Informational Geometry and Conservation
When you treat information as the substrate, geometry becomes your primary tool. You stop asking what things are and start asking how they are positioned, oriented, and transformed within an informational manifold. This shift does not remove physics; it reframes it. Physical laws become constraints on informational transformations, and conservation laws become invariants of informational geometry.
Vectors as States
Every entity is represented as a vector in a high-dimensional space. Each dimension encodes a property—mass, energy, charge, momentum, semantic meaning, relational context. A state of a system is a collection of such vectors. Interactions are operations on these vectors: rotations, translations, projections, and reorientations.
When you think this way, the language of physics merges with the language of computation. A force is not a push but a transformation. A field is not a substance but a geometric gradient within the informational manifold. The equations you already know become operators acting on informational states.
Conservation as Invariance
In an informational geometry, conservation laws are invariants. An invariant is a property that remains unchanged under a transformation. When you apply a rotation to a vector, its magnitude remains constant. When you apply a transformation to a system, certain relational properties are preserved. These preserved properties correspond to conservation laws.
- Energy conservation becomes invariance of informational magnitude under a defined set of transformations.
- Momentum conservation becomes invariance of vector sums under interaction operations.
- Charge conservation becomes invariance of a specific dimension or subspace.
This reframing suggests that conservation laws are not imposed from outside; they are intrinsic to the geometry of information space. The universe does not conserve energy because it chooses to; it conserves energy because informational transformations preserve certain invariants.
Symmetry as Geometry
Symmetry is a central concept in physics. In informational geometry, symmetry is the property that certain transformations leave the informational structure unchanged. If you can rotate a system and its informational configuration is the same, that system has rotational symmetry. These symmetries generate conservation laws. This is the informational version of Noether’s theorem: every symmetry corresponds to an invariant.
In practical terms, you can search for symmetries in data to discover invariants. When you detect a stable pattern across transformations, you have found a conserved property. This method generalizes beyond physics: symmetries in social systems, biological networks, or linguistic structures can reveal conserved informational laws within those systems.
Projection and Observation
Observation is projection. You never see the full vector; you see its projection onto a lower-dimensional subspace. The projection depends on your measurement tools, cognitive filters, or sensing apparatus. This explains why two observers can see different properties of the same system: they project it into different subspaces.
In informational geometry, the act of projection is itself a transformation. It reduces dimensionality and imposes constraints. The observed reality is not the full reality; it is a slice defined by your projection. This does not make reality subjective, but it makes your access to it relative.
Mass–Energy as Rotation
One of the most compelling insights of informational geometry is that mass and energy can be modeled as orthogonal dimensions within the same space. The equation E = mc² becomes a rotation or reorientation: you are turning a vector so that its mass component aligns with its energy component, scaled by a conversion factor.
This reinterprets physical transformations as geometric operations. It also suggests that other physical equivalences may be rotations in higher-dimensional informational space. What appears as a transformation between different substances may be a change in vector orientation within the same manifold.
Curvature and Forces
In general relativity, gravity is curvature of spacetime. In informational geometry, forces correspond to curvature of the informational manifold. A vector moving through the manifold follows geodesics—the paths of least informational action. When the manifold is curved, the geodesics bend, and you observe what you call a force.
This generalizes to non-physical systems. A social system might have informational curvature created by norms, incentives, or cultural constraints. An individual’s trajectory through that system follows geodesics shaped by those informational forces. The same geometry applies at every scale.
Informational Manifolds and Topology
The informational manifold is not necessarily flat or uniform. It has topology: holes, folds, clusters, and singularities. These topological features correspond to stable structures, emergent phenomena, or hidden constraints. When you map data into an informational manifold, you are effectively mapping its topology.
Clusters become basins, regions of stability. Residuals reveal hidden dimensions. Voids indicate missing structure. The geometry of the manifold becomes a tool for discovery: by studying its shape, you can infer underlying laws that are not obvious in the raw data.
Informational Metrics
To work with informational geometry, you need metrics: ways to measure distance, angle, and curvature within the space. In a physical system, distance might correspond to spatial separation. In a semantic system, distance corresponds to conceptual divergence. The same geometric tools apply, but the meaning of distance depends on the domain.
This allows you to compare systems by their shape rather than by their content. Two systems with similar metrics may behave similarly even if their surface details differ. This opens the door to cross-domain translation: you can move from one system to another by aligning their informational metrics.
The Role of Complexity
Complexity in informational geometry is not chaos; it is structure at higher resolution. When you zoom in, new patterns emerge. When you zoom out, patterns compress into simpler forms. The manifold is fractal in the sense that similar structures appear at multiple scales.
This means you can model large systems without tracking every detail. You can focus on the stable geometry—symmetries, invariants, attractors—and predict the system’s behavior without exhaustive computation. This is why large-scale systems often appear simpler: their geometry compresses complexity into stable patterns.
Implications for Modeling
Informational geometry provides a unified modeling language. You can represent physical interactions, cognitive processes, and social dynamics within the same space. You can discover invariants that apply across domains. You can use geometric transformations to simulate how systems evolve.
For example:
- In physics, you can treat particles as vectors and forces as transformations.
- In biology, you can treat organisms as vectors and evolutionary pressures as curvature.
- In cognition, you can treat thoughts as vectors and attention as projection.
- In social systems, you can treat individuals as vectors and norms as constraints.
The same geometry applies; only the interpretation of the dimensions changes.
The Informational Conservation Principle
An informational conservation principle states that the total informational magnitude in a closed system remains constant. It can be redistributed, reoriented, or projected, but it is not destroyed. This parallels the physical conservation of energy but extends it to information as the primary substrate.
This principle has philosophical implications. If information is conserved, then every transformation is a reconfiguration, not a loss. What appears as destruction is often reorganization into a form you cannot directly observe. This provides a conceptual bridge to phenomena like black holes or memory loss: information may not disappear; it may shift into a different projection.
Going Deeper
- Symmetry and Invariance Across Domains
- Curvature as Informational Force
- Projection Matrices and Perception