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|0|"not-signal" is not defined and not definable.|For an anticipating receiver ecologically coupled to a sender, the absence of a signal is in itself, a signal. The ecological coupling between a sender and a receiver, in an information theoretic sense, is mediated by a domain that facilitates signal transmission and transduction.|
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|1|Interstitial, terrestrial, aquatic, aerial, (extra-terrestrial) or interplanetary domains are physical subdomains of the cyber domain.|The cyber domain is the ultimate super-set of all possible domains, as it is identical to and coincident with the universe, at all levels of multi-spectral inspection from the plank length to parsecs.|
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|2|A system S is the triplet (N, R, G): N, a set of nodes; R, a set of relationships among nodes, including reflexive self-relationships; G, a set of ecological embeddings that defines the spatio-temporal adjacency of N and R within a hyper-dimensional space. G mediates R.|*Code 0* establishes that ecological coupling between things and presupposes at least one node (N) with at least one mediated relationship (R), i.e. a single node coupled to itself through a reflexive relation. *Code 1* establishes that all such coupled systems are subdomains of the cyber domain. G formalizes this locally: it is the ecological embedding that positions N and R within the cyber domain, encodes their adjacency, and makes "memory" of S possible within the stipulated N, R, and G structure. Specifically, when G is non-trivially structured, S retains persistent state. When G is absent or unstructured, S is transfer-capable but memoryless — theoretically possible, ecologically intangible. Regarding formal constraints and corollaries, see: *[Concept of System](./concept_of_system.md)*.|
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|2|A system S is the triplet (N, R, G): N, a set of nodes; R, a set of relationships among nodes, including reflexive self-relationships; G, a set of ecological embeddings that defines the spatio-temporal adjacency of N and R within a hyper-dimensional space. G mediates R.|*Code 0* establishes that ecological coupling between things happen to exist, and presupposes at least one node (N) with at least one mediated relationship (R), i.e. a single node coupled to itself through a reflexive relation. *Code 1* establishes that all such coupled systems are subdomains of the cyber domain. G formalizes this locally: it is the ecological embedding that positions N and R within the cyber domain, encodes their adjacency, and makes "memory" of S possible within the stipulated N, R, and G structure. Specifically, when G is non-trivially structured, S retains persistent state. When G is absent or unstructured, S is transfer-capable but memoryless — theoretically possible, ecologically intangible. Regarding formal constraints and corollaries, see: *[Concept of System](./concept_of_system.md)*.|
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|3|A structured G — without the loss of generality and by extension, any structured subdomain of the cyber domain — has three minimum properties: (i) potential for information transfer via momentum transfer or energy transduction at feasible rates; (ii) partitionability into subdomains that inherit these same properties; (iii) a finite "rate of flux" (second order measure of flux) within any conceivable subdomain, defining that subdomain's parametric bounds on minimum and maximum information transfer.|*Property (i)* grounds the ecological coupling of *Code 0* physically: transfer requires a medium capable of momentum transfer or energy transduction at rates sufficient to sustain coupling. *Property (ii)* extends *Code 1* recursively: every subdomain of a structured G is itself a structured G satisfying all three properties — the minimum properties are scale-invariant from the Planck length to parsecs. *Property (iii)* makes subdomains distinguishable from one another: each has characteristic flux bounds, intrinsic to its constitution or inherited from its parent domain, that parametrize the types of relationships defined as R can be sustained within the domain under consideration. Together, *Properties i–iii* are mutually self-reinforcing and recursive: any subdomain of a structured G completely satisfies *Code 3*.|
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|4|The flux across surfaces in G defines vectors; the fundamentally independent and mutually exclusive directions of those vectors yield Principal Axes; the count of Principal Axes is the dimensionality of G or its constituent subdomain; the span or magnitude of a quantity along a single Principal Axis is its size. Degrees of freedom in a domain or subdomain coincide with its dimensionality. Consequently, uncertainty in information transfer is a function of the available degrees of freedom.|Flux (*Premise ii*) requires a surface and a direction of movement perpendicular to that surface. As the area of that surface contracts toward a single-dimensional form, the perpendicular direction becomes a vector: a quantity with magnitude (the flux rate, bounded by *Code 3 Property iii*) and direction. Reiterated for emphasis: the set of all independent directions in which flux can occur within G yields the Principal Axes of G. The count of those independent axes is the dimensionality of G and is equal to the number of degrees of freedom available within G. Each subdomain of G (*Code 3 Property ii*) inherits the same Principal Axes but may have reduced sizes along each. Therefore, uncertainty within any measure of information transfer in a subdomain is a function of dimensionality: more Principal Axes means more directions along which flux can vary, and therefore greater uncertainty in the specified form of information transfer from sender to receiver. *Note: for formal definitions of dimension, size, dimensionality, and degrees of freedom, and their distinction from common architectural usage, see [Concept of System](./concept_of_system.md) Premise 3.*|
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