Reporting-Pyramid Undercount

one_liner: A true event population passes through ordered, non-random reporting filters whose layerwise capture rates multiply, so the recorded count is a biased fraction and changes in any filter can masquerade as changes in incidence.

core_idea:
Reporting-Pyramid Undercount is the structural pattern in which a true population of countable events must pass through several ordered detection, recognition, classification, escalation, and recording stages before it appears in an official dataset. Each stage admits only a fraction of the events that reached it. The final capture fraction is therefore the product of the layerwise fractions, and the recorded count can be a small remainder even when no single layer looks catastrophically lossy.

The loss is normally non-random. Severe, visible, well-documented, easy-to-classify, low-cost, or institutionally safe events cross more gates than mild, ambiguous, stigmatized, inaccessible, or costly ones. The recorded set is consequently not a uniformly shrunken image of the true set. Its composition is systematically displaced toward the kinds of events that survive the pipeline.

The prime has a second load-bearing commitment: the gates can move. Awareness campaigns, new reporting tools, changes in definitions, expanded testing, exhausted staff, political pressure, and altered incentives all change layerwise capture. A rise in the recorded count can therefore be caused by more true events, by a more permeable reporting pipeline, or by both. A fall can indicate improvement or reporting decay. The observed series is jointly generated by the event process and the observation process.

structural_signature:
**Sig role-phrases:**

- **the true event population** — the events that meet the target definition whether or not any reporting system sees them
- **the ordered inclusion gates** — sequential stages an event must cross before entering the dataset
- **the layerwise capture fractions** — the conditional probabilities of surviving each gate
- **the multiplicative attenuation** — the product of the layer fractions that determines aggregate capture
- **the non-random loss** — systematic variation in capture by event severity, visibility, population, place, access, or incentive
- **the recorded remainder** — the biased subset that appears in the official count
- **the moving-filter confound** — changes in capture that imitate changes in the underlying incidence
- **the external calibration apparatus** — audits, population surveys, linked records, capture-recapture studies, or sentinel systems used to estimate what the routine pipeline misses

The strict signature requires both an ordered pipeline and differential inclusion. A single independent sampling gate is ordinary Selection Bias. A sequence of purely random gates produces undercount but not the systematic compositional distortion at the center of this prime.

what_it_is_not:
It is not a generic claim that every official number is wrong. A reporting system can have high or stable capture, and its count can be fit for a bounded purpose. The prime identifies a mechanism that must be demonstrated or plausibly specified layer by layer.

It is not reporting delay. Delayed events are expected to enter the dataset later and can often be handled by nowcasting or censoring corrections. Reporting-pyramid losses are events that never cross one or more gates and therefore never arrive.

It is not simple missing data. A dataset can have blank fields without losing events, and events can be absent without passing through a sequential reporting chain. The pyramid adds ordered conditional gates and a product structure.

It is not merely measurement error. Measurements may be noisy or miscalibrated for every recorded event while the event count remains complete. Reporting-Pyramid Undercount changes which events become visible and usually changes their composition.

It is not capture-recapture. Capture-recapture is one estimator that can help recover a hidden population from overlapping imperfect lists. It is an instrument for some undercount problems, not the filtering pattern itself.

broad_use:
**Epidemiology and public health.** A true infection may have to produce symptoms, cause care-seeking, prompt testing, satisfy a case definition, and reach a surveillance registry. The familiar surveillance pyramid is one concrete domain instance.

**Pharmacovigilance.** An adverse drug reaction must be experienced, recognized as potentially drug-related, communicated, reported through an approved channel, accepted as a valid case, and entered into a database. Severe and unusual reactions have different capture probabilities from common or ambiguous ones.

**Occupational safety.** An injury may need worker disclosure, supervisor recognition, recordability classification, and submission to a regulator. Incentives around safety performance, compensation, employment security, and administrative burden can change several gates.

**Crime, abuse, and discrimination statistics.** An experienced event may need to be recognized, disclosed, accepted by an authority, classified under a legal category, and entered into a record. Social risk and institutional trust make the missing population compositionally unlike the observed one.

**Cybersecurity and operations.** A failure or intrusion must be instrumented, detected, triaged, escalated, declared, and logged. Changes in telemetry or incident policy can create apparent increases without any change in the underlying event rate.

**Public administration.** A person entitled to a benefit, remedy, or service may need to learn of eligibility, reach an office, complete documentation, pass an administrative definition, and remain in the process until entry. Administrative counts then measure successful traversal as much as underlying need.

clarity:
The prime separates four quantities that a raw count silently fuses:

1. the number of true events;
2. the fraction crossing each gate;
3. the kinds of events preferentially lost;
4. changes in the gates themselves.

This is more precise than saying “the data undercount reality.” It asks where the loss occurs, whether the losses compound, which attributes predict survival, and whether the observation system changed during the period being compared. Once those questions are explicit, a count is no longer mistaken for a transparent window onto incidence.

The distinction between *level* and *composition* is especially important. Multiplying a recorded count by an estimated inverse capture rate may recover a plausible total while leaving the distribution of severity, geography, demographics, or type badly distorted. Scaling a biased sample does not unbias its composition.

manages_complexity:
A reporting ecosystem can contain dozens of institutions, forms, incentives, thresholds, and handoffs. The pyramid compresses that machinery into a short ordered chain of conditional gates. For \(k\) stages, the analyst estimates or bounds \(k\) capture fractions rather than inventing an undifferentiated “dark figure.”

The decomposition also gives a routing rule for intervention. If most loss occurs before recognition, a better submission portal will not recover it. If reports are made but rejected during classification, awareness campaigns target the wrong layer. The prime directs effort toward the binding coefficient while predicting which apparent count changes the intervention itself will create.

abstract_reasoning:
Let \(N_t\) be the true number of target events during period \(t\). Let \(p_{it}(x)\) be the conditional probability that an event with attributes \(x\), having survived gates \(1,\ldots,i-1\), crosses gate \(i\). The expected recorded count is

\[
R_t=\sum_{e \in N_t}\prod_{i=1}^{k} p_{it}(x_e).
\]

If capture is treated as homogeneous, this reduces to \(E[R_t]=N_t\prod_i p_{it}\). The reciprocal \(1/\prod_i p_{it}\) is an aggregate multiplier, but it is valid for recovering the total only under assumptions about the population and the estimation of the component fractions.

Two consequences follow. First, moderate losses compound: five gates each capturing 70 percent leave only about 17 percent of true events in the final count. Second, \(R_t\) is not identified with \(N_t\) when the \(p_{it}\) change. The same recorded rise can result from increased incidence or increased permeability at any gate. Auxiliary measures are required to separate the two.

The model also exposes irreversibility. A downstream reform cannot recover events already lost upstream unless it creates a new route around the earlier gate. Better classification cannot count events never detected; mandatory submission cannot count events never recognized.

knowledge_transfer:
The abstraction transfers exactly across systems with an in-principle countable event class and a sequential inclusion path. A drug reaction, workplace injury, assault, software incident, and infection have different substantive content, yet each can be mapped to the same roles: true events, recognition, disclosure, institutional acceptance, classification, and entry.

Transfer fails when the “pyramid” is only a picture of decreasing quantities with no conditional traversal by the same event units. A food chain, organizational hierarchy, or funnel of unrelated populations may have a pyramid shape without instantiating Reporting-Pyramid Undercount. What travels is the event-level passage through non-random gates, not the geometry.

formal_abstract:
Suppose 10,000 events occur. Eighty percent are detectable, half of those are recognized, 60 percent are submitted, and 75 percent of submissions are accepted and entered. The final count is \(10{,}000 \times .8 \times .5 \times .6 \times .75 = 1{,}800\). No layer alone loses more than half, but 82 percent of the events disappear through compounding.

If a new portal raises submission from .6 to .9 while incidence stays fixed, the recorded count rises to 2,700—a 50 percent apparent increase caused entirely by one observation-layer change.

**Mapped back:** the 10,000 are **the true event population**; the four conditional rates are **the layerwise capture fractions**; 1,800 is **the recorded remainder**; and the portal creates **the moving-filter confound**.

title: Incidence versus ascertainment

body: A reported trend is the product of events and capture. Increasing detection can look like worsening incidence; collapsing reporting can look like improvement. **Diagnostic:** Which independent indicators can distinguish a change in true events from a change in one or more gates?

title: Total correction versus compositional correction

body: An aggregate multiplier may recover a plausible total while preserving severity, demographic, geographic, or access bias. **Diagnostic:** Is the inference only about how many events occurred, or does it also claim who or what kinds were affected?

title: Parsimony versus localization

body: A single multiplier is easy to communicate, but an intervention needs layer-specific coefficients. **Diagnostic:** Does the proposed repair act on the layer that accounts for the loss, or merely on the most visible downstream stage?

title: Stable comparison versus changing infrastructure

body: Better reporting is desirable, yet it breaks comparability with the older series. **Diagnostic:** Was the apparent change contemporaneous with a new definition, tool, incentive, capacity, or disclosure environment?

title: Upstream loss versus downstream control

body: The largest losses may occur before the institution has any contact with the event. **Diagnostic:** Can a downstream mandate reach events lost before detection or disclosure, or is a new upstream sampling route needed?

title: Estimation versus governance

body: Measuring the hidden population may require linking sensitive records or intensifying surveillance, creating privacy, trust, and incentive costs that themselves alter reporting. **Diagnostic:** Will the calibration apparatus change the gates it is intended to estimate?

prose:
- **Map the gates:** name the sequential stages and the conditional denominator at each one.
- **Triangulate capture:** combine routine records with audits, population surveys, sentinel sites, linked datasets, or capture-recapture where assumptions permit.
- **Estimate by stratum:** allow capture to vary by severity, group, place, access, and time rather than imposing one universal multiplier.
- **Target the binding layer:** match outreach, tooling, capacity, classification reform, or submission policy to the gate actually causing loss.
- **Bridge regime changes:** overlap old and new reporting systems long enough to estimate a conversion between series.
- **Separate event and observation models:** report uncertainty about incidence and ascertainment independently.
- **Preserve trust:** design calibration and reporting channels so that efforts to observe the hidden population do not further suppress disclosure.

notes:
This entry was drafted during mixed-DAG puzzle pass Recursion 100 because the domain-specific Outbreak Underascertainment entry repeatedly named the cross-domain reporting-pyramid mechanism as its missing parent. It is intentionally compact and requires Claude style re-authoring, density expansion, FACT anchors, and independent source verification before publication.

The node should remain below Selection Bias rather than receive parallel flattened heads to Missing Data, Pipeline, or Measurement Error. Those concepts are neighbors or possible mechanisms. The exact genus is systematic selection into the recorded set; the ordered product of reporting gates is the differentiator.


Representation

type: prime

slug: representation

one_liner: Model complex ideas.

status: draft

origin_domain: mathematics

core_idea:
**Representation is the structured mapping of one system of entities-and-relations (the target) onto a second system of entities-and-relations (the medium) such that selected features of the target correspond to features of the medium under a stated convention, making the represented system available for manipulation, reasoning, communication, or storage via the representing substrate.** The essential commitment is to a *faithfulness claim*: the representation preserves some explicitly stated structure of the target so that operations on the medium correspond — exactly, approximately, or heuristically — to operations on the target, while other features are deliberately dropped, distorted, or left implicit.

Every representation specifies (1) the *target*: what is being represented (a physical system, a dataset, an abstract structure, an idea, an experience); (2) the *medium*: the second system that does the representing (symbols on paper, pixels on a screen, numbers in memory, words in a language, a neural network's hidden activations, a gesture in a dance); (3) the *mapping*: which target entities correspond to which medium entities and how target relations map to medium relations; and (4) the *faithfulness specification*: which features the representation preserves and which it systematically drops, distorts, or leaves under-determined. **A representation claim is operationally complete only when the faithfulness specification is explicit; without it, users will draw inferences the representation never licensed and miss inferences the representation actually supports.**

structural_signature:
A system is a representation of another when each of the following holds:

1. *Target system:* a system of entities and relations identifiable independently of the representation — the world, a dataset, a physical process, a mental state, a conversation, an abstract mathematical structure.
2. *Representing medium:* a second system of entities and relations distinct from the target — symbols on paper, pixels on a screen, numbers in memory, words in a natural language, gestures in a dance, an embedding vector inside a neural network.
3. *Structure-preserving mapping:* specified elements and relations of the target correspond to specified elements and relations of the medium; the mapping may be total or partial, exact or approximate, injective or many-to-one.
4. *Faithfulness specification:* an explicit or implicit commitment to preserve certain features (topology, quantity, order, causal structure, semantic role, group-theoretic action) and to disclaim others (absolute scale, geographic accuracy, sub-resolution detail, off-axis sensitivities).
5. *Operational use:* operations on the medium correspond to operations on the target — rotating a map head-up rotates the encoded spatial relations; multiplying matrices composes the represented linear transformations; rearranging algebraic symbols manipulates the represented mathematical relations[^frege-1892].
6. *Interpretation convention:* a shared rule or practice by which users of the representation recover the target features from the medium features — a legend, a decoding rule, a reading practice, a usage norm in a community.

what_it_is_not:
- **Not the target itself.** The map is not the territory. Treating the representation as identical to the target discards its constructed, feature-selective nature and obscures its failure modes — this is the fundamental error that semiotics, philosophy of science, and information theory all warn against.
- **Not duplication.** A representation preserves *selected* structure, not all structure. A complete duplicate would be another instance of the target, not a representation of it; the representational relation requires that some features be dropped or transformed.
- **Not [`abstraction`](../prime_abstractions/v2/abstraction.md) alone.** Abstraction is the selective dropping of detail to retain purpose-relevant structure; representation is the *mapping* of the abstracted target structure into a second medium. A representation is typically abstraction plus encoding into a new substrate; the two acts are distinguishable even when they happen together.
- **Not communication alone.** Communication transmits a representation; representation is the structured artefact being transmitted. Many representations exist without being communicated (working memory, internal cognitive maps, transient intermediate computations inside a learned model).
- **Not [`duality`](../prime_abstractions/v2/duality.md).** Duality pairs two structures so each fully determines the other; representation is a *directional* mapping from a target onto a medium with an explicit faithfulness loss. Dualities can license representations (linear-algebraic representation of group actions exploits the duality between operators and their adjoints) but are not identical with them.
- **Not [`approximation`](../prime_abstractions/v2/approximation.md).** Approximation substitutes a tractable surrogate for the target with a bounded-error claim; representation maps the target into a different substrate with a faithfulness claim. The two often co-occur — a representation can be approximate, an approximation can be encoded as a representation — but the structural commitments are distinct (bounded-error vs structure-preservation).
- **Common misclassifications:** using a representation as if it perfectly preserved features it never claimed to preserve (reading causal claims off a correlational graph, inferring topology from a connectivity-only schematic, reading precision into a diagram drawn for communication); confusing the representation with the represented (treating a model's parameters as features of the world rather than features of the model); mistaking artefacts of the encoding for features of the target.

broad_use: Representation is the structural commitment that organises a remarkable range of formal, cognitive, computational, and artistic disciplines — and across each of them the four-component structure (target, medium, mapping, faithfulness) is operative even when the local vocabulary is different. *Mathematics* uses representation for coordinate representations of geometric objects, matrix representations of linear transformations and group actions, character tables of finite groups, combinatorial encodings of algebraic structures, and the entire field of *representation theory* (Frobenius 1890s, Schur, Weyl, Brauer) which studies how abstract groups act on linear vector spaces. *Computer science and software engineering* uses representation for data structures, schemas, encodings (JSON, XML, Protocol Buffers), serialisation formats, knowledge-representation languages (OWL, RDF), and the learned representations inside neural networks where the *representation-learning* programme[^bengio-2013] explicitly studies how models construct their own intermediate encodings. *Cognitive science and philosophy of mind* uses representation for mental representations, the symbol-grounding problem, the propositional-vs-depictive debate, the representational theory of mind, and the three-level analysis (computational / algorithmic / implementational) introduced by David Marr's *Vision*[^marr-1982]. *Linguistics and semiotics* uses representation for the sign / signifier / signified triad of Saussure[^saussure-1916] and the icon / index / symbol typology of Peirce; modern formal semantics inherits the Frege programme on sense vs reference[^frege-1892]. *Maps, diagrams, and visualisations* uses representation for cartographic projections, engineering drawings, statistical graphics, flow diagrams, network visualisations, and the entire data-visualisation literature that studies how chart designs preserve and distort data structure. *Engineering and measurement* uses representation for schematics, CAD models, digital sampling of analogue signals, sensor encodings, and the discrete-vs-continuous representation question that recurs in signal processing. *Art and literature* uses representation for the depictive vs expressive vs symbolic distinctions that organise aesthetic theory, with Goodman's *Languages of Art* (1968) the canonical analytic treatment. The cross-domain pervasiveness reflects that representation is not a local technical concept but a near-universal structural relation between any system that needs to be reasoned about and the medium in which the reasoning takes place.

clarity: Representation clarifies by making a representation's *faithfulness claim* explicit: what structure is preserved, what is dropped, and what operations in the medium correspond to operations on the target. A loose claim like "this chart shows sales declining" resolves into a checkable proposition: this chart preserves year-over-year ordering and relative magnitudes within the chosen time-bin, with absolute values shown to two-significant-figure precision; it does not preserve sub-bin seasonal fluctuations, does not encode confidence intervals on the displayed point estimates, and does not distinguish revenue from order count. The clarifying force is to surface the design choices every representation embodies and the inferential licenses it does and does not grant. Naming the act of representation — rather than treating it as transparent — also forces the question *whose representation is this?*, since every representation reflects the purposes and constraints of its designers.

manages_complexity:
- *Substrate substitution:* replaces direct manipulation of the target with manipulation of a more tractable medium — a mathematical equation can be manipulated in ways the underlying physical system cannot, and a database query can be issued in ways the underlying records cannot.
- *Granularity selection:* a representation can be designed to make the relevant structure salient and the irrelevant invisible, allowing reasoning at the granularity that fits the task without forcing the reasoner to navigate the target's full complexity.
- *Composition:* representations compose — overlays on a map, joins between database tables, embeddings concatenated into a single feature vector — to produce derived representations that reveal relationships across sub-targets that no single representation surfaces alone.
- *Shared inference:* once a representation is agreed upon, multiple actors can reason about the same target without each re-encoding it; standards (the metric system, ISO date format, RFC-defined network protocols) are the institutionalised form of shared representational conventions.
- *Design-space exposure:* choosing a representation is often the central design act; alternative representations of the same target make different reasoning easy and other reasoning hard (row-store vs column-store databases for analytic vs transactional workloads, declarative vs imperative programming for what-vs-how reasoning, list vs tree vs graph for sequential-vs-hierarchical-vs-relational data).

abstract_reasoning:
Representation trains a reasoner to ask:

- What is the target, what is the medium, and what is the mapping between them? Is the mapping total or partial, exact or approximate, deterministic or stochastic?
- Which features of the target are preserved in the medium, and which are systematically dropped or distorted? What was the faithfulness claim, stated or implicit?
- What operations in the medium correspond to what operations on the target? Where does the correspondence break down, and what is the cost of misreading the breakdown?
- Whose representation is this — what purpose shaped the selection of preserved features? What inferential agenda does the chosen medium quietly favour?
- Is there a competing representation of the same target that would make the relevant inference easier? What is the cost of switching, and what new failure modes does the alternative introduce?
- When the medium has its own structure (coordinate axes, encoding format, plotting defaults), are we treating *representational coincidences* (artefacts of the encoding) as if they were *features of the target*?

knowledge_transfer:
Role mappings across domains:

- **Mathematics →** target = abstract group / vector space / topological space; medium = matrix / linear operator / coordinate chart; mapping = homomorphism preserving structure (Cayley's theorem on group representation; matrix representation of linear maps).
- **Cognitive science →** target = perceived world / abstract concept; medium = mental representation (propositional, imagistic, distributed); mapping = encoding under perceptual and conceptual constraints (Marr's three levels[^marr-1982]).
- **Linguistics and semiotics →** target = referent (object, action, idea); medium = sign (signifier in Saussure[^saussure-1916]; symbol/icon/index in Peirce); mapping = the conventional relation that constitutes meaning.
- **Logic and formal semantics →** target = mathematical or empirical claim; medium = formal sentence; mapping = sense/reference (Frege[^frege-1892]); model-theoretic interpretation (Tarski).
- **Computer science (data structures) →** target = problem-domain entity; medium = struct, record, object, table row, document; mapping = schema specification.
- **Computer science (machine learning) →** target = task-relevant content of the input; medium = learned embedding vector; mapping = the network's compositional pipeline (representation learning programme[^bengio-2013]).
- **Maps and visualisation →** target = geographic or data-domain structure; medium = projected diagram; mapping = cartographic projection or chart-encoding choice with explicit distortion budget.
- **Engineering and CAD →** target = physical artefact or system; medium = CAD model / schematic / wireframe; mapping = the modelling convention (orthographic projection, exploded view, single-line diagram).
- **Music notation →** target = sounded musical structure; medium = staff notation / tablature / time-frequency representation; mapping = the notational convention (pitch on staff, duration on note shape, dynamics on markings).
- **Art and depiction →** target = depicted scene or concept; medium = painted canvas / sculpture / photograph; mapping = depictive convention (perspective, colour, framing) within an aesthetic tradition.

A cartographer choosing a projection for a world map, a software engineer designing a database schema, a physicist formalising a phenomenon as an equation, a cognitive scientist proposing a mental-representation theory, and a deep-learning researcher training a representation-learning model are all doing the same structural work: select what to preserve, choose a medium that naturally represents those features, name the mapping, and make the interpretation convention explicit. The same diagnostic — *what structure is preserved, what is dropped, and what inferences are licensed by the correspondence?* — applies across their otherwise different substrates, with the same representational failure modes (mistaking artefacts for features, drifting interpretation, faithfulness over- or under-claim).

The transfer to machine-learning settings is particularly forceful: a learned embedding is a representation in exactly the structural sense — a target (the input's task-relevant content), a medium (a vector in `ℝ^d`), a mapping (the network's learned compositional pipeline), and a faithfulness claim (downstream-task accuracy as a proxy for structure preservation). The representation-learning literature[^bengio-2013] is, in essence, the discipline of designing learned mappings whose faithfulness claims are operationally useful, and the same questions semioticians have asked of pictures since Saussure (what is preserved? what is conventional? what is the interpretation community?) translate directly into questions ML researchers ask of embeddings (probe-classifier accuracy on linguistic features, geometry of the embedding space, transfer performance across tasks).

prose:
_(Pass B — pending; see `tracker/v2_upgrade.md`.)_

### Catalog cross-references (v1, batches 02–29)

Solution archetypes that reference this prime (per `solution_archetypes/_catalog/catalog_v1_batches_02_29.yaml`):

**As source prime (11):** `canonical_classification`, `contrastive_differentiation`, `dimensional_consistency_check`, `emergent_formalization`, `essential_structure_extraction`, `explicit_state_modeling`, `functional_specification`, `index_based_retrieval`, `representation_fit_selection`, `system_scope_definition`, `task_relevant_compression`

**As related prime (28):** `boundary_permeability_control`, `boundary_reframing`, `bounded_approximation`, `coarse_graining`, `completeness_audit`, `compositional_assembly`, `core_model_first`, `discrete_continuous_model_selection`, `equivalence_class_consolidation`, `focal_emphasis_design`, `hidden_path_discovery`, `hierarchical_decomposition`, `interoperability_standardization`, `layered_model_validation`, `layered_record_accumulation`, `metasystem_integration`, `objective_function_alignment`, `parameter_rescaling`, `phase_space_mapping`, `sandboxing`, `scale_appropriate_modeling`, `source_of_truth_assignment`, `strategic_juxtaposition`, `system_archetype_diagnosis`, `traceability_linking`, `uncertainty_explicitness`, `virtual_resource_abstraction`, `whole_system_alignment`

notes:
This prime carries the `multi_origin_equal` flag — representation as a structural concept emerges parallel-independently across mathematics (Cayley's group representation theorem 1854; Frobenius-Schur-Brauer representation theory of finite groups; matrix and coordinate representations across linear algebra), philosophy of language and semiotics (Frege 1892[^frege-1892] on sense vs reference; Saussure 1916[^saussure-1916] on the sign as signifier-signified pair; the Peircean icon/index/symbol typology), cognitive science (the representational theory of mind; Marr 1982[^marr-1982] on the three levels of analysis from computation through algorithm to implementation), computer science (data structures, knowledge representation languages, learned embeddings via the representation-learning programme[^bengio-2013]), and philosophy of art (Goodman's *Languages of Art* 1968 on symbol systems and exemplification). The four-component structural signature (target, medium, mapping, faithfulness) is operative in each of these traditions even when the local vocabulary and emphasis differ, which is why the multi-origin-equal flag is preferred over a single primary-origin attribution. Origin domain `mathematics` is listed as primary in frontmatter as a pragmatic choice (representation theory in pure math is the most formally developed instance with the longest paper trail), but cognitive_science, computer_science_software_engineering, and philosophy are each capable of carrying the conceptual weight on their own and should be read as co-equal in any deeper analytical use of this prime.

Tight cross-links: [`abstraction`](../prime_abstractions/v2/abstraction.md) — the definitional contrast in `What It Is Not` is structural rather than merely terminological; representation is typically abstraction *plus* encoding into a new substrate, so the two often appear together in any modelling work; [`duality`](../prime_abstractions/v2/duality.md) (DP-03) — duality structures often *license* representations (linear-algebraic representation of group actions exploits the operator-adjoint duality), and the cross-reference is reciprocal; [`approximation`](../prime_abstractions/v2/approximation.md) (DP-04 G1) — approximation and representation are related but structurally distinct, with approximation carrying a bounded-error claim and representation carrying a structure-preservation claim. Cross-batch tight relationships (carry-forward): when [`abstraction`](../prime_abstractions/v2/abstraction.md) is revised in its DP batch, its `What It Is Not` should reciprocate the cross-reference planted here.

Pass B Solution Archetypes (suggested starting set): *name-target-medium-mapping-faithfulness* (always articulate all four explicitly when introducing a representation); *audit-the-faithfulness-spec* (list what the representation preserves and what it drops, with stated tolerance); *artefact-vs-feature audit* (separate properties of the encoding from properties of the target); *substrate-drift checklist* (when migrating representations across substrates, audit for accumulated semantic change); *competing-representation evaluation* (consider at least one alternative representation and articulate the trade-off); *interpretation-convention documentation* (publish the legend, decoding rules, and design intent alongside the artefact rather than holding it in the designer's head).

Citation reuse and cross-batch ledger: this prime cites `frege-1892`, `saussure-1916`, `marr-1982`, `bengio-2013`, and `goodman-1968` (cited in Broad Use prose only; no `[^marker]` planted because the citation is contextual rather than load-bearing).

There is no `origin_predates_discipline` flag — representation as a named structural concept is roughly coeval with each of the disciplines that hosts it (the formal mathematical representation theory is 19th-century; the formal semiotics is late-19th to early-20th century; the cognitive-representation programme is mid-20th century; the computer-science representation programme is late-20th century). The multi-origin-equal flag captures the core observation about provenance: there is no single field from which the others borrowed.


Representational Modality

one_liner: Choice of medium fundamentally shapes what can be expressed.

core_idea: The choice of *medium* through which information is encoded and transmitted—visual, auditory, tactile, olfactory, or multimodal—fundamentally shapes what can be expressed, what is easily understood, and what actions become possible, as Larkin and Simon (1987) argued in their analysis of how informationally equivalent representations differ in computational efficiency. <!-- FACT-D53-406 -->[^larkin-simon-1987] Modality is not neutral; the same information encoded differently carries different cognitive load, retention, and behavioral consequences. The structure is: **modality choice → encoding/decoding properties → differential cognitive load and retention → behavioral outcomes**, a pipeline systematized by Mayer (2009) in his cognitive theory of multimedia learning. <!-- FACT-D53-407 -->[^mayer-2009]

structural_signature:
Representational modality encodes a pattern: **medium properties → information transformation → cognitive and behavioral differentiation**. Medium properties (salience, bandwidth, accessibility to different sensory systems, cultural conventions) determine what can be expressed economically and what is lost or distorted in translation, as Goodman (1968) developed in his analysis of how distinct symbol systems (notational, dense, articulate) afford different kinds of expression. <!-- FACT-D53-408 -->[^goodman-1968] The concept separates the *content* of a message from the *channel* carrying it, and names the irreducible effects of channel choice on understanding and action.

**Recurring features:**

- Medium properties shape encoding and decoding
- Cognitive load varies by modality independent of content
- Same information, different media, different outcomes
- Multimodal redundancy enhances robustness; complementarity adds dimension
- Accessibility demands show which modalities are assumed and which neglected
- Translation cost between modalities is real and measurable
- Modality choice reflects and reinforces cognitive style, cultural norm, and ability

what_it_is_not:
Representational modality is not mere *medium* in the physical sense. A medium is the physical carrier—paper, screen, air, ink, neural tissue. Modality encompasses the *semiotic and cognitive properties* of a carrier and the conventions governing how meaning is made and transmitted through it. A chalkboard and a whiteboard are similar physical media, but they create different modalities: chalkboard erasure leaves traces that can be recovered; whiteboard erasure is permanent. Both convey information, but the cognitive and social affordances differ. More importantly, the same medium can carry vastly different modalities. A computer screen can present visual diagrams, scrolling text, animated video, or interactive widgets—each a distinct modality with different cognitive demands. Confusing medium with modality leads to false conclusions (e.g., "digital is inherently better" when modality, not medium, determines effectiveness).

Nor is representational modality identical to sensory modality alone. Sensory modality refers to the *perceptual channel*—visual (seen), auditory (heard), tactile (felt), olfactory (smelled). Representational modality encompasses sensory modality but also the semiotic and cognitive conventions governing how meaning is made. A musical score is a *visual* sensory modality (perceived through the eye) but a *musical* representational modality (governed by the conventions of staff notation). A hummed melody is an *auditory* sensory modality but also a *musical* representational modality (governed by conventions of phrasing and ornamentation). Confusing sensory with representational modality leads designers to assume that "visual modality is more intuitive" when what they mean is visual *sensory* input, without recognizing that visual modality still requires learned semiotic conventions (how to read a chart, map, or diagram).

Representational modality is also not identical to format or style. Format refers to the superficial structure—how text is arranged, whether it is in columns or paragraphs, whether lists are bulleted or numbered. Style refers to aesthetic properties—font choice, color, visual design. These interact with modality but are not equivalent. The same textual modality can be formatted and styled in many ways; modality concerns the deeper semiotic structure (how meaning is made, not how it is decorated). A technical manual can be formatted as prose paragraphs, bullet-point lists, or decision trees—each format is a superficial variation within the textual modality. But translating the same content to a visual flowchart or an interactive tool crosses into a different modality because the cognitive and semiotic mechanisms have changed fundamentally.

It is not a claim that any modality is inherently superior or "the best." Different modalities serve different purposes for different audiences in different contexts. Visual modality excels for showing spatial relationships and overall patterns but fails at communicating precise numerical values (where tabular modality excels) or capturing temporal dynamics (where narrative modality excels). There is no universal best modality; optimality is task-dependent and audience-dependent. Naming modality is not prescriptive about which should be used; it is analytical about trade-offs and consequences of each choice.

Finally, representational modality is not equivalent to *accessibility* alone, though modality is central to accessibility. Accessibility concerns whether users with different abilities can access information; modality is a tool for designing accessible systems through multimodal redundancy. But modality affects not just people with disabilities—it affects all users. A visual graph is less accessible to blind users but highly efficient for sighted users; it also has different cognitive properties than the same data in tabular form for sighted users. Modality differences create divergence in efficiency and ease across all users and contexts, not only for people with specific disabilities.

broad_use:
**Education:** Presenting mathematical proofs visually (Cartesian plots) versus algebraically (equations) versus kinesthetically (building physical models) yields different learning outcomes and accommodates different cognitive styles, an effect Paivio (1986) grounded in dual-coding theory's separate verbal and nonverbal mental representation systems. <!-- FACT-D53-412 -->[^paivio-1986] Some learners excel with visual intuition but struggle with symbolic algebra; others require explicit notation to avoid perceptual misinterpretation. The modality is not a cosmetic choice but a structural determinant of what is learned and retained.

**Interface Design:** Critical warnings as red icons, haptic pulses, or auditory alerts have vastly different salience, memorability, and behavioral impact. The modality determines whether users notice them during routine tasks, whether they interpret urgency correctly, and whether they act. A visual warning on a silent screen may be missed; a haptic pulse from a wearable device may cut through inattention—a pattern Wickens (2008) formalizes in multiple-resource theory, which predicts that alerts in unused modalities cut through workload more effectively than additions to an already-loaded channel. <!-- FACT-D53-413 -->[^wickens-2008]

**Medicine and Patient Communication:** Explaining diagnosis outcomes verbally versus showing survival curves versus providing written summaries produces different patient comprehension, emotional processing, and treatment compliance, as Lipkus (2007) documents in his synthesis of numeric, verbal, and visual formats for conveying health risks. A 10% survival rate stated verbally invokes different cognition than a histogram showing 10 out of 100 patients surviving; one emphasizes scarcity, the other emphasizes the distribution. <!-- FACT-D53-414 -->[^lipkus-2007]

**Accessibility and Inclusion:** Screen-reader narration requires different document structure than visual layout; captioning for video audiences requires different detail and emphasis than audio-only presentation. The modality determines who can access information and what aspects they can grasp. A navigation diagram is inaccessible without visual perception; a purely textual description of the same route may be accessible to a blind user but incomprehensible to a visual learner who has never read detailed spatial prose.

**Data Journalism and Visualization:** Showing election outcomes as maps versus bar charts versus time-series animations emphasizes different patterns and shapes reader conclusions, an effect Cleveland and McGill (1984) quantified by ranking the perceptual accuracy of different graphical encodings (position, length, angle, area). A map emphasizes spatial clustering; a time series emphasizes trend direction and volatility; a bar chart emphasizes magnitude comparison. The same data, different modalities, different stories. <!-- FACT-D53-415 -->[^cleveland-mcgill-1984]

**Music Performance and Pedagogy:** A melody communicated as staff notation, by ear (aurally), or through kinesthetic imitation produces different reproduction fidelity, stylistic interpretation, and embodied understanding. Notation encourages precision and analysis; aural learning emphasizes phrasing and ornamentation by ear; kinesthetic learning builds muscle memory and embodied feel. Each modality routes to a different skill set.

**Legal and Technical Documentation:** Laws written as prose versus flowcharts versus decision trees versus interactive tools create different interpretability and compliance rates. Prose is precise but demanding; flowcharts are intuitive but sometimes ambiguous; decision trees are algorithmic but potentially opaque to non-specialists. The modality choice shapes who can understand and apply the rule, echoing Iverson's (1980) Turing-lecture argument that notation is a tool of thought whose properties enable or block reasoning about a domain. <!-- FACT-D53-416 -->[^iverson-1980]

clarity:
Naming modality as a prime lets practitioners see that "the information itself" is inseparable from "the medium carrying it." This dissolves apparent mysteries: Why do some students excel with diagrams but struggle with text? Because diagrams and text demand different cognitive operations—different working-memory loads, different perceptual parsing, different prior experience. Why do safety warnings fail? Not because they're wrong, but because the modality chosen (visual) is low-salience during the target activity (driving while fatigued) or inaccessible to the target audience (colorblind user, deaf user, user in gloved environment), a class of mismatches Norman (1991) analyzes in his account of how cognitive artifacts reshape rather than amplify the tasks their users perform. <!-- FACT-D53-417 -->[^norman-1991] The mismatch between modality and user/task creates a gap that no increase in message intensity or repetition can close. An inaccessible modality remains inaccessible regardless of volume.

The prime also clarifies why "universal design" is necessary but insufficient. No single modality serves all users equally; making one modality more accessible does not serve users who cannot or do not use that modality. Accessibility requires *multimodal redundancy*—information available in visual, auditory, and tactile channels, so that loss of one channel does not mean loss of information. It also requires recognizing that people differ not only in sensory ability but in cognitive style, cultural background, educational training, and context. A modality accessible to one person is opaque to another; inclusion demands intentional multiplicity.

Modality clarity also reframes failure modes. When communication fails or learning stalls, the default explanation is often deficit: "the student doesn't get it," "the user is confused," "the audience is inattentive." Modality thinking inverts this: failure often signals mismatch between information structure and delivery channel, or between channel and audience capability. The error is not the audience but the design—the modality choice. This shift in attribution redirects effort from blame to redesign.

manages_complexity:
The pattern separates *what* is communicated from *how* it is communicated, enabling systematic optimization of each independently. It compresses diverse phenomena—learning styles, accessibility barriers, user attention failures, cross-cultural communication barriers—into a single design variable, in the spirit of Hutchins's (1995) distributed-cognition framing in which cognitive work is reorganized by redistributing representational media across people, tools, and external structures. <!-- FACT-D53-418 -->[^hutchins-1995] Rather than asking "Why is this student struggling?" (which invites deficit thinking), modality invites asking "Which modality routes to understanding for this learner?" A student who struggles with text may thrive with spoken explanation; one who struggles with listening may thrive with visual annotation; one who struggles with both may thrive with kinesthetic or multimodal input.

This abstraction also bridges individual differences and systemic barriers. Some users are blocked by physical accessibility (blind users cannot use visual-only interfaces); others by cognitive style (abstract thinkers find symbolic modality natural, visual thinkers find it opaque); others by training (domain experts can extract meaning from dense notation that novices cannot); others by context (in a loud environment, auditory modality fails; in a low-light environment, visual modality fails). Modality thinking makes these barriers visible and remediable, not as individual deficits but as design problems.

It also manages the complexity of translation and code-switching. When must information be re-encoded for a different audience or context? What is lost or distorted in each translation? A researcher translating findings for a lay audience might move from formal statistics (symbolic modality, high precision, low accessibility) to visual charts (visual modality, intuitive for many, still abstract) to narrative stories (narrative modality, emotionally engaging but potentially anecdotal). Each step loses precision and gains accessibility; the modality choice makes this tradeoff explicit. Understanding modality enables conscious choices rather than inadvertent losses—knowing what you're trading away and for whom.

abstract_reasoning:
Recognizing modality enables reasoning about information conversion and translation cost. When must information be re-encoded for a different audience? What information is lost or distorted in translation? How do multimodal redundancy and complementarity affect robustness? Stenning and Oberlander (1995) gave this a formal treatment, showing that graphical and linguistic representations are logically inter-translatable but differ systematically in the inferences they make easy or hard. <!-- FACT-D53-419 -->[^stenning-oberlander-1995] For instance, a mathematical proof in symbolic form can be translated into visual form (graphical proof), but some aspects—edge cases, boundary conditions, exceptional behavior—may be harder to express graphically and easier in symbols. The translation is not loss-free; the modality choice shapes what remains visible and what is obscured. Understanding this enables practitioners to diagnose and document translation loss, building accountability into cross-modality communication.

Modality also enables reasoning about cognitive load and attention. High-bandwidth modalities (visual, which can convey much information in a glance if well-designed) reduce cognitive load for some tasks; low-bandwidth modalities (text, which requires sequential reading) increase load but permit precise specification. Attention is a scarce resource; modality choice determines what demands attention and what can be processed peripherally. A driver who is fatigued may miss a visual warning on the dashboard but feel a haptic pulse from a steering-wheel alert. The same message (danger ahead) encoded in different modalities creates vastly different attention footprints.

Counterfactual reasoning about modality is particularly generative. "What if we presented this information visually instead of textually?" "What if we added audio narration to this visual diagram?" "What if we offered kinesthetic engagement (building, manipulating, dragging) instead of passive reading?" These thought experiments often reveal new design possibilities or expose hidden assumptions about how information "should" be presented in a domain.

knowledge_transfer:
Modality effects transfer across domains. The principle that visual patterns are easier to grasp than long sequences of numbers applies in education, interface design, and scientific communication, as Mayer and Moreno (2003) document across nine cognitive-load reduction strategies that all exploit cross-modal complementarity. <!-- FACT-D53-420 -->[^mayer-moreno-2003] The accessibility principle that no single modality works for all users recurs in healthcare (some patients understand verbal explanations, others need visual aids or written summaries), emergency response (some alerts must be visual, others auditory, others haptic, depending on context), and organizational communication (some employees prefer synchronous meetings, others asynchronous written updates, others visual dashboards).

The principle of modality *complementarity*—that different modalities highlight different aspects of a phenomenon—transfers as well. In music, a melody heard and a melody notated reveal different structures: hearing reveals phrasing, dynamics, and ornamentation; notation reveals harmonic structure and large-scale form. In architecture, a floor plan reveals function and spatial relationships; a perspective drawing reveals experience and proportion; a physical model reveals embodied scale. No single modality is complete; rich understanding often requires multiple modalities.

formal_abstract:
**Mathematical proof, multiple modalities:** Consider proving the Pythagorean theorem. A symbolic proof works through algebraic manipulation (a² + b² = c²), precise and rigorous but requiring symbolic literacy. A geometric proof visualizes the relationship (squares constructed on each side of a right triangle, rearranged), making the intuition immediate but leaving edge cases and generalization less clear. A kinesthetic proof uses rope or physical models, creating embodied understanding but not generalizing to abstract cases. Each modality conveys the same truth (the theorem holds) but reveals different aspects: symbols reveal structure, geometry reveals spatial intuition, kinesthetic reveals embodied concept. No modality is complete; rich mathematical understanding often requires all three.

2. **Modality Matching:** Diagnose the task, learner, and context; match modality to the cognitive operation required. For building spatial intuition, use visual or kinesthetic modality; for precise specification, use symbolic or textual modality; for emotional engagement, use narrative or multimodal modality. Different learners may require different modalities for the same content.

3. **Modality Complementarity:** Use multiple modalities not for redundancy but for dimension-adding. Pair a visual graph (showing overall trend) with a data table (showing precise values); pair a narrative account (showing human stakes) with statistical analysis (showing population patterns). Together, modalities reveal what neither alone could.

4. **Progressive Disclosure Across Modalities:** Start with the modality most accessible and engaging (visual, narrative, kinesthetic), then offer deeper modalities for users who want precision or rigor (symbolic, formal, textual). Example: begin with an interactive visualization, then offer the underlying data and technical documentation for users who want to verify and extend.

5. **Modality Translation with Fidelity Assessment:** When translating information across modalities (e.g., converting a technical specification to user-facing language), explicitly document what is lost, distorted, or simplified. This creates accountability and helps users understand the translation's limitations.

6. **Contextual Modality Adaptation:** Offer modality choice based on context. Example: provide a visual dashboard for casual browsing, a detailed report for formal review, and an API for programmatic access. Let users select the modality that fits their task.

applied_industry:
**User documentation redesign:** A software company documentation was purely textual (written instructions with occasional screenshots). Users complained of confusion and repeated support tickets. The company redesigned using modality complementarity: step-by-step text (precise, searchable, scannable), annotated screenshots (visual confirmation of what to look for), short videos (kinesthetic demonstration of the action sequence), and interactive tutorials (hands-on practice with immediate feedback). Modality analysis revealed that different users gravitated to different channels: rushed users preferred videos, detail-oriented users preferred text, visual learners preferred annotated screenshots. By offering multimodal access, support tickets dropped 30%, and user satisfaction increased. The content (how to complete the task) was unchanged; the modality changed everything.

**Accessibility compliance with dignity:** A hospital improved emergency protocols by moving beyond a single modality. Critical alerts had been auditory (overhead paging), which failed for deaf staff and in loud environments. The redesign used multimodal signals: a vibration alert on staff pagers (haptic), a visual status board (visual), a text message to mobile devices (visual-textual), and an audio page (auditory). Each staff member could access via their preferred or accessible channel. Critically, the redesign was not presented as "accessibility accommodation" (implying burden) but as "resilience through modality redundancy" (benefiting all staff). In a noisy environment, the haptic alert reaches deaf and hearing staff equally. The modality reframe shifted from "helping the deaf" to "working better for everyone."

title: Modality choice privileges some dimensions of information and obscures others

body: A visual graph makes magnitude comparison and trend direction salient but may obscure absolute values or data-point uncertainty. A data table makes exact values salient but obscures patterns. A narrative account emphasizes causality and human meaning but may obscure statistical rigor. No modality is neutral; every choice amplifies some aspects and shadows others. This creates a dilemma: which dimensions matter most for this audience and purpose? Practitioners often optimize for salience at the expense of comprehensiveness. The risk is that users who need the obscured dimensions will misunderstand or make poor decisions based on incomplete visibility.

title: Modality accessibility varies by audience ability and prior experience, creating equity tensions

body: Visual modality is efficient for sighted users with spatial training but inaccessible to blind users and potentially confusing to spatial-reasoning novices. Auditory modality is natural for hearing people but requires captioning or interpretation for deaf or hard-of-hearing users. Symbolic modality is precise for trained mathematicians but intimidating for novices. No modality is universally accessible; inclusive design requires multimodal redundancy, which carries cost and complexity. The tension: how much redundancy is sufficient, and who bears the cost? Organizations with limited resources must make painful choices about which modalities to prioritize, potentially excluding some users by design necessity.

title: Modality effects depend on context and task, creating inconsistent transfer

body: Visual graphs excel at showing overall trend but fail at precise-value lookup (where tables excel). Oral explanation excels at building intuition but fails at precise specification (where written definitions excel). The "best" modality depends on the learner, the content, the task, and the context. This inconsistency makes it hard to establish universal best practices; instead, practitioners must diagnose the specific task and match modality to it. The cost is high: it requires flexibility, experimentation, and responsiveness rather than standardized one-size-fits-all solutions.

title: Multimodal redundancy enhances robustness but can create cognitive overload or contradiction

body: Pairing a visual graph with a verbal explanation helps learners who prefer different modalities and provides fallback if one channel fails. But excessive multimodality—a graph with every data point labeled, narrated aloud, accompanied by written text, plus animated transitions—can overwhelm attention and obscure the main point. Additionally, if modalities contradict each other (a verbal claim that contradicts what the visual shows), cognitive load increases and trust erodes. Finding the optimal level of redundancy is difficult; too little leaves some users without sufficient access, too much creates noise.

title: Modality choice reflects cultural and institutional norms, constraining what feels natural

body: In scientific communication, symbolic and visual modalities dominate; narrative and embodied modalities are often dismissed as "less rigorous." In legal communication, written prose dominates; visual and narrative modalities are treated with skepticism. In education, textbooks and lectures dominate; kinesthetic and peer-learning modalities are often sidelined. These norms are historical and contingent, not intrinsic, yet they shape what modalities are available and valued. Changing modality norms requires institutional effort and faces resistance from stakeholders invested in the status quo. What feels "natural" or "professional" is often just habituation.

title: Modality conversion is irreversible and lossy; once information is encoded in one modality, some aspects of the original become inaccessible

body: An idea verbally explained can be notated, but the notation may lose the emotional tone, embodied gestures, and immediate responsiveness to audience reaction. A photograph captures visual appearance but loses smell, temperature, and temporal dimension. A poem translated to another language gains accessibility for new audiences but loses sound patterns and cultural resonance. The loss is sometimes acceptable; sometimes it is tragic. Practitioners must weigh the cost of translation against the benefit of broader accessibility, knowing that no solution is perfect and every choice involves loss.

prose: 1. **Multimodal Redundancy:** Present critical information in multiple modalities so that loss of one channel does not mean loss of information. Example: a video with captions (visual text) and audio description (auditory detail) reaches hearing-sighted, deaf, blind, and deafblind users. Each user can access via their preferred or accessible modality; if one channel is lost or unclear, others remain.

notes:
Modality effects depend on cultural and educational background. A user trained in reading graphs will extract information from a graph faster than from a table; a user without such training may find the graph mystifying. Modality "accessibility" is always relative to the audience's prior experience. This means that claims about "intuitive" or "natural" modalities must be interrogated: they often reflect the creator's or dominant group's assumptions, not universal truths.

The concept of *semiotic modality*—the conventions governing how meaning is made and transmitted—is broader than *sensory modality*. A musical score is a visual modality (perceived through the eye) but a musical semiotic modality (governed by the conventions of staff notation). A spoken melody is an auditory sensory modality but also a musical semiotic modality. Confusing the two (sensory channel vs. semiotic convention) leads to errors in design and pedagogy. A designer might assume that "visual is more intuitive" without recognizing that visual modality also requires learned conventions (how to read a chart, map, or diagram); the learning is often invisible to experts.

Modality is often conflated with *medium*, but they are distinct. A book is a medium (physical carrier); black-and-white printed text is a modality (semiotic convention); a novel is a genre, which can be instantiated in any medium or modality (printed book, audiobook, e-book, oral storytelling). Clarity about these distinctions prevents design and pedagogical errors. A "digital transformation" that moves content from paper to screen but does not rethink modality (e.g., a scanned PDF is still a textual modality, not a redesigned interactive or visual modality) often fails to capture the benefits of the new medium.

Modality interacts with *cognitive load theory*. High cognitive load in one modality (reading dense text, for instance) might be reduced by shifting to a lower-load modality (visual diagram) or by leveraging multimodal complementarity (text + diagram, so working memory is distributed across visual and verbal systems). However, modality choice also interacts with domain expertise: an expert may have low cognitive load from dense symbolic notation (because pattern recognition is automatic), while a novice has high load from the same notation. Expertise reshapes modality effects; what is efficient for an expert may be opaque to a learner.


Representational Structure Mismatch

one_liner: A representation preserves its target's elements but misstates one or more task-relevant relations among them, so content can appear complete while its shape licenses the wrong traversal, inference, or action.

origin_domain: synthesized

core_idea:
Representational Structure Mismatch is the failure in which a representation
carries the relevant elements of its target but organizes those elements under
relations that do not preserve a structure required for the representation's
intended use. An inventory can therefore be complete while the representation is
operationally unfaithful. The words, controls, values, steps, or coded fragments
are all present, yet their order, hierarchy, adjacency, scope, grouping,
containment, or dependency differs from the target relation a consumer must
follow.

The prime concerns relational organization, not resemblance. A representation
need not look like its target, and it may deliberately omit almost everything,
provided the relations it claims to preserve remain usable. Conversely, a
visually rich or token-complete medium can fail when one load-bearing relation is
wrong. Nested callback code contains every operation in a temporal chain but
represents succession as syntactic containment. A flat stream can contain every
item in a dependency graph but impose an order unrelated to prerequisite depth.
A coded transcript can retain every phrase while severing the narrative sequence
that determines what the phrases mean.

The identity is fixed by a purpose-relative correspondence test. Name the target
relation the task requires, name the relation the medium actually exposes, and
run the operation the consumer must perform. If the operation succeeds on the
target but is blocked, redirected, or made misleading by the medium's relational
organization, the representation has a structural mismatch. Restoring the
elements without restoring the relation cannot repair it because the elements
were never the missing object.

structural_signature:
*the structured target* — *the representing medium* — *the task-relevant target
relation* — *the medium's encoded relation* — *the purpose-relative faithfulness
claim* — *the consumer operation* — *the divergence under content completeness*
— *the structural realignment*

- **Structured target.** A process, artifact, dataset, experience, task, or
  conceptual object contains identifiable relations among elements.
- **Representing medium.** A second carrier makes the target available for
  reading, manipulation, storage, communication, or action.
- **Task-relevant target relation.** At least one relation—order, dependency,
  hierarchy, adjacency, scope, containment, grouping, or role—must be preserved
  for a specified use.
- **Encoded medium relation.** The medium organizes corresponding elements under
  its own syntax, layout, traversal, schema, or decomposition.
- **Faithfulness claim.** Explicitly or implicitly, consumers are licensed to
  treat the medium relation as carrying the target relation for the task.
- **Divergence.** The two relations are non-corresponding in a way that changes
  a permitted traversal, inference, interpretation, or action.
- **Completeness trap.** Elements or surface content can all remain present, so
  inventory, syntax, and local-validity checks can pass.
- **Realignment test.** Changing the relational encoding while holding the
  elements substantially fixed restores the failed consumer operation.

The last two roles distinguish a structural mismatch from ordinary missing
content. The failure is often silent because a medium can be locally well formed
and element-complete while violating the cross-structure correspondence on which
its usefulness depends.

what_it_is_not:
- **Not Representation generally.** `representation` supplies a target, medium,
  mapping, and faithfulness specification. This prime names the failure of that
  specification at the level of a task-relevant relation.
- **Not missing content.** Tokens, values, controls, operations, or fragments may
  all be present. The missing or incorrect object is the relation among them.
- **Not Isomorphism failure in the full formal sense.** An isomorphism requires
  a bijective, two-sided preservation of all structure named by its formal
  category. Ordinary representations are selective and lossy. This prime fires
  when even one relation the stated task relies on is not preserved.
- **Not Proxy–Target Fidelity.** A proxy is an observable standing in for a
  latent or expensive target, and its fidelity is usually graded by co-movement.
  Here both target elements and medium elements can be explicit; the question is
  whether their relational organization corresponds.
- **Not Grain of Analysis.** Grain compares an operation's decomposition level
  with the level at which a phenomenon's structure exists. A relation can be
  represented at the correct grain and still be ordered, nested, grouped, or
  scoped incorrectly.
- **Not Interface Mismatch.** Interface Mismatch concerns two locally workable
  systems whose offered and required contracts disagree at an exchange seam.
  Representational Structure Mismatch concerns a target-to-medium
  correspondence; no component handoff is required.
- **Not style or legibility alone.** A representation may be ugly but
  structurally faithful, or elegant but structurally wrong. The test is whether
  a named relation needed for use survives.

broad_use:
In software, nested continuation callbacks represent a linear temporal process
as a syntactic containment tree. Every operation is present, but written
structure no longer follows executed structure; error propagation, lexical
scope, and refactoring all inherit the mismatch. In accessibility and document
engineering, presentation may carry heading hierarchy, grouping, state, or
reading order that an explicit semantic model omits. Model-bound consumers then
receive a different relation even though they receive the same text and values.

In responsive interface design, a viewport transformation can preserve every
content token while breaking layout-encoded priority, adjacency, and hierarchy.
In qualitative analysis, overcoding preserves excerpts while fragmenting below
the level at which sequence and relation make the phenomenon legible. In
curricula, checklists, legal briefs, documentation, and build systems, a
multi-relational source is forced into one serial order whose principle differs
from the one the consumer uses. In each case the carrier, relation, and consumer
change, but the failure is literal: element completeness coexists with
relation-level unfaithfulness, and repairing the organization rather than adding
content restores use.

clarity:
The prime makes “the content is all there” an insufficient certification. A
content inventory answers whether elements survived. It does not answer whether
the representation preserved the order in which they must be encountered, the
hierarchy that scopes them, the adjacency that binds them, or the dependencies
that license action. Naming the mismatch forces those relational commitments
into the audit.

It also localizes disputes that otherwise collapse into aesthetic judgment. An
analyst must state which target relation the task depends on, how the medium
encodes it, and what consumer operation fails. If no changed inference or action
can be identified, the complaint may be stylistic. If the operation changes when
the relation is realigned while elements remain fixed, the defect is structural.

manages_complexity:
A large catalog of medium-specific failures collapses to one correspondence
table:

1. enumerate the task-relevant relations in the target;
2. locate the corresponding relation in the medium;
3. identify the consumer and operation that rely on it;
4. test the two structures under the same use;
5. distinguish absent elements from absent or altered relations; and
6. choose whether to realign the medium, add explicit relational metadata,
   change the consumer contract, or narrow the declared faithfulness boundary.

This reduces an open-ended search for “bad design” to a bounded comparison.
Because the prime is purpose-relative, it also prevents over-repair. A map need
not preserve chemical composition, a transcript codebook need not preserve
prosody, and a program listing need not mirror processor microarchitecture.
Only relations required by the declared use enter the fidelity contract.

abstract_reasoning:
The central move is *hold elements constant and vary organization*. If changing
only order, nesting, scope, grouping, adjacency, or dependency changes what a
consumer can infer or do, the relation is load-bearing. Compare that medium
relation with the target relation directly.

A second move is *operation correspondence*. Ask which operation on the
representation is supposed to stand in for which operation on the target.
Following written order should recover temporal order; traversing headings should
recover conceptual hierarchy; following focus should recover task order; joining
codes should recover the phenomenon's sequence. When the paired operations do
not commute, the mismatch is operational rather than merely visual.

A third move is *repair discrimination*. Adding missing elements repairs content
loss. Increasing resolution repairs collapsed distinctions. Changing
decomposition grain repairs level mismatch. Reorganizing relations repairs
representational structure mismatch. Selecting the wrong intervention is itself
evidence that the failure family has not been identified precisely.

knowledge_transfer:
The transferable diagnostic has fixed roles: target, medium, task-relevant
relation, represented relation, consumer operation, and divergence. A software
engineer trained to ask whether written order tracks run order can carry the
same question to a checklist designer asking whether list order tracks
prerequisite order, or to a document engineer asking whether heading structure
tracks visual hierarchy.

The repair moves transfer too. Promise chains and `async`/`await` realign syntax
with execution; semantic markup realigns explicit hierarchy with visual
hierarchy; breakpoint-specific layout restores priority and adjacency; narrative
or episode-level recoding restores relationships severed by fragment-level
coding; alternate linearizations let different consumers traverse the same
material under different reasoning principles. These are not the same local
technique, but they perform the same structural operation: change the medium's
relations so the task-relevant target relation survives.

notes:
Authored when Callback Hell supplied the inverse-dimensional case that
Linearization-Meaning Mismatch could not parent. Reopening the neighborhood
showed independent co-instances in layered semantics, responsive layout, and
qualitative coding, making a general representation-level mismatch prime more
informative than a one-child software umbrella.