Disjunct matrix¶
A binary nonadaptive group-testing design in which every column has a row containing 1 while any chosen set of at most d other columns all contain 0.
Core Idea¶
A d-disjunct matrix assigns items to pooled tests so the Boolean union of any d columns cannot cover another column, enabling unique recovery of up to d positives under the noiseless model. Each item owns a distinguishing test against every competing d-set; observed positive-test unions can therefore eliminate nondefective items or support simple decoding. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.
Scope of Application¶
Disjunct matrix belongs to combinatorial group testing and is useful where the analyst can specify the typed combinatorial group testing carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate for every column and every set of at most d other columns, a row witnesses the column with 1 and all competitors with 0 under the stated Boolean outcome model. The scope is broad within that domain but bounded by the need for for every column and every set of at most d other columns, a row witnesses the column with 1 and all competitors with 0 under the stated Boolean outcome model. Conceptual combinatorial identity only; no biomedical testing or laboratory pooling protocol is provided.
Clarity¶
The abstraction clarifies a crowded vocabulary by making for every column and every set of at most d other columns, a row witnesses the column with 1 and all competitors with 0 under the stated Boolean outcome model the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test.
Manages Complexity¶
Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Disjunct matrix. Disjunct matrix compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.
Abstract Reasoning¶
- Identify the carrier. State what the elements, states, objects, or observations are: the typed combinatorial group testing carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express for every column and every set of at most d other columns, a row witnesses the column with 1 and all competitors with 0 under the stated Boolean outcome model independently of one notation or implementation.
Knowledge Transfer¶
Knowledge transfers strongly among subfields of combinatorial group testing because they reuse the typed combinatorial group testing carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, Each item owns a distinguishing test against every competing d-set; observed positive-test unions can therefore eliminate nondefective items or support simple decoding., and type the carrier, state every parameter and convention in the definition, test that for every column and every set of at most d other columns, a row witnesses the column with 1 and all competitors with 0 under the stated Boolean outcome model, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.
Relationships to Other Abstractions¶
Current abstraction Disjunct matrix Domain-specific
Parents (1) — more general patterns this builds on
-
Disjunct matrix is a kind of Constraint Prime
The proposed strict upward parent is
prime:constraint.
Hierarchy path (1) — routes to 1 parentless root
- Disjunct matrix → Constraint
Neighborhood in Abstraction Space¶
Disjunct matrix sits in a crowded region of the domain-specific corpus (25th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Enumerative Combinatorics & Partitions (24 abstractions)
Nearest neighbors
- Discrepancy theory — 0.92
- Representation theory of the symmetric group — 0.91
- Addition principle — 0.91
- Maximum satisfiability problem — 0.91
- Permutation group — 0.90
Computed from structural-signature embeddings · 2026-09-08