Dimensional Consistency Check¶
Check that quantities, units, and scale relationships are compatible before trusting equations, comparisons, or transfers.
Essence¶
Dimensional Consistency Check prevents formal-looking calculations, metrics, comparisons, and model transfers from being trusted before their quantities are structurally compatible. It asks whether the numbers being combined or compared share compatible units, dimensions, denominators, time bases, scale bases, and meanings.
The archetype is broader than a unit check. A unit check catches inches mixed with centimeters or seconds mixed with milliseconds. Dimensional consistency also catches subtler failures: totals compared with rates, one-time costs compared with recurring costs, nominal dollars compared with present-value dollars, ratios whose units cancel but whose meaning is arbitrary, or pilot metrics reused at a different scale without a transfer boundary.
Compression statement¶
When formulas, metrics, comparisons, or model transfers combine quantities, verify dimensional consistency so invalid operations, misleading normalizations, and scale-transfer errors are caught before they influence decisions.
Canonical formula: quantity_set + unit_dimension_map + operation_compatibility_rule + consistency_test + correction_or_transfer_boundary → dimensionally_valid_expression_or_rejected_comparison
When to Use This Archetype¶
Use this archetype when a decision depends on a formula, metric, dashboard, model output, benchmark, normalized score, rate, ratio, or cross-context transfer. It is especially important when quantities come from multiple systems, when normalized metrics hide denominators, when a calculation appears precise, or when a result will influence safety, finance, policy, operations, or automated decisions.
It is less useful when the main problem is corrupt source data, missing evidence, representational aesthetics, or disagreement about values. In those cases, dimensional checking may still be a supporting review, but it is not the parent intervention.
Structural Problem¶
The structural problem is that formal operations are easier than meaningful operations. A spreadsheet can add columns whose units do not match. A dashboard can rank percentages, totals, and rates side by side. A model can accept inputs with hidden unit conventions. A policy memo can compare regions of very different size without exposing denominators. These artifacts produce numbers that look usable even when the underlying quantities are incompatible.
The dangerous feature is that the error is often invisible. The output may be precise, visually polished, or mathematically sophisticated. Without an explicit compatibility check, reviewers may focus on the conclusion rather than the structure of the quantities that produced it.
Intervention Logic¶
The intervention starts by naming all quantities involved. Then it maps their units, dimensions, denominators, time bases, measurement conventions, and reference frames. Next, it states which operations are allowed: what may be added, compared, divided, averaged, normalized, aggregated, or transferred. The consistency test then checks whether the proposed operation preserves dimensional meaning.
If the expression fails, the archetype does not merely flag an error. It routes the problem through a correction rule: convert units, normalize the denominator, separate stocks from flows, construct a meaningful dimensionless ratio, redefine the metric, reject the comparison, or escalate for redesign. Finally, it records the transfer boundary so future users know when the expression must be checked again.
Key Components¶
Dimensional Consistency Check makes the hidden structure of quantities visible before any calculation, comparison, or metric is trusted. The work begins with a Variable List that names every quantity, input, denominator, and intermediate value, since anything that goes unnamed cannot be checked. The Unit / Dimension Map assigns each variable its unit, dimension, denominator, time basis, and reference frame, distinguishing values that look similar but are not commensurable. Quantity Semantics goes one layer deeper: two values can share units while representing different constructs — nominal dollars versus present-value dollars, prevalence versus incidence — and the semantics layer keeps that distinction explicit. The Scale Basis records whether each quantity is a raw total, per-capita rate, percentage, indexed value, or log-scale measure, exposing a basis that is otherwise implicit and the source of many bad cross-comparisons.
The remaining components turn that structural map into protective action. The Operation Compatibility Rule states which operations are allowed for which mapped quantities, since addition, comparison, division, aggregation, and transfer each demand different compatibility conditions. The Consistency Test then checks whether a proposed equation, metric, or transfer actually preserves dimensional meaning under those rules. A Dimensionless Group can extend valid comparison across scales when constructed with a meaningful numerator, denominator, and interpretation rather than as units-cancel happenstance. When the test fails, the Correction Rule routes the mismatch to a specific response — convert, normalize, separate stocks from flows, redesign, reject, or escalate — so that the archetype is protective rather than merely advisory. Finally, the Transfer Boundary records where the validated expression remains trustworthy, so a formula proven valid in one unit system, population, time window, or scale is flagged for rechecking when moved elsewhere.
| Component | Description |
|---|---|
| Variable List ↗ | The variable list names every quantity, metric, input, output, denominator, and intermediate value. This prevents hidden variables from entering through formulas, dashboards, or prose claims. If a quantity is not named, it cannot be checked. |
| Unit / Dimension Map ↗ | The unit/dimension map assigns each quantity a unit, dimension, denominator, time basis, scale basis, and reference frame. It distinguishes values that look similar but are not commensurable, such as dollars versus dollars per month or counts versus rates. |
| Quantity Semantics ↗ | Quantity semantics define what each measured quantity means in context. Two values can share units while representing different constructs, such as nominal dollars, present-value dollars, replacement-cost dollars, or annual recurring dollars. |
| Operation Compatibility Rule ↗ | The operation compatibility rule states which operations are allowed for the mapped quantities. Addition, comparison, division, aggregation, and transfer each require different compatibility conditions. |
| Scale Basis ↗ | The scale basis records whether a quantity is a raw total, per-unit rate, per-capita rate, percentage, indexed value, normalized score, log-scale value, or time rate. Many bad comparisons arise because this basis is implicit. |
| Consistency Test ↗ | The consistency test verifies whether the proposed equation, comparison, metric, policy rule, or model transfer preserves compatible dimensional meaning. It can be a formal dimensional analysis or a practical review checklist. |
| Dimensionless Group ↗ | A dimensionless group is a ratio, index, or coefficient whose units cancel or whose scale basis is intentionally defined. It can make cross-scale comparison possible, but only if the numerator, denominator, and interpretation are meaningful. |
| Correction Rule ↗ | The correction rule says what happens when a mismatch appears. The answer may be conversion, normalization, separation, redesign, rejection, or escalation. Without a correction rule, dimensional review becomes advisory rather than protective. |
| Transfer Boundary ↗ | The transfer boundary states where the checked expression remains valid. A formula or metric that is valid in one unit system, population, time window, or scale may need rechecking when moved elsewhere. |
Common Mechanisms¶
| Mechanism | Description |
|---|---|
| Unit Check ↗ | The first-line check that every input, output, and intermediate expression carries a compatible unit label before a calculation is trusted. |
| Dimensional Analysis Table ↗ | A worksheet that lists every quantity with its units and base dimensions and tracks how they cancel through a formula, so no scale-dependent term is ever combined blindly. |
| Unit Conversion Workflow ↗ | Converts every quantity into one shared unit convention before combining them, keeping the conversion factors and assumptions attached to the result. |
| Normalized Metric Design ↗ | Designs a metric on a comparable basis — indexed, standardized, or denominator-adjusted — so entities of different size or context can be set side by side honestly. |
| Per-Capita or Per-Unit Conversion ↗ | Divides a total by a clearly chosen denominator — people, units, transactions — turning raw counts into rates so differently sized things can be compared. |
| Dimensionless Ratio Construction ↗ | Combines quantities into a ratio whose units cancel — a pure number that carries meaning across scales, but only when its parts are chosen to mean something. |
| Stock / Flow Separation Check ↗ | Separates accumulated stocks from the flows that fill or drain them — balances from rates, prevalence from incidence — so a level is never compared directly with a speed. |
| Spreadsheet Unit Audit ↗ | Walks an actual spreadsheet cell by cell — columns, hidden intermediate cells, and formula chains — to surface the unlabeled unit and denominator slips that spreadsheets breed. |
| Model Input / Output Contract ↗ | A written specification pinning the unit, dimension, scale basis, and valid range of every model input and output, so an upstream change cannot silently corrupt a downstream result. |
| Cross-Scale Transfer Review ↗ | Rechecks whether a metric, rule, or equation that held at one scale still holds after it moves to another — and draws the boundary where its validity ends. |
| Finance / Accounting Unit Check ↗ | Separates money-like quantities that share a currency symbol but are not interchangeable — nominal vs. real, one-time vs. recurring, cash vs. accrual — before they are added or compared. |
| Engineering Equation Sanity Check ↗ | Checks a formula and its computed result for both dimensional coherence and physically plausible magnitude before the number is trusted. |
Parameter / Tuning Dimensions¶
Strictness can range from advisory warnings to hard blocks. High-stakes safety, finance, or automated decisions usually need hard rejection of incompatible expressions. Exploratory work may tolerate advisory flags.
Granularity can range from unit labels only to full quantity semantics. A simple conversion may be enough for routine unit-system mismatches; cross-scale policy comparisons may require denominator, time-window, construct, and transfer-boundary review.
Normalization basis is a central tuning choice. Raw totals, rates, indexed scores, and dimensionless ratios can all be valid, but they answer different questions. The archetype should expose the choice rather than hiding it inside a single score.
Automation level also matters. Automated unit contracts are useful for routine pipelines, but semantic quantity alignment often requires human judgment.
Invariants to Preserve¶
The most important invariant is quantity meaning. The same symbol or metric name should not change meaning during a calculation or transfer. Units and dimensions must remain compatible for the operation being performed. Denominators and time bases must remain visible. Conversions must be traceable. Transfer boundaries must be explicit enough that future users know when the expression must be checked again.
Target Outcomes¶
The target outcome is not mathematical elegance for its own sake. The goal is to prevent invalid formal structures from driving decisions. A successful dimensional consistency check produces interpretable metrics, auditable calculations, safer model transfers, clearer benchmarks, and fewer decisions based on incompatible quantities.
Tradeoffs¶
The main tradeoff is speed versus validity. Checking dimensions slows the production of numbers, but the delay is often smaller than the cost of acting on invalid numbers. Another tradeoff is simplicity versus semantic fidelity: a single normalized score can make communication easier while hiding denominator choices or absolute magnitude.
There is also a tradeoff between standardization and local meaning. Common unit conventions help systems interoperate, but they can obscure context-specific quantities. Finally, automation improves coverage but can miss cases where two quantities share a unit label but represent different constructs.
Failure Modes¶
Unit-label tunnel vision occurs when reviewers verify unit labels but ignore denominator, time basis, or construct meaning. False equivalence through normalization occurs when a normalized metric makes unlike cases look comparable. Dimensionless nonsense occurs when a ratio is accepted because units cancel even though the ratio has no useful interpretation.
Stock-flow confusion occurs when accumulated quantities are compared with rates. Transfer overreach occurs when a locally valid expression is reused universally. Bureaucratic checklist decay occurs when dimensional review is completed but has no power to change the decision. Hidden conversion assumptions occur when currency bases, fiscal periods, or rounding rules are applied invisibly.
Neighbor Distinctions¶
Dimensional Consistency Check is distinct from Parameter Rescaling. Parameter Rescaling changes a model’s parameterization or numeric range; Dimensional Consistency Check verifies compatibility before trusting a calculation or comparison.
It is distinct from Conservation Accounting. Conservation Accounting tracks a conserved quantity across transformations. Dimensional Consistency Check asks whether quantities may be combined or compared at all.
It is distinct from Stationarity Validation. Stationarity Validation asks whether statistical properties remain stable over time or context. Dimensional Consistency Check asks whether units, dimensions, denominators, and scale meanings are compatible.
It is distinct from False Precision Avoidance. False Precision Avoidance prevents overstating certainty or resolution. Dimensional Consistency Check prevents structurally incompatible quantities from being made precise in the first place.
It is distinct from Representation Fit Selection. Representation Fit Selection chooses the representational form. Dimensional Consistency Check validates the quantity structure inside or across representations.
Cross-Domain Examples¶
In engineering, a design calculation is checked to ensure its output has the intended dimension. In finance, one-time capital cost is separated from annual operating cost and discounted lifecycle cost. In public health, incidence, prevalence, raw case counts, and test positivity are not treated as interchangeable. In operations, backlog size, arrival rate, service time, and throughput are separated before staffing decisions. In product analytics, active users, retention, total revenue, and revenue per user are labeled with their distinct bases before a growth narrative is presented.
Non-Examples¶
Adding more decimal places to a number is not dimensional consistency. Choosing a more attractive chart is not dimensional consistency unless the underlying quantities are incompatible. Fixing missing records is data integrity work. Rejecting a KPI because it incentivizes the wrong behavior is objective alignment or incentive design unless the KPI also combines incompatible quantities.
Related Abstractions¶
Abstractions this archetype builds on — directly (a source ingredient) or as a related pattern. Links follow the typed catalog namespace.
Built directly on (3)
- Dimensional Analysis: Ensures consistency in units.
- Representation: Model complex ideas.
- Scale: Properties change with size.
Also references 6 related abstractions
- Correspondence Principle: New theories match old limits.
- Engineering Tolerances: Acceptable variation.
- Equivalence Principle: Gravity indistinguishable from acceleration.
- Gauge Invariance
- Scale Invariance: Behavior unchanged under scaling.
- Uncertainty: Incomplete knowledge.
Variants¶
Narrower or domain-specific specializations that share this archetype's core structure. Recognized variants are established; candidate variants are provisional.
Unit Consistency Check · mechanism family variant · recognized
A narrower variant focused on compatible unit labels and conversions across inputs, formulas, and outputs.
- Distinct from parent: The parent includes units, dimensions, denominators, scale basis, and transfer boundaries. This variant concentrates on the unit-compatibility layer.
- Use when: {'condition': 'A calculation or workflow combines values from multiple sources or unit systems.', 'why_it_matters': 'Unit conversion errors can produce formally precise but invalid outputs.'}; {'condition': 'Inputs are entered manually or imported from systems that use different conventions.', 'why_it_matters': 'Implicit unit assumptions can change the magnitude of a result without changing the formula structure.'}.
- Typical domains: engineering, data pipelines, finance, operations
- Common mechanisms: Unit Check, Unit Conversion Workflow
Scale Normalization Check · scale variant · recognized
A variant that verifies totals, rates, percentages, indexed values, and normalized metrics are being compared on a compatible basis.
- Distinct from parent: The parent covers dimensional compatibility generally; this variant focuses on the scale basis selected for comparison.
- Use when: {'condition': 'Entities of different sizes are compared using raw totals.', 'why_it_matters': 'Raw totals can hide per-unit intensity or per-person burden.'}; {'condition': 'Normalized scores or indices replace original quantities.', 'why_it_matters': 'Normalization can clarify comparison but may hide absolute magnitude or denominator choice.'}.
- Typical domains: policy, benchmarking, public health, operations
- Common mechanisms: Normalized Metric Design, Per-Capita or Per-Unit Conversion
Dimensionless Ratio Design · mechanism family variant · recognized
A variant focused on constructing ratios, coefficients, or indices that remove units or expose scale-invariant structure.
- Distinct from parent: The parent can reject, convert, normalize, or redesign quantities. This variant emphasizes ratio construction as the chosen correction.
- Use when: {'condition': 'A comparison must transfer across sizes, units, contexts, or scales.', 'why_it_matters': 'Dimensionless ratios can reveal comparable structure where raw units cannot.'}; {'condition': 'A model needs a compact parameter that combines multiple quantities.', 'why_it_matters': 'Ratio design can compress dimensional structure, but only if the numerator and denominator are meaningful.'}.
- Typical domains: engineering, finance, public health, benchmarking
- Common mechanisms: Dimensionless Ratio Construction, Cross-Scale Transfer Review
Semantic Quantity Alignment · other · candidate
A candidate variant that checks whether quantities with compatible units actually represent the same construct or decision-relevant meaning.
- Distinct from parent: The parent includes semantic checks as one component. This variant would foreground construct meaning and correspondence.
- Use when: {'condition': 'Values share units but come from different definitions, accounting bases, populations, or measurement methods.', 'why_it_matters': 'Unit compatibility alone may create false equivalence.'}; {'condition': 'A metric is reused in a new domain because its label looks familiar.', 'why_it_matters': 'Same label does not guarantee same measurement semantics.'}.
- Typical domains: finance, survey research, clinical metrics, product analytics
- Common mechanisms: Model Input / Output Contract, Finance / Accounting Unit Check
Cross-Scale Transfer Check · scale variant · recognized
A variant that verifies whether an equation, rule, metric, or comparison remains valid when moved across scale, population, scope, or context.
- Distinct from parent: The parent checks consistency generally; this variant makes transfer across scale the central hazard.
- Use when: {'condition': 'A pilot metric, model, or calculation is being generalized to a larger setting.', 'why_it_matters': 'Dimensional validity and denominator meaning may change with scale.'}; {'condition': 'An equation or benchmark from one domain is imported into another.', 'why_it_matters': 'Transfer can fail even when the algebraic form is unchanged.'}.
- Typical domains: pilot to rollout, public policy, engineering, organizational metrics
- Common mechanisms: Cross-Scale Transfer Review, Normalized Metric Design
Stock / Flow Separation · subtype · recognized
A variant that prevents accumulated quantities from being confused with rates or flows over time.
- Distinct from parent: The parent covers many dimensions; this variant focuses on temporal dimensionality.
- Use when: {'condition': 'A metric mixes totals, balances, incidence rates, throughput, or time-indexed costs.', 'why_it_matters': 'Stocks and flows answer different questions and cannot be substituted without transformation.'}; {'condition': 'Dashboards or policies compare a one-time amount with a recurring rate.', 'why_it_matters': 'The comparison can drive wrong investment, staffing, or risk decisions.'}.
- Typical domains: finance, operations, public health, system dynamics
- Common mechanisms: Stock / Flow Separation Check, Finance / Accounting Unit Check
Near names: Dimensional Analysis Check, Unit Sanity Check, Apples-to-Apples Comparison, Unit Check, Dimensional Analysis Table, Per-Capita Conversion, Normalized Metric, Scale-Invariant Ratio.