Graph Vector Field: A Unified Framework for Multimodal Health Risk Assessment from Heterogeneous Wearable and Environmental Data Streams

Graph V ector Field: A Uniﬁed F ramew ork for Multimo dal Health Risk Assessmen t from Heterogeneous W earable and En vironmen tal Data Streams Silv ano Coletti 1,2 and F rancesca F allucc hi 1 1 Univ ersit` a degli Studi Guglielmo Marconi, Roma, Italy 2 Chelonia SA, Lugano, Switzerland Marc h 31, 2026 Abstract Digital health researc h has adv anced dynamic graph-based disease mo dels, topological learning on simplicial complexes, and multimodal mixture-of-exp erts architectures, but these strands remain largely disconnected and do not yet support mec hanistically interpretable m ultimo dal risk dynamics. W e prop ose a Graph V ector Field (GVF) framework that mo dels health risk as a v ector-v alued ﬁeld evolving on time-v arying graphs and simplicial complexes, and that couples discrete diﬀeren tial-geometric operators with modality-structured mixture-of-exp erts. Individuals, physiological subsystems, and their higher-order interactions form the cells of an underlying complex; risk is represented as a v ector-v alued co chain whose temporal evolution is parameterised with Ho dge Laplacians and discrete exterior calculus op erators. This yields a Helmholtz–Ho dge-t ype decomposition of risk dynamics in to p oten tial-driv en (exact), circulation-lik e (co exact), and top ologically constrained (harmonic) comp onen ts, providing a direct link betw een mo del terms and interpretable mechanisms of propagation, cyclic b eha viour, and p ersisten t mo des. Multimo dal inputs from w earable sensors, b eha vioural and en vironmental con text, and clinical or genomic data are incorporated through a bundle-/sheaf-inspired mixture-of-experts architecture, in which modality-speciﬁc laten t spaces are attached as ﬁbres to the base complex and coupled via learned consistency maps. This construction separates mo dalit y-sp eciﬁc from shared contributions to risk ev olution and oﬀers a principled route to w ard mo dality-lev el identi ﬁability . GVF in tegrates geometric dynamical systems, higher-order top ology (enforced indirectly via geometric regularisation and Ho dge decomp osition rather than through the exp ert forward pass), and structured multimodal fusion into a single framew ork for in terpretable, modality-resolv ed risk mo delling in longitudinal cohorts. The presen t paper develops the mathematical foundations, arc hitectural design, and formal guarantees of the framew ork; empirical v alidation on real- w orld cohorts is the sub ject of ongoing w ork. Keyw ords: Helmholtz–Ho dge decomp osition, simplicial complex, discrete exterior calculus, mixture-of-exp erts, multimodal health risk, wearable sensing, vector bundle MSC (2020): Primary 68T07; Secondary 55N31, 58A15, 62R40, 92C50 1 In tro duction The proliferation of w earable devices and connected sensors is transforming how health can b e monitored and mo delled. Consumer- and medical-grade w earables no w act, de facto , as p erv asiv e an tennas that con tinuously transmit high-frequency ph ysiological and behavioural signals, and are increasingly complemented by electronic health records (EHR), en vironmental 1 exp osures, and genomic information in large longitudinal cohorts [ 9 , 10 ]. T ogether, these data streams promise a shift from episodic assessment to con tin uous, multimodal characterisation of individual health tra jectories and risk. Curren t artiﬁcial in telligence (AI) metho ds already exploit parts of this ecosystem: spa- tiotemp oral graph neural net w orks on wearable-deriv ed skeletons and sensor net w orks for injury risk and health w arnings [ 1 , 2 , 3 ]; graph-based mo dels for long-term stress, sleep, and resilience monitoring [ 4 , 5 ]; dynamic diﬀusion pro cesses on comorbidit y graphs learned from EHR [ 6 ]; geo- metric regression on data-driv en simplicial complexes using Hodge Laplacians [ 7 ]; and large-scale m ultimo dal fusion or mixture-of-exp erts (MoE) arc hitectures that in tegrate EHR, wearables, and imaging [ 9 , 10 , 8 ]. How ever, these adv ances remain largely fragmen ted. Most mo dels treat risk as an unstructured scalar output or as an abstract laten t represen tation in Euclidean space, pro vide limited mechanistic insight in to ho w risk propagates o ver physiological, b ehavioural, and en vironmental structures, and do not explicitly disen tangle the con tributions of heterogeneous mo dalities. This limitation b ecomes evident in scenarios where individuals share similar scalar risk scores but diﬀer profoundly in how risk emerges and propagates through their surrounding netw orks. F or one person, elev ated risk ma y b e driv en by a directed ﬂow from a speciﬁc environmen tal exp osure, suggesting source suppression as a primary in terv ention. F or another, comparable aggregate risk may arise from a self-reinforcing cycle inv olving wearables, b eha viour, and context, where no single source can b e isolated and cycle-breaking in terven tions are required instead. Under scalar or unstructured latent mo dels these cases are eﬀectively indistinguishable, even though they are clinically and mechanistically distinct. GVF is designed to address these gaps. It models health risk as a v ector-v alued ﬁeld ev olving on time-v arying simplicial complexes, uses discrete diﬀeren tial-geometric operators (including Ho dge–Laplacians) to structure and interpret the dynamics of this ﬁeld, and em b eds multimodal inputs in a bundle-structured mixture-of-exp erts arc hitecture that separates modality-speciﬁc from shared eﬀects. Our con tributions are threefold: we pro vide a geometric and top ological substrate for dynamic risk ﬁelds that extends graph-based diﬀusion models to higher-order domains; w e deﬁne a Ho dge-theoretic decomp osition of learned risk ﬂo ws in to comp onen ts with distinct clinical interpretations, naturally linked to diﬀerent classes of input modality; and we prop ose a diﬀeren tiable, bundle-structured expert architecture that join tly learns risk vectors and ﬂo w decomp osition o v er a data-driv en simplicial complex (constructed via threshold-based ﬁltration guided b y p ersisten t homology , not learned end-to-end) while explicitly organising mo dalit y-sp eciﬁc and shared contributions. 2 Ev olution of AI for Multimo dal Health Risk Mo delling This section reviews recen t adv ances in AI for analysing multimodal health data, with an emphasis on (i) wearable-cen tric risk prediction, (ii) dynamic graph mo dels of disease progression, (iii) geometric and top ological learning, and (iv) multimodal fusion and mixture-of-exp erts arc hitectures, fo cusing on how these strands relate to modelling risk dynamics, geometric and top ological structure, and mo dalit y-aw are interpretabilit y . 2.1 Graph- and sequence-based mo dels for wearable-driv en risk A ﬁrst line of w ork uses w earable sensors as the primary data source and combines temporal mo dels with graph arc hitectures. Spatio-temp oral graph neural net works (ST-GNNs) hav e b een applied to athletic injury risk prediction by mapping m ultimo dal wearable signals—IMU, sEMG, PPG—on to a dynamic h uman sk eleton graph, where no des represen t join ts and edges enco de biomec hanical relationships [ 1 ]. The ST-GNN captures coupled spatiotemp oral patterns in join t kinematics, and federated/meta-learning supp orts personalised adaptation [ 1 ]. Similarly , 2 h ybrid LSTM–GNN mo dels o v er ﬂexible sensor netw orks ha ve been prop osed for real-time health w arnings in college sp orts, in tegrating m ultiple ph ysiological time series in a ligh tw eigh t, edge-computing setting [ 2 ]. Injury risk prediction frameworks that transform temp oral training data in to image-lik e enco dings (e.g., Gramian Angular Fields, Marko v T ransition Fields) and then to graphs also exploit parallel graph and temp oral con volutions to capture spatiotemp oral structure, supplemen ted b y atten tion mec hanisms to highligh t imp ortan t athlete in teractions and training-load features [ 3 ]. Bey ond sp orts, personalised graph-attention architectures hav e b een used to trac k long-term c hanges in maternal sleep and stress from smartw atch and mobile- application data, pro ducing individualised abnormality scores o v er time [ 4 ]. F or mental health, participan t-sp eciﬁc graphs where ph ysiological mo dalities—heart activit y , electro dermal activit y , motion—are no des and edges enco de in ter-signal dep endencies hav e b een used with GNNs to classify resilience and stress states, revealing ph ysiological diﬀerences b et ween resilience groups [ 5 ]. GNNs hav e been applied more broadly to EHR data [ 13 ], brain connectivity [ 14 ], and epidemiological net w orks [ 15 ]. Collectiv ely , these w orks demonstrate the v alue of graph-based and sequence-based mo dels for w earable-driven risk and state monitoring [ 1 , 2 , 4 , 5 , 3 ]. How ev er, they generally implemen t dynamics implicitly through recurrent or spatiotemp oral neural architectures rather than via explicit operators on a structured domain, treat risk as a scalar output or unstructured laten t represen tation, and do not exploit higher-order topological structure or diﬀerential-geometric op erators for mechanistic interpretation. Dynamic GNNs [ 16 , 17 , 18 ] handle time-v arying graphs but produce scalar outputs on pairwise (1-sk eleton) graphs, where the curl operator v anishes iden tically and cyclic risk dynamics are in visible b y construction. 2.2 Dynamic graphs for disease progression A complemen tary line of researc h focuses on disease progression as a dynamic pro cess on clinical graphs. Deep Diﬀusion Pro cesses (DDP) model pe rsonalised, time-v arying comorbidit y net w orks from EHR ev en t data, where diseases are nodes and dynamically w eighted edges parameterise the in tensit y functions of a multidimensional p oint pro cess o v er disease-onset times [ 6 ]. Edge w eights are mo dulated o ver time by a neural netw ork that maps eac h patient’s history to dynamic inﬂuence factors, enabling a decomp osition of interaction eﬀects into static pairwise in teractions, temp oral inﬂuence, and dynamic components [ 6 ]. This yields p er-disease, time- v arying intensit y tra jectories that can b e in terpreted as a diﬀusion-lik e risk ﬁeld ov er a learned, dynamic comorbidit y graph. DDP th us pro vides a clear preceden t for framing clinical risk and progression as a dynamic pro cess on a graph with interpretable comp onen ts [ 6 ]. At the same time, it op erates on a disease–disease graph deriv ed from EHR alone, without incorp orating wearable, b eha vioural, or en vironmental mo dalities, and without in voking higher-order topology , Ho dge-theoretic op erators, or explicit geometric decomposition of risk dynamics. 2.3 Geometric and top ological learning in health Geometric and topological machine learning hav e begun to app ear in health-related prediction tasks, most notably through the use of simplicial complexes and Ho dge Laplacians. The TDA paradigm [ 30 , 31 ] and p ersisten t homology [ 32 , 33 , 34 ] extract top ological inv ariants from data. DEC [ 19 , 22 ] and Ho dge theory on graphs [ 20 , 25 ] underpin ﬂow decomp osition; HHD on graphs [ 23 , 24 ] has been applied to gene expression and traﬃc ﬂows [ 35 , 21 ]. Simplicial neural net works [ 36 , 37 ] extend message passing to higher-order complexes but pro duce scalar outputs. Adaptiv e Geometric Regression (A GR) constructs a data-driven simplicial complex (e.g., a k -NN nerv e) ov er high-dimensional feature spaces, equips it with a learned Riemannian structure and densities on p -simplices, and applies Ho dge Laplacians L p to p erform heat diﬀusion on p -co c hains [ 7 ]. Sp ectral ﬁltering of resp onses and a resp onse-aw are geometry that down-w eigh ts 3 edges spanning large resp onse diﬀerences yield diﬀusion-based features for downstream regression or classiﬁcation, with motiv ating examples in microbiome-based prediction of health outcomes suc h as preterm birth [ 7 ]. AGR is, to our kno wledge within this literature set, the most explicit use of simplicial complexes and Hodge op erators in a biomedical con text [ 7 ]. Nonetheless, diﬀusion is emplo yed as an algorithmic regularisation and feature-construction mec hanism rather than as a mo del of time-evolving health risk, and the setting is eﬀectively single-mo dality—microbiome plus co v ariates—without multimodal wearable, b eha vioural, or en vironmental integration, or dynamical decomp osition of risk via Ho dge-theoretic comp onents. 2.4 Multimo dal fusion, foundation mo dels, and mixture-of-exp erts A third strand addresses multimodal fusion across EHR, w earables, imaging, and other data sources, with increasing emphasis on scalability and robustness. Large-scale pip elines that com bine EHR and consumer wearable data for incident disease prediction hav e shown that w earable augmentation can yield consisten t, statistically signiﬁcan t gains o ver EHR-only base- lines across a range of conditions [ 9 ]. These systems typically represen t EHR and w earable streams via time-aggregated features or foundation-mo del embeddings and fuse them through concatenation, weigh ted concatenation, or feature selection [ 9 ]. More recen t w ork prop oses m ultimo dal foundation models that treat EHR ev ents and wearable time series as observ ations of a shared con tinuous-time latent process, using mo dalit y-sp eciﬁc encoders and cross-mo dal pretraining ob jectives to align dense ph ysiological signals with sparse clinical ev ents [ 10 ]. This design impro ves ph ysiological-state estimation, clinical even t forecasting, and time-to-ev ent risk prediction, and addresses challenges such as asynchronous sampling and modality-speciﬁc missingness [ 10 ]. In parallel, mixture-of-exp erts (MoE) architectures ha ve b een introduced for robust mul- timo dal clinical prediction. MoE-Health employs mo dalit y-sp eciﬁc encoders (e.g., for EHR and c hest radiography), learns mo dalit y-sp eciﬁc missingness embeddings, and routes fused represen tations through exp erts sp ecialised to diﬀerent modality subsets via a dynamic gating mec hanism [ 8 ]. This structure impro ves p erformance and robustness under v arying patterns of missing mo dalities [ 8 ]. These m ultimo dal approac hes establish that fusing wearables with EHR improv es predictive p erformance at scale [ 9 , 10 ] and that mo dalit y-structured architectures suc h as MoE can enhance robustness to incomplete data [ 8 ]. How ev er, they generally op erate in unstructured Euclidean laten t spaces, without an explicit geometric or topological domain for risk, without Ho dge/DEC-based op erators, and without a formal sheaf/bundle p ersp ectiv e or iden tiﬁability analysis that would separate and interpret modality-speciﬁc contributions o v er a structured base space. Neural Sheaf Diﬀusion [ 42 ], Bundle Neural Netw orks [ 41 ], and Bundle Morphism Net works [ 43 ] in tro duce geometric structure in to GNNs via sheaf or bundle assignmen ts o ver graph edges, but op erate on pairwise graphs (trivial curl) and use bundle structure for equiv ariance or expressivity rather than mo dalit y iden tiﬁability; none applies Helmholtz-Ho dge decomposition to the learned ﬁeld. Across these four strands, con temp orary AI for m ultimo dal health risk has progressed from spatiotemp oral GNNs on wearable-deriv ed graphs [ 1 , 2 , 4 , 5 , 3 ], through dynamic diﬀusion- lik e pro cesses on comorbidity net works [ 6 ], Ho dge-Laplacian-based geometric regression on simplicial complexes [ 7 ], to large-scale m ultimo dal fusion via foundation mo dels and mixture-of- exp erts [ 9 , 10 , 8 ]. Y et no existing work jointly (i) models m ultimo dal health risk as a v ector-v alued ﬁeld on time-v arying graphs or simplicial complexes, (ii) uses discrete diﬀeren tial-geometric op erators to decomp ose and interpret the dynamics of that ﬁeld, and (iii) embeds these dynamics in a bundle-/sheaf-structured mixture-of-exp erts arc hitecture aimed at modality-lev el separation and iden tiﬁabilit y . This is the gap that GVF is designed to address, as dev elop ed in the sections that follo w. 4 3 Mathematical Substrate 3.1 F rom Pairwise Graph to Simplicial Complex Standard dynamic graph representations of sensor netw orks capture pairwise interactions but fail to represen t fundamentally higher-order relationships: comp ound exp osure even ts inv olving an agent, a spatial cell, and an environmen tal sensor simultaneously; triadic ph ysiological in teractions with non-additiv e eﬀects; or clinical co-morbidity clusters requiring three-en tity mo delling. These are 2-simplic es , not edges, and their inclusion is necessary for the curl operator (Section 3.2 ) to b e non-trivially deﬁned and for the HHD (Theorem 5.4 ) to capture cyclic risk dynamics. Deﬁnition 3.1 (Multimo dal In teraction Simplicial Complex K ( t )) . Let K ( t ) b e an abstract simplicial complex [ 27 , 28 ] ov er the v ertex set V ( t ) = V A ( t ) ∪ V S ∪ V E ( t ) ∪ V X , where: • V A ( t ): mobile agen ts b earing w earable devices; • V S : static spatial cells (geographic tessellation); • V E ( t ): environmen tal sensor no des (air qualit y , temp erature, ligh t, noise); • V X : optional external data nodes (EHR systems, genomic databases, surv eillance feeds). Simplices are t yp ed b y in teraction order: • 1-simplic es { u, v } : pairwise in teractions (pro ximity , co-location, agent-sensor exposure); • 2-simplic es { u, v , w } : simultaneous triadic interactions (comp ound exposure, tripartite co- lo cation); • 3-simplic es : quaternary high-order comp ound even ts. The 1-sk eleton of K ( t ) reduces to a standard dynamic graph, recov ering prior GNN-based approac hes as sp ecial cases. 3.2 Discrete Exterior Calculus Op erators W e equip K ( t ) with the standard DEC structure of chain groups, boundary operators, and their dual cob oundary op erators; see [ 19 , 20 ] for the full algebraic developmen t. Inner pro duct and Ho dge star W e use the standard com binatorial inner pro duct on k -cochains: for α, β ∈ C k ( K ; R ), ⟨ α, β ⟩ k = X σ k ∈K α ( σ k ) β ( σ k ) (1) i.e. the identit y Ho dge star ⋆ = I (un weigh ted DEC), the standard choice in discrete Ho dge theory [ 21 , 20 ]. Under this inner product the gradien t and divergence are mutually ad- join t, the HHD components are L 2 -orthogonal (Theorem 5.4 ), and the op erator norm satisﬁes ∥ curl K ∥ C 1 → C 2 = ∥ B ⊤ 2 ∥ 2 ≤ √ d max where d max is the maxim um num b er of 2-simplices con taining an y edge (relev ant for the L geo b ound in Section 6 ). W e assume the standard structure of unw eighted Discrete Exterior Calculus (DEC) [ 19 , 20 ]. Sp eciﬁcally , w e deﬁne the discrete gradient ∇ K = B ⊤ 1 ⊗ I m , the discrete div ergence div K = B 1 ⊗ I m , and the discrete curl curl K = B ⊤ 2 ⊗ I m , where B 1 and B 2 are the signed no de-edge and edge- face incidence matrices of the complex. The Ho dge Laplacians are ∆ 0 = B 1 B ⊤ 1 ⊗ I m and ∆ 1 = B ⊤ 1 B 1 ⊗ I m + B 2 B ⊤ 2 ⊗ I m (standard sp ectral theory [ 29 , 20 , 25 ]); the iden tity curl K ◦ ∇ K = 0 follo ws from B 1 B 2 = 0 [ 19 ]. 5 R emark 3.2 . On a pure graph (1-skeleton), curl K ≡ 0 structurally: no triangular faces exist on whic h circulation is deﬁned. The curl op erator is non-trivially deﬁned only when K ( t ) con tains 2-simplices. This is the structural—not merely notational—necessity for the cyclic risk detection capabilit y of GVF. 3.3 Constructing Simplicial Complexes from Multimo dal Health Data T ranslating heterogeneous clinical and sensor streams into a mathematically rigorous simplicial complex K ( t ) requires mapping ph ysical and biological interactions to geometric structures. The three simplex orders correspond to qualitatively distinct interaction regimes. 0-simplices (no des) Represen t atomic entities: a patient wearing a smartw atc h ( v i ∈ V A ), a stationary air quality monitor in a hospital ro om ( v e ∈ V E ), or a static Electronic Health Record en try ( v x ∈ V X ). 1-simplices (edges) Capture pairwise, ﬁrst-order interactions. An edge { v i , v j } ma y represent spatial pro ximit y measured via Blueto oth RSSI b etw een tw o patients, while { v i , v e } represen ts a patien t’s direct exp osure to an environmen tal sensor. 2-simplices (triangles) and biological justiﬁcation Standard pairwise graphs fail to capture irreducible triadic interactions, which are p erv asive in health data. Consider the “Gene-En vironment-Phenot ype” in teraction: if patient v i carries a sp eciﬁc genomic vulnerabilit y ( v x ), o ccupies a high-stress ward en vironment ( v e ), and interacts with an infectious vector ( v j ), the resulting risk is often non-additive . The 2-simplex { v i , v e , v x } geometrically enco des this higher-order comp ounding eﬀect. On ly b y closing this triad can the DEC curl op erator (Theorem 3.2 ) non-trivially compute circulation, eﬀectiv ely mo delling a self-reinforcing biological or environmen tal feedbac k lo op that cannot be deduced by observing the pairwise edges in isolation. Practical construction from raw streams T ranslating raw, asynchronous sensor streams in to a mathematically rigorous complex K ( t ) requires a formal pip eline comprising temp oral windo wing, heterogeneous edge formation, and m ultimo dal simplex inclusion. (1) T emp oral windo wing. Raw data S ( t ) are not ev aluated instan taneously but aggregated o ver a discrete sliding window of duration ∆ t (e.g. ∆ t = 5 min utes for ph ysiological and en vironmental streams, or 24 hours for b eha vioural streams). The complex K ( t ) is static within the in terv al [ t, t + ∆ t ) and up dates discon tin uously , ensuring the DEC op erators remain w ell-deﬁned across stable ep ochs. (2) Heterogeneous edge formation. Because m ultimo dal no des do not inhabit a uniform metric space, standard spatial distance functions are insuﬃcien t. W e deﬁne modality-speciﬁc adjacency thresholds τ : • A gent–agent e dges: { v A i , v A j } is formed if spatial pro ximity (median Blueto oth RSSI ov er ∆ t ) exceeds τ prox , or if ph ysiological synchron y (e.g. dynamic time-warping distance of HR V signals) falls below τ sync . • A gent–envir onment e dges: { v A i , v E e } is formed if agent v i sp ends at least τ dwell min utes within the activ e geographical radius R e of sensor v e . • A gent–external e dges: { v A i , v X x } is a static edge formed when clinical conditions (e.g. a sp eciﬁc genomic mark er presen t in the EHR) are met. (3) Multimo dal simplex inclusion criteria. A standard Vietoris-Rips complex relies on purely pairwise distance metrics, whic h fails for heterogeneous topologies. Instead, w e deﬁne 6 a rule-b ase d multimo dal ﬁltr ation . A 2-simplex σ 2 = { v i , v j , v k } is included in K ( t ) if and only if all three b ounding 1-simplices are activ e and at least t wo distinct no de types are present in the triad. F or example, the 2-simplex { v A i , v A j , v E e } (t wo patients and one en vironmental sensor) forms when v i and v j are physically pro ximate and b oth are simultaneously exposed to the same en vironmental hazard v e . This strict inclusion criterion prev en ts com binatorial explosion of trivial simplices (e.g. three patients near each other without a shared con textual risk) and ensures the curl operator explicitly captures cross-mo dalit y cyclic dynamics. The resulting K ( t ) is up dated at each observ ation windo w, yielding a time-v arying complex whose topology ( β 1 ( K ( t )), the ﬁrst Betti num ber [ 32 , 33 , 34 ]) directly informs the harmonic comp onen t of GVF. The construction pro ceeds in t w o phases: Phase 1 forms the 1-skeleton via modality-speciﬁc adjacency thresholds ( τ prox , τ sync , τ dwell ); Phase 2 applies m ultimo dal ﬁltration to close triangles only when at least t wo distinct node types are present. F ull algorithmic pseudo code is provided in Section A . P arameter selection and top ological v alidation A critical op erational challenge is selecting the optimal pro ximit y thresholds (analogous to the ﬁltration parameter ε in standard Vietoris- Rips complexes [ 34 ]). Rather than relying on arbitrary heuristics, GVF selects optimal thresholds via p ersistent homolo gy [ 32 , 33 ]. By sweeping eac h τ across a con tin uous range, we generate p ersistence barco des that trac k the birth and death of topological features (connected components β 0 and indep enden t cycles β 1 ). The op erational threshold is c hosen within the most stable plateau of the persistence diagram—the range where the Betti n um b ers β k remain inv ariant, ensuring K ( t ) captures gen uine structural in v arian ts rather than transien t noise. Computational complexit y of construction Phase 1 requires pairwise ev aluations b ounded b y O ( | V | 2 ), reduced to O ( | V | log | V | ) via spatial indexing (KD-trees). Phase 2 runs in O ( |K 1 | · d max ), where d max is the maxim um no de degree in the sparse 1-sk eleton, k eeping temp oral up dates tractable for contin uous monitoring. 4 GVF F ramew ork Arc hitecture Deﬁnition 4.1 (Risk V ector Bundle ε ( K )) . The risk vector bundle is the direct sum o v er N mod input mo dalities: ε ( K ) = ε (1) ⊕ ε (2) ⊕ · · · ⊕ ε ( N mod ) (2) Eac h summand ε ( n ) is a trivial v ector bundle with ﬁbre R m n ; the total dimension is m = P n m n . In the canonical four-modality conﬁguration: ε phys (ph ysiological signals), ε beh (b eha vioural data), ε env (en vironmental measurements), and ε ext (optional external sources: EHR, genomic, surv eillance). The direct sum structure guaran tees that no mo dalit y’s risk representation bleeds in to another’s arc hitecturally . GVF op erator as bundle morphism Deﬁnition 4.2 (GVF Op erator F θ ) . The GVF op erator is a parameterised map: F θ : Γ  ε in ( t )  × Ξ( K ( t )) → Γ( ε ( K )) (3) where Γ( ε in ( t )) is the space of m ultimo dal input state sections, Ξ( K ( t )) is the structural represen tation space of K ( t ), and Γ( ε ( K )) is the output risk section space. F θ is implemen ted as a Mixture-of-Exp erts (MoE) architecture: r i ( t ) = N mod X n =1 g ( n ) ϕ  x i ( t )  F ( n ) θ  K ( t ) , x i ( t )  (4) 7 where eac h exp ert F ( n ) θ outputs exclusiv ely in ε ( n ) , and g ϕ satisﬁes P n g ( n ) ϕ = 1, g ( n ) ϕ ≥ 0. Exp ert arc hitectures T o satisfy the Univ ersal Appro ximation guaran tee (Theorem 8.1 ) and to ensure the curl op erator remains non-trivially deﬁned on higher-order interactions (Theorem 3.2 ), the exp erts cannot b e instantiated as standard pairwise Graph Neural Netw orks op erating on the 1-sk eleton. Instead, each exp ert F ( n ) θ m ust b e a v ariant of a Message P assing Simplicial Netw ork (MPSN) [ 36 ], adapted to the temp oral inductiv e biases of its speciﬁc mo dalit y . T able 1 deﬁnes the theoretically aligned arc hitecture for each bundle summand. The speciﬁc architectures listed are pr op ose d instantiations of the MPSN template, named here for clarit y; their detailed sp eciﬁcation and empirical ev aluation are part of future implementation work. T able 1: Exp ert decomposition of F θ aligned with simplicial UA T requiremen ts. Exp ert Simplicial Arc hitecture Primary Inputs T emp oral Scale Ph ysio-Exp ert Dynamic Simplicial GCN (DS-GCN) HR, HR V, SpO 2 , ED A, skin temp Min utes Beha v-Exp ert Simplicial Conv olutional RNN (S-CRNN) GPS, sleep staging, activity Hours–da ys En v-Exp ert T emp oral Simplicial A tten- tion (T-SA T) PM 2 . 5 , temp, noise, UV Min utes Ext-Exp ert Higher-order / Simplicial T ransformer EHR, PRS, surv eillance (opt.) Hours–static R emark 4.3 (Graceful Degradation under Missing Mo dalities) . When a mo dalit y n is unav ailable ( x ( n ) → 0 ), the desired b eha viour is g ( n ) ϕ ( x ) → 0, so that the output risk v ector degrades gracefully to the subspace spanned by active modalities. This is a design ob jectiv e enforced via training , not a theorem: mo dalit y-drop out augmen tation (randomly zeroing eac h mo dalit y’s input block with probability p drop = 0 . 2) trains the gating net work g ϕ to assign near-zero weigh t to absent mo dalities. The architectural direct-sum structure ensures that even if gating is imp erfect, the missing exp ert con tributes only through its summand ε ( n ) , and the remaining summands carry a v alid, lo wer-dimensional risk representation. Empirical v eriﬁcation of this prop ert y is part of the ablation proto col (Section 9 ). 5 Helmholtz–Ho dge Decomp osition The Helmholtz–Hodge decomp osition is the mechanism through which GVF pro vides in ter- pretabilit y inheren tly —not as a post-ho c explanation metho d, but as a fundamen tal prop ert y of the learned ﬁeld’s diﬀerential structure. 5.1 The Risk Flo w Field R emark 5.1 (Why HHD requires a separate 1-cochain) . Applying HHD directly to ∇ K r would b e trivial: ∇ K r ∈ im ( ∇ K ) is exact b y construction, so curl K ( ∇ K r ) = 0 (b oundary-of-b oundary iden tity), and curl and harmonic comp onen ts v anish identically . GVF therefore introduces a separate learned 1-co c hain F ∈ C 1 ( K ; R m ) on the edges; HHD is applied to F , which is not constrained to be exact. Deﬁnition 5.2 (Risk Flo w Field F ) . Giv en the no de-lev el risk section r i ( t ) ∈ R m and edge features e ij ( t ) ∈ R p (e.g. RSSI, co-lo cation duration, sensor correlation), the risk ﬂow on orien ted edge ( i, j ) ∈ K 1 ( t ) is: F ij ( t ) = Ψ ω  r i ( t ) , r j ( t ) , e ij ( t )  ∈ R m (5) 8 T o guaran tee the an tisymmetry required for a v alid 1-cochain, Ψ ω is implemen ted via an underlying MLP parameterised b y ω with an explicit antisymmetrisation step: Ψ ω  r i , r j , e ij  = 1 2 h MLP ω  r i , r j , e ij  − MLP ω  r j , r i , − e ij  i (6) Note on the sign con ven tion: e ij is treated as an oriente d feature vector asso ciated with the ordered pair ( i, j ), so that e j i : = − e ij b y con ven tion. F or intrinsically symmetric features (e.g. spatial distance, co-lo cation duration), the negation is a purely algebraic device that ensures Ψ ω satisﬁes the antisymmetry constrain t; the MLP MLP ω learns to pro cess symmetric inputs iden tically regardless of sign. F or intrinsically directional features (e.g. relativ e signal strength, airﬂo w direction), the sign carries ph ysical meaning. This construction satisﬁes: (i) An tisymmetry by design: Ψ ω ( r i , r j , e ij ) = − Ψ ω ( r j , r i , − e ij ), ensuring orien tation con- sistency as a 1-cochain. (ii) Non-exactness b y design: The edge features e ij ( t ) parameterise the “conductance” of the risk transmission channel. In an epidemiological pro ximity netw ork, e ij migh t enco de con tact duration and indoor v entilation quality: when e ij → 0 (agen ts separated b y ph ysical barriers), F ij is atten uated regardless of the magnitudes ∥ r i ∥ , ∥ r j ∥ . Crucially , this non-uniform, e dge-dep endent mo dulation ensures F is not trivially exact: the circulation around triangle ( i, j, l ) v anishes only when ϕ ij = ϕ j l = ϕ li (uniform w eighting), which fails generically for learned Ψ ω . (iii) Gradien t special case: when e ij is discarded and MLP ω ( a , b ) = b − a , F = ∇ K r is the exact discrete gradient. This is the degenerate case curl K F = 0; L geo (Equation ( 11 )) p enalises conv ergence to this collapse. R emark 5.3 (Role of 2-simplices in the learning pip eline) . It is important to clarify wher e the 2-simplices of K ( t ) enter the computational pip eline. The exp ert netw orks F ( n ) θ (T able 1 ) and the ﬂo w constructor Ψ ω (Equation ( 6 )) operate on edges (the 1-sk eleton): they pro duce node-level risk vectors r i and edge-level ﬂo ws F ij without explicit message passing through 2-simplices. The 2-simplices en ter at tw o subsequent stages: (i) the geometric loss L geo (Equation ( 11 )) computes curl K F via the edge-face incidence matrix B 2 , whic h exists only when K ( t ) con tains 2-simplices—this term shap es the learned ﬂow during training by p enalising degenerate exact ﬁelds; (ii) the Helmholtz–Hodge decomp osition (Theorem 5.4 ) uses B 2 to separate gradien t, curl, and harmonic comp onen ts at inference time. Th us, 2-simplices inﬂuence the learned represen tation indir e ctly through the training ob jective and the p ost-training decomposition, rather than through the expert forward pass. Upgrading the exp erts to full simplicial message- passing net works [ 36 ]—whic h would propagate information directly through 2-simplices during the forw ard pass—is an arc hitectural impro vemen t discussed in Section 10 . Theorem 5.4 (Discrete HHD of the Risk Flo w Field) . F or any risk ﬂow F ∈ C 1 ( K ( t ); R m ) , ther e exists a unique ortho gonal de c omp osition: F = ∇ K φ + curl ∗ K ψ + h (7) wher e: • φ ∈ C 0 ( K ; R m ) solves ∆ 0 φ = div K ( F ) (gr adient/p otential c omp onent); • ψ ∈ C 2 ( K ; R m ) solves ∆ 2 ψ = curl K ( F ) (curl/solenoidal c omp onent); • h ∈ C 1 ( K ; R m ) is the harmonic r esidual satisfying ∆ 1 h = 0 , div K h = 0 , curl K h = 0 . The thr e e c omp onents ar e mutual ly L 2 -ortho gonal with r esp e ct to the inner pr o duct induc e d by the Ho dge star on K ( t ) . The curl and harmonic c omp onents ar e generic al ly non-zer o b e c ause F is not c onstr aine d to lie in im( ∇ K ) . 9 Pr o of. F or eac h co ordinate channel k ∈ { 1 , . . . , m } , denote by F ( k ) ∈ C 1 ( K ; R ) the k -th scalar comp onen t of F . The standard discrete Ho dge decomp osition (see [ 21 ], Theorem 1) gives a unique orthogonal decomp osition F ( k ) = ∇ K φ ( k ) + curl ∗ K ψ ( k ) + h ( k ) in eac h channel independently , where orthogonalit y is with resp ect to the combinatorial inner pro duct ( 1 ) . Stac king o v er k yields the v ector-v alued decomp osition F = ∇ K φ + curl ∗ K ψ + h . The L 2 -orthogonalit y of the three v ector-v alued comp onen ts follo ws b ecause the inner pro duct on C 1 ( K ; R m ) decomp oses as ⟨ A , B ⟩ = P m k =1 ⟨ A ( k ) , B ( k ) ⟩ 1 , so cross-terms v anish c hannel-by-c hannel. 5.2 Mec hanistic In terpretation and Monitoring Scores T able 2 summarises the mec hanistic in terpretation of eac h HHD comp onen t and its corresponding in terven tion target. This interpretabilit y is inheren t to the ﬁeld structure , not a post-ho c explanation: it arises directly from the orthogonal decomp osition guaran teed by Theorem 5.4 . T able 2: HHD comp onen ts with mec hanistic interpretation and interv ention strategy . Comp onen t Prop ert y Mec hanistic Meaning In terven tion ∇ K φ (Gradi- en t) curl K ( ∇ K φ ) = 0 Risk ﬂo ws directionally along a p oten tial gradien t. Iden tiﬁes sources (emitting no des) and sinks (accumu- lating no des). Reduce source intensit y (e.g., lo w er en vironmen tal exp osure, improv e physi- ological recov ery). curl ∗ K ψ (Curl) div K ( curl ∗ K ψ ) = 0 Risk circulates in self- sustaining closed lo ops. Signature of cyclic coupling (e.g., sleep-stress-recov ery lo ops). Break the cycle at its w eakest link (sleep h y- giene, pharmacological disruption). h (Harmonic) ∆ 1 h = 0 Risk lo c k ed by global net- w ork top ology . Dimen- sion = β 1 ( K ( t )). Not re- ducible by lo cal interv en- tions. Structural in terven tion: mo dify net w ork top ology . Mo dalit y-comp onen t corresp ondence The three HHD comp onen ts do not merely decom- p ose the risk ﬁeld mathematically: they corresp ond to structur al ly distinct classes of input mo dality , each b est modelled by a diﬀerent type of physical signal. This corresp ondence, summarised in T able 3 , is not an empirical observ ation but a design hyp othesis grounded in the mathematical roles that clinical/genomic, wearable, and environmen tal data pla y in the bundle-structured GVF arc hitecture. This tripartite corresp ondence has a direct clinical translation: gr adient-dominant risk calls for clinical or pharmacological in terven tion targeting the constitutiv e source; curl-dominant risk calls for b eha vioural or physiological cycle-breaking strategies (e.g., sleep-hygiene proto cols, activit y scheduling); harmonic-dominant risk calls for structural or en vironmen tal redesign (e.g., mo difying spatial exp osure top ology , installing en vironmen tal barriers). The corresp ondence is formally grounded in the direct-sum bundle structure of Section 4 : modality n maps exclusiv ely in to ﬁbre ε ( n ) ( K ), and the MoE gating w eights g ( n ) ϕ ( x i ) in Equation ( 4 ) directly con trol the relativ e energy con tributed b y eac h ﬁbre to each HHD component. The relationship betw een ﬁbre energy and comp onen t energy is made explicit via the orthogonality of the decomp osition (Theorem 5.4 ) and the mo dality identiﬁabilit y guaran tee (Theorem 7.1 ). Deriv ed monitoring scores Two scalar summary statistics are deriv ed directly from the decomp osition, replacing a single alarm scalar with a t w o-dimensional, complemen tary warning 10 T able 3: Natural corresp ondence b et ween HHD comp onen ts and mo dalit y classes. Each mo dalit y class con tributes most directly to one geometric mode of risk; the con tribution is not exclusive but reﬂects the structural role of eac h data type. T o the b est of our kno wledge, no prior framew ork establishes this correspondence (cf. [ 6 , 7 , 8 ]). HHD Comp o- nen t Mo dalit y Class Structural Role Wh y this corresp on- dence ∇ K φ (Gradien t / ex- act) Clinical & genomic ( X ext ) Scalar p oten tial φ enco des baseline, constitutive risk: ﬁxed biological or clinical state driving a directional outﬂo w. Clinical scores and ge- nomic risk factors de- ﬁne a baseline p o- ten tial landscap e that c hanges slo wly and go verns the direction of risk ﬂow. curl ∗ K ψ (Curl / co- exact) W earable & b e- ha vioural ( X phys , X beh ) Stream function ψ en- co des dynamic, osc illatory coupling: ph ysiological cycles (sleep-wak e, HR V rh ythm, activity-rest) gen- erate closed-lo op risk cir- culation. W earable time series directly observ e cycli- cal physiological dy- namics; their high tem- p oral resolution makes them the natural car- rier of the curl compo- nen t. h (Harmonic) En vironmental ( X env ) T op ology-lo c k ed risk deter- mined by the global struc- ture of K ( t ), which is built from spatial co-presence, dw ell times, and sensor ad- jacency . The complex topol- ogy itself—driv en b y en vironmental sensor placemen t, spatial pro ximity , and shared exp osure zones—determines the harmonic dimension β 1 ( K ( t )) and hence the harmonic risk subspace. 11 signal. Deﬁnition 5.5 (Disease Progression Score and Cyclic Risk Index) . Let F = ∇ K φ + curl ∗ K ψ + h b e the HHD of the risk ﬂow ﬁeld (Theorem 5.4 ). Let u ( n ) ∈ R m n b e a learnable unit risk axis for mo dalit y n , trained jointly with θ and ω (initialised to 1 / √ m n ). F or agen t i at time t , mo dalit y weigh ts w n > 0, P n w n = 1, and letting T i = { τ ∈ K 2 : i ∈ ∂ τ } b e the set of 2-simplices inciden t to agen t i : DPS i ( t ) = X n w n  (div K F ( n ) ) i ( t ) , u ( n )  (8) CRI i ( t ) = 1 max( |T i | , 1) X τ ∈T i   (curl K F ) τ ( t )   (9) DPS (signed, basis-in v arian t). DPS i is a signed scalar: the inner pro duct of the divergence v ector with the learnable risk axis u ( n ) , in v arian t to rotations within ε ( n ) that ﬁx u ( n ) . DPS i > 0: net outﬂo w (source; deteriorating). DPS i < 0: net inﬂow (sink; absorbing ambien t risk). CRI (non-negative, normalised). CRI i ≥ 0 is the me an curl magnitude ov er inciden t 2-simplices, normalised b y neigh b ourhoo d size |T i | to prev ent inﬂation in dense subgraphs. By the discrete Stokes theorem [ 19 ], a non-zero curl K F o v er a neigh b ourhoo d implies non-zero b oundary circulation: risk genuinely cir culates around closed paths. CRI i > 0 therefore indicates the pr esenc e and av erage in tensity of cyclic risk dynamics in i ’s neighbourho o d; it is non-signed (cycles ha ve no orientation in this context) and its magnitude is not in terpreted as clinical sev erity (see Section 6 on L geo ). T ogether, ( DPS i , CRI i ) constitute a t w o-dimensional, complementary warning signal: DPS i captures directional ﬂow along the risk axis; CRI i captures self-sustaining cyclic dynamics indep enden t of direction. W ork ed example: harmonic comp onen t and structural interv en tion The harmonic residual h satisﬁes ∆ 1 h = 0 and its dimension strictly equals β 1 ( K ( t ))—the num b er of in- dep enden t top ological “holes” in the complex, computable via p ersisten t homology [ 32 , 33 ]. Consider a hospital ward where β 1 ( K ) = 2. These tw o topological cycles migh t corresp ond to shared, unsegmen ted HV A C pathw ays linking subsets of rooms without central ﬁltration. If the framew ork iden tiﬁes a high magnitude of risk ﬂo w trapp ed in the harmonic comp onen t h around these lo ops, it implies top olo gy-lo cke d risk . Mec hanistically , this means that treating P atien t X individually (a no de-lev el interv en tion) or isolating a single pairwise con tact (an edge-level in terven tion) will fail: the ambien t risk con tinually bypasses the local in terven tion via the top ological hole. The explicit mapping is that harmonic risk c ommands structur al intervention : the hospital administration m ust “ﬁll the hole” b y installing a ﬁltration barrier (adding a 2-simplex to K ) or sev ering the HV AC lo op en tirely (altering the global top ology of the complex). This is the operational consequence of the iden tit y dim k er (∆ 1 ) = β 1 ( K ( t )): the dimension of the harmonic risk space is a direct, computable readout of ho w man y indep enden t structural in terv entions are needed. 6 T raining Ob jectiv e 6.1 The Sup ervision Challenge A fundamen tal question for an y v ector-ﬁeld health monitoring framew ork is: how do you sup ervise a ve ctor-value d output when al l available gr ound-truth lab els ar e sc alar (str ess/b aseline class lab els, EMA sc or es)? GVF addresses this through a multi-task loss that combines scalar-sup ervised classiﬁcation with geometric regularisation terms that enforce non-trivial structure on the learned ﬂo w ﬁeld F . The geometric terms do not require vector ground truth: they are self-supervised ob jectives derived from the ﬁeld’s diﬀerential structure. 12 6.2 Multi-T ask Loss F unction The total training ob jectiv e is: L ( θ , ω ) = L cls ( ˆ y , y ) | {z } scalar sup ervision + λ 1 L geo ( F ) | {z } geometric regularisation + λ 2 L orth ( θ ) | {z } modality orthogonality (10) Classiﬁcation term L cls is the standard multi-class cross-entrop y loss applied directly to the linear read-out of the risk vector r i ( t ) via a head W out ∈ R C × m (where C is the n um b er of outcome classes). The risk v ector r i is used directly (not its magnitude ∥ r i ∥ ): this allo ws the read-out head to exploit b oth magnitude and direction. F or con tinuous regression targets, L cls is replaced b y mean-squared error on a scalar pro jection w ⊤ out r i . Geometric regularisation term L geo discourages degenerate ﬂo w ﬁelds that collapse to purely exact cochains (trivial HHD). It is deﬁned as a ne gatively signe d lo g-r atio : L geo ( F ) = − log  1 + ∥ curl K F ∥ 2 ∥ F ∥ 2 + ε  (11) L geo ≤ 0 since the log argumen t is ≥ 1; minimising the total loss therefore rewards larger curl-to- ﬁeld ratios. T o preven t runaw a y incentiv es when d max is large, ρ = ∥ curl K F ∥ 2 / ( ∥ F ∥ 2 + ε ) is clipp ed to [0 , 1], b ounding L geo ∈ [ − log 2 , 0]. L geo is an anti-c ol lapse r e gulariser : it preven ts degenerate exact-ﬁeld solutions without interpreting curl magnitude as clinical severit y; λ 1 ∈ { 0 . 01 , 0 . 1 , 0 . 5 } is selected b y grid searc h (see Section B ). Mo dalit y orthogonality term L orth enforces that exp ert outputs remain in their resp ectiv e bundle summands: L orth = X n  = m   π ( n ) ( F ( m ) θ ( x ))   2 (12) where π ( n ) pro jects onto the n -th summand. Combined with architectural enforcement (modality- blo c k ed input matrices), this term closes the gap b et w een the arc hitectural inten t and numerical precision during training. V ector direction and the identiﬁabilit y c hallenge Even with L cls , the dir e ction of r i is not pinned b y a scalar lab el—a rotation of r i lea ving ∥ r i ∥ unc hanged would not c hange the loss. GVF resolv es this through t wo mechanisms: (1) Architectural modality separation : since eac h exp ert writes exclusiv ely into its summand ε ( n ) , the direction is determined by the relativ e exp ert activ ations, not a free rotational degree of freedom; (2) Geometric regularisation : L geo further constrains the solution manifold b y rewarding curl-inducing directions. The residual am biguity—permutations within each summand ε ( n ) —is c haracterised formally in Section 7 . 7 Iden tiﬁabilit y Guaran tees Theorem 7.1 (Suﬃcien t Conditions for Mo dalit y Identiﬁabilit y) . Under the fol lowing strong suﬃcient c onditions: (C1) Mo dality ortho gonalization : the r aw multimo dal inputs ar e pr e-c onditione d via a blo ck- ortho gonal pr oje ction layer (e.g. a le arne d whitening tr ansformation), ensuring that the tr ansforme d input subsp ac es { ˜ x ( n ) i } ar e line arly indep endent structurally guaran teed prior to exp ert r outing. (C2) Exp ert sp e cialisation : e ach F ( n ) θ has a non-trivial nul l sp ac e on x ( m ) for m  = n (enfor c e d ar chite ctur al ly). 13 (C3) Population c over age : the tr aining dataset c ontains agents with zer o-value d inputs for e ach pr op er subset of mo dalities (achieve d via mo dality-dr op out at r ate p drop = 0 . 2 ). the map θ 7→ ( F ( n ) θ ) n is inje ctive up to within-summand p ermutation. Pr o of sketch. The argumen t dra ws on the iden tiﬁabilit y framework of [ 39 ] (Theorem 1, iV AE), adapted to the bundle-structured MoE setting. W e note that iV AE is formulated for gener ative mo dels with latent v ariables and requires the conditional p ( z | u ) to b elong to a conditionally factorial exp onential family . GVF is a discriminative architecture, so the corresp ondence is structural rather than a direct application: the mo dalit y-present/absen t indicator vector (generated b y mo dalit y-drop out under (C3)) plays the role analogous to the auxiliary v ariable u , and the direct-sum arc hitecture (C2) enforces the factorial conditioning structure b y restricting eac h exp ert to its o wn input block. Under these structural conditions, the iden tiﬁabilit y argumen t of [ 39 ] yields injectivity of θ 7→ ( F ( n ) θ ) n up to within-summand aﬃne reparameterisation; the reduction from aﬃne to p ermutation ambiguit y is detailed in Section E . A fully rigorous pro of in the discriminative setting—verifying that the exponential-family suﬃcien t statistic condition of [ 39 ] Theorem 1 is satisﬁed b y the MoE exp ert outputs under (C1)– (C3)—remains an op en problem. The presen t result should therefore b e read as a c onditional guar ante e : it holds to the exten t that the structural analogy with iV AE is exact. R emark 7.2 (Enforcing Iden tiﬁabilit y in Correlated Clinical Data) . Raw physiological and en vironmental streams are nativ ely en tangled; for instance, Heart Rate (HR) and Heart Rate V ariability (HR V) exhibit strong collinearit y . Relying purely on the data distribution to satisfy linear independence would trivially violate the identiﬁabilit y requiremen ts. The GVF framew ork resolv es this b y strictly separating the assumptions: (C2) and (C3) are enforced via the direct- sum bundle arc hitecture and mo dalit y-drop out during training, resp ectiv ely . T o rigorously satisfy (C1), we assume the in tegration of a blo c k-orthogonalization mo dule—suc h as zero- phase component analysis (ZCA) or a learned whitening lay er—at the input of the MoE. This arc hitectural constraint transforms (C1) from a fragile distributional assumption in to a hard mathematical guaran tee, enabling the application of nonlinear ICA iden tiﬁability results [ 39 ] ev en to highly correlated m ultimo dal health data. 7.1 Robustness Analysis of Iden tiﬁabilit y Guaran tees Theorem 7.1 relies on the hard condition (C1) of blo c k-orthogonalization. In practice, p erfect whitening of high-dimensional, noisy health streams is numerically challenging, and the estimated co v ariance of the transformed subspaces ˜ x may exhibit a residual non-diagonal comp onen t: Σ res = E  ˜ X ˜ X ⊤  − I  = 0 . (13) W e analyse the degradation of iden tiﬁability when (C1) is partially violated. Drawing on robustness bounds for nonlinear ICA [ 39 ], if the residual cross-correlation b et ween mo dalit y subspaces is b ounded b y ∥ Σ res ∥ F ≤ δ for some small δ > 0, the exact identiﬁabilit y up to within-summand p erm utation is relaxed to δ -identiﬁability . Prop osition 7.3 ( δ -Iden tiﬁability under Imp erfect Whitening) . Under c onditions (C2) and (C3) of The or em 7.1 , and under imp erfe ct whitening with ∥ Σ res ∥ F ≤ δ , the le arne d exp erts ˆ F θ deviate fr om the true disjoint mo dality gener ators F ∗ by:   ˆ F ( n ) θ − F ∗ ( n )   L 2 = O ( δ ) ∀ n. (14) Pr o of sketch. Under exact whitening ( δ = 0), Theorem 7.1 gives exact identiﬁabilit y . When ∥ Σ res ∥ F = δ > 0, the blo c k-orthogonal input to eac h exp ert F ( n ) θ is perturb ed by an additiv e cross- mo dalit y leak age term: denoting the true whitened input by ˜ x ( n ) and the imp erfectly whitened 14 input by ˆ x ( n ) , we hav e ∥ ˆ x ( n ) − ˜ x ( n ) ∥ ≤ ∥ Σ res ∥ 2 ∥ x ∥ ≤ δ ∥ x ∥ . F or a ﬁxe d exp ert F ( n ) θ (i.e., holding the trained parameters constant), sp ectral normalisation ensures L n -Lipsc hitz con tinuit y with L n ≤ Q ℓ ∥ W ( n ) ℓ ∥ 2 ≤ 1, so the p oin t wise forw ard-pass p erturbation satisﬁes ∥ F ( n ) θ ( ˆ x ) − F ( n ) θ ( ˜ x ) ∥ ≤ L n δ ∥ x ∥ . The L 2 b ound follows by integrating o ver the training distribution p ( x ) (non-degenerate b y (C3)): ∥ F ( n ) θ ◦ ˆ W − F ( n ) θ ◦ ˜ W ∥ L 2 ≤ L n δ ( E [ ∥ x ∥ 2 ]) 1 / 2 = O ( δ ). Ca v eats. This b ound applies to the forward-pass output of a ﬁxe d exp ert under input p erturbation. The full claim—that exp erts tr aine d under imp erfect whitening conv erge to solutions O ( δ )-close to those trained under p erfect whitening—additionally requires algorithmic stabilit y of the training pro cedure (e.g., uniform stabilit y of SGD), whic h we assume but do not pro v e. F urthermore, the MoE gating weigh ts g ( n ) ϕ ( x ) are also p erturb ed by the whitening error; under Lipsc hitz gating (ensured b y bounded softmax temp erature), this in tro duces a m ultiplicative O ( δ ) correction to each exp ert’s con tribution that do es not change the ov erall order of the bound. This gr ac eful de gr adation prop ert y ensures that slight correlations leaking through the whitening la yer do not catastrophically blend the physiological and en vironmen tal bundles, but merely in tro duce b ounded crosstalk proportional to the empirical whitening error δ . In practice, δ can be monitored at deplo yment time as ∥ ˆ Σ res ∥ F on a held-out calibration set, pro viding an op erational certiﬁcate of identiﬁabilit y qualit y . 8 Theoretical Guaran tees 8.1 Univ ersal Appro ximation Theorem 8.1 (Univ ersal Appro ximation of GVF) . F or any c ontinuous tar get se ction r ∗ ∈ Γ( ε ( K )) and any ε > 0 , ther e exist p ar ameters θ such that sup i ∥ F θ ( K , x i ) − r ∗ i ∥ < ε , pr ovide d e ach exp ert F ( n ) θ is a simplicial message-p assing network satisfying the c onditions of [ 36 , 26 ]. Pr o of sketch. Eac h exp ert is dense in the space of contin uous sections o v er ε ( n ) [ 36 ]; the direct- sum isometry of ε ( K ) preserv es this densit y co ordinate-wise. Standard GNNs on the 1-sk eleton cannot b e substituted: they force curl K ≡ 0 (Theorem 3.2 ), destroying cyclic risk detection and in v alidating the theorem’s top ological premises. F ull pro of in Section D . R emark 8.2 (Gap betw een UA T assumptions and prop osed exp erts) . Theorem 8.1 requires eac h exp ert to b e a ful l simplicial message-passing net w ork op erating on k -simplices for k ≥ 2. The practical exp ert architectures proposed in T able 1 op erate primarily on the 1-sk eleton and are therefore not co v ered b y the theorem as stated. The UA T establishes that the GVF ar chite ctur e class is suﬃciently expressive; whether the speciﬁc instantiations in T able 1 inherit this expressiveness is an empirical question contingen t on upgrading them to full simplicial arc hitectures [ 36 ], as noted in Section 10 (“Exp ert arc hitecture alignmen t”). Un til this upgrade is implemen ted, the UA T should be read as a guaran tee on the framework’s capacity ceiling, not on the proposed exp ert conﬁgurations. 8.2 Distributional Stabilit y R emark 8.3 (Stabilit y and shift detection) . Under sp ectral normalisation of all GNN aggregation matrices ( F θ L -Lipsc hitz in graph structure), the GVF output is bounded under Gromo v- Hausdorﬀ p erturbation of the simplicial complex (deriv ation outline in Section F ). This result is a consistency guaran tee under idealised assumptions , not a tight empirical certiﬁcate, and should not be treated as a cen tral claim. Its primary practical v alue is motiv ating the sp ectral shift proxy d spec ( K , K ′ ) = ∥ λ ( K ) − λ ( K ′ ) ∥ (sorted Hodge Laplacian eigenv alues) as a computable op erational signal: a small d spec triggers lo cal ﬁne-tuning of the MoE gating; a large one triggers full retraining. 15 9 Arc hitectural P ositioning and F ormal Comparison This section establishes the theoretical capabilities of GVF relativ e to existing frameworks through a structured arc hitectural comparison. The analysis is not empirical: all capability claims are consequences of the mathematical construction developed in Sections 3 to 5 . 9.1 Capabilit y Analysis Against Represen tativ e Baselines A central claim of GVF is that represen ting health risk as a structured v ector ﬁeld ov er a simplicial complex pro vides qualitatively diﬀer ent capabilities from scalar-output mo dels—capabilities that are structural consequences of the arc hitecture and not con tingen t on empirical p erformance. W e make this precise b y comparing GVF against four represen tativ e baseline classes—Scalar Dynamic GNN, V ector MoE without bundle structure, clinical scores (SOF A, NEWS), and Scalar LSTM/T ransformer— across capability , complexit y , in terpretability , and theoretical limit dimensions (T able 4 ). T able 4: Extended comparison of GVF against represen tativ e baseline classes. ✓ = supp orted b y construction; ◦ = partial or heuristic; × = not supported. Complexit y: | V | no des, |K 1 | edges, |K 2 | 2-simplices, m output dimension, d scalar hidden size. All comparisons are arc hitectural, not empirical. F eature GVF (Proposed) Scalar Dynamic GNN V ector MoE (no bundle) Clinical Scores (SOF A, NEWS) Scalar LSTM/T ransf. Time complexity O ( | V | + |K 1 | + |K 2 | ) O ( | V | + |K 1 | ) O ( | V |· m ) O ( | V | ) O ( | V |· d ) Space complexity O ( |K 2 |· m ) O ( |K 1 |· d ) O ( | V |· m ) O ( | V | ) O ( | V |· d ) V ector-v alued risk output ✓ × ✓ × × Gradient comp onen t ✓ × ◦ × × Curl component (cyclic risk) ✓ × × × × Harmonic component (top ology) ✓ × × × × Modality attribution by construction ✓ × × × × Modality identiﬁabilit y guarantee ✓ × × × × Higher-order interactions ✓ † × × × × Distributional shift detection ✓ ◦ × × × Missing modality robustness ✓ × ◦ × × Interpretabilit y Intrinsic geometric (HHD) P ost-hoc (attention/saliency) Post-hoc feature attribution Linear weigh ts (direct read-out) Post-hoc only Data requirements Higher-order simplices Pairwise graphs T abular/multimodal arrays Tabular/static snapshots T abular/sequential Theoretical limits Requires d max clipping for stability Curl ≡ 0 (misses cyclic risk) No identiﬁabilit y guarantee F ails on non-linear interactions No mechanistic decomp osition † 2-simplices enter through the training loss L geo and the HHD, not through the exp ert forw ard pass; see Theorem 5.3 . W e now discuss eac h capabilit y class, using the hospital shift w ork er scenario from Section 1 as a running example. Scalar output mo dels (LSTM/T ransformer [ 11 ]; scalar GNN [ 16 , 17 ]; scalar MoE [ 12 ]). These arc hitectures map multimodal inputs to a single real-v alued risk score. They can achiev e strong predictiv e accuracy on binary or ordinal outcomes but collapse all mec hanistic informa- tion in to a single num b er. Given a risk score of 0.78, a clinician cannot determine whether it reﬂects directional propagation from so cial exp osure, a self-sustaining physiological lo op, or a net work-structural eﬀect—all three require fundamentally diﬀeren t interv entions. Scalar mo dels lac k this distinction architecturally; an y attempt to decomp ose the output p ost hoc (e.g. via gradien t-based attribution or saliency metho ds) requires additional assumptions not enco ded in the mo del and lacks the algebraic guarantees of HHD. Scalar GNN (dynamic) Dynamic GNNs suc h as Evolv eGCN [ 16 ] or TGA T [ 17 ] impro ve on scalar models by incorp orating netw ork structure, but pro duce scalar no de-lev el outputs and op erate on pairwise graphs (1-skeleton). Without 2-simplices, the curl op erator is identically zero b y deﬁnition (Section 3.2 ), so cyclic risk dynamics are in visible. The sp ectral shift pro xy d spec is structurally a v ailable to an y GNN using graph Laplacians, but without a Lipsc hitz guaran tee the b ound (Theorem 8.3 ) do es not apply; shift detection reduces to a heuristic. 16 V ector MoE without bundle structure A vector-output MoE [ 12 ] pro duces a v ector but without the direct-sum constrain t: exp ert outputs are mixed by unconstrained gating. This means: (a) no guarantee that mo dalit y n ’s contribution is conﬁned to an y subspace—attribution is p ost hoc only; (b) no identiﬁabilit y guarantee (Theorem 7.1 requires the arc hitectural separation); (c) the output v ector is not a section of a structured bundle, so gradien t/curl/div ergence op erators are not deﬁned on it. The HHD cannot b e applied to an unconstrained vector output b ecause the notion of “1-co c hain on a simplicial complex” requires b oth the simplicial substrate and the an tisymmetric edge-netw ork Ψ ω . GVF: what the structure pro vides GVF’s capabilities in the last column of T able 4 are not empirical claims; they are consequences of the architecture: • Gradien t comp onen t : alwa ys deﬁned because F ∈ C 1 ( K ; R m ) and ∇ K , div K are deﬁned on an y simplicial complex. It iden tiﬁes which agents are net risk sources and sinks, directly actionable as “reduce the source” interv en tions. • Curl component : non-trivial only when K ( t ) con tains 2-simplices and F is non-exact. It identiﬁes self-sustaining feedback loops that will not resolv e without cycle-breaking in terven tion—a qualitativ ely diﬀeren t target from directional ﬂo w. • Harmonic comp onen t : its dimension equals β 1 ( K ( t )), the ﬁrst Betti n umber—a top ological in v arian t of the in teraction net w ork. Risk lock ed in the harmonic comp onen t cannot be reduced b y any lo cal interv ention; it requires c hanging the netw ork top ology itself (e.g. shift sc heduling, team restructuring). • Mo dalit y attribution : eac h exp ert writes exclusiv ely into its summand ε ( n ) , so the con tribution of physiological vs. b eha vioural vs. environmen tal signals to the total risk v ector is iden tiﬁable structurally , not p ost hoc. • Missing mo dalit y robustness : the direct-sum arc hitecture degrades gracefully—if envi- ronmen tal sensors go oﬄine, the remaining summands carry a v alid, lo w er-dimensional risk represen tation without retraining. None of these prop erties are ac hiev able by retroﬁtting a scalar mo del with a p ost-ho c explanation la yer: they require the simplicial substrate, the 1-cochain ﬂow ﬁeld, and the bundle-structured output to be built into the architecture from the start. 10 Discussion 10.1 Summary of Con tributions GVF mak es one cen tral contribution enabled by four architectural prop erties. Cen tral contribution T o the best of our knowledge, GVF is the ﬁrst framework to apply Helmholtz–Ho dge decomp osition to a le arne d multimodal risk vector ﬁeld for intrinsic clinical in terpretability . The higher-order top ology of the simplicial complex en ters the learned repre- sen tation indirectly—through the geometric regulariser L geo during training and the HHD at inference (Theorem 5.3 )—rather than through the exp ert forward pass, whic h curren tly operates on the 1-sk eleton. Despite this indirectness, the decomp osition is not a p ost-ho c explanation metho d: it is a structural consequence of represen ting risk as a section of a vector bundle ov er a simplicial complex equipped with Discrete Exterior Calculus op erators. This produces three mec hanistically distinct, orthogonal comp onen ts with separate interv en tion targets (gradien t: directional propagation; curl: cyclic dynamics; harmonic: top ology-lo c k ed p ersistence) and t wo derived scalar scores (DPS, CRI) that complemen t a scalar alarm with a t wo-dimensional 17 w arning signal. Existing multimodal fusion frameworks do not pro vide this decomp osition directly: recov ering cyclic and harmonic comp onen ts requires introducing edge-lev el ﬂow ob jects and 2-simplices that are absent from standard graph or scalar architectures. Enabling architectural prop erties The central contribution is made p ossible by four prop erties of the GVF architecture: 1. Geometric ﬁdelity : risk is represented as a vector ﬁeld o v er a time-v arying simplicial complex, equipping the output space with rigorous DEC diﬀerential structure (gradien t, div ergence, curl op erators on K ( t )). 2. Mo dalit y identiﬁabilit y : the direct-sum bundle structure assigns each mo dalit y a dedicated ﬁbre, ensuring mo dality-speciﬁc risk components are identiﬁable structurally rather than p ost-hoc attribution (Theorem 7.1 ). 3. Domain agnosticism : any com bination of w earable, b eha vioural, en vironmental, or external data sources is accommo dated b y adding or remo ving bundle summands and their exp ert; the framework do es not assume a ﬁxed modality set. 4. F ormal guarantees : univ ersal approximation (Theorem 8.1 ), mo dalit y identiﬁabilit y (Theorem 7.1 ), and a consistency bound under simplicial complex p erturbation motiv ating the sp ectral shift proxy d spec (Theorem 8.3 ; full deriv ation in Section F ). 10.2 Implications for Practice The three HHD comp onen ts map directly to three op erationally distinct in terv ention strategies, summarised in T able 5 . Practically , GVF pro duces a per-agent (DPS i , CRI i ) pair at eac h time step, whose thresholds can be calibrated indep enden tly (e.g. alert on high DPS, escalate on concurren t high DPS and CRI). The sp ectral shift proxy d spec signals when the netw ork top ology w arrants mo del recalibration. T able 5: HHD comp onen ts and their operational interv en tion targets. Comp onen t Signal In terpretation In terven tion Gradien t ∇ K φ DPS i > 0 Net risk source; directional propagation Reduce source (ex- p osure, shift pat- tern) Curl curl ∗ K ψ CRI i > 0 Self-sustaining loop; persists if source remov ed Break cycle (sleep h ygiene, pharma- cological) Harmonic h dim k er ∆ 1 > 0 T op ology-lo c k ed; lo cal actions ineﬀectiv e Restructure net- w ork (scheduling, zoning) 10.3 Limitations and F uture W ork Empirical v alidation. The primary direction for future w ork is ev aluation of GVF on real-w orld m ultimo dal w earable cohorts. A suitable v alidation study requires: (a) multimodal ph ysiological signals (cardiac, motion, and at least one environmen tal channel); (b) a lab elled outcome v ariable for classiﬁcation supervision; (c) a pro ximity or in teraction netw ork that yields a non-trivial K ( t ) with 2-simplices and β 1 > 0. V alidation should span datasets with v arying net work top ologies, mo dalit y conﬁgurations, and clinical outcomes b efore an y claim of clinical utilit y . 18 Ablation on simplicial complex order. A con trolled comparison of GVF on the 1- sk eleton versus the full K ( t ) with 2-simplices is needed to empirically verify the con tribution of higher-order in teractions; real pro ximity netw ork data is required for this ablation. Exp ert architecture alignment. The theoretical UA T (Theorem 8.1 ) is stated for suﬃcien tly expressive simplicial netw orks, whereas the practical exp ert architectures in T able 1 op erate primarily on the 1-sk eleton. Upgrading exp erts to full simplicial architectures [ 36 ] is planned. Priv acy and federated deploymen t. The federated arc hitecture (lo cal gradien t computa- tion, diﬀeren tial priv acy noise) is sk etched but not implemented. Non-I ID simplicial topologies across federated clien ts presen t op en c hallenges for Hodge Laplacian aggregation. Causal in terpretation. GVF learns correlational s tructure. Conv erting DPS and HHD outputs into causal eﬀect estimates requires embedding the framew ork within a structural causal mo del—an imp ortant but non-trivial extension. 11 Conclusion W e ha ve introduced the GVF framew ork, recasting m ultimo dal health risk assessment as a v ector ﬁeld learning problem o ver a simplicial complex equipp ed with Discrete Exterior Calculus op erators. By modelling risk as a directional, geometrically structured ob ject rather than a scalar, GVF preserv es propagation path wa ys, cyclic dynamics, and top ology-lo c k ed p ersistence—eac h with a mec hanistically distinct, operationally actionable interpretation. The core design principle is that interpretabilit y should be intrinsic , not a p ost-hoc ov erla y . The Helmholtz–Hodge decomp osition—applied to a learned risk ﬂo w ﬁeld on the edges of the complex, not to the (trivially exact) gradient of the no de-v alued section—provides exactly this: three orthogonal comp onen ts that map to three distinct clinical in terv ention strategies (directional source reduction, cycle breaking, structural netw ork reconﬁguration). This shift from scalar alarm to decomp osed v ector signal is the cen tral contribution of GVF. The bundle-structured Mixture-of-Experts architecture enforces mo dalit y iden tiﬁabilit y by design; formal guaran tees of univ ersal appro ximation and distributional stabilit y establish theoretical soundness. The arc hitectural comparison in Section 9 demonstrates that GVF’s capabilities are structural—not empirical—consequences of the architecture. Empirical ev aluation on real-w orld cohorts constitutes the primary direction for future w ork. GVF provides a mathematically principled, domain-agnostic foundation for the next generation of mechanistically in terpretable multimodal health monitoring systems. Declaration of Comp eting In terests The authors declare that they ha ve no known competing ﬁnancial interests or personal relation- ships that could ha v e app eared to inﬂuence the work rep orted in this paper. F unding This researc h did not receive an y sp eciﬁc gran t from funding agencies in the public, commercial, or not-for-proﬁt sectors. Use of AI Assistance AI-assisted to ols were used for language editing and literature organisation during man uscript preparation. No generativ e AI w as used to pro duce results, ﬁgures, or mathematical pro ofs. 19 A Algorithmic Sp eciﬁcations Algorithm 1 Multimo dal Simplicial Complex Construction for K ( t ) Require: Raw streams S ( t ), windo w ∆ t , thresholds τ prox , τ sync , τ dwell Ensure: Simplicial complex K ( t ) = ( K 0 , K 1 , K 2 ) 1: K 0 ← V A ( t ) ∪ V S ∪ V E ( t ) ∪ V X ▷ Initialise 0-simplices 2: K 1 ← ∅ , K 2 ← ∅ — Phase 1: 1-Sk eleton F ormation (Edges) — 3: for each pair of no des ( u, v ) ∈ K 0 × K 0 do 4: if u, v ∈ V A and ( d spatial ( u, v ) < τ prox or DTW ( u, v ) < τ sync ) then 5: K 1 ← K 1 ∪  { u, v }  6: else if u ∈ V A , v ∈ V E and exp osure time( u, v ) > τ dwell then 7: K 1 ← K 1 ∪  { u, v }  8: end if 9: end for — Phase 2: Multimo dal 2-Simplex Inclusion — 10: for each triad { u, v , w } with all three edges in K 1 do 11: if | unique t yp es( { u, v , w } ) | ≥ 2 then ▷ Require m ultimo dalit y 12: K 2 ← K 2 ∪  { u, v , w }  13: end if 14: end for 15: return K ( t ) = ( K 0 , K 1 , K 2 ) B Discrete Diﬀeren tial Op erators and T raining Details The signed incidence matrices of K ( t ) are B 1 ∈ {− 1 , 0 , +1 } | V |×|K 1 | (no de-edge) and B 2 ∈ {− 1 , 0 , +1 } |K 1 |×|K 2 | (edge-face). The three discrete operators of Section 3.2 in explicit form: ( ∇ K r ) ij = r j − r i , ∇ K = B ⊤ 1 ⊗ I m (15) (div K F ) i = P ( i,j ) F ij − P ( j,i ) F j i , div K = B 1 ⊗ I m (16) (curl K F ) ij l = F ij + F j l + F li , curl K = B ⊤ 2 ⊗ I m (17) Div ergence is signed (p ositiv e = net outﬂow); curl norm is bounded ∥ curl K ∥ C 1 → C 2 ≤ √ d max where d max is the maxim um n umber of 2-simplices p er edge. L geo clipping and λ 1 selection. The ratio ρ = ∥ curl K F ∥ 2 / ( ∥ F ∥ 2 + ε ) is b ounded ab o v e b y d max (not b y 1 in general); explicit clipping to [0 , 1] ensures L geo ∈ [ − log 2 , 0]. In proximit y graphs d max ≤ 6, so clipping is rarely activ e. λ 1 is selected b y grid searc h o ver { 0 . 01 , 0 . 1 , 0 . 5 } . C HHD Computation via Sparse Linear Solv ers The HHD of the risk ﬂo w F ∈ C 1 ( K ; R m ) reduces to t w o sparse linear systems (solved indep en- den tly for eac h of the m co ordinate channels): 1. Solv e ∆ 0 φ = div K F for φ ∈ C 0 ( K ; R m ) (size | V | × | V | , symmetric positive semi-deﬁnite). Unique mo dulo a constant; computed via preconditioned conjugate gradien t using the Mo ore-P enrose pseudoin v erse restricted to k er(∆ 0 ) ⊥ . 2. Solve ∆ 2 ψ = curl K F for ψ ∈ C 2 ( K ; R m ) (size |K 2 | × |K 2 | , sparse). Same solv er. 3. Harmonic residual: h = F − ∇ K φ − curl ∗ K ψ ∈ ker(∆ 1 ). 20 Note: the righ t-hand side in step (3) uses F (not ∇ K r ), consistent with Theorem 5.4 . F or | V | ≤ 10 4 and |K 1 | ≤ 5 × 10 4 , b oth systems conv erge in < 50 conjugate gradien t iterations. D F ull Pro of: Univ ersal Appro ximation (Theorem 8.1 ) Note on the tw o-ob ject arc hitecture. The GVF framew ork learns t w o distinct ob jects: (1) the no de-lev el risk section r i ( t ) ∈ R m , pro duced b y the MoE op erator F θ and sup ervised via L cls ; (2) the edge-level risk ﬂow F ij ( t ) ∈ R m , produced by the antisymmetric function Ψ ω and constrained via L geo . The UA T concerns (1); the non-trivialit y of the HHD concerns (2). These are distinct and complemen tary guaran tees. W e provide the full pro of of the univ ersal appro ximation theorem for the GVF operator on simplicial complexes. Let ε ( K ) = L N n =1 ε ( n ) b e the modality-structured risk v ector bundle ov er K ( t ), with each ε ( n ) a trivial bundle of ﬁbre dimension m n . Let π ( n ) : Γ( ε ( K )) → Γ( ε ( n ) ) denote the canonical pro jection onto the n -th summand. Pr o of. Fix ε > 0 and a target section r ∗ ∈ Γ( ε ( K )). W rite r ∗ = L n r ∗ ( n ) with r ∗ ( n ) = π ( n ) ( r ∗ ). Step 1: approximation within each ﬁbre. Each exp ert F ( n ) θ is a simplicial message- passing netw ork (SMPN) with non-p olynomial activ ations. By Theorem 3 of [ 36 ] (applied comp onen t wise to each of the m n output co ordinates), for each n there exist parameters θ ( n ) suc h that sup i   F ( n ) θ ( n ) ( K , x i ) − r ∗ ( n ) i   < ε. Step 2: the bundle morphism do es not reduce capacity . The GVF operator outputs in Γ( ε ( K )) via the direct-sum map ι : L n Γ( ε ( n ) ) → Γ( ε ( K )), which is a linear isometric embedding. Since ι is an isometry , the appro ximation b ound from Step 1 carries through:   ι  M n F ( n ) θ ( n ) ( K , x i )  − r ∗ i   =  X n   F ( n ) θ ( n ) − r ∗ ( n ) i   2  1 / 2 < ε √ N . Rescaling ε ← ε/ √ N in Step 1 reco v ers the original ε b ound. Step 3: MoE gating recov ers the direct-sum output. Set gating w eights g ( n ) ϕ ≡ 1 and g ( m ) ϕ ≡ 0 for m  = n . These are ac hiev able b y the gating net w ork g ϕ (a softmax-normalised atten tion classiﬁer, UA T from [ 38 ]). The MoE output with these w eights equals ι ( L n F ( n ) θ ( n ) ), completing the proof. E F ull Pro of: Mo dalit y Iden tiﬁability (Theorem 7.1 ) Pr o of. W e adapt the iV AE identiﬁabilit y framew ork [ 39 ] to the bundle-structured MoE setting. As noted in the pro of sketc h of Theorem 7.1 , the corresp ondence with [ 39 ] Theorem 1 is structural: GVF is discriminativ e rather than generativ e, and the exp onen tial-family condition on the conditional laten t distribution is replaced b y the arc hitectural constrain ts (C1)–(C3). The argumen t b elo w proceeds under the assumption that this structural analogy is suﬃciently tigh t to preserv e the identiﬁabilit y conclusion. Deﬁne the eﬀectiv e parameter map Φ( θ ) = ( F (1) θ , . . . , F ( N ) θ ); w e show Φ is injective up to within-summand p erm utation. Condition (C1) separates the training-loss gradient b y mo dalit y (the guaranteed linear indep endence of the orthogonalized input subspaces ensures ∇ θ ( n ) L dep ends only on ˜ x ( n ) ). Condition (C2) pro vides the auxiliary-v ariable conditioning required by [ 39 ] Theorem 1: each exp ert F ( n ) θ outputs zero on all other modality blocks by arc hitectural construction. Condition 21 (C3) ensures the non-degeneracy cov erage required b y the same theorem: mo dalit y-drop out guaran tees training examples for ev ery prop er mo dality subset. F ollowing the structure of [ 39 ] Theorem 1 in this setting, Φ is injective up to within-summand aﬃne reparameterisation: for each n , there exists an in vertible aﬃne map A ( n ) suc h that ˆ F ( n ) θ = A ( n ) ◦ F ∗ ( n ) . The key adaptation that reduces aﬃne to p erm utation: (C2) enforces that eac h F ( n ) θ maps in to a ﬁxed subspace ε ( n ) with blo c k ed input structure. The aﬃne map A ( n ) m ust therefore preserve this subspace, constraining it to act within R m n . Since the output co ordinates within each summand are architecturally symmetric (no preferred basis is imp osed b y the net work), the remaining am biguity is precisely a p ermutation of the m n co ordinates within ε ( n ) , completing the proof. F F ull Deriv ation: Stabilit y Bound (Theorem 8.3 ) Pr o of. The pro of has tw o components: (a) Lipschitz con tin uity of F θ with resp ect to graph structure; (b) the bridge b et w een Gromo v-Hausdorﬀ distance and Ho dge Laplacian sp ectral p erturbation. (a) Lipschitz con tinuit y . Spectral normalisation enforces ∥ W ℓ ∥ 2 ≤ 1 for all GNN ag- gregation matrices, so the Lipsc hitz constant satisﬁes L ≤ Q ℓ ∥ W ℓ ∥ 2 b y induction o ver lay ers. Standard matrix p erturbation theory then gives ∥ F θ ( K , x ) − F θ ( K ′ , x ) ∥ ≤ L · ∥ ∆ k ( K ) − ∆ k ( K ′ ) ∥ F . (b) GH-to-spectral bridge. The Hodge Laplacian eigenv alue p erturbation b ound ∥ λ k ( K ) − λ k ( K ′ ) ∥ ∞ ≤ C · d GH ( K , K ′ ) is motiv ated by the contin uous Laplacian stabilit y results of [ 40 ] (Theorem 6). 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Graph Vector Field: A Unified Framework for Multimodal Health Risk Assessment from Heterogeneous Wearable and Environmental Data Streams

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