{ "artifactId": "artifact.library.mathlib.ostrowski-theorem.v001", "candidateOnly": false, "category": "Number theory", "claimBoundary": "This target indexes Mathlib's classification of nontrivial absolute values f : AbsoluteValue ℚ ℝ. The conclusion is equivalence, not equality: f is equivalent to the standard real absolute value, or there is a unique natural number p carrying a prime instance for which f is equivalent to the p-adic absolute value. The declaration concerns ℚ with values in ℝ; it does not classify absolute values over arbitrary global fields or valuations with arbitrary codomains.", "collectionSlug": "landmark-theorems-in-mathlib", "exactFormalStatement": "theorem Rat.AbsoluteValue.equiv_real_or_padic (f : AbsoluteValue ℚ ℝ) (hf_nontriv : f.IsNontrivial) : f ≈ real ∨ ∃! p, ∃ (_ : Fact p.Prime), f ≈ (padic p)", "family": "Absolute values and places", "id": "library.mathlib.ostrowski-theorem.v001", "links": { "collection": "/collections/landmark-theorems-in-mathlib/", "collectionData": "/data/collections/mathlib-landmarks.json", "entry": "/library-theorems/ostrowski-theorem/", "entryData": "/data/library-theorems/ostrowski-theorem.json", "evidence": "/proofs/artifact.library.mathlib.ostrowski-theorem.v001.html", "evidenceData": "/data/proofs/artifact.library.mathlib.ostrowski-theorem.v001.evidence.json", "source": "/sources/upstream/ostrowski-theorem/" }, "landmark": { "completeFormalType": "theorem Rat.AbsoluteValue.equiv_real_or_padic (f : AbsoluteValue ℚ ℝ) (hf_nontriv : f.IsNontrivial) : f ≈ real ∨ ∃! p, ∃ (_ : Fact p.Prime), f ≈ (padic p)", "concepts": [ "Ostrowski's theorem", "absolute values", "equivalent absolute values", "Archimedean absolute value", "p-adic absolute value", "rational numbers" ], "plainLanguage": "Every nontrivial real-valued absolute value on the rational numbers measures size in one of two ways up to rescaling: like the usual real absolute value, or like the p-adic absolute value for one uniquely determined prime p.", "poster": { "alt": "An ivory number-theory poster states that every nontrivial absolute value from the rationals to the reals is equivalent, not equal, to the standard real absolute value or to the p-adic absolute value for one unique prime.", "byteSize": 2481378, "caption": "Over ℚ with values in ℝ, every nontrivial absolute value belongs to the standard real equivalence class or to one uniquely determined p-adic class.", "description": "The sparse engraved poster distinguishes equivalence from equality before splitting a rational fraction-wheel into an unbounded real-distance branch and a bounded nested p-adic branch centered on one gold prime. Its footer keeps the statement within ℚ and ℝ.", "derivatives": [ { "byteSize": 93122, "height": 960, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/ostrowski-theorem/theorem-poster-v1-640.webp", "sha256": "sha256:463978c65f7e7e649ceed44a3a98311764d503287197f698f784181e8a98f6df", "width": 640 }, { "byteSize": 197218, "height": 1536, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/ostrowski-theorem/theorem-poster-v1-1024.webp", "sha256": "sha256:45813f0a363a84b35ac9ba2fb67cf5443e7603137913e00c8af14cabdd9d6153", "width": 1024 } ], "generatedAt": "2026-07-25T16:29:09-04:00", "height": 1536, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/ostrowski-theorem/theorem-poster-v1.png", "sha256": "sha256:37b1d73d8107a553e14f97c015511f020eb8c754fa9f04c727e640a6df1e8876", "title": "Ostrowski's theorem — the real class or one unique p-adic class", "width": 1024, "kind": "theorem_poster", "transcript": "NUMBER THEORY · ABSOLUTE VALUES ON ℚ\nOSTROWSKI'S THEOREM\nEvery nontrivial absolute value f : ℚ → ℝ\nis equivalent to the standard real absolute value\nor to the p-adic absolute value for one unique prime p.\nEQUIVALENT, NOT EQUAL\nThe same ordering of sizes, up to a positive-power rescaling.\nTHE CHECKED DICHOTOMY\nUNBOUNDED · STANDARD REAL CLASS\nBOUNDED · UNIQUE p-ADIC CLASS\nEXACT SCOPE\nℚ to ℝ only. No arbitrary global-field or codomain claim." }, "visual": { "alt": "A dark engraved schematic sends one rational absolute-value lens along an unbounded arc to the real class or along bounded nested rings to one highlighted p-adic prime class.", "byteSize": 2248476, "caption": "The unbounded branch reaches the standard real class; the bounded branch determines one unique p-adic class.", "description": "A double-struck rational medallion feeds one rescaling lens and splits into two structurally different branches. An expanding calibrated arc represents the Archimedean real class, while concentric emerald valuation rings contract onto one selected gold prime, with alternative prime medallions left outside the chosen route.", "derivatives": [ { "byteSize": 31148, "height": 427, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/ostrowski-theorem/theorem-schematic-v1-640.webp", "sha256": "sha256:8925906ecf09cb3b415aff6ed22c2c69ce4573b0820fe13141d0894483294f26", "width": 640 }, { "byteSize": 84080, "height": 800, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/ostrowski-theorem/theorem-schematic-v1-1200.webp", "sha256": "sha256:d0b4f766d28166dec742f045c5934a0034261446ad406b3f01af692c8df6b218", "width": 1200 } ], "generatedAt": "2026-07-25T16:29:09-04:00", "height": 1024, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/ostrowski-theorem/theorem-schematic-v1.png", "sha256": "sha256:e37b7c862fd435c9796469b0f002d58c9090e3cc8b294b8b9523250e65c157c9", "title": "The rational absolute-value dichotomy", "width": 1536 }, "walkthrough": null, "whyLandmark": "Ostrowski's theorem gives the definitive classification of nontrivial real-valued absolute values on ℚ, separating the familiar Archimedean size from the p-adic sizes that organize local number theory." }, "nonClaims": [ "Equivalent absolute values need not be equal; equivalence preserves comparisons of sizes and, here, corresponds to positive-power rescaling.", "The theorem assumes the absolute value is nontrivial.", "The selected declaration is over ℚ with codomain ℝ, not over arbitrary global fields or arbitrary valuation codomains.", "It is not a product formula, completion theorem, or algorithm for finding a prime from numerical samples.", "ProofAtlas is indexing an existing Mathlib theorem, not claiming new mathematics." ], "publicExportEligible": true, "relationship": "upstream_direct", "schemaVersion": "library-theorem-entry-data.v1", "statementId": "statement.library.mathlib.ostrowski-theorem.v001", "statusAxes": { "acceptedAtlasResult": "Not recorded for the preferred artifact", "locallyReproduced": "Exact upstream declaration replayed", "reviewedPage": "Current public presentation reviewed", "upstreamIndexed": "Pinned source bytes verified locally" }, "summary": "Every nontrivial real-valued absolute value on ℚ is equivalent to the standard real absolute value or to a p-adic absolute value for one unique prime.", "targetId": "target.library.mathlib.ostrowski-theorem.v001", "title": "Ostrowski's Theorem", "upstreamOrigin": { "declarationName": "Rat.AbsoluteValue.equiv_real_or_padic", "immutableSourceUrl": "https://github.com/leanprover-community/mathlib4/blob/5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib/NumberTheory/Ostrowski.lean#L466", "packageName": "mathlib", "repositoryUrl": "https://github.com/leanprover-community/mathlib4", "sourceArtifactHash": "sha256:4cb972c590d300b3be4c4a8092c522cc200f499b7940c845494957cbdbcd784d", "sourceByteLength": 21866, "sourceCommit": "5e932f97dd25535344f80f9dd8da3aab83df0fe6", "sourceFile": "Mathlib/NumberTheory/Ostrowski.lean", "sourceLine": 466, "verificationKind": "git_worktree" }, "publicPresentationReview": { "status": "reviewed" } }