Type S Error

one_liner: Quantify the risk that a statistically significant estimate points the wrong way by computing, before data collection, the probability that a two-sided significance filter is cleared from the opposite tail when the true effect is near zero relative to noise.

core_idea:
A Type S (sign) error is the failure mode in which a statistically significant estimate has the wrong sign — claiming a positive effect when the true effect is negative, or vice versa — because the study's power is so low relative to the true effect size that the significance threshold is passed almost exclusively by large-magnitude draws from the sampling distribution, and when the true effect is near zero relative to sampling noise, both tails of the distribution can generate significant results with similar probability. Gelman and Carlin (2014) introduced Type S alongside Type M to reframe inferential risk away from the classical null-versus-alternative binary and onto the question: given that I will only act on a significant result, how likely is my significant result to correctly identify the direction of the true effect? In extreme low-power settings — a study powered at 6% to detect a true effect of d = 0.05 — the probability of a sign-correct result given significance can fall to roughly 0.76, meaning roughly one in four significant results points the wrong way. At power levels near 50%, the conditional probability of a correct sign is still well below 1 if the true effect is small relative to sampling variation.

The mechanism is the same selection process that produces Type M exaggeration, but evaluated at the sign rather than the magnitude. A two-sided significance filter selects estimates whose absolute value exceeds a threshold; when the true effect is small, both positive and negative values above that threshold are reachable from the sampling distribution, and the relative frequencies with which each tail is reached depend on how far the true effect is from zero relative to the threshold. The closer the true effect is to zero, the more evenly the two tails contribute significant observations, and the closer Pr(sign correct | significant) is to 0.5. Sign flips between high-profile original studies and their high-powered replications — multiple reversal cases in behavioral science — are the downstream observable signature of Type S errors materializing across a literature. The diagnostic use of Type S is prospective: compute Pr(wrong sign | significant) at design time, before data collection, to determine whether a proposed study is informative about the direction of the effect even if it achieves significance.

scope_of_application:
Type S error lives across the significance-testing sciences that share the substrate of estimation from a two-sided significance-filtered sample with true effects near zero relative to noise; its reach is within that one statistical domain, the mechanism recurring in identical form across them (always evaluated jointly with its magnitude-sibling Type M). The broader pattern — select noisy estimates near zero through a gate and the selected extremum can carry the wrong sign — travels under `winner_s_curse` and the `selection_bias` family, not under this construct, which needs a two-sided NHST filter, a p-value, and power to be the thing it is.

- **Behavioural-science replication crisis** — priming and embodied-cognition reversals between high-profile originals and their high-powered replications, read as materialized Type S risk.
- **Health-policy A/B tests** — underpowered studies reporting a harmful or beneficial direction when the true effect is opposite.
- **Genome-wide association studies** — top discovery-stage SNP hits flipping sign at the replication stage when the true effect is small.
- **Election-poll subgroup analyses** — small-cell subgroup swings reversing direction across consecutive surveys.

structural_signature:
**Sig role-phrases:**

- **the near-zero noisy estimator** — a sampling distribution whose true effect sits close to zero relative to sampling noise, the low-power regime that makes sign error possible
- **the two-sided significance filter** — the gate admitting estimates whose absolute value clears a threshold, reachable from *both* tails when the truth is near zero
- **the two-tail reachability** — the geometry whereby, the closer the true effect is to zero relative to the standard error, the more evenly both tails contribute significant draws
- **the conditional sign-flip** — the failure: a significant estimate carrying the wrong direction because the opposite tail retains non-trivial mass (a 6%-power study against d=0.05 points wrong ~24% of the time)
- **the sign-error probability** — the prospective diagnostic scalar Pr(sign correct | significant), computable at design time from the effect-to-noise ratio and power
- **the near-zero-versus-far fork** — the boundary: far from zero only the correct tail is reachably significant and the sign is safe; near zero the probability slides toward 0.5 and the direction is barely better than a coin flip
- **the power-not-precision remedy** — the diagnosis-fixed prescription that the cure is enough power to push opposite-tail mass toward negligible, not better measurement of magnitude
- **the sign-reversal signature** — the retrospective tell: a direction flip between an original and its higher-powered replication, read as a materialized Type S risk rather than an unstable phenomenon
- **the joint pairing with Type M** — the operational unit: exaggeration ratio and Pr(wrong sign) computed together from one design, jointly judging whether a significant result's size and direction are both worth believing

what_it_is_not:
- **Not a Type I or Type II error.** Those frame error around the null-versus-alternative *decision*; Type S frames it around the *sign of the estimate conditional on rejection* — "given that I will only report a significant result, how often does it point the wrong way?" It presupposes the same hypothesis-testing scaffolding but asks a question about direction that the reject/fail-to-reject binary never poses.
- **Not the same as Type M error.** Type M is the *magnitude* sibling — right direction, exaggerated size — and is the more common failure; Type S is the sharper, rarer alarm of a *reversed* direction. They share the conditional-on-significance machinery and are computed jointly, but a wrong sign and an inflated magnitude are distinct verdicts on the same significant result.
- **Not regression to the mean.** Regression to the mean shrinks extreme follow-up measurements toward the average but does *not*, in general, flip the sign of a true effect that is far from zero. Type S flips signs precisely in the *near-zero* regime, where both tails of the filtered distribution are reachable — a different phenomenon with a different trigger.
- **Not evidence of fraud or an inherently unstable phenomenon.** A sign reversal between a high-profile original and its high-powered replication is the *predictable materialization* of a Type S risk already present in the underpowered original design, not a scandal and not proof that the effect is capricious. The framework reads direction-reversals as a diagnostic of underpowered sign claims rather than as instability.
- **Not fixable by better measurement of magnitude.** The danger is opposite-tail mass under low power, so the remedy is *enough power to push that mass toward negligible*, not more precise measurement of the effect's size. Improving magnitude precision without addressing power leaves the sign risk untouched.
- **Not a risk when the true effect is far from zero.** When the true effect is large relative to sampling noise, only the correct tail is reachably significant and the sign is safe — Pr(sign correct | significant) sits near one. The sign-flip danger is specific to true effects near zero relative to the standard error; it is not a universal indictment of all significant direction claims.

clarity:
Naming the Type S error punctures the rhetorical comfort of "at least the direction is right" — the implicit fallback a researcher reaches for when an effect size looks shaky but the sign of a significant result seems safe. Type S shows this fallback is exactly the assumption that fails under low power: when the true effect is small relative to sampling noise, a two-sided significance filter is reachable from *both* tails, so a significant result can carry almost no information about which way the effect actually points. The distinction it sharpens is between two questions the classical null-versus-alternative frame fuses — "is there an effect?" versus "given that I will only act on a significant result, is its direction trustworthy?" — and it makes vivid that significance can be achieved while the sign is barely better than a coin flip.

Its clarifying force is to convert "we found a positive effect" from an apparently safe claim into one with a *computable* failure probability, Pr(wrong sign | significant), available at design time before any data exist. This localizes the danger to a specific regime — true effects near zero relative to standard error — and tells the analyst that the remedy is not better measurement of magnitude but enough power to push the opposite-tail mass toward negligible. It also names the downstream signature precisely: a sign reversal between a high-profile original and its high-powered replication is not a scandal or a fluke but the predictable materialization of a Type S risk that was present in the original design, which lets a reader read direction-reversals in a literature as a diagnostic of underpowered sign claims rather than as evidence that the phenomenon is simply unstable.

manages_complexity:
Whether the *direction* of a reported effect can be trusted is, in practice, a worry that proliferates wherever estimates are read off small or noisy cells: an interaction term, a subgroup contrast, an exploratory comparison, a discovery-stage SNP, a subgroup swing in an election poll. Each such direction claim seems to demand its own judgment about how seriously to take "the effect went this way," and across a literature the sign reversals between high-profile originals and their replications can look like an unruly collection of one-off embarrassments or evidence that the phenomena themselves are simply unstable. Type S error compresses all of these onto a single mechanism and a single prospective scalar. The mechanism is the same significance-filter selection that drives magnitude inflation, evaluated at the sign instead of the size; the quantity that governs it is one number — Pr(sign correct | significant) — computable from the design before any data are collected. A scattered set of direction-of-effect anxieties collapses to one probability of pointing the wrong way.

What the analyst tracks reduces to essentially one thing: how large the true effect is relative to sampling noise, because that ratio alone sets how the two tails of the distribution contribute significant observations, and the qualitative verdict on a direction claim reads off it through a fixed regime split. A two-sided significance filter selects estimates whose absolute value clears a threshold; when the true effect is far from zero relative to the standard error, only the correct tail is reachably significant and the sign is safe, so Pr(sign correct | significant) sits near one and "we found a positive effect" can be believed. When the true effect is near zero relative to noise — the low-power regime — both tails clear the threshold with comparable probability, Pr(sign correct | significant) slides toward 0.5, and a significant result carries almost no information about which way the effect actually points; in extreme cases roughly one significant result in four points the wrong way. The verdict thus turns on a single boundary (true effect near versus far from zero relative to standard error), and the remedy the branch prescribes is specific: not better measurement of magnitude but enough power to push the opposite-tail mass toward negligible. The structure also fixes the downstream reading — a sign reversal between an original and its higher-powered replication is the predictable materialization of a Type S risk already present in the original design, so direction-reversals across a literature are read as a diagnostic of underpowered sign claims rather than as proof of an unstable phenomenon. A high-dimensional "can I trust the direction of each of these many contrasts" problem becomes a low-dimensional "compute one Pr(wrong sign) from the effect-to-noise ratio" problem with a single near-zero-versus-far-from-zero fork. The pair with Type M is the operational unit — exaggeration ratio and sign-error probability computed together from the same design — so one prospective calculation answers, jointly, whether a significant result's magnitude and its direction are both worth believing.

abstract_reasoning:
The defining move is *conditioning the sign on the filter* — asking not "is the estimate's direction right?" but "given that I will only report a significant estimate, how often does that significant estimate point the wrong way?" The analyst reasons from a two-sided significance gate to the geometry of which tails are reachable: the filter selects estimates whose absolute value clears a threshold, so when the true effect sits far from zero relative to sampling noise only the correct tail is reachably significant and the sign is safe, but when the true effect is near zero *both* tails clear the threshold with comparable probability. The characteristic inference runs from *the effect-to-noise ratio* to *Pr(sign correct | significant)*: the closer the truth is to zero relative to the standard error, the more evenly the two tails contribute significant draws, and the closer that probability slides toward 0.5 — in extreme low-power cases roughly one significant result in four points the wrong way.

This converts an apparently safe claim into one with a *computable failure probability available before data exist*. The analyst computes Pr(wrong sign | significant) at design time from the assumed effect and the power, turning "we found a positive effect" into a statement carrying a known chance of being directionally false. The inference is from *design parameters alone* to *whether a significant result will be trustworthy about direction*, which licenses a prospective verdict: a study can be near-certain to reach significance yet provide almost no information about which way the effect points, so the analyst judges its informativeness about *direction* separately from its probability of *detection*.

A *boundary-drawing* move punctures the specific fallback the classical frame invites — "at least the direction is right" — by locating exactly the regime where it fails. The analyst reasons that this comfort holds when the true effect is far from zero relative to noise and collapses when it is near zero, so the move is to identify which side of that boundary a given contrast sits on before trusting its sign. Crucially this also fixes the *remedy* by diagnosis: because the danger is opposite-tail mass and not mismeasurement, the prescribed fix is not better measurement of magnitude but *enough power to push the opposite-tail mass toward negligible* — the inference runs from the cause of the sign risk to the one intervention that removes it, ruling out precision improvements that would not touch it.

The *retrospective diagnostic* move reads direction-reversals across a literature as signal rather than scandal: a sign flip between a high-profile original and its high-powered replication is inferred to be the *predictable materialization of a Type S risk already present in the original design*, not evidence that the phenomenon is inherently unstable nor that the original was fraudulent. The analyst reasons backward from observed reversals to underpowered sign claims in the originals. And because the same significance-filter selection drives both magnitude and direction failures, the framework licenses a *joint* move: compute the exaggeration ratio and Pr(wrong sign) together from one design, so a single prospective calculation answers whether a significant result's size and its direction are both worth believing — with the natural intervention being to prefer interval estimation and meta-analytic pooling over a dichotomous single-study sign claim whenever the effect-to-noise ratio puts the study in the danger regime.

knowledge_transfer:
Type S error is a *named statistical bias and its prospective diagnostic quantity* — the probability that a significant estimate points the wrong way, summarised by Pr(sign correct | significant) — not a causal mechanism in the world, so "mechanism within / metaphor beyond" applies only loosely. Within statistics and the significance-testing sciences it transfers as full mechanism, and what carries is the whole apparatus: the condition-the-sign-on-the-filter move, the two-tail reachability geometry, the prospective Pr(wrong sign) computed at design time from an assumed effect and the power, the near-zero-versus-far-from-zero regime fork, the puncturing of the "at least the direction is right" fallback, the power-up (not better-measurement) remedy, and the sign-reversal-on-replication retrospective signature. The mechanism recurs in identical form across fields that share the substrate of *estimation from a two-sided significance-filtered sample with true effects near zero relative to noise*: the *behavioural-science replication crisis* (priming and embodied-cognition reversals between originals and high-powered replications), *health-policy A/B tests* (underpowered studies reporting a harmful or beneficial direction when the truth is opposite), *genome-wide association studies* (top discovery-stage SNP hits flipping sign at replication), and *election-poll subgroup analyses* (small-cell subgroup swings reversing direction across consecutive surveys). It is the sign-sibling of Type M, always evaluated jointly with it — one prospective calculation answering whether a significant result's magnitude and its direction are both worth believing. The transfer is literal because the substrate — a noisy estimator near zero, filtered by a two-sided significance threshold — is held fixed.

Beyond the significance-testing scaffolding the situation is the *shared-abstract-mechanism* case, with the same exact parent Type M inherits: the cross-domain structural pattern is the `winner_s_curse` — *selection on a noisy estimator produces a biased conditional distribution that, when the true value is near zero, has non-trivial mass on the opposite tail*, so the selected extremum can carry the wrong sign. That pattern recurs wherever noisy estimates are selected near zero, with `selection_bias` as the broader parent (here the selector is specifically a two-sided NHST filter). But that recurrence is winner's-curse-near-zero travelling, not Type S: strip the frequentist hypothesis-testing scaffolding and the significance gate, and the residual content is selection-on-noisy-estimates with an opposite-tail flip. What does *not* travel under Type S's own steam are the inference-domain particulars that make it Type S — the specific quantity Pr(wrong sign | significant), the specific two-sided NHST filter, and the design-stage retrospective inside frequentist testing; and note the boundary to its cousin `regression_to_the_mean`, which shrinks extreme follow-ups but does *not* in general flip the sign of true effects far from zero, whereas Type S flips signs precisely in the near-zero regime.

So the cross-domain lesson — *select the extremum of noisy estimates near zero and you can get not just an exaggerated magnitude but a reversed direction* — should be carried by `winner_s_curse` (and the broader `selection_bias` family), not by "Type S error," whose contribution is to operationalise that flip for the binary significance filter inside frequentist hypothesis testing. The portable structural reasoning belongs to the parent; the teachable, computable, replication-crisis-relevant operational content — Pr(wrong sign | significant), the joint Type M/Type S design-stage calculation, the NHST-conditioned sign diagnosis — is what Type S uniquely contributes and what stays in statistics and experimental design (see *Structural Core vs. Domain Accent*).


Type System

one_liner: A discipline that assigns every value and expression a type and uses that classification to constrain which operations may legally apply, verifying via compositional typing rules that well-typed programs cannot reach a wrong-kinded state (soundness).

status: draft

core_idea:
A type system is a programming-language-theoretic discipline that assigns every value and expression a type — a category drawn from a finite vocabulary — and uses that classification to constrain which operations may be legally applied to which expressions, verifying conformance either at compile time (static type systems) or at runtime (dynamic type systems). The load-bearing guarantee is type soundness, expressed by Milner's slogan "well-typed programs cannot go wrong": a typing judgment (in the static case, a formal proof derivable by the language's type rules) certifies that a program will not reach a state of applying an operation to an argument of the wrong kind — no integer treated as a function pointer, no null pointer dereferenced as a record, no float passed where a string is expected — and two meta-theorems, progress and preservation, formalize this guarantee as a structural induction over the operational semantics.

The mechanism by which a type system enforces this is a set of compositional typing rules. Each syntactic form in the language has a rule that assigns it a type given the types of its subexpressions: a function application `f x` is well-typed with result type `B` if `f` has type `A → B` and `x` has type `A`; a conditional `if b then e1 else e2` is well-typed with type `T` if `b` has type `Bool` and both `e1` and `e2` have type `T`. Compositionality means that a type checker can verify an entire program by checking each constituent expression once, propagating types inward and outward through the syntax tree — a linear-time check in most practical systems. Where the type language is richer, the checking problem becomes harder: parametric polymorphism requires unification; subtyping requires a subsumption relation; dependent types require definitional equality checking between type expressions that may involve arbitrary term computation, which can be undecidable.

The design space of type systems is defined by the tradeoff between expressiveness and decidability of checking. Simple types (C's primitive types) are fast to check but cannot express invariants about list lengths, sorted order, or ownership. Parametric polymorphism (Haskell, ML, Rust) allows functions to operate over any type satisfying a specified interface without sacrificing static checking. Linear and affine types (Rust's ownership and borrow system) track how many times a value may be used and who currently holds access to it, statically preventing data races and use-after-free errors — a security and correctness guarantee otherwise achievable only at runtime. Refinement types (Liquid Haskell) attach logical predicates to types, so a type like `{x : Int | x > 0}` constrains an integer to be positive and the type checker verifies this statically using an SMT solver. Dependent types (Coq, Agda, Lean, Idris) allow types to depend on term values, making a type like `Vec a n` (a vector of exactly `n` elements of type `a`) expressible and checkable — at the cost of requiring programs to carry proof terms and the checker to decide definitional equality. The Curry-Howard correspondence reveals that this hierarchy has a unified logical reading: a type is a proposition, a well-typed program is a proof of that proposition, type checking is proof verification, and the richness of the type language is exactly the expressiveness of the logic it encodes.

scope_of_application:
A type system lives across the formal-language and rule-driven-classification subfields anchored in programming-language theory; its reach is within that domain, wherever a formal symbolic system has classifiable values and a need to constrain operations syntactically. The cross-domain "compatibility rules" reach (drug interactions, legal status, materials) is the conjoined parent pattern `classification` + `constraint`, not literal type-system habitats here.

- **Programming languages** — the home turf, spanning the whole expressiveness frontier: C's coarse primitives, Java's nominal class hierarchy, ML/Haskell parametric polymorphism, Rust's linear/affine ownership, Liquid Haskell's SMT-checked refinements, and Coq/Agda/Lean/Idris dependent types.
- **Static analysis and verification** — refinement types, session types for concurrent protocols, and linear/affine types for resource use that turn correctness proofs into type checks.
- **Schema and serialization languages** — JSON Schema, XML Schema, Protocol Buffers, and Avro impose a type discipline on interchange formats.
- **Database schemas** — column types, constraints, and referential integrity form a type system over relational data.
- **Type theory and proof assistants** — Martin-Löf type theory, the calculus of constructions, and homotopy type theory turn the type system, via Curry-Howard, into a foundation for mathematics where checking is proof verification.

structural_signature:
**Sig role-phrases:**

- **the value-and-expression space** — the programs and values to which the discipline assigns types
- **the type vocabulary** — a finite set of category names and constructors from which every type is drawn
- **the compositional typing rules** — one rule per syntactic form tying each operation to admissible argument and result types (e.g. `f x : B` iff `f : A → B` and `x : A`)
- **the typing judgment** — a compositionally-derived certificate (a formal proof, in the static case) that an expression conforms
- **the checking procedure** — a checker (or runtime check) that propagates types through the syntax tree, typically a single linear-time pass, accepting or locating a mismatch
- **the soundness meta-theorem** — progress and preservation, proved by structural induction over the operational semantics, certifying that a well-typed program cannot reach a wrong-kinded state
- **the absence-certificate guarantee** — acceptance rules out an entire error class wholesale ("well-typed programs cannot go wrong"), inverting effort from hunting bugs to pushing invariants into types
- **the expressiveness-vs-decidability frontier** — the single design axis trading what types can express against what the checker can decide, with a known cost gradient (linear check → unification → SMT discharge → definitional-equality, possibly undecidable)
- **the Curry-Howard reading** — type as proposition, well-typed program as proof, checking as proof verification, unifying the feature hierarchy as increasing logical strength

what_it_is_not:
- **Not a heuristic bug-finder or linter.** A sound type checker does not flag suspicious-looking code; it *certifies* the absence of an entire error class. Progress and preservation prove that a well-typed program cannot reach a wrong-kinded state, so acceptance is a guarantee, not a warning. Reading "well-typed" as "probably fine" misses that the checker rules the failure mode out wholesale rather than estimating its likelihood.
- **Not a guarantee that a program is correct.** Milner's "well-typed programs cannot go wrong" means a precise, narrow thing: no operation is applied to an argument of the wrong kind. It does not certify that the program computes the right answer, terminates, or is free of logic errors. A well-typed program can be completely wrong about its intended behavior; type soundness bounds one class of failure, not correctness in general.
- **Not necessarily static.** A type system need not check at compile time. Dynamic type systems classify values and constrain operations at runtime, raising type errors as the program executes. Equating "type system" with "static checking" excludes dynamically-typed languages, which have genuine type systems enforced at a different time.
- **Not "richer is always better."** Adding expressive power to the type language is not a free win. Expressiveness trades against decidability of checking along a single frontier: simple types check in linear time, polymorphism needs unification, refinements need an SMT solver, and dependent types require deciding equalities that can be undecidable. A "more powerful type system" buys more statable invariants at a known cost in checking difficulty, not unalloyed improvement.
- **Not the type-inference algorithm.** A type system *specifies* what is well-typed — the rules and the soundness guarantee; type inference is the separate algorithmic problem of *reconstructing* the types a program left unannotated. A language can have a rich type system with little inference, or pervasive inference over a simple one; the specification and the reconstruction algorithm are distinct concerns.
- **Not merely classification.** Sorting values into a finite vocabulary of categories is only half the structure. A type system adds the operation-permission layer — which categories may legally combine with which — and, in the static case, the soundness meta-theorem tying well-typedness to runtime behavior. Collapsing it to "classification" drops exactly the constraint-on-operations and the certificate that distinguish a type system from a bare taxonomy.

clarity:
Naming the type system makes legible the relation between the *category* a value belongs to and the *operations* that category permits, and that single move recasts a whole class of bugs. The question "is this combination of code legal?" reduces to "do these types match?", so an entire family of errors — applying an operation to the wrong kind of argument — becomes detectable as a type mismatch rather than discovered as a runtime crash. The soundness guarantee gives this teeth: because a typing judgment is a proof that the program cannot reach a wrong-kinded state, the type checker is not heuristically flagging suspicious code but certifying the *absence* of an error class. That is what lets a designer push representable invariants into the type itself — a non-empty-list type, a positive-integer refinement, an ownership discipline — so that the violation is no longer a bug to test for but a program the checker simply refuses to accept.

The concept also makes the field's central design tension legible as a knob rather than a mystery: expressiveness against decidability of checking. A richer type language can state more invariants, but checking it grows harder, until at the dependent-type end the checker must decide equalities that can require arbitrary computation. Holding "what the types can express" distinct from "what the checker can decide" is what turns a vague wish for a "more powerful type system" into the sharp question of where on that frontier a language should sit and what each step costs — linear-time checking for simple types, unification for polymorphism, an SMT solver for refinements, proof terms for dependent types. Every familiar annoyance and capability becomes a position on one designable axis instead of an arbitrary fact about a particular language.

manages_complexity:
The sprawl a type system tames is the combinatorial explosion of "what may legally combine with what." In a language of any size, the number of ways expressions can be plugged into operations is enormous, and the number of those combinations that would misbehave at runtime — an integer used as a function pointer, a null dereferenced as a record, a float passed where a string is expected — is a vast, scattered space no one could enumerate. The type system collapses that space to a small grammar of type categories plus one compositional rule per syntactic form. Legality stops being a global property of the whole assembled program and becomes a local check at each node of the syntax tree: an application is well-typed if the argument's type matches the function's domain, a conditional if its branches agree, and so on. Because the rules compose, a checker verifies the entire program by visiting each expression once, propagating types through the tree — typically a linear-time pass. The practitioner stops reasoning about the exponential combination space and tracks only whether each local typing rule is satisfied, reading global legality off the accumulation of local checks.

Two sharper reductions ride on that. First, soundness turns an open-ended testing burden into a certificate. Progress and preservation establish that a well-typed program cannot reach a wrong-kinded state, so an entire class of bugs — every misapplication of an operation to the wrong kind of argument — is not something to hunt for case by case but something the checker's acceptance rules out wholesale; the analyst's effort shifts from enumerating failures to pushing invariants into the types (a non-empty-list type, a positive-integer refinement, an ownership discipline), after which a violation is simply a program the checker refuses. Second, the whole design space of type systems compresses to a single governing axis — expressiveness against decidability of checking — and the Curry-Howard correspondence collapses what would otherwise be a heterogeneous pile of features (polymorphism, subtyping, linearity, refinement, dependency) to one unified reading: a type is a proposition, a well-typed program its proof, checking the verification, and the richness of the type language exactly the strength of the logic it encodes. A designer no longer weighs each feature on its own terms but reads its cost off one position on that frontier: linear-time checking for simple types, unification for polymorphism, an SMT solver for refinements, proof terms and definitional-equality checking (possibly undecidable) for dependent types. A high-dimensional space of legality, error classes, and language features reduces to a small grammar of rules, one soundness guarantee, and a single expressiveness-versus-decidability parameter with a known cost gradient.

abstract_reasoning:
A type system licenses reasoning moves about program legality and safety, all anchored to two facts: legality reduces to a compositional check of category-against-permitted-operation, and a typing judgment is a *proof* — by progress and preservation — that an entire error class cannot occur.

The signature *compositional-checking* move reasons from local rule-satisfaction to global legality. The reasoner does not ask whether a whole assembled program is sound as one question; it checks each syntactic node against its rule — an application `f x` is well-typed at result type `B` exactly when `f : A → B` and `x : A`; a conditional is well-typed at `T` exactly when its guard is `Bool` and both branches are `T` — and concludes global legality from the accumulation of these local checks, propagating types inward and outward through the syntax tree in (typically) a single linear-time pass. So the reasoner reduces "is this combination of code legal?" to "do these types match at each node?", and treats a node whose rule is unsatisfied as the located fault.

The *soundness-as-certificate* move is what gives the check its force and licenses a claim of *absence* rather than suspicion. Because progress and preservation establish, by structural induction over the operational semantics, that a well-typed program cannot reach a wrong-kinded state, the reasoner concludes that an accepted program is *certified* free of an entire class of errors — no integer applied as a function pointer, no null dereferenced as a record, no float where a string is required — rather than merely lacking flagged suspicion. This inverts the reasoner's effort: instead of enumerating possible runtime failures and testing for them, the reasoner pushes the invariant it wants into the type (a non-empty-list type, a positive-integer refinement, an ownership discipline) and concludes that any violation is now not a bug to hunt but a program the checker simply refuses to accept. The reasoner reads "well-typed" as "this failure mode is ruled out wholesale."

The *expressiveness-versus-decidability* move fixes the design frontier and predicts the cost of any feature from its position on it. The reasoner holds two things distinct — what the types can *express* versus what the checker can *decide* — and treats them as trading off along one axis, so a wish for a "more powerful type system" is converted into the sharp question of where on the frontier the language should sit and what each step costs. From a candidate feature the reasoner predicts the checking burden directly: simple types check in linear time but cannot state list-length or ownership invariants; parametric polymorphism requires unification; subtyping requires a subsumption relation; refinement types (a type like `{x : Int | x > 0}`) require discharging logical predicates with an SMT solver; dependent types (a type like `Vec a n`) require deciding definitional equality between type expressions that may involve arbitrary term computation, which the reasoner recognizes can become undecidable. Each familiar capability or annoyance is thus reasoned about as a designable position on one axis with a known cost gradient, not an arbitrary fact about a particular language.

The *Curry-Howard* move supplies a unifying re-reading the reasoner uses to collapse an otherwise heterogeneous feature pile. Reading a type as a proposition, a well-typed program as a proof of it, and type checking as proof verification, the reasoner treats the richness of the type language as exactly the strength of the logic it encodes — and so reasons about polymorphism, linearity, refinement, and dependency not as unrelated mechanisms but as positions of increasing logical expressiveness. This licenses importing logical reasoning into type design (a vector-length-indexed type *is* a constructive proof obligation) and predicting that a richer logic carries a correspondingly heavier verification cost, tying the move directly back to the decidability frontier.

knowledge_transfer:
Within formal-language and rule-driven-classification substrates the type system transfers *as mechanism*, and the transfer is real and wide because the core machinery — a finite type vocabulary, compositional typing rules tying each operation to admissible argument and result types, a typing judgment that compositionally certifies conformance, and (where static) a soundness meta-theorem connecting well-typedness to runtime behavior — operates on any formal symbolic system with classifiable values and a need to constrain operations syntactically. So it carries across the whole programming-language spectrum (C's coarse primitives, Java's nominal class hierarchy, ML/Haskell parametric polymorphism, Rust's linear/affine ownership, Liquid Haskell's SMT-checked refinements, Coq/Agda/Lean/Idris dependent types), into schema and serialization languages (JSON Schema, XML Schema, Protocol Buffers, Avro) which impose a type discipline on interchange formats, into database schemas (column types, constraints, referential integrity as a type system over relational data), and into proof assistants and type-theoretic foundations (Martin-Löf type theory, the calculus of constructions, homotopy type theory) where the type system *becomes* a foundation for mathematics via Curry-Howard. The expressiveness-versus-decidability frontier, the progress/preservation soundness pattern, and the proposition-as-type reading all carry intact across these; only the leaf vocabulary changes. Within this substrate the transfer is full and literal — it is recognizably "the same thing" across its instances.

Beyond formal symbolic systems the honest report is **(B) shared abstract mechanism**, and the portable content is the *conjunction of two parent primes*, not the type system itself. The genuinely recurring cross-domain pattern is *classification plus constraint*: sort entities into discrete categories by explicit rules (`classification`), then restrict which operations or combinations those categories permit (`constraint`). That conjoined pattern really does recur as co-instances — drug-interaction systems classify medications and refuse class-incompatible co-prescriptions; legal-status regimes classify entities and constrain which transactions they may enter; material-compatibility tables classify materials and forbid certain joinings; role-authorization systems classify principals and gate which actions they may perform. Each is a faithful instance of classification-plus-constraint, and *that* is what the cross-domain lesson should carry. What does **not** travel is the type system's *named substantive cargo*: progress and preservation, parametricity and Reynolds' free theorems, the Curry-Howard correspondence, decidable compositional checking, unification for polymorphism, SMT discharge of refinements, definitional-equality checking for dependent types. All of that presupposes a formal expression language with a compositional operational semantics, and it has no referent in a drug-interaction table or a legal capacity rule. So outside CS the thing to carry is the pair `classification` + `constraint` (with `abstract_data_type` as a behavioral-specification cousin), of which a type system is the programming-language-theoretic realization.

The boundary to mark is therefore against over-claiming: invoking "a type system" for a legal compatibility regime or a materials table is **(A) analogy** if it imports the soundness-theorem or decidable-checking connotations — those domains have classification and constraint but no progress/preservation guarantee, no compositional proof of error-class absence, and no checker whose acceptance certifies anything. The conjoined parent pattern is what is genuinely at work there; the type-theoretic machinery is the home-bound accent. (The algorithmic sibling `type_inference` stands in the same relation: type systems specify what is well-typed, type inference reconstructs the types — both home-bound in their machinery, both instances of the same parents cross-domain.) See *Structural Core vs. Domain Accent* for why this profile places the entry as a domain-specific discipline rather than a prime.


Type-Token Ratio

one_liner: Estimate a language sample's lexical diversity as distinct word types over total tokens, while correcting for the sample-size confound that makes vocabulary grow sublinearly and raw TTR fall with length.

core_idea: The type-token ratio (TTR) is the ratio of distinct word types to total word tokens in a language sample, used as a surface proxy for lexical diversity: a sample of 1,000 tokens containing 400 distinct words has a TTR of 0.40. The ratio is computationally simple and theoretically transparent, but carries a well-documented sample-size pathology: vocabulary grows sublinearly with text length (following the curve Heaps' law describes), so TTR systematically declines as a text gets longer even when the underlying lexical richness is unchanged, making raw TTR non-comparable across samples of unequal length. The consequence is a proliferation of corrected measures — Guiraud's root TTR (V / √N), Herdan's C (log V / log N), MATTR (moving-average TTR computed over fixed-size windows), and vocd-D (a theoretical-curve fit to the TTR-versus-sample-size function developed by Malvern et al.) — each an attempt to produce a length-independent estimate of the diversity underlying the sample. In speech-language pathology TTR and its variants are used as screening indices for lexical-access impairment in aphasia and dementia: a patient producing a low-diversity lexical sample relative to matched controls is a candidate for further assessment, with the sample-size correction essential because spontaneous-speech samples differ in length between patients and between sessions. In stylometry, L2 writing assessment, and child language acquisition research, the same basic ratio (type count over token count, with correction applied) serves as a profiling statistic for vocabulary breadth, with the specific correction chosen depending on the sample-size distribution of the corpus under study.

scope_of_application:
The type-token ratio is a *measure*, not a causal mechanism, and its precondition is narrow: a sample of *word types drawn from a token stream*, where vocabulary grows sublinearly. So its literal reach is bounded to the linguistic substrate — unlike its sibling Heaps' law, TTR does *not* travel beyond language, because the general diversity-versus-volume problem is met off-substrate by other fields' own measures (ecology's Shannon/Simpson and rarefaction, genetics' effective number of types, information theory's entropy), which would not accept TTR as their name. Those parallel measures, and the general `diversity` parent, are therefore off this map; the habitats below are the within-linguistics subfields where TTR and its corrections (root TTR, Herdan's C, MATTR, vocd-D, Maas) apply literally.

- **Stylometry** — authorship attribution and register comparison, profiling vocabulary breadth with the length-correction applied so unequal-length texts are comparable.
- **Child language-acquisition research** — lexical-diversity growth curves over development, where the sublinear-growth confound must be controlled across samples of differing length.
- **Aphasia and dementia screening** — TTR and its variants as indices of lexical-access impairment, with the length correction essential because spontaneous-speech samples differ between patients and sessions.
- **L2 writing assessment** — the corrected ratio as a proficiency proxy for vocabulary range across learner texts of unequal length.
- **Corpus genre profiling** — comparing the lexical diversity of registers and genres, reading a difference as signal only after it survives length control.

structural_signature:
**Sig role-phrases:**

- **the language sample** — a stretch of text or speech drawn from a speaker, register, or session
- **the type count** — the number of distinct word types in the sample (the numerator)
- **the token count** — the total number of word tokens in the sample (the denominator)
- **the headline ratio** — V/N, the single scalar offered as a surface proxy for lexical diversity
- **the property-vs-measurement gap** — the construct's core caution: the ratio estimates an underlying lexical *richness* it must not be identified with
- **the sample-size pathology** — the load-bearing flaw: because vocabulary grows sublinearly (Heaps' curve), raw TTR falls with length even at constant richness, confounding any unequal-length comparison
- **the corrected-variant family** — the enumerated length-independent estimators (Guiraud's root TTR, Herdan's C, MATTR, vocd-D, Maas), each embedding a different closed-form growth-curve assumption
- **the correction-selection rule** — the design move: choose the variant from the corpus's sample-length distribution, not by which index is "best" in the abstract
- **the confound gate** — the diagnostic discipline: a TTR difference counts as a diversity finding only if it survives length control, else it is a sample-length artifact

what_it_is_not:
- **Not lexical diversity itself.** TTR is a *measurement*, a surface proxy, not the property it estimates. The construct's whole caution is to break the identification of the ratio with the underlying richness, so a TTR number is read as an estimate that may be confounded — never as the diversity it stands in for.
- **Not comparable across samples of unequal length.** Because vocabulary grows sublinearly, raw TTR *declines as a sample lengthens even at constant richness*, so a TTR gap between two texts is confounded with a length gap. The predicted artifact is sharp: the *longer* sample shows the *lower* raw TTR, so a longer transcript read naively looks like a poorer vocabulary.
- **Not a length-independent statistic.** Raw TTR is not stable across sample sizes; the proliferation of corrected measures (Guiraud's root TTR, Herdan's C, MATTR, vocd-D, Maas) exists precisely *because* TTR is length-dependent. These are not competing definitions of diversity but length-controlled estimators, each assuming a different growth curve.
- **Not a substrate-general diversity index.** TTR's types-over-tokens formula and its linguistic corrections (vocd-D especially) are bound to lexical sampling. Other fields built their own measures for their own substrates — ecology's Shannon/Simpson and rarefaction, genetics' effective number of types, information theory's entropy — none of which would accept TTR as the canonical name for what they compute. The *diversity-versus-volume problem* generalizes; this *solution* does not.
- **Not a clinical finding on its own.** A low-diversity sample relative to matched controls makes a patient a *candidate* for lexical-access impairment only *after length is controlled*. A later sample with lower raw TTR is not evidence of deterioration if it is merely longer; a real change is one that survives the correction, and a difference that vanishes under it is a sample-length artifact, not a finding.

clarity:
Naming the type-token ratio, and naming its pathology with it, makes legible the gap between a *measurement* and the *property* it is meant to estimate. Read naively, TTR looks like lexical diversity itself — a number you can compare across any two samples. The construct's clarifying force is to break that identification: because vocabulary grows sublinearly, the ratio falls as a sample lengthens even when the underlying richness is unchanged, so a difference in TTR between two texts is *confounded with* a difference in their length. The sharp question the practitioner can then ask is no longer "which sample is more lexically diverse?" but *is this difference signal or sample-size artifact?* — and that question has a determinate answer where the raw comparison did not. It is exactly what prevents a longer transcript from being misread as a poorer vocabulary.

The concept also clarifies what the proliferation of corrected measures — root TTR, Herdan's C, MATTR, vocd-D — actually *is*: not a menu of competing diversity definitions but a family of attempts to recover a single length-independent estimate of the diversity behind the sample, each making a different assumption about the vocabulary-growth curve. Seeing them this way sharpens the design choice, since the right correction depends on the sample-size distribution of the corpus rather than on which index is "best" in the abstract. And it pinpoints where the measure does decision-relevant work: in speech-language pathology, the construct makes clear that a low-diversity sample is interpretable as evidence of lexical-access impairment *only after* length is controlled — so a patient whose later sample is merely longer is not deteriorating, and a real change is one that survives the correction. The clarity is in localizing the inference to the property and refusing to read the artifact as the finding.

manages_complexity:
The full lexical character of a language sample is a high-dimensional object: the complete vocabulary-growth curve, the frequency profile of every word, the way new types arrive as tokens accumulate. To compare two speakers, two registers, or one patient across two sessions on "how rich is the vocabulary?" by inspecting those curves directly would be intractable. The type-token ratio compresses that object to a single scalar — distinct types over total tokens — so the analyst tracks one number per sample and reads a first-pass diversity ordering off it. That is the primary compression, and it is what makes large-corpus profiling, real-time clinical screening, and cross-text stylometry feasible at all: the headline lexical-richness question reduces to a quotient.

The construct's deeper management of complexity is that it also packages the *correction* for the one confound that would otherwise wreck the comparison. Because vocabulary grows sublinearly, raw TTR is entangled with sample length, so the single quotient would mislead across unequal samples — and the field's response is not an open-ended modelling problem but a small, enumerated family of length-controlled variants (root TTR, Herdan's C, MATTR, vocd-D), each a different closed-form assumption about the growth curve. The analyst therefore tracks two things: the ratio, and which correction the corpus's sample-size distribution calls for. The branch structure is correspondingly tight. Faced with a difference in TTR, the practitioner routes through one gate — is this signal or a sample-length artifact? — by asking whether the difference survives length control; only the surviving difference is read as a diversity finding. In the clinical case this collapses an open interpretive question ("is this patient's lower-diversity sample evidence of lexical-access impairment, or just a longer transcript?") to a single conditional: control length, then compare to matched norms. What would be a comparison of full vocabulary-growth curves reduces to one scalar plus a correction selected from a short menu, with a single confound-gate deciding whether any observed difference counts.

abstract_reasoning:
The construct's governing move is a *confound diagnosis* that separates the measurement from the property it estimates. Facing a difference in TTR between two samples, the analyst does not read it as a difference in lexical diversity but asks *is this signal or a sample-length artifact?* — because vocabulary grows sublinearly, the raw ratio falls as a sample lengthens even when underlying richness is unchanged, so a TTR gap is confounded with a length gap. The inference therefore runs from a raw TTR difference, through a check of whether it survives length control, to a verdict: only the surviving difference counts as a diversity finding. This has a sharp *predictive* corollary that lets the analyst anticipate the artifact before computing anything — given two samples of unequal length drawn from speakers of equal richness, predict that the *longer* one will show the *lower* raw TTR, so a longer transcript read naively will be mistaken for a poorer vocabulary. Recognising the direction of the artifact is what disarms it.

A second move is *correction selection reasoned from the growth curve*. The proliferation of variants — Guiraud's root TTR (V/√N), Herdan's C (log V / log N), MATTR over fixed windows, vocd-D from a theoretical TTR-versus-sample-size fit — is not a menu of competing diversity definitions but a family of length-independent estimators, each embedding a different closed-form assumption about how vocabulary grows with tokens. The analyst therefore reasons from the corpus's sample-size distribution to the appropriate correction: which transform recovers a length-stable estimate *for this distribution of sample lengths* rather than which index is "best" in the abstract. The choice is an inference about the growth curve the data follow, and a mismatch between the assumed curve and the corpus is itself a source of residual artifact.

The construct's *clinical-diagnostic* move chains these together into a guarded inference. A low-diversity sample relative to matched controls makes a patient a candidate for lexical-access impairment (in aphasia or dementia screening), but the inference is licensed *only after length is controlled*, because spontaneous-speech samples differ in length between patients and across sessions. So from a patient's later sample showing lower raw TTR, the analyst must not infer deterioration: control length first, and a real change is one that survives the correction (MATTR over a window, say), while a difference that vanishes under correction is attributed to the longer transcript, not the lexicon. A *boundary-drawing* move fixes the scope of all this: the specific corrections are linguistic-internal estimators of the diversity behind a *word-type-over-token* sample, so they apply to stylometry, L2 assessment, child-language growth curves, and clinical screening within language, and are not assumed to carry to other diversity-with-volume problems (which field their own substrate-specific corrections), even though the underlying diversity-versus-sample-size confound is shared. The reasoning stays inside "estimate the lexical diversity behind this sample, net of its length," and refuses to let the artifact be read as the finding.

knowledge_transfer:
The type-token ratio is a *measure*, not a causal mechanism, so "mechanism within, metaphor beyond" does not apply; the question is how far the instrument reaches before it is over-read. And its reach is narrower than its sibling Heaps' law, in an instructive way: what transfers literally is bounded to the *word-type-over-token* substrate, while the general problem it addresses transfers as a parent prime.

Within linguistics the construct and its correction family transfer literally across subfields, because all of them sample word types from token streams: stylometry (authorship attribution, register comparison), child language-acquisition growth curves, aphasia and dementia screening, L2 writing assessment, and corpus genre profiling all compute types over tokens and all face the identical sublinear-growth confound, so the confound-diagnosis, the predicted direction of the artifact (longer sample, lower raw TTR), the length-controlled estimators (Guiraud's root TTR, Herdan's C, MATTR, vocd-D, the Maas index), and the guarded clinical inference (control length before reading a low-diversity sample as impairment) all carry without change. Only the corpus's sample-length distribution decides which correction to use.

Beyond the linguistic substrate the honest reading separates two things the surface conflates. The general *diversity-versus-volume problem* — a count of distinct categories over total observations, which must be corrected for sample size because new categories accrue sublinearly — genuinely recurs across domains, and it is carried at the catalog level by `diversity` (with `variability` as a broader sibling). That parent is what travels, and it is where any cross-domain lesson lives. But TTR's *own machinery* does not travel: the cross-domain analogues are not applications of the type-token ratio at all but independent measures that each field built for its own substrate — ecology's Shannon and Simpson indices and rarefaction, genetics' effective-number-of-types, information theory's entropy over identifier distributions in code analysis — and none of these would accept TTR, or its linguistic corrections like vocd-D, as the canonical name for what they compute. So the over-reading to avoid is treating TTR (or one of its variants) as a substrate-general diversity index: the *problem* generalizes, the *solution* is one of many, each tuned to its own growth curve and sampling regime. The disciplined move when a diversity-with-volume question arises off the linguistic substrate is to reach for the general `diversity` parent and adopt that field's own sample-size correction (rarefaction, entropy, effective number), not to port TTR's types-over-tokens formula and its linguistics-internal estimators, which are exactly the part bound to lexical sampling. (See *Structural Core vs. Domain Accent*.)


Ultimatum Game

one_liner: A two-player, one-shot bargaining test in which a proposer splits a fixed stake and a responder may accept or reject for nothing, isolating costly punishment of unfairness by pitting the selfish equilibrium against the offers and rejections people actually make.

core_idea: The ultimatum game is a two-player, one-shot bargaining paradigm — introduced by Werner Güth, Rolf Schmittberger, and Bernd Schwarze (1982) — in which a proposer divides a fixed monetary stake between herself and a responder, who must either accept the proposed split (both receive their shares) or reject it (both receive nothing). The subgame-perfect equilibrium under the assumption of purely self-interested, forward-looking players predicts that the proposer offers the smallest positive amount and the responder accepts rather than receive zero — any positive share is better than none. The empirical finding, replicated across hundreds of laboratory studies and a wide range of cultural settings, is that proposers typically offer 40–50 percent of the stake and that responders reliably reject offers below roughly 20 percent even when doing so costs them a substantial absolute sum. The mechanism the game isolates is costly punishment of perceived unfairness: responders forgo a positive payoff specifically to deny the proposer a reward they deem inequitably large, and proposers — anticipating this — offer near-equal splits to avoid rejection. Ernst Fehr and Klaus Schmidt's (1999) inequity-aversion model formalised this by adding a disutility from payoff differences into the responder's utility function, capturing why rejection is rational given those preferences. The game became foundational in behavioural game theory because it provides the minimal laboratory instrument that separates the standard rational-choice prediction from the social-preference deviation with a single clear empirical test: if responders reject positive offers, the selfish model is falsified for that population. Joseph Henrich and colleagues (2001, 2005) extended the paradigm across small-scale societies — hunter-gatherer, pastoralist, and horticulturalist groups — and found that both the frequency of rejection and the level of offers co-vary with the degree of market integration and collective cooperation in each society, establishing that the fairness threshold itself is culturally calibrated rather than a fixed parameter of human psychology.

scope_of_application:
Because the ultimatum game is a controlled experimental instrument rather than a causal mechanism, it is not bounded by a domain but runs wherever its precondition holds: a valuing human (or human-like) agent who can be seated in the proposer or responder role. The fields below are genuine literal deployments of the same paradigm, returning comparable offer/threshold/punishment numbers; what travels *beyond* an actual seatable agent is not the game but the findings it measures (fairness, reciprocity, costly punishment), which belong to those parent primes, not to this instrument.

- **Behavioural game theory / experimental economics** — the home laboratory; the canonical minimal test that pits the selfish subgame-perfect benchmark against social-preference behaviour by measuring willingness to bear cost to punish unfairness.
- **Sibling-paradigm family** — the instrument transfers cleanly among its near kin: the dictator game (strip the rejection right to isolate altruism from strategic fairness), the trust game (add a move to isolate reciprocity), and the public-goods game (test cooperation at group scale).
- **Cross-cultural anthropology** — the Henrich et al. small-scale-society studies run the identical game across hunter-gatherer, pastoralist, and horticulturalist groups, operationalizing fairness-norm variation and showing the rejection threshold co-varies with market integration.
- **Neuroeconomics** — fMRI of ultimatum rejection localizes the neural correlates of refusing an inequitable offer (insular-cortex activation), reading the same game's decisions off brain activity.
- **Evolutionary / cooperation theory** — used to argue that strong reciprocity and altruistic punishment co-evolved with human-scale cooperation, the game supplying the behavioural signature.
- **Negotiation pedagogy** — a vivid classroom demonstration that take-it-or-leave-it bargaining destroys surplus when the responder's outside option is to tear up the deal.

structural_signature:
**Sig role-phrases:**

- **the fixed stake** — a known monetary sum to be divided between the two players
- **the proposer** — the player who unilaterally offers a split of the stake
- **the responder** — the player who must either accept (both get their shares) or reject (both get nothing)
- **the one-shot take-it-or-leave-it scaffolding** — anonymous, single-round play that strips out reputation and repetition, leaving rejection as the only response move
- **the selfish-equilibrium benchmark** — the subgame-perfect prediction under purely self-interested play: minimal offer, accept anything positive (something beats nothing)
- **the rejection-of-positive-offers deviation** — the empirical finding that responders reliably reject offers below ~20% at real cost, falsifying the selfish model for that population and measuring its departure
- **the costly-punishment / inequity-aversion read** — rejection modelled as a disutility-from-payoff-difference term that makes punishing an inequitable split rational, not irrational
- **the culturally calibrated threshold** — the rejection bar treated as a quantity co-varying with a society's market integration and cooperative scale, not a fixed psychological constant

what_it_is_not:
- **Not a real-world mechanism.** The ultimatum game is a controlled *instrument* — a stake, a proposer, a responder, one-shot anonymity — built to isolate a finding, not a force operating out in bargaining. What travels into wage disputes or treaties is the *content* it measures (fairness, reciprocity, costly punishment), cited *as evidence for* those patterns; "the ultimatum game" itself stays in the lab.
- **Not evidence that responders are irrational.** Rejecting a positive offer looks like leaving money on the table only under a purely selfish utility function. Add a disutility-from-payoff-difference term and rejecting an inequitable split is the *rational* maximization of that utility. The result reframes the objective, not the rationality.
- **Not a measure of altruism or generosity.** Proposers' near-equal offers are largely *strategic anticipation* of rejection, not kindness; the responder's veto is doing the work. Stripping that veto — the dictator-game variant — is the move that isolates pure altruism, and offers fall sharply when it is removed, showing the ultimatum game was never measuring generosity in the first place.
- **Not a fixed human constant.** The rejection threshold is not hard-wired psychology: across small-scale societies it co-varies with market integration and cooperative scale, so the fairness bar is a *calibrated* quantity set by social conditions, not a universal parameter read off "human nature."
- **Not described by its subgame-perfect equilibrium.** The "offer the minimum, accept anything positive" prediction is the *selfish benchmark the game is read against*, not an account of how people play. The whole point is the systematic departure from it; treating the equilibrium as a behavioral forecast inverts what the instrument was built to expose.

clarity:
The paradigm's clarifying force is that it makes a previously diffuse dispute — do people actually behave as the self-interested rational-choice model says? — into a single, unmissable, falsifiable test. Fairness-versus-self-interest had been a tangle of competing intuitions with no clean adjudicating instrument; the ultimatum game supplies the minimal scaffolding (one-shot, anonymous, take-it-or-leave-it, a stake to divide) under which the two accounts make *contradictory* predictions about one observable. The subgame-perfect equilibrium says responders accept any positive offer; the data say they reject offers below roughly a fifth of the stake at real cost. The gap is not noise to be explained away — it is the result the game was built to produce, and its size is a direct measurement of how far a population departs from the selfish benchmark.

In doing so the game sharpens distinctions that ordinary talk of "fairness" runs together. It separates the *proposer's strategic anticipation* (offering near-equal splits to dodge rejection) from the *responder's costly punishment* (forgoing a payoff to deny an inequitable reward), letting an experimenter localize where social preferences bite and parametrically vary it. It lets the field ask whether the deviation is best modelled as a taste — Fehr and Schmidt's disutility-from-payoff-difference, which renders rejection *rational* given those preferences — rather than as irrationality, reframing the anomaly as a question about the utility function instead of a failure of reason. And the cross-cultural extensions make the sharper question explicit: is the rejection threshold a fixed parameter of human psychology, or is it *calibrated* by a society's market integration and cooperative scale? Naming the instrument is what lets that be settled by measurement rather than asserted.

manages_complexity:
Fairness is otherwise a sprawling and slippery object of study: it surfaces in wage disputes, divorce settlements, treaty negotiations, allocations of windfalls, and countless everyday divisions of surplus, each laden with context, history, relationship, and stakes, and any attempt to ask "do people behave as the self-interested model predicts?" across that range drowns in confounds. The ultimatum game compresses the entire question to a single controlled instrument with four moving parts — a fixed stake to divide, a proposer's offer, a binary accept-or-reject, and one-shot anonymity that strips out reputation and repetition — under which the selfish rational-choice account and the social-preference account make flatly contradictory predictions about one observable. Behavioural game theory then no longer needs to track the full texture of each fairness situation; it tracks a small set of quantities the game exposes: the offer fraction the proposer puts on the table, the threshold below which responders reject, and the cost responders will absorb to punish. The qualitative verdict reads off these directly — if responders reject positive offers at real cost, the selfish model is falsified for that population, and the magnitude of the gap between the predicted penny-offer equilibrium and the observed 40–50 percent offers with rejection-below-a-fifth measures exactly how far that population departs from the self-interested benchmark.

That reduction also resolves the otherwise tangled space of explanations into a few separable, tunable parameters. The proposer's strategic anticipation and the responder's costly punishment, ordinarily fused in talk of "fairness," become distinct quantities an experimenter can localize and parametrically vary by changing stakes, information, anonymity, or adding third-party punishment. The deviation itself collapses from "people are irrational" to a single modelling choice — Fehr and Schmidt's disutility-from-payoff-difference term — that, once added to the responder's utility function, makes rejection rational and turns the anomaly into a one-parameter question about preferences rather than an open-ended failure of reason. And the rejection threshold, rather than being a fixed constant of human psychology, becomes a calibrated quantity the analyst can predict from a society's market integration and cooperative scale, after Henrich and colleagues found both offers and rejection co-varying with those variables across hunter-gatherer, pastoralist, and horticulturalist groups. So an enormous, context-saturated domain reduces to: run the minimal game, read off offer, threshold, and punishment-cost, and from that small set settle whether a population is selfish, how strong its inequity aversion is, and what social conditions set its fairness bar — a high-dimensional dispute rendered as a few measurable parameters with a clear falsification branch.

abstract_reasoning:
The ultimatum game licenses a distinctive set of moves in behavioural game theory, all generated by its status as the minimal instrument under which the selfish and social-preference accounts make contradictory predictions about one observable.

*Diagnostic (falsify the selfish model for a population, and measure its departure, from rejection behavior).* The defining move is to read a single observable — whether responders reject positive offers at real cost — as a decisive test between two theories. The subgame-perfect prediction under purely self-interested play says responders accept any positive offer (something beats nothing) and proposers offer the minimum; the observation that responders reliably reject offers below roughly a fifth of the stake *falsifies the selfish model for that population*. The reasoning then turns quantitative: the gap between the predicted minimal-offer equilibrium and the observed 40–50 percent offers *measures* how far the population departs from the self-interested benchmark, so the magnitude of the deviation is itself the datum. The inference runs from observed choices (offer fraction, rejection threshold, punishment cost absorbed) to a verdict on which utility function the population is operating under — a model test conducted by reading three numbers off a controlled instrument rather than by argument.

*Interventionist (localize where social preferences bite by parametrically varying the game).* Because the paradigm has few moving parts — a fixed stake, the proposer's offer, a binary accept/reject, one-shot anonymity — the experimenter can intervene on each and predict how behavior shifts, separating components ordinary talk of "fairness" fuses. Stripping the responder's rejection right (the dictator variant) is predicted to isolate pure altruism from strategic fairness, since the proposer no longer faces punishment; adding a stake increase, an information change, or third-party punishment is predicted to move the rejection threshold or offer in identifiable directions. The move is "change one feature of the minimal game, predict the change in offer or rejection," which lets the analyst localize *where* social preferences enter — the proposer's strategic anticipation of rejection versus the responder's costly punishment — and tune that locus deliberately rather than observing fairness as an undifferentiated lump.

*Boundary-drawing (taste versus irrationality; fixed parameter versus calibrated threshold).* The game draws two consequential boundaries. First, it separates a *social preference* from a *failure of reason*: rejection is not ruled irrational once a disutility-from-payoff-difference term is added to the responder's utility function, after which rejecting an inequitable offer is the rational maximization of that utility — so the move "responders are behaving irrationally" is out of bounds, replaced by "responders have a different objective." This reframes the anomaly as a question about the utility function, not a breakdown of optimization. Second, it draws the line between a fixed parameter of human psychology and a culturally calibrated one: the cross-cultural extensions establish that the rejection threshold is *not* a universal constant but co-varies with a society's market integration and cooperative scale, so the move "this fairness bar is hard-wired" is ruled out, and the threshold must be treated as a quantity set by social conditions. Deciding which boundary a given result sits on — taste or irrationality, universal or calibrated — is the interpretive judgment the instrument makes decidable.

*Predictive (forecast a population's fairness behavior from its social structure, and a proposer's offer from anticipated rejection).* The paradigm licenses two forward predictions. From a society's degree of market integration and collective cooperation, the analyst predicts the level of offers and the frequency of rejection — the fairness threshold becomes something forecast from social variables rather than assumed. And within the game, the proposer's behavior is predicted by backward reasoning from the responder's anticipated rejection: a proposer who expects offers below a fifth to be rejected is predicted to offer near-equal splits to avoid the joint-zero outcome, so the observed generosity is read as strategic anticipation, not altruism. The reasoning runs from social structure to fairness norms, and from anticipated punishment to equilibrium offers, turning a context-saturated phenomenon into predictions keyed to a few measurable parameters.

knowledge_transfer:
The ultimatum game is an *experimental instrument* — a controlled paradigm, not a causal mechanism out in the world — so its transfer has two distinct channels that must be kept separate, and "mechanism within / metaphor beyond" does not straightforwardly apply. The first channel is the instrument itself: the scaffolding (a fixed stake, a proposer's offer, a binary accept-or-reject, one-shot anonymity) transfers *literally* wherever its precondition holds — namely, wherever there are human (or human-like) agents who can be put in front of it. This is what lets the same minimal game be run across hundreds of laboratories and, in Henrich and colleagues' work, across hunter-gatherer, pastoralist, and horticulturalist societies with stakes scaled to months of income, returning comparable, calibratable numbers (offer fraction, rejection threshold, punishment cost). Within behavioural game theory and experimental economics the instrument also sits in a tight family of sibling paradigms it transfers cleanly among — the dictator game (strip the rejection right to isolate altruism from strategic fairness), the trust game (add a move to isolate reciprocity), the public-goods game (test cooperation at group scale) — and into **cross-cultural anthropology** (operationalizing fairness-norm variation), **neuroeconomics** (fMRI of rejection localizing insular activation), and **negotiation pedagogy** (a vivid demonstration that take-it-or-leave-it bargaining destroys surplus). Across all of these the construct is the same controlled test, meaning the same thing, wherever there is a valuing agent to seat in the proposer or responder role.

The second channel is the game's *findings*, and here honesty requires marking that what travels is not the paradigm but the structural content it measures — content already named by more general patterns. The load-bearing result ("humans will pay to punish allocations they perceive as inequitable, and anticipating this, allocators offer near-equal splits") really does reach into wage negotiations, divorce settlements, and international treaties, but it reaches *as* `fairness`, `reciprocity`, costly punishment, and the credible threat to destroy joint surplus — the genuine cross-domain mechanisms — not as "the ultimatum game." The tell is in the citation pattern: ultimatum-game results are cited *as evidence for* strong reciprocity and inequity aversion, not *as* those patterns themselves, which is exactly the signature of a paradigm rather than a portable structure. So the cross-domain lesson should carry the parent content (fairness preferences, reciprocity, costly punishment, the commitment logic of a credible rejection), while the game's own machinery — the proposer/responder roles, the one-shot take-it-or-leave-it scaffolding, the penny-offer subgame-perfect benchmark it is read against, the dictator/trust sibling variants — stays home as the apparatus that *produced* the evidence. The boundary to police is therefore twofold: instrument-reach (the game transfers wherever a valuing human agent can be tested, and over-reading begins when "ultimatum game" is invoked for systems with no such agent, no rejection option, and no surplus to destroy) and content-attribution (the durable lesson belongs to fairness/reciprocity/costly-punishment, not to the instrument that revealed them). The full split between the home-bound instrument and the traveling content is drawn in *Structural Core vs. Domain Accent*.


Uncanny Valley

one_liner: Human affinity for a human-like artefact does not rise steadily with realism but climbs, then plunges into a valley of unease in the narrow almost-human band, and recovers only when realism reads as genuinely human — because near-human stimuli recruit human-specific recognition routines that flag the remaining mismatches.

core_idea: The uncanny valley is the hypothesis, introduced by roboticist Masahiro Mori in 1970, that human affective response to human-like artefacts does not rise monotonically with human-likeness. Instead, affinity increases with realism up to a threshold, then drops sharply into a *valley* of unease, eeriness, or revulsion for near-but-not-quite-human figures, and recovers toward high affinity only when realism is sufficient to read as genuinely human. Movement amplifies the effect: a near-human still image is mildly unsettling; a near-human animated figure with subtly mismatched motion is sharply more so. The structural shape of the curve — rise, sharp dip in a narrow "almost human" band, recovery — is the load-bearing abstraction; it makes the valley a *design-relevant threshold phenomenon* rather than a simple preference gradient. The proposed mechanisms involve mismatch detection between expected and observed cues across appearance, motion, and social-signal channels: the brain recruits human-specific recognition routines for near-human stimuli, and those routines detect anomalies (subtly wrong micro-expressions, implausible skin texture, mismatched motion dynamics) that coarser object-recognition would not, producing a heightened negative signal precisely because high-level expectations were engaged and violated. The effect was operationalised in animation studies (cited in post-mortems of *Polar Express*, 2004, and *Final Fantasy: The Spirits Within*, 2001) and informs deliberate stylisation decisions in character design: targeting either full abstraction or full photorealism, and avoiding the in-between band where the dip lives.

scope_of_application:
The uncanny valley lives across perception-and-design wherever a human-like (or conspecific-like) artefact is varied along a realism axis and recruits human-specific recognition routines past a threshold; these are different industrial settings of one trigger, not distinct substrates, so its reach is bounded there — the rise-dip-recovery *shape* travels further only under the `non_monotonic_relationships` and `phase_transitions` primes, and "uncanny valley of mind/voice" coinages are analogy that drop the perceptual trigger.

- **Robotics and human-robot interaction** — Mori's original domain, an industrial-design heuristic for android development.
- **Computer animation** — the diagnosed failure mode of *The Polar Express* and *Final Fantasy: The Spirits Within*, and the spur for deliberate Pixar/Disney stylisation.
- **Virtual agents, deepfakes, and generative video** — near-realistic AI-generated faces and voices trigger documented uncanny responses, an ongoing concern in trust calibration for human-AI interaction.
- **Prosthetics and medical mannequins** — a too-lifelike prosthetic hand or training mannequin can disturb users and clinicians.
- **Taxidermy, wax figures, and dolls** — the long-standing folk-art intuition that the genre concentrates uncanny figures.

structural_signature:
**Sig role-phrases:**

- **the human-like artefact** — a robot, avatar, animation, doll, or prosthetic representing a human (or living conspecific), the stimulus whose affinity is at issue
- **the realism axis** — degree of human-likeness, the single continuous parameter along which the artefact is varied
- **the recruitment threshold** — the point at which crossing into near-human territory switches on human-specific recognition routines that coarser object perception never engages
- **the mismatch detector** — those engaged routines flagging anomalies (mis-timed micro-expressions, dead eyes, wrong skin, motion that does not track the face) precisely because high-level expectations were raised and violated
- **the non-monotonic curve** — affinity rising with realism, dropping sharply into a valley across the narrow "almost but not quite" band, then recovering once realism reads as genuinely human
- **the amplifier terms** — motion deepening the dip, and cross-modal inconsistency (photoreal face, cartoon voice) dropping an artefact in even where each channel alone would clear it
- **the safe-direction heuristic** — the design consequence: inside the trough, retreat to full stylisation or push to full photorealism, never polish in place

what_it_is_not:
- **Not a monotonic preference gradient.** The whole point is non-monotonicity: affinity does *not* rise steadily with human-likeness. It rises, drops sharply into a valley across the narrow "almost but not quite" band, then recovers — so the default design assumption that more realism always means more appeal is false in a specific, costly region, and inside the trough added realism makes things worse before they get better.
- **Not unease at realism as such.** The eeriness lives in the *inconsistencies* — mis-timed micro-expressions, dead eyes, wrong skin, motion that does not track the face — not in fidelity itself. Crossing into near-human territory is what recruits the scrutiny that surfaces those flaws; remove the mismatch or push realism far enough to read as genuinely human and affinity is restored.
- **Not a settled empirical law.** It is a hypothesis with a contested empirical base: meta-analyses find the dip in some populations and stimulus sets but not others. The load-bearing abstraction is the proposed curve *shape* (rise, dip, recovery), not a universally confirmed quantitative law of affect.
- **Not produced by non-human artefacts approaching realism.** The valley appears only for stimuli that recruit human-specific (or conspecific) recognition routines along a realism axis — no one reports an uncanny-valley dip for a car or abstract object approaching photorealism, because those routines were never engaged. The trigger regime is the boundary the concept itself draws.
- **Not the general non-monotonic-response pattern.** The uncanny valley is the *human-perception instance* of a broader family — non-monotonic response to a continuous design parameter — whose other members (phase transitions in matter, hormetic dose-response, the Yerkes-Dodson arousal-performance curve) share the rise-dip-or-peak *form* but each run on a different mechanism. The shape travels under `non_monotonic_relationships` and `phase_transitions`; this concept is one keyed to near-human artefacts.

clarity:
Naming the uncanny valley overturns the default design assumption it was coined against: that affinity toward a human-like artefact rises with realism, so the path to a more appealing android, avatar, or animated character is simply *more* human-likeness. Once the non-monotonic curve has a name, that assumption is exposed as false in a specific, costly band — and the practitioner's question changes from "how realistic can we make it?" to the sharper "which side of the valley are we on, and is incremental realism moving us toward recovery or down into the dip?" This reframes a vague worry that a character looks "creepy" into a locatable position on a known curve, with a clear implication: near the dip, adding fidelity can make things worse, and the safe targets are the two extremes — full stylisation or full photorealism — not the plausible-looking middle.

The label also makes legible *why* the dip is sharp and *what* deepens it, which a flat preference-gradient picture cannot. By tying the unease to mismatch detection within human-specific recognition routines, it tells the designer that the eeriness lives in the *inconsistencies* — dead eyes, mis-timed micro-expressions, skin that reads wrong, motion that does not match the face — rather than in realism as such, and that crossing into near-human territory is precisely what recruits the scrutiny that surfaces those flaws. That yields concrete, otherwise-invisible diagnostics: motion amplifies the dip, so a still that passes may fail once animated; and cross-modal mismatch (a photoreal face with a cartoon voice, or vice versa) drops an artefact into the valley even when each channel alone would not. The concept thereby separates "not yet realistic enough" from "realistic enough to trigger the routines that catch its remaining errors" — two states that look alike from a monotonic vantage but call for opposite design moves.

manages_complexity:
Character and artefact design confronts an open-ended catalogue of "this reads as creepy" reports that resist a unified account: an animated lead whose photoreal skin and proportions land but whose micro-expressions are mis-timed; a service android that disturbs precisely as its face grows more lifelike; a deepfake face that unsettles where a cartoon would charm; a too-lifelike prosthetic or training mannequin that clinicians shrink from; wax figures and dolls that a whole folk tradition flags as eerie. Without an organising principle, each is its own design accident to be diagnosed and patched, and the field's tacit rule — *more human-likeness yields more appeal* — actively misleads, predicting that pushing realism will always help. The uncanny valley compresses the whole catalogue into one curve: affinity as a non-monotonic function of human-likeness, rising, dropping sharply into a narrow dip in the "almost but not quite human" band, then recovering. The sprawling question "why is this particular artefact unsettling?" collapses to "where on the affinity-versus-realism curve does it sit?"

That single relocation lets the designer stop enumerating creepy cases and track one continuous parameter — degree of human-likeness — plus a small set of amplifiers, and read the qualitative outcome off the curve's shape. The branch structure is explicit and actionable. Below the dip, on the rising stretch, an artefact reads as appealingly stylised and added realism helps. Inside the narrow near-human band, affinity is in the trough: here the counterintuitive branch bites, because incremental realism moves the artefact *deeper* before it recovers, so the correct move is to retreat to stylisation or push past to full photorealism rather than polish in place. Above the dip, sufficient realism reads as genuinely human and affinity is restored. The mechanism — human-specific recognition routines, recruited only once a stimulus crosses into near-human territory, detecting the mismatches coarser object perception would miss — supplies the amplifier terms the analyst also tracks: motion deepens the dip, so a still that passes can fail once animated; and cross-modal inconsistency (a photoreal face with a cartoon voice) drops an artefact into the valley even when each channel alone would clear it. So instead of modelling the full perceptual and affective response to every artefact, the designer locates it on one curve, checks the realism-match across channels and whether motion is involved, and reads off appeal, repulsion, or recovery — and, critically, which direction along the realism axis is safe to move.

abstract_reasoning:
The uncanny valley turns design reasoning about human-like artefacts into *position-on-a-curve* reasoning, where the curve is non-monotonic and a recruitment threshold marks the point past which human-specific scrutiny switches on. Every characteristic move depends on knowing both the artefact's location along the realism axis and whether that threshold has been crossed.

*Diagnostic (locate the artefact, name the cause of unease).* Faced with a report that an artefact "reads as creepy," the analyst infers its position: a negative affective response to a near-but-not-quite-human figure places it in the trough, not on the rising stretch. From there the inference runs to the specific cause — because the unease is produced by mismatch detection within human-specific recognition routines, the eeriness is read as living in the *inconsistencies* (mis-timed micro-expressions, dead eyes, wrong skin, motion that does not track the face) rather than in realism as such. This separates two states that look identical from a monotonic vantage but demand opposite responses: "not yet realistic enough" (still climbing toward the dip, below the recruitment threshold) versus "realistic enough to recruit the routines that catch its remaining flaws" (in the dip, past the threshold). The diagnostic move is surface revulsion → location in the near-human band + a specific cross-cue mismatch, never surface revulsion → "needs more realism."

*Interventionist (move along the axis with sign-aware predictions).* The signature intervention is direction-sensitive in a way the default assumption gets backwards. Inside the trough, adding fidelity is predicted to move the artefact *deeper* before any recovery, so the licensed moves are to retreat toward full stylisation or push past to full photorealism — both predicted to raise affinity — while polishing in place is predicted to worsen it. Each manipulation pairs a move along the realism axis with a forecast sign on appeal, and the sign flips depending on which side of the dip the artefact starts. The mechanism also supplies amplifier interventions with their own predictions: introduce motion and the dip must deepen, so a still that passed is predicted to fail once animated; mismatch realism across channels (photoreal face, cartoon voice) and the artefact must drop into the valley even where each channel alone would clear it. Removing the inconsistency, or aligning the channels, is predicted to lift it back out.

*Boundary-drawing (fix the trigger regime).* The curve applies only within a specific trigger condition — artefacts that recruit human-specific (or conspecific) recognition routines along a realism axis. That condition draws the regime: a near-photoreal human face is inside and shows the dip; a car or abstract object approaching photorealism is outside and shows no valley, because the human-specific routines were never engaged. The recruitment threshold is the internal boundary that matters most — below it the artefact reads as appealingly stylised and ordinary "more realism helps" reasoning is valid; above it the scrutiny switches on and the counterintuitive branch takes over. Naming the regime tells the designer when uncanny-valley reasoning governs at all and when a plain preference gradient does.

*Predictive / branch-ordering.* From one continuous parameter (degree of human-likeness) plus the amplifier checks (motion present? channels matched?), the qualitative outcome is read off the curve's shape before testing: below the dip, appealing and improvable by realism; inside the narrow near-human band, in the trough and worsened by incremental realism; above the dip, read as genuinely human and affinity restored. The forecast names not just the affective valence but the safe direction of travel along the axis.

knowledge_transfer:
Within perception-and-design the hypothesis transfers *as mechanism*, because every place it applies is the same trigger condition — a human-like (or, more generously, conspecific-like) artefact varied along a realism axis, recruiting human-specific recognition routines past a threshold. So the curve, the recruitment-threshold boundary, the amplifier terms (motion deepens the dip; cross-modal mismatch drops an artefact in even where each channel alone would clear it), and the sign-aware design heuristics (retreat to stylisation or push to full photorealism, never polish in the trough; match realism across modalities; test affinity before committing) all carry intact as the context changes: from Mori's original robotics and human-robot interaction to computer animation (the diagnosed failure of *The Polar Express* and *Final Fantasy: The Spirits Within*, and the deliberate stylisation of Pixar/Disney), to virtual agents and deepfakes and generative video, to too-lifelike prosthetics and medical training mannequins, to the folk-art cluster of taxidermy, wax figures, and dolls. These are different industrial settings, not different substrates — one trigger, many applications — which is exactly why the same reasoning governs all of them and the transfer is genuinely mechanistic.

Beyond that trigger condition the transfer splits into two cases worth marking separately. The first is *analogy by metaphorical extension*: coinages like "the uncanny valley of mind" or "of voice" lift the rise-dip-recovery picture into territory where the load-bearing trigger — human-specific perceptual recognition routines detecting near-human anomalies — is absent or far less crisp, so the mechanism that makes the visual valley sharp and predictable does not come along; the shape is borrowed, the engine is not. (Tellingly, no one reports an uncanny-valley dip for a car approaching photorealism, because the human-specific routines were never engaged — the regime boundary the concept itself draws.) The second is the genuinely broader structural pattern the uncanny valley *instantiates*: a *non-monotonic response to a continuous design parameter*, a real and widely-travelling abstraction whose other instances — phase transitions in matter, hormetic dose-response in pharmacology, the Yerkes-Dodson arousal-performance curve — share the rise-dip-or-peak *form* but each have their own, different mechanism. That pattern does transfer cross-domain, but as the general primes (`non_monotonic_relationships`, or `phase_transitions`), not as the uncanny valley, which is merely the human-perception instance of the family. The honest report, then, is that "uncanny valley" travels mechanistically only within the near-human-artefact trigger; where the cross-domain lesson is the *shape*, it belongs to the non-monotonic-response prime, and where the lesson is the *cause*, it belongs upstream to expectancy-violation and mismatch-detection — not to this concept as named (see *Structural Core vs. Domain Accent*).