Strong CP as Higher-Categorical Coherence: Structural Selection of θ = 0 and SM-Only Global Relaxation

Strong CP as Higher-Categorical Coherence: Structural Selection of θ= 0 and SM-Only Global Relaxation Andrei T. Patrascu 1 1 FAST Foundation, Destin FL, 32541, USA email: andrei.patr[email protected] The strong–CP problem is revisited from two complementary angles that together remain within the Standard Model (SM). First, we show that the QCD vacuum angle θ is a superselection parameter: the algebra of gauge–invariant local observables extends by a central unitary from C∗ ( Z ), with character einθ fixed by the topology π3 ( SU (3)) = Z . Equivalently, θ –vacua are Bloch superpositions of Chern–Simons sectors and the Euclidean functional Z ( θ ) = PQeiθQZQ weights disconnected topological sectors; no SM spacetime dynamics can change θ . Second, we propose a structural selection principle: among the family {QCDθ} , impose CP–equivariance with full higher coherence, reflection positivity, absence of a CP coherence anomaly, and uniqueness of the vacuum. These axioms force θ to a CP fixed point and, under mild conditions excluding the Dashen scenario at θ=π, select θ= 0. Within any fixed θphys sector, we derive a coarse–grained SM evolution on FRW backgrounds (via Schwinger–Keldysh → Mori–Zwanzig) for CP–odd deviations from equilibrium. Euclidean susceptibilities (notably χ ( T )) define a positive metric G ( T )that makes a Lyapunov functional explicit; local decay, quasi–Lax spectral transport on adiabatic time patches, and higher–categorical sheaf coherence across overlaps yield a global contraction theorem (operational CP conservation inside the chosen sector). Across sectors, a coherence functional on theory space produces a gradient flow (a “categorical PDE”) whose stable fixed points coincide with the axiomatic selection, reconciling selection with superselection. An analogy with Galilean mass superselection and its resolution by enlarging symmetry to the Poincaré group motivates the role of higher–categorical coherence as the appropriate enlargement for QCD: it trivializes the topological extension that underlies θ –superselection and singles out θ = 0, while SM dynamics ensures global relaxation within that sector. No new particles are introduced. I. INTRODUCTION The strong–CP problem encapsulates a striking tension at the heart of quantum chromodynamics (QCD): the gauge– and Lorentz–invariant CP–odd density Q ( x ) = g2 s 32π2Ga µν ˜ Ga µν admits the θ –term θRd4x Q ( x ), yet experimental bounds on hadronic CP violation—most notably the neutron electric dipole moment (EDM)—are consistent with zero at the level |θphys|. 10 −10 [ 6 ]. Theoretically, instanton physics and large– N arguments suggest that the vacuum energy E ( θ )is 2 π –periodic and nontrivial in θ [ 1 – 4 ], while the structure of finite–temperature QCD implies a temperature–dependent topological susceptibility χ ( T ) = ∂2 θE ( θ ) |θ=0 [ 5 ]. The longstanding puzzle is then: why does Nature realize QCD arbitrarily close to a CP–conserving branch? Two logically distinct layers. A key point, sharpened in this work, is the separation between: (i) Within–sector dynamics in physical spacetime. In the Standard Model (SM), θ is a superselection parameter: local, finite–energy operations act inside (not across) a fixed θ –sector. Consequently, spacetime dynamics can relax CP–odd observables toward their sector–dependent equilibrium, but cannot move θitself [4]. (ii) Across–sector selection at the level of theory space. One may, however, impose structural consistency requirements on the family {QCDθ}θ∈S1 . We show that demanding CP–equivariance with full higher coherence, reflection positivity, absence of a CP coherence anomaly, and uniqueness of the vacuum selects the CP fixed point θ= 0 among admissible members [23–27]. Layer (i) is dynamical—we derive an effective (parabolic) evolution for CP–odd deviations on Friedmann–Robertson–Walker (FRW) backgrounds using Schwinger–Keldysh and a Mori–Zwanzig reduction [ 14 – 21 ]. Layer (ii) is structural—it does not posit spacetime evolution of θ ; rather, it is a criterion on which members of the θ –family admit a coherent CP (“Real”) extension compatible with Euclidean reconstruction [23, 24]. Historical context. Soon after the recognition of instantons in nonabelian gauge theory [ 1 ], the θ –dependence of QCD was analyzed in large– N chiral frameworks [ 2 , 3 ] and through QCD sum rules [ 4 ]. Finite–temperature effects on topological fluctuations were studied in [ 5 ]. The neutron EDM bound [ 6 ] precipitated two canonical classes of solutions: (a) a new anomalous global symmetry U (1) PQ with an axion 2 field dynamically relaxing θeff →0[7–9]; (b) special flavor structures (Nelson–Barr) rendering θphys = 0 at tree level [ 10 , 11 ]. Independently, Vafa and Witten [ 12 ] proved that CP is not spontaneously broken at θ = 0. Broad reviews of θ –dependence and its nonperturbative features appear in [ 13 ]. The present article pursues a strictly SM–content approach that is orthogonal to axions and flavor model building: the novelty lies in (i) a clean dynamical statement within a fixed sector, and (ii) a higher–categorical structural selection across sectors. Topological origin of θ superselection. The superselection nature of θ is ultimately topological. Gauge configurations in Euclidean 4D decompose into disconnected sectors labeled by the integer Pontryagin index Q∈Z , since π3 ( SU (3)) = Z . The partition function is a 2 π –periodic sum Z ( θ ) = PQeiθQZQ , with ZQ the path integral restricted to sector Q [ 1 , 4 ]. Algebraically, the group of large gauge transformations π0 ( G ) ∼ =Z induces a central extension of the algebra of gauge–invariant local observables by a central unitary whose spectrum is S1 ; irreducible representations carry a central character einθ , and the Hilbert space decomposes as a direct integral R⊕Hθdθ on which Aloc acts diagonally. No local observable connects θ6 = θ0 . Canonically, θ –vacua are Bloch states |θi = Pn∈Zeinθ|ni built from sectors with integer Chern–Simons number; matrix elements of local gauge–invariant operators vanish between distinct sectors. Hence, in the SM, θdoes not evolve in physical time. Analogy with Galilei mass superselection and the Poincaré resolution. There is a helpful precedent. In Galilean quantum mechanics the Bargmann algebra—a central extension of the Galilei algebra— contains the mass M as a central charge, producing a mass superselection rule [ 28 , 29 ]. The resolution does not modify Galilei by fiat; rather, it embeds nonrelativistic kinematics into the larger Poincaré group, where mass is a Casimir PµPµ = m2 of the representation (Wigner classification) and particle creation/annihilation renders mass dynamical at the QFT level [ 30 ]. The moral is structural: a superselection parameter tied to a central extension can disappear when the theory is embedded into a larger symmetry setting in which the extension trivializes. The present work advances the view that higher–categorical coherence plays an analogous role for QCD: among the θ–family, only θ= 0 admits a CP–equivariant, reflection–positive extension with full higher coherence and a unique vacuum, thereby structurally selecting the CP–conserving member of the family. Dynamical statement within a fixed sector. While θ itself is superselected, observables do evolve. We derive, from the Schwinger–Keldysh functional and a Mori–Zwanzig projection, a coarse–grained SM evolution for a fixed set of CP–odd gauge–invariant observables, written for the deviations δY := Y−Yeq(θphys, T)on an FRW background: ∂tδY=−Γ(T)δY+a(t)−2∇·D(T)∇δY+N(δY;T) + ξ−˙ T ∂TYeq, with Kubo expressions for the damping and diffusion matrices Γ , D [ 16 – 18 ], and noise consistent with fluctuation–dissipation [ 19 – 21 ]. Static Euclidean susceptibilities, notably χ ( T ), define a positive metric G ( T ) = Σ( T ) −1 making a Lyapunov functional explicit. A G –polar split of the linearized generator A = S + K isolates a dissipative part S and a reversible mixer K . This enables a local quasi–Lax description ˙ L = [ M, L ] + R with small adiabatic remainder R [ 22 ], and—together with higher–categorical sheaf coherence across adiabatic time patches—yields a global contraction theorem: CP–odd deviations decay exponentially in physical time inside the chosen θphys sector. Structural selection across sectors. At the level of Euclidean field theories viewed as symmetric monoidal functors Zθ : Bordor 4→VectC [ 23 ], CP acts by orientation reversal. A CP–equivariant (“Real”) structure is a monoidal natural isomorphism compatible with gluing, together with higher coherence data [ 25 – 27 ]. Comparing Zθ ( M )with Zθ(M) forces eiθQ = e−iθQ for all Q∈Z , so θ∈ { 0 , π} . Reflection positivity [ 24 ], compatibility with Euclidean reconstruction, and uniqueness of the vacuum (excluding Dashen–type degeneracy at θ = π [ 13 , 31 ]) eliminate π , leaving θ = 0. We also introduce a coherence functional on theory space whose gradient flow (a “categorical PDE”) has stable fixed points precisely at the theories satisfying the foregoing structural axioms, thereby reconciling “dynamical selection” in theory space with superselection in spacetime. What this article contributes. The present work (i) gives a transparent, SM–only derivation of observable–space relaxation on cosmological backgrounds, with Lyapunov control, quasi–Lax spectral transport, and higher–categorical gluing; (ii) formulates and proves a structural selection principle that singles out θ = 0 among {QCDθ} without introducing new particles; and (iii) clarifies the conceptual status of θ –superselection through an explicit analogy with Galilei mass superselection and its resolution by enlarging symmetry to Poincaré [ 28 – 30 ]. Throughout, CP violation in the weak sector is immaterial: the CP–equivariant structure is applied to the strong subsector regarded autonomously; within any fixed θphys sector, the nonequilibrium analysis and its global contraction are unaffected. The remainder of the paper develops these points in detail. 3 II. THE STRONG–CP PROBLEM AND THE θSUPERSELECTION RULE This section develops, in a single place and with complementary perspectives, the structural reason why the QCD vacuum angle θ is a superselection parameter in the Standard Model (SM). The argument is topological in origin and manifests algebraically, canonically, and in the Euclidean path integral. We also fix notation for the observable–space dynamics used later and explain the physical consequences: spacetime evolution in the SM relaxes CP–odd observables within a fixed θphys sector, but it does not evolve θ itself. The selection of θ = 0 across sectors, when attempted without new fields, must therefore be a structural principle (Sec. IV), not a spacetime equation of motion. A. The θ–term, physical content, and CP With Ga µν the gluon field strength and ˜ Ga µν =1 2µνρσGa ρσ its dual, the CP–odd density Q(x) = g2 s 32π2Ga µν ˜ Ga µν (1) allows the Lorentz– and gauge–invariant θ–term θRd4x Q(x). The physically invariant combination is θphys =θ+ argdet Mq,(2) with Mq the quark mass matrix. Instanton analyses and large– N arguments imply a 2 π –periodic, nontrivial vacuum energy density E ( θ )[ 1 – 4 ], and at finite temperature the topological susceptibility χ ( T ) = ∂2 θE ( θ ) |θ=0 controls small– θ curvature [ 5 , 13 ]. Under CP: Q7→ −Q , so the action is CP–even only at θphys = 0 (mod π). Empirically, neutron EDM bounds point to |θphys|1[6]. B. Topological sectors and large gauge transformations For finite–action configurations on (compactified) Euclidean spacetime, S4 , the integral of (1) is an integer: Q=ZS4 d4x Q(x)∈Z.(3) Equivalently, the space of gauge connections modulo small gauge transformations decomposes into disconnected components labeled by Q , a reflection of the homotopy group π3 ( SU (3)) = Z [ 1 ]. Large gauge transformations shift between components, and their connected components form a group isomorphic to Z , denoted π0 ( G ) ∼ =Z . In the canonical (Hamiltonian) description on spatial R3 , classical vacua are labeled by the Chern–Simons number NCS ∈Z; large gauge transformations shift NCS 7→NCS +n. C. Algebraic formulation: central character and direct–integral decomposition Let Aloc be the C ∗ –algebra generated by gauge–invariant local SM observables. Let Un denote the implementers of large gauge transformations (winding number n∈Z ) on the physical Hilbert space H . Local gauge invariance implies [O, Un]=0 for all O∈ Aloc, n ∈Z.(4) Completing the group algebra of Z yields C∗ ( Z ) ∼ =C ( S1 ), generated by a central unitary V with spectrum S1 . The extended observable algebra Aext := Aloc b ⊗C∗ ( Z )therefore has a center containing V . In any physical representation π, the center acts by a character χθ:Z→U(1), π(V)ψ=eiθ ψ, χθ(n) = einθ, θ ∈[0,2π).(5) By the spectral theorem and (4), πdecomposes into a direct integral of superselection sectors: H ≃ Z⊕ S1 Hθdµ(θ), π(Aloc) = Z⊕ S1 πθ(Aloc)dµ(θ),(6) with the action diagonal in θ . If ψθ0∈Hθ0 and φθ∈Hθ with θ06 = θ , then hψθ0, π ( O ) φθi =0 for all O∈Aloc . Thus θ labels superselection sectors: local SM operations cannot change it. This is the algebraic heart of the superselection rule for θ. 4 D. Canonical picture: Bloch vacua and orthogonality Let {|ni}n∈Z denote (idealized) vacua in sectors of definite Chern–Simons number. Quantum tunneling (instantons) mixes these sectors; the true vacua are Bloch states [1] |θi=X n∈Z einθ |ni,hθ0|θi= 2π δ(θ0−θ).(7) Any gauge–invariant local operator O commutes with the shift S|ni = |n +1 i generated by a large gauge transformation, hence Ohas no matrix elements between |niand |miwith m6=n. It follows that hθ0|O|θi= 0 for θ06=θ, (8) which is the canonical manifestation of superselection: Oacts within a fixed θ. E. Euclidean path integral: θas a weight, not a field The Euclidean partition function at fixed θis a weighted sum over topological sectors: Z(θ) = X Q∈Z eiθQ ZQ, ZQ=Ztop. charge Q DADψD¯ ψ e−SQCD[A,ψ, ¯ ψ].(9) Here θ is a c–number parameter. It is not integrated over, carries no conjugate momentum, and satisfies no Euler–Lagrange equation. Nonperturbative transitions (instantons, sphalerons) change Q within the θ –weighted ensemble but do not evolve θ itself [ 1 , 5 , 13 ]. Differentiations of log Z ( θ )yield sector–dependent equilibrium CP–odd expectations, e.g. hQiT=−∂θF(θ;T)≈χ(T)θ(for |θ|  1),(10) but there is no mechanism in the SM that changes θ. F. Consequences for dynamical modeling Equations (6) – (9) imply that any honest spacetime PDE “driving θ→ 0” is not an SM equation. What evolves in physical time are state observables. In particular, within a fixed θphys sector the coarse–grained SM dynamics on FRW backgrounds (derived later from Schwinger–Keldysh → Mori–Zwanzig) acts on CP–odd deviations from equilibrium δY(t, x) := Y(t, x)−Yeq(θphys, T(t)),(11) and drives δY→ 0(global contraction), while Yeq ( θphys, T )itself depends on the fixed θphys via (10) . This distinction underpins the two–layer structure of the paper: structural selection across sectors (Sec. IV) versus dynamical relaxation within a sector (Sec. V). G. Intuitive picture: topological islands and central phase It is helpful to visualize configuration space as a chain of islands labeled by the integer Q . Local, finite–energy deformations roam freely on an island but cannot ferry the system to a different island; only “bridges” corresponding to large gauge transformations identify adjacent islands. The θ –parameter is a global phase that weights the islands in the Euclidean sum (9) and labels the central character (5) in the algebraic language. Local observables are insensitive to which island was used to define a state—they see only the relative phases within a fixed θ. This is why physical time evolution cannot alter θ. 5 H. Standing notation for later sections We fix a finite collection of gauge–invariant, CP–odd local observables {Oi ( x ) }N i=1 (e.g. Q , EDM– sensitive composites). Coarse–graining at scales long compared to microscopic correlation times yields slow variables Yi(t, x) = hOi(t, x)icg, δYi=Yi−Yi,eq(θphys, T ).(12) Let Σ ij ( T )denote the static Euclidean susceptibility matrix of {Oi} at temperature T (with Σ QQ = χ ( T )); define the positive weight matrix G(T) := Σ(T)−10,(13) used as a Lyapunov metric in the observable–space evolution. We shall derive in Sec. V a Markovian parabolic equation for δY on FRW backgrounds, identify the dissipative and reversible parts of the linearized generator, and prove a global contraction theorem using a quasi–Lax transport on adiabatic time patches glued by higher coherence. I. Historical remarks and pointers The topological origin of (3) , its relation to instantons, and the Bloch form (7) go back to the seminal work of ’t Hooft and subsequent developments in large– N and chiral dynamics [ 1 – 4 ]. Finite– T susceptibilities and their role in θ –dependence were clarified in [ 5 ]; modern reviews include [ 13 ]. The fact that CP is not spontaneously broken at θ = 0 is rigorously established in [ 12 ]. These inputs, together with the nonequilibrium formalism of Schwinger–Keldysh and Mori–Zwanzig [ 14 – 21 ], will be the backbone of the dynamical analysis in later sections. J. Summary of Section II • The integer topology of gauge fields ( π3 ( SU (3)) = Z ) produces disconnected sectors labeled by Q , and large gauge transformations form π0(G)∼ =Z. • Algebraically, this yields a central unitary with spectrum S1 ; physical representations are labeled by a central character einθ , producing a direct–integral decomposition into superselection sectors Hθon which Aloc acts diagonally. • Canonically, θ –vacua are Bloch superpositions of Chern–Simons sectors, and local gauge–invariant operators have no off–diagonal matrix elements between different θ. • In the Euclidean path integral, θ is a weight, not a field; it has no conjugate momentum and no spacetime equation of motion. • Therefore: SM spacetime dynamics cannot change θ . The correct dynamical statement is the global relaxation of CP–odd deviations within a fixed θphys sector (Sec. V). The cross–sector selection of θ= 0 is a structural criterion developed in Sec. IV. III. ANALOGY: GALILEAN MASS SUPERSELECTION AND THE POINCARÉ RESOLUTION This section develops the analogy between the topological superselection of the QCD vacuum angle θ (Sec. II) and the group–cohomological mass superselection in Galilean quantum mechanics. The analogy is mathematically precise: in both cases a nontrivial central extension produces a superselection label. The resolution in the nonrelativistic case came from embedding Galilean kinematics into the larger Poincaré symmetry, where the central extension trivializes and mass becomes a Casimir invariant rather than a central charge. This suggests, at the conceptual level, why an enlarged structural framework (here, the higher–categorical coherence of Sec. IV) can select θ= 0 without violating SM dynamics. 6 A. Projective representations and central extensions Let G be a Lie group of kinematical symmetries. Quantum states are rays in a Hilbert space and symmetries act by projective unitary representations U:G→PU(H), obeying U(g)U(h) = eiω(g,h)U(gh), g, h ∈G, (14) for some phase ω:G×G→R/ 2 πZ . Associativity of the group action forces ω to satisfy a 2–cocycle condition; two projective representations are equivalent if their cocycles differ by a 2–coboundary. Equivalence classes are classified by the group cohomology H2 ( G, U (1)); a nontrivial class corresponds to acentral extension 1→U(1) −→ b G−→ G→1,(15) and (14) lifts to a true unitary representation of b G [ 32 – 34 ]. At the Lie–algebra level, a nontrivial 2–cocycle adds a central generator to the commutators. B. The Bargmann (centrally extended Galilei) algebra and mass The (unextended) Galilei Lie algebra in 3+1 dimensions is generated by spatial rotations Ji , spatial translations Pi, Galilean boosts Ki, and time translations H, with the familiar commutators [Ji, Jj] = iijkJk,[Ji, Pj] = iijkPk,[Ji, Kj] = iijkKk,(16) [Pi, Pj]=0,[Ki, Kj]=0,[Pi, H]=0,[Ki, H] = iPi,[Ki, Pj]=0.(17) However, quantum mechanically the Galilei group admits a nontrivial projective ambiguity with H2(Gal, U(1)) 6= 0, and the physically realized algebra is the Bargmann algebra: [Ki, Pj] = i M δij,[M, ·]=0,(18) where M is a central generator (the mass) [ 32 , 36 ]. One way to see (18) is to consider finite boosts U(v) = e−iv·Kand finite translations U(a) = e−ia·P: a projective multiplier of the form U(v)U(a) = eim v·aU(a)U(v)(19) implies, by differentiating at the identity, the commutator [ Ki, Pj ] = im δij . Here m is the eigenvalue of Min an irreducible representation (irrep). In particular: •Mis central, hence by Schur’s lemma acts as m1in any irrep. • Observables constructed from {J, P, K, H} commute with M ; therefore, the Hilbert space decomposes into a direct sum/integral of mass sectors on which all local observables act diagonally. This is the mass superselection rule in Galilean quantum mechanics [36]. Intuitive narrative. Equation (18) encodes a quantum anomaly of boosts and translations: consecutive infinitesimal operations differ by a phase proportional to m. The central generator Mtracks this phase and enforces a rigid decomposition into superselected mass sectors. Within the Galilean framework, there is no local operation that changes min a given irrep. C. Poincaré symmetry: trivial projective ambiguity and Wigner classification For the connected Poincaré group P = R1,3oSO+ (1 , 3) (or its double cover R1,3oSL (2 ,C )), the projective ambiguity is trivial once the double cover is taken: every projective unitary representation lifts to a true unitary representation (aside from the universal cover for spin). In particular, there is no analogue of (18) with a nonzero central generator [ 32 , 35 ]. Irreducible unitary representations (UIRs) are classified by the eigenvalues of the Casimir operators: PµPµ=m2(mass squared), WµWµ=−m2s(s+ 1) (spin via Pauli–Lubanski Wµ).(20) Thus, in the relativistic setting, mass is not a central charge but an invariant labeling UIRs. In quantum field theory (QFT), fields of different masses can interact, particle–antiparticle pairs can be created, and bound states can have masses different from their constituents; the rigid Galilean mass superselection dissolves in the relativistic (field–theoretic) framework. 7 Galilei as a contraction of Poincaré. The Inönü–Wigner contraction c→ ∞ maps the Poincaré algebra to the Galilei algebra. In particular, [Ki, Pj] = iH c2δij c→∞ −−−−→ i M δij, M := lim c→∞ H c2central,(21) so the central mass of Galilean kinematics emerges from the relativistic energy via H≈Mc2 + P2/ 2 M + · · · [ 37 , 38 ]. This mathematically clarifies why the mass superselection is a feature of the Galilean (nonrelativistic) regime and disappears in the fully relativistic theory. D. Lessons for the θsuperselection The structural parallels with QCD are clear: Context Superselection origin Trivialization by enlargement Galilean QM central extension (Bargmann mass M) embed into Poincaré (mass →Casimir) QCD (θ) central character einθ from π3(SU(3)) = Zimpose higher–categorical CP–coherence In both cases a superselection label is tied to a nontrivial central structure: •For Galilei, H2(Gal, U(1)) 6= 0 and the central generator Myields mass superselection [32, 36]. • For QCD, the large–gauge group π0 ( G ) ∼ =Z induces a central unitary with spectrum S1 acting by the character einθ on each sector (Sec. II C). The Galilei example teaches that a superselection parameter need not be an absolute obstruction: it can disappear when the theory is embedded into a larger, more coherent structure in which the relevant central extension trivializes. In the strong–CP problem, Sec. IV shows that augmenting the strong sector with higher–categorical CP–equivariance, reflection positivity, and unique–vacuum requirements (all physically motivated consistency conditions) trivializes the CP–induced phase eiθQ coherently only at θ = 0 (CP fixed point with no Dashen degeneracy). In this sense, higher–categorical coherence plays the conceptual role of the “Poincaré upgrade”: it is an enlarged structural framework in which the superselection label θ is not allowed to vary arbitrarily; the only admissible member is θ= 0. Intuitive narrative. In Galilean QM, the projective ambiguity forces a universal “mass phase” whenever boosts and translations are interleaved; promoting to Poincaré removes the phase ambiguity and upgrades mass to a kinematical invariant compatible with particle creation. In QCD, the Euclidean strong sector carries a universal “topological phase” eiθQ upon orientation reversal (CP). Imposing higher–coherence and positivity removes the ambiguity (up to fixed points), and the only fully coherent, reflection–positive, unique–vacuum theory is the CP–even member θ= 0. E. Mathematical interlude: from cocycles to central charges For completeness, the link between projective multipliers and central generators can be made explicit. Consider the Galilean ray representation with U ( a ) = e−ia·P , U ( v ) = e−iv·K . Assume the multiplier (19) . Expand to first order: U(v)U(a) = 1−iv·K+· · · 1−ia·P+· · · = 1 −i(v·K+a·P)−1 2[v·K,a·P] + · · · , whereas eim v·aU(a)U(v) = 1 + im v·a+· · · 1−i(a·P+v·K) + · · · . Equating both sides at order O(va)yields −1 2X ij viaj[Ki, Pj] = im v·a⇒[Ki, Pj] = im δij, i.e. the central generator M with eigenvalue m . This is the Lie–algebra avatar of a nontrivial class in H2(Gal, U(1)) [32, 33]. For the Poincaré group, after passing to the double cover, the relevant H2 ( P, U (1)) is trivial (in 3+1 dimensions), so no additional central generator appears; the only projective ambiguity is removed by the spin cover [35]. 8 F. Summary of Section III • Projective quantum symmetries are classified by H2 ( G, U (1)) and correspond to central extensions. For the Galilei group this yields the Bargmann algebra with central mass M. • Mass is therefore a superselection parameter in Galilean QM. In the Poincaré framework the projective ambiguity is trivial (after the double cover), and mass becomes a Casimir invariant. The Galilei algebra arises from Poincaré by Inönü–Wigner contraction, with the central mass emerging as M= limc→∞ H/c2. • The QCD θ superselection similarly arises from a central structure (the large–gauge character einθ ). An enlarged structural framework—higher–categorical CP coherence with reflection positivity and unique vacuum—selects the CP–even fixed point θ = 0, mirroring how Poincaré resolves Galilean mass superselection at the structural level. IV. HIGHER–CATEGORICAL COHERENCE AS STRUCTURAL SELECTION ACROSS θ This section formulates the strong sector of the Standard Model (SM) as a symmetric monoidal Euclidean field theory functor and states a structural selection principle that singles out θ = 0 among the family {QCDθ}θ∈S1 . The key inputs are: (i) a CP–equivariant (“Real”) structure with full higher coherence; (ii) reflection positivity; (iii) absence of a CP coherence anomaly; and (iv) uniqueness of the vacuum object. Mathematically, these data live in a 2–category whose objects are functors, 1–morphisms are monoidal natural isomorphisms, and 2–morphisms are coherence maps. Physically, they encode the compatibility of CP and Euclidean gluing with unitarity (Osterwalder–Schrader positivity) and the absence of spontaneous CP breaking. Under these assumptions, only the CP fixed point θ = 0 ( mod 2 π ) is admissible. A. Euclidean strong sector as a symmetric monoidal functor Let Bordor 4 denote the symmetric monoidal category whose objects are closed oriented 3–manifolds Σand whose morphisms are oriented bordisms W : Σ in → Σ out , with monoidal structure given by disjoint union t . The (nonextended) Euclidean strong sector at fixed vacuum angle θ will be viewed as a symmetric monoidal functor Zθ: Bordor 4−→ VectC,Zθ(W2◦W1) = Zθ(W2)Zθ(W1),Zθ(Σ1tΣ2) = Zθ(Σ1)⊗Zθ(Σ2),(22) assigning to a closed 4–manifold M a partition function Zθ ( M ) ∈C and to a closed 3–manifold Σa state space Zθ(Σ). For QCD, the θ–term shifts the partition function by a topological phase, Zθ(M) = eiθ Q(M)Z0(M), Q(M) = 1 8π2ZM tr(F∧F)∈Z,(23) reflecting the integer Pontryagin index. The functorial viewpoint parallels Atiyah–Segal axioms and their higher–categorical refinements [39–41]. B. CP equivariance and reflection positivity as Real structure with higher coherence Let CP : Bordor 4→Bordor 4 denote orientation reversal (Euclidean CP). A CP–equivariant (“Real”) structure on Zθis a monoidal natural isomorphism η:Zθ=⇒ Zθ◦CP,(24) together with higher coherence data: specified 2–morphisms witnessing compatibility on disjoint unions and compositions (gluing), and satisfying coherence diagrams. Reflection positivity (Osterwalder–Schrader) is realized by a Real structure for reflection across a codimension–one split, ensuring that the induced sesquilinear form on Zθ (Σ) is positive and that Lorentzian reconstruction yields a unitary theory [ 42 , 43 ]. We will assume throughout that Z0 admits such a Real structure and that for admissible θ a Real structure exists on Zθ. 9 Coherence and anomalies. Failure of the higher coherence diagrams to commute defines a projective defect—a 2–cocycle in the relevant coherence group—interpretable as a ’t Hooft anomaly obstructing the implementation of CP as an honest symmetry in the extended functorial setting. At θ = π , pure Yang–Mills exhibits a mixed anomaly between CP and the Z(1) N one–form center symmetry [ 44 – 46 ], which precisely manifests as a coherence obstruction. In full QCD with fundamentals, the one–form symmetry is explicitly broken, but we will keep the “no CP coherence anomaly” hypothesis explicit. C. Fixed points of CP equivariance The Real structure (24) imposes a phase constraint comparing Zθ ( M )with Zθ(M) . Since Q ( M ) = −Q(M), (23) gives Zθ(M) = eiθQ(M)Z0(M)⇐⇒ Zθ(M) = eiθQ(M)Z0(M).(25) Equivariance demands Zθ(M)∼ =Zθ(M), hence eiθQ(M)=e−iθQ(M)for all integers Q(M),(26) which is possible iff θ∈ {0, π}(mod 2π).(27) Lemma IV.1 (CP fixed points) . If Zθ admits a CP–equivariant Real structure with higher coherence and reflection positivity, then θis a CP fixed point, i.e. (27) holds. This is a categorical restatement of the familiar fact that CP flips Q7→−Q ; only at fixed points can the θ–phase be coherently trivialized. D. Excluding θ=π: anomaly and vacuum structure Two structural mechanisms obstruct θ=π: (a) Coherence/anomaly obstruction. In pure SU ( N )Yang–Mills, θ = π has a mixed anomaly between CP and the Z(1) N one–form center symmetry, visible as a nontrivial element in the relevant cohomology controlling the coherence of the Real structure [ 45 , 46 ]. In the functorial language, the CP 1–morphism cannot be lifted to a strictly coherent Real structure: a projective defect remains on certain bordisms. This violates our “no CP coherence anomaly” hypothesis. (b) Vacuum degeneracy (Dashen phenomenon). Independently of (a), many QCD–like theories exhibit spontaneous CP breaking at θ = π (Dashen phenomenon): the vacuum energy E ( θ )develops a cusp and the vacuum becomes degenerate. In the functorial setting, reflection positivity and uniqueness of the vacuum correspond to the existence of a distinguished terminal object in the 0–dimensional sector; degeneracy at θ = π violates this axiom. Effective field theory analyses and large– N arguments support this picture across broad classes of theories [ 47 – 49 ]. We therefore adopt unique vacuum as an explicit structural hypothesis. E. Structural selection theorem We combine Lemma IV.1 with the obstructions above. Theorem IV.2 (Categorical selection of the strong sector) . Let {Zθ}θ∈[0,2π) be the θ –family of Euclidean strong–sector functors. Assume: (i) CP equivariance with higher coherence: a Real structure η : Zθ⇒ Zθ◦CP compatible with gluing. (ii) Reflection positivity: Real structure for reflection across a codimension–one split ensuring OS positivity and unitary reconstruction. (iii) No CP coherence anomaly: the higher coherence diagrams commute (no projective defect). 16 this frame. Along the θ –direction, CP equivariance furnishes a family Aθ ( θ )implementing the Real lift (whenever it exists). The pair ( At,Aθ )forms a 2–connection whose 2–curvature has a mixed component Ftθ =∂tAθ−∂θAt+ [At,Aθ] + (coherence 2–morphisms).(51) Holonomy around a rectangular loop in ( t, θ )(evolve in t , apply CP, reverse in t , reverse CP) measures failure of path independence of adiabatic transport. The CP–induced topological phase eiθQ appears as a nontrivial holonomy unless θ is a CP fixed point; coherence data make this precise at the bicategorical level [67]. Define the curvature penalty H[θ] = 1 2X αZIα×S1 Ftθ 2dt dθ, (52) with a suitable G –compatible operator norm. A Real lift with full coherence exists if and only if H [ θ ] = 0, which occurs only at θ∈ { 0 , π} ; adding the vacuum penalty near π reproduces (49) . The gradient flow of His another avatar of (44). Intuition. In physical terms, (52) demands that the result of: (1) evolving in time, (2) implementing CP, (3) evolving back, (4) undoing CP, be the identity. The θ –phase obstructs this unless θ is a fixed point; higher coherence ensures the obstruction vanishes on all gluings. This is a “flatness” condition analogous to integrability, now in the parameter direction. E. Relationship to classical gradient flows Equation (44) is a gradient flow on a compact 1–manifold ( S1 ), so existence, uniqueness, and global convergence to a (stable) critical point are immediate. The usefulness of the analogy is that it clarifies the status of the selection: it is like harmonic map heat flow reducing the Dirichlet energy on maps [ 68 ], or the L2 –gradient flow of the entropy in Ricci flow [ 69 ], or Wasserstein gradient flows of free energies [ 70 – 72 ]. In all cases a nonnegative functional decreases monotonically under a flow in a space of objects (maps, metrics, probability measures). Here the objects are theories; the functional is C. F. A toy model Choose charges {Qk} = { 1 , 2 } and equal weights wk = 1, with a simple smooth bump b ( θ ) = δsin2((θ−π)/2) and parameter γ > 0. Then C(θ) = sin2θ+ sin2(2θ) + 1 2γδ sin2θ−π 2∆2 π,(53) ∂θC(θ) = sin(2θ) + 2 sin(4θ) + 1 2γδ sin(θ−π) ∆2 π.(54) The flow ˙ θ = −∂θC has equilibria at 0and π when ∆ π = 0; for ∆ π6 = 0 the term proportional to sin ( θ−π ) shifts the π critical point away and breaks its stability, while leaving θ = 0 a strict minimum. Linearization near 0gives ∂2 θC (0) = 2 + 8 > 0, so θ = 0 is asymptotically stable. Numerical phase portraits (not shown) display global attraction to 0for generic initial data when γδ∆2 π>0. G. Orthogonality of flows: state vs. theory It is essential to distinguish: (i) the state/observable flow in physical time t of Sec. V, which drives δY→ 0inside the selected θphys sector; and (ii) the theory–space flow in coherence time s of (44) , which selects θ= 0 across sectors. The two flows live on different spaces and commute in the sense that ∂t∂sobservables at fixed θ=∂s∂tobservables at fixed θ, because ∂s acts only on the label of the theory, while ∂t acts on expectations within that theory. This orthogonality ensures conceptual clarity and preserves superselection. 17 H. Limits, scope, and choices The functional C is not unique. Different choices of test bordisms {Mk} , weights wk , and vacuum– uniqueness diagnostics b ( θ )produce equivalent flows with the same stable fixed points, provided they separate CP fixed points and detect vacuum degeneracy near π . This is fully analogous to multiple Lyapunov functionals producing the same basin of attraction in state space. The construction is structural: it does not require computing F ( θ )nor imposing an explicit microscopic model of the Dashen phenomenon. It suffices to encode the axiom of unique vacuum near πvia a positive penalty. I. Summary of Section VI • Acoherence defect functional C [ θ ]on theory space measures failure of CP equivariance and vacuum uniqueness. Its zeros are exactly the members of {Zθ} admitting a CP–equivariant, reflection– positive extension with higher coherence and a unique vacuum. • The gradient flow ∂sθ = −∂θC is a categorical PDE in theory space. Its stable fixed points coincide with the structural selection set; under the uniqueness axiom, the unique stable fixed point is θ = 0. • A( t, θ )–space 2–connection on local Lax data yields an equivalent holonomy penalty functional whose vanishing is 2–flatness; only the CP fixed points admit a Real lift, and vacuum uniqueness excludes π. • The selection flow in theory space is orthogonal to the observable flow in physical time: no conflict with superselection arises. Together, they provide a coherent SM–only account: selection across sectors and relaxation within the selected sector. VII. HIGHER–CATEGORICAL LAX PAIRS: INTEGRABILITY AS COHERENCE–FLATNESS This section develops a higher–categorical generalization of the Lax formalism that unifies: (i) local spectral transport in time for the observable dynamics of Sec. V, and (ii) structural flatness in the parameter direction θ underpinning the selection principle of Secs. IV–VI. The outcome is a rigorous formulation in which “integrability” is promoted from classical isospectrality to coherence–flatness: the vanishing of appropriate 1– and 2–curvatures for a ( t, θ )–family of Lax data with higher coherence. Physically, the reversible part of the SM coarse–grained evolution is a gauge connection in a moving frame that transports spectral data, while the CP–equivariance and vacuum–uniqueness requirements appear as flatness in the θdirection with trivial holonomy only at θ= 0. A. Classical Lax pairs and zero–curvature equations (recap) Classically, a Lax pair for a nonlinear evolution ∂tu = F [ u ]consists of operators L [ u ]and M [ u ]such that ∂tL[u] = [ M[u], L[u] ],(55) so that the spectrum of L is conserved (isospectral flow) [ 73 , 74 ]. Equivalently, for matrix–valued connections At, Ax depending on u and a spectral parameter λ , the zero–curvature (Zakharov–Shabat) condition reads ∂tAx−∂xAt+ [At, Ax] = 0,(56) and encodes integrable PDEs such as KdV, NLS, sine–Gordon, etc. via AKNS/Zakharov–Shabat schemes [ 75 – 80 ]. Gauge transformations g act by Aµ7→ g−1Aµg + g−1∂µg , preserving (56) . The invariants are monodromies and scattering data, which evolve trivially under the flow. Intuition. The Lax pair encodes “parallel transport” of spectral data along time: the evolution is a pure gauge in the spectral bundle, hence isospectral. Our generalized setting will keep this picture but allow (i) dissipation (no exact isospectrality), and (ii) higher coherence across patch overlaps (time) and parameter changes (θ). 18 B. Quasi–Lax form of the SM observable dynamics (time direction) From Sec. V, the linearized generator of the deviation dynamics is A(t) = −Γ(T(t)) −S(H(t)) + a(t)−2∇·D(T(t)) ∇ · ,hu, viG=Zu>G(T)v d3x, (57) with G ( T ) = Σ −1 ( T )  0. The G –polar split A = S + K (symmetric dissipative S ≤ 0, skew reversible K† G = −K ) induces a G –unitary U solving ∂tU = KU , U ( t0 ) = 1 . Conjugating to the moving frame f δY=U−1δYyields (cf. (40)) ∂tf δY=e S(t)f δY+e R(t),e S:= U−1SU≤0,(58) where e R collects small adiabatic remainder terms ( ˙ T , ˙ H , ∇a , etc.). Define the Gramian operator in the moving frame, L(t) := Z∞ 0 ese S(t)Πese S(t)ds, Πfinite–rank projector selecting the slow modes,(59) which solves (formally) the Lyapunov equation e S† GL + Le S = − Π. Differentiating (59) and using ∂te S = [Ω ,e S ] + O ( η )(with Ω := U−1˙ U G –skew and η 1an adiabatic parameter) gives the quasi–Lax evolution ∂tL= [ M, L ] + R, M := Ω + 1 2L−1∂tLsym,kRk=O(η).(60) Here ( · ) sym is the G –symmetric part. Equation (60) shows that, to leading adiabatic order, the spectrum of L is transported isospectrally by the reversible connection M ; dissipation sits in the construction of L (Gramian) rather than in its transport. This is the time–direction Lax picture of our dissipative system: we cannot make the evolution isospectral for the state, but we can make it isospectral for a diagnostic operator that encodes the slow dissipative geometry. Physical meaning. The moving frame U removes non-dissipative mixing; in that frame the object L measures how the dissipative directions are weighted (the “shape” of decay ellipsoids). Adiabaticity ensures that L is transported mostly by a commutator with a connection M (rigid rotation in the G –metric), so its eigenvalues are essentially constant on each time patch, and changes are controlled by the small remainder R. C. Sheaf of local Lax data and higher coherence (time gluing) Let {Iα} be the adiabatic cover of the time axis (Sec. V F). On each Iα choose local Lax data ( Lα, Mα ) satisfying (60) with remainder kRαk = O ( η ). On overlaps Iαβ := Iα∩Iβ , pick G –unitary identifications (gauge transformations) gαβ relating the two pairs: Lβ=g−1 αβ Lαgαβ, Mβ=g−1 αβ Mαgαβ +g−1 αβ ∂tgαβ,(61) so that the zero–curvature defect ∂tAx−∂xAt + [ At, Ax ]remains small and consistent across the overlap (here ‘ x ’ is a formal label; there is no spatial scattering only the time transport). On triple overlaps Iαβγ , demand higher coherence: gαβ gβγ =zαβγ gαγ, zαβγ ∈U(1) (phase 2–cocycle),(62) and require that the 2–cocycle be cohomologically trivial in the adiabatic limit (or be absorbed into a redefinition of the local Mα ), so that the sheaf of local Lax data defines a global object up to O ( η ) ambiguities. The coherence penalty used in Sec. V F precisely measures the failure of (62) to hold with zαβγ = 1, and its control yields global decay. Mathematical remark. This is a standard descent problem: a prestack of local Lax objects with 1–morphisms gαβ on overlaps and 2–morphisms zαβγ on triple overlaps; triviality of the 2–cocycle in H2 achieves global gluing [ 81 – 83 ]. Our adiabatic estimates ensure that any residual 2–cocycle is small and can be absorbed into the error Rof (60). 19 D. Parameter direction (θ): Real lifts and 2–flatness Across the parameter circle S1 θ , CP acts by θ7→ −θ and orientation reversal on bordisms. Consider the bundle of local Lax data ( Lα ( θ ) , Mα ( θ )) over Iα×U , where U⊂S1 is a small arc. A Real lift is an anti–linear involution Rsuch that Lα(−θ) = RLα(θ)R−1, Mα(−θ) = − R Mα(θ)R−1+R∂θR−1,(63) compatible with the θ–space gauge identifications. Introduce the (t, θ)–connection At:= Mα(θ),Aθ:= Bα(θ),(64) where Bα generates CP transport in θ (the analog of Mα along the parameter direction). The ( t, θ ) 2–curvature has the mixed component Ftθ =∂tBα−∂θMα+ [Mα,Bα] + (coherence 2–morphisms).(65) In the strong sector, the CP–induced topological phase eiθQ appears as a holonomy obstruction for the loop: evolve in t , perform CP ( θ7→ −θ ), evolve back, undo CP. Precisely at CP fixed points θ∈ { 0 , π} the phase is its own inverse; only there can one achieve Ftθ = 0 with full higher coherence. The unique–vacuum axiom then excludes π, leaving θ= 0. Physical meaning. Parallel transport in ( t, θ )must be path–independent to be “integrable.” The topological θ –phase obstructs this unless θ is a fixed point; the Real structure is the anti–linear lift that identifies CP–related data. The higher–categorical coherence ensures that this flatness is global on all gluings; otherwise a residual 2–cocycle persists (an anomaly). E. Coherence–flatness as generalized integrability We can now formalize the integrability claim. Definition VII.1 (Coherence–flat Lax system) . A family { ( Lα, Mα,Bα ) } over {Iα×Uα} is coherence–flat if: (i) Time quasi–Lax: ∂tLα= [Mα, Lα] + Rαwith kRαk=O(η)and Rαcoherent under (61). (ii) θReal lift: there exist anti–linear Rαsuch that (63) holds compatibly with overlaps. (iii) 2–flatness: the mixed 2–curvature (65) vanishes up to O(η)and the coherence 2–cocycle on triple overlaps is trivial. Theorem VII.2 (Selection by coherence–flatness) . If a coherence–flat Lax system exists globally for the strong sector with unique vacuum, then θ = 0 ( mod 2 π ). Conversely, at θ = 0 such a system exists with η as small as the adiabaticity permits. Sketch. Assume a global coherence–flat system. Then 2–flatness forces θ to be a CP fixed point (otherwise the CP holonomy is nontrivial). Unique vacuum excludes π . Conversely, at θ = 0 the Real lift is trivial, the CP holonomy vanishes, and the time direction admits local quasi–Lax pairs by Sec. VII B; gluing is possible because the overlap 2–cocycle can be killed by choosing gαβ and Rαcoherently. F. GENERIC structure and the S+Ksplit The G –polar decomposition A = S + K is the operator–theoretic avatar of the GENERIC formalism in nonequilibrium thermodynamics [ 84 – 86 ]: a dissipative gradient part generated by a positive operator (here S ≤ 0) and a reversible Poisson part generated by a skew operator (here K† G = −K ). The moving frame U integrates the Poisson geometry, yielding a pure gradient flow in the transported variables (cf. (58) ). The quasi–Lax construction (60) can be seen as the representation–theoretic content of GENERIC: the dissipative metric induces a Gramian L whose transport is generated by the reversible connection M . Intuition. GENERIC says “reversible is Hamiltonian, irreversible is gradient”; our higher–Lax says “reversible is a connection (gauge), irreversible is encoded in a Lyapunov operator whose spectrum is transported by that connection.” 20 G. Spectral stability, pseudospectra, and robustness Non–selfadjoint generators can exhibit large transient growth unrelated to eigenvalues [ 87 ]. The moving frame suppresses such transients by removing the skew mixing, and the G –Lyapunov identity controls growth. Quantitatively, let σε ( L )be the ε –pseudospectrum. Under the quasi–Lax evolution ∂tL= [M, L] + Rwith kRk ≤ η, distσε(L(t)), σε+O(η)(L(t0))≤C η (t−t0),(66) so spectral diagnostics of dissipation are stable on adiabatic patches. This robustness is the operator–level input behind the global contraction result. H. Intuitive narrative: moving frames, flat parameter loops The physical picture is simple. In time, pass to the moving frame where all reversible couplings are gauged away; what remains is pure decay, with a shape (set by L ) that is merely rotated by the reversible connection as time passes. Across θ , demand that if we traverse a small rectangle in ( t, θ )—evolve, CP, evolve back, inverse CP—we return to the same point. The topological θ –phase prevents that unless θ is a CP fixed point; among the fixed points, only θ = 0 supports a unique vacuum and full positivity. That is “integrability as coherence–flatness.” I. Summary of Section VII • The SM observable dynamics admits a quasi–Lax formulation on adiabatic patches: a dissipative Gramian L is transported isospectrally to leading order by a reversible connection M , with small remainders controlled by adiabaticity. • Local Lax data on time patches assemble into a sheaf with 1–morphisms on overlaps and 2– morphisms on triple overlaps. Controlling the 2–cocycle (coherence) produces global decay, matching the Lyapunov theory. • In the parameter ( θ ) direction, a Real lift and vanishing mixed 2–curvature are possible only at CP fixed points; unique vacuum excludes π, leaving θ= 0. • Thus, integrability in our sense is coherence–flatness: time–direction isospectral transport (up to small errors) + parameter–direction 2–flatness. This recovers the structural selection and the within–sector relaxation in a single higher–Lax framework. VIII. DIAGNOSTICS, PHENOMENOLOGY, AND RELATION TO AXION MECHANISMS The previous sections established two logically distinct layers of our proposal: a structural selection across the family {QCDθ} (Sec. IV, Sec. VI) that singles out θ = 0, and a dynamical relaxation of CP–odd deviations within the selected sector on FRW backgrounds (Sec. V, Sec. VII). This section develops (i) concrete diagnostics—primarily lattice–Euclidean and effective theory checks—that can probe our assumptions; (ii) phenomenological implications for EDMs, heavy–ion observables, and cosmology; and (iii) a careful comparison with axion and Nelson–Barr frameworks. The goal is to show how the structural selection principle can be confronted with data and computations without adding new particles, and to clarify the scope and limitations of this SM–only approach. A. What is observable if the selection fixes θ= 0? If θ is structurally fixed to zero, the strong sector is CP–even in equilibrium and the leading SM contribution to hadronic EDMs arises from weak CP phases and higher–order electroweak effects. In chiral effective theory, the neutron EDM dninduced by a nonzero θobeys dn(θ)≃κnθ e ·cm, κn∼ O(10−16),(67) 21 with κn depending on low–energy constants and πN loops [ 88 – 91 ]. The current neutron EDM limit |dn|.few × 10 −26 e·cm then implies |θphys|. 10 −10 , consistent with θ = 0 as a structural choice. In our framework the dynamic statement is stronger: within the θphys = 0 sector, nonequilibrium CP–odd deviations in QCD operators relax according to Sec. V, so that operational CP violation in the strong subsector is suppressed both statically and dynamically. The weak sector does generate tiny EDMs ( dweak n∼ 10 −32 –10 −31 e·cm , model dependent), well below foreseeable sensitivities [ 90 , 91 ]. Therefore, our SM–only selection predicts a null result in hadronic EDM searches at current and near–future precision, in contrast to many BSM frameworks that anticipate signals. Intuition. Structural selection fixes the vacuum label to θ = 0; dynamical relaxation guarantees that any CP–odd displacement of slow QCD observables damps away on cosmic backgrounds. Operationally, both the equilibrium value and the trajectory are CP–even in the strong subsector. B. Lattice–Euclidean diagnostics of the structural axioms The selection theorem of Sec. IV rests on CP equivariance with higher coherence, reflection positivity, the absence of a CP coherence anomaly, and unique vacuum. While these are formulated functorially, several are testable with Euclidean lattice techniques: (i) CP equivariance at θ = 0.At θ = 0, CP requires that the Euclidean measure be invariant under orientation reversal, up to complex conjugation of observables. Numerically one checks that expectation values of CP–odd operators vanish and that CP–even correlators coincide with their reflected counterparts within statistical errors, after defining the topological charge Q with gradient flow or improved operators [ 93 – 95 ]. This is standard but conceptually important: it asserts the existence of a Real structure at θ = 0 in the sense of Sec. IV. (ii) Detecting a CP coherence anomaly at θ = π .In pure Yang–Mills, a mixed anomaly between CP and the one–form center symmetry at θ = π obstructs a Real lift. On the lattice, this shows up as an unavoidable sign problem or a nontrivial phase in CP–related loop operators [ 95 ]; more systematically, one can test for inconsistencies in CP–gluing identities on simple bordisms (tori, lens spaces) by comparing log Z ( θ ; M )and log Z(θ;M) on CP–pairs M, M . The obstruction should vanish only at CP fixed points and, in QCD with light fundamentals, the remaining test is vacuum uniqueness near π. (iii) Vacuum uniqueness near θ = π .A cusp in F ( θ )at π indicates Dashen degeneracy. Lattice determinations of χ ( T ) = ∂2 θF (0) are mature [ 94 , 95 ]; while a direct simulation at θ = π is afflicted by severe sign problems, analytic continuation from imaginary θ combined with spectral methods can bound the possibility of a cusp. Our structural axiom is that QCD–like theories with light quarks admit a unique vacuum at θ = 0 and do not develop a first–order transition at π ; this can be falsified (or supported) by improved constraints on the θ–dependence of F. (iv) A practical proxy for the coherence functional. The functional C [ θ ]of Sec. VI B can be estimated by choosing a small basis of manifolds with known Q and computing (via reweighting or imaginary θ ) the combinations 1 −cos (2 θQk ), then adding a penalty for any sign of nonuniqueness near π . This provides a concrete lattice test of the theory–space picture: the unique minimum at θ = 0 is tantamount to our selection. C. Finite–temperature and cosmological considerations At high temperature, the topological susceptibility decreases rapidly, χ(T)∼T−bfor TΛQCD, b > 0,(68) in broad agreement with dilute instanton gas expectations and modern lattice results [ 94 , 95 ]. In our dynamical analysis, GQQ = χ−1 ( T )weights the deviation energy; thus, as T increases, fluctuations in Q are suppressed in equilibrium but deviations in Q are more strongly penalized by G , leading to faster decay of Q –dominated modes in the Lyapunov metric. The FRW dilution term S ( H )further damps homogeneous modes. A simple homogeneous estimate for a single slow variable with linearized rate γ ( T ) yields ∂tδY =−γ(T) + s HδY, δY (t) = δY (t0) exp−Zt t0γ(T(t0)) + s H(t0)dt0,(69) 22 so the integrated damping rate is the sum of microscopic and cosmological contributions. On subhorizon comoving scales, diffusion adds an a ( t ) −2D ( T ) k2 rate. The upshot is that in any adiabatic epoch, CP–odd deviations relax rapidly compared to Hubble time for standard QCD transport numbers [56]. Domain walls and initial conditions. Because selection is structural, not spontaneous symmetry breaking, there are no θ –domain walls: the theory space has a unique admissible member rather than degenerate vacua within a fixed theory. Therefore, the usual cosmological domain wall constraints associated with a field rolling to different θvalues do not apply. D. Heavy–ion collisions and CP–odd fluctuations Event–by–event CP–odd fluctuations in heavy–ion collisions (e.g. the chiral magnetic effect) have long been proposed as windows into topological charge dynamics [ 96 , 97 ]. Our framework is consistent with transient CP–odd fluctuations: Sec. V describes precisely how such deviations relax under QCD transport. The structural selection at θ = 0 says that the equilibrium strong sector is CP–even; it does not prohibit nonequilibrium CP–odd correlators sourced by initial geometry, magnetic fields, or vorticity. Quantitative comparison requires realistic Γ( T ), D ( T ), and background evolution, which can be imported from hydrodynamic modeling; our Lyapunov control ensures that these deviations decay with rates fixed by Euclidean susceptibilities and Kubo coefficients. E. Relation to axion and Nelson–Barr mechanisms The Peccei–Quinn (PQ) mechanism promotes θ to a dynamical field a ( x ) /fa via an anomalous global symmetry; classical evolution and quantum effects drive θeff to zero, and the axion is a dark matter candidate [ 98 – 100 ]. Nelson–Barr (NB) models retain θ = 0 at tree level by construction of the quark Yukawas. Our proposal differs: • No new fields. We retain the SM field content; selection is imposed by structural coherence conditions (Sec. IV). There is no axion dark matter in this minimal version; cosmological predictions differ. • Superselection respected. In PQ, θeff changes dynamically in spacetime. Here, theory–space flow (Sec. VI) is a meta–dynamics on the label of the theory and does not contradict superselection. • Phenomenology. Axion searches (haloscopes, helioscopes, NMR techniques) probe a new particle; null results there are compatible with our SM–only selection. Gradient flows vs. c – and a –theorems. It is instructive to compare our coherence gradient flow with known monotonic functionals in QFT, such as Zamolodchikov’s c –function in 2D and the a –theorem in 4D [ 101 , 102 ]. In all cases a nonnegative functional decreases monotonically along an abstract flow (RG, here theory–space coherence), selecting fixed points with special structure (conformal invariance, here CP equivariance with positivity). The analogy clarifies that our “PDE” lives on theories, not in spacetime. F. Limitations and falsifiability The selection theorem is conditional on explicit axioms: (a) CP equivariance with higher coherence. Violations would appear as persistent CP–odd Euclidean effects at θ= 0 or inconsistencies under CP gluing. (b) Reflection positivity. A failure would obstruct OS reconstruction; lattice tests can probe positivity of correlators and reflection symmetry. (c) No CP coherence anomaly. In pure YM at θ = π a mixed anomaly is known; with light fundamentals, this obstruction is believed absent, but improved anomaly matching could, in principle, reveal hidden obstructions. (d) Unique vacuum. A cusp at θ = π would violate the axiom; improved constraints on F ( θ )would test this. 23 Any future evidence of a nonzero θphys (e.g. an unambiguously hadronic contribution to EDMs inconsistent with θ = 0) would falsify the selection as stated. Conversely, precise lattice–Euclidean checks of CP equivariance and vacuum uniqueness, combined with increasingly strong EDM null limits, would support the structural hypothesis. G. A practical program of tests We summarize a feasible roadmap: 1. Lattice CP checks at θ = 0 . Use gradient flow definitions of Q to verify CP–odd correlators vanish and CP–even correlators are invariant under reflection to high precision; verify OS positivity numerically [93]. 2. θ–dependence of F ( θ ) . Use imaginary– θ reweighting and analytic continuation to constrain the curvature and potential cusp near π[94, 95]. Bound any Dashen–type degeneracy. 3. Coherence functional proxy. Compute a truncated C [ θ ]using a basis of manifolds with Q = 1 , 2 and verify a unique minimum at θ= 0 within uncertainties. 4. Transport coefficients. Determine Γ( T ), D ( T )from Kubo formulas for relevant CP–odd operators and currents; insert into (34) to simulate relaxation in homogeneous FRW and in simple expanding flows. 5. EDM phenomenology. Combine chiral EFT estimates (67) with latest experimental bounds [ 92 ] to maintain the θ= 0 hypothesis; update as limits improve. H. Intuitive narrative The picture is this. The strong sector lives on a topological circle of theories labeled by θ , a parameter superselected in the SM. Demanding that CP act coherently with Euclidean gluing and positivity pins us to the CP fixed points; demanding a unique vacuum removes π , leaving θ = 0. This is not time evolution; it is a consistency filter on admissible theories. Once the theory is fixed, the Universe expands; QCD transport and cosmological dilution damp out any CP–odd deviation that the early cosmos or a heavy–ion collision might briefly create. Lattice computations can probe the structural assumptions directly, and experiments can keep pushing EDM bounds. In this way, the proposal is both physically modest (no new fields) and practically testable. I. Summary of Section VIII • If selection fixes θ = 0, hadronic EDMs receive only tiny weak contributions. This aligns with current null results and yields clear expectations for future EDM searches. • Lattice–Euclidean diagnostics can test CP equivariance, reflection positivity, and vacuum uniqueness, and can even approximate the theory–space coherence functional. • Finite–temperature behavior ( χ ( T )decreasing) and FRW expansion imply rapid decay of CP–odd deviations; there are no θ–domain walls because selection is structural. • The framework is complementary to axion/Nelson–Barr: it does not add particles nor violate superselection; it replaces dynamical relaxation of θ by structural selection of the admissible theory. • The proposal is falsifiable: evidence for θ6 = 0, a CP coherence anomaly with fundamentals, or a cusp at π would contradict the axioms; conversely, improving Euclidean and experimental constraints strengthens it. 24 IX. CONCLUSION We close by assembling the mathematical and physical threads of this work into a single, coherent picture. The strong–CP problem has two logically distinct faces in the Standard Model (SM): an across– sector question about which member of the family {QCDθ} is physically admissible, and an in–sector question about how strong–sector observables evolve in time once that member is chosen. Our proposal answers the first by a structural selection principle grounded in higher–categorical coherence, and the second by a dynamical relaxation theory on cosmological backgrounds. Both layers are SM–only and respect the superselection of θ. Main statements (S1) Topological superselection of θ. The integer topology π3 ( SU (3)) = Z yields a central unitary in the extended algebra of gauge–invariant local observables, acting by the character einθ . The Hilbert space decomposes as a direct integral R⊕Hθdµ ( θ )on which all local SM observables act diagonally. Equivalently, Z ( θ ) = PQeiθQZQ with Q∈Z ; θ is a label, not a field. Thus, no SM spacetime dynamics can change θ. (S2) Structural selection across sectors. Viewing Zθ as a symmetric monoidal Euclidean field theory functor, impose: (i) CP–equivariance with full higher coherence; (ii) reflection positivity; (iii) no CP coherence anomaly; (iv) unique vacuum. CP equivariance forces the CP fixed points by eiθQ =e−iθQ (∀Q∈Z) =⇒θ∈ {0, π}(mod 2π), and (iii)–(iv) eliminate θ=π. Hence only θ= 0 is admissible (§IV). (S3) Dynamical relaxation within a sector. Fix θphys and let δY collect coarse–grained CP–odd slow variables. On FRW, the Schwinger–Keldysh → Mori–Zwanzig reduction yields a parabolic evolution, ∂tδY=−Γ(T) + S(H)δY+a−2∇·D(T)∇δY+N(δY;T) + ξ−˙ T ∂TYeq(θphys, T), with Kubo relations for Γ , D and G ( T ) = Σ( T ) −1 0a Lyapunov metric. One obtains a global contraction inequality (E=1 2RδY>GδY) ˙ E≤ − c?E−a−2Z(∇δY)>(GD)(∇δY)d3x+adiabatic remainders, c?>0, so CP–odd deviations relax in physical time inside the chosen sector (§V). (S4) Coherence flow and higher–Lax unification. Acoherence defect functional C [ θ ]penalizes failure of CP equivariance and vacuum uniqueness. Its gradient flow in theory space, ∂sθ(s) = −∂θC[θ(s)], has the unique stable fixed point at θ = 0 and is orthogonal to physical time (§VI). In a higher–Lax formulation, local time–direction transport is (quasi)isospectral after removing reversible mixing, and the parameter–direction flatness condition Ftθ =∂tAθ−∂θAt+ [At,Aθ] + (coherence 2–morphisms) = 0 is achievable only at CP fixed points; unique vacuum then selects θ= 0 (§VII). Physical narrative The structural picture mirrors a familiar story: in Galilean quantum mechanics, mass is a central charge and thus a superselection label; embedding into the larger, more coherent Poincaré symmetry trivializes the central extension and renders mass a Casimir. Here, the CP–induced topological phase eiθQ plays the role of a central twist on the strong sector. Embedding the strong sector into the larger higher–categorical framework—demanding CP equivariance with higher coherence and positivity—trivializes the twist only 25 at the CP fixed points, and unique vacuum removes the π branch. No spacetime dynamics crosses superselection sectors; rather, coherence filters the family of sectors to a single admissible theory, QCDθ=0 . Within that theory, ordinary SM transport on FRW drives any CP–odd displacement back to equilibrium with a Lyapunov control fixed by Euclidean susceptibilities. Mathematics in one page For the reader’s convenience, the four critical equations are: CP fixed–point condition: eiθQ =e−iθQ (∀Q∈Z)⇐⇒ θ∈ {0, π}(mod 2π). (70) Lyapunov decay (in sector): ˙ E≤ − c?E−a−2Z(∇δY)>(GD)(∇δY)d3x+· · · . (71) Coherence gradient flow (on theories): ∂sθ=−∂θC(θ),Critstable(C) = {θ= 0}.(72) 2–flatness (higher–Lax): Ftθ = 0 only at CP fixed points; unique vacuum excludes θ=π. (73) Equations (70) and (73) encapsulate the structural selection; (71) encodes dynamical relaxation; (72) provides a meta–dynamical avatar that neither violates nor circumvents superselection. Consistency, scope, and falsifiability Our selection theorem is conditional on explicit axioms. It can be falsified by any of the following: (i) an unambiguous hadronic EDM signal consistent with θ6 = 0; (ii) a robust, lattice–Euclidean indication of CP– incoherence or reflection–positivity failure at θ = 0; (iii) incontrovertible evidence for vacuum degeneracy (a cusp in F ( θ )) at θ = π in QCD with light quarks; (iv) a CP coherence anomaly persisting with fundamentals. Conversely, precise Euclidean tests of CP equivariance and positivity, increasingly strong EDM null bounds, and improved constraints on F ( θ )support the structural hypothesis. Importantly, selection by coherence is complementary to axion and Nelson–Barr solutions: it adds no new fields and preserves superselection; it is a filter on admissible theories rather than a dynamical field that rolls. Outlook Several technical directions naturally follow: •Extended field theories. Upgrade from non–extended to fully extended functors to sharpen the role of defects, boundaries, and one–form symmetries within the coherence data. •Cohomological classification. Make precise the obstruction class underlying CP coherence at θ=πwith fundamentals; identify sufficient conditions for its vanishing. •Lattice proxies for C [ θ ] . Implement truncated versions of the coherence functional on small sets of manifolds with controlled topology. •Transport numerics. 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