Axioms for the Property Structure of Realized Axes in Dimensional-Structural Describability

Abstract

This paper develops a typed axiom system for the property structure of realized axes overthe Stage-VI channel-complete truncation of a Dimensional-Structural Describability formation model. The formation layer determines admitted operational channels and preservesconfiguration, material item, quantity-kind, assigned value, and role in channel identity.The present extension begins after operational channel formation and before any analyticrealization of component terms or channel aggregation; it asks what properties may be assigned to realized axes, to typed families of realized axes and auxiliary geometric inputs,and to property-dependent closure structures.The system distinguishes a bare realized axis line from the tagged channel that realizes it.Property kinds are selected from a fixed candidate universe and carry typed input profilesrather than arity alone. The profiles permit homogeneous unary, binary, and higher-orderaxis records as well as mixed inputs such as one normal record with two tagged channels.Property assignments are partial maps. Undeclared property, undefined application, definedvalue, and—for zero-bearing kinds—defined zero and defined nonzero value remain distinct.A tag-based property may be line invariant or tag sensitive, so two channels that realizethe same line may still carry different property values when their channel coordinates arerelevant.Self, relational, higher-order, and closure-associated candidate labels are optional property kinds rather than universally imposed physical quantities. Their mathematical meaningexists only after a model supplies a typed profile, value carrier, and assignment map. Matrix, tensor, operator, and quaternion expressions are optional representations and are notidentified with the abstract properties they encode. Bilinear, normal, and closure-associatedstructures form designated property subclasses inside the wider axis-property system.Two residual primitive axioms govern total realization of selected axis channels and compatibility of bilinear-dependent claims with supplied symmetric bilinear data. All optionalrepresentation and closure choices belong to dependent primitive layers; only functionallydetermined records are added by explicit completion. The two primitive axioms are mutuallynon-derivable in the admitted class of typed axis-property presentations. Every layered primitive axis-property presentation satisfying the two residual axioms has a unique full-modelcompletion, every Stage-VI truncation induced by a full formation model admits a trivialaxis-property extension, and every compatible presentation satisfying the stated finite-datahypotheses has a finitely specified realization. The complete descriptor retains the inheritedStage-VI record together with global and per-configuration coordinates, and strict Stage-VIfixed, signature-fixed, representation-inclusive axis-property isomorphism is an equivalencerelation. Equal realized-axis rank and equal matrix size are not complete classifiers, andthe displayed finite-coordinate summary can likewise identify strictly nonisomorphic modelsunder the typed collision hypothesis established below.