Formal Structure of NDR
This page presents the mathematical skeleton of NDR with both formal notation and plain-language explanations. The mathematics is intentionally minimal—designed to be clear, reviewable, and extensible.
The Core NDR Equation
Your access state s is mapped by routing function R to domain d. The projection π then determines what portion of underlying reality Ω becomes observable.
Formal Definitions
Access States
SLet S be the set of access states. An access state s ∈ S represents the combined biological, technological, environmental, and instrumental conditions that constrain what an observer can detect, survive, and decode.
S is the list of 'ways you can be set up' (no gear, scuba gear, spacesuit, microscope, etc.).
Domains
DLet D be the set of domains. Each domain d ∈ D is a coherent regime of description in which processes, measurements, and regularities are well-defined.
D is the list of 'worlds you can experience' (surface world, underwater world, space environment, microscopic world, etc.).
Underlying Reality
ΩLet Ω (Omega) denote the underlying reality. NDR remains agnostic about the ultimate nature of Ω; it is introduced as a placeholder for 'whatever is really there' prior to selection and projection.
Ω is 'everything that exists underneath,' whether we can see it or not.
Nesting Relation
dᵢ < dⱼWe write dᵢ < dⱼ to mean 'domain dᵢ is nested within domain dⱼ.' This is a partial order expressing containment of description or operation, not necessarily literal spatial containment.
Smaller worlds fit inside bigger worlds, like Russian dolls. The smaller world does not disappear; it just lives inside a bigger context.
Routing Function
R : S → DDefine the routing function R : S → D. For any access state s ∈ S, the value R(s) is the domain that becomes observable, survivable, and operational under that access state.
Routing is like selecting a channel on a TV. All channels exist simultaneously, but your current setup determines which one you see.
Projection Mapping
π : Ω → Ω(d)For each domain d ∈ D, define a projection mapping π : Ω → Ω(d), where Ω is the underlying state space of reality, and Ω(d) is the set of observable states available within domain d.
Even inside a chosen world, you never see everything. You see a 'picture' of reality shaped by that world.
Routing Is Selection, Not Transport
In Nested Domain Reality, routing does not move an observer to a different universe. Instead, it selects which domain of a single, continuous reality becomes the experienced world.
Let Ω denote the underlying state space of reality, assumed to remain fixed. Changing the access state s changes the value of R(s) and therefore changes the experienced domain, even though Ω itself does not change.
You don't leave reality to go somewhere else. You change how you are set up. That changes which part of reality becomes your world.
Information Loss and Coarse-Graining
Projection mappings need not be one-to-one. Multiple underlying states in Ω may map to the same observable state in Ω(d). This reflects information loss or coarse-graining introduced by the constraints of a domain.
This loss is not an error—it expresses the practical limits of measurement, survival, and interpretation within a given domain. Projection mappings produce simplified, workable descriptions appropriate to the conditions of access.
Different situations in reality can look the same to you because your "window" into reality is limited. This is normal in science: models and measurements simplify the world so we can understand and work with it.