Dimensional Consistency Check¶
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 ↗ | A unit check verifies compatible unit labels across inputs, outputs, and intermediate calculations. It is a mechanism under the archetype, not the archetype itself, because dimensional consistency also includes denominators, quantity meaning, scale basis, and transfer validity. |
| Dimensional Analysis Table ↗ | A dimensional analysis table records variables, dimensions, units, conversions, and cancellation logic. It is useful when equations, model formulas, or repeated calculations need systematic review. |
| Unit Conversion Workflow ↗ | A unit conversion workflow converts quantities into a shared convention before they are combined. It must preserve conversion factors and assumptions; conversion alone does not prove that the quantities mean the same thing. |
| Normalized Metric Design ↗ | Normalized metric design creates per-unit, per-capita, indexed, or standardized metrics so comparisons are made on a compatible basis. It implements the archetype when the denominator and interpretation are explicit. |
| Per-Capita or Per-Unit Conversion ↗ | This mechanism converts totals into rates against a clear denominator. It is useful for benchmarking differently sized entities, but it can mislead if absolute burden or capacity still matters. |
| Dimensionless Ratio Construction ↗ | Dimensionless ratio construction builds ratios or coefficients whose dimensions cancel or whose reference basis is explicit. It supports transfer across scale only when the ratio has a real interpretation. |
| Stock / Flow Separation Check ↗ | A stock/flow check distinguishes accumulated quantities from rates over time. It prevents comparisons such as one-time cost versus monthly cost or prevalence versus incidence from being treated as direct equivalents. |
| Spreadsheet Unit Audit ↗ | A spreadsheet unit audit reviews columns, formulas, labels, denominators, and hidden conversions. It is valuable because spreadsheet errors often arise from unlabeled intermediate values. |
| Model Input / Output Contract ↗ | A model input/output contract defines units, dimensions, scale bases, expected ranges, and meanings for model fields. It helps prevent upstream data changes from silently invalidating downstream models. |
| Cross-Scale Transfer Review ↗ | A cross-scale transfer review checks whether a metric, rule, or equation remains valid after moving from pilot to rollout, local to global, lab to field, or one domain to another. |
| Finance / Accounting Unit Check ↗ | This mechanism separates currencies, time periods, accounting bases, nominal versus real values, and one-time versus recurring amounts. It catches money-like quantities that are not actually interchangeable. |
| Engineering Equation Sanity Check ↗ | This familiar mechanism verifies that formulas, constants, coefficients, and calculated outputs are dimensionally coherent and plausible. It anchors the archetype but does not exhaust it. |
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.
Variants and Near Names¶
Unit Consistency Check is the unit-focused variant. Scale Normalization Check focuses on raw totals, rates, percentages, indices, and denominator-adjusted comparisons. Dimensionless Ratio Design focuses on ratios or coefficients whose dimensions cancel or whose basis is explicitly defined. Semantic Quantity Alignment checks cases where unit labels match but meanings differ. Cross-Scale Transfer Check focuses on whether expressions remain valid after moving across scale or context. Stock / Flow Separation focuses on accumulated quantities versus rates over time.
Near names include Dimensional Analysis Check, Dimensional Sanity Check, Unit Sanity Check, Scale Consistency Review, and apples-to-apples comparison. Unit check, dimensional analysis table, per-capita conversion, and normalized metric should remain mechanisms or variants unless future reconciliation shows distinct standalone structure.
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.