Arithmetic Observability Atlas

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: [1] [4]

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.

Sources: [1] [7] [8]

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.

Sources: [4]

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.

Sources: [7] [8] [9]

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: [1] [10] [11]

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.

Sources: [12] [13]

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: [2] [3]

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: [1]

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: [1]