From Geometric Complexity to Informational Dimensionality in Scaffold-Guided Tissue Regeneration

Abstract

Scaffold architecture shapes tissue regeneration through the mechanical, topographical, and biochemical cues it presents to cells, yet geometrically elaborate scaffolds do not reliably produce more organized tissues, while comparatively simple architectures can exert strong organizational effects. We argue that scaffold performance is better understood by distinguishing geometric complexity from effective informational dimensionality: a relational property of the scaffold–cell system, defined as the number of independently manipulated architectural directions that produce distinguishable, above-noise changes in a jointly measured mechanotransductive response. Unlike structural entropy, fractal dimension, or feature-counting metrics, this construct depends on cellular accessibility, cue persistence, and non-redundancy. Mechanotransduction supplies its biological basis, integrating scaffold-derived cues through focal adhesions, cytoskeletal organization, nuclear deformation, and YAP/TAZ signaling, and we distinguish early resolvability from later organizational stabilization. We outline an operational strategy for estimating both from factorial scaffold libraries, common readout panels, and rank-based analysis of the response mapping, illustrated with selected experimental precedents rather than a systematic evidence sample. Positioned relative to biomimetic, mechanobiology-guided, and morphospace approaches, it yields testable predictions on dimensional compression, redundancy, and the resolvability–stability dissociation. Scaffold design is thus reframed from maximizing complexity or native resemblance toward engineering stable, cell-readable dimensions of organization.

1. The Problem of Organization in Tissue Engineering

The central challenge of tissue engineering is not simply keeping cells alive, but guiding cellular collectives toward stable, functional, and architecturally coherent forms [1]. Although cells possess intrinsic capacities for adhesion, migration, differentiation, and extracellular matrix deposition [2], these processes do not spontaneously converge toward organized tissue architectures under arbitrary physical conditions [3]. Cells cultured on smooth or weakly structured substrates frequently generate disordered, metastable organizations despite remaining viable and metabolically active [4,5]. The fundamental problem is organizational: physical environments must somehow bias cells toward coherent architectures rather than the disordered, weakly coordinated states they default to in vitro.

Over the past decades, biomaterial scaffolds have evolved from passive supports into active microenvironments capable of influencing cellular behavior through biochemical, mechanical, and topographical cues [6,7,8,9,10,11,12]. The native extracellular matrix (ECM) represents a dynamically structured and multiscale instructive environment integrating fibrillar architecture, anisotropy, curvature, stiffness gradients, ligand accessibility, viscoelasticity, and spatial heterogeneity across multiple scales to regulate tissue homeostasis, mechanotransduction, and regenerative morphogenesis [13,14,15]. Importantly, biological organization emerges not from isolated geometric features alone, but from the coordinated interaction of partially independent mechanical, spatial, and biochemical constraints acting across scales [16,17,18,19].

Contemporary scaffold fabrication has consequently emphasized biomimicry, employing electrospinning, lithography, additive manufacturing, and related techniques to reproduce aspects of ECM geometry and hierarchy [20,21,22,23]. Yet for all the empirical progress, scaffold design has remained largely heuristic [24]. Although contemporary mechanobiology increasingly recognizes that scaffold geometry regulates collective cellular organization through mechanotransduction [14], many studies still focus on individual cue-response relationships. What remains difficult to explain is why certain scaffold architectures reliably stabilize coherent organization, whereas others, despite comparable geometric richness, generate heterogeneous, metastable, or weakly coordinated outcomes [25].

Existing mechanobiological models successfully describe many of the signaling pathways through which cells respond to physical stimuli, including integrin engagement, focal adhesion dynamics, cytoskeletal tension, and mechanosensitive transcriptional regulators such as YAP/TAZ [26,27]. Recent studies further demonstrate that scaffold curvature, pore geometry, and nanoscale topographical accessibility can directly regulate cytoskeletal organization, YAP/TAZ signaling, stemness, and lineage specification [28,29,30], suggesting that scaffold architecture does not merely modulate isolated signaling pathways but restructures the organizational conditions under which cellular decisions unfold [17]. Scaffold architecture, in other words, does not merely present physical cues—it shapes whether those cues can be coherently integrated at all [31].

The relevant question is therefore not simply how much structure a scaffold contains, but whether its architecture supplies cues that cells can reliably integrate into coordinated behavior [32]. This distinction is defined formally below as the difference between geometric complexity and effective informational dimensionality [33].

Here we propose an informational interpretation of scaffold-guided tissue organization based on effective dimensionality and morphogenetic separability. The central claim is that scaffold architecture matters insofar as it increases the number of stable, partially independent, mechanotransductively interpretable variables available to cells [4]. The remainder of the paper defines this claim operationally, relates it to mechanotransduction, and derives experimentally testable consequences.

Mechanotransduction provides the biological mechanism through which this interpretation becomes meaningful. Cells do not construct an explicit representation of the surrounding environment; instead, they continuously integrate complex spatial and mechanical inputs into states of cytoskeletal tension, nuclear deformation, and downstream signaling activity [26,34]. These integrated signals bias cellular trajectories toward specific organizational outcomes, while feedback mechanisms convert graded mechanical responses into stabilized morphological states such as alignment, differentiation, or coordinated tissue assembly [35,36].

The aim of this paper is deliberately narrower than a general theory of morphogenesis, biological possibility, or biological computation, where such a theoretical approach is commoner. The focus is scaffold-guided tissue organization at the cell–material interface: how engineered architectures alter the dimensional structure of mechanosensory environments and thereby influence the separability and stabilization of cellular organizational states. Likewise, the term morphogenesis is used here primarily in the context of scaffold-guided tissue regeneration and engineered tissue organization rather than as a universal theory of embryonic development. Scaffold fabrication can thus be reframed not primarily as the imitation of extracellular morphology, but as the design of mechanotransductively interpretable constraints capable of stabilizing desired tissue architectures [37].

This article is a conceptual work based on a purposive narrative examination of literature on scaffold architecture, mechanotransduction, and tissue organization. References were selected for their relevance to the development, biological grounding, and illustration of the proposed framework rather than through an exhaustive systematic-review protocol. Supplementary Material S3 presents selected experimental examples spanning the principal architectural variables and mechanotransductive outcomes considered in the framework. These examples illustrate how existing experiments relate to the proposed constructs; they are not intended as a representative sample or quantitative evidence map.

2. Geometric Complexity and Effective Informational Dimensionality

2.1. Definitions: Complexity, Dimensionality, and Cell-Readable Dimensions

Because the central claim of this paper rests on a distinction between two kinds of architectural “richness,” the relevant terms must be fixed before the argument proceeds further. We distinguish three constructs that are frequently conflated in the scaffold-design literature: geometric complexity, biological complexity, and effective informational dimensionality.

Geometric complexity (${\mathit{C}}_{\mathit{g}}$) is a property of the scaffold considered as a material object, independent of any cell. It denotes the quantity and heterogeneity of structural features present in the architecture, and increases with feature count and density, the number of constituent materials, the number of hierarchical levels, surface irregularity, pore-size dispersity, and orientational disorder. It is quantified using conventional structural descriptors, including the developed interfacial area ratio ($\mathbf{S}\mathbf{d}\mathbf{r}$), specific surface area, box-counting fractal dimension, porosity and pore-size distribution, tortuosity, and the orientational entropy of constituent fibers [38,39]. Critically, ${\mathit{C}}_{\mathit{g}}$ is scale-dependent and cell-agnostic: a randomly irregular surface and a hierarchically ordered one may score identically on these descriptors despite presenting entirely different information to a cell [40].

Biological complexity (${\mathit{C}}_{\mathit{b}}$), by contrast, is a property of the resulting tissue rather than of the scaffold. It denotes the diversity of cellular states, phenotypes, spatial compartments, and interactions that the construct ultimately supports, and is assessed through histology, immunophenotyping, and transcriptional profiling rather than through any architectural measurement. ${\mathit{C}}_{\mathit{b}}$ is therefore an outcome variable. Scaffolds are frequently praised for “complexity” when what is actually meant is ${\mathit{C}}_{\mathit{g}}$, while being evaluated on outcomes that properly belong to ${\mathit{C}}_{\mathit{b}}$; the implicit assumption that the former reliably produces the latter is precisely the assumption this paper questions (Table 1).

Table 1. Geometric complexity ($C_g$), biological complexity ($C_b$), and effective informational dimensionality ($D_{eff}$): property of, definition, drivers, measurement, and worked examples across low-, intermediate-, and high-$D_{eff}$ scaffolds.

	Geometric Complexity (${\mathit{C}}_{\mathit{g}}$)	Biological Complexity (${\mathit{C}}_{\mathit{b}}$)	Effective Informational Dimensionality (${\mathit{D}}_{\mathit{e}\mathit{f}\mathit{f}}$)
Property of	The scaffold, as a material object	The regenerated tissue	The mapping from scaffold to cellular response
Definition	Quantity and heterogeneity of structural features present in the architecture	Diversity of cellular states, phenotypes, and spatial organization supported by the construct	Number of mutually distinguishable, stable, mechanotransductively coupled axes along which architecture alters cellular state
Increases with	Feature count and density; number of materials; hierarchical levels; surface irregularity; pore-size dispersity; orientational disorder	Number of coexisting lineages; regional specialization; hierarchical tissue architecture; functional integration	Addition of cues that are cell-accessible, temporally and spatially stable, and non-redundant with existing cues
Measured via	$Sdr$, specific surface area, fractal dimension, porosity and pore-size distribution, tortuosity, orientational entropy	Histology, immunophenotyping, lineage markers, transcriptional profiling, functional assays	Thresholded effective rank, above a permutation-derived noise floor, of the response matrix relating architectural parameters to the four early co-acquired readouts: focal adhesion maturation, cytoskeletal order, nuclear deformation, and YAP/TAZ localization. Coupling strength $G$ is reported separately, and the participation ratio is retained only as a diagnostic (Appendix A).
Low-$D_{eff}$ example	Flat culture dish; smooth glass	Monolayer of a single cell type	Isotropic substrate: no privileged axis; organized and disorganized states poorly separable
Intermediate example	Aligned electrospun fibrous mat	Aligned myotube array	Strong, stable directional axis; lower $C_g$ than a random 3D gel, but higher $D_{eff}$
High-$D_{eff}$ example	Hierarchical anisotropic scaffold with a coherent stiffness gradient	Zonal osteochondral or tendon-to-bone interface	Several stable, partially independent axes
Instructive counter-example	Random collagen gel: high $C_g$	—	Random collagen gel: low $D_{eff}$—geometric heterogeneity without stable, separable, interpretable cues
Relation to regeneration	Weak, non-monotonic predictor	The outcome to be explained	Proposed proximate determinant of organizational stabilization

Effective informational dimensionality $\left({\mathit{D}}_{\mathbf{e}\mathbf{f}\mathbf{f}}\right)$ occupies a different logical position: it is a property not of the scaffold or the tissue alone, but of the mapping between them. Let $\mathit{x}\in {\mathbb{R}}^{\mathit{n}}$ denote the independently manipulable architectural parameters introduced in Section 2.2. Let ${\mathit{r}}_{\mathbf{e}\mathbf{a}\mathbf{r}\mathbf{l}\mathbf{y}}\in {\mathbb{R}}^4$ denote the joint early mechanotransductive response vector, comprising focal-adhesion maturation, cytoskeletal orientational order, nuclear deformation, and YAP/TAZ nuclear-to-cytoplasmic localization [41]. These four readouts are acquired on the same specimens, under the same conditions, and at the same timepoint. Effective informational dimensionality is the number of mutually distinguishable directions along which variation in $\mathit{x}$ produces separable variation in ${\mathit{r}}_{\mathbf{e}\mathbf{a}\mathbf{r}\mathbf{l}\mathbf{y}}$ above the experimental noise floor. Formally, ${\mathit{D}}_{\mathbf{e}\mathbf{f}\mathbf{f}}$ is the thresholded effective rank of the local response mapping $\mathit{J}=\partial {\mathit{r}}_{\mathbf{e}\mathbf{a}\mathbf{r}\mathbf{l}\mathbf{y}}/\partial \mathit{x}$, and therefore counts the directions in scaffold-parameter space that the measured cell–readout system can resolve as distinct [42]. Extracellular-matrix organization is not included in ${\mathit{r}}_{\mathbf{e}\mathbf{a}\mathbf{r}\mathbf{l}\mathbf{y}}$ because it becomes informative at a later timescale. It is treated separately as a late, cumulative probe of stabilized organization and therefore of ${\mathit{D}}_{\mathbf{s}\mathbf{t}\mathbf{a}\mathbf{b}}$.

This definition yields three admissibility criteria that give the relation

$${\mathit{D}}_{\mathbf{e}\mathbf{f}\mathbf{f}}\le \mathit{n}$$

a precise content, since $\mathit{n}$ counts the parameters an engineer can vary while ${\mathit{D}}_{\mathbf{e}\mathbf{f}\mathbf{f}}$ counts only those a cell can separate. A parameter contributes a dimension only if it satisfies all three. It must be cell-accessible, varying at length and force scales resolvable by the mechanosensory apparatus and at amplitudes above the relevant detection threshold, since the mechanical environment perceived at cellular length scales can diverge substantially from properties measured in bulk [43,44]. It must be stable in the sense of cue persistence, persisting over the timescale of the cellular decision it is meant to influence and correlated over distances comparable to or greater than the cell itself—a cue-persistence sense of stability that is distinct from the organizational-state stability quantified later by ${\mathit{D}}_{\mathbf{s}\mathbf{t}\mathbf{a}\mathbf{b}}$; a cue whose correlation length is shorter than the cell, or whose correlation time is shorter than the decision it should bias, is registered as noise rather than as information [45]. And it must be non-redundant, producing a mechanotransductive consequence not already produced by the parameters already counted, since two parameters that remain indistinguishable in $\mathbf{r}$ constitute one dimension rather than two, however distinct they may appear structurally.

${\mathit{D}}_{\mathbf{e}\mathbf{f}\mathbf{f}}$ is therefore not a complexity metric, an entropy measure, or a fractal dimension, and the distinction is worth making explicit. Entropy-based and information-theoretic measures of architecture are computed on the material and are maximized by disorder, whereas ${\mathit{D}}_{\mathbf{e}\mathbf{f}\mathbf{f}}$ is not: a randomly oriented fiber mesh has high structural entropy and low ${\mathit{D}}_{\mathbf{e}\mathbf{f}\mathbf{f}}$, because its cues fail the stability criterion [46]. Fractal dimension describes how features scale across length scales but is indifferent to whether those features are mechanotransductively coupled and can be identical for architectures with very different ${\mathit{D}}_{\mathbf{e}\mathbf{f}\mathbf{f}}$. Feature-counting complexity metrics such as $\mathbf{S}\mathbf{d}\mathbf{r}$, feature density, or description length quantify how much structure is present rather than how many independent axes of it a cell can resolve. Mechanical degrees of freedom describe the configuration space of the material system itself, of which ${\mathit{D}}_{\mathbf{e}\mathbf{f}\mathbf{f}}$ is the image after projection through the cell’s sensing apparatus, and to which it is therefore bounded above.

A direct consequence is that the relationship between geometric complexity and cell-readable organization need not be monotonic, but the quantity that varies non-monotonically depends on how complexity is varied. Along a single-parameter series, the scaffold–cell coupling strength, defined below as $\mathit{G}$, may rise and then fall while the resolvable rank changes only discretely, when a singular direction crosses the noise floor. Across a multi-parameter architectural space, ${\mathit{D}}_{\mathbf{e}\mathbf{f}\mathbf{f}}$ itself may rise while added features remain accessible, stable, and non-redundant, and fall once further complexity contributes only redundant or fluctuating cues. Observations that increasing surface complexity promotes focal adhesion maturation and cellular elongation up to a point, beyond which further complexity impairs proliferation, are consistent with this predicted non-monotonicity of coupling strength rather than with a smooth inverted-U in dimensionality itself [47].

The definition is operational rather than purely conceptual because it prescribes the kind of measurement required. Across a scaffold library in which architectural parameters are varied systematically, the four early readouts are acquired jointly and used to estimate the local response mapping $\mathit{J}=\partial {\mathit{r}}_{\mathbf{e}\mathbf{a}\mathbf{r}\mathbf{l}\mathbf{y}}/\partial \mathit{x}$. ${\mathit{D}}_{\mathbf{e}\mathbf{f}\mathbf{f}}$ is then estimated as the thresholded rank of the standardized response matrix, with coupling strength $\mathit{G}$ reported separately. The estimation procedure is summarized in Appendix A and specified in full in Supplementary Material S2.

2.2. From Definition to Illustration: The Feature Vector and Representative Cases

The definitions above can be made more concrete by locating representative culture and scaffold systems along a qualitative continuum of geometric complexity. This classification considers dimensional organization, the number and heterogeneity of structural features, compositional complexity, spatial irregularity, and the number of hierarchical length scales. As summarized in Table 2, this continuum extends from smooth conventional two-dimensional substrates to patterned surfaces, fibrous and porous scaffolds, architected three-dimensional constructs, hierarchical composites, and decellularized native matrices. The classification is deliberately geometric and cell-agnostic: it does not constitute a ranking of regenerative performance or an estimate of effective informational dimensionality, which remains specific to the scaffold–cell–readout mapping.

Table 2. Heuristic classification of representative culture and scaffold systems according to relative geometric complexity. Geometric complexity is classified qualitatively according to dimensional organization, the number and heterogeneity of structural features, spatial irregularity, compositional heterogeneity and the number of hierarchical length scales. Architectural characteristics of engineered scaffold classes are based principally on Carotenuto et al. [22]. The classification does not represent regenerative performance or measured effective informational dimensionality, $D_{\mathrm{e}\mathrm{f}\mathrm{f}}$, which remains specific to the scaffold–cell–readout mapping.

Representative System	Characteristic Architectural Organization	Relative Geometric Complexity	Candidate Cell-Readable Variables	Interpretation Within the Present Framework
Smooth conventional 2D substrate	Planar, materially homogeneous and topographically unpatterned surface; no intrinsic directional axis or internal pore architecture [48]	Low	Local stiffness, ligand density and surface chemistry	Few geometric variables are present. Low geometric complexity will often coincide with low $D_{eff}$, unless non-geometric variables such as ligand patterning or a stiffness gradient introduce additional separable responses.
Micro- or nanopatterned 2D substrate	Planar surface carrying grooves, ridges, gratings, pits or other ordered features with defined size, spacing and orientation [22]	Low–intermediate	Feature orientation, periodicity, width, spacing, local curvature and roughness	A small addition to geometric complexity can generate a strong and persistent directional cue. Such a substrate may therefore be geometrically simple but informationally effective.
Aligned electrospun fibrous scaffold	Fibrous architecture with controlled fiber diameter, spacing and coherent orientation; usually dominated by one principal anisotropic axis [49]	Intermediate	Fiber orientation, orientational order, fiber diameter, local modulus, pore spacing and ligand presentation	Commonly provides one strong cell-readable directional dimension. Its organizational value arises from coherence and persistence rather than from a large number of structural features.
Random fibrous mesh or structurally disordered 3D gel	Volumetric fibrillar environment with heterogeneous local fiber orientation, pore geometry and adhesion configurations but no coherent global axis	Intermediate–high	Local fiber orientation, pore confinement, ligand distribution and microscale stiffness heterogeneity	High local heterogeneity need not generate high $D_{eff}$. If cues vary over distances shorter than the cell, fluctuate across the sampled region or produce redundant responses, the architecture can remain informationally low-dimensional.
Porous scaffold produced by freeze-drying, phase separation or related methods	Three-dimensional pore network characterized by pore size, shape, distribution, porosity and interconnectivity; architecture may be isotropic or directionally organized [10]	Intermediate–high	Pore geometry, interconnectivity, confinement, transport accessibility, local curvature and mechanical heterogeneity	Several geometric descriptors are present, but they need not represent independent informational dimensions. Pore size, interconnectivity and transport may be strongly coupled, while poorly accessible pores may not contribute to the cellular response mapping.
Architected 3D-printed or bioprinted scaffold	Precisely defined three-dimensional struts, channels, pores and surfaces generated layer by layer; potentially includes controlled curvature, pore orientation and regional geometry [50]	High	Pore orientation, channel direction, local curvature, confinement, local stiffness, surface topography and transport pathways	Provides multiple independently designable architectural parameters. Nevertheless, nominal printing freedom sets only the upper bound $n$; $D_{eff}$ depends on how many of those parameters generate separable mechanotransductive responses.
Multilayered, multizone or hierarchical composite scaffold	Multiple materials or regions organized across distinct length scales; may combine graded composition, oriented fibers, region-specific pores and mechanically distinct layers [22]	High	Regional composition, fiber orientation, pore hierarchy, stiffness or mineral gradients, interface geometry and cue persistence	Has the potential to generate several partially independent cell-readable dimensions. Hierarchy contributes additional dimensions only when its structural levels produce non-redundant cellular responses. Visually distinct zones may still collapse onto the same mechanotransductive direction.
Decellularized native tissue or whole-organ matrix	Tissue-derived three-dimensional architecture retaining multiscale fibrillar, vascular, ductal or zonal organization together with heterogeneous ECM composition and mechanics	High	Tissue-specific fiber organization, regional mechanics, vascular or ductal geometry, ligand distribution, viscoelasticity and biochemical composition	Represents high geometric and compositional complexity, but high $D_{eff}$ cannot be assumed. Decellularization may modify mechanics, accessibility and cue persistence, and many retained features may be redundant or undetectable to a particular cell type.

Table 2 also illustrates why geometric complexity and effective informational dimensionality cannot be placed on the same ordinal scale. A comparatively simple patterned or aligned architecture may provide a strong, stable, and separable directional cue, whereas a more heterogeneous three-dimensional environment may present cues that are redundant, spatially incoherent, or inaccessible to the cell. To identify the variables that may become biologically informative, the local environment sensed by a cell can be represented as a multidimensional collection of biophysical and biochemical parameters [37,51], including adhesion density, curvature, stiffness, stiffness gradients, topographical roughness, fiber orientation, anisotropy, ligand presentation, degradation dynamics, diffusible signaling gradients, and scaffold-mediated delivery of proliferative or differentiation cues [6,34].

Formally, these variables can be represented schematically as a mechanosensory feature vector:

$$x=\left[a,c,E,\nabla E,\rho ,\theta ,p,b,d,\dots \right]\in R^n$$

where $a$ denotes adhesion-related variables, $c$ curvature, $E$ stiffness, $\nabla E$ stiffness gradients, $\rho$ topographical organization, $\theta$ anisotropy and orientation, $p$ porosity or connectivity, $b$ biochemical signaling variables including ligand or growth-factor presentation, $d$ degradation-dependent environmental remodeling, etc. The specific composition of this vector may vary across experimental systems; its purpose here is to make explicit that cells operate within environments defined by multiple potentially informative dimensions.

Not all measurable scaffold variables contribute equally to cellular organization. Some environmental features may provide overlapping information, vary across spatial and temporal scales, or couple only indirectly to mechanotransductive pathways [26,52]. A scaffold parameter contributes to effective informational dimensionality only when it produces a stable, distinguishable, and mechanotransductively coupled change in the cellular readout vector. This relationship can be expressed qualitatively as:

$$D_{\mathrm{e}\mathrm{f}\mathrm{f}}\le n$$

where $D_{\mathrm{e}\mathrm{f}\mathrm{f}}$ denotes the effective informational dimensionality of the mechanosensory landscape. In this sense, two scaffold features contribute independently only insofar as they produce distinguishable and biologically interpretable mechanotransductive consequences [4,32].

The distinction is visible in standard culture systems. Smooth two-dimensional substrates support survival and proliferation, but provide little directional information for coordinated migration, force transmission, or matrix deposition [4,48,53,54]. Conversely, disordered collagen or fibrin gels may be geometrically heterogeneous without supplying stable directional constraints: fibers vary locally, adhesion geometries fluctuate, and stiffness gradients may lack coherent spatial correlation [55,56,57]. In both cases, cells may remain viable while collective organization remains weakly constrained. These examples illustrate why dimensionality, as used here, refers to stable, cell-readable axes of response rather than spatial richness alone.

The analogy with kernel methods [52,58] is used here only to clarify separability. Just as a transformation can make previously entangled states distinguishable in a richer representation, scaffold architecture may make alternative cellular organizational trajectories more separable by introducing stable variables such as anisotropy, curvature, alignment, and spatial gradients [51,59]. The analogy is bounded explicitly in Section Scope and Limits of the Kernel Analogy and Appendix B; here it serves only to motivate the question of which scaffold-derived variables become resolvable to the cell.

The argument that follows uses a small number of formal expressions, introduced once here and summarized in Table 3. Throughout, x ∈ ℝⁿ denotes the vector of architectural parameters an experimenter can independently manipulate, and r ∈ ℝ^m the vector of mechanotransductive readouts defining the cellular organizational state introduced above. The two must not be confused: x is a property of the material, r a property of the cell, and this paper is concerned with the map between them. Φ denotes the mechanosensory map by which architectural parameters become cell-resolvable cues; $K_{bio}$ measures the mechanotransductive indistinguishability of two architectures; V(r; x) is an organizational potential over cellular states, parameterized by the scaffold; λ is the gain with which mechanotransduction converts cue coherence into organizational bias; and κ denotes curvature of the potential in organizational space.

Table 3. Mathematical notation used throughout the theoretical framework.

Symbol	Meaning	Lives in
$x$	Architectural parameter vector (anisotropy, curvature, pore geometry, stiffness gradient, ligand density, …)	Scaffold
$n$	Number of independently manipulable architectural parameters	Scaffold
$\Phi$	Mechanosensory transduction map; lossy, non-invertible, cell-type-dependent	Scaffold → Cell
$r$	Cellular readout vector comprising four early co-acquired mechanotransductive components—focal adhesion maturation, cytoskeletal order, nuclear deformation, and YAP/TAZ N:C ratio—used to estimate $D_{\mathrm{e}\mathrm{f}\mathrm{f}}$, plus late ECM alignment, acquired on parallel specimens and used to probe $D_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}$.	Cell
$K_{bio}(x_i,x_j)$	Mechanotransductive indistinguishability of two scaffold architectures	Scaffold × Scaffold
$V(r;x)$	Organizational potential over cellular states, parameterized by scaffold architecture	Cell (parameterized by scaffold)
$E_{int}\left(r\right)$	Intrinsic organizational cost, independent of scaffold	Cell
$r^{\ast }\left(x\right)$	Organizational state best supported by scaffold architecture	Cell
$K_{org}(r,r^{\ast })$	Alignment between the current organizational state and the scaffold-favoured state	Cell × Cell
$\lambda$	Mechanotransductive coupling gain	Cell
$C(x)={\displaystyle \frac{{\partial }^{2}V}{\partial r \partial x}}$	Scaffold–cell coupling matrix	Scaffold × Cell
$H\left(x\right)={\nabla }_r^{2}V$	Hessian of the organizational potential in state space; eigenvalues ${\kappa }_i$ quantify local stability	Cell
$J={\displaystyle \frac{\partial \hat{r}}{\partial x}}=-H^{-1}C$	Response mapping from scaffold parameters to predicted organizational state	Relation
$D_{eff}$	Effective rank of $J$; number of resolvable informational dimensions	Relation
$D_{stab}$	Number of directions within the resolvable subspace of $J$, that is, singular directions above the noise floor, along which the restoring curvature ${\kappa }_i$ exceeds biological-noise amplitude. $D_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}$ counts only directions the architecture both reaches and holds, so $D_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}\le D_{\mathrm{e}\mathrm{f}\mathrm{f}}$.	Relation

No expression in this paper is intended as a predictive quantitative model. None is fitted to data, none returns a number, and none should be used to compute a regenerative outcome. Their function is organizational: they fix the types of the objects involved, make the paper’s claims falsifiable in principle, and expose which quantities would need to be measured to test the framework. We do not repeat this qualification at each subsequent equation; it applies throughout.

2.3. Relation to Existing Scaffold-Design Frameworks

The present framework builds on, but is distinct from, three established approaches to scaffold design. The first is biomimetic design, which seeks to reproduce the composition, architecture, mechanical properties, and hierarchical organization of native extracellular matrices [1,18,22]. Biomimicry provides an important source of design principles, but structural resemblance does not by itself establish which scaffold features are detected or distinguished by a particular cell type. The present framework therefore treats biomimetic fidelity as neither necessary nor sufficient for effective informational dimensionality. A non-native architecture may provide several coherent cell-readable dimensions, whereas a highly tissue-like architecture may present features that are inaccessible, transient, or redundant in the cellular response.

The second approach is mechanobiology-guided scaffold design, which relates individual material variables—such as stiffness, curvature, anisotropy, pore geometry, or ligand presentation—to adhesion, cytoskeletal organization, nuclear deformation, mechanosensitive signalling, and cell fate [33,51]. The present framework does not replace these cue–response relationships; it asks a different question about their joint structure. Rather than asking only whether a parameter changes a cellular outcome, it asks how many independently manipulated scaffold parameters generate mutually distinguishable directions in a jointly acquired mechanotransductive response vector. This introduces a distinction between coupling strength, which describes the magnitude of scaffold-dependent response, and effective informational dimensionality, which describes the number of separable response directions.

The closest conceptual precursor is morphospace engineering and morphological computation, in which scaffold architecture is understood as constraining the space of cellular forms and redistributing part of the organizational work from the cell to the environment [37]. The present framework narrows and operationalizes that proposal. It specifies the architectural parameter vector $\mathrm{x}$, the cellular response vector $\mathrm{r}$, and the local mapping between them; defines $D_{\mathrm{e}\mathrm{f}\mathrm{f}}$ as the thresholded rank of that mapping; and distinguishes early resolvability from later organizational stabilization through $D_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}$. Morphological constraint is therefore translated into a measurable question about dimensional compression, redundancy, and response stability.

The novelty of the present proposal is consequently not the general claim that scaffold geometry influences cells, nor the claim that multiple cues interact. It lies in treating informational dimensionality as a relational, cell-type- and measurement-dependent property of the scaffold–cell mapping; providing an operational procedure for estimating it from jointly acquired responses; distinguishing dimensionality from response magnitude and stability; and predicting that structurally different architectures may be biologically equivalent when they generate indistinguishable mechanotransductive response vectors. These commitments generate falsifiable comparisons that biomimetic resemblance, single-cue response models, and geometric complexity metrics do not provide on their own.

3. Scaffold Architecture as Informational Expansion

If weakly structured environments provide only a limited set of mechanotransductively meaningful constraints, then one important function of scaffold engineering is to introduce additional biologically interpretable dimensions into the cellular microenvironment (Figure 1).

Figure 1. From engineered scaffold parameters to cell-resolvable informational dimensions. (A) A factorial electrospun scaffold library independently varies fiber alignment $\theta$, fiber diameter $d$, and local stiffness $E$, defining a three-dimensional engineered parameter space $x$. (B) The cell’s mechanosensory apparatus, represented by $\Phi$, converts these physical variables into a joint early response vector comprising focal-adhesion maturation, cytoskeletal order, nuclear deformation, and YAP/TAZ localization. Colors distinguish these four response components throughout the figure: blue, focal adhesions; orange/red, cytoskeletal order; purple, nuclear deformation; and green, YAP/TAZ localization. (C) The framework predicts dimensional compression: fewer singular directions of the scaffold–cell response matrix may exceed the experimental noise floor than the number of parameters manipulated, giving $D_{eff}<n$. The figure illustrates Design A; it represents a proposed experiment rather than data generated in the present study.

Formally, this transformation can be represented schematically as:

$$\mathsf{\Phi }:X\to \mathcal{M}$$

where $X$ represents the initial low-dimensional mechanosensory environment and $\mathcal{M}$ represents the expanded informational landscape generated by scaffold architecture. Three properties of Φ carry the argument, and none is incidental. First, Φ is a property of the cell–material pair, not of the material: the same architecture presents different cue configurations to a fibroblast and to an osteoblast, whose adhesion machinery and characteristic length scales differ. Informational dimensionality is therefore not an intrinsic property of a scaffold, and reporting it as a material specification would be a category error. Second, Φ is lossy: it is many-to-one and non-invertible. Architectural variation below mechanosensory detection thresholds, or fluctuating faster than the cellular decision it might bias, or correlated over distances shorter than a cell, is mapped to the same cue configuration as no variation at all. Third, Φ is not chosen: it is fixed by the cell’s mechanosensory apparatus and cannot be optimized by the designer, who manipulates x while Φ constrains what any manipulation of x can achieve. The lossiness of Φ is the mechanism generating the central asymmetry of this paper: geometric complexity is measured on x, while regenerative performance depends on Φ(x). Because Φ collapses distinctions, architectures may differ arbitrarily in $C_g$ while remaining indistinguishable in the image of Φ—the case of the random collagen gel, geometrically rich and informationally impoverished (Table 1).

Scaffold fabrication can therefore be viewed as a process of dimensional expansion when it introduces mechanotransductively relevant variables that reshape the informational structure of the cellular environment. Section 2 defined the criteria for such variables; this section examines how common scaffold architectures instantiate them.

Engineered scaffold architectures progressively structure the cellular microenvironment through anisotropy [60], curvature [61], pore geometry [62], stiffness gradients [63], and hierarchical spatial organization. In doing so, they do not simply increase geometric complexity, but generate more differentiated mechanotransductive conditions under which previously unstable morphogenetic trajectories may become reproducible and dynamically stable [64]. The scaffold therefore acts as an active mechanosensory environment that reshapes the range of organizational trajectories available to cellular collectives.

The native extracellular matrix exemplifies this principle: it coordinates spatial and mechanical constraints across scales, from fibrillar assemblies to macroscopic tissue organization [3]. Scaffold engineering can be interpreted as an attempt to reproduce selected aspects of this coordination, not the full structural complexity of the ECM [65,66].

Different fabrication techniques can be interpreted as different strategies for introducing informational dimensionality into the cellular environment. Electrospinning, for example, generates fibrous architectures whose orientation can range from random to highly aligned depending on collector geometry and processing conditions [21,67]; highly aligned fibers introduce strong directional constraints that bias migration, elongation, and cytoskeletal tension along privileged axes [60]; while randomly deposited fibers provide substantially weaker orientational information even when overall material composition remains identical [68]. Additive manufacturing illustrates a different route to the same end, enabling precise regulation of pore orientation, curvature, and hierarchical connectivity across the full construct architecture [69]. Freeze-drying, thermally induced phase separation, and soft lithography introduce comparable classes of constraint at other length scales [70]. Despite their mechanical and chemical diversity, these approaches share a common organizational function: each introduces a distinct category of stable, partially independent mechanosensory variable that cells can detect, integrate, and respond to collectively.

The critical issue, however, is not the fabrication method itself, but the type of informational variable introduced into the cellular environment. Each architectural modification changes the set of mechanical relationships available to mechanosensory interpretation. In this sense, fabrication parameters act indirectly: they do not determine cellular behavior directly but reshape the informational geometry through which cellular decisions emerge. Different scaffold architectures therefore correspond to different Φ-transformations acting on the mechanosensory landscape available to the cell.

Experimental observations support this interpretation. Rabel et al. demonstrated that osteoblast responses correlate more strongly with the developed interfacial area ratio (Sdr) than with mean roughness alone, suggesting that the organization of available adhesion states is more important than simple geometric irregularity. Increasing Sdr promoted focal adhesion maturation, cytoskeletal tension, and elongation, consistent with a Φ-mediated expansion of interpretable adhesion configurations. Yet excessive surface complexity impaired proliferation, indicating that dimensional expansion is beneficial only when it remains biologically coherent rather than informationally noisy [40].

Similarly, Dalby and colleagues showed that nanoscale disorder could induce osteogenic differentiation even in the absence of biochemical induction factors, implying that specific forms of topographical heterogeneity enrich the set of mechanically interpretable states available to the cell [71]. Subsequent studies demonstrated that nanoscale architecture influences cytoskeletal tension, nuclear deformation, and gene expression through coordinated alterations in adhesion geometry and force distribution [34].

These examples motivate the next question: how are scaffold-derived variables integrated by cells into organizational responses?

4. Mechanotransduction as Integrative Evaluation

Scaffold architecture affects cellular behavior only when its features are detected at the cell–material interface and converted into an integrated mechanical response. Integrin engagement and focal-adhesion maturation depend on the spatial organization of adhesive sites, local curvature, stiffness, and the capacity of the substrate to support tension. Architectural variables such as anisotropy, pore geometry, confinement, and surface topography therefore act by reorganizing adhesion formation and force transmission rather than as independent instructions to the cell [27,72,73,74,75].

The actin cytoskeleton integrates these local interactions across the cell. Actomyosin tension couples focal adhesions into a mechanical network, so coherent directional or mechanical cues promote stress-fiber organization, cell elongation, polarized migration, and aligned force propagation. Conversely, isotropic, spatially incoherent, or rapidly fluctuating cues generate fragmented and less reproducible responses [76,77,78,79]. Cytoskeletal forces are transmitted to the nucleus through the LINC complex and activate mechanosensitive pathways including FAK, RhoA/ROCK, and YAP/TAZ, thereby linking scaffold architecture to nuclear deformation, transcriptional state, proliferation, migration, lineage commitment, and extracellular-matrix production [80,81,82,83,84] (Figure 2).

Figure 2. Mechanotransductive conversion of scaffold architecture into cellular behavior and its experimental perturbation. (A) Random and aligned electrospun scaffolds differ in fiber orientational order while material composition, fiber diameter, ligand presentation, and local stiffness are held approximately constant. (B) Coherent alignment reorganizes integrin engagement and focal-adhesion maturation, actomyosin force transmission, nuclear deformation through the LINC complex, and YAP/TAZ localization. These processes are measured through focal-adhesion, cytoskeletal, nuclear, and transcriptional readouts acquired jointly. Colors distinguish the four components of the early mechanotransductive response vector: blue, focal-adhesion maturation; orange/red, cytoskeletal order; purple, nuclear deformation; and green, YAP/TAZ localization. (C) The framework predicts that viable-dose inhibition of ROCK, myosin II, or integrin-mediated coupling will reduce scaffold-dependent response magnitude and move singular directions of the response matrix toward the noise floor. Persistence of high coupling strength and high $D_{eff}$ under effective mechanotransductive blockade would challenge the proposed mechanism.

YAP/TAZ localization is particularly useful within the present framework because it reflects the integrated effects of substrate stiffness, adhesion geometry, spreading area, anisotropy, and force-transmission efficiency rather than any single scaffold parameter [41,85,86]. Geometrically different architectures may therefore be biologically equivalent when they generate indistinguishable joint responses across focal adhesions, cytoskeletal organization, nuclear deformation, and YAP/TAZ signaling; this representational equivalence is formalized by $K_{\mathrm{b}\mathrm{i}\mathrm{o}}$ in Appendix B.2 and tested experimentally in Design B. Conversely, the same architecture may generate different response states in different cell types. This is the biological basis for treating effective informational dimensionality as a property of the scaffold–cell mapping rather than of the scaffold alone.

Scaffold-induced responses often begin as graded changes in adhesion, cytoskeletal tension, and mechanosensitive signaling, but intracellular and tissue-level feedback can stabilize them as persistent differences in alignment, migration, differentiation, and matrix organization [35,36,87,88]. Mechanotransduction therefore provides the bridge between scaffold architecture and cellular behavior: it determines which architectural variables generate distinct and coherent cellular states, while subsequent feedback and matrix remodeling determine whether those states persist [89]. The corresponding experimental readouts are specified in Section 7.1, while Section 5 considers how architecture-dependent response states may become stabilized.

Scope and Limits of the Kernel Analogy

The kernel analogy is retained only to describe redundancy and separability. Two architectures may differ substantially as material objects yet become indistinguishable after projection through the cell’s mechanosensory map $\mathsf{\Phi }$ if they generate the same cellular response. Conversely, scaffold architecture may make previously entangled response states more separable by introducing stable, cell-readable variables. The analogy does not provide the dimensional count itself: $D_{\mathrm{e}\mathrm{f}\mathrm{f}}$ is estimated from the local scaffold–cell response matrix $J$, whereas $K_{\mathrm{b}\mathrm{i}\mathrm{o}}$ is only a pairwise measure of representational similarity.

The analogy also has strict biological limits. Cells are not classifiers: there is no training set, label structure, objective function, separating hyperplane, or margin. The mapping $\mathsf{\Phi }$ is imposed by the cell’s mechanosensory machinery rather than optimized by the designer, and $K_{\mathrm{b}\mathrm{i}\mathrm{o}}$ is an empirical similarity measure rather than a proven Mercer kernel. The computational kernel trick has no biological counterpart because cells physically instantiate mechanosensory transformation through adhesion formation, cytoskeletal tension, nuclear deformation, and signaling. What survives of the analogy is therefore the geometry of redundancy and separability, not the machine-learning algorithm. The formal details are provided in Appendix B.

5. From Resolvable Dimensions to Stabilized Organization

If mechanotransduction integrates scaffold-derived cues into coherent cellular tendencies, a further question follows: which of those tendencies are actually maintained over time? This distinction is central to the framework. $D_{\mathrm{e}\mathrm{f}\mathrm{f}}$ describes what the cell can resolve early; it does not by itself show that the resulting state is robust. Regenerative organization also requires stabilization against biological noise, fluctuating adhesion dynamics, stochastic gene expression, and mechanical perturbation [36,82].

This point is especially important because scaffold-guided organization is not simply a matter of making alternative cellular trajectories distinguishable. A scaffold may introduce a readable directional cue, increase cytoskeletal alignment, or bias YAP/TAZ localization, while the induced state remains shallow, reversible, or easily disrupted. Conversely, a cellular system may contain intrinsically stable states that are not reached by any scaffold parameter. The framework therefore distinguishes resolvability from stabilization.

We describe this distinction using the language of attractor-like organizational landscapes. Cellular states can be conceived as occupying a dynamic state space shaped by adhesion, cytoskeletal tension, nuclear deformation, signaling activity, matrix deposition, and cell–cell interactions [83,84]. Within this space, some states are more stable than others: perturbations are more likely to return to deep, well-defined basins than to shallow or poorly specified ones. Scaffold architecture does not deterministically impose a final tissue form, but it can reshape this landscape by making some organizational trajectories more accessible and more robust.

Aligned electrospun fibers in peripheral nerve guidance, anisotropic substrates in skeletal muscle engineering, and graded mechanical or mineralized interfaces in tendon-to-bone models are consistent with this interpretation. In each case, the scaffold does not encode a complete mature tissue architecture. Rather, it biases cellular organization by supplying coherent directional, mechanical, or spatial constraints. These examples should be read as illustrations of the framework, not as proof of the landscape formalism.

The formal version of this argument is given in Appendix B. There, cellular states are represented by an organizational potential $V(r;x)$, where $r$ denotes the cellular organizational state and $x$ denotes scaffold architecture. A locally stable state satisfies a stationarity condition, and differentiating that condition gives the local response relation

$$J=-H^{-1}C,$$

where $J$ is the scaffold–cell response mapping, $H$ is the Hessian of the organizational potential in cellular state space, and $C$ is the scaffold–cell coupling matrix. The main text does not require the derivation, but the relation clarifies the biological distinction: coupling determines whether scaffold variation reaches the cell, while curvature determines whether the induced state is held.

This yields the central inequality:

$$D_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}\le D_{\mathrm{e}\mathrm{f}\mathrm{f}}\le n.$$

The second inequality reflects dimensional compression: the cell cannot resolve more independent scaffold directions than the experimenter varies and often resolves fewer because $\mathsf{\Phi }$ is lossy. Some architectural parameters may be inaccessible, unstable at the relevant cellular scale, or redundant with other parameters. The first inequality reflects stabilization: some directions may be resolved early but not maintained against biological noise or perturbation.

$D_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}$ is therefore defined as the number of directions within the resolvable subspace of $J$ along which the induced organizational state is sufficiently stable. The restriction to the resolvable subspace is necessary. A cellular state may be intrinsically stable, but if no architectural parameter reaches it, that stability should not be attributed to the scaffold. $D_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}$ counts only those directions that the architecture both reaches and holds.

This distinction has an immediate experimental implication. A scaffold can have high $D_{\mathrm{e}\mathrm{f}\mathrm{f}}$ but low $D_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}$: cells initially resolve several architectural directions, but the induced organization fails to persist. Conversely, two scaffolds may have similar $D_{\mathrm{e}\mathrm{f}\mathrm{f}}$ but different robustness if one produces deeper or more persistent organizational basins (Figure 3). Design D in Section 7.4 and Supplementary Material S1 is intended to test precisely this dissociation.

Figure 3. Experimental interpretation of scaffold-guided stabilization of organizational states. (A) Two aligned scaffold systems are matched for initial orientational order, local stiffness, and ligand presentation but differ in cue persistence or stress-relaxation behavior. (B) Both may initially generate comparable focal-adhesion, cytoskeletal, nuclear, and YAP/TAZ responses and therefore similar early effective informational dimensionality. (C) Their long-term outcomes may nevertheless diverge. A persistent architecture is represented by a deeper organizational basin associated with maintained alignment, organized extracellular-matrix deposition, and recovery after perturbation; a rapidly relaxing architecture is represented by a shallower basin associated with loss of alignment and heterogeneous matrix organization. The prediction is that $D_{stab}$ can differ between scaffolds with similar $D_{eff}$. The landscapes are conceptual representations of Design D, not fitted energy functions.

The framework also predicts that reducing mechanotransductive coupling should reduce both resolvability and stabilization. Contractility inhibition, ROCK or myosin-II inhibition, integrin blockade, or loss of adhesion competence should weaken the coupling through which scaffold architecture affects cellular organization. If an architecture retained high $D_{\mathrm{e}\mathrm{f}\mathrm{f}}$ and high $D_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}$ under effective mechanotransductive blockade, the framework would be strongly challenged.

The practical conclusion is that scaffold design should not aim only to increase the number of cell-readable dimensions. It should also ask which of those dimensions are robustly maintained. Informational dimensionality is therefore necessary but not sufficient for organized regeneration: the architecture must generate cues that cells can resolve, integrate, and stabilize over the timescale of tissue formation.

6. Designing Informationally Structured Regenerative Environments

The distinction between geometric complexity and effective informational dimensionality has important implications for biomaterial design, including which architectural features create environmental conditions that cells can reliably detect, integrate, and exploit during tissue formation.

This perspective shifts attention away from complexity as an objective in itself and toward the informational structure of the cellular microenvironment. A growing body of evidence suggests that regenerative outcomes emerge from the interaction of multiple mechanical, geometric, topographical, transport, and temporal variables rather than from isolated parameters [90]. Regenerative outcomes emerge from the interaction of multiple environmental variables rather than from isolated parameters [91,92].

An important implication is that the properties measured by engineers are not always identical to those experienced by cells. Experimental studies have shown that the local mechanical environment perceived at cellular length scales may diverge substantially from bulk material properties [43], while transport accessibility and pore interconnection can exert stronger biological effects than nominal pore size alone [93]. Likewise, hierarchical pore architectures frequently outperform simpler structures not because any individual pore dimension is optimal, but because multiple scales cooperate to support transport, infiltration, mechanical support, and extracellular matrix deposition simultaneously [90]. These observations suggest that the biologically relevant variables are often those that remain locally accessible and interpretable to cellular mechanosensing rather than those most easily quantified at the material scale.

The admissibility criteria of Section 2.1 convert this general observation into a constraint on scaffold design, because they specify what an architectural parameter must satisfy in order to contribute a dimension at all. A parameter earns a column in the coupling matrix C only if it is cell-accessible, stable over the timescale of the decision it is meant to bias, and non-redundant with the parameters already counted. Table 4 applies these criteria to the architectural variables most frequently manipulated in the literature, stating for each the descriptor by which it is measured, the components of the readout vector r into which it is expected to couple, the parameters with which it is predicted to be redundant, and—most usefully for design—the conditions under which it would be disqualified as a dimension despite being present in the material.

Three consequences of this table are worth stating explicitly, because each contradicts a common design assumption. First, bulk stiffness is not a dimension: it lies largely outside the image of Φ, and cellular responses should track local, cell-scale mechanics instead [43,44]. Second, hierarchical organization is the parameter most often conflated with complexity, and it earns its dimensions only if each structural level loads a distinct region of r; a hierarchy whose removal of one level leaves r unmoved along any new direction was never contributing a dimension, however visible it is in a micrograph [90,94]. Third, roughness descriptors such as Sdr are measures of $C_g$, not of $D_{eff}$, and their relation to organizational outcome is the inverted-U described in Section 2.1 rather than a monotonic gain [40].

Table 4. Architectural parameters as candidate informational dimensions. For each parameter, the table reports the descriptor by which it can be quantified, the components of the readout vector r into which it is expected to couple, its predicted redundancy partners, and the conditions under which it fails the admissibility criteria defined in Section 2.1 and therefore contributes no informational dimension despite being physically present.

Parameter	Descriptor and Measurement	Coupled Components of $\mathit{r}$	Predicted Redundancy	Disqualified as a Dimension if…
Anisotropy ($\theta$)	Nematic order parameter (S) derived from FFT analysis of SEM or confocal images; directional variance [60,91]	Cytoskeletal order; nuclear aspect ratio; ECM alignment [68,95]	With lithographic groove orientation: same cue class, with predicted $K_{bio}\approx 1$	The correlation length is shorter than the cell diameter, so the cue is registered as noise rather than as a stable direction
Curvature ($c$)	Mean and Gaussian curvature obtained from $\mu$CT surface reconstruction; exactly specified in TPMS and gyroid lattices [61,96]	Nuclear deformation; YAP/TAZ nuclear-to-cytoplasmic ratio [41,97]; focal adhesion distribution	Partial redundancy with spatial confinement	The radius of curvature is much larger than the cell and is therefore effectively invisible to mechanosensing
Pore interconnectivity ($p$)	$\mu$CT-derived throat diameter, connectivity density, and tortuosity, rather than nominal pore size [62,93]	Infiltration depth; ECM organization; weak coupling to focal adhesion organization	With transport accessibility	It affects transport without altering adhesion geometry and loads too few components of $r$ to constitute an independent axis
Stiffness and stiffness gradient ($E,\nabla E$)	Local AFM or optical-tweezer microrheology rather than bulk rheometry [43,44]	Focal adhesion maturation; YAP/TAZ nuclear-to-cytoplasmic ratio; migration bias, including directional migration through durotaxis above a cell-type-specific threshold [63]	Bulk modulus is not a separate informational dimension because it lies outside the image of $\Phi$ when it is not locally cell-accessible	The gradient is below the durotactic threshold characteristic of the cell type
Ligand density and presentation ($a$)	RGD surface density; nanoscale ligand spacing; tethered versus diffusible presentation	Focal adhesion number and maturation; spreading area [73,74]	With adhesive area fraction	Ligand spacing is coarser than the range over which integrins cluster into nascent adhesions, making the cue subthreshold [73,74]
Hierarchical organization	Number of structural length scales that vary independently [69,90,94]	Predicted to load distinct components of $r$ at different scales	This is the parameter most frequently conflated with geometric complexity	Removal of one hierarchical level does not displace $r$ along a direction that is not already spanned by the remaining levels
Degradation and stress relaxation ($d$)	Mass-loss rate; crosslink half-life; viscoelastic relaxation time $\tau$ [98]	Persistence of all components of $r$; ECM organization	—	$\tau$ is shorter than the cellular decision timescale it is intended to bias, so the cue functions as noise rather than as information
Roughness ($\rho$)	$Sdr$, $S_a$, and $S_q$ measured by profilometry [40]	Focal adhesion maturation; cellular elongation	—	Roughness functions only as a descriptor of $C_g$ rather than as an independent contributor to $D_{eff}$; its relationship with organizational outcome is non-monotonic, as discussed in Section 2.1 [40]

Table 4. Architectural parameters as candidate informational dimensions. For each parameter, the table reports the descriptor by which it can be quantified, the components of the readout vector r into which it is expected to couple, its predicted redundancy partners, and the conditions under which it fails the admissibility criteria defined in Section 2.1 and therefore contributes no informational dimension despite being physically present.

Parameter	Descriptor and Measurement	Coupled Components of $\mathit{r}$	Predicted Redundancy	Disqualified as a Dimension if…
Anisotropy ($\theta$)	Nematic order parameter (S) derived from FFT analysis of SEM or confocal images; directional variance [60,91]	Cytoskeletal order; nuclear aspect ratio; ECM alignment [68,95]	With lithographic groove orientation: same cue class, with predicted $K_{bio}\approx 1$	The correlation length is shorter than the cell diameter, so the cue is registered as noise rather than as a stable direction
Curvature ($c$)	Mean and Gaussian curvature obtained from $\mu$CT surface reconstruction; exactly specified in TPMS and gyroid lattices [61,96]	Nuclear deformation; YAP/TAZ nuclear-to-cytoplasmic ratio [41,97]; focal adhesion distribution	Partial redundancy with spatial confinement	The radius of curvature is much larger than the cell and is therefore effectively invisible to mechanosensing
Pore interconnectivity ($p$)	$\mu$CT-derived throat diameter, connectivity density, and tortuosity, rather than nominal pore size [62,93]	Infiltration depth; ECM organization; weak coupling to focal adhesion organization	With transport accessibility	It affects transport without altering adhesion geometry and loads too few components of $r$ to constitute an independent axis
Stiffness and stiffness gradient ($E,\nabla E$)	Local AFM or optical-tweezer microrheology rather than bulk rheometry [43,44]	Focal adhesion maturation; YAP/TAZ nuclear-to-cytoplasmic ratio; migration bias, including directional migration through durotaxis above a cell-type-specific threshold [63]	Bulk modulus is not a separate informational dimension because it lies outside the image of $\Phi$ when it is not locally cell-accessible	The gradient is below the durotactic threshold characteristic of the cell type
Ligand density and presentation ($a$)	RGD surface density; nanoscale ligand spacing; tethered versus diffusible presentation	Focal adhesion number and maturation; spreading area [73,74]	With adhesive area fraction	Ligand spacing is coarser than the range over which integrins cluster into nascent adhesions, making the cue subthreshold [73,74]
Hierarchical organization	Number of structural length scales that vary independently [69,90,94]	Predicted to load distinct components of $r$ at different scales	This is the parameter most frequently conflated with geometric complexity	Removal of one hierarchical level does not displace $r$ along a direction that is not already spanned by the remaining levels
Degradation and stress relaxation ($d$)	Mass-loss rate; crosslink half-life; viscoelastic relaxation time $\tau$ [98]	Persistence of all components of $r$; ECM organization	—	$\tau$ is shorter than the cellular decision timescale it is intended to bias, so the cue functions as noise rather than as information
Roughness ($\rho$)	$Sdr$, $S_a$, and $S_q$ measured by profilometry [40]	Focal adhesion maturation; cellular elongation	—	Roughness functions only as a descriptor of $C_g$ rather than as an independent contributor to $D_{eff}$; its relationship with organizational outcome is non-monotonic, as discussed in Section 2.1 [40]

A final caveat belongs here, because it follows from the definition rather than from any limitation of technique. $D_{eff}$ is not a material specification. It is a property of the scaffold–cell mapping, and is therefore specific to the cell type, the readout panel, and the timescale over which the response is measured. It cannot be printed on a datasheet, and two laboratories reporting $D_{eff}$ for nominally identical scaffolds will not obtain comparable numbers unless they share a readout panel. We regard this as a consequence of the concept rather than a defect in it: informational dimensionality is relational, and a framework that claimed otherwise would be claiming that scaffolds instruct cells irrespective of which cells they are.

This interpretation also helps explain why regenerative performance is often poorly predicted by geometric sophistication alone. Geometrically elaborate scaffolds may contain numerous structural features while providing relatively little additional organizational information if those features are redundant, unstable, or mechanotransductively incoherent [99]. Conversely, comparatively simple architectures may exert strong organizational effects when they introduce robust directional, mechanical, transport, or biochemical constraints [100]. Evidence from aligned fibrous scaffolds, anisotropic architectures, and hierarchically interconnected porous systems suggests that coherent organizational cues can be more influential than geometric richness per se [91,93,94]. The relevant design criterion is therefore not the amount of structural complexity, but the extent to which complexity generates stable and biologically interpretable dimensions for cellular organization.

A further implication is that scaffold design cannot be treated as a purely static optimization problem. Cell–scaffold systems continuously remodel themselves through matrix deposition, degradation, force generation, and local restructuring of the microenvironment [98]. Mechanical properties perceived by cells may evolve during regeneration, while the organization of the developing tissue progressively modifies the very environment from which it emerged [101]. Effective scaffold architectures must therefore do more than provide useful initial cues: they must preserve sufficient informational coherence as regeneration unfolds. The challenge is not simply to create an initial template, but to maintain organizational accessibility throughout the regenerative process.

The scope and limits of the kernel analogy invoked throughout this framework are set out in Section Scope and Limits of the Kernel Analogy and Appendix B and are not repeated here. Biological systems remain non-deterministic in ways that analogy does not capture: cellular outcomes are influenced by stochastic gene expression, local matrix heterogeneity, variable adhesion dynamics, and continuous cell-mediated remodeling [102], so scaffold architecture biases cellular populations toward organizationally coherent regions of state space rather than determining a unique outcome.

These considerations also suggest a role for computational and AI-assisted scaffold design. Current machine-learning approaches increasingly predict material properties, degradation kinetics, mechanical behavior, manufacturability, and, in some cases, proxies of cellular growth [103]. The perspective developed here suggests an additional objective: identifying architectural configurations that generate stable and biologically interpretable mechanosensory dimensions. Rather than searching exclusively for biomimetic geometries, future computational approaches could seek scaffold architectures that maximize mechanotransductive coherence while minimizing redundancy and informational noise. Although such approaches remain largely prospective, they point toward a shift from optimizing scaffold geometry alone to optimizing the informational structure of the cellular microenvironment.

More broadly, the perspective developed here suggests that successful regenerative biomaterials do not function because they reproduce the extracellular matrix in exhaustive structural detail. Rather, they appear to create sufficiently structured environments within which organized regeneration becomes more accessible. Scaffold architecture contributes to this process by shaping the mechanical, spatial, transport, and temporal dimensions available to cellular collectives. Tissue engineering can therefore be understood not simply as the fabrication of materials, but as the design of environments capable of supporting robust biological self-organization.

7. Testing the Framework: Readouts, Predictions, and Experimental Designs

A framework that fixes the types of its objects and derives relations between them must specify how it could be tested. This section identifies the readouts required by the framework, summarizes what the cited literature has and has not measured, and states the predictions that would falsify the proposal. No experiments were performed in this study. The designs below are proposed validation strategies: their logic is summarized here, their full experimental specification is provided in Supplementary Material S1, and the statistical implementation is provided in Supplementary Material S2.

7.1. Early Readout Vector and Late Stabilization Outcome

The formalism of Section 5 makes the required measurement unambiguous. Because $D_{eff}$ is the effective rank of the response mapping J, and rank is a property of a joint response, the components of r that enter the estimate must be acquired on the same specimens under the same conditions and at the same timepoint. This is a genuine methodological constraint rather than a convenience: an experiment that measures focal adhesion maturation in one cohort and YAP/TAZ localization in another cannot estimate a rank, because it never observes the covariance between them. The four early components of the panel—focal adhesion maturation, cytoskeletal orientational order, nuclear deformation and YAP/TAZ localization—are simultaneously accessible by multichannel immunofluorescence on a single specimen, and it is these four that constitute the readout vector from which $D_{eff}$ is estimated. Extracellular matrix organization is acquired at a later timepoint on parallel specimens from the same fabrication batch and condition, and enters the framework in a different role, as set out below. Table 5 specifies the panel.

Table 5. Early components of the mechanotransductive response vector $r_{\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{l}\mathrm{y}}$ and the separate late ECM outcome, with representative assays, quantitative metrics, and characteristic timescales. The four early components used to estimate $D_{\mathrm{e}\mathrm{f}\mathrm{f}}$ must be acquired jointly on common specimens. ECM organization is measured later on parallel specimens and is used as a probe of $D_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}$, not as a fifth component of the rank estimate.

Component of $\mathit{r}$	Representative Assay	Quantitative Metric	Characteristic Timescale	What It Reports
Focal adhesion maturation	Immunofluorescence for vinculin, paxillin, and phospho-FAK (Y397)	Adhesion count; mean adhesion area; elongation ratio; fraction of adhesions exceeding 2 μm	Minutes to hours	Local force transmission and accessibility of nanoscale adhesive cues
Cytoskeletal orientational order	Phalloidin staining of F-actin	Nematic order parameter (S); coherency index	Hours	Coherence of directional mechanical cues
Nuclear deformation	DAPI staining; lamin A/C immunofluorescence	Nuclear aspect ratio; projected area; nuclear volume; chromatin condensation index	Hours	Force transmission through the LINC complex; sensing of curvature and spatial confinement
YAP/TAZ localization	Immunofluorescence; qPCR for CTGF and CYR61	Nuclear-to-cytoplasmic fluorescence intensity ratio	Hours to one day	The integrated mechanical state experienced by the cell
Extracellular matrix (ECM) organization	Collagen I and fibronectin immunofluorescence; second-harmonic generation imaging	Fiber alignment order parameter; matrix coverage; layer thickness	Days to weeks	Whether the organizational state persisted and was written back into the extracellular matrix

The final component of the panel plays a distinct role, and the fact that it cannot be co-acquired with the others is a consequence of the biology rather than a limitation of technique. Focal adhesion maturation, cytoskeletal alignment, nuclear deformation and YAP/TAZ localization are early and largely reversible, and their joint variation at a common timepoint estimates what the cell can resolve—that is, $D_{eff}$. Extracellular matrix organization is late and cumulative, requires the specimen to be carried to a timepoint at which the early readouts have long since relaxed, and reports something the early panel cannot: what the cell was able to hold. It is therefore the natural probe of $D_{stab}$ rather than a fifth component of the rank estimate. The framework predicts that these two quantities can dissociate, and Design D (Section 7.4; Supplementary Material S1.6) is constructed to detect that dissociation.

7.2. Experimental Precedents and the Measurement Structure Required by the Framework

The framework developed here is grounded in experimental studies showing that scaffold and substrate properties can influence adhesion, cytoskeletal organization, nuclear mechanics, mechanosensitive signalling, and later tissue organization. The relevant literature was assembled through a purposive narrative process rather than through an exhaustive systematic-review protocol. The examples discussed in this section and presented in Supplementary Material S3 are therefore illustrative experimental precedents, not a representative sample from which the prevalence of particular parameters, readouts, or experimental practices can be inferred.

These precedents show that the individual components required by the framework are experimentally accessible. Dynamic and static topographies can alter focal-adhesion organization and cell geometry; anisotropic fibrous matrices can direct cellular and extracellular-matrix alignment; stiffness and stiffness gradients can regulate migration, adhesion-dependent signalling, and cellular phenotype; and curvature can couple cellular geometry to nuclear deformation and YAP/TAZ localization [4,17,26,35,44,46,53,58,80,91]. Hierarchical and porous scaffold systems further illustrate how several spatial scales can influence cell organization and tissue-level outcomes [54,88]. Collectively, these studies establish the biological plausibility of treating scaffold architecture as an input to a measurable mechanotransductive response.

Their role in the present argument is illustrative rather than evidentially exhaustive. Each example demonstrates one or more relations between an architectural variable and a cellular or matrix response, but such relations are not themselves estimates of effective informational dimensionality. A strong response to one scaffold variable establishes coupling along that comparison; it does not determine how many independently manipulated architectural directions remain distinguishable in the joint cellular response.

Estimating $D_{\mathrm{e}\mathrm{f}\mathrm{f}}$ requires a different experimental structure. Several scaffold parameters must be manipulated independently within the same library, their final cell-relevant values must be verified directly, and the components of ${\mathbf{r}}_{\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{l}\mathrm{y}}$—focal-adhesion maturation, cytoskeletal orientational order, nuclear deformation, and YAP/TAZ localization—must be acquired through a common measurement panel across the same parameter space. The resulting parameter-to-readout sensitivity matrix must then be standardized and compared with an experimentally estimated noise floor. Measurements distributed across unrelated studies cannot be combined to recover this rank because cell type, scaffold system, parameter range, timepoint, assay definition, and experimental noise differ among studies.

The distinction does not imply that conventional single-cue experiments are incomplete relative to their own objectives. Such designs are often the clearest way to identify causal mechanisms, thresholds, or nonlinear responses. They answer a different question. The framework asks not only whether a scaffold parameter changes a cellular outcome, but whether multiple parameters generate independent or redundant directions in a jointly observed response space.

The experimental precedents in Supplementary Material S3 therefore serve two purposes. First, they connect the abstract objects of the framework to real scaffold systems and measurable biological responses. Second, they clarify the additional design requirements imposed by an estimate of $D_{\mathrm{e}\mathrm{f}\mathrm{f}}$ or $D_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}$. They are not used to claim that one parameter or readout predominates across the field, nor to establish that no published study has implemented the required design. The predictions and proposed experiments that follow specify how these quantities could be tested directly.

7.3. Predictions and Their Falsifiers

The framework generates predictions that are more specific than the claim that structured scaffolds influence cells. Each prediction links a formal quantity to a measurable experimental design and to a result that would weaken or falsify the framework. Table 6 summarizes these tests; full experimental details are provided in Supplementary Material S1.

Table 6. Predictions generated by the framework, the design that tests each, and the result that would falsify it. Predictions concern three distinct quantities—resolvable dimensionality ($D_{eff}$), coupling strength (G), and stabilized dimensionality ($D_{stab}$)—which respond differently to perturbation. Full protocols are given in Supplementary Material S1.

Prediction	Quantity	Experimental Design	Falsified if…
The mechanosensory map is lossy in the regions sampled: cells resolve fewer directions than engineers manipulate.	$D_{eff}<n$	A. Factorial electrospun scaffold library; joint acquisition of the readout vector $r$ at a common timepoint; estimation of the thresholded rank of the regional sensitivity matrix.	Full local rank is recovered repeatedly across diverse scaffold libraries, cell types, and operating regions. This would argue against the prevalence of local dimensional compression, although it would not establish that $\Phi$ is globally invertible.
Architectures that differ structurally but not in the image of $\Phi$ are representationally equivalent to the cell.	$K_{bio}$	B. Comparison of aligned electrospun fibers and lithographic micro-grooves, matched for order parameter and local modulus; equivalence testing against a pre-specified margin, with a positive control.	The architectures are reliably separable in $r$ at above-chance cross-validated accuracy, with $K_{bio}$ outside the equivalence margin. This would indicate that anisotropy presented by fibers and by grooves is read differently and that the feature vector omits a variable accessible to the cell.
Coupling is carried by mechanotransduction: as $\lambda \to 0$$, C\to 0$, and hence $J\to 0$.	$G$$, D_{\mathrm{e}\mathrm{f}\mathrm{f}}$, and $D_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}$	C. Convergent suppression of contractility using para-nitroblebbistatin, Y-27632, and ${\beta }_1$-integrin blockade, applied to an architecture with high $D_{eff}$.	Coupling strength and resolvable dimensionality persist under effective contractility blockade at viable doses. This would indicate that $D_{eff}$ is not mechanotransductive in origin and would undermine the framework at its foundation.
Resolvability and durability are distinct: architectures may be equally readable yet unequally stable.	$D_{stab}$ versus $D_{\mathrm{e}\mathrm{f}\mathrm{f}}$	D. Comparison of two architectures matched for $D_{eff}$ but differing in basin curvature; controlled perturbation followed by measurement of return to the prior organizational state.	Architectures matched for $D_{eff}$ are invariably matched for robustness. This would render the inequality $D_{stab}\le D_{eff}$ empirically empty.
Coupling strength is a non-monotonic function of geometric complexity.	$G$ [40]	E. Graded-$Sdr$ ladder based on Rabel [40]. A one-parameter series cannot exhibit non-monotonic rank: with (n = 1), the sensitivity matrix has a single column, whose rank is (1) wherever it is non-zero.	$G$ rises monotonically with $C_g$ across the ladder.
Bulk material properties lie largely outside the image of $\Phi$	$G$ and $D_{eff}$ [43]	F. Hydrogels matched for bulk storage modulus $G^{\prime }$ but differing in cell-scale mechanics, following Ciccone et al. (2020) [43].	The readout vector $r$ tracks bulk modulus as closely as local cell-scale mechanics.

7.4. The Designs

Table 6 specifies the six designs (A–F) that operationalize these predictions, together with the result that would falsify each; full scaffold systems, readout panels, controls, and statistical procedures are provided in Supplementary Material S1. Design A estimates $D_{\mathrm{e}\mathrm{f}\mathrm{f}}$ directly from a factorial scaffold library; Designs B and C test, respectively, redundancy between architectures and the mechanotransductive origin of coupling; Design D dissociates resolvability ($D_{\mathrm{e}\mathrm{f}\mathrm{f}}$) from stabilization ($D_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}$); and Designs E and F test the predicted non-monotonicity of coupling strength and the limited role of bulk descriptors. The full logic of each design is given in Table 6 and is not repeated here.

7.5. What Would Not Count as a Test

Several common results would not, by themselves, validate the framework. Showing that a structured scaffold improves one cellular outcome over a flat control would not be sufficient, because many scaffold-design theories predict such an effect. Showing that one readout changes, for example, YAP/TAZ localization or cytoskeletal alignment alone, would also not estimate $D_{\mathrm{e}\mathrm{f}\mathrm{f}}$, because dimensionality is a property of a joint response mapping. Finally, showing that a geometrically complex scaffold improves regeneration would not establish informational dimensionality unless the relevant cell-readable directions, their redundancy, and their stabilization were measured. The framework is tested only where it makes a prediction that simpler cue-response or biomimetic accounts would not, and the designs summarized above and specified in Supplementary Material S1 were selected on that basis.

8. Conclusions: From Complexity to Informational Dimensionality

This paper has argued that scaffold-guided regeneration should be evaluated not only by geometric sophistication or biomimetic resemblance, but by the number of stable, cell-readable dimensions an architecture introduces into the mechanosensory environment. The practical consequence is a shift in scaffold design criteria. Architectures should be assessed by whether their variables are accessible at cellular length and force scales, stable over the relevant decision timescale, non-redundant in the mechanotransductive response, and capable of supporting robust return after perturbation.

This reframing does not replace established scaffold descriptors such as porosity, anisotropy, roughness, stiffness, curvature, ligand density, or degradation rate. Rather, it asks which of these descriptors remain biologically meaningful after projection through cellular mechanosensing. A scaffold feature that is geometrically present but does not alter focal adhesion maturation, cytoskeletal order, nuclear deformation, YAP/TAZ localization, or later extracellular matrix organization may be structurally measurable but informationally inactive for the cell type and timepoint considered.

The framework is therefore most useful as an experimental design tool. It encourages factorial scaffold libraries, multiplexed readout panels, local rather than bulk material characterization, redundancy tests between distinct fabrication routes, and perturbation assays that distinguish resolvability from durability. These approaches would allow scaffold performance to be evaluated not only by the magnitude of a cellular response, but by the dimensional structure of the scaffold–cell mapping and by the stability of the organizational states that mapping supports.

The proposal remains conceptual and requires experimental testing. Its value will depend on whether cell-resolvable dimensionality predicts organized and robust tissue formation better than geometric complexity alone. If it does, scaffold optimization could move beyond the search for ever more elaborate architectures toward the design of environments that provide coherent, stable, and mechanotransductively interpretable conditions for tissue self-organization.

Appendix A.1. Purpose of the Estimator

Effective informational dimensionality, $D_{\mathrm{e}\mathrm{f}\mathrm{f}}$, is defined in the main text as the number of scaffold-derived directions that a given cell–readout system can resolve as distinct. It is not a property of the scaffold alone, nor of the cell alone, but of the local mapping between experimentally manipulated scaffold parameters and early mechanotransductive cellular responses.

The purpose of this appendix is to specify the minimal operational procedure required to estimate $D_{\mathrm{e}\mathrm{f}\mathrm{f}}$. The procedure is deliberately local and relational. A reported value of $D_{\mathrm{e}\mathrm{f}\mathrm{f}}$ is meaningful only for a declared scaffold library, cell type, readout panel, timepoint, scaling convention, and noise-floor criterion. It should therefore not be reported as an intrinsic material property, in the way one might report porosity, elastic modulus, or mean fiber diameter.

The detailed statistical implementation, including batch structure, randomization, bootstrap uncertainty, interaction terms, and reporting conventions, is provided in Supplementary Material S2. The aim here is narrower: to show how the conceptual definition in the main text becomes a measurable quantity.

Appendix A.2. Scaffold Parameters and Early Readout Vector

Let

$$x\in {\mathbb{R}}^n$$

denote the vector of independently manipulable scaffold parameters. These may include, for example, anisotropy, curvature, pore geometry, stiffness, stiffness gradients, ligand presentation, roughness, degradation behavior, or hierarchical organization. The entries of $x$ should be verified architectural descriptors rather than fabrication settings. Collector speed, polymer concentration, etching time, or crosslinking protocol are useful fabrication variables, but $D_{\mathrm{e}\mathrm{f}\mathrm{f}}$ concerns the scaffold properties actually presented to the cell.

Let

$$r_{\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{l}\mathrm{y}}\in {\mathbb{R}}^4$$

denote the early mechanotransductive response vector. In the present framework this vector contains four jointly acquired readouts:

• focal-adhesion maturation;
• cytoskeletal orientational order;
• nuclear deformation;
• YAP/TAZ nuclear-to-cytoplasmic localization.

These four readouts are acquired on the same specimens, under the same culture conditions, and at a common early timepoint. This requirement is not merely technical. $D_{\mathrm{e}\mathrm{f}\mathrm{f}}$ is a rank property of the joint scaffold–cell response. If focal adhesions are measured in one cohort, cytoskeletal order in another, nuclear morphology in a third, and YAP/TAZ localization in a fourth, the covariance structure of the response vector cannot be reconstructed.

Extracellular-matrix organization is treated separately. ECM alignment, matrix continuity, and late tissue-level organization are biologically central, but they are late and cumulative outcomes. They are therefore not included in $r_{\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{l}\mathrm{y}}$ and do not enter the rank estimate for $D_{\mathrm{e}\mathrm{f}\mathrm{f}}$. Instead, they are used as probes of stabilized organization, denoted in the main text as $D_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}$.

Appendix A.3. The Scaffold–Cell Response Matrix

At a specified operating point or sampled region of scaffold-parameter space, the local scaffold–cell response mapping is written as

$$J={\displaystyle \frac{\partial r_{\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{l}\mathrm{y}}}{\partial x}}$$

The matrix $J$ has four rows, corresponding to the four early readouts, and $n$ columns, corresponding to the independently manipulated scaffold parameters. Each column describes how the early mechanotransductive state changes when one architectural parameter is varied.

In practice, $J$ is estimated from a scaffold library. For each scaffold condition, the architectural parameters are measured, and the four early readouts are quantified. Each component of $r_{\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{l}\mathrm{y}}$ is then modeled as a function of the architectural parameters. In the simplest regional linear case,

$$r_{\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{l}\mathrm{y},i}={\beta }_{0,i}+{\sum }_{j=1}^n{\beta }_{ij}{x}_j+{\epsilon }_i$$

where $i$ indexes the readout and $j$ indexes the scaffold parameter. The fitted coefficients ${\beta }_{ij}$ form the empirical estimate of $J$.

The interpretation of $J$ depends on the design. If a parameter is sampled at only two levels, the corresponding coefficient is a finite-difference estimate across the sampled region, not a pointwise derivative. A true local Jacobian requires multiple levels around a specified operating point. Therefore, every estimate of $D_{\mathrm{e}\mathrm{f}\mathrm{f}}$ should specify whether it is regional or local.

Appendix A.4. Standardization

The raw sensitivity matrix cannot be decomposed directly, because both the scaffold parameters and the readouts have different units and numerical ranges. Fiber diameter, stiffness, curvature, and ligand density are not commensurable. Likewise, focal-adhesion area, cytoskeletal order, nuclear aspect ratio, and YAP/TAZ nuclear-to-cytoplasmic ratio have different scales.

The readouts are therefore scaled by their replicate-level noise. Let

$$D_r=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}({\sigma }_{\mathrm{F}\mathrm{A}},{\sigma }_{\mathrm{C}\mathrm{S}\mathrm{K}},{\sigma }_{\mathrm{N}\mathrm{U}\mathrm{C}},{\sigma }_{\mathrm{Y}\mathrm{A}\mathrm{P}})$$

where each $\sigma$ is the pooled within-condition noise scale for the corresponding early readout. This expresses each response in units of its own experimental resolvability.

The scaffold parameters are scaled by their experimentally accessible manipulation ranges. Let

$$D_x=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(\Delta x_1,\Delta x_2,\dots ,\Delta x_n)$$

where $\Delta x_j$ is the prespecified range over which parameter $j$ can be reliably varied and biologically interpreted in the chosen scaffold system.

The doubly standardized sensitivity matrix is then

$$\tilde{J}=D_r^{-1}J{D}_x.$$

This matrix asks how many readout-noise units of response are produced when each scaffold parameter is varied across its usable experimental range. Scaling by accessible range is biologically important: a parameter that can be varied only within a narrow window offers less resolvable variation to the cell than one that can be varied broadly.

Appendix A.5. Singular Values and Noise Floor

The singular values of $\tilde{J}$ are computed as

$${\sigma }_1\ge {\sigma }_2\ge \cdots \ge {\sigma }_{\mathrm{m}\mathrm{i}\mathrm{n}(4,n)}.$$

These singular values describe how scaffold variation is distributed across independent directions of early cellular response. A large leading singular value with much smaller subsequent values indicates that the scaffold library mainly drives one dominant mechanotransductive direction. Several singular values above noise indicate that multiple independent directions are resolved.

A singular value should be counted only if it exceeds the experimental noise floor. The threshold is denoted

$${\tau }_{\mathrm{n}\mathrm{o}\mathrm{i}\mathrm{s}\mathrm{e}}.$$

In the full implementation, ${\tau }_{\mathrm{n}\mathrm{o}\mathrm{i}\mathrm{s}\mathrm{e}}$ is estimated by permutation: the association between scaffold parameters and cellular responses is destroyed while preserving the replicate and block structure of the experiment, the sensitivity matrix is re-estimated, and the null singular-value distribution is computed. The 95th percentile of the leading null singular value provides a conservative threshold. Other prespecified thresholds may be used, but the criterion must be reported.

Effective informational dimensionality is then defined as the thresholded numerical rank:

$$D_{\mathrm{e}\mathrm{f}\mathrm{f}}=\#\left\{{\sigma }_i\right(\tilde{J})>{\tau }_{\mathrm{n}\mathrm{o}\mathrm{i}\mathrm{s}\mathrm{e}}\}.$$

Thus, $D_{\mathrm{e}\mathrm{f}\mathrm{f}}$ is the number of scaffold-parameter directions that produce early cellular responses distinguishable from replicate-level noise. Because $\tilde{J}$ has one column per independently manipulated architectural parameter,

$$D_{\mathrm{e}\mathrm{f}\mathrm{f}}\le n.$$

Because the proposed early readout vector has four components,

$$D_{\mathrm{e}\mathrm{f}\mathrm{f}}\le 4$$

for this empirical panel. This upper bound reflects the resolution of the measurement panel, not the true resolving capacity of the cell.

Appendix A.6. Coupling Strength and Spectral Evenness

Dimensionality and response magnitude are different quantities. The framework therefore reports coupling strength separately:

$$G={\parallel \tilde{J}\parallel }_F,$$

where ${\parallel ·\parallel }_F$ is the Frobenius norm. $G$ measures the total strength with which scaffold variation couples to the measured early cellular state, regardless of how that coupling is distributed across directions.

This distinction is essential. A scaffold may produce a strong response along one direction and therefore have high $G$ but low $D_{\mathrm{e}\mathrm{f}\mathrm{f}}$. Another scaffold may produce weaker responses across several independent directions and therefore have lower $G$ but higher $D_{\mathrm{e}\mathrm{f}\mathrm{f}}$. These are different design outcomes.

A participation ratio may also be reported as a diagnostic of spectral evenness:

$$D_{\mathrm{P}\mathrm{R}}={\displaystyle \frac{{\left({\sum }_i{\sigma }_i^2\right)}^2}{{\sum }_i{\sigma }_i^4}}.$$

The participation ratio indicates whether sensitivity is concentrated in one dominant direction or distributed more evenly across several directions. It should not replace $D_{\mathrm{e}\mathrm{f}\mathrm{f}}$ as the primary estimator, because it does not by itself determine whether the singular directions exceed noise.

Appendix A.7. Relationship to Stabilized Organization

$D_{\mathrm{e}\mathrm{f}\mathrm{f}}$ estimates early resolvability: the number of architectural directions that produce distinguishable early mechanotransductive responses. It does not measure whether those responses are later stabilized. That second question is captured by $D_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}$, which concerns the subset of resolvable directions along which the induced organizational state is held against perturbation or biological noise.

Late ECM organization is therefore analyzed separately. ECM alignment, matrix continuity, and tissue-level organization are not added to $r_{\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{l}\mathrm{y}}$, because doing so would mix early interpretation with late stabilization. Instead, late ECM outcomes are used to ask whether the dimensions resolved early are also maintained over time.

The conceptual relation is

$$D_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}\le D_{\mathrm{e}\mathrm{f}\mathrm{f}}\le n.$$

The first inequality expresses the possibility that a cell can resolve more dimensions than it can stably maintain. The second inequality expresses the lossiness of the scaffold–cell mapping: the cell cannot resolve more independent scaffold directions than the experimenter has actually varied and may resolve fewer.

Appendix A.8. Minimal Reporting Requirements

Any empirical estimate of $D_{\mathrm{e}\mathrm{f}\mathrm{f}}$ should report at least the following information: the scaffold library; the independently manipulated architectural parameters; the verified architectural values; the cell type; the early timepoint; the four readouts used to construct $r_{\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{l}\mathrm{y}}$; confirmation that the readouts were acquired jointly on common specimens; the number of independent scaffold batches; the model used to estimate $J$; the scaling matrices $D_r$ and $D_x$; the singular values of $\tilde{J}$; the noise-floor procedure; the resulting $D_{\mathrm{e}\mathrm{f}\mathrm{f}}$; the coupling strength $G$; and any late ECM outcome used to probe $D_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}$.

Without these elements, a reported value of $D_{\mathrm{e}\mathrm{f}\mathrm{f}}$ is not interpretable. A single mechanotransductive marker does not estimate rank. A scaffold that improves one outcome does not necessarily increase informational dimensionality. The framework is tested only when the dimensional structure of the scaffold–cell response mapping is measured explicitly.

Appendix B.1. Purpose and Status of the Formalism

This appendix summarizes the formal structure underlying the framework developed in the main text. The purpose is not to introduce a predictive quantitative model of tissue regeneration. None of the expressions below is fitted to data in the present article, none returns a predicted regenerative outcome, and none should be interpreted as a mechanistic simulation of cellular behavior.

The purpose is more limited but still important. The formalism fixes the domains of the objects involved, distinguishes scaffold variables from cellular states, clarifies the relation between resolvability and stabilization, and identifies the quantities that would have to be measured to test the framework empirically. In this sense, the equations are organizational rather than predictive: they provide a disciplined grammar for the argument.

The central distinction is between three objects. The scaffold is described by a vector of manipulable architectural parameters, $x$. The cell is described by a vector of organizational or mechanotransductive states, $r$. The framework concerns the relation between them: how variation in scaffold architecture changes the cellular state, and whether the resulting state is later stabilized against noise or perturbation.

Appendix A describes how $D_{\mathrm{e}\mathrm{f}\mathrm{f}}$ can be estimated operationally from the early readout vector $r_{\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{l}\mathrm{y}}$. This appendix addresses the complementary formal question: why the distinction between effective informational dimensionality and stabilized organization is coherent, and why the kernel analogy must remain bounded.

Appendix B.2. Notation and Domains

Let

$$x\in {\mathbb{R}}^n$$

denote the vector of independently manipulable scaffold parameters. These parameters may include anisotropy, curvature, pore geometry, stiffness, stiffness gradients, ligand density or presentation, roughness, degradation behavior, or hierarchical organization. The dimension $n$ is the number of architectural directions the experimenter varies.

Let

$$r\in {\mathbb{R}}^m$$

denote the cellular organizational state. This is a broader object than the measured early response vector used for estimating $D_{\mathrm{e}\mathrm{f}\mathrm{f}}$. In empirical implementation, $r_{\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{l}\mathrm{y}}\in {\mathbb{R}}^4$ contains focal-adhesion maturation, cytoskeletal orientational order, nuclear deformation, and YAP/TAZ nuclear-to-cytoplasmic localization. Late ECM organization is treated separately as an outcome of stabilization.

Let $\mathsf{\Phi }$ denote the mechanosensory map through which scaffold parameters are converted into cell-resolvable cues. $\mathsf{\Phi }$ is not a designed mathematical transformation. It is a biological mapping imposed by the cell’s adhesion machinery, cytoskeletal organization, nuclear mechanics, and mechanotransductive signaling pathways. It is cell-type-specific, context-dependent, and lossy.

Let

$$K_{\mathrm{b}\mathrm{i}\mathrm{o}}({\mathbf{x}}_i,{\mathbf{x}}_j)=S \left({\tilde{\mathbf{r}}}_{\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{l}\mathrm{y}}\left({\mathbf{x}}_i\right),{\tilde{\mathbf{r}}}_{\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{l}\mathrm{y}}\left({\mathbf{x}}_j\right)\right)\in \left[\mathrm{0,1}\right]$$

denote an empirical similarity between the standardized early mechanotransductive response vectors generated by two scaffold architectures. Here, $S$ is a prespecified similarity function and ${\tilde{\mathbf{r}}}_{\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{l}\mathrm{y}}$ comprises the jointly acquired focal-adhesion, cytoskeletal, nuclear-deformation, and YAP/TAZ readouts after scaling. Values approaching one indicate increasing response similarity, but representational equivalence should be determined against a prespecified experimental margin rather than by an exact numerical equality. $K_{\mathrm{b}\mathrm{i}\mathrm{o}}$ is used to compare architectures after cellular projection; it is not used to estimate $D_{\mathrm{e}\mathrm{f}\mathrm{f}}$, and no assumption is made that it satisfies the conditions of a Mercer kernel.

Collecting $K_{\mathrm{b}\mathrm{i}\mathrm{o}}$ across several architectures produces a global pairwise similarity matrix. Its rank should not be identified with $D_{\mathrm{e}\mathrm{f}\mathrm{f}}$, because it may reflect both local response directions and curvature of the response manifold across widely separated scaffold conditions. Effective informational dimensionality is instead a local quantity estimated from the standardized scaffold–cell sensitivity matrix $J$ at a specified operating point. Accordingly, redundancy between architectural parameters is assessed from collinearity among the corresponding columns of $J$, whereas $K_{\mathrm{b}\mathrm{i}\mathrm{o}}$ is retained only for pairwise tests of representational equivalence.

Let

$$K_{\mathrm{o}\mathrm{r}\mathrm{g}}(r,r^{\ast }(x\left)\right)$$

denote the alignment between the current cellular organizational state $r$ and the organizational state $r^{\ast }\left(x\right)$ favored by scaffold architecture $x$. This is distinct from $K_{\mathrm{b}\mathrm{i}\mathrm{o}}$. $K_{\mathrm{b}\mathrm{i}\mathrm{o}}$ compares scaffold architectures through their cellular effects; $K_{\mathrm{o}\mathrm{r}\mathrm{g}}$ compares a cellular state with the state favored by a given architecture.

Finally, let

$$\lambda$$

denote the mechanotransductive coupling gain. This parameter summarizes how efficiently scaffold-derived cue coherence is converted into cellular organizational bias. It is a cellular and relational property, not a purely architectural one. Contractility inhibition, ROCK inhibition, myosin-II inhibition, integrin blockade, or loss of adhesion competence are expected to reduce $\lambda$.

Appendix B.3. Organizational Potential

The landscape formalism is introduced to express how scaffold architecture can bias the stability of cellular organizational states. We write an organizational potential over cellular states, parameterized by scaffold architecture:

$$V(r;x)=E_{\mathrm{i}\mathrm{n}\mathrm{t}}\left(r\right)-\lambda K_{\mathrm{o}\mathrm{r}\mathrm{g}}(r,r^{\ast }(x\left)\right).$$

Here, $E_{\mathrm{i}\mathrm{n}\mathrm{t}}\left(r\right)$ denotes the intrinsic organizational cost of cellular state $r$. It includes the cytoskeletal, adhesive, deformational, and regulatory constraints the cell incurs irrespective of the scaffold. The second term expresses the scaffold-dependent contribution: the architecture favors some organizational states over others, and the strength of this bias depends on the mechanotransductive coupling gain $\lambda$.

Lower values of $V(r;x)$ correspond to organizational states that are more stable within this formal representation. The scaffold does not prescribe a unique state in a deterministic way. Rather, it reshapes the landscape of possible states, making some cellular configurations more likely to be reached and maintained.

A locally stable cellular state $\widehat{r}\left(x\right)$ satisfies the stationarity condition

$${\nabla }_{r}V\left(\widehat{r}\right(x);x)=0.$$

This expression says that, for a given scaffold architecture, the cell occupies a local minimum or stationary point of the organizational potential. The framework does not claim that real cells literally compute or minimize this potential. The potential is a compact way to represent the idea that cellular organizational states are constrained by both intrinsic cellular costs and scaffold-derived mechanotransductive bias.

Appendix B.4. Response Mapping and Scaffold–Cell Coupling

The response of the cellular organizational state to changes in scaffold architecture is obtained by differentiating the stationarity condition with respect to $x$. Starting from

$${\nabla }_{r}V\left(\widehat{r}\right(x);x)=0,$$

differentiation gives

$$H{\displaystyle \frac{\partial \widehat{r}}{\partial x}}+C=0,$$

where

$$H={\nabla }_r^{2}V\left(\widehat{r}\right(x);x)$$

is the Hessian of the organizational potential with respect to the cellular state, and

$$C={\nabla }_x{\nabla }_{r}V\left(\widehat{r}\right(x);x)$$

is the scaffold–cell coupling matrix. $H$ describes local landscape curvature in cellular state space. $C$ describes how scaffold variation changes the cellular force or bias field.

If $H$ is locally invertible, the cellular response mapping is

$$J={\displaystyle \frac{\partial \widehat{r}}{\partial x}}=-H^{-1}C.$$

This relation is local. It holds only around a differentiable stationary state, under the assumption that the local minimum is non-degenerate and that $H$ is invertible. It should not be generalized to the entire scaffold space without local estimation.

The relation nevertheless clarifies the framework. Cellular responsiveness to scaffold architecture depends on two distinct objects. The first is coupling, $C$: whether scaffold variation actually reaches the cellular organizational machinery. The second is curvature, $H$: whether the induced cellular state lies in a steep or shallow region of the organizational landscape. The same scaffold perturbation can generate a large response in a shallow direction but a small response in a stiff direction.

Because multiplication by an invertible matrix does not change rank,

$$\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\left(J\right)=\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\left(C\right)$$

under the local assumptions above. The directions resolved by the cell are therefore the directions along which scaffold variation couples into the cellular organizational landscape. This is the formal counterpart of the admissibility criteria in the main text: a parameter counts as informational only if it is cell-accessible, persistent at the scale of the cellular decision, and non-redundant with already counted parameters.

Appendix B.5. Resolvability and Stabilization

Effective informational dimensionality, $D_{\mathrm{e}\mathrm{f}\mathrm{f}}$, counts the number of scaffold-derived directions that produce separable early cellular responses above the experimental noise floor. It is an estimate of resolvability. It does not, by itself, guarantee that the induced cellular state will be maintained over time.

Stabilization depends on curvature. Along a cellular response direction $q_i$, the relevant local restoring curvature can be written schematically as

$${\kappa }_i=q_i^{T}H{q}_i,$$

where $q_i$ is a normalized cellular response direction associated with a resolvable singular direction of the scaffold–cell response mapping. A large ${\kappa }_i$ indicates a steep local basin and strong resistance to perturbation. A small ${\kappa }_i$ indicates a shallow basin and weak resistance to noise.

We therefore define $D_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}$ as the number of directions within the resolvable response subspace along which restoring curvature exceeds the relevant biological-noise amplitude. In words, $D_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}$ counts only those directions that the scaffold both reaches and holds.

This gives

$$D_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}\le D_{\mathrm{e}\mathrm{f}\mathrm{f}}\le n.$$

The second inequality follows because the cell cannot resolve more independent scaffold directions than the experimenter has varied, and often resolves fewer because $\mathsf{\Phi }$ is lossy. The first inequality follows because not every resolvable direction is necessarily stabilized. Some scaffold-derived directions may be detected by the cell but remain too shallow to resist biological noise, mechanical perturbation, or temporal drift.

This distinction is central to the framework. A scaffold may be highly readable but poorly stabilizing. Conversely, a cellular system may have intrinsically stable organizational directions that no scaffold parameter reaches. Such directions should not be counted as scaffold-stabilized dimensions. $D_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}$ is therefore defined within the resolvable subspace, not across the entire cellular state space.

Appendix B.6. Consequence of Reducing Mechanotransductive Coupling

The formalism also clarifies why contractility or adhesion blockade provides a strong falsification test. If mechanotransductive coupling is reduced, then the scaffold-dependent contribution to the organizational potential weakens. In the limiting case,

$$\lambda \to 0,$$

the scaffold–cell coupling matrix tends toward zero:

$$C\to 0.$$

Under the same local assumptions, this implies

$$J=-H^{-1}C\to 0.$$

The singular values of the response mapping should therefore approach the experimental noise floor, and $D_{\mathrm{e}\mathrm{f}\mathrm{f}}$ should decline. Because the scaffold-dependent contribution to the landscape is also weakened, scaffold-induced stabilization should decline as well.

This prediction does not require the scaffold architecture to change. The architecture remains present; what changes is the cell’s ability to convert scaffold structure into mechanotransductive organization. This is why pharmacological or functional suppression of contractility, adhesion, or mechanosensitive signaling is a stringent test of the framework. If high $D_{\mathrm{e}\mathrm{f}\mathrm{f}}$ and high stabilization persist despite effective disruption of mechanotransductive coupling, the framework would be strongly challenged.

Appendix B.7. Limits of the Kernel Analogy

The kernel analogy is useful only within strict limits. It helps express two ideas. First, scaffold architectures that are structurally different may become equivalent after projection through cellular mechanosensing. Second, a transformation of the cellular microenvironment can make previously entangled organizational trajectories more separable.

However, the analogy should not be extended beyond this role.

Cells are not classifiers. There is no training set, no label structure, no objective function, no separating hyperplane, and no margin in the machine-learning sense. Cellular outcomes are dynamical trajectories within a biological state space, not discrete decisions made by an algorithm.

$\mathsf{\Phi }$ is not optimized. In machine learning, a feature map may be chosen or engineered to improve classification. In the biological setting considered here, $\mathsf{\Phi }$ is imposed by the cell’s mechanosensory apparatus. The scaffold designer manipulates $x$, but does not directly choose $\mathsf{\Phi }$.

$K_{\mathrm{b}\mathrm{i}\mathrm{o}}$ is not a proven Mercer kernel. It is an empirical similarity measure between scaffold architectures after cellular projection. The framework does not assume that it is positive semidefinite or that it satisfies the formal requirements of kernel methods.

The computational kernel trick has no biological counterpart here. In machine learning, the kernel trick allows inner products in a transformed space to be computed without explicitly evaluating the transformation. Cells do not use such a shortcut. They physically instantiate mechanosensory transformation by forming adhesions, generating cytoskeletal tension, deforming nuclei, and activating downstream signaling.

Therefore, what survives of the analogy is not the algorithmic structure of support vector machines or kernel learning. What survives is the geometry of separability, redundancy, and projection. The analogy is useful because it makes explicit that geometric difference is not the same as biological distinguishability. It should not be read as claiming that cells perform machine learning.

Appendix B.8. Summary

The formal structure supports three claims used in the main text.

First, effective informational dimensionality is a property of the scaffold–cell mapping, not of the scaffold alone. It depends on which scaffold parameters are varied, which cell type reads them, which early readouts are measured, and which noise floor is applied.

Second, resolvability and stabilization are distinct. $D_{\mathrm{e}\mathrm{f}\mathrm{f}}$ counts directions the cell can distinguish early; $D_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}$ counts the subset of those directions that are later held against noise or perturbation.

Third, the kernel analogy is heuristic and bounded. It clarifies why geometrically different architectures may be biologically equivalent and why additional scaffold structure matters only when it becomes cell-readable. It does not imply that cells classify, optimize, compute kernels, or implement a machine-learning algorithm.