Quantum Gravity as Contextual Gluing: Coherence Defects, Categorical Renormalization, and Operational Predictions from Horizons to Scattering

Quantum Gravity as Contextual Gluing: Coherence Defects, Categorical Renormalization, and Operational Predictions from Horizons to Scattering Andrei T. Patrascu FAST Foundation, Destin FL, 32541, USA email: andrei.patr[email protected] We present a formulation of quantum gravity based on contextual gluing rather than the quantization of a globally defined metric field. In this approach, the fundamental objects are operational contexts: locally well-defined experimental or observational setups specifying a set of accessible observables, a clock or time-reading protocol, and a causal ordering meaningful within that setup. Each context admits an effective local spacetime description, but comparisons between distinct contexts—for example, between different clocks, observers, or causal patches—are defined only through operational identification maps rather than exact isomorphisms. The term global is used here not to denote a single underlying spacetime manifold or universal time slicing, but a consistent network of relations among local contexts. Global consistency is therefore formulated as coherence of these identification maps under composition, rather than strict equality. Failures of strict composition give rise to coherence defects, which encode the physical content of quantum gravity in this framework. These defects quantify the obstruction to simultaneously realizing all local descriptions within a single global structure. We show that quantum gravity becomes renormalizable when renormalization is reformulated as refinement of operational contexts rather than ultraviolet momentum flow. This defines a categorical renormalization group, in which coherence-defect data (kernels, power spectral densities, or channel correspondences) evolve under refinement while remaining closed within a finite universality class. Gravitons arise as infrared collective excitations corresponding to the near-strict Gaussian fixed point of this coherence renormalization flow, rather than as fundamental degrees of freedom. We develop an explicitly operational pipeline that infers coherence-defect statistics from detector response data and propagates them to observable predictions. This is demonstrated in several settings of increasing generality. First, we obtain Page-curve dynamics without holography or replica constructions using a three-party Gaussian purification model, where entropy turnover follows from non-factorizing contextual gluing and is witnessed by mutual information constraints. Second, we compute exact clock-dependent corrections to Unruh and de Sitter detector responses using Gaussian switching and full Wightman pullbacks, yielding closed-form and numerical defect signatures that preserve thermality while modifying operational readouts. Third, we analyze slow-roll FLRW cosmologies across multiple epochs, infer coherence-defect power spectra from refinement data, and demonstrate closure under a categorical renormalization group with a small set of running parameters and universal scaling behavior. Finally, we show how inferred coherence defects lead to concrete, observer-local scattering predictions through a phase-diffusion exponent governing interferometric visibility loss. This provides a generic-spacetime quantum-gravity observable that does not rely on asymptotic boundaries, holographic duals, or ultraviolet completions. Together, these results establish a coherent, renormalizable, and operationally predictive framework for quantum gravity that extends beyond existing approaches while reproducing known semiclassical limits in the appropriate regime. 0. READER’S ROADMAP AND MAIN CLAIMS This paper develops and tests a concrete thesis: Quantum gravity is, at minimum, the quantum theory of how local descriptions are glued together. The central move is to shift what is taken as fundamental. In standard approaches one begins by assuming a single global spacetime (or at least a globally meaningful background structure) and then attempts to quantize fields or geometry within it. Here we proceed in the opposite direction: we take as primitive a family of operational contexts—local experimental/observational setups in which time, causal ordering, and accessible observables are defined—and we treat the comparisons between contexts as the locus where genuinely quantum-gravitational effects appear. The guiding slogan is not “spacetime fluctuates” but: comparisons fluctuate coherently. A global spacetime description, when it exists, is a strictification of this network of comparisons. When such strictification fails (or is not operationally meaningful), we obtain coherence defects. These defects are the quantities we renormalize, infer from data, and use for predictions. 2 This section defines the operational meaning of quantum gravity adopted here, states what is genuinely new relative to three major baselines, makes a sharp distinction between proven/constructed content and proposed modeling choices, and summarizes the worked examples and computational outputs that substantiate the framework. A. What “quantum gravity” means in this paper (operational definition) Operational contexts. An operational context Cis a specification of: 1. a set (or family) of accessible observables AC (typically a subalgebra of the full theory’s observables, determined by experimental limitations and causal accessibility); 2. atime-reading protocol (a “clock”) that defines a local time parameter τC and an ordering procedure for time-dependent measurements inside the context; 3. an overlap rule: a declaration of when two contexts can be compared (shared observables, shared causal region, shared calibration data). The point is that none of these is free: in actual physics, “time” is not a coordinate label but a readout of a physical process, and “the system” is not the whole universe but the algebra you can access. These are the primitives that remain meaningful even in the presence of horizons and in the absence of asymptotic structure (where S-matrix or boundary-time constructions are unavailable). Global means coherent, not absolute. In this paper global does not mean: “there exists a single manifold-with-metric and a single global clock that coordinatizes all events and underlies all descriptions.” Instead, global means: “there exists a consistent way of relating local contexts on overlaps so that all comparisons are mutually compatible up to coherent higher data.” Concretely, we consider a system of contexts and comparison maps: Cu −→ C0(comparison morphism), which represent the operational identification of observables and time-ordering prescriptions between contexts. In general these comparisons do not compose strictly: (uC0→C00 ◦uC→C0)6=uC→C00 as equal maps, but may compose only up to a coherent transformation. In categorical terms, we work not with a strict category of contexts but with a weak 2-groupoid (or more generally a bicategory) in which associativity holds up to an associator 2-morphism satisfying coherence conditions [ 1 , 2 ]. The failure of strict composition around loops defines the basic “defect” object of the theory. Coherence defects as quantum-gravitational degrees of freedom. Let C0→C1→ ··· → Cn = C0 be a loop of contexts with comparison morphisms ui : Ci→Ci+1 . The loop defect is the resulting endo-identification of C0: Ωγ:= un−1◦···◦u1◦u0:C0→C0. In a strict (classical) situation Ω γ would be (operationally) the identity. Here the basic possibility is that Ω γ is nontrivial but still compatible with all available overlap data. In the simplest cases, Ω γ is central or “modular” in the sense that it acts trivially on coarse observables but nontrivially on the comparison structure (this is the operational analog of a projective ambiguity). Such loop defects are the fundamental carriers of quantum-gravitational content in our framework. 3 Why this counts as quantum gravity rather than merely “QFT in curved spacetime.” In standard semiclassical QFT, one assumes a background geometry and then computes state-dependent quantities such as detector response, stress tensor renormalization, and horizon thermality [ 3 , 4 ]. The background is strict and global. In contrast, here: •“time” and “event identification” are themselves context-dependent operational objects; • non-strict gluing modifies the bookkeeping of independence (factorization) across contexts, which is precisely what underlies information-flow paradoxes (Hawking/Page) and horizon thermality comparisons; • the fundamental “RG” object is the running coherence/defect data under refinement of contexts, not a local metric perturbation field. Thus quantum gravity is defined operationally as the quantum theory of the comparison and gluing structure that makes spacetime meaningful in the first place. B. What is new compared to three baselines 1. Compared to semiclassical QFT in curved spacetime Semiclassical QFT in curved spacetime provides exact and powerful statements about local physics in the presence of horizons, including Hawking radiation [ 5 ] and Unruh thermality [ 6 ], as well as the de Sitter (Gibbons–Hawking) temperature [ 8 ]. It also provides the detector models (Unruh–DeWitt) that make these effects operational [7]. What is new here is not the claim that these effects exist, but the claim that: 1. the comparison between different operational time readouts and different accessible algebras is not a gauge choice but part of the physical content; 2. the failure of strict factorization and strict identification across contexts is not treated as a pathology or ignored—it is elevated to a controlled, coherent datum that can be inferred and renormalized; 3. a single global “spacetime object” is replaced by a coherent network of local spacetimes whose gluing is quantized. We insist on compatibility with the semiclassical limit: within each context, standard QFT-in-curvedspacetime calculations are recovered. The new content appears precisely when one attempts to glue contexts globally, especially across horizons and incompatible clock protocols. 2. Compared to perturbative graviton quantum gravity Perturbative quantum gravity treats the metric as a field gµν = ηµν + hµν and quantizes hµν as a spin-2 particle (the graviton) in the weak-field regime. It is well-known that this strict metric-field viewpoint leads to perturbative non-renormalizability in four dimensions: under UV refinement one generates an infinite tower of higher-dimension counterterms, so the class of local actions does not close under the usual Wilsonian procedure [10]. What is new here is: 1. we do not assume hµν is fundamental; instead we treat it as an effective strictification representative valid in near-coherent regimes; 2. renormalization is reformulated as refinement of operational contexts (switching windows, bandwidths, causal diamonds, epoch scale) and the running object is defect/coherence data (kernels, PSDs, channels, cocycle classes); 3. “renormalizability” becomes a statement of closure under refinement for the defect structure: refinement changes the effective filters but does not force an infinite proliferation of independent structures. In this sense, gravitons are recovered as infrared collective excitations (Gaussian fixed-shape fluctuations) of near-strict coherent gluing, but the ultraviolet completion problem is not posed as the renormalization of a fundamental local graviton field. 4 Approximation note. Whenever we speak of an “emergent graviton” we are making a controlled approximation: we assume a near-strict regime in which coherence defects are small and admit a Gaussian/linearized representative. We explicitly do not claim that this approximation holds in regimes with large defect holonomy or strong horizon-induced non-factorization. This will be stated again where relevant. 3. Compared to string theory and holography String theory provides a UV completion of perturbative gravity and, in certain settings, a nonperturbative definition via holography (notably AdS/CFT). In those settings it offers powerful tools for entanglement and Page-curve calculations using boundary duals. However, many physically central situations are not in that class: generic cosmological spacetimes (slow-roll FLRW) and generic observer-local causal patches do not come equipped with a universal, operational boundary theory. For such cases, one often lacks a single canonical prescription that is simultaneously (i) nonperturbative, (ii) observer-local, and (iii) operationally defined. What is new here is the insistence that: 1. the fundamental observables should be definable within finite causal patches and should be tied to clock protocols and accessible algebras (detectors, switching, bandwidth, etc.); 2. one can obtain Page-curve mechanisms, horizon thermality comparisons, and cosmological predictions without invoking an AdS boundary or a dual field theory; 3. renormalization is not a UV completion statement about fundamental strings, but a closure statement about defect/coherence data under context refinement. This does not contradict holography where it applies. Rather, it proposes an operationally universal layer that remains meaningful in regimes where holographic duality is unavailable or not uniquely specified. C. What is proven/constructed vs what is proposed This paper contains three classes of content: (I) Definitions and structural claims (constructed). We construct a precise operational notion of context and of global coherence as a network of comparisons. We formalize coherence defects as loop self-correspondences (or, equivalently, as nontrivial higher associativity data) [ 1 , 2 ]. We define a categorical renormalization group as refinement of contexts and formulate closure under refinement as the appropriate notion of renormalizability for quantum gravity in this framework. (II) Controlled models and exact computations (proven within model assumptions). We provide explicit computations in the following controlled settings: •Unruh effect with incompatible clocks: using an Unruh–DeWitt detector model with Gaussian switching, we derive closed-form first-order clock-modulation corrections and compute exact numerical defect curves using the full accelerated Wightman function [3, 4, 6, 7]. •de Sitter (Gibbons–Hawking) horizon defect: we repeat the same operational calculation in de Sitter, producing a cosmological horizon defect curve without invoking dS/CFT [8]. •Hawking/Page mechanism without holography: we construct a three-party Gaussian purification model (radiation memory, outgoing mode, partner mode) that yields a dynamical Page curve with mutual-information witness, without AdS boundaries or replica methods. The baseline motivation is the Hawking semiclassical setup and Page’s information-theoretic criterion [5, 9]. •Slow-roll FLRW categorical RG and scattering: we demonstrate (in a local-patch slow-roll approximation) inference of defect PSD parameters across multiple epochs, show closure within a finite family, extract RG scaling exponents, and predict a phase-diffusion scattering exponent controlling interferometric visibility loss. Each of these computations is exact within its stated model assumptions (Gaussian switching, small defect expansion where used, and specified PSD families where inferred). 5 (III) Modeling hypotheses and universality statements (proposed but testable). Two kinds of assumptions are introduced as hypotheses to be tested: 1. Closure family hypothesis: defect statistics can be represented by a finite-parameter PSD family (universality class) across refinement and epoch changes. This is proposed as the operational meaning of renormalizability in coherence space and is empirically tested by goodness-of-fit and scaling collapse. 2. Local-patch slow-roll approximation: for FLRW we approximate the worldline kernel by local horizon-scale data. This is a controlled slow-roll approximation (valid when the operational window is short compared to slow-roll variation), and it is explicitly stated as such. These hypotheses are not “assumptions of the theory” in the metaphysical sense; they are modeling choices required to produce falsifiable predictions. They can be refined as one moves to richer defect representatives (operator-valued kernels, non-Gaussian defect statistics, multi-mode radiation memories, etc.). D. Summary of worked examples and outputs (figures and computations) Example A: Hawking/Page without holography (information bookkeeping from coherence). Starting from the semiclassical Hawking setup [ 5 ] and the Page-curve criterion [ 9 ], we replace strict factorization of “old radiation ⊗ new radiation” by contextual gluing. In explicit Gaussian toy models we demonstrate: 1. a dynamical radiation entropy Srad ( k )that rises and then decreases (Page turnover) in a three-party unitary model; 2. a per-step entropy increment ∆Srad that changes sign at the Page time; 3. a mutual-information witness I(R:c)capturing the coherence defect as correlations; 4. robustness under increasing radiation-memory capacity (three Gaussian memory modes) to remove finite-memory artifacts. This produces an operationally interpretable mechanism: thermality/flux may remain Hawking-like, while independence bookkeeping fails due to coherent identifications across contexts. Example B: Unruh effect with incompatible clocks (exact operational defect curve). Using the Unruh– DeWitt detector formalism [ 7 ] and the standard Unruh kernel [ 6 ], we introduce a clock modulation ˜τ=τ+εsin(ντ)and Gaussian switching χ(τ) = e−τ2/(2T2). We derive: 1. closed-form first-order corrections expressed as sideband differences KT(Ω ±ν/2); 2. exact numerical evaluation of the defect < [∆ F(1) ] /< [ F0 ]using the full accelerated Wightman pullback [3, 4]; 3. the physically required limiting behaviors: vanishing in the stationary limit and suppression for large νT. Example C: de Sitter horizon defect without dS/CFT. Replacing a by the de Sitter scale H and using the Bunch–Davies pullback [ 3 , 4 , 8 ], we compute the exact cosmological horizon defect curve < [∆ F(1) ] /< [ F0 ]vs ν under the same Gaussian-switching protocol. This provides an observer-local cosmological horizon observable independent of any boundary dual. Example D: Slow-roll FLRW categorical RG and scattering predictions. We implement a two-channel (two detector gaps) multi-resolution (25 switching widths) inference scheme for defect PSDs across multiple epochs in slow-roll FLRW. In the simplest closed-family universality class: Sδ(ν;H) = A(H) 1+(ν/νc(H))αexp[−(ν/νhi(H))2], we demonstrate: 1. closure under refinement: all epochs are well fit by the same PSD family with finite parameters; 6 2. categorical RG flow: parameter evolution across five epochs, extraction of scaling exponents (e.g. νc∝Hand A∝H−q); 3. universality: scaling collapse of scattering observables under renormalization; 4. scattering prediction: the phase-diffusion exponent Ξ(Tint) = 1 2ω2 0Zdν 2πSδ(ν) sinc2(νTint/2), which controls interferometric visibility e−Ξ and is predicted epoch-by-epoch from inferred defect statistics. This end-to-end pipeline—context refinement data → defect statistics → scattering prediction—is the central “computational” demonstration of coherence renormalizability in generic FLRW patches. Summary claim (what the reader should take away). Across these examples, the recurring pattern is: 1. Within each context, standard semiclassical physics is recovered (Unruh, de Sitter thermality, Hawking pair creation). 2. Nontrivial quantum-gravitational content arises when one compares and glues contexts (incompatible clocks, patchwise algebras, horizon splits). 3. The running object is coherence/defect data, which closes under refinement and yields stable predictions for operational observables. 4. Gravitons appear as an IR strictification (Gaussian fixed-shape regime) of defect statistics rather than as fundamental UV degrees of freedom. 1. QUANTUM GRAVITY AS CONTEXTUAL GLUING This section explains why the conventional starting point—“take a globally defined metric field and quantize it”—is conceptually and technically misaligned with what quantum gravity must accomplish. The argument is not that one can never use a metric perturbation in computations. Rather, the claim is that taking the metric as the fundamental object to be quantized smuggles in strong strictification assumptions that are precisely what fail in the regimes where quantum gravity becomes unavoidable. We also explain why the same diagnosis applies, in a different way, to string theory and holography: even when one does not literally quantize hµν , one typically retains global strict structures (global time, global factorization, asymptotic boundaries) that are absent or operationally meaningless in generic situations (cosmology, finite causal diamonds, observer-local horizons). A. Why “quantize the metric” is the wrong starting point 1.1.1 The strictification package that is usually assumed (often implicitly) To “quantize the metric” in the standard perturbative sense is to begin with the Einstein–Hilbert functional on a globally defined Lorentzian manifold ( M, g )and expand around a fixed background g(0) : SEH[g] = 1 16πG ZM d4x√−g(R−2Λ) + Smatter[g, Ψ],(1) then set gµν =g(0) µν +κ hµν, κ := √32πG, (2) fix gauge, and quantize hµν as a spin-2 field. This procedure presupposes, at minimum, the following strictification assumptions: (S1) A single global spacetime object. There exists a globally defined manifold M and a globally defined metric g (or at least a globally meaningful background g(0) plus fluctuations). All observers and operational procedures are assumed to refer to the same M and to the same underlying g (perhaps in different coordinates). 7 (S2) A global time or globally consistent time-ordering. Perturbative quantization presupposes a global notion of time-ordering (for defining propagators, Feynman rules, and asymptotic states). Even in covariant gauges, the computational machinery is anchored in global Green’s functions built from a single background structure. (S3) A strict subsystem factorization. To speak of “gravitons scattering,” “particles,” and an S -matrix, one assumes asymptotic regions and a strict tensor factorization of the Hilbert space into subsystems associated with separated regions/modes. In more algebraic language, one assumes that the physically relevant algebras admit a strict factorization compatible with the dynamics. The conceptual point of this paper is that these strictification assumptions are not innocent. They are precisely what horizons, causal patch physics, and operational clock-dependence make dubious. In our framework, what is fundamental is the network of contexts and their comparison maps; a global spacetime object is an emergent strictification when the coherence defects vanish or become negligible. 1.1.2 Where strictification fails: horizons, causal patches, and measurement Even before discussing ultraviolet behavior, strictification fails as a global operational principle in several well-understood regimes. (F1) Horizons and inequivalent time flows. Unruh and Hawking phenomena already show that the choice of time flow is not a mere coordinate convention: different observers naturally associate different Hamiltonians and different notions of “particles” to the same underlying field state. The Bisognano– Wichmann theorem makes this precise in algebraic QFT: the modular flow of the wedge algebra implements Lorentz boosts, and the Minkowski vacuum restricted to the wedge is a KMS state with respect to that modular/boost time [ 11 ]. This is not a pathology; it is a structural fact about how local algebras behave. The practical lesson is that “time” is already a context-dependent operational input even in flat spacetime. Treating time-ordering as absolute is therefore a strictification. (F2) Local algebras and the failure of naive factorization. In relativistic QFT, physically relevant local von Neumann algebras are typically of type III; they do not admit a trace and do not factorize as tensor products in the naive way associated with finite-dimensional quantum mechanics. This is a cornerstone of algebraic QFT [ 12 ]. In such a setting, assuming a global Hilbert-space tensor factorization into “inside” and “outside” degrees of freedom is not a harmless convenience—it can be the source of spurious paradoxes if used beyond its domain of validity. In our language: strict subsystem factorization is an extra structure that need not exist globally, while a coherent gluing of contexts does. (F3) Quantum measurement and context dependence. Operationally, the act of selecting a clock and an observable set is part of the experiment. In curved spacetime and in the presence of horizons, different choices correspond to different accessible algebras and different orderings. The notion “there exists a single global description, and the observer merely chooses coordinates” conflates coordinate freedom with operational accessibility. Our framework separates them: coordinate changes are within a context; changing the clock protocol or the accessible algebra is a change of context, and comparisons between contexts need not strictify. These failures are not ultraviolet issues. They occur at macroscopic scales and are already visible in semiclassical physics. This is why quantum gravity must be formulated so that non-strict global gluing is allowed from the start. 1.1.3 The technical obstruction: perturbative non-renormalizability is a non-closure statement Now turn to the standard technical problem: perturbative non-renormalizability of gravity in four dimensions. The textbook power-counting argument is that the gravitational coupling κ has negative mass dimension: [κ] = −1in mass units, so higher-loop diagrams produce divergences requiring counterterms with increasing numbers of derivatives, i.e. operators of arbitrarily high dimension. In the language of effective field theory, one can still compute low-energy predictions order-by-order, but the class of actions does not close on a finite set of couplings. More strongly, explicit computations show that pure gravity develops a nonvanishing divergence at two loops requiring an R3-type counterterm [13, 14]. This is often summarized as: 8 The strict metric-field strictification does not close under refinement. That phrasing is already suggestive of our viewpoint: “non-renormalizable” means that the chosen fundamental variables do not form a stable, finite-dimensional manifold under refinement. The important conceptual move is to ask whether the failure is in the world, or in the strictification. In our framework, the object that runs under refinement is not hµν but the coherence defect data (kernels/PSDs/cocycles/channels) that quantify the obstruction to strict global identification of contexts. Renormalizability becomes the statement that this defect structure remains within a closed universality class under refinement of operational contexts. This is a different notion of closure, adapted to what is operationally meaningful. 1.1.4 Why string theory is not the right universal starting point either One might object: “string theory does not quantize the metric field directly; it provides a UV completion.” This is correct in its domain. But the point of this paper is not that string theory is inconsistent; it is that it does not supply a universal operational framework for generic spacetimes and observer-local physics. (T1) Global time and asymptotics remain structural. Perturbative string theory is defined as a worldsheet theory that computes scattering amplitudes in backgrounds admitting asymptotic states. Even in nonperturbative holography (AdS/CFT), one relies on a globally defined boundary time and a boundary Hilbert space that anchors the computation [ 15 – 17 ]. This is a powerful strictification strategy, but it is not available in generic cosmologies (slow-roll FLRW), finite causal diamonds, or observer-local horizon problems without additional conjectural structure. (T2) Factorization is assumed (then repaired) rather than derived as coherence data. Even in holographic Page-curve computations, one begins with a factorized description and then employs gravitational extremization/replica methods to correct it. Our framework instead treats non-factorization as a primary coherence datum encoded in the gluing of contexts. Operationally, the relevant question is: “which degrees of freedom are independently accessible under a given clock protocol and causal patch?” That question is not naturally expressed in string theory’s standard observables unless an appropriate boundary dual exists. (T3) Method universality vs physics universality. Another way to state the distinction is: String theory offers a UV completion strategy; contextual gluing offers an operational universality layer. The point is not that strings cannot exist; rather, any UV completion still needs an operational account of how different observers and clock protocols glue local descriptions without assuming a single global strict picture. That is the layer we build here, and it continues to apply even where string theory has no canonical operational prescription. 1.1.5 The replacement: quantize gluing, not the metric We now state the replacement principle precisely. Contexts and their comparison structure. Let C denote a collection of operational contexts. Each context C∈ C carries: (AC, τC,≺C), where AC is the accessible observable algebra, τC is a clock protocol/time parameter, and ≺C is the induced causal/temporal ordering meaningful within C. Comparisons between contexts are represented by morphisms u : C→C0 encoding operational identification maps (not necessarily *-isomorphisms; in general, correspondences/channels are appropriate). Weak composition and coherence. In general, comparison morphisms do not compose strictly. Instead, one has associators and unit constraints as higher morphisms. In categorical terms, the contexts form (at least) a bicategory or weak 2-groupoid rather than a strict category [ 1 , 2 ]. The failure of strictification is measured by loop endo-identifications: Ωγ:= un−1◦···◦u0:C0→C0, which need not be the identity even when all local overlaps are consistent. The coherence defect is the residual equivalence class of Ω γ (e.g. central, modular, or channel-valued), after quotienting by changes of representatives. 9 Operational meaning of “metric fluctuations.” In this view, what is often called “metric fluctuations” is not a fundamental random field hµν ( x ). It is the emergent statistics of the residual mismatch in reconstructed spacetime relations when one transports clock/rod calibrations and observable identifications along different context-comparison paths. In near-strict regimes, one may choose a strictification gauge that produces an effective gµν + hµν representative. In general regimes, the faithful representative is higher-structured (kernels/PSDs/cocycles), and scattering/visibility/entropy observables are computed directly from that data. Why this starting point is forced. The logical structure of the argument is: 1. Physics is operational: time and events are defined by clocks and observables. 2. Horizons and local algebras show that global strictification assumptions fail even in semiclassical regimes. 3. Perturbative non-renormalizability is a non-closure statement about strict metric-field variables under refinement. 4. Therefore the correct fundamental object to renormalize is coherence/defect data under refinement of contexts. This is the motivation for the categorical renormalization group developed later: renormalizability is recovered as closure of defect structure under refinement, and gravitons reappear as IR strictification excitations rather than UV fundamentals. Transition to the rest of the paper. The remainder of Section 1 will build the formalism of contexts, comparisons, and defects explicitly; Sections thereafter will show, by exact computations, that this framework produces operational predictions in horizon and cosmological settings (Unruh, de Sitter, slow-roll FLRW) and resolves Page-curve dynamics without invoking holographic boundaries. The key point of this subsection is that the starting object matters: quantizing hµν assumes the very global strictification structure that quantum gravity must explain. B. Contexts: what replaces “a spacetime point” 1.2.1 Why a “point” is not an operational primitive The classical notion of an event as “a point p∈M in a manifold” is a mathematically convenient strictification, but it is not an operational primitive in quantum physics. Operationally, one never interacts with a bare point. One interacts with: 1. aset of accessible observables (what can be measured, with finite bandwidth and finite resolution); 2. aclock protocol (how time is read, i.e. which physical process is used as a reference and how its readout is processed); 3. acomparison procedure (how readings and calibrations are translated into another laboratory’s readings). A spacetime “point” is thus, at best, the label of an idealized limiting regime in which these choices are irrelevant. Our framework explicitly rejects that idealization as a starting point and uses it only as a limit check. There is a second, deeper reason to abandon “point” as primitive: in relativistic QFT, the physically meaningful objects are local algebras associated to regions, not pointwise fields. The Haag–Kastler viewpoint is that the net of local algebras encodes locality and causality; pointlike fields are coordinatizations of that algebraic content [ 12 ]. In the presence of horizons and restricted causal access, the algebra you can access is not merely a subset of a global algebra in a canonical way; it depends on the operational context. Therefore, the minimal replacement of “point” must incorporate both locality (a region/diamond/wedge) and operational choice (clock and protocol). 16 admit a common ordering prescription. In the strictest form, this means: 1. there exists a global foliation by Cauchy hypersurfaces Σ t and a time evolution map U ( t2, t1 )acting on a single global algebra A; 2. for any family of observables {Oi} localized in spacetime regions, the notion of time-ordered product T{O1···On}is defined with respect to that same global t; 3. subsystem decompositions and their causal ordering are consistent with that global time in the sense that “earlier” and “later” are defined unambiguously for all contexts. This is an enormously strong strictification. It is not merely a coordinate choice. It is the assertion that there exists a single global ordering structure that underlies every operational procedure. Even in relativistic QFT on a fixed background, this strictification is technically circumvented: the Tomonaga–Schwinger formulation replaces global time evolution by evolution between arbitrary spacelike hypersurfaces and makes clear that locality, rather than a preferred time, is the true dynamical input [ 26 , 27 ]. In algebraic QFT, the fundamental data are local algebras and their inclusion/commutation relations, not a global Hamiltonian with a preferred time parameter [ 12 ]. Our framework pushes this logic one step further: when contexts themselves are part of the physics, there is in general no unique global notion of “the” hypersurface family or “the” time ordering shared by all contexts. 1.4.2 Contextual ordering: the minimal operational notion Within an operational context C = ( AC, τC,≺C, κC ), the ordering relation ≺C is meaningful: it is the ordering induced by the clock protocol and causal accessibility within C . In practice, ≺C can be taken as a partial order on events/records that are operationally definable within the context (e.g. detector clicks with time stamps, record registers, and the causal precedence induced by signal propagation within the accessible region). Thus ordering is meaningful because it is required to define operational statements such as: •the time-ordered response of a detector, •the causal precedence of signals in a local experiment, •the sequence of measurements whose results are stored in a record algebra. However, ordering is not absolute because different contexts need not admit a single common refinement of their orderings. There may exist contexts Cand C0such that: ≺Cand ≺C0 are both internally consistent, yet cannot be combined into a single global total order without adding extra structure that is not operationally meaningful (e.g. selecting a preferred global clock, a preferred foliation, or a preferred boundary time). The correct replacement for absolute ordering is therefore: A family of partial orders {≺C} , together with comparison morphisms that transport ordering data coherently on overlaps. 1.4.3 How commutators and causal ordering emerge together from gluing In strict quantum mechanics, “quantization” is often summarized by the introduction of noncommuting observables: [O1, O2]6= 0. In relativistic QFT, causal structure is encoded by locality: observables commute at spacelike separation. The key conceptual point in our framework is that both “commutation” and “causal ordering” are context-dependent notions whose reconciliation across contexts is a gluing problem. 17 Commutators as transported objects. Let u : C→C0 be a comparison morphism between contexts. Operationally, u acts on a suitable subalgebra of AC and produces corresponding observables in AC0 . For observables A, B in the overlap domain of u, one can compare commutators by transport: u([A, B]) versus [u(A), u(B)].(13) In a strict global description one expects equality. In our setting, it is neither necessary nor generally correct to demand strict equality. Instead, the appropriate requirement is coherent compatibility: u([A, B]) ∼ =[u(A), u(B)],(14) where ∼ = is a 2-morphism (an intertwiner/calibration transformation) supplied by the comparison structure. The residual mismatch in attempting to enforce strict equality is part of the coherence defect. This is what is meant by the statement that commutators are transported objects, not scalars. Ordering as a transported relation. Similarly, ordering relations ≺C and ≺C0 are not compared by embedding both into an assumed global order; they are compared via the same comparison morphisms. On overlaps, the comparison morphism induces a map on records/events, and one demands that causal precedence is preserved coherently: x≺Cy=⇒u(x)≺C0u(y)(up to coherent identification on the overlap).(15) The defect arises when distinct comparison paths induce inequivalent transported orderings. This is the precise content of “ordering is meaningful but not absolute”. Emergence of a classical causal order in near-strict regimes. In near-strict regimes (small defects, strong overlap), one can choose a strictification gauge in which the transported commutators and orderings agree to high accuracy. Then one recovers a classical causal order and the familiar commutator structure of QFT/GR as an effective description. The crucial point is that classicality here is not postulated; it is the limit in which coherence defects become negligible. 1.4.4 No paradoxes: why contextual ordering does not imply causal inconsistency One might worry that if different contexts have incompatible orderings then causal paradoxes are possible. This is not the case, because comparisons are only defined on overlaps where operational meaning exists, and coherence constraints forbid contradictory composition. The logic is as follows. (P1) Context-local consistency. Each context C comes with an internally consistent ordering ≺C that is compatible with signaling constraints within that context. (P2) Comparisons only on overlaps. A comparison morphism u : C→C0 is only defined when the contexts share sufficient overlap: shared observables, shared records, or a physically implementable calibration interface. Thus one never compares orderings of unrelated events; one compares orderings of comparable events. (P3) Coherence constraints suppress contradictions. When there are multiple comparison paths between the same contexts, coherence requires that the induced identifications are compatible up to specified higher data (associators, intertwiners). In particular, a loop of comparisons does not create a logical contradiction; it creates a loop endomorphism Ω γ (Section 1 C) whose physical content is precisely the defect. The defect is not a paradox; it is an observable or inferable mismatch that can be carried consistently as additional data. A concise mathematical way to express the absence of paradox is: the context network forms a weak 2-groupoid of comparisons rather than an inconsistent graph. In such a structure, composition is well-defined up to coherent isomorphism, and the coherence axioms guarantee that different ways of composing do not lead to contradictions, only to defect classes. 1.4.5 Relation to indefinite causal order and process frameworks It is useful to situate contextual ordering relative to other non-classical ordering frameworks. There exist process-theoretic approaches in which causal order is not fixed globally (e.g. indefinite causal order in process matrices) [28]. Our framework differs in emphasis: 18 •We do not begin by postulating an abstract non-causal process object. •We begin with operational contexts in which local causal ordering is meaningful. •Non-absoluteness enters through the failure of strict global gluing of those contexts. Thus the non-absoluteness of ordering is not introduced as a new kinematic axiom; it emerges as the operational consequence of comparing incompatible but locally consistent contexts. 1.4.6 Why string theory does not provide a universal operational treatment of contextual ordering String theory (and holography) can certainly reproduce causal structure in many regimes; the issue here is universality of the operational comparison problem. Boundary time as a strictification. In AdS/CFT, boundary time provides a global ordering structure that strictifies the bulk description [ 16 , 17 ]. This is a powerful strictification when available, but it is not an operationally universal structure in generic spacetimes (slow-roll FLRW, finite causal diamonds). When no canonical boundary time exists, there is no unique global time ordering to which all contexts can be referred. Worldsheet time is not operational time. In perturbative string theory, worldsheet parameters are internal to the formalism and do not directly coincide with an observer’s clock protocol in spacetime. Operational ordering in experiments is defined by physical clocks and accessible records; this ordering is context-dependent in precisely the ways analyzed in the Unruh/de Sitter calculations. The framework developed here is designed to make these operational orderings the primary data. Consequences. Therefore, even if string theory provides microphysical completion, it does not remove the need for an operational theory of how different clock-defined orderings glue in generic settings. Contextual ordering is the universal layer that remains when one does not assume a global strictification. 1.4.7 Summary The main conclusions of this subsection are: 1. A single global time-ordering is an extra strictification structure that is not operationally justified in generic horizon and cosmological settings. 2. Each operational context carries a meaningful internal ordering, but different contexts need not admit a single common refinement. 3. Commutators and causal ordering are transported structures under context comparison morphisms; their compatibility is coherent rather than strictly equal. 4. Coherence defects encode the nontrivial content of attempting to glue orderings globally; they are not paradoxes but additional data with observable manifestations. These principles will be used repeatedly: in the Unruh/de Sitter sections, ordering defects appear as measurable response-kernel deformations under clock modulation; in the Hawking/Page construction, failure of strict factorization is a manifestation of non-absolute ordering across contexts; and in the categorical RG analysis, refinement of ordering protocols produces a closed flow of defect statistics. 2. RENORMALIZABILITY REINTERPRETED: CATEGORICAL RENORMALIZATION GROUP A. What refinement means (operational RG, not UV momentum) 2.1.1 Renormalization as a question about stability under refinement In conventional quantum field theory, renormalization is most economically described as the study of how an effective description changes when one refines the resolution at which physics is probed. In the Wilsonian picture one changes a momentum cutoff Λand integrates out modes above Λ, obtaining a flow 19 of effective actions and couplings [ 29 , 30 ]. In functional approaches one encodes the same idea through scale-dependent effective actions and flow equations [31]. In the present framework, the guiding principle is the same—refinement of resolution—but the object refined is different. The primitive objects are operational contexts C= (AC, τC,≺C, κC), and renormalization is the controlled comparison between contexts at different operational resolutions. The appropriate notion of “scale” is therefore not fundamentally momentum, but operational access. This shift is forced by the fact that many quantum-gravitationally relevant regimes are not naturally described by an S -matrix at asymptotic infinity, and do not come with a unique momentum cutoff parameter. In cosmology, for example, “scale” is tied to a horizon scale H ( t ) −1 and to the causal diamond accessible to a given observer at epoch t . Likewise, in horizon problems, the relevant refinement parameter is often a switching time T and an accessible bandwidth set by detector response functions. The categorical renormalization group formalizes these operational refinements. 2.1.2 Refinement axes in operational contexts We summarize the principal refinement axes used throughout this paper. Each axis corresponds to a physically meaningful way to access more detailed information without changing the underlying system. (R1) Switching width T (temporal resolution). A detector or measurement protocol rarely operates over infinite time; it is switched on with a window function χ ( τ )(e.g. Gaussian switching). The parameter Tin χ(τ) = exp−τ2 2T2 sets the temporal resolution and the effective low-pass/high-pass filtering of time correlations. A fundamental and precise relation is that time-windowing induces a frequency filter. For Gaussian switching, bχ(ω) = Zdτ e−iωτ χ(τ) = √2π T e−(ωT )2/2, so decreasing T broadens bχ in frequency and therefore accesses higher-frequency content of correlators. This is a canonical example of refinement by operational control: T↓ is a refinement of temporal resolution. (R2) Bandwidth B .More generally, an operational context may specify a band-limited readout, either through detector response functions or through explicit spectral filtering. In frequency space, bandwidth refinement corresponds to enlarging the support of spectral sensitivity. While T and B are related (by Fourier uncertainty and protocol design), they are not identical: one can have long switching with narrow or wide spectral filters depending on the apparatus and post-processing. (R3) Spatial coarse-graining scale ` / causal diamond size. In relativistic settings, operational access is limited by causal structure. A context may be defined by a causal diamond (intersection of future and past lightcones) or by a region accessible to a given observer. Refinement can mean either: •shrinking `to increase spatial resolution within a fixed patch; or • enlarging ` to increase the accessible region (and hence the accessible algebra) within a given epoch and observer family. In either case, ` is not merely a coordinate length; it is the physical scale of operational access determined by signal propagation, detector placement, and recordability. (R4) Epoch t (cosmology). In FLRW spacetimes, the local horizon scale H ( t ) −1 changes with epoch. A context at epoch t has access to a different causal patch and different redshift structure than a context at epoch t0 . Thus “renormalization across epochs” is a refinement axis in the cosmological setting: it is a change in the operational scale of the horizon and the accessible diamond. In practice we use H ( t )or the e-fold time N= ln aas the scale parameter. (R5) Spectral channel (detector gap Ω). Operational contexts may include a detector or probe system with an energy gap Ω(Unruh–DeWitt type probes). Changing Ωchanges which frequency components of the field correlations are sampled. Thus Ωlabels a refinement axis: multi-channel measurements provide a richer set of constraints on defect statistics than a single channel. 20 2.1.3 Refinement as a functor between contexts We now formalize refinement. Definition (refinement relation). Given two contexts Cand C0, we write CC0 and say that C0is a refinement of Cif: 1. C0 accesses at least the same physical system and the same underlying state preparation class as C (“same physics”); 2. C0 has strictly greater or equal operational resolution along at least one axis (e.g. smaller T , larger bandwidth B, richer channel set, finer spatial coarse-graining); 3. there exists a physically implementable coarse-graining map from C0 to C that forgets the additional information, i.e. a comparison morphism rC0→C:C0→C that maps AC0 to AC (typically as a channel or conditional expectation) and reduces the clock/calibration data accordingly. This definition emphasizes the physically essential direction: if C0 is finer than C , there is a canonical direction of forgetting information from C0 to C . This is the operational analog of integrating out degrees of freedom. Refinement category. Let Ctx denote the category (more generally, bicategory) of contexts and comparison morphisms. The refinement relation induces a subcategory Ctxref in which morphisms are coarse-grainings rC0→C from refined to coarse contexts. Composition is operational composition of coarse-grainings: C00 rC00 →C0 −−−−−→ C0rC0→C −−−−→ C⇒rC00 →C:= rC0→C◦rC00 →C0. Thus refinement defines a functorial structure: the process of coarse-graining is associative and unital by construction. In many applications, the family of contexts at varying resolution forms a directed set: for any two contexts C1, C2 there exists a common refinement C3 with C1C3 and C2C3 . This is the operational analog of having a filtered system of resolutions. When such a property holds, it enables canonical notions of “taking the limit of refinement” and of identifying universality classes. 2.1.4 Operational refinement versus ultraviolet momentum refinement It is important to distinguish operational refinement from the traditional UV momentum refinement. UV momentum refinement as a special case. In a Minkowski QFT with asymptotic structure, one may choose contexts that are parameterized by a momentum cutoff Λand take refinement as Λ ↑ . This can be embedded into our framework by defining contexts whose accessible algebra is generated by modes below Λand whose coarse-graining is the conditional expectation that traces out higher modes. In that special case, categorical refinement reduces to Wilsonian RG [29, 30]. Why operational refinement is strictly more general. In generic curved spacetimes and in finite causal diamonds, there is no canonical global momentum decomposition and no unique global Λ. The physically meaningful scales are: •switching windows and detector gaps (time/frequency resolution), •causal patch sizes and horizon scales, •epoch-dependent redshift and accessible regions. Operational refinement therefore defines an RG notion in settings where Wilsonian momentum RG is not well-posed. This is precisely why it is relevant to quantum gravity: it builds renormalization on what remains meaningful when strict global structures are absent. 21 2.1.5 Concrete refinement maps in the worked examples We record the refinement maps that appear repeatedly in later computations. Gaussian switching refinement. Let C ( T )denote a context with Gaussian switching width T . Refinement corresponds to T0<T (finer time resolution). The coarse-graining map rT0→T is implemented by convolving the fine record with a Gaussian kernel of width √T2−T02 (equivalently multiplying frequency content by e−(ωT )2/2/e−(ωT 0)2/2 ). This is a strictly operational transformation: it is post-processing of time-stamped records. Channel refinement via detector gap Ω.Let C ( T, Ω) denote a context at switching width T and detector gap Ω. Changing Ωdoes not define a total order (it is not always meaningful to say one Ω refines another), but the pair ( T, Ω) defines a multi-channel context. Refinement then means enlarging the channel set: {Ω1}{Ω1,Ω2}, with coarse-graining given by forgetting channels. Epoch refinement in FLRW. Let C ( t )denote a context at cosmological epoch t for a fixed comoving observer family. “Refinement” across epochs is not a refinement of resolution in the same sense as T , but it is a controlled change of the horizon scale and accessible patch. In the categorical RG developed later, we treat t (or H ( t )) as the scale parameter: defect statistics are inferred at one epoch and transported to another via the flow of a finite set of parameters. This is a renormalization concept adapted to cosmology. 2.1.6 Summary: the refinement functor C→C0 We summarize the operational RG principle as follows. Refinement functor. A refinement step is a morphism in Ctxref : rC0→C:C0−→ C, (16) where C0 has finer operational access than C and rC0→C implements the controlled loss of information. This functorial structure replaces the role of momentum cutoffs in settings where they are unavailable. In the subsequent subsections we will define what it means for defect structure to close under such refinement and how this yields a renormalizability notion appropriate to quantum gravity in the present framework. B. What runs under categorical RG 2.2.1 The guiding principle: renormalize the comparison data, not a fundamental metric field In a strict metric-field approach one expands gµν =g(0) µν +κhµν and interprets renormalization as the flow of couplings in an effective action for hµν under UV refinement. As reviewed in Section 1 A, this strictification does not close on a finite set of couplings in four dimensions, producing the familiar non-renormalizability problem [13, 14]. In the present framework, the fundamental object is not hµν ( x )but the comparison and gluing structure between operational contexts. Renormalization is therefore the evolution of the data that quantify (i) how contexts are related and (ii) how those relations fail to strictify globally. This subsection identifies the concrete objects that run under categorical RG, from the most operational representations (kernels/PSDs) to the most structural ones (cocycle classes and correspondences). 2.2.2 The defect kernel H(ν;C): running of operational response filters The most direct running object arises whenever a context includes a detector or a measurement protocol with a specified switching window and channel structure. In the Unruh/de Sitter computations, the 22 operational content of the context Cis summarized by: C= (T, Ω,clock protocol,observer worldline, κC), and the leading-order sensitivity of detector statistics to a clock defect is captured by a frequency-domain kernel H(ν;C). Definition (defect response kernel). Let δ ( τ )be a small clock-protocol deformation within a context, e.g. ˜τ=τ+δ(τ)with δ(τ) = εsin(ντ). The detector response functional F(Ω) changes by ∆F(1)(Ω; C) = εR(ν;C) + O(ε2), where R( ν ; C )is a calculable functional of the pullback Wightman distribution and the switching profile (as derived in the Unruh section). We define the defect kernel H(ν;C)by H(ν;C) := ∂ ∂ε∆F(1)(Ω; C)ε=0 ,(17) which is a context-dependent response filter. (In concrete models, H is expressed in terms of sideband differences of the windowed Fourier kernel KT(Ω ±ν/2).) Why H runs under refinement. Refinement changes T ,Ω, the clock protocol, and potentially the causal patch/epoch. Each such change modifies the effective windowing and therefore the kernel. Concretely, for Gaussian switching one obtains exponential suppression for large νT , so changing T is a controlled transformation of H. Thus C→C0⇒H(ν;C)→H(ν;C0). This is a precise, operational notion of RG running: the response filter changes when operational resolution changes. 2.2.3 The defect PSD Sδ(ν;C): running of defect statistics While H ( ν ; C )is a filter, it is not itself the defect. The defect is the mismatch between contexts; in a statistical description it is encoded by the power spectral density (PSD) of the clock/coherence defect process. Definition (defect PSD). Let δτC ( τ )denote the stochastic component of the clock/context mismatch relevant in C(mean zero, stationary within C). The defect PSD is defined by hδτC(ν)δτC(ν0)i= (2π)δ(ν+ν0)Sδ(ν;C),(18) equivalently by the Fourier transform of the autocorrelation function. The pair ( H, Sδ )determines the variance of response fluctuations: Var∆F(C)≈Zdν 2π|H(ν;C)|2Sδ(ν;C),(19) which is the central measurement equation used in the defect tomography and scattering predictions. The quantity Sδ ( ν )is not postulated as a phenomenological noise spectrum. Rather, it is an operationally defined object encoding the spectral distribution of coherence defects arising from non–strict comparison morphisms between locally valid quantum contexts. All observable effects discussed in this work — including detector response shifts, entropy bookkeeping, and interferometric visibility loss — depend on Sδ ( ν )only through protocol–dependent linear functionals. As such, Sδ ( ν )constitutes the minimal spectral data required to characterize departures from strict factorization and strict temporal ordering. Running and closure. Under refinement C→C0 , both the filter H and the inferred defect PSD Sδ may change: (H, Sδ)C−→ (H, Sδ)C0. Renormalizability in the coherence sense will be the statement that the family of admissible Sδ ( ν ; C ) remains within a finite universality class across refinement and epoch changes, so that predictions for observables such as scattering exponents are stable. 23 2.2.4 The cocycle class [ω]C: running in cohomology The most structural description of defects is in terms of cocycle classes in the comparison bicategory. Fix a family of contexts {Ci} covering the operational domain of interest and comparison morphisms uij : Ci→Cj on overlaps. As described in Section 1 C, on triple overlaps one has coherence 2-morphisms αijk :ujk ◦uij =⇒uik. The equivalence class of {αijk}under changes of representatives defines a defect class [ω]. Context dependence. The defect class depends on the chosen refinement: when contexts are refined (e.g. smaller T , finer causal diamonds), the overlap structure changes and so does the groupoid/bicategory on which cocycles are defined. Thus one should label the defect class by the refinement scale: [ω]C∈H2(GC;Z×),(20) where GC is the comparison groupoid restricted to the refined context system and Z× indicates the relevant target of units (central phases, center-valued units, or modular automorphisms depending on regime). The categorical RG is then the evolution of [ω]Cunder refinement: C→C0⇒[ω]C→[ω]C0. Relation to operational representatives. The operational objects ( H, Sδ )are representative projections of [ ω ] C onto measurable quantities. In near-factor regimes, [ ω ] C reduces effectively to a phase-like or central mismatch, and Sδ captures its statistics. In general regimes, [ ω ] C may carry higher structure not fully captured by a scalar PSD; nevertheless, closure at the level of observables is expressed by the stability of these projected representatives under refinement. 2.2.5 Channels and correspondences EC: running of comparison dynamics Comparison morphisms between contexts are naturally implemented as channels or correspondences rather than isomorphisms. Thus another natural running object is the comparison channel itself. Definition (comparison channel). Let EC→C0 denote a completely positive unital map (or, more generally, a correspondence) implementing the operational comparison from AC to AC0 . Under refinement, these channels change because the accessible algebras and their calibrations change: EC→C0−→ Ee C→e C0. In many operational refinement families, repeated coarse-graining defines a semigroup, and the corresponding “beta function” is a generator LCof a CP flow: ∂ ∂ln `E`=L`◦E`, where `is the refinement scale. Why channels are the correct level of structure in quantum gravity. Channels encode the fact that comparisons can be irreversible and protocol dependent, which is unavoidable in finite causal patches and detector-defined contexts. They also encode the operational fact that local algebras in QFT are type III, so strict tensor factorization is not canonical. Thus running of channels/correspondences is the correct abstract mechanism for contextual renormalization. 2.2.6 The hierarchy of running objects and their mutual consistency We summarize the running objects as a hierarchy of representatives: 1. Structural defect class: [ ω ] C (cohomological/coherence obstruction in the comparison bicategory). 2. Comparison dynamics: channels/correspondences ECimplementing context identification. 3. Operational filters: defect kernel H(ν;C)derived from response to protocol perturbations. 24 4. Statistical defect content: defect PSD Sδ(ν;C)inferred from refinement data via (19). The categorical RG is most fundamentally a flow of [ ω ] C and EC under refinement. In practice, the computations in this paper use the operationally robust representatives H and Sδ because they can be computed and inferred directly in the horizon and cosmological settings of interest. The central renormalizability claim is that these representatives close within a finite universality class under refinement and epoch change, enabling stable predictions for scattering observables. 2.2.7 Why hµν reappears only as an emergent representative Finally, we clarify the relationship to the usual graviton/metric fluctuation language. In near-strict regimes one may choose a strictification gauge in which the gluing of contexts is nearly trivial and defect statistics are small and approximately Gaussian. In that regime, it is consistent to encode the defect by an effective metric perturbation field hµν , in the same sense that a phonon field encodes collective excitations of a lattice. However, hµν is not the object that runs fundamentally under refinement; it is a derived, IR-appropriate representative. Demanding renormalizability at the level of a fundamental hµν ( x ) field is therefore the demand that an emergent strictification remain valid at all scales—the origin of the traditional non-renormalizability problem. The categorical RG instead renormalizes the coherence/defect data that remain meaningful beyond strictification. C. “Closure under refinement” as renormalizability 2.3.1 Renormalizability as a closure property: the conceptual shift In standard Wilsonian renormalization, “renormalizable” means (roughly) that under refinement of resolution the effective description can be kept within a controlled class of theories: either a finite set of couplings (strict renormalizability) or a controlled asymptotic expansion with finitely many relevant directions near a fixed point (modern effective-field-theory viewpoint) [ 29 , 30 ]. The practical meaning is that refinement does not force an uncontrolled proliferation of independent structures required to make predictions. Our framework adopts precisely this closure principle, but it applies it to the correct primitive objects: operational contexts and their coherence/defect data, rather than a globally strict metric perturbation field. The renormalizability question therefore becomes: As we refine operational access (switching width, bandwidth, causal diamond size, epoch, channel set), does the defect/coherence structure remain within a stable structural class, or does refinement generate an unbounded tower of new independent data? 2.3.2 Refinement as a functor and defect data as a presheaf (structural formulation) Let Ctxref be the refinement category introduced in Section 2 A, whose objects are contexts and whose morphisms are coarse-grainings rC0→C from refined contexts C0 to coarser contexts C . Refinement thus forms a directed/fibered structure over operational scales. A natural way to package “what runs” is as a presheaf (or pseudofunctor) on Ctxref: D:Ctxop ref −→ Str,(21) where Str is a category of structured representatives (kernels, PSDs, cocycles, channels). Concretely: D(C)∈ {kernels} ∪ {PSDs} ∪ {cocycle classes} ∪ {channels/correspondences}. The contravariant direction reflects the operational fact that refinement CC0 corresponds to a canonical coarse-graining map rC0→C, while the defect data naturally lifts to the refined context. Because comparisons between contexts compose only up to coherent 2-data, it is typically more accurate to treat D as a pseudofunctor: functoriality holds up to specified coherent isomorphisms. This is consistent with the bicategorical structure of comparisons described in Section 1 C. 25 2.3.3 Definition: closure under refinement We now state the central definition. Definition (closure under refinement). Fix a class of defect representatives K (kernels), P (PSDs), H (cohomology classes), or E (channels/correspondences). We say the defect structure is closed under refinement if for any refinement morphism rC0→Cthere exists a well-defined refinement map RC→C0:D(C)−→ D(C0)(22) such that: 1. Type stability: if D ( C ) ∈ K then D ( C0 ) ∈ K (kernel → kernel); similarly PSD → PSD, cocycle → cocycle, correspondence→correspondence. 2. Finite description: the refinement does not require introducing an unbounded tower of new independent structures. In the strongest form, D ( C )lies in a finite-dimensional parameter family stable under refinement (universality class). 3. Predictive compatibility: for any operational observable O in the overlap of C and C0 , the predictions computed from D ( C )and from D ( C0 )agree after applying the corresponding coarsegraining map rC0→C, up to the stated coherent equivalences. Interpretation. Type stability ensures we are renormalizing the same kind of object throughout refinement. Finite description ensures predictivity: refinement does not endlessly enlarge the set of independent parameters required. Predictive compatibility ensures that refinement is physically meaningful: a refined description reduces to the coarse one when one forgets the extra access. 2.3.4 Concrete closure statements for the running objects of Section 2 B We connect the abstract definition to the running objects identified previously. (C1) Kernel closure. In detector-based contexts, refinement T↓ modifies the windowed Fourier kernel KT ( ω )by an explicit filter determined by the switching profile. For Gaussian switching, the frequencydomain weight is e−(ωT )2/2 , so the refinement map is multiplicative in frequency space. Thus the defect kernel H(ν;C)remains a kernel under refinement: H(ν;C)7→ H(ν;C0)(kernel→kernel). No new tower of structures is created; only the kernel parameters change. (C2) PSD closure. The defect PSD Sδ ( ν ; C )inferred from variance data remains a PSD under refinement. Concretely, in the basic measurement equation V(C) = Zdν 2π|H(ν;C)|2Sδ(ν;C), refinement changes the filter H but does not change the functional type of Sδ . Closure is the claim that Sδ ( ν ; C0 )can be represented within the same universality class as Sδ ( ν ; C )(e.g. a finite-parameter family), so that inference and prediction remain stable across contexts. (C3) Cocycle closure. At the structural level, refinement changes the context groupoid/bicategory and hence the domain of cocycle classes. Closure means that the defect class [ ω ] C pulls back to a well-defined class [ ω ] C0 under refinement, modulo coboundaries induced by changes of representatives. This is the cohomological analog of stability: refinement produces new representatives but does not destroy the classification principle of defects. (C4) Channel/correspondence closure. Refinement maps are implemented by channels/correspondences. Closure means that composition of such maps remains within the same structural class (CP maps, correspondences) and that the family of comparison dynamics admits a semigroup-like structure under repeated refinement. In that case the “beta function” is a generator on the space of channels, which is well-defined precisely because the class closes. 2.3.5 Contrast with perturbative gravity: non-closure of metric-field strictification We now make precise the contrast stated in the outline. 32 2.5.5 Universality classes and emergent gravitons Universality also clarifies the status of gravitons and metric fluctuations. In near-strict regimes where defect statistics are small and Gaussian-like, one expects a Gaussian fixed point for defect fluctuations. In that regime: •defect statistics can be encoded by a two-point function (PSD) to leading order; •coarse-grained observables depend primarily on second moments; •the effective strictification representative behaves like a linear metric perturbation field. Thus gravitons arise as IR collective excitations associated with the Gaussian universality class of coherence defects. This does not require that a fundamental hµν theory be renormalizable in the traditional sense; it requires that the defect statistics approach a stable Gaussian fixed shape under refinement, which is precisely the universality statement. 2.5.6 Summary Universality and fixed points provide the correct conceptual bridge between microscopic coherence/defect data and macroscopic gravitational behavior: 1. Universality class means stable functional form of defect statistics under operational refinement. 2. Fixed points are understood as fixed-shape representatives modulo scaling, and can be formulated at the level of PSDs, channels, or cocycle classes. 3. Scaling collapse of operational observables is direct evidence of universality and enables predictive transport across scales. 4. In near-strict regimes, a Gaussian universality class yields emergent linearized gravitational behavior (graviton-like excitations) as an IR strictification representative. 3. EMERGENT GRAVITONS FROM COHERENCE RG A. What “graviton” means here 3.1.1 The conceptual shift: gravitons are not fundamental variables In the conventional perturbative approach, a graviton is introduced by strictifying the global geometry first and then quantizing the resulting metric perturbation field hµν ( x )around a chosen background. In that view, the graviton is a fundamental quantum of the metric field. In the present framework this is not the starting point. The fundamental objects are operational contexts and their comparison/coherence data (Sections 1–2). The metric perturbation language is recovered only as an effective strictification representative in regimes where coherence defects are small and admit a Gaussian/linear description. Thus, in this paper the term “graviton” denotes: An infrared collective excitation of a near-strict coherence regime, represented by the linearized fluctuations of an effective strictification metric gµν + hµν obtained from defect/coherence data. This definition has two essential components: 1. Near-strict gluing: the context-comparison network is close to strictifiable, so loop defects are small in a suitable sense (e.g. small kernel/PSD amplitude; small cohomology representative in a chosen gauge slice). 2. Gaussian/linear universality: under coherence RG, the relevant defect statistics approach a Gaussian fixed-shape universality class (Section 2 E), so second moments dominate and a linear field description becomes valid. 33 It is crucial that neither component is assumed globally. In horizon-dominated regimes or in strongly non-factorizing settings, the graviton description is not expected to be valid; the correct variables are the higher-structured defect representatives (kernels, PSDs, channels, cocycles). The graviton is therefore a derived object, not the fundamental carrier of quantum-gravitational degrees of freedom. 3.1.2 Analogy: phonons and collective excitations The appropriate analogy is the emergence of phonons in a crystal. The fundamental degrees of freedom are atoms and their interactions; under suitable conditions (near equilibrium, long wavelengths), collective normal modes appear and can be quantized as phonons. The phonon is not fundamental; it is an effective excitation of a near-equilibrium phase of the underlying system. Likewise, in our framework: •the fundamental “micro” structure is the network of contexts and their coherence defects; •refinement under categorical RG drives these defects toward universality classes; • in the near-strict Gaussian universality class, linear collective excitations exist and can be represented by a spin-2 field. The graviton is the analog of a phonon: a collective excitation of a phase (the near-strict coherent phase) of the underlying contextual gluing structure. 3.1.3 Operational meaning: what is being excited? A potential ambiguity must be resolved immediately. If the fundamental objects are contexts and comparisons, what does it mean to “excite” them? The correct operational answer is: one excites the statistics of loop mismatches and comparison kernels. Concretely: • In detector-based contexts, the defect is represented by a kernel H ( ν ; C )and a defect PSD Sδ ( ν ; C ) (Section 2 B). • An “excitation” corresponds to a change in these objects relative to a reference coherent state: e.g. increasing the variance of the clock defect, modifying the low-frequency scaling, or changing the correlation length. • In the Gaussian universality class, such changes can be summarized by a two-point function; this is precisely when a linear field description becomes meaningful. Thus “gravitational waves” correspond, in our framework, to propagating coherence-mode fluctuations: long-wavelength variations in the defect statistics that affect time delay, phase diffusion, and related operational observables. 3.1.4 Mathematical emergence: from defect PSD to an effective metric perturbation We now state the minimal mathematical mechanism by which an effective hµν arises. Clock/rod reconstruction and linear response. Within a context, geometry is reconstructed operationally from clocks, radar signaling, and algebraic causal structure. Suppose one has a reference coherent regime in which the reconstructed geometry is g(0) µν . A small change in the comparison/calibration data induces small changes in reconstructed proper times and radar distances. In linear response, such changes can be encoded by a symmetric tensor perturbation hµν defined implicitly by: δτ ≈1 2Zdλ hµν(x(λ)) uµ(λ)uν(λ),(35) for a worldline with tangent uµ = dxµ/dλ . Equation (35) is the standard linearization of proper time in GR, but in our framework it is interpreted as a pushforward map: it maps the fundamental defect observable δτ (clock mismatch statistics) to an effective metric perturbation representative. 34 PSD relation and Gaussian regime. If the defect statistics are stationary in a given regime and described by a PSD Sδ ( ν ), then, under mild regularity assumptions, (35) implies a corresponding effective PSD for the projected metric fluctuation huu := hµνuµuν along the worldline. The precise relation depends on the modeling of the integration in (35) (and on gauge conventions in the reconstruction map), but the essential point is structural: in the Gaussian universality class, second moments of defect statistics determine second moments of the effective metric representative, and those second moments are sufficient to predict linear scattering observables (phase diffusion exponents, time delay variances, etc.). This is exactly the regime in which a graviton-like description is valid. Why spin-2? The effective perturbation hµν that reproduces the leading variation of proper time and radar distance is necessarily a symmetric rank-2 object, because the leading variation of the interval is quadratic in the tangent vector. This is the same kinematic reason that linearized gravity is described by a symmetric tensor representation. Thus the graviton is not introduced by postulate; it is the unique linear representative compatible with the operational reconstruction map in near-coherent regimes. 3.1.5 Coherence RG and the graviton as a Gaussian fixed point The categorical RG described in Section 2 renormalizes defect data under operational refinement. In many physical settings, repeated coarse-graining leads to Gaussian fixed points by central-limit mechanisms: the sum of many weakly correlated contributions tends toward Gaussian statistics. In our language, this means the defect PSD approaches a fixed-shape universality class dominated by second moments (Section 2 E). When this happens: 1. defect statistics are effectively encoded by a two-point function (PSD); 2. scattering and response observables become linear/quadratic functionals of that PSD; 3. an effective hµν representative becomes meaningful as the linearized encoding of these second moments. The graviton is precisely this Gaussian fixed-point excitation: it is the collective linear mode of the near-strict coherence phase. The traditional perturbative graviton approach fails when it attempts to elevate this IR effective representative to a UV fundamental variable; in the present framework it remains what it should be: an emergent IR description, valid when coherence defects are small. 3.1.6 Relation to conventional gravity waves and consistency checks A consistency requirement of any quantum gravity framework is that it recover classical GR and its weak-field gravitational-wave sector in appropriate limits. In our framework, this requirement becomes: In a near-strict coherent regime, the effective strictification representative g(0) µν + hµν should reproduce the standard linearized gravitational-wave phenomenology when interpreted through operational observables. The present paper does not attempt to re-derive the full linearized Einstein equation from first principles; rather, it establishes the logical and operational mechanism by which the graviton description emerges as a consistent representative. Later sections demonstrate that the same defect statistics which produce observable response defects (Unruh/de Sitter) and scattering exponents (FLRW) can be organized by categorical RG in a way that yields universal scaling behavior. This is the expected signature of an emergent field description in the IR. 3.1.7 Summary The graviton in this framework is: 1. not a fundamental quantum of a globally defined metric field; 2. an infrared collective excitation of near-strict coherence gluing; 35 3. a linear representative of Gaussian defect statistics obtained by operational reconstruction (clocks/radar/algebraic causality); 4. the appropriate effective strictification variable for describing long-wavelength coherence modes (gravitational waves) in regimes where defects are small. This viewpoint reconciles the empirical success of the graviton picture at long wavelengths with the failure of strict metric-field renormalizability in the ultraviolet: one uses hµν where it is justified by coherence RG (near-strict Gaussian universality), and one uses higher-structured defect data where strictification is not stable. B. From defect PSD to effective metric fluctuation correlators 3.2.1 Objective and scope This subsection explains how a coherence-defect power spectral density (PSD) inferred from operational context comparisons yields an effective correlator for metric fluctuations in regimes where a strictified metric representative is meaningful. The goal is not to postulate a fundamental stochastic metric field, but to derive an effective hµν -correlator as a pushforward of defect statistics under operational reconstruction maps (clocks, radar procedures, and record comparisons). We emphasize at the outset: 1. The defect observable δτ is fundamental and operational (it is directly inferred from comparison data). 2. The effective metric perturbation hµν is derived and only meaningful in near-strict coherent regimes. 3. The mapping from δτ to hµν is not unique; it depends on the chosen reconstruction protocol (a gauge in the strictification sense). The predictions we ultimately use are gauge-invariant operational quantities (phase diffusion, timing jitter, scattering exponents). 3.2.2 Linearized proper time and the projected perturbation huu Consider a timelike worldline xµ ( λ )with tangent uµ = dxµ/dλ . In a strictified Lorentzian geometry (M, g), the proper time between two parameter values λ1, λ2is τ=Zλ2 λ1 dλ q−gµν(x(λ)) uµ(λ)uν(λ). Let gµν =g(0) µν +hµν with hsmall in a near-strict regime. Write −gµνuµuν=−g(0) µν uµuν | {z } =:N2 −hµνuµuν, where N(λ) := q−g(0) µν uµuνis the background norm. Expanding the square root to first order gives qN2−hµνuµuν=N1−1 2N2hµνuµuν+O(h2). Thus the first-order proper-time perturbation is δτ =−1 2Zλ2 λ1 dλ hµν(x(λ)) uµ(λ)uν(λ) N(λ)+O(h2).(36) In the particularly important case that λ is chosen as background proper time, N ( λ ) ≡ 1and uµuµ = − 1 with respect to g(0). Then δτ =−1 2Zτ2 τ1 dτ huu(τ) + O(h2),(37) 36 where we define the projected perturbation along the worldline huu(τ) := hµν (x(τ)) uµ(τ)uν(τ).(38) Equation (37) is the central operational mapping: within a near-strict regime, any clock defect δτ induced by comparison mismatches can be represented as the integral of an effective projected metric perturbation huu along the worldline. 3.2.3 From time-domain statistics to PSDs: Sδ(ν)versus Shuu (ν) We now connect the defect PSD to an effective metric correlator. Treat δτ ( t )as a stationary stochastic process in a given context (or within a sufficiently local patch where stationarity is a controlled approximation). Define its PSD by hδτ(ν)δτ(ν0)i= (2π)δ(ν+ν0)Sδ(ν).(39) To relate Sδ to Shuu , we must specify the measurement protocol. Operationally, a measured clock discrepancy is not an indefinite integral over all time; it is a finite-time measurement with a window function. Introduce a measurement window w ( t )normalized so that w represents the temporal support of the comparison, and define the measured defect as δτw:= −1 2Zdt w(t)Zt ds huu(s),(40) or equivalently (by integrating by parts) as a linear functional of huu with a known kernel. The precise form of w depends on the protocol (switching, record averaging, differencing). The key point is that δτ is a linear functional of huu in the near-strict regime, hence their second moments are related by a standard spectral relation. For concreteness, consider the simplest and most common case: δτ is measured as a finite-time integral of huu over an interval of duration T, i.e. w(t) = 1[0,T ](t). Then (37) gives δτ(T) = −1 2ZT 0 dt huu(t), so the Fourier-domain relation is δτ(ν) = −1 2 1−e−iνT iν huu(ν), and therefore Sδ(ν) = 1 4 1−e−iνT iν  2 Shuu (ν) = 1 4 4 sin2(νT/2) ν2Shuu (ν).(41) Equivalently, Shuu (ν) = ν21 sin2(νT/2) Sδ(ν)(for the boxcar measurement protocol). (42) General window. For a general window w(t), the relation takes the universal form: Sδ(ν) = |Kw(ν)|2Shuu (ν),(43) where Kw ( ν )is a known transfer function determined by the measurement protocol. In particular, if δτ is an integral of huu against a window w , then Kw ( ν ) ∼bw ( ν ) /ν up to normalization. The precise form is fixed by the operational definition of the defect observable, not by a gauge choice. Calibration and gauge in strictification. Equations (41) – (43) encode the statement “ Sδ ( ν )corresponds to Shuu ( ν )up to integration factors.” The remaining ambiguity is calibration: different strictification gauges and different clock reconstruction conventions correspond to multiplicative rescalings and mixing between components of hµν . This does not affect the operational predictions we ultimately compute (phase diffusion exponents, timing jitter, visibility), because those are expressed directly in terms of Sδ or in terms of Shuu together with the same window transfer function. 37 3.2.4 From projected correlators to effective tensor correlators The projected object huu is sufficient for many operational predictions along a given worldline (e.g. clock noise, phase diffusion). To connect to the conventional graviton picture one may wish to reconstruct an effective tensor correlator hhµνhρσi . This requires additional structure: multiple worldlines/observers and/or multiple independent projections. Multiple projections. If one has a family of worldlines with distinct uµ , then huu = hµνuµuν samples different components of hµν . In a near-isotropic regime one may assume an effective Gaussian tensor correlator of the form hhµν(x)hρσ(y)i ≈ Pµνρσ C(x, y),(44) where P is the appropriate spin-2 projector (depending on gauge and background) and C is a scalar correlation function. Then Shuu determines Calong worldlines up to a known contraction with P: Shuu (ν)∼(uµuνuρuσPµνρσ)SC(ν). This is the standard reconstruction step in stochastic gravity and gravitational-wave phenomenology [ 44 ]. In our framework, this reconstruction is optional and should be performed only when justified by near-strict Gaussian regimes. The operational predictions do not require it. 3.2.5 Near-strict Gaussian fixed point and “graviton-like” statistics The coherence RG developed in Section 2 renormalizes defect statistics under refinement. In many physical settings, repeated coarse-graining drives defect statistics toward Gaussian fixed points, in the same sense that central-limit behavior drives sums of weakly correlated contributions to Gaussian distributions. In our language, the statement is: In near-strict regimes, the defect PSD approaches a stable fixed-shape universality class, and higher cumulants become irrelevant under refinement. When this happens, the defect statistics are fully characterized (to leading order) by Sδ ( ν ), and the effective metric fluctuation correlators are determined by (43) . This is the precise sense in which the IR limit is “graviton-like”: the effective fluctuations are Gaussian, linear response applies, and the resulting tensor structure is the unique symmetric rank-2 representative compatible with the operational reconstruction of proper time and radar distance. 3.2.6 Practical consequence: scattering and decoherence observables depend only on Sδ Because δτ is the operational primitive, many experimentally meaningful observables can be written directly in terms of Sδ without ever invoking a tensor hµν . For example, the phase diffusion exponent used in the scattering pipeline is Ξ(Tint) = 1 2ω2 0Zdν 2πSδ(ν) sinc2(νTint/2), which is a bounded functional of Sδ . In the near-strict regime one may equivalently express it in terms of Shuu using (43) . Thus the effective graviton correlator is a convenient representative, not an additional assumption. 3.2.7 Summary The mapping from defect PSD to effective metric correlators proceeds in three steps: 1. Operationally infer Sδ ( ν )from context refinement data (Sections on detector response and categorical RG). 2. In a near-strict regime, represent δτ as the linear functional of huu given by (37) ; convert PSDs by the protocol transfer function (43). 38 3. When justified by isotropy/near-Gaussian behavior, reconstruct an effective tensor correlator hhµνhρσifrom multiple projections. This establishes a precise sense in which “metric fluctuations” are induced by coherence defect statistics, and clarifies why gravitons arise as IR collective modes in the Gaussian universality class. C. Why this avoids traditional non-renormalizability 3.3.1 The statement to be explained The traditional non-renormalizability of perturbative quantum gravity is the statement that, when one takes the metric perturbation field hµν ( x )as fundamental and refines ultraviolet resolution in the usual Wilsonian sense, the effective action does not close on a finite set of couplings; instead one generates an infinite tower of higher-derivative counterterms [ 10 , 13 , 14 ]. This subsection explains why that conclusion does not obstruct the present framework. The central claim is: The graviton description is an infrared strictification representative of a near-coherent phase of contextual gluing. One does not demand that this representative be ultraviolet-complete. Renormalization is instead performed on coherence/defect data, which closes under operational refinement. Gravitons appear only after this closure is established. We now justify this claim in precise terms. 3.3.2 Non-renormalizability as failure of closure of a strictification representative From the perspective developed in Section 2, renormalizability is a closure property under refinement. The standard perturbative approach imposes a strong strictification: 1. a single global spacetime object with a globally defined metric field; 2. a fundamental local tensor field hµν(x)representing quantum fluctuations; 3. a Wilsonian refinement procedure based on a global momentum decomposition and a UV cutoff. Renormalization then asks whether the class of actions expressed in terms of hµν closes under refinement. The classical Einstein–Hilbert action (1) contains two derivatives. Power counting implies that loop corrections generate divergences requiring counterterms with increasing numbers of derivatives; explicit computations confirm that pure gravity requires an R3 counterterm at two loops [ 14 ]. In modern EFT language, one can treat gravity as a low-energy effective theory and compute predictive corrections orderby-order, but strict renormalizability fails because infinitely many independent couplings are generated [10]. In the closure language of Section 2 C, this is precisely: The strictification “hµν(x)is fundamental” is not closed under ultraviolet refinement. This diagnosis is essential: it identifies which object fails to close under refinement, rather than concluding that quantum gravity is computationally inaccessible in principle. 3.3.3 The replacement: closure of coherence data under operational refinement In our framework, the primitive running objects are the defect/coherence representatives described in Section 2 B: H(ν;C), Sδ(ν;C),[ω]C,EC, and refinement is operational (Section 2 A). The renormalization problem is therefore: do these objects close under refinement? In the worked examples, they do: 39 • In Unruh and de Sitter settings, refinement in switching width T modifies the response kernel through a known filter dependence; the object remains a kernel. • In slow-roll FLRW, defect PSDs inferred across multiple epochs close within a finite-parameter universality class, with scaling collapse of scattering observables under renormalization. • In the Page-curve constructions, the relevant “defect witness” is mutual information and its evolution under memory refinement; increasing radiation-memory capacity removes artifacts while preserving the qualitative turnover structure. This is the operational content of “renormalizable quantum gravity” in our sense: there exists a stable universality class of coherence defects under refinement, and predictions are computed directly from these defects, not from a fundamental hµν field. 3.3.4 Gravitons appear only after closure is established (IR strictification) The graviton is introduced in Section 3 A as an infrared collective excitation of a near-strict coherence regime. The logical order is: 1. Define and renormalize coherence/defect data under refinement of operational contexts. 2. Identify universality classes and fixed-shape behavior under categorical RG (Section 2 E). 3. In a near-strict Gaussian universality regime, push forward defect statistics to an effective metric representative hµν (Section 3 B). 4. Interpret the resulting linear excitations as gravitons. This order matters. One does not start with hµν and demand UV completeness of its perturbation series. One obtains hµν as a strictification representative that is valid precisely when the defect RG indicates Gaussian/linear behavior. The graviton is therefore a derived IR variable. Demanding that it remain closed under arbitrary refinement is analogous to demanding that phonons remain a UV-complete description of the atomic lattice: it is not the correct notion of completion. 3.3.5 Mathematical mechanism: why the IR representative can be simple even when UV is complex The emergence of a simpler IR description from a complex UV structure is a standard phenomenon in renormalization. In our setting, the complexity is carried by higher-structured defects (kernels/channels/cocycles). Under refinement/coarse-graining, higher cumulants and detailed microstructure may become irrelevant, leaving a Gaussian fixed point characterized by second moments (PSD). This is the precise regime in which the mapping of Section 3 B is valid: second moments of defect statistics determine effective two-point correlators of the projected metric perturbation huu , and (with additional assumptions) of hµν itself. Thus the IR graviton description is the standard renormalization phenomenon of a Gaussian fixed point, but with the crucial difference that the fixed point is reached in the space of coherence defects rather than in the space of fundamental metric-field couplings. 3.3.6 Relation to effective field theory of gravity and to other nonperturbative programs It is important to clarify what is, and is not, being claimed. Compatibility with gravitational EFT. Gravitational EFT remains a valid tool for computing lowenergy quantum corrections around backgrounds in regimes where a strict global geometry is a good approximation [ 10 ]. Our framework does not negate EFT; it provides a broader operational structure in which: •the EFT graviton field is an IR strictification variable; • coherence defects quantify the regime of validity of strict factorization and strict global identification assumptions; 40 • operational predictions can be made even when an EFT graviton description is not canonically available (finite causal diamonds, observer-local horizons, generic cosmologies). Relation to asymptotic safety. Asymptotic safety approaches seek a nonperturbative UV fixed point for metric couplings [ 35 – 37 ]. Our viewpoint differs in the fundamental object being renormalized: rather than requiring a fixed point in a metric-coupling space, we require closure of coherence/defect structure under operational refinement. The two viewpoints are not logically incompatible; they operate on different levels. If a metric-coupling fixed point exists, it would correspond in our language to a regime where the strictification representative hµν is stable at all scales. The coherence framework does not require that; it requires only that the defect data itself closes. 3.3.7 Summary Traditional perturbative non-renormalizability is reinterpreted here as the non-closure of a particular strictification representative—the fundamental local graviton field hµν ( x )—under ultraviolet refinement. The present framework avoids this obstruction by: 1. renormalizing coherence/defect data under operational refinement of contexts; 2. establishing closure and universality in that defect space (categorical RG); 3. recovering gravitons as IR collective excitations only in near-strict Gaussian regimes, after closure is established. In this sense, quantum gravity is renormalizable in the coherence sense: it admits a controlled, closed flow of the objects that remain operationally meaningful in generic spacetimes and finite causal patches, while reproducing the graviton description as an effective IR strictification where appropriate. 4. WORKED EXAMPLE I: HAWKING RADIATION AND PAGE CURVE WITHOUT ADS/ISLANDS A. Standard Hawking factorization assumption (strict QFT) 4.1.1 Goal of this subsection The purpose of this subsection is to isolate, with full mathematical clarity, the strict factorization assumption that underlies the standard “ever-growing entanglement” conclusion in the Hawking evaporation argument. The subsequent subsections will replace this assumption by contextual gluing. Here we emphasize that the factorization assumption is not a tautology; it is a strong structural input. Once stated precisely, it becomes clear why it is the natural point at which a contextual (non-strict) quantum gravity framework can modify the information bookkeeping without modifying local semiclassical emission. Throughout this subsection we work in a standard semiclassical setting: quantum fields on a classical black hole background. We will explicitly separate what is (i) an uncontested local QFT statement (pair creation and thermality) from what is (ii) an additional global strictification (global tensor factorization of degrees of freedom into “old radiation” and “new radiation” across time). The phrase “one new independent qubit per emission” is a shorthand for that strictification. 4.1.2 Hawking pair creation as a two-mode squeezed state Consider a quantum field on an evaporating black hole spacetime. In the semiclassical treatment, the outgoing Hawking radiation arises from a Bogoliubov transformation between “in” modes and “out” modes. For each effective frequency (or wavepacket label) k , one obtains a pair of modes: an outgoing mode bk accessible at future null infinity I+ and an interior partner mode ck supported behind the horizon (or on the complementary algebra in the algebraic formulation). The local vacuum structure near the horizon implies that the state of the pair is, to good approximation, a two-mode squeezed vacuum: |Ψki=1 cosh rk ∞ X n=0 (tanh rk)n|nibk|nick,(45) 41 where the squeeze parameter rk is determined by the Hawking temperature and the effective frequency of the mode. Tracing out the partner yields a thermal state for the outgoing mode: ρbk= Trck|ΨkihΨk|= ∞ X n=0 pn,k |nibkhn|, pn,k = (1 −e−βωk)e−nβωk,(46) with inverse temperature β = 1 /TH (Hawking temperature). This is the local semiclassical content: each emitted mode is thermally populated when its interior partner is ignored. Nothing in (45) – (46) requires a global factorization across emission times. It is a local statement about the near-horizon vacuum and the mode decomposition used to describe outgoing/partner degrees of freedom. 4.1.3 The strict global factorization step: “old” ⊗“new” The information-loss tension is not produced by (45) alone. It requires an additional assumption: that each emission step adds a new tensor factor to the Hilbert space (or algebra) of the radiation, independent of all previously emitted radiation. To state this precisely, let Rn denote the algebra (or Hilbert space factor) associated with the radiation that has escaped to I+ up to emission step n . The strict factorization assumption asserts that there exists a decomposition HRn+1 ∼ =HRn⊗Hbn+1 ,(47) where bn+1 is the outgoing mode produced at step n + 1, and where the identification is strict in the sense that: 1. Hbn+1 is independent of HRn; 2. the global state update is described by appending a fresh factor Hbn+1 and a partner Hcn+1 to the remaining black-hole degrees of freedom; 3. the identification (47) is consistent across all steps, i.e. it defines a strict global factorization of the radiation Hilbert space as a growing tensor product. In a qubit caricature, one replaces each mode pair by a pair of qubits and says: “each emission step adds one new (approximately maximally entangled) qubit to the radiation.” This is the origin of the slogan: One new independent qubit per emission. However, the slogan hides the key mathematical assumption: the existence of a strict growing tensorproduct decomposition of the radiation degrees of freedom across time. Algebraic formulation of the same assumption. Instead of (47) , one may state the assumption at the level of von Neumann algebras: ARn+1 ∼ =ARn⊗Abn+1 ,(48) where ⊗ is the von Neumann tensor product. This is an even stronger statement in AQFT contexts because local algebras are typically type III and do not admit canonical tensor factorization. In semiclassical discussions one implicitly uses an effective factorization by selecting mode subspaces and treating them as independent subsystems. The tension arises precisely when one elevates such effective factorizations into a global strict statement across the entire evaporation history. 4.1.4 Entropy growth in the strict factorization picture Assume strict factorization (47) and that each emission step produces a state close to (45) between the new outgoing mode bn+1 and its partner cn+1 , with negligible correlations between bn+1 and earlier radiation Rn. Then the radiation entropy update obeys (schematically) S(Rn+1) = S(Rn⊗bn+1) = S(Rn) + S(bn+1),(49) 48 In the strict factorization picture, c is inaccessible and plays no role in the radiation bookkeeping; the record R would continue to accumulate entropy. In the contextual-gluing picture, the channel Upur makes c operationally relevant through coherent identification constraints; the correlation budget measured by I ( R : c )tracks the onset and strength of purification. In the explicit numerical models constructed later in this paper, I ( R : c )changes sharply around the turnover scale, providing a direct, basis-independent witness of the mechanism. 4.3.5 Why this does not use “min” or external entropy clamps A frequent heuristic construction of Page curves is to impose Srad(n) = min{Snaive rad (n), SBH,rem(n)}, which enforces a turnover by hand. In contrast, the three-party model above produces turnover dynamically because the record is not updated by appending independent factors; it is updated by coherent gluing transformations that generate and transfer correlations. The only inputs are: 1. local Hawking pair creation as a Gaussian pure state (b, c); 2. contextual gluing represented as a coherent comparison channel between record and partner degrees of freedom; 3. a record subsystem Rrepresenting operationally accessible radiation memory. Thus the Page curve arises from the same logic as the Unruh/de Sitter defects: local thermality is preserved while global identification/factorization is modified by coherent comparison structure. 4.3.6 Upgrading the radiation memory: from one mode to three modes The three-party model above still requires a choice of how to represent the radiation record. A single-mode R is the smallest Gaussian memory, but it can produce finite-memory artifacts (e.g. spurious oscillations in ∆ S ) because it cannot store sufficient independent information about the radiation history. To remove such artifacts while retaining analytic control, we upgrade R to a multi-mode Gaussian memory: R=R1⊕R2⊕R3, with sequential storage couplings (mixing b into each Ri ) and distributed purification couplings (mixing c into each Ri with prescribed weights). This increases the record capacity and smooths per-step increments: ∆Srad(n) := S(Rn)−S(Rn−1). In explicit computations, this upgrade produces a robust, smooth sign change in ∆ Srad and a stable Page turnover in S(Rn), while preserving the local Hawking marginal structure for b. 4.3.7 Relation to contextual gluing and to coherence defects The Gaussian models above instantiate the general categorical picture: 1. The strict extension ARn+1 ∼ =ARn⊗Abn+1 is replaced by a correspondence implementing a coherent identification between new and old records. 2. The strength of the gluing is encoded by parameters (beam-splitter angles, weights) which represent operationally measurable overlap/mismatch data (analogous to the response kernels and PSDs in Unruh/de Sitter/FLRW). 3. The defect is not a change in diagonal occupation numbers; it is the cross-correlation structure, whose operational witness is mutual information. Thus Gaussian models provide the minimal explicit realization of the statement made in Section 4 B: independence is not absolute; it is constrained by coherent gluing, and the constraints are carried by correlations. 49 D. What is uniquely achieved vs. string theory 4.4.1 The comparison to be made The objective of this subsection is to state, precisely and without polemic, what is uniquely achieved by the contextual-gluing mechanism developed in Sections 4 A–4 C when compared to the dominant string-theoretic strategy for Page-curve physics. The comparison is not about mathematical sophistication or ultraviolet completion in general. Rather, it is about the domain of applicability and the operational content of the Page-curve calculation. We therefore fix the following criterion: A Page-curve mechanism is considered achieved in the present sense if it produces a dynamical turnover of the entropy of an observer-local radiation record in a generic evaporating spacetime, with an operational witness of the mechanism, and without requiring a boundary dual or a special asymptotic structure. 4.4.2 Page curve as an observer-local statement The semiclassical Hawking description is formulated in terms of local emission near the horizon and local QFT propagation to the exterior [ 3 – 5 ]. The Page curve, however, is not a local emission statement; it is a global information statement about the entropy of a radiation subsystem. In generic evaporation settings, the only operationally meaningful “radiation subsystem” is the record accessible to some observer (or family of observers) within a finite causal domain. Our construction makes this operational point explicit by defining radiation entropy as Srad(n) := S(Rn), where Rn is the Gaussian memory subsystem representing accessible records after n emission steps (Section 4 C). This definition is observer-local in two senses: 1. Rn encodes the record algebra and storage protocol available to the observer, rather than an assumed global tensor factor of a universal Hilbert space; 2. the update dynamics Rn→Rn+1 is defined by context comparisons (storage and gluing channels) that represent operational procedures, not by a boundary-time evolution. Within this observer-local formulation, we obtain a proper Page turnover: Srad(n)increases for early steps and decreases for late steps, with a sign change in the incremental entropy ∆ Srad ( n ), and with a mutual-information witness (Section 4 C). This satisfies the criterion above without assuming asymptotic AdS structure. 4.4.3 The mechanism is intrinsic: non-factorization from contextual gluing The essential input in our construction is the replacement of strict factorization by contextual gluing (Section 4 B). In strict factorization, the radiation algebra grows by tensor-product extension and the entropy increment is additive: ∆Sstrict rad (n+ 1) = S(bn+1). In contextual gluing, the new record is not strictly independent of the old record; the deficit in factorization is witnessed by correlations: ∆Srad(n+ 1) = S(bn+1)−I(Rn:bn+1), so the mechanism of turnover is a growth of mutual information induced by coherent identifications. This is operationally captured in the three-party Gaussian model by the mutual information witness I(R:c) and by the effective purification channel that transfers correlations into the record. Importantly, the 50 outgoing marginal of b may remain approximately thermal; the bookkeeping change is in independence, not in local flux. This is an intrinsic mechanism: it is formulated entirely in terms of local QFT states (Gaussian Hawking pairs), operational record algebras, and coherent gluing maps. No additional global strictification is required. 4.4.4 Contrast with the string-theoretic strategy in AdS/CFT and related settings String theory provides two broad routes to Page-curve physics: 1. microscopic counting of black hole states in special supersymmetric or near-extremal regimes, where the black hole can be embedded into a controlled string background [62]; 2. holographic computations (AdS/CFT) in which a boundary theory provides a global Hilbert space and a global time, enabling entanglement entropy computations by geometric extremization and replica-like constructions [16, 17]. Both routes are powerful in their regimes, but they rely on structural strictifications that are absent in generic evaporating spacetimes: •The existence of a canonical asymptotic boundary and boundary time (AdS/CFT). •A canonical global factorization anchored in a boundary algebra. • A geometric extremization prescription that is formulated in the presence of the above global structure. By contrast, the present construction is deliberately formulated without assuming these strictifications. The Page curve is computed for an observer-local record subsystem in a generic setting, where the comparison of contexts is the fundamental structure and where the defect witness is operational (mutual information and record entropy). 4.4.5 No replica method, no islands: what is and is not being claimed The statement “no replica, no islands” means the following precise claim: The Page turnover in our model arises from explicit unitary dynamics on ( R, b, c )together with contextual gluing (non-strict factorization) and does not require introducing an auxiliary replica construction or an extremization principle over additional geometric saddles. This does not deny that replica and island techniques can be valid computational tools in certain settings. It states that the turnover mechanism itself can be obtained as an intrinsic, operational property of coherent gluing and record updating. The advantage is that the mechanism remains meaningful in settings where: •no boundary dual exists; •no canonical asymptotic time is available; •the radiation subsystem is defined operationally rather than by global factorization. 4.4.6 The operational witness is intrinsic and basis-independent A distinctive feature of the present approach is that the mechanism is accompanied by an explicit operational witness. In Gaussian models, the witness is mutual information, which is invariant under local basis changes and quantifies non-independence in a way that does not depend on a particular mode decomposition. The Page turnover is accompanied by: 1. a sign change in ∆Srad (a direct operational criterion); 2. a nontrivial evolution of I(R:c)(a basis-independent correlation witness). This witness is conceptually aligned with the framework: it is a statement about coherence constraints in gluing, not about special global geometries. 51 4.4.7 Summary: what is uniquely achieved The uniquely achieved content of the present Page-curve construction is therefore: 1. A Page turnover is obtained in a non-AdS, observer-local setting with a radiation subsystem defined operationally as a record algebra. 2. The mechanism is intrinsic: it follows from contextual gluing replacing strict factorization and is encoded by correlations (mutual information) rather than by altering local Hawking thermality. 3. The construction is fully local in inputs (Hawking pair creation) and does not require a holographic boundary, a replica method, or an island extremization prescription. 4. The result integrates naturally with the categorical renormalization viewpoint: the relevant running object is coherence/defect data, and the graviton description is recovered only as an IR strictification in near-coherent regimes. These points are not presented as a criticism of string theory; rather, they identify an operationally universal layer of Page-curve physics that remains meaningful in generic spacetimes without special asymptotic structure. E. Figures and outputs This subsection lists the concrete numerical/graphical outputs associated with the Hawking/Page worked example developed in Sections 4 A–4 D. Throughout, the radiation “subsystem” is defined operationally as the Gaussian radiation memory R used to store accessible radiation records (Section 4 C). The Page turnover is therefore represented as a change in the entropy of that record, S(Rn), under the unitary three-party update map on (R, b, c). Figure A: Radiation entropy only (proper Page curve) 52 Figure 1. Radiation entropy Srad ( n ) := S ( Rn )in the three-party Gaussian model with three-mode radiation memory R = R1⊕R2⊕R3 . The curve exhibits Page-type behavior: an initial increase, a maximum (Page time), and a decrease driven by coherent gluing/purification. The baseline curve (no purification channel) is included. Entropy of the operational radiation record Rn(Gaussian memory proxy) What this figure demonstrates. This figure is the central Page-curve output: it shows that a turnover occurs dynamically in the entropy of the operational radiation record, without imposing any external min{·,·} clamp and without using replica/island constructions. The turnover is generated by correlation transfer mediated by the contextual-gluing channel between Rand the partner mode c(Section 4 C). Figure B: Per-step entropy increment (sign change) 53 Figure 2. Per-step radiation entropy increment ∆ Srad ( n ) := S ( Rn ) −S ( Rn−1 )for the three-mode radiationmemory model. The increment changes sign near the Page time: ∆ Srad > 0at early steps (record entropy growth) and ∆Srad <0at late steps (purification of the record), providing a direct operational signature of turnover. What this figure demonstrates. The sign change of ∆ Srad is the sharpest operational indicator that the Page turnover is not a plotting artifact: it shows that the record becomes less entropic at late times due to coherent identification constraints, despite local Hawking-like pair creation remaining present at each emission step. Figure C: Mutual-information witness 54 Figure 3. Mutual information witness I ( R : c )between the radiation memory R and the partner mode c in the three-party Gaussian model. The evolution of I ( R : c )tracks the redistribution of correlations induced by contextual gluing and accompanies the turnover in S(Rn). What this figure demonstrates. In the contextual-gluing framework, the factorization deficit is carried by correlations rather than by diagonal occupation numbers. Mutual information provides a basisindependent witness of that deficit. In the present model, I ( R : c )diagnoses the coupling of the record to the partner degrees of freedom that enables late-time purification of the radiation record. Figure D: Purification schedule 55 Figure 4. Purification-channel schedule θpur ( n )used in the three-party Gaussian model. This function controls the strength of the coherent mixing between the radiation memory R and the partner mode c and represents the onset and growth of contextual gluing constraints in the operational update rule. What this figure demonstrates. The schedule plot is included for transparency of modeling assumptions: it specifies when the gluing/purification channel becomes effective in the toy model and therefore how the Page time emerges dynamically. The subsequent sections generalize this notion by treating the gluing strength as an inferred, running object under categorical renormalization (Sections 2 and following). 5. WORKED EXAMPLE II: UNRUH EFFECT WITH INCOMPATIBLE CLOCKS (EXACT ANALYTICS + NUMERICS) A. Standard Unruh response and KMS structure 5.1.1 Purpose and scope of the standard result This subsection recalls the standard Unruh effect in a form suited to our later contextual analysis. The Unruh effect has two complementary formulations: 1. Detector formulation: a uniformly accelerated Unruh–DeWitt detector coupled to a quantum field in the Minkowski vacuum responds as if immersed in a thermal bath at temperature TU = a/ (2 π ) [3, 6, 7]. 2. Algebraic/modular formulation: the Minkowski vacuum restricted to the right Rindler wedge algebra is a KMS state with respect to the boost modular flow, with inverse temperature βU = 2 π/a [11, 12, 21]. The central point for this paper is that Unruh thermality is structural (it is robust and does not depend on fine details of detector modeling) and therefore serves as an ideal baseline: our later “incompatible clocks” effects must preserve the KMS structure while modifying only the operational comparison between clock protocols and response readouts. 56 5.1.2 Geometry: Rindler wedge and uniformly accelerated worldlines In (1 + 1)-dimensional Minkowski spacetime for notational simplicity (the generalization to (3 + 1) is straightforward), define Minkowski coordinates ( t, x )with metric ds2 = −dt2 + dx2 . The right Rindler wedge is WR={(t, x) : x > |t|}. Uniformly accelerated observers with proper acceleration afollow worldlines in WR: t(τ) = a−1sinh(aτ), x(τ) = a−1cosh(aτ),(53) where τ is the observer’s proper time. The Rindler (boost) time flow corresponds to translations in τ along these worldlines. 5.1.3 Wightman pullback and the thermal KMS periodicity Let φ be a (say, massless) scalar field in the Minkowski vacuum |0i . The Wightman two-point function is W(x, x0) := h0|φ(x)φ(x0)|0i. Pulled back to the accelerated worldline (53) , the Wightman function becomes a function of the proper-time difference u:= τ−τ0: W(u) = −a2 16π2 1 sinh2a 2(u−i).(54) (Here  > 0specifies the usual distributional boundary condition.) The crucial property is the KMS periodicity in imaginary time: W(u+iβU) = W(−u), βU=2π a.(55) Equation (55) is the KMS condition for a thermal state at temperature TU=β−1 U=a 2π.(56) Thus the accelerated observer’s restriction of the Minkowski vacuum exhibits thermality in the precise KMS sense: correlation functions are thermal with respect to the proper-time/boost evolution. 5.1.4 Detector formulation: Unruh–DeWitt response as a thermal response Consider an Unruh–DeWitt detector with energy gap Ω > 0and monopole operator m ( τ )coupled to the field along the worldline (53) with switching function χ(τ): HI(τ) = λ χ(τ)m(τ)φ(x(τ)).(57) To leading order in λ, the excitation probability is P(Ω) = λ2Fχ(Ω),Fχ(Ω) = Zdτ Zdτ0χ(τ)χ(τ0)e−iΩ(τ−τ0)W(τ−τ0).(58) For sufficiently long and smooth switching (so that transient effects are negligible), one obtains a transition rate: ˙ F(Ω) = Z∞ −∞ du e−iΩuW(u),(59) 57 which, using the KMS property (55), yields the detailed balance relation ˙ F(−Ω) ˙ F(Ω) =e−βUΩ,(60) i.e. thermal response at temperature TU = a/ (2 π ). In particular, for Ω > 0the excitation rate has the Planck form ˙ F(Ω) = Ω 2π 1 e2πΩ/a −1(up to conventional normalizations),(61) which is the detector statement of the Unruh effect. Robustness. The key point is that thermality here is not an artifact of a particular detector model; it is a consequence of the KMS property of the restricted vacuum state with respect to the appropriate time flow. The detector merely provides an operational probe of that KMS structure. 5.1.5 Algebraic formulation: wedge restriction and modular flow The Unruh effect admits a deeper structural formulation in algebraic QFT. Let A ( WR )be the local von Neumann algebra associated with the right Rindler wedge. The Bisognano–Wichmann theorem states that, for relativistic QFT satisfying standard axioms, the modular automorphism group of ( A ( WR ) ,|0i ) is implemented by Lorentz boosts that preserve WR [ 11 , 12 , 21 ]. In particular, the vacuum restricted to A ( WR )is a KMS state with respect to this modular flow, with inverse temperature βU = 2 π/a once one normalizes the boost parameter to the proper time of an observer with acceleration a. This is the most robust statement of Unruh thermality: it is a theorem about the local algebraic structure and the vacuum, independent of any particle interpretation. 5.1.6 Significance for contextual gluing The results above have two consequences that will be used in the following subsections: 1. Thermality is structural. The KMS property is encoded in the analytic structure of correlators and in modular theory. It is therefore stable under small operational modifications of how time is read (clock protocol changes), as long as one does not alter the underlying local state and algebra. 2. Time is already context-dependent. The relevant Hamiltonian for thermality is the boost/- modular generator, not the Minkowski time translation generator. This provides the ideal setting to study incompatible clock protocols: different operational time readouts can be compared within the wedge, and their comparison need not strictify even though the underlying KMS structure remains intact. In the next subsection we introduce clock-protocol deformations ˜τ = τ + δ ( τ )and show that they induce measurable response defects while preserving the underlying thermality, thereby isolating quantumgravitational comparison effects at the level of operational gluing rather than at the level of local field dynamics. B. Clock protocol as context; reparametrized time readout 5.2.1 Why clock protocols belong to the definition of context Section 5 A established that Unruh thermality is a KMS property of the Minkowski vacuum restricted to the wedge algebra, with respect to the boost/proper-time flow. This already shows that “time” is not an absolute label: it is the parameter associated with a specific physical flow (boost evolution) and a specific observer family. The present subsection pushes the point one step further: even when the worldline is fixed, the way proper time is read and processed is a protocol choice. In our framework, such protocol choices define distinct operational contexts. Concretely, the operational context of a detector includes: 64 Figure 5. Relative first-order clock-protocol defect R ( ν ) = < [∆ F(1) ] /< [ F0 ]for a uniformly accelerated Unruh– DeWitt detector with Gaussian switching. The curve is computed by exact numerical evaluation of the windowed kernel KT ( ω )using the full accelerated Wightman pullback (79) and by finite-difference evaluation of ∂ΩKT as in (80) . The suppression for large ν reflects the factor e−(νT/2)2 , while the intermediate-frequency structure reflects the sideband dependence on Ω±ν/2. and the relative first-order defect: R(ν) := <∆F(1)(Ω) <F0(Ω).(81) By (74) ,∆ F(1) (Ω) carries an overall factor εAT ( ν ); in the plotted curve we fix ε and report the resulting relative response shift as a function of ν. The dimensionless parameters are: •acceleration a(sets the Unruh scale), •detector gap Ω, •switching width T, •modulation amplitude εand modulation frequency ν. The qualitative features of R(ν)are robust: 1. R(ν)→0for large νT due to the Gaussian factor e−(νT/2)2. 2. R(ν)→0for T→ ∞ (stationary limit), consistent with Section 5 C. 3. R ( ν )exhibits a characteristic dip/peak structure at intermediate ν , reflecting the sideband difference KT(Ω −ν/2) −KT(Ω + ν/2). These are direct consequences of the closed-form structure (73) – (77) and do not rely on further approximations. The main numerical output of this subsection is the plot of R ( ν )in (81) computed from the full Wightman integral. 65 5.4.6 Interpretation The numerical curve R ( ν )has a clear operational meaning: it quantifies the response difference between two contexts defined by distinct clock protocols (ideal proper time τ versus modulated readout ˜τ ) while keeping the underlying local algebra and Minkowski vacuum fixed. The existence of a nonzero but controlled R(ν)demonstrates that: 1. Unruh thermality (KMS structure) remains intact at the level of W(u); 2. operational comparisons of clock protocols induce measurable defects in detector response; 3. these defects vanish in the stationary limit and are suppressed at high modulation frequencies, as required for consistency. This is the prototype for later sections, where analogous defect kernels and PSDs are inferred and renormalized in cosmological settings. E. What is uniquely achieved vs. string theory 5.5.1 The comparison principle The Unruh effect is a benchmark phenomenon: it is a horizon-type thermal behavior arising in ordinary QFT on Minkowski spacetime. Any proposed quantum-gravitational framework that claims operational universality should (i) reproduce the Unruh KMS structure and (ii) provide a principled account of what changes when operational contexts are compared and refined. The present worked example does precisely this: it preserves the KMS structure while producing a computable, protocol-dependent observable associated with comparing clock contexts. The comparison to string theory is therefore not about ultraviolet completion of gravity in general. It is about whether a framework provides a universal operational calculus for context comparisons in a finite-causal-domain setting without relying on asymptotic boundaries. 5.5.2 What is computed here: a protocol-dependent horizon observable The central observable in Sections 5 C–5 D is the relative first-order response defect R(ν) := <∆F(1)(Ω) <F0(Ω), computed exactly from: 1. the accelerated Wightman pullback W(u)(which encodes the KMS structure), 2. a finite-time switching protocol χ(τ), 3. a clock-protocol deformation ˜τ=τ+εsin(ντ), 4. and the two distinct operational meanings of protocol change (phase-readout and switching-schedule deformations). This observable has a direct operational interpretation: It measures how much the detector response changes when the same physical worldline and field state are read out using two different clock protocols, while keeping the local KMS structure fixed. It is therefore a pure context-comparison observable: it vanishes for trivial protocol changes (constant offsets), vanishes in the stationary limit (as shown by the factor e−(νT/2)2 ), and is controlled for finite-time contexts by the sideband structure Ω±ν/2. 66 5.5.3 Why this is a quantum-gravitationally relevant observable in the present framework Although the underlying background is flat Minkowski spacetime, the Unruh effect is already a horizon phenomenon in the operational sense: acceleration restricts the observer to the wedge and replaces global time translation by modular/boost flow. In our framework, this is precisely the regime where strict global ordering and strict global subsystem identification become nontrivial. The new element introduced here is that the clock protocol is treated as part of the context rather than as an internal gauge choice. The defect R(ν)is then interpreted as a representative of coherence mismatch between contexts: C(τ)−→ C(˜τ), with the comparison implemented by the explicit deformation ˜τ = τ + δ ( τ ). This makes the Unruh setting the simplest arena in which the general conceptual claims of Sections 1–2 can be tested without any gravitational UV assumptions. The significance is therefore twofold: 1. The KMS/thermality structure is preserved exactly, showing that contextual gluing does not require modifying local QFT dynamics. 2. The comparison defect is explicitly computable and depends on operational refinement parameters ( T ,Ω) and on the protocol deformation frequency ν , providing the prototype for defect kernels and PSD inference in later cosmological sections. 5.5.4 Domain distinction: boundaries, asymptotics, and operational universality String theory and holography are fundamentally oriented toward UV completion and, in the holographic case, toward settings where a boundary theory provides a global algebra and a global time. The Unruh setting is qualitatively different: •It is a finite causal-domain phenomenon (wedge restriction). •It is defined by an observer-local modular flow (boost generator). • The operational observable of interest is a comparison between clock protocols on the same worldline with finite switching. The calculation performed here is therefore not naturally expressed as an asymptotic scattering amplitude or as a boundary correlation function. It is an intrinsically operational detector observable in a horizon setting, evaluated with finite-time switching and protocol dependence. The unique achievement of the present framework in this context is that it provides: 1. an explicit analytic defect formula in terms of KT(Ω ±ν/2) and ∂ΩKT(Section 5 C), 2. an exact numerical evaluation based solely on the Wightman pullback and the switching protocol (Section 5 D), 3. a natural interpretation of this defect as a context-comparison object that runs under refinement (switching width, channel selection), forming the prototype of categorical renormalization. 5.5.5 Independence from UV completion assumptions A further distinctive feature is that the entire analysis requires no ultraviolet completion assumptions: •The Wightman function is fixed by standard QFT on Minkowski spacetime. •The detector model is standard and minimal. • The only new input is the operational protocol deformation of clock readout, which is defined entirely at the level of context comparison. Thus the computed defect is not a claim about Planckian microphysics; it is a claim about the operational comparison structure that remains meaningful even when the global strictification assumptions of quantum gravity fail. In later sections, the same structure is transported to cosmological horizons and slow-roll FLRW, where boundary duals are unavailable and the operational calculus becomes essential. 67 5.5.6 Summary The uniquely achieved content of the Unruh worked example is the explicit construction of a protocoldependent, operational observable in a horizon setting that: 1. preserves the underlying KMS/thermality structure exactly, 2. isolates a context-comparison defect induced by incompatible clock protocols, 3. is computable analytically and numerically without boundary constructions and without UV completion assumptions, 4. naturally integrates into the categorical RG viewpoint as a running kernel/filter under refinement of operational contexts. 6. WORKED EXAMPLE III: COSMOLOGICAL HORIZON DEFECT WITHOUT DS/CFT A. de Sitter horizon as cosmological Unruh 6.1.1 Motivation The Unruh effect (Section 5) shows that horizon-type thermality arises from restricting a global state to an observer-accessible algebra and comparing it with respect to the observer’s natural time flow. De Sitter spacetime provides the cosmological analog: every geodesic observer possesses a cosmological event horizon, and the Bunch–Davies (Euclidean) vacuum restricted to an observer’s static patch is thermal at the Gibbons–Hawking temperature [3, 4, 8]. This section uses de Sitter as the simplest cosmological laboratory for the contextual-gluing framework. The goal is to: 1. recall the KMS structure of the static patch (thermality is robust and structural), 2. set up the operational detector response in the Bunch–Davies vacuum, 3. prepare the ground for a protocol-dependent “horizon defect” computation analogous to the Unruh clock-defect curve, but now with the horizon scale set by Hrather than acceleration a. Importantly, no dS/CFT or boundary dual structure is used; all quantities are defined intrinsically in the static patch and via operational detector protocols. 6.1.2 Geometry: de Sitter static patch and horizon Four-dimensional de Sitter spacetime can be described in static coordinates adapted to a geodesic observer at the origin: ds2=−1−H2r2dt2+1−H2r2−1dr2+r2dΩ2 2,(82) where H is the Hubble parameter. The coordinate patch covers the observer’s static region 0 ≤r < H−1 , bounded by the cosmological horizon at r = H−1 . The natural time flow for the observer is t , generated by the timelike Killing vector ∂t in the static patch. The associated surface gravity at the horizon is κ=H, leading to the temperature stated below. 6.1.3 Bunch–Davies vacuum and KMS thermality The Bunch–Davies vacuum is the de Sitter-invariant state selected by analytic continuation from the Euclidean sphere and by regularity at the horizon. Restricting this state to the algebra accessible in the static patch yields a thermal (KMS) state with respect to the static time flow. The temperature is the Gibbons–Hawking temperature: TdS =H 2π, βdS =2π H.(83) 68 This is the cosmological counterpart of the Unruh temperature (56) . In both cases thermality is not imposed; it is a consequence of restricting a globally defined state to an observer-accessible algebra and evolving with respect to the observer’s natural time flow. Wightman KMS property on a comoving/geodesic worldline. For suitable free fields, the pullback of the Bunch–Davies Wightman function to a geodesic observer worldline depends only on the proper-time difference u = τ−τ0 and has the same analytic structure as the Unruh pullback with a→H . For example, for a conformally coupled massless scalar the pullback takes the form WdS(u) = −H2 16π2 1 sinh2H 2(u−i),(84) which satisfies the KMS condition WdS(u+iβdS) = WdS(−u), βdS =2π H. Thus the static patch thermality is encoded directly in the same analytic structure used in the Unruh analysis. 6.1.4 Detector formulation: response in the Bunch–Davies vacuum Let a detector follow a geodesic worldline in the static patch. An Unruh–DeWitt coupling as in (57) , with proper time τand switching function χ(τ), yields the response functional Fχ(Ω) = Zdτ Zdτ0χ(τ)χ(τ0)e−iΩ(τ−τ0)WdS(τ−τ0).(85) In the long-time stationary limit, the KMS property implies the detailed balance relation ˙ F(−Ω) ˙ F(Ω) =e−βdSΩ, so the detector behaves as if immersed in a thermal bath at temperature H/ (2 π ). This is the cosmological Unruh effect. 6.1.5 Significance for contextual gluing The significance of the de Sitter Unruh analogy for the present framework is twofold: 1. Thermality is structural and horizon-local. The KMS property is encoded in the analytic structure of correlators and does not depend on asymptotic boundaries. This makes de Sitter an ideal setting to test protocol-dependent defects while preserving thermality. 2. Cosmology lacks canonical boundary strictification. Unlike AdS, generic cosmological spacetimes do not come equipped with a universally accepted boundary dual that provides a global time and global factorization. Therefore an operational, observer-local calculus of context comparisons is required to define and compute protocol-dependent observables. In the next subsection we introduce incompatible clock protocols in de Sitter exactly as in the Unruh case, derive the corresponding closed-form defect expressions in terms of a windowed kernel KT ( ω )built from (84) , and evaluate the defect curve numerically. The resulting observable is a cosmological-horizon analog of the Unruh clock defect, obtained without invoking dS/CFT. B. Exact numerical defect curve in de Sitter 6.2.1 Objective and relation to the Unruh computation This subsection repeats the Unruh protocol-defect computation (Sections 5 C–5 D) in a cosmological horizon setting: a detector in de Sitter spacetime in the Bunch–Davies vacuum. The purpose is to obtain 69 a fully explicit, operational, and numerically exact defect curve RdS(ν) := <∆F(1)(Ω) <F0(Ω) as a function of modulation frequency ν, using: 1. Gaussian switching χ(τ) = e−τ2/(2T2), 2. clock protocol modulation ˜τ=τ+εsin(ντ)with |εν|  1, 3. the exact de Sitter Wightman pullback WdS(u), 4. and the same separation into phase-readout and switching-schedule contributions. The resulting curve is a cosmological-horizon analog of the Unruh clock defect. It is constructed entirely within the static patch and is independent of any boundary dual. 6.2.2 Closed-form structure and kernel definition Because the de Sitter pullback WdS ( u )depends only on u = τ−τ0 , the Gaussian-switching derivation in Section 5 C applies verbatim with W(u)→WdS(u). We therefore define the windowed kernel KdS T(ω) := Z∞ −∞ du exp−u2 4T2e−iωu WdS(u),(86) and the baseline response F0(Ω) = √π T KdS T(Ω).(87) With AT ( ν )defined as in (74) , the first-order phase and switching contributions retain the same sideband structure: ∆F(1) phase(Ω) = Ω ε AT(ν)hKdS TΩ−ν 2−KdS TΩ + ν 2i,(88) ∆F(1) switch(Ω) = −ε AT(ν)ν 2KdS TΩ−ν 2+KdS TΩ + ν 2+1 2∂ΩKdS TΩ−ν 2−∂ΩKdS TΩ + ν 2. (89) The total first-order defect is their sum: ∆F(1)(Ω) = ∆F(1) phase(Ω) + ∆F(1) switch(Ω). 6.2.3 Exact Wightman pullback used numerically For the numerical evaluation we use the de Sitter pullback on the relevant worldline. In the simplest (and standard) free-field setting, this pullback has the same functional form as the Unruh pullback with a→H: WdS(u) = −H2 16π2 1 sinh2H 2(u−i).(90) This expression exhibits the KMS periodicity at inverse temperature βdS = 2 π/H (Section 6 A), ensuring that local thermality is preserved. As in the Unruh case, the i prescription is implemented numerically by choosing  small compared to all other time scales and checking stability of the resulting RdS ( ν )curve. 6.2.4 Numerical evaluation: truncation, discretization, and derivative The numerical evaluation proceeds as follows. 70 Figure 6. Cosmological-horizon protocol defect in de Sitter: the relative first-order response shift RdS ( ν ) = < [∆ F(1) ] /< [ F0 ]computed from Gaussian switching, sinusoidal clock modulation, and the exact de Sitter Wightman pullback (90) . The suppression at large ν reflects the Gaussian factor e−(νT/2)2 , while the intermediate-frequency structure reflects the sideband dependence on Ω±ν/2. (N1) Truncation of the integral. Because of the Gaussian factor e−u2/(4T2) , the integral (86) is rapidly convergent. One truncates to u∈ [ −Umax, Umax ]with Umax T (typically Umax ≈ 10 T –15 T ) and verifies that further increases do not change results beyond tolerance. (N2) Discretization. The oscillatory factor e−iωu requires a grid fine enough to resolve oscillations at frequencies ω∼ Ω ±ν/ 2. A uniform grid with sufficiently many points is used; convergence is checked by doubling the grid density. (N3) Numerical derivative. The switching term (89) requires ∂ΩKdS T ( ω ), evaluated by symmetric finite differences as in (80): ∂ΩKdS T(ω)≈KdS T(ω+h)−KdS T(ω−h) 2h, with hchosen to balance accuracy and numerical stability. Stability under variation of his checked. (N4) Realness and consistency checks. The baseline response F0 and the defect ∆ F(1) are real; numerically, small imaginary parts due to finite discretization should be negligible compared to the real part. This provides a practical diagnostic for numerical accuracy. 6.2.5 Relative defect curve We define the relative first-order defect curve exactly as in the Unruh case: RdS(ν) := <∆F(1)(Ω) <F0(Ω).(91) 6.2.6 Interpretation The de Sitter defect curve is the cosmological analog of the Unruh defect curve: 71 1. The underlying thermality is robust: it is encoded in the KMS structure of WdS ( u )and in the static patch modular flow (Section 6 A). 2. The defect is operational and protocol dependent: it quantifies the mismatch between contexts defined by distinct clock readouts, without changing the underlying state. 3. The defect is controlled: it vanishes in the stationary limit and is suppressed for large νT by the switching envelope. These properties make RdS ( ν )a well-defined observer-local cosmological horizon observable that requires no boundary dual and naturally integrates into the categorical RG program developed later for slow-roll FLRW. C. Interpretation 6.3.1 What the defect curve means The numerical curve in Figure 6 represents the relative first-order response shift RdS(ν) = <[∆F(1)(Ω)] <[F0(Ω)] , computed for a detector in de Sitter with Gaussian switching and a sinusoidally modulated clock protocol. The essential conceptual point is that this curve is not a correction to de Sitter thermality itself. Instead it is a correction to the comparison between two operational contexts that assign time stamps and time ordering differently while probing the same local algebraic structure and the same underlying state (Bunch–Davies vacuum restricted to the static patch). In the language developed earlier: •the local state and algebra define a KMS structure at temperature TdS =H/(2π)(Section 6 A); •the clock protocol belongs to the context data (AC, τC,≺C, κC); •changing τCby ˜τ=τ+δ(τ)defines a comparison morphism between contexts; • the defect is the operational residue of this comparison as measured by a concrete observable (detector response). Thus the curve is the simplest cosmological-horizon instantiation of the central framework thesis: global strict identification of contexts is not assumed; instead, comparisons are treated as physical operations that can carry coherent defects. 6.3.2 Thermality is preserved: why this is not a temperature shift De Sitter thermality is encoded in the KMS property of the Wightman function and in the modular flow of the static patch. Concretely, the pullback Wightman function (90) obeys the KMS relation with inverse temperature βdS = 2 π/H . This is a statement about analytic continuation and equilibrium with respect to the static time flow. The clock protocol modification ˜τ = τ + δ ( τ )does not alter the underlying two-point function WdS ( u ); it only alters how the detector samples it through the oscillatory phase and the switching profile. In particular: 1. The KMS periodicity of WdS(u)remains unchanged. 2. The baseline response F0(Ω) continues to satisfy detailed balance in the stationary limit. 3. The defect ∆ F(1) vanishes in the stationary limit (Section 5 C, applied with a→H ), meaning that the equilibrium thermal rate is unaffected at first order by a sinusoidal modulation averaged over infinite time. Therefore, the computed defect should not be interpreted as a new physical temperature, nor as a violation of de Sitter thermality. It is a controlled finite-time protocol effect: it quantifies the mismatch between two equally valid operational definitions of “time” within the same horizon patch. 72 6.3.3 Coherent difference: sidebands and finite-time context dependence The closed-form expressions (88)–(89) exhibit a characteristic sideband structure: KdS TΩ−ν 2, KdS TΩ + ν 2, which is the analytic fingerprint of a time-modulated clock protocol. Physically, it expresses the fact that modulating the time readout induces frequency mixing in the detector phase accumulation and in the switching schedule. The resulting difference is coherent: it is not a random disturbance, but a deterministic shift associated with a well-defined comparison morphism between contexts. The Gaussian switching profile introduces the factor e−(νT/2)2 , which suppresses high-frequency modulation effects and enforces the vanishing in the stationary limit. This is a generic and physically necessary behavior: a finite-time context has finite bandwidth, and clock modulation at frequencies far above that bandwidth averages out. Thus the defect curve as a function of νis naturally interpreted as: •small for νT 1(nearly rigid time reparametrizations), •maximal at intermediate νwhere sideband sampling is strongest, •exponentially suppressed for νT 1. 6.3.4 Contextual meaning: two descriptions, one coherent network The cosmological-horizon setting highlights the meaning of “global” used in this paper (Section 0 A). There is no single global observer who accesses all of de Sitter. Instead, each observer has a static patch with its own natural time flow and accessible algebra. The Bunch–Davies vacuum restricted to any static patch is thermal, but the operational comparison between distinct clock protocols (or between distinct observers’ protocols) requires a comparison structure. The defect curve demonstrates that even within a fixed patch and fixed worldline, there can be multiple valid time readouts whose comparison is not strictly trivial. This is the simplest instance of the general phenomenon: “global” physics is the coherent pattern of relations among contexts, not a single absolute description. 6.3.5 Relation to categorical RG and to cosmological inference The de Sitter defect curve is also the simplest example of a running object under context refinement. The response depends on: •switching width T(temporal resolution), •detector gap Ω(channel), •horizon scale H(epoch scale in the FLRW generalization), •modulation frequency ν(protocol deformation). Therefore the defect is naturally treated as a kernel H ( ν ; C )in the categorical RG sense (Section 2 B). In later sections, the same structure is used for slow-roll FLRW: 1. infer defect statistics (PSD) from refinement data across multiple Tand Ωchannels, 2. transport these statistics across epochs, 3. predict scattering observables via bounded functionals of the inferred PSD. The de Sitter curve provides the clean horizon-scale anchor for this program: it is the cosmological analog of the Unruh defect and is computed without any boundary dual input. 73 6.3.6 Summary The de Sitter horizon defect curve supports the central interpretive claim of the framework: 1. The underlying horizon thermality is preserved; it is encoded in the KMS structure of the restricted state. 2. Distinct clock protocols define distinct operational contexts; their comparison produces a coherent, controlled defect in finite-time detector response. 3. The defect is not a violation of semiclassical physics; it is a measurable consequence of non-strict contextual identification in a horizon setting. 4. The result integrates naturally into categorical RG: the defect kernel is a running object under refinement and under horizon/epoch scaling, enabling later inference and scattering predictions. 7. WORKED EXAMPLE IV: GENERIC SLOW-ROLL FLRW: DEFECT + CATEGORICAL RG + SCATTERING A. Why “generic spacetime” matters 7.1.1 Motivation: beyond special backgrounds and boundary strictifications The preceding worked examples (Hawking/Page, Unruh, and de Sitter) already establish the core operational logic of the framework: local semiclassical physics is preserved, while the comparison and gluing of contexts produces controlled defects that run under refinement. However, these examples still involve backgrounds with special symmetry or special structure: • Unruh physics occurs in Minkowski spacetime but with a wedge restriction and a Killing boost flow. • de Sitter is maximally symmetric and possesses a static patch with a Killing time and exact KMS structure. • The Hawking/Page toy models are constructed in an effective emission-step framework designed to isolate factorization/gluing logic. The ultimate arena for quantum gravity, however, is generic spacetime: settings where no global Killing time exists, no canonical asymptotic boundary algebra is available, and the relevant observables must be defined within finite causal domains. Slow-roll FLRW cosmology is the canonical example: it is physically central, observationally constrained, and does not naturally admit a universal boundary strictification analogous to AdS/CFT. For this reason, slow-roll FLRW is an essential testbed for any operationally universal formulation of quantum gravity. 7.1.2 What “generic spacetime” means operationally In this paper, “generic spacetime” does not mean mathematically arbitrary metrics; it means a physical setting where the following strictifying structures are absent or not operationally universal: (G1) No canonical asymptotic boundary with global time. In asymptotically AdS settings, boundary time and boundary operator algebras provide a strict global reference structure. In slow-roll FLRW, there is no analogous canonical boundary theory that supplies a unique global time parameter and a unique global factorization of the Hilbert space. Operationally, the relevant information is contained in finite causal diamonds and in observer-local records. (G2) No global timelike Killing flow. In de Sitter static patches or stationary black hole exteriors, one can define time evolution with respect to a Killing vector. In slow-roll FLRW, the expansion rate H ( t ) varies with time; there is no exact global time-translation symmetry. Consequently, “frequency” and “particle” notions become inherently context-dependent, and operational probes must be defined with explicit switching windows, gaps, and local clock readouts. 80 7.3.7 Conceptual conclusion The five-epoch analysis provides the strongest evidence of categorical renormalizability in the operational sense developed in Section 2: 1. The defect statistics close within a single PSD universality class across five epochs. 2. The running of parameters with Hdefines a finite-dimensional categorical RG flow. 3. The extracted scaling slopes are consistent with horizon-set scaling νc∝H and a definite amplitude exponent q. 4. Scattering observables computed from the inferred PSD exhibit scaling collapse, confirming fixedshape universality in a generic slow-roll cosmological setting. This completes the demonstration that coherence renormalization yields renormalizable quantum gravity predictions in the sense appropriate to observer-local, non-boundary, non-stationary spacetimes. D. What is the categorical RG claim here 7.4.1 Statement of the claim in one paragraph The categorical renormalization-group (RG) claim in the slow-roll FLRW setting is the following: There exists a closed structural class of coherence-defect representatives (here: defect PSDs) such that (i) the same class describes all operational contexts obtained by refinement in switching width T and by channel selection Ω, and (ii) the parameters of this class evolve with cosmological epoch t (equivalently local Hubble scale H ( t )) according to a finite-dimensional flow. Moreover, after rescaling by the inferred flow exponents, operational observables (in particular the scattering exponent Ξ) exhibit scaling collapse, providing evidence for fixedshape universality. This subsection makes each part of this claim precise and explains why it is the appropriate notion of “renormalizable quantum gravity” in the present framework. 7.4.2 Closure: the same PSD universality family across contexts and epochs Recall the fundamental measurement equation for context refinement data: V(t;T, Ω) = Zdν 2πH(ν;t, T, Ω)2Sδ(ν;t),(110) where His a known protocol filter and Sδis the defect PSD. The central closure statement is: Closure across contexts (within a fixed epoch). For fixed t , changing T and Ωchanges the context C ( t, T, Ω) and therefore the filter H ( ν ; t, T, Ω). The defect statistics, however, remain of the same type: a PSD on the same frequency axis. Thus contextual refinement does not change the structural class; it changes only the filter through which the same defect statistics are interrogated. Closure across epochs (cosmological refinement). For different epochs t , the accessible causal patch and horizon scale change. The claim is that the defect statistics remain describable within a single PSD universality family across these epochs. Concretely, we work with the closed family Sδ(ν;H) = A(H) 1 + ν/νc(H)αexph−ν/νhi(H)2i,(111) with either fixed αand fixed ratio νhi/νc, or a weakly running extension. Closure under categorical RG means that all inferred Sδ(ν;t)lie within (111) for the entire refinement family considered. This closure statement is the operational analog of renormalizability: refinement does not generate an unbounded tower of new independent structures; it changes the parameters of a stable family. 81 7.4.3 Finite-dimensional flow: beta functions for defect parameters Once the defect PSD is assumed (and verified) to lie in a finite-parameter family, renormalization becomes a flow on the parameter space. Let θ(H) := A(H), νc(H), α(H), νhi(H), and choose a scale coordinate. In cosmology the natural choice is the e-fold time N:= ln a, for which the slow-roll parameter is  = −dln H/dN . The categorical RG claim is that θ obeys a finite-dimensional flow: βθ(H) := dθ dN =B(θ;H),(112) for a smooth vector field B determined by the universality class and by the operational refinement structure. In the fixed-shape scaling regime, the flow reduces to scaling laws: νc(H)∝H, νhi(H)∝H, A(H)∝H−q, α(H)≈α, (113) so that the beta functions are (using dln H/dN =−): dln νc dN =−, dln νhi dN =−, dln A dN =q , dα dN ≈0.(114) Equations (114) are the precise meaning of “finite-dimensional flow” in this setting: the cosmological renormalization group is captured by a small set of running parameters, rather than by an infinite series of higher-curvature counterterms. 7.4.4 Universality and fixed-shape behavior: scaling collapse of observables The strongest operational evidence for a universality class is scaling collapse. In this FLRW setting, we test fixed-shape behavior not by comparing microscopic fields, but by comparing operational observables computed from the inferred defect statistics. The primary scattering/decoherence observable used in this paper is the phase diffusion exponent Ξ(Tint;H) = 1 2ω2 0Zdν 2πSδ(ν;H) sinc2 νTint 2.(115) Assume fixed-shape scaling of defect statistics: Sδ(ν;H) = H−qe Sν H,(116) for a universal function e S. Substituting (116) into (115) and changing variables ν=Hx yields Ξ(Tint;H) = H1−q1 2ω2 0Zdx 2πe S(x) sinc2 x H Tint 2.(117) Therefore, after rescaling the integration time by the horizon scale, Tint := H Tint, and rescaling the exponent by the inferred amplitude scaling, Ξ := Hq−1Ξ, one obtains a master curve Ξ(Tint) = X(Tint),(118) independent of H . This is the precise mathematical expression of scaling collapse in our setting. The multi-epoch figures in Section 7 C confirm this collapse empirically, providing evidence that the defect statistics indeed lie in a fixed-shape universality class. 82 7.4.5 The categorical content of the claim The term “categorical RG” emphasizes that what is being renormalized is not a field on a fixed global spacetime but the comparison/gluing structure of contexts. In the FLRW setting this has a clean interpretation: 1. Context refinement (changing T and channel Ω) corresponds to morphisms in the refinement category of contexts, with coarse-graining maps that forget resolution. 2. Epoch flow (changing t ) changes the base object: the accessible causal patch and its natural time scale. 3. The defect PSD family (111) is a representative of coherence data (a projection of loop defects/cocycle structure to a measurable statistic). 4. Closure and finite-dimensional flow mean that the coherence data form an invariant manifold under refinement: they are stable objects of the renormalization functor. In categorical terms, the claim is that the defect data define a stable pseudofunctorial assignment on the refinement structure, and that its evolution across epoch scales factors through a finite-parameter moduli space (the universality class). 7.4.6 Summary of the categorical RG claim in FLRW The categorical RG claim in the FLRW worked example consists of three precise components: 1. Closure: defect statistics at all epochs and refinement settings are describable within a single PSD universality family. 2. Finite-dimensional flow: the dependence of the defect PSD on epoch is captured by a small set of running parameters with beta functions, rather than by an infinite tower of couplings. 3. Universality: after rescaling by the inferred exponents, scattering observables derived from the defect statistics collapse to a master curve, evidencing fixed-shape behavior. Taken together, these statements implement “renormalizability” in the coherence sense developed in Section 2: quantum gravity in generic slow-roll FLRW is predictive because the objects that run under refinement are closed and universal. E. Figures and outputs (explicit) This subsection lists the concrete numerical and graphical outputs associated with the slow-roll FLRW categorical RG and scattering analysis developed in Sections 7 B–7 D. The figures are organized to isolate (i) the running of defect statistics across epochs, (ii) the extraction of scaling exponents (beta data), (iii) the predicted epoch-dependent scattering observable, and (iv) the universality test via scaling collapse. All plots correspond to the five-epoch demonstration of Section 7 C. Figure I: Parameter flow vs. H(categorical RG flow) 83 Figure 7. Categorical RG flow of defect-PSD parameters across five cosmological epochs parameterized by the local Hubble scale H = H ( t )(early → late). The plotted quantities are the fitted amplitude A ( H )and characteristic frequency scales νc ( H )and νhi ( H )within the closed PSD family used in Section 7 C. This figure visualizes closure of the defect statistics under epoch refinement: the same structural class persists, while parameters run smoothly with H. Interpretation. Figure 7 is the most direct visualization of the categorical RG flow: it shows that the defect statistics at all five epochs are describable within the same PSD family, with a finite set of running parameters. In the coherence-renormalization viewpoint, this is the operational meaning of renormalizability: refinement across epochs does not generate an unbounded tower of new independent structures. Figure II: Log–log slope plots (extraction of scaling exponents) 84 Figure 8. Log–log scaling relations extracted from the five-epoch fit: ln ( νc/νc,0 )and ln ( A/A0 )plotted against ln ( H/H0 ). Linear behavior supports the scaling laws νc ( H ) ∝Hsν and A ( H ) ∝H−q , which define the finitedimensional categorical RG exponents in the closed-family truncation. Interpretation. Figure 8 provides the operational analog of beta-function extraction: rather than tracking running couplings of a metric field, one tracks the running of defect statistics (here, the PSD parameters). The linearity of these log–log plots supports the fixed-shape scaling hypothesis and provides numerical values of the exponents controlling the RG flow, as described in Section 7 C and formalized in Section 2 D. Figure III: Scattering exponent vs. Tint for five epochs 85 Figure 9. Predicted scattering/decoherence observable across five epochs: the phase diffusion exponent Ξ( Tint ; H ) computed from the inferred defect statistics via the bounded functional (115) . The separation of curves across epochs reflects the epoch dependence of the inferred defect PSD. Interpretation. Figure 9 is the direct operational prediction of the five-epoch analysis: it maps inferred coherence-defect statistics into an observer-local scattering/decoherence measure. This figure makes explicit that the categorical RG is not purely formal: it yields quantitatively distinct predictions for the same operational protocol applied at different cosmological epochs. Figure IV: Scaling collapse (universality/fixed-shape test) 86 Figure 10. Universality test by scaling collapse: the scattering exponent curves from Figure 9 after rescaling Tint by ( H/H0 )and rescaling the amplitude by the inferred exponent q (as described in Section 2 E and specialized to FLRW in Section 7 D). Collapse onto a single master curve provides operational evidence for fixed-shape behavior of defect statistics under categorical RG. Interpretation. Figure 10 is the strongest universality output: it demonstrates that, after applying the RG rescaling extracted from the parameter flow, the epoch-dependent scattering predictions become approximately epoch-independent in scaled variables. This is the operational signature that the defect statistics belong to a universality class and that the categorical RG exhibits fixed-shape behavior. 8. DEFECT-STATISTICS TOMOGRAPHY →SCATTERING PREDICTION (OPERATIONAL QG PIPELINE) A. The “measurement equation” 8.1.1 Purpose The preceding worked examples establish three recurring facts: 1. Operational contexts are defined by switching windows, spectral channels, clock protocols, and causal accessibility. 2. Context comparisons induce coherent defects that are measurable through response differences, even when local thermality (KMS structure) is preserved. 3. In generic spacetimes (slow-roll FLRW), these defects run under refinement and across epochs, and can be used to predict scattering/decoherence observables. This section isolates the shared mathematical backbone of these computations: the measurement equation that maps defect statistics into observable response variances. This equation is the core of the operational quantum-gravity pipeline: it allows one to infer defect PSDs from refinement data and to transport them to scattering predictions. 87 8.1.2 From a deterministic protocol deformation to stochastic defect statistics In Sections 5 and 6, we introduced a deterministic protocol deformation ˜τ=τ+δ(τ), δ(τ) = εsin(ντ), and derived an explicit first-order shift ∆ F(1) in detector response. For tomography and renormalization, it is natural to allow the defect to be stochastic at the level of unresolved microscopic comparison data. Operationally, this corresponds to the fact that clock comparisons, calibrations, and gluing procedures accumulate unresolved fluctuations due to finite resolution, environmental noise, and microscopic degrees of freedom that are not tracked in the coarse description. Accordingly, we treat the (zero-mean) defect δτ ( t )as a stationary stochastic process in a given context, characterized by its power spectral density (PSD) hδτ(ν)δτ(ν0)i= (2π)δ(ν+ν0)Sδ(ν),(119) where δτ(ν)denotes the Fourier transform of δτ(t)in the time variable appropriate to the context. 8.1.3 Linear response: the defect filter H(ν) Let ∆ F denote the response difference between a baseline protocol and a perturbed protocol. In a weak-defect regime, linear response takes the form ∆F(t)≈Zds G(t−s)δτ(s),(120) for some deterministic kernel G determined by the underlying QFT correlations and by the protocol (switching window, detector gap, and worldline). In frequency space this becomes ∆F(ν)≈H(ν)δτ(ν),(121) where H ( ν )is the defect response filter. In the Unruh/de Sitter calculations, H is expressed in terms of the windowed Wightman kernel KT ( ω )evaluated at sidebands Ω ±ν/ 2and (for switching deformation) its Ω-derivative. In FLRW contexts, Hdepends additionally on epoch through the local horizon scale. Importantly, H is not a fitted object: it is computable from the protocol and the underlying semiclassical input. It is therefore a known kernel in the measurement equation. 8.1.4 Variance data as a linear functional of the PSD Given (121), the variance of the response difference is Var(∆F) = Zdν 2πh∆F(ν) ∆F(−ν)i. Using (121) and (119) yields V:= Var(∆F) = Zdν 2π|H(ν)|2Sδ(ν).(122) Equation (122) is the fundamental measurement equation: it expresses the observed variance as a linear functional of the defect PSD, with a known nonnegative weight |H|2. Multiple contexts: indexed measurement equations. In practice, one collects variance data across a family of contexts indexed by protocol parameters (switching widths Ti , detector gaps Ω i , epochs ti , and possibly different worldlines). Denote the i -th context by Ci and the corresponding filter by Hi(ν) := H(ν;Ci). Then the measurement equation becomes a linear system of integral constraints: Vi=Zdν 2π|Hi(ν)|2Sδ(ν), i = 1, . . . , M. (123) This is the operational tomography problem: infer Sδ ( ν )from the data {Vi} and the known kernels {|Hi|2}. 88 Discretization. For numerical inference one discretizes ν on a grid {νk} and approximates the integral by a quadrature. Writing Sk≈Sδ ( νk )and Wik ≈ |Hi ( νk ) |2 ∆ ν/ (2 π ), one obtains the standard linear inverse form Vi≈X k Wik Sk.(124) Since Sδ ( ν ) ≥ 0, the inverse problem includes a positivity constraint Sk≥ 0. The conditioning of W depends on the diversity of contexts (in particular, the diversity of switching widths and channels), which is why multi-channel and multi-Tdata are central to stable inference. 8.1.5 Interpretation as a categorical object Equation (123) can be interpreted as the evaluation of a presheaf-like assignment on contexts. Each context Cidefines a deterministic functional ΦCion defect statistics: ΦCi[Sδ] := Zdν 2π|H(ν;Ci)|2Sδ(ν), and the variance datum is Vi = Φ Ci [ Sδ ]. Refinement of contexts (changing T , enlarging channel sets, changing epoch) corresponds to moving in the refinement category Ctxref , while the family of functionals { Φ C} provides a systematic way to probe the same underlying defect statistics across contexts. This is the operational content of the categorical viewpoint: contexts are objects, refinements are morphisms, and measured data are evaluations of a functorial assignment. 8.1.6 Structural consequences Several structural consequences follow immediately from (123): (i) Separation of “physics input” and “defect statistics.” All semiclassical QFT input enters through Hi ( ν )(Wightman pullbacks, switching, channel gaps). The quantum-gravitational content is isolated in Sδ ( ν ). This separation is precisely what allows inference of defect statistics without postulating a UV completion. (ii) Closure under refinement becomes testable. Because the same Sδ must explain data across multiple contexts, closure hypotheses (finite-parameter PSD families, scaling laws across epochs) can be tested quantitatively by goodness-of-fit and scaling collapse, as done in Section 7 C. (iii) Direct pipeline to scattering predictions. Once Sδ is inferred, any observable that is a bounded functional of Sδ is immediately predictable. The phase diffusion exponent Ξused throughout this paper is of this form, enabling the end-to-end pipeline from refinement data to scattering predictions. 8.1.7 Summary The measurement equation Vi=Zdν 2π|Hi(ν)|2Sδ(ν) is the central operational bridge between contextual gluing (defect statistics) and experimentally meaningful observables (response variances). It is the basis for defect-statistics tomography, for categorical RG extraction in cosmology, and for scattering/decoherence predictions in generic spacetimes. B. Inference methods 8.2.1 The inverse problem implied by the measurement equation The measurement equation derived in Section 8 A can be written abstractly as Vi=Zdν 2πWi(ν)Sδ(ν), Wi(ν) := |Hi(ν)|2≥0,(125) 89 or after discretization (Section 8 A) as the linear system Vi≈X k Wik Sk, Sk≥0,(126) where Sk≈Sδ(νk). The inference problem is to reconstruct the nonnegative spectrum Sδ (or a parametrized representative thereof) from the variance data {Vi} and known kernels {Wi} . This is a classical inverse problem: the mapping is linear but typically smoothing, and therefore can be ill-conditioned. In this subsection we describe two inference strategies: 1. Nonparametric inversion (NNLS + smoothness regularization): conceptually direct, but sensitive to conditioning. 2. Parametric inference (MLE/least squares in a universality family): robust and directly aligned with categorical RG extraction. 8.2.2 Ill-conditioning is expected: why the inverse problem is nontrivial Before specifying estimators, it is important to understand why the inversion is inherently delicate. Each kernel Wi ( ν )is determined by switching windows and detector gaps and therefore acts as a broad frequency filter. In typical operational settings: • changing T changes the effective width of Wi ( ν )(Gaussian switching produces exponential suppression at high ν), • changing Ωshifts and reshapes spectral sensitivity but does not yield delta-function localization in ν, •the integral (125) is therefore a smoothing transform. Smoothing transforms generically suppress high-frequency information and amplify noise upon inversion. This is not a flaw; it is the operational statement that finite protocols cannot resolve arbitrarily fine defect structure. Hence some form of regularization or structural restriction is necessary for stable inference. 8.2.3 Nonparametric inversion: NNLS with smoothness regularization NNLS formulation. In discretized form (126) , the most direct nonparametric estimator is the nonnegative least squares (NNLS) solution: b S= arg min Sk≥0X iPkWikSk−Vi σi2 ,(127) where σi represent measurement uncertainties. Positivity Sk≥ 0is physically required because Sδ is a PSD. Need for smoothness. Because W is typically ill-conditioned, (127) may admit many near-equivalent solutions in the presence of noise. The standard stabilization is to add a smoothness penalty, yielding Tikhonov-type regularization: b S= arg min Sk≥0"X iPkWikSk−Vi σi2 +λkLSk2#,(128) where L is a discrete derivative operator (e.g. first or second difference) enforcing smoothness of S ( ν )and λ > 0sets the regularization strength. Interpretation. The penalty kLSk2 is not an ad hoc mathematical trick. It encodes an operational prior: defect statistics inferred from finite-time protocols are expected to be band-limited and smooth on scales below the protocol’s intrinsic resolution. In categorical RG terms, it encodes the expectation that refinement reveals a controlled flow rather than arbitrary spectral oscillations. Choosing λ corresponds to choosing the degree of refinement that the data can support. 96 8.4.7 Summary The tomography-to-scattering pipeline is method-independent in content because: 1. it uses only operationally defined variance data across contexts and standard linear statistics of stationary processes, 2. it does not require a fundamental gauge-dependent metric perturbation field to be defined or quantized, 3. category theory enters only to encode the refinement/gluing structure of contexts and the closure/universality claims, not to impose a particular microscopic quantization technology, 4. its predictions are expressed in terms of directly measurable scattering/decoherence observables (phase diffusion exponent and visibility). E. What other QG methods cannot do 8.5.1 The content of the claim The purpose of this subsection is to identify, in a precise and non-rhetorical way, what is structurally distinctive about the operational tomography → scattering pipeline developed here. The statement is not that other approaches to quantum gravity are incorrect, nor that they cannot address related questions in special settings. The statement is that the present pipeline achieves an end-to-end, observer-local inference-and-prediction construction in generic slow-roll FLRW spacetimes that is not provided as a universal procedure by existing quantum gravity frameworks. The specificity of the claim is important: “generic slow-roll FLRW” means no canonical asymptotic boundary, no global Killing time, and observer-local contexts defined by switching windows, channels, and epoch dependence (Section 7 A). “End-to-end operational inference → prediction” means that the theory provides a concrete map {V(t;T, Ω)}T,Ω,t −→ Sδ(ν;t)−→ Ξ(Tint;t), with explicit kernels, explicit inference procedures, and explicit scattering observables. 8.5.2 What must be supplied to complete the pipeline To complete the pipeline in a generic FLRW context, a theory must provide: 1. A definition of observer-local contexts (what is accessible, what is the clock protocol, what is the ordering). 2. A computable comparison kernel H ( ν ; C )(how a protocol deformation changes operational response). 3. A stable running object (here: defect PSD) that closes under refinement and across epochs. 4. A prescription for inference of this running object from multi-context data. 5. A direct observable prediction expressed as a bounded functional of the inferred running object (here: Ξand e−Ξ). The present framework supplies each item explicitly: •contexts are defined by (t, T, Ω) and protocol data; •His computed from worldline pullback correlators and switching windows; •Sδis inferred by constrained inversion or parametric universality fitting; •categorical RG provides closure and scale transport across epochs; •Ξis computed from Sδvia a universal filter functional. Therefore, the distinctive feature is not a single formula; it is the existence of a complete, operationally grounded chain. 97 8.5.3 Limitations of strict metric quantization for this task Perturbative graviton quantum gravity begins by strictifying a global metric field and expanding gµν = g(0) µν + κhµν . The natural outputs are perturbative correlation functions and (when asymptotic states exist) scattering amplitudes. In generic FLRW, several obstacles arise: 1. There is no canonical S -matrix in a cosmological spacetime, so scattering observables must be defined operationally within finite causal domains. 2. Gauge dependence of hµν complicates the definition of local observables; operational clock comparisons are not naturally expressed as local h -field correlators without introducing further reconstruction conventions. 3. The strictification variables do not close under ultraviolet refinement (Sections 1 A and 2 C); the result is not a universal inference framework for defect statistics across contexts. The present approach bypasses these issues by avoiding hµν as a fundamental variable and working directly with operationally defined context mismatch statistics. 8.5.4 Limitations of holographic/boundary-based methods in generic FLRW Holographic methods in AdS/CFT provide a nonperturbative definition of quantum gravity in settings with asymptotic AdS boundaries. In that regime, boundary correlators provide a global algebra and global time, and many entanglement and Page-curve computations are available. In generic slow-roll FLRW: 1. There is no canonical boundary dual that supplies a unique global time and a unique global Hilbert space factorization analogous to AdS/CFT. 2. Even when speculative dualities are proposed, the operational mapping from local detector protocols and switching windows to boundary data is not given as a universal, model-independent prescription. 3. The inference problem of reconstructing a defect PSD from multi-context variance data is intrinsically observer-local and protocol-defined; it is not naturally expressed as a boundary observable without additional assumptions. The present framework supplies an intrinsic cosmological procedure precisely in the absence of boundary strictification: it defines the running object (defect statistics) in the bulk operational language and transports it across epochs by categorical RG. 8.5.5 Limitations of purely semiclassical treatments Semiclassical QFT in curved spacetime provides the local ingredients required to compute response kernels H ( ν ; C ): Wightman pullbacks and detector response functionals. However, semiclassical theory by itself does not provide: 1. a principle that elevates protocol mismatch to a fundamental, renormalized object, 2. a closure/universality hypothesis for defect statistics across context refinement and cosmological epochs, 3. a categorical RG structure that transports inferred defect statistics between contexts and scales. Thus semiclassical theory supplies the local filter, but not the global gluing and not the renormalization of coherence that turns the defect statistics into a predictive quantum-gravity object. The present framework integrates semiclassical local physics with global coherence constraints: local KMS/thermality remains intact, while global comparisons and gluing produce defect statistics that run and can be inferred and used predictively. 98 8.5.6 The distinctive achievement: operational renormalizability and predictive transport across epochs The categorical RG claim is that defect statistics close under refinement and flow in a finite-dimensional universality class. The unique achievement of the pipeline is that it turns this closure statement into concrete computations in generic FLRW: •infer Sδ(ν;t)at multiple epochs from data V(t;T, Ω), •extract scaling exponents and demonstrate universality by scaling collapse, • compute scattering exponents Ξ( Tint ; t )that differ across epochs in unscaled variables but collapse under RG rescaling. This is a complete renormalization-and-prediction program in a setting where global strictification is not available. The central novelty is that the running object is not a metric coupling; it is coherence defect statistics inferred from operational comparisons. 8.5.7 Summary The present operational QG pipeline provides a fully explicit end-to-end map {V(t;T, Ω)} −→ Sδ(ν;t)−→ Ξ(Tint;t), implemented in generic slow-roll FLRW with multi-epoch categorical RG and scaling collapse. Existing approaches may address parts of this chain in special regimes (e.g. boundary duals in AdS, local semiclassical response in stationary settings), but they do not supply a universal observer-local procedure that simultaneously (i) defines the running object operationally, (ii) closes under refinement, and (iii) yields scattering predictions across cosmological epochs without asymptotic boundary assumptions. 9. INFLATIONARY CORRELATORS: POWER SPECTRUM AND BISPECTRUM FROM CONTEXTUAL TIME ORDERING A. Why the bispectrum is the right target 9.1.1 Contextual time ordering and cosmological correlators Inflationary predictions are formulated in terms of correlation functions of the comoving curvature perturbation ζ (or related gauge-invariant variables) evaluated at late times. In standard single-field slow-roll inflation, the leading predictions are: •the power spectrum Pζ(k)(two-point function) and its tilt ns−1, •the bispectrum Bζ(k1, k2, k3)(three-point function) and its non-Gaussianity amplitude/shape. In the present framework, the new ingredient is not a modification of the inflaton potential or the introduction of extra fields. The new ingredient is contextual time ordering: the claim that the operational time parameter used to order quantum processes is a protocol-defined object and need not strictify globally. This affects precisely those observables that depend sensitively on time ordering and on the composition of quantum histories. The bispectrum is the cleanest target for this reason: it is computed at leading order from time-ordered interaction integrals and therefore is directly sensitive to any controlled deformation of the time-ordering prescription. The power spectrum, by contrast, is dominated by free-field evolution and is comparatively insensitive to such deformations. 9.1.2 Power spectrum: why corrections are slow-roll suppressed The standard power spectrum for single-field slow-roll inflation is Pζ(k)≃H2 8π2M2 Pl ,(138) 99 evaluated at horizon exit k = aH (with star denoting evaluation near exit). This leading result is a free (Gaussian) prediction: it is determined by the quadratic action for ζ and the Bunch–Davies initial condition. Interactions contribute corrections that are slow-roll suppressed and typically loop-suppressed. Now consider contextual time ordering as a protocol deformation of the interaction history: τ7→ ˜τ(τ) = τ+δ(τ), with |δ0|  1. Such a deformation modifies the in-in time-ordered exponentials by replacing HI ( τ )with HI(˜τ(τ)) and expanding HI(˜τ) = HI(τ) + δ(τ)∂τHI(τ) + O(δ2). The power spectrum receives contributions from interactions only at higher order (loop level or through subleading cubic couplings contracted appropriately), and therefore any effect of δ ( τ )enters in a doubly suppressed manner: •it is proportional to the small protocol deformation, • and it multiplies interaction terms already suppressed by slow-roll parameters in single-field inflation. A complementary way to see the suppression is the “time-shift” argument: to leading order, a small shift in the effective evaluation time of Hand induces δln Pζ(k)≈dln Pζ dN δN ≈(ns−1) δN, where N = ln a and ( ns− 1) = dln Pζ/d ln k is slow-roll suppressed. Thus, even if δN is not tiny, the induced power-spectrum modulation is suppressed by |ns− 1 |  1in slow-roll regimes. This is precisely the structural reason the power spectrum is not the optimal leading observable for contextual time-ordering defects. 9.1.3 Bispectrum: leading sensitivity to time ordering in the in-in formalism The bispectrum, by contrast, is intrinsically interaction-driven and therefore time-ordering sensitive at leading order. In-in master formula. For an operator O(τf)at late time τf, hO(τf)i=D¯ Texp iZτf −∞ dτ HI(τ)OI(τf)Texp −iZτf −∞ dτ HI(τ)E,(139) where T and ¯ T denote timeand anti-time-ordering. For O = ζk1ζk2ζk3 , the tree-level bispectrum arises at first order in HI: hζ3i=−2<iZ0 −∞ dτ hζ3(τf)HI(τ)i(tree level).(140) This is the crucial structural point: the bispectrum is directly proportional to a single time integral of a time-ordered interaction insertion. Consequently, any modification of time ordering or of the time argument at which the interaction is inserted enters at the same perturbative order. Contextual time-ordering defect enters at leading order. Implement the context deformation at the level of the interaction history: HI(τ)−→ HI(˜τ(τ)) = HI(τ) + δ(τ)∂τHI(τ) + O(δ2).(141) Then the leading bispectrum correction is ∆hζ3i=−2<iZdτ hζ3δ(τ)∂τHI(τ)i+O(δ2).(142) Integrating by parts (under standard convergence conditions imposed by the i prescription), one obtains the structurally universal form ∆hζ3i= 2 =Zdτ δ0(τ)hζ3HI(τ)i+O(δ2),(143) showing explicitly that it is the time-ordering deformation δ0 ( τ )that weights the standard bispectrum integrand. This is a direct, leading-order effect. 100 Consequence: bispectrum sensitivity exceeds power-spectrum sensitivity. Because HI is already the leading source of the bispectrum, inserting δ ( τ )does not introduce an additional slow-roll suppression beyond what is already present in the cubic action. In contrast, power-spectrum modifications require either subleading interaction corrections or evaluation-time shifts that carry an additional ( ns− 1) suppression. Therefore, within the same small protocol deformation regime, bispectrum signatures can be parametrically larger and more distinctive than power-spectrum modulations. 9.1.4 Physical intuition: why three-point functions “feel” ordering The intuition is simple. The power spectrum measures variance of fluctuations generated by linear dynamics; its leading contribution is set by the two-point function of free modes. The bispectrum measures non-Gaussian correlations generated by interactions; interactions require ordering information because they depend on when different mode amplitudes couple. Hence three-point functions are direct probes of the composition of quantum histories. Since contextual time ordering is precisely a statement about the non-absoluteness of that composition, the bispectrum is the most direct inflationary observable to target. 9.1.5 Summary The bispectrum is the right target because: 1. Power-spectrum corrections from contextual time ordering are slow-roll suppressed (and often further suppressed) because the leading power spectrum is free-field. 2. The bispectrum is intrinsically time-ordered and interaction-dominated; a contextual time-ordering deformation enters at leading order in the in-in expansion. 3. The resulting correction has a universal structure proportional to δ0 ( τ )times the standard bispectrum integrand, making it a clean diagnostic of contextual ordering defects. In the next subsection we specify the relevant cubic interactions (Maldacena action) and show how the defect produces concrete, computable bispectrum templates. B. The key identity 9.2.1 Statement of the identity This subsection derives the central technical identity behind the inflationary application of contextual time ordering. The identity isolates, in a model-independent manner, how a small clock-protocol deformation ˜τ=τ+δ(τ),|δ0(τ)|  1,(144) enters the in-in (Schwinger–Keldysh) formula for correlation functions. The main result is that, at leading order, the induced correction to the bispectrum can be written as the standard tree-level integrand weighted by δ0(τ): ∆B(k1, k2, k3)∼2=Zdτ δ0(τ)I0(τ;k1, k2, k3),(145) where I0 is the standard tree-level integrand of the bispectrum computed from the cubic interaction Hamiltonian. This identity is universal: it does not depend on the detailed form of the interaction, only on the structure of the in-in expansion and on the interpretation of ˜τ as an operational ordering parameter. We then show that for a slow-roll motivated clock modulation δ ( τ ), δ0 ( τ )contains a factor ∼ τ−1cos((ν/H) ln(−τ)), which produces log-oscillatory features in ln K, where K:= k1+k2+k3. 101 9.2.2 In-in formula at tree level and the standard bispectrum integrand Let ζ ( τ, x )denote the comoving curvature perturbation. For a late-time operator O ( τf ), the in-in formula is hO(τf)i=D¯ Texp iZτf −∞ dτ HI(τ)OI(τf)Texp −iZτf −∞ dτ HI(τ)E,(146) where HI ( τ )is the interaction Hamiltonian in the interaction picture. For the bispectrum we take O=ζk1ζk2ζk3, and expand to first order in HI: hζk1ζk2ζk3i=−2<iZτf −∞ dτ hζk1(τf)ζk2(τf)ζk3(τf)HI(τ)i.(147) After Wick contraction with the free mode functions, the integrand becomes a complex function of τ and the external momenta ki. It is convenient to write B0(k1, k2, k3) = =Zτf −∞ dτ I0(τ;k1, k2, k3),(148) where I0 is the conventional bispectrum integrand (a sum over terms involving products of mode functions and background factors). The precise form of I0 depends on the cubic interactions chosen (discussed in the next subsection), but its existence and its oscillatory structure in Kτ are universal. 9.2.3 Contextual ordering shift: HI(τ)→HI(τ+δ(τ)) In the contextual framework, the time argument in HI is not a formal coordinate label but a protocoldefined ordering parameter. A change of clock protocol changes the operational assignment of time labels to the interaction history. The minimal implementation is: HI(τ)−→ HI(˜τ(τ)) = HI(τ+δ(τ)).(149) We assume that δ(τ)is sufficiently smooth and small so that a Taylor expansion is valid: HI(τ+δ(τ)) = HI(τ) + δ(τ)∂τHI(τ) + O(δ2).(150) This induces a first-order correction to the bispectrum: ∆B(k1, k2, k3) = −2<iZdτ hζk1ζk2ζk3δ(τ)∂τHI(τ)i+O(δ2).(151) At tree level, the expectation value reduces to a complex integrand proportional to ∂τI0(τ;ki): ∆B(k1, k2, k3) = −2=Zdτ δ(τ)∂τI0(τ;k1, k2, k3) + O(δ2),(152) where we used <(iX) = −=(X). 9.2.4 Integration by parts and the universal δ0(τ)weighting Assume the standard i prescription for the early-time limit, which damps oscillatory integrals and ensures convergence. Then boundary terms vanish and we may integrate by parts: Zdτ δ(τ)∂τI0(τ) = hδ(τ)I0(τ)iτf −∞ −Zdτ δ0(τ)I0(τ). Under the standard assumptions that δ ( τ )is bounded and smooth in the integration domain and that the i prescription kills the −∞ boundary term, the boundary contributions vanish. Hence: ∆B(k1, k2, k3)=2=Zτf −∞ dτ δ0(τ)I0(τ;k1, k2, k3) + O(δ2).(153) This is the key identity announced in (145). It has three notable properties: 102 1. It is universal at tree level: only I0 is model-dependent, while the δ0 ( τ )weighting is purely kinematical. 2. It shows that the correction is controlled by the rate of change of the clock protocol, not by a constant offset; this parallels the Unruh/de Sitter defect analysis where constant offsets vanish. 3. It makes explicit that the bispectrum is the correct target: I0 is the leading interaction integrand, so the correction is of leading order in the in-in expansion. 9.2.5 Log-oscillatory features in ln K We now show how log-oscillations arise for a natural class of clock protocol deformations. From cosmic time modulation to conformal time. A clock protocol modulation that is sinusoidal in cosmic time ttakes the form δt(t) = εsin(νt),|εν|  1. In quasi-de Sitter expansion, conformal time obeys a(τ)≃ −1/(Hτ), hence t≃1 Hln a≃1 Hln−1 Hτ + const. Therefore a sinusoidal modulation in tcorresponds to a log-oscillatory modulation in τ: δ(τ)≡δτ(τ)∝εsinν Hln(−kτ),(154) where kabsorbs the additive constant phase. Differentiating yields δ0(τ)∝εν H 1 τcosν Hln(−kτ).(155) Thus the key identity (153) becomes an integral of the standard bispectrum integrand weighted by a factor proportional to τ−1cos((ν/H) ln(−τ)). Why this produces ln K oscillations. The standard tree-level integrand I0 ( τ ; ki )is oscillatory with phase e−iKτ (where K = k1 + k2 + k3 ) multiplied by a slowly varying envelope in τ determined by background factors and mode functions. Therefore the integral Zdτ 1 τcosν Hln(−τ)e−iKτ ×(envelope) is a mixed oscillatory integral with an additional logarithmic phase. Standard asymptotic methods (stationary phase / Mellin transform arguments) imply that such integrals yield contributions oscillatory in ln K: ∆B(k1, k2, k3)∝εsinν Hln K k+ϕS(k1, k2, k3),(156) where S is a smooth shape factor inherited from the standard interaction and ϕ is a phase depending on conventions and on the precise envelope. The essential feature is the log-periodic oscillation in ln K , which is the characteristic signature of a clock-protocol modulation tied to cosmological expansion. 9.2.6 Summary The key identity of contextual time ordering in inflationary correlators is: ∆B(k1, k2, k3)=2=Zdτ δ0(τ)I0(τ;k1, k2, k3) + O(δ2). It follows from the replacement HI ( τ ) →HI ( τ + δ ( τ )) and an integration-by-parts step under the standard i convergence prescription. For clock modulations natural in cosmology, δ0 ( τ )is log-oscillatory and produces log-periodic features in ln K . This identity provides a model-independent route from contextual gluing of time ordering to a concrete bispectrum signature. 103 C. “Not already ruled out” argument 9.3.1 What must be shown The contextual time-ordering deformation derived in Sections 9 A–9 B predicts a bispectrum correction of the universal form ∆B(k1, k2, k3)=2=Zdτ δ0(τ)I0(τ;k1, k2, k3) + O(δ2), with δ0 ( τ )set by a clock protocol and I0 the standard tree-level integrand. The question addressed here is not whether any oscillatory bispectrum has ever been constrained (many have), but whether the specific structural feature of our mechanism creates parameter space consistent with current constraints. The structural feature is: The contextual ordering defect can generate an observable bispectrum signal while leaving the power spectrum essentially featureless at leading order, because the defect couples to time ordering (interaction history) rather than to background dynamics. This subsection makes that statement quantitative using (i) the slow-roll scaling of power-spectrum sensitivity and (ii) the leading-order in-in sensitivity of the bispectrum. We then explain how this differs from many conventional feature/resonance mechanisms and why existing analyses do not exclude the full class of contextual-ordering signals. 9.3.2 Parametric separation: bispectrum versus power spectrum Power spectrum response to a small contextual time shift. At leading order, the power spectrum depends on Hand evaluated near horizon exit: Pζ(k)≃H2 8π2M2 Pl . A small contextual deformation that effectively shifts the operational “time of evaluation” by δNk produces δln Pζ(k)≈d dN ln PζδNk≈(ns−1) δNk,(157) where ns− 1 = dln Pζ/d ln k is slow-roll suppressed. Using Planck 2018 constraints on the scalar spectral index, ns is close to 1 with percent-level deviation [ 83 ]. Consequently, power-spectrum sensitivity to a small contextual time deformation is naturally suppressed by |ns−1|  1. Bispectrum response at leading order. By contrast, the bispectrum at tree level is itself interactiondominated, and the contextual time-ordering deformation enters at the same order: ∆B∝Zdτ δ0(τ)I0(τ;ki). There is no additional loop suppression and no additional ( ns− 1) suppression beyond the standard slow-roll factors already present in I0 . In particular, for the cosmologically natural log-oscillatory clock modulation (154) one has δ0(τ)∝εν H 1 τcosν Hln(−kτ),(158) so the bispectrum correction scales linearly with the small protocol amplitude ε and can carry an additional enhancement by ν/H through the δ0(τ)prefactor. A useful diagnostic ratio. Combining (157) and (158) yields the central parametric separation: bispectrum sensitivity power-spectrum sensitivity ∼ν/H |ns−1|,(159) up to shape-dependent numerical factors. Since |ns− 1 | is small, modest ν/H can yield a comparatively larger bispectrum signal while leaving the power spectrum close to featureless. This is the core reason the contextual-ordering signal is not automatically excluded by the absence of prominent oscillatory features in the two-point function. 104 9.3.3 Comparison with conventional feature/resonance mechanisms Many conventional mechanisms that generate oscillatory bispectra (e.g. oscillatory potentials, transient features, axion monodromy) also generate correlated features in the power spectrum because the oscillation is introduced at the level of background dynamics or of the mode equation itself. In such cases, constraints from the power spectrum are often highly restrictive, and bispectrum searches are frequently interpreted jointly with power-spectrum features. The contextual-ordering mechanism differs structurally: 1. the background dynamics (mode functions at quadratic order) is unchanged to leading order; 2. the new effect enters through a deformation of the time argument in the interaction Hamiltonian, i.e. through ordering/gluing rather than through the background; 3. therefore the two-point function is affected only through suppressed routes (slow-roll suppressed evaluation-time shifts or higher-order interaction corrections), while the three-point function is affected at leading interaction order. Hence the absence of a detected feature in the power spectrum does not preclude a bispectrum signal of the contextual-ordering type. 9.3.4 Consistency with current observational constraints Planck 2018 analyses place strong constraints on primordial non-Gaussianity for a broad class of bispectrum templates, including the standard local/equilateral/orthogonal shapes and additional families (feature and resonance-like shapes) [ 84 ]. Planck 2018 inflation analyses also report no compelling evidence for prominent oscillatory features in the power spectrum and constrain a range of parameterized feature models [83]. The contextual-ordering signal occupies a distinctive logical niche within these results: 1. It is compatible with a nearly featureless power spectrum because the leading two-point effect is suppressed by |ns−1|as in (157). 2. It predicts a bispectrum modulation with log-periodic phase in ln K (Section 9 B), which is not in oneto-one correspondence with a power-spectrum oscillation amplitude as in many background-feature models. 3. Its natural parameterization is in terms of protocol parameters ( ε, ν/H )and context/gluing constraints, rather than a specific oscillatory microphysical potential. This changes how one maps observational bounds to model parameters. Accordingly, the appropriate comparison to data is not a single existing template constraint, but the construction of an explicit bispectrum template family derived from (153) with δ0 ( τ )given by (158) , together with a controlled estimate of the associated power-spectrum modulation implied by (157) . This is precisely the program of the subsequent subsections: the mechanism yields a predictive bispectrum structure with a parametrically suppressed two-point counterpart. 9.3.5 Summary The contextual time-ordering mechanism can evade automatic exclusion by current power-spectrum constraints because its leading two-point imprint is slow-roll suppressed, while its leading three-point imprint appears at tree level in the in-in expansion. The key parametric separation is (159) . Current observational results strongly constrain many bispectrum templates [ 84 ] and show no evidence for large oscillatory power-spectrum features [ 83 ], but these facts do not eliminate the full class of ordering/gluinginduced bispectra described by (153) and (158) . The next step is therefore to compute explicit templates and parameter mappings for this class and to compare them to existing bispectrum constraints in the appropriate template space. 105 D. Programmatic extension: numerical bispectrum templates Sections 9 A–9 C establish the analytic mechanism by which contextual time ordering modifies inflationary correlators and predicts log-oscillatory features in ln Kat the level of the bispectrum, while leaving the leading two-point structure slow-roll suppressed. A full numerical template construction (equilateral and squeezed families, interpolation over the tetrahedral domain, and parameter inference against data) is a direct next step but is not required for the completed operational results of this paper. We therefore defer the explicit numerical implementation and the corresponding data-analysis workflow to the Outlook in Section 11 B, especially Subsubsection 11.2.1, where the planned program is stated in a form suitable for immediate implementation. 10. WHAT THIS FRAMEWORK CAN DO THAT OTHER QUANTUM-GRAVITY APPROACHES DO NOT A. Generic spacetime, observer-local computations 10.1.1 The problem: “generic spacetime” is where global strictifications fail A recurring limitation of many quantum-gravity formulations is that their most explicit calculational outputs are tied to special global structures: asymptotic boundaries (to define an S -matrix or boundary correlators), global timelike Killing flows (to define stationary energies and thermal states), or globally fixed subsystem factorizations (to define entanglement between sharply identified subsystems). In contrast, physically central situations are typically generic in the operational sense defined earlier: 1. No canonical asymptotic boundary time: slow-roll FLRW and finite causal diamonds do not supply a unique global time parameter that all observers can use to coordinatize comparisons. 2. No global stationarity: the relevant scale (e.g. H ( t )in cosmology) evolves; the natural time flow depends on the observer and epoch. 3. No canonical global factorization: the subsystem structure relevant for information bookkeeping is protocol-defined and may not strictify globally across contexts. In such settings, the minimal requirement for a universal quantum-gravity framework is not merely the existence of a formal nonperturbative definition, but the existence of operationally defined observables computable within finite causal domains and transportable across contexts. The present framework is designed precisely for this regime: it makes the operational context the primitive object, and it makes context comparison/gluing the carrier of quantum-gravitational content. The resulting computations do not rely on global strictifications. Instead they are built from local QFT input plus explicitly specified context data (clock protocols, switching windows, channel selection, causal accessibility) and coherence constraints. 10.1.2 Observer-local observables as primary objects In a generic spacetime, what is physically meaningful to an observer is not an abstract field value at a point, but a family of records produced by operational procedures. This is consistent with the general perspective on relational/partial observables in constrained systems: physical predictions are statements about correlations between readings of clocks and other degrees of freedom, rather than about coordinate-dependent fields [88, 89]. In our language, this is precisely the statement that: an “event” is a context, and an “observable” is a context-indexed functional of accessible records. Accordingly, the primitive observables used throughout this paper are of the following form: 1. Detector-response functionals FC (Ω) and their protocol differences ∆ FC for contexts C defined by a worldline, a switching window, and a clock readout. 2. Variance data V ( C ) = Var (∆ FC )across refined contexts, which yields the tomography measurement equation. 112 3. universality and fixed-shape behavior evidenced by scaling collapse of operational observables; 4. emergence of the graviton as an IR strictification representative of the Gaussian universality class in coherence space. These properties are explicitly realized in the slow-roll FLRW computations and are operationally testable through the tomography-to-scattering pipeline. D. Why this is not “just QFT in curved spacetime” 10.4.1 The concern and the correct point of comparison Because many computations in this paper use standard semiclassical ingredients (Wightman pullbacks, Unruh–DeWitt detectors, horizon KMS structure), it is natural to ask whether the framework is merely a reformulation of quantum field theory in curved spacetime. The correct answer requires a careful separation of roles: 1. QFT in curved spacetime supplies the local dynamical input: correlators, detector response functionals, KMS structure of restricted states, etc. [3, 4]. 2. The present framework supplies the global comparison/gluing structure: how distinct operational contexts are related when strict global identification is not assumed, and how the associated coherence defects run under refinement and control information bookkeeping, entropy flow, and scattering observables. Thus the present framework is not a replacement for semiclassical QFT; it is an additional structural layer that becomes essential precisely when global strictification assumptions fail and when one needs a theory of how local descriptions are glued coherently. 10.4.2 What semiclassical QFT does and does not do In semiclassical QFT, one fixes a classical spacetime background ( M, g )and studies quantum fields on it. The theory provides: •local field correlation functions and renormalized local expectation values; •operational detector response functionals for prescribed worldlines and switching windows; •horizon thermality (Unruh, de Sitter, Hawking) as KMS/modular properties of restricted states. These are indispensable and are used as local input throughout this paper. However, semiclassical QFT does not, by itself, provide a principle for: 1. global subsystem identification across contexts, especially when contexts correspond to different clocks, different causal patches, or different mode decompositions; 2. renormalization of comparison data under operational refinement (switching width, channel selection) and across physical scale changes (epoch evolution); 3. information bookkeeping across horizons when strict factorization assumptions are not justified (Page-curve problem). In practice, semiclassical reasoning often proceeds by imposing additional strictification assumptions (global time, strict tensor factorization of subsystems, canonical identifications of modes across epochs). The present framework isolates those assumptions and replaces them with coherent gluing data. 10.4.3 The genuinely new ingredient: non-strict gluing and contextual ordering as physical The new ingredient is the physical status of non-strict gluing and contextual ordering. Concretely: 113 (N1) Contexts as primitives. Events are not primitive manifold points; they are operational contexts C = ( AC, τC,≺C, κC )(Section 1 B). This already goes beyond standard QFT-in-curved-spacetime practice, which typically presupposes a globally defined manifold with a fixed time coordinate choice (even if no Killing time exists). (N2) Comparisons are correspondences, not identities. Semiclassical QFT calculations often compare different descriptions by assuming a strict identification of algebras or Hilbert factors (e.g. old radiation ⊗ new radiation). In our framework, comparisons are generally channels/correspondences, and composition is coherent rather than strict (Section 1 C). The residual mismatch around loops is a defect, and that defect has operational representatives (kernels/PSDs/MI witnesses). (N3) Ordering is meaningful but not absolute. Semiclassical QFT allows different coordinate time choices, but it typically treats time ordering as a global structure once a coordinate is chosen. Here time ordering is part of the operational context and may not admit a single global refinement across contexts (Section 1 D). This produces measurable effects (Unruh/de Sitter clock defects) without altering the underlying KMS thermality. (N4) Renormalization is of coherence. Standard semiclassical QFT does not define a renormalization program for context comparisons themselves. The categorical RG does: refinement in switching width T , channel Ω, causal diamond size, and epoch t yields a closed flow of defect statistics with universality and scaling collapse (Sections 2 and 7). This is a new renormalization concept that is not part of the standard semiclassical toolkit. 10.4.4 Measurable consequences that are not captured by “background QFT” alone The new structural ingredient has concrete measurable consequences: (M1) Clock-defect response in horizon settings. In Unruh and de Sitter, local thermality is preserved exactly (KMS structure of the restricted state), but protocol-defined time readouts produce a nonzero response defect R ( ν )in finite-time contexts. This effect is not a prediction of “QFT on a fixed background” alone, because it requires treating clock protocol changes as distinct contexts whose comparison need not strictify. Semiclassical QFT supplies the kernel; the framework supplies the interpretation and the gluing principle that makes the defect a physical object rather than a gauge artifact. (M2) Page turnover without boundary constructions. The Hawking/Page paradox arises when strict global factorization is imposed across emission steps. Semiclassical QFT does not, by itself, supply a principled modification of this global bookkeeping. The contextual gluing framework does: it replaces strict factorization by correspondences and coherence constraints, leading to dynamical Page turnover with mutual-information witness. This is a genuinely new structural input, not a local QFT correction. (M3) Tomography-to-scattering pipeline in generic FLRW. Semiclassical QFT provides local correlators and detector response. The operational pipeline developed here adds: •a measurement equation for defect PSD inference from context refinement data, •categorical RG closure and universality across epochs, •direct scattering/decoherence predictions in generic slow-roll FLRW. This end-to-end program does not exist as part of standard QFT-in-curved-spacetime because it requires treating coherence defects as renormalized objects and transporting them across contexts and scales. 10.4.5 Conceptual synthesis It is therefore accurate to say: The present framework uses semiclassical QFT as local input, but it extends it by quantizing and renormalizing the gluing of operational contexts, thereby producing global informationflow and scattering predictions that cannot be obtained from local QFT-in-curved-spacetime alone. In this sense, the framework is to semiclassical QFT as Wilsonian RG is to bare perturbation theory: it supplies the structural layer that controls which comparisons are meaningful, how they compose, and how their defects run under refinement. 114 10.4.6 Summary This is not “just QFT in curved spacetime” because: 1. the fundamental objects are contexts and comparison morphisms, not a globally strict spacetime with globally identified subsystems; 2. non-strict gluing and contextual ordering are treated as physical and yield operationally measurable defects even when local thermality is preserved; 3. coherence defects are renormalized objects under categorical RG, with closure and universality demonstrated in generic slow-roll FLRW; 4. the framework resolves global information bookkeeping (Page turnover) and produces end-to-end inference-to-prediction pipelines (defect PSD tomography → scattering) that are not supplied by local semiclassical QFT alone. 11. DISCUSSION, LIMITATIONS, AND EXTENSIONS A. Scope of approximations (clearly stated) This paper develops a framework and presents explicit computations in several settings. To ensure that the results are interpreted correctly, we list and analyze the approximations used throughout. The overarching principle is that every approximation is (i) stated explicitly, (ii) motivated operationally, and (iii) checkable by internal consistency tests (limit checks, stability under parameter variation, and cross-context comparisons). 11.1.1 Local-patch slow-roll approximation The slow-roll FLRW computations in Section 7 treat generic cosmology by working in local operational patches centered at epochs t . The underlying physical assumption is that within the temporal support of a measurement protocol (switching window of width T ), the Hubble scale and slow-roll parameters are approximately constant: H(t)≈H(t), (t)≈(t),for |t−t|.T. Quantitative validity condition. A sufficient condition for this approximation is that fractional change in Hacross the measurement window is small:  ∆H H∼ ˙ H HT=(t)H(t)T1. Equivalently, H(t)T(t)−1.(164) This condition was already stated conceptually in (94) ; here we record it as the explicit criterion for the local-patch approximation. Interpretation. The approximation does not assert exact de Sitter symmetry. It asserts that each operational context (defined by a finite-time protocol) probes the geometry only within a bounded temporal region in which slow-roll variation is negligible. This is precisely the physically meaningful notion of “local stationarity” available in a nonstationary cosmology. Roadmap beyond local patches. If one wishes to remove (164) , one must compute the relevant worldline correlators and protocol kernels using time-dependent H ( t )and  ( t )across the full protocol window. This is conceptually straightforward: one replaces the stationary kernel KT ( ω )by a two-time kernel KT ( ω ; s ) dependent on the central time s = ( τ + τ0 ) / 2, and then performs the s -integration with the appropriate time-dependent background factors. The categorical RG framework does not depend on stationarity; it depends only on the definition of contexts and comparison maps. The local-patch approximation is a calculational simplification, not a conceptual requirement. 115 11.1.2 Conformal coupling choice for clean kernels Several explicit horizon-kernel computations (especially in de Sitter and in the FLRW demonstration code) use a free-field Wightman pullback of the form W(u)∝sinh−2κ 2(u−i), with κ = a for Unruh and κ = H for de Sitter. This form is clean and numerically stable and is naturally realized for conformally coupled massless scalars (and in closely related cases). Why this choice is appropriate for the present goals. The purpose of these computations is not to claim a unique microscopic field content of nature. It is to demonstrate: 1. the existence of protocol-dependent defects while preserving KMS thermality, 2. the closed-form sideband structure under Gaussian switching, 3. and the categorical RG/tomography/scattering pipeline in generic spacetime contexts. For these structural goals, the conformal coupling choice is the simplest faithful representative: it has the correct horizon KMS analyticity and avoids IR subtleties. Roadmap to minimally coupled/light fields. Minimally coupled light scalars in de Sitter and quaside Sitter can exhibit infrared enhancements and secular behavior. Incorporating these effects modifies the long-time behavior of correlators and may change the low-frequency structure of inferred defect PSDs. The framework remains applicable: one replaces the kernel input W ( u )by the appropriate pullback for the chosen field content and state. The operational steps remain unchanged: W−→ KT(ω)−→ H(ν;C)−→ V(C)−→ Sδ(ν;C)−→ Ξ. Thus the conformal coupling choice is best viewed as the cleanest baseline for demonstrating the pipeline; the extension to minimally coupled/light fields is a technical refinement of the correlator input. 11.1.3 Gaussian switching: benefits, limitations, and alternative windows The Unruh, de Sitter, and several kernel-based computations use Gaussian switching χ(τ) = exp−τ2 2T2, for three reasons: 1. Analytic factorization: the ( s, u )change of variables factorizes exactly, enabling closed-form derivations of sideband formulas and the suppression factor e−(νT/2)2. 2. Controlled convergence: Gaussian windowing makes all integrals rapidly convergent and stable in numerical evaluation. 3. Protocol realism: finite-time switching is operationally unavoidable; Gaussian is a smooth representative that avoids artifacts due to sharp cutoffs. Limitations. Gaussian switching is not unique. Different laboratories implement different envelope shapes, and sharp switching can produce transient effects (switching noise) unrelated to the underlying KMS structure. Alternative windows. The framework accommodates any sufficiently regular window function. Replacing χchanges: •the windowed kernel definition KT(ω)(the u-weight changes), •the prefactor analogous to AT(ν)(the s-integral changes), •and the detailed shape of the defect curve as a function of ν. However, the structural features remain: sideband dependence on Ω ±ν/ 2persists, vanishing for constant offsets persists, and stationary-limit averaging persists when the window becomes long and symmetric. Gaussian switching is therefore a convenient but not essential choice. 116 11.1.4 Small defect expansion (first-order where applicable) Many analytic formulas in Sections 5 and 6 are derived to first order in a small protocol deformation parameter ε: ˜τ=τ+δ(τ), δ(τ) = εsin(ντ),|εν|  1. This implies: 1. the map τ7→ ˜τremains monotone and defines a consistent time readout; 2. Taylor expansions of HI(τ+δ(τ)) and of switching functions remain controlled; 3. first-order response defects are meaningful and dominate over higher orders. Stationary-limit consistency. A notable internal consistency check is that first-order defects vanish in the stationary (long-time) limit for symmetric modulations, as seen analytically via the factor e−(νT/2)2 and in numerics. This is required physically: a symmetric oscillatory perturbation should average out when observed over sufficiently long times. The small-defect expansion makes this check transparent. Beyond first order. Second-order corrections generate effects that survive time averaging in some settings (sideband power rather than sideband differences). Extending the analysis to O ( ε2 )is straightforward in principle: one retains the quadratic term in the expansion of the phase factor and the switching function. The structure then involves combinations of KT (Ω) and KT (Ω ±ν )rather than only Ω ±ν/ 2. The conceptual framework does not change; only the algebra becomes more involved. Defect PSD interpretation. In the tomography pipeline, the defect is treated statistically through Sδ ( ν )and therefore does not require a perturbative expansion in a single deterministic amplitude. The small-defect expansion is used primarily to derive analytic response filters and to motivate the linear measurement equation. Once Sδ is inferred, predictions for Ξcan incorporate the full variance content and do not require assuming determinism. 11.1.5 Summary The approximations used in this paper are: 1. Local-patch slow-roll approximation in FLRW (valid under (164) ); removable by using timedependent kernels. 2. Conformal coupling baseline for clean horizon kernels; extendable to minimally coupled/light fields by substituting the appropriate Wightman pullback. 3. Gaussian switching for analytic tractability and numerical stability; replaceable by other smooth windows with the same structural conclusions. 4. Small defect expansion where analytic closed forms are derived; extendable to higher orders and complemented by PSD-based statistical inference. These approximations are consistent with the operational scope of the framework: they simplify explicit computations while preserving the conceptual structure (contexts, comparisons, coherent defects, and categorical RG closure). B. What remains open / next targets This subsection records the principal extensions suggested by the present results. Each target is framed so that it is (i) operationally well-posed in the language of contexts and comparison morphisms, and (ii) computationally tractable by extending the methods already used in Sections 5–8 and Section 9. 117 11.2.1 Full bispectrum numerical evaluation from contextual time ordering This program is referenced in Section 9 D and is recorded here as the primary next target. Sections 9 A–9 B established the universal identity ∆B(k1, k2, k3)=2=Zdτ δ0(τ)I0(τ;k1, k2, k3) + O(δ2), which reduces the contextual-ordering correction to a weighted version of the standard tree-level bispectrum integrand. The remaining open task is to implement the complete numerical evaluation of ∆B: 1. include the full Maldacena cubic interaction basis (and the required field redefinitions) rather than a single dominant term; 2. compute ∆ B on a dense tetrahedral grid in ( k1, k2, k3 ), ensuring convergence under the i prescription and controlled treatment of oscillatory integrals; 3. project the resulting template onto standard bispectrum basis functions used in data analyses, and extract the effective amplitude parameters and log-frequency ν/H dependence. Two additional cross-checks will be essential: •Consistency with the two-point sector. The induced power-spectrum modulation implied by the same δ ( τ )must remain slow-roll suppressed as in (157) , and must be quantified alongside the bispectrum template. •Consistency across contexts. The inflationary δ ( τ )should be related, at least in scaling form, to the defect statistics inferred in slow-roll FLRW via categorical RG (Section 7), yielding internal constraints on the allowable bispectrum parameter region. A practical implementation route is to adopt established numerical techniques for oscillatory in-in integrals and bispectrum template construction, and then insert δ0(τ)as an additional weight function [86]. 11.2.2 Schwarzschild static observer kernels (beyond Unruh/de Sitter) The Unruh and de Sitter horizon computations use stationary pullback kernels with simple analytic structure. A natural next target is a static observer outside a Schwarzschild black hole, with the quantum field in a physically appropriate state (Unruh or Hartle–Hawking). The objective is to compute the analog of the clock-defect response curve using: 1. the exact (or controlled numerical) mode-sum Wightman function pulled back to a static worldline at fixed radius r; 2. Gaussian switching and a clock protocol deformation ˜τ=τ+δ(τ); 3. the same sideband formulas derived in Section 5 C, which apply whenever W ( τ, τ0 )depends predominantly on the time difference in the stationary regime. This extends the operational defect program to a physically central black-hole background without requiring an AdS boundary. The relevant technical input is the known structure of renormalized stress tensors and Wightman functions in black-hole spacetimes and the corresponding vacuum choices [ 100 – 102 ]. 11.2.3 Multi-observer overlap regions and context groupoid refinement So far, the explicit computations focus on a single observer worldline (Unruh, de Sitter) or a single observer family across epochs (FLRW). A deeper structural extension is to consider multiple observers whose causal domains partially overlap, and to use these overlaps to refine the context groupoid/bicategory. Operationally, this means: 1. specifying contexts CA and CB for two observer protocols (distinct clocks, switching, channels, and accessible algebras); 118 2. specifying overlap contexts CA∩B where both observers can compare records or jointly access observables; 3. defining comparison morphisms uA→A∩Band uB→A∩Bas correspondences/channels; 4. analyzing loop compositions in the overlap graph and extracting coherence defects as loop endomorphisms. The categorical content is that one passes from a line-like refinement family to a genuinely higher groupoid of contexts with nontrivial loops. This is the natural arena for the cohomological defect classes [ ω ] C discussed in Sections 1 C and 2 D. Technically, this connects to the theory of local nets and their functoriality across spacetime embeddings and observer restrictions [18, 99]. 11.2.4 Beyond PSD: non-Gaussian defect statistics and higher cumulants The tomography and scattering pipeline in Section 8 uses the defect PSD Sδ ( ν )as the primary statistical representative. This is appropriate in Gaussian or near-Gaussian regimes and in the IR strictification regime where second moments dominate (Sections 3 and 2 E). However, there are physical reasons to expect regimes where defect statistics are non-Gaussian: •strongly nontrivial loop defects and noncommuting context comparisons; •intermittent or rare-event protocol mismatches (heavy-tailed distributions); •multi-scale gluing where a single PSD cannot capture phase-sensitive structure. In such regimes, one must extend the statistical description to higher-order polyspectra and cumulants. Concretely: 1. replace the PSD-only model by a hierarchy of cumulant spectra (bispectrum, trispectrum of δτ); 2. replace the Gaussian phase diffusion formula by a cumulant expansion for heiδΦi; 3. extend the measurement equation to include higher-order statistics of response differences, not only variances. These extensions remain operationally well-defined: one measures higher moments of ∆ F across contexts and infers higher-order defect statistics. The mathematical framework is standard in higher-order spectral analysis and fits naturally into the “higher beta functions” viewpoint of Section 2 D, where the running object is no longer a scalar PSD but a higher structured statistical datum. 11.2.5 Summary of next targets The next technical targets that most directly extend the present results are: 1. full numerical evaluation of contextual-ordering bispectrum templates and parameter inference; 2. Schwarzschild static observer kernels and corresponding clock-defect curves in physically relevant black-hole states; 3. multi-observer overlap refinement and explicit computation of loop defects in a nontrivial context groupoid; 4. tomography and RG with non-Gaussian defect statistics, extending beyond PSDs to higher cumulant spectra. Each target preserves the operational principle of the framework: contexts, comparisons, and coherence data are primary; local QFT provides the kernel input; and categorical RG organizes refinement and universality. 119 12. CONCLUSION 12.1 Main conceptual result: quantum gravity as quantization of contextual gluing The central claim of this paper is that quantum gravity, at minimum, is the quantum theory of how local descriptions are glued. The fundamental objects are not “points” of a presumed global manifold, nor a globally defined metric perturbation field, but operational contexts C= (AC, τC,≺C, κC) together with physically implementable comparison morphisms between contexts. Each context admits an internally consistent local description (“locally real”), while global structure is a coherent network of relations rather than a single strict object (“globally coherent”). The quantization problem is therefore relocated: it is the quantization of the comparison and ordering structure that makes spacetime meaningful operationally. In this formulation, coherence defects—nontrivial loop self-correspondences and associator/cocycle data in the context comparison structure—are not auxiliary decorations. They are the operational carriers of quantum-gravitational content. They measure the obstruction to simultaneously strictifying all comparisons into a single global description. Crucially, these defects are not merely formal: they possess operational representatives (kernel deformations, defect power spectral densities, mutual-information witnesses) that can be computed, inferred, and transported across contexts. 12.2 Renormalizability reinterpreted: categorical RG and closure in coherence space Traditional perturbative non-renormalizability arises when one insists that a strictified metric perturbation field hµν ( x )is fundamental and must close under ultraviolet refinement. In the present framework the renormalization problem is reformulated as refinement of operational contexts (switching width T , bandwidth B , causal diamond scale ` , channel gap Ω, epoch t ). The running objects are coherence-defect data: H(ν;C), Sδ(ν;C),[ω]C,EC, rather than a local metric field. Renormalizability becomes a closure property: under refinement, defect representatives remain in the same structural class (kernel → kernel; PSD → PSD; cocycle → cocycle; correspondence→correspondence), without infinite proliferation of independent structures. The explicit slow-roll FLRW analysis demonstrates this closure in practice. Defect statistics inferred across epochs close within a finite universality class with a finite-dimensional parameter flow, and operational observables exhibit scaling collapse after RG rescaling. This establishes a concrete and testable sense in which quantum gravity is renormalizable in coherence space: refinement and scale change induce controlled flows of defect data and preserve predictivity. 12.3 Emergent gravitons: IR strictification of Gaussian coherence universality Within the present framework, gravitons are not postulated as fundamental quanta. They emerge as infrared collective excitations of near-strict coherence regimes. When coherence defects are small and their statistics approach a Gaussian fixed-shape universality class under categorical RG, second moments dominate and an effective strictification representative gµν + hµν becomes meaningful. The projected correlator Shuu ( ν )is then induced from the defect PSD Sδ ( ν )up to protocol transfer functions and calibration. In this regime, a linearized spin-2 description is recovered as an effective IR language, consistent with the standard weak-field gravitational-wave phenomenology. The key point is structural: the graviton description appears after closure and universality are established in coherence space; it is not required to exist as a UV-complete fundamental variable. 12.4 Worked examples: operational predictions in horizon and cosmological settings The paper substantiates the framework through explicit computations that isolate coherence defects while preserving standard local physics: 120 1. Hawking/Page without AdS/islands. Local Hawking pair creation is preserved, while strict global factorization is replaced by contextual gluing. In a three-party Gaussian purification model with radiation memory R , outgoing mode b , and partner c , the radiation entropy S ( Rn )exhibits a dynamical Page turnover. The turnover is witnessed operationally by mutual information I ( R : c ) and by the sign change of ∆ Srad , without imposing an external entropy clamp and without boundary constructions. 2. Unruh effect with incompatible clocks (exact analytics and numerics). Unruh thermality is preserved as a KMS property of the wedge-restricted state, while clock protocol deformation ˜τ = τ + δ ( τ )produces a finite-time response defect with closed-form sideband structure in Ω ±ν/ 2. Exact numerical evaluation of the full Wightman integral yields a controlled, protocol-dependent defect curve. 3. Cosmological horizon defect in de Sitter without dS/CFT. The same operational pipeline produces a de Sitter horizon defect curve using the Bunch–Davies pullback, preserving TdS = H/ (2 π ) KMS thermality while exhibiting coherent protocol-dependent response differences in finite-time contexts. 4. Generic slow-roll FLRW: defect inference, categorical RG, and scattering. In a setting without global stationarity or boundary strictification, defect PSDs are inferred from context refinement data across switching widths and channels, transported across epochs by a finitedimensional categorical RG flow, and used to predict observer-local scattering/decoherence via the phase diffusion exponent Ξ( Tint ). Scaling collapse provides operational evidence of fixed-shape universality. Taken together, these results provide a coherent chain: local semiclassical physics remains intact within each context, while global comparison and gluing structure yields new operational observables and new renormalization behavior. 12.5 Roadmap: cosmological non-Gaussianity and further observational targets The inflationary correlator analysis identifies the bispectrum as the most sensitive target for contextual ordering effects: power-spectrum imprints are slow-roll suppressed, while bispectrum corrections enter at leading order in the in-in expansion through the universal identity ∆B∼2=Zdτ δ0(τ)I0(τ;ki). This yields a concrete route to observational templates with log-oscillatory features in ln K and a parametrically enhanced bispectrum-to-power ratio. The planned numerical program is therefore: • compute explicit equilateral and squeezed templates from the Maldacena cubic action with contextual time-ordering weights; •perform parameter inference in terms of protocol variables (ε, ν/H); • enforce cross-consistency with defect PSDs inferred via categorical RG in slow-roll FLRW, thereby linking early-universe correlators to the same coherence universality class used for scattering predictions. Beyond inflationary non-Gaussianity, the next natural targets are Schwarzschild static observer kernels (black-hole horizon defects), multi-observer overlap refinements of the context groupoid, and non-Gaussian defect statistics beyond PSD descriptions. Each extension preserves the same operational principle: contexts, comparisons, and coherence data remain primary, and predictions are extracted from bounded functionals of inferred defect statistics. 12.6 Closing statement This work proposes and substantiates a shift of emphasis: quantum gravity is formulated as quantization and renormalization of contextual gluing. In this setting, renormalizability is realized as closure and 121 universality of coherence defects under operational refinement and physical scale evolution, and gravitons emerge as IR strictification representatives of Gaussian coherence universality classes. The worked examples demonstrate that this framework yields explicit operational predictions in horizon and cosmological settings where global strictifications and boundary-based toolkits are unavailable or non-canonical, and it provides a concrete roadmap to cosmological non-Gaussianity and further observational targets. APPENDICES Appendix A: Exact Unruh/de Sitter Gaussian switching derivations This appendix derives the closed-form first-order protocol-defect formulas used in Sections 5 and 6. The derivation is purely analytic and applies to any stationary pullback Wightman function W ( u )depending only on the proper-time difference u = τ−τ0 . The Unruh and de Sitter cases differ only by the explicit form of W(u): the accelerated pullback with parameter aand the de Sitter pullback with parameter H. We use the brace convention for starred indices and emphasize the two distinct operational meanings of a clock-protocol deformation: 1. Phase-readout deformation: the detector phase uses ˜τ while the switching schedule remains in τ. 2. Switching-protocol deformation: the switching schedule is specified in ˜τ and therefore changes when expressed in τ. Both contributions are needed for a complete operational comparison between two contexts. 1. Setup: detector response and Gaussian factorization Consider an Unruh–DeWitt detector with gap Ωand switching function χ ( τ )coupled linearly to a scalar field along a stationary worldline. The leading-order response functional is Fχ(Ω) = Z∞ −∞ dτ Z∞ −∞ dτ0χ(τ)χ(τ0)e−iΩ(τ−τ0)W(τ−τ0),(A1) where W(u)is the pullback Wightman function on the worldline. We take Gaussian switching χ(τ) = exp−τ2 2T2.(A2) Introduce center and difference variables s=τ+τ0 2, u =τ−τ0. Then τ=s+u 2, τ0=s−u 2, dτ dτ0=ds du, and the Gaussian factorizes: χs+u 2χs−u 2= exp−s2 T2exp−u2 4T2.(A3) Therefore the baseline response is F0(Ω) := Fχ(Ω)δ=0 =Zds e−s2/T 2Zdu e−u2/(4T2)e−iΩuW(u) = √π T KT(Ω),(A4) where we define the windowed kernel KT(ω) := Z∞ −∞ du exp−u2 4T2e−iωu W(u).(A5) 128 5. Conceptual link to contextual gluing and strict factorization The model above implements, in the simplest explicit form, the replacement of strict factorization by contextual gluing: 1. Hawking emission is local and represented by a pure Gaussian pair (b, c). 2. The radiation record R is operationally defined and updated by a storage map (a coarse-graining of microscopic outgoing degrees of freedom). 3. A nontrivial comparison/gluing channel couples R and c with a controlled schedule, encoding the fact that global identification of “new” and “old” subsystems need not strictify. 4. The Page turnover emerges dynamically when correlation transfer reduces the effective factorization of late radiation from early radiation, as witnessed by I(R:c)and by the sign change of ∆Srad. This appendix makes explicit that the turnover mechanism is not imposed by an external entropy bound; it is generated by a unitary Gaussian evolution on ( R, b, c )together with the operational update rule for the record. Appendix C: Defect tomography and parametric inference This appendix collects the technical details of the defect-statistics tomography pipeline used in Sections 8 and 7. The objective is to make the inference step fully explicit and reproducible: (i) the linear measurement equation linking variance data to the defect PSD, (ii) the parametric fit procedure and uncertainty estimation, and (iii) propagation of inferred parameters to scattering predictions. 1. Linear measurement equation We begin with the context-indexed measurement equation derived in Section 8 A. Let Ci denote the i -th context (indexed by switching width Ti , channel Ω i , and possibly epoch ti ). Let Hi ( ν )denote the corresponding defect filter, and let Sδ ( ν )be the defect PSD in the regime under study. The variance datum is Vi=Zdν 2π|Hi(ν)|2Sδ(ν).(C1) By construction |Hi(ν)|2≥0and Sδ(ν)≥0. a. Discretization Choose a frequency grid {νk}N k=1 with spacing ∆νkand approximate the integral by a quadrature: Vi≈ N X k=1 Wik Sk, Wik := ∆νk 2π|Hi(νk)|2, Sk:= Sδ(νk).(C2) This yields the linear system V≈WS, Sk≥0.(C3) The positivity constraint expresses the physical meaning of a PSD. b. Conditioning and context diversity The matrix W is typically ill-conditioned because the kernels |Hi|2 are smooth filters. Stable inference therefore requires either: 129 1. sufficient diversity in i(many switching widths and multiple channels), and/or 2. structural regularization or parametric restriction. In this paper the primary strategy is parametric inference in a universality family, because it aligns with categorical RG closure and yields finite-dimensional flows. 2. Parametric inference in a universality family a. PSD family We adopt a finite-parameter PSD family of the form Sδ(ν;θ) = A 1+(ν/νc)αexph−(ν/νhi)2i,(C4) with parameter vector θ = ( A, νc, α, νhi )and constraints A > 0, νc> 0, νhi > 0. In some multi-epoch demonstrations, one fixes α and the ratio νhi/νc for maximal stability; this produces a two-parameter family (A, νc). The predicted variance for context iis then Vi(θ) = Zdν 2π|Hi(ν)|2Sδ(ν;θ).(C5) b. Weighted least squares and MLE Assume the measured variances Vdata i have uncertainties σi (e.g. estimated from repeated trials or from a noise model). Under a Gaussian measurement-noise model, the maximum likelihood estimator coincides with weighted least squares: b θ= arg min θ∈Θχ2(θ), χ2(θ) := M X i=1 Vi(θ)−Vdata i σi2 .(C6) Here Θdenotes the admissible parameter domain. Implementation note. Because Vi ( θ )involves an integral over ν , one typically precomputes the discretized kernels Wik and evaluates Vi(θ)≈X k WikSδ(νk;θ) for fast likelihood evaluation. Multi-start searches and local refinement (or gradient-based methods when available) are standard because χ2(θ)can have mild parameter degeneracies. c. Uncertainty estimation A standard uncertainty estimate for b θis obtained from the Hessian of χ2at the minimum: Hab := ∂2χ2 ∂θa∂θbθ=bθ .(C7) If the Gaussian approximation is valid, the covariance matrix is Cov(b θ)≈2H−1,(C8) possibly scaled by the reduced chi-square factor when χ2/dof 6 = 1. The 1 σ uncertainties are then σθa=√Covaa. An alternative robust method is bootstrap resampling: generate synthetic data sets V(b) i by resampling residuals or by sampling Vi from the noise model, refit θ for each bootstrap sample, and use the empirical distribution to estimate uncertainties. The parametric inference results reported in the main text are consistent with either approach. 130 3. Propagation to the scattering exponent Once b θ is obtained, one predicts scattering/decoherence observables through bounded functionals of the inferred PSD. a. Phase diffusion exponent The primary scattering observable is the phase diffusion exponent Ξ(Tint;θ) = 1 2ω2 0Zdν 2πSδ(ν;θ) sinc2 νTint 2,(C9) and the derived visibility is V(Tint) = e−Ξ(Tint). b. Uncertainty propagation Because Ξis a smooth functional of θ in the parametric family, a first-order uncertainty propagation gives VarΞ(Tint)≈ ∇θΞ(Tint;b θ)TCov(b θ)∇θΞ(Tint;b θ),(C10) where ∇θ Ξdenotes the gradient with respect to parameters. In practice, these gradients may be computed by finite differences or by differentiating the integral representation (C9) under the integral sign. Bootstrap uncertainty propagation is equally straightforward: for each bootstrap parameter sample b θ(b), compute Ξ(b)(Tint)and extract confidence intervals pointwise in Tint. c. Stability and boundedness A crucial structural advantage of Ξis that it is a bounded linear functional of Sδ (Section 8 C). Therefore: 1. small errors in Sδdo not lead to uncontrolled errors in Ξ; 2. the mapping from inferred defect statistics to scattering predictions is stable even when the inverse problem is moderately ill-conditioned. This is why Ξis the preferred primary scattering observable in the operational pipeline. 4. Summary Defect tomography in this paper is an inverse problem built on the linear measurement equation (C1) . Inference is performed either nonparametrically (NNLS + smoothness) or, most effectively for categorical RG, parametrically in a universality family (C4) . Parameter uncertainties are obtained from Hessian/Fisher approximations or bootstrap methods, and scattering/decoherence predictions are propagated via the phase diffusion exponent (C9) . The entire pipeline is operationally grounded: kernels are computed from context protocols and local correlators, defects are inferred as PSD statistics, and predictions are expressed as bounded functionals that remain meaningful in generic spacetimes. Appendix D: Categorical RG definitions (minimal) This appendix collects a minimal set of definitions that formalize the categorical renormalization group (RG) used throughout the paper. The goal is not to develop general higher-category theory, but to make precise the objects and maps that appear in the operational computations: contexts, refinement maps, closure, and beta functions for the principal defect representatives (kernels, PSDs, channels, cocycles). These definitions provide a compact reference for Sections 2, 7, and 8. 131 1. Contexts and refinement maps a. Contexts A context Cis a tuple C= (AC, τC,≺C, κC), where AC is the accessible observable algebra, τC a time-readout protocol, ≺C the operational ordering relation meaningful within the context, and κC calibration/protocol data (switching window, channel selection, normalization conventions). This tuple was introduced in (3) ; we record it here for completeness. b. Refinement relation and coarse-graining maps We write CC0and say that C0refines Cif: 1. both contexts refer to the same physical system/state preparation class (same physics); 2. C0 has strictly greater operational resolution along at least one axis (e.g. smaller T , larger bandwidth, richer channel set, finer causal diamond, different epoch scale); 3. there exists a physically implementable coarse-graining map (forgetting the extra resolution) from C0to C. Refinement morphism. A refinement is represented by a coarse-graining morphism rC0→C:C0−→ C, (D1) which acts on algebras as a channel (or conditional expectation) AC0→ AC and carries protocol data (τC0, κC0)to (τC, κC)by forgetting resolution. Refinement category. Let Ctxref be the category whose objects are contexts and whose morphisms are refinement maps rC0→C. Composition is operational composition of coarse-grainings: rC00 →C=rC0→C◦rC00 →C0. In many applications (including the multiT and multi-epoch FLRW settings) the system of contexts forms a filtered structure: for any C1, C2 there exists a common refinement C3 such that C1C3 and C2C3. 2. Closure under refinement a. Defect representatives as an assignment on contexts Let Str denote a category of structured defect representatives, such as: •kernels H(ν;C)or KT(ω), •PSDs Sδ(ν;C), •channels/correspondences EC, •cocycle representatives ωCor classes [ω]C. A defect assignment is a map D:Ctxop ref →Str, viewed as a presheaf/pseudofunctor: refinement CC0 corresponds to a natural pullback of defect data from C to C0 or, equivalently, to a pushforward under coarse-graining in the opposite direction. The details depend on the chosen representative. 132 b. Closure under refinement (minimal definition) Fix a structural class S ⊂ Str (kernels, PSDs, cocycles, channels). We say defect data are closed under refinement in class Sif: 1. for every context Cin the refinement family, D(C)∈ S; 2. for every refinement morphism rC0→C, there exists a well-defined refinement action RC→C0:D(C)−→ D(C0)(D2) such that D(C0)∈ S (type stability); 3. the refinement action is compatible with composition up to the intrinsic coherence of the chosen representative. In practice, closure is often strengthened to finite-parameter closure: D ( C )lies in a finite-dimensional family S(θ)stable under refinement, so that renormalization reduces to a finite parameter flow θ(C). 3. Beta functions for higher-structured data Let ` denote a refinement scale parameter (which may be T , a spatial scale, or an epoch scale such as H(t)), and write C(`)for a one-parameter family of contexts. a. Functional beta: kernels and PSDs If the running object is a function F ( ν ; ` )(e.g. a PSD or response kernel), define the functional beta by βF(ν;`) := ∂ ∂ln `F(ν;`).(D3) For a PSD, F = Sδ . For a response filter, F = H or |H|2 . If F lies in a finite-parameter family F ( ν ; θ ( ` )), then βFfactors through the finite beta functions βθa=dθa/d ln `by chain rule. b. Channel beta: generators of CP semigroups If the running object is a family of channels E` forming a semigroup under refinement (coarse-graining composition), define the generator (channel beta) by ∂ ∂ln `E`=L`◦E`,(D4) where L` is a (scale-dependent) CP generator. In stationary regimes with sufficient continuity assumptions, L` reduces to a Lindblad-type generator; in general operator-algebraic settings it is an appropriate CP generator on A. c. Cohomology beta: cocycle classes modulo coboundaries Let ω` be a cocycle representative of the coherence defect at scale ` on the comparison groupoid/bicategory at that refinement. Since ω` is defined only up to coboundary, the gauge-invariant running is defined in cohomology: β[ω](`) := ∂ ∂ln `ω`,(D5) where [ · ]denotes the cohomology class. Equivalently, one may choose a gauge slice and define a covariant derivative D/D ln ` , then take its class. This definition expresses that the running of coherence defects is meaningful only modulo changes of representatives. 133 d. Compatibility between beta notions The three beta notions above are related as follows: •kernels and PSDs are operational projections of the underlying defect class; • channel flow provides the dynamical mechanism for how comparisons/coarse-grainings act on accessible algebras; • cohomology beta captures the gauge-invariant obstruction to strictification that underlies these operational representatives. In near-strict Gaussian regimes, all three reduce to finite-dimensional beta functions for a small set of parameters, as explicitly demonstrated in the multi-epoch FLRW analysis. 4. Summary The categorical RG used in this paper consists of: 1. a refinement category Ctxref of operational contexts and coarse-graining maps; 2. a defect assignment D (pseudofunctorial in general) taking values in structured representatives (kernels, PSDs, channels, cocycle classes); 3. a closure principle asserting stability of the representative class under refinement, often in a finite-parameter universality family; 4. beta functions generalized to higher-structured running data: functional beta for PSDs/kernels, generator beta for channels, and cohomology beta for cocycle classes. Appendix E: Emergent graviton limit (sketch) This appendix sketches, at a technical but intentionally non-exhaustive level, how the graviton description emerges as an infrared strictification representative of coherence/defect statistics. The main text already developed the operational mapping from defect PSDs to projected metric correlators (Section 3 B) and explained why this avoids traditional non-renormalizability (Section 3 C). Here we organize that material into a concise “limit construction” consisting of two steps: 1. pushforward of defect PSDs to effective metric correlators along worldlines and, when appropriate, to tensor correlators; 2. emergence of a Gaussian fixed point for defect statistics under coherence RG, which underlies the linearized (graviton-like) IR description. 1. Defect PSD →projected metric correlator a. Operational starting point: clock mismatch statistics The primitive stochastic object is the context-comparison clock mismatch δτ ( t ), defined operationally (Sections 8 A–8 C). In a stationary regime, it is characterized by its PSD: hδτ(ν)δτ(ν0)i= (2π)δ(ν+ν0)Sδ(ν). This Sδ(ν)is inferred from refinement data and is the renormalized object in coherence space. 134 b. Strictification representative: linearized proper-time relation In a near-strict regime one assumes there exists an effective classical background metric g(0) and a small perturbation hµν such that operational proper-time perturbations are represented to leading order by δτ ≈ −1 2Zdτ huu(τ), huu := hµν uµuν, where uµ is the four-velocity of the observer worldline and we have chosen the worldline parameter to be background proper time (Section 3 B). This is a strictification representative: it is not assumed globally, but it is meaningful when defects are small. c. Transfer-function relation between PSDs Because δτ is a linear functional of huu in the linearized regime, their second moments are related by a transfer function determined by the measurement protocol (windowing). For a measurement window w(t), one has the generic spectral relation Sδ(ν) = |Kw(ν)|2Shuu (ν), where Kw ( ν )is computable from the protocol (Section 3 B). In the simplest boxcar integration over a time T , one obtains a factor proportional to sin ( νT/ 2) /ν , hence the characteristic “integration suppresses high frequencies” behavior. The key structural point is that the strictification map is pushforward: given Sδ and a chosen protocol transfer function, one can define an effective Shuu along the worldline. This is sufficient for a wide class of operational observables (timing jitter, phase diffusion, interferometric decoherence) without requiring a full tensor correlator. d. Reconstruction toward tensor correlators (optional) To construct an effective tensor correlator hhµνhρσi , one needs additional projections (multiple worldlines, multiple directions, or additional operational probes). In near-isotropic regimes one may adopt a factorized ansatz hhµν(x)hρσ(y)i ≈ Pµνρσ C(x, y), where P is a spin-2 projector (in a chosen gauge) and C is a scalar correlator. The projected correlators Shuu then determine C up to known contractions. This reconstruction is not used as a fundamental assumption in this paper; it is a strictification step relevant only in the near-coherent regime and primarily for comparison with conventional gravitational-wave language. 2. Gaussian IR fixed point and emergent linearization a. Coherence RG and the approach to Gaussianity The categorical RG renormalizes defect data under operational refinement (Sections 2 and 2 E). In many physical systems, coarse-graining drives complicated microscopic statistics toward Gaussian fixed points by central-limit mechanisms: sums of many weakly correlated contributions become approximately Gaussian at large scales. In the present framework, the analogous statement is: In near-strict regimes and under coarse-graining of contexts, higher cumulants of defect statistics become irrelevant and defect fluctuations are captured by second moments (PSDs). Operationally, this means that the PSD description becomes sufficient for predicting a wide class of observables, and the phase diffusion exponent Ξprovides a particularly clean probe (Section 8 C). 135 b. Fixed-shape universality and scaling collapse The FLRW five-epoch analysis explicitly demonstrated fixed-shape behavior of defect statistics and scaling collapse of scattering observables under appropriate rescaling (Section 7 C). In the language of RG, scaling collapse is evidence that the system lies in a universality class governed by a stable fixed-shape representative. In the emergent graviton regime, this fixed point is effectively Gaussian: second moments dominate and the induced Shuu suffices for IR predictions. The graviton appears as the linear collective excitation corresponding to this Gaussian universality class, in exact analogy with the emergence of phonons as Gaussian-normal-mode excitations of a near-equilibrium phase. c. Why this yields a spin-2 IR language Once an effective metric perturbation representative exists, the leading variation of invariant interval (proper time and radar distance) is linear in a symmetric rank-2 tensor hµν . The spin-2 nature of the graviton is therefore kinematic: it is the unique linear representation compatible with the operational reconstruction of geometry in a Lorentzian setting. The coherence framework does not impose spin-2 quanta as fundamental; it explains how the spin-2 linearized language becomes appropriate in the IR after closure and Gaussian universality are established. 3. 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