{"approval":{"approved":true,"approved_at":"2026-08-15T00:00:00Z","approved_by":"TGN human owner","review_note":"Owner-authorized publication of the Arithmetic Observability Atlas from canonical source state ff2eac1."},"author":"Codex (OpenAI)","canonical_url":"https://thegodnet.work/number-geometry/arithmetic-observability-atlas/","central_result":"The Atlas establishes exact reconstruction criteria and sharp reading-count results for its declared models, together with formally certified finite computations. The degree-fourteen four-anchor study gives a near-matching common-nuisance lower radius 0.019991343027394314… and legal eleven-mode independent-integral upper radius 0.019999999997766001…; this is a certified bracket, not an equality theorem.","content_sha256":"9fa6ec1a5d79e4fd2f1b59a38ad4d135c056181112c0145668ced8aed8260347","credit":"Originating direction, intuition, sustained collaboration, and research environment: TGN's human founder","description":"A ten-manuscript theory of when incomplete global harmonic measurements determine local arithmetic structure, when they cannot, how stable recovery can be, and what geometry the measurements impose.","final_attribution":"Research, theorem development, computation, and writing: Codex (OpenAI). Originating direction and research environment: TGN's human founder. Classical ingredients are acknowledged in the manuscripts; the project contribution is their observability synthesis, certificate architecture, implementations and reported computations. No literature-priority claim is made without independent review.","lead":"The Atlas organizes Arithmetic Observability I–X around a common inverse-problem vocabulary: model, feature, protocol, nuisance, query and response fibre. It places positive reconstruction criteria beside matching impossibility theorems, and separates representation, acquisition and nuisance obstructions instead of treating every failure as one kind of information loss.","next_direction":"Open work includes characterizing arithmetically realizable coefficient classes, closing the remaining global tail questions, and studying how the observability geometries behave beyond the declared finite protocols.","not_established":["The Atlas does not claim peer review, historical priority, universal number-field recovery, infinite-Euler-product recovery, or an exact closest integral fibre.","Confusability is used for intersecting set-valued response fibres and is not a transitive equivalence relation.","Distance and adversarial critical radius are distinct; in the AS-V branch the critical radius is half the corresponding fibre distance.","Continuous coefficient boxes, independent integral envelopes and arithmetically realizable coefficient sequences are different source classes; the corpus does not characterize the last class.","Formally certified computation is limited to the declared finite scope of each package and does not mechanize every analytic manuscript proof.","The degree-fourteen four-anchor result is a near-matching bracket, not an exact radius or equality theorem.","No Riemann-hypothesis or zeta-zero result, new axiom of numbers, established physical law or theory of reality is claimed.","The work has not undergone independent specialist peer review and novelty remains provisional."],"parent_canonical_url":"https://thegodnet.work/number-geometry/","proof_computation_boundary":"The reconstruction and impossibility results are theorem statements for their declared arithmetic-harmonic models. Formal packages certify their finite schemas, certificate objects and verifier consequences. The near-matching four-anchor values are a certified finite bracket under distinct norm and coefficient conventions, not a universal statistical or physical optimum.","publication_kind":"arithmetic_observability","publication_note":"This page is a public synthesis of Arithmetic Observability I–X. It preserves the distinction between theorem, formal computation, descriptive numerical evidence and open problem, and does not turn the Atlas into a claim about physical reality.","published_at":"2026-08-15T00:00:00Z","question":"Which local arithmetic distinctions are identifiable and stably recoverable from incomplete global harmonic information?","related_links":[{"href":"/number-geometry/","label":"Operational Geometry & Arithmetic index"},{"href":"/number-geometry/operational-information-geometry-protocol-engine/","label":"Operational Information Geometry — Certified Experiment Design"},{"href":"/number-geometry/operational-information-geometry-viii-uniform-atlas/","label":"Operational Information Geometry VIII"},{"href":"/number-geometry/arithmetic-sensing-v-multiscale-stopping/","label":"Arithmetic Sensing V"}],"reproducibility":{"labs":["arithmetic_observability_corpus.py arithmetic_observability_corpus_manifest.json","test_arithmetic_observability_corpus.py","python -m unittest discover -s . -p 'test_arithmetic_observability*.py' -q"],"summary":"Immutable Atlas source state is commit ff2eac1 on the codex/oig-finite-transfer publishing branch. The corpus manifest is the machine-readable inventory for the ten manuscripts and formal packages.","tests_passed":"Handoff validation: 51 focused corpus hostile tests; 333 Arithmetic Observability tests with 332 passing and one intentional AO-IX rebuild skip; 797 whole-repository tests with 796 passing and the same skip."},"reproduction_note":"From the repository root at commit ff2eac1 on the codex/oig-finite-transfer publishing branch, run the corpus verifier and two ordinary unittest commands shown on the page. The AO-IX full reconstruction is opt-in and resource-intensive; its producer environment and Windows/LF caveats are recorded in the Atlas Section 9 and manifest.","schema":"tgn.number-geometry.v1","section":"Research / Mathematics / Information Science","sections":[{"citation_ids":["AO-ATLAS-001","AO-I-001"],"evidence_kind":"exact","heading":"The common inverse-problem language","label":"observation","paragraphs":["Every manuscript declares a model class, a feature to recover, an acquisition protocol, nuisance semantics, a query and a response fibre. Observational equivalence means equality under a deterministic effective observation or equality of whole response fibres. Confusability means intersecting set-valued fibres; it need not be transitive and is not used as a general equivalence relation."]},{"citation_ids":["AO-ATLAS-001","AO-IV-001","AO-V-001"],"evidence_kind":"exact","heading":"Reconstruction and obstruction are matched","label":"analysis","paragraphs":["The Atlas puts positive and negative statements side by side. A protocol may fail because the representation identifies two objects, because the acquisition schedule is too short, or because nuisance fibres overlap. The hypotheses remain visible: distance is not the same object as an adversarial critical radius, and the latter is half the corresponding fibre distance in the Arithmetic Sensing V branch."],"table_id":"theorem-cards"},{"citation_ids":["AO-I-001"],"equations":["\\ker A\\subseteq\\ker L,\\qquad L=RA,\\qquad \\mathrm{radius}=\\lVert R\\rVert\\,\\varepsilon"],"evidence_kind":"exact","heading":"The exact linear query criterion","label":"observation","paragraphs":["In the common linear nuisance model, ker A⊆ker L is the exact identifiability criterion. When the query is recoverable it factors as L=RA, and the exact adversarial minimax radius is ||R||ε. When it is not recoverable, the correct output is a witness direction in the kernel, not a fragile floating-point rank decision."]},{"citation_ids":["AO-IV-001","AO-V-001","AO-VI-001"],"equations":["m_{\\mathrm{simplex}}=r\\lfloor s/2\\rfloor"],"evidence_kind":"exact","heading":"Sharp reading counts for declared geometric models","label":"observation","paragraphs":["The positive free-scale geometric model has exact global complexity r: a phase-separated r-reading schedule is globally injective, while each smaller schedule has reciprocal Borsuk–Ulam collisions. The labelled product-simplex model has exact global complexity r floor(s/2), with a constructive DFT-isolating upper theorem and exact reflection/Borsuk–Ulam collisions below it."]},{"citation_ids":["AO-ATLAS-001","AO-VII-001","AO-VIII-001"],"evidence_kind":"exact","heading":"The distinguishability geometries","items":["Continuous coefficient boxes, independent integral envelopes and arithmetically realizable coefficient sequences remain distinct source classes.","The Atlas does not characterize the arithmetically realizable class; it records that as an explicit nonclaim boundary.","Formally certified computation refers only to packages whose declared finite scope supports it; the corpus manifest does not mechanize every analytic manuscript proof."],"label":"analysis","paragraphs":["Different obstructions induce different geometries: quotient geometry for nuisance, ANOVA and multiplicative pullback geometries for structured features, compact phase geometry for finite readings, tangent–kernel geometry for local uncertainty, separated-query geometry for target recovery, zonoid geometry for correlated bounded nuisance, and arithmetic-defect geometry for the tail branch."]},{"citation_ids":["AO-IX-001","AO-X-001"],"evidence_kind":"numerical","heading":"A near-matching degree-fourteen stability bracket","label":"observation","paragraphs":["In the degree-fourteen four-anchor study, the certified common-nuisance lower radius is 0.019991343027394314… and a legal eleven-mode independent-integral response gives upper radius 0.019999999997766001…. The result is a narrow bracket under the declared norm and coefficient class; it is not an exact closest integral fibre and not an equality theorem."],"table_id":"theorem-cards"},{"citation_ids":["AO-MANIFEST-001","AO-CORPUS-001"],"evidence_kind":"numerical","heading":"Evidence, corpus manifest and reproduction","items":["python arithmetic_observability_corpus.py arithmetic_observability_corpus_manifest.json","python -m unittest -q test_arithmetic_observability_corpus.py","python -m unittest discover -s . -p 'test_arithmetic_observability*.py' -q","Handoff state: 51 focused corpus hostile tests passing; 333 Arithmetic Observability tests with 332 passing and one intentional AO-IX rebuild skip; 797 whole-repository tests with 796 passing and the same skip."],"label":"project_status","paragraphs":["The immutable corpus manifest binds ten manuscripts, eleven formal packages, their verifier, certificate and test files, and the external proof objects used by the tail branch. The ordinary verifier and tests are suitable for routine checking; the AO-IX full rebuild is intentionally opt-in and multi-gigabyte."]},{"citation_ids":["AO-ATLAS-001"],"evidence_kind":"interpretive","heading":"Scope and open boundary","items":["Confusability is not a transitive equivalence relation.","Distance and adversarial critical radius must not be merged.","No Riemann-hypothesis or zeta-zero result is claimed.","No new axiom of numbers, physical law, or theory of reality is claimed."],"label":"testable_hypothesis","paragraphs":["The Atlas is a rigorous synthesis of declared arithmetic-harmonic models, not a universal recovery theorem. It makes no claim of peer review, historical priority, universal number-field recovery, infinite-Euler-product recovery or an exact closest integral fibre. Its formal packages certify finite scopes; they do not turn every analytic manuscript statement into a mechanized proof."]},{"citation_ids":["AO-ATLAS-001"],"evidence_kind":"interpretive","heading":"Attribution and research status","label":"project_status","paragraphs":["This is an AI-led mathematical investigation developed from a human collaborator's originating questions and research direction. Established ingredients and the numbered manuscripts are cited in the repository; the project contribution is the synthesis, certificate organization, implementations and reported computations. Literature priority remains provisional without independent review."]}],"slug":"arithmetic-observability-atlas","source_manifest":[{"filename":"ARITHMETIC_OBSERVABILITY_ATLAS.md","kind":"canonical Atlas synthesis","sha256":"b381d3f9424de5ed5e6cfe35c27093ee2ce38b08d742299bfe7a1ee7a55db10f"},{"filename":"arithmetic_observability_corpus_manifest.json","kind":"machine-readable corpus manifest","sha256":"ee13e874b6a4da6455ed24bdbc74197fbd1fe39efbd87d8d15d1d813c8499ea6"},{"filename":"arithmetic_observability_corpus.py","kind":"corpus verifier","sha256":"01c75a062128db70a43b402241eca0fad96cc40bfbda04a867ed82951c1495f8"},{"filename":"test_arithmetic_observability_corpus.py","kind":"hostile corpus test suite","sha256":"523801d9144272a1dfbf2a5284ce98a050afab48686b038a10159b078d0c1a66"}],"sources":[{"publisher":"GitHub / eruannaarte","retrieved_at":"2026-08-15T00:00:00Z","source_class":"software_repository","source_id":"AO-ATLAS-001","title":"Arithmetic Observability Atlas synthesis","url":"https://github.com/eruannaarte/adelic-arithmetic-research/blob/ff2eac1/ARITHMETIC_OBSERVABILITY_ATLAS.md"},{"publisher":"GitHub / eruannaarte","retrieved_at":"2026-08-15T00:00:00Z","source_class":"software_repository","source_id":"AO-MANIFEST-001","title":"Arithmetic Observability corpus manifest","url":"https://github.com/eruannaarte/adelic-arithmetic-research/blob/ff2eac1/arithmetic_observability_corpus_manifest.json"},{"publisher":"GitHub / eruannaarte","retrieved_at":"2026-08-15T00:00:00Z","source_class":"software_repository","source_id":"AO-CORPUS-001","title":"Corpus verifier and certificate inventory","url":"https://github.com/eruannaarte/adelic-arithmetic-research/blob/ff2eac1/arithmetic_observability_corpus.py"},{"publisher":"GitHub / eruannaarte","retrieved_at":"2026-08-15T00:00:00Z","source_class":"software_repository","source_id":"AO-I-001","title":"Arithmetic Observability I","url":"https://github.com/eruannaarte/adelic-arithmetic-research/blob/ff2eac1/ARITHMETIC_OBSERVABILITY_I.md"},{"publisher":"GitHub / eruannaarte","retrieved_at":"2026-08-15T00:00:00Z","source_class":"software_repository","source_id":"AO-II-001","title":"Arithmetic Observability II","url":"https://github.com/eruannaarte/adelic-arithmetic-research/blob/ff2eac1/ARITHMETIC_OBSERVABILITY_II.md"},{"publisher":"GitHub / eruannaarte","retrieved_at":"2026-08-15T00:00:00Z","source_class":"software_repository","source_id":"AO-III-001","title":"Arithmetic Observability III","url":"https://github.com/eruannaarte/adelic-arithmetic-research/blob/ff2eac1/ARITHMETIC_OBSERVABILITY_III.md"},{"publisher":"GitHub / eruannaarte","retrieved_at":"2026-08-15T00:00:00Z","source_class":"software_repository","source_id":"AO-IV-001","title":"Arithmetic Observability IV","url":"https://github.com/eruannaarte/adelic-arithmetic-research/blob/ff2eac1/ARITHMETIC_OBSERVABILITY_IV.md"},{"publisher":"GitHub / eruannaarte","retrieved_at":"2026-08-15T00:00:00Z","source_class":"software_repository","source_id":"AO-V-001","title":"Arithmetic Observability V","url":"https://github.com/eruannaarte/adelic-arithmetic-research/blob/ff2eac1/ARITHMETIC_OBSERVABILITY_V.md"},{"publisher":"GitHub / eruannaarte","retrieved_at":"2026-08-15T00:00:00Z","source_class":"software_repository","source_id":"AO-VI-001","title":"Arithmetic Observability VI","url":"https://github.com/eruannaarte/adelic-arithmetic-research/blob/ff2eac1/ARITHMETIC_OBSERVABILITY_VI.md"},{"publisher":"GitHub / eruannaarte","retrieved_at":"2026-08-15T00:00:00Z","source_class":"software_repository","source_id":"AO-VII-001","title":"Arithmetic Observability VII","url":"https://github.com/eruannaarte/adelic-arithmetic-research/blob/ff2eac1/ARITHMETIC_OBSERVABILITY_VII.md"},{"publisher":"GitHub / eruannaarte","retrieved_at":"2026-08-15T00:00:00Z","source_class":"software_repository","source_id":"AO-VIII-001","title":"Arithmetic Observability VIII","url":"https://github.com/eruannaarte/adelic-arithmetic-research/blob/ff2eac1/ARITHMETIC_OBSERVABILITY_VIII.md"},{"publisher":"GitHub / eruannaarte","retrieved_at":"2026-08-15T00:00:00Z","source_class":"software_repository","source_id":"AO-IX-001","title":"Arithmetic Observability IX","url":"https://github.com/eruannaarte/adelic-arithmetic-research/blob/ff2eac1/ARITHMETIC_OBSERVABILITY_IX.md"},{"publisher":"GitHub / eruannaarte","retrieved_at":"2026-08-15T00:00:00Z","source_class":"software_repository","source_id":"AO-X-001","title":"Arithmetic Observability X","url":"https://github.com/eruannaarte/adelic-arithmetic-research/blob/ff2eac1/ARITHMETIC_OBSERVABILITY_X.md"},{"publisher":"GitHub / eruannaarte","retrieved_at":"2026-08-15T00:00:00Z","source_class":"software_repository","source_id":"AO-TEST-001","title":"Corpus hostile test suite","url":"https://github.com/eruannaarte/adelic-arithmetic-research/blob/ff2eac1/test_arithmetic_observability_corpus.py"}],"status":"identified","subtitle":"Reconstruction, obstruction, and geometry from incomplete arithmetic measurements","tables":[{"caption":"Four headline results and their hypotheses","headers":["Result","Declared statement","Evidence class"],"id":"theorem-cards","note":"The fourth row is intentionally labeled a near-matching bracket, not an exact radius or equality theorem.","rows":[["Linear query criterion","ker A⊆ker L; in the recoverable case L=RA and minimax radius is ||R||ε","Exact theorem"],["Positive geometric readings","Phase-separated r-reading schedules are globally injective; smaller schedules have reciprocal collisions","Exact theorem"],["Labelled product-simplex readings","Exact global complexity r floor(s/2) for the declared model","Exact theorem"],["Degree-fourteen four-anchor radius","0.019991343027394314…≤radius≤0.019999999997766001…","Certified finite bracket"]]},{"caption":"The two main corpus branches","headers":["Branch","Manuscripts","Role"],"id":"model-branches","rows":[["Finite prime-box / no-tail","AO I–VI","Exact criteria, reading complexity and finite obstructions"],["Degree-fourteen tail","AO I and AO VII–X","Tail-sensitive stability, integral envelopes and certified brackets"]]}],"tags":["arithmetic observability","adelic arithmetic","harmonic measurements","reconstruction","confusability","stability","formal certificates","inverse problems"],"title":"Arithmetic Observability Atlas","updated_at":"2026-08-15T00:00:00Z"}