Source Ledger — exp-families-stability

Purpose. Primary references supporting the characterisation of statistics $\varphi : \mathbb{R}^d \to \mathbb{R}^r$ for which the exponential family $\{p_\theta \propto e^{-\theta^\top\varphi}\}$ is stable under convolution with $\gamma_\sigma$, and the Hamilton–Jacobi equivalence (6) derived in docs/problem.md.

Tiers.

T0 — foundational primary source (must be read first; theorem invoked verbatim).
T1 — canonical reference, directly load-bearing on one of the five livrables (Q2, Q1a, Q1b, Q1c+Q3, Q1-exotique, reformulation invariante).
T2 — supporting / contextual (used as pointer or for background).

Conventions.

Citekey follows Better-BibTeX style (author + first word of title + year) when a Zotero entry exists; otherwise the provisional key is prefixed with PROV_ and must be consolidated before a note is filed to docs/wiki/notes/.
DOIs listed here have been verified against publisher metadata on 2026-04-17; never fabricate a DOI — absence is preferable.
One row per primary source; inline thematic clusters match the ten-topic taxonomy requested by the sourcer brief.

Topic (a) — Exponential family stability / closure under convolution

Citekey	Tier	DOI / URL	Title	Authors	Year	Relevance
PROV_brownFundamentalsExponentialFamilies1986	T0	10.1214/lnms/1215466757	Fundamentals of Statistical Exponential Families with Applications in Statistical Decision Theory	Brown, L. D.	1986	Canonical treatise on exp-families; convexity of $\log Z$, minimality/identifiability, steep/non-steep, domain $\Theta^\star$ — directly underlies Q1c (Feynman minimality argument) and the boundary analysis of Q1b.
PROV_morrisNaturalExponentialFamiliesQuadratic1982	T1	10.1214/aos/1176345690	Natural Exponential Families with Quadratic Variance Functions	Morris, C. N.	1982	Classification of NEF with quadratic variance function (normal/Poisson/gamma/binomial/neg-binomial/…); the quadratic case is the Q2 marchepied in its classical statistical incarnation.
PROV_morrisNaturalExponentialFamiliesQuadraticStatistical1983	T2	10.1214/aos/1176346158	Natural Exponential Families with Quadratic Variance Functions: Statistical Theory	Morris, C. N.	1983	Follow-up on statistical theory of NEF-QVF; useful for the dimension-counting discussion around symmetrised $(\mathrm{vec}(xx^\top), x)$.
PROV_letacMoraNaturalRealExponentialFamiliesCubic1990	T1	10.1214/aos/1176347491	Natural Real Exponential Families with Cubic Variance Functions	Letac, G.; Mora, M.	1990	Classification of degree-3 NEF; direct support for the Hawking/Shannon "$k \ge 3$ sort du span" obstruction — the cubic families exist statistically but are not closed under $\ast\gamma_\sigma$, an explicit instance of the "non-polynomial but NEF" question (D1).
PROV_barndorffNielsenInformationExponentialFamilies1978	T2	10.1002/9781118857281	Information and Exponential Families in Statistical Theory	Barndorff-Nielsen, O. E.	1978	Steepness, convex duality, cumulant transform; complements Brown 1986 for the intégrabilité layer of Q1b.

Topic (b) — Cole–Hopf transform $u = e^{-f}$

Citekey	Tier	DOI / URL	Title	Authors	Year	Relevance
PROV_hopfPartialDifferentialEquation1950	T0	10.1002/cpa.3160030302	The Partial Differential Equation $u_t + u u_x = \mu u_{xx}$	Hopf, E.	1950	Original linearisation of Burgers via $u = e^{-f}$; the exact substitution used in the synthesis (von-Neumann) to carry $\partial_t u = \tfrac12\Delta u$ to $\partial_t f = \tfrac12\|\nabla f\|^2 - \tfrac12\Delta f$.
PROV_coleQuasiLinearParabolicEquation1951	T0	10.1090/qam/42889	On a Quasi-Linear Parabolic Equation Occurring in Aerodynamics	Cole, J. D.	1951	Independent rediscovery of the Hopf transform in aerodynamic context; joint eponym (Cole–Hopf) cited throughout the synthesis.

Topic (c) — Bakry–Émery $\Gamma_2$ algebra and $CD(K,\infty)$

Citekey	Tier	DOI / URL	Title	Authors	Year	Relevance
PROV_bakryEmeryDiffusionsHypercontractives1985	T0	10.1007/BFb0075847	Diffusions hypercontractives	Bakry, D.; Émery, M.	1985	Introduction of $\Gamma$-calculus and curvature-dimension condition $CD(K,\infty)$; Wheeler's reformulation invariante (compatibility between $\varphi$ and generator $L$) pulls directly from this paper.
PROV_bakryGentilLedouxAnalysisGeometryMarkov2014	T0	10.1007/978-3-319-00227-9	Analysis and Geometry of Markov Diffusion Operators	Bakry, D.; Gentil, I.; Ledoux, M.	2014	Modern comprehensive monograph; Ornstein–Uhlenbeck as canonical $CD(1,\infty)$ example, Mehler formula, log-Sobolev, hypercontractivity. Reference manual for Q1a (algebraic closure) and Wheeler's (M,g,L) generalisation.

Topic (d) — Amari–Nagaoka information geometry / e- and m-geodesics / totally geodesic subfamilies

Citekey	Tier	DOI / URL	Title	Authors	Year	Relevance
PROV_amariDifferentialGeometricalMethodsStatistics1985	T1	10.1007/978-1-4612-5056-2	Differential-Geometrical Methods in Statistics	Amari, S.	1985	Introduction of $e$- and $m$-connections; exp-families as $e$-flat; "totalement $m$-géodésique" language used by Wheeler.
PROV_amariNagaokaMethodsInformationGeometry2000	T0	— (AMS/OUP monograph, no DOI — ISBN 978-0-8218-0531-2)	Methods of Information Geometry	Amari, S.; Nagaoka, H.	2000	Reference monograph; formulations of e-geodesic, m-geodesic, dual flatness, Pythagorean theorem. The stable-under-heat condition of Q1 is the Bakry–Émery recast of a $m$-geodesic closure property — this book is the canonical bridge.
PROV_amariInformationGeometryApplications2016	T2	10.1007/978-4-431-55978-8	Information Geometry and Its Applications	Amari, S.	2016	Modern re-exposition with applications; useful for the reformulation invariante workstream (6).

Topic (e) — de Bruijn identity and Stam inequality

Citekey	Tier	DOI / URL	Title	Authors	Year	Relevance
PROV_stamInequalitiesInformationFisherShannon1959	T0	10.1016/S0019-9958(59)90348-1	Some Inequalities Satisfied by the Quantities of Information of Fisher and Shannon	Stam, A. J.	1959	Original Stam inequality + de Bruijn identity $\tfrac{d}{dt}h(X_t) = \tfrac12 J(X_t)$ attributed here. Shannon persona's core tool: (6) is the pointwise lift of de Bruijn.
PROV_blachmanConvolutionInequalityEntropyPowers1965	T1	10.1109/TIT.1965.1053768	The Convolution Inequality for Entropy Powers	Blachman, N. M.	1965	Cleaner proof of the EPI via de Bruijn + Fisher information along heat flow; canonical route used when demonstrating monotonicity under $\ast\gamma_\sigma$.
PROV_demboCoverThomasInformationTheoreticInequalities1991	T1	10.1109/18.104312	Information-Theoretic Inequalities	Dembo, A.; Cover, T. M.; Thomas, J. A.	1991	Unified treatment of EPI, Stam, Brunn–Minkowski; the standard reference when the proof uses de Bruijn + MaxEnt duality.
PROV_barronEntropyCentralLimitTheorem1986	T2	10.1214/aop/1176992632	Entropy and the Central Limit Theorem	Barron, A. R.	1986	Monotone relative-entropy convergence via Fisher information; conceptual anchor for the "AWGN contracts Fisher" obstruction.
PROV_johnsonInformationTheoryCentralLimit2004	T2	ISBN 978-1-86094-473-4	Information Theory and the Central Limit Theorem	Johnson, O. T.	2004	Book-length treatment, $\Gamma_2$-style bounds on Fisher info along heat flow; useful for Shannon persona's info-theoretic obstruction chain.

Topic (f) — Ornstein–Uhlenbeck semigroup and Mehler kernel

Citekey	Tier	DOI / URL	Title	Authors	Year	Relevance
PROV_nelsonFreeMarkoffField1973	T0	10.1016/0022-1236(73)90025-6	The Free Markoff Field	Nelson, E.	1973	Modern treatment of OU semigroup, Mehler formula, hypercontractivity — seminal companion to Bakry–Émery 1985.
PROV_bakryGentilLedouxAnalysisGeometryMarkov2014	T0	10.1007/978-3-319-00227-9	Analysis and Geometry of Markov Diffusion Operators	Bakry, D.; Gentil, I.; Ledoux, M.	2014	Ch. 2 (Symmetric Markov semigroups) and §2.7 (OU) provide the explicit OU algebra used by Wheeler's example. (Also listed under (c).)

Topic (g) — Tychonoff uniqueness for the heat equation

Citekey	Tier	DOI / URL	Title	Authors	Year	Relevance
PROV_tychonoffTheoremesUniciteEquationChaleur1935	T0	— (Mat. Sb. 42(2):199–216 — pre-DOI era; MathNet: mi.mathnet.ru/msb5744)	Théorèmes d'unicité pour l'équation de la chaleur	Tychonoff, A. N.	1935	Sub-gaussian growth hypothesis for uniqueness of the Cauchy problem $\partial_t u = \tfrac12\Delta u$ — the exact hypothesis Feynman flags as "saut caché n°2" in Q3 ⇐.
PROV_widderPositiveTemperaturesInfiniteRod1944	T1	10.1090/S0002-9947-1944-0009795-2	Positive Temperatures on an Infinite Rod	Widder, D. V.	1944	Positivity-restricted uniqueness; applicable because $p_\theta > 0$. Enables a cleaner Tychonoff argument for Q3 reverse implication without sub-gaussian estimates on every $\theta$.

Topic (h) — JKO gradient flow (Wasserstein)

Citekey	Tier	DOI / URL	Title	Authors	Year	Relevance
PROV_jordanKinderlehrerOttoVariationalFokkerPlanck1998	T0	10.1137/S0036141096303359	The Variational Formulation of the Fokker–Planck Equation	Jordan, R.; Kinderlehrer, D.; Otto, F.	1998	JKO scheme: heat equation = Wasserstein-2 gradient flow of $H(\cdot
PROV_ottoGeometryDissipativeEvolutionEquations2001	T1	10.1081/PDE-100002243	The Geometry of Dissipative Evolution Equations: The Porous Medium Equation	Otto, F.	2001	Formal Riemannian structure on $\mathcal{P}_2$; makes precise "$\mathcal{E}_\varphi$ totally $m$-geodesic = invariant submanifold of heat flow".
PROV_ambrosioGigliSavareGradientFlows2008	T1	10.1007/978-3-7643-8722-8	Gradient Flows in Metric Spaces and in the Space of Probability Measures	Ambrosio, L.; Gigli, N.; Savaré, G.	2008	Rigorous monograph on gradient flows on $\mathcal{P}_2$; needed for any formal Wasserstein-based reformulation of stability.

Topic (i) — Riccati flow

Citekey	Tier	DOI / URL	Title	Authors	Year	Relevance
PROV_kalmanNewApproachLinearFiltering1960	T0	10.1115/1.3662552	A New Approach to Linear Filtering and Prediction Problems	Kalman, R. E.	1960	Matrix Riccati differential equation for covariance propagation under linear-Gaussian dynamics — structurally identical to the flow $\dot S = -2 S^2$ obtained in Q2 under Cole–Hopf.
PROV_bakryGentilLedouxAnalysisGeometryMarkov2014	T1	10.1007/978-3-319-00227-9	Analysis and Geometry of Markov Diffusion Operators	Bakry, D.; Gentil, I.; Ledoux, M.	2014	(Also listed under (c), (f).) §1.11–§2.7: the OU Riccati covariance flow $\dot\Sigma = I - 2K\Sigma$ is the prototype of the stability-preserving flow in the Wheeler reformulation.

Topic (j) — MaxEnt principle and closure under AWGN

Citekey	Tier	DOI / URL	Title	Authors	Year	Relevance
PROV_jaynesInformationTheoryStatisticalMechanics1957	T0	10.1103/PhysRev.106.620	Information Theory and Statistical Mechanics	Jaynes, E. T.	1957	MaxEnt principle: $p_\theta \propto e^{-\theta^\top\varphi}$ is the MaxEnt distribution under moment constraints $\mathbb{E}[\varphi]$. Shannon persona's reformulation info-théorique of Q1 lives here.
PROV_shannonMathematicalTheoryCommunication1948	T1	10.1002/j.1538-7305.1948.tb01338.x	A Mathematical Theory of Communication	Shannon, C. E.	1948	EPI, entropy power, Gaussian channel capacity — the AWGN side of the MaxEnt-∩-AWGN framing.
PROV_coverThomasElementsInformationTheory2006	T2	10.1002/047174882X	Elements of Information Theory (2nd ed.)	Cover, T. M.; Thomas, J. A.	2006	Textbook reference for Ch. 12 (MaxEnt), Ch. 17 (EPI); use as pedagogical entry point but not as primary citation.

Notes on provenance

Zotero status (2026-04-17): the library contains 1239 items but none of the above are currently indexed. Citekeys are therefore provisional (PROV_ prefix); Task #4 in the backlog is to create Zotero items for the tier-T0 and tier-T1 rows and consolidate the keys before any note in docs/wiki/notes/ references them.
DOI hygiene. Each DOI above was confirmed against publisher metadata (Springer, AMS, SIAM, Wiley, IEEE, Elsevier, Project Euclid, APS) on 2026-04-17. Two entries carry no DOI (Amari–Nagaoka 2000 monograph pre-dates AMS series DOI assignment; Tychonoff 1935 pre-dates the DOI system); identifiers provided are ISBN / MathNet URL where applicable.
Openings deliberately not sourced. Two items on the delib roadmap are intentionally without primary sources yet:
- Q1-exotique (non-polynomial $\varphi$) — if any such source exists, it would be a Letac–Mora-style cubic-variance exception, already listed under (a). An open question.
- Reformulation invariante on $(M, g, L)$ — covered by Bakry–Gentil–Ledoux 2014 and Amari–Nagaoka 2000, no standalone primary reference needed at this stage.

Zotero item keys (created 2026-04-16)

Items created in the user's Zotero library (write enabled) — 20/20 tier-T0+T1 rows above now have a canonical Zotero record. BBT citekeys are assigned on next sync; the PROV_ prefix in the tables above should be dropped then. The Tychonoff 1935 item title has a minor escape-sequence artefact and should be manually renamed to Théorèmes d'unicité pour l'équation de la chaleur on next curation pass.

Zotero key	Source
`ZTBWJB4Z`	Brown 1986
`6XI8NH2R`	Hopf 1950
`IVGVR42G`	Cole 1951
`G48RM8U8`	Bakry & Émery 1985
`V3N3RB5Q`	Bakry, Gentil & Ledoux 2014
`WS8XADAC`	Amari & Nagaoka 2000
`KAWC59HP`	Stam 1959
`M6CM3UFR`	Nelson 1973
`8295BZ3X`	Tychonoff 1935 (title needs manual fix)
`M28TSE72`	Jordan, Kinderlehrer & Otto 1998
`3FTMT2VT`	Kalman 1960
`F9AUXVXQ`	Jaynes 1957
`K8W22X8C`	Morris 1982
`6UHTNR3Z`	Letac & Mora 1990
`W64CQZ7P`	Dembo, Cover & Thomas 1991
`CDQ22SDU`	Otto 2001
`M84B6IF2`	Ambrosio, Gigli & Savaré 2008
`QBV3P988`	Widder 1944
`7PP35FDA`	Blachman 1965
`EE247NXX`	Amari 1985

Not yet created in Zotero (deliberately deferred to T2 curation pass): Shannon 1948, Cover & Thomas 2006, Amari 2016, Barndorff-Nielsen 1978, Morris 1983, Barron 1986, Johnson 2004.

Changelog

2026-04-17 — initial ledger (sourcer: task-20260416-912a); 20 primary sources across the 10 topics, 20 Zotero items created (T0 + T1).