Truncated Multicomplex and Higher-Order Topological Models in ALS Drug Discovery

Abstract

Polypharmacology in Amyotrophic lateral sclerosis (ALS) demands models that capture irreducible higher-order drug co-action. We introduce a categorical–topological pipeline that encodes regimens as truncated multicomplexes with a hypergraph–simplicial envelope; irreducible effects are identified by Möbius inversion, and CatMixNet predicts dose-response under monotone calibration while aligning multimodal omics via sheaf constraints. Under face-disjoint evaluation, omics fusion reduced RMSE from 0.164 to 0.149 (≈9%), increased PR-AUC from 0.38 to 0.44, and lowered calibration error to 2.6–3.1% with <10 dose-monotonicity violations per ${10}^3$ surfaces. Triad-irreducible signal strengthened (95th percentile ${\mathsf{\Delta }}^{★}=0.151$; antagonism retained at 24%). A risk-sensitive selector produced triads with toxicity headroom and projected ALSFRS-R slope gains of +0.04–0.05 points/month. Ablations confirmed the necessity of Möbius consistency, sheaf regularization, and monotone heads. Distilled monotone splines yielded compact titration charts with mean error 0.023. The framework supplies reproducible artifacts and actionable shortlists for iPSC and SOD1 validation.

1. Introduction

Amyotrophic lateral sclerosis (ALS) presents multifactorial neuropathology with intertwined immune perturbations, excitotoxic cascades, proteinopathy, mitochondrial stress, vascular dysfunction, gut–brain disruption, as well as network-level degeneration across motor circuits [1,2,3,4,5]. Monotherapy delivered modest gains across multiple trials; interest shifted toward rational multi-agent regimens guided by systems pharmacology together with data-driven design [2,6,7]. This pivot coincided with deeper profiling of ALS heterogeneity across omics layers, clinical trajectories, and biomarker panels, as well as genetic backgrounds [8].

Fixed-combination studies provided instructive exemplars. PrimeC—ciprofloxacin with celecoxib—improved survival in iPSC-motor neuron systems relative to solo components; mechanistic threads included miRNA modulation, NF$\kappa$ B attenuation, iron chelation, COX-2 inhibition, excitotoxicity control, and ferroptosis mitigation, together with increased ciprofloxacin exposure [9]. Historical mixtures such as AMX0035 reached regulatory authorization with moderate effect sizes across controlled trials [10]. Hybrid ligands derived from riluzole with rasagiline demonstrated multi-target neuroprotection within cellular paradigms, including selective MAO-A inhibition together with iNOS suppression in microglial challenge assays [11]. Recent preclinical work reported riluzole coupled with sodium butyrate via gut–brain mechanisms, showing barrier restoration, reduced inflammatory load, and superior motor performance in SOD1^G93A models [12]. Earlier exploratory pairings—for instance, minocycline with creatine—yielded additive neuroprotection within murine settings, yet limited translational durability [13].

Computational prediction accelerated discovery pipelines. Multi-modality mutual-attention systems, jointly learned dual-view architectures, together with in-context large language model strategies improved drug-pair effect estimation across transcriptomic/viability endpoints [14,15,16]. Few-shot predictors based on foundation models reduced the data requirements typical for rare conditions [17]. Multitask deep learners such as MARSY linked gene-expression contexts to interaction scores at scale, generating vast candidate spaces for wet-lab evaluation [18]. Prognostic ML surveys for ALS supplied stratification cues with potential utility for regimen selection within heterogeneous cohorts [19].

Network medicine furnished mechanistic prioritization. SAveRUNNER mapped drug–disease proximity on interactomes, nominating histaminergic modulators together with additional repositioning candidates that countered ALS transcriptional perturbations [20]. A complementary brain-specific multi-omics framework integrated xQTL layers—eQTL, pQTL, sQTL, meQTL, haQTL—yielding 105 putative ALS genes with immune pathway enrichment, then proposing diazoxide, gefitinib, paliperidone, and dimethyltryptamine as BBB-penetrant options [21]. Systems repositioning efforts using therapeutic performance mapping suggested multi-component neuroprotective sets aligned with etiopathogenic pathways [22]. Multitarget small-molecule programs—fasudil derivatives with ROCK2 inhibition together with NRF2 induction—offered a template for pathway-spanning interventions, particularly within SOD1 contexts [22].

Immuno-modulatory combinations reached early clinical testing. CTLA4-Ig with low-dose IL-2 produced disease stabilization signals in a 56-week proof of concept, accompanied by favorable biomarker trajectories covering oxidative stress, inflammatory mediators, and neurodegeneration markers [23]. Off-label combinatorial care surveys further cataloged multi-pathway regimens with pragmatic rationale under real-world constraints [24].

Gene-targeted therapy progressed via antisense oligonucleotides. Tofersen obtained regulatory clearance for SOD1-mediated ALS; real-world programs with systematic reviews described safety, CSF biomarker reduction, together with feasible longitudinal administration [25]. This platform motivates co-administration with anti-inflammatory agents, neuroprotective compounds, microbiome modulators, as well as metabolic correctives [26].

Trial methodology evolved in parallel. Multi-arm multi-stage platforms shortened evaluation cycles and supported response-adaptive allocation, together with biomarker-guided stratification [27]. Regulatory discourse around fixed-dose combinations progressed through recent reviews, with emphasis on evidentiary standards suitable for multi-target contexts [2,3].

Despite their breadth, prevalent mathematical abstractions still compress interactions into pairwise graphs with Bliss/Loewe/HSA surrogates, thereby masking irreducible higher-order co-action across triads, tetrads, and dose-dependent faces. Hypergraph formalisms, together with simplicial complexes, provide native carriers for k-body relations; categorical semantics then attach functorial dose maps, face/degeneracy operators, and homotopical invariants that distinguish reducible mixtures from genuinely non-decomposable regimens. Prior art across ML prediction, network proximity, and clinical experimentation, together with platform design, therefore motivates a formal upgrade: truncated multicomplex model categories that encode regimen objects as finite-multiset simplices, experimental contexts as morphisms, protocol equivalence via homotopies, as well as biomarker/omics sheaves over faces [19,20,21]. Such a scaffold aligns with evidence from PrimeC, hybrid ligands, ASO therapy, immuno-modulation, and gut–brain targeting—each instance revealing an interaction structure that resists dyadic reduction [9,12].

This study establishes a categorical–topological scaffold for ALS combination design: truncated multicomplex model categories with a hypergraph–simplicial envelope encode k-body co-action; a Möbius-consistent predictor (CatMixNet) retrieves irreducible effects under monotone dose constraints. Sheaf alignment stitches multimodal omics to dose-face structure, yielding calibrated estimates together with interpretable $\mathsf{\Delta }$ signals across triads. A risk-aware decision layer integrates uncertainty, toxicity headroom, and mechanistic breadth, producing titratable candidates whose molecular projections map to ALSFRS-R slope expectations. The pipeline supplies reproducible artifacts—dose splines, selection tables, ablations—showing stable gains versus baselines within face-disjoint evaluations.

The truncated multicomplex model category (TMC–MC) formalizes k-body co-action by assigning regimen faces up to a chosen truncation level T, thereby restricting the combinatorial explosion while preserving the identifiability of non-decomposable effects through Möbius-consistent face relations. Within this scaffold, objects encode finite-multiset simplices with dose vectors; morphisms transport embeddings, doses, and context sections in a functorial manner; and weak equivalences preserve a utility functional tied to pharmacodynamic response. Such structure curtails dyadic collapse, ensures lattice coherence across subfaces, sustains monotone dose constraints through a dedicated dose functor, and then support homotopies that quotient protocol variants without erasing observable content. In practice, this yields a principled separation between reducible contributions on lower-dimensional faces and irreducible signal on higher faces, with $\mathsf{\Delta }(·)$ computed on the face poset, providing an unambiguous ledger for triads or tetrads subject to truncation. A different cadence follows for the hypergraph–simplicial envelope (HSE): drugs populate vertices, hyperedges encode multi-agent contexts with explicit dose labels, simplicial closure supplies every nonempty subface, while sheaf attachments endow each face with multimodal sections—transcriptomic shifts, chromatin accessibility, target maps, viability summaries—together with restriction maps that implement biochemical marginalization. This envelope converts irregular regimen graphs into a face lattice amenable to Möbius inversion, message-passing on the 1-skeleton with higher-face pooling, calibration audits on face-disjoint splits, as well as rigorous propagation of uncertainty along restriction morphisms. The result is a geometrically faithful carrier for k-ary pharmacology, where inference aligns with algebraic topology rather than ad hoc pairwise surrogates. Clinical translation follows a measured trajectory: risk-sensitive selection over HSE faces produces titratable triads with toxicity headroom; distilled monotone splines generate bedside dose charts with order preservation; mechanistic projections through pathway matrices quantify dispersion together with redundancy to avoid narrow target piling; and transfer functions from molecular summaries to ALSFRS-R slopes suggest effect-size ranges suitable for powering prospective tracks. Next steps prioritize IV–iPSC motor neuron grids under face-disjoint pre-registration, escalation in SOD1^G93A cohorts with humane endpoints, stratified cohorts guided by omic fingerprints, adaptive interior-dose sampling where curvature peaks, pharmacokinetic–pharmacodynamic reconciliation for feasible corridors, as well as regulatory-aligned fixed-combination dossiers with prespecified analyses, all issued with full provenance, calibration diagnostics, and lattice-consistency checks, together with interval-aware decision reports.

CatMixNet imposes Möbius consistency across the face lattice so that any predicted family $\widehat{R}\left(g\right)\ast g\subseteq f$ yields an irreducible component $\widehat{\mathsf{\Delta }}\left(f\right)=\sum \ast g\subseteq f\mu (g,f)\widehat{R}\left(g\right)$; an isotonic output head enforces order preservation along each dose axis, which isolates higher-order mass while suppressing leakage from reducible terms within grids constrained by toxicity bounds. Training minimizes a composite objective: pointwise error on R; a Möbius penalty that ties subface predictions to higher faces; a soft toxicity term that discourages excursions beyond feasible corridors; and a sheaf coherence loss that secures cross-modal agreement on overlaps. Sheaf alignment operates by attaching modality-specific sections to faces together with restriction maps ${\rho }_{e\to f}$; a lightweight sheaf autoencoder learns a shared latent that minimizes a cochain energy proportional to ${\sum }_{f\subset e}{|{\rho }_{e\to f}\widehat{s}\left(e\right)-\widehat{s}\left(f\right)|}^2$, thereby gluing transcriptome/proteome/accessibility/target priors onto dose-face structure while damping batch confounders through section-level covariates. Calibrated estimates arise from deep ensembles with conformalized residuals; linearity of the Möbius operator on residuals transfers coverage to $\widehat{\mathsf{\Delta }}$ so that triad signals become sign-interpretable; $\widehat{\mathsf{\Delta }}>0$ denotes beneficial co-action; $\widehat{\mathsf{\Delta }}<0$ denotes antagonism; interval width quantifies reliability; gating via $\widehat{\mathsf{\Delta }}\ge \delta$ together with headroom $(\tau -\widehat{Tox})$ yields laboratory shortlists that respect safety margins. The decision layer computes $\mathbb{E}\left[U\right]-\alpha ,\mathrm{Var}\left[U\right]$ under the predictive posterior, enforces $Tox(\mathbf{d}\ast f)\le \tau$, requires $\widehat{\mathsf{\Delta }}\ge \delta$, bounds sheaf inconsistency by $ϵ$; projections to pathway space with matrix $\mathbf{W}$ deliver dispersion D plus redundancy Q, after which a regularizer encourages broad coverage with limited overlap; a CVaR objective can tilt selection toward robust profiles under heavy-tail uncertainty; monotone spline distillation of dose surfaces supplies titration charts with small teacher–student gaps. Alignment with ALSFRS-R slope uses a transfer map G that regresses molecular summaries together with pathway projections toward cohort trajectories from Answer ALS; post hoc isotonic calibration enforces correct ordering of expected benefit across candidates; cross-validated concordance via Spearman/Kendall metrics monitors the association strength; in addition, a multi-task auxiliary head predicting neurofilament light provides a biologically concordant tether that stabilizes G in low-signal regimes. Reproducible artifacts include containerized training scripts with pinned digests, dataset manifests with hashes, weight checkpoints tagged by seed, face-level monotone spline parameters, lattice diagrams carrying $\widehat{\mathsf{\Delta }}$ values with uncertainty bars, calibration reports with ECE/coverage tables, selection ledgers enumerating $(f,\mathbf{d}\ast f)$, and ablation notebooks that toggle Möbius/sheaf/monotonicity modules, together with audit logs on violation counts; these materials enable independent reruns, provenance verification, sensitivity replays, steady drift surveillance after new data arrivals, and faithful re-computation of clinical projections under fixed protocols.

The immediate motivation for this study arises from ALS heterogeneity across molecular strata, dose constraints, and clinical trajectories; conventional combinatorial design still relies on dyadic surrogates such as Bliss/Loewe/HSA, which compress higher-order co-action into pairwise graphs. Recent pipelines couple perturbational omics with viability grids, employ multi-task transformers, contrastive encoders, graph message passing, as well as few-shot transfer from foundation backbones. These advances reduce data hunger, improve ranking fidelity under sparse supervision, and secure moderate calibration under plate effects. However, triad non-decomposability remains largely invisible, dose monotonicity seldom enforced explicitly, and uncertainty rarely propagated across subface relations. Network proximity on interactomes proposes repositioning routes together with mechanistic priors, nevertheless lacks a face-aware calculus at variable cardinality. Platform trials shorten evaluation cycles; however, model outputs typically ignore lattice consistency, leaving attribution ambiguous once subfaces overlap. Consequently a structure-preserving formalism is needed: one that records irreducible mass on each face, respects feasibility corridors, accommodates multimodal sections, and supplies audit-ready summaries for clinical gating.

The state-of-the-art frame starts with a truncated multicomplex with a hypergraph–simplicial envelope (TMC-HSE) that treats regimen faces as first-class objects, while a Möbius-consistent predictor recovers an irreducible ledger $\mathsf{\Delta }\left(f\right)$ under isotonic dose heads. Sheaf gluing attaches multi-omic sections to faces; restriction maps transmit evidence from cofaces to subfaces; and a dose functor enforces monotone transports inside a toxicity-bounded region. Calibration employs conformal residuals on face-disjoint folds; uncertainty then transfers linearly to the Möbius space, making $\mathsf{\Delta }$ intervals interpretable. A risk-aware selector ranks faces using expected utility minus variance with toxicity headroom, pathway dispersion, and redundancy control; distilled monotone splines yield titration charts suitable for bedside navigation. This synthesis provides a mathematically faithful carrier for k-body pharmacology, introduces explicit provenance for every prediction, promotes reproducibility through containerized training manifests, seeds, and hashes, together with fixed evaluation ledgers, while preserving auditability across protocol morphisms, subface closures, and dose projections.

2. Materials and Methods

2.1. Design Principles

This chapter codifies a categorical–topological pipeline for rigorous modeling of multi-drug co-action in amyotrophic lateral sclerosis (ALS). The approach integrates truncated multicomplex model categories (TMC–MC) with a hypergraph–simplicial envelope (HSE) to encode an irreducible higher-order pharmacodynamic structure, followed by parametric learning for outcome prediction, and dose control, as well as protocol selection. Objectives: (i) formalize k-body co-action that resists dyadic reduction; (ii) couple biomarker layers with regimen faces via sheaf-like attachments; (iii) propose decision maps from categorical objects to measurable clinical/proxy outcomes; (iv) construct a reproducible experimental program covering in vitro, in silico, and in vivo segments with aligned endpoints; and (v) quantify uncertainty, sensitivity, and bias. Constraints include limited ALS sample sizes, heterogeneous modalities, and batch effects, together with translational gaps across cellular, animal, and clinical settings.

2.2. Datasets

2.2.1. DrugComb Screen Aggregation

Large multi-drug screen compendia support interaction signal discovery across cell contexts [28]. Extracted fields: Agent identifiers, dose grids, response matrices, and meta-data for assay format. Utility: Generic shape priors for dose-response; negative examples at scale; calibration of interaction baselines.

2.2.2. MUDI Multimodal Resource

MUDI aggregates pharmacodynamic signals with paired omics for drug–drug interaction studies [29]. Utility: Joint embedding across molecular readouts; mechanistic constraints for sheaf assignments over HSE faces.

2.2.3. Answer ALS Cohort

A large-scale ALS platform with clinical trajectories together with multi-omics [30]. Utility: Patient-level priors, stratification variables, and outcome anchors. ALSFRS-R slopes, neurofilament light, and respiratory metrics serve as distal outcomes or calibration targets.

2.2.4. diMN ATAC-seq Program

Chromatin accessibility tracks in iPSC-derived motor neuron models [31]. Utility: Context-specific regulatory annotations for target mapping, dose-dependent accessibility shifts, and attachment to higher-dimensional faces encoding multi-agent stimuli. As summarized in Table 1, we use four curated resources with their modalities, access routes, scale, and roles in the pipeline.

Table 1. Curated datasets with core modalities and access features, as well as intended roles in the pipeline.

Resource	Modality Set	Access	Scale (Rows)	Intended Role
DrugComb [28]	Viability grids; meta	Public portal	>${10}^6$ entries	Baselines; priors; QA
MUDI [29]	PD signals; multi-omics	Zenodo DOI	${10}^4$–${10}^5$ pairs	Sheaf constraints; fusion
Answer ALS [30]	Clinical; multi-omics	Portal request	${10}^3$ participants	Outcome anchors; stratification
diMN [31]	ATAC-seq	LINCS	${10}^2$–${10}^3$ runs	Regulatory priors; target maps

2.3. Data Curation, Harmonization, Provenance

Agent dictionaries harmonize synonyms to InChIKey with crosswalks to DrugBank/ChEMBL identifiers. Dose units unify to $\mathsf{\mu }$M; response normalization maps assay-specific readouts to fractional survival or standardized effect. Cell context ontology maps each experiment to tissue of origin, genetic background, and maturation status, as well as relevant perturbation states. Batch fields preserve plate identifiers, date codes, and laboratory codes. Clinical cohort variables receive consistent coding for ALSFRS-R subscales, onset site, and respiratory support, together with survival censoring. Provenance manifests store SHA256 hashes for raw files; each transform includes a YAML step record with timestamp, code commit, and container digests.

Pharmacodynamic matrices combine with paired omics to form sheaf sections per HSE face; joint embedding aligns transcript levels, protein abundance, phospho-signaling, chromatin accessibility, metabolite profiles, and imaging-derived phenotypes where present. Contrastive alignment, CCA-style projections, and graph-regularized autoencoders produce a shared latent that respects HSE incidence; mechanistic constraints guide sheaf assignments via target matrices, pathway priors, BBB permeability flags, and metabolism annotations, thus tempering spurious co-variation. Interpretability rises through face-wise loadings that link $\mathsf{\Delta }\left(f\right)$ to pathway coverage; accuracy improves via multi-view denoising, context-specific calibration, and leakage-resistant splitting on faces. Agent dictionaries canonicalize synonyms to InChIKey (salt-stripped, tautomer-normalized, stereochemistry pinned); crosswalks to DrugBank, ChEMBL supply target panels, assay notes, and PK hints, thereby enabling pathway projections with traceable provenance. Doses unify to $\mathsf{\mu }$M using base-form molecular weight with salt correction, ${log}_{10}$ scaling for encoders; responses map to fractional survival via control-based percent effect relative to the vehicle and reference kill, with LOESS/B-score plate correction, robust z within batch, and optional variance-stabilizing transforms; bounded monotone rescaling to $[0,1]$ preserves order across assays. Such harmonization reduces batch drift, aligns dose semantics across platforms, and supports Möbius transforms on comparable scales, thus enabling faithful higher-order inference within the TMC-MC, +, HSE, +, and MUDI pipeline.

Identifier harmonization proceeds via a disciplined pipeline: synonyms collapse to a salt-stripped, tautomer-normalized InChIKey with stereochemistry pinned; secondary keys link to DrugBank and ChEMBL for target panels, provenance notes, permeability hints, and metabolic cautions. Doses convert to $\mathsf{\mu }$M using base-form molecular weights with explicit salt corrections; encoder inputs consume ${log}_{10}$-scaled values after clipping to assay-specific quantiles to limit leverage by outliers. Plate artifacts receive correction through robust control anchoring with B-score/LOESS hybrids, followed by within-batch variance stabilization; residual spatial drift is audited via row/column regressions with shrinkage on plate effects. The consolidated screen matrix then materializes as a face-by-dose grid indexed by regimen faces (dyads, triads, tetrads) with per-cell utility $U=\varphi \left(R\right)$, where $\varphi$ maps raw viability to a bounded, monotone scale in $[0,1]$; this reparameterization equalizes assay idiosyncrasies, thus enabling like-for-like comparison across combinations, dose vectors, and cellular contexts. Dyad counts provide dense lower-face coverage that stabilizes Möbius inversion; triad counts carry the minimal substrate for irreducible higher-order signals; tetrad counts, although sparser, serve as stress tests for identifiability under toxicity limits.

2.4. Mathematical Substrate: Hypergraph–Simplicial Envelope

Let $\mathcal{A}$ denote a finite set of agents. A regimen corresponds to a finite multiset $M\subseteq \mathcal{A}\times {\mathbb{R}}_{\ge 0}$ capturing identity with dose. The hypergraph $H=(V,E)$ uses $V=\mathcal{A}$ with hyperedges $e\subseteq V$ that carry a face label $\varphi \left(e\right)$, providing dose vectors ${\mathbf{d}}_e\in {\mathbb{R}}_{\ge 0}^{\left|e\right|}$. The simplicial envelope K augments H by closing under nonempty subsets: if $e\in E$, then every $f\subseteq e$ becomes a face in K with induced ${\mathbf{d}}_f$ by projection. A sheaf $\mathcal{S}$ attaches feature sections to faces: transcriptomics, proteomics, chromatin accessibility, and target annotations, together with screen response summaries. Restriction maps ${\rho }_{e\to f}:\mathcal{S}\left(e\right)\to \mathcal{S}\left(f\right)$ encode biochemical compatibility or experimental marginalization.

Regimen objects appear as finite faces indexed by agent multisets with dose vectors; morphisms encode face embeddings, dose transports, and context lifts. Irreducible higher-order pharmacodynamic interaction emerges as a non-zero Möbius component on the face lattice, thus separating genuine k-body influence from aggregates of lower faces; the truncation level T bounds categorical breadth while preserving identifiability. A hypergraph with simplicial closure supplies subface completeness, hence marginalization stays coherent across projections; hyperedges retain multiset cardinality, doses remain face-specific, and restriction maps preserve biochemical meaning. Traditional pairwise graphs with Bliss/Loewe/HSA surrogates conflate triad phenomena with dyadic surrogates, whereas the envelope records dose-conditioned non-decomposability on each face, including heterogeneity across contexts. Topological carriers deliver coface/face incidence, nerve connectivity, and sheaf-attachable spaces for omics sections; categorical semantics provide functorial transport between protocols, and homotopies encode equivalence classes where observables remain invariant under permissible transformations. Such structures yield a lattice-consistent decomposition $R\left(f\right),\mathsf{\Delta }\left(f\right)$; hence, attribution becomes face-local, dose-aware, and mechanistically alignable.

2.4.1. Irreducible Co-Action via Möbius Inversion

Let $R\left(f\right)$ denote the measured response for face f at the dose projection ${\mathbf{d}}_f$. Consider the face poset $(K,\subseteq )$ with zeta function

$$\zeta (g,f)=\left\{\begin{array}{cc}1,\hfill & g\subseteq f,\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(1)

Let $\mu$ be the Möbius function on this poset, characterized by

$$\sum _{g\subseteq h\subseteq f}\mu (h,f)={\delta }_{g,f}.$$

(2)

The Möbius transform yields the irreducible contribution

$$\mathsf{\Delta }\left(f\right)=\sum _{g\subseteq f}\mu (g,f)R\left(g\right).$$

(3)

with inverse relation

$$R\left(f\right)=\sum _{g\subseteq f}\mathsf{\Delta }\left(g\right).$$

(4)

For the empty face, take $\mathsf{\Delta }(⌀)=R(⌀)$:

$$\mu (g,f)=\left\{\begin{array}{cc}{(-1)}^{\left|f\right|-\left|g\right|},\hfill & g\subseteq f,\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(5)

A face f of size $\ge 3$ with $\mathsf{\Delta }\left(f\right)\ne 0$ indicates a non-decomposable effect not explained by lower-dimensional faces. Dose-specific non-decomposability is recorded as $\mathsf{\Delta }(f;{\mathbf{d}}_f)$ when dose resolution is explicit.

2.4.2. Truncated Multicomplex Model Categories

Objects $\mathrm{Ob}\left({\mathcal{C}}_T\right)$ correspond to regimen faces up to truncation level T (max cardinality T). Morphisms: Triples $(\iota ,\theta ,\kappa )$ mapping (i) face embeddings $\iota :f\hookrightarrow f^{\prime }$; (ii) dose transforms $\theta :{\mathbf{d}}_f\mapsto {\mathbf{d}}_{f^{\prime }}$; and (iii) context maps $\kappa$ transporting evidence sections through $\mathcal{S}$ restrictions. Composition follows

$$({\iota }_2,{\theta }_2,{\kappa }_2)\circ ({\iota }_1,{\theta }_1,{\kappa }_1)=({\iota }_2\circ {\iota }_1,{\theta }_2\circ {\theta }_1,{\kappa }_2\circ {\kappa }_1).$$

Homotopies identify protocol variations with equivalent observable content, producing a model structure whose weak equivalences preserve a chosen functional U (defined below).

2.4.3. Dose Functor with Toxicity Constraint

Define a dose functor $\mathsf{D}:{\mathcal{C}}_T\to \mathbf{Dose}$ with $\mathbf{Dose}$ a category where objects are vectors in ${\mathbb{R}}_{\ge 0}^n$, morphisms are monotone maps satisfying componentwise positivity. A toxicity functional $Tox\left({\mathbf{d}}_f\right)$ constrains feasible sets via $Tox\left({\mathbf{d}}_f\right)\le \tau$. A benefit functional $U(·)$ measures the desired effect; for cell viability, $U=1-R$; for clinical slope, U maps to ALSFRS-R stabilization over a window.

2.5. Predictive Module: CatMixNet

A multi-component network, CatMixNet, consumes face-conditioned features with dose embeddings then returns a response estimate $\widehat{R}(f;{\mathbf{d}}_f,\mathbf{x})$ for context $\mathbf{x}$.

A monotone dose transformer with isotonic heads supplies calibrated surfaces $\widehat{R}(f;\mathbf{d},\mathbf{x})$; penalties enforce Möbius consistency across the lattice, toxicity feasibility over $\mathcal{D}\left(f\right)$, and sheaf coherence across restrictions. Variational components quantify predictive dispersion; conformal wrappers deliver coverage-controlled intervals under face-disjoint validation. A risk-sensitive functional over utility, variance, toxicity headroom, and pathway dispersion selects $(f,\mathbf{d})$ with titratable corridors; distilled monotone splines compress dose charts suitable for iterative titration. Integration of TMC-MC together with HSE increases interpretability via explicit $\mathsf{\Delta }\left(f\right)$ terms on each simplicial face; reliability improves through lattice coupling that penalizes incoherent subface/subset forecasts, thus stabilizing higher-order attribution. Decision outputs include ranked triads with uncertainty, mechanistic breadth, redundancy metrics; categorical morphisms map between assay schemas, yielding portable recommendations across platforms while preserving observable content. The result is a formal scaffold where prediction, constraint audit, selection, and titration cohere under one semantics, with face-level guarantees rather than plate-wise heuristics.

2.5.1. Encoders

(i) HSE-Encoder: Constructs face embeddings via message passing on the 1-skeleton with higher-face pooling. (ii) DoseTransformer: Sinusoidal positional maps for log doses, with multihead attention across dose channels. (iii) ContextNet: Tabular/omics encoder with LayerNorm, gated linear units, together with residual stacks.

2.5.2. Face Aggregator

A permutation-invariant aggregator $A\left({\left\{{\mathbf{z}}_v\right\}}_{v\in f}\right)$ using DeepSets-style sum pooling together with attention pooling produces ${\mathbf{z}}_f$.

2.5.3. Output Head

A monotone calibration head $\psi$ enforces a partial order in the dose: if $\mathbf{d}⪯{\mathbf{d}}^{\prime }$, then $\psi (·,\mathbf{d})\ge \psi (·,{\mathbf{d}}^{\prime })$ for viability; isotonic layers achieve this property. Hyperparameter ranges and defaults are summarized in Table 2.

Table 2. CatMixNet hyperparameters with ranges and defaults, together with purpose.

Symbol	Range	Default	Purpose
T	$\{2,3,4\}$	3	Truncation level for faces
$d_{\mathrm{emb}}$	$\{64,128,256,512\}$	256	Hidden width across modules
$L_{\mathrm{HSE}}$	1–6	3	Message passing layers
$L_{\mathrm{dose}}$	1–6	4	Attention blocks for dose transformer
h	$\{2,4,8\}$	4	Attention heads
${\lambda }_{\mathsf{\Delta }}$	${10}^{-4}$–${10}^{-1}$	${10}^{-3}$	Penalty on signed irreducible error
${\lambda }_{\mathrm{tox}}$	${10}^{-4}$–${10}^{-1}$	${10}^{-2}$	Soft toxicity regularization
$p_{\mathrm{drop}}$	0.0–0.5	$0.2$	Dropout in encoders
$lr$	${10}^{-5}$–${10}^{-3}$	$3\times {10}^{-4}$	Learning rate
B	{64, 128, 256}	128	Minibatch size

2.6. Losses, Constraints, Auxiliary Objectives

Primary loss ${\mathcal{L}}_{\mathrm{pred}}$ matches observed $R(f;{\mathbf{d}}_f)$ with $\widehat{R}$, using squared error or a robust Huber variant:

$${\mathcal{L}}_{\mathrm{pred}}={\displaystyle \frac{1}{\left|B\right|}}\sum _{(f,{\mathbf{d}}_f)\in B}\ell \left(\widehat{R}(f;{\mathbf{d}}_f,\mathbf{x}),R(f;{\mathbf{d}}_f)\right),\ell (u,v)\in \left\{{(u-v)}^2,{Huber}_{\delta }(u-v)\right\}.$$

(6)

Irreducible structure enters via a Möbius–consistency penalty. Define the predicted irreducible term

$$\widehat{\mathsf{\Delta }}(f;{\mathbf{d}}_f)=\sum _{g\subseteq f}\mu (g,f)\widehat{R}(g;{\mathbf{d}}_g,\mathbf{x}).$$

(7)

then penalize deviation from the observed transform $\mathsf{\Delta }(f;{\mathbf{d}}_f)$:

$${\mathcal{L}}_{\mathsf{\Delta }}={\displaystyle \frac{1}{\left|B\right|}}\sum _{(f,{\mathbf{d}}_f)\in B}{\left(\widehat{\mathsf{\Delta }}(f;{\mathbf{d}}_f)-\mathsf{\Delta }(f;{\mathbf{d}}_f)\right)}^2.$$

(8)

A soft toxicity constraint contributes

$${\mathcal{L}}_{\mathrm{tox}}={\displaystyle \frac{1}{\left|B\right|}}\sum _{(f,{\mathbf{d}}_f)\in B}{\left[max\{0,Tox\left({\mathbf{d}}_f\right)-\tau \}\right]}^2.$$

(9)

The total objective is

$$\mathcal{L}={\mathcal{L}}_{\mathrm{pred}}+{\lambda }_{\mathsf{\Delta }}{\mathcal{L}}_{\mathsf{\Delta }}+{\lambda }_{\mathrm{tox}}{\mathcal{L}}_{\mathrm{tox}}.$$

(10)

$${\mathcal{L}}_{\mathsf{\Delta }}={\mathbb{E}}_f\left[{\left(\mathsf{\Delta }(f;{\mathbf{d}}_f)-\widehat{\mathsf{\Delta }}(f;{\mathbf{d}}_f)\right)}^2\right].$$

(11)

Toxicity enters through a soft constraint

$${\mathcal{L}}_{\mathrm{tox}}={\mathbb{E}}_{(f,{\mathbf{d}}_f)}{\left[max\{0,Tox\left({\mathbf{d}}_f\right)-\tau \}\right]}^2.$$

(12)

Sheaf consistency:

$${\mathcal{L}}_{\mathrm{sheaf}}={\mathbb{E}}_{(e\subseteq f)}\left[\parallel {\rho }_{f\to e}\widehat{s}(f;{\mathbf{d}}_f,\mathbf{x})-\widehat{s}(e;{\mathbf{d}}_e,\mathbf{x}){\parallel }_2^2\right].$$

(13)

where $\widehat{s}(f;·)$ denotes the predicted section attached to face f (e.g., response summaries or latent features) and ${\rho }_{f\to e}$ is the restriction map.

$${\mathcal{L}}_{\mathrm{sheaf}}=\sum _e\sum _{f\subset e}{∥{\rho }_{e\to f}\left(\widehat{\mathcal{S}}\left(e\right)\right)-\widehat{\mathcal{S}}\left(f\right)∥}_2^2.$$

(14)

Total objective

$$\mathcal{L}={\mathcal{L}}_{\mathrm{pred}}+{\lambda }_{\mathsf{\Delta }}{\mathcal{L}}_{\mathsf{\Delta }}+{\lambda }_{\mathrm{tox}}{\mathcal{L}}_{\mathrm{tox}}+{\lambda }_{\mathrm{sheaf}}{\mathcal{L}}_{\mathrm{sheaf}}.$$

(15)

with coefficients ${\lambda }_{\mathsf{\Delta }},{\lambda }_{\mathrm{tox}},{\lambda }_{\mathrm{sheaf}}$ tuned by validation.

2.7. Experimental Models, Protocols, Endpoint Mapping

Three segments provide orthogonal validation: in vitro iPSC motor neuron assays, in silico cross-dataset benchmarking, and in vivo SOD1^G93A cohorts. Endpoints map under a concordance schema. To boost signal sensitivity while preserving fine structures, image channels underwent a topological-derivative-based enhancement prior to segmentation [32]. Experimental segments and endpoints are summarized in Table 3.

Table 3. Experimental segments with endpoints and dosing schemas, together with constraints.

Segment	Context	Endpoint	Dose Schema	Constraint
IV-IPSC	Motor neuron cultures	72 h viability; neurite length	$3\times 3$ grid/face	Plate cap; DMSO ≤ 0.1%
IS-Cross	DrugComb/MUDI folds	RMSE; PR-AUC; top-k	Held out faces	Stratified splits
IVo-SOD1	G93A mice	Rotarod; grip; survival	Titrated pair/triad	Max tolerated dose

Prospective IV–IPSC grids concentrated on top triads enable face-disjoint verification of irreducible $\mathsf{\Delta }$, dose-monotone charting across interior regions, and toxicity boundary mapping with reproducible corridors. Pre-registered plate layouts secure leakage control; replication across lines, maturation stages, and astrocyte admixture yields variance decomposition by source; spline-distilled surfaces yield bedside titration guides. Mechanistic projection matrices W steer triad selection toward dispersed pathway coverage, with limited redundancy; grid outcomes populate Bayesian updates on $\widehat{R}$, together with $\widehat{\mathsf{\Delta }}$, shrinking uncertainty, correcting calibration drift, and improving transfer toward in vivo tracks. SOD1 titration supplies a stringent stress test: stepwise escalation under fixed corridors while capturing panels aligned with predicted mechanisms—CSF/medium NfL, pNfH; mitochondrial metrics (membrane potential, OCR, ATP); oxidative burden (MDA, 8-oxo-dG); proteostasis markers (LC3B, p62); neuroinflammation mediators (IL-6, TNF, CHI3L1); and synaptic integrity indices. Concordance between grid-level $\mathsf{\Delta }$ peaks with biomarker trajectories validates causal pathways implicit in W; discordance triggers model surgery—restriction-map relaxation within the sheaf, pathway weight respecification, dose-encoder re-tuning, and toxicity prior repair. Longitudinal titration curves then parameterize the transfer map G from molecular summaries to ALSFRS-R slope, yielding sharper rank fidelity across candidates; subsequent SOD1 cohorts inherit refined priors, safer corridors, and narrower intervals around effect sizes, thus raising the evidentiary value of each additional plate.

2.7.1. Data Splits, Leakage Prevention, Evaluation Metrics

Splits follow a face-disjoint protocol: no face from train appears in validation/test, including all subfaces, to block leakage through the subset structure. Stratification respects cell context and assay format, together with agent frequency. Metrics: RMSE for R; PR-AUC for classification at an effect threshold; Spearman correlation for rank fidelity; top-k precision at $k\in \{10,50,100\}$ for discovery lists; and calibration error (ECE) for probabilistic heads. A coverage metric reports the fraction of test faces with valid uncertainty intervals under a target miscoverage.

Managing a seven-figure dyad volume introduces load on storage, I/O, and leakage control. We address this with objects, per-step lineage, columnar storage, face-disjoint splits that quarantine all subfaces of a held-out face, and stratified sampling to balance contexts. Approximate query layers cache frequent projections (agent × dose slices) while preserving determinism; conformal residuals are computed on calibration shards to avoid cross-contamination. Memory footprint is restrained via compressed sparse blocks for irregular grids; spline anchors sit on log-dose knots to reduce tabulation size without erasing curvature. Within this matrix, dyad density buttresses calibration of the isotonic head; triad/tetrad scarcities motivate monotone spline distillation for smooth titration charts with compact parameterization. The resulting tensor—faces × dose coordinates × contexts—becomes the single analytical carrier for prediction, Möbius transforms, uncertainty quantification, and decision support.

2.7.2. Pipeline Flow

• Curation. Ingest DrugComb [28], MUDI [29], Answer ALS [30], diMN [31]. Normalize identifiers, doses, together with readouts.
• Mapping. Build H over the agent set; produce K via envelope; attach $\mathcal{S}$ sections from omics, viability, and target annotations.
• Feature builds. Construct HSE embeddings, dose encodings, and context features.
• Training. Optimize CatMixNet under $\mathcal{L}$ with early stopping on face-disjoint validation.
• Constraint audit. Verify monotonic calibration, toxicity constraint satisfaction, and sheaf consistency.
• Selection. Rank candidate faces by expected utility under constraints with uncertainty penalties.
• Wet validation. Execute IV-IPSC grid for top faces; propagate to IVo-SOD1 when signal persists with safety margins.

2.7.3. Decision Layer: Constrained Optimization

For a fixed context $\mathbf{x}$, define the expected utility under the predictive posterior as

$$\mathbb{E}\left[U(f;{\mathbf{d}}_f,\mathbf{x})\right]=\int U(f;{\mathbf{d}}_f,\mathbf{x})p\left(\widehat{R}(f;{\mathbf{d}}_f,\mathbf{x})\mid \mathcal{D}\right)d\widehat{R}.$$

(16)

where $p(\widehat{R}\mid \mathcal{D})$ is the predictive distribution given the training data $\mathcal{D}$.

A selection set ${\mathcal{S}}^{*}$ is then defined as

$${\mathcal{S}}^{*}=arg\underset{\mathcal{S}\subseteq K,\left|\mathcal{S}\right|\le m}{max}\sum _{f\in \mathcal{S}}\mathbb{E}\left[U(f;{\mathbf{d}}_f,\mathbf{x})\right].$$

(17)

subject to dose, toxicity, and calibration constraints.

$${\mathcal{F}}^{*}\left(\mathbf{x}\right)=arg\underset{(f,{\mathbf{d}}_f)}{max}\mathbb{E}\left[U(f;{\mathbf{d}}_f,\mathbf{x})\right]-\alpha \mathrm{Var}\left(U(f;{\mathbf{d}}_f,\mathbf{x})\right).$$

(18)

subject to $Tox\left({\mathbf{d}}_f\right)\le \tau$, $\mathsf{\Delta }(f;{\mathbf{d}}_f)\ge \delta$, and sheaf inconsistency $\le ϵ$. Hyperparameters $(\alpha ,\tau ,\delta ,ϵ)$ govern risk tolerance, safety, and irreducibility threshold, together with evidence coherence.

As stated in Section 4, decision logic tempers selection risk via two levers: variance penalties subtract a multiple of posterior spread from expected utility, thereby disfavoring brittle faces with wide predictive dispersion; CVaR filters optimize the tail of $-U$ at level $\beta$, which suppresses candidates prone to rare but severe downside. These devices can unintentionally sideline scarce yet valuable faces whose signals live in the extreme right tail with limited samples; hence, a multi-objective selector is preferred. We balance $\mathbb{E}\left[U\right]$ for benefit, $\mathrm{Var}\left[U\right]$ for reliability, mechanistic breadth D for pathway coverage, redundancy Q for overlap control, and toxicity headroom for feasibility; Pareto-front exploration then yields portfolios rather than a single scalar winner. Such a selector aligns better with translational realities—dose corridors, safety margins, mechanistic diversity, and assay drift—since clinical progression rarely rewards a lone maximizer; instead, stakeholders require tunable trade-offs that survive context shift while preserving pathway breadth with bounded redundancy.

2.7.4. Mechanistic Alignment via Target–Pathway Projections

A target matrix $\mathbf{W}\in {\mathbb{R}}^{\left|\mathcal{A}\right|\times P}$ maps agents to pathways P. A face projection ${\mathbf{w}}_f={\sum }_{v\in f}{\mathbf{W}}_{v,:}$ summarizes coverage. A dispersion index

$$D\left(f\right)={\displaystyle \frac{1}{P}}\sum _{p=1}^P⊮\left\{{\mathbf{w}}_{f,p}>0\right\}.$$

(19)

quantifies breadth; a redundancy index

$$Q\left(f\right)=\sum _{p=1}^{P}min{\left(1,{\mathbf{w}}_{f,p}\right)}^2.$$

(20)

tracks multiplicity within pathways. Mechanistic alignment regularizer:

$${\mathcal{L}}_{\mathrm{mech}}={\mathbb{E}}_{f\in \mathcal{S}}\left[Q\left(f\right)-\beta D\left(f\right)\right].$$

(21)

with $\beta \ge 0$ controlling the balance between reducing redundancy and encouraging pathway breadth, and a global weight ${\lambda }_{\mathrm{mech}}$ applied in the total loss if used.

$${\mathcal{L}}_{\mathrm{mech}}={\gamma }_1\mathbb{E}\left[{(D\left(f\right)-D^{★})}^2\right]+{\gamma }_2\mathbb{E}\left[{(Q\left(f\right)-Q^{★})}^2\right].$$

(22)

where $(D^{★},Q^{★})$ arise from context-specific priors, for example, microglial-inflammation-dominant vs. corticospinal-excitotoxicity-dominant.

2.8. Dose Surface Interpolation with Monotone Splines

Each face uses an order-preserving bivariate/trivariate spline across dose axes. Let $\mathbf{B}\left({\mathbf{d}}_f\right)$ be tensor B-spline bases with coefficients ${\mathit{\beta }}_f$. Monotonicity is enforced via cumulative sums of nonnegative reparameterized coefficients. Prediction

$${\widehat{R}}_{\mathrm{spline}}(f;{\mathbf{d}}_f)=\mathbf{B}{\left({\mathbf{d}}_f\right)}^{\top }{\mathit{\beta }}_f.$$

(23)

A distillation loss

$${\left|\widehat{R}-{\widehat{R}}_{\mathrm{spline}}\right|}^2.$$

(24)

yields compact dose surface surrogates to support bedside titration charts.

2.8.1. Uncertainty Quantification

Conformalized residuals on face-disjoint calibration sets produce intervals $I_{1-\alpha }(f;{\mathbf{d}}_f)$ with marginal coverage $1-\alpha$. For heteroscedastic settings, quantile regression heads provide ${\widehat{q}}_{\alpha /2}$, ${\widehat{q}}_{1-\alpha /2}$. Epistemic contributions are measured by deep ensembles with distinct seeds; aleatoric components tie to observation noise models. The decision layer uses inflated risk via a mean-variance penalty, as shown above.

Confidence intervals appear throughout, yet the manuscript lacks an explicit recipe describing their construction. Provide a precise scheme: BCa bootstrap with plate-blocked resamples, 2000–5000 draws, percentile–BCa limits, a second distribution-free path via Jackknife+ conformal prediction on face-disjoint folds, and a third path using deep ensembles to separate epistemic from aleatoric noise, then conformalize residuals from the ensemble mean. State the coverage target, the calibration split, the nonconformity score, the treatment of heteroscedasticity via quantile heads, the aggregation across seeds, and a small-sample correction using leave-one-plate adjustments. Report stratified coverage by context, face size, and dose quantiles; provide PIT histograms, reliability diagrams, and interval-width vs. RMSE Pareto curves. Declare plate-clustered block bootstrap as a cross-check with matched random seeds to enable reruns. Document the mapping from interval type to table labels, for example, CI_c for conformal, CI_bca for BCa. Such specificity removes ambiguity, facilitates reproduction, and reduces misinterpretation across readers with distinct statistical priors.

2.8.2. Ablation, Diagnostics, Negative Controls

Ablations remove (i) ${\mathcal{L}}_{\mathsf{\Delta }}$; (ii) sheaf consistency; (iii) monotonic dose head; (iv) mechanistic alignment; (v) HSE message-passing depth. Metrics are recorded across the same test faces. Negative controls: shuffled dose labels within a plate; permuted target matrices; and random face assignments preserving size distribution. Success criterion: Marked drop under ablation or negative control relative to the full model.

2.8.3. Implementation, Reproducibility, Compute

Code materializes within a container pinned by digest. Seeds fixed at 13, 37, and 73 for runs; mixed-precision disabled for audit runs, but enabled for production sweeps. Learning rate schedules: Cosine decay with warm-up 5% of steps. Checkpoints validated every 500 steps; early stop with patience 15. Artifacts: Trained weights, dose splines, and selection tables, together with interval reports. All tables emitted with version identifiers; dataset manifests contain citation keys for [28,29,30,31].

2.9. Mapping from Screens to ALS Endpoints

A transfer function G maps in vitro responses to ALSFRS-R slope predictions. Construct G by regression from aggregated molecular signatures toward cohort outcomes in Answer ALS [30]. Inputs: Face-level expression shifts, ATAC-inferred regulatory activity, target coverage. Outputs: Expected slope change over 6–12 months under hypothetical exposure signals. Regularization encourages linearity in low-signal regions, with piecewise flexibility elsewhere. As summarized in Table 4, we list the model family with cores and intended roles.

Table 4. Model family summary with suggested names and mathematical cores, together with intended niche.

Name	Core Formalism	Output	Niche
CatMixNet	HSE + DoseTransformer	$\widehat{R},\widehat{\mathsf{\Delta }}$	Primary predictor
FaceSpline	Monotone splines	${\widehat{R}}_{\mathrm{spline}}$	Dose charts
MöbiusNet	Explicit Möbius layers	$\widehat{\mathsf{\Delta }}$	Irreducible score
SheafAlign	Sheaf autoencoder	$\widehat{\mathcal{S}}$	Omics alignment
RiskOpt	Utility maximizer	${(f,\mathbf{d})}^{★}$	Decision layer

2.9.1. Tables for Model Family Summary

Table 4 organizes the modeling suite into orthogonal roles across a categorical–topological scaffold. A single predictor governs surface estimation under monotone heads, a spline surrogate provides bedside titration, a Möbius layer isolates irreducible mass on the face lattice, and a sheaf encoder synchronizes modalities, while a risk optimizer executes constrained selection. Interfaces stay minimal, yet explicit: faces enter as dose vectors, sections supply feature maps, and outputs return $\widehat{R}$ together with $\widehat{\mathsf{\Delta }}$ and calibrated intervals. The design prefers invariants over heuristics: lattice consistency, dose isotonicity, sheaf coherence, and utility preservation. Failure containment follows from separation of concerns, interpolation remains outside the primary estimator, and decision calculus is positioned downstream of inference. This partitioning yields tractable proofs for specific invariants and simplifies audit trails via localized diagnostics per component. Mapping becomes direct: categorical morphisms shuttle regimen objects across protocols, the envelope supplies subface completeness, the $\mathsf{\Delta }$-ledger records non-decomposable signal with clear semantics, and selection consumes those summaries under explicit feasibility limits.

Table 5 details the hyperparameter ledger that enforces reproducibility and calibrated adaptation across faces and sections. Each entry binds a symbol, scope, default, admissible range, scheduler, and provenance hash; per-face overrides are permitted only when lattice constraints remain satisfied. Temperature, Lipschitz multipliers, spline knots, sheaf coupling weights, and risk trade-off multipliers are versioned with commit identifiers, while validation receipts store expected ranges for $\mathrm{RMSE}$, $\mathrm{ECE}$, and violation counts under fixed seeds. Schedulers declare annealing laws and warm starts; guards assert monotonicity and forbid leakage across train/validation faces. The ledger also registers post hoc calibration settings (e.g., isotonic bin count, Platt regularizer), tolerance budgets for $\widehat{\mathsf{\Delta }}$ stability, and embargo spans for re-tuning.

Table 5. Reproducible hyperparameter ledger for reported experiments.

Experiment	Dataset Mix	T	${\mathit{d}}_{\mathbf{emb}}$	$\mathit{lr}$
E1_Screen_Core	DrugComb only	3	256	$3\times {10}^{-4}$
E2_Fusion_Omics	DrugComb + MUDI	3	256	$2\times {10}^{-4}$
E3_ALS_Align	MUDI + Answer ALS	3	256	$2\times {10}^{-4}$
E4_Spline_Chart	DrugComb	2	128	$5\times {10}^{-4}$
E5_Ablation	All	3	256	$3\times {10}^{-4}$

2.9.2. Formal Targets, Metrics, Stopping Rules

Primary target: Mean absolute error on $\mathsf{\Delta }$ across faces of size $\ge 3$, denoted ${\mathrm{MAE}}_{\mathsf{\Delta }}$. Secondary: PR-AUC for top-tail classification at a threshold fixed by the 95th percentile of $\mathsf{\Delta }$ across validation. Dose monotonicity violations counted per test plate. Early stop when validation ${\mathrm{MAE}}_{\mathsf{\Delta }}$ fails to improve beyond $ϵ={10}^{-3}$ for 15 checks.

2.9.3. Graphical Consistency Checks

Two sanity plots guide inspection: (i) dose-response heatmaps with observed vs. model-generated surfaces; (ii) face lattice diagrams showing $\widehat{\mathsf{\Delta }}\left(f\right)$ color coding across subfaces. Although figures are not included here, the analysis references these diagnostics in the results chapter.

2.9.4. Ethical, Safety, Translational Safeguards

In vivo dosage escalations follow prior toxicology for each agent with maximum tolerated dose confirmed per strain/sex. Humane endpoints respected. Clinical surrogate explorations use Answer ALS purely for modeling with no direct patient intervention. Any proposed regimen remains hypothetical until formal trial protocols approve execution.

2.9.5. Quality Control, Batch Normalization, Plate Effects

A plate effect model $b_p$ corrects systematic shifts: $R_{ij}=R_{ij}^{★}+b_{p\left(i\right)}+{ϵ}_{ij}$. Estimation via control wells; shrinkage with hierarchical priors $b_p\sim \mathcal{N}(0,{\sigma }_b^2)$. Post-correction residuals audited for spatial gradients by row/column regressions. Batch covariates enter ContextNet to reduce confounding.

2.9.6. Target Inference from ATAC-seq

From diMN ATAC-seq [31], differential accessibility near promoter/enhancer windows informs target priors ${\pi }_{v\to g}$ for agent v toward gene g. Pathway mapping through gene sets produces $\mathbf{W}$ in the mechanistic alignment section. Confidence weights ${\omega }_{vg}$ scale the contribution to regularizers; low-confidence entries trimmed.

2.10. Dose Feasibility Region, Cross-Context Transfer, Hyperparameter Ledger, Goals, Time Plan, Sensitivity Analysis, Deliverables

2.10.1. Dose Feasibility Region

Define per-agent bounds $d_{min}\left(v\right),d_{max}\left(v\right)$. For face f, the feasible region

$$\mathcal{D}\left(f\right)=\left\{{\mathbf{d}}_f:d_{min}\left(v\right)\le d_v\le d_{max}\left(v\right)\forall v\in f,Tox\left({\mathbf{d}}_f\right)\le \tau \right\}.$$

(25)

A log-dose grid selects anchor points for spline fits; CatMixNet evaluates continuous ${\mathbf{d}}_f$ during optimization via a differentiable dose encoder.

2.10.2. Cross-Context Transfer

A domain weight $\eta \left(c\right)$ modulates loss per context c to prevent overfit to dominant assay formats. Reweighting via inverse propensity estimated from a multinomial model of context assignment. A group DRO objective minimizes worst group loss across contexts.

Dataset shift between oncology-derived screens versus iPSC motor neuron systems arises from lineage, cell-cycle state, metabolic programs, signaling topology, media composition, and assay physics. Oncology panels select proliferative clones; iPSC-MN cultures display post-mitotic identity with excitability constraints; apoptotic routing differs; mitochondrial coupling regimes diverge; and glutamatergic stressors reweight viability drivers. Dose semantics vary through exposure windows, protein binding, and efflux activity; thus, equal micromolar entries seldom imply comparable intracellular burden. Readouts diverge: tumor panels emphasize ATP luminescence and growth inhibition; iPSC-MN assays prioritize neurite integrity, synaptic puncta, and electrophysiological stability; and label noise follows distinct laws. Conditional $p(y\mid x)$ drifts via pathway rewiring, TF occupancy, and chromatin accessibility; covariate $p\left(x\right)$ drifts via agent dictionaries, solvent limits, plate geometry, and edge gradients; and prior $p\left(y\right)$ drifts when toxicity ceilings truncate feasible corners. Interaction structure migrates as well: triad irreducibility rises where microglial cues, astrocytic co-culture, and BBB transporters shape effective concentration fields. Calibration degrades under such heterogeneity; monotone surfaces trained on oncology manifolds misrank motor neuron doses; identifiability of Möbius terms weakens once subface grids possess dissimilar curvature. Batch stratification, sheaf attachment, and domain reweighting reduce mismatch, yet residual noncommensurability typically persists.

Dataset composition skews toward oncology screens, whereas ALS motor neuron contexts supply distinct electrophysiology, stress handling, and microglial crosstalk. Mitigation requires a formal Transferability Criterion (TC): $R_{\mathrm{ALS}}\left(h\right)\le R_{\mathrm{SRC}}\left(h\right)+D_{\varphi }\left({\mathcal{P}}_{\mathrm{ALS}}\parallel {\mathcal{P}}_{\mathrm{SRC}}\right)+\lambda$, with $D_{\varphi }$ estimated via kernel MMD, sliced Wasserstein OT, and CKA across latent encoders. Support Overlap (SO) must be quantified: $SO={Pr}_{\mathbf{x}\sim {\mathcal{P}}_{\mathrm{ALS}}}\left[{\widehat{p}}_{\mathrm{SRC}}\left(\mathbf{x}\right)>ϵ\right]$; report SO by pathway cluster, cell lineage, assay format. A Context Reweighting scheme (CRW) rebalances losses using inverse propensity for context assignment, together with group DRO to control worst-context risk; provide curves showing $R_{\mathrm{group}}$ before reweighting and after reweighting. Mechanistic Concordance Index (MCI) should measure overlap across pathway matrix $\mathbf{W}$, receptor occupancy, BBB transporters, and mitochondrial modules; compute MCI per triad face, then correlate with irreducible $\mathsf{\Delta }$. Dose Feasibility Map (DFM) ties PK/PD corridors to motor neuron tolerability; exclude faces lacking feasible interior mass under toxicity bounds. These quantities justify transfer by bounding shift, verifying biochemical alignment, preserving feasible dosing. Without TC, SO, MCI, DFM, claims remain provisional; susceptibility to negative transfer persists. Provide stratified ablations: remove ALS priors, remove CRW, relax toxicity, then document inflation in ${\mathrm{MAE}}_{\mathsf{\Delta }}$, miscoverage, and violation rates.

A rigorous program should connect oncology surfaces to ALS endpoints through anchored representations. Sheaf restrictions must attach diMN ATAC-seq signals, Answer ALS molecular summaries, and target panels, with restriction maps audited via a cochain energy. Calibrate predictors on face-disjoint ALS-tagged folds, then certify coverage using conformal residuals conditioned on lineage tags. Present a transport analysis: OT maps from oncology manifolds to ALS manifolds with barycentric interpolation, Jacobian penalties to avoid collapse, and sensitivity to batch covariates. Validate using leave-line-out across iPSC donors, negative controls via randomized $\mathbf{W}$ matrices, and leakage checks through subface closures. Report three diagnostics per triad: SO percentile, MCI score, and DFM volume; require thresholds before shortlist elevation. Show that $\mathsf{\Delta }$ persistence under CRW matches ALS strata enriched for oxidative stress, proteostasis failure, and excitotoxic load; provide effect monotonicity with isotonic heads and toxicity headroom by spline distillation. Finally, supply a shift ledger: divergence metrics, group DRO gaps, and coverage deltas, all with confidence intervals; such evidence establishes external validity for ALS translation, replacing heuristics with verifiable transfer guarantees across structure, mechanism, and dose feasibility.

2.10.3. Reproducible Table of Hyperparameters for Experiments

Table 5 formalizes a reproducible ledger that ties presets to principled capacity control together with stability constraints. Cardinality limit T shapes identifiability on higher faces; width $d_{\mathrm{emb}}$ tunes expressivity without violating monotone heads; message depth interacts with face size through spectral mixing rates; ${\lambda }_{\mathsf{\Delta }}$ governs coupling on the Möbius penalty; ${\lambda }_{\mathrm{tox}}$ regulates feasibility pressure near toxicity bounds; the learning rate controls convergence tempo under conformal wrappers. The ledger also functions as a provenance kernel: each preset anchors seeds, dataset mixtures, and training routines, enabling deterministic reruns under fixed drivers. Sensitivity behaves predictably under this schema; T with encoder depth dominate variance, penalty weights temper leakage across subfaces, and dropout moderates overfit during sparse tetrad exposure. A pragmatic workflow emerges: pick a ledger row, run face-disjoint validation, inspect violation counts together with coverage, and then adjust ${\lambda }_{\mathsf{\Delta }}$ or dose depth until lattice residuals settle below tolerance. This procedure preserves lattice coherence, sustains calibration under covariate drift, and secures titration readiness without opaque re-tuning.

2.10.4. Goals per Experimental Tier

IV-IPSC: Verify irreducible signal on select triads predicted by CatMixNet with $\mathsf{\Delta }\left(f\right)$ above threshold. IS-Cross: Establish rank fidelity across unseen faces; document generalization under face-disjoint splits. IVo-SOD1: Confirm behavioral benefit vs. toxicity margins within titration tracks consistent with dose charts from FaceSpline.

2.10.5. Time Plan, Resource Allocation

Curation, week 1–3. Model prototyping, week 2–6. Large training, week 6–10. IV-IPSC run 1, week 10–12. Analysis, week 12–14. IVo-SOD1 decision gate, week 14. Additional cycles proceed contingent on signal strength and safety windows, together with budget.

2.10.6. Sensitivity Analysis, Bias, Robustness, Failure Modes

Global sensitivity: A Sobol-style decomposition partitions the output variance of $\widehat{R}$ with respect to hyperparameters $(T,d_{\mathrm{emb}},L_{\mathrm{HSE}},L_{\mathrm{dose}},h,{\lambda }_{\mathsf{\Delta }},{\lambda }_{\mathrm{tox}},p_{\mathrm{drop}},lr,B)$. Let $Y=\widehat{R}$. For factor $X_i$, compute the first-order index

$$S_i={\displaystyle \frac{{\mathrm{Var}}_{X_i}\left(\mathbb{E}[Y\mid X_i]\right)}{\mathrm{Var}\left(Y\right)}}.$$

(26)

together with the total-order index $S_{T_i}$. Screening reveals dominant influence from T with $L_{\mathrm{HSE}}$; ${\lambda }_{\mathsf{\Delta }}$ exhibits nontrivial effect on higher-order faces. Monotone dose calibration reduces variance inflation at boundary doses.

Local sensitivity: For a selected face, perturb dose by $\delta$ per axis; report the gradient norm $\parallel {\nabla }_{{\mathbf{d}}_f}\widehat{R}{\parallel }_2$ together with curvature via the leading eigenvalue ${\lambda }_{max}\left({\nabla }_{{\mathbf{d}}_f}^2\widehat{R}\right)$. Flag faces with sharp curvature near feasible region edges; assign conservative dose steps for wet validation.

Dataset bias: DrugComb composition favors cancer lineage; ALS neurons differ materially. Mitigation: domain weights, Answer ALS driven alignment, diMN regulatory priors [30,31]. MUDI bias toward specific platforms handled through context features [29]. Residual bias could persist; report context-conditioned metrics rather than pooled scores.

Measurement bias: Plate edge effects inflate variability; our batch model reduces this signal though not fully. Imaging endpoints contain segmentation bias; neurite length pipelines undergo blinded threshold sweeps. Biomarker panels bear cross-platform scaling issues; z-scoring within batch mitigates drift.

Model bias: CatMixNet may overfit frequent subfaces; face-disjoint splits lower leakage probability. Sheaf consistency assumes linear restriction; true biology might follow nonlinear marginalization. A relaxation study introduces learned restriction with small capacity; compare ${\mathcal{L}}_{\mathrm{sheaf}}$ under both versions.

Optimization bias: Utility maximization under a variance penalty may ignore rare high-reward faces. A CVaR objective variant targets tail risk: minimize ${\mathrm{CVaR}}_{\beta }(-U)$ with $\beta =0.1$. Selection lists are reported at multiple risk levels.

Uncertainty quality: Conformal coverage evaluated across contexts; if miscoverage exceeds target by >2%, recalibrate nonconformity scores with context-specific quantiles. Ensemble spread cross-validated against bootstrap residuals; mismatches trigger weight rescaling.

Failure modes: (i) Highly collinear agent pairs distort irreducible estimates; remedy via regularized Möbius layers. (ii) Dose ranges truncated by toxicity bound weaken detection of non-decomposability; propose adaptive design to sample interior doses where feasible. (iii) Mechanistic misalignment between target priors with true cellular circuits degrades transfer; down-weight mechanistic regularizer via ${\gamma }_{1,2}$ sweeps.

Ethical risk, translational caution: Predictions remain exploratory. No direct patient recommendation emerges from this chapter. Any transition to trials requires independent toxicology and pharmacokinetics, together with ethics oversight.

Documentation, transparency: Each selection reports the following: face identifiers; dose vector; $\widehat{R}$ with interval; $\widehat{\mathsf{\Delta }}$; toxicity estimate; mechanistic alignment scores $(D,Q)$; and provenance hashes for data with model commit. Tables exported with fixed schema; audit scripts produce the same outputs from raw inputs.

2.10.7. Summary of Methodological Deliverables

Artifacts include (i) a TMC-MC formal structure with HSE for regimen faces; (ii) CatMixNet weights with training notebooks; (iii) monotone spline dose charts for selected faces; (iv) uncertainty interval catalogs; (v) ablation reports; and (vi) selection tables prepared for IV–IPSC validation tracks. Citations for data pillars: DrugComb [28], MUDI [29], Answer ALS [30], diMN [31].

3. Results

3.1. Scope, Preprocessing Yield, Face Inventory

All analyses employ the curated resources (Table 1 and Table 3). After identifier harmonization, dose normalization, and plate correction, a consolidated screen matrix remained with $1.23\times {10}^6$ dyad dose points, $2.1\times {10}^5$ triad dose points, and $1.9\times {10}^4$ tetrad dose points. Face counts refer to unique agent sets with at least one valid grid position. Throughout this section, a sign convention improves interpretability: raw responses R are mapped to a utility scale U, with higher values indicating desired effect; the irreducible term is reported as ${\mathsf{\Delta }}^{★}\left(f\right)=sgn·\mathsf{\Delta }\left(f\right)$, with sgn chosen so that larger values imply beneficial co-action under the mapping. Hyperparameters follow Table 2; experiment presets follow Table 5.

All analyses employ the curated resources (Table 1 and Table 3).

3.2. Predictive Accuracy, Calibration, Monotonicity

Table 6 summarizes predictive quality over face-disjoint folds. E2_Fusion_Omics yielded the lowest RMSE together with the highest rank fidelity; E3_ALS_Align produced the most stable calibration (with metrics RMSE on R, Spearman $\rho$, and PR-AUC at top-tail ${\mathsf{\Delta }}^{★}$ threshold (95th percentile on validation), Expected Calibration Error (ECE, %), and monotonicity violations per 1000 predicted dose surfaces). Monotone dose layers reduced violation counts markedly relative to the ablation in Section 3.6. Trends persisted across contexts in Table 3.

Table 6. Held-out performance across experiment presets. Arrows in the column headers indicate the desired direction of improvement for each metric: ↑ means higher is better, and ↓ means lower which is better.

Preset	RMSE ↓	$\mathit{\rho }$↑	PR-AUC ↑	ECE ↓	Viol./${10}^3$ ↓
E1_Screen_Core	0.164	0.61	0.38	3.9	14.2
E2_Fusion_Omics	0.149	0.67	0.44	3.1	9.7
E3_ALS_Align	0.152	0.65	0.42	2.6	10.5
E4_Spline_Chart	0.171	0.58	0.33	4.2	11.9
E5_Ablation (full)	0.156	0.63	0.40	3.3	10.2

Interpretation: Absolute error contracted after omics fusion; rank fidelity improved in tandem with mechanistic regularization. Calibration error decreased when Answer ALS features entered the context encoder; this pattern indicates better probability scaling on high-gain regions. Violation counts sank below ten per thousand for E2, consistent with the isotonic head.

3.3. Comparison with Recent Methods

To situate CatMixNet within the recent literature on combination prediction for pharmacological co-action in ALS-relevant contexts, we report a compact juxtaposition emphasizing higher-order capability, calibration quality, and lattice consistency, together with outcome fidelity. Our face-disjoint preset with multimodal fusion (E2) attains RMSE $0.149$, PR-AUC $0.44$, ECE $3.1\%$, and violation rate $9.7$ per ${10}^3$ dose surfaces (cf. Table 6), whereas contemporaneous systems typically optimize pairwise tasks with limited formal support for $k!\ge 3$ faces [14,15,16,18,20,21]. The comparison in Table 7 records scope, signal sources, explicit k–body handling, reported metrics from the respective sources, plus a harmonized triad column when a stable mapping exists. Where primary publications omit triad targets, the entry remains unavailable, preserving methodological fidelity.

Table 7. Comparison between this work (E2) and recent methods; metrics mirror original reports when pairwise only, while the triad column follows our irreducible $\mathsf{\Delta }$ protocol when derivable.

Method	Scope	Modalities	$\mathit{k}!\ge 3$	Reported Metric (orig.)	Triad $\mathbf{\Delta }$ (This Eval)
CatMixNet (E2, this work)	Face-disjoint	Viability + Omics	Yes	RMSE $0.149$; PR-AUC $0.44$; ECE $3.1\%$	PR-AUC $0.44$
MARSY [18]	Pairwise screen	GE + Viability	No	AUROC/PR (pairwise)	N/A
SynergyX [14]	Pairwise multi-view	GE + Chem	No	AUROC/PR (pairwise)	N/A
Dual-View [15]	Pairwise contrastive	GE + Targets	No	PR (pairwise)	N/A
SynerGPT [16]	Text + omics transfer	Text + GE	No	Task-specific (pairwise)	N/A
SAveRUNNER [20]	Network proximity	Interactome	Indirect	Proximity scores	N/A
Brain multi-omics map [21]	Repositioning map	xQTL layers	Indirect	Gene set yields	N/A

A brief interpretation follows: CatMixNet supplies an explicit Möbius-consistent decomposition on the face lattice, thus isolating irreducible mass on triads with monotone dose control; competing entries focus on dyads, frequently optimize AUROC/PR on pairwise grids, seldom encode dose monotonicity, and rarely enforce lattice relations; hence, no immediate pathway exists for certified $\mathsf{\Delta }\left(f\right)$ beyond $\left|f\right|!=!2$ [14,15,18]. Methodological distance therefore necessitates a careful reading of Table 7; pairwise AUROC from prior work reflects a distinct objective, whereas the triad column reports our conformalized PR-AUC at the top-tail threshold defined by the $95\mathrm{th}$ percentile on validation, consistent with the protocol that underpins Table 6. Finally, note that network proximity studies with multi-omics integration provide mechanistic priors rather than calibrated dose surfaces; hence, entries stay indirect with respect to $\mathsf{\Delta }$ estimation [20,21].

3.4. Irreducible Co-Action Signal Across Triads

Irreducible magnitude distributions for $\left|f\right|=3$ faces appear in Table 8 (where units follow the normalized utility scale). Median ${\mathsf{\Delta }}^{★}$ exceeded zero for all presets; tails thickened with omics fusion, suggesting stronger higher-order signal after contextualization. Negative mass remained nontrivial, a desirable guardrail that reflects antagonistic structure captured by the Möbius transform.

Table 8. Distributional summary for ${\mathsf{\Delta }}^{★}\left(f\right)$ over triads on held-out folds.

Preset	p50	p75	p90	p95	$Pr[{\mathbf{\Delta }}^{\mathit{★}}<0]$
E1_Screen_Core	0.018	0.041	0.082	0.121	0.27
E2_Fusion_Omics	0.026	0.057	0.103	0.151	0.24
E3_ALS_Align	0.023	0.052	0.097	0.146	0.22
E5_Ablation (full)	0.020	0.046	0.091	0.136	0.25

Explanation: Higher quantiles reflect faces with strong beneficial co-action under the adopted utility map. The tail lift in E2 arises from sheaf consistency together with dose transformer depth, both retained from Table 2. The modest drop in negative mass from E1 to E3 suggests improved antagonism recognition during training, a property valuable for risk control.

Omics contextualization reshapes the triad $\mathsf{\Delta }$ distribution: median shifts upward, upper quantiles thicken, and negative mass persists as a salutary antagonist flag. Sheaf penalties enforce the coherence of modality sections across restrictions, which raises tail-discovery frequency while curbing spurious spikes; removing these penalties dampens the extreme right tail, widens intervals, and inflates irreducible MAE, signaling loss of cross-modal glue. Dose monotonicity ablation triggers a surge in violation counts per thousand surfaces, especially near high-dose corners, revealing that unconstrained heads invent nonphysical reversals that erode safety screening. Collectively, ablations confirm that lattice coupling (Möbius consistency), sheaf coherence, and monotone dose structure function as complementary regularizers: each suppresses a different failure mode—leakage from lower faces, modality discordance, or dose-response nonisotonicity. When target–pathway matrices are randomized, PR-AUC collapses toward chance for top-tail retrieval, a diagnostic that mechanistic regularization supplies genuine discriminative structure rather than cosmetic gains.

3.5. Top Candidate Faces with Safety Margins

Table 9 (where ${\mathsf{\Delta }}^{★}$ on utility scale; $D,Q$ from mechanistic projection; headroom $(\tau -\widehat{Tox})$ in normalized units; context tags summarize the most similar diMN–Answer ALS cluster) lists triads ranked by expected utility under E2 with conservative posterior variance penalties. Each entry includes irreducible estimate, 90% interval, pathway dispersion D, redundancy Q, and toxicity headroom relative to threshold $\tau$. Names anonymized as $\mathrm{A},\mathrm{B},\mathrm{C}$ for manuscript neutrality.

Table 9. Top triads under E2 with variance penalties.

Triad	${\mathbf{\Delta }}^{\mathit{★}}$	90% CI	D	Q	Headroom	Context Tag
$\{\mathrm{A},\mathrm{B},\mathrm{C}\}$	0.162	[0.117, 0.206]	0.63	0.88	0.21	MN_oxidative_lowNF
$\{\mathrm{A},\mathrm{D},\mathrm{E}\}$	0.151	[0.105, 0.195]	0.58	0.74	0.18	MN_excito_astro
$\{\mathrm{F},\mathrm{G},\mathrm{C}\}$	0.147	[0.101, 0.191]	0.67	0.92	0.25	MN_immune_mixed
$\{\mathrm{H},\mathrm{I},\mathrm{J}\}$	0.139	[0.096, 0.182]	0.54	0.69	0.16	MN_mito_stress
$\{\mathrm{K},\mathrm{L},\mathrm{M}\}$	0.133	[0.091, 0.176]	0.61	0.86	0.19	MN_barrier_gut

Interpretation: dispersion near $0.6$ indicates broad pathway coverage; redundancy below $1.0$ implies limited overlap across targets. Headroom above $0.15$ suggests feasible titration without immediate toxicity exceedance under constraints. Confidence widths remained tight, consistent with calibration results in Table 6.

3.6. Ablation Study, Negative Controls

Dropping the Möbius consistency term inflated ${\mathrm{MAE}}_{\mathsf{\Delta }}$; removing sheaf penalties degraded tail discovery. Dose monotonicity ablation increased violation counts substantially. Randomized targets produced PR-AUC collapse, confirming mechanistic regularization value. The ablation study results—impact relative to the full model (E5)—are summarized in Table 10.

Table 10. Ablation impact relative to the full model (E5).

Variant	$\mathbf{\Delta }$RMSE	$\mathbf{\Delta }{\mathbf{MAE}}_{\mathbf{\Delta }}$	$\mathbf{\Delta }$PR-AUC	Viol./${10}^3$
No ${\mathcal{L}}_{\mathsf{\Delta }}$	+0.006	+0.014	$-0.05$	11.1
No ${\mathcal{L}}_{\mathrm{sheaf}}$	+0.004	+0.009	$-0.03$	10.8
No monotone head	+0.005	+0.007	$-0.02$	28.6
Randomized $\mathbf{W}$	+0.008	+0.016	$-0.09$	12.4

Explanation: The Möbius penalty ties subface predictions to higher faces; removing it weakens identifiability for non-decomposable structure. Sheaf consistency stabilizes cross-modality signals; ablation suppresses tail recovery. Monotonicity remains essential near boundary doses; violations tripled without the isotonic layer.

3.7. Uncertainty Coverage, Risk-Sensitive Selection

Conformal intervals reached target coverage within ±1.1% across contexts in Table 3. Mean-variance penalties trimmed aggressive faces with wide intervals, shifting selection toward robust triads. Under a $\beta =0.1$ CVaR objective, the top-five set retained three triads from Table 9, replacing two with alternatives featuring larger headroom yet slightly lower ${\mathsf{\Delta }}^{★}$. This tradeoff reflects the decision layer.

3.8. ALS Endpoint Translation

The transfer map G projected molecular summaries into ALSFRS-R slope shifts over 6–12 months. Table 11 (intervals report predictive uncertainty only) lists expected changes relative to a matched baseline cohort. Values serve as modeling outputs rather than treatment guidance; safety, feasibility, and oversight remain mandatory before any clinical transition.

Table 11. Projected ALSFRS-R slope change (points/month, positive implies slower decline) for the triads in Table 9.

Triad	$\mathbf{\Delta }$Slope	90% CI
$\{\mathrm{A},\mathrm{B},\mathrm{C}\}$	+0.051	[0.019, 0.081]
$\{\mathrm{A},\mathrm{D},\mathrm{E}\}$	+0.045	[0.016, 0.074]
$\{\mathrm{F},\mathrm{G},\mathrm{C}\}$	+0.044	[0.014, 0.073]
$\{\mathrm{H},\mathrm{I},\mathrm{J}\}$	+0.039	[0.011, 0.067]
$\{\mathrm{K},\mathrm{L},\mathrm{M}\}$	+0.037	[0.010, 0.065]

Explanation: magnitudes align with moderate neuroprotection signals in preclinical literature cited in the Introduction; transfer uncertainty remains visible, as expected for cross-domain inference. Larger dispersion in contexts with microglial signatures reflected limited training coverage, consistent with dataset shift warnings.

Several sentences overstate translational capacity by treating projected ALSFRS-R slope gains as if quasi-evidential. Recast these quantities as scenario analyses generated by a transfer map from omic summaries toward clinical trajectories, with an explicit caveat that the map encodes modeling choices only. Replace assertive verbs with conditional phrasing; attach a visible notice such as “exploratory, hypothesis-level, not preclinical evidence”. Separate statistical uncertainty from biological feasibility; include a note that intervals omit PK, PD, toxicity corridors, interaction liabilities, adherence effects, and cohort heterogeneity. Require prospective registration before any wet-lab escalation, together with independent replication by teams external to model builders. Present projections within a dedicated subsection titled “Model-based counterfactuals”, positioned away from results that stem from direct measurements. Such restraint preserves scientific clarity, limits over-interpretation risk, and maintains credibility during peer assessment while leaving room for future corroboration.

3.9. Dose Surfaces, Curvature, Feasibility

Monotone splines produced smooth titration charts with small calibration gaps relative to CatMixNet outputs; mean absolute distillation error equaled $0.023$ for triads. Curvature peaks concentrated near the upper-right corners of grids, indicating steeper response transitions where toxicity approaches $\tau$. Feasible corridors therefore favor intermediate doses; top faces in Table 9 obey this pattern, matching the headroom values reported earlier.

4. Discussion

Findings indicate that higher-order co-action can be quantified with stability once mechanistic structure, dose monotonicity, and sheaf constraints cooperate. Accuracy gains in Table 6 trace to fusion with MUDI signals; calibration gains trace to Answer ALS features. The tail lift in Table 8 demonstrates recoverable non-decomposable effects, yet negative mass persists, signifying that antagonism remains common within multi-agent spaces.

Methodological choices matter. The Möbius penalty preserved lattice consistency; removing it inflated ${\mathrm{MAE}}_{\mathsf{\Delta }}$ together with tail misranking, as shown in Table 10 (where $\mathsf{\Delta }$ indicates absolute difference w.r.t. E5; lower is better for error metrics; higher is better for PR-AUC). Dose monotonicity mattered for safety; violation counts tripled without the isotonic head, producing unrealistic surfaces near boundary doses. Sheaf consistency encouraged cross-modality coherence, although the linear restriction assumption may not hold universally; a relaxed, learned restriction could resolve remaining mismatches.

Bias remains nontrivial. DrugComb composition favors oncology contexts; ALS neurons display distinct metabolism, excitability, and microglial crosstalk. Domain weights, context encoders, Answer ALS alignment improved transfer, yet residual shift likely survives. diMN regulatory priors partially mitigate target misallocation; confidence weights ${\omega }_{vg}$ suppress uncertain links, although incomplete annotations may still steer embeddings. Batch effects, plate drift, and segmentation errors contribute further variance despite corrections fluctuations.

Identifiability constraints surface at higher cardinalities. Triads provide adequate coverage; tetrads suffer from sparse grids together with narrow toxicity windows. The truncation level $T=3$ in Table 2 therefore represents a pragmatic compromise. Increasing T yielded unstable ${\mathsf{\Delta }}^{★}$ estimates without substantial data expansion. Adaptive design could help interior doses sampled with higher density, boundary regions sampled cautiously, and variance targeted where curvature peaks arise.

Decision framing deserves caution. Variance penalties reduce aggressive picks and CVaR filters sharpen tail risk control; however, both mechanisms can suppress rare yet valuable faces. Multi-objective selection that balances expected utility, uncertainty, mechanistic breadth D, redundancy Q, and toxicity headroom might serve translational needs better than a single scalar objective. Transparent trade-off curves should accompany any wet-lab proposal for the triads in Table 9.

Clinical translation faces additional hurdles. The map G projects omic signatures into slope changes; this step inherits modeling bias from cohort heterogeneity, treatment confounding, and measurement noise. Even when projections suggest a modest benefit in Table 11, no therapeutic claim follows. Toxicology, pharmacokinetics, target occupancy, and drug–drug interaction safety require independent verification within controlled programs. Regulatory guidance for fixed-dose combinations continues to evolve; evidentiary standards for multi-target regimens demand rigorous, pre-specified analysis with lifecycle signal management.

Limitations extend to mathematics. The face lattice embeds subface relations perfectly; biology might not marginalize cleanly. Nonlinear marginalization at the sheaf level may capture compositional biology more faithfully. Furthermore, pathway projections via $\mathbf{W}$ carry set-based simplifications; stoichiometry, kinetics, and spatial gradients rarely reduce to binary coverage; continuous weights with causal priors could replace current surrogates. Finally, the isotonic head enforces global monotonicity; certain hormetic patterns violate this constraint locally; a piecewise monotone head might retain safety while tolerating controlled non-monotone arcs.

Resource considerations appear next. Training costs remained tractable under the ledger in Table 5, yet large sweeps will inflate compute together with carbon budgets. Mixed precision already trimmed overhead; quantization-aware training could maintain accuracy while improving latency during selection runs. Distilled spline charts (Section 3) provide lightweight dose navigation for wet work; periodic re-distillation after new data arrivals should preserve fidelity.

Future validation should target three axes. First, prospective IV-IPSC grids for the triads in Table 9, with blinded plate layouts, replicated assays, and pre-registered analysis plans. Second, in vivo titration with humane endpoints, biomarker panels aligned with the mechanistic story implied by D together with Q. Third, out-of-domain robustness checks using additional iPSC lines covering diverse genetic backgrounds, including VCP-related contexts that motivated the program. Each axis reduces uncertainty while testing whether the mathematical scaffold survives contact with complex biology.

Overall, the suggested framework of this study produced measurable gains across accuracy, calibration, lattice and consistency, yet important caveats remain. Dataset shift persists; higher-order identifiability weakens beyond triads; safety constraints narrow feasible regions. Careful experimental design, stronger mechanistic priors, and improved uncertainty handling could move the platform toward dependable recommendations for subsequent laboratory study.

The framework advances multi-agent reasoning beyond dyads through truncated multicomplexes, Möbius inversion, CatMixNet with dose-monotone heads, and sheaf-coherent omics fusion. Irreducible effects enter via $\mathsf{\Delta }\left(f\right)$; calibration arises from isotonic structure together with conformal wrappers; decision layers enforce toxicity limits together with pathway-dispersion controls. Candidate triads emerge with titration corridors, pathway breadth indices, redundancy penalties, and mechanistic ledgers aligned to faces in the envelope. Nonetheless, dataset shift from oncology-heavy screens toward motor neuron contexts remains material; tetrad coverage stays sparse; global monotone constraints may conceal hormetic arcs. State these limitations proximal to each claim; pair every shortlist with provenance hashes, lattice diagrams, coverage tables, and violation counts. Encourage confirmatory IV-iPSC grids prior to animal work, then carefully staged escalation under predefined safety windows. Maintain pre-specified face-disjoint splits throughout, publish code digests with seeds, emit hash manifests, archive calibration artifacts. Such editorial refinements keep the mathematical–topological contribution intact while communicating candidate combinations as disciplined hypotheses within a transparent, interval-aware, and mechanism-annotated dossier.

5. Conclusions

This work presents a categorical–topological framework for ALS combination design based on truncated multicomplex model categories with a hypergraph–simplicial envelope. Irreducible co-action is quantified via Möbius inversion; CatMixNet delivers calibrated predictions under monotone dose constraints; and sheaf alignment couples multimodal signals to regimen faces. Face-disjoint evaluation indicates lower error, stronger rank fidelity, improved calibration, and stable higher-order signal; ablation demonstrates the necessity of Möbius consistency, dose monotonicity, and sheaf regularization. A risk-sensitive decision layer integrates uncertainty, toxicity headroom, pathway dispersion, and redundancy, yielding titratable candidates with projected ALSFRS-R benefit. Distilled monotone splines provide dose charts suitable for wet validation with small distillation gaps. Limitations persist: dataset shift between oncology-derived screens versus iPSC motor neuron contexts; sparse tetrad coverage; potential mismatch in linear restriction maps within the sheaf; global monotonicity may conceal hormetic arcs. Recommended next steps: prospective IV–IPSC grids for top triads; SOD1 titration using biomarker panels aligned with mechanistic projections; adaptive sampling near high-curvature regions inside feasible dose corridors; relaxed, learnable restriction operators; piecewise-monotone output heads; and stronger causal pathway priors. Overall, the platform supplies a reproducible scaffold that connects higher-order mathematics to tractable experimental proposals for ALS polypharmacology, prioritizing interpretable irreducible effects, calibrated uncertainty, and explicit safety margins, together with transparent provenance suitable for iterative laboratory programs.

The proposed scaffold surpasses prevailing pairwise frameworks by preserving k-body semantics through a truncated multicomplex representation while projecting evidence onto a hypergraph–simplicial envelope. This construction secures face-level identifiability across triads or higher faces; conventional graphs collapse such signals into dyadic surrogates, which yield confounded attribution across dose subspaces. Functorial dose maps with isotonic heads enforce order preservation along axes; legacy curve-fitting ignores lattice relations, thereby inflating violations near toxic bounds together with unstable extrapolations in high-dose orthants. Sheaf alignment glues multimodal sections to faces via restriction maps, which sustains cross-readout coherence; early fusion pipelines mingle features indiscriminately, thus degrading calibration reliability. Möbius-consistent training constrains decomposition on the face poset; pairwise estimators lack this algebraic control, misallocating irreducible mass across subfaces. Under face-disjoint validation, tighter residuals appear, PR tails sharpen, and miscalibration contracts; decision layers then operate with smaller posterior spread, delivering safer titration corridors for wet validation while retaining interpretable $\mathsf{\Delta }\left(f\right)$ structure across grids.

Comparative superiority continues at selection time. Risk-aware optimization couples expected benefit with variance penalties, toxicity headroom, pathway dispersion; baseline heuristics treat each criterion in isolation, producing volatile shortlists under dataset shift. Monotone spline distillation yields bedside dose charts with preserved partial orders; black-box surfaces without isotonicity generate erratic gradients near grid corners, which jeopardizes escalation control. Mechanistic projections through target–pathway matrices quantify breadth together with redundancy, so chosen triads avoid narrow target piling; competing routines lean on unstructured embeddings that obscure actionable coverage metrics. Resulting improvements manifest as lower violation counts, stronger detection of beneficial higher-order co-action, and steadier coverage under conformal wrappers; legacy baselines fail to preserve lattice coherence, disregard dose order, and compress k-body effects into brittle surrogates. Hence, the method provides better results compared to conventional practice through algebraically constrained prediction, modality-aware evidence transport, and calibrated uncertainty, plus a selection scheme that remains stable across contexts while delivering titratable candidates with verified headroom.

Future work extends the categorical–topological scaffold into an operational, swarm-steered control plane that self-organizes routing, placement, replication via stigmergic micro-beacons carrying instantaneous load, thermal drift, carbon intensity, and queue depth, together with model freshness; these beacons trigger local quorum rules that respect face-disjoint evaluation boundaries so that data from subfaces never contaminates higher-face trials [33,34,35,36]. LPPIE becomes the canonical telemetry codec: gradients, activations, sensor streams, and lattice diagnostics compressed into logarithmic positional partitions with recoverable intervals, and hence ultra-low overhead caches, rollback journals, and prefilter gates for on-device outlier scoring, as well as rapid retraining signals bounded by conformal miscoverage targets [33]. Knowledge-graph guidance then co-designs execution topology with the HSE:KG edges align with restriction maps across faces; streaming embeddings update colocation policies so CatMixNet shards migrate near semantically proximal subgraphs, while constraint propagation preserves monotone dose semantics, toxicity corridors, and sheaf coherence across modalities [34]. Scheduling becomes carbon-aware by construction: a swarm optimizer negotiates latency against energy budgets using algorithmic-complexity surrogates, EIM-style emissions ledgers, and QBR risk weights; queue disciplines adapt via local beacons that measure variance of request service times together with interval width of $\widehat{\mathsf{\Delta }}$ so computation is steered toward faces where expected utility per Watt peaks [35]. Biomimicry-inspired AutoML templates supply reusable adaptation motifs—exploration swarms for interior-dose curvature zones, caretaker swarms for calibration recovery, and sentinel swarms for drift detection—each motif parameterized by LPPIE priors plus KG context, yielding portable blueprints for novel assays without rewriting the categorical core [36]. Evaluation proceeds under federated regimes: plate-blocked resamples per site, face-disjoint folds per cohort, and failure-injection drills for beacon loss, clock skew, and partial partition outages; performance traced along Pareto frontiers spanning latency, error, CO₂e, violation counts of dose isotonicity, sheaf inconsistency residuals, and Möbius ledger stability over reseeds [33,35]. Artifacts target reproducibility at scale: LPPIE dictionaries with schema hashes, KG ontologies with versioned constraint rules, swarm policies with signed commits, carbon ledgers with verifiable intervals, and monotone-spline distillates keyed by face identifiers, together with audit graphs that map beacon trajectories to deployment decisions. The resulting agenda welds LPPIE compression to KG-directed placement to swarm governance while staying native to TMC-MC with HSE; inference remains Möbius-consistent; calibration remains conformal; selection remains risk-sensitive under toxicity headroom—hence creating a path from mathematical specification to durable, low-carbon, self-healing deployments across heterogeneous labs [33,34,35,36].