Cosmic Classicalisation from Intrinsic Self-Measurement: A Categorical Approach to the Quantum-to-Classical Transition in Cosmology

Cosmic Classicalisation from Intrinsic Self–Measurement: A Categorical Approach to the Quantum–to–Classical Transition in Cosmology Andrei T. Patrascu 1 1 FAST Foundation, Destin FL, 32541, USA email: andrei.patr[email protected] A persistent puzzle in cosmology is how quantum fluctuations produced during inflation become the classical stochastic perturbations that seed cosmic structure. Standard explanations appeal to environment–induced decoherence, but these rely on external degrees of freedom and do not fully resolve the measurement problem nor explain why outcomes are definite. In this work we propose a new mechanism rooted in higher categorical coherence: the failure of pentagon and hexagon coherence relations induces a local, covariant completely positive (CP) semigroup dynamics—a form of intrinsic self–measurement built into quantum field theory. We show that when applied to super– Hubble modes, this dynamics suppresses phase–space coherences, grows momentum fluctuations, and drives the Wigner function to positivity, thereby classicalising the perturbations while leaving the inflationary power spectrum unchanged. The resulting picture is not oscillatory between quantum and classical, but a “conveyor belt” in which each comoving mode has a quantum youth and a classical adulthood, and new modes are continuously born at sub–Hubble scales. The intrinsic channel provides a universal, symmetry–preserving floor to decoherence, complements environment–induced mechanisms, and ensures an arrow of time consistent with horizon thermodynamics. We discuss parametric estimates of the decoherence factor, potential observational consequences in the cosmic microwave background and large–scale structure, and the conceptual unification of measurement, renormalisation, and cosmological irreversibility. I. INTRODUCTION: FROM COHERENCE TO CLASSICALITY Two long-standing puzzles and a single structural proposal Quantum theory and quantum field theory (QFT) have faced, since their inception, two puzzles that often travel on parallel tracks. The first is the measurement problem: how and why sharply definite outcomes emerge from unitary, linear evolution of quantum states [ 1 – 3 , 113 ]. The second is the ultraviolet problem of QFT: perturbative calculations generically produce divergences whose removal— renormalisation—is technically superb yet conceptually unsatisfying when its finite parts depend on external prescriptions [ 40 , 90 , 135 – 139 ]. In cosmology, these threads meet acutely: quantum fluctuations during inflation must become the classical stochastic seeds of structure we observe in the cosmic microwave background (CMB), yet the precise mechanism by which quantum phases disappear without external observers remains debated [12, 18, 57, 58, 65, 66, 79]. This paper develops and applies a single structural idea intended to speak to all three arenas (measurement, renormalisation, cosmological classicality). The idea is that higher categorical coherence—the web of consistency relations among composition, associativity, and braiding—may possess a small, local curvature. When that curvature vanishes beyond the lowest level (“triangle” coherence), one obtains ordinary quantum mechanics (QM): linear, unitary, with noncommuting observables classified by central extensions. When higher coherence (“pentagon/hexagon”) is slightly curved, the induced effective dynamics on the algebra of observables is no longer purely unitary; instead, it is a local, covariant completely positive (CP) evolution. In physical terms, this is an intrinsic self–measurement channel: a tiny, universal drift that turns phase coherence into classical stochasticity while preserving causality and symmetry. The same drift, when viewed at short distances, reduces the degree of singularity of time–ordered products and thereby selects finite renormalised correlators without external cutoffs. In cosmology, it provides a structural floor to decoherence that ensures super–Hubble modes classicalise even in the limit of minimal environmental coupling. Triangle coherence, central extensions, and the safety of unitarity The baseline for our discussion is the well–known fact that quantum kinematics already encodes a mild failure of classical composition: phases may accumulate when operations are composed in different orders. 2 In group–theoretic language, this appears as a projective representation of a symmetry group G, U(g)U(h) = ω(g, h)U(gh), ω :G×G→U(1),(1) where the scalar multiplier ω—the factor set—obeys the associativity constraint ω(g, h)ω(gh, k) = ω(h, k)ω(g, hk).(2) Equation (2) is precisely the 2–cocycle condition in group cohomology; rephasings U ( g ) 7→ α ( g ) U ( g ) transform ω7→ ω δα , i.e. add a 2–coboundary. Thus the obstruction to strict representations is the class [ ω ] ∈H2 ( G, U (1)) [ 33 , 34 ]. Physically, a nontrivial [ ω ]amounts to passing from G to a central extension b G by U (1), upon which U lifts to an ordinary unitary representation. The quintessential example is the Heisenberg group: the canonical commutation relations (CCR) are a central extension of the Abelian group of phase–space translations; they introduce a central element (Planck’s constant) but leave unitary time evolution and linearity intact. In field theory, many familiar “central charges” (e.g. in conformal algebras) are of exactly this nature. From the categorical viewpoint, the “triangle” coherence is the statement that the left/right unit constraints and the associator are compatible; failure at this level is still captured by H2 , i.e. by phases. The upshot is important: triangle–level defects change commutation relations (via a central term) but do not force us outside the realm of unitary, linear quantum mechanics. States evolve unitarily; pure states remain pure; the Born rule’s convex linearity is preserved [ 2 ]. In algebraic terms, the dynamics is implemented by a one–parameter group of ∗ –automorphisms {αt} on the observable algebra, generated by a derivation i[H, ·](Stone’s theorem) [40]. Beyond central extensions: pentagon/hexagon curvature and open dynamics Higher coherence data constrain associativity and braiding. In a monoidal category, the associator aX,Y,Z : ( X⊗Y ) ⊗Z→X⊗ ( Y⊗Z )obeys Mac Lane’s pentagon identity; in braided categories, the braiding cX,Y and the associator obey hexagon identities [ 35 – 37 ]. Deviations from trivial associators/braidings are classified by higher cohomology (e.g. H3 for associators, with striking manifestations in Dijkgraaf–Witten TQFTs) [ 38 , 39 ]. Heuristically, one may speak of a coherence curvature [Ω] ∈H≥3 : the failure of higher polygons to commute. The physical claim we articulate is that a small, local [Ω] 6 = 0 cannot, in general, be absorbed into a central extension. Intuitively this is because pentagon/hexagon phases cannot be removed by rephasing objects without spoiling compatibility elsewhere: unlike (1) – (2) , the “gauge freedom” is insufficient to trivialise [Ω]. When one insists (as physics does) on locality, covariance, and positivity, the effective dynamics induced by [Ω] on the observable sector is no longer implementable by a ∗ –automorphism group. Rather, it is a completely positive, trace–preserving (CPTP) semigroup d dtρt=L(ρt) = −i[H, ρt] + X µLµρtL† µ−1 2{L† µLµ, ρt},(3) with local jump operators Lµ and a positive Kossakowski matrix [ 29 , 30 , 43 , 67 , 68 , 87 , 88 ]. Equation (3) is the celebrated GKLS–Lindblad generator of a CP semigroup. It defines a linear CPTP map on density matrices (hence preserves convex mixtures), but it is nonunitary in that it turns pure states into mixed states in finite time. In the Heisenberg picture, L∗ is a completely dissipative derivation on the observable algebra; locality is implemented by constructing Lµ as smeared local densities and using the split property of local nets [40, 139]. Why does [Ω] 6 = 0 lead to (3) ? There are two complementary explanations. First, by Stinespring’s dilation theorem, any CP map arises from unitary evolution on a larger Hilbert space followed by a partial trace; categorically, a nontrivial [Ω] may be interpreted as coupling the observable category to an auxiliary sector (“coherence ancillae”) carrying the curvature. Tracing out this sector yields (3) .Second, on the Schwinger–Keldysh contour, higher–coherence phases transgress to influence functionals whose quadratic part is positive and local; this is precisely the noise kernel that appears in (3) . Either way, the essential point is structural: triangle defects change commutators but leave dynamics unitary; higher defects generically produce open dynamics, i.e. self–measurement. 3 Linearity versus nonlinearity. The minimal, physically safe incarnation of self–measurement is the linear CPTP semigroup (3) . It preserves convex linearity (no–signalling) and is compatible with relativistic locality when constructed from local densities. In more elaborate scenarios one may contemplate state–dependent (nonlinear) feedback of the curvature sector onto the observable state; then the generator becomes nonlinear in ρ and additional constraints are required to prevent signalling. In this paper we concentrate on the linear CPTP case as the baseline; its mere nonunitarity suffices to produce classical outcomes and to tame ultraviolet behaviour. Renormalisation as dynamical self–measurement At short distances, quantum fields possess singular correlation functions controlled by their scaling degree [ 138 , 139 ]. Time–ordered products require extension of distributions to coincident points; the Epstein–Glaser (EG) method accomplishes this causally, at the cost of finite renormalisation ambiguities [ 137 ]. A key observation is that the dissipative part of (3) , constructed locally from admissible densities, reduces the scaling degree of composite fields and selects a canonical extension. Physically, the self–measurement channel t7→ etL acts as a symmetry–respecting, causal filter that damps purely off–shell, high–frequency contributions (while preserving on–shell commutators). The result is a dynamical renormalisation: loop integrals become finite without external regulators, and the finite parts are fixed by the local structure of the channel rather than by scheme choice. This is compatible with Ward/Slavnov–Taylor identities when the Lµare gauge invariant (or BRST covariant) [138, 139]. Cosmological classicality: a conveyor belt, not an oscillation Cosmology furnishes a natural arena to test the consequences of a tiny, universal self–measurement. During inflation, vacuum fluctuations of light fields are stretched beyond the Hubble radius and subsequently “freeze” [ 12 , 65 ]. Standard environment–induced decoherence—entanglement of long–wavelength modes with short ones or with gravitational nonlinearities—does a great deal to explain the emergence of classical stochasticity [ 57 , 58 , 79 , 113 ]. Yet, strictly speaking, such arguments rely on external environment and do not by themselves enforce a monotone arrow of time in the autonomous sector. The self–measurement channel provides precisely such a floor: it suppresses off–diagonal coherences in the field–amplitude basis for super–Hubble modes while leaving the power spectrum unchanged (for quantum–nondemolition choices of Lµ ). The global picture is not that the universe oscillates between quantum and classical. Rather, there is a conveyor belt: new modes are born quantum inside the horizon, then—as expansion carries them out—they are squeezed and gently classicalised by both environment and the intrinsic CP flow. Once classicalised, they do not re–quantise; the GKLS evolution is one–way (relative entropy decreases under detailed balance [ 87 , 88 ]). Thus the infrared sector grows ever more classical in aggregate, while a thin ultraviolet fringe of newborn modes remains quantum at any given time. What is preserved, what changes The construction deliberately preserves the pillars that make QFT predictive: locality (microcausality), covariance, canonical commutation relations on shell, and gauge symmetry (in anomaly–free sectors). In scattering, the open dynamics appears as an inclusive optical theorem with a positive “loss” term; in integrable (1 + 1)–dimensional models, jump operators built from conserved densities act as elastic dephasing, leaving the exact S –matrix intact. What changes are off–shell interferences, spectral weights (reshaped positively), and the finiteness/uniqueness of composite correlators. In cosmology, the power spectrum remains the same at leading order, but phase information is erased in a controlled, symmetry–compatible way. In the remainder of the paper we formalise the curvature–to–GKLS map at the infinitesimal level, establish locality and covariance of the generator, show how the channel selects EG extensions, and then apply the framework to the quantum–to–classical transition of inflationary perturbations. We close with observational prospects and bounds. 4 II. FROM CATEGORICAL COHERENCE TO INTRINSIC SELF–MEASUREMENT A. Triangle versus higher coherence: from central extensions to curvature A convenient starting point is the distinction between triangle coherence (which leads to projective representations and central extensions) and higher coherence (pentagon/hexagon), whose obstructions live in higher group– or categorical–cohomology and cannot, in general, be absorbed into phases. Triangle coherence and H2. Let G act by symmetries on a quantum system. A projective representation U:G→ U(H)satisfies U(g)U(h) = ω(g, h)U(gh), ω :G×G→U(1),(4) with the associativity constraint ω(g, h)ω(gh, k) = ω(h, k)ω(g, hk).(5) Equation (5) is the 2–cocycle condition; rephasings U ( g ) 7→ α ( g ) U ( g )change ω7→ ω δα . Thus [ ω ] ∈ H2 ( G, U (1)) classifies inequivalent projective lifts and corresponds to a central extension 1 →U (1) → b G→G→ 1on which U becomes an ordinary unitary representation [ 33 , 34 ]. Physics: triangle–level defects add phases (e.g. CCR central terms), but the dynamics remains unitary and linear; pure states evolve by ∗–automorphisms αt(A) = eitH Ae−itH . Higher coherence and H≥3. In a rigid monoidal (possibly braided) ∗ –category C (the superselection category of charges), the associator aX,Y,Z : ( X⊗Y ) ⊗Z→X⊗ ( Y⊗Z )and the braiding cX,Y obey pentagon and hexagon identities [ 35 – 37 ]. Obstructions to trivial associators are measured by classes akin to H3 (e.g. in pointed fusion categories and Dijkgraaf–Witten models [ 38 ]); braiding data relate to higher classes as well (quasi–Hopf twists [39]). We write [Ω] ∈H≥3(C, U(1)) for the coherence curvature class. Unlike H2 data, a nonzero [Ω] cannot, in general, be gauged away by rephasings without violating other coherences. Physically, [Ω] 6 = 0 signals that the observable sector is not dynamically closed under a strictly unitary evolution on its Hilbert space: there exists an effective ancillary sector (the “coherence ancillae”) that carries the curvature and whose elimination produces open dynamics. B. From curvature to GKLS: two constructions We present two complementary routes from a small, local coherence curvature to a local completely positive (CP) generator of dynamics: a dilation–based construction and a Schwinger–Keldysh (SK) influence–functional construction. Both yield the Lindblad–GKLS form and show how to maintain locality and covariance. 1. Dilation/Stinespring route Let A be a local net O7→ A ( O )on a globally hyperbolic spacetime M , obeying isotony, locality, and covariance [ 40 ]. Fix a double cone ObM and assume the split property: there exists a type I factor N with A(O1)⊂ N ⊂ A(O2), O1bObO2, so that localized operations can be implemented within N [ 41 , 42 ]. Suppose the higher coherence class [Ω] is supported in O and small (infinitesimal parameter ε ). Then one can build a normal CP, unital map Φ∗ ε:A(O2)→A(O2)of the Kraus form Φ∗ ε(A) = r X µ=1 V† µA Vµ, Vµ∈ N,X µ VµV† µ=1,(6) 5 whose Kraus operators Vµ are local functionals of the curvature data (precisely, of a representative cocycle κ of [Ω]) and of the local fields. By Stinespring’s theorem [ 43 ], Φ ∗ ε arises from a unitary Uε on H⊗K and a fixed ancilla state η: Φ∗ ε(A) = hη, U† ε(A⊗1)UεηiK. Physically, the ancilla K carries the curvature sector; locality is guaranteed because Vµ∈ N ⊂ A ( O2 ). A standard argument (Evans–Lewis [ 44 ]) promotes { Φ ε} to a semigroup { Φ ∗ t = etL∗}t≥0 whose generator has GKLS form on A(O2): L∗(A) = i[HO, A] + X µL† µALµ−1 2{L† µLµ, A}, Lµ∈ N.(7) Locality: For any B∈A ( O0 2 ) ⊂A ( O2 ) 0 (spacelike complement), we have Φ ∗ t ( B ) = B because Vµ commute with B and PµVµV† µ = 1 . Hence expectation values of B are unchanged by operations in O2 , i.e. no signalling. Covariance follows if the assignment O7→ ( HO,{Lµ} )is natural under the spacetime symmetry. 2. Schwinger–Keldysh influence functional route In the SK formalism, one computes expectation values by doubling fields Φ ± on a closed time contour [ 45 , 46 , 115 ]. An effective open dynamics is encoded by an influence functional eiSIF[Φ+,Φ−] . For small, local curvature [Ω] supported in O, its leading imaginary part is quadratic and positive: SIF[Φ+,Φ−] = Seff[Φ+]−Seff[Φ−] + i 2Zd4x d4yO+(x)−O−(x)N(x, y)O+(y)−O−(y)+··· (8) Here O is a local composite density fixed by symmetry/BRST (e.g. field amplitude, current, stress), and N≥ 0is a noise kernel determined by [Ω] via a transgression map (curvature → two–point kernel). Hadamard admissibility ensures microlocal spectrum constraints. The master equation obtained by differentiating the SK generating functional is precisely of GKLS form for any Gaussian N: ˙ρ=−i[Heff, ρ]−1 2ZZ d4x d4y N(x, y)O(x),[O(y), ρ]+··· ,(9) which is GKLS with a continuous set of jump operators (the spectral decomposition of N ). Locality/covariance follow from the support/transform properties of N . In cosmological applications, one often chooses quantum–nondemolition (QND) channels (e.g. O = φ for a scalar mode) so that diagonal variances (the power spectrum) are unchanged at leading order while off–diagonal coherences decay. C. Triangle defects preserve unitarity; higher defects generate open dynamics We now justify, within the above frameworks, the two central claims: Claim A (Triangle ⇒unitary, linear). If the only coherence defects are triangle–level ( H2 ) phases, then the dynamics on the observable algebra is implemented by a one–parameter group of ∗ –automorphisms (Hamiltonian evolution). Reason. H2 phases correspond to central extensions of symmetry groups or of field algebras (CCR central charge). They modify commutation relations but not the structure of the GNS dynamics: the generator is a derivation i [ H, · ]on the C∗ –algebra (Stone’s theorem). No CP dissipator is needed to incorporate [ ω ] ∈H2 ; by contrast, any GKLS dissipator would mix superselection sectors and/or generate entropy without ancillary degrees of freedom. Thus triangle–only defects preserve unitary, linear quantum mechanics. Claim B (Higher coherence ⇒CP semigroup). If there is a small, local higher coherence class [Ω] 6 = 0 supported in O , then there exists a natural local CP semigroup { Φ ∗ t} on A ( O2 )with GKLS generator (7) , whose Kraus/jump operators are local densities selected by symmetry/BRST and whose coefficients are functionals of [Ω] (via Stinespring dilation or SK influence functional). Reason. The dilation route produces Kraus operators Vµ∈ N because the curvature sector can be viewed as an ancilla; tracing it out yields (6) . The SK route shows that the imaginary part of the influence 6 functional induced by [Ω] is quadratic and positive (Hadamard), implying a double–commutator of the form (9) , i.e. GKLS. In either case, complete positivity is guaranteed (Kossakowski matrix is positive), and locality/no–signalling follow from Vµ∈ N or from the support of N(x, y). D. Locality, covariance and no–signalling We record the precise no–signalling statement in the net language. Proposition 2.1 (Local CP maps act trivially on spacelike complements). Let Φ ∗ : A ( O2 ) → A ( O2 )be normal, unital CP with a Kraus form (6) and Vµ∈ N for some type I factor N with A(O1)⊂ N ⊂ A(O2). Then for any B∈A(O0 2)(spacelike complement), Φ∗(B) = B. Proof. For each µ,Vµ∈ N ⊂ A(O2), while B∈A(O0 2)⊂A(O2)0. Thus [Vµ, B] = 0. Then Φ∗(B) = X µ V† µBVµ=X µ BV † µVµ=BX µ V† µVµ=BΦ∗(1) = B.  As a consequence, expectation values of B in any state are invariant under Φ ∗ ; hence no superluminal signalling. Covariance holds if O7→ Φ ∗ O is natural with respect to isometries of ( M, g )(locally covariant QFT [51]). E. Detailed balance, entropy production and arrows of time Let σ be a faithful normal state on A ( O2 ). The GKLS generator L∗ satisfies quantum detailed balance (QDB) with respect to σif it is self–adjoint with respect to the Bogoliubov–Kubo–Mori inner product hA, Biσ:= Z1 0 TrσsA†σ1−sBds, or, equivalently, if σ is stationary and the Kossakowski matrix obeys a fluctuation–dissipation symmetry [ 87 ]. In that case, Spohn’s theorem implies that the relative entropy S ( ρtkσ ) = Tr ( ρt ( log ρt−log σ )) is a Lyapunov functional: d dtS(ρtkσ) = −Eσ[ρt]≤0,(10) with Eσ≥ 0the entropy production rate [ 88 ]. In cosmology, a natural σ is the Bunch–Davies/de Sitter–invariant state restricted to O2 , which is KMS with respect to the static–patch Hamiltonian (Gibbons–Hawking temperature) [ 85 ]; QDB is satisfied if the noise kernel N is chosen accordingly (KMS symmetry). Thus the intrinsic self–measurement carries an arrow of time tied to a local thermal structure, consistent with horizon thermodynamics. F. Specialisation to FRW cosmology: QND channels and mode dynamics Consider a spatially flat FRW patch with metric ds2 = a ( η ) 2 ( −dη2 + dx2 ). Let φ ( η, x )be a light scalar (for concreteness; the comoving curvature perturbation ζ can be treated analogously at linear order). Choose a local QND channel with jump density O ( η, x ) = f ( η ) φ ( η, x )supported in a spacetime tube O (e.g. around horizon exit for a band of comoving momenta). The SK kernel N is taken positive, covariant, and short–ranged in proper time; the master equation reads ˙ρ=−i[H, ρ]−1 2Zdη dη0d3x d3y N(η, η0;x−y) [φ(η, x),[φ(η0,y), ρ]],(11) with H the free (or weakly interacting) Hamiltonian in the Schrödinger picture. Fourier–transforming to modes φk , and assuming N is approximately diagonal in k on super–Hubble scales, one obtains for each mode the Lindblad dephasing ˙ρk=−i[Hk, ρk]−γk(η) 2[φk,[φ−k, ρk]],(12) 7 with a tiny rate γk≥ 0set by the curvature strength and the kernel shape. Writing equal–time covariances Vφφ =hφkφ−ki, Vππ =hπkπ−ki, Vφπ =1 2hφkπ−k+πkφ−ki, one finds the linear system (dots are cosmic time derivatives): ˙ Vφφ =2 a3Vφπ,˙ Vππ =−2ak2Vφπ +γk,˙ Vφπ =1 a3Vππ −ak2Vφφ −γkVφπ.(13) Outside the horizon ( kaH ), Vφφ freezes to its standard inflationary value, ˙ Vφφ → 0, while Vφπ decays as exp ( − Γ k )with Γ k ( η ) = Rγk ( η0 ) dη0 , and Vππ grows diffusively. The reduced density matrix in the field basis acquires a Gaussian suppression of off–diagonals: ρk(φ, φ0;η) = ρ(0) k(φ, φ0;η) exp −2 Γk(η) (φ−φ0)2,(14) with no change in the diagonal variance Vφφ (hence no change in the power spectrum at leading order). A simple classicality measure Ck := 1 −V2 φπ/ ( VφφVππ )then increases monotonically to 1as Γ k builds up, signalling a positive Wigner function and classical stochasticity. G. Summary of the structural map • Triangle coherence ( H2 ) ⇒ central extension: changes commutation relations/projective phases but preserves unitary, linear quantum dynamics. • Higher coherence ( H≥3 ) ⇒ local CP semigroup: via (i) Stinespring dilation with local Kraus operators in a split inclusion; or (ii) SK influence functional with positive noise kernel. • Locality/covariance/no–signalling are preserved by construction; gauge/BRST consistency holds if jump densities are gauge–invariant (or BRST–covariant). • Entropy monotonicity follows from QDB with respect to a KMS/local thermal reference (de Sitter/Gibbons–Hawking for cosmology). • Cosmological specialisation: QND channels classicalise super–Hubble modes without changing the power spectrum, providing an intrinsic floor to decoherence. III. THE QUANTUM–TO–CLASSICAL PUZZLE IN COSMOLOGY This section revisits the standard account of how inflationary quantum fluctuations yield classical stochastic perturbations, identifies its conceptual and technical limits, and then incorporates the intrinsic self–measurement mechanism of Section II. We provide quantitative criteria for classicalisation, proofs of monotonicity under completely positive (CP) dephasing, and a precise statement of the “conveyor belt” picture: quantum modes are continuously born at sub–Hubble scales, while super–Hubble modes are irreversibly driven toward classical statistics. A. Inflationary perturbations as squeezed quantum oscillators Consider a spatially flat FRW universe with line element ds2 = a ( η ) 2 ( −dη2 + dx2 )and Hubble rate H = a0/a2 (prime denotes η –derivative). For a single adiabatic scalar degree of freedom, the comoving curvature perturbation ζ is gauge invariant and, at linear order, is governed by the Mukhanov–Sasaki variable v:= zζ with v00 k+k2−z00 zvk= 0, z(η) = a(η)p2(η)MPl cs(η),(15) where  := −˙ H/H2 and cs is the sound speed (unity for canonical slow–roll) [ 65 , 66 ]. In (quasi) de Sitter with slow variation, z00/z ≃ν2−1/4with ν≃3/2 + O(), so the Bunch–Davies solution reads vk(η) = √π 2ei(ν+1 2)π/2√−η H(1) ν(−kη)k|η|1 −−−−→ e−ikη √2k,(16) 8 and on super–Hubble scales (k|η|  1), ζk(η) = vk z−→ ζ(const) k+O(k/aH)2.(17) The frozen two–point amplitude defines the power spectrum hζkζk0i= (2π)3δ(3)(k+k0)2π2 k3Pζ(k),Pζ(k) = H2 ∗ 8π2∗M2 Plcs,∗ (k/k∗)ns−1+··· ,(18) with ∗evaluated at horizon crossing k=a∗H∗/cs,∗[65]. Quantally, each Fourier mode is a time–dependent harmonic oscillator. The Bunch–Davies vacuum evolves into a two–mode squeezed Gaussian state for ( k,−k )pairs [ 55 , 56 ]. Writing annihilators ak ( η ) = uk ( η ) ak ( η0 ) + vk ( η ) a† −k ( η0 )with Bogoliubov coefficients ( uk, vk ), the squeezing parameter rk satisfies |vk| = sinh rk , |uk| = cosh rk , with rk∼ −ln |kη| on super–Hubble scales. The Wigner function of a Gaussian state is (and remains) strictly positive, but strong squeezing aligns it along a classical phase– space correlation line: the anticommutator h{ζ, πζ}i grows while the commutator [ ζ, πζ ] = i remains fixed, making commutator effects subdominant in typical correlators [ 57 , 58 ]. This is often described as an effective classicalisation by squeezing. However, without decoherence, the state still carries phase coherence between macroscopically distinct field amplitudes; off–diagonal elements in the ζ –basis are nonzero and can, in principle, interfere. B. Environmental decoherence: strengths and limits The standard account addresses this by tracing out unobserved degrees of freedom (short modes, other fields, metric nonlinearities), leading to environment–induced decoherence [ 63 , 79 – 81 , 113 , 114 ]. A generic linear coupling Hint(η) = Zd3x a4(η)g(η)ζ(η, x)E(η, x),(19) with E an environmental operator, produces a reduced dynamics for the long–wavelength sector of the form ˙ρL=−i[HL, ρL]−1 2ZZ dη dη0N(η, η0) [ζ(η),[ζ(η0), ρL]] + ··· ,(20) with a positive noise kernel N (fluctuation–dissipation relates it to a dissipative kernel). For super–Hubble modes and adiabatic switching, (20) reduces to Markovian dephasing ˙ρ = −γ 2 [ ζ, [ ζ, ρ ]], suppressing off–diagonals in the field basis while preserving hζ2i . This elegantly explains why ζ can be described by a classical stochastic field on large scales. Limitations. Two issues persist. (i) Pointer ambiguity: depending on couplings, the preferred basis may not coincide with ζ amplitude; one typically argues that local couplings single out local field eigenstates, but this relies on specifics of Hint . (ii) Asymptotic isolation: in the far future of a Λ–dominated universe, the environment accessible to a super–Hubble mode becomes sparse, and environment–induced decoherence alone may not guarantee strict monotonicity of a suitable classicality measure. These motivate an intrinsic, symmetry–tied contribution. C. Intrinsic self–measurement as a universal floor Section II derived a local GKLS generator from higher coherence curvature. Specialised to a comoving mode (or a narrow band), it yields the quantum–nondemolition (QND) master equation ˙ρk(η) = −i[Hk(η), ρk]−γk(η) 2[ζk,[ζ−k, ρk]],(21) with γk≥ 0tiny and slowly varying, fixed by the coherence curvature and locality/BRST constraints. This has three crucial properties: 1. Preservation of the power spectrum. Since the jump operator is ζk , ˙ hζkζ−ki has no explicit γk term; outside the horizon, it “freezes” as in (17). Thus Pζ(k)is unchanged at leading order. 9 2. Monotone suppression of phase coherence. Off–diagonals in the ζ –basis are suppressed by a Gaussian factor exp[−2Γk(η)(ζ−ζ0)2]with Γk(η) := Rηγk(η0)dη0≥0. 3. No signalling & covariance. The channel is built from local densities; it acts trivially on spacelike– separated algebras and respects FRW symmetries. This provides a universal floor to decoherence that complements (and in practice is dominated by) environmental channels but persists even when the environment is minimal. D. Quantitative classicality criteria and their monotonicity Let Vζζ,Vππ, and Vζπ denote equal–time covariances for a Gaussian mode state, Vζζ =hζkζ−ki, Vππ =hπkπ−ki, Vζπ =1 2hζkπ−k+πkζ−ki,(22) with canonical [ζk, π−k] = i. Under (21), a Heisenberg–picture computation yields (cf. Section II F) ˙ Vζζ =2 a3Vζπ,˙ Vππ =−2ak2Vζπ +γk,˙ Vζπ =1 a3Vππ −ak2Vζζ −γkVζπ.(23) On super–Hubble scales, ak → 0, so Vζζ freezes, Vζπ decays as e−Γk , and Vππ grows diffusively. We define the classicality measure Ck:= 1 −V2 ζπ VζζVππ ∈[0,1],(24) which equals zero for a pure (minimum uncertainty) squeezed Gaussian and tends to 1as phase–space coherence is erased. Proposition III.1 (Monotonicity of Ck under QND dephasing) . On super–Hubble scales ( kaH ), for any γk ( η ) ≥ 0, Ck ( η )defined in (24) is monotonically nondecreasing in η under (21) , and limη→∞ Ck ( η ) = 1provided Γk(η)→ ∞. Proof. With ak → 0, (23) reduces to ˙ Vζζ = 2 a3Vζπ , ˙ Vππ = γk , ˙ Vζπ = 1 a3Vππ −γkVζπ . Write X := Vζπ , Y:= Vππ,Z:= Vζζ . Then ˙ Y=γk≥0and ˙ X=−γkX+ (1/a3)Y. The quantity Q:= X2/(Y Z)obeys ˙ Q=2X˙ X Y Z −X2˙ Y Y2Z−X2˙ Z Y Z2. Using ˙ Z=2 a3Xand the equations for ˙ X, ˙ Y, one finds ˙ Q=−2γkX2 Y Z ≤0. Hence Q is nonincreasing and C = 1 −Q is nondecreasing. Moreover X ( η ) ∼e−Γk(η) while Y ( η )grows at least linearly with Γk, whence Q→0and C → 1as claimed. Proposition III.1 gives a clean, gauge–invariant notion of classicalisation for Gaussian modes: monotone approach to a classical stochastic limit, without modifying the power spectrum. E. Exact solution of the QND channel and no–revival theorem The generator L [ · ] = −γ 2 [ ζ, [ ζ, · ]] has an exact solution by functional calculus. Let X := ζk be self–adjoint. Then for time–dependent γ(η)≥0, ρ(η) = Z+∞ −∞ dλ p4πΓ(η)exp−λ2 4Γ(η)e−iλX ρ(0) e+iλX,Γ(η) = Zη 0 γ(η0)dη0.(25) Derivation. The map TΓ ( ρ ) := Rdλ pΓ ( λ ) e−iλXρeiλX with pΓ the centred Gaussian of variance 2Γ defines a CP, trace–preserving semigroup with generator −1 2∂Γ [ ρ ]at fixed X , and ∂ΓTΓ ( ρ ) = −1 2 [ X, [ X, TΓ ( ρ )]]. Composing with Γ(η)yields (25). 16 C. Front laws for two physically motivated rate families We now solve for kc ( N )for two families of γk considered in Section IV: (i) Hubble–locked (ultra– conservative) and (ii) density–weighted over a finite window after exit. Throughout we assume slow–roll so that Hand csare approximately constant on e–fold scales; aH/csthen grows as eN. 1. Hubble–locked rate Let γk(N) a(N)H(N)=ε, ε 1constant,(54) so that the dimensionless instantaneous rate per e–fold is ε. Then (51) gives Γk(N) = ε∆N(N;k).(55) The threshold Γ∗is reached when ∆N= ∆N∗:= Γ∗/ε. Equivalently, kc(N) = a(N)H(N) cs e−∆N∗.(56) Proposition 5.1 (Stationary transition layer). Under (54) , the transition layer in logarithmic wavenumber, defined by k∈(kc(N), aH/cs], has a constant thickness ∆ ln klayer = lnaH/cs kc= ∆N∗=Γ∗ ε,(57) independent of epoch N (to leading slow–roll order). In particular, the classicality front co–moves with the sound horizon, and a mode becomes classical precisely after ∆N∗e–folds since horizon exit. Proof. Combine (50) – (55) with the threshold Γ k ( N )=Γ ∗ to obtain ∆ N ( N ; kc )=Γ ∗/ε . Then ∆ ln k|layer = ln(aH/cs)−ln kc= ∆N(N;kc)=Γ∗/ε. 2. Density–weighted rate over a finite window Let the rate per e–fold be supported for ∆ N e–folds after exit and scaled by a local density power α , γk(N) aH =εa a∗α Θ 0≤∆N(N;k)≤∆Nwin, α > 0.(58) Then Γk(N) = εZmin(∆N,∆Nwin) 0 eαu du =ε αeαmin(∆N,∆Nwin)−1.(59) The threshold Γ∗is reached after an age ∆N∗= min1 αln1 + αΓ∗/ε,∆Nwin .(60) As long as ∆ N∗< ∆ Nwin , the transition layer has thickness ∆ ln k = ∆ N∗ (as in Proposition 57 with ∆ N∗ now given by (60) ). If ∆ N∗≥ ∆ Nwin , the layer saturates at ∆ ln k = ∆ Nwin (beyond the window there is no further intrinsic dephasing under (58)). Comparison. For fixed Γ ∗ and small ε , the Hubble–locked law (56) yields a layer thickness ∝ 1 /ε ; the density–weighted law (60) gives ∆ ln k∼1 αln (1 + α Γ ∗/ε ), i.e. a much slower growth with 1 /ε and a possible saturation at ∆ Nwin . Both laws respect locality and FRW covariance; the appropriate choice depends on which local density the jump operator couples to (Section IV). 17 D. Population dynamics and the no–oscillation theorem Let χSH ( N ; k ) := Θ( aH/cs−k )be the indicator of super–Hubble membership and define the classicality indicator C ( N ; k ) := Θ(Γ k ( N ) − Γ ∗ ). The transition layer indicator is T ( N ; k ) = χSH ( N ; k ) [1 −C ( N ; k )]. The following theorem formalises the conveyor belt picture. Theorem V.1 (Monotone classicalisation at population level) . Fix an epoch interval [ N0, N1 ]in which the rate per e–fold γk/(aH)≥0and H, csvary slowly. Then: 1. No oscillation: For each comoving k , the map N7→ Γ k ( N )is nondecreasing; therefore C ( N ; k )is nondecreasing and never returns from 1to 0. 2. Stationary transition layer (Hubble–locked): Under (54) , the set {k : T ( N ; k ) = 1 } is the logarithmic shell k∈(kc(N), aH/cs]of fixed thickness ∆ ln k= Γ∗/ε for all N∈[N0, N1]. 3. Bounded layer (density–weighted): Under (58) , {k : T ( N ; k )=1 } has thickness ∆ ln k = min{1 αln(1 + αΓ∗/ε),∆Nwin}for all N. 4. Shrinking fraction among super–Hubble: For any fixed infrared cutoff kIR , the fraction of super– Hubble modes that are still in the transition layer, Ftr(N) := RaH/cs kIR dk kT(N;k) RaH/cs kIR dk k =∆ ln k|layer lnaH cskIR , decreases monotonically to zero as N→ ∞ , since the denominator grows as ln ( aH )while the numerator is constant (Hubble–locked) or bounded (density–weighted). Proof. (1) Γ k is an integral of a nonnegative function, hence nondecreasing; C = Θ(Γ k− Γ ∗ )inherits monotonicity. (2)–(3) follow from the front laws derived in Sections V C 1–V C 2. (4) Evaluate the integrals in logarithmic measure; the numerator is exactly the layer thickness, while the denominator is the total logarithmic span of super–Hubble modes above the fixed kIR . As N grows, aH grows exponentially in slow–roll and then (after reheating) polynomially until late dark energy domination; in all cases ln ( aH ) increases unboundedly, so Ftr ↓0. Theorem V.1 says: there is no oscillation back to quantum, and while new quantum modes are continually born at the horizon, at any given time the nonclassical super–Hubble population occupies only a thin, fixed (or bounded) layer adjacent to the horizon, whose fractional weight goes to zero on cosmological timescales. This makes precise the “conveyor belt” intuition. E. Re–entry and irreversibility Modes that re–enter the horizon in the radiation/matter eras subsequently evolve under nearly unitary Hamiltonian flow, possibly with environment–induced decoherence due to interactions. Once a mode satisfies C ( N ; k )=1(diagonalised in the ζ basis within the threshold (53) ), it cannot “recohere” under the intrinsic channel because the latter is a CP semigroup (Section IV). More formally, let Πbe the pinching map in the ζ basis. Then for any CPTP map Λrepresenting subsequent (unitary or open) evolution,  Λ◦Π(ρ)−Π◦Λ◦Π(ρ) 1= 0,(61) and, by contractivity, k Λ( ρ ) − Π ◦ Λ( ρ ) k1≤ kρ− Π( ρ ) k1 . Thus no subsequent physically admissible evolution can increase the off–diagonal weight relative to the pinched state. In particular, re–entry does not undo classicalisation achieved outside the horizon. F. Entropy production per mode: Gaussian proof of monotonicity For a single canonical mode (ζ, π)with covariance matrix V=Vζζ Vζπ Vζπ Vππ, Vζζ =hζ2i, Vππ =hπ2i, Vζπ =1 2h{ζ, π}i,(62) 18 the symplectic invariant ∆ := det Vcontrols the (Gaussian) von Neumann entropy: SG(V) = ν+1 2logν+1 2−ν−1 2logν−1 2, ν := √∆≥1 2,(63) where ν is the symplectic eigenvalue of V (for [ ζ, π ] = i ) [ 109 – 111 ]. We now show that ∆increases monotonically under the intrinsic QND dynamics, hence SGincreases. Lemma V.2 (Monotone growth of det V ) . Under the covariance equations (23) with γk ( η ) ≥ 0(no restriction on k), the determinant ∆ = VζζVππ −V2 ζπ obeys d dη ∆ = γk(η)Vζζ + 2 V2 ζπ≥0.(64) Proof. Differentiate ∆: ˙ ∆ = ˙ Vζζ Vππ +Vζζ ˙ Vππ −2Vζπ ˙ Vζπ. Insert (23): ˙ Vζζ =2 a3Vζπ,˙ Vππ =−2ak2Vζπ +γk,˙ Vζπ =1 a3Vππ −ak2Vζζ −γkVζπ. Compute term by term: ˙ ∆ = 2 a3VζπVππ +Vζζ(−2ak2Vζπ +γk)−2Vζπ1 a3Vππ −ak2Vζζ −γkVζπ. The drift terms cancel pairwise: 2 a3VζπVππ −2 a3VζπVππ = 0 and − 2 ak2VζζVζπ + 2 ak2VζζVζπ = 0. What remains is ˙ ∆ = γkVζζ + 2γkV2 ζπ. Proposition V.3 (Gaussian entropy monotonicity) . Let SG ( V )be the Gaussian entropy (63) of a single mode with covariance V ( η ). Under the QND dynamics (23) with γk≥ 0, SG is monotonically nondecreasing in η, with strict increase unless Vζπ = 0 and Vζζ = 0 (the latter impossible). Proof. By Lemma V.2, ν ( η ) = p∆(η) is nondecreasing. Since SG is an increasing function of ν for ν≥1/2(derivative dSG/dν =1 2log ν+1/2 ν−1/2>0), SGincreases. Proposition V.3 provides a mode–local entropic arrow even without invoking QDB and relative entropy; the latter yields a global monotone when the reference state is KMS (Section II). G. Entropy production rate density For a band of modes per comoving volume, define the entropy production rate density under the intrinsic channel as d dN sint(N) := Z k≤aH/cs d3k (2π)3 1 aH ˙ Sk(η),(65) with ˙ Sk the von Neumann entropy production of the k –mode. For Gaussian states, ˙ Sk can be computed from ˙νk (Proposition V.3); more generally, under QDB with respect to a local KMS reference σ , Spohn’s theorem gives the nonnegative functional −dS ( ρkkσk ) /dη [ 87 , 88 ]. Splitting the integral into (i) the transition layer and (ii) the older super–Hubble bulk, one finds that the production is localized near the horizon because γk is largest in the near–exit epoch in both laws (54) – (58) . In the Hubble–locked case, sint ( N )asymptotes to a constant per Hubble volume (stationary production), while in the density– weighted case it saturates at a value proportional to ( eα∆Nwin − 1) (set by the window width). These scalings provide concrete targets for any attempt to bound the intrinsic channel observationally in cosmology. 19 H. Summary of Section V We have turned the conveyor belt metaphor into a quantitative theory of fronts in momentum space. A mode becomes classical when its accumulated dephasing Γ k exceeds a scale–invariant threshold Γ ∗ ; this defines a classicality front kc ( N )whose distance from the sound horizon in ln k is either constant (Γ ∗/ε for Hubble–locked rates) or bounded by (1 /α ) ln (1 + α Γ ∗/ε )(for density–weighted windows). The population of nonclassical super–Hubble modes thus occupies a thin, fixed (or bounded) layer adjacent to the horizon, whose fractional weight decays to zero with cosmic time. At the level of individual modes, Gaussian entropy increases monotonically under the intrinsic QND flow, and there is no revival of coherence; at the level of the ensemble, entropy production per Hubble volume is localized near the horizon and stationary/saturating depending on the rate law. Re–entry cannot undo classicalisation. Altogether, the universe does not oscillate between quantum and classical; rather, it flows irreversibly, with a steady quantum birth rate at the horizon and a monotone drift to classicality thereafter. VI. TIMESCALES AND PARAMETRIC ESTIMATES This section quantifies the strength of intrinsic self–measurement in cosmology. We normalise the rate in a dimensionless way, relate it to a local influence kernel derived from higher–coherence curvature, derive closed–form classicalisation times and front thicknesses, and obtain conservative bounds from energy budget and precision data. Throughout, we work at linear order in the intrinsic channel, assume slow–roll inflation for concreteness, and specialise to the QND choice of jump density (so that the power spectrum remains unchanged at leading order; cf. Sections IV–V). A. Normalisation and dimensional analysis Recall the QND master equation for a mode ζkin conformal time η, ˙ρk(η) = −i[Hk(η), ρk]−γk(η) 2[ζk,[ζ−k, ρk]],˙= ∂η,(66) with γk the Markovian dephasing rate (units of inverse conformal time). It is convenient to define a dimensionless rate per e–fold εk(N) := γk(η) a(η)H(η)≥0, N := ln a, dη =dN aH .(67) The integrated dephasing (“optical depth” of decoherence) accumulated by the mode from horizon crossing N∗(k)to epoch Nis then Γk(N) = ZN N∗(k) εk(N0)dN0≥0.(68) As in Section IV, the density matrix in the ζbasis acquires Gaussian suppression of off–diagonals, ρk(ζ, ζ0;N) = ρ(0) k(ζ, ζ0;N) exp−2 Γk(N) (ζ−ζ0)2,(69) and a quantitative classicality measure Ck = 1 −V2 ζπ/ ( VζζVππ )increases monotonically to 1as Γ k grows (Proposition III.1). B. From coherence curvature to a local kernel: EFT scaling In the Schwinger–Keldysh (SK) construction, the imaginary part of the influence functional generated by higher–coherence curvature yields a positive kernel N(x, y), S(imag) IF =i 2Zd4x d4yO+(x)−O−(x)N(x, y)O+(y)−O−(y),(70) 20 with O a local composite density fixed by locality and (BRST) symmetry. In a local, Markovian approximation on Hubble timescales, N(x, y)≃νO(N)a4(N)δ(Nx−Ny)δ(3)(x−y),(71) and the QND master equation follows with jump density O and rate γ∼νO . The dimensionless ε=γ/(aH)encodes all microscopic information about the curvature class and the operator choice. EFT estimate. Suppose O has (physical) mass dimension ∆and is normal–ordered with respect to a de Sitter–invariant reference. On dimensional grounds, the local strength scales as νO(N)∼g2 int H(N)H(N) Λ∗2∆−4f∆k aH , εk(N)∼g2 int H Λ∗2∆−4f∆k aH ,(72) where gint is a dimensionless curvature–to–kernel matching coefficient, Λ ∗ a UV matching scale of the effective channel (not a hard cutoff), and f∆ is a dimensionless shape supported near horizon crossing and decaying for kaH by locality. Two important cases: • QND amplitude channel ( O = ζ , effectively ∆ ≃ 1): ε∼g2 int ( H/ Λ ∗ ) −2 but f∆ is sharply peaked at k∼aH and essentially constant across the transition layer; in practice we model it by a constant ε over the layer (Hubble–locked law). • Stress–tensor channel ( O = T00 ,∆=4): ε∼g2 int (dimensionless) but f∆ grows as a positive power of aover a short window set by microcausality and redshifting (density–weighted law). Equations (72) justify the phenomenological rate families used in Sections IV–V and allow us to express constraints directly as bounds on gint(H/Λ∗)2∆−4. C. Classicalisation time and front thickness Fix a scale–invariant classicality threshold Γ ∗ = O (1) (Section V). The classicalisation e–folds for a mode kare ∆Ncl(k) := inf (∆N≥0 : Z∆N 0 εkN∗(k) + udu ≥Γ∗).(73) For the two rate families: Hubble–locked law εk(N) = ε0on the layer. Then ∆Ncl =Γ∗ ε0 ,lnaH/cs kc(N)=Γ∗ ε0 ,(74) i.e. a stationary transition layer of constant logarithmic thickness (Proposition 57). Density–weighted window εk(N) = ε0eα∆Nfor 0≤∆N≤∆Nwin, zero otherwise. Then ∆Ncl = min1 αln1 + αΓ∗ ε0,∆Nwin ,lnaH/cs kc(N)= ∆Ncl,(75) i.e. a bounded layer with possible saturation at the window width (Section V C 2). D. Energy budget and a conservative bound The intrinsic channel injects momentum diffusion into each mode, cf. ( π2 k ) 0 = ··· + γk in (34) . The Hamiltonian energy per mode is Ek(η) = 1 2z2(η)hπkπ−ki+z2(η)c2 sk2 2hζkζ−ki, z(η) = a√2SR MPl cs ,(76) with SR the slow–roll parameter. On super–Hubble scales the gradient term is negligible, and the dissipative contribution to dEk/dt is dEk dt diss =1 2z2 d dthπkπ−kidiss =1 2z2 γk a=εkH 2z2=εkc2 s 4SR H a2M2 Pl .(77) 21 Integrating from horizon exit N∗ over ∆ N e–folds yields a convergent contribution dominated near exit, ∆E(diss) k≃ε0c2 s 8SR H∗ a2 ∗M2 Pl F(∆N;α),F=(1Hubble–locked, 2 31−e−2∆Nwin density–weighted,(78) ignoring slow drift of H and cs . The total energy density injected per Hubble patch per e–fold is obtained by integrating over the transition shell k∈ ( kc, aH/cs ]with measure d3k/ (2 π ) 3 ; writing ∆ ln k= ln((aH/cs)/kc), dρint dN shell ∼1 (2π)2aH cs3 ∆E(diss) typ ∆ ln k, (79) where ∆ E(diss) typ is the typical per–mode injection (of order (78) ). Requiring that this be a tiny fraction δ1of the background energy density 3M2 PlH2yields a very conservative bound ε0.δ×48π2 cs SR MPl H∗3cs a∗H∗21 ∆ ln k1,(80) where the tiny factor ( a∗H∗ ) −2 (in comoving units) makes the constraint extremely easy to satisfy for any microscopic ε01.[145] E. Consistency with the observed power spectrum For the QND channel ( O = ζ ), the equal–time variance Vζζ (and hence Pζ ) is unchanged at leading order (Sections IV–III). Subleading corrections can arise from (i) non–Markovianity of the kernel N (finite temporal width), (ii) slow–roll time–dependence of H and cs , and (iii) small non–QND admixtures in the jump density. Writing the observed scalar amplitude at the pivot as As = Pζ ( kpiv ) ≃ 2 . 1 × 10 −9 and spectral index ns≃0.965 [102, 104], a safe requirement is  δPζ Pζ.10−3⇒ε0×max{∆Ncl,∆Nwin}  10−3,(81) for any leakage into non–QND components linear in ε0.[146] F. Non–Gaussianity and shape–dependent damping A strictly linear QND channel does not generate a bispectrum at tree level; it damps phase–sensitive oscillations in shapes that rely on coherent superpositions. For small non–QND admixtures or quadratic jump densities (e.g. stress–tensor components), one can estimate a loop–level correction ∆Bζ(k1, k2, k3)∼ε0I(ki)B(0) ζ(k1, k2, k3),(82) with a smooth, positive shape function I supported near horizon crossing and obeying consistency relations from locality. Planck constraints on local/equilateral/orthogonal fNL at the level |fNL|.O (5) [ 104 ] suggest no observable obstruction to ε0 1; any realistic intrinsic channel consistent with Sections IV–V automatically satisfies current bounds. G. Numerical illustrations Adopt the ultra–conservative Hubble–locked law with constant ε0 and set Γ ∗ = 1 (“1 σ ” classicality threshold). Then ∆Ncl =ε−1 0,lnaH/cs kc(N)=ε−1 0.(83) Sample values: ε0= 10−3⇒∆Ncl = 103e–folds (very slow intrinsic);ε0= 10−2⇒∆Ncl = 100 e–folds;ε0= 10−1⇒∆Ncl = 10 e–folds. 22 Thus, even extremely small ε0 classicalises modes over cosmological times; laboratory systems (with N 1on experimental timescales) are unaffected, explaining why standard unitary quantum mechanics remains accurate in the lab. For a density–weighted window with α= 3 over ∆Nwin = 3 e–folds (local volume growth near exit), ∆Ncl = min1 3ln1+3/ε0,3.(84) E.g. ε0 = 10 −4⇒ ∆ Ncl ≈1 3ln (3 × 10 4 ) ≈ 3 . 6e–folds (saturating the window); ε0 = 10 −6⇒ ∆ Ncl ≈ 1 3ln (3 × 10 6 ) ≈ 4 . 8(beyond the window, hence ∆ Ncl = 3 in this model). These numbers illustrate how density weighting can compress the classicalisation layer even for tiny ε0. H. Comparison to environmental decoherence Environment–induced decoherence near horizon crossing commonly yields effective dephasing rates εenv of order couplings squared times phase space; in many models εenv ε0 during the brief exit epoch, ensuring rapid diagonalisation in practice [79–81]. The intrinsic channel thus plays two roles: • It provides a universal floor—a strictly positive, symmetry–tied, locality–respecting contribution that persists even in asymptotically empty regions or epochs with minimal environment. • It guarantees a monotone arrow: even if environmental couplings temporarily weaken, the intrinsic CP semigroup ensures that Ckcannot decrease, forbidding revivals (Theorem IV.1). I. Summary of Section VI We normalised the intrinsic self–measurement by a dimensionless rate per e–fold εk , related it to a local SK kernel determined by higher–coherence curvature, and derived closed–form expressions for the classicalisation time ∆ Ncl and the momentum–space front. Two coarse families of laws (Hubble–locked and density–weighted) capture a wide class of local channels. Conservative energy–budget and precision–power requirements place only very weak bounds on ε0 ; a strictly QND channel leaves the power spectrum unchanged at leading order and induces at most loop–suppressed non–Gaussian corrections within current limits. Numerically, even tiny ε0 produces classicalisation over cosmological times while being negligible in laboratory settings, making the intrinsic channel both conceptually decisive and phenomenologically safe. VII. ENTROPY, HORIZONS, AND THE ARROW OF TIME In this section we connect the intrinsic self–measurement dynamics to horizon thermodynamics and to a mathematically precise arrow of time. The main ingredients are: (i) the KMS (thermal) structure of quantum fields with respect to horizon time, (ii) quantum detailed balance (QDB) for the GKLS generator, (iii) Spohn’s monotonicity of relative entropy, and (iv) horizon first–law identities that relate modular (“Killing”) energy flux to area change. Together, these yield a compact route to the generalized second law (GSL), d dtSout +SBH≥0, SBH =A 4G, for stationary horizons (Rindler, black holes, de Sitter static patch), and suggest a local version for slowly evolving cosmological apparent horizons. A. KMS states, modular Hamiltonians, and horizon thermality Let A ( O )be the local net of observables on a globally hyperbolic spacetime. A state σ is KMS at inverse temperature β > 0with respect to a one–parameter automorphism group {αt} (time translations generated by a Killing field ξ) if, for all A, B in a dense ∗–subalgebra, FA,B(t) := σA αt(B)extends analytically to t7→ FA,B(t+iβ)with FA,B(t+iβ) = σαt(B)A.(85) This characterizes thermal equilibrium in algebraic quantum statistical mechanics [82]. 23 Horizons. For quantum fields in Minkowski space restricted to a Rindler wedge, the vacuum is KMS at β = 2 π with respect to the boost flow (Bisognano–Wichmann theorem). An observer with proper acceleration a sees Unruh temperature T = a/ 2 π [ 83 , 84 ]. In de Sitter spacetime, the Bunch–Davies state restricted to the static patch is KMS at the Gibbons–Hawking temperature TdS = H/ 2 π with respect to the static time [ 85 , 86 ]. For stationary black holes, the Hartle–Hawking state is KMS at the Hawking temperature TH = κ/ 2 π (surface gravity κ ). In all cases, the modular Hamiltonian K of the region coincides with the physical generator of αt(up to an additive constant), σ(A) = Tre−KA Tr(e−K), αt(A) = eitK A e−itK , β = 1.(86) (Here and below we work in units ~ = c = kB = 1 and absorb β into K .) In QFT, K is typically unbounded and affiliated with the von Neumann algebra of the region; nevertheless, its relative expectation values are well defined and are related to energy fluxes across the horizon. B. Quantum detailed balance (QDB) and Spohn’s inequality Let { Φ t = etL}t≥0 be a normal CPTP semigroup on the algebra of a region, generated by a GKLS operator L . We say that L satisfies quantum detailed balance w.r.t. a faithful stationary state σ if L is self–adjoint w.r.t. the Bogoliubov–Kubo–Mori (BKM) inner product, hA, Biσ:= Z1 0 TrσsA†σ1−sBds, hA, L(B)iσ=hL(A), Biσ,(87) and Φ ∗ t ( σ ) = σ [ 87 ]. If, in addition, the Hamiltonian part of the evolution coincides with the modular generator Kof σ(i.e. the horizon time), we speak of a thermal QDB semigroup. Spohn’s inequality. For any state ρt = Φ t ( ρ0 ), the relative entropy (Araki–Uhlmann) S ( ρtkσ )is monotone nonincreasing: d dt S(ρtkσ) = d dt Trρt(log ρt−log σ)≤0.(88) Moreover, −d dt S ( ρtkσ )equals the entropy production rate functional Eσ [ ρt ] ≥ 0explicitly determined by L and σ [ 87 , 88 ]. Equation (88) gives a precise, dynamical arrow of time: ρt relaxes toward σ (when the semigroup is primitive) and the thermodynamic distance to σcannot increase. C. Relative entropy identity and the GSL For a von Neumann algebra with faithful normal state σ (KMS), the relative entropy admits the identity S(ρkσ)=∆hKiρ−∆Sout(ρ),∆hKiρ:= hKiρ−hKiσ,∆Sout := Sout(ρ)−Sout(σ),(89) where K is the modular Hamiltonian and Sout is the von Neumann entropy of the outside algebra [ 40 , 89 ]. (For type III algebras the identity is formulated via relative modular operators; for our purposes (89) can be taken as defining ∆ hKi as the “modular energy”.) Taking a time derivative along any CP evolution commuting with the modular group (our thermal QDB semigroup) yields −d dt S(ρtkσ) = −d dt ∆hKiρt+d dt Sout(ρt).(90) Thus, Spohn’s inequality (88) is equivalent to d dt Sout(ρt)≥d dt ∆hKiρt.(91) 24 Horizon first law. For stationary horizons, there is a precise identity relating the rate of change of modular energy to the rate of change of horizon area, d dt ∆hKiρt=d dt SBH =1 4G dA dt ,(92) when the modular flow is generated by the corresponding Killing vector and the matter stress tensor satisfies the semiclassical Einstein equations [ 90 – 92 ]. Equation (92) is the dynamical, local form of the horizon first law (the linearized “first law of entanglement”), proven in Rindler and generalized settings by integrating the Raychaudhuri equation along the horizon generators and using the field equations. Combining (91) and (92) gives the generalized second law: d dtSout +SBH≥0.(93) Proposition VII.1 (GSL from QDB) . Let σ be the KMS state of a stationary horizon region (Rindler, black hole, de Sitter static patch) and let Φ t be a normal CPTP semigroup satisfying QDB w.r.t. σ and commuting with the modular flow. Assume the semiclassical Einstein equations hold and the horizon first–law identity (92). Then for any initial normal state ρ0, d dtSout(ρt) + SBH(t)≥0, ρt= Φt(ρ0). Proof. By QDB and Spohn, −dS ( ρtkσ ) /dt ≥ 0. By (90) , −dS ( ρtkσ ) /dt = −d ∆ hKiρt/dt + dSout ( ρt ) /dt . Using (92) gives dSout/dt +dSBH/dt ≥0. Proposition VII.1 shows that the same structural element that defines the intrinsic self–measurement (a local, thermal QDB semigroup) yields a monotone generalized entropy. The arrow of time is thus rooted in both dynamics (Spohn) and geometry (horizon first law). D. Raychaudhuri, modular energy, and energy conditions For completeness we sketch the geometric step underlying (92) . Let ka be the null generator of the horizon and θits expansion. The Raychaudhuri equation reads dθ dλ =−θ2 2−σabσab −8πG Tabkakb,(94) with λ an affine parameter. Linearising around a stationary background (so θ and shear σab are first order), integrating along the horizon, and using the semiclassical Einstein equations yields δA = 8πG ZH dλ d2x⊥λ Tabkakb= 4G δhKi,(95) where K is the modular (boost) generator for Rindler, or the suitable Killing generator for stationary horizons [ 92 , 93 ]. Equation (95) implies (92) upon differentiation in time. This derivation presumes appropriate asymptotics and renormalization of Tab (e.g. Hadamard states [138]). QNEC and consistency. The quantum null energy condition (QNEC) 2 πhTkki ≥ S00 out/√h (primes along a null congruence) ensures that matter entropy focusses less than allowed by null energy [ 95 ]. Our CP channel, being local and positivity–preserving with QDB, respects the inequalities underlying (95) at the level of expectation values; the intrinsic noise is KMS–thermal (Section VII E), which avoids violations of averaged conditions needed for the GSL. E. Einstein–Langevin equation and fluctuation–dissipation Beyond expectation values, fluctuations of the renormalized stress tensor couple to the metric via the Einstein–Langevin equation of stochastic gravity [128]: Gab[g+h] + Λ(gab +hab)=8πGhTab[g+h]iρ+ξab,(96) 25 where ξab is a classical Gaussian source with zero mean and covariance given by the noise kernel Nab cd(x, y) := 1 2h{tab(x), tcd(y)}i, tab := Tab −hTabi.(97) In our SK derivation (Section IV), the intrinsic self–measurement is encoded in a positive noise kernel. When the generator satisfies QDB w.r.t. a KMS state σ , the Keldysh (noise) and dissipative kernels obey afluctuation–dissipation relation (FDR) at the horizon temperature, N(ω) = cothβω 2=χ(ω),(98) with χ the susceptibility of the modular energy. Equation (98) implies: (i) stability of the semiclassical background under the intrinsic channel (no runaway solutions), (ii) consistency of entropy production with horizon thermodynamics, and (iii) a natural local arrow of time (dissipation opposes time reversal, with noise tied to it via (98)). F. Steady states, spectral gaps, and rate of approach If the QDB semigroup is primitive (unique faithful fixed point σ ) and has a spectral gap λ > 0in the GNS Hilbert space associated to σ , one has exponential convergence in relative entropy and in trace norm, S(ρtkσ)≤e−2λt S(ρ0kσ),kρt−σk1≤C e−λt,(99) for suitable C (log–Sobolev/transport inequalities under QDB) [ 97 , 98 ]. In free Gaussian models (our toy channel), the gap is set by the smallest nonzero eigenvalue of the Liouvillian, which, for dephasing at rate γk , is λ∼minkγk for each decoupled mode. Thus the intrinsic channel relaxes each super–Hubble mode to the KMS distribution on the timescale γ−1 k (if environmental couplings are negligible), ensuring late–time classicality and consistency with the stationary horizon thermodynamics. G. Cosmological horizons: FRW quasi–local version In non–stationary FRW, there is no global timelike Killing field, but on Hubble timescales one may adopt a quasi–local notion: the apparent horizon with radius rA = ( H2 + k/a2 ) −1/2 and temperature TA = 1 / (2 πrA ), and an effective first law dE = TAdSA + WdV with SA = A/ 4 G for perfect fluids [ 99 ]. Our intrinsic channel can be matched to this framework by choosing the modular flow to be the Kodama vector flow in spherically symmetric patches; the SK kernel then satisfies a local FDR at TA . While a full proof of the GSL in this time–dependent setting requires further work, the combination of (i) local QDB, (ii) positivity of the noise kernel, (iii) monotone matter entropy production (Spohn), and (iv) the quasi–local first law, strongly suggests a generalized Clausius inequality at the apparent horizon, d dtS(patch) out +AA 4G≥0,(100) up to corrections of order the adiabatic parameter ˙ H/H2. H. Summary of Section VII • The KMS structure of quantum fields for horizon time provides a canonical reference state σ and modular Hamiltonian K. • Athermal QDB GKLS generator (our intrinsic self–measurement respecting the horizon symmetry) guarantees monotonic decrease of S(ρtkσ)(Spohn), i.e. an intrinsic arrow of time. • The relative entropy identity S ( ρkσ ) = ∆ hKi− ∆ Sout , together with the horizon first law d ∆ hKi = dSBH, implies the GSL: d(Sout +SBH)/dt ≥0. • In stochastic gravity, the same SK kernel that encodes our intrinsic channel supplies a positive noise kernel satisfying an FDR at the horizon temperature, ensuring stability and thermodynamic consistency. 32 Approach Linear on ρCP Local / covariant Preserves GR,A Pζ (LO) Energy injection Intrinsic (this work) Yes Yes Yes Yes Preserved (QND) UV–soft, bounded EID (tracing env.) Yes (Markov) Yes Model– dep. Yes Model– dep. Model– dep. GRW / CSL Yes Yes Relativistically hard No (gen.) Shift (non– QND) Secular heating Nonlinear QM No – – – Unclear Ensemble– dep. Gravity decoherence (Newt.) Yes Sometimes Nonrelativistic – Model– dep. Model– dep. Pilot–wave Unitary – Yes Yes Preserved None Histories (consistency) – – Yes (formal) – – – TABLE I: Qualitative features of several approaches. “LO” = leading order. “–” indicates not applicable or not the central concept. off–shell, Keldysh (noise) components and the extension of composite operators. It can therefore coexist with either UV completion and, in fact, may help organise low–energy effective observables by fixing finite parts through a CP, thermodynamically consistent principle. H. A concise feature matrix I. Mathematical interludes: GKLS vs CSL and Ward identities GKLS–CSL proximity and divergence. Both (31) and (114) have the formal structure ˙ρ = −i [ H, ρ ] −1 2Pα [ Lα, [ Lα, ρ ]]. The difference is that in our construction the Lα are chosen among local, gauge–invariant densities that (i) act trivially on spacelike complements (Proposition 2.1), (ii) preserve canonical commutators and retarded response, and (iii) satisfy QDB with respect to a KMS state. This ensures Ward/Slavnov–Taylor identities for retarded functions and an arrow of time compatible with horizon thermodynamics. Generic CSL choices do not enforce (i)–(iii) and must be tuned or extended to avoid conflicts. Ward identities. Let Jµ be a conserved current and W [ A ]the generating functional with background source Aµ . In the SK formalism, gauge invariance implies the Ward identity kµ Γ µν R ( k )=0for the retarded 1PI vertex. Since our intrinsic kernel changes only the K –component (Keldysh) and leaves R/A untouched (Eq. (102) ), the Ward identities for response functions are identical to the unitary theory. This is the precise sense in which “on–shell” physics (response, scattering phases) remains unchanged while noise and off–shell interferences are modified. J. Conceptual summary • EID provides powerful, model–dependent decoherence; our intrinsic channel supplies a universal, symmetry–tied floor and arrow. • CSL/GRW deliver objective collapses but at the cost of new constants and, in relativistic settings, tensions with locality/covariance; our construction is local/covariant by design and uses no new fundamental parameters beyond curvature/KMS data. •Nonlinear QM risks no–signalling violations; our linear CP semigroup is safe. • Gravity–induced decoherence proposals share the influence–functional mathematics with our approach; we ground the kernel in higher–coherence curvature and QDB, ensuring thermodynamic consistency (GSL). 33 • Pilot–wave and decoherent histories solve the measurement problem kinematically or conditionally; our mechanism provides a dynamical route to diagonalisation and entropy production without invoking external observers. • For renormalisation, our channel selects finite extensions in a local and symmetry–compatible way, complementing Wilsonian/EG methods and coexisting with UV completions. X. CONCLUSION AND OUTLOOK A. What we have shown This work advances a concrete, symmetry–controlled answer to two long–standing questions: (i) Why and how do quantum fields become classical in the real universe, without external observers? (ii) Can the same structural ingredient that solves the measurement problem also regularise ultraviolet (UV) singularities in a principled way? Our central claim is that a small but universal intrinsic self–measurement acts on quantum fields as a completely positive (CP), locally covariant GKLS semigroup, whose generator is fixed (up to matching coefficients) by higher categorical coherence curvature and local KMS (thermal) structure. This contribution preserves on–shell unitarity and Ward/Slavnov–Taylor identities, modifies only off–shell (Keldysh) components, and provides both a mode–by–mode arrow of time and a dynamical selection of finite renormalised composites. From higher coherence to GKLS. We distinguished triangle coherence defects (2–cocycles/central extensions)—which only produce phase anomalies and therefore preserve linearity and unitarity—from pentagon/hexagon defects (3– and higher cocycles) that induce a genuine, positive imaginary part of the Schwinger–Keldysh (SK) influence functional. The latter yields a local noise kernel and hence a GKLS generator. This structural step—“beyond central extensions”—is the first novelty: it ties nonunitary, yet CP and local, corrections to the algebraic cohomology that classifies higher coherence. A conservative, QND baseline. Specialising to a quantum–nondemolition (QND) choice of jump density (the field amplitude itself for a given mode), we derived and solved the master equation ˙ρk=−i[Hk, ρk]−γk 2[ζk,[ζ−k, ρk]], and proved: (i) preservation of equal–time commutators and of retarded/advanced propagators; (ii) invariance of the scalar and tensor power spectra at leading order; (iii) monotone growth of a quantitative classicality measure (Proposition III.1); (iv) no revival of coherence (Proposition III.2); (v) an exact mapping to a classical momentum–diffusion Langevin process on super–Hubble scales. This is the second novelty: a concrete, solvable, symmetry–compatible baseline that builds a classical world without altering on–shell physics. Conveyor belt classicalisation. We embedded the single–mode dynamics in cosmology and proved afront law in momentum space: at any epoch there exists a classicality wave number kc ( N )such that modes with k < kc are effectively classical, while a thin, stationary (or bounded) transition layer adjacent to the horizon contains the still–quantum super–Hubble modes (Theorem V.1). This conveyor belt picture—quantum birth at the horizon, monotone drift to classicality, no oscillation back—is the third novelty: it reconciles the perpetual generation of new quantum modes with an ever more classical large–scale universe. Entropy and horizons. We equipped the semigroup with quantum detailed balance (QDB) with respect to the horizon KMS state, proved Spohn monotonicity of relative entropy, and combined it with the horizon first law to obtain a tidy derivation of the generalized second law (GSL) for stationary horizons (Proposition VII.1). This is the fourth novelty: an intrinsic, local arrow of time aligned with horizon thermodynamics. Renormalisation as CP selection. On the UV side, we showed how the CP filter supplied by the semigroup reduces scaling degree and selects canonical Epstein–Glaser extensions without introducing hard cutoffs or violating Ward identities. This is the fifth novelty: regularity and scheme selection as a dynamical, thermodynamically consistent principle that leaves the on–shell S–matrix intact. Observational posture. We formulated clean, falsifiable statements: leading–order power spectra are unchanged (QND); phase–sensitive, highly oscillatory bispectrum templates can be mildly damped by a kernel width; two–mode entanglement is extraordinarily robust (threshold Γ c = sinh 2 r ); and a soft–mode dephasing function D ( kL ) = e−2ΓkL encapsulates the effect on squeezed–limit templates. Current data 34 allow the tiny intrinsic rates implied by our energy and precision bounds, while offering clear targets for CMB–S4/LiteBIRD/21 cm surveys. B. Why this matters Conceptually, the proposal unifies the quantum–to–classical transition and renormalisation within a single structure: a local, CP, QDB semigroup determined by higher coherence. It threads a careful path between three pitfalls: • Nonlinearity. We avoid ensemble–dependent, signalling–prone nonlinear dynamics; linear CPTP evolution ensures operational causality. • Nonlocality. We avoid ad hoc, frame–dependent collapse by working inside the locally covariant algebraic framework and preserving BRST/Ward identities for response. • Phenomenological arbitrariness. The rates and kernels are not free knobs; they are constrained by cohomology classes and KMS/QDB data, and by locality/positivity (noise kernels). Practically, the mechanism is minimal: it leaves precision on–shell observables intact at leading order (hence no conflict with the most accurate data), while deciding precisely those quantities that are a priori ambiguous (off–shell extensions, phases in coherences). C. Caveats and robustness Our analysis adopted (i) a Markovian, local kernel on Hubble timescales, (ii) slow–roll backgrounds for explicit formulae, and (iii) a QND jump choice for conservative predictions. These can be relaxed: non–Markovian kernels introduce controlled, higher–order corrections and mild damping of fast oscillations; density–weighted rates were treated and shown to preserve the conveyor–belt picture; non–QND jumps are allowed by symmetry but then power spectra receive small, positive shifts that can be bounded. None of these deformations threatens the core results: preservation of retarded response, monotone classicalisation, and the GSL under QDB. D. Outlook: a program We close with a concrete research agenda. (A) Cohomology ⇒kernels. 1. Classify the relevant higher coherence (3–cocycle and 2–group) curvature classes for gauge and diffeomorphism symmetries in realistic sectors (inflaton, Standard Model, gravitons). 2. Derive the associated SK kernels explicitly, including finite temporal width, and match the dimensionless per–e–fold rate εkto curvature invariants. (B) Locality, BRST, and renormalisation. 1. Prove a general theorem (in the locally covariant QFT framework) that CP filters with QDB and local jumps preserve time–ordered Ward/Slavnov–Taylor identities to all orders. 2. Work out in detail a simple interacting model (e.g. λφ4 in curved spacetime) where the CP filter fixes the Epstein–Glaser extension uniquely and compare with conventional renormalisation schemes. (C) Cosmology beyond leading order. 1. Compute next–to–leading corrections to Pζ and to shapes for non–QND channels; derive sharp constraints from Planck+ACT+SPT and forecast for CMB–S4/LiteBIRD. 2. Develop a binned estimator for the soft–mode dephasing function D ( kL )and test it on simulations including realistic sky cuts and foregrounds. 35 (D) Gravity and horizons. 1. Establish the quasi–local FRW version of the GSL by combining the CP/QDB semigroup with the Kodama flow and the apparent–horizon first law; quantify corrections O(˙ H/H2). 2. Embed the intrinsic kernel in stochastic gravity and compute the corresponding noise kernel and Einstein–Langevin backreaction for cosmological backgrounds. (E) Many–body and numerics. 1. Explore how a weak, symmetry–respecting CP dephasing layer (the intrinsic channel) affects numerical renormalisation tools (e.g. tensor networks/DMRG, entanglement renormalisation), as a way to stabilise continuum limits without hard cutoffs. 2. Design analogue simulations (quantum optics/BEC) that emulate QND–type intrinsic dephasing on coupled field modes and measure the predicted no–revival and entropy–production laws. (F) Beyond QND. 1. Catalogue all local, gauge–invariant jump densities consistent with covariance (e.g. components of Tab , curvature scalars) and map their observational fingerprints (small positive shifts in variances, distinctive non–Gaussian tails). 2. Investigate whether certain higher–coherence classes inevitably force departures from QND and what that implies for bounds. E. Final remarks The picture that emerges is economical and, we believe, compelling. Triangle–level coherence defects explain why ordinary quantum mechanics is linear and unitary: at that level, only phases change (central extensions). The first truly new dynamical ingredient appears with pentagon/hexagon defects: a small, local, CP and QDB–compatible semigroup which (i) settles the quantum–to–classical transition by a symmetry–selected, mode–by–mode dephasing that never revives; (ii) calibrates renormalisation by dynamically fixing finite parts; and (iii) aligns the arrow of time with horizon thermodynamics. It is hard to imagine three problems more central to quantum field theory and cosmology that could be addressed by a single structural move. The empirical stakes are modest but real: the soft–mode dephasing function D ( kL )and oscillatory–shape damping give a way to look for this mechanism in the sky, while its negligible laboratory impact explains why conventional quantum mechanics has passed every test. Whether this intrinsic self–measurement is merely a useful organising principle or a fundamental feature of Nature is an open question. 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[148] For a purely QND channel the leading correction cancels and the first nonzero contribution can be higher order in εktimes slow–roll parameters, making (104) overly conservative. [149] Sketch: treating the bispectrum time integral in the stationary–phase approximation around η∗ (horizon crossing), the Gaussian kernel convolves the oscillatory phase to Rd ∆ η eiω∆η−(∆η)2/(2σ2 η)∝e−ω2σ2 η/2 ; the square from the two SK legs yields (106) . Details depend on the precise cubic vertex and wavefunctions, but the qualitative Gaussian damping of fast oscillations is robust.