Align README G/E

Author: claude-subagent

README.md

@@ -13,19 +13,19 @@ An agent that passes the test of abiding by these ecological codes, is said to b
 
 **Premise:** 
 
-1. Let, G is the collection of all feasible and veridical aspects of The Universe. Then, all ecological embeddings (symbol-meaning bindings) that describe things and relationships among those things within G, have geometric properties. Consequently, algebraic operations on any measurable quantities are feasible within structured subdomains of G.
+1. Let, G is the mathematically generalized [rank](https://en.wikipedia.org/wiki/Rank_(linear_algebra)#Generalization) of the collection of all feasible and veridical aspects of The Universe. Let **E** denote the space of all such ecological embeddings; G = rank(E). Then, all ecological embeddings (symbol-meaning bindings) that describe things and relationships among those things within E, have geometric properties. Consequently, algebraic operations on any measurable quantities are feasible within structured subdomains of maximum rank G.
 
-2. Flux denotes the rate of information transfer across a surface within G, [in an information theoretic sense](https://en.wikipedia.org/wiki/Entropy_(information_theory)#Definition).
+2. Flux denotes the rate of information transfer across a surface within E, [in an information theoretic sense](https://en.wikipedia.org/wiki/Entropy_(information_theory)#Definition).
 
 3. *Degrees of freedom* of a system coincide with its dimensionality.
 
 |**Code**|**Description**|**Explanation**|
 |---|---|---|
 |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.|
 |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.|
-|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)*.|
-|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. *Code 3* parametrizes the types of relationships defined as R, that 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*.|
-|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 subdomains. 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. Other properties of the medium (subdomain) such as its density and thermodynamic phase also impact expected rate of flux through any of its cross-sections. *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.*|
+|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, the mathematically generalized rank of E — a scalar integer; E, the space of all ecological embeddings that defines the spatio-temporal adjacency of N and R within a hyper-dimensional space. E 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. E formalizes this locally: it is the ecological embedding space 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 E is non-trivially structured (G > 0), S retains persistent state. When R ≠ Ø, G > 0 is required — a system with relationships but no structured E is formally excluded under this definition. Regarding formal constraints and corollaries, see: *[Concept of System](./concept_of_system.md)*.|
+|3|A structured E — 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 E is itself a structured E 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. *Code 3* parametrizes the types of relationships defined as R, that can be sustained within the domain under consideration. Together, *Properties i–iii* are mutually self-reinforcing and recursive: any subdomain of a structured E completely satisfies *Code 3*.|
+|4|The flux across surfaces in E defines vectors; the fundamentally independent and mutually exclusive directions of those vectors yield Principal Axes; the count of Principal Axes is the dimensionality of E or its constituent subdomains. 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 E yields the Principal Axes of E. The count of those independent axes is the dimensionality of E and is equal to the number of degrees of freedom available within E. Each subdomain of E (*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. Other properties of the medium (subdomain) such as its density and thermodynamic phase also impact expected rate of flux through any of its cross-sections. *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.*|
 
 ## 4. Examples of Ecologically Designed User Prompts