corroboration asks how many complete units support the selected action. Together, these operations prevent repeated agreement from substituting for an additional evidential origin under the supplied partition. Copy interventions vary multiplicity while preserving the numerical opinions, and equal-cardinality reassignment shows that retained evidence depends on component assignment, not on component count alone. The two operations also act at different stages of the decision. After removing the constructed adversarial-consensus arm, provenance-partition aggregation retains a 0.0557 ncsAURC reduction relative to singleton aggregation, whereas the corroboration contrast disappears. On the full controlled benchmark, corroboration produces the larger ncsAURC contrast because it targets that arm. Provenance-partition aggregation governs evidence accumulation, component corroboration tests whether separately countable units provide the required support, and typed admission maps the first unmet release condition to a response. Budget conservation follows the partition lattice, but score ordering and probability quality need not (Section 7.1.1). False splitting can improve probability metrics while worsening selective ordering, so aggregate metrics need not expose a miscounted evidential unit. Conservation is a structural constraint, not a calibration theorem. 8.2 Implications for multimodal foundation models and embodied systems Foundation models can generate many predictions without acquiring new evidence. Self-consistency samples multiple reasoning paths from one input [1], and outputs from related language models can share correlated errors [2]. When outputs share a derivation, output multiplicity alone should not be treated as separately countable support. In the tested assignments, prompt variants of one image represent repeated inference, while distinct synchronized camera identities represent separate acquisitions. Architectural grouping and empirical error association yield different checkpoint partitions (Section 7.2.2). Model-family labels and output similarity did not establish countability in this panel, so the assignment remained explicit and was tested against observed dependencies. PACT carries this counting constraint into typed admission. Selective prediction abstains outside a retained subset [9], while safety-oriented embodied-agent methods constrain VLA policies or screen candidate actions for physical hazards [33, 34]. An exact computational copy that inherits its evidence and acquisition parent leaves the final response unchanged, provided eligibility and the remaining admission conditions do not change. Fusion-rule rankings remain empirical, and physical safeguards remain necessary. Temporal evidence exposes another limit of static confidence. Across the six offline HABIT tasks, a fixed absolute threshold produces task-dependent errors; a within-episode change instead compares each event with its own reference. That comparison becomes admissible evidence only after the earlier observation exists. Until then, the temporal comparison cannot authorize release. The ordered policy returns confirm for a target mismatch and hold otherwise. Separating countable support, temporal comparison, and admission creates a testable interface for studying predictability, meaningful human control, and perceived safety. Behavioral evaluation is needed to establish those human-facing effects. 8.3 Scope, limitations, and research directions The experiments evaluate provenance-conditioned accounting and offline admission. PIE measures the budget response, whereas the HRC study measures reference consistency rather than physical success. Hotel Towel exposes a task-level boundary. Episode-relative scores retain cross-task discrimination, yet in the camera-grouped, four-prompt setting PACT recalls 0 of 10 reference-ready cases for both tested Qwen3-VL checkpoints. The absence of observed reference-inconsistent admissions does not imply zero underlying risk. Closed-loop and behavioral studies must evaluate task success, response burden, perceived safety, and meaningful human control. The simulation benchmark specifies provenance by construction, enabling alternative assignments to be compared over the same source opinions. The external studies move toward observed assignment structure through camera acquisition identity and compare model-family groupings with empirical error association. The guarantees require a supplied partition and commensurate evidence under the stated aggregation assumptions. The held-out protocols specify empirical assignments in advance and prohibit post-outcome assignment selection. Missing-source semantics remain first-order modeling choices. In the fusion-only comparison, excluding an unavailable source from the component meet changes ncsAURC from 0.6294 to 0.7748. The next methodological problem is to infer partially observed provenance while calibrating source scale and decision utility. 9 Conclusion Computation can multiply agreement without multiplying evidence. PACT turns this distinction into a constraint on accumulation. Support is conserved within a provenance component and added only across components licensed as separately countable. For a supplied provenance partition under the stated evidence assumptions, the construction yields the greatest budget satisfying singleton fidelity and insertion non-amplification, together with exact-copy invariance and coarsening monotonicity. Provenance reassignment over the same learned predictions preserves the predicted budget response across multi-view and multi-camera settings. Within-acquisition copies likewise leave offline HRC admission unchanged when eligibility and the remaining admission conditions are unchanged. Calibration and risk–coverage ordering remain empirical properties. Temporal admission exposes a different constraint. Within-episode change transfers more consistently than absolute score levels across the evaluated tasks, yet becomes usable only after a reference observation exists. Provenance conditions how support may accumulate, component corroboration tests whether agreement spans the required separately countable units, and typed admission determines what that evidence permits. Within PACT, repeated computation does not create a new countable evidential origin. Appendix A Proofs and Additional Stability Results A.1 Notation Table 11 summarizes the symbols used in the main text and appendix. Table 11: PACT notation from source representation and provenance to fusion, scoring, and typed admission. Symbol Meaning Symbol Meaning K finite set of K≥2K≥ 2 action contracts with a fixed tie-breaking order Cb_C component evidence budget, the meet of its members x decision instance ΠE_ cumulative evidence under partition Π J size of the expected source set BΠB_ total budget ∥Π∥1 _ _1 ξi _i, Φi _i source representation and its evidence adapter W, a prior strength and base rate ip_i categorical opinion or zero vector π projected categorical posterior qiq_i, did_i source quality and conflict scores k k predicted contract ziz_i, viv_i unavailability and required-field structural-validity indicators u, s vacuity and selection score s=1−us=1-u Pi(x)P_i(x) provenance parent set FΠF_ PACT fusion map ℐobsI_obs, ℐkI_k observed sources and sources supporting k τ score-selection threshold GprovG_prov, ℰPE_P, ΠP _P parent-overlap graph, its edge set, and induced component partition ν, 3U_3 required number of complete supporting components and the indicator that all three expected sources are observed and support the selected candidate MopM_op, MevM_ev numbers of supporting outputs and components, without the admission completeness requirement ζbase _base eligibility condition ρi _i, κi _i reliability factor and source scale τA_τ, V score-selection and admission maps ie_i nonnegative evidence vector of source i ⪯ , ∧ coordinatewise order and meet σ(0)σ^(0), ϱ(0) ^(0) preliminary selection status and typed response y(0)y^(0) preliminary selection decision fcurf_cur, fstructf_struct command-consistency and source-validity failure indicators friskf_risk, fprovf_prov risk-support and component-corroboration failure indicators A.2 Proofs of the main-text propositions A.2.1 Auxiliary definitions and decision maps The numbers of supporting outputs and supporting provenance components are Mop(k,x):=|ℐk(x)|,Mev(k,x):=|C∈ΠP(x):C∩ℐk(x)≠∅|.M_op(k;x):=|I_k(x)|, M_ev(k;x):= | \C∈ _P(x):C _k(x)≠ \ |. (13) Here MevM_ev counts each provenance component containing at least one supporting opinion once, regardless of completeness or retained positive budget. A non-supporting bridge remains part of its component and cannot create another supporting component. MevM_ev describes how supporting opinions are distributed over the partition; retained support magnitude is determined separately by the component meet. Definition 1 (Agreement under shared provenance). Given the provenance partition ΠP(x) _P(x), an instance has agreement under shared provenance for contract k when Mop(k,x)>Mev(k,x).M_op(k;x)>M_ev(k;x). (14) The contract then has more supporting opinions than supporting provenance components. For the benchmark evaluation, the reliability factor is ρi:=[qi]01(1−[di]01),[a]01:=min1,max0,a. _i:=[q_i]_0^1 (1-[d_i]_0^1 ), [a]_0^1:= \1, \0,a\\. (15) The reliability mapping in Eq. 15 is fixed rather than learned. The shared fusion concentration and the source-specific scaling used only in the sensitivity analysis are selected on training data and then transferred to held-out instances. Definition 2 (Provenance-aware evidence conservation). Consider any extension of the source set that preserves all existing evidence vectors and parent assignments. A fusion operator satisfies provenance-aware evidence conservation when two conditions hold for every such extension. Adding a source to one or more existing components cannot increase cumulative evidence coordinatewise, and adding an exact duplicate with matching parents preserves cumulative evidence. A source that forms a disconnected component may contribute through cross-component accumulation. The exact preliminary selection map is τ(k^,s,ζbase):=(selected,k^,none),s≥τ∧ζbase=1,(withheld,k^,hold),otherwise.A_τ ( k,s, _base ):= cases(selected, k,none),&s≥τ\ \ _base=1,\\ (withheld, k,hold),&otherwise. cases (16) Write the preliminary decision as y(0)=(σ(0),k^,ϱ(0))y^(0)=(σ^(0), k, ^(0)), where σ(0)σ^(0) is the selection status and ϱ(0) ^(0) is the typed response. The indicators fcurf_cur, fstructf_struct, friskf_risk, and fprovf_prov correspond, respectively, to failures of command consistency, source validity, risk support, and component corroboration in Table 3. Each equals one exactly when its associated condition fails. The typed admission map is (y(0),x):=y(0),σ(0)≠selected,(withheld,k^,hold),fcur(x)=1,(withheld,k^,fallback),fstruct(x)=1,(withheld,k^,hold),frisk(x)=1,(withheld,k^,confirm),fprov(k^,x)=1,(admitted,k^,none),otherwise.V(y^(0),x):= casesy^(0),&σ^(0) ,\\ (withheld, k,hold),&f_cur(x)=1,\\ (withheld, k,fallback),&f_struct(x)=1,\\ (withheld, k,hold),&f_risk(x)=1,\\ (withheld, k,confirm),&f_prov( k;x)=1,\\ (admitted, k,none),&otherwise. cases (17) This case map formalizes the decision path summarized in Algorithm 1. If an exact within-component copy preserves its evidence vector, parent assignment, base eligibility, and all remaining admission conditions, then ΠE_ , π, s, k k, y(0)y^(0), and the final decision Y(x)Y(x) remain unchanged. Let PΣ=∑i|Pi(x)|P_ = _i|P_i(x)| and Pmax=maxi|Pi(x)|P_ = _i|P_i(x)|. Reading and indexing the parent sets requires O(PΣ)O(P_ ) time and space. With indexed parent sets, overlap discovery and aggregation require O(PΣ+∑i<jmin|Pi(x)|,|Pj(x)|+JK)O\! (P_ + _i<j \|P_i(x)|,|P_j(x)|\+JK ) time and O(J2+JK+JPmax)O(J^2+JK+JP_ ) space. After overlap discovery, storing only the discovered edges reduces connected-component traversal and evidence aggregation to O(J+|ℰP(x)|+JK)O(J+|E_P(x)|+JK) time and space. The bounded-cardinality and sparse-graph reductions stated in Section 3.5 follow directly. A.2.2 Proofs Proof of Proposition 1. Both inputs present ,∈ℳ()\ α, α\ (A) to H and therefore yield the same output. ∎ Proof of Proposition 2. Fix ∈ℰv . Starting from \v\ and inserting every other member yields g(ℰ)⪯g(E) . Since v was arbitrary, g(ℰ)⪯⋀∈ℰg(E) _v v. The meet itself satisfies both requirements and is therefore the unique pointwise greatest admissible rule. ∎ Proof of Proposition 3. For each component C, admissibility gives C⪯Cg_C _C, and Cb_C itself is admissible. Summing over components proves greatestness. If an inserted source connects a family joinC_join of existing components, their previous contribution is ∑C∈joinC _C _joinb_C. The new merged-component budget is the meet of the inserted source with all members of these components. It is coordinatewise no greater than any Cb_C, for C∈joinC _join, and therefore no greater than their nonnegative sum. An exact copy with matching parents remains in the original component. Every source sharing one of those parents was already connected to the original source, so the copy cannot merge a previously distinct component. Its duplicate vector preserves the meet. A disconnected component contributes the nonnegative term Db_D. ∎ Proof of Corollary 1. Each block of Π2 _2 is a union of blocks of Π1 _1. Its coordinatewise meet is no greater than any constituent fine-block meet and therefore no greater than their nonnegative sum. Summing over the disjoint coarse blocks yields (7). ∎ A.3 Stability under a given partition Theorem 1 (Distribution and score stability under a given partition). Fix the source index set, provenance partition ΠP _P, prior strength W>0W>0, and base rate a. For two evidence collections i\e_i\ and i′\e_i \, define ΔΠ=∑C∈ΠP∑k=1Kmaxi∈C|eik′−eik|. _ = _C∈ _P _k=1^K _i∈ C|e_ik -e_ik|. (18) Let Z=W+∥Π∥1Z=W+ _ _1 and Z′=W+∥Π′∥1Z =W+ _ _1. Then ∥Π′−Π∥1≤ΔΠ,∥′−∥1≤2ΔΠmaxZ,Z′≤2ΔΠW, _ -E_ _1≤ _ , π - π _1≤ 2 _ \Z,Z \≤ 2 _ W, (19) and |s′−s|≤WΔΠZZ′≤ΔΠW.|s -s|≤ W _ Z ≤ _ W. (20) Proof. For each component, the scalar minimum is nonexpansive under perturbations of its members. Summing these coordinatewise bounds gives the first inequality. Let =W+Πx=Wa+E_ and ′=W+Π′x =Wa+E_ . Because a is a base-rate vector and all evidence vectors are nonnegative, the coordinate sums of x and ′x are Z and Z′Z , respectively. Their numerator difference is exactly the cumulative-evidence difference and is therefore bounded by ΔΠ _ . The reverse triangle inequality gives |Z−Z′|≤∑k∈|xk−xk′||Z-Z |≤ _k |x_k-x_k |. If Z≥Z′Z≥ Z , then ‖Z−′Z′‖1≤∥−′∥1Z+|Z−Z′|Z≤2∥−′∥1Z. xZ- x Z _1≤ -x _1Z+ |Z-Z |Z≤ 2 -x _1Z. (21) The case Z′≥Z ≥ Z follows symmetrically. Writing Etot=∥Π∥1E_tot= _ _1 gives s=Etot/(W+Etot)s=E_tot/(W+E_tot) and |s′−s|=W|Etot′−Etot|/[(W+Etot)(W+Etot′)]|s -s|=W|E _tot-E_tot|/[(W+E_tot)(W+E _tot)]. The first evidence bound gives |Etot′−Etot|≤ΔΠ|E _tot-E_tot|≤ _ , which yields the stated score bound. ∎ With the source set and provenance partition fixed, the posterior and selection score are Lipschitz in ΔΠ _ with global constants 2/W2/W and 1/W1/W, respectively. These global constants decrease as the prior strength W increases; the sharper score bound above need not be monotone in W. For ∈ℝKx ^K, let ()+(x)_+ denote its coordinatewise positive part, with [()+]k=maxxk,0[(x)_+]_k= \x_k,0\. Corollary 2 (Within-component near-copy insertion). Insert +e_+ into an existing component C without merging previously distinct components, and define ε+=∥(C−+)+∥1 _+= (b_C-e_+)_+ _1. Then Π+=Π−(C−+)+,∥+−∥1≤2ε+W,s+≤s.E_ ^+=E_ -(b_C-e_+)_+, π^+- π _1≤ 2 _+W, s^+≤ s. (22) If ∥+−j∥1≤ϵ _+-e_j _1≤ε for some j∈Cj∈ C, then ε+≤ϵ _+≤ε. The budget identity follows coordinatewise from C∧+=C−(C−+)+b_C _+=b_C-(b_C-e_+)_+. The normalization bound in the proof of Theorem 1 depends only on the cumulative-evidence numerators and their coordinate sums, so it applies to this budget change. Monotonicity gives s+≤s^+≤ s, and C⪯jb_C _j yields the final inequality. Corollary 3 (Margin-conditional prediction and score-selection stability). Suppose k k is the unique maximizer of π with margin μπ=πk^−maxj≠k^πj>0 _π= _ k- _j≠ k _j>0. Sufficient conditions for preserving both the predicted contract and score-selection status are 2ΔΠmaxZ,Z′<μπ,WΔΠZZ′<|s−τ|. 2 _ \Z,Z \< _π, W _ Z <|s-τ|. (23) The first condition preserves the predicted contract. For fixed τ and ζbase _base, the second preserves select–withhold status. Together they preserve the selection-stage decision y(0)y^(0). The posterior condition keeps every top-versus-competitor gap positive under the stated perturbation bound, and the score condition prevents s from crossing τ. Appendix B Additional Experimental Details B.1 Benchmark and comparator specifications The benchmark maps language, handover geometry, and path risk to opinions over the five action contracts. Close-to-hand geometry and increasing path risk shift support from unconstrained transfer toward slower, withheld, or corrective responses. Occlusion lowers source quality and increases conflict. Near-copy perturbations move a bounded amount of probability mass between the two leading geometry contracts while preserving provenance; the multiplicity study repeats geometry within the same component. Missing-source sensitivity compares retaining an unavailable source as zero evidence with omitting that slot before component construction. In this benchmark, omission, source removal, and null-source isolation are equivalent because the unavailable source bridges no observed components. The primary benchmark specification uses K=W=5K=W=5 and equal source scales. Nested Dirichlet concentration is selected within each outer training fold and then shared with PACT. Hierarchy-matched cautious fusion participates in the outer-fold comparison but not concentration selection. Source-specific rescaling is evaluated separately as a sensitivity. Quality-weighted fusion averages available distributions with weights proportional to qiq_i. Product fusion discounts each distribution toward the uniform base rate, rik=ρipik+(1−ρi)/Kr_ik= _ip_ik+(1- _i)/K, multiplies class probabilities, and normalizes across classes. Nested Dirichlet composition forms source evidence using the fold-selected concentration κ; PACT uses the same concentration under equal source scaling: eik=ρiκpik,bik=eikK+∑ℓeiℓ,ui=K+∑ℓeiℓ.e_ik= _iκ p_ik, b_ik= e_ikK+ _ e_i , u_i= KK+ _ e_i . Available opinions are composed sequentially in the fixed order L,G,RL,G,R using c=∑k≠ℓbkbℓ′,b~k=bkbk′+bku′+bk′u1−c,u~=uu′1−c,c= _k≠ b_kb _ , b_k= b_kb _k+b_ku +b _ku1-c, u= u 1-c, followed by πk=b~k+u~/K _k= b_k+ u/K under the TMC parameterization; 1−u~1- u is the corresponding selection score [4]. Global cautious applies Denœux’s cautious rule to all available masses without a provenance partition [7]. Hierarchy-matched cautious fusion takes the cautious minimum within each provenance component before conjunctive combination across components. Unavailable sources contribute vacuous mass, and the pignistic transformation supplies decision probabilities [41]. For pooling, source i receives log-weight 2qi−2di−4mi+logδi2q_i-2d_i-4m_i+ _i, where mim_i marks an unavailable source and δi=(1+ηi)−1 _i=(1+ _i)^-1 discounts the total registered edge weight ηi _i to connected sources with the same predicted class. Normalized weights define the weighted log pool; the value-only counterpart sets δi=1 _i=1. The remaining comparators omit unavailable sources, while nonmissing invalid distributions are mapped to the uniform distribution. Score comparisons use posterior peak, top-two margin, inverse normalized entropy, and a regularized correctness model fitted on outer-training posterior summaries and observed-source count. Method identity is excluded, and the fitted score transfers to the remaining analytic operators without refitting. B.2 Threshold and learned-comparator details Concentration selection averages candidate costs across the target coverages. These costs assign penalties of 5 to wrong admission and 0.5 to non-release, an equivalent rescaling of C10C_10 that leaves all rankings unchanged. Ties are resolved by fewer wrong admissions, greater correct retention, and then the prespecified candidate order. For method b, outer fold f, and stage a∈pre,posta∈\pre,post\, let gbg_b denote the selection score, ftrD_f^tr the full outer-training set, and f,b,elig,atrD_f,b,elig,a^tr its base-eligible subset, additionally restricted to admission-eligible instances under method b when a=posta=post. At target coverage γtar _tar, the cutoff is rf=roundeven(γtar|ftr|),jf,b,a=min|f,b,elig,atr|,rf,τf,b,a=gb,(jf,b,a),jf,b,a≥1,+∞,jf,b,a=0,r_f=round_even\! ( _tar|D_f^tr| ), j_f,b,a= \|D_f,b,elig,a^tr|,r_f\, _f,b,a= casesg_b,(j_f,b,a),&j_f,b,a≥ 1,\\ +∞,&j_f,b,a=0, cases (24) Here gb,(jf,b,a)g_b,(j_f,b,a) is the jf,b,aj_f,b,ath largest method- and stage-eligible score. The target retain count uses the full outer-training set as its denominator and is capped by the number of eligible cases. Half-integers are rounded to even; ties at τf,b,a _f,b,a are retained, and +∞+∞ denotes no selection. A Set Transformer with two four-head self-attention blocks (hidden width 96) and a learned pooling query jointly predicts the action contract and a selection score from source opinions, evidence attributes, source roles, and provenance indicators. Its objective combines selection-weighted classification, a coverage penalty, and full-sample classification. The model is optimized with AdamW and selected by validation loss. Each held-condition–outer-fold result averages three independent initializations; training and cutoff selection exclude its test scenes and held-out source condition. Each of the 13 source conditions contributes 2,400 of the 31,200 instances. The conditions cover source loss or invalidity, degraded quality, stale context, unsafe urgency, adversarial agreement, missing conflict information, partial disagreement, and high-confidence contradiction. This design separates local evidence degradation from errors in how support is counted. B.3 Learned-prediction evaluation details HandWritten/Mfeat supports budget, posterior, accuracy, and ordering analyses for TMC and RCML; PIE–RCML supports budget only. Scene15–TMC adds a second dataset for the full provenance, accuracy, and selective-ordering analysis. For HABIT, log odds under both label mappings are oriented as ready minus not ready and averaged; their sigmoid preadyp_ready yields evidence 2(1−pready,pready)2(1-p_ready,p_ready). Across the six tasks, the evaluation comprises 120 episode-target and 600 counterfactual-target queries. The checkpoint panel pairs the 8B/32B Qwen3-VL, 2B/8B InternVL3, 0.5B/7B LLaVA-OneVision, and 0.5B/2.2B SmolVLM2 variants. The 105 partitions into four checkpoint pairs are ranked by mean within-pair binary-error ϕφ minus mean between-pair binary-error ϕφ; bootstrap resampling draws episodes within tasks while retaining all windows and checkpoints. Event proximity is estimated by regularized logistic regression over frozen ResNet-50 visual features [49] and text TF–IDF. Regularization is selected by episode-grouped cross-validation; the primary model uses all development episodes, and each task-held-out model excludes the evaluated task. Absolute interfaces use 0.5, development-selected thresholds, or Platt scaling. Positive change is sproximal>searly1\s_proximal>s_early\ with ties at zero. Estimates average within episodes and equally across tasks. Admission and temporal-control intervals use task-stratified episode bootstraps. For joint fusion and admission, each checkpoint supplies 2,400 outputs from 60 episodes, two temporal windows, five cameras, and four prompt variants. Prompt multiplicity, camera removal, and exact-copy multiplicity vary repeated computation and acquired viewpoints separately. Task-stratified bootstrap intervals keep episodes intact. Temporal tests use fixed-point-free permutations of complete early-score profiles within each task and exact event-occurrence-profile stratum. Removing the sole singleton block leaves 1,126 events in 695 episodes. The statistic is the equal-task mean of episode-level strict concordance. Cross-task sensitivity uses the largest outcome-blind exact-profile subset with different-task donors and no episode reuse. Task-dependent baseline analyses use all 1,128 pairs from 696 episodes. In the separate 60-event sequential control, the pre-early frame is fixed at 1.0 s before the designated early frame for each of the five cameras. B.4 Sensitivity analysis and Scene15 evaluation The 256-point scrambled Sobol design varies language peak probability over 0.65,0.75,0.85,0.90,0.95\0.65,0.75,0.85,0.90,0.95\, opinion-generation concentration over 50,100,200\50,100,200\, and occlusion strength over 0.5,1.0,1.5\0.5,1.0,1.5\, while sampling the quality and conflict multipliers within [0.8,1.2][0.8,1.2]. Geometry and risk perturbations preserve the original top class, and fusion concentration remains fixed within each fold. PACT source scales have unit geometric mean and are selected by training NLL; every fold selects (0.63,0.63,2.52)(0.63,0.63,2.52) in language–geometry–risk order. The corresponding ncsAURC values are 0.6323 versus 0.7749 for nested Dirichlet before admission and 0.1147 versus 0.1479 after admission; equal-scale PACT yields 0.6294 and 0.0861. Scene15–TMC uses 30 stratified random 80/20 train–test realizations with 897 held-out instances each. Following the preprocessing used by the released TMC implementation, MinMax scaling is fitted separately to each training and test partition. Mean accuracy is 0.6679 (population SD 0.0222), 0.95 percentage points below the published value of 0.6774. The provenance interventions are applied to held-out predictions from five realizations, leaving the learned outputs unchanged (4,485 predictions; 897 per realization). Original grouping, multiplicity-eight false splitting, and all-view merging yield budgets 5.9374, 19.7381, and 0.3161; accuracies 0.6640, 0.6178, and 0.5790; and full-retained-range ncsAURC values 0.2313, 0.2896, and 0.2865. A 2,000-draw two-stage paired bootstrap, resampling realizations and then held-out instances within realizations, gives signed budget changes relative to original grouping of [12.7965,14.7817][12.7965,14.7817] and [−5.9439,−5.3008][-5.9439,-5.3008] for splitting and merging. In a separate train-fitted normalization sensitivity, each scaler is fitted on the training partition and applied unchanged to test; mean accuracy is 71.82% across five realizations (population SD 1.88 percentage points; 4,485 held-out predictions). Latency measurement. For the decision-layer benchmark reported in Section 6.3, the fold-specific cutoff is determined before timing and applied without refitting. Measurements use one process on an Intel Core Ultra 9 285K. For each instance, the benchmark constructs the graph from supplied parent sets, prepares source evidence, applies meet-within/sum-across fusion, selects an action contract, compares 1−u1-u with the cutoff, and evaluates base eligibility and typed admission. Threshold fitting, provenance acquisition, perception, model inference, communication, and physical execution are excluded. Increasing the source count from three to 64 raises median connected-component discovery on sparse provenance graphs with disjoint source pairs from 1.01.0 to 150.8μs150.8\, ; the corresponding 95th percentiles are 1.11.1 and 166.4μs166.4\, . Data availability Materials supporting the reported analyses, including source code, experimental settings, aggregate result summaries, and redistributable evaluation resources, are available at https://github.com/ZekaiJ/PACT. Data redistribution and third-party asset boundaries are documented in the repository. Declaration of competing interest Zhen Dong is a faculty member at the University of California, Santa Barbara (UCSB), and is also affiliated with NVIDIA Corporation as a researcher. The remaining authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. CRediT authorship contribution statement Zekai Jin: Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Software, Validation, Visualization, Writing – original draft. Hanrong Zhang: Formal analysis, Writing – review & editing. Yihong Tang: Formal analysis, Methodology, Writing – original draft. Fei Hu: Data curation, Writing – original draft. Zhen Dong: Conceptualization, Supervision, Writing – review & editing. Yi Shao: Conceptualization, Funding acquisition, Supervision, Writing – review & editing. Funding This work was supported by the McGill Engineering Doctoral Award (MEDA). The funder had no role in study design, data generation, analysis, interpretation, manuscript preparation, or the decision to submit the article. References [1] X. Wang, J. Wei, D. Schuurmans, Q. V. Le, E. H. Chi, S. Narang, A. Chowdhery, and D. Zhou (2023) Self-consistency improves chain of thought reasoning in language models. 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