Basis Sensitivity Review¶
Robustness review — instantiates Generated Span Closure Design
Swaps the generator set and compares the resulting spans, exposing which downstream claims are robust to basis choice and which are not.
The same span can often be generated by very different sets of primitives — and the choice between them, which looks like a matter of taste, quietly shapes what is easy to reach, what is efficient to express, and what a downstream consumer can safely assume. Basis Sensitivity Review is the what-if across generating sets: it holds the problem fixed and varies the basis — swap a generator, add one, drop one, adopt a rival kit entirely — then compares the resulting spans and coverage records to see what changes. Its output is a robustness verdict: which claims hold for any reasonable basis and which are artifacts of the particular one you happened to pick. Where the elimination test minimizes a single set from the inside, this review compares competing sets from the outside — a comparative, not a reductive, mechanism.
Example¶
A country is redesigning its coinage and must pick a set of denominations. Two candidate bases are on the table: the familiar {1, 5, 10, 25} and a theoretically tidier {1, 3, 11, 37}. Basis Sensitivity Review compares them on the thing that matters — the "span" of amounts they can make and how efficiently. Both bases can make every amount (thanks to the 1-cent coin), so reachability is robust to the choice. But efficiency is not: {1, 5, 10, 25} lets a cashier make most everyday amounts in few coins under the usual greedy rule, while {1, 3, 11, 37} produces surprising cases where the greedy "biggest coin first" rule uses more coins than necessary. The review reports exactly this split: the claim "any amount is makeable" survives the basis swap; the claim "greedy change-making is optimal" does not.
The decision-makers now know which of their assumptions were about the problem and which were about the coins — and they can pick a basis knowing what each choice quietly buys or costs downstream.
How it works¶
- Fix the problem, vary the basis. Hold the target space and admissible operations constant; enumerate the candidate generating sets to compare.
- Recompute the span per basis. For each candidate set, obtain its coverage record — what it reaches, how efficiently, with what redundancies.
- Diff the results. Line up the coverage records and mark where they agree (robust) and where they diverge (basis-dependent), including which generators are redundant under one basis but load-bearing under another.
- Report robustness and re-scope the contract. State which downstream claims survive any reasonable basis and which are conditional, and revise the use contract to flag the conditional ones.
The distinguishing move is comparison across sets: it never asks "is this element reachable" in isolation, only "does the answer depend on which primitives we chose."
Tuning parameters¶
- Basis candidate breadth — how many and how different the compared generating sets are. Wider comparison catches more fragility but costs a full span recomputation per candidate.
- Perturbation size — small edits (swap one generator) versus wholesale rival bases. Small perturbations probe local stability; large ones test whether a claim is basis-specific at all.
- Comparison metric — reachability only, or also efficiency, redundancy, or expression length. Richer metrics reveal more sensitivity but demand more from each coverage record.
- Robustness threshold — how much divergence across bases counts as "fragile." A strict threshold flags more claims as conditional; a loose one may bless a fragile assumption.
When it helps, and when it misleads¶
Its strength is that it separates problem truth from basis artifact — the single most valuable thing to know before you commit downstream decisions to a particular set of primitives, because it tells you which of your guarantees will survive a refactor of the basis and which will silently break.
Its failure mode is the sampling trap: you can only compare the bases you thought to try, so "robust across the ones we tested" is not "robust across all reasonable bases," and a fragility that lives between your samples goes unseen.[n1] The classic misuse is declaring a claim basis-independent after checking two similar sets, then being blindsided when a genuinely different basis breaks it. A subtler error is comparing bases whose spans were computed under different truncation depths, so an apparent difference in coverage is really a difference in how far each enumeration ran. The guarding discipline is to compare bases that are meaningfully different (not near-duplicates), to hold truncation depth constant across the comparison, and to state robustness verdicts as "over the bases reviewed," never as universal.
How it implements the components¶
Basis Sensitivity Review realizes the comparative machinery, aimed across generating sets:
dependence_redundancy_map— it reports how redundancy shifts between bases: a generator that is load-bearing under one set and redundant under another.generator_set_specification— it operates on whole alternative specifications, treating the generator set itself as the variable.coverage_or_reachability_record— it diffs one basis's coverage record against another's to locate divergence.downstream_use_contract— it re-scopes that contract, tagging which downstream claims are conditional on the chosen basis.
It does NOT test whether a single specific target is reachable (span_membership_criterion) — that is Span Membership Certificate. And it compares rival bases from the outside rather than minimizing one basis from within — that inward minimization is Dependency Elimination Test, its nearest twin: compare across alternative sets versus reduce a single set.
Related¶
- Instantiates: Generated Span Closure Design — the review supplies the robustness verdict that tells the appraisal which claims survive a change of generators.
- Consumes: Reachability Matrix or Table supplies the per-basis coverage records this review diffs across candidate sets.
- Sibling mechanisms: Generator Inventory · Closure Generation Workflow · Span Membership Certificate · Dependency Elimination Test · Normal Form Reduction Procedure · Reachability Matrix or Table · Bounded Depth Generation Template
Editorial Notes¶
Form Classification¶
Form family: Assessment, Review & Assurance
Rationale: The mechanism holds the target problem fixed, recomputes coverage under alternative bases, and reports which downstream claims remain robust or depend on basis choice, so its operative output is a robustness assessment.
Nearest alternative: Analysis, Modeling & Optimization — Span calculations produce the evidence, but the mechanism culminates in a bounded finding about claim sensitivity rather than a new predictive model.
Review outcome: Adjudicated after independent review; high confidence.
Origin Attribution¶
Primary origin: Mathematics
Origin pattern: Cross-disciplinary synthesis
Present-day reach: Specialized
Rationale: Linear algebra and numerical analysis establish that basis choice changes coordinates, conditioning, and ease of representation even when spans coincide.
Related originating lineages:
- Engineering & Design — Engineering robustness reviews test conclusions against alternative representations and assumptions.
- Physics — Changes of basis and representation are materially formative in physical modeling.
- Statistics & Experimental Design — Model and specification sensitivity analysis supplies the robustness-review logic.
Review resolution: Mathematics is the agreed primary lineage because the mechanism varies a generator set and compares the resulting spans. Physics, statistics, and engineering independently made basis choice and sensitivity consequential, so the generalized review is a cross-disciplinary Encyclopedia synthesis.
Encyclopedia synthesis: The exact catalogued form synthesizes established practice rather than reproducing a single standard historical label.
Review outcome: Reconciled after independent review; high confidence.
Notes¶
[n1] In numerical linear algebra the condition number measures how sensitive a computation's answer is to small changes in its inputs, and it depends on the chosen basis: the same space represented in a well-conditioned versus an ill-conditioned basis can give wildly different numerical stability. It is the standard reminder that basis choice, not just the underlying space, materially changes downstream behavior — which is exactly what this review exists to surface. ↩