# Source ledger

Last audited: 2026-07-23

This ledger records the primary or authoritative sources that later papers may
cite. A source supports only the claim written under its entry. It does not
support a stronger reconstruction claim by association. DOI URLs are used when
available; otherwise the author, institutional, or archival URL is retained.

## Categorical locality and descent

### CAT-ATIYAH-1988

- Title: *Topological quantum field theory*
- Author: Michael F. Atiyah
- Stable identifier: DOI `10.1007/BF02698547`
- URL: https://doi.org/10.1007/BF02698547
- Access date: 2026-07-23
- Exact claim supported: Atiyah gives axioms in which bordisms compose and a
  topological field theory assigns compatible algebraic data. This is an
  established categorical model of composition, but its domain already
  contains manifolds and therefore does not establish pre-geometric locality.
- Locator: pp. 175–186, axioms in §2.

### CAT-LURIE-HTT-2009

- Title: *Higher Topos Theory*
- Author: Jacob Lurie
- Stable identifier: arXiv `math/0608040`; ISBN `978-0-691-14049-0`
- URL: https://arxiv.org/abs/math/0608040
- Access date: 2026-07-23
- Exact claim supported: higher topoi provide an infinity-categorical setting
  for Grothendieck-style local-to-global structures and descent. The source
  supplies mathematical descent machinery, not a choice of physical contexts
  or a physical interpretation of covers.
- Locator: Chapters 6–7; arXiv abstract and table of contents.

### CAT-MACLANE-MOERDIJK-1992

- Title: *Sheaves in Geometry and Logic: A First Introduction to Topos
  Theory*
- Authors: Saunders Mac Lane; Ieke Moerdijk
- Stable identifier: DOI `10.1007/978-1-4612-0927-0`; ISBN
  `978-0-387-97710-2`
- URL: https://doi.org/10.1007/978-1-4612-0927-0
- Access date: 2026-07-23
- Exact claim supported: a basis for a Grothendieck topology can be presented
  through covering families satisfying identity, pullback stability, and
  composition axioms. This supplies the standard categorical provenance for
  the Paper 1 coverage definition, not a physical principle selecting covers.
- Locator: Chapter III, Section 2, Definition 2.

### CAT-COSTELLO-GWILLIAM-2017

- Title: *Factorization Algebras in Quantum Field Theory, Volume 1*
- Authors: Kevin Costello; Owen Gwilliam
- Stable identifier: DOI `10.1017/9781316678626`
- URL: https://doi.org/10.1017/9781316678626
- Access date: 2026-07-23
- Exact claim supported: factorization algebras encode observables as
  local-to-global objects with structure maps for disjoint opens. They provide
  a controlled comparator for locality and gluing, but the construction is
  normally carried by opens of an already specified space or manifold.
- Locator: Introduction and Chapters 1–3.

### DESC-STACKS-023N

- Title: *Descent, Proposition 35.3.9 (Tag 023N)*
- Author: The Stacks Project Authors
- Stable identifier: Stacks Project Tag `023N`
- URL: https://stacks.math.columbia.edu/tag/023N
- Access date: 2026-07-23
- Exact claim supported: for a faithfully flat ring map, every module descent
  datum is effective, and base extension is an equivalence with the category of
  descent data.
- Locator: Proposition 35.3.9.

### DESC-STACKS-08KE

- Title: *Non-effective descent data for projective schemes*
- Author: The Stacks Project Authors
- Stable identifier: Stacks Project Tag `08KE`
- URL: https://stacks.math.columbia.edu/tag/08KE
- Access date: 2026-07-23
- Exact claim supported: there is an étale cover carrying projective descent
  data that is not effective in schemes, although it is effective in algebraic
  spaces. Descent therefore requires a declared ambient category and
  effectiveness hypotheses.
- Locator: Lemma 110.68.1.

## Condensed mathematics

### COND-CLAUSEN-SCHOLZE-2026

- Title: *Lectures on Condensed Mathematics*
- Author: Peter Scholze
- Collaboration statement: the arXiv abstract and the lecture-note preface say
  that the material is joint work with Dustin Clausen.
- Stable identifier: arXiv `2605.03658v1`; DOI
  `10.48550/arXiv.2605.03658`; author-declared version pin
  `stable-citable-2026-05`
- URL: https://people.mpim-bonn.mpg.de/scholze/Condensed.pdf
- ArXiv URL: https://arxiv.org/abs/2605.03658
- Access date: 2026-07-23
- Pinning method: the official arXiv record lists Peter Scholze as author,
  identifies version 1, and says that the material is joint work with Dustin
  Clausen. The downloaded arXiv PDF has title metadata
  *Lectures on Condensed Mathematics*, author metadata Peter Scholze, and 78
  pages. The preface calls the document the stable citable version and dates
  it May 2026.
- Exact claim supported: a condensed set is, at the introductory level, a
  sheaf on the pro-étale site of a point (profinite sets with finite jointly
  surjective covers); condensed abelian groups form an abelian category with
  strong exactness properties; compactly generated weak Hausdorff spaces embed
  fully faithfully in quasiseparated condensed sets. None of these statements
  selects physical covers, spacetime geometry, or dynamics.
- Locator: Definition 1.2; Theorem 1.10; Theorem 2.16 and the paragraph
  following it.

## Representation reconstruction and algebraic quantum theory

### TAN-TANNAKA-1939

- Title: *Über den Dualitätssatz der nichtkommutativen topologischen Gruppen*
- Author: Tadao Tannaka
- Stable identifier: *Tohoku Mathematical Journal* 45 (1939), 1–12
- URL: https://www.jstage.jst.go.jp/article/tmj1911/45/0/45_0_1/_article
- Access date: 2026-07-23
- Exact claim supported: the full structured finite-dimensional
  representation data of a bicompact (compact) topological group supports the
  source's double-dual reconstruction. This is not a theorem about an
  arbitrary modern monoidal category, an affine group scheme, or a supergroup.
- Locator: Section 1 for the dual-semigroup operations; first `Satz 1`,
  printed p. 2; `Satz 2`, printed pp. 9-10; `Satz 3`, printed pp. 10-11.

### TAN-DELIGNE-MILNE-1982

- Title: *Tannakian Categories*
- Authors: Pierre Deligne; James S. Milne
- Stable identifier: Lecture Notes in Mathematics 900 (1982), pp. 101–228
- URL: https://www.jmilne.org/math/xnotes/tc2018.pdf
- Access date: 2026-07-23
- Exact claim supported: under the neutral Tannakian hypotheses, including a
  suitable fibre functor, an affine group scheme is recovered as tensor
  automorphisms of that functor and the category is represented as its
  finite-dimensional representations. Without rigidity the construction
  gives an affine monoid scheme. Without neutralization, fibre functors form
  an affine gerbe rather than selecting a canonical base-field-valued fibre
  functor.
- Locator: corrected November 2018 version, retaining original numbering:
  Proposition 2.8 and Remark 2.10, p. 20; Theorem 2.11, p. 21; proof
  conclusion, p. 24; Definition 2.19, p. 25; Theorem 3.2, pp. 30-31;
  Definition 3.7 and Theorem 3.9, pp. 32-33.

### TAN-DOPLICHER-ROBERTS-1989

- Title: *A new duality theory for compact groups*
- Authors: Sergio Doplicher; John E. Roberts
- Stable identifier: DOI `10.1007/BF01388849`
- URL: https://doi.org/10.1007/BF01388849
- Access date: 2026-07-23
- Exact claim supported: every strict symmetric monoidal C-star category with
  conjugates, subobjects, direct sums, and scalar endomorphisms of the tensor
  unit is isomorphic to a category `H(G)` of finite-dimensional continuous
  unitary representations of a compact group unique up to isomorphism. Each
  listed hypothesis is substantive. The output is a compact group, not an
  affine group scheme or supergroup, and the statement does not require a
  separately supplied fibre functor.
- Locator: official Springer summary only. Full text was not available in the
  source audit, so no theorem-number or page-level locator is asserted.

### TAN-DOPLICHER-ROBERTS-ANNALS-1989

- Title: *Endomorphisms of C*-algebras, cross products, and duality for compact
  groups*
- Authors: Sergio Doplicher; John E. Roberts
- Stable identifier: DOI `10.2307/1971477`
- URL: https://doi.org/10.2307/1971477
- Access date: 2026-07-23
- Exact claim supported: the official abstract starts with a C-star algebra
  and a selected semigroup/category of endomorphisms equipped with permutation
  symmetry and conjugates. It supports an observable-to-gauge reconstruction
  comparator, not the reverse claim that bare representation data selects an
  observable algebra.
- Locator: official Annals abstract only. The full theorem hypotheses were not
  pinned in this audit, so no detailed theorem claim is made from the
  whole-article page range.

### TAN-DELIGNE-SUPER-2002

- Title: *Catégories tensorielles*
- Author: Pierre Deligne
- Stable identifier: *Moscow Mathematical Journal* 2 (2002), 227-248
- URL: https://publications.ias.edu/book/export/html/434
- Access date: 2026-07-23
- Exact claim supported: over an algebraically closed field of characteristic
  zero, a finitely tensor-generated tensor category satisfying the stated
  Schur-finiteness condition is tensor-equivalent to `Rep(G,epsilon)` for an
  affine supergroup scheme with parity datum. This is not a compact group or a
  compact supergroup conclusion.
- Locator: Definition 0.1, printed p. 2; Proposition 0.5 and Theorem 0.6,
  printed p. 3; Corollaries 0.7-0.8, printed pp. 3-4.

### AQFT-DOPLICHER-ROBERTS-FIELD-1990

- Title: *Why there is a field algebra with a compact gauge group describing
  the superselection structure in particle physics*
- Authors: Sergio Doplicher; John E. Roberts
- Stable identifier: DOI `10.1007/BF02097680`
- URL: https://doi.org/10.1007/BF02097680
- Access date: 2026-07-23
- Exact claim supported: the official abstract describes a compact gauge
  group and field-algebra result starting from local observables and
  superselection data. It is a reverse-direction physical comparator and does
  not derive observable algebras from bare representation data.
- Locator: official *Communications in Mathematical Physics* abstract only;
  volume 131 (1990), pp. 51-107. No detailed theorem locator is asserted.

### CSTAR-COMPACT-ERGODIC-1981

- Title: *Compact ergodic groups of automorphisms*
- Authors: Raphael Hoegh-Krohn; Magnus Landstad; Erling Stormer
- Stable identifier: DOI `10.2307/1971377`
- URL: https://doi.org/10.2307/1971377
- Access date: 2026-07-23
- Exact claim supported: for the source's stronger compact ergodic
  automorphism-action setting, the unique invariant state is a trace. Paper 3
  uses this only as a comparator; its elementary compact averaging statement
  neither assumes ergodicity nor claims uniqueness.
- Locator: *Annals of Mathematics* 114 (1981), 75-86; source-level main-result
  scope.

### AMEN-VON-NEUMANN-1929

- Title: *Zur allgemeinen Theorie des Masses*
- Author: John von Neumann
- Stable identifier: DOI `10.4064/fm-13-1-73-116`
- URL: https://doi.org/10.4064/fm-13-1-73-116
- Access date: 2026-07-23
- Exact claim supported: invariant means mark the classical amenability
  boundary. Paper 3 uses this to reject unconditional invariant-state
  existence for arbitrary group actions.
- Locator: *Fundamenta Mathematicae* 13 (1929), 73-116.

### AMEN-DAY-1957

- Title: *Amenable semigroups*
- Author: Mahlon M. Day
- Stable identifier: DOI `10.1215/ijm/1255380675`
- URL: https://doi.org/10.1215/ijm/1255380675
- Access date: 2026-07-23
- Exact claim supported: amenability supplies invariant-mean and fixed-point
  hypotheses that are absent from a bare automorphic action.
- Locator: *Illinois Journal of Mathematics* 1 (1957), 509-544.

### AMEN-DAY-1961

- Title: *Fixed point theorems for compact convex sets*
- Author: Mahlon M. Day
- Stable identifier: DOI `10.1215/ijm/1255631582`
- URL: https://doi.org/10.1215/ijm/1255631582
- Access date: 2026-07-23
- Exact claim supported: appropriate amenability and compact-convex
  hypotheses yield fixed points. These hypotheses cannot be omitted when
  inferring invariant states.
- Locator: *Illinois Journal of Mathematics* 5 (1961), 585-590.

### DYN-STONE-1932

- Title: *On one-parameter unitary groups in Hilbert space*
- Author: Marshall H. Stone
- Stable identifier: DOI `10.2307/1968538`
- URL: https://doi.org/10.2307/1968538
- Access date: 2026-07-23
- Exact claim supported: a strongly continuous one-parameter unitary group
  has a self-adjoint generator. This result requires a Hilbert-space
  representation and strong continuity; neither follows from representation
  category monoidality.
- Locator: *Annals of Mathematics* 33 (1932), 643-648.

### AQFT-NOETHER-BDL-1986

- Title: *On Noether's theorem in quantum field theory*
- Authors: Detlev Buchholz; Sergio Doplicher; Roberto Longo
- Stable identifier: DOI `10.1016/0003-4916(86)90086-2`
- URL: https://www-library.desy.de/preparch/desy/postpr/1985/desy85-023.pdf
- Access date: 2026-07-23
- Exact claim supported: the source proves a weak AQFT Noether result after
  assuming a Hilbert-space realization, bounded Minkowski-region algebras,
  unitary symmetry implementation, vacuum/localization properties, and the
  split property. It does not turn pre-geometric categorical symmetry into a
  conserved current.
- Locator: assumptions, printed pp. 1-2; local-implementer warning, printed
  p. 4; Section 3; current-reconstruction limitation, printed p. 12.

### OUT-ABRAMSKY-HEUNEN-2019

- Title: *Operational theories and categorical quantum mechanics*
- Authors: Samson Abramsky; Chris Heunen
- Stable identifier: DOI `10.1017/CBO9781139519687.007`; arXiv
  `1206.0921v3`
- URL: https://arxiv.org/abs/1206.0921v3
- Access date: 2026-07-23
- Exact claim supported: preparations, measurements, finite outcome labels,
  and evaluation are separately declared operational interfaces. Outcome
  labels and evaluation do not follow from monoidality alone.
- Locator: Section 3, printed pp. 3-5; Equation (2), p. 5; Sections 3.3 and
  3.5, pp. 6-7; Section 5.3, p. 13; Proposition 3, p. 14.

### REF-BARTLETT-RUDOLPH-SPEKKENS-2007

- Title: *Reference frames, superselection rules, and quantum information*
- Authors: Stephen D. Bartlett; Terry Rudolph; Robert W. Spekkens
- Stable identifier: DOI `10.1103/RevModPhys.79.555`; arXiv
  `quant-ph/0610030v3`
- URL: https://arxiv.org/abs/quant-ph/0610030v3
- Access date: 2026-07-23
- Exact claim supported: this physical quantum-reference-frame framework
  starts with Hilbert spaces, unitary representations, density operators,
  operations, POVMs, and probabilities. Its twirling construction is a
  comparator, not input to Paper 3, and it explicitly distinguishes induced
  superselection restrictions from conservation laws.
- Locator: Section II.A, printed p. 4; Equations (2.14)-(2.15), printed p. 6;
  accompanying conservation-law warning, printed p. 6.

### AQFT-HAAG-KASTLER-1964

- Title: *An algebraic approach to quantum field theory*
- Authors: Rudolf Haag; Daniel Kastler
- Stable identifier: DOI `10.1063/1.1704187`
- URL: https://doi.org/10.1063/1.1704187
- Access date: 2026-07-23
- Exact claim supported: local quantum physics can be axiomatized through
  algebras of observables associated with regions, with inclusion and
  locality relations. The framework assumes a spacetime region structure and
  does not reconstruct it.
- Locator: historical provenance only, *Journal of Mathematical Physics* 5
  (1964), 848-861. The source audit did not pin theorem-level pages; use
  `AQFT-BFV-2003`, Proposition 2.2, for a precise modern fixed-spacetime-net
  comparison.

### AQFT-BFV-2003

- Title: *The generally covariant locality principle — A new paradigm for
  local quantum physics*
- Authors: Romeo Brunetti; Klaus Fredenhagen; Rainer Verch
- Stable identifier: DOI `10.1007/s00220-003-0815-7`; arXiv
  `math-ph/0112041`
- URL: https://arxiv.org/abs/math-ph/0112041v1
- Access date: 2026-07-23
- Exact claim supported: a locally covariant quantum field theory can be
  formulated as a covariant functor from globally hyperbolic spacetimes with
  admissible embeddings to unital C-star algebras with faithful unital
  star-homomorphisms. The source begins with Lorentzian spacetimes rather than
  deriving them. It also supplies a no-natural-state warning and a
  metric-response conservation theorem only after substantial geometric,
  representation, domain, and differentiability assumptions.
- Locator: arXiv v1 Section 2.2 and Definition 2.1, printed pp. 6-8;
  Proposition 2.2, pp. 10-12; state definition and no-natural-state
  discussion, pp. 15-16; Definition 3.1, pp. 17-18; relative Cauchy
  evolution and the hypotheses preceding Theorem 4.2, pp. 22-24;
  Theorem 4.2, pp. 24-25.

## Lorentzian and gravitational reconstruction

### LOR-HKM-1976

- Title: *A new topology for curved space-time which incorporates the causal,
  differential, and conformal structures*
- Authors: S. W. Hawking; A. R. King; P. J. McCarthy
- Stable identifier: DOI `10.1063/1.522874`
- URL: https://doi.org/10.1063/1.522874
- Access date: 2026-07-23
- Exact claim supported: on a smooth time-orientable Lorentzian spacetime, the
  path topology is the finest topology inducing the manifold topology on every
  timelike curve; under the paper's dimensional and regularity hypotheses,
  suitable path-topology homeomorphisms and null-geodesic preservers are smooth
  conformal maps. This is rigidity for an existing spacetime.
- Locator: §§2.7, 2.9, and 3; Theorems 1, 2, 5, 6, and 7; printed pp. 174–181.

### LOR-MALAMENT-1977

- Title: *The class of continuous timelike curves determines the topology of
  spacetime*
- Author: David B. Malament
- Stable identifier: DOI `10.1063/1.523436`
- URL: https://doi.org/10.1063/1.523436
- Access date: 2026-07-23
- Exact claim supported: a chronological or isochronal bijection between
  already smooth Lorentzian spacetimes, with both past- and
  future-distinguishing hypotheses in the chronological branch, is a
  homeomorphism and hence a smooth conformal map with the associated HKM
  result. The causal-bijection version is due to Levichev.
- Locator: Theorems 1 and 2 and the distinguishing counterexample, printed
  pp. 1399–1404; exact modern restatement in `LOR-MINGUZZI-2019`.

### LOR-EPS-1972

- Title: *The geometry of free fall and light propagation*
- Authors: Jürgen Ehlers; Felix A. E. Pirani; Alfred Schild
- Stable identifier: republication DOI `10.1007/s10714-012-1353-4`
- URL: https://doi.org/10.1007/s10714-012-1353-4
- Access date: 2026-07-23
- Exact claim supported: the program begins with events, light rays, massive
  particle paths, a dimension-four assumption, and substantial incidence and
  smoothness axioms. Light and free fall yield separate conformal and
  projective structures; compatibility leads first to Weyl geometry, while a
  clock/no-second-clock input is needed for metric geometry. The authors state
  a rigor caveat for the full axiomatic derivation.
- Locator: original pp. 63–84, especially pp. 63–70 and §2; republication
  pp. 1587–1609.

### GR-LOVELOCK-1971

- Title: *The Einstein tensor and its generalizations*
- Author: David Lovelock
- Stable identifier: DOI `10.1063/1.1665613`
- URL: https://doi.org/10.1063/1.1665613
- Access date: 2026-07-23
- Exact claim supported: in four dimensions, the space of symmetric,
  divergence-free, second-order natural metric `(0,2)` tensors is spanned by
  the metric and Einstein tensors. No variational origin is required, so this
  is a field-tensor classification rather than uniqueness of an action.
- Locator: original abstract and theorem, printed pp. 498–501; full modern
  formulation and proof in `GR-NAVARRO-LOVELOCK-2011`, Theorem 2.6.

### GR-STELLE-1977

- Title: *Renormalization of higher-derivative quantum gravity*
- Author: K. S. Stelle
- Stable identifier: DOI `10.1103/PhysRevD.16.953`
- URL: https://doi.org/10.1103/PhysRevD.16.953
- Access date: 2026-07-23
- Exact claim supported: generally covariant gravitational actions with
  curvature-squared, higher-derivative terms exist and have field equations
  different from Einstein’s. Covariance and locality alone therefore do not
  select the Einstein-Hilbert action.
- Locator: *Physical Review D* 16 (1977), 953–969.

### LOR-LEVICHEV-1987

- Title: *The causal structure of a Lorentzian manifold determines its
  conformal geometry*
- Author: A. V. Levichev
- Stable identifier: MathNet record `dan8158`
- URL: https://www.mathnet.ru/eng/dan8158
- Access date: 2026-07-23
- Exact claim supported: a causal bijection between distinguishing smooth
  Lorentzian spacetimes is a conformal diffeomorphism. It is a rigidity result
  for existing spacetimes, not an order-to-manifold construction.
- Locator: *Doklady Akademii Nauk SSSR* 293 (1987), 1301–1305; modern
  restatement in `LOR-MINGUZZI-2019`, Theorem 4.64.

### LOR-MINGUZZI-2019

- Title: *Lorentzian causality theory*
- Author: Ettore Minguzzi
- Stable identifier: DOI `10.1007/s41114-019-0019-x`
- URL: https://doi.org/10.1007/s41114-019-0019-x
- Access date: 2026-07-23
- Exact claim supported: gives the exact modern comparison of HKM,
  Malament chronological rigidity, and Levichev causal rigidity, including
  distinguishing and bidirectional preservation hypotheses.
- Locator: Definitions 4.53–4.59 and Theorems 4.60–4.66, printed pp. 127–129.

### LOR-PARRIKAR-SURYA-2011

- Title: *Causal topology in future and past distinguishing spacetimes*
- Authors: Onkar Parrikar; Sumati Surya
- Stable identifier: arXiv `1102.0936v2`
- URL: https://arxiv.org/abs/1102.0936v2
- Access date: 2026-07-23
- Exact claim supported: causal bijections between already given
  distinguishing spacetimes yield dimension-equality and conformal/topological
  rigidity results under the paper's exact hypotheses. They do not create a
  manifold dimension from an arbitrary order.
- Locator: Theorem 2, Corollary 3, Proposition 4, and Corollary 18.

### LOR-EPS-TRAUTMAN-2012

- Title: *Editorial note to: The geometry of free fall and light propagation*
- Author: Andrzej Trautman
- Stable identifier: DOI `10.1007/s10714-012-1352-5`
- URL: https://doi.org/10.1007/s10714-012-1352-5
- Access date: 2026-07-23
- Exact claim supported: identifies the incomplete affine/Weyl existence
  proof in the original EPS presentation.
- Locator: printed pp. 1582–1585.

### LOR-MATVEEV-TRAUTMAN-2014

- Title: *A criterion for compatibility of conformal and projective structures*
- Authors: Vladimir S. Matveev; Andrzej Trautman
- Stable identifier: DOI `10.1007/s00220-013-1850-7`; arXiv `1302.2329v1`
- URL: https://arxiv.org/abs/1302.2329v1
- Access date: 2026-07-23
- Exact claim supported: gives necessary and sufficient local conditions for a
  metric in a conformal class to have a Levi-Civita connection in a projective
  class and exhibits a nonintegrable case.
- Locator: theorem and conditions (7)–(8), printed pp. 2–4.

### LOR-MATVEEV-SCHOLZ-2020

- Title: *Light cone and Weyl compatibility of conformal and projective
  structures*
- Authors: Vladimir S. Matveev; Erhard Scholz
- Stable identifier: DOI `10.1007/s10714-020-02716-9`; arXiv `2001.01494v1`
- URL: https://arxiv.org/abs/2001.01494v1
- Access date: 2026-07-23
- Exact claim supported: for indefinite signature and dimension at least
  three, a torsion-free projective class whose unparametrized geodesics contain
  every conformal null geodesic is Weyl-compatible.
- Locator: Definitions 1–2, Theorem 1, and Discussion, printed pp. 2–6.

### LOR-PERLICK-1987

- Title: *Characterization of standard clocks by means of light rays and
  freely falling particles*
- Author: Volker Perlick
- Stable identifier: DOI `10.1007/BF00759142`
- URL: https://doi.org/10.1007/BF00759142
- Access date: 2026-07-23
- Exact claim supported: supplies the standard-clock definition used in the
  modern Weyl proper-time completion.
- Locator: *General Relativity and Gravitation* 19 (1987), 1059–1073.

### LOR-AVALOS-DAHIA-ROMERO-2018

- Title: *A note on the problem of proper time in Weyl space-time*
- Authors: R. Avalos; F. Dahia; C. Romero
- Stable identifier: DOI `10.1007/s10701-017-0134-z`; arXiv `1611.10198v2`
- URL: https://arxiv.org/abs/1611.10198v2
- Access date: 2026-07-23
- Exact claim supported: establishes equivalence of EPS and Perlick standard
  clocks and proves the local no-second-clock-effect integrability result,
  with simple connectedness added for the global conclusion.
- Locator: Proposition 1, equation (15), and §V, printed pp. 11–18.

### LOR-HOBSON-LASENBY-2021

- Title: *Weyl gauge theories of gravity do not predict a second clock effect*
- Authors: Michael P. Hobson; Anthony N. Lasenby
- Stable identifier: arXiv `2009.06407`
- URL: https://arxiv.org/abs/2009.06407
- Access date: 2026-07-23
- Exact claim supported: gives a counter-position in which Weyl-covariant
  matter weights and a scalar mass compensator change the standard
  second-clock-effect inference, demonstrating clock-model dependence.
- Locator: §§II.E, III, and V.

### DG-GEROCH-TETRAD-1968

- Title: *Spinor structure of space-times in general relativity. I*
- Author: Robert Geroch
- Stable identifier: DOI `10.1063/1.1664507`
- URL: https://doi.org/10.1063/1.1664507
- Access date: 2026-07-23
- Exact claim supported: in the noncompact four-dimensional scope of the
  theorem, spin structure is equivalent to existence of a global orthonormal
  tetrad. A global coframe is therefore extra topological data.
- Locator: §II theorem, printed pp. 1740–1741.

### DG-LEFLOCH-MARDARE-2007

- Title: *Definition and stability of Lorentzian manifolds with distributional
  curvature*
- Authors: Philippe G. LeFloch; Cristinel Mardare
- Stable identifier: DOI `10.4171/PM/1794`; arXiv `0712.0122`
- URL: https://arxiv.org/abs/0712.0122
- Access date: 2026-07-23
- Exact claim supported: a uniformly nondegenerate
  `H^1_loc ∩ L^infinity_loc` metric has an `L^2_loc` Levi-Civita connection
  and distributional curvature in the source's precise sense.
- Locator: Proposition 4.4 and Theorem 4.6, printed pp. 547–550.

### GR-TRAUTMAN-EC-2006

- Title: *Einstein-Cartan theory*
- Author: Andrzej Trautman
- Stable identifier: arXiv `gr-qc/0606062v1`
- URL: https://arxiv.org/abs/gr-qc/0606062v1
- Access date: 2026-07-23
- Exact claim supported: supplies coframe, connection, torsion, curvature,
  Bianchi, and modified conservation formulas for a local torsionful
  gravitational alternative.
- Locator: equations (7)–(12) and (19)–(29), printed pp. 1–6.

### GR-NAVARRO-LOVELOCK-2011

- Title: *Lovelock's theorem revisited*
- Authors: Alberto Navarro; José Navarro
- Stable identifier: DOI `10.1016/j.geomphys.2011.05.004`; arXiv `1005.2386v4`
- URL: https://arxiv.org/abs/1005.2386v4
- Access date: 2026-07-23
- Exact claim supported: precisely defines second-order natural metric tensors
  and proves that the Lovelock tensors form the basis of symmetric,
  divergence-free tensors in that class. Variational origin is not a theorem
  hypothesis.
- Locator: Definitions 1.1 and 1.8 and Theorem 2.6, printed pp. 2–8.

### GR-GIBBONS-HAWKING-1977

- Title: *Action integrals and partition functions in quantum gravity*
- Authors: G. W. Gibbons; S. W. Hawking
- Stable identifier: DOI `10.1103/PhysRevD.15.2752`
- URL: https://doi.org/10.1103/PhysRevD.15.2752
- Access date: 2026-07-23
- Exact claim supported: the Einstein-Hilbert Dirichlet problem for a smooth
  non-null boundary requires the extrinsic-curvature boundary term when the
  induced metric is fixed but normal metric derivatives are not.
- Locator: §II, equation (2.1), and following boundary discussion.

### GR-IYER-WALD-1994

- Title: *Some properties of Noether charge and a proposal for dynamical black
  hole entropy*
- Authors: Vivek Iyer; Robert M. Wald
- Stable identifier: DOI `10.1103/PhysRevD.50.846`; arXiv `gr-qc/9403028`
- URL: https://arxiv.org/abs/gr-qc/9403028
- Access date: 2026-07-23
- Exact claim supported: gives the general covariant variational identity,
  Noether-current framework, and exact-form ambiguity of the Lagrangian.
- Locator: equations (20)–(22), Lemma 3.1, and equations (41)–(43).

### GR-HAYWARD-1993

- Title: *Gravitational action for spacetimes with nonsmooth boundaries*
- Author: Geoff Hayward
- Stable identifier: DOI `10.1103/PhysRevD.47.3275`
- URL: https://doi.org/10.1103/PhysRevD.47.3275
- Access date: 2026-07-23
- Exact claim supported: nonsmooth spacetime boundaries require joint terms
  beyond the smooth non-null GHY contribution.
- Locator: printed pp. 3275–3280.

### GR-LEHNER-ETAL-2016

- Title: *Gravitational action with null boundaries*
- Authors: Luis Lehner; Robert C. Myers; Eric Poisson; Rafael D. Sorkin
- Stable identifier: DOI `10.1103/PhysRevD.94.084046`; arXiv `1609.00207`
- URL: https://arxiv.org/abs/1609.00207
- Access date: 2026-07-23
- Exact claim supported: null boundary segments and null joints require
  additional terms and carry parametrization choices.
- Locator: §§II–III.

### GR-BUCHDAHL-1970

- Title: *Non-linear Lagrangians and cosmological theory*
- Author: Hans A. Buchdahl
- Stable identifier: DOI `10.1093/mnras/150.1.1`
- URL: https://doi.org/10.1093/mnras/150.1.1
- Access date: 2026-07-23
- Exact claim supported: metric `f(R)` actions have the displayed general
  field equation. For `f(R)=R-2 Lambda+alpha R^2` with nonzero `alpha`, the
  generic equation has a non-Einstein fourth-order correction.
- Locator: introduction, equation (2.1), printed pp. 1–3, and Appendix.

### GR-BRANS-DICKE-1961

- Title: *Mach's principle and a relativistic theory of gravitation*
- Authors: Carl Brans; Robert H. Dicke
- Stable identifier: DOI `10.1103/PhysRev.124.925`
- URL: https://doi.org/10.1103/PhysRev.124.925
- Access date: 2026-07-23
- Exact claim supported: a local generally covariant extra-scalar
  gravitational action gives coupled metric/scalar equations and tests the
  metric-only field-content restriction.
- Locator: equations (6), (9), and (11), printed pp. 929–930.

### GR-DESER-WOODARD-2007

- Title: *Nonlocal cosmology*
- Authors: Stanley Deser; R. P. Woodard
- Stable identifier: DOI `10.1103/PhysRevLett.99.111301`; arXiv `0706.2151`
- URL: https://arxiv.org/abs/0706.2151
- Access date: 2026-07-23
- Exact claim supported: supplies an explicit generally covariant nonlocal
  gravity contrast when locality is relaxed.
- Locator: action and field-equation construction in the full article.

## Quantum reconstruction and categorical quantum mechanics

### QM-GLEASON-1957

- Title: *Measures on the closed subspaces of a Hilbert space*
- Author: Andrew M. Gleason
- Stable identifier: DOI `10.1512/iumj.1957.6.56050`
- URL: https://doi.org/10.1512/iumj.1957.6.56050
- Access date: 2026-07-23
- Exact claim supported: for real or complex separable Hilbert spaces of
  dimension at least three, countably additive measures on closed subspaces
  have the trace-operator form. The theorem presupposes Hilbert-space and
  orthogonality structure and excludes the two-dimensional case.
- Locator: Theorem 3.5 and Theorem 4.1, printed pp. 892-893.

### QM-SOLER-1995

- Title: *Characterization of Hilbert spaces by orthomodular spaces*
- Author: Maria Pia Solèr
- Stable identifier: DOI `10.1080/00927879508825218`
- URL: https://doi.org/10.1080/00927879508825218
- Access date: 2026-07-23
- Exact claim supported: an orthomodular Hermitian space over an involutive
  division ring that contains an infinite orthonormal sequence has scalar
  field `R`, `C`, or `H` and is the corresponding Hilbert space. Infinite
  dimension without the sequence hypothesis is insufficient, and the theorem
  does not select the complex field.
- Locator: *Communications in Algebra* 23 (1995), 219–243.

### QM-HARDY-2001

- Title: *Quantum Theory From Five Reasonable Axioms*
- Author: Lucien Hardy
- Stable identifier: arXiv `quant-ph/0101012v4`
- URL: https://arxiv.org/abs/quant-ph/0101012v4
- Access date: 2026-07-23
- Exact claim supported: finite and countably infinite distinguishability
  cases are reconstructed from five axioms, while continuously dimensional
  Hilbert spaces are excluded. The finite parameter-counting core uses
  simplicity and continuous reversible transformations. The source derives
  no particular Hamiltonian.
- Locator: Section 6 axioms; Section 8.1 for `K=N^r`; Section 8.5 for the
  continuity/classical split; Equation (90); Sections 8.8-8.9. Version 2
  corrected Axiom 2 and its proof; version 4 revised the main proofs and
  state-update discussion.

### QM-CDP-2011

- Title: *Informational derivation of quantum theory*
- Authors: Giulio Chiribella; Giacomo Mauro D’Ariano; Paolo Perinotti
- Stable identifier: DOI `10.1103/PhysRevA.84.012311`; arXiv `1011.6451v3`
- URL: https://arxiv.org/abs/1011.6451v3
- Access date: 2026-07-23
- Exact claim supported: within a finite-dimensional
  operational-probabilistic framework, causality, perfect distinguishability,
  ideal compression, local distinguishability, pure conditioning, and
  purification with essential uniqueness reconstruct complex density
  matrices, effects, and completely positive trace-nonincreasing
  transformations. These are independent inputs, not categorical
  consequences.
- Locator: Section II framework; Axioms 1-5; Postulate 1; Theorem 20;
  Corollary 52. Corollary 52 and the printed-page-7 definitions say general
  operations are trace-nonincreasing; one proof sentence saying
  trace-preserving is treated as a typographical slip.

### QM-ABRAMSKY-COECKE-2004

- Title: *A categorical semantics of quantum protocols*
- Authors: Samson Abramsky; Bob Coecke
- Stable identifier: DOI `10.1109/LICS.2004.1319636`; arXiv
  `quant-ph/0402130`
- URL: https://arxiv.org/abs/quant-ph/0402130
- Access date: 2026-07-23
- Exact claim supported: compact closed categorical structure with biproducts
  captures structural features of teleportation, entanglement swapping, and
  related protocols. This is a semantics of quantum processes; it does not
  prove that arbitrary compositional categories force complex quantum theory.
- Locator: abstract and §§3–7.

### QM-ADLER-1995

- Title: *Quaternionic Quantum Mechanics and Quantum Fields*
- Author: Stephen L. Adler
- Stable identifier: DOI `10.1093/oso/9780195066432.001.0001`
- URL: https://doi.org/10.1093/oso/9780195066432.001.0001
- Access date: 2026-07-23
- Exact claim supported: quantum mechanics admits a systematic formulation on
  quaternionic Hilbert space. Complex scalars are therefore not forced by the
  bare Hilbert-space idea or compositional language alone.
- Locator: Chapters 1–3.

### QM-BARNUM-WILCE-2012

- Title: *Local tomography and the Jordan structure of quantum theory*
- Authors: Howard Barnum; Alexander Wilce
- Stable identifier: DOI `10.1007/s10701-014-9777-1`; arXiv `1202.4513v2`
- URL: https://arxiv.org/abs/1202.4513v2
- Access date: 2026-07-23
- Exact claim supported: finite-dimensional state-complete homogeneous
  self-dual models with factorizable self-duality, or the stated dagger-HSD
  categorical structure, plus suitable non-signalling locally tomographic
  composites and a qubit select complex finite-dimensional C-star/Jordan
  systems. Direct sums and superselection remain.
- Locator: Proposition 1 and Proposition 16.

### QM-PIRON-1976

- Title: *Foundations of Quantum Physics*
- Author: Constantin Piron
- Stable identifier: W. A. Benjamin, 1976
- URL: https://search.worldcat.org/title/1529675
- Access date: 2026-07-23
- Exact claim supported: the quantum-logic representation route starts from
  a strong lattice hypothesis and yields a generalized Hilbert space, not
  complex Hilbert space from outcome labels.
- Locator: representation theorem; use the modern exposition below for the
  exact rank and lattice formulation.

### QM-PIRON-EXPOSITION-2007

- Title: *Propositional systems, Hilbert lattices and generalized Hilbert spaces*
- Authors: Isar Stubbe; Bart Van Steirteghem
- Stable identifier: arXiv `0710.2098`
- URL: https://arxiv.org/abs/0710.2098
- Access date: 2026-07-23
- Exact claim supported: an irreducible complete atomistic orthomodular
  lattice with the covering law and rank at least four is represented by the
  biorthogonally closed subspaces of a generalized Hilbert space over an
  involutive division ring.
- Locator: modern statement of Piron's representation theorem.

### QM-HOLLAND-SOLER-1995

- Title: *Orthomodularity in infinite dimensions; a theorem of M. Soler*
- Author: S. S. Holland Jr.
- Stable identifier: arXiv `math/9504224`
- URL: https://arxiv.org/abs/math/9504224
- Access date: 2026-07-23
- Exact claim supported: the orthonormal-sequence condition is essential to
  the Soler route; Keller-type infinite-dimensional examples survive over
  nonclassical involutive division rings when it is absent.
- Locator: Theorem 1.3, Keller discussion, and Theorem 4.1.

### QM-BUSCH-2003

- Title: *Quantum states and generalized observables: a simple proof of Gleason's theorem*
- Author: Paul Busch
- Stable identifier: DOI `10.1103/PhysRevLett.91.120403`; arXiv
  `quant-ph/9909073v3`
- URL: https://arxiv.org/abs/quant-ph/9909073v3
- Access date: 2026-07-23
- Exact claim supported: a positive normalized countably additive valuation
  on all effects has trace form and includes two-dimensional Hilbert space.
  This strengthens the measurement domain beyond projection-only Gleason.
- Locator: properties P1-P3, theorem, and dimension-two discussion.

### QM-CFMR-2004

- Title: *Gleason-Type Derivations of the Quantum Probability Rule for Generalized Measurements*
- Authors: Carlton M. Caves; Christopher A. Fuchs; Kiran K. Manne; Joseph M. Renes
- Stable identifier: arXiv `quant-ph/0306179`
- URL: https://arxiv.org/abs/quant-ph/0306179
- Access date: 2026-07-23
- Exact claim supported: in finite dimension, a normalized noncontextual frame
  function on the full effect set, additive across finite POVMs, has the
  unique trace form and includes qubits.
- Locator: Theorem 1 and Section IV.

### QM-HANCHE-OLSEN-1985

- Title: *JB-algebras with tensor products are C-star algebras*
- Author: Harald Hanche-Olsen
- Stable identifier: DOI `10.1007/BFb0074886`
- URL: https://doi.org/10.1007/BFb0074886
- Access date: 2026-07-23
- Exact claim supported: under the stated Jordan tensor-product identities
  with the self-adjoint two-by-two complex matrices, a JB-algebra is the
  self-adjoint part of a complex C-star algebra.
- Locator: Lecture Notes in Mathematics 1132 (1985), pp. 223-229.

### QM-BGW-2020

- Title: *Composites and Categories of Euclidean Jordan Algebras*
- Authors: Howard Barnum; Matthew A. Graydon; Alexander Wilce
- Stable identifier: DOI `10.22331/q-2020-11-08-359`; arXiv `1606.09331v3`
- URL: https://arxiv.org/abs/1606.09331v3
- Access date: 2026-07-23
- Exact claim supported: coherent categories can contain real, complex, and
  quaternionic Euclidean Jordan systems with nonstandard,
  non-locally-tomographic composites. Complex composites can carry an extra
  classical bit, and quaternionic composites can be real.
- Locator: Definition 2.1, universal tensor-product table, Proposition 3.10,
  Theorem 4.15, Corollary 4.16, and examples.

### QM-HARDY-WOOTTERS-2012

- Title: *Limited Holism and Real-Vector-Space Quantum Theory*
- Authors: Lucien Hardy; William K. Wootters
- Stable identifier: arXiv `1005.4870`
- URL: https://arxiv.org/abs/1005.4870
- Access date: 2026-07-23
- Exact claim supported: real-vector-space quantum theory fails local
  tomography but is bilocally tomographic, so failure of local tomography is
  a discriminator rather than an inconsistency.
- Locator: main parameter-counting and bilocal-tomography results.

### QM-FINKELSTEIN-ETAL-1962

- Title: *Foundations of Quaternion Quantum Mechanics*
- Authors: David Finkelstein; Josef Jauch; Samuel Schiminovich; David Speiser
- Stable identifier: DOI `10.1063/1.1703794`
- URL: https://doi.org/10.1063/1.1703794
- Access date: 2026-07-23
- Exact claim supported: quaternionic quantum mechanics is a systematic
  formal alternative whose composite-system tensor product is a central
  difficulty because the scalar field is noncommutative.
- Locator: *Journal of Mathematical Physics* 3 (1962).

### QM-DAVIES-LEWIS-1970

- Title: *An operational approach to quantum probability*
- Authors: E. B. Davies; J. T. Lewis
- Stable identifier: DOI `10.1007/BF01647093`
- URL: https://doi.org/10.1007/BF01647093
- Access date: 2026-07-23
- Exact claim supported: an instrument is the operation-valued object that
  combines outcome probabilities with conditional state change.
- Locator: *Communications in Mathematical Physics* 17 (1970), 239-260.

### QM-OZAWA-1984

- Title: *Quantum measuring processes of continuous observables*
- Author: Masanao Ozawa
- Stable identifier: DOI `10.1063/1.526000`
- URL: https://doi.org/10.1063/1.526000
- Access date: 2026-07-23
- Exact claim supported: completely positive instruments correspond to state
  changes produced by measurement processes in the declared Hilbert-space
  framework; this is not an ontological collapse theorem.
- Locator: *Journal of Mathematical Physics* 25 (1984), 79-87.

### QM-STINESPRING-1955

- Title: *Positive functions on C-star-algebras*
- Author: W. Forrest Stinespring
- Stable identifier: DOI `10.1090/S0002-9939-1955-0069403-4`
- URL: https://doi.org/10.1090/S0002-9939-1955-0069403-4
- Access date: 2026-07-23
- Exact claim supported: completely positive maps admit a dilation
  representation. Dilation nonuniqueness does not choose a measurement
  ontology.
- Locator: *Proceedings of the American Mathematical Society* 6 (1955),
  211-216.

### QM-EVERETT-1957

- Title: *"Relative State" Formulation of Quantum Mechanics*
- Author: Hugh Everett III
- Stable identifier: DOI `10.1103/RevModPhys.29.454`
- URL: https://doi.org/10.1103/RevModPhys.29.454
- Access date: 2026-07-23
- Exact claim supported: unitary quantum mechanics admits a relative-state
  interpretation without a primitive collapse law.
- Locator: *Reviews of Modern Physics* 29 (1957), 454-462.

### QM-BOHM-1952

- Title: *A Suggested Interpretation of the Quantum Theory in Terms of "Hidden" Variables. I*
- Author: David Bohm
- Stable identifier: DOI `10.1103/PhysRev.85.166`
- URL: https://doi.org/10.1103/PhysRev.85.166
- Access date: 2026-07-23
- Exact claim supported: nonrelativistic quantum predictions admit an
  empirically matching hidden-variable completion in quantum equilibrium.
- Locator: *Physical Review* 85 (1952), 166-179.

### QM-GRW-1986

- Title: *Unified dynamics for microscopic and macroscopic systems*
- Authors: GianCarlo Ghirardi; Alberto Rimini; Tullio Weber
- Stable identifier: DOI `10.1103/PhysRevD.34.470`
- URL: https://doi.org/10.1103/PhysRevD.34.470
- Access date: 2026-07-23
- Exact claim supported: a literal collapse theory requires a new stochastic
  dynamics beyond the operational quantum formalism.
- Locator: *Physical Review D* 34 (1986), 470-491.

### QM-SCHACK-2002

- Title: *Quantum theory from four of Hardy's axioms*
- Author: Rudiger Schack
- Stable identifier: arXiv `quant-ph/0210017`
- URL: https://arxiv.org/abs/quant-ph/0210017
- Access date: 2026-07-23
- Exact claim supported: Hardy's limiting-frequency axiom can be removed
  after a Bayesian reformulation and a replacement linearity argument.
- Locator: main reconstruction argument.

## Kinetic limits, propagation of chaos, and the Boltzmann equation

### KIN-BOLTZMANN-1872

- Title: *Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen*
- Author: Ludwig Boltzmann
- Stable identifier: *Sitzungsberichte der Kaiserlichen Akademie der
  Wissenschaften* 66 (1872), 275–370; reprint DOI
  `10.1017/CBO9781139381420.023`
- URL: https://doi.org/10.1017/CBO9781139381420.023
- Access date: 2026-07-23
- Exact claim supported: the 1872 work introduces the kinetic equation and
  monotonic H functional under the collision-statistical assumptions used
  there. It is the historical source, not by itself a modern many-particle
  limit theorem. The modern classical balance and dissipation formulas used
  in Paper 6 are located separately in GST Section 2.2 and require collision
  symmetries, integrability, and regularity.
- Locator: original pp. 275–370; reprint pp. 316–402; modern formula locator
  `KIN-GALLAGHER-SAINT-RAYMOND-TEXIER-2013`, Section 2.2, equations
  (2.2.1) through (2.2.5).

### KIN-KAC-1956

- Title: *Foundations of Kinetic Theory*
- Author: Mark Kac
- Stable identifier: MR `MR0084985`
- URL: https://digicoll.lib.berkeley.edu/record/112857
- Access date: 2026-07-23
- Exact claim supported: Kac introduces a stochastic N-particle collision
  model and a program relating chaos of marginals to a nonlinear homogeneous
  kinetic equation. The simplified scalar-velocity rotation model drops
  physical momentum and is spatially homogeneous. Kac explains that the
  propagation estimate used there does not transfer to the physically
  significant hard-sphere case.
- Locator: Proceedings of the Third Berkeley Symposium, vol. III,
  equations (3.1) through (3.10), the Basic Theorem, and Section 6,
  pp. 171–197.

### KIN-LANFORD-1975

- Title: *Time evolution of large classical systems*
- Author: Oscar E. Lanford III
- Stable identifier: DOI `10.1007/BFb0017183`
- URL: https://doi.org/10.1007/BFb0017183
- Access date: 2026-07-23
- Exact claim supported: in the Boltzmann–Grad limit for a dilute hard-sphere
  system with suitable chaotic initial data, the hierarchy converges to the
  Boltzmann hierarchy for a short time. The theorem does not justify arbitrary
  long-time relaxation.
- Companion primary exposition: Oscar E. Lanford III, *On a derivation of
  the Boltzmann equation*, Astérisque 40 (1976), 117–137,
  https://www.numdam.org/item/AST_1976__40__117_0/.
- Locator: Lecture Notes in Physics 38 (1975), pp. 1–111; 1976 exposition for
  the three-dimensional hypotheses and the at-most-one-fifth-of-a-mean-free-
  time scope.

### KIN-SZNITMAN-1991

- Title: *Topics in propagation of chaos*
- Author: Alain-Sol Sznitman
- Stable identifier: DOI `10.1007/BFb0085169`
- URL: https://doi.org/10.1007/BFb0085169
- Access date: 2026-07-23
- Exact claim supported: propagation of chaos formalizes asymptotic
  independence of finite marginals in large interacting-particle systems and
  supplies equivalences and methods under stated probabilistic hypotheses.
  It is a limiting property to prove, not an automatic result of
  coarse-graining. The standard definition concerns normalized symmetric laws
  on fixed product spaces; a conversion is required before identifying it
  with grand-canonical rescaled correlation factorization.
- Locator: Lecture Notes in Mathematics 1464, definition of `u`-chaos and
  Proposition 2.2, pp. 165–251.

### KIN-GALLAGHER-SAINT-RAYMOND-TEXIER-2013

- Title: *From Newton to Boltzmann: Hard Spheres and Short-Range Potentials*
- Authors: Isabelle Gallagher; Laure Saint-Raymond; Benjamin Texier
- Stable identifier: DOI `10.4171/129`; arXiv `1208.5753v2`
- URL: https://arxiv.org/abs/1208.5753v2
- Access date: 2026-07-23
- Exact claim supported: for hard spheres, Theorem 8 proves short-time
  Boltzmann-Grad convergence in the mode of Definition 6.2.1, with explicit
  collision-tree and recollision control. Theorem 11 treats only the stated
  radial, repulsive, monotone, compactly supported, regular potential class
  satisfying its additional scattering condition, not arbitrary short-range
  interactions. Section 1.2 distinguishes mean-field from Boltzmann-Grad
  scaling.
- Locator: Definition 6.2.1; Theorem 8; Assumption 1.2.1; condition (8.3.1);
  Theorem 11; Section 2.2 for classical collision invariants and entropy.

### KIN-DIPERNA-LIONS-1989

- Title: *On the Cauchy problem for Boltzmann equations: Global existence and
  weak stability*
- Authors: Ronald J. DiPerna; Pierre-Louis Lions
- Stable identifier: DOI `10.2307/1971423`
- URL: https://annals.math.princeton.edu/1989/130-2/p03
- Access date: 2026-07-23
- Exact claim supported: renormalized solutions give global existence and weak
  stability for broad Boltzmann Cauchy problems. This PDE existence result is
  distinct from deriving the equation from N-particle dynamics. In the
  bounded-cutoff GST presentation used by Paper 6, the continuity equation is
  an equality, the momentum equation carries a nonnegative symmetric defect
  measure, and entropy is an inequality; these are not the classical local
  equalities.
- Locator: official abstract and main theorems, pp. 321–366; GST Theorem 3
  for the exact cutoff renormalized/defect formulation quoted in Paper 6.

### KIN-DENG-HANI-MA-2024

- Title: *Long time derivation of the Boltzmann equation from hard sphere
  dynamics*
- Authors: Yu Deng; Zaher Hani; Xiao Ma
- Stable identifier: arXiv `2408.07818v3`
- URL: https://arxiv.org/abs/2408.07818v3
- Access date: 2026-07-23
- Pinning method: the official arXiv submission history identifies `v3` as
  revised 2025-07-18. A header request to the versioned PDF returned content
  length 7,682,019 bytes and the server-declared ETag
  `sha256:a5abbd14c294df399e2c625fbfe26d7748a7a3f2908566064316dfde761c290d`.
- Exact claim supported: for deterministic hard spheres in dimension at
  least two with the version 3 grand-canonical initial law and expected
  Boltzmann-Grad scaling, Theorem 1 gives a quantitative L1 bound between the
  rescaled `s`-point correlation and the product Boltzmann solution times the
  hard-core exclusion indicator. The bound is uniform on a selected finite
  interval and for `s <= |log epsilon|`, after assuming the regular Boltzmann
  solution, its uniform Maxwellian bound, the initial `Bol_(2 beta)` spatial
  norm bounds, and epsilon sufficiently small depending on
  `(d,tfin,beta,A,B0)`. It is not an unconditional `t=infinity` theorem.
- Locator: Definition 1.3(2) through (5), equations (1.9) through (1.12);
  hard-sphere equation and kernel (1.13) through (1.14); norm (1.15);
  Theorem 1, equations (1.16) through (1.18); Remark 1.5.

## Coverage audit

| Required domain | Source keys |
|---|---|
| Categorical locality | `CAT-ATIYAH-1988`, `CAT-COSTELLO-GWILLIAM-2017` |
| Descent | `CAT-LURIE-HTT-2009`, `DESC-STACKS-023N`, `DESC-STACKS-08KE` |
| Condensed mathematics | `COND-CLAUSEN-SCHOLZE-2026` |
| Representation reconstruction | `TAN-TANNAKA-1939`, `TAN-DELIGNE-MILNE-1982`, `TAN-DOPLICHER-ROBERTS-1989`, `TAN-DELIGNE-SUPER-2002` |
| Algebraic quantum theory | `AQFT-HAAG-KASTLER-1964`, `AQFT-BFV-2003`, `TAN-DOPLICHER-ROBERTS-ANNALS-1989`, `AQFT-DOPLICHER-ROBERTS-FIELD-1990`, `AQFT-NOETHER-BDL-1986` |
| Invariant states and dynamics | `CSTAR-COMPACT-ERGODIC-1981`, `AMEN-VON-NEUMANN-1929`, `AMEN-DAY-1957`, `AMEN-DAY-1961`, `DYN-STONE-1932` |
| Outcome and frame interfaces | `OUT-ABRAMSKY-HEUNEN-2019`, `REF-BARTLETT-RUDOLPH-SPEKKENS-2007` |
| Lorentzian reconstruction | `LOR-HKM-1976`, `LOR-MALAMENT-1977`, `LOR-EPS-1972` |
| Quantum reconstruction | `QM-GLEASON-1957`, `QM-SOLER-1995`, `QM-HARDY-2001`, `QM-CDP-2011`, `QM-BARNUM-WILCE-2012` |
| Categorical quantum mechanics | `QM-ABRAMSKY-COECKE-2004` |
| Kinetic limits | `KIN-LANFORD-1975`, `KIN-GALLAGHER-SAINT-RAYMOND-TEXIER-2013`, `KIN-DENG-HANI-MA-2024` |
| Propagation of chaos | `KIN-KAC-1956`, `KIN-SZNITMAN-1991` |
| Boltzmann equation | `KIN-BOLTZMANN-1872`, `KIN-DIPERNA-LIONS-1989` |

## Evidence rule

A later artifact may cite a key only for the exact claim above or for a weaker
statement. Any stronger statement requires a new ledger entry with a checked
locator. Search snippets, tertiary summaries, and the existence of a DOI are
discovery aids, not evidence.