Gel'fand, Vilenkin — *Generalized Functions* Vols. 1–6 (Fast Track 2.04–2.09) — Combined Audit + Gap Plan

Books: I. M. Gel'fand and N. Ya. Vilenkin (with co-authors I. M. Shilov for Vols. 1–3 and M. I. Graev for Vol. 5), Generalized Functions, six-volume series. Russian originals: Fizmatgiz, Moscow 1958–1962. English translation: Academic Press, New York 1964–1968. AMS Chelsea reprint set (complete six-volume box) released 2015 (AMS Chelsea Publishing, ISBN 978-1-4704-2658-3 series, individual ISBNs per volume).

Canonical volume titles and the FT-booklist mapping (docs/catalogs/FASTTRACK_BOOKLIST.md lines 64–69):

FT entry	Vol	Title	Year (E)	Year (R)
2.04	1	Properties and Operations	1964	1958
2.05	2	Spaces of Fundamental and Generalized Functions	1968	1958
2.06	3	Theory of Differential Equations	1967	1958
2.07	4	Applications of Harmonic Analysis	1964	1961
2.08	5	Integral Geometry and Representation Theory	1966	1962
2.09	6	Representation Theory and Automorphic Forms	1966	1962

Vol. 1 lists Shilov as co-author (it is "Gel'fand-Shilov" in the literature); Vols. 4–6 list Vilenkin and Graev — the series is universally cited as "Gel'fand-Shilov" for the foundational half (Vols. 1–3) and "Gel'fand-Vilenkin" / "Gel'fand-Graev-Vilenkin" for the harmonic-analysis and representation-theory half (Vols. 4–6). The FT booklist collapses the authorship to "Gel'fand, Vilenkin" for shelving.

Fast Track entries: 2.04–2.09, the SOURCE / BUY entries for the entire distribution-theory + harmonic-analysis-on-groups block in Tier 2. Per the AGENTIC_EXECUTION_PLAN.md Wave 6 recommendation, these six volumes are audited as a single combined plan rather than six per-volume stubs: the series is one editorial programme, the volume splits are arbitrary from a curriculum standpoint (Vol. 1 ends mid-thread; Vol. 4 begins mid-thread), and the cross-references between volumes are dense enough that a per-volume audit would duplicate ~40% of its content. Audit completeness will be reported per-volume in §7.

Purpose: lightweight audit-and-gap pass (P1-lite + P2 + P3-lite of the orchestration protocol). Output is a concrete punch-list of new units to write so that the Gel'fand series is covered to the equivalence threshold (≥95% effective coverage of theorems, key examples, exercise pack, notation, sequencing, intuition, applications — see docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §3.4).

Audit completeness: reduced for all six volumes. No local PDFs found in reference/fasttrack-texts/, reference/textbooks-extra/, or reference/book-collection/. AMS Chelsea reprints (2015) are copyright-active and not redistributed on Archive.org or author pages. The older Academic Press scans (1964–1968) are intermittently available on Archive.org but most copies are restricted-access lending. Two WebFetch probes (Archive.org search; Wikipedia Israel Gelfand / Generalized function articles) confirmed volume titles and series structure but no free PDF surface. The §1, §2, §3 inventories below are built from the publicly visible TOC, the canonical peer-reference network (Schwartz, Hörmander, Reed-Simon, Stein-Shakarchi, Helgason, Bump, Folland), and the citation graph in already-audited Codex plans (Chatterjee 2.03, Woit 2.02, Lawson-Michelsohn 1.07). P5 verification on this audit cannot mark any volume equivalence-covered; only equivalence-partial until each is re-audited from a full source. Re-audit upgrades queued in docs/catalogs/NEED_TO_SOURCE.md (entry already exists at lines 97 of that file, "2.04–2.09 ... SOURCE (6 vols); AMS Chelsea; older vols may be on Archive.org").

This is the largest single audit in the FT corpus at the time of writing (six volumes, ~2500 pages combined). The plan is structured so that each volume's punch-list is independently actionable; the production agent should expect to fan out volume-by-volume even though the audit is combined.

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§1 What the Gel'fand series is for

The Gel'fand-Shilov-Vilenkin Generalized Functions series is the Soviet-school synthesis of distribution theory and the harmonic analysis of Lie groups, written between 1958 and 1962 (Russian) and translated 1964–1968 (English) as a six-volume programme that begins from L. Schwartz's 1944–1950 axiomatisation of distributions and ends at the unitary representation theory of $SL(2,\mathbb{R})$, integral geometry, adelic representations, and the spectral theory underlying what later became the Langlands programme. The series is to distribution-theoretic analysis what Bourbaki Topologie générale is to point-set topology: an editorial project that defines and fixes a vocabulary which becomes the lingua franca of the next half-century.

Distinctive contributions, in roughly the order the series develops them:

1. A Soviet-school presentation of distribution theory (Vols. 1–2). Where Schwartz [Sch] Théorie des distributions (1950–1951) develops distributions abstractly as continuous linear functionals on $\mathcal{D}(\Omega )=C_c^{\infty }(\Omega )$ with the inductive-limit topology, Gel'fand-Shilov organise the theory around explicit families of test-function spaces $K\{M_p\}$ (later called Gel'fand-Shilov spaces ${\mathcal{S}}_{\alpha }^{\beta }$), each tuned to a specific growth rate of test functions. This gives a finer-grained theory than Schwartz's: distributions of arbitrary order, ultra- distributions, hyperfunctions all arise as duals of $K\{M_p\}$ for suitable $M_p$. The Schwartz tempered distributions ${\mathcal{S}}^{\prime }({\mathbb{R}}^n)$ are one example among many.
2. Operations on distributions (Vol. 1). Differentiation, multiplication by smooth functions, change of variables, direct product, convolution, support, singular support, restriction and extension. The Vol. 1 treatment is the canonical operations table that Hörmander [Hör, Vol. 1, Ch. 2–3] later refines with the wave-front-set microlocal calculus.
3. Topological vector spaces and locally convex spaces (Vol. 2). Bornologies, inductive and projective limits, nuclear spaces (in the sense of Grothendieck, parallel to Produits tensoriels topologiques 1955). Vol. 2 ships the TVS prerequisites that Schwartz takes for granted and that Reed-Simon [RS, Vol. 1, App. to §V.3] treats only in passing.
4. Fundamental solutions and the theory of linear differential equations (Vol. 3). Existence of fundamental solutions for hypoelliptic operators (Malgrange-Ehrenpreis, 1954–1955), the Cauchy problem in distributional form, propagation of singularities (foreshadowing the wave-front-set calculus of Hörmander 1971), regularity theorems. Vol. 3 is the "what distributions are for in PDE" volume.
5. Generalised stochastic processes and the Bochner-Minlos theorem (Vol. 4). Characteristic functionals on nuclear spaces; the Minlos theorem extending Bochner's theorem from ${\mathbb{R}}^n$ to nuclear ${\mathcal{S}}^{\prime }$, foundational for constructive QFT (Glimm-Jaffe, Wightman reconstruction). Also: spectral theory of operators in rigged Hilbert spaces (the "Gel'fand triple" $\Phi \subset \mathcal{H}\subset {\Phi }^{\prime }$), the basis for Dirac's bra-ket formalism made rigorous.
6. Integral geometry: Radon transform, John transform, horocycle transform (Vol. 5). Inversion formulas, Plancherel formulas, the Gel'fand programme of "integral geometry" — recovering a function on a homogeneous space from its integrals over a family of submanifolds. The Vol. 5 treatment is the canonical foundation that Helgason [Hel] later extends to general symmetric spaces.
7. Representation theory of $SL(2,\mathbb{R})$ and $SL(2,\mathbb{C})$ (Vol. 5). Principal, complementary, and discrete series; Plancherel formula. The Vol. 5 treatment is the originating exposition that every subsequent rep-theory text (Knapp, Vogan, Wallach, Bump) follows or reformulates.
8. Automorphic forms, the adelic viewpoint, $L$-functions, and the seed of the Langlands programme (Vol. 6). Vol. 6 is the volume that reframes Tate's 1950 thesis as a representation-theoretic statement over $G{L}_1$ and points toward the $G{L}_2$ generalisation. The Vol. 6 exposition predates Jacquet-Langlands (1970) by roughly a decade and is the canonical Soviet-school precursor.

The series is not a first introduction to functional analysis. It assumes the reader has had a measure-theory course, basic functional analysis (Banach and Hilbert spaces, the closed graph theorem), and is willing to work with locally convex spaces from the start. Vol. 2's TVS material is a serviceable refresher but not a substitute for, e.g., Rudin Functional Analysis or Schaefer Topological Vector Spaces.

Peer references for the §1 framing (cited per the rubric of §1 of sibling FT plans; ≥3 sources required):

• [Sch] L. Schwartz, Théorie des distributions, 2 vols., Hermann Paris, 1950–1951 (revised 1966). The Western-school synthesis; Schwartz's $\mathcal{D}, \mathcal{D}', \mathcal{S}, \mathcal{S}', \mathcal{E}, \mathcal{E}'$ notation is the dual notation to Gel'fand-Shilov's $K\{M_p\}$ programme and Vols. 1–2 of Gel'fand are best read as the Soviet-school answer to Schwartz.
• [Hör] L. Hörmander, The Analysis of Linear Partial Differential Operators, Vol. 1: Distribution Theory and Fourier Analysis, Grundlehren 256, Springer, 1983 (2nd ed. 1990). Hörmander's Vol. 1 is the modern canonical distribution-theory text, replacing Schwartz and Gel'fand for most pedagogical purposes; the wave-front-set calculus it introduces in Ch. 8 is the microlocal refinement Gel'fand Vol. 3 almost anticipates.
• [RS] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. 1: Functional Analysis (1972, rev. 1980) and Vol. 2: Fourier Analysis, Self-Adjointness (1975). Reed-Simon Vol. 1 §V (Locally Convex Spaces) and Vol. 2 §IX (the Fourier transform, tempered distributions, Sobolev spaces, Bochner's theorem) is the mathematical- physicist's compressed version of Gel'fand Vols. 1–4. Reed-Simon Vol. 1 App. V.3 explicitly cites Gel'fand-Vilenkin Vol. 4 for the Minlos theorem and the construction of Gaussian measures on nuclear duals.
• [SS] E. Stein and R. Shakarchi, Functional Analysis: Introduction to Further Topics in Analysis, Princeton Lectures in Analysis IV, Princeton University Press, 2011. Ch. 3 (Distributions) and Ch. 4 (Generalised Functions) is the modern undergraduate-master's-bridge treatment; the historical notes there credit Gel'fand-Shilov for the theory of $K\{M_p\}$ test-function spaces.
• [Hel] S. Helgason, Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators, and Spherical Functions, Academic Press 1984 (reprinted AMS 2000). Helgason Ch. 1 and the historical notes credit Gel'fand-Graev-Vilenkin Vol. 5 as the originating source for the modern integral-geometry programme on symmetric spaces.
• [Bu] D. Bump, Automorphic Forms and Representations, Cambridge Studies in Advanced Math. 55, 1997. The historical notes there credit Gel'fand-Graev-Piatetski-Shapiro (Vol. 6) for the adelic / representation-theoretic reformulation of automorphic forms.

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§2 Coverage table (Codex vs Gel'fand series, by volume)

Cross-referenced against the current Codex corpus. The home chapters are 02-analysis/11-functional-analysis/ for distribution theory (Vols. 1–4) and 07-representation-theory/ for rep theory + automorphic forms (Vols. 5–6); a new sub-chapter 02-analysis/13-distributions/ (or 02-analysis/14-harmonic-analysis/) is implied by the punch-list below and should be created as part of priority-1 production.

✓ = covered, △ = partial / different framing, ✗ = not covered.

Vol. 1 — Properties and Operations (FT 2.04)

Gel'fand Vol. 1 topic	Codex unit(s)	Status	Note
Definition of generalised function as continuous linear functional on $K=\mathcal{D}(\Omega )$	—	✗	Gap (foundational).
Order of a distribution, support, singular support	—	✗	Gap.
Differentiation of distributions	—	✗	Gap.
Multiplication by smooth functions	—	✗	Gap.
Change of variables; pull-back, push-forward of distributions	—	✗	Gap.
Direct (tensor) product of distributions	—	✗	Gap.
Convolution of distributions	—	✗	Gap.
Fourier transform on ${\mathcal{S}}^{\prime }({\mathbb{R}}^n)$ (cross-ref Vol. 4)	—	✗	Gap; loaded by Chatterjee 2.03 punch-list 02.XX.YY (Schwartz / tempered distributions).
The delta function $\delta$, its derivatives, and the fundamental solutions table	—	✗	Gap. Already silently invoked in 05.05.04-hamilton-jacobi.md, 12.03.01-bosonic-fock-space-and-second-quantisation.md (the $\delta (p-p^{\prime })$ normalisation), 06.04.05-dbar-hilbert-pde.md (the $\bar{\partial }$ Green's function).
Principal-value, finite-part regularisations (Hadamard, Riesz)	—	✗	Gap.

Vol. 1 aggregate coverage estimate: ~0%. The gap is total. Several existing Codex units (listed above) silently invoke Vol. 1 content without a unit to anchor it.

Vol. 2 — Spaces of Fundamental and Generalized Functions (FT 2.05)

Gel'fand Vol. 2 topic	Codex unit(s)	Status	Note
Locally convex topological vector spaces	—	✗	Gap. 02.11.04-banach-spaces.md covers Banach only.
Seminorms, gauges, Minkowski functionals	—	✗	Gap.
Inductive and projective limits of TVS	—	✗	Gap.
Nuclear spaces (Grothendieck)	—	✗	Gap. Reed-Simon Vol. 1 App. to V.3 anchor.
Spaces $\mathcal{D},\mathcal{E},\mathcal{S}$ and their duals	—	✗	Gap.
Gel'fand-Shilov spaces ${\mathcal{S}}_{\alpha }^{\beta }$, $K\{M_p\}$	—	✗	Gap. Soviet-school distinctive.
Schwartz kernel theorem	—	✗	Gap. Foundational for the operator-valued-distribution framing in Chatterjee 2.03 / Woit 2.02.
Tensor products of locally convex spaces; topological tensor products	—	✗	Gap.

Vol. 2 aggregate coverage estimate: ~0%.

Vol. 3 — Theory of Differential Equations (FT 2.06)

Gel'fand Vol. 3 topic	Codex unit(s)	Status	Note
Linear PDE with constant coefficients	—	✗	Gap. Codex has no PDE chapter; 02-analysis/12-ode/ is ODE only.
Fundamental solutions (Malgrange-Ehrenpreis theorem)	—	✗	Gap.
Hypoellipticity	—	✗	Gap. Loaded by 03.09.22-pseudodifferential.md.
Cauchy problem for linear PDE in distributional form	—	✗	Gap.
Sobolev spaces $H^s({\mathbb{R}}^n)$, $H^s(\Omega )$	—	✗	Gap. Silently invoked in 03.09-spin-geometry/ for the index theorem (Sobolev embeddings, elliptic regularity).
Elliptic regularity	△	△	Stated as a fact in 03.09.22-pseudodifferential.md; no foundational unit.
Wave-front set (Hörmander 1971; not in Gel'fand Vol. 3, but the immediate successor microlocal-analysis theory the Codex should pointer-link)	—	✗	Gap (Master-tier pointer).

Vol. 3 aggregate coverage estimate: ~0%. The Codex's lack of a PDE chapter is the single largest gap touched by this audit; Vol. 3 cannot ship until a PDE foundation is laid.

Vol. 4 — Applications of Harmonic Analysis (FT 2.07)

Gel'fand Vol. 4 topic	Codex unit(s)	Status	Note
Fourier transform on $\mathcal{S},{\mathcal{S}}^{\prime },L^1,L^2$	—	✗	Gap. No Fourier-analysis chapter exists in the Codex.
Plancherel theorem on ${\mathbb{R}}^n$	—	✗	Gap.
Paley-Wiener theorem (Schwartz version for compactly supported distributions)	—	✗	Gap.
Bochner theorem	—	✗	Gap.
Minlos theorem (Bochner on nuclear spaces)	—	✗	Gap. Foundational for measure theory on ${\mathcal{S}}^{\prime }$.
Rigged Hilbert space (Gel'fand triple)	—	✗	Gap. Loaded by 12.02.01-hilbert-space-formalism.md (Dirac bra-ket rigorous foundation).
Generalised eigenfunctions; spectral theorem in rigged form	△	△	02.11.03-unbounded-self-adjoint.md covers the spectral theorem but not the rigged-Hilbert-space refinement.
Generalised stochastic processes; characteristic functional	—	✗	Gap. Foundational for constructive QFT (Glimm-Jaffe); silently invoked by Chatterjee 2.03 punch-list.

Vol. 4 aggregate coverage estimate: ~5% (only the bare spectral theorem is in place).

Vol. 5 — Integral Geometry and Representation Theory (FT 2.08)

Gel'fand Vol. 5 topic	Codex unit(s)	Status	Note
Radon transform on ${\mathbb{R}}^n$	—	✗	Gap.
John transform; X-ray transform	—	✗	Gap.
Funk transform on $S^2$	—	✗	Gap.
Inversion formulas; range characterisation	—	✗	Gap.
Plancherel for the Radon transform	—	✗	Gap.
Horocycle transform on hyperbolic plane	—	✗	Gap.
Unitary representations of $SL(2,\mathbb{R})$ — principal, complementary, discrete series	—	✗	Gap. 07-representation-theory/ covers compact and Lie-algebraic cases only; non-compact semisimple is absent.
Unitary representations of $SL(2,\mathbb{C})$	—	✗	Gap.
Plancherel formula for $SL(2,\mathbb{R})$ (Harish-Chandra)	—	✗	Gap.
Generalised matrix coefficients; spherical functions	—	✗	Gap. Helgason cross-reference.

Vol. 5 aggregate coverage estimate: ~0%. Codex 07 covers compact / finite-dim semisimple rep theory only; the entire infinite-dimensional unitary side is absent.

Vol. 6 — Representation Theory and Automorphic Forms (FT 2.09)

Gel'fand Vol. 6 topic	Codex unit(s)	Status	Note
Automorphic forms on the upper half-plane	—	✗	Gap. 06.06-jacobians/ touches modular curves implicitly; no $\Gamma$-invariant function theory.
Hecke theory; Hecke $L$-functions	—	✗	Gap.
$p$-adic numbers ${\mathbb{Q}}_p$	—	✗	Gap. No $p$-adic content in the Codex.
Adeles $\mathbb{A}$ and ideles $\mathbb{I}$ of $\mathbb{Q}$	—	✗	Gap.
Tate's thesis: $\zeta (s)$ as a $G{L}_1(\mathbb{A})$ integral	—	✗	Gap. Originator: Tate 1950 PhD thesis.
Adelic $G{L}_2$ automorphic representations	—	✗	Gap.
Whittaker functions and the Whittaker model	—	✗	Gap.
Pointer: the Langlands programme	—	✗	Gap (Master-tier pointer only — full Langlands is FT-out-of-scope).

Vol. 6 aggregate coverage estimate: ~0%.

Aggregate series coverage estimate: ~0–2% of the six-volume series has corresponding Codex units. This is the largest single coverage gap in the FT corpus and is unsurprising — the Codex's 02-analysis/ chapter currently has no measure theory, no Lebesgue integration beyond a casual mention, no Fourier analysis, no PDE, and no distribution theory. Closing this gap requires net-new sub-chapters within 02-analysis/ before Vol. 1 of Gel'fand can ship.

Existing Codex units that silently depend on Gel'fand-style distribution theory (load-bearing gap):

1. 12.03.01-bosonic-fock-space-and-second-quantisation.md — the $[a(p),a^{†}(p^{\prime })]={\delta }^3(p-p^{\prime })$ commutator and the smearing $a(f)=\int f(p)a(p)dp$ are operator-valued-distribution statements that need Vol. 1 (delta function) + Vol. 4 (rigged Hilbert space) for rigorous interpretation.
2. 02.11.03-unbounded-self-adjoint.md — the spectral theorem in the form needed for QM is the rigged-Hilbert-space spectral theorem of Vol. 4, not just the von Neumann statement.
3. 06.04.05-dbar-hilbert-pde.md — invokes the $\bar{\partial }$ fundamental solution (Cauchy kernel as a distribution); Vol. 3 anchor missing.
4. 03.09.22-pseudodifferential.md — pseudodifferential operators act on distributions; the entire framework needs Vol. 1–3 as substrate.
5. 03.09.06-fredholm-operators.md and the index theorem track — elliptic regularity / Sobolev embeddings are Vol. 3 content, currently unstated.
6. 12.02.01-hilbert-space-formalism.md — Dirac bra-ket made rigorous is the Gel'fand-triple statement of Vol. 4.
7. Chatterjee 2.03 punch-list item 13 (Schwartz space + tempered distributions) and item 12 (spectral theorem + Stone's theorem) are both Gel'fand-loaded — currently parked on the Sternberg Semi-Classical Analysis sub-list.
8. Woit 2.02 Ch. 11 row (Fourier analysis; distributions; Schwartz space) is explicitly flagged as a gap in woit-quantum-theory-groups-representations.md line 172 — same dependency.

These silent dependencies make the priority-1 Vol. 1–2 punch-list items a hard prerequisite for marking Chatterjee 2.03, Woit 2.02, and the QFT units in 12-quantum/ as equivalence-covered.

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§3 Gap punch-list (P3-lite — units to write, priority-ordered, per volume)

The punch-list is organised by volume block (Vol. 1–2 foundational, Vol. 3 PDE, Vol. 4 harmonic analysis, Vol. 5 integral geometry + rep theory, Vol. 6 automorphic forms). Within each volume block, priorities P1 (load-bearing) → P4 (Master-tier deepening) follow the convention of sibling FT plans.

Priority 0 — chapter-creation blockers (must ship before any Vol. 1 units):

• Create 02-analysis/13-distributions/ sub-chapter directory. Home for all Vol. 1–4 distribution-theory units (~12 units).
• Create 02-analysis/14-pde/ sub-chapter directory. Home for Vol. 3 PDE units (~6 units) and the existing pseudodifferential / elliptic regularity content currently orphaned in 03.09-spin-geometry/.
• Create 02-analysis/15-fourier-analysis/ sub-chapter directory. Home for Vol. 4 Fourier-analysis units (~6 units).
• Create 02-analysis/06-measure-theory/ sub-chapter directory. Home for the Lebesgue / Borel / abstract-measure prerequisites the Codex currently lacks; this is also a strict prereq for any serious Vol. 1–6 unit. Approximately 8–10 prerequisite units (sigma-algebra, Lebesgue measure, Lebesgue integral, monotone / dominated convergence, Fubini, Radon-Nikodym, $L^p$ spaces, weak-* convergence). This is the single biggest production cost in the audit.

Vol. 1 — Properties and Operations (FT 2.04) — punch-list

P1 (load-bearing):

1. 02.13.01 Test function $\phi \in \mathcal{D}(\Omega )=C_c^{\infty }(\Omega )$; distribution as continuous linear functional; the spaces $\mathcal{D},{\mathcal{D}}^{\prime },\mathcal{E},{\mathcal{E}}^{\prime }$.
2. 02.13.02 Order of a distribution; support; singular support.
3. 02.13.03 The Dirac delta $\delta$, its derivatives, and the worked examples (point masses, surface measures on submanifolds, principal-value distributions).
4. 02.13.04 Differentiation of distributions; the derivative of $\log |x|$, the Heaviside step, $\text{p.v.}(1/x)$.
5. 02.13.05 Multiplication by smooth functions; the Schwartz impossibility theorem for multiplication of distributions.
6. 02.13.06 Convolution of distributions; convolution with $\delta$ and its derivatives; the associativity issue.

P2 (operations on distributions):

7. 02.13.07 Pull-back / change of variables; the chain rule for $\delta (f(x))$ when $f$ has simple zeros (the "delta-function composition formula").
8. 02.13.08 Direct (tensor) product of distributions; the kernel theorem statement (full proof in Vol. 2 unit).
9. 02.13.09 Hadamard / Riesz finite-part regularisation; ${\chi }_{+}^{\lambda }$ and the analytic continuation in $\lambda$. Master-tier.

Vol. 1 total: 9 units, ~3 hours each = 27 hours.

Vol. 2 — Spaces of Fundamental and Generalized Functions (FT 2.05) — punch-list

P1 (TVS prerequisites):

10. 02.11.10 Locally convex topological vector space; seminorm families; Minkowski functionals.
11. 02.11.11 Fréchet space; inductive-limit topology; $\mathcal{D}(\Omega )$ as LF space.
12. 02.13.10 The Schwartz space $\mathcal{S}({\mathbb{R}}^n)$ and the tempered distributions ${\mathcal{S}}^{\prime }({\mathbb{R}}^n)$. This is the item Chatterjee 2.03 and Woit 2.02 both reference as a missing prerequisite.

P2 (nuclear spaces and Schwartz kernel):

13. 02.11.12 Nuclear space (Grothendieck); statement of the kernel theorem.
14. 02.13.11 Schwartz kernel theorem: continuous bilinear maps $\mathcal{D}({\Omega }_1)\times \mathcal{D}({\Omega }_2)\to \mathbb{C}$ are integration against a distribution on ${\Omega }_1\times {\Omega }_2$. Loaded by the operator-valued-distribution framing of Chatterjee 2.03.

P3 (Gel'fand-Shilov spaces — Master-only):

15. 02.13.12 Gel'fand-Shilov spaces ${\mathcal{S}}_{\alpha }^{\beta }$, $K\{M_p\}$; ultradistributions as the Soviet-school refinement of Schwartz. Master-only pointer unit.

Vol. 2 total: 6 units, ~3 hours each = 18 hours.

Vol. 3 — Theory of Differential Equations (FT 2.06) — punch-list

P0 (foundation — depends on a PDE chapter not yet existing):

The Codex has no PDE chapter. Vol. 3 units depend on a baseline of 4–5 PDE units that should ship first:

16. 02.14.01 Linear partial differential operator; principal symbol; classification (elliptic, parabolic, hyperbolic).
17. 02.14.02 Sobolev space $H^s({\mathbb{R}}^n)$; Sobolev embedding; Rellich-Kondrachov compactness.
18. 02.14.03 Weak solutions; energy estimates.

P1 (Vol. 3 proper):

19. 02.14.04 Fundamental solution of a linear PDE; the Malgrange-Ehrenpreis existence theorem (any non-zero constant-coefficient operator on ${\mathbb{R}}^n$ admits a fundamental solution).
20. 02.14.05 Hypoelliptic operators; the Hörmander characterisation (Master-tier).
21. 02.14.06 The Cauchy problem in distributional form; well-posedness for hyperbolic equations (wave equation as worked example).
22. 02.14.07 Elliptic regularity (interior); statement and use of the pseudodifferential calculus from 03.09.22. Promote 03.09.22-pseudodifferential.md to depend on 02.14.07 once shipped.

P2 (microlocal pointer):

23. 02.14.08 Wave-front set $\text{WF}(u)$ (Hörmander 1971); propagation of singularities. Master-only pointer unit; full theory is FT-out-of-scope (Hörmander Vols. 1, 4).

Vol. 3 total: 8 units, ~3.5 hours each = 28 hours. Higher per-unit estimate because PDE content requires careful coordinate work and worked examples.

Vol. 4 — Applications of Harmonic Analysis (FT 2.07) — punch-list

P1 (Fourier-analysis foundation):

24. 02.15.01 Fourier transform on $L^1({\mathbb{R}}^n)$; Riemann-Lebesgue; inversion under hypotheses.
25. 02.15.02 Fourier transform on $\mathcal{S}({\mathbb{R}}^n)$ and the isomorphism $\mathcal{F}:\mathcal{S}\to \mathcal{S}$ (Plancherel's theorem in Schwartz form).
26. 02.15.03 Fourier transform on ${\mathcal{S}}^{\prime }({\mathbb{R}}^n)$; distributional Fourier transform of $\delta$, of $1$, of $x^n$, of $\text{p.v.}(1/x)$ (Sokhotski-Plemelj).
27. 02.15.04 Fourier transform on $L^2$; Plancherel's theorem.
28. 02.15.05 Paley-Wiener theorem (Schwartz version): characterisation of $\mathcal{F}({\mathcal{E}}^{\prime })$ as entire functions of exponential type.

P2 (Vol. 4 distinctives — Bochner-Minlos, rigged Hilbert):

29. 02.15.06 Bochner theorem: a continuous positive-definite function on ${\mathbb{R}}^n$ is the Fourier transform of a finite positive measure.
30. 02.13.13 Rigged Hilbert space / Gel'fand triple $\Phi \subset \mathcal{H}\subset {\Phi }^{\prime }$; the generalised-eigenfunction spectral theorem. Promote 02.11.03-unbounded-self-adjoint.md to depend on this once shipped; this is the rigorous foundation of Dirac bra-ket.
31. 02.13.14 Minlos theorem (Bochner on nuclear spaces); construction of Gaussian measure on ${\mathcal{S}}^{\prime }$. Master-tier; loaded by constructive QFT (Glimm-Jaffe, Wightman reconstruction).

Vol. 4 total: 8 units, ~3.5 hours each = 28 hours.

Vol. 5 — Integral Geometry and Representation Theory (FT 2.08) — punch-list

P1 (integral geometry):

32. 02.15.07 Radon transform on ${\mathbb{R}}^n$; definition, inversion formula (odd $n$ vs even $n$), Plancherel.
33. 02.15.08 Range characterisation (Helgason-Ludwig); the moment conditions.
34. 02.15.09 X-ray / John transform; the John ultrahyperbolic equation.
35. 02.15.10 Funk transform on $S^2$; integral geometry on the sphere. Master-tier.

P1 (non-compact rep theory — load-bearing):

36. 07.08.01 Unitary representation of a locally compact group; Haar measure; the regular representation.
37. 07.08.02 Principal series of $SL(2,\mathbb{R})$; induction from parabolic.
38. 07.08.03 Discrete series of $SL(2,\mathbb{R})$; holomorphic discrete series and the upper half-plane realisation.
39. 07.08.04 Complementary series of $SL(2,\mathbb{R})$.
40. 07.08.05 Plancherel formula for $SL(2,\mathbb{R})$ (Harish-Chandra). Master-tier.
41. 07.08.06 Unitary representations of $SL(2,\mathbb{C})$; pointer to the general semisimple case (Master).

P2 (horocycle / spherical functions):

42. 02.15.11 Horocycle transform on the hyperbolic plane / upper half-plane; the duality with the principal series.
43. 07.08.07 Spherical function on $SL(2,\mathbb{R})/SO(2)$; the Harish-Chandra $c$-function. Master-tier.

Vol. 5 total: 12 units, ~4 hours each = 48 hours. Higher per-unit estimate because both the integral-geometry and the non-compact-rep-theory side require new sub-chapters with no Codex prior; the Master tier needs careful Lie-theoretic machinery.

Vol. 6 — Representation Theory and Automorphic Forms (FT 2.09) — punch-list

P0 (foundational $p$-adic / adelic prerequisites — no current Codex home):

44. 02.16.01 $p$-adic absolute value; $p$-adic numbers ${\mathbb{Q}}_p$; completion of $\mathbb{Q}$ at $p$.
45. 02.16.02 Local-global / Ostrowski's theorem; the places of $\mathbb{Q}$.
46. 02.16.03 Adeles ${\mathbb{A}}_{\mathbb{Q}}$ and ideles ${\mathbb{I}}_{\mathbb{Q}}$; the restricted direct product topology; strong approximation.

P1 (Vol. 6 proper):

47. 07.09.01 Automorphic form on $\Gamma \backslash \mathbb{H}$; Maass forms vs holomorphic modular forms; Hecke operators.
48. 07.09.02 Tate's thesis: $\zeta (s)$ as a $G{L}_1(\mathbb{A})$ integral. Originator: J. Tate, PhD thesis 1950, "Fourier analysis in number fields and Hecke's zeta-functions," Princeton. Master-tier.
49. 07.09.03 Automorphic representation of $G{L}_2(\mathbb{A})$; cuspidal vs Eisenstein.
50. 07.09.04 Whittaker function / Whittaker model; uniqueness (multiplicity-one).

P2 (Langlands pointer — Master-only):

51. 07.09.05 Pointer to the Langlands programme: automorphic representations of $G{L}_n(\mathbb{A})$, the principle of functoriality, the connection to Galois representations. Survey unit only; full Langlands is FT-out-of-scope.

Vol. 6 total: 8 units, ~4 hours each = 32 hours.

Punch-list aggregate counts

By volume:

Volume	Units	Hours
Vol. 1 (2.04)	9	~27
Vol. 2 (2.05)	6	~18
Vol. 3 (2.06)	8	~28
Vol. 4 (2.07)	8	~28
Vol. 5 (2.08)	12	~48
Vol. 6 (2.09)	8	~32
Chapter-creation P0 (measure theory + chapter dirs)	~10	~30
Total	~61 units	~211 hours

By priority across the series:

Priority	Unit count
P0 chapter-creation + measure-theory prereqs	~10
P1 load-bearing	~32
P2 operations / deepenings	~12
P3–P4 Master / pointer	~7
Total	~61

This is roughly 3× the size of the next-largest combined FT audit (Hörmander Vols. 1–4 if/when audited, or Reed-Simon Vols. 1–4) and ~6× a typical single-book FT audit. The production roll-out should fan out into three parallel mini-tracks (distributions / PDE / Fourier; integral geometry

• rep theory; automorphic forms) once the P0 chapter scaffolding is in place.

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§4 Implementation sketch (P3 → P4) and originator-prose targets

Hour estimates per volume: see the table in §3 (Vol. 5 is the largest at ~48 hours; the P0 chapter-creation cost is ~30 hours dominated by the measure-theory mini-chapter).

Total realistic production: ~211 hours of focused unit production, i.e. 5–7 weeks of dedicated work at the corpus' historical 30–35 hour/ week sustained pace, or 3–4 months at a sustainable 15 hour/week side-cycle pace. Per AGENTIC_EXECUTION_PLAN.md §2 (concurrency budget), this is too large for a single Wave; it should be split into three Waves:

• Wave 6 (Vols. 1–2 + measure-theory P0): ~85 hours (~3 weeks focused). Unblocks Chatterjee 2.03 and Woit 2.02 equivalence.
• Wave 7 (Vols. 3–4): ~56 hours (~2 weeks focused). Unblocks the PDE / Fourier prerequisites silently invoked across the QFT and spin-geometry tracks.
• Wave 8 (Vols. 5–6): ~80 hours (~3 weeks focused). New representation-theory and automorphic-forms sub-chapters; this Wave also unblocks the Master-tier connections to Langlands and to Helgason FT 1.06.

Originator-prose targets per docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §10. Originator citations (the historical sources whose statements should be quoted or near-quoted in the §Originator section of the relevant units):

• S. L. Sobolev (1936), "Méthode nouvelle à résoudre le problème de Cauchy pour les équations linéaires hyperboliques normales," Mat. Sb. 1, 39–72. Originator of functionals as generalised solutions (the distributional pre-history). Cite in 02.13.01 and 02.13.04.
• L. Schwartz (1944–1950), "Sur certaines familles non fondamentales de fonctions continues," Bull. Soc. Math. France 72 (1944), and the two-volume Théorie des distributions (Hermann 1950–1951). Originator of the formal theory: the spaces $\mathcal{D}, \mathcal{D}', \mathcal{S}, \mathcal{S}'$ and the systematic operations calculus. Cite in 02.13.01, 02.13.10, 02.15.03.
• I. M. Gel'fand and G. E. Shilov (1953–1958), Generalized Functions Vols. 1–3 (Russian original Fizmatgiz, English Academic Press 1964). The Soviet-school synthesis; introduces the $K\{M_p\}$ test-function-space programme. Cite in 02.13.01, 02.13.10, 02.13.12.
• L. Hörmander (1955, 1971), "On the theory of general partial differential operators," Acta Math. 94, and "Fourier integral operators I," Acta Math. 127. Originator of the modern PDE framework (Hörmander's theorem on hypoellipticity, the wave-front set). Cite in 02.14.05 and 02.14.08.
• J. Tate (1950), "Fourier analysis in number fields and Hecke's zeta-functions," PhD thesis Princeton (published 1967 in Cassels-Fröhlich, Algebraic Number Theory). Originator of the adelic / representation-theoretic formulation of $\zeta (s)$. Cite in 07.09.02.
• Harish-Chandra (1950s–1960s), the long sequence of papers on the representation theory of real semisimple Lie groups; the Plancherel formula for $SL(2,\mathbb{R})$ in Trans. AMS 1952. Cite in 07.08.05 and 07.08.07.

Notation crosswalk. Gel'fand-Shilov use $K,K^{\prime }$ where Schwartz uses $\mathcal{D},{\mathcal{D}}^{\prime }$ and the modern literature (Hörmander, Reed-Simon, Stein-Shakarchi) uses $\mathcal{D},{\mathcal{D}}^{\prime }$. The Codex should follow the modern convention ($\mathcal{D},{\mathcal{D}}^{\prime }$, $\mathcal{S},{\mathcal{S}}^{\prime }$, $\mathcal{E},{\mathcal{E}}^{\prime }$) and note the Gel'fand-Shilov $K\{M_p\}$ notation only in 02.13.12 (the Gel'fand-Shilov-spaces pointer unit). Record this decision in a §Notation paragraph of 02.13.01.

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§5 What this plan does NOT cover (non-goals)

• The Tate-Wiles / modularity-theorem material. Vol. 6 points toward Langlands; the modularity theorem (Wiles 1995, Taylor-Wiles 1995, Breuil-Conrad-Diamond-Taylor 2001) is FT-out-of-scope. Pointer in 07.09.05 only.
• Full Langlands programme. Local Langlands for $G{L}_n$ (Harris-Taylor / Henniart for $p$-adic; Langlands for real Archimedean) and global functoriality conjectures are FT-out-of-scope. Pointer in 07.09.05 only.
• Gel'fand-Manin Algebra V (FT 3.02). The sibling Gel'fand series-entry on derived categories, D-modules, and Riemann-Hilbert is FT 3.02 and gets its own audit plan. This plan does not cover D-modules even though they are the natural successor to the Vol. 3 distribution / PDE framework. (Riemann-Hilbert: the equivalence between regular holonomic D-modules and constructible sheaves is a later refinement of the Vol. 3 fundamental-solutions programme; FT 3.02 plan owns it.)
• Constructive / axiomatic QFT. Glimm-Jaffe construction of ${\varphi }_{2,3}^4$ from the Bochner-Minlos / characteristic-functional framework is a Vol. 4 application mentioned only in pointers; full constructive-QFT coverage is FT-out-of-scope. Chatterjee 2.03 plan touches this lightly.
• Hyperfunctions (Sato 1959–1960). A successor refinement to Gel'fand-Shilov $K\{M_p\}$; mentioned in 02.13.12 Master-tier only. Full Sato-Kashiwara hyperfunction theory FT-out-of-scope.
• Microlocal analysis / Fourier integral operators beyond pointer. Hörmander Vols. 3–4 are the canonical reference; FT-out-of-scope. 02.14.08 is a pointer only.
• Line-number-level inventory of every named theorem across the six volumes. A full P1 audit at line-number granularity is a multi-week job per volume and is queued in docs/catalogs/NEED_TO_SOURCE.md for re-audit once full PDFs are obtained.
• Exercise pack production. Gel'fand-Vilenkin volumes are notoriously thin on exercises (Soviet-school style: examples are integrated into the body). Exercise-pack production for these units is P4 priority and follows the priority-1 unit ship.

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§6 Acceptance criteria for FT equivalence (Gel'fand series)

Per docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §3.4, each volume is at equivalence-coverage when:

• The P0 chapter-creation + measure-theory prerequisite block has shipped (strict prereq for all six volumes).
• For Vols. 1–2 (FT 2.04, 2.05): the P1 units 02.13.01–02.13.06, 02.11.10–02.11.11, 02.13.10 have shipped. Vol. 2's P2 nuclear-space
  • Schwartz kernel block (02.11.12, 02.13.11) brings coverage to ~95%; Vol. 2 Master-only 02.13.12 brings it to ≥95%.
• For Vol. 3 (FT 2.06): the P0 PDE foundation (02.14.01–02.14.03) plus Vol. 3 P1 (02.14.04–02.14.07) brings coverage to ~90%; the wave-front-set pointer (02.14.08) brings it to ≥95% for FT purposes.
• For Vol. 4 (FT 2.07): the P1 Fourier block (02.15.01–02.15.05) plus the rigged-Hilbert / Minlos block (02.13.13–02.13.14) brings coverage to ~95%.
• For Vol. 5 (FT 2.08): the P1 integral-geometry block (02.15.07–02.15.10) plus the P1 non-compact-rep-theory block (07.08.01–07.08.06) brings coverage to ~85%; the horocycle / spherical-function block (02.15.11, 07.08.07) brings it to ≥95%.
• For Vol. 6 (FT 2.09): the P0 $p$-adic / adelic block (02.16.01–02.16.03) plus the P1 automorphic-forms block (07.09.01–07.09.04) brings coverage to ~85%; the Langlands pointer (07.09.05) brings it to ≥95%.
• Notation decisions are recorded (see §4).
• Pass-W weaving connects the new units (a) within the new 02-analysis/13-distributions/, 02-analysis/14-pde/, 02-analysis/15-fourier-analysis/, 02-analysis/16-adeles-p-adic/ sub-chapters, and (b) lateral-link back to silently dependent units listed in §2 (Chatterjee 2.03 punch-list items, Woit 2.02 Ch. 11 row, 12.03.01, 02.11.03, 06.04.05, 03.09.22, 03.09.06, 12.02.01).

P5 verification will run per-volume, not series-wide. Six independent verification runs, one per volume — this is the largest single audit in the FT corpus and the verification cost reflects that. Per-volume P5 sign-off allows partial-equivalence reporting in the FT roadmap as each volume's punch-list closes.

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§7 Sourcing

Local copies (sought, all paths checked):

• reference/fasttrack-texts/00-prereqs/ — no Gel'fand titles.
• reference/fasttrack-texts/01-fundamentals/ — no Gel'fand titles.
• reference/fasttrack-texts/02-quantum-stat/ — no Gel'fand titles.
• reference/textbooks-extra/ — checked, no Gel'fand titles.
• reference/book-collection/ — checked, no Gel'fand titles.
• General ~/Documents + ~/Downloads find — no PDFs match Gel'fand / Vilenkin / "Generalized Functions" (the only hit is a marketing thumbnail image in reference/fast-track/images/gelfand4-672x1024__9a16cf9de3.png).

Per-volume audit completeness:

Volume	FT	Local PDF?	Audit	Notes
Vol. 1	2.04	No	reduced	Russian + English on Archive.org intermittently (1964 Academic Press scan).
Vol. 2	2.05	No	reduced	1968 Academic Press scan partially on Archive.org.
Vol. 3	2.06	No	reduced	1967 Academic Press scan partially on Archive.org.
Vol. 4	2.07	No	reduced	1964 Academic Press scan partially on Archive.org.
Vol. 5	2.08	No	reduced	1966 Academic Press scan partially on Archive.org.
Vol. 6	2.09	No	reduced	Same; the Gel'fand-Graev-Piatetski-Shapiro edition.

All six volumes are marked reduced. P5 cannot mark any volume equivalence-covered from this plan alone; each volume is queued for re-audit in docs/catalogs/NEED_TO_SOURCE.md (existing entry at line 97 of that file covers all six).

Acquisition options for full audit:

• AMS Chelsea reprint set (2015). Complete six-volume box set, hardcover, ~$250 retail (AMS member discount available). Canonical reprint with original pagination and corrections; ISBN 978-1-4704-2658-3 for the set.
• Academic Press scans (1964–1968). Intermittently available on Archive.org under restricted-access lending; the older the volume, the more likely it has dropped into accessible scans. Worth a re-check at each Wave kickoff.
• Russian originals (Fizmatgiz 1958–1962). Public domain in Russia; available on Russian mathematical archives (e.g. mat.net.ua, eqworld.ipmnet.ru); language barrier but the notation is largely consistent.

License. AMS Chelsea reprints are copyright-active (AMS holds rights to the English translation; original Russian rights expired). Local PDF storage of AMS Chelsea is not licit; either purchase the print volumes for local audit reference or use restricted-access scans of the older Academic Press edition.

Recommendation: purchase the AMS Chelsea six-volume set (~$250) and add to reference/fasttrack-texts/02-quantum-stat/ as Gelfand-Vilenkin-GeneralizedFunctionsVol{1,2,3,4,5,6}.pdf (scanned from purchased prints) only for personal-audit fair-use purposes. The cost is justified by the audit scope: this is the largest single audit in the FT corpus and unblocks Chatterjee 2.03, Woit 2.02, the QFT / spin-geometry silent dependencies (§2), and the entire non-compact-rep-theory and automorphic-forms tracks of FT Tier 2.

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§8 Coordination notes (cross-plan)

This audit is the load-bearing prerequisite for several other FT plans, listed below for orchestrator-status tracking:

• Chatterjee 2.03 punch-list items 12 (spectral theorem + Stone's theorem) and 13 (Schwartz space + tempered distributions) are satisfied by Vols. 1–2 + Vol. 4 P1 units in this plan. Coordinate with plans/fasttrack/chatterjee-qft-lecture-notes.md.
• Woit 2.02 Ch. 11 row (Fourier analysis; distributions; Schwartz space) is satisfied by the same Vols. 1–2 + Vol. 4 P1 block. Coordinate with plans/fasttrack/woit-quantum-theory-groups-representations.md.
• Lawson-Michelsohn 1.07 spin-geometry / index-theorem track silently depends on Vol. 3 Sobolev / elliptic-regularity content (02.14.02, 02.14.07). Coordinate when those units enter production.
• Sternberg Semi-Classical Analysis (related-only free text, not a canonical FT entry) sub-list parks the Schwartz / tempered-distribution
  • spectral-theorem prereqs; this plan owns those prereqs. Sternberg sub-list should be updated to reference Vols. 1–2 + Vol. 4 of this plan.
• Helgason FT 1.06 (Differential Geometry, Lie Groups, and Symmetric Spaces) has a downstream relationship: Helgason's later text Groups and Geometric Analysis (1984) is the modern extension of Vol. 5's integral-geometry programme; the FT 1.06 plan should cross-link to Vols. 5 units of this plan.
• Gel'fand-Manin Algebra V (FT 3.02) sibling — explicit non-goal per §5. Both plans audit Gel'fand-named series but cover disjoint material (this plan: distributions, harmonic analysis, rep theory, automorphic forms; FT 3.02: derived categories, D-modules, Riemann-Hilbert).

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§9 Plan provenance

Plan stub generated 2026-05-18 as part of the AGENTIC_EXECUTION_PLAN.md Wave 6 audit pass, combined-audit recommendation. Audit agent: Claude Opus 4.7 (1M context). Reduced-audit basis: no local PDFs found across three reference-folder paths; WebFetch probes (Archive.org search, Wikipedia Generalized function, Wikipedia Israel Gelfand) returned volume-title confirmation only, no free PDF surface. Punch-list inventory built from the publicly visible volume TOC, the canonical peer-reference network (Schwartz, Hörmander, Reed-Simon, Stein-Shakarchi, Helgason, Bump), and the silent-dependency analysis of the current Codex corpus listed in §2. Re-audit queued upon acquisition of AMS Chelsea reprint set.