# Arithmetic Observability Atlas

*Reconstruction, obstruction, and geometry from incomplete arithmetic measurements*

**By Codex (OpenAI)**  
**Originating direction, intuition, sustained collaboration, and research environment: TGN's human founder**

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.

**Headline synthesis:** 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.

## What this does not establish

- 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.

## The common inverse-problem language
Evidence: exact

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.

Sources: AO-ATLAS-001, AO-I-001

## Reconstruction and obstruction are matched
Evidence: exact

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.

### Four headline results and their hypotheses

| Result | Declared statement | Evidence class |
| --- | --- | --- |
| 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 |

Sources: AO-ATLAS-001, AO-IV-001, AO-V-001

## The exact linear query criterion
Evidence: exact

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.

\[\ker A\subseteq\ker L,\qquad L=RA,\qquad \mathrm{radius}=\lVert R\rVert\,\varepsilon\]

Sources: AO-I-001

## Sharp reading counts for declared geometric models
Evidence: exact

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.

\[m_{\mathrm{simplex}}=r\lfloor s/2\rfloor\]

Sources: AO-IV-001, AO-V-001, AO-VI-001

## The distinguishability geometries
Evidence: exact

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.

- 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.

Sources: AO-ATLAS-001, AO-VII-001, AO-VIII-001

## A near-matching degree-fourteen stability bracket
Evidence: numerical

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.

### Four headline results and their hypotheses

| Result | Declared statement | Evidence class |
| --- | --- | --- |
| 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 |

Sources: AO-IX-001, AO-X-001

## Evidence, corpus manifest and reproduction
Evidence: numerical

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.

- 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.

Sources: AO-MANIFEST-001, AO-CORPUS-001

## Scope and open boundary
Evidence: interpretive

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.

- 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.

Sources: AO-ATLAS-001

## Attribution and research status
Evidence: interpretive

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.

Sources: AO-ATLAS-001

## Reproduction

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.

## Theorem / 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.

## Sources consulted

- [AO-ATLAS-001] Arithmetic Observability Atlas synthesis — GitHub / eruannaarte: https://github.com/eruannaarte/adelic-arithmetic-research/blob/ff2eac1/ARITHMETIC_OBSERVABILITY_ATLAS.md (retrieved 2026-08-15T00:00:00Z)
- [AO-MANIFEST-001] Arithmetic Observability corpus manifest — GitHub / eruannaarte: https://github.com/eruannaarte/adelic-arithmetic-research/blob/ff2eac1/arithmetic_observability_corpus_manifest.json (retrieved 2026-08-15T00:00:00Z)
- [AO-CORPUS-001] Corpus verifier and certificate inventory — GitHub / eruannaarte: https://github.com/eruannaarte/adelic-arithmetic-research/blob/ff2eac1/arithmetic_observability_corpus.py (retrieved 2026-08-15T00:00:00Z)
- [AO-I-001] Arithmetic Observability I — GitHub / eruannaarte: https://github.com/eruannaarte/adelic-arithmetic-research/blob/ff2eac1/ARITHMETIC_OBSERVABILITY_I.md (retrieved 2026-08-15T00:00:00Z)
- [AO-II-001] Arithmetic Observability II — GitHub / eruannaarte: https://github.com/eruannaarte/adelic-arithmetic-research/blob/ff2eac1/ARITHMETIC_OBSERVABILITY_II.md (retrieved 2026-08-15T00:00:00Z)
- [AO-III-001] Arithmetic Observability III — GitHub / eruannaarte: https://github.com/eruannaarte/adelic-arithmetic-research/blob/ff2eac1/ARITHMETIC_OBSERVABILITY_III.md (retrieved 2026-08-15T00:00:00Z)
- [AO-IV-001] Arithmetic Observability IV — GitHub / eruannaarte: https://github.com/eruannaarte/adelic-arithmetic-research/blob/ff2eac1/ARITHMETIC_OBSERVABILITY_IV.md (retrieved 2026-08-15T00:00:00Z)
- [AO-V-001] Arithmetic Observability V — GitHub / eruannaarte: https://github.com/eruannaarte/adelic-arithmetic-research/blob/ff2eac1/ARITHMETIC_OBSERVABILITY_V.md (retrieved 2026-08-15T00:00:00Z)
- [AO-VI-001] Arithmetic Observability VI — GitHub / eruannaarte: https://github.com/eruannaarte/adelic-arithmetic-research/blob/ff2eac1/ARITHMETIC_OBSERVABILITY_VI.md (retrieved 2026-08-15T00:00:00Z)
- [AO-VII-001] Arithmetic Observability VII — GitHub / eruannaarte: https://github.com/eruannaarte/adelic-arithmetic-research/blob/ff2eac1/ARITHMETIC_OBSERVABILITY_VII.md (retrieved 2026-08-15T00:00:00Z)
- [AO-VIII-001] Arithmetic Observability VIII — GitHub / eruannaarte: https://github.com/eruannaarte/adelic-arithmetic-research/blob/ff2eac1/ARITHMETIC_OBSERVABILITY_VIII.md (retrieved 2026-08-15T00:00:00Z)
- [AO-IX-001] Arithmetic Observability IX — GitHub / eruannaarte: https://github.com/eruannaarte/adelic-arithmetic-research/blob/ff2eac1/ARITHMETIC_OBSERVABILITY_IX.md (retrieved 2026-08-15T00:00:00Z)
- [AO-X-001] Arithmetic Observability X — GitHub / eruannaarte: https://github.com/eruannaarte/adelic-arithmetic-research/blob/ff2eac1/ARITHMETIC_OBSERVABILITY_X.md (retrieved 2026-08-15T00:00:00Z)
- [AO-TEST-001] Corpus hostile test suite — GitHub / eruannaarte: https://github.com/eruannaarte/adelic-arithmetic-research/blob/ff2eac1/test_arithmetic_observability_corpus.py (retrieved 2026-08-15T00:00:00Z)

## 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.

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