Knowledge as a Hidden Dimension: A Geometric Framework for Modeling Awareness Through Metric Deformation

Knowledge as a Hidden Dimension: A Geometric Framework for Modeling Awareness Through Metric Deformation Bora Arcak 5 November 2025 Abstract Human perception is rarely uniform. Familiar objects appear crisp and meaningful, while unfamiliar ones blur into undifferentiated shapes. This suggests that knowledge does more than add information—it reshapes the space in which information is represented. In this paper, I introduce a minimal geometric framework formalizing this idea. Knowledge is modeled as a scalar coordinate Kthat acts as a hidden dimension: not an additional axis of sensory data, but a parameter that deforms the intrinsic metric along task-relevant directions. I motivate this concept using a historical analogy: Einstein’s extension of classical threedimensional space by adding time as a fourth dimension did not create new spatial directions, but reinterpreted motion through a geometric coordination rule. Similarly, I propose that the knowledge variable Kgoverns the geometry of the existing manifold. By modeling this as an anisotropic deformation of a Riemannian metric, I derive mathematical guarantees showing that increasing Kmonotonically improves discrimination thresholds and reduces Bayes-optimal error. This framework connects the philosophy of awareness with the rigor of information geometry. Preprint. ©2025 Bora Arcak. Released on Zenodo for academic discussion. The author retains all intellectual property rights. . 1 Introduction When we recognize the silhouette of a dog, spot a friend from afar, or read text in a language we know well, perception feels sharp and structured. In contrast, unfamiliar objects appear “flat”—difficult to parse, lacking clear internal distinctions. This everyday asymmetry suggests that perception is not purely a matter of sensory input. Rather, it is modulated by what we know. A useful way to think about knowledge is to view it not as a symbol or fact stored in memory, but as something that changes the geometry of perception. Familiar features become easier to discriminate because the cognitive system effectively “stretches” the perceptual space along meaningful directions. Unfamiliar dimensions contract or blur. This geometric intuition echoes a major moment in the history of physics. Classical mechanics operated in three spatial dimensions until Einstein introduced spacetime: four dimensions, where time acted not as another spatial direction but as a hidden coordinate that governed transformations. The fourth dimension did not add new material content; it restructured the geometry of what already existed. 1 The central claim of this paper is analogous: knowledge acts as a hidden coordinate, not because it enlarges the representational manifold, but because it determines how distances—hence discriminability, awareness, and meaning—are computed. 2 Knowledge as a Hidden Dimension The phrase “dimension” can be misleading. I do not propose that the representational manifold gains another physical axis when someone learns a new concept. Instead, I use “dimension” in the same sense that physics uses time: as a governing coordinate that modulates geometric structure. 2.1 Dimensions as coordination rules A dimension is not always a direction. In relativity, time is coupled with space by a metric that determines how intervals contract or stretch depending on velocity and gravitational potential. What makes time a dimension is not spatial extension but its role in defining how the geometry transforms. Knowledge plays an analogous role. It does not supply new raw inputs; it determines how strongly the system distinguishes points along certain directions in the existing representation. Thus, Kis a hidden dimension because the mapping (u, K)7−→ gK(u) defines the geometry of the perceptual manifold. This interpretation is conceptually clean, philosophically defensible, and mathematically precise. 3 A Geometric Framework for Knowledge To formalize this intuition, let S⊂Rdbe a smooth stimulus manifold, and let TsSdenote its tangent space at s∈S. An observer perceives Sthrough an internal representation whose precision is modulated by the knowledge parameter K∈R≥0. 3.1 Task-Relevant Directions Perception is rarely isotropic. Let Us⊆TsSdenote the subspace of task-relevant directions at stimulus s. We model this subspace via the orthogonal projector Ps:TsS→Us. The complementary projector is P⊥ s=I−Ps, representing irrelevant directions. 3.2 The Knowledge-Dependent Metric Let g0denote a baseline Riemannian metric on S(often Euclidean). We define the anisotropic K-dependent metric gKby its action on any two tangent vectors u, v ∈TsS: gK(u, v) = g0(u, v) + β(K)g0(Psu, v),(1) where β:R≥0→R≥0is a smooth, monotonically increasing function with β(0) = 0. Since Psis an orthogonal projector with respect to g0, this ensures that distances along taskrelevant directions are stretched by a factor of p1+β(K), while orthogonal directions remain invariant. 2 3.3 Schematic Illustration The effect of Kon the local geometry is illustrated below. x z K ss′ small separation stretched at high K Figure 1: With increasing knowledge K, the metric gKstretches task-relevant directions, increasing the effective distance between sand s′. 4 Theoretical Guarantees We now prove that this geometric deformation leads to concrete improvements in discrimination. Lemma 1 (Increasing Discriminability).For any fixed s∈Sand any nonzero v∈Us, the gK-norm ∥v∥gKis strictly increasing on any interval where β′(K)>0. Proof. Since v∈Us, we have Psv=v. Using the definition in (1): ∥v∥2 gK=gK(v, v)=g0(v, v) + β(K)g0(v, v) = (1 + β(K))∥v∥2 g0. Differentiating with respect to Kyields ∂ ∂K ∥v∥2 gK=β′(K)∥v∥2 g0>0. 4.1 Reduction of Bayes Error Consider a discrimination task under Gaussian noise. We assume the internal representation adheres to the Cram´er-Rao bound, such that the noise covariance Σ(K) is proportional to the inverse of the Fisher Information metric: Σ(K)∝g−1 K=⇒Σ(K)−1∝I+β(K)Ps. Theorem 1 (Knowledge Reduces Bayes Error).Let δs ∈Us. Under the assumption above, the Bayes-optimal discrimination error Pe(K)is strictly decreasing in K. Proof. The discrimination performance is governed by the Mahalanobis distance d2 Mah =δs⊤Σ(K)−1δs. Substituting the metric dependence: d2 Mah ∝δs⊤(I+β(K)Ps)δs =∥δs∥2 g0+β(K)∥δs∥2 g0. Since β(K) is increasing, the signal-to-noise ratio increases. Because the error probability Pe(K) = Φ(−dMah/2) is a monotonically decreasing function of distance, the error strictly decreases. 3 5 Operationalizing the Knowledge Coordinate The scalar Kis not merely a theoretical construct; it can be estimated via: 1. Mutual Information: K≈I(S;X), where higher knowledge implies the representation retains more stimulus-relevant information. 2. Fisher Information: Metrics of the form gKresemble Fisher information matrices J(θ). We can define K= tr J(θ) or K= log det J(θ). 3. Intrinsic Dimensionality: As learning proceeds, neural representations often collapse irrelevant variation [2]. This reduction in effective dimension corresponds to an increase in metric anisotropy β(K). 6 Discussion We return to the central question: Is knowledge really a dimension? If we define a dimension solely as a spatial axis, the answer is no. But if we define a dimension as a coordinate that governs geometric structure, the answer is yes. Knowledge shares the fundamental property of time in relativity: it does not exist as an object within the space, but rather as a parameter that dictates the curvature and distance relations of the space itself. By modeling knowledge as an information dimension K, we gain a unified language to describe how learning transforms the ”flat” noise of a novice into the structured, high-resolution manifold of an expert. References [1] S. Amari. Information Geometry and Its Applications. Springer, 2016. [2] Ansuini, A., Laio, A., Macke, J. H., & Zoccolan, D. (2019). Intrinsic dimension of data representations in deep neural networks. NeurIPS. [3] T. Cover and J. Thomas. Elements of Information Theory. Wiley, 2006. [4] Crutchfield, J. P. (1990). Information and its metric. In L. Lam & H. C. Morris (Eds.), Nonlinear Structures in Physical Systems. Springer. [5] Mohammad-Djafari, A. (2015). Information Geometry and Bayesian Inference. Entropy, 17(7), 3989–4027. [6] Oizumi, M., Tsuchiya, N., & Amari, S. (2016). Unified framework for information integration based on information geometry. Proceedings of the National Academy of Sciences, 113(51), 14817–14822. 4