Coherent Linear Space Design¶

Declare a carrier, scalars, and linear operations so adding, scaling, decomposing, and interpolating elements have stable meaning.

Draft status¶

Generated as a full gap-fill draft for the accepted target prime vector_space from queue position 43.

Disposition rationale¶

Disposition check found no accepted archetype, pilot accepted gap-fill archetype, pilot variant addition, prior uploaded-queue output, alias-map entry, duplicate/merge-map entry, component, or mechanism that directly covers vector_space as the carrier-plus-operation structure in which addition and scaling are coherent and closed under linear combination. Accepted neighbor proportional_response_design covers linearity as proportional response intensity, not a reusable vector-space representation. Accepted neighbors such as closure_preserving_operation, canonical_classification, representation_fit_selection, dimensionality_reduction_for_signal, degrees_of_freedom_reduction, and metric/measurement-oriented patterns each cover one face of the structure but do not define the full algebra of vectors, scalars, addition, scalar multiplication, zero, additive inverse, closure, basis/dimension metadata, and coordinate semantics. Prior uploaded-queue outputs generated_span_closure_design, independent_generator_validation, additive_measure_space_design, reversible_operation_structure_design, exhaustive_disjoint_partition_design, shared_subset_intersection_mapping, and inclusive_membership_union_design are close algebraic neighbors. They supply span, independence, measure, group-like reversibility, partition, intersection, and union boundaries, but none is the parent intervention of declaring and governing a coherent linear carrier space. A distinct full archetype is warranted.

Practitioner note¶

Use this archetype when a representation or formal system is already behaving as though elements can be added, scaled, interpolated, decomposed, projected, or combined. The draft keeps the parent object narrow: it is not every numerical representation, not every metric space, and not every generated span. It is the governance pattern for a coherent linear carrier space.

Boundary note¶

The recent queue outputs for span, linear independence, measure, group, partition, union, intersection, complement, and unity are treated as neighbors. This draft gives those neighbors a shared parent structure without absorbing their distinct proof obligations.

Common Mechanisms¶

basis_coordinate_table
change_of_basis_review
linear_axiom_verification_checklist
linear_combination_membership_test
linear_embedding_diagnostics
nonlinear_boundary_stress_test
vector_space_specification_sheet
zero_span_linearity_check

Compression statement¶

Coherent Linear Space Design is the intervention pattern of representing elements inside a governed vector-like space: a scoped carrier with an admissible scalar domain, vector addition, scalar multiplication, zero, inverses, closure and distributive checks, coordinate semantics, and exception boundaries. It turns informal claims such as “combine these signals,” “move in this direction,” “interpolate between options,” or “decompose this state” into auditable linear-space commitments.

Canonical formula: For carrier V and scalars F, V is closed under u + v and a·v, with zero 0, additive inverses, and distributive/associative/identity laws; linear combinations Σ a_i v_i remain in V.

Abstractions this archetype builds on — directly (a source ingredient) or as a related pattern. Links follow the typed catalog namespace.

Built directly on (4)

Closure: Ensures operations remain within a set.
Linear Combination: Scale each of several objects by a weight and add them together.
Set and Membership: Groups and categorizes elements.
Vector Space: A collection closed under linear combination, where adding and scaling are coherent.

Also references 24 related abstractions

Abstraction: Focus on core elements.
Associativity: Grouping does not affect result.
Basis: A minimal independent generating set — the smallest collection from which every element of a space can be produced, with no member derivable from the others.
Boundedness: Values remain within limits.
Category: Describe a system by its arrows and their composition, not by what its objects are.
Commutativity: Order of inputs does not affect output.
Composition: Arranges components into a cohesive whole.
Constraint: Limits possibilities to guide outcomes.
Dense Set: A smaller set reaches arbitrarily close to every point of a larger one.
Dimension: Degrees of freedom in a system.

▸ Show 14 more

Variants¶

Narrower or domain-specific specializations that share this archetype's core structure. Recognized variants are established; candidate variants are provisional.

Finite-Dimensional Coordinate Space · other · recognized

A vector-space design where every element is represented by a finite coordinate tuple relative to a declared basis.

Distinct from parent: Narrower because it assumes finite coordinate representation rather than any vector space.
Use when: Finite coordinates drive computation or communication; Dimension and basis choices are stable enough to version.
Typical domains: linear algebra, data science analytics, engineering design
Common mechanisms: basis coordinate table, change of basis review

Subspace Restriction Design · other · recognized

A vector-space variant that restricts operation to a closed linear subset for safety, feasibility, or interpretability.

Distinct from parent: Narrower because it operates inside a selected linear subset.
Use when: Only a subset of the full space is valid; Constraint or feasibility boundaries matter.
Typical domains: control theory, operations research, machine learning
Common mechanisms: linear combination membership test, nonlinear boundary stress test

Semantic Embedding Space Governance · domain variant · recognized

A vector-space governance variant for learned or designed embeddings whose vector arithmetic has domain-dependent meaning.

Distinct from parent: Narrower because the carrier is a semantic or learned embedding space.
Use when: Embeddings support similarity, interpolation, analogy, clustering, or retrieval; Model version, normalization, and coordinate semantics affect decisions.
Typical domains: machine learning, search and retrieval, semantic representation
Common mechanisms: linear embedding diagnostics, change of basis review

Near names: Linear Combination Space Architecture, Additive Scaling Space Design, Vector Representation Space Design.