---
slug: interpolation
name: Interpolation
one_liner: "Infer the value or state at a query point that lies *within* the span of known reference points, licensed by a continuity assumption — the bounded, well-supported pole of inference-from-anchors whose beyond-the-span twin is extrapolation."
type: prime
schema_version: 2
status: density_v2_draft
categories: [structural, epistemic, computational]
origin_domain: mathematics
alternate_origin_domains: [numerical_analysis, signal_processing, statistics_and_machine_learning, computer_graphics, cartography]
aliases: [within_range_inference, in_sample_inference, tweening, resampling, gap_filling_between_anchors]
related: []
solution_archetypes: []
review_flags:
- "REFLEXIVELY DISCOVERED (2026-07-04). While documenting the discovery-operator set, `interpolation` was the ONE operator (op-12) with no corresponding prime in the corpus. Screening that gap under the standard existence-check revealed that interpolation itself is genuinely absent (search_prime + broad grep: no interpolation / betweenness / intermediacy / within-range-inference prime). The discovery machine's own missing part pointed at a real hole in the catalog it was built to search — a clean instance of the reflexivity noted in the operator compendium (the operators are themselves primes; this one had no home). STATUS: OWNER-ACCEPTED 2026-07-04 (decision made in conversation — strong, clean, structural). Mechanical incorporation deferred to the end-of-search batch curation pass, since the corpus is frozen at 1,325 for the duration of the operator sweep."
- "DUALIZATION GAP: `extrapolation_beyond_sampled_regime` IS present (though framed as a failure-mode — a calibrated apparatus blind to its own regime exit), while its safe twin, interpolation, is absent. Operator 01 (dualization) should have surfaced this pair and did not — the asymmetry (people name the *risky* pole) is itself the kind of gap the sweep exists to catch. Consider whether the present extrapolation prime should be reframed to the clean structural pole so the {interpolation, extrapolation} duality is symmetric."
- "Created 2026-07-04; STUB ONLY (frontmatter + Core Idea + Notes). SI/SF/similarity are PROPOSER ESTIMATES pending triangulation. Body sections (Structural Signature onward) NOT written."
- "GENUS QUESTION — RESOLVED (owner, 2026-07-04). The question was whether interpolation is a prime in its own right or merely a species of a deeper `betweenness`/`intermediacy` primitive. It is a prime in its own right, and the two are not competing for the same slot because they are different KINDS of thing: **betweenness is a static relation** ('B lies between A and C' — needs no assumption or procedure, like `order`), whereas **interpolation is an inference/operation** (given the anchors, the betweenness fact, AND a smoothness assumption, produce the value at the between-point). Betweenness is one ingredient interpolation uses; interpolation is one thing you do with betweenness. The natural conceptual stack is: betweenness (relation) → built on it: `convexity` (a region contains every point between any two members — ALREADY a prime) and interpolation (infer at a between-point) → and interpolation's complement `extrapolation_beyond_sampled_regime` (infer at a NOT-between point — ALREADY a prime). The corpus already holds convexity and extrapolation but names neither the underlying relation (betweenness) nor the safe inference (interpolation) — it reached the branches without naming the trunk. DECISION: accept interpolation (this stub); **do NOT mint `betweenness` for now** — it is a genuine deeper primitive but is already partly embodied by `convexity` + `order`, rarely invoked bare, and minting it risks a thin/overlapping entry; treated as a matter of taste and deferred (not a priority). The two decisions are DECOUPLED: interpolation stands on its own whether or not betweenness is ever added, so no genus dependency blocks it."
- "Phase-C action: re-similarity-check vs `extrapolation_beyond_sampled_regime` (the dual), `imputation` (missing-ENTRY filling, not necessarily bounded-between), `pattern_completion` (general gap-fill), `convexity` (defined via the same within-the-hull relation), and `gradient` — confirm interpolation is distinct from each (its anatomy = bounded-between + smoothness prior + exact-at-anchors). Draw the dualization edge to `extrapolation_beyond_sampled_regime`; consider a shared parent 'inference of the unobserved from the observed' over {interpolation, extrapolation, imputation, pattern_completion}. Optionally reframe the present extrapolation prime (currently a failure-mode framing) to the clean structural pole so the dual pair is symmetric. Sync one_liner v1↔v2 per CLAUDE.md before merge."
substrate_independence:
  composite_substrate_independence: 5
  domain_breadth: 5
  structural_abstraction: 5
  transfer_evidence: 5
  reasoning: 'The bare structure — given a set of known anchor points and a continuity/smoothness assumption about what lies between them, posit the value or state at a query point inside their span, with the estimate passing through the anchors and its error bounded because the query is within the support — is a pure relational/inferential form with no commitment to any medium. It recurs with exact transfer across numerical analysis (polynomial and spline interpolation), signal processing (resampling and upsampling), computer graphics (keyframe tweening), cartography (contour infilling), statistics and machine learning (the within-range / in-distribution regime where prediction is reliable), music (glissando/portamento), and colour (gradient stops), and it is recognized rather than translated in each. It sits at the same substrate-independence tier as random_walk or projection: a canonical structural 5.'
similarity_to_nearest_existing_prime:
  slug: extrapolation_beyond_sampled_regime
  similarity: 0.6
structural_framed:
  label: structural
  aggregate: 0.05
  criteria: {vocab_travels: 0.0, evaluative_weight: 0.0, institutional_origin: 0.0, human_practice_bound: 0.0, import_vs_recognize: 0.25}
  rationale: 'Interpolation is a bare relational/inferential form — anchors plus a smoothness prior yield an intermediate — operative across substrates with no normative or institutional content; nearly every diagnostic reads zero. It carries no home lexicon that must travel, no evaluative weight, and a formal mathematical origin. The one mild lift is that the term is recognizably imported from numerical mathematics, but the structure is recognized rather than translated wherever a value between known values is inferred, so it reads clearly structural.'
phase_b_passes_completed: []
---
# Interpolation

## Core Idea

Interpolation is the inference you make when you have several *known reference points* and want the value or state at some *query point that lies between them*, and you get there by assuming the thing varies smoothly enough that what happens in the gap is constrained by what happens at the anchors. Given temperatures measured at noon and at 2 p.m., interpolation supplies a defensible estimate for 1 p.m.; given two keyframes of an animation, it supplies the frames in between; given a coarse grid of elevation samples, it fills the contour lines that run through them. The move has three load-bearing commitments. First, **boundedness** — the query point lies *within* the span (formally, the convex hull) of the anchors; this is what separates interpolation from its twin. Second, a **continuity or smoothness prior** — some assumption (linearity, a spline, a kernel, a distributional shape) that licenses reading the unobserved gap off the observed edges; without it, the anchors say nothing about the between. Third, **exactness at the anchors** — a genuine interpolant passes *through* the known points, reproducing them rather than merely fitting near them.

The characteristic payoff is **reliable, bounded inference**. Because the query sits inside the support of what you actually observed, the smoothness prior is doing work it has evidence for, and interpolation error stays controlled — this is precisely why "interpolate freely, extrapolate at your peril" is one of the most load-bearing heuristics in every quantitative field. The characteristic limit is the mirror image of that strength: interpolation is *silent beyond the anchors*, and its very reliability inside the span is what makes its beyond-the-span twin, **extrapolation**, so dangerous — the same smoothness prior, carried past the last known point, is suddenly unsupported and its errors grow without bound. Interpolation and extrapolation are therefore not two topics but the two poles of a single axis: *inference within the observed range* versus *inference beyond it*. This is the in-sample / out-of-sample distinction that organizes forecasting, curve-fitting, and machine-learning generalization; the difference between a model you can trust and one you cannot is very often just which side of the last data point the question falls on.

Read structurally rather than as a numerical-methods technique, the same move recurs wherever a gap between knowns is filled under a continuity assumption. In *signal processing* it is resampling — reconstructing a continuous signal between its samples (Nyquist–Shannon is the statement of when this is exact). In *computer graphics* it is tweening between keyframes and blending between colour stops. In *statistics and machine learning* it is the safe regime of prediction — inside the training distribution, where the learned function is well-constrained. In *cartography* it is drawing the contour that must pass between two spot heights. In *music* it is the glissando that slides continuously between two pitches. In *reasoning* it is positing the intermediate case between two cases you already understand. Across all of these the recognizable skeleton is identical: fix the anchors, assume smoothness, and read off the between — bounded, exact at the knowns, and trustworthy exactly to the edge of the support and no further.

## Notes

**Provenance.** Reflexively discovered on 2026-07-04. The operator-search program is built from fourteen discovery operators, and while documenting them it emerged that thirteen correspond to existing corpus primes (dualization ↔ `duality`, self-application ↔ `reflexivity_self_reference`, generalization ↔ `abstraction`, composition ↔ `composition`, analogy ↔ `analogy`, and so on) — but **operator 12, interpolation, had no home prime.** Screening that gap under the standard existence-check confirmed interpolation is genuinely absent from the catalog. The discovery machine's own missing part turned out to point at a real hole in the catalog it was built to search.

**For owner evaluation (keep/cut).** *Owner decision 2026-07-04: ACCEPTED.* The case is strong:
interpolation is a canonical, domain-general structural form with exact cross-domain transfer
(numerical analysis, signal processing, graphics, statistics, cartography, music), it clears the
prime bar cleanly (its own load-bearing anatomy — bounded, smoothness-licensed, exact-at-anchors),
and its absence is conspicuous *because its dual is present* — the corpus already holds
`extrapolation_beyond_sampled_regime`, so the safe pole of a standard two-pole distinction was simply
never named. The in-sample/out-of-sample heuristic it anchors is one of the most invoked ideas in all
of quantitative practice.

The one substantive question raised and resolved in discussion was the **genus question** — is
interpolation a prime in its own right, or a species of a deeper `betweenness`/`intermediacy`
primitive? It is a prime in its own right. Betweenness and interpolation are different *kinds* of
thing: betweenness is a static *relation* ("B is between A and C," needing no assumption or
procedure, like `order`), while interpolation is an *inference* that uses betweenness plus a
smoothness assumption to produce a value. They do not compete for one slot; the encyclopedia already
holds relation-primes and operation-primes side by side. The conceptual stack is betweenness
(relation) → `convexity` and interpolation (both built on it) → `extrapolation` (interpolation's
complement); the corpus already names convexity and extrapolation but neither betweenness nor
interpolation. **Decision: accept interpolation; do not mint `betweenness` for now** — it is a real
deeper primitive but is already partly embodied by `convexity` + `order`, rarely invoked on its own,
and minting it risks a thin, overlapping entry. Betweenness is deferred as a matter of taste, not a
priority, and — crucially — the two calls are decoupled: interpolation stands on its own whether or
not betweenness is ever added.

Two smaller items for the incorporation pass (below): screen the family boundary against `imputation`
/ `pattern_completion` / `convexity`, and decide whether the present `extrapolation_beyond_sampled_regime`
(currently a *failure-mode* framing) should be reframed to the clean structural pole so the dual pair
reads symmetrically.

**At incorporation.** Re-run similarity against `extrapolation_beyond_sampled_regime`, `imputation`, `pattern_completion`, `convexity`, and `gradient`. Resolve the betweenness/interpolation genus–species question. Draw the dualization edge to `extrapolation_beyond_sampled_regime`, and — if the family is judged real — a subsumption edge up to a shared parent along the lines of *inference of the unobserved from the observed* over {interpolation, extrapolation, imputation, pattern_completion}. Write a v1 draft and sync the one_liner across v1 and v2 per CLAUDE.md before merge.
