Carryover Effect

one_liner: The validity threat in crossover and within-subject designs where residual influence from an earlier treatment persists into a later measurement window, biasing the contrast — its magnitude set by the unit's relaxation time against the inter-treatment gap.

origin_domain: statistics

core_idea:
The carryover effect is the validity threat in crossover, repeated-measures, and within-subject experimental designs in which residual influence from an earlier treatment or condition persists into the measurement window of a later one, so that the response attributed to the later condition is contaminated by the earlier condition's lingering effect. The mechanism requires two features. First, the experimental unit — a patient, participant, animal, or system — has an internal state that the earlier treatment modifies and that does not return to baseline before the later treatment's measurement window begins: a pharmacokinetic drug concentration that has not cleared, a practised skill, muscle fatigue, sensitisation of a receptor population, an updated belief or expectation. Second, the design assumes that each measurement window samples an independent, clean realisation of the treatment-outcome relationship; when the internal state is still influenced by the prior treatment, that independence assumption is violated, and the estimated contrast between conditions is biased by whatever amount of the prior treatment's effect persists.

The magnitude of the bias depends on the relaxation time of the internal state relative to the inter-treatment gap: a drug with a 36-hour half-life whose carry-in concentration after a 24-hour washout still occupies a substantial fraction of the receptor pool contaminates the second treatment period by an amount proportional to that residual concentration. The standard design interventions are: *washout periods* — intervals between treatments long enough for the internal state to return to a negligible residual, commonly operationalised as five half-lives for pharmacokinetic carryover; *counterbalancing* — assigning different participants (or units) to different treatment orders in a Latin-square or Williams design, so that carryover residues are distributed symmetrically across treatment conditions and cancel at the contrast level; and *explicit modelling* — including a carryover term in the statistical model (often as a lag of the prior treatment in a mixed-effects framework) to estimate and remove its contribution. The concept applies across crossover clinical trials (where it is a major concern in ICH E9 regulatory guidance), behavioural and psychological experiments (where practice, fatigue, sensitisation, and demand-characteristics produce carryover), product-analytics A/B tests run sequentially on the same users, and sensory evaluation panels where palate carryover from one sample to the next requires cleansing protocols and order counterbalancing.

scope_of_application:
The carryover effect lives across the within-subject and crossover application contexts of statistics and experimental-design practice; its reach is within that one methodological domain, restaging identically across substantive settings that differ only in *which* internal state carries the residue. The substrate-independent state-from-history primitive (ferromagnetic remanence, control-system memory, ecological legacy effects) travels under `hysteresis`, not under this construct.

- **Clinical crossover trials** — the home: residual drug or metabolite from period A biasing the response in period B, codified in ICH E9 regulatory guidance and Senn's crossover-trials literature.
- **Pharmacology and PK/PD studies** — half-life-driven washout calculation, residual receptor occupancy from a prior dose, and desensitisation hysteresis.
- **Behavioural and psychological experiments** — practice, fatigue, sensitisation, and demand-characteristics carryover, with counterbalanced order (Latin-square, Williams designs) the standard control.
- **Educational research** — skill, vocabulary, habit, or motivation retained from an earlier instructional method contaminating measurement of a later one.
- **Sequential A/B testing on the same users** — learning effects, novelty fatigue, and anchoring from variant A distorting the variant-B window, recovered by randomising direction at a cost in power.
- **Policy and programme evaluation under sequential reforms** — behavioural adaptation and institutional learning persisting from reform A into the estimate of reform B.
- **Sensory-evaluation and panel studies** — palate residue from one sample biasing the next, cleared by cleansing protocols and order counterbalancing.

structural_signature:
**Sig role-phrases:**

- **the unit of analysis** — a subject, participant, animal, or system carrying an internal state-variable that treatments modify (a drug concentration, a practised skill, a fatigued muscle, an updated belief)
- **the successive treatments** — the within-subject/crossover design in which the same unit is measured under one condition then another
- **the independence-of-windows assumption** — the design's commitment that each measurement window samples a clean, independent realisation of the treatment-outcome relationship
- **the earlier treatment's perturbation** — the move whereby the first condition writes a residue into the unit's internal state
- **the incomplete relaxation** — the dynamic that the state does not return to baseline before the later treatment's measurement window opens, set by the relaxation time-constant versus the inter-treatment gap
- **the contaminated contrast** — the outcome: the later-condition response is a function of "later treatment plus residual of the earlier," biasing the estimated contrast by an amount proportional to the surviving residual
- **the remedy menu** — the relaxation-indexed fixes: washout (a gap sized to the state, five half-lives when pharmacokinetic), counterbalancing (Latin-square/Williams order distributing the residue symmetrically so it cancels), and an explicit carryover term modelled and subtracted

what_it_is_not:
- **Not random noise or measurement error.** Carryover is a *deterministic residue inside the unit of analysis* left by its own prior treatment, not stochastic scatter and not a property of the instrument. It biases the contrast in a signable direction proportional to the surviving residual, where noise would merely add variance — so the analyst looks at the carried state, not the measuring device.
- **Not external confounding.** A confounder is an outside variable that co-varies with treatment; carryover is internal, a state the *earlier condition itself* wrote into the subject and failed to erase. Assignment can be perfectly correct and confounder-free, and carryover still contaminates the later window — the locus is the unit's own history, not a third variable.
- **Not selection bias.** Selection bias concerns *who* lands in which condition; carryover assumes correct, even randomised, assignment and concerns *within-condition* contamination from a prior treatment. The threat is to the independence of measurement windows, not to the comparability of groups.
- **Not regression to the mean.** Regression to the mean is an artefact of *extreme baselines* reverting on remeasurement; carryover is an artefact of *prior exposure* persisting forward. One arises because a starting value was unusually high or low; the other because the unit's state has not relaxed — different causes, different remedies.
- **Not hysteresis itself.** Hysteresis — a system's state depending on its history, not just current input — is the substrate-independent primitive; carryover is the experimental-design profession's name for that primitive when it appears in the unit of a crossover or repeated-measures design and biases a contrast. A ferromagnet shows hysteresis but runs no biased estimator and has no washout or counterbalancing; the cross-domain lesson travels via hysteresis, not via "carryover."
- **Not neutralised by any washout whatever.** A washout works only when it is *long relative to the state's relaxation time* (commonly five half-lives for pharmacokinetic residue); a too-short gap leaves substantial residual and a still-biased contrast. Nor is counterbalancing a guaranteed fix — it cancels the residue only when it is *symmetric across orders*, an assumption that must itself be checked, failing which the carryover must be modelled explicitly.

clarity:
Naming the carryover effect isolates a validity threat that casual analysis tends to dissolve into its neighbours. A contaminated within-subject contrast is easy to mislabel as noise, as measurement error, or as confounding by some external variable — but carryover is none of these: it is a *deterministic residue inside the unit of analysis*, left by the unit's own prior treatment, not a property of the instrument or of an outside cause that happens to co-vary. That precision separates it cleanly from selection bias (which concerns *who* lands in which condition, given here a correct assignment), from confounding (an external variable moving with treatment), and from regression to the mean (an artefact of extreme baselines rather than of prior exposure). The single word tells the methodologist where to look: not at the assignment mechanism or the measuring device, but at the state the earlier condition wrote into the subject and failed to erase.

The construct's sharper payoff is the set of questions it makes mandatory. To invoke carryover is to be forced to specify three things the design otherwise leaves implicit: *which internal state-variable* carries the residue (a drug concentration, a practised skill, a fatigued muscle, an updated expectation), *over what timescale* that state relaxes, and *whether the later measurement window opens before it has relaxed*. Posing the threat in those terms converts a vague worry that "earlier conditions might still be affecting things" into a quantitative comparison between a relaxation time-constant and an inter-treatment gap — which is exactly what makes the standard remedies decidable rather than ritual. A washout is no longer a fixed waiting period but a length chosen against the state's own relaxation (five half-lives, when the residue is pharmacokinetic); counterbalancing is recognised as the move that distributes the residue symmetrically across conditions so it cancels in the contrast rather than biasing it; and an explicit carryover term becomes the option of last resort when the design cannot purge the residue and the analysis must estimate and subtract it. The framework also makes legible what each remedy *costs* and *cannot* do — counterbalancing buys an unbiased contrast at the price of statistical power and assumes the residue is symmetric across orders, an assumption that itself becomes a thing to check — so the practitioner reasons about the trade-off deliberately instead of trusting that any washout at all has settled the matter.

manages_complexity:
Within-subject and crossover designs are threatened by a long, heterogeneous list of ways an earlier condition can poison a later one — residual drug in the bloodstream, a skill practised in the first arm, accumulated fatigue, a sensitised receptor population, an updated expectation, palate residue, novelty wearing off — each arising in a different substantive setting and each, untheorised, looking like its own separate problem demanding its own ad hoc fix. The carryover effect compresses that whole list to one diagnosis and a single reasoning move. Every item is the same structure: the unit carries an internal state that the earlier treatment modified and that has not relaxed to baseline before the later treatment's measurement window opens, violating the design's assumption that each window samples an independent realisation of the treatment-outcome relationship. So the methodologist stops enumerating substrate-specific pathologies and tracks just three things the diagnosis makes mandatory — which internal state-variable carries the residue, the timescale on which that state relaxes, and whether the later window opens before it has relaxed. The qualitative verdict reads off the comparison of two numbers: a relaxation time-constant against the inter-treatment gap. Gap long relative to relaxation, and the contrast is clean; gap short, and the contrast is biased by an amount proportional to the surviving residual. A vague worry that "earlier conditions might still be affecting things" becomes a single quantitative inequality.

That same compression makes the remedy menu decidable rather than ritual, and that is the second sprawl it tames. Because the bias is a function of relaxation-time versus gap, the three standard interventions stop being independent tricks and become readable responses to one parameter: a *washout* is a gap chosen against the state's own relaxation (five half-lives when the residue is pharmacokinetic), not a fixed waiting period; *counterbalancing* (Latin-square or Williams order) is the move that distributes the residue symmetrically across conditions so it cancels at the contrast level rather than biasing it; and an *explicit carryover term* in a mixed-effects model is the last-resort option of estimating and subtracting the residue when the design cannot purge it. The framework even prices each choice — counterbalancing buys an unbiased contrast at the cost of statistical power and on the assumption that the residue is symmetric across orders, itself now a checkable thing — so the practitioner reasons about the trade-off deliberately. The construct further keeps the diagnosis from collapsing into its neighbours, which is part of how it manages the space: it is a deterministic residue *inside* the unit, so it is not noise, not measurement error, not external confounding, not regression to the mean, and the single word points the analyst at the subject's own carried state rather than at the assignment mechanism or the instrument. A scattered catalogue of design failures across pharmacology, behavioural experiments, sequential A/B tests, and sensory panels thus reduces to one state-relaxation parameter and a three-option remedy menu indexed by it — from which the presence of bias, its magnitude, and the appropriate fix all read off.

abstract_reasoning:
The carryover effect licenses *state-space reasoning about the unit of analysis*: the subject carries an internal state at the start of each measurement window, the treatment perturbs that state, and the outcome is read before the state has relaxed. The defining *diagnostic* move runs from a contaminated within-subject contrast to a specific locus, against the pull to mislabel it. A biased crossover contrast is read not as noise, not as measurement error, not as external confounding, and not as regression to the mean, but as a *deterministic residue inside the unit* left by its own prior treatment. The surface signature is the off contrast; the inferred cause is a carried internal state. Reasoning FROM "the B-period response looks wrong" TO "look at the state the A treatment wrote into the subject and failed to erase" is what points the methodologist at the subject's carried state rather than at the assignment mechanism or the instrument, and it is the move that separates carryover from every neighbor it is casually fused with.

The quantitative heart is a *predictive* move comparing two timescales. Once the threat is posed as carryover, the reasoner must name which internal state-variable carries the residue (a drug concentration, a practised skill, a fatigued muscle, an updated expectation), the relaxation time-constant of that state, and whether the later window opens before it has relaxed — and then predicts the bias by comparing the relaxation time-constant against the inter-treatment gap. Gap long relative to relaxation, and the contrast is clean; gap short, and the contrast is biased by an amount *proportional to the surviving residual* (a 36-hour-half-life drug after a 24-hour washout still occupies a substantial fraction of the receptor pool, contaminating the second period by roughly that fraction). Reasoning FROM "relaxation time versus inter-treatment gap" TO "is there bias, and how much" converts a vague worry that earlier conditions might still matter into a single quantitative inequality with a signable magnitude.

The *interventionist* moves are three, each selected by and tuned against that one parameter rather than applied by rote. Lengthen the *washout* to a gap chosen against the state's own relaxation (five half-lives when the residue is pharmacokinetic), predicting the residual shrinks to negligibility. *Counterbalance* the order (Latin-square or Williams design), predicting the residue is distributed symmetrically across conditions so it cancels at the contrast level — at the cost of statistical power and on the assumption, itself now checkable, that the residue is symmetric across orders. Or, when the design cannot purge the residue, add an *explicit carryover term* (a lag of the prior treatment in a mixed-effects model) and estimate-and-subtract it. Reasoning FROM "the bias is a function of relaxation-versus-gap" TO "purge it with a washout sized to the state, cancel it by symmetry, or model it away" makes the remedy menu decidable, and pricing each option (counterbalancing trades power for an unbiased contrast under a symmetry assumption) lets the practitioner choose the trade-off deliberately rather than trusting that any washout at all has settled the matter.

A *boundary-drawing* move scopes when the diagnosis even applies and which remedy is admissible, by checking the two structural preconditions: does the unit have an internal state the earlier treatment modifies, and does that state fail to return to baseline before the later window opens? Where the inter-treatment gap already exceeds the relaxation time, there is no carryover and no remedy is needed; where the residue is *asymmetric* across orders, counterbalancing will not fully cancel it and explicit modelling becomes necessary. Reasoning FROM "is the state still perturbed when the window opens, and is its residue symmetric" TO "does carryover bite here, and which fix is valid" keeps the analysis from invoking carryover where independence already holds, and from trusting counterbalancing where the symmetry it relies on fails.

knowledge_transfer:
Within statistics and experimental-design practice the carryover effect transfers as full mechanism, and the cargo that carries is the whole apparatus: the state-relaxation diagnosis, the relaxation-time-versus-gap inequality, the three-precondition scoping, and the washout / counterbalancing / explicit-modelling remedy menu. The concept restages identically across substrates that differ only in *what* state carries the residue: *clinical crossover trials* (residual drug or metabolite biasing the next period — the home, codified in ICH E9 and Senn's crossover-trials literature), *pharmacology and PK/PD studies* (half-life-driven washout, residual receptor occupancy, desensitisation hysteresis), *behavioural and psychological experiments* (practice, fatigue, sensitisation, demand-characteristics, with counterbalanced order the standard control), *educational research* (skill, vocabulary, or motivation retained from an earlier method), *sequential A/B tests on the same users* (learning effects, novelty fatigue, anchoring), *policy and programme evaluation under sequential reforms* (behavioural adaptation and institutional learning persisting from reform A into reform B), and *sensory-evaluation panels* (palate residue cleared by cleansing protocols and order counterbalancing). Across all of these the diagnosis, the timescale comparison, and the remedy menu are the same objects — and concrete templates port directly: pharmacology's "five half-lives" washout rule, psychology's Latin-square and Williams counterbalancing designs, and mixed-effects carryover terms all travel as ready-made tools. Even within human-subject work the concept reaches across substantive settings as mechanism, since `learning_curve_effects` (skill from condition A persisting into B) is simply one of its most common instances. The transfer is literal because the substrate — a unit measured under successive treatments, carrying an internal state that does not relax in time — is held fixed.

Beyond the experimental-design substrate the situation is the *shared-abstract-mechanism* case with an unusually crisp parent. The carryover effect is the experimental-design profession's name for what happens when `hysteresis` — *a system's state at time t depends on its history before t, not just the current input* — appears in the unit of analysis of a crossover or repeated-measures design. Strip the statistics vocabulary and "carryover" reduces exactly to the hysteresis primitive: the unit's state at the start of measurement is a function of its history. That primitive genuinely recurs across domains — magnetic and mechanical hysteresis in materials science, lag and memory in control systems, alternative stable states and legacy effects in ecology — but those recurrences are *hysteresis* travelling, not the carryover effect. None of them inherits the design-validity machinery; a ferromagnet exhibiting remanent magnetisation is not running a "biased contrast estimator," and there is no washout period, no counterbalancing, and no independence-of-measurement-windows assumption to violate.

So the cross-domain lesson — *a system carrying state from its past will contaminate a present reading unless the past has had time to relax* — should be carried by `hysteresis`, the substrate-independent primitive of which carryover is one domain instance, not by "carryover effect," whose value as a named construct lives inside statistics and experimental design. What stays home-bound is everything that makes this entry methodological: the violated independence-of-measurement-windows assumption, the biased-and-inconsistent contrast estimator, the washout/counterbalancing/explicit-term remedy family, and the regulatory and design literature (ICH E9, Williams designs, mixed-effects carryover modelling) built around it. The honest framing is therefore: the state-from-history *shape* is general and travels via `hysteresis`; the crossover-design validity threat and its remedy menu are the domain accent that stays in statistics and data science (see *Structural Core vs. Domain Accent*).


Case-Definition Drift

one_liner: The surveillance failure in which the operational definition of 'a case' silently changes across time, place, or data source while counts are still reported and compared under a stable label — conflating real epidemiological change with an artefact of what the instrument now counts.

status: draft

core_idea: Case-definition drift is the epidemiological and public-health surveillance failure in which the operational definition of "a case" of a condition changes — across time, across reporting jurisdictions, or across data sources — while aggregate counts and trends derived under that definition continue to be reported and compared as though the definition were stable. The drift may be deliberate and formally announced (a revised ICD code, an updated CSTE case definition, a new screening guideline that catches earlier-stage disease), administrative and implicit (a billing-code split, an EHR documentation template change, a shift in clinicians' threshold for ordering a confirmatory test), or methodological (an upgrade in assay sensitivity that reclassifies subclinical cases as positive). In every variant the mechanism is the same: the *operational categorisation instrument* that the surveillance system uses to map events in the world to counts in a dataset has changed, while the *output time-series label* has not. The apparent change in disease frequency, severity, or geographic distribution thus conflates two inseparable contributions — real epidemiological change in the underlying population and artefactual change in what the measurement instrument counts as an event. The artefact component is invisible without deliberate audit of the definition history, because the surveillance system's published output gives no internal signal of the definitional shift. Within epidemiology, two worked examples define the pattern's scope: the decades-long apparent rise in autism prevalence, where successive DSM revisions (DSM-III to DSM-IV to DSM-5) progressively broadened the diagnostic perimeter while awareness and diagnostic intensity also increased, making it methodologically nontrivial to partition the observed trend between real change and definitional expansion; and the "Will Rogers phenomenon" in cancer staging, where refined staging technology moves borderline patients to a higher stage, improving the observed survival rates of both the stage they left and the stage they entered without any change in actual patient outcomes — stage-migration as a threshold-shift form of case-definition drift. The intervention vocabulary the pattern unlocks is specific to surveillance engineering: freezing the operational definition within a surveillance window, maintaining parallel old-definition and new-definition counts during a bridging period, applying retrospective re-coding where feasible, computing bridging coefficients for trend analysis that crosses a definitional change, and publishing the definition-history alongside the time-series so that downstream analysts can audit the commensurability of cross-period comparisons.

scope_of_application:
Case-definition drift lives across the surveillance and trend-analysis subfields of epidemiology and public health, enumerated here by the kind of condition and definitional instrument that drifts; its reach is within that domain. The structurally identical siblings in ML annotation, psychometric rating, and manufacturing metrology are true co-instances of the parent `instrument_interpretive_drift`, carried there rather than by the epidemiology label.

- **Infectious-disease surveillance** — the salient recent habitat: COVID-19 case counts under repeatedly revised CSTE definitions (symptom-only → PCR-confirmed → antigen-inclusive → reinfection rules), producing apparent step-changes that conflate real waves with definitional moves.
- **Notifiable-disease reporting** — CSTE/CDC case-definition revisions across the reportable conditions, where the agencies publish definition-update histories precisely to support trend analysis across drift.
- **Psychiatric and developmental epidemiology** — the multi-decade autism-prevalence debate under successive DSM revisions (DSM-III → IV → 5), the field's canonical problem of apportioning an observed rise between real change and definitional broadening.
- **Cancer staging and survival epidemiology** — the Will Rogers phenomenon under successive AJCC editions, where refined staging migrates borderline patients and lifts the apparent survival of both stages with no real outcome change (a threshold-shift special case).
- **Critical-care / sepsis surveillance** — the Sepsis-1/2/3 transitions, which partly disagree on who counts as septic and make cross-definition incidence and mortality comparison non-trivial.
- **Chronic-disease threshold setting** — hypertension and diabetes guideline updates (JNC-8 vs. ACC/AHA 2017; ADA thresholds) that reclassify tens of millions overnight, the threshold-reset form of drift.
- **Mortality / cause-of-death coding** — ICD-9 → ICD-10 → ICD-11 revisions producing step-discontinuities in cause-specific mortality that require bridging coefficients.

structural_signature:
**Sig role-phrases:**

- **the surveillance system** — an apparatus mapping events in the world through an operational case definition to a count or rate
- **the operational case definition** — the categorization instrument (criteria, ICD/CSTE/DSM codes, thresholds, forms) that decides what counts as "a case"
- **the stable output label** — the time-series name and reporting frame that stay fixed even as the definition underneath them moves
- **the definitional shift** — a change to the instrument across time, jurisdiction, or data source: deliberate (revised criterion), administrative (billing-code split), or methodological (assay-sensitivity upgrade)
- **the conflated observed trend** — the published series read as real epidemiological signal *plus* a definitional-boundary function *plus* sampling noise
- **the invisible artefact** — the boundary-shift contribution gives no internal signal in the output, so commensurability is never given and must be audited from the definition history
- **the both-directions sign rule** — a broadened definition inflates counts (false alarm); a tightened one deflates them and can mask real worsening (false reassurance)
- **the bridging toolkit** — freeze the definition within a window, dual old/new counting across a bridging period, retrospective re-coding, bridging coefficients, and publish the definition history

what_it_is_not:
- **Not necessarily a real change in disease.** An apparent rise or fall may be partly an artefact of how "a case" came to be counted, not biology, behavior, or an intervention's effect. The observed series is real epidemiological signal *plus* a shifting definitional boundary *plus* noise, and crediting a causal story before auditing the definition history mistakes the instrument's move for the world's.
- **Not Goodhart distortion.** Case-definition drift is *upstream* of agents gaming a measure: the operational definition itself changes, not the strategic response of people to being measured. No one need be optimizing against the metric — the categorization criteria simply moved under a stable label.
- **Not selection bias.** Drift alters which events are *counted as cases*, irrespective of how the sample was drawn; it is a change in the categorization instrument, not a skew in sampling. A perfectly representative sample still yields incomparable counts across a definitional change.
- **Not inherently inflationary.** Drift does not only manufacture spurious increases. A *broadened* definition inflates counts (a false alarm), but a *tightened* one deflates them and can mask real worsening (a false reassurance) — so the direction of the definitional shift sets the sign of the artefact, and the analyst must reason both ways.
- **Not visible in the published output.** The artefact component gives no internal signal in the time-series; the system reports counts as commensurable whether or not its criteria changed. Comparability across a definitional change is never given by the data and must be established by deliberate audit of the definition history.
- **Not regression to the mean or survivorship bias.** It is not a statistical artefact of extreme baselines, nor a selection pathology acting on outcomes. Case-definition drift operates on the *input categorization* — what gets counted as a case — which is why threshold-shift forms like the Will Rogers phenomenon fall under it as boundary-move special cases.

clarity:
Naming case-definition drift reframes an apparent change in disease as a possible change in the *measuring instrument*, and that single reframe is its central clarifying force. Without the label, a rising prevalence curve is read as biology, behavior, or the effect of an intervention, and the epidemiologist reaches for causal explanations of a trend that may be partly an artifact of how "a case" came to be counted. With it, the analyst is obliged to ask, before any causal story, whether the operational case definition shifted, where, and how its magnitude compares to the observed move — a discipline that has both deflated false alarms and surfaced false reassurances, as when a tightened definition masked real worsening. The concept thereby decomposes a surveillance time-series into separable contributions — genuine epidemiological signal versus a definitional-boundary function that maps events to counts — and insists they cannot be read off the published output alone, because the system gives no internal signal that its categorisation instrument changed under a stable label.

The label also sharpens the distinction between drift and its neighbors, which determines the correct remedy. It is *upstream* of Goodhart distortion: the definition itself changes, not agents' strategic response to being measured. It is not selection bias, since it alters which events are *counted as cases* irrespective of how the sample was drawn. And it subsumes threshold-shift forms — the Will Rogers phenomenon in cancer staging, where reclassifying borderline patients lifts the apparent survival of both the stage they leave and the stage they enter with no change in any patient — as a boundary-move special case rather than a separate puzzle. Localizing the breakdown to the definition, and recognizing that comparability across a definitional change is not given but must be engineered, is exactly what licenses the surveillance-specific toolkit the concept implies: freezing the definition within a window, dual old/new counting across a bridging period, retrospective re-coding, bridging coefficients, and publishing the definition history so downstream analysts can audit whether two periods are even commensurable.

manages_complexity:
Surveillance data is beset by a long and heterogeneous catalogue of pathologies that can each bend a reported trend: a revised ICD code, an updated CSTE or DSM case definition, a billing-code split, an EHR documentation-template change, a screening-program rollout that catches earlier-stage disease, an assay-sensitivity upgrade that reclassifies subclinical cases, a guideline that resets a diagnostic threshold and reclassifies tens of millions overnight, a registry-form revision, a shift in clinicians' test-ordering threshold. Confronting an apparent rise or fall in disease without the concept, the epidemiologist faces a sprawl of possible causes and a standing temptation to reach immediately for a biological, behavioral, or interventional story for a movement that may be partly an artifact of how "a case" came to be counted — re-litigating each surveillance series from scratch with no shared structure linking the autism-prevalence debate to the sepsis-incidence transition to the cause-of-death discontinuity at an ICD revision.

Case-definition drift collapses that catalogue into a single diagnosis by reducing any surveillance time-series to a fixed compositional decomposition: the observed series equals a real underlying epidemiological signal, plus a definitional-boundary function that maps events in the world to counts in the dataset, plus sampling noise. Every item in the heterogeneous catalogue — code revision, template change, threshold reset, assay upgrade — enters that decomposition through the same middle term, because each is just another way the operational categorisation instrument changed while the output time-series label stayed fixed. So instead of inventing a bespoke explanation for each anomaly, the analyst tracks one thing: the definition history of the series, and the magnitude of its boundary-shift contribution measured against the size of the observed move. That single regularity — instrument changed under a stable label — is what the analyst reads, and it carries a hard methodological consequence the framing makes explicit: the artefact component is invisible in the published output, so commensurability across a definitional change is never given by the data and must be established by deliberate audit of the definition history.

The decomposition then fixes a compact branch structure for both diagnosis and remedy. Diagnostically, before any causal story, the analyst asks whether the operational definition shifted, where, and how its magnitude compares to the observed change — a fork that has both deflated false alarms (an apparent rise that was definitional broadening) and surfaced false reassurances (a tightened definition masking real worsening). The framing also sorts drift from its neighbors in a way that selects the correct fix: it is upstream of Goodhart distortion (the definition changes, not agents' strategic response to being measured), distinct from selection bias (it changes which events are counted as cases irrespective of sampling), and it subsumes threshold-shift forms such as the Will Rogers phenomenon in cancer staging as a boundary-move special case rather than a separate puzzle. Once drift is identified, the remedy menu is fixed and shares one shape regardless of which catalogue item caused it: freeze the operational definition within a surveillance window, maintain parallel old-definition and new-definition counts across a bridging period, apply retrospective re-coding where feasible, compute bridging coefficients for trend analysis that crosses the change, and publish the definition history alongside the series so downstream analysts can audit commensurability. The epidemiologist thereby reasons from one tracked quantity — the definition history against the observed move — straight to whether a trend is real, which neighbor-effect it is not, and which bridging tool restores comparability, replacing a pathology-by-pathology, series-by-series investigation with a single low-dimensional decomposition and a fixed corrective toolkit.

abstract_reasoning:
The concept's master move is a *compositional decomposition* the epidemiologist runs before any causal story: read an observed surveillance series as real underlying epidemiological signal *plus* a definitional-boundary function that maps events to counts *plus* sampling noise, and ask which term moved. The reasoning is FROM "autism prevalence rose across decades" or "COVID cases fell 40% year-on-year" *not* to "biology changed" but to "decompose the series — how much of this move is the boundary function shifting under a stable label?" The decisive *diagnostic* discrimination is *real change versus instrument change*: the epidemiologist treats an apparent epidemiological shift as a candidate *measurement-instrument* shift until the definition history is audited, reasoning that the artefact component is *invisible in the published output* (the system gives no internal signal that its categorisation criteria changed), so commensurability across a definitional change is never given by the data and must be established by deliberate audit.

This diagnostic runs in *both directions*, which is the move's predictive teeth. The epidemiologist predicts that a *broadened* definition will inflate counts with no real change (deflating a false alarm — an apparent rise that was definitional expansion), and that a *tightened* definition will deflate counts and can *mask real worsening* (surfacing a false reassurance). So the reasoner does not assume drift always manufactures spurious increases; it reasons from the *direction* of the definitional shift to the *sign* of the artefact, and weighs that artefact's magnitude against the size of the observed move before crediting any trend.

The *boundary-drawing* move sorts drift from its neighbors in a way that selects the correct remedy. The epidemiologist reasons that drift is *upstream* of Goodhart distortion (the definition changes, not agents' strategic response to being measured), *distinct* from selection bias (it changes which events are *counted as cases* irrespective of sampling), and that it *subsumes* threshold-shift forms — classifying the Will Rogers phenomenon in cancer staging, where reclassifying borderline patients lifts the apparent survival of *both* the stage they leave and the stage they enter with no change in any patient, as a boundary-move special case rather than a separate puzzle. Each classification routes the case to a different analytic treatment.

The *interventionist* move follows a fixed toolkit, each tool predicted to restore comparability across a definitional change: freeze the operational definition within a surveillance window, maintain parallel old-definition and new-definition counts across a bridging period, apply retrospective re-coding where feasible, compute bridging coefficients for trend analysis that crosses the change, and publish the definition history alongside the series so downstream analysts can audit whether two periods are even commensurable. The reasoner predicts that a naive cross-period comparison spanning a definitional change is *not interpretable* without one of these, and restricts cross-site or cross-time comparison to windows where the operational definitions are commensurable — reasoning from one tracked quantity, the definition history against the observed move, straight to whether a trend is real, which neighbor-effect it is not, and which bridging tool the situation requires.

knowledge_transfer:
Within epidemiology and public-health surveillance the concept transfers *as mechanism*, carrying its compositional decomposition and its bridging toolkit intact across every condition and every kind of definitional change. The same real-signal-plus-boundary-function-plus-noise decomposition, the same both-directions diagnostic (broadened definitions inflate, tightened definitions deflate and can mask worsening), and the same remedy menu (freeze the definition within a window, dual old/new counting across a bridging period, retrospective re-coding, bridging coefficients, publish the definition history) carry across infectious-disease case counts under shifting COVID-19 and CSTE definitions, the multi-decade autism-prevalence debate under successive DSM revisions, cancer stage-migration under successive AJCC editions (the Will Rogers phenomenon as a threshold-shift special case), the Sepsis-1/2/3 transitions, hypertension and diabetes threshold resets that reclassify tens of millions overnight, and cause-of-death step-discontinuities at each ICD revision. They carry because the substrate is constant: a surveillance system mapping events through an operational case definition to a count, with a downstream comparison machinery that presents counts as commensurable across the change. Within this surveillance range it is mechanism, not analogy — *operational case definition*, *bridging coefficient*, and *commensurability across a definitional change* are literal everywhere in the field. (Two whole disciplines also transfer literally *into* this domain on the same grounds: gauge-drift statistical-process-control from metrology, and rater-drift / inter-rater-reliability methodology from psychometrics, because they describe the identical instrument-stability problem.)

Beyond epidemiology the honest verdict is **shared abstract mechanism — a strong, genuine recurrence**, not metaphor. The pattern that recurs is *a categorical measurement instrument that maps events through an operational definition to a category, whose definition silently changes over time or across implementers while the output continues to be reported as a commensurable time-series*. That mechanism really recurs as a *co-instance* — not a borrowed shape — in annotation drift in ML datasets (the instrument is a labeling guideline), rater drift in psychometrics, performance-review rubric drift in HR, gauge drift in manufacturing metrology, photometric drift in astronomical surveys, and regulatory-category drift in compliance. Because it is the *same mechanism* in each, the right unit to carry the cross-domain lesson is the general parent, which the seed names explicitly: the emergent candidate `instrument_interpretive_drift` (the categorical instrument producing a time-series changes while the series is reported as if it had not). "Case-definition drift" is the public-health surveillance profession's name for that broader primitive in one substrate — sibling to annotation drift, rater drift, gauge drift, and photometric drift in theirs. What stays home-bound is the surveillance accent: the *operational case definition* vocabulary, the *CSTE / DSM / AJCC / ICD* revision machinery, the *CDC bridging-coefficient* methodology, and the specific worked cases (autism prevalence, Will Rogers staging, sepsis transitions). A drifting ML label set has no CSTE archive; an HR rubric has no ICD revision — yet all share the instrument-interpretive-drift mechanism, which is why the lesson belongs to that parent. A caution the seed flags directly: the tempting "four-domain" sketch (cybersecurity signature-set updates, aviation occurrence-classification updates, food recall-categorisation updates, financial audit-rule updates) names *parallel substrates of the broader pattern*, not transfers of the case-definition-drift named concept — so the correct move there is to invoke `instrument_interpretive_drift`, not to stretch the epidemiology label onto signatures and audit rules. This is exactly the boundary drawn in *Structural Core vs. Domain Accent*: the categorical-instrument-changes-under-a-stable-label skeleton lifts to `instrument_interpretive_drift` and recurs as true co-instances across measurement substrates; the public-health-surveillance accent — case definitions, CSTE/ICD revisions, bridging coefficients — stays home.


Causal Inference

one_liner: Infer the effect of changing X on Y from data by fixing a causal estimand and defending an identification design or assumption that separates that effect from noncausal association, then quantify its uncertainty and scope.

core_idea:
Causal inference is the disciplined practice of learning the effect of changing a treatment, exposure, action, policy, or variable on an outcome from necessarily incomplete data. It starts by fixing a causal estimand: a contrast between outcomes under at least two possible treatment or intervention states for a named population and time horizon. The observed data cannot reveal every alternative outcome for the same unit or system. An observed association therefore becomes a causal effect only through an explicit identification argument connecting the data to that missing contrast.

That argument may be manufactured by randomized assignment or defended from observational structure. Conditional exchangeability licenses adjustment for measured confounders; a valid instrument isolates exogenous treatment variation; continuity supports comparison around a cutoff; parallel trends supports a difference-in-differences contrast. Each strategy makes different claims about which noncausal paths are blocked and which effect is recovered. The estimator comes after this identification work. It maps the identifying contrast to the estimand, quantifies uncertainty, and exposes the result to diagnostics, falsification checks, sensitivity analysis, and scope limits.

The difference between causal and merely statistical inference is thus not a special regression coefficient or vocabulary. It is the combination of a directed change question, a counterfactual target, and a defended warrant for treating a data contrast as that target. A perfectly estimated association remains noncausal if this warrant is absent. Conversely, a responsible causal analysis may conclude that the effect is only bounded, locally identified, highly assumption-sensitive, or not identified at all.

scope_of_application:
The abstraction transfers literally across domains that share empirical causal questions and an identification apparatus:

- **Medicine and public health** — treatment, exposure, policy, and screening effects in trials and observational cohorts.
- **Economics and policy evaluation** — program, price, education, labor-market, and institutional effects.
- **Political and social science** — campaign, media, governance, conflict, and social-intervention effects.
- **Education and psychology** — instructional, behavioral, and environmental interventions.
- **Digital systems and product experimentation** — randomized feature tests, encouragement designs, spillover-aware experiments, and policy evaluation.
- **Biology and ecology** — manipulated or naturally varying exposures when a defined intervention and identification argument are available.

The named abstraction remains domain-specific because the literal practice depends on statistical estimands, observed-data distributions, assignment mechanisms, causal models, estimators, and uncertainty. Analogies in everyday reasoning may borrow its discipline, but without that empirical apparatus they belong under broader primes such as Causality, Counterfactuals, or Inference.

structural_signature:
**Sig role-phrases:**

- **the directed change question** — what would happen to outcome Y if treatment, exposure, action, or variable X were set differently?
- **the causal estimand** — the exact contrast among potential or interventional outcomes, attached to a population, time, treatment versions, and aggregation rule
- **the incomplete observation** — only some relevant alternative states are observed for any unit or system, creating the fundamental inferential gap
- **the causal representation** — a potential-outcomes, structural, graphical, or equivalent model that makes the target and relevant causal paths explicit
- **the identification warrant** — the design feature or substantive assumption that makes an observed-data contrast equal, bound, or otherwise inform the causal estimand
- **the estimator** — the rule translating the identified contrast into a numerical result with uncertainty
- **the threat audit** — tests, diagnostics, negative controls, falsification probes, and sensitivity analyses directed at the identification warrant
- **the scoped conclusion** — the effect actually supported for the population, treatment margin, setting, and intervention represented by the evidence

**Relations:**

- **Causal Inference → Statistical Inference** — strict subsumption: the genus is inference from finite, noisy observations; the differentia are the causal estimand and identification burden.
- **Causal Inference → Counterfactuals** — strict presupposition: alternative treatment-state outcomes define the missing contrast that the analysis seeks to recover.
- **Natural Experiment → Causal Inference** — strict subsumption: found as-if-random assignment supplies the specialized warrant.
- **Instrumental Variable → Causal Inference** — strict subsumption: relevance, exclusion, and exogeneity supply the specialized warrant.
- **Selection on Observables → Causal Inference** — strict presupposition: the assumption has its identifying role only inside an observational causal inference.
- **Internal Validity → Causal Inference** — strict presupposition: the property audits whether the study's causal conclusion is warranted in its own setting.

what_it_is_not: Causal inference is not an observed correlation with a causal label, a predictive model that forecasts accurately, temporal precedence by itself, or a plausible mechanism story without an identification bridge. It is also not identical to randomized experimentation: experiments are one powerful route, while natural experiments and observational strategies can supply different warrants. Nor does the label promise point identification. Bounds, sensitivity regions, and demonstrations of non-identification are legitimate outputs when the evidence cannot support a unique effect.

clarity:
A compact audit asks six questions:

1. What intervention or treatment contrast is being claimed?
2. For which population, time horizon, and treatment versions is the effect defined?
3. Which outcomes are missing by construction?
4. What exact design feature or assumption connects the observed contrast to the target?
5. What would violate that warrant, and how was it probed?
6. How far does the estimate travel beyond the units and margin actually identified?

If the fourth question has no answer, the analysis has not crossed from association to causal inference. If the other answers remain vague, it may have crossed only rhetorically.

manages_complexity: Causal inference decomposes an otherwise open-ended causal claim into separately inspectable layers: target definition, representation, identification, estimation, validation, and transport. This prevents a strong estimator from laundering a weak design and prevents substantive plausibility from replacing uncertainty quantification. It also localizes disagreement. Analysts can agree on the estimand and estimator while contesting exchangeability, or agree on identification while contesting generalization.

abstract_reasoning: The central abstraction is a missing-data problem constrained by causal structure. For each unit, outcomes under mutually exclusive treatment states cannot all be observed at once. Identification supplies invariances or independence relations that permit information from observed units, times, thresholds, or assignment mechanisms to stand in for those missing outcomes. Different designs are variations on this bridge-building problem, not merely a catalog of estimators.

knowledge_transfer: Transfer works by preserving the inference contract rather than copying a favored method. A policy threshold, a genetic variant, a lottery, and staggered adoption look substantively different, yet each can be analyzed by asking what variation is exogenous, what paths remain open, what population is moved, and what counterfactual contrast the data identify. The transfer fails when institutional or biological details invalidate the borrowed warrant—for example, when a cutoff is manipulated, an instrument has a direct outcome path, or a comparison group lacks parallel trends.

notes: _(New domain genus; queued for Claude style harmonization, FACT-anchor treatment, and independent citation verification.)_


Centipede Game

one_liner: A sequential finite-horizon game in which each player can take the larger share or pass to grow the pot, where backward induction prescribes taking on the first move — isolating common knowledge of rationality as the load-bearing assumption cooperation depends on.

core_idea: The Centipede Game, introduced by Robert Rosenthal in 1981, is a sequential two-player game with perfect information and a known finite horizon in which cooperation unravels by backward induction despite generating larger total payoffs the longer it is sustained. The structure is as follows: at each node one player chooses either to *take* — ending the game and claiming the larger share of the current pot, leaving the smaller share for the opponent — or to *pass* — handing the move to the opponent and doubling (or otherwise growing) the total pot, giving the opponent the same take-or-pass choice at a larger stake. The game ends either when a player takes or when the last node is reached. Because the pot grows with each pass, continued cooperation is collectively beneficial; but subgame-perfect equilibrium analysis via backward induction prescribes *take on the very first move*. The logic is inexorable: at the final node the mover should take; knowing this, the second-to-last mover should take pre-emptively at the penultimate node rather than pass and receive less; that reasoning propagates backward through every node until the first mover, anticipating defection at every subsequent node, takes immediately. The predicted outcome is immediate defection and the minimum possible payoff for both players. The empirical finding, first documented systematically by McKelvey and Palfrey in 1992, is that experimental subjects routinely pass for multiple rounds, often reaching the end or near-end of the game and splitting much larger total stakes than the backward-induction equilibrium would predict. This gap between equilibrium prescription and observed behavior is the game's diagnostic value: it isolates the *common knowledge of rationality* assumption as the load-bearing element, since the backward-induction chain requires not only that each player is rational but that each player knows the other is rational, knows the other knows, and so on without bound. When that common-knowledge chain is broken — by bounded rationality, by uncertainty about the opponent's reasoning depth, or by other-regarding preferences that attach value to the opponent's welfare — cooperation can be sustained for several rounds before unraveling. The game is the canonical laboratory illustration of finite-horizon backward-induction failure, and it closely parallels the chain-store paradox (Selten 1978), in which a monopolist facing a sequence of potential entrants should, by backward induction, never fight entry — yet empirically and strategically often does.

scope_of_application:
The Centipede Game lives across game theory and strategic-interaction analysis wherever a situation is genuinely centipede-shaped — a sequential, perfect-information, common-knowledge finite horizon with a growing joint payoff and a known last move; its reach is one substrate (reasoning agents) at many social scales, and the broader unravelling family travels under the parent `finite_horizon_unravelling` / `backward_induction`, not under this named game.

- **Game theory proper** — the canonical illustration of finite-horizon backward-induction failure, studied beside the chain-store paradox and finitely-repeated prisoner's dilemmas with a common-knowledge endpoint.
- **Behavioral economics** — the standard testbed where equilibrium concepts diverge from experimental play (the McKelvey–Palfrey pass-for-rounds finding), a staple of teaching.
- **Bounded-rationality modeling** — the proving ground for level-k reasoning, quantal-response equilibrium, and other refinements predicting the observed pass-then-take pattern.
- **Reputation and trust experiments** — designs letting reputation accumulate across rounds, framing the centipede as a repeated trust game under backward-induction pressure.
- **Finite-horizon negotiation and treaty analysis** — the unravelling of long-running cooperative arrangements once the end-date becomes common knowledge.

structural_signature:
**Sig role-phrases:**

- **the two players** — sequential movers with strict turn order and perfect information
- **the take-or-pass move** — at each node: *take* (end the game, claim the larger share of the current pot) or *pass* (hand the move to the opponent, growing the pot, granting them the same choice at a larger stake)
- **the growing pot** — the joint payoff that increases with each pass, making continued cooperation collectively beneficial
- **the known finite horizon** — a common-knowledge last move, both players knowing exactly how many nodes remain
- **the common-knowledge-of-rationality assumption** — each player rational, each knowing the other is, each knowing the other knows, without bound — the isolated load-bearing premise
- **the backward-induction chain** — solving from the terminal node: the last mover takes, so the penultimate takes pre-emptively, propagating back to the first mover
- **the unravelling equilibrium** — the unique subgame-perfect prescription of *take on the very first move*, the minimum joint payoff despite cooperation being collectively better at every step
- **the empirical departure** — the robust finding that subjects pass for multiple rounds, splitting much larger stakes than equilibrium predicts
- **the three-branch diagnosis** — the only ways the remaining assumption can fail: bounded reasoning depth, belief-uncertainty about the opponent's depth, or other-regarding preferences, each implying a distinct testable model

what_it_is_not:
- **Not a prisoner's dilemma.** There is no dominant-strategy defection here. The centipede is *sequential, finite, perfect-information*, and cooperation fails through the *backward-induction* chain — the last mover would take, so the penultimate takes pre-emptively, and the logic propagates to move one — not through single-shot dominance. The failure mode is unravelling from a known last move, not a temptation that pays regardless of the partner.
- **Not evidence that players who pass are irrational.** Passing for several rounds is the robust empirical finding, and the game's value is to force a *specific* reading of it rather than a wave at "irrationality": players reason to bounded depth, or reason fully but doubt the opponent does, or value the opponent's payoff. The deviation indicts the unbounded *common-knowledge-of-rationality* chain, not the rationality of the people who cooperate.
- **Not a recommendation to "take immediately."** Backward induction *prescribes* taking on move one as the unique subgame-perfect equilibrium, but that outcome is the minimum joint payoff while passing is collectively better at every step. The prescription is a diagnostic about what the rationality chain implies, not strategic advice to emulate; in practice, cooperating past the equilibrium is what splits the larger stakes.
- **Not driven by greed or temptation.** Nothing in the structure makes defection dominant; the unravelling is produced purely by the *known finite horizon* plus mutual rationality being common knowledge. This is exactly why obscuring the horizon, adding uncertainty about the last move, or building reputation across rounds can sustain cooperation — the collapse is a property of the common-knowledge chain, not of a payoff temptation.
- **Not a cross-substrate "centipede dynamic."** The mechanism requires reasoning agents propagating a common-knowledge chain; it has no counterpart in chemistry, ecology, or engineering, which contain no such agents. What recurs across the family — the chain-store paradox, finitely-repeated prisoner's dilemmas with a known endpoint — is the parent pattern of finite-horizon unravelling / backward-induction collapse, of which this game is the cleanest two-player illustration, not a structure that travels under its own name.

clarity:
The centipede game's clarifying force is that it isolates a single load-bearing assumption and makes its failure observable. Many cooperation breakdowns are overdetermined — information asymmetry, coordination problems, risk aversion, and temptation to defect all press at once, and one cannot say which is doing the work. The centipede strips those away: perfect information, a known horizon, no dominant-strategy temptation (passing is collectively better at every step), so the *only* thing driving the unique subgame-perfect prescription of "take on move one" is the backward-induction chain, and that chain in turn rests entirely on *common knowledge of rationality* — each player rational, each knowing the other is, each knowing the other knows, without end. Naming the game gives a theorist a clean instrument for that assumption: it converts an abstract worry about how much idealized common knowledge equilibrium reasoning smuggles in into a concrete, replicable measurement of how far real reasoning actually propagates.

That isolation sharpens what an observed deviation can mean. When subjects pass for several rounds — the robust McKelvey–Palfrey finding — the game forces the modeller to commit to a specific reading rather than waving at "irrationality": either players do not reason backward all the way (bounded depth), or they reason fine but do not *believe* their opponent reasons all the way (uncertainty about the other's depth), or they value the opponent's payoff (other-regarding preferences). Each branch carries distinct, testable content, and the centipede is the setting where they are pried apart — which is exactly why level-k models, quantal-response equilibrium, and reputation designs are tried *here*. The sharper question it licenses is not "are people rational?" but "how deep does the common-knowledge-of-rationality chain actually run before it breaks, and which break is operating?" — and by exhibiting the same unravelling skeleton as the chain-store paradox, it tells the field that the diagnosis travels to any finite-horizon cooperation with a known last move, not just to this toy.

manages_complexity:
Cooperation breakdowns, taken as they come, are overdetermined messes: information asymmetry, coordination failure, risk aversion, and the temptation to defect all press at once, and an analyst confronting a real unravelling rarely knows which force is responsible. The centipede game compresses that tangle by constructing a setting in which all but one of the forces is shut off by design — perfect information, a known finite horizon, and no dominant-strategy temptation, since passing is collectively better at every node — so that exactly one mechanism remains to drive the unique "take on move one" prescription: the backward-induction chain, which itself rests entirely on common knowledge of rationality. The sprawl of candidate causes collapses to a single load-bearing assumption, isolated and made observable, so the analyst no longer has to disentangle competing explanations of a given breakdown but can instead measure how far one chain propagates.

That isolation imposes a clean, small branch structure on what an observed deviation can mean. When subjects pass for several rounds against the equilibrium — the robust empirical finding — the explanation space is not open-ended; it is exactly three branches, because the only assumption left to fail can fail in only a few ways: players do not reason backward all the way (bounded depth), or they reason fine but do not believe the opponent does (uncertainty about the other's depth), or they value the opponent's payoff (other-regarding preferences). The modeller tracks which branch operates and reads off the rest — which is precisely why level-k models, quantal-response equilibrium, and reputation designs are all tested in this one setting, each a commitment to a specific branch. The compression also travels: because the same unravelling skeleton appears in the chain-store paradox and in any finite-horizon cooperation with a known last move, recognizing a situation as centipede-shaped imports the whole apparatus — the single load-bearing assumption, the three-way diagnosis, the depth-of-reasoning parameter — without re-deriving it. A high-dimensional, overdetermined question about why cooperation fails reduces to one assumption, one measurable quantity (how deep the common-knowledge-of-rationality chain runs before it breaks), and a three-way branch from which the qualitative behavior follows.

abstract_reasoning:
The centipede game licenses reasoning moves a game theorist runs on any finite-horizon cooperation problem, all flowing from the backward-induction chain and the common-knowledge-of-rationality assumption it rests on.

The constructive move is *backward-induction prediction from the last node*. Given a sequential game with a known finite horizon, the analyst reasons from the end: identify what the final mover does (take), then propagate that determination backward — if the last mover takes, the penultimate mover should take pre-emptively rather than pass into a smaller share; if the penultimate takes, so should the one before; and the chain runs unbroken to the first mover, who takes immediately. The predicted outcome is immediate defection at the minimum joint payoff, derived not by looking forward from the start but by solving from the terminal node and folding the solution back. This is the move that generates the unique subgame-perfect prescription, and its characteristic signature is that *more cooperation is collectively better at every step* yet the equilibrium still prescribes none — the analyst predicts the welfare-destroying outcome precisely because each link in the chain is individually sound.

The second move is *isolation-by-design diagnosis*: using the game's stripped structure to pin a deviation on one assumption. Because perfect information, a known horizon, and the absence of any dominant-strategy temptation are built in, the analyst reasons that *nothing else* can be driving the unique prescription except the backward-induction chain, which rests entirely on common knowledge of rationality. So when behavior departs from "take on move one," the deviation cannot be blamed on information asymmetry or coordination failure (those are designed out) and must implicate the common-knowledge chain itself. This converts a vague worry about idealized equilibrium reasoning into a concrete measurement: the game becomes an instrument for asking *how far the common-knowledge-of-rationality chain actually propagates before it breaks*, with the round at which taking typically occurs reading off the effective depth of mutual reasoning.

The third move is *three-branch attribution* of an observed deviation. When subjects pass for several rounds against the equilibrium, the analyst does not wave at "irrationality" but commits to exactly one of three readings, because the single remaining assumption can fail in only a few ways: either players do not reason backward all the way (bounded reasoning depth), or they reason fully but do not *believe* the opponent reasons all the way (uncertainty about the other's depth), or they attach value to the opponent's payoff (other-regarding preferences). Each branch carries distinct, testable content and implies a different model — bounded depth points to level-k reasoning, belief-uncertainty to quantal-response equilibrium, other-regarding preferences to social-utility designs — so the analyst's move is to design the manipulation that pries the branches apart (vary the opponent's known sophistication, vary the payoff structure's distributional stakes) and read which branch the data support. The reasoning is interventionist: each candidate fix is simultaneously a hypothesis about which break is operating, confirmed or refuted by whether it shifts the observed unraveling point.

The fourth move is *skeleton-recognition transfer within the domain*: identifying a real situation as centipede-shaped and importing the whole apparatus. The analyst reasons that any finite-horizon cooperation with a known last move and a growing joint payoff carries the same unraveling pressure — most directly the chain-store paradox, where a monopolist facing a known finite sequence of entrants should by backward induction never fight, yet strategically and empirically often does. So recognizing the shape lets the analyst predict the equilibrium (unravel to immediate defection), anticipate the empirical departure (cooperation sustained well past the equilibrium), and reach for the same three-way diagnosis and the same depth-of-reasoning parameter — without re-deriving any of it. The interventionist payoff travels too: to *sustain* cooperation in a finite-horizon setting, the analyst predicts one must break the common-knowledge chain on purpose — obscure the horizon, introduce uncertainty about the last move, or build reputation across rounds — because as long as the known last move and mutual rationality are common knowledge, the backward-induction collapse is the predicted fate.

knowledge_transfer:
Within game theory and strategic-interaction analysis the centipede game transfers *as model* — functioning here as transfer-as-mechanism — wherever a situation is genuinely centipede-shaped: a sequential, perfect-information, common-knowledge finite horizon with a growing joint payoff and a known last move. Where that precondition holds, the whole apparatus applies literally: the backward-induction prediction (unravel to immediate defection), the anticipated empirical departure (cooperation sustained well past the equilibrium), the three-way attribution of any deviation (bounded depth / belief-uncertainty about the opponent's depth / other-regarding preferences), the depth-of-reasoning parameter, and the interventionist corollary that *sustaining* cooperation requires breaking the common-knowledge chain on purpose (obscure the horizon, introduce uncertainty about the last move, build reputation). So it carries without translation to the chain-store paradox (a monopolist facing a known finite sequence of entrants), to finite-horizon negotiation and treaty exits once the end-date is known, to end-of-tournament cooperation, and to finitely-repeated prisoner's dilemmas with a common-knowledge endpoint — each a real instance of the same unravelling pressure, recognized rather than likened.

The honest qualification is that this apparent breadth is *one substrate at many social scales*, not transfer across structurally distinct domains. Every one of those cases — bilateral bargaining, business-cycle endings, employee retention near retirement, treaty termination — is still *cooperation among reasoning agents under a finite horizon*, so what looks like cross-domain reach is the model applying within a single substrate (strategically reasoning agents) at varying scales, which is why the transfer is mechanism-preserving rather than analogical there. Where the model genuinely stops is the boundary of that substrate: stripped of game-theoretic vocabulary (sequential moves, payoffs, equilibrium, backward induction), the centipede is the substrate-specific observation that "in a finite cooperative situation with a known last move, the rational-prescription chain unravels cooperation from the back" — and that does **not** port to chemistry, ecology, or engineering, because those have no reasoning agents propagating a common-knowledge chain. Any invocation of "a centipede dynamic" outside reasoning-agent cooperation would be borrowing the unravelling *shape* without the mechanism, and should be marked as analogy.

The genuinely portable content is one level up: the broader pattern *finite-horizon unravelling / backward-induction collapse of cooperation*, of which the centipede game is the cleanest two-player illustration and the chain-store paradox and finitely-repeated PD are siblings. That parent (the side-captured `finite_horizon_unravelling`, kin to `backward_induction`) is the level at which the lesson recurs across the family, and it is what should be carried — with the centipede cited as the canonical instrument that isolates *common knowledge of rationality* as the load-bearing assumption — rather than the name "centipede game," whose specific cargo (Rosenthal's construction, the McKelvey–Palfrey pass-for-rounds finding, the level-k/quantal-response/social-utility refinements tested on it) is a game-theory artifact best kept as the famous illustration. Mechanism within reasoning-agent cooperation at every scale; the genuine cross-family reach resident in the *parent unravelling pattern*, not in this named game. This is exactly the boundary *Structural Core vs. Domain Accent* draws.


Central Executive Network

one_liner: Coordinate goal-directed, externally oriented cognition through a distributed frontoparietal control network that maintains task rules, working-memory contents, and top-down attention while flexibly coupling to other systems as demands change.

origin_domain: neuroscience

core_idea:
The central executive network (CEN) is a distributed functional brain network,
commonly centered on dorsolateral prefrontal and posterior parietal regions,
that supports externally directed, goal-maintaining cognition. It coordinates
working-memory contents, task rules, top-down attention, response selection,
and flexible adjustment when demands change. Its identity is relational rather
than local: no single cortical region is "the executive"; executive control
arises from coordinated activation and task-dependent coupling across the
network and with sensory, motor, mnemonic, and salience systems.

The network is commonly discussed within a triple-network model. The
default-mode network supports internally directed and self-referential
processing; the central executive network supports externally directed,
goal-guided processing; and the salience network detects behaviorally relevant
events and helps recruit or disengage the other two. This division is a
functional architecture, not a claim that the networks are mutually exclusive
pieces of tissue. Nodes can participate in more than one task-defined network,
and coupling changes with context.

scope_of_application:
The central executive network applies literally within cognitive, systems,
and clinical neuroscience. It organizes findings from working memory,
selective attention, cognitive control, task switching, inhibition,
problem-solving, and disorders involving impaired executive coordination.
Claims about organizational or software "executive networks" are analogies;
their portable structure belongs to Network, Attention, state transition, and
control abstractions.

structural_signature:
**Sig role-phrases:**

- **the frontoparietal node set** — distributed prefrontal and parietal regions
  whose coordinated activity tracks executive demand
- **the maintained task model** — goals, rules, priorities, and relevant
  working-memory contents held available across intervening events
- **the top-down allocation** — selective attention directed according to the
  task model rather than only bottom-up salience
- **the flexible coupling** — task-dependent coordination with sensory, motor,
  memory, and other control systems
- **the demand response** — recruitment rising when conflict, updating,
  inhibition, or multi-step control becomes harder
- **the control-failure signature** — impaired maintenance, switching,
  inhibition, or allocation rather than loss of one content representation

what_it_is_not:
- **Not a single executive center.** The construct is a functional network,
  not a modern version of one homuncular command region.
- **Not all executive function.** Executive functions are a family of
  capacities and tasks; the CEN is one neural-network architecture implicated
  in coordinating them.
- **Not the salience network.** The salience network detects relevant events
  and helps initiate a network switch. The CEN is the externally oriented
  goal-control configuration recruited by that switch.
- **Not the default-mode network.** The default-mode network is relatively
  prominent during internally directed cognition; the CEN is relatively
  prominent during externally directed goal pursuit, though real tasks can
  recruit or couple both.
- **Not a permanently fixed anatomical partition.** Functional-network
  boundaries depend on measurement, parcellation, and task, and terminology
  overlaps with "frontoparietal control network" and "executive control
  network." The re-authoring pass should preserve distinctions supported by
  current neuroscience rather than asserting perfect synonymy.

clarity:
The network frame replaces a search for the brain's single command center with
a coordination question: which distributed regions are coupled, what goal or
rule do they maintain, and how does that coupling change as task demands
change? It also separates content from control failure. A person may retain
the relevant representations yet fail to keep them active, prioritize them,
switch appropriately, or suppress competitors; those are candidate failures
of executive coordination rather than evidence that the content itself is
absent.

manages_complexity:
Many task-specific activation findings compress into a small architecture:
distributed nodes, a maintained task model, top-down allocation, flexible
coupling, and demand-sensitive recruitment. Instead of assigning each
executive task to a unique region, the analyst asks how the same network
reconfigures across tasks and which connection or control operation fails.

abstract_reasoning:
The model supports forward prediction: increasing conflict, working-memory
load, rule switching, or distractor pressure should increase executive-network
recruitment and alter its coupling to task-relevant systems. It also supports
diagnosis by dissociation: intact content with poor maintenance or switching
points toward control-network dysfunction, while missing content despite
appropriate control points elsewhere.

Within the triple-network architecture, a salient event predicts engagement of
the salience network followed by recruitment of the CEN and relative
disengagement of default-mode activity when the event demands externally
directed action. A failure can therefore lie in detection, switching, or
executive maintenance; the framework keeps those stages separate.

knowledge_transfer:
Within neuroscience, the complete architecture transfers among working-memory,
attention, cognitive-control, task-switching, and clinical-connectivity
research. Across those subfields, researchers refill the same roles—node set,
task model, allocation rule, coupling pattern, and failure signature—rather
than treating every activation map as an unrelated result.