nucleation

TERM-043: fidelity, as a profile of directed distortions compared component by component

5 min read

Properties6
StatusActive
Version2
Tagsphilosophy-of-language, information-theory
Linepragmatic-transport
Used byCLAIM-128 — No single ranking of translations holds across tasks: where distortion profiles trade off, only a task's priorities rank them, and the theory of fidelity fixes no weighting, CLAIM-133 — Communicative identity is relational continuity under admissible transformation: translation transports relationally organized observational structure, correspondence fixes what can be compared, admissibility which transformations count, and fidelity which communicatively consequential distinctions remain recoverable, CLAIM-139 — A communicative kind's identity is grounded in its relational organization rather than in any expression or encoding; statistical methods recover aspects of that organization from restricted observations, translation transports it, and fidelity is which independently identifiable, decision-relevant relational distinctions persist, CLAIM-147 — A trope is a family of observational profiles joined by admissible transformations around a core of preserved relations, so trope instances can separate structural preservation from pragmatic preservation, CLAIM-218 — The task-paired risk difference the record uses in place of Blackwell's order, D_Q = sup_q |R_q(U) − R_τ(q)(V)|, is a different quantity: it reduces each risk set to one number, needs the losses of paired tasks on one scale, and its absolute value counts improvement as distortion, CLAIM-256 — D_obs sums over source contexts only, so a target context outside the image of τ appears in no term: a rendering that adds a relation pays nothing, and a change of cover registers only as loss, CLAIM-201 — Over a fixed scenario, a change of global compatibility at small local distortion needs contextual judgements, which the behavioural data read so far lack; the form of the headline those data leave open is a change of cover, in which a rendering makes some of the source's questions undecidable for its readers and decides others the source left open, CLAIM-204 — A communicative kind's identity is grounded in its relational organization, for which competent participants' uptake is the standard; statistical methods recover aspects of it from restricted observations; translation transports it; and what persists is what the receivers can still recover, of which the analyst can certify the independently identifiable part, CLAIM-210 — Sharing a kind and being a rendering are different relations: kind-sharing is symmetric and says what is to be preserved, while fidelity belongs to a transport from a given source, so an utterance of the same kind made independently of the source shares its kind and has no fidelity to it, CLAIM-228 — D_obs restated over a rendering procedure, with one kernel family fitted on some source items and scored on held-out ones and the target term the readers' response averaged over the procedure's output, separates rendering procedures that receivers' task-indexed recovery separates, so restated it measures observational fidelity informatively, CLAIM-231 — On observed responses, a rendering that keeps the response distribution of each order of two framings within ε of the source's, under an outcome correspondence, keeps the observed order effect within 2ε, a bound identified from data, while the intertwining identity bounds it only through defects of an unidentified state model, CLAIM-240 — A holistic fidelity judgement is fidelity under the judges' default task, a weighting of the profile's components that can be estimated on some renderings and tested on others, so it is well defined and consistent with task-relativity, and a defeat condition that uses it must name the population and fix the weighting before the test, and across populations with unequal access to the source must fit the weighting only on the components each can perceive, CLAIM-257 — Observational distortion should be summed over the contexts both covers can host, with the target's new questions asked of the source too, and the questions only one side can decide reported as a separate change of cover, so that a rendering that adds a relation pays for it, CLAIM-259 — For report fidelity, the decision term should be the positive part of the risk increase on the same decision problem, with losses in [0, 1] and a stated prior; so restated it is directed, counts no improvement as loss, and vanishes for every problem and prior exactly when the rendering is at least as informative as the source in Blackwell's order, CLAIM-319 — Over a scenario of n contexts in an odd cycle, changing the support of one context takes the contextual fraction from 1 to 0 while the other n − 1 contexts and every marginal are kept exactly, at a weighted observational distortion of 1/n, so within a scenario the fraction's Lipschitz constant against that distortion is at least n; and any change of global extension needs a contextual model on one side, CLAIM-371 — Under the componentwise order on distortion profiles, two profiles are incomparable exactly when they trade off, and then positive weightings can rank them either way, while every positive weighting agrees with a dominance
Refined byTERM-010 — fidelity, as a weighted three-term directed distortion

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Definition

A125 §3 proposed that the two Active fidelity lineages be unified: “The first is a measurement criterion; the second is the general relation.” The owner answered at U41: “agreed.” A129 §3: “the solution is not to add more meanings to the word fidelity. We should distinguish the transformation being evaluated from the tests applied to it.”

A transport T from a source system to a target system “specifies how realizations, contexts, observations, and relevant inference tasks correspond” (A129 §3). Its fidelity is the profile of its distortions. A151 §5 gives five components, D(T) = (D_obs(T), D_struct(T), D_dec(T), D_causal(T), D_dyn(T)), and glosses each:

ComponentWhat it measures (A151 §5)
ObservationalDifferences between corresponding context-indexed empirical distributions
StructuralFailures to preserve restriction maps and specified relational constraints
Decision-theoreticLoss of ability to perform relevant communicative inference tasks
CausalDifferences in responses to corresponding interventions
DynamicalDifferences in subsequent interpretation and reconstruction trajectories

A129 §3 had four components, without D_causal, which it added after them as “Pearl adds another dimension”. A178 and A218 keep five. A203 §22 adds a sixth, D_χ: “The character component D_χ is optional and meaningful only when the relevant transformation representations have been defined”.

Transports are compared component by component: T_1 ⪯ T_2 when D_j(T_1) ≤ D_j(T_2) for every j. “When the profiles trade off, the transports are incomparable without selecting additional priorities” (A129 §3). A weighted sum D_w = Σ_j w_j D_j is “introduced for optimization” once a task fixes w. A151 §5: “two translations can preserve different structures without being forced into a universal ranking. We can introduce task-specific weights when a scalar objective is useful.” The consequence for ranking translations is CLAIM-128.

How the older terms sit in the profile

  • TERM-021 is D_obs, with the source distributions pushed forward by the transport: D_obs(T) = Σ_C w_C d_C(T_C# e_C^o, e_τ(C)^t) (A125 §3).
  • TERM-015 is the directed, graded relation that the components measure. A129 §3: “TERM-015 supplies the general directed, graded transport relation.”
  • TERM-033 is the zero-distortion limit over response trajectories. A129 §3: “TERM-033 supplies the limiting case of trajectory-based behavioral equivalence.”
  • TERM-028 and TERM-034 are D_dec. The profile drops TERM-034’s rate constraint and its interpreter, which A108 had put in the definition.
  • TERM-036, fidelity as preservation of selected symmetries, is not placed. A203’s optional D_χ comes nearest to it, and no turn connects them.

What it is not

  • Not TERM-010’s scalar. TERM-010 minimized a weighted sum of three distortions under a rate budget. This term supersedes it as the foundation. The scalar survives as one weighting, chosen per task.
  • Not a metric. Each component is directed, and only D_obs has a formula.
  • D_causal is not cleanly a further component. A129 §3 defines it as Σ_a π(a) d(T_# P_o(Y | do(a)), P_t(Y′ | do(τ(a)))). That is TERM-033’s comparison under corresponding interventions at one step, and it overlaps D_dyn (“Does the reconstruction respond similarly to subsequent framing interventions?”). A129 says that D_causal “makes the relationship between dynamic fidelity and causal fidelity precise”, but does not separate them (R1). D_causal also needs a correspondence τ on interventions as well as on contexts (CLAIM-127), and how it would be estimated when corresponding interventions cannot be identified is QUESTION-005.
  • Character distance is a diagnostic, not a definition of fidelity. A198 §VIII: “I would resist turning character distance into the universal definition of communicative fidelity.” A218 §8: “Character similarity is neither a general sufficient condition nor a universal definition of fidelity.”
  • Not a partial order on transports. Componentwise comparison is a partial order on profile vectors, but only a preorder on transports: distinct transports can share a profile (R1).

Note of 2026-10-10: the replies

After the audit: D_obs is restated over a class of items, with one kernel family fitted on some and scored on others (CLAIM-228), and over both covers (CLAIM-257); D_dec as the directed positive part of the risk increase on the same problem (CLAIM-259); D_dyn measured on observed order effects (CLAIM-231).

Note of 2026-10-21: D_causal and causal abstraction

The causal-abstraction papers were read on 2026-10-21 (NOTE-1037, NOTE-1032, NOTE-1045). D_causal is Rubenstein et al.’s exact transformation made graded: a π-average of distances between the push-forward of the source’s interventional law and the target’s, without their requirement that the correspondence on interventions be surjective and order-preserving, which their Theorem 6 and Examples 7–8 need for composed interventions. So it inherits Beckers and Halpern’s Example 3.4 permissiveness (a well-chosen distribution lets unrelated models match). Beckers, Eberhardt and Halpern’s d_τ (THEORY-843) is the worked graded relation in that literature, worst-case over interventions and context by context, with composition bounds that are a template for composing transports.

One consequence for an entry of the same day. The governance-response profile, TERM-067, was first filed as refining this term, as D_causal with τ the identity. It is not: for the hidden governance bit D_causal is zero, since marginalising the hidden bit is exact (THEORY-848), while the profile does not factor through the coarse state (CLAIM-399). The link was removed. The definition here is unchanged.

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