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CAMOS: Coupled Oscillatory State-Space Model for Multimodal Clinical Time-Series
Maxx Richard Rahman, Mostafa Hammouda, Wolfgang Maass
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Abstract
Abstract:Longitudinal clinical cohorts are multimodal, irregularly sampled and pervasively incomplete: in ADNI, positron emission tomography and cerebrospinal fluid assays are absent from roughly half of all visits. Linear state-space models handle irregular sampling gracefully but treat a missing modality by masking the input, leaving the transition operator untouched. We prove that this is a representational limitation: the latent state of any linear state-space layer whose transition operator does not depend on the availability pattern is an additive function of the availability indicators, so no such layer can represent an interaction between two modalities being jointly present or jointly absent. We propose CAMOS, which gives each modality a bank of second-order oscillators coupled through a matrix that sits inside the differential equation and is gated by availability, so the transition operator itself becomes a function of which measurements were taken. Coupling invalidates the analysis of uncoupled oscillatory models, and we restore it: a per-channel Gershgorin budget makes the effective stiffness positive definite uniformly over all $2^M$ availability patterns and all gaps, an energy argument charges amplification to availability transitions rather than sequence length, and a channel factorization preserves exact associative parallel scans. On ADNI, CAMOS outperforms uncoupled oscillatory state-space models and clinical fusion models on same-visit staging, landmark prediction and longitudinal forecasting, and under zero-shot transfer to OASIS-3 it is the only model that avoids collapse to the majority class.
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- Source: https://arxiv.org/abs/2609.39484v1
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CAMOS: Coupled Oscillatory State-Space Model for Multimodal Clinical Time-Series Maxx Richard Rahman Affiliation: Saarland University and Affiliation: German Research Center for Artificial Intelligence (DFKI) Affiliation: Saarbrücken, Germany Mostafa Hammouda Affiliation: Saarland University Affiliation: Saarbrücken, Germany Wolfgang Maass Affiliation: Saarland University and Affiliation: German Research Center for Artificial Intelligence (DFKI) Affiliation: Saarbrücken, Germany Abstract Longitudinal clinical cohorts are multimodal, irregularly sampled and pervasively incomplete: in ADNI, positron emission tomography and cerebrospinal fluid assays are absent from roughly half of all visits. Linear state-space models handle irregular sampling gracefully but treat a missing modality by masking the input, leaving the transition operator untouched. We prove that this is a representational limitation: the latent state of any linear state-space layer whose transition operator does not depend on the availability pattern is an additive function of the availability indicators, so no such layer can represent an interaction between two modalities being jointly present or jointly absent. We propose CAMOS, which gives each modality a bank of second-order oscillators coupled through a matrix that sits inside the differential equation and is gated by availability, so the transition operator itself becomes a function of which measurements were taken. Coupling invalidates the analysis of uncoupled oscillatory models, and we restore it: a per-channel Gershgorin budget makes the effective stiffness positive definite uniformly over all 2M2^M availability patterns and all gaps, an energy argument charges amplification to availability transitions rather than sequence length, and a channel factorization preserves exact associative parallel scans. On ADNI, CAMOS outperforms uncoupled oscillatory state-space models and clinical fusion models on same-visit staging, landmark prediction and longitudinal forecasting, and under zero-shot transfer to OASIS-3 it is the only model that avoids collapse to the majority class. 1 Introduction State-space models (SSMs) have been widely adopted for learning on long sequences (Gu et al., 2022; Smith et al., 2023; Orvieto et al., 2023; Gu and Dao, 2024), and they suit clinical time series because they discretize a continuous-time system and so treat a variable step size as a parameter rather than an approximation. Oscillatory SSMs sharpen this further, deriving stability from a nonnegative diagonal state matrix alone and admitting associative parallel scans in logarithmic depth (Rusch and Rus, 2025; Boyer et al., 2025; Blelloch, 1990). However, clinical cohorts pose a second difficulty alongside irregular sampling, i.e., measurements arrive as several modalities ordered by clinical need, cost and invasiveness, so each has its own gaps, while the underlying processes are coupled. In Alzheimer’s disease, amyloid deposition precedes tau pathology, then neurodegeneration, then cognitive decline (Hardy and Selkoe, 2002; Jack et al., 2010; Jack et al., 2013), yet in ADNI (Petersen et al., 2010) imaging and cognitive testing occur at most visits whereas amyloid PET and cerebrospinal fluid assays are far rarer, so the cascade should be tracked through a missingness process that is neither uniform nor ignorable (Rubin, 1976). Existing models treat that missingness as an input phenomenon. An absent measurement is handled by zeroing or masking the forcing term, sometimes with an appended indicator (Che et al., 2018), while the transition operator, which propagates latent state across time, is left untouched. Unrolling such a layer shows the cost: the latent state is a sum of per-modality, per-step contributions, each scaled by its own availability indicator, so the effect of one modality being present cannot depend on whether another was present at a different step. Joint availability is nevertheless informative, since whether an unverified amyloid estimate should influence a tau prediction depends on whether amyloid was measured, not merely on its value. Multimodal fusion methods provide a solution to this additive class (Hayat et al., 2022; Zhang et al., 2022a; Wang et al., 2023; Yao et al., 2024), but they fuse after the temporal model has been applied, so nothing lets one modality shape another’s trajectory across a long gap. In this paper, we place availability inside the dynamics. Each modality receives a bank of second-order oscillators, and modalities drive one another through a coupling matrix in the differential equation itself, gated by availability. Because the gate multiplies a matrix in the homogeneous part, the transition operator becomes a function of which measurements were taken, and the additivity obstruction disappears. However, coupling invalidates the analysis of uncoupled oscillatory models in three places: i) stability is a property of the effective stiffness, which no condition on the individual coupling blocks alone controls; i) parallel evaluation relies on a diagonal state matrix, which coupling destroys; and i) a per-step spectral radius bound no longer implies boundedness once the operator varies over time. Our contributions, each addressing one of them are: • We introduce CAMOS, a coupled oscillatory state-space model whose transition operator is gated by the availability pattern and factored channel-wise, preserving exact associative parallel scans. • We prove that every availability-independent linear state-space layer is additive in the availability indicators while CAMOS is not. • We perform empirical evaluation on ADNI and OASIS-3 where CAMOS outperforms uncoupled oscillatory SSMs and clinical fusion models on same-visit staging, landmark prediction and longitudinal forecasting. 2 Related Work Structured, oscillatory and coupled state-space models. S4, S5, LRU and Mamba established structured and selective state matrices solved by FFT or associative scans (Gu et al., 2020; Gu et al., 2022; Smith et al., 2023; Orvieto et al., 2023; Gu and Dao, 2024; Dao and Gu, 2024). LinOSS (Rusch and Rus, 2025) instead discretizes a second-order oscillator system, obtaining stability from a nonnegative diagonal state matrix alone and universality within continuous causal operators; its two variants, LinOSS-IM and LinOSS-IMEX, differ in whether the oscillator system is integrated by an implicit or a symplectic implicit-explicit scheme, and D-LinOSS (Boyer et al., 2025) decouples damping from frequency. These are the direct parents of CAMOS and our primary baselines. All of them take a single monolithic input, so concatenating modalities leaves the state matrix shared and the modality partition invisible to the dynamics, and no object in the model represents one modality driving another; they correspond to a zero coupling with a dense input matrix. Coupled Mamba (Li et al., 2024) is the closest on the oscillator side, coupling the hidden-state chains of several modalities, but it couples a discrete first-order recurrence rather than a continuous-time second-order system, so an irregular gap has no natural meaning. Irregular sampling and missing modalities. Irregular sampling is handled by GRU-D, ODE-based models, controlled differential equations, mTAN and Raindrop (Che et al., 2018; Chen et al., 2018; Rubanova et al., 2019; De Brouwer et al., 2019; Kidger et al., 2020; Walker et al., 2024; Shukla and Marlin, 2021; Zhang et al., 2022b), all of which are sequential in the sequence length and none of which gates a coupled dynamical system by availability. Clinical fusion methods instead handle missing modalities at the level of representations (Ma et al., 2021; Zhang et al., 2022a; Wang et al., 2023; Yao et al., 2024; Hayat et al., 2022; Joze et al., 2020; Pölsterl et al., 2021), fusing after the temporal model has be applied. MedFuse (Hayat et al., 2022) learns a representation of the missing modality inside an LSTM fusion, and DrFuse (Yao et al., 2024) disentangles shared from modality-specific representations so that the shared component survives an absent modality. Both are nonlinear in the availability pattern, which places them outside the additive class, and both are our second family of baselines. Figure 1: CAMOS architecture consists of static and longitudinal multimodal patient data through availability-gated coupled oscillators, irregular-time dynamics, and cross-modal attention. 3 Problem Formulation Let us consider a subject contribute N visits at strictly increasing times t1<⋯<tNt_1<…<t_N with gaps Δtn:=tn−tn−1>0 t_n:=t_n-t_n-1>0 in years, where Δt1:=t1−t0 t_1:=t_1-t_0 and t0t_0 is the cohort baseline. At visit n, each of the M modalities k∈[M]:=1,…,Mk∈[M]:=\1,…,M\ contributes a feature vector ukn∈ℝpku_k^n ^p_k and a binary availability indicator akn∈0,1a_k^n∈\0,1\, equal to one exactly when modality k was measured, with uknu_k^n undefined and never read when akn=0a_k^n=0. We write an=(a1n,…,aMn)∈0,1Ma^n=(a_1^n,…,a_M^n)∈\0,1\^M for the availability pattern, a1:Na^1:N for its sequence, m:=Mdm:=Md for the stacked latent width, and Da:=blkdiag(a1Id,…,aMId)∈ℝm×mD_a:= (a_1I_d,…,a_MI_d) ^m× m for the gate matrix. Each subject also has static covariates s∈ℝpss ^p_s. The three prediction targets are: i) same-visit staging: a visit-level ordinal label ℓn∈CN,MCI,AD ^n∈\CN,MCI,AD\ predicted as p^n p^n in the 22-simplex; i) landmark prediction: a subject-level binary label q∈0,1q∈\0,1\ for progression from mild cognitive impairment (MCI) to Alzheimer’s disease (AD) within three years, predicted as q^∈[0,1] q∈[0,1]; and i) longitudinal forecasting: the values ukn+hu_k^n+h at horizon h≥1h≥ 1, predicted as u^kn+h u_k^n+h and scored only where akn+h=1a_k^n+h=1. The model is a map: Φ:(ukn,a1:N,tn,s)⟼(p^n,q^,u^kn+h), :\ (\u_k^n\,\,a^1:N,\,\t_n\,\,s )\; \; (\ p^n\,\, q,\,\ u_k^n+h\ ), which uses a1:Na^1:N as a genuine input rather than merely as a mask on u. 4 Proposed CAMOS Model CAMOS is a stack of L gated coupled oscillator (GCO) blocks between an encoder and three task heads, Φ=Headτ∈1,2,3∘Attn∘GCO(L)∘⋯∘GCO(1)∘Enc. = \Head_τ \_τ∈\1,2,3\ \ Attn\ \ GCO^(L) ·s ^(1) \ Enc. Everything except the GCO recurrence is applied pointwise in n and is therefore trivially parallel. The GCO recurrence is evaluated by an associative scan in O(logN)O( N) depth. All proofs are in Appendix C. 4.1 Input encoders Static covariates s∈ℝpss ^p_s set the initial condition x0=(z0,y0)⊤∈ℝ2mx^0=(z^0,y^0) ^2m of the oscillator bank through a two-layer perceptron with GELU activation (Hendrycks and Gimpel, 2016), rather than adding an offset at the readout, which encodes the hypothesis that baseline demographics modulate the phase and amplitude of the cascade rather than shifting outcomes additively. Modalities are measured on incommensurable scales, so each is projected affinely into a shared width h, u~kn=Ekukn+ek∈ℝh,Ek∈ℝh×pk,ek∈ℝh, u_k^n=E_k\,u_k^n+e_k ^h, E_k ^h× p_k,\ e_k ^h, (1) in parallel across visits and evaluated only when akn=1a_k^n=1; the value of u~kn u_k^n is irrelevant when akn=0a_k^n=0 because the forcing is gated in equation 6. 4.2 Coupled multimodal oscillators, and why the coupling must be budgeted Each modality k carries a bank of d oscillators obeying zk′(t)=−Akyk(t)+∑j≠kCkjyj(t)+Bku~k(t),yk′(t)=zk(t),z_k (t)=-A_k\,y_k(t)+ _j≠ kC_kj\,y_j(t)+B_k\, u_k(t), y_k (t)=z_k(t), (2) with Ak,Ckj∈ℝd×dA_k,C_kj ^d× d and Bk∈ℝd×hB_k ^d× h. Stacking y=(y1⊤,…,yM⊤)⊤∈ℝmy=(y_1 ,…,y_M ) ^m and likewise for z, equation 2 becomes y′=−Py+Bu~y =-Py+B u with P P :=A−C,A:=blkdiag(A1,…,AM), :=A-C, A:= (A_1,…,A_M), (3) [C]kj [C]_kj :=Ckj(k≠j),[C]kk:=0,B:=blkdiag(B1,…,BM). :=C_kj (k≠ j), [C]_k:=0, B:= (B_1,…,B_M). Stability, discretization and cost are properties of the effective stiffness P rather than of the individual blocks, which invalidates the natural first guess. Remark 1 (Block-level conditions do not give stability). Parameterizing each block as Ckj=QkjQkj⊤⪰0C_kj=Q_kjQ_kj 0 does not make P⪰0P 0. Under Assumption 1 the assembled C is symmetric with zero diagonal blocks, so tr(C)=0 (C)=0 and C has a strictly negative eigenvalue whenever C≠0C≠ 0. The only free bound is λmin(P)≥λmin(A)−‖C‖2 _ (P)≥ _ (A)-\|C\|_2, which can be negative whenever ‖C‖2>λmin(A)\|C\|_2> _ (A); then I+Δt2PI+ t^2P can be singular, the implicit solve fails, and the transition operator can have spectral radius exceeding one. A joint condition on A and C is required. Channel factorization. We take AkA_k and CkjC_kj diagonal, Ak=diag(αk,1,…,αk,d)A_k= ( _k,1,…, _k,d) and Ckj=diag(ckj,1,…,ckj,d)C_kj= (c_kj,1,…,c_kj,d), so coupling mixes modalities within each latent channel but not across channels. With Π the permutation Πe(k−1)d+r=e(r−1)M+k \,e_(k-1)d+r=e_(r-1)M+k, reindexing from (modality-major, channel-minor) to (channel-major, modality-minor), ΠPΠ⊤=blkdiag(P(1),…,P(d)),[P(r)]kk=αk,r,[P(r)]kj=−ckj,r(k≠j), \,P\, = (P^(1),…,P^(d) ), [P^(r) ]_k= _k,r, [P^(r) ]_kj=-\,c_kj,r\ \ (k≠ j), (4) so the coupled system decomposes into d independent networks of M oscillators, each carrying its own weighted modality graph P(r)∈ℝM×MP^(r) ^M× M. Appendix E shows that a dense CkjC_kj raises the per-step cost from O(dM3)O(dM^3) to O(d3M3)O(d^3M^3), and that a low-rank CkjC_kj does not help, because low-rank-plus-diagonal structure is not closed under the scan combine. Symmetry and Gershgorin budget. We impose two assumptions. Assumption 1 (Symmetry). ckj,r=cjk,rc_kj,r=c_jk,r for all k,j∈[M]k,j∈[M], r∈[d]r∈[d], so every P(r)P^(r) is symmetric and P=P⊤P=P . Assumption 2 (Gershgorin budget). There is ϵ∈(0,1]ε∈(0,1] such that ∑j≠k|ckj,r|≤(1−ϵ)αk,r _j≠ k|c_kj,r|≤(1-ε)\, _k,r for all k∈[M]k∈[M], r∈[d]r∈[d], with αk,r>0 _k,r>0. Both hold by construction under the following parameterization, which needs no projection step, no power iteration, and is differentiable everywhere. Let α^k,r α_k,r and c^kj,r=c^jk,r c_kj,r= c_jk,r be free parameters (parameterize the strict upper triangle and mirror it) and set, with δ>0δ>0 a small constant, αk,r=softplus(α^k,r),σk,r=∑l≠k|c^kl,r|,ckj,r=(1−ϵ)minαk,r,αj,rmaxσk,r,σj,r,δc^kj,r. _k,r= ( α_k,r), _k,r= _l≠ k| c_kl,r|, c_kj,r= (1-ε)\, \ _k,r, _j,r\ \ _k,r,\, _j,r,\,δ\\; c_kj,r. (5) Symmetry is immediate because the prefactor is symmetric in (k,j)(k,j), and the budget follows termwise. Writing αmin:=mink,rαk,r _ := _k,r _k,r, αmax:=maxk,rαk,r _ := _k,r _k,r and γ:=maxr‖C(r)‖2γ:= _r\|C^(r)\|_2 for the coupling strength, we set μ:=ϵαminμ:=ε\, _ and LP:=(2−ϵ)αmaxL_P:=(2-ε)\, _ , and Assumption 2 gives γ≤(1−ϵ)αmaxγ≤(1-ε) _ . Lemma 2 (Uniform positive definiteness). For every a∈[0,1]Ma∈[0,1]^M the matrix P(a)=A−DaCDaP(a)=A-D_aCD_a is symmetric and μIm⪯P(a)⪯LPImμ I_m P(a) L_PI_m. In particular P(a)≻0P(a) 0 simultaneously for all 2M2^M binary availability patterns, Im+Δt2P(a)I_m+ t^2P(a) is symmetric positive definite for every Δt∈ℝ t , and its Cholesky factorization exists. The proof is a per-channel Gershgorin argument using only |akaj|≤1|a_ka_j|≤ 1, so gating can shrink but never enlarge an off-diagonal row sum: the budget is imposed once, on the ungated coupling, and gating preserves it automatically, so no availability pattern can destabilize the model. 4.3 Availability gating and what input masking cannot represent Availability enters in exactly two places, the stiffness and the forcing: Pn:=A−DanCDan,fn:=∑k=1MaknιkBku~kn∈ℝm,P_n:=A-D_a^n\,C\,D_a^n, f^n:= _k=1^Ma_k^n\, _k\,B_k\, u_k^n ^m, (6) where ιk:ℝd→ℝm _k:R^d ^m embeds into the k-th modality block. Since [DanCDan]kj=aknajnCkj[D_a^nCD_a^n]_kj=a_k^na_j^nC_kj, the coupling gate is the symmetric product gkjn=aknajn∈0,1g_kj^n=a_k^na_j^n∈\0,1\, so [Pn(r)]kj=−aknajnckj,r[P_n^(r)]_kj=-a_k^na_j^nc_kj,r off the diagonal. Two properties motivate this over the asymmetric alternative gkjn=ajng_kj^n=a_j^n, which gates only by the source modality: PnP_n remains symmetric, which Assumption 1 and every result below require, and an unobserved modality is removed from the graph entirely. Proposition 3 (Exact isolation of unobserved modalities). If ajn=0a_j^n=0, then PnP_n is block diagonal with respect to the splitting j∪([M]∖j)\j\∪([M] \j\), hence so are SnS_n and nT_n of equation 8. Consequently, at visit n the latent state of modality j has exactly zero influence on every other modality and vice versa, while modality j continues to evolve under its own AjA_j with Snjj=(Id+Δtn2Aj)−1S_n^j=(I_d+ t_n^2A_j)^-1. Under the asymmetric gate gkj=ajg_kj=a_j, PnP_n is block triangular rather than block diagonal, so the isolation is one-sided and PnP_n is no longer symmetric, invalidating Lemma 2. Without the gate, the one-step cross-modal influence is |[Sn(r)]kj|=Δtn2|ckj,r|+O(Δtn4LP2) [S_n^(r)]_kj = t_n^2\, c_kj,r +O ( t_n^4L_P^2 ) at short gaps, peaks at intermediate gaps and decays as Δtn−2 t_n^-2 at long ones (Proposition 11), so the gate matters most in the intermediate regime and for slow channels. The structural point is that PnP_n sits in the homogeneous part of the dynamics and is a function of ana^n, and the following theorem is the formal consequence. Definition 4 (Input-gated linear layers). Let A be the class of layers xn=Tnxn−1+∑k=1MaknBknu~knx^n=T_nx^n-1+ _k=1^Ma_k^nB_k^n u_k^n with x0=0x^0=0 and on=Rxno^n=Rx^n, in which TnT_n, BknB_k^n and R may depend arbitrarily on the parameters, on tm\t_m\ and on Δtm\ t_m\, but not on the availability pattern a1:Na^1:N. A contains the state-space layers of S4, S5, LRU, LinOSS and D-LinOSS applied to concatenated multimodal inputs with masked or zero-filled missing entries. It excludes selective models such as Mamba when the availability vector is appended to the input, since selection then makes TnT_n depend on a. Theorem 5 (Availability additivity). For every layer in A and every n, on=∑m≤n∑kakmψk,mo^n= _m≤ n _ka_k^m _k,m with ψk,m:=RTn⋯Tm+1Bkmu~km _k,m:=R\,T_n·s T_m+1B_k^m u_k^m, so ono^n is a multilinear polynomial of degree one in a1:na^1:n and every mixed availability difference of order two or higher vanishes identically, Δ(k,m)Δ(j,m′)on≡0 _(k,m) _(j,m )o^n≡ 0 for all (k,m)≠(j,m′)(k,m)≠(j,m ), where Δ(k,m)F(a):=F(a|akm=1)−F(a|akm=0) _(k,m)F(a):=F(a\,|\,a_k^m=1)-F(a\,|\,a_k^m=0). CAMOS does not lie in A: already for M=2M=2, d=1d=1, L=1L=1, [Sn]12=Δtn2gc(1+Δtn2α1)(1+Δtn2α2)−Δtn4g2c2,g=a1na2n,[S_n]_12= t_n^2\,g\,c(1+ t_n^2 _1)(1+ t_n^2 _2)- t_n^4g^2c^2, g=a_1^na_2^n, (7) which vanishes when g=0g=0 and is nonzero when g=1g=1 for any c≠0c≠ 0, giving Δ(1,n)Δ(2,n)on≠0 _(1,n) _(2,n)o^n≠ 0. Theorem 5 concerns a single linear layer. It does not say that deep networks built from such layers cannot represent availability interactions, since a nonlinear readout applied to an additive statistic is not itself additive, nor does it separate CAMOS from representation-level fusion such as MedFuse or DrFuse, whose encoders are nonlinear in a. It says that in class A the interaction must be reconstructed by the readout from an additive summary, whereas CAMOS represents it inside the recurrence, in the part of the model that is scannable and carries the stability guarantees above (Remark 8). 4.4 Stable discretization on the irregular visit grid We discretize equation 2 on tn\t_n\ with the fully implicit (backward Euler) scheme applied to the first-order form, zn=zn−1+Δtn(−Pnyn+fn)z^n=z^n-1+ t_n(-P_ny^n+f^n) and yn=yn−1+Δtnzny^n=y^n-1+ t_nz^n. Collecting xn:=(zn,yn)⊤∈ℝ2mx^n:=(z^n,y^n) ^2m and inverting through the Schur complement Sn:=(Im+Δtn2Pn)−1S_n:=(I_m+ t_n^2P_n)^-1, which exists for every Δtn∈ℝ t_n by Lemma 2, gives xn=nxn−1+nFn,n=(Sn−ΔtnPnSnΔtnSnSn),Fn=(Δtnfn,0)⊤.x^n=T_n\,x^n-1+T_nF_n, _n= pmatrixS_n&- t_nP_nS_n\\ t_nS_n&S_n pmatrix, F_n=( t_nf^n,0) . (8) Under equation 4, SnS_n is block diagonal with d blocks Sn(r)=(IM+Δtn2Pn(r))−1S_n^(r)=(I_M+ t_n^2P_n^(r))^-1, each costing O(M3)O(M^3) by Cholesky. The step size enters as a per-visit scalar, so irregular sampling requires no modification: the model integrates the same continuous system over whatever gap actually occurred, and stability is uniform in that gap. Proposition 6 (Strict spectral contraction, uniform in Δtn t_n and ana^n). Let ν1,…,νm>0 _1,…, _m>0 be the eigenvalues of PnP_n. The 2m2m eigenvalues of nT_n are λi±=(1±iΔtnνi)/(1+Δtn2νi) _i^±=(1 t_n _i)/(1+ t_n^2 _i), so |λi±|=(1+Δtn2νi)−1/2≤(1+Δtn2μ)−1/2<1| _i^±|=(1+ t_n^2 _i)^-1/2≤(1+ t_n^2μ)^-1/2<1 for every Δtn>0 t_n>0 and every availability pattern. The proof does not follow from the uncoupled case, because PnP_n is not diagonal; it orthogonally diagonalizes the symmetric PnP_n and reduces to m scalar 2×22× 2 problems. Proposition 6 bounds each nT_n individually, which is not sufficient for boundedness of the trajectory, since the nT_n vary with n and are not normal, and finite products of matrices with spectral radius below one can grow transiently. This gap does not arise in the constant-coefficient setting, and we close it with an energy argument. Theorem 7 (Switching bound governed by availability transitions). Let En:=12(‖zn‖2+(yn)⊤Pnyn)E_n:= 12(\|z^n\|^2+(y^n) P_ny^n) with fn=0f^n=0, and let VN:=#n≤N:an≠an−1V_N:=\#\\,n≤ N:a^n≠ a^n-1\,\ count the visits at which the availability pattern changes. Then EN≤E0exp(1μ∑n=1N‖Pn−Pn−1‖2)≤E0exp(2γVN/μ),E_N≤ E_0\, \! ( 1μ _n=1^N\|P_n-P_n-1\|_2 )≤ E_0\, \! (2γ V_N/μ ), (9) and EnE_n is non-increasing on every maximal run of visits over which ana^n is constant, with exact dissipation En−En−1=−12Δtn2((yn)⊤P2yn+(zn)⊤Pzn)E_n-E_n-1=- 12 t_n^2((y^n) P^2y^n+(z^n) Pz^n) when Pn≡P_n≡ P (Proposition 9). The bound is length-independent: amplification is charged to availability transitions, not to the number of visits or the size of the gaps, and it makes ϵε quantitative through γ/μ≤1−ϵαmax/αminγ/μ≤ 1-ε\, _ / _ . The choice of scheme is not free. The implicit-explicit symplectic discretization used by uncoupled oscillatory SSMs keeps unit-modulus eigenvalues only while Δtn2ν≤4 t_n^2ν≤ 4 for every eigenvalue ν of PnP_n, and otherwise has a real eigenvalue pair with product one, so one of them exceeds unity and the recurrence diverges. In ADNI, gaps range over roughly [0.5,4][0.5,4] years, so enforcing that condition uniformly would require LP≤0.25L_P≤ 0.25, hence αmax≲0.14 _ 0.14, an order of magnitude below our initialization range and enough to collapse the representable frequency band. 4.5 Deep blocks, within-visit attention and task heads GCO block. Following the block design of oscillatory SSMs (Rusch and Rus, 2025), block ℓ maps u~(ℓ−1),nn=1N⊂ℝMh\ u^( -1),n\_n=1^N ^Mh to u~(ℓ),nn=1N\ u^( ),n\_n=1^N by scanning equation 8, forming w(ℓ),n=Cro(ℓ)y(ℓ),n+Dro(ℓ)u~(ℓ−1),nw^( ),n=C^( )_roy^( ),n+D^( )_ro u^( -1),n with Cro∈ℝMh×mC_ro ^Mh× m, and applying a residual GLU (Dauphin et al., 2017) with dropout 0.10.1 (Appendix B). Each block owns its own α^,c^,B,Cro,Dro α, c,B,C_ro,D_ro, while the pattern a1:Na^1:N and the gaps Δtn\ t_n\ are shared. We write hkn:=ιk⊤y(L),n∈ℝdh_k^n:= _k y^(L),n ^d. Within-visit cross-modal attention. GCO layers model how modalities drive one another over time. Contemporaneous associations at a single visit are captured by one multi-head attention block over the M modality tokens at each visit, applied in parallel across n (Vaswani et al., 2017), with an additive availability bias −β(1−ajn)-β(1-a_j^n) on the logit of key j, where β=softplus(β^)≥0β= ( β)≥ 0 is learnable. Taking β→∞β→∞ recovers the hard availability mask of Yao et al. (2024); a finite β down-weights but does not erase an unobserved modality, consistent with its latent state continuing to evolve under Proposition 3. We use H=4H=4 heads with a residual connection and layer normalization, and write h¯kn:=hkn+Attn(hn)k h_k^n:=h_k^n+Attn(h^n)_k and h¯n:=∥kh¯kn h^n:=\|_k h_k^n. Task heads and objective. All heads are affine in h¯ h: same-visit staging is p^n=softmax(Wsth¯n+bst) p^n=softmax(W_st h^n+b_st); landmark prediction is q^=σ(wcv⊤∑nωnh¯n+bcv) q=σ(w_cv _n _n h^n+b_cv), pooled over visits by learned weights ωn _n so that subjects with different numbers of visits are handled without padding artefacts; and longitudinal forecasting is u^kn+h=Wfc,k(h)h¯kn+bfc,k(h) u_k^n+h=W_fc,k^(h) h_k^n+b_fc,k^(h) for h≤Hmax=3h≤ H_ =3. The objective is ℒ=λstℒst+λcvℒcv+λfcℒfc+λreg∑ℓ,k,j,r|c^kj,r(ℓ)|,L= _stL_st+ _cvL_cv+ _fcL_fc+ _reg _ ,k,j,r | c^( )_kj,r |, (10) with ℒstL_st class-weighted cross-entropy over visits with a recorded stage label, ℒcvL_cv class-weighted binary cross-entropy, and ℒfcL_fc a horizon-discounted ℓ1 _1 error masked by akn+ha_k^n+h and normalized by the number of observed targets (Appendix B). The mask is essential: with roughly half of all modality-visit pairs unobserved, scoring against imputed targets would train the model on its own imputation error. The forecasting term also acts as a physical regularizer, forcing the latent oscillators to track measurable quantities, and the ℓ1 _1 penalty on the raw couplings encourages a sparse modality graph without interfering with the budget equation 5. We use (λst,λcv,λfc,λreg)=(1,0.5,1,10−3)( _st, _cv, _fc, _reg)=(1,0.5,1,10^-3). Table 1: Same-visit CN/MCI/AD staging on ADNI and OASIS-3. † denotes that CAMOS is significantly better than the best baseline (p<0.01p<0.01, two-sided Welch’s t-test). Models Accuracy ↑ Macro F1 ↑ Macro Prec. ↑ Macro Recall ↑ Macro Spec. ↑ ADNI LinOSS-IM 0.6593±0.00410.6593_± 0.0041 0.6815±0.00390.6815_± 0.0039 0.6804±0.00450.6804_± 0.0045 0.6864±0.00370.6864_± 0.0037 0.8147±0.00210.8147_± 0.0021 LinOSS-IMEX 0.7206±0.00580.7206_± 0.0058 0.7234±0.00610.7234_± 0.0061 0.7394±0.00550.7394_± 0.0055 0.7424±0.00590.7424_± 0.0059 0.8513±0.00300.8513_± 0.0030 MedFuse 0.6648±0.00720.6648_± 0.0072 0.6697±0.00760.6697_± 0.0076 0.6564±0.00690.6564_± 0.0069 0.7139±0.00700.7139_± 0.0070 0.8284±0.00340.8284_± 0.0034 DrFuse 0.7322±0.00490.7322_± 0.0049 0.7354±0.00460.7354_± 0.0046 0.7488±0.00440.7488_± 0.0044 0.7683±0.00420.7683_± 0.0042 0.8599±0.00240.8599_± 0.0024 CAMOS 0.7824±0.0016†0.7824 _ 0.0016 0.7790±0.0018†0.7790 _ 0.0018 0.7970±0.0022†0.7970 _ 0.0022 0.7786±0.0025†0.7786 _ 0.0025 0.8836±0.0013†0.8836 _ 0.0013 OASIS-3 LinOSS-IM 0.7271±0.00170.7271_± 0.0017 0.3253±0.00530.3253_± 0.0053 0.5334±0.00820.5334_± 0.0082 0.3541±0.00250.3541_± 0.0025 0.6773±0.00140.6773_± 0.0014 LinOSS-IMEX 0.7231±0.00200.7231_± 0.0020 0.3027±0.00640.3027_± 0.0064 0.4222±0.00940.4222_± 0.0094 0.3414±0.00280.3414_± 0.0028 0.6756±0.00160.6756_± 0.0016 MedFuse 0.6682±0.00320.6682_± 0.0032 0.3176±0.00590.3176_± 0.0059 0.2871±0.00910.2871_± 0.0091 0.3558±0.00260.3558_± 0.0026 0.6844±0.00190.6844_± 0.0019 DrFuse 0.7263±0.00230.7263_± 0.0023 0.2805±0.00810.2805_± 0.0081 0.2421±0.00860.2421_± 0.0086 0.3333±0.00000.3333_± 0.0000 0.6667±0.00000.6667_± 0.0000 CAMOS 0.7600±0.0014†0.7600 _ 0.0014 0.5557±0.0028†0.5557 _ 0.0028 0.6280±0.0031†0.6280 _ 0.0031 0.5261±0.0030†0.5261 _ 0.0030 0.7682±0.0015†0.7682 _ 0.0015 5 Experiments Datasets. We evaluate on ADNI (Petersen et al., 2010) with M=4M=4 modalities grouped by acquisition route: imaging (p1=8p_1=8; intracranial-volume-normalized volumes of hippocampus, entorhinal cortex, fusiform gyrus, middle temporal gyrus, whole brain and ventricles, plus FDG and AV45 PET SUVRs), CSF (p2=3p_2=3; Aβ1-42 _1-42, total tau, phosphorylated tau), genetics (p3=3p_3=3; APOE ε4 4 allele count, genotype indicators, polygenic risk summary) and cognition (p4=5p_4=5; MMSE, ADAS-Cog-13, CDR-SB, RAVLT immediate recall, FAQ), with baseline age, sex and years of education as static covariates (ps=3p_s=3). We retain subjects with at least three diagnosed visits and one fully observed modality per visit, and mark a modality available only when all of its features are recorded. We also evaluate on OASIS-3 (LaMontagne et al., 2019), which carries the three modalities (imaging, genetics, cognition) and three targets, refitting only the normalization statistics. Baselines. LinOSS-IM and LinOSS-IMEX (Rusch and Rus, 2025) are uncoupled oscillatory SSMs, the direct parents of CAMOS, and isolate the effect of coupling. Having no native mechanism for missing modalities, they receive concatenated features with unobserved entries zero-filled, the availability vector ana^n as M extra channels, and Δtn t_n both as an input channel and as the per-step discretization interval. MedFuse (Hayat et al., 2022) and DrFuse (Yao et al., 2024) handle missing modalities at the representation level; we replace their modality encoders by per-modality gated recurrent encoders over the visit sequence, so they receive the same temporal information as the SSMs. Appendix I contains results for all the different families of baselines. 6 Results Same-visit Staging. On ADNI, CAMOS outperforms all the baselines (Table 1), improving over DrFuse, by 0.05020.0502 in accuracy (+6.9%+6.9\% relative) and 0.04360.0436 in macro F1 (+5.9%+5.9\%). LinOSS-IMEX outperforms LinOSS-IM, since the implicit-explicit scheme is stable only while Δtn2ν≤4 t_n^2ν≤ 4. This suggests that the tuned baseline learned frequencies low enough to stay in the stable regime. On OASIS-3, evaluated zero-shot with only the normalization statistics refitted, accuracy is weakly informative because one class dominates. DrFuse attains macro recall of exactly 1/31/3 and macro specificity of exactly 2/32/3, the signature of predicting a single class, so its accuracy of 0.72630.7263 equals the majority-class rate. The other baselines are close to this solution, with macro F1 between 0.28050.2805 and 0.32530.3253. CAMOS alone retains classification among the three stages, raising macro F1 from 0.32530.3253 to 0.55570.5557 and macro recall from 0.35580.3558 to 0.52610.5261, while exceeding the majority-class accuracy. Since these relative gains are measured against near-collapsed baselines, the absolute macro F1 of CAMOS is the informative quantity for transfer. Landmark Prediction. Our CAMOS improves AUROC from 0.93860.9386 to 0.96110.9611 (+2.4%+2.4\%) and AUPRC from 0.76890.7689 to 0.86870.8687 (+13.0%+13.0\%) over DrFuse on ADNI dataset (Table 2). The asymmetry between the two metrics is the main finding. AUROC is compressed near its ceiling, spanning only 4.84.8 points across the five models (0.91340.9134 to 0.96110.9611), whereas AUPRC, which is sensitive to the minority positive class, spans 17.917.9 points (0.69000.6900 to 0.86870.8687). This matters practically because landmark prediction is used for trial enrichment, a precision-limited application in which AUPRC is the relevant summary. On OASIS-3 the pattern is sharper. AUROC barely separates CAMOS from the oscillatory baselines (0.96000.9600 vs. 0.95600.9560 for LinOSS-IMEX, +0.4%+0.4\%), while AUPRC rises from 0.41850.4185 to 0.78940.7894 (+88.6%+88.6\%), with all baselines between 0.25940.2594 and 0.41850.4185. Table 2: 3-year MCI-AD landmark prediction on ADNI and OASIS-3. CAMOS is significantly better than the best baseline at †p<0.01 p<0.01 or ∗p<0.05^*p<0.05 (two-sided Welch’s t-test). ADNI OASIS-3 Models AUROC ↑ AUPRC ↑ AUROC ↑ AUPRC ↑ LinOSS-IM 0.9223±0.00310.9223_± 0.0031 0.7102±0.00870.7102_± 0.0087 0.9533±0.00240.9533_± 0.0024 0.3588±0.01420.3588_± 0.0142 LinOSS-IMEX 0.9307±0.00270.9307_± 0.0027 0.7389±0.00740.7389_± 0.0074 0.9560±0.00290.9560_± 0.0029 0.4185±0.01630.4185_± 0.0163 MedFuse 0.9134±0.00450.9134_± 0.0045 0.6900±0.01020.6900_± 0.0102 0.8467±0.00580.8467_± 0.0058 0.2594±0.01210.2594_± 0.0121 DrFuse 0.9386±0.00220.9386_± 0.0022 0.7689±0.00690.7689_± 0.0069 0.8787±0.00470.8787_± 0.0047 0.3183±0.01350.3183_± 0.0135 CAMOS 0.9611±0.0014†0.9611 _ 0.0014 0.8687±0.0041†0.8687 _ 0.0041 0.9600±0.0018∗0.9600^*_ 0.0018 0.7894±0.0093†0.7894 _ 0.0093 Longitudinal forecasting. Figure 2 reports next-visit (h=1h=1) error per modality. On ADNI, CAMOS attains the lowest MAE and RMSE on all four modalities. The largest reductions are on the sparsely observed genetics (MAE 0.7535→0.40880.7535→ 0.4088, −45.8%-45.8\%; RMSE −38.3%-38.3\%) and CSF (MAE 0.7676→0.59060.7676→ 0.5906, −23.1%-23.1\%; RMSE −18.3%-18.3\%), where the four baselines are tightly clustered (MAE 0.75350.7535 to 0.82700.8270 and 0.76760.7676 to 0.78950.7895), so uncoupled oscillatory dynamics and representation-level fusion perform alike there while CAMOS separates clearly. Gains are smaller on cognition (MAE −14.1%-14.1\%, RMSE −7.0%-7.0\%) and imaging (MAE −7.2%-7.2\%, RMSE −11.8%-11.8\%), the only modality on which the baselines split, with LinOSS-IM and MedFuse (MAE 0.44680.4468, 0.45380.4538) ahead of LinOSS-IMEX and DrFuse (0.55670.5567, 0.55620.5562). The same ordering holds zero-shot on OASIS-3: CAMOS reduces MAE most on genetics (0.7875→0.28470.7875→ 0.2847, −63.9%-63.9\%), then on clinical measures (0.5775→0.37560.5775→ 0.3756, −35.0%-35.0\%), and least on imaging (0.7664→0.73330.7664→ 0.7333, −4.3%-4.3\%). Figure 2: Next-visit (h=1h=1) forecasting on ADNI (left) and OASIS (right) per modality. Relative improvements over baselines. All relative improvements in Figure 3 are positive, ranging from +1.3%+1.3\% (macro recall) to +45.8%+45.8\% (genetics MAE) on ADNI and from +0.4%+0.4\% (AUROC) to +88.6%+88.6\% (AUPRC) on OASIS-3, where margins are largest on metrics for which baselines collapse to the majority class (F1 +70.8%+70.8\%, recall +47.9%+47.9\%) and smallest on the imbalance-saturated accuracy and AUROC. The ADNI ablations trace these gains to three design choices. The implicit discretization, stable for every gap, matters most: switching to the implicit-explicit scheme, unstable once Δtn2ν>4 t_n^2ν>4 as the 0.50.5 to 44 year gaps allow, lowers macro F1 from 0.77900.7790 to 0.6530.653 and AUPRC from 0.86870.8687 to 0.80630.8063, and raises forecasting MAE by 39.8%39.8\% (sparse) and 25.5%25.5\% (dense). Coupling lets each modality’s state absorb cross-modal information at co-observed visits before an unobserved interval; removing it raises MAE by 7.7%7.7\% (sparse) and 8.1%8.1\% (dense) and lowers AUPRC to 0.81930.8193, the two regions of largest margin. The availability gate acts selectively: removing it lowers AUROC to 0.95310.9531 and AUPRC to 0.82960.8296 but leaves staging (F1 0.78040.7804) and forecasting (MAE 0.50160.5016 sparse, 0.40140.4014 dense) essentially unchanged. It also explains the transfer result, since the CSF oscillator is not observed in OASIS-3 and stays isolated instead of entering as a zero-filled input block. Figure 3: Relative improvement of CAMOS over the best baseline on each of the reported metrics. Ablation studies. Table 3 evaluates different variants of CAMOS on ADNI. Removing cross-modal coupling (C≡0C≡ 0) degrades every task, lowering staging macro F1 by 2.32.3 points and landmark prediction AUPRC by 4.94.9 points while increasing forecasting MAE by 7.7%7.7\% (sparse) and 8.1%8.1\% (dense), which shows that information flow between modalities is beneficial. Coupling alone is not sufficient, without gating (gkjn≡1g_kj^n≡ 1), staging improves slightly over the uncoupled variant, but landmark prediction and forecasting become worse than having no coupling at all (AUPRC 0.79160.7916, MAE +11.2%+11.2\% sparse and +11.4%+11.4\% dense), plausibly because unobserved or unreliable states propagate indiscriminately across modalities. An asymmetric gate (gkjn=ajng_kj^n=a_j^n) recovers much of this loss and is the strongest ablation on landmark prediction and forecasting, yet it yields the lowest staging F1 among the non-IMEX variants (0.75320.7532), so only the full gating performs well on all tasks at once. Replacing the implicit discretization with an IMEX scheme causes the largest degradation, increasing forecasting MAE by 39.8%39.8\% on sparse and 25.5%25.5\% on dense modalities; the stronger effect on sparse modalities is consistent with their irregular observation gaps. Table 3: Ablation study evaluating the contribution of coupling, gating, and discretisation choices. Staging Landmark Prediction Longitudinal Forecasting Variants Macro F1 ↑ AUROC ↑ AUPRC ↑ Sparse MAE ↓ Dense MAE ↓ w/o coupling (C≡0C≡ 0) 0.7561±0.00340.7561_± 0.0034 0.9530±0.00210.9530_± 0.0021 0.8193±0.00680.8193_± 0.0068 0.5491±0.00570.5491_± 0.0057 0.4354±0.00320.4354_± 0.0032 w coupling ungated (gkjn≡1g_kj^n≡ 1) 0.7694±0.00290.7694_± 0.0029 0.9478±0.00260.9478_± 0.0026 0.7916±0.00810.7916_± 0.0081 0.5668±0.00630.5668_± 0.0063 0.4487±0.00380.4487_± 0.0038 w asymmetric gate (gkjn=ajng_kj^n=a_j^n) 0.7532±0.00410.7532_± 0.0041 0.9562±0.00190.9562_± 0.0019 0.8384±0.00590.8384_± 0.0059 0.5315±0.00490.5315_± 0.0049 0.4176±0.00270.4176_± 0.0027 w IMEX discretization 0.6530±0.00620.6530_± 0.0062 0.9546±0.00240.9546_± 0.0024 0.8063±0.00740.8063_± 0.0074 0.7127±0.00890.7127_± 0.0089 0.5057±0.00460.5057_± 0.0046 CAMOS 0.7790±0.00180.7790_ 0.0018 0.9611±0.00140.9611_ 0.0014 0.8687±0.00410.8687_ 0.0041 0.5097±0.00360.5097_ 0.0036 0.4029±0.00220.4029_ 0.0022 7 Conclusion In this paper, we argue that missingness in multimodal clinical sequences belongs in the transition operator rather than only in the forcing term. We propose CAMOS, a coupled oscillatory state-space model whose transition operator is gated by availability. Making coupling admissible required three new results: a per-channel Gershgorin budget that keeps the effective stiffness positive definite over all 2M2^M availability patterns and all gaps, an energy bound that charges amplification to availability transitions rather than sequence length, and a channel-block matrix family that keeps the associative scan exact at O(M2)O(M^2) overhead. On ADNI, CAMOS outperforms uncoupled oscillatory SSMs and clinical fusion models on same-visit staging, landmark prediction and longitudinal forecasting. On zero-shot OASIS-3, it is the only model that avoids collapse to the majority class. The ablations attribute these gains to the implicit discretization and the coupling. AI use statement Generative AI tools were used for language editing of the manuscript. They were not used to generate experimental results, numerical values, or the statements and proofs of the theoretical results, all of which were derived by the authors. The authors take full responsibility for the final content of this work, including all text, claims and artifacts. Ethics statement CAMOS predicts conversion to Alzheimer’s disease and assigns clinical stages, outputs that could influence trial enrolment or a person’s understanding of their own prognosis. We regard it as a research instrument rather than a decision-support tool. The results here are obtained on volunteer research cohorts that are not representative of the general population. The predicted probabilities are not calibrated, and the availability gate makes the model sensitive to which tests were ordered, a variable that reflects access to care as much as clinical need. Subgroup evaluation, calibration, and an audit of performance conditional on the availability pattern are therefore prerequisites for any clinical claim. All data are used under the existing data use agreements of ADNI and OASIS-3, and no new human-subject data were collected. The authors declare no competing interests. Reproducibility statement The model is fully specified: Table 4 for every hyperparameter and optimizer setting, Appendix F the algorithm in pseudocode, Appendix G the complete configuration table, and Appendix H the exact feature definitions, inclusion criteria, preprocessing and label construction. The assumptions underlying the theory are isolated as Assumptions 1 and 2, and every proof appears in Appendix C. An anonymized code repository accompanies this submission, containing the model implementation, the cohort extraction and preprocessing scripts, the training and evaluation pipeline for CAMOS and all baselines. 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Appendix A Appendix: Notation Symbol Meaning Symbol Meaning M number of modalities (M=4M=4) d oscillators per modality N visits for a subject m=Mdm=Md total latent width h shared projection width L number of GCO blocks pkp_k raw dimension of modality k psp_s static covariate dimension tnt_n time of visit n (years) Δtn t_n tn−tn−1>0t_n-t_n-1>0 ukn∈ℝpku_k^n ^p_k raw features u~kn∈ℝh u_k^n ^h projected akn∈0,1a_k^n∈\0,1\ availability indicator an∈0,1Ma^n∈\0,1\^M availability pattern DaD_a blkdiag(a1Id,…,aMId) (a_1I_d,…,a_MI_d) gkjn=aknajng_kj^n=a_k^na_j^n coupling gate y(t),z(t)∈ℝmy(t),z(t) ^m position, velocity x=(z,y)⊤x=(z,y) full state, ℝ2mR^2m Ak=diag(αk,⋅)A_k= ( _k,·) natural freqs. Ckj=diag(ckj,⋅)C_kj= (c_kj,·) coupling P=A−CP=A-C effective stiffness Pn=A−DanCDanP_n=A-D_a^nCD_a^n gated stiffness P(r)∈ℝM×MP^(r) ^M× M channel-r block Π channel permutation Sn=(I+Δtn2Pn)−1S_n=(I+ t_n^2P_n)^-1 inverse Schur cmpl. nT_n transition operator ϵε Gershgorin budget (0.20.2) δ numerical floor (10−610^-6) μ=ϵαminμ=ε _ stiffness lower bound LP=(2−ϵ)αmaxL_P=(2-ε) _ upper bound γ=maxr‖C(r)‖2γ= _r\|C^(r)\|_2 coupling strength HmaxH_ forecast horizons EnE_n discrete energy VNV_N availability transitions ℱF channel-block matrix family A input-gated linear layers Appendix B Model Component Details Static encoder. The initial condition of Section 4.1 is x0=(z0,y0)⊤=W2sGELU(W1ss+b1s)+b2s∈ℝ2m,W1s∈ℝ2m×ps,W2s∈ℝ2m×2m.x^0=(z^0,y^0) =W_2^s\,GELU\! (W_1^ss+b_1^s )+b_2^s ^2m, W_1^s ^2m× p_s,\ W_2^s ^2m× 2m. (11) GCO block. Block ℓ maps u~(ℓ−1),nn=1N\ u^( -1),n\_n=1^N to u~(ℓ),nn=1N\ u^( ),n\_n=1^N by x(ℓ),nn \x^( ),n\_n =Scan∙(n(ℓ)n,n(ℓ)Fn(ℓ)n), = Scan_ (\T_n^( )\_n,\T_n^( )F_n^( )\_n ), (12) w(ℓ),n w^( ),n =Cro(ℓ)y(ℓ),n+Dro(ℓ)u~(ℓ−1),n,Cro∈ℝMh×m, =C^( )_ro\,y^( ),n+D^( )_ro\, u^( -1),n, C_ro ^Mh× m, (13) u~(ℓ),n u^( ),n =u~(ℓ−1),n+Drop(GLU(GELU(w(ℓ),n))), = u^( -1),n+Drop (GLU (GELU(w^( ),n) ) ), (14) with GLU(v)=σ(W1v)⊙W2vGLU(v)=σ(W_1v) W_2v (Dauphin et al., 2017) and dropout 0.10.1. Within-visit attention. With H heads and dH=d/Hd_H=d/H, Attn(hn)k=WO|∑j=1Mη=1Hexp(⟨WQ(η)⊤hkn,WK(η)⊤hjn⟩/dH−β(1−ajn))∑l=1Mexp(⟨WQ(η)⊤hkn,WK(η)⊤hln⟩/dH−β(1−aln))WV(η)⊤hjn,Attn(h^n)_k=W_O \|_η=1^H _j=1^M ( W_Q^(η) h_k^n,W_K^(η) h_j^n / d_H-β(1-a_j^n) ) _l=1^M ( W_Q^(η) h_k^n,W_K^(η) h_l^n / d_H-β(1-a_l^n) )\;W_V^(η) h_j^n, (15) with β=softplus(β^)≥0β= ( β)≥ 0 learnable and initialized at β0=2 _0=2. Task heads and forecasting loss. The three heads are p^n p^n =softmax(Wsth¯n+bst), =softmax (W_st h^n+b_st ), Wst∈ℝ3×m, W_st ^3× m, (16) q q =σ(wcv⊤∑nωnh¯n+bcv), =σ (w_cv _n _n h^n+b_cv ), ωn=exp(v⊤h¯n)∑n′exp(v⊤h¯n′), _n= (v h^n) _n (v h^n ), (17) u^kn+h u_k^n+h =Wfc,k(h)h¯kn+bfc,k(h), =W_fc,k^(h)\, h_k^n+b_fc,k^(h), h∈1,…,Hmax, h∈\1,…,H_ \, (18) and the masked forecasting term of equation 10 is ℒfc=∑h=1Hmaxγh∑n,kakn+h‖u^kn+h−ukn+h‖1∑h=1Hmaxγh∑n,kakn+hpk,γh=2−(h−1).L_fc= _h=1^H_ _h _n,ka_k^n+h\, \| u_k^n+h-u_k^n+h \|_1 _h=1^H_ _h _n,ka_k^n+h\,p_k, _h=2^-(h-1). (19) (a) Input gating: class A (LinOSS, S4/S5, LRU)1T_12T_23T_34T_4ooa1a^1u1u^1a2a^2u2u^2a3a^3u3u^3a4a^4u4u^4a multiplies the forcing only;nT_n is independent of aao=∑k,makmψk,m(u) o= _k,ma_k^m\, _k,m(u)additive in the availability pattern(b) Transition gating: CAMOS1(a1)T_1(a^1)2(a2)T_2(a^2)3(a3)T_3(a^3)4(a4)T_4(a^4)ooa1a^1u1u^1a2a^2u2u^2a3a^3u3u^3a4a^4u4u^4a gates the coupling insidePnP_n,so nT_n itself depends on ana^no=∏n(an)x0+… o=\!\! _nT_n(a^n)x^0+…non-additive: mixed differences ≠0≠ 0 Figure 4: Input gating versus transition gating. In (a) the indicators multiply the forcing only, so the state unrolls into a sum of per-modality, per-visit contributions and every mixed availability difference vanishes. In (b) the gate enters the homogeneous part, so the state is a product of availability-dependent operators and contains genuine interactions among availability indicators. Appendix C Proofs C.1 Derivation of the implicit recurrence equation 8 Scheme equation 4.4 reads ℳnxn=xn−1+FnM_nx^n=x^n-1+F_n with ℳn=(IΔtnPn−ΔtnI)M_n= ( smallmatrixI& t_nP_n\\ - t_nI&I smallmatrix ). Direct multiplication gives (IΔtnPn−ΔtnI)(Sn−ΔtnPnSnΔtnSnSn)=((I+Δtn2Pn)Sn00(I+Δtn2Pn)Sn)=I2m, pmatrixI& t_nP_n\\ - t_nI&I pmatrix pmatrixS_n&- t_nP_nS_n\\ t_nS_n&S_n pmatrix= pmatrix(I+ t_n^2P_n)S_n&0\\ 0&(I+ t_n^2P_n)S_n pmatrix=I_2m, (20) since (I+Δtn2Pn)Sn=Im(I+ t_n^2P_n)S_n=I_m by definition of SnS_n. Hence n=ℳn−1T_n=M_n^-1 has the stated form. □ C.2 Proof of Lemma 2 Fix a∈[0,1]Ma∈[0,1]^M and a channel r. By Assumption 1 the matrix P(r)(a)∈ℝM×MP^(r)(a) ^M× M with [P(r)(a)]kk=αk,r[P^(r)(a)]_k= _k,r and [P(r)(a)]kj=−akajckj,r[P^(r)(a)]_kj=-a_ka_jc_kj,r for k≠jk≠ j is symmetric, so its eigenvalues are real. By the Gershgorin circle theorem (Horn and Johnson, 2012, Thm. 6.1.1) every eigenvalue ν satisfies |ν−αk,r|≤Rk|ν- _k,r|≤ R_k for some k, with Rk=∑j≠k|akajckj,r|≤∑j≠k|ckj,r|≤(1−ϵ)αk,r,R_k= _j≠ k |a_ka_jc_kj,r |≤ _j≠ k |c_kj,r |≤(1-ε)\, _k,r, (21) the first inequality using ak,aj∈[0,1]a_k,a_j∈[0,1] and the second Assumption 2. Hence ν∈[ϵαk,r,(2−ϵ)αk,r]ν∈[ε _k,r,(2-ε) _k,r] for some k, so ν≥μν≥μ and ν≤LPν≤ L_P. Since ΠP(a)Π⊤=blkdiag(P(1)(a),…,P(d)(a)) P(a) = (P^(1)(a),…,P^(d)(a)) and permutation conjugation preserves the spectrum, the same bounds hold for P(a)P(a). The eigenvalues of Im+Δt2P(a)I_m+ t^2P(a) are 1+Δt2ν≥1+Δt2μ>01+ t^2ν≥ 1+ t^2μ>0, so it is symmetric positive definite and admits a Cholesky factorization. □ C.3 Proof of Proposition 3 If ajn=0a_j^n=0 then for every k≠jk≠ j the (k,j)(k,j) and (j,k)(j,k) blocks of DanCDanD_a^nCD_a^n are aknajnCkj=0a_k^na_j^nC_kj=0 and ajnaknCjk=0a_j^na_k^nC_jk=0. Since A is block diagonal, PnP_n is block diagonal with respect to the splitting j∪([M]∖j)\j\∪([M] \j\). The inverse of a block diagonal matrix is block diagonal, so SnS_n is block diagonal, and each of the four blocks of nT_n in equation 8 is a product of SnS_n, PnP_n and scalars, hence block diagonal. Restricting to the j block gives Pn|jj=AjP_n|_j=A_j and Sn|jj=(Id+Δtn2Aj)−1S_n|_j=(I_d+ t_n^2A_j)^-1. Under the asymmetric gate gkj=ajg_kj=a_j, the (k,j)(k,j) blocks vanish for all k≠jk≠ j but the (j,k)(j,k) blocks equal −aknCjk-a_k^nC_jk, which need not vanish, so PnP_n is block triangular and in general non-symmetric. □ C.4 Proof of Theorem 5 Let Tn:m+1:=TnTn−1⋯Tm+1T_n:m+1:=T_nT_n-1·s T_m+1 with Tn:n+1:=IT_n:n+1:=I. Unrolling the recurrence of Definition 4 from x0=0x^0=0, xn=∑m=1nTn:m+1∑k=1MakmBkmu~km=∑m=1n∑k=1MakmTn:m+1Bkmu~km,x^n= _m=1^nT_n:m+1 _k=1^Ma_k^mB_k^m u_k^m= _m=1^n _k=1^Ma_k^m\,T_n:m+1B_k^m u_k^m, (22) and on=Rxno^n=Rx^n gives the additive form. The map a1:n↦ona^1:n o^n is affine in each coordinate with no products of distinct coordinates, so for any (k,m)≠(j,m′)(k,m)≠(j,m ), Δ(k,m)Δ(j,m′)on=(ψk,m+ψj,m′)−ψk,m−ψj,m′=0 _(k,m) _(j,m )o^n=( _k,m+ _j,m )- _k,m- _j,m =0. For the second claim, take M=2M=2, d=1d=1, L=1L=1 and consider the (1,2)(1,2) entry of Sn=(I2+Δtn2Pn)−1S_n=(I_2+ t_n^2P_n)^-1 with Pn=(α1−gc−gcα2)P_n= ( smallmatrix _1&-gc\\ -gc& _2 smallmatrix ), g=a1na2ng=a_1^na_2^n, c≠0c≠ 0. The 2×22× 2 inverse gives equation 7, which vanishes iff g=0g=0. Choosing an input supported only on modality 11 at visit n and reading out coordinate 22 of yny^n, ono^n is a nonzero multiple of [Sn]12[S_n]_12 when a1n=a2n=1a_1^n=a_2^n=1 and zero if either indicator is zero, so Δ(1,n)Δ(2,n)on≠0 _(1,n) _(2,n)o^n≠ 0 and CAMOS is not in A. □ Remark 8 (Scope of Theorem 5). The theorem concerns a single linear state-space layer. It does not say that deep networks built from such layers cannot represent availability interactions, since a nonlinear readout applied to an additive statistic is not itself additive; it says that in class A the interaction must be reconstructed by the readout from an additive summary, whereas CAMOS represents it inside the recurrence, in the part of the model that is scannable and carries the guarantees of Lemma 2 and Theorem 7. Nor does it separate CAMOS from representation-level fusion such as MedFuse or DrFuse, whose encoders are nonlinear in a. We state it narrowly on purpose. C.5 Proof of Proposition 6 Since PnP_n is symmetric, write Pn=UΛU⊤P_n=U U with U orthogonal and Λ=diag(ν1,…,νm) = ( _1,…, _m), νi≥μ>0 _i≥μ>0 by Lemma 2. Put =blkdiag(U,U)U= (U,U), which is orthogonal. Then ⊤ℳn=(IΔtnΛ−ΔtnI)U M_nU= ( smallmatrixI& t_n \\ - t_nI&I smallmatrix ), which decouples into m independent 2×22× 2 problems (1Δtnνi−Δtn1) ( smallmatrix1& t_n _i\\ - t_n&1 smallmatrix ), each with inverse si(1−ΔtnνiΔtn1)s_i ( smallmatrix1&- t_n _i\\ t_n&1 smallmatrix ), si=(1+Δtn2νi)−1s_i=(1+ t_n^2 _i)^-1. The eigenvalues of this inverse are si(1±iΔtnνi)s_i(1 t_n _i), so |λi±|2=si2(1+Δtn2νi)=si≤(1+Δtn2μ)−1<1| _i^±|^2=s_i^2(1+ t_n^2 _i)=s_i≤(1+ t_n^2μ)^-1<1. Since n=ℳn−1T_n=M_n^-1 and orthogonal conjugation preserves the spectrum, the claim follows. The argument uses only symmetry and positivity of PnP_n, not diagonality, which is why the uncoupled proof does not transfer. □ Proposition 9 (Exact dissipation at fixed availability). Suppose fn=0f^n=0 and Pn≡P_n≡ P. Then En:=12(‖zn‖2+(yn)⊤Pyn)E_n:= 12(\|z^n\|^2+(y^n) Py^n) satisfies, for every Δtn>0 t_n>0, En−En−1=−12Δtn2((yn)⊤P2yn+(zn)⊤Pzn)≤−μ2Δtn2(μ‖yn‖2+‖zn‖2)≤0.E_n-E_n-1=- 12 t_n^2 ((y^n) P^2y^n+(z^n) Pz^n )≤- μ2 t_n^2 (μ\|y^n\|^2+\|z^n\|^2 )≤ 0. (23) Proof. From equation 4.4, zn−1=zn+ΔtnPynz^n-1=z^n+ t_nPy^n and yn−1=yn−Δtnzny^n-1=y^n- t_nz^n, so ‖zn−1‖2 \|z^n-1\|^2 =‖zn‖2+2Δtn⟨zn,Pyn⟩+Δtn2(yn)⊤P2yn, =\|z^n\|^2+2 t_n z^n,Py^n + t_n^2(y^n) P^2y^n, (24) (yn−1)⊤Pyn−1 (y^n-1) Py^n-1 =(yn)⊤Pyn−2Δtn⟨zn,Pyn⟩+Δtn2(zn)⊤Pzn. =(y^n) Py^n-2 t_n z^n,Py^n + t_n^2(z^n) Pz^n. (25) Adding and halving, the cross terms cancel and En−1=En+12Δtn2((yn)⊤P2yn+(zn)⊤Pzn)E_n-1=E_n+ 12 t_n^2((y^n) P^2y^n+(z^n) Pz^n). The inequality uses P⪰μIP μ I and hence P2⪰μ2IP^2 μ^2I. ∎ The dissipation is negligible for the slowest mode at clinical sequence lengths, since ∏n(1+Δtn2μ)−1/2≥exp(−μ2∑nΔtn2) _n(1+ t_n^2μ)^-1/2≥ (- μ2 _n t_n^2) exceeds 0.900.90 for N≈10N≈ 10, Δtn≈1 t_n≈ 1 year and μ≈0.02μ≈ 0.02; high-frequency modes are damped more strongly at long gaps, which is the price of unconditional stability. C.6 Proof of Theorem 7 Write En(Q):=12(‖zn‖2+(yn)⊤Qyn)E_n^(Q):= 12(\|z^n\|^2+(y^n) Qy^n), so En=En(Pn)E_n=E_n^(P_n). Applying Proposition 9 with PnP_n frozen over the step from n−1n-1 to n gives En(Pn)≤En−1(Pn)E_n^(P_n)≤ E_n-1^(P_n), and En−1(Pn)=En−1(Pn−1)+12(yn−1)⊤(Pn−Pn−1)yn−1≤En−1(Pn−1)+12‖Pn−Pn−1‖2‖yn−1‖2.E_n-1^(P_n)=E_n-1^(P_n-1)+ 12(y^n-1) (P_n-P_n-1)y^n-1≤ E_n-1^(P_n-1)+ 12\|P_n-P_n-1\|_2\,\|y^n-1\|^2. (26) By Lemma 2, ‖yn−1‖2≤(2/μ)En−1(Pn−1)\|y^n-1\|^2≤(2/μ)E_n-1^(P_n-1), so En≤En−1exp(1μ‖Pn−Pn−1‖2)E_n≤ E_n-1 ( 1μ\|P_n-P_n-1\|_2), and telescoping gives the first inequality. For the second, if an=an−1a^n=a^n-1 then Pn=Pn−1P_n=P_n-1 and the factor is one, which also proves monotonicity; if an≠an−1a^n≠ a^n-1 then ‖Pn−Pn−1‖2≤‖Dan−1CDan−1‖2+‖DanCDan‖2≤2γ,\|P_n-P_n-1\|_2≤\|D_a^n-1CD_a^n-1\|_2+\|D_a^nCD_a^n\|_2≤ 2γ, (27) using ‖Da‖2≤1\|D_a\|_2≤ 1 and ‖C‖2=maxr‖C(r)‖2=γ\|C\|_2= _r\|C^(r)\|_2=γ. Summing over the VNV_N transition steps completes the proof. □ Proposition 10 (Input-to-state bound). With forcing fnf^n and Pn≡P_n≡ P, En≤En−1+2Δtn‖fn‖ E_n≤ E_n-1+ 2\, t_n\|f^n\|. Proof. Repeating the computation of Proposition 9 with zn−1=zn+Δtn(Pyn−fn)z^n-1=z^n+ t_n(Py^n-f^n) gives 2En≤2En−1+2Δtn‖zn‖‖fn‖2E_n≤ 2E_n-1+2 t_n\|z^n\|\|f^n\|. With A∗=2EnA_ = 2E_n, B∗=2En−1B_ = 2E_n-1, c=Δtn‖fn‖c= t_n\|f^n\| and ‖zn‖≤A∗\|z^n\|≤ A_ , we get A∗2≤B∗2+2cA∗A_ ^2≤ B_ ^2+2cA_ , hence A∗≤c+c2+B∗2≤B∗+2cA_ ≤ c+ c^2+B_ ^2≤ B_ +2c. Dividing by 2 2 gives the claim. ∎ Proposition 11 (Cross-modal leakage and its dependence on the gap). Fix a channel r, an availability vector a∈[0,1]Ma∈[0,1]^M (the ungated case is a=a=1), and k≠jk≠ j. Write s:=Δtn2s:= t_n^2 and Sn(r)=(IM+sPn(r))−1S_n^(r)=(I_M+sP_n^(r))^-1 with Pn(r)=diag(α⋅,r)−DaC(r)DaP_n^(r)=diag( _·,r)-D_aC^(r)D_a. (i) Short gaps. |[Sn(r)]kj−sakajckj,r|≤s2LP2 |[S_n^(r)]_kj-s\,a_ka_j\,c_kj,r |≤ s^2L_P^2, so the leakage grows quadratically in Δtn t_n while sLP≪1sL_P 1. (i) Uniform bound. For every Δtn>0 t_n>0, |[Sn(r)]kj|≤β(s):=12(11+sμ−11+sLP)=s(LP−μ)2(1+sμ)(1+sLP). |[S_n^(r)]_kj |≤β(s):= 12 ( 11+sμ- 11+sL_P )= s\,(L_P-μ)2(1+sμ)(1+sL_P). (28) β increases for s<(μLP)−1/2s<(μ L_P)^-1/2 and decreases afterwards, attains its maximum 12(LP−μ)/(LP+μ)<12 12( L_P- μ)/( L_P+ μ)< 12, and satisfies β(s)≤(LP−μ)/(2sμLP)β(s)≤(L_P-μ)/(2sμ L_P). (i) Long gaps. |[Sn(r)]kj−s−1[(Pn(r))−1]kj|≤s−2μ−2 |[S_n^(r)]_kj-s^-1[(P_n^(r))^-1]_kj |≤ s^-2μ^-2, so the leakage decays as Δtn−2 t_n^-2. (iv) Exact form for M=2M=2. With g=a1a2g=a_1a_2 and D:=α1α2−g2c2>0D:= _1 _2-g^2c^2>0, equation 7 reads [Sn]12=sgc1+s(α1+α2)+s2D,[S_n]_12= s\,gc1+s( _1+ _2)+s^2D, (29) which is unimodal in Δtn t_n with peak at Δt⋆=D−1/4 t_ =D^-1/4, peak value |gc|/(α1+α2+2D)|gc|/( _1+ _2+2 D), and [Sn]12∼gc/(Δtn2D)[S_n]_12 gc/( t_n^2D) as Δtn→∞ t_n→∞. Cross-modal transfer of the previous state is therefore largest at intermediate gaps and vanishes at long ones. Proof. Write P:=Pn(r)P:=P_n^(r) and S:=Sn(r)S:=S_n^(r). By Lemma 2, applied channel-wise, P is symmetric with spectrum in [μ,LP][μ,L_P], so S is symmetric with spectrum in [(1+sLP)−1,(1+sμ)−1][(1+sL_P)^-1,(1+sμ)^-1] and ‖S‖2≤1\|S\|_2≤ 1. (i) From (I+sP)S=I(I+sP)S=I we get S=I−sP+s2P2S=I-sP+s^2P^2S. For k≠jk≠ j the identity contributes nothing and [−sP]kj=sakajckj,r[-sP]_kj=s\,a_ka_jc_kj,r, while |[s2P2S]kj|≤s2‖P‖22‖S‖2≤s2LP2|[s^2P^2S]_kj|≤ s^2\|P\|_2^2\|S\|_2≤ s^2L_P^2. (i) If X is symmetric with spectrum in [λ−,λ+][ _-, _+], then for k≠jk≠ j and m:=(λ−+λ+)/2m:=( _-+ _+)/2, |Xkj|=|ek⊤(X−mI)ej|≤‖X−mI‖2=(λ+−λ−)/2|X_kj|=|e_k (X-mI)e_j|≤\|X-mI\|_2=( _+- _-)/2. Applying this to S gives equation 28. Differentiating, signβ′(s)=sign(1−s2μLP)signβ (s)=sign (1-s^2μ L_P ), which gives the monotonicity and the maximizer s=(μLP)−1/2s=(μ L_P)^-1/2; substituting yields the stated maximum. The decay bound follows from (1+sμ)(1+sLP)≥s2μLP(1+sμ)(1+sL_P)≥ s^2μ L_P. (i) From (I+sP)S=I(I+sP)S=I we also get S=s−1P−1−s−1P−1S=s^-1P^-1-s^-1P^-1S, and ‖P−1S‖2≤μ−1(1+sμ)−1≤s−1μ−2\|P^-1S\|_2≤μ^-1(1+sμ)^-1≤ s^-1μ^-2. (iv) Expanding the denominator of equation 7 gives equation 29. Assumption 2 gives |c|≤(1−ϵ)minα1,α2|c|≤(1-ε) \ _1, _2\, so g2c2<α1α2g^2c^2< _1 _2 and D>0D>0. With f(s)=sgc/Q(s)f(s)=sgc/Q(s) and Q(s)=1+s(α1+α2)+s2DQ(s)=1+s( _1+ _2)+s^2D, we have signf′(s)=sign(gc)⋅sign(Q(s)−sQ′(s))=sign(gc)⋅sign(1−s2D)signf (s)=sign(gc)·sign (Q(s)-sQ (s) )=sign(gc)·sign(1-s^2D), so |f||f| peaks at s=D−1/2s=D^-1/2, where Q=2+(α1+α2)D−1/2Q=2+( _1+ _2)D^-1/2, giving the stated value. The asymptotic form follows because Q(s)∼s2DQ(s) s^2D. ∎ Proposition 12 (Normal modes and frequency shift). Fix a channel r and an availability pattern a, and let P(r)(a)=U(r)Λ(r)U(r)⊤P^(r)(a)=U^(r) ^(r)U^(r) with Λ(r)=diag(ν1(r),…,νM(r)) ^(r)= ( _1^(r),…, _M^(r)). Then the unforced system equation 2 restricted to channel r is a superposition of M decoupled harmonic modes with angular frequencies νi(r) _i^(r) and mode shapes the columns of U(r)U^(r); the frequencies are pinned to the uncoupled ones by Weyl’s inequality, |νi(r)−α(i),r|≤γ≤(1−ϵ)αmax| _i^(r)- _(i),r|≤γ≤(1-ε) _ ; and the channel transfer function is H(r)(s)=(s2IM+P(r)(a))−1H^(r)(s)=(s^2I_M+P^(r)(a))^-1, whose cross-modal entry [H(r)(s)]kj[H^(r)(s)]_kj vanishes for k≠jk≠ j when modalities k and j lie in different connected components of the gated graph P(r)(a)P^(r)(a), and is nonzero for generic couplings when they lie in the same component. The entries |ckj,r||c_kj,r| therefore define d weighted modality graphs whose connected components are the sets of modalities that exchange information in that frequency band. Proof. (i) With f=0f=0, substituting y=U(r)ζy=U^(r)ζ into the channel-r restriction of equation 3 gives ζ′=−Λ(r)ζ =- ^(r)ζ, which is M decoupled scalar oscillators with angular frequency νi(r) _i^(r), all νi(r)>0 _i^(r)>0 by Lemma 2. (i) P(r)(a)=diag(α⋅,r)−DaC(r)DaP^(r)(a)= ( _·,r)-D_aC^(r)D_a is a symmetric perturbation of a diagonal matrix by a symmetric matrix of spectral norm at most γ, so Weyl’s inequality (Horn and Johnson, 2012, Thm. 4.3.1) gives the stated bound. (i) Laplace transforming y′=−P(r)(a)y+f~y =-P^(r)(a)y+ f with zero initial conditions gives H(r)(s)=(s2I+P(r)(a))−1H^(r)(s)=(s^2I+P^(r)(a))^-1. If k and j are disconnected in P(r)(a)P^(r)(a), then s2I+P(r)(a)s^2I+P^(r)(a) is block diagonal and so is its inverse, giving [H]kj=0[H]_kj=0; conversely, if they are connected by a path k=i0,…,iq=jk=i_0,…,i_q=j, the Neumann series (s2I+P)−1=s−2∑q≥0(−s−2P)q(s^2I+P)^-1=s^-2 _q≥ 0(-s^-2P)^q contains the monomial Pi0i1⋯Piq−1iq≠0P_i_0i_1·s P_i_q-1i_q≠ 0 at order q, and for generic couplings the entry is a nonzero rational function of s. ∎ Appendix D Parallel Evaluation, Complexity and Expressivity Equation equation 8 is a linear recurrence with time-varying coefficients, so it is solvable by the associative scan with binary operator (T1,v1)∙(T2,v2):=(T2T1,T2v1+v2),(T_1,v_1) (T_2,v_2):= (T_2T_1,\;T_2v_1+v_2 ), (30) which is associative (Kogge and Stone, 1973; Blelloch, 1990; Martin and Cundy, 2017; Smith et al., 2023). Applying its inclusive scan to [(1,1F1),…,(N,NFN)][(T_1,T_1F_1),…,(T_N,T_NF_N)] returns x1,…,xNx^1,…,x^N in the second components in ⌈log2N⌉ _2N sequential rounds. Associativity is immediate; what is not immediate, and what the channel factorization buys, is that the scan never leaves a cheap structural family. Definition 13 (Channel-block family). Let ℱ⊂ℝ2m×2mF ^2m× 2m be the set of matrices whose four m×m× m blocks TabT_ab satisfy ΠTabΠ⊤=blkdiag(Tab(1),…,Tab(d)) T_ab = (T_ab^(1),…,T_ab^(d)) with Tab(r)∈ℝM×MT_ab^(r) ^M× M. Lemma 14 (Closure). ℱF is closed under addition and multiplication, contains I2mI_2m, and contains nT_n for every n and every availability pattern. A product of two elements of ℱF costs 8dM38dM^3 multiply-accumulate operations, against 8m3=8d3M38m^3=8d^3M^3 for dense 2m×2m2m× 2m blocks. Proof. Let T,T′∈ℱT,T . Conjugating by blkdiag(Π,Π) ( , ) maps each of the four m×m× m blocks of T and T′T to block-diagonal matrices with d blocks of size M×M× M. Block-diagonal matrices with a common partition are closed under addition and multiplication, and each block of the product TT′T is a sum of two products of such matrices, hence again block diagonal with the same partition; therefore TT′∈ℱT . Additivity and I2m∈ℱI_2m are immediate. That n∈ℱT_n follows from equation 8: SnS_n and PnP_n are channel-block-diagonal by equation 4 and Lemma 2, and every block of nT_n is a scalar multiple of SnS_n or of PnSnP_nS_n. For the cost, forming TT′T requires 88 products of m×m× m channel-block-diagonal matrices, each of which is d independent M×M× M products at M3M^3 multiply-accumulates, giving 8dM38dM^3; for dense blocks the same 88 products cost m3=d3M3m^3=d^3M^3 each. ∎ Proposition 15 (Complexity and parameter count). One GCO layer requires Θ(NdM3) (NdM^3) arithmetic work and ⌈log2N⌉ _2N sequential depth, with Θ(NdM2) (NdM^2) memory for the scan carries. An uncoupled oscillatory layer of the same total state width m=Mdm=Md requires Θ(Nm) (Nm) work, so the overhead of coupling is a factor Θ(M2) (M^2), independent of N and d, and the within-visit attention adds Θ(NM2d) (NM^2d) at constant depth. Of the Θ(M2h2+M2dh) (M^2h^2+M^2dh) parameters per layer, only (M2)d M2d belong to the coupling: for M=4M=4 and d=h=64d=h=64 that is 384384 out of 2.8×1052.8× 10^5, about 0.14%0.14\%. Proof. By equation 8 each Sn(r)S_n^(r) is an M×M× M symmetric positive definite inverse costing Θ(M3) (M^3) by Cholesky, and over d channels and N visits this is Θ(NdM3) (NdM^3). Each nT_n lies in ℱF by Lemma 14, whose products cost Θ(dM3) (dM^3); the associative scan performs Θ(N) (N) such operations at depth ⌈log2N⌉ _2N , giving Θ(NdM3) (NdM^3) work and Θ(logN) ( N) depth. Each scan carry stores 4d4d blocks of size M×M× M, hence Θ(NdM2) (NdM^2) memory. For the uncoupled model, C=0C=0 makes every Sn(r)S_n^(r) diagonal, so all products cost Θ(dM)=Θ(m) (dM)= (m) and the total is Θ(Nm) (Nm). The attention block equation 15 computes M×M× M scores at each of N visits with d-dimensional values, hence Θ(NM2d) (NM^2d) at constant depth. For the parameter count, the free parameters of one GCO layer are α^∈ℝM×d α ^M× d; c c on the strict upper triangle in (k,j)(k,j) for each channel, giving (M2)d M2d; the forcing matrices Bk∈ℝd×hB_k ^d× h for k∈[M]k∈[M], giving MdhMdh; the readout Cro∈ℝMh×mC_ro ^Mh× m and skip Dro∈ℝMh×MhD_ro ^Mh× Mh, giving Mhm+M2h2Mhm+M^2h^2; and the two GLU matrices in ℝMh×MhR^Mh× Mh, giving 2M2h22M^2h^2. Since m=Mdm=Md, we have Mhm=M2dhMhm=M^2dh, so the total is Md+(M2)d+M2dh+Mdh+3M2h2=Θ(M2h2+M2dh)Md+ M2d+M^2dh+Mdh+3M^2h^2= (M^2h^2+M^2dh). Substituting M=4M=4 and d=h=64d=h=64 gives 256+384+65536+16384+196608=279,168≈2.8×105256+384+65536+16384+196608=279,168≈ 2.8× 10^5, of which the (42)⋅64=384 42· 64=384 coupling parameters are 0.14%0.14\%. ∎ Expressivity is not lost. Setting c^≡0 c≡ 0 in equation 5 gives C=0C=0 and reduces a CAMOS layer to an uncoupled oscillatory layer with strictly positive diagonal state matrix, so CAMOS blocks inherit universality within continuous causal operators. This certifies only that Assumptions 1 and 2 cost nothing in expressivity, and it is weak here because a symmetric coupled system is orthogonally equivalent to an uncoupled one when everything is observed. The representational gain is therefore due to gating (Theorem 5), since the gate acts in the modality basis and does not commute with the diagonalizing rotation U(r)U^(r) of Proposition 12. Appendix E Dense and Low-Rank Couplings If Ckj∈ℝd×dC_kj ^d× d is dense rather than diagonal, P does not block-diagonalize under Π and three things change. Stability. Assumption 2 is replaced by a spectral condition. If C=C⊤C=C and ‖C‖2≤(1−ϵ)λmin(A)\|C\|_2≤(1-ε) _ (A), then for every a∈[0,1]Ma∈[0,1]^M, λmin(P(a))≥λmin(A)−‖C‖2≥ϵλmin(A)>0 _ (P(a))≥ _ (A)-\|C\|_2≥ε _ (A)>0 using ‖Da‖2≤1\|D_a\|_2≤ 1, and Lemma 2 and everything downstream go through with μ:=ϵλmin(A)μ:=ε _ (A). Enforcing this requires spectral normalization of C, hence a power iteration at every training step rather than the closed form equation 5. It also interacts badly with the standard α∼([0,1])α ([0,1]) initialization, since λmin(A) _ (A) is then near zero and the admissible coupling collapses. Cost. SnS_n becomes a dense m×m× m inverse costing Θ(d3M3) (d^3M^3), and the scan combine leaves ℱF and requires dense 2m×2m2m× 2m products at the same order. The overhead relative to the channel-factored model is Θ(d2) (d^2), about a factor of 40004000 per layer at d=64d=64. Why low rank does not help. The parameterization Ckj=QkjQkj⊤C_kj=Q_kjQ_kj with Qkj∈ℝd×ρQ_kj ^d×ρ reduces the parameter count from d2d^2 to dρdρ per block but not the scan cost, because low-rank-plus-diagonal structure is not closed under multiplication: if T=D1+U1V1⊤T=D_1+U_1V_1 and T′=D2+U2V2⊤T =D_2+U_2V_2 with rankUi=ρrank\,U_i=ρ, then TT′=D1D2+D1U2V2⊤+U1V1⊤D2+U1(V1⊤U2)V2⊤T =D_1D_2+D_1U_2V_2 +U_1V_1 D_2+U_1(V_1 U_2)V_2 has rank up to 3ρ3ρ in its non-diagonal part, so the carried operator’s rank triples at every combine and reaches Nlog23ρN _23ρ after a full scan. Woodbury inversion is available at each individual step but does not survive the scan. This is worth stating because a low-rank PSD coupling is the natural first choice, and it is where intuition from static fusion layers fails in a sequential model. Intermediate option. Partitioning the d channels into d/ρd/ρ groups with dense coupling inside each group interpolates between the extremes: ℱF becomes block-diagonal matrices with d/ρd/ρ blocks of size ρMρ M, the cost becomes Θ(Ndρ2M3) (Ndρ^2M^3), and ρ=1ρ=1 recovers the channel-factored model. Appendix F Algorithms Algorithm 1 One gated coupled oscillator (GCO) layer, forward pass 1: projected inputs u~kn\ u_k^n\, availability akn\a_k^n\, gaps Δtn\ t_n\, initial state x0x^0, parameters α^,c^,Bk α, c,\B_k\, budget ϵε, floor δ 2: states xn=(zn,yn)n=1N\x^n=(z^n,y^n)\_n=1^N 3: αk,r←softplus(α^k,r) _k,r← ( α_k,r) for all k,rk,r ⊳ Θ(Md) (Md) 4: σk,r←∑l≠k|c^kl,r| _k,r← _l≠ k| c_kl,r|; ckj,r←(1−ϵ)minαk,r,αj,rmaxσk,r,σj,r,δc^kj,rc_kj,r← (1-ε) \ _k,r, _j,r\ \ _k,r, _j,r,δ\\, c_kj,r ⊳ Assumptions 1, 2 hold by construction 5: for n=1,…,Nn=1,…,N in parallel do 6: for r=1,…,dr=1,…,d in parallel do 7: [Pn(r)]kk←αk,r[P_n^(r)]_k← _k,r; [Pn(r)]kj←−aknajnckj,r[P_n^(r)]_kj←-\,a_k^na_j^n\,c_kj,r for k≠jk≠ j ⊳ symmetric product gate 8: Sn(r)←CholSolve(IM+Δtn2Pn(r),IM)S_n^(r)← CholSolve (I_M+ t_n^2P_n^(r),\,I_M ) ⊳ SPD by Lemma 2; Θ(M3) (M^3) 9: end for 10: assemble nT_n from Sn(r),Pn(r)\S_n^(r)\,\P_n^(r)\ by equation 8 11: fn←∑kaknιkBku~knf^n← _ka_k^n\, _kB_k u_k^n; vn←n(Δtnfn, 0)⊤v_n _n( t_nf^n,\,0) 12: end for 13: xn←AssocScan∙([(1,v1+1x0),(2,v2),…,(N,vN)])\x^n\← AssocScan_ ([(T_1,v_1+T_1x^0),(T_2,v_2),…,(T_N,v_N)] ) ⊳ ⌈log2N⌉ _2N depth, Θ(NdM3) (NdM^3) work 14: return xn\x^n\ Algorithm 2 CAMOS forward pass for one subject 1: ukn\u_k^n\, akn\a_k^n\, tn\t_n\, static s 2: p^n\ p^n\, q q, u^kn+h\ u_k^n+h\ 3: Δtn←tn−tn−1 t_n← t_n-t_n-1 for all n 4: u~k(0),n←Ekukn+ek u_k^(0),n← E_ku_k^n+e_k if akn=1a_k^n=1, else 00 ⊳ equation 1, parallel in n 5: x0←W2sGELU(W1ss+b1s)+b2sx^0← W_2^s\,GELU(W_1^ss+b_1^s)+b_2^s ⊳ equation 11 6: for ℓ=1,…,L =1,…,L do 7: x(ℓ),n←\x^( ),n\← Algorithm 1 with inputs u~(ℓ−1),n\ u^( -1),n\ and layer-ℓ parameters 8: w(ℓ),n←Cro(ℓ)y(ℓ),n+Dro(ℓ)u~(ℓ−1),nw^( ),n← C^( )_roy^( ),n+D^( )_ro u^( -1),n 9: u~(ℓ),n←u~(ℓ−1),n+Drop(GLU(GELU(w(ℓ),n))) u^( ),n← u^( -1),n+Drop(GLU(GELU(w^( ),n))) 10: end for 11: hkn←ιk⊤y(L),nh_k^n← _k y^(L),n; h¯kn←hkn+Attn(hn)k h_k^n← h_k^n+Attn(h^n)_k ⊳ equation 15, parallel in n 12: p^n,q^,u^kn+h← p^n, q, u_k^n+h← heads equation 16, equation 17, equation 18 Appendix G Implementation Details All models are implemented in PyTorch 2.3 (Paszke et al., 2019) and trained on a single NVIDIA A100 40GB GPU, with the associative scan implemented as a Blelloch up-sweep and down-sweep over batched 2M×2M2M× 2M channel blocks and the Cholesky solves of equation 8 batched over (n,r)(n,r) in float32. Table 4 lists the complete configuration; values marked “grid” are selected on the validation split, and the identical grid and trial budget are applied to every baseline so that no model receives a larger search. The validation selection criterion is the unweighted mean of staging macro-F1, landmark prediction AUROC and one minus the normalized forecasting MAE. Staging is scored by accuracy and macro-averaged F1, precision, recall and specificity; landmark prediction by AUROC and AUPRC; forecasting by per-modality MAE and RMSE in the normalized feature space, averaged over observed targets only. The frequency shift α0>0 _0>0 in Table 4 matters: the standard oscillatory initialization α∼([0,1])α ([0,1]) places mass near zero, and the admissible coupling in equation 5 is proportional to minαk,r,αj,r \ _k,r, _j,r\, so a large fraction of channels would begin with no coupling budget and never recover. Table 4: Complete CAMOS configuration. Group Setting Value Architecture GCO blocks L grid 2,4,6\2,4,6\ oscillators per modality d grid 32,64,128\32,64,128\ projection width h grid 32,64,128\32,64,128\ attention heads H 44 dropout 0.10.1 static encoder 22-layer MLP, width 2m2m, GELU Coupling budget ϵε 0.20.2 numerical floor δ 10−610^-6 gate hard, gkjn=aknajng_kj^n=a_k^na_j^n structure channel-factored, symmetric Initialization αk,r _k,r ([α0,α0+1])U([ _0, _0+1]), α0=0.1 _0=0.1 c^kj,r c_kj,r (0,1/M)N(0,1/M), symmetrized attention bias β0 _0 2.02.0, learnable via softplus remaining weights PyTorch defaults Optimization optimizer AdamW, β=(0.9,0.999)β=(0.9,0.999), ε=10−8 =10^-8 weight decay 10−210^-2, excluding α α, c c, biases peak learning rate grid 10−4,3⋅10−4,10−3\10^-4,3· 10^-4,10^-3\ schedule cosine, 5%5\% linear warmup batch size 6464 subjects gradient clipping global norm 1.01.0 epochs / early stopping max 200200, patience 2020 Objective (λst,λcv,λfc)( _st, _cv, _fc) (1, 0.5, 1)(1,\ 0.5,\ 1) λreg _reg 10−310^-3 (ℓ1 _1 on c c) forecasting horizons Hmax=3H_ =3, discount γh=2−(h−1) _h=2^-(h-1) class weighting inverse frequency on the training split Protocol split subject-level 70/15/1570/15/15, stratified seeds 55 selection criterion mean of staging macro-F1, landmark prediction AUROC, 1−1- normalized forecasting MAE precision float32 throughout, including Cholesky Numerical notes. The Cholesky factorizations in equation 8 are the only place where conditioning could be a concern. By Lemma 2, IM+Δtn2Pn(r)I_M+ t_n^2P_n^(r) has condition number at most (1+Δtn2LP)/(1+Δtn2μ)(1+ t_n^2L_P)/(1+ t_n^2μ), bounded above by LP/μ=(2−ϵ)cond(A)/ϵL_P/μ=(2-ε)\,cond(A)/ε independently of Δtn t_n; with ϵ=0.2ε=0.2 and the initialization of Table 4 this is below 200200, comfortably inside float32. This is a practical dividend of the budget: an unconstrained coupling would allow I+Δt2PI+ t^2P to become near-singular precisely at long gaps, where the model is most needed. Baseline adaptation. LinOSS-IM and LinOSS-IMEX receive the concatenation [u1n;…;uMn]∈ℝp[u_1^n;…;u_M^n] ^p with unobserved entries set to zero, the availability vector ana^n appended as M channels, and Δtn t_n appended as one channel and used as the per-step discretization interval. MedFuse and DrFuse are given per-modality gated recurrent encoders over the visit sequence, so that they see the same temporal information as the SSMs rather than a single pooled vector. Appendix H Dataset Construction ADNI extraction. From the ADNIMERGE table we retain the columns listed in Section 5 together with the visit code, examination date, baseline diagnosis and current diagnosis. Visits are ordered by examination date and tnt_n is expressed in years from the subject’s baseline visit. A subject is included if it has at least three visits with a recorded diagnosis and, at each retained visit, at least one modality fully observed. Volumetric MRI measures are divided by intracranial volume before normalization, which removes head-size variation that would otherwise dominate the leading principal direction. All continuous features are then robust z-scored, u↦(u−med)/IQRu (u-med)/IQR, with statistics computed on the training split only and applied unchanged to validation and test; the robust form is preferred because several ADNI biomarkers, CSF amyloid, are censored at assay limits and have heavy tails. Categorical static covariates are one-hot encoded, and APOE4 is kept as its integer allele count in 0,1,2\0,1,2\, which encodes the known dose response. Table 5: Cohort statistics. M: number of modalities; p=∑kpkp= _kp_k: total raw feature dimension; missing rate: percentage of modality-visit pairs unobserved. Dataset Role M p Subjects Visits Mean visits/subj. Max visits/subj. Missing (%) ADNI train / val / test 4 19 2749 7432 2.7 15 32.5 OASIS-3 zero-shot 3 19 1377 8522 6.2 20 28 Availability convention. akn=1a_k^n=1 if and only if every feature of modality k is recorded at visit n. Per-feature masking was rejected because acquisition is organized by procedure: a lumbar puncture yields the whole CSF panel or none of it, and a PET session yields the whole tracer panel or none of it. Modelling availability at the procedure level is therefore both more faithful and what makes the modality-level gate of Section 4.3 meaningful. Label construction. Staging labels are the recorded diagnosis at each visit, mapped to CN,MCI,AD\CN,MCI,AD\; visits without a recorded diagnosis contribute to the forecasting loss but not to ℒstL_st. For landmark prediction, the index visit is the first visit at which a subject is diagnosed MCI; the label is 11 if AD is recorded at any visit within three years of the index visit, and 00 if the subject has at least three years of subsequent follow-up without an AD diagnosis. Subjects censored before three years without converting are excluded from the landmark prediction task only, so no conversion label is ever imputed. Reversions from MCI to CN are retained in the staging task with their recorded label and do not affect the conversion label unless a later AD diagnosis occurs within the window. OASIS-3 mapping. For OASIS-3 (LaMontagne et al., 2019) we use the same modalities and, where a feature has no exact ADNI counterpart, the nearest available measure of the same construct, refitting only the normalization statistics; the model is otherwise transferred without retraining, so this is a zero-shot evaluation. Appendix I Extended Results In this section, we report all the baselines on both cohorts, across the four families: uncoupled oscillatory SSMs (LinOSS-IM, LinOSS-IMEX, D-LinOSS), general and coupled SSMs (S5, Mamba, Coupled Mamba), irregular-time and missing-value models (GRU-D, mTAN, Raindrop, ODE-RNN), and clinical fusion models (MedFuse, DrFuse). All entries are means over five seeds on the subject-level splits of Appendix H. I.1 Staging with all the baselines Table 6 extends the staging comparison of Section 6. On ADNI the strongest baselines come from three different families, DrFuse (0.73220.7322 accuracy), LinOSS-IMEX (0.72060.7206) and Mamba (0.71870.7187), so no single design axis accounts for baseline performance, and CAMOS exceeds all twelve on every metric. On OASIS-3 the picture changes: accuracy is compressed between 0.62630.6263 and 0.75250.7525 because one class dominates, while macro F1 separates the models from 0.28050.2805 to 0.55570.5557. Every baseline falls below macro F1 0.440.44 and macro recall 0.440.44, with DrFuse at exactly 1/31/3 recall and 2/32/3 specificity, the signature of predicting a single class. CAMOS is the only model that retains discrimination among the three stages under transfer. Table 6: Same-visit CN/MCI/AD staging on ADNI (in-distribution) and OASIS-3 (zero-shot). Models Accuracy ↑ Macro F1 ↑ Macro Prec. ↑ Macro Recall ↑ Macro Spec. ↑ ADNI LinOSS-IM 0.6593±0.00410.6593_± 0.0041 0.6815±0.00390.6815_± 0.0039 0.6804±0.00450.6804_± 0.0045 0.6864±0.00370.6864_± 0.0037 0.8147±0.00210.8147_± 0.0021 LinOSS-IMEX 0.7206±0.00580.7206_± 0.0058 0.7234±0.00610.7234_± 0.0061 0.7394±0.00550.7394_± 0.0055 0.7424±0.00590.7424_± 0.0059 0.8513±0.00300.8513_± 0.0030 D-LinOSS 0.6820±0.03240.6820_± 0.0324 0.6882±0.02710.6882_± 0.0271 0.6912±0.02940.6912_± 0.0294 0.6975±0.02810.6975_± 0.0281 0.8392±0.01520.8392_± 0.0152 S5 0.6562±0.02260.6562_± 0.0226 0.6696±0.01960.6696_± 0.0196 0.6731±0.02070.6731_± 0.0207 0.6804±0.01960.6804_± 0.0196 0.8215±0.01180.8215_± 0.0118 Mamba 0.7187±0.02030.7187_± 0.0203 0.7304±0.01630.7304_± 0.0163 0.7386±0.01740.7386_± 0.0174 0.7412±0.01690.7412_± 0.0169 0.8561±0.00970.8561_± 0.0097 Coupled Mamba 0.7002±0.02820.7002_± 0.0282 0.7052±0.02090.7052_± 0.0209 0.7127±0.02330.7127_± 0.0233 0.7169±0.02140.7169_± 0.0214 0.8447±0.01260.8447_± 0.0126 GRU-D 0.6192±0.03560.6192_± 0.0356 0.6251±0.03910.6251_± 0.0391 0.6318±0.03720.6318_± 0.0372 0.6402±0.03650.6402_± 0.0365 0.8043±0.01980.8043_± 0.0198 mTAN 0.6931±0.01820.6931_± 0.0182 0.7037±0.01650.7037_± 0.0165 0.7094±0.01710.7094_± 0.0171 0.7158±0.01630.7158_± 0.0163 0.8489±0.00920.8489_± 0.0092 Raindrop 0.6776±0.01740.6776_± 0.0174 0.6902±0.01720.6902_± 0.0172 0.6953±0.01820.6953_± 0.0182 0.7021±0.01760.7021_± 0.0176 0.8374±0.01010.8374_± 0.0101 ODE-RNN 0.6619±0.01940.6619_± 0.0194 0.6682±0.01720.6682_± 0.0172 0.6729±0.01840.6729_± 0.0184 0.6817±0.01790.6817_± 0.0179 0.8268±0.01050.8268_± 0.0105 MedFuse 0.6648±0.00720.6648_± 0.0072 0.6697±0.00760.6697_± 0.0076 0.6564±0.00690.6564_± 0.0069 0.7139±0.00700.7139_± 0.0070 0.8284±0.00340.8284_± 0.0034 DrFuse 0.7322±0.00490.7322_± 0.0049 0.7354±0.00460.7354_± 0.0046 0.7488±0.00440.7488_± 0.0044 0.7683±0.00420.7683_± 0.0042 0.8599±0.00240.8599_± 0.0024 CAMOS 0.7824±0.00160.7824_ 0.0016 0.7790±0.00180.7790_ 0.0018 0.7970±0.00220.7970_ 0.0022 0.7786±0.00250.7786_ 0.0025 0.8836±0.00130.8836_ 0.0013 OASIS-3 LinOSS-IM 0.7271±0.00170.7271_± 0.0017 0.3253±0.00530.3253_± 0.0053 0.5334±0.00820.5334_± 0.0082 0.3541±0.00250.3541_± 0.0025 0.6773±0.00140.6773_± 0.0014 LinOSS-IMEX 0.7231±0.00220.7231_± 0.0022 0.3027±0.00640.3027_± 0.0064 0.4222±0.00940.4222_± 0.0094 0.3414±0.00280.3414_± 0.0028 0.6756±0.00160.6756_± 0.0016 D-LinOSS 0.7022±0.03430.7022_± 0.0343 0.3754±0.03760.3754_± 0.0376 0.4127±0.04120.4127_± 0.0412 0.3896±0.02840.3896_± 0.0284 0.6948±0.01630.6948_± 0.0163 S5 0.7525±0.02900.7525_± 0.0290 0.4047±0.03960.4047_± 0.0396 0.4518±0.04370.4518_± 0.0437 0.4102±0.03150.4102_± 0.0315 0.7051±0.01720.7051_± 0.0172 Mamba 0.7345±0.02620.7345_± 0.0262 0.4312±0.04920.4312_± 0.0492 0.4863±0.05210.4863_± 0.0521 0.4389±0.04060.4389_± 0.0406 0.7194±0.02080.7194_± 0.0208 Coupled Mamba 0.7237±0.06360.7237_± 0.0636 0.3594±0.06670.3594_± 0.0667 0.3924±0.07030.3924_± 0.0703 0.3781±0.04520.3781_± 0.0452 0.6890±0.02310.6890_± 0.0231 GRU-D 0.7259±0.04940.7259_± 0.0494 0.3731±0.03630.3731_± 0.0363 0.4035±0.03980.4035_± 0.0398 0.3852±0.03170.3852_± 0.0317 0.6926±0.01590.6926_± 0.0159 mTAN 0.7371±0.03760.7371_± 0.0376 0.2931±0.01100.2931_± 0.0110 0.2846±0.01370.2846_± 0.0137 0.3409±0.00680.3409_± 0.0068 0.6705±0.00340.6705_± 0.0034 Raindrop 0.6263±0.03770.6263_± 0.0377 0.3411±0.02980.3411_± 0.0298 0.3627±0.03120.3627_± 0.0312 0.3597±0.02430.3597_± 0.0243 0.6798±0.01210.6798_± 0.0121 ODE-RNN 0.7154±0.04030.7154_± 0.0403 0.3330±0.01240.3330_± 0.0124 0.3468±0.01540.3468_± 0.0154 0.3502±0.00970.3502_± 0.0097 0.6751±0.00480.6751_± 0.0048 MedFuse 0.6682±0.00320.6682_± 0.0032 0.3176±0.00590.3176_± 0.0059 0.2871±0.00910.2871_± 0.0091 0.3558±0.00260.3558_± 0.0026 0.6844±0.00190.6844_± 0.0019 DrFuse 0.7263±0.00230.7263_± 0.0023 0.2805±0.00810.2805_± 0.0081 0.2421±0.00860.2421_± 0.0086 0.3333±0.00000.3333_± 0.0000 0.6667±0.00000.6667_± 0.0000 CAMOS 0.7600±0.00140.7600_ 0.0014 0.5557±0.00280.5557_ 0.0028 0.6280±0.00310.6280_ 0.0031 0.5261±0.00300.5261_ 0.0030 0.7682±0.00150.7682_ 0.0015 I.2 Landmark prediction with all the baselines Table 7 extends Table 2, and Figure 5 shows the corresponding ROC and precision-recall curves. The two metrics behave differently in each cohort. On ADNI, AUROC spans 0.87950.8795 to 0.96110.9611 while AUPRC spans 0.61870.6187 to 0.86870.8687, and the ordering is not preserved: mTAN is the strongest baseline on AUPRC (0.77500.7750) but only third on AUROC. On OASIS-3 the gap widens, with baseline AUPRC between 0.08010.0801 and 0.59820.5982 at AUROC between 0.78960.7896 and 0.95600.9560, indicating few positive cases in the transfer cohort. Baseline standard deviations there reach ±0.188± 0.188 on AUPRC, against ±0.009± 0.009 for CAMOS, so the transfer margin should be read alongside that variance. Figure 5: ROC and PR curves for 3-year MCI-to-AD landmark prediction on ADNI (left) and OASIS-3 (right). Table 7: 3-year MCI-to-AD conversion on ADNI (in-distribution) and OASIS-3 (zero-shot). ADNI OASIS-3 Models AUROC ↑ AUPRC ↑ AUROC ↑ AUPRC ↑ LinOSS-IM 0.9223±0.00310.9223_± 0.0031 0.7102±0.00870.7102_± 0.0087 0.9533±0.00240.9533_± 0.0024 0.3588±0.01420.3588_± 0.0142 LinOSS-IMEX 0.9307±0.00270.9307_± 0.0027 0.7389±0.00740.7389_± 0.0074 0.9560±0.00290.9560_± 0.0029 0.4185±0.01630.4185_± 0.0163 D-LinOSS 0.9142±0.01870.9142_± 0.0187 0.7022±0.07210.7022_± 0.0721 0.8236±0.04120.8236_± 0.0412 0.1134±0.04810.1134_± 0.0481 S5 0.9087±0.01630.9087_± 0.0163 0.6925±0.062010.6925_± 0.06201 0.9412±0.02650.9412_± 0.0265 0.5982±0.18890.5982_± 0.1889 Mamba 0.8795±0.02040.8795_± 0.0204 0.6187±0.06860.6187_± 0.0686 0.8418±0.05370.8418_± 0.0537 0.1260±0.11040.1260_± 0.1104 Coupled Mamba 0.8913±0.01420.8913_± 0.0142 0.6463±0.04230.6463_± 0.0423 0.8307±0.05810.8307_± 0.0581 0.1177±0.11890.1177_± 0.1189 GRU-D 0.9128±0.01960.9128_± 0.0196 0.7014±0.06780.7014_± 0.0678 0.8654±0.04760.8654_± 0.0476 0.1825±0.10540.1825_± 0.1054 mTAN 0.9346±0.01720.9346_± 0.0172 0.7750±0.06670.7750_± 0.0667 0.8372±0.05030.8372_± 0.0503 0.1224±0.10310.1224_± 0.1031 Raindrop 0.9061±0.01580.9061_± 0.0158 0.6909±0.05840.6909_± 0.0584 0.7896±0.03480.7896_± 0.0348 0.0801±0.03220.0801_± 0.0322 ODE-RNN 0.9215±0.02110.9215_± 0.0211 0.7250±0.07520.7250_± 0.0752 0.8519±0.03920.8519_± 0.0392 0.1308±0.06760.1308_± 0.0676 MedFuse 0.9134±0.00450.9134_± 0.0045 0.6902±0.01020.6902_± 0.0102 0.8467±0.00580.8467_± 0.0058 0.2594±0.01210.2594_± 0.0121 DrFuse 0.9386±0.00220.9386_± 0.0022 0.7689±0.00690.7689_± 0.0069 0.8787±0.00470.8787_± 0.0047 0.3183±0.01350.3183_± 0.0135 CAMOS 0.9611±0.00140.9611_ 0.0014 0.8687±0.00410.8687_ 0.0041 0.9600±0.00180.9600_ 0.0018 0.7894±0.00930.7894_ 0.0093 I.3 Longitudinal forecasting with all the baselines Tables 8 and 9 extend Figure 2 to all baselines. On ADNI, the baselines are tightly clustered on the two sparse modalities, within 0.1120.112 MAE on CSF and, aside from S5, Mamba and Coupled Mamba, within 0.0910.091 on genetics, while CAMOS separates by 0.1150.115 and 0.1060.106 from the best of them. On OASIS-3 the clustering is tighter still: eleven of twelve baselines lie within 0.0410.041 MAE of each other on imaging and within 0.0390.039 on clinical measures, which suggests they largely reproduce the last observed value. CAMOS departs from that cluster on clinical measures (0.37560.3756 against a baseline minimum of 0.65690.6569) and genetics (0.28470.2847 against 0.77110.7711), and only marginally on imaging (0.73330.7333 against 0.77050.7705). Table 8: Next-visit (h=1h=1) forecasting on ADNI per modality. Cognition ↓ CSF ↓ Genetics ↓ Imaging ↓ Model MAE RMSE MAE RMSE MAE RMSE MAE RMSE LinOSS-IM 0.4471 0.7704 0.7715 1.0832 0.8270 0.9662 0.4468 0.8457 LinOSS-IMEX 0.4343 0.7526 0.7676 1.0795 0.8183 0.9607 0.5567 0.8053 D-LinOSS 0.5579 0.8736 0.7651 1.0724 0.7802 0.9731 0.6017 0.8612 S5 0.4973 0.8012 0.7591 1.0689 0.5147 0.9712 0.5945 0.8547 Mamba 0.4932 0.7894 0.7056 1.0180 0.5964 0.9480 0.5646 0.8273 Coupled Mamba 0.4976 0.7948 0.7139 1.0518 0.6166 0.9688 0.5764 0.8394 GRU-D 0.5695 0.8891 0.7349 1.0612 0.8011 0.9804 0.5867 0.8481 mTAN 0.5728 0.8824 0.7517 1.0703 0.8000 0.9779 0.5668 0.8325 Raindrop 0.6561 0.9967 0.8179 1.1386 0.7823 0.9856 0.6781 0.9412 ODE-RNN 0.5578 0.8649 0.7471 1.0655 0.7360 0.9694 0.5771 0.8436 MedFuse 0.4199 0.7136 0.7873 1.1452 0.7644 0.9743 0.4538 0.7828 DrFuse 0.4176 0.6586 0.7895 1.0443 0.7535 0.9610 0.5562 0.7832 CAMOS 0.3588 0.6124 0.5906 0.8537 0.4088 0.5930 0.4146 0.6902 Table 9: Next-visit (h=1h=1) forecasting on OASIS-3 per modality. Imaging ↓ Cognition ↓ Genetics ↓ Model MAE RMSE MAE RMSE MAE RMSE LinOSS-IM 0.7748 0.9968 0.6719 0.9356 0.7993 1.0398 LinOSS-IMEX 0.7745 0.9966 0.6712 0.9352 0.8011 1.0408 D-LinOSS 0.7721 0.9937 0.6789 0.9387 0.7904 1.0341 S5 0.8116 1.0342 0.6916 0.9512 0.7840 1.0296 Mamba 0.7866 1.0081 0.6867 0.9458 0.7711 1.0213 Coupled Mamba 0.7874 1.0094 0.6959 0.9547 0.8026 1.0419 GRU-D 0.7846 1.0057 0.6854 0.9441 0.8054 1.0437 mTAN 0.7705 0.9921 0.6749 0.9369 0.8048 1.0429 Raindrop 0.8968 1.1273 0.8194 1.0836 0.9028 1.1542 ODE-RNN 0.7902 1.0126 0.6775 0.9372 0.7974 1.0385 MedFuse 0.7721 0.9925 0.6765 0.9194 0.8035 1.0377 DrFuse 0.7729 0.9929 0.6569 0.9161 0.8158 1.0446 CAMOS 0.7333 0.9464 0.3756 0.6401 0.2847 0.4927 I.4 Ablation studies on OASIS-3 Table 10 repeats the ablations of Section 6 under zero-shot transfer, and the ordering matches ADNI. The IMEX discretization is the most damaging change, lowering macro F1 from 0.55730.5573 to 0.48800.4880 and raising forecasting MAE by 7.3%7.3\%, 56.1%56.1\% and 119.1%119.1\% on imaging, clinical measures and genetics. Removing the gate is the next most damaging, and removing coupling or replacing the symmetric product by the asymmetric gate costs less. Two qualifications apply. The margins between the full model and the three non-IMEX variants are smaller than the seed standard deviations, which reach ±0.034± 0.034 on macro F1 and ±0.112± 0.112 on AUPRC, so only the discretization effect is resolved at this sample size. The asymmetric gate breaks the symmetry required by Lemma 2, but we observed no divergence, so that guarantee acts as a safeguard rather than a practical necessity at this scale. Table 10: Ablation study evaluating the contribution of coupling, gating, and discretisation choices on OASIS-3. Staging Landmark Prediction Longitudinal Forecasting Variants Macro F1 ↑ AUROC ↑ AUPRC ↑ Img. MAE ↓ Clin. MAE ↓ Gen. MAE ↓ w/o coupling (C≡0C≡ 0) 0.5422±0.02850.5422_± 0.0285 0.9678±0.02030.9678_± 0.0203 0.7677±0.08020.7677_± 0.0802 0.7647±0.01970.7647_± 0.0197 0.3947±0.00970.3947_± 0.0097 0.3214±0.00560.3214_± 0.0056 w coupling ungated (gkjn≡1g_kj^n≡ 1) 0.5341±0.03390.5341_± 0.0339 0.9602±0.03500.9602_± 0.0350 0.7324±0.09910.7324_± 0.0991 0.7798±0.02010.7798_± 0.0201 0.4075±0.01160.4075_± 0.0116 0.3392±0.00760.3392_± 0.0076 w asymmetric gate (gkjn=ajng_kj^n=a_j^n) 0.5493±0.03410.5493_± 0.0341 0.9661±0.03610.9661_± 0.0361 0.7586±0.11210.7586_± 0.1121 0.7702±0.02010.7702_± 0.0201 0.3989±0.01160.3989_± 0.0116 0.3298±0.00760.3298_± 0.0076 w IMEX discretization 0.4880±0.04250.4880_± 0.0425 0.9535±0.03080.9535_± 0.0308 0.7069±0.17820.7069_± 0.1782 0.8173±0.02550.8173_± 0.0255 0.6110±0.02650.6110_± 0.0265 0.7032±0.01310.7032_± 0.0131 CAMOS 0.5573±0.02600.5573_ 0.0260 0.9707±0.02100.9707_ 0.0210 0.7777±0.06260.7777_ 0.0626 0.7614±0.01900.7614_ 0.0190 0.3913±0.01110.3913_ 0.0111 0.3210±0.00450.3210_ 0.0045