Linear-Axiom Verification Checklist¶

Verification checklist — instantiates Coherent Linear Space Design

Walks a declared space through the vector-space axioms — closure, identity, inverse, associativity, distributivity — to catch operations that only look linear.

Version: v1 · 2026-08-24 · History
Mechanism #: 4854
Type: Verification Checklist
Form family: Assessment, Review & Assurance
Solution family: Representation & Modeling
Problem family: Correctness, Conformance & Formal Validity Failure
Problem subfamily: Generator, Basis & Operation Structure
Origin domain: Mathematics
Instantiates: Coherent Linear Space Design

A specification sheet says what the operations are supposed to be; it cannot tell you whether they actually satisfy the laws that make linear reasoning valid. The Linear-Axiom Verification Checklist is the step that tests each law in turn. It takes the declared carrier, addition, scaling, and zero, and runs them through the vector-space axioms one at a time — closure under addition and scaling, an additive identity, additive inverses, associativity and commutativity of addition, distributivity of scalars over sums, and compatibility of scalar multiplication — checking each with both representative and deliberately adversarial elements. Its distinguishing move is that it tests structure rather than asserting it: where the sheet declares, the checklist tries to break, so that a mapping which merely resembles a vector space is caught before downstream code trusts it with interpolation, averaging, or superposition.

Example¶

A numerical-computing library author wants to reuse a generic linear-algebra routine on a new Quantity type that carries a physical unit alongside a magnitude. The routine will only be correct if Quantity forms a vector space, so the author runs the checklist. Closure under addition fails immediately in the adversarial case: adding a length to a mass does not produce a Quantity at all — the type is only closed within a single unit dimension, so the "space" must be scoped per-unit. Within one unit, associativity of addition then fails in a subtler way: (a + b) + c and a + (b + c) give different results in floating point because rounding is not associative,^[n1] so the axiom holds only approximately. Distributivity, identity (a true zero of the right unit), and inverses pass. The outcome is not a yes/no stamp but a precise caveat the author writes into the docs: Quantity is a vector space per unit dimension and only up to floating-point tolerance, and the generic routine is safe exactly within those bounds.

How it works¶

Enumerate the axioms for the declared scalars and operations, not a generic list — a restricted scalar set changes which laws even apply.
Choose witnesses per axiom. Each axiom is checked against representative elements and adversarial ones: boundary values, mismatched units, extreme magnitudes, near-cancellations.
Classify failures. A failure is sorted as structural (the set is genuinely not closed — fatal) or numerical (the law holds symbolically but not in finite precision — a documented tolerance).
Report the valid regime. The output states not "it is a vector space" but "it is a vector space within these bounds," which is the honest answer whenever the check is by example rather than proof.

The checklist consumes the specification sheet's declared rules; it produces the verdict the sheet cannot produce for itself.

Tuning parameters¶

Axiom coverage — which laws are actually tested. Skipping commutativity when addition is known-symmetric saves effort but assumes what it should check.
Witness strategy — representative-only, boundary, adversarial, or randomized fuzzing. More adversarial witnesses catch more real violations at more cost.
Tolerance — exact equality versus an epsilon band for numeric types. Too tight flags harmless rounding; too loose hides real structural breaks.
Proof depth — spot-check by examples, exhaustive check over a finite carrier, or a formal proof. Deeper is more conclusive but far slower.

When it helps, and when it misleads¶

Its strength is catching silent structure violations — the mapping that behaves linearly on the inputs anyone tried and betrays that assumption only on the input that ships to production. It converts a hopeful "this should be linear" into a scoped, defensible claim.

Its failure mode is intrinsic to checking by example: passing on a sample of witnesses does not prove an axiom holds universally, and one unprobed adversarial element can hide a counterexample. The superposition principle that all the downstream math relies on is only as trustworthy as the closure the checklist could actually exercise. The classic misuse is declaring victory after checking only "nice" inputs — small, same-unit, well-conditioned — and thereby missing floating-point non-associativity or integer overflow that appears at scale.^[n1] The guarding discipline is to lead with adversarial witnesses and to state the regime in which the axioms were confirmed, never a bare pass.

How it implements the components¶

The checklist fills the validation face of the archetype — it takes declared operations and certifies (or refutes) that they obey the laws:

closure_axiom_checklist — its core output: the axiom-by-axiom pass/fail record, with closure under addition and scaling checked first.
vector_addition_rule — tests the declared addition for associativity, commutativity, and closure rather than merely restating it.
scalar_multiplication_rule — tests scaling for distributivity and scalar compatibility.
zero_element_and_inverse_policy — confirms a genuine additive identity and that every element has an inverse.

It does not declare the carrier or scalar domain (the Vector-Space Specification Sheet does), fix a basis (the Basis & Coordinate Table), rule on admissible combinations (the Linear-Combination Membership Test), interpret coordinates (the Change-of-Basis Review), or map nonlinear exceptions (the Nonlinear-Boundary Stress Test).

Instantiates: Coherent Linear Space Design — the checklist certifies that a declared space actually obeys the linear axioms.
Consumes: Vector-Space Specification Sheet — it verifies exactly the operations the sheet declares.
Sibling mechanisms: Vector-Space Specification Sheet · Basis & Coordinate Table · Linear-Combination Membership Test · Change-of-Basis Review · Zero-Span Linearity Check · Linear Embedding Diagnostics · Nonlinear-Boundary Stress Test

Editorial Notes¶

Form Classification¶

Form family: Assessment, Review & Assurance

Rationale: Linear-Axiom Verification Checklist operates as a bounded evaluation of existing evidence or work that produces a finding or disposition because it walks a declared space through the vector-space axioms — closure, identity, inverse, associativity, distributivity — to catch operations that only look linear.

Independent corroboration: The frozen evidence defines Linear-Axiom Verification Checklist as 'Walks a declared space through the vector-space axioms — closure, identity, inverse, associativity, distributivity — to catch operations that only look linear', so its operative form is Assessment, Review & Assurance.

Review outcome: Independent reviewer agreement; high confidence.

Origin Attribution¶

Primary origin: Mathematics

Origin pattern: Single lineage

Present-day reach: Specialized

Rationale: Vector-space axioms and their verification are a pure linear-algebra construction in mathematics.