---
slug: observational_equivalence
name: Observational Equivalence
one_liner: "Two or more distinct internal configurations that produce identical observable behavior, so no observation of that kind can tell them apart — the many-to-one, indistinguishable pole opposite identifiability."
type: prime
schema_version: 2
status: density_v2_draft
categories: [epistemic, structural, computational]
origin_domain: statistics
alternate_origin_domains: [control_theory, computer_science, philosophy_of_science]
aliases: [indistinguishability, unidentifiability, equivalence_of_models, empirical_equivalence]
related: []
solution_archetypes: []
review_flags:
- "OPERATOR-DERIVED (op-01 dualization, sub-flavor 1c) — the structural dual of `identifiability` along the internals-to-observation mapping (one-to-one vs. many-to-one) axis. Discovered in the op-01 full sweep 2026-07-03. Verified absent via search_prime + slug check (manifest note: 'Verified absent. STRONG structural dual.'). STATUS: HOLD — pending owner curation."
- "Created 2026-07-03; STUB ONLY (frontmatter + Core Idea + Notes). Did NOT go through Phase-A triage. SI/SF/similarity grades are PROPOSER ESTIMATES pending triangulation. Body sections (Structural Signature onward) NOT yet written."
- "Phase-C action: re-similarity check vs `identifiability` (the input/dual), `underdetermination`, `degeneracy`, `abstraction`, and any equivalence-relation primes to confirm it does not collapse; draw the dual edge to `identifiability` at incorporation. one_liner must be synced to a v1 draft per CLAUDE.md before merge."
substrate_independence:
  composite_substrate_independence: 4
  domain_breadth: 4
  structural_abstraction: 5
  transfer_evidence: 4
  reasoning: 'The relation — distinct internal configurations mapping to the same observable output, so the observation cannot separate them — is a pure statement about a many-to-one map between a hidden space and a visible one, with no commitment to a medium, earning structural abstraction a full 5. It recurs across a wide span: unidentified parameters in statistics and econometrics, indistinguishable state-space realizations in control theory, observationally equivalent programs/automata and equivalent circuits in computer science, empirically equivalent theories in philosophy of science, and bisimilar processes in concurrency. Domain breadth is 4 spanning statistics, engineering, CS, and philosophy. Transfer evidence is 4 because the same fixptheorem-level result — that only a structural intervention (new channel, restriction, prior) can break the equivalence — is independently proved and used across these fields. Composite 4: a strongly substrate-independent epistemic/structural relation whose instances are genuine and cross-disciplinary.'
similarity_to_nearest_existing_prime:
  slug: identifiability
  similarity: 0.7
structural_framed:
  label: structural
  aggregate: 0.08
  criteria: {vocab_travels: 0.1, evaluative_weight: 0.0, institutional_origin: 0.0, human_practice_bound: 0.0, import_vs_recognize: 0.1}
  rationale: 'A many-to-one map from internal configurations to observable outputs is a pure relational structure, medium-neutral and carrying no normative or institutional content, so nearly every diagnostic reads at or near zero. It has a formal origin (statistics/logic/model theory), no evaluative weight — being observationally equivalent is neither good nor bad, only a fact about what the data can resolve — and no human-practice binding, since a control system, a circuit, or an automaton exhibits the relation with no human in the loop. The faint non-zeros reflect only that the exact phrase is mildly a term of art one carries in; the underlying indistinguishability structure is recognized wherever two hidden things produce the same visible signal.'
phase_b_passes_completed: []
---
# Observational Equivalence

## Core Idea

Observational equivalence is the relation that holds between two or more *distinct internal configurations* — parameters, mechanisms, hidden states, latent structures, or whole models — when they produce the *same observable output*, so that no observation of the available kind can tell them apart. It is one pole of a single structural axis, *the mapping from internals to observations*, whose opposite pole is `identifiability` (the condition under which that mapping is one-to-one, so a hidden configuration can in principle be uniquely recovered). Where identifiability asserts uniqueness — every distinguishable internal object leaves a distinguishable observable trace — observational equivalence is the exact failure of that uniqueness: the map from internals to observations is *many-to-one*, collapsing a set of genuinely different underlying objects onto a single visible signature. When two internal configurations are observationally equivalent, the object of interest is *unidentified*, and the equivalence is the reason why.

The load-bearing invariant is that no additional data *of the same kind* can break the equivalence. This is what makes observational equivalence a structural relation and not a transient shortage of information. Because the two configurations produce identical observable distributions, more samples drawn through the same channel will, in expectation, look identical under both — the likelihood is flat across the equivalence class, and the class does not shrink with sample size. What *can* break it is only a *structural* change to what is observed or assumed: opening a new measurement channel that the two configurations differ on, performing an experimental intervention that pushes the system somewhere the equivalence does not hold, imposing a parametric restriction that admits only one member of the class, or committing to a prior that selects among them. The characteristic payoff-and-cost is thus double-edged: observational equivalence *bounds inference* (it marks precisely the questions the current observations cannot answer, no matter how much data accrues), and it also *licenses substitution* (any member of an equivalence class is, for prediction through the same channel, as good as any other — a simpler surrogate can replace a complex mechanism without observable penalty). The same relation that frustrates the scientist trying to recover a true mechanism rewards the engineer content with any adequate model.

Across substrates the pattern is one relation. In **statistics and econometrics** it is *non-identifiability* — distinct parameter values yielding the same likelihood (a rotated factor-analysis loading, a mixture with unlabeled components, a simultaneous-equations system without exclusion restrictions). In **control theory** it is *indistinguishable state-space realizations* — different internal matrices producing identical input-output behavior, so no experiment on the terminals separates them. In **computer science** it appears as *observationally equivalent programs* and *bisimilar processes* — different code or automata that no context can distinguish by behavior, the very relation that justifies replacing one implementation with another. In **philosophy of science** it is the *empirical equivalence* of rival theories that make identical predictions, the underdetermination that no crucial experiment through existing observations can settle. In every case the recognizable move is the same: several distinct hidden things share one visible fingerprint, so the observation defines an *equivalence class* rather than a point, and only a structurally new kind of look can split it.

## Notes

**Provenance.** Produced by the op-01 dualization operator (sub-flavor 1c) as the structural dual of `identifiability` along the internals-to-observation mapping axis: identifiability is the *one-to-one* (uniquely recoverable) pole, observational equivalence is the *many-to-one* (indistinguishable) pole. Discovered in the op-01 full sweep on 2026-07-03; the manifest marks it a STRONG structural dual and records it as verified absent from the corpus.

**For owner evaluation (keep/cut).** The case FOR: this is a clean, medium-neutral relational structure — a many-to-one map from a hidden space to a visible one — that recurs under independent names across statistics (non-identifiability), control theory (indistinguishable realizations), CS (observational/behavioral equivalence, bisimulation), and philosophy of science (empirical equivalence / underdetermination), which is strong convergence evidence for a genuine prime. It is the natural dual of an existing prime (`identifiability`) and completes that pair on the recoverability axis, and it carries independent content (the double-edged bound-inference / license-substitution consequence) rather than being a mere negation. The manifest reports it verified absent, so there is no direct slug collision. The specific RISK to weigh: **screen overlap with `identifiability` itself** — a candidate concern is whether observational equivalence is genuinely a distinct prime or merely the negative statement of identifiability that the existing prime already implies (identifiability *is defined by* the absence of observational equivalence). The decision-relevant question for Kurt is whether the two poles each earn a prime — as with `eager`/`lazy` and `risk`/`knightian_uncertainty`, the argument is that the equivalence pole has its own load-bearing structure (equivalence classes, the substitution license, the "only a structural intervention breaks it" invariant) that is not merely "not identifiable." A secondary screen: check against any existing `underdetermination`, `degeneracy`, `abstraction`, or equivalence-relation primes, since observational equivalence borders all of them.

**At incorporation.** Re-run similarity against `identifiability` (dual, primary overlap check), `underdetermination`, `degeneracy`, `abstraction`, and any equivalence-relation primes to confirm it does not collapse into identifiability's negation or into a more general equivalence prime. If kept, draw the dual edge to `identifiability` and consider edges to `underdetermination` / `abstraction` if those exist. Sync the final `one_liner` to a v1 draft per CLAUDE.md before merge.
