Theoretical Gap Matrix¶
Planning matrix — instantiates Theory-Responsive Case Sampling Design
Maps the model's open gaps against candidate cases to rank which case would teach the most next.
A Theoretical Gap Matrix is a prospective grid whose rows are the model's open gaps — ambiguous categories, missing relations, untested contexts, unresolved rival explanations — and whose columns are candidate cases, with each cell rating how much a given case would teach about a given gap. It plans forward across many candidates and many gaps at once, so the next case is chosen by scanning for the highest-value cell rather than by convenience or habit. Its defining move is comparison among not-yet-collected cases: it turns selection into a portfolio decision, making visible that some available cases would teach far more than others before a single one is fielded.
Example¶
A researcher is building a theory of why nonprofit boards become dysfunctional. The current model half-suspects that founder dominance drives member disengagement, but several things are unsettled. She lays them out as rows: does founder dominance cause disengagement or the reverse? Does the pattern hold on a board with no founder present? What happens on an all-volunteer board with no paid staff to defer to? Down the columns she lists candidate boards she could realistically access: a founder-led arts nonprofit, a founderless food bank, a large hospital board, a tiny all-volunteer animal shelter.
She scores each cell for expected learning. The founderless food bank scores highest against the "is founder dominance necessary?" gap — it is the one candidate that could show the pattern surviving without a founder — while the hospital board scores high on the paid-staff-deference gap but low on the founder question. The grid makes the choice legible: the food bank goes next, not because it is interesting in the abstract, but because it sits on the highest-value cell in the matrix. After it is analyzed and the founder-necessity gap closes, that row drops out and the grid is re-scored for the next pick.
How it works¶
- List open gaps as rows. Enumerate the model's unresolved concepts, relations, contexts, and rival explanations.
- List candidate cases as columns. Include only cases realistically reachable, classified by the role each would play.
- Score the cells. Rate expected learning value (and feasibility) for each gap-by-case pairing.
- Pick the maximum cell, then re-score. Select the highest-value case, and after it is analyzed, drop closed gaps and re-score the grid for the next selection.
Tuning parameters¶
- Gap granularity — coarse gaps versus finely split ones. Fine gaps target selection precisely but multiply rows and invite false precision in scoring.
- Candidate breadth — a short list of obvious cases versus an exhaustive enumeration. More candidates surface high-value cases that habit would miss, at the cost of effort.
- Scoring scheme — pure expected learning versus learning discounted by access and feasibility. Weighting feasibility keeps the plan realistic but can bury a high-value case that is merely hard to reach.
- Re-score cadence — after every case versus every few. Re-scoring often keeps the plan current; re-scoring rarely is cheaper but drifts from the model state.
When it helps, and when it misleads¶
Its strength is making case selection an explicit portfolio decision: the grid surfaces, before any fielding, that candidate cases differ enormously in what they would teach, and it points selection toward the case that best resolves a live anomaly rather than the one nearest to hand.[n1] It is the loop's planning instrument.
Its central failure mode is false precision — a tidy grid of scores implying quantified learning where the numbers are educated guesses, lending unearned objectivity to a judgment call. A subtler misuse is gap-gaming: defining the gaps so that the case the analyst already wants always tops the grid. The guarding discipline is to fix the gap definitions independently of the candidate one hopes to pick, and to read the scores as a structured argument to be interrogated, not a verdict to be obeyed.
How it implements the components¶
A Theoretical Gap Matrix fills the prospective-planning slice of the archetype, not its analysis or single-selection slices:
analytic_gap_register— the matrix's rows are the gap register, made comparative and rankable.case_contrast_palette— its columns classify candidates by role (typical, deviant, boundary, contrasting), keeping multiple case roles in view at once.case_learning_question— each high-scoring cell implies the learning question the chosen case will carry.
It does not work already-collected cases into model revisions — that retrospective engine is Constant Comparison Matrix via analysis_sampling_loop and model_revision_register; nor does it author the single next case's narrative rationale, which is Grounded Theory Sampling Memo via selection_rationale_record.
Related¶
- Instantiates: Theory-Responsive Case Sampling Design — supplies the forward-looking plan that ranks candidate cases by expected learning.
- Consumes: Constant Comparison Matrix — supplies the revised model whose fresh gaps become the matrix rows.
- Sibling mechanisms: Boundary Case Probe · Case Selection Audit Trail · Constant Comparison Matrix · Grounded Theory Sampling Memo · Maximum Variation Case Round · Negative Case Sampling Pass · Rival Explanation Discriminator · Saturation Review Memo · Transferability Claim Check
Editorial Notes¶
Form Classification¶
Form family: Analysis, Modeling & Optimization
Rationale: Theoretical Gap Matrix is defined in the frozen evidence as: Maps the model's open gaps against candidate cases to rank which case would teach the most next. Its operative deployed or enacted form is therefore Analysis, Modeling & Optimization.
Nearest alternative: Decision, Gate & Allocation — Decision, Gate & Allocation can support this mechanism, but the evidence centers the concrete operation described above rather than the alternative family's defining operation.
Review outcome: Adjudicated after independent review; high confidence.
Origin Attribution¶
Primary origin: Statistics & Experimental Design
Origin pattern: Convergent development
Present-day reach: Multi-domain
Rationale: The defining operation is: Maps the model's open gaps against candidate cases to rank which case would teach the most next. In the statistics_experimental_design lineage, that operation is specifically evidenced by authoritative or primary work that requires explicit objectives, variables, levels, and experimental designs chosen to expose model effects and support testing, revision, and confirmation. This makes statistics_experimental_design the best historical origin, while the retained alternates document contributing methods and later applications rather than being mistaken for coequal origins.
Related originating lineages:
- Data Science & Analytics — Data science, analytics, and operational monitoring supplies a parallel or contributing lineage for the mechanism's defining operation: maps the model's open gaps against candidate cases to rank which case would teach the most next.
- Ethnography & Qualitative Methods — Ethnography and qualitative comparative inquiry supplies a parallel or contributing lineage for the mechanism's defining operation: maps the model's open gaps against candidate cases to rank which case would teach the most next.
- Mathematics — Mathematics' formal structure, proof, and transformation tradition provides a formative adjacent lineage for the same theoretical gap matrix operation.
- Philosophy — Philosophical logic, epistemology, and normative reasoning supplies a parallel or contributing lineage for the mechanism's defining operation: maps the model's open gaps against candidate cases to rank which case would teach the most next.
Review resolution: The blind reviewers disagree on primary lineage (philosophy versus statistics_experimental_design), so I adjudicated the mechanism rather than inheriting either label. The defining operation is: Maps the model's open gaps against candidate cases to rank which case would teach the most next. In the statistics_experimental_design lineage, that operation is specifically evidenced by authoritative or primary work that requires explicit objectives, variables, levels, and experimental designs chosen to expose model effects and support testing, revision, and confirmation. This makes statistics_experimental_design the best historical origin, while the retained alternates document contributing methods and later applications rather than being mistaken for coequal origins. The cited NIST Handbook: Design of Experiments directly supports the mechanism-specific operation and its disciplinary lineage. I retain all independently explained historical alternates without a numeric cap. origin_mode=convergent records how the mechanism arose; domain_reach=multi_domain separately records how broadly it can now be applied.
Encyclopedia synthesis: The exact catalogued form synthesizes established practice rather than reproducing a single standard historical label.
Review outcome: Researched adjudication after independent review; high confidence.
Sources consulted:
Notes¶
[n1] Abduction — C. S. Peirce's mode of inference to the best explanation, which reasons from a surprising observation to the hypothesis that would render it unsurprising. A gap matrix is abductive in spirit: it ranks candidate cases by how much each would help resolve the account's live anomalies. ↩