Status legend (used in every cell of every Status column).

Token	Meaning
PROVEN	A classification theorem with a completeness statement exists in the stated scope.
CLAIMED	An announced result whose supporting artifacts are not publicly checkable.
OPEN	No complete result; partial or parametric results noted in the row.
IMPOSSIBLE	A theorem proves that no classification of the stated kind can exist.

Citation convention: bracketed citations link to the Sources section below. Inferences are marked conjectural or open.

(a) Fundamental structural families and their defining invariants

Families I–X are the candidate organizing families of the founding document §4 (DEF-026–035). They are candidates, not a completed classification.

Family	Structure	Representative systems	Defining invariants	Status	Sources
I	Symplectic and Poisson structures	Classical mechanics	Symplectic form ω; locally: none (Darboux normal form); globally: topology of ω. Poisson: Weinstein splitting (symplectic × transverse), symplectic-leaf foliation, coadjoint orbits for KKS	PROVEN local; global classification topological	[6][7]
II	Hilbert spaces and operator algebras	Quantum mechanics	Dimension vector (finite-dim C*: A ≅ ⊕ M_{n_i}(C)); Murray–von Neumann type + Connes invariants (amenable factors); Kraus rank (channels)	PROVEN islands; general factors IMPOSSIBLE [8]; fusion cats OPEN [14]	[8][9][10][11][14]
III	Smooth manifolds and geometric structures	Relativity	Holonomy group (Lorentzian, Berger-list analogue); causal class (complete causal ladder); Thurston geometry type (3-manifolds)	PROVEN in stated regimes; dim-4 unrecognizable instances IMPOSSIBLE	[50][51][54][60]
IV	Principal bundles and connections	Gauge theories	[X, BG]; characteristic classes in H*(BG) (SW, Pontryagin, Euler, Chern, DD); anomaly classes H¹⊕H²(CE(P))	PROVEN (bundles; anomalies as obstructions)	[17][18][19]
V	Probability and measure-theoretic structures	Statistical physics	Fisher metric, unique up to constant (Chentsov, classical)	PROVEN unique classical; quantum parametric (Petz–Sudar)	[52]
VI	Topological and higher-categorical structures	TQFTs	Fully dualizable objects (cobordism hypothesis); Atiyah axioms	CLAIMED for extended TQFTs (Lurie proof sketch) [35]; functorial TQFT PROVEN tractable [36]	[34][35][36]
VII	Dynamical systems and operator semigroups	Dissipative and open systems	Generator data; Stinespring minimal dilation of CP maps	OPEN as a classification; normal form only	[10][11]
VIII	Tensor categories and compositional structures	Quantum information	Symmetric structure (load-bearing); conjugates, subobjects, End(1)=C (DR hypotheses); pivotal structure (fusion cats)	PROVEN islands (DR, Deligne); fusion cats OPEN	[12][13][14][16]
IX	Discrete combinatorial structures	Lattice and network models	Lattice data; tame/wild class	PROVEN regime-restricted theorems [30][31][32]; wild sectors IMPOSSIBLE to classify	[30][31][32][55][56][57][58]
X	Derived and homotopical structures	Higher gauge and derived field theories	Derived-tame vs derived-wild dichotomy	OPEN beyond the dichotomy	[55]

Cross-family correspondences. Classical–quantum connection via deformation quantization is conjectural (Founding Document §4). Gelfand/Stone/DR/Deligne links are proved — see region (c). The 2024–2026 supermap-representability line [40][41] and its possible unification with Gelfand/Stone/Naimark is open.

(b) Exceptional structures

The CFSG template suggests families + sporadics. The founding document warns: one should not assume that physical theories possess a finite collection of exceptional objects analogous to the 26 sporadic groups (Founding Document 03). The entries below are the proven exceptional mathematics the table must accommodate.

Exceptional object	Region	Defining data	Status	Sources
26 sporadic finite simple groups (27 counting Tits)	Family II/group theory	Order, Schur multiplier, Out(G), presentations, character tables with power maps (ATLAS companions)	PROVEN to exist and be exhaustive (CFSG); no uniform parameterization known	[1][2][3]
Exceptional Lie algebras G2, F4, E6, E7, E8	Family I–II boundary (symmetry)	Dynkin diagrams outside A–D series; Satake diagrams for their real forms	PROVEN (Killing–Cartan complete)	[4]
Real forms and compact/split forms	Family III (symmetry layer)	Order-≤2 automorphism classes; split and compact forms always exist	PROVEN	[4]
Amenable type III₀ and III₁ factors	Family II	III₀ = ergodic flows; III₁ completed by Haagerup; III_λ (0<λ≤1) continuous parameter	PROVEN within the amenable class	[8]
Hyperfinite (AFD) II₁ factor	Family II	Unique amenable II₁ factor	PROVEN; contrast: uncountably many non-amenable separable II₁ factors	[8]
Thurston's 8 model geometries	Family III	S³, E³, H³, S²×R, H²×R, Nil, Sol, SL₂(R)~	PROVEN (geometrization)	[60]
Filipkiewicz's dim-4 model geometries	Family III	18 + one countably infinite family	PROVEN to exist; only restricted 4-manifold classes decompose	[60]
Isocategorical group pairs (Etingof–Gelaki)	Family VIII	Non-isomorphic groups with equivalent plain tensor representation categories	PROVEN to exist — the symmetric layer is not recoverable without them	[12]
Extra Z₂ factor in higher-form DHR (Casini–Magan)	Families IV, VIII	Sector not realizable by Lagrangian gauge models	OPEN [single source]	[20]

(c) Proven reconstruction relationships

Objects determined by probes — the pattern CFSG-class classification rarely achieves but reconstruction theorems routinely do.

Reconstruction theorem	Recovers	From	Scope limits	Status	Sources
Gelfand–Naimark	A concrete representation of a C*-algebra in B(H)	Abstract C*-algebra	Representation theorem, not a classification; only the commutative case is fully classified	PROVEN	[9][42]
Gelfand duality	Compact Hausdorff space	Commutative C*-algebra	Commutative side only	PROVEN	[43]
Stone duality	Spectrum (zero-dimensional compact space)	Boolean algebra	Boolean side only	PROVEN	[44]
Doplicher–Roberts reconstruction	Compact supergroup (unique up to isomorphism)	Symmetric tensor *-category with conjugates, direct sums, subobjects, End(1)=C	Requires the symmetric structure; DHR selection criteria; Haag duality in the QFT reading	PROVEN under stated hypotheses	[12][15][16]
Deligne's theorem	Supergroup representation category	Char-0 symmetric tensor category of subexponential growth	Hypotheses essential	PROVEN	[13]
Stinespring dilation	Minimal dilation (unique up to unitary)	Completely positive map	Normal form; not an isomorphism classification of channels	PROVEN	[10][11]
Classifying-space construction	Principal G-bundle from a map X → BG	Map to classifying space	Homotopy-classified; characteristic classes are pullbacks	PROVEN	[17][18]
Supermap representability	Concrete representation of categorical supermaps	Theory with suitable channel–state duality (2024–2026)	Settings with channel–state duality; unification with Gelfand/Stone/Naimark open	PROVEN in stated settings	[40][41]

(d) Established equivalences and dualities

Standing warning (Etingof–Gelaki). Tensor-category equivalence without symmetric structure provably fails to recover the group: non-isomorphic finite groups can carry equivalent plain tensor representation categories [12]. Any equivalence, duality, or classification in the series that forgets the monoidal/symmetric layer is false (DEF-018). The table marks this row IMPOSSIBLE as a recovery principle.

Equivalence / duality	Identifies	Preserves	Status	Sources
Tensor-cat equivalence without symmetric structure → group recovery	—	nothing usable	IMPOSSIBLE (Etingof–Gelaki isocategorical groups)	[12]
DR equivalence: symmetric tensor *-cat ≃ Rep(compact supergroup)	Superselection data and compact supergroup	Symmetric tensor structure, conjugates, End(1)=C	PROVEN	[12][16]
Deligne equivalence: subexponential char-0 symmetric tensor cats ≃ supergroup reps	Tensor category and supergroup representations	Symmetric structure, growth condition	PROVEN	[13]
Gelfand duality	Commutative C*-algebras and compact Hausdorff spaces	Algebraic/topological structure	PROVEN	[43]
Stone duality	Boolean algebras and zero-dimensional compact spectra	Boolean structure	PROVEN	[44]
-isomorphism of finite-dim C-algebras	Dimension vectors and algebras	*-algebra structure	PROVEN	[9]
Channel equivalence via minimal Stinespring dilation	CP maps and dilations	Complete positivity; unitary freedom	PROVEN (up to unitary)	[10]
Müger's Galois theory for braided tensor categories	Fixed-point-net representation categories	Braided structure plus extra "soliton" data	PROVEN; extra data required	[21]
Holographic duality (AdS/CFT) as applied to RT	Bulk geometry and boundary entanglement	—	OPEN — RT verified exactly only in AdS₃/2D CFT [45]; entanglement-wedge reconstruction is a PROVEN quantum-information theorem [46]; HaPPY codes are tensor-network toys, not AdS/CFT [48]	[45][46][48]
Yoneda embedding Hom(−, P)	Objects and their relational profiles	Categorical structure	PROVEN mathematically; the Yoneda qualification binds: "Yoneda does not, by itself, establish physical equivalence, empirical adequacy, or a classification theorem."	Founding Document §5

(e) Open classification problems

Problem	Region	What is known	Status	Sources
Fusion categories	Family VIII	Every fusion category is Rep of a weak Hopf algebra (Ostrik); ENO pivotal-structure conjecture (Annals 162, 2005)	OPEN	[14]
Elliott program (simple nuclear C*-algebras, UCT)	Family II	Active unfinished program	OPEN	[9]
Holographic entropy cone, n ≤ 5	Families III, VI	Complete inequality sets for 2, 3, and 4 boundary regions follow from strong subadditivity and monogamy of mutual information [47]; the five-region cone is completely constructed by exhibiting graph models for every extreme ray orbit of the cone defined by all proven inequalities, which settles the existence of further five-party inequalities in the negative (Hernández-Cuenca, Phys. Rev. D 100, 026004 (2019))	PROVEN	[47]; Hernández-Cuenca 2019
Holographic entropy cone, n ≥ 6	Families III, VI	For fixed n finitely many independent inequalities exist; only one infinite family known; a 2026 "multi-entropy cone" preprint is unverified	OPEN	[47]
Lurie proof gap (extended TQFTs)	Family VI	Baez–Dolan proposal [34]; Lurie 2009 is an "informal sketch of a proof" [35]; exact range of completed cases unverified	OPEN (claim CLAIMED)	[34][35]
Yang–Mills mass gap	Family VII/QFT	Clay: "no proof of this property is known"; explicitly not claimed in the 2026 announcements	OPEN	[23][25][26]
4D YM/QCD existence	Family VII/QFT	Existence claimed 2026-10-26 (QCD, GUT completion, pure YM subproblem, Balaban-based); 1147-page manuscript unpublished; claimed Lean formalization withheld; KITP verification pending	CLAIMED — never conflated with mass gap	[24][25][26]
Quantum monotone metrics	Family V	Classified parametrically by operator monotone functions (Petz–Sudar) — not a finite list	OPEN (classification not finite)	[52]
General II₁ / III factors	Family II	Coarse Connes-spectrum strata only; fundamental groups vary	IMPOSSIBLE globally [8]; residual strata OPEN	[8]
Uniform parameterization of finite simple groups	Group theory	CFSG complete; "no clean uniform description… is known" [1]	OPEN	[1]
Freed–Hopkins anomaly classification statement	Family IV	Cobordism-spectrum classification of topological anomalies; scope as stated in the cited source	OPEN	[19] context
Type-III factor structure beyond Connes spectrum	Family II	Coarse strata; examples with countable and trivial fundamental groups	OPEN	[8]

(f) Obstructions region — where no table exists

A periodic table must mark where no table can exist. Each entry below is a theorem, not a difficulty.

Obstruction	Statement	What it forbids	Status	Sources
Adyan–Rabin	Every Markov property of finitely presentable groups is algorithmically undecidable — including finiteness and simplicity (Collins extended beyond Markov properties)	Any uniform recognition procedure for group-flavored sectors	IMPOSSIBLE	[53]
McDuff	Uncountably many non-isomorphic separable type II₁ factors	CFSG-style total classification of factors outside the amenable case	IMPOSSIBLE	[8]
Wildness (derived-wild; Buchweitz–Leuschke; Grevstad–May)	Rank-2 isomorphism classes of the generic determinant's Cohen–Macaulay modules are not parametrizable by a finite-dimensional variety; "no meaningful classification" of compact C_p-equivariant HF_p-modules at odd primes; wildness is degeneration-robust	Finite-variety parameterization of wild representation categories	IMPOSSIBLE	[55][56][58]
Tame/wild tractability boundary (Grochow–Qiao)	Polynomial-time group isomorphism for tame extensions; required bounds fail in the wild case	Algorithmic classification in wild regimes	IMPOSSIBLE (wild side)	[57]
Borel reducibility / turbulence (Sasyk–Tørnquist)	Von Neumann algebra isomorphism admits no classification by countable invariants	Countable-invariant classifications of vN algebras	IMPOSSIBLE [single source; announcement-level]	[59]
Dimension-4 unrecognizability (Tancer; Gordon)	#₉(S²×S²) homeomorphism is algorithmically unrecognizable (improving #₁₂)	Effective classification of 4-manifolds in general	IMPOSSIBLE	[54][60]
Etingof–Gelaki	Isocategorical groups: tensor equivalence without symmetric structure does not determine the group	Equivalence notions that discard the monoidal/symmetric layer	IMPOSSIBLE	[12]
Unknotting problem	Unknot recognition is in NP and coNP; P-membership open	— (a "decidable but unclassified" middle regime)	OPEN	[61]
Darboux triviality	Symplectic geometry has no local invariants	Local classification content on the symplectic side (classification must be global)	PROVEN (trivially classified locally)	[6]

Regime contrast. Geometrization decomposes 3-manifolds into exactly 8 Thurston geometries [60] — finite tables exist, but only in tame regimes (dimension 3, not 4 [54][60]). The synthesis offered to the program is that one should expect restricted-class classification theorems plus explicit obstruction theorems, not one global periodic table; this synthesis is conjectural.

Reading the table

PROVEN cells are citations to completed theorems; IMPOSSIBLE cells are citations to impossibility theorems — both are knowledge.
CLAIMED appears exactly once as a classification-status cell (4D YM/QCD existence) [24][25][26]; existence and mass gap are never conflated [23][26].
OPEN cells mark where the series' own conjecture (T-CMRPS-01) and its five obligations live; the table is the chart that conjecture would complete.
Every equivalences entry names the structure it preserves, or records the failure that follows from not naming it [12].

Sources

[1] Classification of finite simple groups — https://en.wikipedia.org/wiki/Classification_of_finite_simple_groups (accessed 2026-10-29)

[2] Solomon, "A brief history of the classification of the finite simple groups", Bull. AMS 38 (2001) 315–352 — https://www.ams.org/journals/bull/2001-38-03/S0273-0979-01-00909-0/ (published 2001-03-27, accessed 2026-10-29)

[3] ATLAS of Finite Groups — https://en.wikipedia.org/wiki/ATLAS_of_Finite_Groups (accessed 2026-10-29)

[4] Simple Lie group — https://en.wikipedia.org/wiki/Simple_Lie_group (accessed 2026-10-29)

[5] Theorem of the highest weight — https://en.wikipedia.org/wiki/Theorem_of_the_highest_weight (accessed 2026-10-29)

[6] Darboux's theorem — https://en.wikipedia.org/wiki/Darboux%27s_theorem (accessed 2026-10-29)

[7] Poisson manifold (WikiJournal of Science 7(1):6, 2024) — https://en.wikipedia.org/wiki/Poisson_manifold (accessed 2026-10-29)

[8] Von Neumann algebra — https://en.wikipedia.org/wiki/Von_Neumann_algebra (accessed 2026-10-29)

[9] C*-algebra — https://en.wikipedia.org/wiki/C*-algebra (accessed 2026-10-29)

[10] Stinespring dilation theorem — https://en.wikipedia.org/wiki/Stinespring_dilation_theorem (accessed 2026-10-29)

[11] Quantum channel — https://en.wikipedia.org/wiki/Quantum_channel (accessed 2026-10-29)

[12] Doplicher–Roberts reconstruction theorem (nLab) — https://ncatlab.org/nlab/show/Doplicher-Roberts+reconstruction+theorem (accessed 2026-10-29)

[13] Deligne's theorem on tensor categories (nLab) — https://ncatlab.org/nlab/show/Deligne%27s+theorem+on+tensor_categories (accessed 2026-10-29)

[14] Fusion category (nLab) — https://ncatlab.org/nlab/show/fusion+category (accessed 2026-10-29)

[15] Haag–Kastler axioms (nLab) — https://ncatlab.org/nlab/show/Haag-Kastler+axioms (accessed 2026-10-29)

[16] DHR superselection theory (nLab) — https://ncatlab.org/nlab/show/DHR+superselection+theory (accessed 2026-10-29)

[17] Classifying space (nLab) — https://ncatlab.org/nlab/show/classifying+space (accessed 2026-10-29)

[18] Characteristic class (nLab) — https://ncatlab.org/nlab/show/characteristic+class (accessed 2026-10-29)

[19] Quantum anomaly (nLab) — https://ncatlab.org/nlab/show/quantum+anomaly (accessed 2026-10-29)

[20] Casini–Magan, arXiv:2511.21810 — https://arxiv.org/abs/2511.21810 (published 2025-11-26, accessed 2026-10-29)

[21] Müger, arXiv:math-ph/0008027 — http://export.arxiv.org/api/query?search_query=all:%22Doplicher-Roberts%22&start=0&max_results=15 (entry arXiv:math-ph/0008027; published 2000-08-21, accessed 2026-10-29)

[22] Pinzari, arXiv:2502.06229 — https://arxiv.org/abs/2502.06229 (v8 2026-09-19, accessed 2026-10-29)

[23] Clay Mathematics Institute, Yang–Mills problem page — https://www.claymath.org/millennium/yang-mills-the-maths-gap/ (accessed 2026-10-29)

[24] OpenAI, "Building scientific understanding with academic communities" — https://openai.com/index/building-scientific-understanding-with-academic-communities/ (published 2026-10-26, accessed 2026-10-29)

[25] KITP Yang–Mills working group statement — https://www.kitp.ucsb.edu/sites/default/files/YM_working_group.pdf (published 2026-10, accessed 2026-10-29)

[26] Scientific American, "Dueling proofs from OpenAI and Anthropic astound scientists" — https://www.scientificamerican.com/article/dueling-proofs-from-openai-and-anthropic-astound-scientists/ (published 2026-10-27, accessed 2026-10-29)

[27] Douglas, Nat. Rev. Phys. 8, 86–97 — https://www.nature.com/articles/s42254-025-00909-2 (published 2026-01-12, accessed 2026-10-29)

[28] Faizal–Shabir, arXiv:2606.19362 — https://arxiv.org/abs/2606.19362 (published 2026-06-09, accessed 2026-10-29)

[29] Jacobsen, arXiv:2506.00284 (withdrawn) — https://arxiv.org/abs/2506.00284 (published 2025-05-30, accessed 2026-10-29)

[30] Nissim, arXiv:2510.22788 — https://arxiv.org/abs/2510.22788 (published 2025-10-26, accessed 2026-10-29)

[31] Cao–Nissim–Sheffield, arXiv:2509.04688 — https://arxiv.org/abs/2509.04688 (published 2025-09-04, accessed 2026-10-29)

[32] Faria da Veiga–O'Carroll, arXiv:2509.03513 — https://arxiv.org/abs/2509.03513 (published 2025-09-03, accessed 2026-10-29)

[33] Nair, arXiv:2608.10133 — https://arxiv.org/abs/2608.10133 (published 2026-08-10, accessed 2026-10-29)

[34] Baez–Dolan, arXiv:q-alg/9503002 — https://arxiv.org/abs/q-alg/9503002 (published 1995-03-05, accessed 2026-10-29)

[35] Lurie, arXiv:0905.0465 — https://arxiv.org/abs/0905.0465 (published 2009-05-04, accessed 2026-10-29)

[36] Topological quantum field theory (nLab) — https://ncatlab.org/nlab/show/topological+quantum+field+theory (accessed 2026-10-29)

[37] Butterfield–Isham, arXiv:quant-ph/9803055 — https://arxiv.org/abs/quant-ph/9803055 (published 1998-03-20, accessed 2026-10-29)

[38] Isham–Döring, arXiv:quant-ph/0703060 — https://arxiv.org/abs/quant-ph/0703060 (published 2007-03-07, accessed 2026-10-29)

[39] Isham, arXiv:1004.3564 — https://arxiv.org/abs/1004.3564 (published 2010-04-20, accessed 2026-10-29)

[40] Wilson–Hefford–Hoffreumon, arXiv:2602.23865 — https://arxiv.org/abs/2602.23865 (published 2026-02-27, accessed 2026-10-29)

[41] Hefford–Wilson, arXiv:2402.02997 — https://arxiv.org/abs/2402.02997 (published 2024-02-05, accessed 2026-10-29)

[42] Gelfand–Naimark theorem (nLab) — https://ncatlab.org/nlab/show/Gelfand-Naimark+theorem (accessed 2026-10-29)

[43] Gelfand duality (nLab) — https://ncatlab.org/nlab/show/Gelfand+duality (accessed 2026-10-29)

[44] Stone duality (nLab) — https://ncatlab.org/nlab/show/Stone+duality (accessed 2026-10-29)

[45] Ryu–Takayanagi, arXiv:hep-th/0603001 — https://arxiv.org/abs/hep-th/0603001 (published 2006-02-28, accessed 2026-10-29)

[46] Dong–Harlow–Wall, arXiv:1601.05416 — https://arxiv.org/abs/1601.05416 (published 2016-01-20, accessed 2026-10-29)

[47] Bao–Nezami–Ooguri–Stoica–Sully–Walter, arXiv:1505.07839 — https://arxiv.org/abs/1505.07839 (published 2015-05-28, accessed 2026-10-29)

[48] Pastawski–Yoshida–Harlow–Preskill, arXiv:1503.06237 — https://arxiv.org/abs/1503.06237 (published 2015-03-20, accessed 2026-10-29)

[49] Van Raamsdonk, arXiv:1005.3035 — https://arxiv.org/abs/1005.3035 (published 2010-05-17, accessed 2026-10-29)

[50] Minguzzi–Sánchez, arXiv:gr-qc/0609119 — https://arxiv.org/abs/gr-qc/0609119 (published 2006-09-26, accessed 2026-10-29)

[51] Galaev, arXiv:1611.01551 — https://arxiv.org/abs/1611.01551 (published 2016-11-04, accessed 2026-10-29)

[52] Petz–Sudar, arXiv:quant-ph/0102132 — https://arxiv.org/abs/quant-ph/0102132 (published 2001-02-27, accessed 2026-10-29)

[53] Adyan–Rabin theorem — https://en.wikipedia.org/wiki/Adian%E2%80%93Rabin_theorem (accessed 2026-10-29)

[54] Tancer, arXiv:2310.07421 — https://arxiv.org/abs/2310.07421 (published 2023-10-11, accessed 2026-10-29)

[55] Bekkert–Drozd, arXiv:math/0310352 — https://arxiv.org/abs/math/0310352 (published 2003-10-22, accessed 2026-10-29)

[56] Buchweitz–Leuschke, arXiv:math/0505315 — https://arxiv.org/abs/math/0505315 (published 2005-05-15, accessed 2026-10-29)

[57] Grochow–Qiao, arXiv:1507.01917 — https://arxiv.org/abs/1507.01917 (published 2015-07-07, accessed 2026-10-29)

[58] Grevstad–May, arXiv:2509.17604 — https://arxiv.org/abs/2509.17604 (published 2025-09-22, accessed 2026-10-29)

[59] Sasyk–Tørnquist, arXiv:0909.2657 — https://arxiv.org/abs/0909.2657 (published 2009-09-14, accessed 2026-10-29)

[60] Geometrization conjecture — https://en.wikipedia.org/wiki/Geometrization_conjecture (accessed 2026-10-29)

[61] Unknotting problem — https://en.wikipedia.org/wiki/Unknotting_problem (accessed 2026-10-29)