Vector-Space Specification Sheet¶
Specification artifact — instantiates Coherent Linear Space Design
Declares a representation as an explicit vector space — its carrier, its scalars, its addition, its scaling, and its zero — so every service that touches it reads and combines the same objects the same way.
Before anyone adds, averages, scales, or interpolates a representation, someone has to say — in writing — exactly what they are combining. The Vector-Space Specification Sheet is that written declaration: a short, versioned contract that pins down the five commitments which turn a bag of numbers into a governed vector space. It names the carrier set (which objects count as elements and what universe they live in), the scalar domain (which coefficients are legal), the addition rule (how two elements combine, in what units, under what normalization), the scalar-multiplication rule (what it means to scale one), and the zero-and-inverse policy (what "no element" means and what subtracting one element from another represents). Its whole job is declaration — it does not prove the axioms hold and it does not fix a basis; it states the intended structure so that the rest of the machinery, and every downstream consumer, works against one agreed object instead of a private guess.
Example¶
A robotics company has a perception team producing state estimates and a control team consuming them. Each has been passing "state vectors" to the other, and the fusion code that averages redundant estimates keeps producing subtly wrong results. The fix is a specification sheet. They declare the carrier: a 9-dimensional state of position (x, y, z), velocity (ẋ, ẏ, ż), and linear acceleration — all in the map frame, all in SI units. Crucially, they exclude orientation, because they represent it as a quaternion, and quaternions are not closed under componentwise addition; putting it in the same vector would license nonsense. They declare the scalars as real-valued fusion weights, the addition rule as componentwise in SI (with an explicit note that frames must match first), the scalar-multiplication rule as confidence weighting, and the zero/inverse policy: the zero vector is the map origin at rest, and negation is used only to form residuals between two estimates.
When a third team later joins to add lidar estimates, they read the sheet and their averaging code interoperates on the first try — and the sheet's explicit exclusion of orientation stops them from naively averaging two quaternions, which was exactly the bug that started it all.
How it works¶
The sheet is a small set of filled slots, treated as a versioned artifact rather than tribal knowledge:
- Carrier — the element set and its ambient universe, including what is deliberately excluded (the orientation-quaternion carve-out above).
- Scalars — the coefficient domain, and whether it is a genuine field (reals) or a restricted set (nonnegative weights) that quietly changes what "vector space" can even mean.
- Addition and scaling — the two operations, written with units, normalization, and any precondition (frames aligned, same basis) spelled out.
- Zero and inverses — what absence/neutrality means and what a difference or residual represents.
Every downstream claim then cites the sheet's version; when carrier, scalars, or operations change, the version bumps and consumers are notified. The sheet is a claim of structure, not a proof of it — validation is delegated to its sibling checklist.
Tuning parameters¶
- Carrier tightness — a broad universe admits more elements but weakens closure guarantees; a narrow one is safer but rejects edge cases. Turning it tighter trades reach for trust.
- Scalar domain — reals, rationals, nonnegative reals, or integers. Restricting scalars can break the vector-space guarantee entirely (nonnegative "scalars" give a cone, not a space) — a high-consequence dial.
- Normalization convention — whether units, scaling, and frame are baked into addition or left to callers. Baking them in prevents mismatched sums but couples the sheet to one convention.
- Zero semantics — whether zero means "origin," "no information," or "cancellation"; each licenses different residual math.
- Versioning granularity — how small a change forces a version bump. Fine granularity catches silent drift but adds churn.
When it helps, and when it misleads¶
Its strength is interoperability and the prevention of category errors: once the carrier and operations are written down, teams stop silently averaging things that were never addable, and every consumer combines the same objects the same way. It is the single artifact that the rest of the archetype's machinery reads first.
Its central failure mode is that a specification sheet can confidently declare a structure that is not actually a vector space. Declaration is not validation: writing "addition is componentwise" does not make the set closed under it, and the sheet has no way to notice. The classic misuse is drafting a sheet for the probability simplex — the set of distributions over outcomes — as though it were a vector space, then adding two distributions and getting a vector that sums to two and is no longer a distribution.[n1] The guarding discipline is to never trust a sheet alone: pair every specification sheet with the Linear-Axiom Verification Checklist, which tests whether the declared operations actually satisfy the axioms.
How it implements the components¶
The specification sheet fills the declaration side of the archetype — the commitments that must exist before anything can be verified, coordinated, or combined:
carrier_set_declaration— its first slot: the element set, its universe, and its deliberate exclusions.scalar_domain_specification— names the legal coefficients and whether they form a field.vector_addition_rule— states how two elements combine, with units, normalization, and preconditions.scalar_multiplication_rule— states what scaling an element means.zero_element_and_inverse_policy— fixes the zero element and the meaning of negation, difference, and residual.
It does not verify closure or the axioms (that is the Linear-Axiom Verification Checklist), fix a basis or coordinates (the Basis & Coordinate Table), rule on which combinations are admissible (the Linear-Combination Membership Test), interpret coordinate semantics (the Change-of-Basis Review), or mark nonlinear exceptions (the Nonlinear-Boundary Stress Test).
Related¶
- Instantiates: Coherent Linear Space Design — the sheet is the root declaration the whole archetype governs.
- Sibling mechanisms: Linear-Axiom Verification Checklist · 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: Representation, Specification & Plan
Rationale: Vector Space Specification Sheet is defined in the frozen evidence as: Declares a representation as an explicit vector space — its carrier, its scalars, its addition, its scaling, and its zero — so every service that touches it reads and combines the same objects the same way. Its operative deployed or enacted form is therefore Representation, Specification & Plan.
Nearest alternative: Rule, Policy & Commitment — Rule, Policy & Commitment 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; medium confidence.
Origin Attribution¶
Primary origin: Mathematics
Origin pattern: Single lineage
Present-day reach: Specialized
Rationale: Both independent reviews identify mathematics as the historical home of the operation—Declares a representation as an explicit vector space — its carrier, its scalars, its addition, its scaling, and its zero — so every service that touches it reads and combines the same objects the same way.. The retained alternates document formative adjacent traditions; the reach field, not the origin field, carries later applicability.
Related originating lineages:
- Computer Science & Software Engineering — Computer science and software-engineering practice supplies a parallel or contributing lineage for the mechanism's defining operation: declares a representation as an explicit vector space — its carrier, its scalars, its addition, its scaling, and its zero — so every service that touches it reads and combines the….
- Operations Research — Operations research's allocation, scheduling, optimization, and decision-analysis tradition contributes a separate formative lineage to the mechanism's vector space specification sheet logic.
Review resolution: Both blind reviewers independently place the defining operation—Declares a representation as an explicit vector space — its carrier, its scalars, its addition, its scaling, and its zero — so every service that touches it reads and combines the same objects the same way.—in mathematics. Their queued differences are secondary: alternate_origin_disagreement, origin_mode_disagreement, domain_reach_disagreement, encyclopedia_synthesis_disagreement. Reviewer A uniquely contributes ['operations_research']; reviewer B uniquely contributes ['computer_science']. I preserve the full evidence-supported union of 2 alternate domain(s), without a numeric cap. origin_mode=single_lineage reflects the more specific lineage judgment in reviewer B's evidence, while domain_reach=specialized separately records present-day portability. The affirmative encyclopedia-synthesis finding is preserved, and confidence=high uses the more conservative reviewer level.
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¶
The sheet is a claim, not a proof. Its value depends entirely on being paired with verification: a beautifully written specification for a non-closed carrier is worse than none, because it lends false authority to an invalid structure. Keep declaration and validation as separate mechanisms so a team can improve one — tighten the carrier, restrict the scalars — without silently re-asserting that the axioms still hold.
[n1] The probability simplex (all nonnegative vectors summing to 1) is not a vector subspace: it is not closed under addition or under scaling by arbitrary reals, and it does not contain the zero vector. It is a canonical example of a set that looks combinable but fails the closure axioms — exactly the trap a specification sheet cannot catch on its own. ↩