# Claim register

Foundation snapshot: 2026-07-23

Claims remain `OPEN` unless an existing repository artifact proves the exact
registered statement and a current independent review covers that artifact.
Literature keys identify prior art and constraints; they do not become project
evidence by association.

Paper 1 proves a narrower repository-local theorem about non-uniqueness of
diagnostic annotations on one finite reduct. That theorem does not realize
causal or Lorentzian geometry and does not prove the stable `P1-C04`
statement. All Paper 1 claims therefore remain open.

## Paper 1 — Categorical locality before spacetime

| Claim ID | Paper | Statement | Status | Assumptions | Evidence path | Source keys | Reviewer state | Counterexamples |
|---|---|---|---|---|---|---|---|---|
| P1-C01 | categorical-locality | The selected physical contexts and processes form the declared weak symmetric monoidal higher category and satisfy its coherence laws. | OPEN | `P1-A01`, `P1-A02` | none | `CAT-ATIYAH-1988` | UNREVIEWED | `CE-P1-02` |
| P1-C02 | categorical-locality | Compatible local data for the selected coverage satisfy an effective-descent equivalence in the declared target category. | OPEN | `P1-A02`, `P1-A03`, `P1-A04` | none | `CAT-LURIE-HTT-2009`, `DESC-STACKS-023N` | UNREVIEWED | `CE-P1-01` |
| P1-C03 | categorical-locality | Geometric and algebraic examples instantiate the axioms through explicit realization functors without being built into the definition of physical context. | OPEN | `P1-A05`, `P1-A06` | none | `CAT-ATIYAH-1988`, `CAT-COSTELLO-GWILLIAM-2017`, `AQFT-HAAG-KASTLER-1964` | UNREVIEWED | `CE-P3-03` |
| P1-C04 | categorical-locality | The Paper 1 axioms alone do not determine causal order, dimension, topology, or metric, as witnessed by admissible non-geometric models. | OPEN | `P1-A07` | none | none | AGY round 5 confirms `OPEN`: `reviews/categorical-locality-agy-review-round-5.md` | `CE-P1-02`, `CE-P1-03` |

## Paper 2 — Condensed physical contexts

| Claim ID | Paper | Statement | Status | Assumptions | Evidence path | Source keys | Reviewer state | Counterexamples |
|---|---|---|---|---|---|---|---|---|
| P2-C01 | condensed-physical-contexts | A declared subcategory of Paper 1 context data admits a fully faithful condensed realization functor. | OPEN | `P2-A01`, `P2-A02`, `P2-A03` | none | `COND-CLAUSEN-SCHOLZE-2026` | AGY `ACCEPT` confirms `OPEN`: `reviews/condensed-physical-contexts-agy-review.md` | `CE-P2-02` |
| P2-C02 | condensed-physical-contexts | The condensed realization preserves the limits, colimits, and descent operations explicitly used by later papers. | OPEN | `P2-A02`, `P2-A04`, `P2-A05` | none | `COND-CLAUSEN-SCHOLZE-2026` | AGY `ACCEPT` confirms `OPEN`: `reviews/condensed-physical-contexts-agy-review.md` | `CE-P2-02` |
| P2-C03 | condensed-physical-contexts | Solidification or the selected analytic localization gives controlled distributional/limit objects for the specified physical data. | OPEN | `P2-A06`, `P2-A07` | none | `COND-CLAUSEN-SCHOLZE-2026` | AGY `ACCEPT` confirms `OPEN`: `reviews/condensed-physical-contexts-agy-review.md` | `CE-P2-01` |
| P2-C04 | condensed-physical-contexts | The Paper 2 construction is semantically conservative: it adds no causal, metric, probabilistic, or dynamical structure without an explicit later realization map. | OPEN | `P2-A08` | none | `COND-CLAUSEN-SCHOLZE-2026` | AGY `ACCEPT` confirms `OPEN`: `reviews/condensed-physical-contexts-agy-review.md` | `CE-P2-01` |

## Paper 3 — Representation-theoretic measurement

| Claim ID | Paper | Statement | Status | Assumptions | Evidence path | Source keys | Reviewer state | Counterexamples |
|---|---|---|---|---|---|---|---|---|
| P3-C01 | representation-theoretic-measurement | The selected representation category and fibre functor reconstruct a symmetry object unique up to the stated equivalence. | OPEN | `P3-A01`, `P3-A02`, `P3-A03` | none | `TAN-TANNAKA-1939`, `TAN-DELIGNE-MILNE-1982`, `TAN-DOPLICHER-ROBERTS-1989`, `TAN-DELIGNE-SUPER-2002` | AGY `ACCEPT` confirms `OPEN`: `reviews/representation-theoretic-measurement-agy-review.md` | `CE-P3-01`, `CE-P3-02`, `CE-P3-08` |
| P3-C02 | representation-theoretic-measurement | The reconstructed symmetry data act on a declared observable algebra/interface compatible with context composition. | OPEN | `P3-A04`, `P3-A05` | none | `TAN-DOPLICHER-ROBERTS-ANNALS-1989`, `AQFT-DOPLICHER-ROBERTS-FIELD-1990`, `AQFT-HAAG-KASTLER-1964` | AGY `ACCEPT` confirms `OPEN`: `reviews/representation-theoretic-measurement-agy-review.md` | `CE-P3-03`, `CE-P3-04` |
| P3-C03 | representation-theoretic-measurement | Under positivity, continuity, and normalization assumptions, the observable interface admits invariant states and conserved quantities with a proved correspondence. | OBSTRUCTED | `P3-A06`, `P3-A07` | `research/counterexample-register.md` | `AQFT-BFV-2003`, `AMEN-VON-NEUMANN-1929`, `AMEN-DAY-1957`, `AMEN-DAY-1961`, `DYN-STONE-1932`, `AQFT-NOETHER-BDL-1986` | AGY `ACCEPT` confirms `OBSTRUCTED`: `reviews/representation-theoretic-measurement-agy-review.md` | `CE-P3-05`, `CE-P3-06` |
| P3-C04 | representation-theoretic-measurement | Monoidal composition produces typed pre-geometric outcome objects while probability remains an independent interface. | OPEN | `P3-A08` | none | `QM-ABRAMSKY-COECKE-2004`, `OUT-ABRAMSKY-HEUNEN-2019` | AGY `ACCEPT` confirms `OPEN`: `reviews/representation-theoretic-measurement-agy-review.md` | `CE-P3-07`, `CE-P3-09`, `CE-P5-03`, `CE-P5-05` |
| P3-C05 | representation-theoretic-measurement | A realization map cleanly separates pre-geometric outcome data from ordinary measurements localized in a realized spacetime. | OPEN | `P3-A09` | none | `AQFT-BFV-2003` | AGY `ACCEPT` confirms `OPEN`: `reviews/representation-theoretic-measurement-agy-review.md` | `CE-P3-03` |

## Paper 4 — Lorentzian geometry as a realization

| Claim ID | Paper | Statement | Status | Assumptions | Evidence path | Source keys | Reviewer state | Counterexamples |
|---|---|---|---|---|---|---|---|---|
| P4-C01 | lorentzian-geometry-realization | A specified relational interface from Papers 1–3 defines a causal order satisfying the hypotheses of the selected reconstruction theorem. | OPEN | `P4-A01`, `P4-A02`, `P4-A03` | none | `LOR-HKM-1976`, `LOR-MALAMENT-1977`, `LOR-LEVICHEV-1987`, `LOR-MINGUZZI-2019` | REVIEWED: `reviews/lorentzian-geometry-realization-agy-review.md` | `CE-P1-03`, `CE-P4-05` |
| P4-C02 | lorentzian-geometry-realization | Causal, conformal, projective, and clock data reconstruct a Lorentzian metric up to the exact equivalence stated in the theorem. | OPEN | `P4-A03`, `P4-A04`, `P4-A05`, `P4-A06` | none | `LOR-EPS-1972`, `LOR-MATVEEV-SCHOLZ-2020`, `LOR-AVALOS-DAHIA-ROMERO-2018` | REVIEWED: `reviews/lorentzian-geometry-realization-agy-review.md` | `CE-P4-01`, `CE-P4-02`, `CE-P4-08` |
| P4-C03 | lorentzian-geometry-realization | The reconstructed realization supports a compatible connection and curvature tensors with the required differentiability and nondegeneracy. | OPEN | `P4-A05`, `P4-A07` | none | `LOR-EPS-1972`, `DG-LEFLOCH-MARDARE-2007`, `GR-TRAUTMAN-EC-2006` | REVIEWED: `reviews/lorentzian-geometry-realization-agy-review.md` | `CE-P4-02`, `CE-P4-06`, `CE-P4-07` |
| P4-C04 | lorentzian-geometry-realization | Einstein’s equations follow conditionally from the stated field content, dimension, derivative order, variational class, conservation law, and boundary assumptions. | OPEN | `P4-A08`, `P4-A09`, `P4-A10`, `P4-A11` | none | `GR-LOVELOCK-1971`, `GR-NAVARRO-LOVELOCK-2011`, `GR-GIBBONS-HAWKING-1977`, `GR-IYER-WALD-1994` | REVIEWED: `reviews/lorentzian-geometry-realization-agy-review.md` | `CE-P4-03`, `CE-P4-04`, `CE-P4-09`, `CE-P4-10`, `CE-P4-11` |
| P4-C05 | lorentzian-geometry-realization | If the restricted variational/derivative assumptions are removed, alternative local covariant dynamics obstruct uniqueness of Einstein gravity. | OBSTRUCTED | `P4-A12` | `research/counterexample-register.md` | `GR-BUCHDAHL-1970`, `GR-STELLE-1977` | REVIEWED: `reviews/lorentzian-geometry-realization-agy-review.md` | `CE-P4-03` |

## Paper 5 — Quantum mechanics from compositional representations

| Claim ID | Paper | Statement | Status | Assumptions | Evidence path | Source keys | Reviewer state | Counterexamples |
|---|---|---|---|---|---|---|---|---|
| P5-C01 | categorical-quantum-mechanics | An explicitly supplied strong-monoidal operational bridge connects the Paper 3 observable/outcome interface to a finite-dimensional operational-probabilistic theory with well-defined composition. | OPEN | `P5-A01`, `P5-A02`, `P5-A03` | none | `QM-CDP-2011`, `OUT-ABRAMSKY-HEUNEN-2019` | AGY round 2 confirms `OPEN`: `reviews/categorical-quantum-mechanics-agy-review.md` | `CE-P5-05`, `CE-P5-06` |
| P5-C02 | categorical-quantum-mechanics | The finite-dimensional operational bridge plus the six separate CDP principles reconstruct complex density-matrix state spaces, effects, and completely positive trace-nonincreasing transformations. | OPEN | `P5-A01`, `P5-A02`, `P5-A03`, `P5-A04`, `P5-A05`, `P5-A06`, `P5-A07`, `P5-A08`, `P5-A09` | none | `QM-CDP-2011` | AGY round 2 confirms `OPEN`: `reviews/categorical-quantum-mechanics-agy-review.md` | `CE-P5-03`, `CE-P5-06` |
| P5-C03 | categorical-quantum-mechanics | In the finite state-complete HSD comparison route, factorizable or dagger-HSD composites, local tomography, and a qubit select complex finite-dimensional C-star/Jordan systems while retaining direct-sum superselection. | OPEN | `P5-A14`, `P5-A15`, `P5-A16`, `P5-A17`, `P5-A18`, `P5-A19` | none | `QM-BARNUM-WILCE-2012`, `QM-HANCHE-OLSEN-1985` | AGY round 2 confirms `OPEN`: `reviews/categorical-quantum-mechanics-agy-review.md` | `CE-P5-04`, `CE-P5-07`, `CE-P5-08` |
| P5-C04 | categorical-quantum-mechanics | Downstream of Hilbert/effect structure, noncontextual additive probability measures have Born trace form: projection measures in dimension at least three and all-effects/POVM valuations for dimension two. | OPEN | `P5-A10`, `P5-A11` | none | `QM-GLEASON-1957`, `QM-BUSCH-2003`, `QM-CFMR-2004` | AGY round 2 confirms `OPEN`: `reviews/categorical-quantum-mechanics-agy-review.md` | `CE-P5-02`, `CE-P5-11` |
| P5-C05 | categorical-quantum-mechanics | One reconstruction theorem classifies or excludes real, quaternionic, low-dimensional, infinite-dimensional, and superselection alternatives. | OBSTRUCTED | `P5-A12` | `research/counterexample-register.md` | `QM-SOLER-1995`, `QM-BGW-2020`, `QM-BARNUM-WILCE-2012` | AGY round 2 confirms `OBSTRUCTED`: `reviews/categorical-quantum-mechanics-agy-review.md` | `CE-P5-01`, `CE-P5-02`, `CE-P5-04`, `CE-P5-08`, `CE-P5-12` |
| P5-C06 | categorical-quantum-mechanics | The compositional reconstruction does not by itself select a measurement interpretation or solve the measurement problem. | OPEN | `P5-A13` | none | `QM-CDP-2011`, `QM-EVERETT-1957`, `QM-BOHM-1952`, `QM-GRW-1986` | AGY round 2 confirms `OPEN`: `reviews/categorical-quantum-mechanics-agy-review.md` | `CE-P5-10`, `CE-P5-13` |

## Paper 6 — Statistical mechanics from categorical coarse-graining

| Claim ID | Paper | Statement | Status | Assumptions | Evidence path | Source keys | Reviewer state | Counterexamples |
|---|---|---|---|---|---|---|---|---|
| P6-C01 | categorical-coarse-graining | For one specified many-body model, the microscopic descriptions and chosen reductions form the registered physical functorial coarse-graining interface. | OPEN | `P6-A01`, `P6-A02`, `P6-A03` | none | `KIN-GALLAGHER-SAINT-RAYMOND-TEXIER-2013`, `KIN-DENG-HANI-MA-2024` | AGY `ACCEPT`: `reviews/categorical-coarse-graining-agy-review.md` | `CE-P6-01`, `CE-P6-02`, `CE-P6-08` |
| P6-C01a | categorical-coarse-graining | The finite rational marginal maps defined in the Lean model compose and preserve nonnegative mass. | PROVED | none | `FoundationalReformulation/Kinetic.lean` | none | AGY `ACCEPT`: `reviews/categorical-coarse-graining-agy-review.md` | none |
| P6-C01b | categorical-coarse-graining | Finite one-particle marginalization is nonfaithful, witnessed by the product and correlated two-bit laws. | PROVED | none | `FoundationalReformulation/Kinetic.lean` | none | AGY `ACCEPT`: `reviews/categorical-coarse-graining-agy-review.md` | `CE-P6-01`, `CE-P6-08` |
| P6-C01c | categorical-coarse-graining | The grand-canonical hard-sphere densities and rescaled correlations form the full claimed categorical interface. | OPEN | `P6-A01`, `P6-A02`, `P6-A03` | none | `KIN-DENG-HANI-MA-2024` | AGY `ACCEPT`: `reviews/categorical-coarse-graining-agy-review.md` | `CE-P6-01`, `CE-P6-08` |
| P6-C02 | categorical-coarse-graining | The bundled executable collision and coarse-graining model preserves every registered physical structure on its stated domain. | OPEN | `P6-A04`, `P6-A05` | none | `KIN-BOLTZMANN-1872`, `KIN-KAC-1956` | AGY `ACCEPT`: `reviews/categorical-coarse-graining-agy-review.md` | `CE-P6-06`, `CE-P6-07` |
| P6-C02a | categorical-coarse-graining | The exact rational hard-sphere collision map preserves vector momentum and squared-speed energy for a supplied unit normal, and the rational Kac rotation preserves pair energy. | PROVED | none | `FoundationalReformulation/Kinetic.lean` | `KIN-KAC-1956`, `KIN-DENG-HANI-MA-2024` | AGY `ACCEPT`: `reviews/categorical-coarse-graining-agy-review.md` | `CE-P6-06` |
| P6-C02b | categorical-coarse-graining | The named Haskell suite passes its finite generated and adversarial floating-point checks at the exported tolerance. | COMPUTATIONAL | `P6-A05` | `src/Foundational/Kinetic.hs`, `test/Spec.hs` | none | AGY `ACCEPT`: `reviews/categorical-coarse-graining-agy-review.md` | `CE-P6-07` |
| P6-C03 | categorical-coarse-graining | The exact correlation-factorization conclusion of Deng-Hani-Ma version 3 holds under its pinned model and full hypothesis package. | CONDITIONAL | `P6-A01`, `P6-A06`, `P6-A07`, `P6-A08`, `P6-A09`, `P6-A12` | `papers/latex/categorical-coarse-graining.tex` | `KIN-DENG-HANI-MA-2024` | AGY `ACCEPT`: `reviews/categorical-coarse-graining-agy-review.md` | `CE-P6-01`, `CE-P6-03`, `CE-P6-08`, `CE-P6-09` |
| P6-C04 | categorical-coarse-graining | The pinned grand-canonical deterministic hard-sphere dynamics converge to the stated hard-sphere Boltzmann solution in the theorem's quantitative L1 correlation mode. | CONDITIONAL | `P6-A01`, `P6-A06`, `P6-A07`, `P6-A08`, `P6-A09`, `P6-A12` | `papers/latex/categorical-coarse-graining.tex` | `KIN-DENG-HANI-MA-2024` | AGY `ACCEPT`: `reviews/categorical-coarse-graining-agy-review.md` | `CE-P6-02`, `CE-P6-03`, `CE-P6-04`, `CE-P6-09` |
| P6-C05 | categorical-coarse-graining | The limiting solution satisfies one bundled family of conservation and entropy equalities or inequalities across all proposed solution classes. | OPEN | `P6-A10`, `P6-A11` | none | `KIN-BOLTZMANN-1872`, `KIN-DIPERNA-LIONS-1989` | AGY `ACCEPT`: `reviews/categorical-coarse-graining-agy-review.md` | `CE-P6-04`, `CE-P6-07`, `CE-P6-10` |
| P6-C05a | categorical-coarse-graining | The exact binary hard-sphere map preserves vector momentum and squared-speed energy for a supplied unit normal. | PROVED | none | `FoundationalReformulation/Kinetic.lean` | `KIN-DENG-HANI-MA-2024` | AGY `ACCEPT`: `reviews/categorical-coarse-graining-agy-review.md` | none |
| P6-C05b | categorical-coarse-graining | Classical local mass, momentum, energy, and entropy balances hold for a sufficiently regular and integrable solution under the stated kernel symmetries and boundary conditions. | CONDITIONAL | `P6-A10`, `P6-A11` | `papers/latex/categorical-coarse-graining.tex` | `KIN-BOLTZMANN-1872`, `KIN-GALLAGHER-SAINT-RAYMOND-TEXIER-2013` | AGY `ACCEPT`: `reviews/categorical-coarse-graining-agy-review.md` | `CE-P6-07`, `CE-P6-10` |
| P6-C05c | categorical-coarse-graining | The renormalized branch has the stated continuity equality, momentum-defect equation, and entropy inequality in its exact solution class. | CONDITIONAL | `P6-A10`, `P6-A11` | `papers/latex/categorical-coarse-graining.tex` | `KIN-DIPERNA-LIONS-1989`, `KIN-GALLAGHER-SAINT-RAYMOND-TEXIER-2013` | AGY `ACCEPT`: `reviews/categorical-coarse-graining-agy-review.md` | `CE-P6-04`, `CE-P6-10` |
| P6-C06 | categorical-coarse-graining | Under P6-A12, Deng-Hani-Ma version 3 Theorem 1 gives its stated correlation convergence on each selected finite interval, with the smallness threshold for epsilon depending on the interval and theorem bounds. | CONDITIONAL | `P6-A12` | `papers/latex/categorical-coarse-graining.tex` | `KIN-DENG-HANI-MA-2024` | AGY `ACCEPT`: `reviews/categorical-coarse-graining-agy-review.md` | `CE-P6-05`, `CE-P6-09` |

## Synthesis — Representation before geometry

| Claim ID | Paper | Statement | Status | Assumptions | Evidence path | Source keys | Reviewer state | Counterexamples |
|---|---|---|---|---|---|---|---|---|
| PS-C01 | synthesis | The complete typed dependency graph records every use of a Paper 1–3 interface by each realization branch. | OPEN | `PS-A01`, `PS-A02` | none | none | UNREVIEWED | `CE-PS-01` |
| PS-C02 | synthesis | Familiar geometric, quantum, and kinetic laws occur as realizations when the branch-specific reconstruction hypotheses hold. | OPEN | `PS-A03`, `PS-A04` | none | `LOR-HKM-1976`, `LOR-MALAMENT-1977`, `LOR-EPS-1972`, `GR-LOVELOCK-1971`, `GR-STELLE-1977`, `QM-GLEASON-1957`, `QM-SOLER-1995`, `QM-HARDY-2001`, `QM-CDP-2011`, `QM-ABRAMSKY-COECKE-2004`, `QM-ADLER-1995`, `QM-BARNUM-WILCE-2012`, `KIN-BOLTZMANN-1872`, `KIN-KAC-1956`, `KIN-LANFORD-1975`, `KIN-SZNITMAN-1991`, `KIN-GALLAGHER-SAINT-RAYMOND-TEXIER-2013`, `KIN-DIPERNA-LIONS-1989`, `KIN-DENG-HANI-MA-2024` | UNREVIEWED | `CE-PS-01` |
| PS-C03 | synthesis | Only claims with compatible versions, evidence, assumptions, and current terminal reviews are composed. | OPEN | `PS-A05`, `PS-A06` | none | none | UNREVIEWED | `CE-PS-01` |
| PS-C04 | synthesis | Every missing bridge is localized as `OPEN` or `OBSTRUCTED`, rather than being hidden inside “emergence” language. | OPEN | `PS-A07` | none | none | UNREVIEWED | `CE-PS-01` |

## Update rules

1. Never delete or reuse an ID. Superseded statements remain in history and
   receive a successor ID if their logical content changes materially.
2. A status upgrade requires a real evidence path and a dependency audit.
3. A changed artifact resets reviewer state to `UNREVIEWED`.
4. A source key may constrain or motivate a claim but cannot replace project
   evidence.
5. Counterexample IDs remain attached after a claim is weakened; the paper
   must explain why the revised assumptions exclude them.