Chapter 4 — Mathematical Preliminaries
RUNNING EXAMPLE — Maya’s Pages
Maya has a draft novel and no idea whether it works. Her agent has asked whether the book “hangs together”, whether the voice is consistent, whether the second act has “shape”. Maya sits with a stack of paragraphs and tries to articulate what these metaphors could possibly mean in non-metaphorical terms. She notices something. When she reads paragraph 14, it feels close to paragraph 9 — they share a texture — and far from paragraph 47, which belongs to a different kind of scene. Close. Far. These are geometric words. She suspects that somewhere, implicit in the act of reading, she is computing distances between points in a space whose coordinates she cannot write down. She does not yet know that a multilingual sentence encoder will later give her an explicit 768-dimensional coordinate for each paragraph, a mean and covariance for the whole book, and a set of divergences measuring how her book differs from the corpus prior. She has intuited the geometry before learning its name. This chapter gives the geometry its name.
This chapter develops the mathematical vocabulary used throughout the book. No prior knowledge of differential geometry or statistical manifold theory is assumed; we require multivariate calculus and linear algebra. Readers with background in either field may skim Sections 4.1–4.3 and focus on Sections 4.4–4.7, which introduce tools less commonly encountered in an aesthetic-theoretic context: divergences between Gaussians, the geometry of symmetric positive-definite matrices, and Lasso regularization for axis discovery.
The chapter shares foundation material with Chapter 4 of Geometric Ethics; readers who have that book in hand may skim Sections 4.1 and 4.2 and concentrate on the material from 4.3 onward, which is specific to the aesthetic setting. Readers who do not need to be talked to as though this is their first contact with geometry should skim anyway — the notation established here is used in every subsequent chapter.
4.1 Euclidean Space and the Inner-Product Structure
The Objects of Representation
A modern encoder — LaBSE for text, MERT for music, CLIP or DINO for images — maps a token, a clip, or a patch to a point in a real vector space $\mathbb{R}^d$. For LaBSE, $d = 768$; for MERT-v1-330M, the layer-7 hidden state is $d = 1024$; for CLIP ViT-L/14, $d = 768$. These are high-dimensional but finite-dimensional real spaces. The space is flat in the sense that familiar Euclidean constructions work point-by-point: we may add vectors, scale them, and form inner products. The geometry of $\mathbb{R}^d$ is the geometry most readers learned as undergraduates, and most of what the book does is done inside such a space. The complication is that the distribution over such a space, induced by a work or a corpus, is not a point — it is a cloud, and clouds have shape.
Inner Products, Norms, Angles
For $x, y \in \mathbb{R}^d$, the standard inner product and Euclidean norm are
$$\langle x, y \rangle = \sum_{i=1}^{d} x_i y_i, \qquad \|x\| = \sqrt{\langle x, x\rangle}.$$
The cosine similarity
$$\cos(x, y) = \frac{\langle x, y\rangle}{\|x\|\,\|y\|}$$
is invariant to rescaling and is the workhorse of sentence-embedding literature. For normalized embeddings (LaBSE outputs are L2-normalized by convention), cosine similarity and dot product coincide; for non-normalized embeddings, a choice must be made and defended. Throughout this book we work with L2-normalized embeddings unless otherwise stated; consequences of this choice (loss of absolute-magnitude information, invariance to embedding-norm artefacts) are discussed in Chapter 5.
Why Flat Space Is Not Enough
If aesthetic objects were single points, Euclidean structure would suffice. But a work $W$ generates a cloud of points $X(W) = \{\phi(t_1), \ldots, \phi(t_N)\}$ — one point per token, per second of audio, per patch of image. The relevant objects are distributions over $\mathbb{R}^d$, not points in it. The shape of such a distribution — mean, spread, orientation, tails — is the subject of Chapter 5 and the source of the four structural channels of Chapter 6. Euclidean space is the ambient; the aesthetic manifold is the space of distributions that live inside it. For that, we need more machinery.
4.2 Riemannian Geometry, Briefly
Manifolds
A smooth manifold $M$ of dimension $n$ is a topological space that locally looks like $\mathbb{R}^n$: every point $p \in M$ has an open neighborhood $U$ with a homeomorphism $\varphi: U \to V \subset \mathbb{R}^n$ (a chart), and overlapping charts are related by smooth transition functions. The surface of the sphere is the textbook example; the space of probability distributions over a finite alphabet (the simplex) is the example most relevant here.
A manifold is specified by an atlas: a collection of charts $\{(U_\alpha, \varphi_\alpha)\}$ covering $M$ such that on each overlap $U_\alpha \cap U_\beta$ the transition function $\varphi_\beta \circ \varphi_\alpha^{-1}$ is a smooth map between open subsets of $\mathbb{R}^n$. Smoothness of transitions is what allows us to do calculus on $M$ — to take derivatives of functions and vector fields in a way that does not depend on which chart we use.
At each point $p$, the tangent space $T_p M$ is an $n$-dimensional real vector space; intuitively, it is the space of infinitesimal directions one can move from $p$. A tangent vector is a velocity, an infinitesimal displacement, or (equivalently) a derivation on smooth functions near $p$. The disjoint union $TM = \bigsqcup_{p \in M} T_p M$ is the tangent bundle, itself a smooth manifold of dimension $2n$. A vector field is a smooth assignment of a tangent vector to each point — a smooth section of $TM$.
The cotangent space $T_p^* M$ is the dual: the space of linear maps $T_p M \to \mathbb{R}$. Cotangent vectors (covectors, 1-forms) at $p$ are the natural objects that weight tangent vectors to produce numbers. The gradient $df$ of a smooth function $f : M \to \mathbb{R}$ is the canonical covector field: $(df)_p(v) = v(f)$, the derivative of $f$ in the direction $v$. The vector/covector distinction is usually invisible in Euclidean space (the metric identifies them), but becomes important on curved or non-Euclidean manifolds — including the SPD manifold of Section 4.5.
The Metric Tensor
A Riemannian metric is a smooth assignment, to each point $p$, of a positive-definite inner product $g_p(\cdot, \cdot)$ on $T_p M$. In local coordinates $(x^1, \ldots, x^n)$, the metric has components $g_{ij}(p)$ satisfying
$$g_p(u, v) = \sum_{i,j} g_{ij}(p)\, u^i v^j, \qquad g_{ij} = g_{ji}, \qquad g_{ij}(p)\, v^i v^j > 0 \text{ for } v \neq 0.$$
The length of a curve $\gamma:[0,1] \to M$ is
$$L(\gamma) = \int_0^1 \sqrt{g_{\gamma(t)}(\dot\gamma(t), \dot\gamma(t))}\, dt,$$
and the distance $d(p, q)$ between two points is the infimum of lengths of curves from $p$ to $q$. Geodesics are length-minimizing curves; they are the “straight lines” of a curved space.
Geodesics, Connections, and Curvature
On a flat Euclidean space, comparing vectors at different points is trivial — translate and compare. On a curved manifold, this fails: there is no canonical way to identify $T_p M$ with $T_q M$ for distinct $p, q$. The resolution is a connection, a rule $\nabla$ for differentiating vector fields along curves. The Levi-Civita connection is the unique metric-compatible, torsion-free connection on a Riemannian manifold. Its Christoffel symbols are
$$\Gamma^k_{ij} = \tfrac{1}{2}\, g^{k\ell}\!\left(\frac{\partial g_{i\ell}}{\partial x^j} + \frac{\partial g_{j\ell}}{\partial x^i} - \frac{\partial g_{ij}}{\partial x^\ell}\right),$$
and the geodesic equation is $\ddot\gamma^k + \Gamma^k_{ij} \dot\gamma^i \dot\gamma^j = 0$. The Riemann curvature tensor $R^{\ell}_{kij} = \partial_i \Gamma^\ell_{kj} - \partial_j \Gamma^\ell_{ki} + \Gamma^\ell_{im}\Gamma^m_{kj} - \Gamma^\ell_{jm}\Gamma^m_{ki}$ measures the path-dependence of parallel transport. A manifold is flat iff its Riemann tensor vanishes identically. On the SPD manifold (Section 4.5) the Riemann tensor does not vanish: covariance space is genuinely curved, and two different “straight-line” paths between covariances give different interpolants.
What This Buys Us
For the aesthetic manifold, the metric will not be the Euclidean one. Two books with the same cosine distance between mean embeddings may be aesthetically very different if their covariance structures differ — the distance that matters is not distance between means, it is a distance in the space of distributions. The divergences of Section 4.4 define such distances. The SPD (symmetric positive-definite) manifold of Section 4.5 gives them a Riemannian home. The curvature of that home matters empirically: the Hellinger-saturation pathology noted in Chapter 17 is a concrete manifestation of ill-conditioning near the boundary of $\mathcal{S}^d_{++}$, where the affine-invariant metric blows up.
4.3 Principal Component Analysis and Spectral Decomposition
The Empirical Covariance
Given samples $\{x_1, \ldots, x_N\} \subset \mathbb{R}^d$, the empirical mean and covariance are
$$\hat\mu = \frac{1}{N}\sum_{n=1}^{N} x_n, \qquad \hat\Sigma = \frac{1}{N-1}\sum_{n=1}^{N}(x_n - \hat\mu)(x_n - \hat\mu)^{\top}.$$
$\hat\Sigma \in \mathbb{R}^{d\times d}$ is symmetric and positive-semidefinite. It is the core second-order object in the book: nearly every geometric quantity we compute about a work’s embedding cloud is a functional of $\hat\Sigma$ (together with $\hat\mu$).
Spectral Decomposition
By the spectral theorem, any real symmetric matrix $\Sigma$ admits an orthogonal eigendecomposition
$$\Sigma = U \Lambda U^{\top}, \qquad U^{\top} U = I_d, \qquad \Lambda = \operatorname{diag}(\lambda_1, \ldots, \lambda_d),$$
with eigenvalues $\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_d \geq 0$ and eigenvectors $u_1, \ldots, u_d$ forming the columns of $U$. The $k$-th principal component is $u_k$; its explained variance is $\lambda_k$; the fraction of total variance is $\lambda_k / \sum_i \lambda_i$.
PCA as Dimensionality Reduction
Projecting onto the top $K$ components yields a low-rank approximation that preserves as much variance as possible among all rank-$K$ approximations (Eckart–Young). The projection matrix is $P_K = U_K U_K^{\top}$ where $U_K$ stacks the first $K$ eigenvectors. For the empirical work in this book, we use $K = 128$ on the English-book corpus — a basis fit once, frozen, and used across all languages as the common coordinate system (Chapter 12, on Noether symmetries, gives the cross-lingual evidence that this basis transfers). The choice $K = 128$ is empirical: it captures roughly 85% of embedding variance on the Gutenberg-LaBSE corpus while keeping the Gaussian fits of Section 4.4 well-conditioned.
The Spectrum Itself Is a Feature
Most uses of PCA discard the eigenvalues after projection. For us, the eigenvalue spectrum $(\lambda_1, \ldots, \lambda_K)$ is itself an aesthetic signature: a work whose embedding cloud has a long, flat spectrum is structurally different from one with a sharply decaying spectrum, even if they share means. Chapter 6 will put Lasso on top of this spectrum and discover 71 non-zero axes that correspond to interpretable genre/form distinctions — a discovered tensor level unavailable to scalar representations.
4.4 Multivariate Gaussian Fits and Divergences
The Gaussian Approximation
Given a cloud $X(W) = \{x_1, \ldots, x_N\}$, the simplest sufficient-statistic description is the fitted multivariate Gaussian
$$p_W(x) = \mathcal{N}(x;\, \hat\mu_W, \hat\Sigma_W) = \frac{1}{(2\pi)^{d/2} |\hat\Sigma_W|^{1/2}} \exp\!\left(-\tfrac{1}{2}(x - \hat\mu_W)^{\top} \hat\Sigma_W^{-1} (x - \hat\mu_W)\right).$$
This is an approximation: the empirical cloud need not be Gaussian. Empirically it often isn’t — embedding clouds exhibit heavy tails, multi-modality, and anisotropy that a single Gaussian cannot capture. But the Gaussian captures the first two moments exactly, and the geometric content of those two moments turns out to carry most of the aesthetic signal documented in Chapter 17. The residual non-Gaussianity is the subject of Level 3 (trajectory geometry) in Chapter 6.
Divergences Between Gaussians — Closed Forms
Let $p_0 = \mathcal{N}(\mu_0, \Sigma_0)$ (the corpus prior) and $p = \mathcal{N}(\mu, \Sigma)$ (the work). The four divergences we use pervasively admit closed forms.
Kullback–Leibler divergence.
$$D_{\mathrm{KL}}(p \,\|\, p_0) = \tfrac{1}{2}\!\left[\operatorname{tr}(\Sigma_0^{-1}\Sigma) + (\mu_0 - \mu)^{\top}\Sigma_0^{-1}(\mu_0 - \mu) - d + \ln\frac{|\Sigma_0|}{|\Sigma|}\right].$$
KL is asymmetric: $D_{\mathrm{KL}}(p\|p_0) \neq D_{\mathrm{KL}}(p_0\|p)$ in general. It measures the expected log-likelihood ratio under $p$, and is finite only when $\operatorname{supp}(p) \subseteq \operatorname{supp}(p_0)$.
Jensen–Shannon divergence.
$$D_{\mathrm{JS}}(p, p_0) = \tfrac{1}{2} D_{\mathrm{KL}}(p \,\|\, m) + \tfrac{1}{2} D_{\mathrm{KL}}(p_0 \,\|\, m), \qquad m = \tfrac{1}{2}(p + p_0).$$
JS is symmetric, bounded above by $\ln 2$, and its square root is a proper metric. For Gaussians we evaluate it numerically via the KL formula above and a Monte-Carlo or moment-matched $m$.
Hellinger distance.
$$H^2(p, p_0) = 1 - \frac{|\Sigma_0|^{1/4}|\Sigma|^{1/4}}{|\bar\Sigma|^{1/2}} \exp\!\left(-\tfrac{1}{8}(\mu_0 - \mu)^{\top}\bar\Sigma^{-1}(\mu_0 - \mu)\right),\qquad \bar\Sigma = \tfrac{1}{2}(\Sigma_0 + \Sigma).$$
$H$ is symmetric, bounded in $[0, 1]$, and is a proper metric.
Bhattacharyya distance.
$$D_B(p, p_0) = \tfrac{1}{8}(\mu_0 - \mu)^{\top}\bar\Sigma^{-1}(\mu_0 - \mu) + \tfrac{1}{2}\ln\frac{|\bar\Sigma|}{\sqrt{|\Sigma_0|\,|\Sigma|}}.$$
Hellinger and Bhattacharyya are related: $H^2 = 1 - e^{-D_B}$, so the two track each other tightly. In practice we report Bhattacharyya because it remains well-behaved when Hellinger saturates near 1 due to ill-conditioned covariances (a pathology we encountered in the MERT music experiments; see Chapter 21).
Total Variation.
$$\mathrm{TV}(p, p_0) = \tfrac{1}{2} \int |p(x) - p_0(x)|\, dx.$$
No closed form for Gaussians, but $\mathrm{TV} \leq \sqrt{\tfrac{1}{2} D_{\mathrm{KL}}}$ (Pinsker) and $\mathrm{TV} \leq H\sqrt{2 - H^2}$ (Le Cam). We compute TV by numerical integration when needed, though the book relies more heavily on the three smooth divergences above.
Mahalanobis distance of means.
$$D_{\mathrm{Mahal}}(\mu, \mu_0; \Sigma_0) = \sqrt{(\mu - \mu_0)^{\top}\Sigma_0^{-1}(\mu - \mu_0)}.$$
This is the distance-from-prior ignoring the work’s own covariance — just how far the mean has moved, measured in units of prior standard deviations. It is a rank-1 quantity and constitutes Level 1 of the tensor hierarchy (Chapter 6).
A Worked Example
Suppose we are given two one-dimensional Gaussians: $p_0 = \mathcal{N}(0, 1)$, the standard prior, and $p = \mathcal{N}(\mu, \sigma^2)$, a work. The closed forms specialize to
$$D_{\mathrm{KL}}(p\|p_0) = \tfrac{1}{2}(\sigma^2 + \mu^2 - 1 - \ln \sigma^2), \qquad H^2(p, p_0) = 1 - \sqrt{\tfrac{2\sigma}{1+\sigma^2}} \exp\!\left(-\tfrac{\mu^2}{4(1+\sigma^2)}\right).$$
For $\mu = 1, \sigma = 1$ (same variance, shifted mean): $D_{\mathrm{KL}} = 0.5$, $H^2 \approx 0.118$. For $\mu = 0, \sigma = 2$ (same mean, doubled standard deviation): $D_{\mathrm{KL}} = \tfrac{1}{2}(4 - 1 - \ln 4) \approx 0.807$, $H^2 \approx 0.056$. The two perturbations are “comparably large” under KL but Hellinger regards the mean shift as the larger change. This is a feature, not a bug: different divergences emphasize different aspects of the difference, and reporting only one hides the asymmetry. We report all of them and let the pattern of agreement and disagreement be part of the signal.
Why All Six
One might ask why we use six different divergences when they all measure “how different the distributions are”. The answer is empirical: they are correlated but not redundant. In the cross-lingual invariance analysis (Chapter 12, §17.7 in the Aesthetics Ch. 17 which is the already-drafted empirical chapter), the mean Spearman correlation across language pairs for Hellinger (ρ = +0.675), Bhattacharyya (ρ = +0.675), and JS (ρ = +0.674) are nearly identical — but the Mahalanobis mean-distance (ρ = +0.710) and the internal-coherence feature pair_sim_mean (ρ = +0.712) are higher, indicating that each divergence family captures a slightly different slice of the aesthetic signal. Using all of them is the empirical counterpart of refusing to collapse information prematurely.
4.5 The SPD Manifold
Covariances Are Not Vectors
The space of $d \times d$ symmetric positive-definite matrices, denoted $\mathcal{S}^d_{++}$, is not a vector space: it is a convex open cone inside the space of symmetric matrices. Adding two covariances gives a covariance; subtracting them does not. Any honest treatment of “distance between covariances” needs the right geometry.
The Affine-Invariant Metric
$\mathcal{S}^d_{++}$ is a Riemannian manifold of dimension $d(d+1)/2$ under the affine-invariant metric
$$g_\Sigma(A, B) = \operatorname{tr}(\Sigma^{-1} A \Sigma^{-1} B), \qquad A, B \in T_\Sigma \mathcal{S}^d_{++}.$$
The induced geodesic distance between $\Sigma_0$ and $\Sigma$ is
$$d_{\mathrm{AI}}(\Sigma_0, \Sigma) = \sqrt{\sum_{i=1}^{d} \log^2 \lambda_i(\Sigma_0^{-1}\Sigma)}.$$
This metric is invariant under congruence transformations $\Sigma \mapsto G \Sigma G^{\top}$ for $G \in GL(d)$, which in our setting means the distance is invariant to linear change of basis in the embedding space — an appropriate invariance for a geometric theory that should not depend on whether we work in the original coordinates, the PCA coordinates, or any other linear reparameterization.
The Log-Euclidean Metric
A computationally cheaper alternative, widely used in diffusion-tensor imaging, is the log-Euclidean metric:
$$d_{\mathrm{LE}}(\Sigma_0, \Sigma) = \|\log \Sigma_0 - \log \Sigma\|_F,$$
where $\log$ is the matrix logarithm and $\|\cdot\|_F$ is the Frobenius norm. It is affine-invariant only under the commuting subgroup but is fast to compute, and it tracks the affine-invariant distance closely in practice. For the book’s empirical work we use the divergences of Section 4.4 rather than raw SPD distances, but the SPD viewpoint is what justifies taking those divergences seriously as measurements on a space of distributions rather than arbitrary numerical summaries.
4.6 Lasso Regularization
The Setup
Given design matrix $X \in \mathbb{R}^{N \times p}$ and target $y \in \mathbb{R}^N$, Lasso (Tibshirani, 1996) solves
$$\hat\beta_\lambda = \arg\min_{\beta \in \mathbb{R}^p}\; \tfrac{1}{2N}\|y - X\beta\|_2^2 + \lambda \|\beta\|_1, \qquad \|\beta\|_1 = \sum_{j=1}^{p}|\beta_j|.$$
The $\ell_1$ penalty induces sparsity: for $\lambda$ above a threshold, many components of $\hat\beta_\lambda$ are exactly zero. The nonzero coordinates constitute the selected features.
Why Lasso Is Load-Bearing Here
The core discovery of Chapter 6 — the fourth channel of aesthetic structure — is that when we apply Lasso to the 128-dimensional PCA spectrum of a work’s embedding cloud against rating, 71 components have nonzero coefficients and the selected axes correspond to interpretable genre/form distinctions. This is axis discovery, not regression in the usual sense: we care less about the predictive coefficient than about which PCA axes light up. Lasso is the tool that does this because of its sparsity property; Ridge regression (which distributes weight across all features) would obscure the discovery.
The Bias-Variance Tradeoff and $\lambda$ Selection
Lasso’s $\lambda$ governs the bias-variance tradeoff. We select $\lambda$ by 5-fold cross-validation, with the split author-disjoint (books in Phase 1, artists in Phase 4): a fold’s training set shares no authors with its validation set, so the model cannot memorize an author-level signal. This is not a statistical nicety — without author-disjoint splitting, apparent $R$ values are inflated by an order of magnitude.
Combined Ridge + Lasso
In practice we often combine Lasso-selected features with a Ridge regression (an $\ell_2$ penalty):
$$\hat\beta = \arg\min_\beta\; \tfrac{1}{2N}\|y - X\beta\|_2^2 + \lambda_1 \|\beta\|_1 + \lambda_2 \|\beta\|_2^2.$$
The combined objective is Elastic Net when $\lambda_2 > 0$, and we use it when the number of nonzero Lasso features would otherwise be unstable across folds. The Phase-1 headline result (books, combined Ridge+Lasso, $R = 0.241$, 17σ, $R^2 = 0.058$ before genre correction) uses this combination.
4.7 Notation and Conventions
We close with notation used throughout the book.
Works and clouds. A work is denoted $W$. Its embedding cloud is
$$X(W) = \{\phi(t_1), \ldots, \phi(t_{N_W})\} \subset \mathbb{R}^d,$$
where $\phi$ is a modality-specific encoder (LaBSE, MERT, CLIP, DINO) and $t_i$ are tokens, frames, or patches.
Corpus prior. The corpus prior is the Gaussian $p_0 = \mathcal{N}(\mu_0, \Sigma_0)$ fitted to the pooled embeddings of a reference corpus (for Phase 1, the 4,998 Gutenberg-matched books).
Per-work Gaussian. The per-work Gaussian is $p_W = \mathcal{N}(\mu_W, \Sigma_W)$, fitted on $X(W)$.
PCA basis. $U_K \in \mathbb{R}^{d \times K}$ is the PCA basis fitted on the prior corpus. All cross-lingual analyses project into this single frozen basis, not into per-language bases — a methodological choice whose empirical consequences appear in Chapter 12.
Indices. Greek indices $\mu, \nu$ range over aesthetic tensor dimensions (defined per modality in Chapter 5). Latin indices $i, j$ range over coordinate components of $\mathbb{R}^d$. Indices $\alpha, \beta$ range over corpus items. Indices $k$ range over PCA components, $1 \leq k \leq K$.
Significance reporting. Every empirical claim in the book reports $(n,\, \text{effect size},\, p\text{-value})$ together. Claims that omit any of these three are conjectural and labeled as such.
4.8 Bibliographic Notes
For differential geometry at the level of this chapter, Lee’s Introduction to Smooth Manifolds and do Carmo’s Riemannian Geometry are standard. For information geometry — the Fisher metric, Bregman divergences, the Gaussian submanifold — Amari and Nagaoka’s Methods of Information Geometry is the canonical reference, and Nielsen’s more recent surveys collect closed forms for the divergences between parametric families. For the SPD manifold specifically, Pennec, Fillard, and Ayache’s work on the affine-invariant metric (originally for diffusion-tensor MRI) and Arsigny et al.’s log-Euclidean framework are the references we rely on implicitly throughout. For Lasso, Hastie, Tibshirani, and Wainwright’s Statistical Learning with Sparsity collects the modern theory. None of these are required reading for the rest of this book; the machinery developed in this chapter is self-contained at the level we use it.
4.9 Bridge to the Rest of the Book
This chapter has given the reader the toolkit. Chapter 5 uses it to construct the aesthetic manifold — the space on which a work $W$ lives as a cloud, and on which the corpus lives as a prior. Chapter 6 stacks tensors on this manifold: Level 1 (mean direction and Mahalanobis distance), Level 2 (covariance divergences and internal coherence), Level 3 (trajectory geometry), Level 4 (Lasso-discovered axes on the 128-d spectrum). Each level uses exactly the machinery of this chapter — PCA, Gaussian fitting, divergences, Lasso — and each level extracts information that the preceding levels cannot represent. The empirical payoff arrives in Chapter 17: 8σ signal from divergences, 8.4σ from internal coherence, 3–6σ from trajectory features, and 71 non-zero axes from Lasso — four channels each contributing signal the others cannot. None of this is visible to a scalar summary; all of it is visible to the tensor hierarchy that this chapter’s mathematics supports.