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Generated Span Closure Design

Declare the primitives and allowed operations, then make the whole generated possibility space explicit and auditable.

Generated Span Closure Design captures the reusable pattern behind span: a declared set of primitives generates a larger reachable or expressible space under declared operations. The draft keeps this parent pattern distinct from closure-preserving operation, basis selection, linear independence, partitioning, and measure assignment.

Disposition

draft_full_archetype — no existing accepted archetype or prior queue output directly covers span as the complete generated space produced from primitives under admissible operations.

Boundary note

This draft should be reviewed alongside the upcoming vector_space, union, and unity_test targets so the algebraic/set-theoretic family remains coherent rather than over-fragmented.

Common Mechanisms

  • basis_sensitivity_review
  • bounded_depth_generation_template
  • closure_generation_workflow
  • dependency_elimination_test
  • generator_inventory
  • normal_form_reduction_procedure
  • reachability_matrix_or_table
  • span_membership_certificate

Compression statement

Generated Span Closure Design is the intervention pattern of turning a small set of generators and operation rules into a governed reachable set: the span. It declares generators, admissible operations, expression forms, closure bounds, membership criteria, redundancy tests, and downstream use limits so claims about what can be produced, expressed, derived, reached, or represented are not merely intuitive.

Canonical formula: Span(G, O) = the smallest closed set S such that G ⊆ S and every admissible operation in O maps applicable elements of S back into S.

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

Built directly on (4)

  • 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.
  • Closure: Ensures operations remain within a set.
  • Linear Combination: Scale each of several objects by a weight and add them together.
  • Span: The complete set reachable by combining primitives under admissible operations.

Also references 20 related abstractions

  • Abstraction: Focus on core elements.
  • Associativity: Grouping does not affect result.
  • Boundedness: Values remain within limits.
  • Category: Describe a system by its arrows and their composition, not by what its objects are.
  • Completeness: No gaps in structure.
  • Composition: Arranges components into a cohesive whole.
  • Compositionality: Meaning from parts.
  • Constraint: Limits possibilities to guide outcomes.
  • Formal System: Symbols, formation rules, axioms, and inference rules closed under mechanical derivation.
  • Group: A set with an associative operation, identity, and inverses — reversible composable transformations.