Clock-Comparison Holonomy: Emergent Geometry, Gravitons, and Non-Geometric Phases from Operational Coherence Defects
Abstract
This work develops an operational framework in which spacetime geometry is not assumed as a fundamental background structure, but instead emerges from the coherence properties of real physical clocks and their mutual comparisons. Rather than starting from a metric and defining clocks as probes of that metric, the approach reverses the logic: clocks are taken as primary physical systems, and geometry is inferred—when possible—from the consistency of clock comparisons. In practice, clocks are not idealized pointlike devices. They are physical systems built from different observables (atomic transitions, electromagnetic cavity modes, quantum circuits), interrogated using different protocols, and interpreted using different estimators and feedback loops. As a result, clock comparisons are not trivial identities but structured inference procedures. This work formalizes clock comparisons as directed transport channels acting on time or frequency readout series. These channels can be composed, and their failure to compose consistently around loops defines an experimentally measurable quantity: clock-comparison holonomy. The central insight is that local clock mismatches are not physically decisive. Any single pair of clocks can always be recalibrated locally by adjusting offsets, scales, or estimator conventions. What carries invariant physical information is loop non-closure: transporting time information around a closed loop of different clocks and returning to the starting clock does not, in general, reproduce the original readout. This loop residual is a gauge-invariant observable, directly analogous to holonomy in differential geometry. The paper introduces the notion of strictification: a compression of comparison data into a smooth spacetime representative, such as a manifold equipped with geometric fields. Strictification is not guaranteed to exist. When it does, it is non-unique, and this non-uniqueness corresponds to a gauge freedom analogous to diffeomorphism invariance. Within this framework, a precise criterion is established: a geometric spacetime description exists if and only if loop defects admit a local, infinitesimal generator in the limit of small loops. Operationally, this means that shrinking the physical support of clock comparisons causes loop non-closure to shrink in a controlled, local manner. When this criterion is satisfied and clocks couple universally, curvature emerges as the unique local encoding of physical defects. This is not assumed but derived: once smoothness, locality, and gauge freedom are imposed, curvature and its derivatives are the only local invariants that survive strictification. Higher-curvature terms then appear naturally as coarse-grained footprints of residual coherence defects, in direct analogy with effective field theory. A key result of the work is that, in the geometric phase, defect statistics generically flow toward a Gaussian infrared fixed point under coarse graining. In this regime, the dominant invariant information is carried by two-point correlations of loop residuals. Universal clock response implies that these fluctuations couple as a symmetric rank-two field in the strictified description. The associated gauge redundancy is exactly the linearized diffeomorphism freedom familiar from gravity. Together with locality, this fixes the quadratic dynamics uniquely to the linearized gravitational theory. In this precise operational sense, gravitons emerge as an infrared universality class of clock-comparison defects, rather than as fundamental quantized excitations of a pre-existing metric. Beyond the leading geometric description, three systematic families of corrections are identified. First, local higher-curvature terms arise as the derivative-expansion footprint of coarse-grained coherence curvature. Second, nonlocal memory terms appear whenever comparison kernels retain finite or long temporal memory, reflecting the fact that clock comparisons are fundamentally transport processes rather than instantaneous relations. Third, stochastic corrections arise naturally: loop residuals are random processes with measurable power spectra, and in the strictified description they act as noise sources for metric fluctuations, leading to Einstein–Langevin–type stochastic dynamics. Importantly, these effects are controlled by coherence, protocol sensitivity, and integration time, not by Planck-scale suppression. The framework also identifies genuinely non-geometric phases of spacetime. In memory-dominated regimes, comparison kernels exhibit long-tailed temporal behavior, leading to non-analytic low-frequency spectra in loop residuals. Such spectra cannot be generated by any local differential operator and therefore obstruct any local geometric description. In these phases, loop holonomies remain well-defined and measurable, but they cannot be encoded by curvature or by any local geometric field. The work further shows that in these regimes higher coherence conditions fail: triangle defects do not glue consistently on tetrahedra, corresponding to a breakdown of higher Bianchi or pentagon identities. This provides a sharp, testable definition of non-geometric spacetime phases. A concrete experimental program is proposed based on a three-clock loop. The clocks are deliberately chosen to be operationally distinct: an optical lattice atomic clock with discrete servo output, a cavity-stabilized laser with continuous phase readout, and a programmable quantum circuit clock based on a flux field coupled to a qubit with tunable switching. Each clock defines a different comparison kernel, and together they form a minimal loop capable of exhibiting nontrivial holonomy. A complete analysis pipeline is specified, including time-series alignment, channel fitting with causal kernels, construction of triangle and tetrahedron residuals, spectral analysis, and a suite of null controls designed to rule out trivial systematics. Crucially, the framework predicts that observable signatures of emergent gravity and of non-geometric behavior need not be Planckian. Precision clock networks are sensitive to accumulated phase and frequency fluctuations over long times, and loop residual statistics can be amplified by protocol design and integration. This opens a route to probing foundational aspects of spacetime using tabletop experiments rather than extreme energies. Overall, the work provides a unified operational perspective on geometry, gravity, and their breakdown. Geometry appears as a phase of coherent clock comparisons; gravitons emerge as an infrared universality class; and non-geometric phases arise naturally when memory and contextuality obstruct local strictification. The framework is mathematically precise, experimentally testable, and designed to connect foundational questions in spacetime physics with modern precision metrology and quantum technologies.
Full text
Clock-Comparison Holonomy: Emergent Geometry, Gravitons, and Non-Geometric Phases from Operational Coherence Defects Andrei T. Patrascu FAST Foundation, Destin FL, 32541, USA email: andrei.patr[email protected] We develop an operational framework in which spacetime geometry is not assumed a priori but emerges from the coherence properties of clock comparisons. Each clock, defined by its physical observable, interrogation protocol, and estimator, induces a comparison channel to other clocks. These channels form a directed transport structure whose composition need not be strictly transitive. We show that the resulting loop non-closure—an experimentally measurable clock-comparison holonomy—provides a gauge-invariant notion of coherence defect. We introduce strictification as the projection from comparison data to a smooth spacetime representative, with strictification gauge corresponding to diffeomorphism freedom. Local clock mismatches are shown to be removable by gauge choice, while nontrivial loop holonomies define physical defects. We prove that a geometric spacetime description exists if and only if loop defects admit a small-loop area law, in which case spacetime curvature emerges as the unique universal low-energy encoding of coherence defects. Under mild coarse-graining assumptions, defect correlators flow to a Gaussian infrared fixed point, yielding an emergent massless spin-2 field whose quadratic dynamics coincide with the Fierz–Pauli theory. Higher-curvature, nonlocal, and stochastic Einstein– Langevin corrections arise naturally as coarse-grained footprints of defect statistics, without requiring Planck-scale suppression. When comparison kernels exhibit long-memory behavior, the framework predicts non-geometric phases of spacetime in which loop holonomies are physical but cannot be represented by any local curvature density. These phases are characterized by fractional low-frequency spectra, path-ordering sensitivity, and violations of higher (tetrahedral) coherence identities, obstructing any local geometric strictification. We further classify geometric but non-universal regimes corresponding to torsion, non-metricity, and multi-metric structures. Finally, we propose concrete clock-loop experiments—combining optical atomic clocks, ultrastable cavity oscillators, and programmable circuit-QED scalar clocks—that directly measure comparison holonomy, defect power spectra, and higher-order coherence. The framework provides a unified operational origin for geometry, gravitons, and their breakdown, and predicts experimentally accessible transitions between geometric and non-geometric spacetime phases at tabletop scales. 1. INTRODUCTION A. From assumed geometry to inferred geometry In the standard presentation of relativity, one begins by postulating a spacetime manifold equipped with a Lorentzian metric gµν and then defines ideal clocks as physical systems whose elapsed time coincides with the metric proper time along a worldline. In this view, geometry is primary and clocks are secondary probes. This stance is mathematically elegant and operationally successful at macroscopic scales, yet it hides a logically nontrivial step: in practice, time is defined by clocks, and different clocks are physical systems built from different observables, with different protocols, different estimators, and different couplings to their environments. If geometry is to be taken as an operational summary of what clocks and rods do, it is natural to ask for a framework in which geometry is inferred from clock comparison data rather than assumed a priori. The present work develops such a framework. The central idea is simple but, we will argue, structurally unavoidable: Clocks define time only through comparisons. The consistent gluing of comparisons is what it means for a global geometric notion of time (and hence spacetime) to exist. This shifts the primitive datum from a metric to a comparison structure on operational contexts. The philosophical slogan is “geometry as a phase”; the technical content is that global geometric structure is equivalent to a certain strictifiability of a higher transport system generated by clock comparisons. This viewpoint resonates with two independent bodies of established knowledge. First, in general relativity, the geometric invariant content of a connection is encoded by holonomy and curvature: local coordinate choices can trivialize connection coefficients at a point, but not loop holonomies, whose infinitesimal limit is curvature [ 1 ]. Second, in quantum physics and precision metrology, measurement outcomes depend not only on the system Hamiltonian but on protocols and estimators; their statistics are naturally expressed via correlation functions, spectra and filters [ 3 , 4 ]. Our framework combines these
2 facts: we treat clock comparisons as transport maps with protocol-dependent kernels, and we identify the loop non-closure of these transports as the operational analogue of curvature. B. Operational stance: clocks, comparisons, and gluing Fix a class of operational “clock contexts” C , each context including: (i) a physical system (atomic transition, cavity mode, qubit, etc.), (ii) an observable used as the clock readout, (iii) an interrogation/switching protocol (windowing, Ramsey/Rabi sequence, continuous lock, etc.), (iv) an estimator producing a time or frequency record from raw data. Rather than treating a clock as “an ideal parameter along a worldline,” we treat it as a channel from physical evolution + protocol to a classical readout record. In metrology practice, the most robust common currency is a fractional frequency (or phase) time series. Thus, to each context C we associate a space XC of admissible readout signals xC ( t )(e.g. fractional frequency yC ( t ), phase φC ( t ), or a discrete-time cycle record). A comparison from C to C0 is then a directed map EC→C0:XC−→ XC0,(1) interpreted operationally as: given a readout produced in context C during a given run, EC→C0 outputs the best-predicted readout in context C0 for the same run, under a specified calibration/estimation procedure. The fact that real comparisons are finite-protocol and estimator-dependent is captured by representing EC→C0as an affine map plus a causal kernel plus a residual: (EC→C0x)(t) = αCC0x(t) + βCC0+Zt −∞ ds KCC0(t, s)x(s) + rCC0(t).(2) Here αCC0 and βCC0 encode calibration scale/offset, KCC0 encodes protocol-induced memory/transport, and rCC0 is the irreducible residual (in practice, the object whose spectrum one measures after fitting). We emphasize that (2) is not a restriction to linear physics: it is the leading operational representation in a weak-defect/near-strict regime and can be systematically extended (e.g. via nonlinear Volterra series) when higher cumulants are relevant. For the conceptual structure of geometry emergence, the linear-kernel regime already contains the key obstructions and invariants. C. What is new: transport, holonomy, phases, and the IR graviton sector (i) Comparison channels as transport. The maps EC→C0compose as ordinary functions: EC0→C00 ◦EC→C0:XC→XC00 .(3) This composition is the operational analogue of parallel transport: it tells you how a readout in C is translated into a prediction in C00 by passing through an intermediate context C0 . In the kernel representation (2) , composition induces convolution of kernels and therefore naturally generates path dependence. (ii) Holonomy as an observable defect. Given a loop of contexts γ:C1→C2→ ··· → Cn→C1,(4) the holonomy is the endomorphism of XC1defined by Ωγ:= ECn→C1◦ECn−1→Cn◦···◦EC1→C2.(5) Operationally, Ω γ is measured by transporting a readout around the loop and comparing it to the original readout in C1 . If all comparisons were strictly glueable into a single global time standard, one would have Ω γ = Id for every loop. In reality, we expect Ω γ6 = Id in general, and the associated loop residual time series (e.g. in phase or fractional frequency) is the primary observable of our framework. A key point—and one we will use repeatedly—is that local redefinitions of the strictified representative act by conjugation on loop holonomy, so the presence or absence of nontrivial holonomy is gauge-invariant.
3 Concretely, suppose one allows a per-context redefinition (a “strictification gauge” choice) represented by invertible maps GC:XC→XC, and define transformed comparison maps by E0 C→C0:= GC0◦EC→C0◦G−1 C.(6) Then the loop holonomy transforms as Ω0 γ=GC1◦Ωγ◦G−1 C1.(7) Therefore, whether Ω γ is the identity (or, more generally, whether it is trivializable for all loops) is a gauge-invariant statement: gauge choices may change representatives but cannot eliminate a genuine loop obstruction. This is the operational analogue of the fact that curvature (as infinitesimal holonomy) is invariant under gauge transformations, while connection coefficients are not. (iii) Geometric versus non-geometric phases. The next step is to clarify what it means for a smooth spacetime geometry to exist in this framework. We will define strictification as the existence of a smooth representative (a manifold M with fields on it) such that the family of comparison channels can be represented as transports induced by local geometric data. In the geometric phase, loop holonomies admit a small-loop expansion with an area law; in that regime the infinitesimal generator is a curvature tensor, and curvature (and its covariant derivatives) is the unique local gauge-invariant encoding of defects in the universal-coupling sector. By contrast, in a non-geometric phase, loop holonomies are physical and unremovable (nontrivial cocycles), yet do not admit a local curvature density. A particularly important and experimentally natural subclass is the memory-dominated phase, in which comparison kernels have long tails, producing nonanalytic infrared spectra and obstructing any local derivative expansion. Such phases are not pathologies; they are the operational analogue of regimes in many-body physics where no local hydrodynamic field exists even though the microscopic dynamics is well-defined. (iv) The IR graviton sector as a Gaussian fixed point. When strictification is possible and defects are weak, the fundamental correlator is the two-point function of the loop residual (or pairwise residual) δ ( t ), typically described by its power spectral density. The combination of protocol windowing and coarse graining leads naturally to an infrared Gaussian fixed point: higher cumulants are suppressed under aggregation, and the defect sector becomes effectively characterized by a two-point function. In that regime, the strictified representative of the defect can be encoded by a symmetric rank-2 field hµν (because universal elapsed-time response is quadratic in four-velocity), with a gauge redundancy corresponding to strictification freedom. Under locality and gauge invariance, the unique quadratic dynamics is the Fierz–Pauli theory [ 2 ], i.e. the linearized graviton sector. This emergence of spin-2 is not imposed as an axiom but follows from the combination of (a) universality of clock response in the geometric phase and (b) gauge redundancy of strictification. We stress that this is conceptually distinct from the conventional approach “quantize the metric.” Rather, we treat clock comparison defects as primary and recover the graviton as an infrared universal description when geometry exists and the defect sector has flowed to a Gaussian regime. D. What experiments would measure: predictions at a glance The framework yields concrete, falsifiable statements in terms of measurable time series. Let yC ( t ) denote fractional frequency readouts for each context C, aligned on a common time grid. Then: •(P1) Loop non-closure (holonomy): for a loop γof contexts, the closure residual ∆γ(t) := ΩγyC1(t)−yC1(t)(8) need not vanish even after optimal pairwise calibration; its statistics (PSD and higher cumulants) are the primary defect observables. •(P2) Protocol dependence via window transfer: changing interrogation windows (Ramsey time, duty cycle, switching envelopes) changes the sampled defect spectrum via a transfer relation of the form S∆,obs(ω) = |W(ω)|2S∆(ω). •(P3) Geometric vs non-geometric diagnosis: in a geometric phase, shrinking the physical loop (when meaningful) reduces ∆ γ in an area-law manner and yields analytic low-frequency behavior consistent with a local generator; in a memory-dominated non-geometric phase, ∆ γ exhibits longmemory scaling and nonanalytic infrared behavior (e.g. fractional power spectra), and does not admit a local curvature density.
4 •(P4) IR graviton sector: in the geometric, weak-defect regime, coarse graining drives the defect statistics toward Gaussianity; the resulting two-point function is representable by a massless spin-2 field in the strictified description, with quadratic dynamics fixed by locality and gauge invariance [2]. •(P5) Beyond deterministic GR: coarse-grained defect statistics naturally generate (a) highercurvature local terms, (b) nonlocal memory terms, and (c) stochastic (Einstein–Langevin type) corrections in the strictified description, providing a home for tabletop signatures that are not Planck-suppressed but controlled by coherence, protocol sensitivity, and integration time [5]. To anchor these predictions experimentally, we will specify a minimal three-clock loop combining (A) an optical lattice atomic clock, (B) an ultrastable cavity oscillator, and (C) a programmable circuit-QED scalar clock, and we will provide a data-analysis pipeline (time series outputs, alignment, kernel fitting, triangle/tetrahedron residuals, and null controls) in metrology-native terms [ 3 , 4 ]. This yields a direct route for experimental tests in precision time and frequency laboratories. E. Clock contexts 1. Why “clock = context” rather than “clock = frequency” A central operational lesson of precision timekeeping is that a “clock” is not merely an oscillator with a nominal frequency; it is a complete measurement-and-inference pipeline. In practice one records a time series (typically fractional frequency or phase) and interprets it via an estimator whose performance depends on interrogation windows, dead time, servo dynamics, and environmental couplings. This is already implicit in standard metrological treatments (e.g. Allan variance and sampling-induced instabilities) [3, 4]. Our framework elevates this to a definitional principle: Aclock context is the minimal operational package needed to define a time readout, and hence the minimal unit from which “time” and “geometry” can be inferred. The reason this is not merely terminological is logical: if different clock constructions define time through inequivalent observables and protocols, then “proper time” cannot be assumed as a primitive shared parameter; it must be inferred (if possible) from the consistency of comparisons. This motivates a formal definition of context that is sufficiently rich to (i) predict the statistics of its readout, and (ii) compare readouts across contexts. 2. Minimal definition of a context We define a clock context Cas a tuple C:= SC, OC, wC,EstC,EnvC,(9) where: • SC is the physical clock system (atoms, cavity field, superconducting qubit, etc.), including the degrees of freedom used for readout. •OC is the readout observable or, more precisely, the operator (or operator-valued distribution) to which the measurement ultimately couples. Examples: – optical atomic clock: an internal transition (projective measurement in the {|gi,|ei} basis, with dynamics generated by the atomic Hamiltonian and the interrogating field); –cavity clock: optical phase/frequency of a mode set by boundary conditions; – circuit-QED “scalar” clock: local flux field Φ( x0, t )coupled to a qubit and read out via Ramsey phase or excitation statistics. •wC is the protocol window/filter. This includes the full time dependence of coupling and interrogation (switching envelopes, Ramsey sensitivity functions, servo sampling, duty cycle), i.e. the object that determines which temporal/frequency components of fluctuations are sampled. In metrology, this is precisely the origin of transfer functions and aliasing effects [3, 4].
5 •EstC is the estimator mapping raw measurement records to a time (or frequency/phase) series. This includes feedback/servo laws. Operationally, EstC is what turns measurement outcomes into a reported time coordinate. • EnvC is an environment class: a specification of relevant external degrees of freedom and their statistical state (thermal, vacuum, squeezed, driven, structured reservoir, etc.), together with boundary conditions (cavity geometry, transmission line impedance, shielding, vibration spectrum, etc.). This is not an optional detail: it determines the correlation functions that enter the clock statistics, and hence the comparison structure. In particular, the possibility of long-memory kernels or non-Gaussian statistics is controlled primarily by EnvC. The tuple (9) is deliberately minimal: each component is necessary to define the operational meaning of a “tick” and to predict the statistics of the readout. To make this precise, we now show how (9) canonically induces a measurement channel and hence a readout random process. 3. From context to readout: instruments, records, and estimators A physical clock is a measurement device acting on a quantum (or classical) dynamical system coupled to an environment. The mathematically correct object describing “measurement with outcomes” is a quantum instrument [ 6 , 7 ]: a family of completely positive maps labeled by outcomes, whose sum is trace-preserving. In the simplest (discrete outcome) setting, an instrument is IC={IC,r}r∈RC,IC,r :ρ7→ IC,r(ρ),X r∈RC Tr IC,r(ρ) = Tr ρ, (10) where RC is the set of raw outcomes (clicks, IQ samples, populations, etc.). In continuous-outcome and continuous-time measurements one uses instrument-valued measures; we keep the notation schematic, since the conceptual structure is the same. The context data (SC, OC, wC,EnvC)determines the statistics of the record rthrough: 1. dynamics of SCcoupled to EnvC; 2. coupling specified by OCand time dependence specified by wC; 3. measurement map producing outcomes r. Standard open-system reductions show that, under weak coupling and suitable timescale separation, the reduced dynamics and measurement statistics are controlled by environment correlation functions projected by the coupling operator; in particular, power spectra and response functions enter naturally [ 8 ]. This is the formal underpinning of why wC and EnvC cannot be ignored: they shape the effective kernel seen by the clock. The estimator EstC then maps the raw record (possibly over a time window) to the reported clock readout. Concretely, let r[0,t]denote the measurement record up to time t. Then ˆτC(t) = EstCr[0,t],(11) or, in metrology-native terms, the estimator outputs a fractional frequency series yC ( t )and/or phase φC ( t ) = RtyC ( t0 ) dt0 . In typical atomic-clock operation, (11) is implemented as a cycle-to-cycle servo update law; in cavity clocks it is a continuous feedback filter; in circuit-QED clocks it can be a Ramsey phase estimator per run. Thus, a context C canonically defines a stochastic process ˆτC ( t )(or yC ( t ), φC ( t )) together with its statistics. This is the object that will be compared between contexts. Key consequence (minimality). If any of the components in (9) is omitted, the mapping from physical dynamics to readout is no longer well-defined: • Without OC , one cannot specify which environment correlations are sampled (different observables project different correlators). • Without wC , one cannot specify how correlations are sampled (window transfer; aliasing; switching). •Without EstC, one cannot define the reported time series (servo memory, nonlinear estimators).
6 •Without EnvC, one cannot specify the correlators at all. Hence (9) is minimal in the precise sense that it is the smallest data set from which a reproducible time readout process can be constructed. 4. Context spaces and the categorical viewpoint (intuitive but exact) Having defined contexts as operational packages, we now explain why category theory is the natural language. The reason is structural: comparisons are directional, composable, and exhibit higher coherence constraints. Objects and morphisms. Let Ctx be the collection of contexts, and for each pair ( C, C0 )let EC→C0 denote an experimentally defined comparison procedure (to be formalized in Sec. 1 F, with holonomy in Sec. 1 H). Then contexts are objects and comparisons are morphisms. Composition of comparisons is the operational procedure “compare C to C0 and then C0 to C00 ,” which is precisely morphism composition (3) . Why higher structure is unavoidable. In ordinary geometry, the obstruction to global trivialization is not seen at the level of a single overlap, but at the level of loops: holonomy. Likewise here, the obstruction to strict global time gluing is not seen in a single pairwise calibration (which can always be locally adjusted), but in triangles and loops of comparisons. This forces a higher-categorical view: •a triangle compares two different composite morphisms C→C00 (direct vs via C0); •the mismatch is naturally a 2-morphism (an associator-type defect); •tetrahedra test higher coherence (pentagon-type constraints). We will not require heavy formalism in the main text; the essential point is that coherence (closure of compositions) is the correct notion of “global consistency” for time comparisons, and its failure is the physical defect we seek to measure. 5. Concrete clock contexts used throughout For definiteness and experimental relevance, we will focus on three canonical contexts that are naturally incompatible: 1. Optical lattice atomic clock CA : OCA is an internal transition read out via population measurement; wCA is the Ramsey/Rabi sensitivity function with dead time; EstCA is a discrete-time servo producing yA[n][3, 4]. 2. Ultrastable cavity oscillator CB : OCB is the cavity field phase/frequency; wCB is a continuous feedback transfer function; EstCBis the lock loop producing yB(t)(continuous time series). 3. Circuit-QED effective-scalar clock CC (“UDW-like”): OCC is a local flux field Φ( x0, t )(scalarlike degree of freedom) coupled to a qubit; wCC is a programmable switching envelope χ ( t ); EstCC is a Ramsey phase estimator producing φC [ n ]or yC [ n ]. This context is designed to provide tunable memory (via structured reservoirs/highQ modes) and tunable windowing, making it a controlled probe of geometric vs memory-dominated regimes. These three contexts are not chosen for exoticity but for complementarity: they differ in observable algebra, protocol structure, estimator memory, and environment coupling, ensuring that a nontrivial holonomy signal is not artificially excluded by construction. Preview of what follows. In Sec. 1 F we define comparison channels EC→C0 on the resulting readout spaces and introduce the affine-plus-kernel model (2) as the operational definition of transport between contexts. In Sec. 1 H we define loop holonomy and the associated defect process δ ( t )whose power spectrum Sδ(ω)is the fundamental correlator used to diagnose geometric vs non-geometric phases.
7 F. Comparison channels 1. Operational meaning: “predict C0from C” A clock context C (Sec. 1 E) produces a readout time series (phase, fractional frequency, or a discretecycle record). The primitive operation that turns many clocks into a geometric or non-geometric structure is not the existence of those series individually, but the ability to compare them. Because clocks are operational devices, comparison is not an abstract equality; it is a calibrated inference procedure: given the readout series in C, what would we have predicted for the readout series in C0for the same run? This motivates the central object of the framework. Definition 1.1 (Directed comparison channel) . Given two contexts C and C0 , a directed comparison channel is a map EC→C0:XC−→ XC0,(12) where XC and XC0 are the admissible readout-signal spaces for C and C0 (e.g. fractional frequency series y ( t ), phase series φ ( t ), or discrete sequences). Operationally, for any realized readout x∈XC from context C, the output ˆx0=EC→C0[x]is the best predicted readout in context C0under a specified calibration/protocol/estimator class. Two remarks are crucial. (1) EC→C0 is directed. The arrow C→C0 matters: the inference “predict C0 from C ” is not generally invertible or symmetric. In metrology, this is routine: predicting a cycle-based atomic servo record from a continuous cavity phase record is not equivalent to the reverse prediction because sampling, dead time, and estimator memory break symmetry [ 3 , 4 ]. In our framework, this directionality is fundamental because it is the seed of non-commutativity and holonomy. (2) EC→C0 is a procedure class, not a metaphysical map. The map depends on what you allow yourself to do when comparing: linear prediction, affine calibration, finite-memory filters, nonlinear estimators, etc. The framework is compatible with any choice; however, to extract invariant content and to connect to experiments, one must fix a comparison class and then test whether loop compositions close within that class. In this paper we emphasize the affine-plus-causal-kernel class because it is (i) the natural first-order operational model in weak-defect regimes, (ii) the standard language of signal processing and system identification, and (iii) directly connected to power spectra and window transfer functions. 2. Channels as conditional laws (Markov kernels) and as optimal predictors Conceptually, there are two levels at which one may define a comparison channel. Level A: full statistical channel. A context C defines a stochastic process XC (the readout series) through the instrument/estimator pipeline in Sec. 1 E 3. The most general comparison object is then a conditional law of XC0given XC, i.e. a Markov kernel on path space: PXC0∈dx0|XC=x=KC→C0(x, dx0).(13) This is the most faithful representation of “predict C0 from C ” in the presence of non-Gaussianity, intermittency, and estimator nonlinearities. Level B: operational predictor (a chosen section of the channel). In experiments, one often needs a single predicted series ˆx0 associated to the observed x , for example a best linear predictor, a maximumlikelihood predictor, or a Bayesian posterior mean. Fix a prediction rule Π(e.g. conditional mean, MAP, etc.). Then EC→C0[x] := ΠKC→C0(x, ·)∈XC0.(14) This is the object we will use in defining holonomy and loop closure, since it yields a concrete endomorphism on readout spaces. In the weak-defect regime relevant for IR/strictified behavior, the most robust choice is the minimum mean-square linear predictor. Classical results show that, under stationarity assumptions, the optimal linear predictor is a causal linear filter determined by second-order statistics (power spectra and crossspectra) [9]. This motivates our standard comparison class.
8 3. Affine-plus-kernel comparison class We now define the comparison class used throughout the paper. Definition 1.2 (Affine plus causal kernel channel) . Fix two contexts C and C0 and suppose their readouts are represented on a common time axis (after timestamp alignment and resampling, Sec. 5 B). A comparison channel in the affine-plus-kernel class is (EC→C0x)(t) = αCC0x(t) + βCC0+Zt −∞ ds KCC0(t, s)x(s),(15) together with a residual (innovation) process rCC0(t) := xC0(t)−(EC→C0xC)(t).(16) Interpretation. •αCC0 and βCC0 are the best-fit scale and offset between readouts (unit conversion, calibration of fractional frequency conventions, etc.). In the strictification language introduced later, these are precisely the degrees of freedom associated with local redefinitions of time origin/scale in a chosen representative (the “strictification gauge” acts on them). •KCC0 ( t, s )is a causal memory kernel capturing protocol and estimator effects: dead time, servo memory, cavity relaxation, switching envelopes, etc. In stationary regimes one often has KCC0 ( t, s ) = KCC0(t−s). •rCC0 ( t )is the irreducible mismatch after optimal fitting in the chosen model class. The statistics of r(PSD, cumulants, history dependence) is the primary carrier of coherence defects. Why causal? A comparison channel should not depend on future input when used as a predictor; hence we impose causality ( s≤t ). This is also the correct operational constraint when the estimator is implemented in real time. Noncausal filters can be used in post-processing, but for diagnosing physical memory and holonomy it is important to distinguish causal transport (what can be realized as a protocol) from acausal smoothing artifacts. Why this class is both general enough and sharp enough. The kernel class includes: •discrete-time servos (atomic clocks) as a causal filter with finite memory in the cycle index; •continuous-time feedback (cavity locks) as an LTI/LTV kernel; • programmable switching (cQED clock) as a designed filter, whose χ ( t )acts as a window transfer function. At the same time it is sharp enough to produce invariant statements: loop closure can be tested as equality (or near-equality) of composed kernels, and the obstruction is encoded in residuals and their spectra. 4. Composition rule and correct indices The most common source of errors in holonomy constructions is inconsistent index ordering. We therefore state the composition rule explicitly and verify domain/codomain matching. Lemma 1.3 (Composition of directed channels) . Let EC→C0 : XC→XC0 and EC0→C00 : XC0→XC00 be two comparison channels. Then the composite channel C→C00 is E(via C0) C→C00 := EC0→C00 ◦EC→C0:XC→XC00 .(17) Proof. For any x∈XC , EC→C0 [ x ] ∈XC0 by definition. Applying EC0→C00 yields an element of XC00 . Hence EC0→C00 ◦EC→C0 is well-defined with domain XC and codomain XC00 . The index ordering in (17) is forced by domain/codomain matching.
9 Kernel-level composition (convolution). In the stationary kernel regime, where (EC→C0x)(t) = αx(t) + β+Z∞ 0 dτ KCC0(τ)x(t−τ), the kernel parts compose by convolution: K(via C0) C→C00 (τ) = Zτ 0 dσ KC0→C00 (τ−σ)KC→C0(σ),(18) and in frequency space e K(via C0) C→C00 (ω) = e KC0→C00 (ω)e KC→C0(ω).(19) This explicit algebra is the operational origin of path dependence: unless kernels satisfy strong compatibility relations, different routes C→C00 (direct versus via intermediates) yield different effective kernels. Categorical reading (kept intuitive). If we collect contexts as objects and comparison channels as morphisms, then Lemma 1.3 is simply the definition of composition in a category [ 10 ]. The highercategorical structure we exploit later arises because there can be multiple morphisms between the same pair of objects (direct versus composite) whose mismatch defines a higher coherence defect. In other words, the failure of strict transitivity is not a bug; it is precisely the data from which curvature/non-geometry will be inferred. 5. Why local mismatches are not decisive but loop mismatches are Before moving to holonomy (Sec. 1 H), it is useful to clarify a common confusion. A single pairwise residual rCC0 can always be reduced by adjusting αCC0 and βCC0 and by enlarging the permissible kernel class. This is the analogue of the fact that connection coefficients can be set to zero at a point by a coordinate choice: it does not diagnose curvature. What does diagnose a physical obstruction is the incompatibility of channel compositions around loops, i.e. the existence of nontrivial holonomy endomorphisms (Sec. 1 H) whose residual statistics persist under model refinement and gauge redefinitions. Thus, the role of comparison channels is twofold: 1. they provide the transport structure needed to define gluing; 2. they define loop invariants (holonomies) whose nontriviality is the operational content of coherence defects. This is the precise sense in which “comparisons define gluing” and sets the stage for the holonomy-based definition of curvature and non-geometry. G. Kernel model (main text version) 1. Why a canonical kernel model is both natural and sufficient The comparison channel EC→C0 (Sec. 1 F) is, in full generality, a map between stochastic processes (or their path-space laws). For the purposes of (i) defining holonomy and (ii) diagnosing whether a local strictified geometric description exists, one does not need the full microscopic channel. What one needs is a class of comparison maps that is: •operationally implementable (predict C0from Cusing past data); •expressive enough to capture protocol-induced memory and estimator dynamics; •compatible with standard spectral diagnostics (PSD, transfer functions); •composable with controlled algebra (so loop closure is a concrete test); •systematically extensible when higher cumulants are important.
16 • {Fa} is a collection of smooth fields on M (at minimum, a Lorentzian metric gµν in the universal geometric phase; more generally, a connection or other geometric structures in non-universal geometric phases); •ι assigns to each context C an operational “placement” ι ( C )in M (a point, region, or worldline segment, depending on the experiment). The role of ι is not to identify contexts with points metaphysically, but to encode the empirical fact that contexts are realized at definite laboratory locations and times, and comparisons are performed along definite physical procedures. Transport induced by a strictified representative. Given S , one can induce geometric transport maps between readout representations by specifying how clock readouts are encoded in the strictified fields (e.g. elapsed time as an integral of a line element, phase as an integral of frequency along a path, etc.). Denote by TS C→C0:XC→XC0(44) the transport map predicted by the strictified representative (possibly including protocol windows and estimator idealizations). The strictification problem is then: Definition 2.1 (Strictification (operational)) . We say the comparison structure is strictifiable (on a class of experiments) if there exists a strictified representative Ssuch that for all relevant pairs (C, C0), EC→C0≃ TS C→C0(within a specified tolerance/model class),(45) and such that loop holonomies computed from E coincide (up to conjugation) with holonomies induced by TS. Here “ ≃ ” denotes equivalence in the operational sense appropriate to the chosen model class (e.g. equality of affine+kernel predictors up to residuals whose statistics are consistent with the strictified model). This definition is deliberately flexible: strictification is not an all-or-nothing statement but a phase-dependent approximation whose quality can be quantified by the residual spectra/cumulants. Why this is a representation problem. The point is that EC→C0 are measured objects (fitted from time series), while TS C→C0 are model-predicted objects derived from smooth fields on M . Strictification asks whether the measured transport structure admits such a smooth representation. This makes the emergence of geometry a falsifiable claim rather than an assumption. 3. Strictification gauge: non-uniqueness of representatives Even if a strictified representative exists, it is not unique. This non-uniqueness is not a flaw; it is precisely what becomes gauge freedom in the strictified description. Basic reason for non-uniqueness. The operational comparison data constrain only those aspects of the strictified fields that affect clock readouts. Many different choices of coordinates, embeddings, and field representatives can yield the same observable transport. In ordinary GR, different coordinate systems describe the same geometry; more generally, different gauge choices describe the same physical holonomies. Definition 2.2 (Strictification gauge transformation) . A strictification gauge transformation is a change of strictified representative S= (M, {Fa}, ι)7→ S0= (M, {F0a}, ι0) such that all predicted comparison transports are observationally equivalent: TS0 C→C0≃ TS C→C0for all relevant (C, C0).(46) In the universal geometric phase, where the strictified description reduces to a Lorentzian metric gµν and proper time, the canonical gauge transformations are diffeomorphisms. Concretely, a diffeomorphism ϕ:M→Macts by pullback on fields: g7→ g0=ϕ∗g, Fa7→ F0a=ϕ∗Fa, ι(C)7→ ι0(C) = ϕ(ι(C)).(47) Because clock observables are diffeomorphism-invariant functionals of the geometric data (e.g. integrals of scalar quantities along worldlines), the induced transports TS C→C0 are unchanged. This is the operational origin of diffeomorphism gauge freedom: it is simply the non-uniqueness of representing the same comparison data on a manifold.
17 Infinitesimal form (linearized gauge). For later use (spin-2 emergence), note that an infinitesimal diffeomorphism generated by a vector field ξµacts on the metric perturbation by hµν 7→ hµν +∇µξν+∇νξµ,(48) which we will interpret as strictification gauge in the IR graviton sector. At this stage it is simply the statement that representative choices are non-unique. 4. Local redefinitions versus global obstructions A crucial conceptual point is that strictification gauge explains why local mismatches do not automatically imply physical curvature. Local absorbability. Given a strictified representative, one can always adjust local conventions: •shift time origin: t7→ t+ const; •rescale units locally within tolerance: t7→ (1 + )tover a finite band; •choose coordinates to set connection coefficients to zero at a point. These are strictification-gauge operations: they change the representative, not the underlying comparison data. Therefore, a single pairwise mismatch between two co-located clocks can typically be absorbed into strictification gauge to leading order; it does not diagnose curvature. Global obstruction. What cannot be absorbed by local gauge is a nontrivial loop holonomy class. If the transport structure has Ω γ6 = Id for some loop γ (Sec. 1 H), then no choice of strictification gauge can trivialize all transports simultaneously. This is the operational analogue of the standard geometric fact: one can gauge away the connection locally, but not its curvature. In our framework, the “curvature” is defined by the part of holonomy that survives all strictification gauge choices. This distinction sets up the criterion used later: Geometry exists when the loop defects admit a local infinitesimal generator (area law); curvature is precisely that generator in the universal geometric phase. 5. Strictification as compression: what information is discarded? Strictification is a compression of operational data into smooth fields. Like any compression, it discards information. In our framework, the discarded information is precisely what becomes invisible in the geometric phase (e.g. nonlocal memory beyond derivative expansion, higher cumulants beyond Gaussian fixed point) and what reappears as non-geometric behavior when it cannot be discarded consistently. This yields a clean interpretation of the later phase structure: • In the geometric phase, the comparison data are well-approximated by local fields; the discarded nonlocalities are small corrections (higher-derivative, weak memory, stochastic noise kernels). • In memory-dominated non-geometric phases, the discarded information is not small: kernels remain long-tailed and nonanalytic in the IR, obstructing a local field representation. Thus strictification is not assumed to always succeed; rather, its success or failure is a phase-dependent statement testable by loop residual scaling and spectra. B. Removable vs physical defects 1. The central distinction Section 1 H defined loop holonomy Ω γ and the measurable loop residual ∆ γ . Section 2 A explained why strictified representatives are non-unique and why this non-uniqueness induces a strictification gauge (diffeomorphism-like freedom in the universal geometric phase). We now formalize the key criterion of the framework:
18 A defect is removable if it can be eliminated by strictification gauge choices (local redefinitions of time representation). A defect is physical if it survives all such redefinitions, i.e. if it is encoded by loop invariants (nontrivial cocycles/holonomies). Operationally, this is the statement that a single local mismatch between clocks is not decisive; what is decisive is the failure of closure around loops. 2. Local mismatch as gauge: a precise statement Consider two contexts C and C0 realized in the same laboratory region and compared over a finite analysis interval. A comparison channel in the canonical class (Sec. 1 G) has the form byC0(t) = αCC0yC(t) + βCC0+Zt −∞ ds KCC0(t, s)yC(s) + rCC0(t).(49) Suppose that, for a given pair ( C, C0 ), the measured residual rCC0 is nonzero. Does this imply a physical defect? In general, no. The reason is that the affine degrees of freedom ( α, β )and, more broadly, per-context reparameterizations can absorb any purely local mismatch. To make this precise, define a per-context redefinition (strictification gauge at the readout level) by an invertible map GC : XC→XC . In the simplest metrology-relevant case, GC is an affine transformation on the readout: (GCy)(t) := aCy(t) + bC, aC6= 0.(50) More general GC (e.g. smooth time reparameterizations, estimator conventions) can be treated similarly; the affine case already captures the essential logic. Under (50), comparison channels transform by conjugation: E0 C→C0:= GC0◦EC→C0◦G−1 C.(51) This transformation precisely encodes the statement “change the per-context convention, then compare.” Since any single pairwise mismatch can be reabsorbed into the choice of ( aC, bC )for the two contexts, pairwise residuals are not invariant diagnostics of physical obstructions. Operational content. In practice, what (50) and (51) say is familiar: one can always recalibrate two clocks locally by adjusting offset and scale, and one can improve the pairwise predictor kernel by adding more filter taps. Therefore, local mismatch is ambiguous: it may be physics, or it may be a choice of representation/protocol class. 3. Loop invariants as the definition of physical defects By contrast, loop holonomy cannot be eliminated by per-object gauge choices unless it is trivial. This is already reflected in conjugation invariance of holonomy triviality (Prop. 1.5). We now sharpen this into a cohomological statement. Cocycle viewpoint (intuitive but exact). Fix a collection of contexts and comparisons forming a groupoid (objects are contexts, morphisms are comparison channels). Choose a “gauge” {GC} as above. The transformation (51) is exactly the categorical analogue of a gauge transformation: it changes the representative of each morphism by conjugation at its endpoints. In such a setting, the obstruction to strictification is encoded in the failure of compositions around loops to equal the identity. Formally, let γbe a loop based at C1as in (32). Under gauge, Ωγ7→ Ω0 γ=GC1ΩγG−1 C1. Hence, the conjugacy class of Ωγis gauge-invariant, and in particular: Ωγ6= Id is a gauge-invariant statement. (52) This motivates the definition:
19 Definition 2.3 (Physical defect) . A coherence defect is physical (non-removable) if there exists a loop γ such that Ω γ is not gauge-trivial, i.e. cannot be transformed to Id by any choice of per-context redefinitions {GC}within the allowed strictification gauge class. In the affine gauge class (50) , “gauge-trivial” means “conjugate to identity by affine reparameterizations.” In the universal geometric phase, the corresponding strictification gauge enlarges to diffeomorphisms acting on the smooth representative, but the logic is the same: local redefinitions cannot remove nontrivial holonomy. 4. Triangle defects and the first nontrivial cocycle The simplest nontrivial loop is a triangle. Let A, B, C be three contexts with channels EA→B , EB→C , EC→A. The triangular holonomy is ΩABC := EC→A◦EB→C◦EA→B∈End(XA).(53) Triviality of Ω ABC is equivalent to the statement that the three pairwise comparison channels glue consistently around the triangle. Operationally, rather than applying Ω ABC to an arbitrary yA , one typically compares the two routes from Ato C: ∆A→B→C(t) := EB→C◦EA→B[yA](t)−EA→C[yA](t),(54) and analyzes its PSD and cumulants (Sec. 1 H 5). If ∆ A→B→C persists under refinement of the pairwise fits and under allowed gauge redefinitions, it is a physical 2-defect (associator obstruction), the first manifestation of a nontrivial cocycle. 5. Cohomological form in the affine-plus-kernel model In the canonical affine-plus-kernel model, one can see explicitly how local gauge changes cancel on loops and why only loop obstructions survive. Work in the stationary regime for clarity and suppress offsets by detrending. Write each channel as an operator with transfer function e Ei→j(ω) = αij +e kij(ω). A per-context affine gauge (50) acts in frequency space as multiplication by aC (and an additive DC term from bCwhich we omit by detrending). Then e E0 i→j(ω) = aje Ei→j(ω)a−1 i. For a loop γ:i1→ ··· → in→i1, the holonomy transfer becomes e Ω0 γ(ω) = ai1e Ωγ(ω)a−1 i1. Thus the loop defect e Ωγ(ω)−1 is invariant up to conjugation and cannot be made to vanish for all ω unless it is identically zero. In particular, any attempt to remove a loop defect by changing the gauges ai cancels telescopically around the loop, leaving only a conjugation at the base point. This is the precise algebraic reason that local gauge choices cannot remove physical defects. 6. Intuitive summary: connection vs curvature, but operational The removable/physical distinction mirrors the connection/curvature distinction in geometry:
20 • Pairwise calibration parameters and local predictor kernels are like a connection: they can be changed by convention and improved locally. • Loop holonomy is like curvature: it measures the failure of global consistency and is invariant under local redefinitions. In our setting, the “connection” lives on the context-comparison network, and the “curvature” lives as holonomy endomorphisms on readout spaces. A geometric spacetime description will exist only when these holonomies admit a local infinitesimal generator (small-loop area law), which we will use as the criterion for the geometric phase in Sec. 2 C. When that criterion fails, the defects remain physical but non-geometric (Sec. 4). C. When geometry exists 1. From loop invariants to a local geometric generator Sections 2 A–2 B established two foundational points: (i) strictified spacetime descriptions are representatives of comparison data and are therefore gauge-nonunique, and (ii) physical defects are precisely those that survive this gauge freedom, i.e. loop holonomies Ωγ(Sec. 1 H). We now address the central existence question: When does there exist a local geometric description (a smooth manifold with fields) that faithfully encodes the physical defects detected by holonomy? The decisive feature that distinguishes a geometric phase from a non-geometric phase is locality. A smooth geometric description is local if the effect of a small loop is controlled by local data near that loop and vanishes as the loop shrinks. In differential geometry, this is encoded in the fact that holonomy around an infinitesimal loop scales with its enclosed area and is generated by curvature. We will now translate this into our operational comparison setting and formulate it as a sharp criterion. 2. Small-loop families and the operational notion of “shrinking” A loop γ in the context network is not automatically associated with a geometric size; it is a sequence of comparisons. To speak of “small loops” we need a family of experiments in which: 1. contexts are realized in a laboratory region that can be continuously varied (e.g. nearby spatial positions, nearby times, or nearby protocol parameters), and 2. comparison procedures are performed in a way that has a clear notion of physical support (time window, spatial separation, signal path). In practice, “small loop” can mean: • shrinking the spacetime support of the comparisons (shorter time windows, smaller spatial separations, shorter signal paths), •shrinking protocol windows while keeping other features fixed, •or shrinking an effective “area” in a two-parameter family of protocols. The criterion we formulate does not depend on which realization is chosen; it depends only on whether holonomy admits a local infinitesimal generator in the appropriate limit. 3. Area law and local generator: the geometric criterion Let {γ}>0 be a family of loops based at some context C , where is a physical smallness parameter (e.g. a length scale, time-window scale, or combined scale). Let Ω γ be the corresponding holonomy endomorphisms on XCand let ∆γbe the measurable loop residuals (Sec. 1 H 4).
21 Definition 2.4 (Small-loop area law (operational)) . We say the loop defects admit a small-loop area law if there exists a function Area ( γ )with Area ( γ ) → 0as → 0and an operator-valued local generator R (depending on the base point/region) such that Ωγ= Id + Area(γ)R+o(Area(γ)) (→0),(55) in the topology relevant to the chosen readout space (e.g. in mean-square for stochastic processes, or in operator norm on a suitable subspace). Equation (55) is the operational analogue of the standard holonomy-curvature relation in differential geometry. The content is twofold: •Locality: the leading deviation from identity is proportional to an area-like quantity and therefore vanishes as the loop shrinks. •Existence of an infinitesimal generator: the defect is controlled to leading order by a single local object R , independent of the detailed shape of the loop at fixed basepoint in the small-loop limit. Interpretation of R .In a metric geometric phase, R will be representable as (a projection of) the Riemann curvature tensor evaluated at the basepoint/region; more generally, it may correspond to curvature of a non-metric connection in non-universal phases (Sec. 2 E 3, later). In non-geometric phases, no such local generator exists: holonomy remains physical but does not admit an area-law expansion. 4. Geometry exists iff a local generator exists We can now state the criterion precisely. Theorem 2.5 (Geometric phase criterion) . A local strictified geometric description exists (in the sense of Def. 2.1 with locality) if and only if the family of physical loop holonomies admits a small-loop area law (55) with a local generator R. Proof sketch (operational and geometric). (Necessity.) Assume a local strictified geometric description exists. Then comparisons can be represented (within the chosen tolerance) by transports induced by smooth fields on a manifold M (Sec. 2 A 2). In any local geometric theory, transport around an infinitesimal loop is generated by the curvature of the underlying connection; equivalently, holonomy admits an expansion whose leading term is proportional to the loop’s enclosed area, with coefficient given by curvature at the basepoint [ 13 , 14 ]. Pushing this geometric fact through the readout functional yields (55) for Ωγacting on readout representations. Hence a local generator must exist. (Sufficiency.) Assume (55) holds for all sufficiently small loops in a neighborhood (in the operational sense above), with a generator R that depends smoothly on the basepoint/region. Then one can define a local curvature density on that neighborhood by identifying R as the infinitesimal holonomy per unit area. Standard reconstruction arguments in differential geometry show that a curvature tensor (or curvature of an appropriate connection) is precisely the local object whose integrals reproduce small-loop holonomies [ 13 ]. Thus there exists a local geometric structure (at least a connection, and in the universal clock-coupling sector a metric) whose curvature reproduces the measured small-loop holonomies to leading order, providing a local strictified representative. Higher-order corrections correspond to higher-derivative and nonlocal terms discussed later. The theorem makes explicit what is meant by “geometry exists”: it is not merely that one can fit some global manifold model, but that the defect structure admits a local infinitesimal generator that controls small-loop behavior. This is the sharp boundary between geometric and non-geometric phases. 5. Curvature as encoding, and why non-geometry is not pathology When Theorem 2.5 holds, the generator R is the operational encoding of curvature. In the universal geometric phase (where all clocks couple universally to a metric), R can be identified with Riemann curvature (up to the projection determined by the readout functional), and local invariants are built from
22 curvature and its covariant derivatives. In that regime, curvature is not an assumption: it is the unique local object left after quotienting by strictification gauge (Sec. 2 A 3). When the criterion fails, holonomy remains gauge-invariant and measurable (Sec. 1 H 3), but it does not admit a local generator. This yields non-geometric phases. Crucially, failure of the criterion does not mean inconsistency or acausality; it means that no local smooth-field compression exists. The appropriate description is then at the level of kernels and higher holonomies, as developed later for memory-dominated phases. 6. How defects survive gauge but fail locality Finally, we emphasize the logical separation: •Gauge survival: A defect is physical if Ω γ is nontrivial up to conjugation (Sec. 2 B 3). This is a global statement about loop invariants. •Locality (geometry existence): A defect is geometric if, in addition, it admits the area-law local generator (55). This is a local statement about small-loop families. Therefore it is entirely consistent (and generically expected) that physical defects exist in regimes where geometry does not. This is the operational meaning of a non-geometric phase of spacetime and is the conceptual bridge to the kernel-based obstructions analyzed in later sections. D. Why curvature (and derivatives) is the only local invariant 1. Statement of the claim in our setting In Sec. 2 C we established the sharp criterion for when a local geometric (strictified) description exists: loop defects admit a small-loop area law and hence a local generator. In the universal geometric phase we further assume that the strictified representative is a smooth Lorentzian metric gµν on a manifold M and that strictification gauge is diffeomorphism freedom (Sec. 2 A 3). The claim of this subsection is then: If the only fundamental strictified field is a metric gµν and the only gauge freedom is diffeomorphisms, then the first nontrivial local gauge-invariant information is curvature. More generally, any local diffeomorphism-invariant functional built from g is an integral of scalar invariants constructed from the Riemann tensor and its covariant derivatives. This is the precise sense in which curvature is the unique universal low-energy encoding of coherence defects when geometry exists. The result is standard in differential geometry and “natural operations” theory; we include a detailed proof sketch because it is conceptually central to our program. 2. Why the connection is not invariant, and why curvature is A metric g determines the Levi–Civita connection ∇ with coefficients Γ µ νρ in a coordinate chart. However, Γ µ νρ are not tensorial; under a coordinate change they transform inhomogeneously. Hence any purported “local observable” built directly from Γis gauge-dependent. This is not an abstract remark; it has an operational counterpart in our framework. Strictification gauge includes reparameterizations of the local representative, which in the metric phase include diffeomorphisms. Therefore any quantity that can be set to zero at a point by a diffeomorphism is not a physical encoding of a defect; it is a choice of representative. The standard geometric fact is the following: Proposition 2.6 (Normal coordinates remove first-derivative data) . Let ( M, g )be a smooth pseudoRiemannian manifold and p∈M . There exist local coordinates around p (Riemann normal coordinates) such that gµν(p) = ηµν , ∂λgµν (p)=0,equivalently Γµ νρ(p)=0.(56)
23 Proof sketch. This is classical: choose an orthonormal basis in TpM , use the exponential map of the Levi–Civita connection to define coordinates by geodesics, and note that geodesics through p have vanishing connection at p, which forces (56). Standard references include [13, 14]. Proposition 2.6 has a sharp implication: no diffeomorphism-invariant local observable can depend on ∂g at a point, because ∂g can be killed by a gauge choice. Thus the first potentially invariant jet data arise at second order in derivatives. The unique tensorial package of second-derivative information of a metric is the Riemann curvature tensor: Rµ νρσ =∂ρΓµ νσ −∂σΓµ νρ + Γµ λρΓλ νσ −Γµ λσΓλ νρ, which transforms tensorially and is invariantly defined. In normal coordinates, Γ( p ) = 0 but ∂ Γ( p ) remains and is exactly what curvature measures. More explicitly, in normal coordinates one has the Taylor expansion gµν(x) = ηµν −1 3Rµανβ(p)xαxβ+O(|x|3),(57) so curvature is precisely the first coefficient in the metric jet that cannot be removed by diffeomorphism gauge. This is the rigorous underpinning of the statement “curvature is the first local invariant.” 3. Local diffeomorphism invariants are built from curvature and its covariant derivatives We now sharpen the statement from “curvature is the first invariant” to “curvature and its covariant derivatives generate all local invariants.” Alocal scalar functional of the metric is an expression of the form S[g] = ZM ddxp|g| Lg, ∂g, ∂2g, . . . , ∂Ng,(58) where L depends on finitely many derivatives at a point (locality), and S [ g ]is required to be diffeomorphism invariant. The standard theorem of “natural operations” (equivalently, the classification of scalar differential invariants of a metric) states that any such L can be expressed as a scalar polynomial (or smooth function) of the curvature tensor and its covariant derivatives, fully contracted with g . In other words, up to total derivatives, L=FRµνρσ,∇αRµνρσ,∇(α∇β)Rµνρσ, . . . ,(59) with all indices contracted using g. Rigorous treatments can be found in [17, 18]. Why this must be true (operational proof sketch). The argument is conceptually straightforward once normal coordinates are used: 1. In normal coordinates at p , the diffeomorphism gauge has been used to set g ( p )and ∂g ( p )to canonical values (56) . Hence any invariant scalar at p cannot depend on g ( p )beyond η and cannot depend on ∂g(p)at all. 2. The remaining coordinate-independent data in the Taylor expansion are precisely the tensors built from ∂2g ( p )and higher derivatives. However, the raw derivatives ∂2g are not tensorial; their tensorial content is extracted by taking combinations that cancel gauge artifacts. The unique second-derivative tensor is Rµνρσ(p). 3. Higher derivatives of g at p can be decomposed into covariant derivatives of curvature plus terms algebraically determined by lower jets; the covariant derivatives ∇kR provide the tensorial higher-jet content. Any diffeomorphism-invariant scalar must be formed by contracting these tensors with g . 4. Therefore, any local diffeomorphism-invariant scalar density is a function of R and its covariant derivatives as in (59), up to divergences. This result is exactly the mathematical version of a physical statement: once geometry exists and gauge redundancies are factored out, local observables depend only on curvature data.
24 4. Higher-curvature terms as the derivative expansion footprint In our framework, coherence defects are primary and curvature is their strictified local encoding in the geometric phase. It follows that coarse-graining the defect sector produces corrections to the strictified description that must be expressible in the curvature basis (59) . In particular, the most general local effective action for the metric at low energies has the schematic form Γeff[g] = Zddxp|g|Λ+c1R+c2R2+c3Rµν Rµν +c4RµνρσRµνρσ+···+(covariant derivatives of curvature). (60) This is nothing but the derivative expansion consistent with locality and diffeomorphism invariance. In conventional effective field theory, the coefficients ciare set by integrating out short-distance degrees of freedom; in our setting, they encode the coarse-grained footprint of coherence-defect dynamics. Either way, the structural reason these terms appear is the same: they are the allowed local invariants. Why the expansion is natural (and when it fails). If the defect sector has a finite correlation length/time after coarse graining, then nonlocalities are suppressed at scales much larger than that length/time and the effective description becomes quasi-local, admitting an expansion in derivatives. This is the standard locality logic used in QFT in curved spacetime and gravitational EFT; for example, vacuum polarization and renormalization in curved backgrounds necessarily generate curvature-squared counterterms and higher invariants [ 19 ]. Conversely, in the memory-dominated non-geometric phases discussed later, the defect kernels remain long-tailed and nonanalytic in the IR, so no convergent derivative expansion exists; precisely then curvature is no longer a sufficient encoding and geometry ceases to exist as a local description (Theorem 2.5). Connection to holonomy (the bridge to Sec. 2 C). The local curvature tensor is the infinitesimal generator of holonomy (Ambrose–Singer type results) [ 16 ]. Our operational criterion for geometry existence (small-loop area law) is exactly the condition that holonomy defects admit such an infinitesimal generator. Once that holds and universality of clock coupling selects a metric representative, (60) is the unique local language in which defect footprints can appear. 5. Takeaway Within the universal geometric phase (smooth metric representative + diffeomorphism gauge), curvature is not assumed to be the carrier of defects; it is forced as the first local invariant that survives gauge. Moreover, all local invariant corrections are necessarily built from curvature and its covariant derivatives. This is the precise sense in which curvature is the universal low-energy encoding of coherence defects when geometry exists, and it sets the stage for deriving both the IR graviton sector and higher-curvature/nonlocal/stochastic corrections from defect statistics. E. Universality vs non-universality 1. Universality as an operational equivalence principle In Sec. 2 D we showed that once a metric strictified representative exists and diffeomorphism gauge is the relevant strictification freedom, local invariant encodings are built from curvature and its covariant derivatives. What remains to be specified is when the strictified representative is in fact metric and when curvature alone suffices. This is controlled by an operational notion of universality: Definition 2.7 (Universal clock coupling (operational)) . A strictified geometric phase is universal if, for all admissible clock contexts C in the class under consideration, the leading response of the inferred elapsed time to the strictified geometry is the same functional of the worldline, up to controllable calibration factors. Equivalently, there exists a single Lorentzian metric gµν such that all clocks define the same proper time τ = Rp−gµν ˙xµ˙xνdλ at leading order, and any residual differences are subleading corrections encoded by the defect sector. Intuitively, universality means that once geometry exists, clock construction details do not matter at leading order. This is the operational form of the equivalence principle for timekeeping. In our framework, it is not postulated; it is a property of a phase: it can hold approximately in the IR and fail outside it.
25 2. Why universality selects metric curvature Assume a local geometric strictification exists (Theorem 2.5) and suppose clocks couple universally in the sense of Def. 2.7. Then: 1. The clock readout is a scalar (elapsed time) attached to a worldline. 2. The only universal local scalar functional that (i) depends on the tangent ˙xµ and (ii) is invariant under reparameterization is determined by a quadratic form in ˙xµ. 3. A quadratic form on each tangent space that varies smoothly is precisely a metric. More concretely, for any local, reparameterization-invariant notion of line element dτ , the leading dependence on velocities must be of the form (dτ)2=−gµν (x)dxµdxν,(61) because any higher-order dependence on dx would define a non-quadratic geometry (Finsler-type) and therefore would not be universal across all clock constructions unless additional structure is postulated. Hence, universality plus locality selects a metric representative, and by Sec. 2 D the unique local invariant encoding of defects is metric curvature (and its derivatives). This is the precise sense in which “curvature is all there is” in the universal geometric phase. 3. If universality fails: geometric but non-universal phases (taxonomy) It is equally important to state what happens when geometry exists (local generator exists) but universality fails. In that case, a smooth strictified description may still exist, but it cannot be captured by a single metric that all clocks agree upon. The appropriate encoding then involves more general geometric structures. We list the main possibilities, keeping details for an appendix. (i) Torsion (Riemann–Cartan geometry). If the relevant transport is governed by an affine connection with torsion Tλ µν 6 = 0, then loop holonomies can be geometric while different clocks couple differently depending on internal spin/structure. Scalar clocks may be insensitive while spinful clocks detect torsioninduced effects. This yields a geometric phase in which curvature alone is insufficient; torsion is additional invariant data. (ii) Non-metricity (metric–affine geometry). If the connection is not metric-compatible, ∇λgµν 6 = 0, then lengths and clock rates depend on how standards are transported. Different clock constructions can disagree even along identical worldlines, violating universality. The defect is geometric (local generator exists) but not representable solely by Riemann curvature of a Levi–Civita connection. (iii) Weyl geometry (scale transport). A special case of non-metricity is Weyl geometry, ∇λgµν = ωλgµν , in which transport induces path-dependent scale changes. This is a natural geometric language for phases where clocks exhibit reproducible path-dependent rescaling rather than a single universal proper time. (iv) Finsler-type geometry (direction dependence). If the effective line element depends non-quadratically on velocities, dτ = F ( x, ˙x ) dλ , then different clocks (or different protocol-defined “time” functionals) can correspond to different effective norms on tangent vectors. This yields a smooth geometric description with a local generator but no single metric quadratic form, hence non-universality. (v) Multi-metric / sectorized geometry. A particularly natural outcome in our context is that different classes of clocks define different strictified metrics (or different effective light cones) that cannot be unified into one. Then geometry exists per sector but not globally: one has multiple effective metrics g(a) µν , with comparison defects measuring the obstruction to unification. Operationally, this corresponds to universality holding within a clock family but failing across families. The key point for the present paper is simply this: Curvature is the unique local invariant encoding of defects only in the universal metric phase . If universality fails, additional geometric structures (torsion, non-metricity, scale transport, direction dependence, or multi-metric sectors) are required. Our later derivation of an emergent spin-2 sector and of GR-like dynamics is explicitly restricted to the universal metric phase; the framework itself is more general and classifies the other phases operationally.
32 The most general diffeomorphism-invariant quadratic correction to the strictified action is nonlocal: ∆Γnonlocal[g]∼Zddx ddyp|g(x)|p|g(y)| O[g](x) Π(x, y)O[g](y),(78) where O [ g ]is a scalar (or tensor contracted to a scalar density) built from curvature, and Π( x, y )is a bi-kernel encoding memory and transport. In momentum space, (78) corresponds to non-polynomial form factors Π(k), i.e. terms like OΠ()O. Why this is natural in our framework. Comparison channels are causal kernel operators (Sec. 1 G), and loop holonomies are compositions of such kernels. Even when a metric representative exists, the induced effective action inherits this kernel nature: integrating out defect degrees of freedom yields an influence functional with memory. Therefore nonlocal terms are not “exotic corrections”; they are the direct strictified image of the fact that defects are fundamentally comparison/transport data. Locality as a special limit. If the kernel Π( x, y )is sharply peaked near x = y (short memory), then (78) reduces to the local derivative expansion (77) . If Πhas long tails or nonanalytic IR behavior, locality fails and the geometric description becomes nonlocal or breaks down altogether (memory-dominated non-geometric phases, Sec. 4). 4. (III) Stochastic Einstein–Langevin form and noise kernels The most experimentally important correction family is stochastic. The reason is simple: in our framework, the defect process δ ( t )is a measured random process with PSD Sδ ( ω )(Sec. 3 A). When strictified, δ is a projected functional of the metric perturbation hµν (Eq. (75) ). Therefore, in the strictified description, defect fluctuations appear as metric noise. A clean and standard way to encode this is the Einstein–Langevin form of semiclassical/stochastic gravity [5]: Gµν[g]+Λgµν +deterministic local/nonlocal corrections from Γeff = 8πG hTµνi+ξµν,(79) where ξµν is a stochastic source with hξµν(x)i= 0,hξµν (x)ξρσ(y)i=Nµνρσ(x, y).(80) In conventional stochastic gravity, N is related to stress-energy fluctuations. In our framework, the noise kernel is determined by defect statistics: the comparison/holonomy sector supplies additional fluctuating degrees of freedom whose projected correlators are measured as Sδ(ω)and higher cumulants. Projection to measurable defect PSD. Since δτ is a line functional of hµν , e.g. (75) , and since hµν responds linearly to ξµν at leading order in the Gaussian regime, the measured Sδ ( ω )constrains projected components of Nµνρσ ( x, y ). Protocol windows act as filters (Sec. 3 A 4), enabling defect-noise tomography: different windows and worldlines probe different projections of the same underlying noise kernel. Why stochasticity is the natural “home” for tabletop signatures. Local higher-curvature deterministic corrections are often difficult to see because they are small and require isolating tiny systematic shifts. By contrast, stochastic corrections produce phase diffusion, visibility loss, and route-dependent closure noise that accumulate with integration time. Precision clock networks are exquisitely sensitive to precisely such accumulated phase statistics. Thus, the Einstein–Langevin form is not a formal flourish; it is the natural effective language for translating measured defect PSDs into strictified geometric fluctuations. Summary. Within the geometric phase, strictification of the defect sector yields: •local higher-curvature operators as the derivative-expansion footprint of coherence curvature; •nonlocal form factors (memory kernels) as the strictified image of comparison transport; • stochastic source terms with noise kernels fixed by defect statistics, providing direct experimental observables via Sδ(ω)and its window dependence. D. Why effects need not be Planckian The standard intuition in quantum-gravity phenomenology is that observable deviations from general relativity are suppressed by powers of E/EPl (or equivalently by curvatures near `−2 Pl ), because one
33 assumes that the only genuinely new gravitational degrees of freedom live at the Planck scale and that low-energy probes couple to them only through irrelevant operators [ 25 ]. Our framework changes the locus of “quantum-gravitational” content: the primitive degrees of freedom are not UV metric modes but comparison holonomies and defect processes δ ( t )generated by finite-protocol, context-dependent clock transport. In the geometric phase, these defects admit a strictified spin-2 description in the IR (Sec. 3 B), but their magnitude is controlled not by E/EPl , rather by (i) proximity to the strictification fixed point (how small the loop holonomies are after optimal gauge choices), (ii) protocol sensitivity (window transfer functions |W ( ω ) |2 selecting portions of the defect PSD, Sec. 3 A 4), and (iii) integration time (phase diffusion and closure noise accumulate, Sec. 3 C 4). Concretely, the experimentally relevant quantity is not a local scattering amplitude at energy E , but the statistics of accumulated mismatch, whose variance is a filtered spectral integral, Varδobs∼Zdω |W(ω)|2Sδ(ω),(81) and which can grow with averaging time even when all participating frequencies are far below EPl . This is why “tabletop” signatures are natural in our setting: precision clock networks probe integrated phase/time with extraordinary sensitivity, and defect PSDs can be large in the infrared due to memory kernels or estimator dynamics without any appeal to Planckian energies. The emergence of a graviton sector is then an IR universality statement about the strictified description (Gaussian fixed point, locality, gauge redundancy), while the observability of deviations is governed by coherence/strictifiability and metrological filtering, not by E/EPl. 4. NON-GEOMETRIC PHASES: MEMORY-DOMINATED OBSTRUCTION A. Memory-dominated kernels ⇒fractional PSD 1. What “memory-dominated” means operationally In the geometric phase (Sec. 2 C), loop holonomies admit a small-loop area law and hence a local generator. This is a locality statement: the effect of a small loop is controlled by data near the loop and vanishes as the loop shrinks. In the kernel language of Sec. 1 G, locality corresponds to kernels with sufficiently short memory so that a derivative expansion exists. Amemory-dominated regime is the opposite limit: comparison/transport kernels have long temporal tails, so the present output depends appreciably on a long history of the input. Operationally this manifests as: •strong dependence of loop residuals on protocol history (preparation and ordering), •persistence of low-frequency power in defect spectra, •failure of derivative-expansion (analytic) behavior near ω= 0. The fundamental diagnostic is a fractional or otherwise nonanalytic infrared power spectrum. In this subsection we show explicitly how long-memory kernels generate fractional-power infrared spectra: K(τ)∼τ−α⇒e K(ω)∼ |ω|α−1⇒Sδ(ω)∼ |ω|−β, and we interpret this as the characteristic signature of a non-geometric phase. 2. Kernel-to-PSD transfer: the general mechanism Work in a stationary regime and consider a defect process δ ( t )generated by a causal transport-withmemory relation: δ(t) = Zt −∞ ds K(t−s)η(s) = Z∞ 0 dτ K(τ)η(t−τ),(82)
34 where η ( t )is a “microscopic” driving/noise process (representing unresolved fluctuations, estimator innovations, or any base process whose detailed origin is not needed here), and K ( τ )is a causal memory kernel. Taking Fourier transforms (in the stationary generalized sense), e δ(ω) = e K(ω)eη(ω),(83) and therefore the power spectral densities satisfy the transfer identity Sδ(ω) = |e K(ω)|2Sη(ω).(84) Thus, infrared structure of Sδ is determined by infrared structure of e K (up to the base spectrum Sη ). We now compute e Kfor a canonical long-memory kernel. 3. Power-law memory kernel and its Fourier transform Consider the prototypical long-memory kernel K(τ) = 1 Γ(1 −α)τ−α1τ>0,0< α < 1.(85) This kernel is the standard fractional-integral kernel: it defines the Riemann–Liouville fractional integral of order (1 −α )acting on η [ 27 , 28 ]. The long tail τ−α means the response does not have a finite memory time scale. We compute its Fourier transform. For a causal kernel, it is natural to use the Laplace/Fourier convention with a small convergence factor; one obtains the standard distributional identity Z∞ 0 dτ τ−αe−iωτ =e−iπ 2(1−α) sgn(ω)Γ(1 −α)|ω|α−1,0< α < 1,(86) so that, with the normalization in (85), e K(ω)∝e−iπ 2(1−α) sgn(ω)|ω|α−1.(87) In particular, the magnitude scales as |e K(ω)|∼|ω|α−1,|e K(ω)|2∼ |ω|2(α−1).(88) The key point is the nonanalytic behavior at ω = 0 (fractional power): no finite-order local differential operator has such a transfer function near ω= 0. 4. Fractional infrared PSD for the defect Insert (88) into the PSD transfer identity (84). If the base process has finite nonzero infrared power, Sη(ω)→Sη(0) as ω→0,(89) then the defect PSD behaves as Sδ(ω)∼ |ω|2(α−1) Sη(0) = const |ω|β, β := 2(1 −α)∈(0,2).(90) Thus, long-memory kernels generate fractional-power infrared spectra: K(τ)∼τ−α⇒e K(ω)∼ |ω|α−1⇒Sδ(ω)∼ |ω|−β. Generalization. If Sη(ω)∼ |ω|−pat low frequencies, then Sδ(ω)∼ |ω|−(p+2(1−α)).(91) Thus, fractional infrared behavior is robust: it arises whenever either the base process has IR enhancement or the transport kernel has long memory, and the exponents simply add. In our context, both sources can occur (servo memory, structured reservoirs, estimator nonlinearities), but the key qualitative signature is the nonanalytic low-frequency scaling.
35 5. Physical and geometric meaning: why fractional PSD signals non-geometry Why does (90) imply a non-geometric phase in our sense? (1) No derivative expansion ⇒ no local generator. In a local strictified geometric phase, small-loop holonomies admit an area-law expansion with a local generator (Def. 2.4). In frequency space, locality implies analyticity (or at least a controlled Taylor/derivative expansion) near ω = 0 for response kernels generated by local operators. By contrast, |ω|α−1 and |ω|−β are nonanalytic at ω = 0 (fractional power). This is the spectral signature that no local differential operator (hence no local curvature density or local 2-form representative) can generate the observed defect correlations. (2) Small-loop shrinking does not tame history dependence. A local curvature generator implies that shrinking the support of comparisons reduces the defect proportionally to an area-like quantity. Memory kernels with long tails do not scale with geometric size in this way: the defect depends on a long protocol history and persists even when the instantaneous “loop size” is reduced. This is precisely the operational meaning of a memory-dominated non-geometric phase. (3) Experimental lever: window transfer tomography. Because the observed defect spectrum obeys window transfer (Sec. 3 A 4), one can vary protocol windows W ( ω )and test whether the inferred underlying Sδ ( ω )exhibits robust fractional scaling. Persistence of a fractional low-frequency tail across window choices is a sharp diagnostic that the obstruction is not a model artifact but a genuine memory-dominated defect. 6. Takeaway In the kernel language, memory-dominated non-geometric phases arise when comparison/transport kernels have long tails. Such tails produce nonanalytic infrared transfer functions and hence fractionalpower defect PSDs. This provides both: •a mathematically explicit mechanism (long-memory kernels ⇒ |ω|-fractional spectra), and •an experimentally direct diagnostic (measure Sδ(ω)and its robustness under window changes). In the next subsection we use this to formulate the obstruction to local curvature encoding and to connect to higher-coherence (tetrahedral) failures. B. Obstruction to local geometry 1. From fractional IR to failure of local generators Section 4 A exhibited the defining spectral signature of memory-dominated regimes: Sδ(ω)∼1 |ω|β(ω→0),(92) arising from long-memory kernels K ( τ ) ∼τ−α . We now show that (92) is not merely a “noisy infrared tail” but a sharp obstruction to the existence of a local geometric strictification in the sense of Theorem 2.5. The logic is the promised chain: nonanalytic IR ⇒no derivative expansion ⇒no local generator ⇒non-geometric phase. 2. Local geometry implies analytic (derivative-expandable) IR response Assume, for contradiction, that a local strictified geometric description exists for the defect sector in a neighborhood and that it is governed (at the linearized level) by a local differential operator acting on the strictified field representative. Concretely, in the universal metric phase this means that the relevant strictified perturbation hobeys an equation of the form D(∂)h=source,(93)
36 where D is a finite-order differential operator (or admits a well-defined derivative expansion at low frequencies), and the measured defect is a linear functional (projection) of h along worldlines/protocol windows (Sec. 3 B 3, Eq. (75) ). In Fourier space, the Green’s function G ( ω, k )of D is a rational (or at least analytic) function of ω and k near ( ω, k ) = (0 , 0), up to the standard massless poles associated with propagation. More precisely, locality implies that the inverse operator admits an expansion in powers of momenta: G(ω, k) = G0(ω, k)1 + a1(ω2,k2) + a2(ω2,k2)2+···,(94) where an are analytic functions near the origin. Upon projecting onto a one-dimensional time series (via protocol windows and worldline sampling), the resulting effective transfer function inherits analyticity in ωnear ω= 0 (again up to standard pole structure). Key point. A local geometric generator corresponds to an analytic low-frequency expansion (a derivative expansion). This is the spectral expression of “small-loop area law”: shrinking loops reduces the effect smoothly and locally. 3. Memory-dominated spectra are nonanalytic: no derivative expansion By contrast, (92) is nonanalytic at ω = 0. In particular, for β /∈ 2 Z≥0 the function |ω|−β has no Taylor expansion at ω = 0, and even for integer β it signals infrared divergence inconsistent with an ordinary local susceptibility. Likewise, the underlying transfer function |e K(ω)|2∼ |ω|2(α−1) is nonanalytic. Therefore, if the measured defect PSD exhibits a robust fractional-power infrared behavior, it cannot be produced by any response kernel that admits a local derivative expansion. Equivalently: Sδ(ω)∼ |ω|−β(robustly) ⇒no local differential operator Dexists whose Green’s function yields δ. (95) This is the spectral (and experimentally testable) obstruction to local geometry. 4. Failure of the small-loop limit We now connect the spectral obstruction to the geometric criterion of Sec. 2 C. Theorem 2.5 states that a local geometric description exists if and only if loop defects admit a small-loop area law: Ωγ= Id + Area(γ)R+o(Area(γ)). In the memory-dominated regime, this fails for the following operational reason: • A small-loop area law requires that the loop residual depends only on local data near the loop and shrinks with the loop’s physical support. • A long-memory kernel means that the residual depends on an integral over a long history of the protocol, not on local support. • Therefore shrinking the geometric support of the loop does not shrink the defect in a manner controlled by area; the defect remains controlled by memory time scales and history. This is the time-domain counterpart of the nonanalytic IR: long memory corresponds to slow decay of correlations and persistent infrared power, which precludes an infinitesimal local generator. 5. No local curvature / no local 2-form representative In Sec. 2 D we established that in the universal metric phase, local invariants are curvature and its covariant derivatives. More generally, in a local geometric 2-connection picture, triangle defects should be representable by integrating a local 2-form over small faces. The memory-dominated obstruction rules out both possibilities:
37 1. No local curvature tensor representation. A local curvature tensor is exactly the infinitesimal generator of holonomy per unit area. Without the area law, curvature is not defined as a local density encoding loop defects. Any attempt to represent the defect by Rµνρσ ( x )fails because the defect depends on history, not on a pointwise tensor. 2. No local 2-form (surface) representative. In higher-gauge language, the triangle associator (2-holonomy) would be generated by a local 2-form B on a manifold. But fractional IR behavior implies that the triangle defect operator (kernel associator) is nonlocal and nonanalytic, hence cannot arise from integrating a smooth 2-form over a small surface. This connects directly to our kernel composition analysis: long-tailed kernels yield nonanalytic transfer functions under convolution, obstructing any local surface generator. Thus, memory-dominated behavior implies not merely “large corrections” but a categorical obstruction: the transport structure cannot be strictified into a local geometric 1and 2-connection. 6. Definition: memory-dominated non-geometric phase We can now state the phase characterization in operational terms. Definition 4.1 (Memory-dominated non-geometric phase) . A regime is a memory-dominated nongeometric phase if: 1. (Physical defect) There exist loops γ with nontrivial holonomy Ω γ6 = Id up to strictification gauge (Sec. 2 B 3); 2. (No local generator) For families of physically shrinking loops γ , no area-law expansion (55) exists (Theorem 2.5 fails); 3. (Spectral signature) The defect PSD exhibits robust nonanalytic IR behavior, typically fractionalpower scaling (92), stable under protocol window changes (Sec. 3 A 4). This definition makes the phase experimentally testable: one measures loop residuals, estimates Sδ ( ω ), varies windows W ( ω ), and tests for persistence of nonanalytic IR structure and failure of shrinkage with loop size. 7. Experimental remark: why this is not “just technical noise” A metrology-minded reader may object that 1 /|ω|β spectra are common in technical systems. Our framework does not deny this; rather, it provides a sharp discriminator: Technical noise becomes a non-geometric spacetime phase only when it appears as a loopinvariant, route-dependent residual that survives pairwise calibration and exhibits protocolhistory dependence incompatible with any local generator. This is why we emphasize loop holonomies and null controls (Sec. 5). A fractional spectrum on a single oscillator is not evidence of non-geometry; a fractional spectrum of loop closure residuals robust under window tomography and ordering tests is. 8. Takeaway Memory-dominated kernels generate nonanalytic infrared defect spectra. Nonanalytic IR behavior implies there is no derivative expansion and therefore no local infinitesimal holonomy generator. Consequently, neither metric curvature nor a local 2-form surface generator can represent the defect: the strictified local geometric description fails. This is the precise operational meaning of a non-geometric phase in our framework.
38 C. Higher obstruction (Bianchi/pentagon failure) 1. From triangle defects (2-curvature) to tetrahedron defects (3-curvature) Sections 4 A–4 B established the primary obstruction: memory-dominated kernels yield nonanalytic IR behavior and hence obstruct any local geometric generator (no curvature/2-form representative). There is a further, higher-level obstruction that becomes visible once we treat the comparison structure as a genuinely higher transport system: even if one attempted to encode triangle (2-simplex) defects by a local surface generator, memory-dominated regimes generically fail the next coherence condition on tetrahedra (3-simplices). This is the analogue of a higher Bianchi identity, or, in categorical language, the failure of the pentagon coherence law for associators. The purpose of this subsection is to state the result crisply and in a testable form; a detailed derivation is deferred to Appendix B 4. 2. Triangle defect as a 2-curvature functional Work in the stationary kernel regime for clarity. Recall that each directed comparison channel in the canonical model induces a causal kernel kij (Sec. 1 G 3) and that composition corresponds to convolution: k(via j) i→k=kjk ∗kij. Thus the triangle (2-simplex) defect for contexts ( i, j, k )is the mismatch between indirect and direct transport: ∆Kijk := kjk ∗kij −kik.(96) This ∆ Kijk is an operator-valued 2-curvature: it vanishes exactly when transport is strictly functorial on the triangle, and it is the kernel-level representative of the associator-type obstruction discussed earlier in categorical terms (Sec. 1 E 4). 3. Tetrahedron composition and the 3-defect (pentagonator) Now consider four contexts ( i, j, k, ` ). There are multiple inequivalent ways to transport from i to ` by composing three edges (different parenthesizations / different intermediate compositions). In a strict 2-functor (i.e. a genuinely local 2-connection), the associators on triangular faces satisfy a coherence condition: the product of associators around the boundary of a tetrahedron is trivial. This is the higher-categorical analogue of a Bianchi identity. In kernel language, this coherence becomes a statement that a certain 3-defect functional built from the ∆ Kijk vanishes (or reduces to a local 3-form generator in a smooth limit). Concretely, define a tetrahedral 3-defect (one convenient choice among equivalent oriented conventions) by ∆Kijk` := ∆Kjk` ∗kij−kk` ∗∆Kijk+ ∆Kij` −∆Kik` ∗kij.(97) The precise sign/orientation convention is not essential for the present statement; what matters is that ∆Kijk` measures the failure of face defects to glue consistently on a tetrahedron. 4. Result: memory-dominated regimes obstruct any local 3-form representative Proposition 4.2 (Higher obstruction in memory-dominated phases) . In a memory-dominated regime, where the comparison kernels kab ( τ )have long tails (e.g. kab ( τ ) ∼τ−α with 0 < α < 1) and hence nonanalytic infrared transfer functions e kab ( ω ) ∼ |ω|α−1 , the tetrahedral 3-defect ∆ Kijk` generically has nonanalytic infrared behavior and cannot be represented as the small-volume limit of any local 3-form curvature H on a manifold. Equivalently, no local “higher Bianchi identity” exists: the pentagon coherence required for a local 2-connection fails in a way that cannot be removed by strictification gauge.
39 Proof sketch. In frequency space, convolutions become products. Long-memory kernels have nonanalytic IR scaling. The triangle defects ∆ Kijk inherit nonanalyticity unless fine-tuned cancellation occurs. The tetrahedral combination (97) is built from sums and products of these nonanalytic objects and therefore generically remains nonanalytic at ω = 0. A local 3-form curvature H would yield an analytic (derivative-expandable) small-volume generator, contradicting the observed nonanalyticity. Full details and the relation to the pentagon identity are given in Appendix B 4. 5. Crisp experimental meaning Proposition 4.2 provides an operational test that goes beyond “no small-loop area law”: •Measure and fit pairwise kernels kij (Sec. 1 G) for a set of four contexts (i, j, k, `). •Compute triangle defects ∆Kijk via (96) (or equivalently the triangle residual time series). • Form two inequivalent triple-composition predictions from i to ` (different parenthesizations/orderings) and compute their difference as a time series; this is the observable counterpart of ∆Kijk`. • Diagnose whether this tetrahedral residual exhibits robust nonanalytic IR behavior and protocolhistory dependence. If the tetrahedral residual is nontrivial and persists under protocol refinements and null controls, it constitutes direct evidence of a higher obstruction: the comparison transport cannot be captured by any local geometric 2-connection (hence, a fortiori, by metric curvature alone). This yields a crisp, testable definition of a non-geometric memory-dominated phase that is more stringent than pairwise or triangle diagnostics alone. 5. EXPERIMENTAL PROGRAM: THREE-CLOCK LOOP A. Three clocks (A, B, C): one page each This section specifies three operationally distinct clock contexts that can be combined into an incompatible comparison loop. The objective is not to give a lab manual, but to define each clock at the level required to (i) produce a well-defined readout time series yX (fractional frequency) or φX (phase), (ii) identify the protocol knobs that change the effective comparison kernel, and (iii) implement the channel fitting and holonomy diagnostics of Secs. 1 G–1 H. Clock A: optical lattice atomic clock (discrete servo output yA[n]) Physical oscillator and readout. Clock A is an optical lattice atomic clock (e.g. Sr/Yb-class) whose oscillator is an ultra-narrow internal transition with angular frequency ωA . Operationally, the “hand” is the phase of the atomic coherence relative to an interrogating local oscillator (LO). The raw measurement per cycle is a population-derived error signal eA [ n ](often obtained by probing on either side of resonance), which a digital servo converts into a frequency correction. Time series output. The primary output series is a discrete-time fractional-frequency record {(tn, yA[n])}NA n=1, yA[n]≈νLO(tn)−νA νA ,(98) where tn is a consistent time tag (recommended: the midpoint of the Ramsey free-evolution interval in cycle n). A phase series can be formed by discrete integration, φA[n] := X m≤n yA[m]Tc,(99) with Tc the cycle time. In standard metrology language, yA [ n ]is the servo output that realizes the clock [3].
40 Protocol knobs (what to vary). The effective kernel of clock A is controlled primarily by: (i) Ramsey (or Rabi) interrogation time TR and pulse shape (changes the sensitivity function), (ii) cycle time Tc and dead time (changes aliasing/Dick effect), (iii) synchronous vs asynchronous interrogation relative to other clocks, (iv) servo law (integrator gains, bandwidth, estimator memory). These are precisely the parameters that change how LO noise and environmental fluctuations are filtered into yA [ n ]; the Dick effect provides the canonical example of protocol-induced kernel reshaping [4]. Kernel viewpoint. Clock A is intrinsically windowed and sampled: it measures the LO through a sensitivity function on each cycle and outputs a discrete-time estimate via a stateful estimator. In the canonical comparison model (20) , this means that comparison channels involving A naturally acquire nontrivial causal kernels even when all underlying physics is Markovian. Clock B: cavity-stabilized laser (continuous yB(t)) Physical oscillator and readout. Clock Bis a cavity-stabilized optical oscillator. The oscillator is an optical cavity resonance νcav ( t ) ∼c/ (2 L ( t )) set by cavity length and boundary conditions; the readout “hand” is the continuous optical phase of a laser locked to the cavity. The cavity is thus a boundary-defined frequency reference; its fluctuations arise from thermal and mechanical processes and are shaped by a feedback lock. Time series output. The primary output is a continuous (or high-rate sampled) fractional-frequency series {(tm, yB(tm))}MB m=1, yB(t)≈νlaser(t)−ν0 B ν0 B ,(100) obtained from calibrated actuator control signals and/or heterodyne beats. A phase series is obtained by integration, φB(t) = Zt yB(t0)dt0.(101) Protocol knobs. The effective kernel of B is controlled by: (i) lock bandwidth and controller parameters (transfer function shaping), (ii) cavity temperature setpoint and stabilization (drift/relaxation memory), (iii) vibration isolation and acceleration sensitivity (low-frequency colored noise), (iv) deliberate thermal/mechanical history protocols (to probe hysteresis/memory). These knobs directly modify the causal response KB·(t, s)appearing in (20) when Bis used as input or output. Kernel viewpoint. Clock B is naturally continuous-time and often exhibits slow relaxation processes; hence it is a natural gateway to memory-dominated behavior if operated in regimes with long correlation times. Conversely, in a well-controlled short-memory regime it approximates a local geometric clock channel. This makes B an essential element of the loop: it is operationally distinct from the sampled atomic context Aand from the programmable switching context Cbelow. Clock C: circuit-QED flux-field clock (programmable switching χ(t)) Physical oscillator and readout. Clock C is an effective-scalar field clock implemented in circuit QED: a superconducting qubit couples locally to the (scalar-like) flux field Φ( x0, t )of a 1D transmission line or to a single resonator mode (mode reduction). The interaction is programmed in time by a switching envelope χ ( t ), making C the cleanest operational realization of a “UDW-like” clock context in the lab. Circuit-QED implementations of qubit–field coupling with high control are standard [31, 32]. Time series output (discrete, two equivalent forms). Operate C in Ramsey phase mode (recommended for loop closure): •each cycle nyields a Ramsey phase estimate b φC[n], •define a fractional-frequency-like increment over cycle duration ∆tCby yC[n] := b φC[n+ 1] −b φC[n] 2πν0 C∆tC ,(102) with timestamp t(C) ntaken as the midpoint of the coupling/interrogation window. Thus the primary series is {(t(C) n, yC[n])}, directly comparable to yA[n]after resampling.
41 Protocol knobs (the “kernel dial”). Clock C provides unusually direct control of the comparison kernel: •Switching envelope χ ( t ): width, rise/fall, multi-pulse structures ⇒ direct control of the window transfer |eχ(ω)|2(Sec. 3 A 4). •Coupling strength g0in g(t) = g0χ(t): tunes linear-response vs nonlinear/non-Gaussian regime. •Reservoir engineering : by coupling to a highQ intermediate mode or structured impedance, one can introduce controlled non-Markov memory (long tails), directly accessing the memory-dominated regime diagnosed in Secs. 4 A–4 B. •Readout definition : phase-clock (Ramsey) vs count-clock (excitation statistics) changes the estimator functional and therefore the induced channel class. Kernel viewpoint. The defining feature of C is that it makes the window transfer relation operationally programmable: one can vary χ ( t )and directly test whether inferred defect spectra are consistent with a local derivative expansion (geometric phase) or exhibit robust fractional IR structure (memory-dominated non-geometry). Because C couples to a different operator algebra and can be tuned between Markovian and non-Markovian regimes, it is the most sensitive “handle” for driving and diagnosing nontrivial holonomy. Summary: why ( A, B, C )form an informative loop. A is sampled, estimator-driven, and interrogationwindow limited; B is continuous boundary-defined phase with environmental relaxation; C is a programmable switching clock with tunable memory and operator content. These three contexts therefore generate genuinely distinct comparison kernels in (20) , enabling a loop holonomy signal whose PSD and cumulants can be tuned and diagnosed using protocol knobs. B. Analysis pipeline (time series, alignment, channel fitting, defects, null controls) This subsection gives a compact, end-to-end pipeline that turns the three clock outputs of Sec. 5 A into (i) fitted comparison channels EX→Y in the canonical model (20) , (ii) triangle and tetrahedron defect observables, and (iii) falsification-grade null controls. The goal is that an experimentalist can immediately identify what to record and what statistics to compute, without adopting any category-theory language beyond “compose channels and check closure.” (i) Time series outputs and common currency Choose fractional frequency as the primary common currency. Output: {(tn, yA[n])},{(tm, yB(tm))},{(tk, yC[k])}, as defined in Sec. 5 A. Optionally also record phase series φX = RyXdt for cross-checks. For A and C (cycle-based), time-tag each sample at the midpoint of the effective interrogation/coupling window; for B use mid-sample tagging. (ii) Timestamp alignment and resampling Use a shared laboratory timebase (common reference clock) for all acquisition systems. Resample all series onto a common analysis grid t`=t0+`∆t: •for B, low-pass filter then decimate to ∆tto avoid aliasing; • for A and C , treat each yX [ n ]as applying over its cycle interval and map to the grid by piecewise hold or by a causal interpolation consistent with the estimator window. Denote the aligned discrete-time sequences by yA[`], yB[`], yC[`].
48 3. Imposing linearized diffeomorphism invariance Under δξhµν = ∂µξν + ∂νξµ , we require δξL be a total derivative (so that the action is invariant). A standard and efficient way to enforce this is to demand that the Euler–Lagrange equations yield a conserved “Bianchi identity” compatible with gauge symmetry, equivalently that the quadratic operator annihilates pure-gauge configurations. Criterion (pure gauge should solve the source-free equations). If L is gauge invariant, then the field equations obtained from δS/δhµν = 0 must be identically satisfied when hµν = ∂µξν + ∂νξµ for arbitrary ξ. This strongly constrains (a, b, c, d). A direct coefficient-level computation (omitted here for brevity but standard) yields the unique solution (up to an overall scale): b=−a, c =a, d =−a. (C3) Substituting (C3) into (C2) gives the Fierz–Pauli Lagrangian: LFP =1 2∂λhµν ∂λhµν −∂µhµν ∂λhλν +∂µhµν ∂νh−1 2∂λh ∂λh, (C4) where we have set the overall normalization a= 1 without loss of generality. Alternative (Noether identity) viewpoint. Gauge invariance implies a Noether identity: the Euler– Lagrange tensor Eµν ( h )satisfies ∂µEµν ≡ 0identically (linearized Bianchi identity). Requiring this identity for the general form (C2) also fixes the coefficients to (C3) . This is the linearized analogue of ∇µGµν ≡0in GR. 4. Equations of motion and the emergence of the linearized Einstein tensor Varying the action SFP =Rd4xLFP yields Eµν(h)=0,(C5) where Eµν(h)is (up to normalization) the linearized Einstein tensor: Eµν(h) = 1 2hµν −∂µ∂λhλν −∂ν∂λhλµ +∂µ∂νh−ηµνh+ηµν ∂α∂βhαβ.(C6) One verifies directly that ∂µEµν ≡ 0(Noether identity) and that Eµν is invariant under (C1) . In harmonic (de Donder) gauge, ∂µ¯ hµν = 0,¯ hµν := hµν −1 2ηµνh, the equations reduce to the wave equation ¯ hµν = 0, exhibiting two propagating degrees of freedom. This is the standard “free graviton” dynamics. 5. Uniqueness statement in the language of Sec. 3 B 5 We can now summarize the uniqueness result used in Proposition 3.1: Proposition C.1 (Fierz–Pauli uniqueness, local inertial form) . Let L be a Lorentz-invariant local quadratic Lagrangian for a symmetric tensor field hµν with at most two derivatives. If L is invariant under the linearized diffeomorphism transformation (C1) (up to total derivatives), then L is uniquely fixed, up to an overall normalization and total derivatives, to be the Fierz–Pauli Lagrangian (C4). Role in the main text. In Sec. 3 B the assumptions “Gaussian IR” and “locality” justify truncation to a quadratic, two-derivative action, while “strictification gauge” yields (C1) . Proposition C.1 then shows that the IR strictified mediator must be a massless spin-2 field governed by the linearized Einstein operator. This is the precise mathematical meaning of “emergent graviton” in our framework.
49 Appendix D: Kernel calculus (convolution, PSD transfer, fractional memory) 1. Causal convolution and composition of comparison channels In the stationary regime (Sec. 1 G 3), the kernel part of a comparison channel can be written as a causal convolution operator on a time series x(t): (Kx)(t) = Z∞ 0 dτ k(τ)x(t−τ), k(τ)=0for τ < 0.(D1) This is the continuous-time analogue of an FIR filter in discrete time. Composition ⇔ convolution. Let K1 and K2 be two causal convolution operators with kernels k1 and k2. Then their composition is again a causal convolution operator: (K2◦K1)x=Z∞ 0 dτ (k2∗k1)(τ)x(t−τ),(D2) where the convolution of kernels is (k2∗k1)(τ) = Zτ 0 dσ k2(τ−σ)k1(σ).(D3) Proof: substitute (D1) twice and change the order of integration (Fubini), using causality to fix integration limits; this yields (D3). This is the algebraic reason memory makes path dependence generic: composite transports correspond to iterated convolutions of kernels, and different routes lead to different effective kernels unless strong compatibility relations hold. 2. Fourier transform: convolution becomes multiplication Let e k ( ω )denote the Fourier transform of a causal kernel (understood distributionally when needed). Then: ^ (k2∗k1)(ω) = e k2(ω)e k1(ω).(D4) This is the standard convolution theorem and is the frequency-domain representation of channel composition used repeatedly in Secs. 1 G 4 and 4 C. 3. PSD transfer for linear filtering (window transfer) Let δ ( t )be a wide-sense stationary process with PSD Sδ ( ω ). Let w be an LTI filter (window/impulse response) and define δobs =w∗δ. Then in frequency space e δobs =ewe δ, and the PSD satisfies Sδobs (ω) = |ew(ω)|2Sδ(ω).(D5) This is the window-transfer relation used in Sec. 3 A 4. Remark (relation to cross-spectra). In full system identification of EX→Y , the optimal linear transfer in the Wiener sense involves cross-spectra SXY ( ω ); the transfer (D5) is the special case when the observable is explicitly filtered (or when δitself is defined as an innovation after fitting). 4. Fractional memory kernel transform The memory-dominated phase analysis in Sec. 4 A relies on the Fourier transform of a power-law causal kernel. Consider K(τ) = 1 Γ(1 −α)τ−α1τ>0,0< α < 1.(D6)
50 This is the Riemann–Liouville fractional-integral kernel of order (1 −α )[ 27 , 28 ]. Its Fourier transform can be computed from the standard integral (understood as an oscillatory integral with a convergence factor): Z∞ 0 dτ τ−αe−iωτ =e−iπ 2(1−α) sgn(ω)Γ(1 −α)|ω|α−1.(D7) Therefore e K(ω) = e−iπ 2(1−α) sgn(ω)|ω|α−1,|e K(ω)|∼|ω|α−1.(D8) If δ=K∗ηfor a base process ηwith finite nonzero infrared PSD, Sη(ω)→Sη(0), then Sδ(ω) = |e K(ω)|2Sη(ω)∼const |ω|β, β = 2(1 −α)∈(0,2),(D9) which is the fractional-power infrared scaling used as the diagnostic of memory-dominated non-geometric phases (Sec. 4 B). Nonanalyticity as obstruction. The crucial feature is that |ω|α−1 is nonanalytic at ω = 0; hence no finite-order local differential operator can reproduce this IR response. This is the precise spectral reason that such memory kernels obstruct local curvature/2-form/3-form representations. [1] A. Einstein, Die Grundlage der allgemeinen Relativitätstheorie, Annalen der Physik 49, 769–822 (1916). [2] M. Fierz and W. Pauli, On relativistic wave equations for particles of arbitrary spin in an electromagnetic field, Proceedings of the Royal Society of London A 173, 211–232 (1939). [3] D. W. Allan, Statistics of atomic frequency standards, Proceedings of the IEEE 54(2), 221–230 (1966). [4] G. J. Dick, Local oscillator induced instabilities in trapped ion frequency standards, in Proceedings of the 19th Annual Precise Time and Time Interval (PTTI) Applications and Planning Meeting, pp. 133–147 (1987). [5] B. L. Hu and E. Verdaguer, Stochastic Gravity: Theory and Applications, Living Reviews in Relativity 11 , 3 (2008). [6] E. B. Davies and J. T. Lewis, An operational approach to quantum probability, Communications in Mathematical Physics 17, 239–260 (1970). [7] A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory, 2nd ed., Edizioni della Normale, Pisa (2011). [8] H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press, Oxford (2002). [9] N. Wiener, Extrapolation, Interpolation, and Smoothing of Stationary Time Series, Wiley, New York (1949). [10] S. Mac Lane, Categories for the Working Mathematician, 2nd ed., Springer, New York (1998). [11] M. B. Priestley, Spectral Analysis and Time Series, Academic Press, London (1981). [12] C. W. Gardiner and P. Zoller, Quantum Noise: A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics, 3rd ed., Springer, Berlin (2004). [13] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vol. I, Wiley, New York (1963). [14] R. W. Sharpe, Differential Geometry: Cartan’s Generalization of Klein’s Erlangen Program, Springer, New York (1997). [15] C. A. Weibel, An Introduction to Homological Algebra, Cambridge University Press, Cambridge (1994). [16] W. Ambrose and I. M. Singer, A theorem on holonomy, Transactions of the American Mathematical Society 75, 428–443 (1953). [17] I. Kolář, P. W. Michor, and J. Slovák, Natural Operations in Differential Geometry, Springer, Berlin (1993). [18] P. B. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah–Singer Index Theorem, 2nd ed., CRC Press, Boca Raton (1995). [19] N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space, Cambridge University Press, Cambridge (1982). [20] F. W. Hehl, P. von der Heyde, G. D. Kerlick, and J. M. Nester, General relativity with spin and torsion: Foundations and prospects, Reviews of Modern Physics 48, 393–416 (1976). [21] E. Scholz, The unexpected resurgence of Weyl geometry in late 20th-century physics, in Beyond Weyl: The Theory of Gravitation, Gauge Theory and Cosmology, ed. H. Goenner et al., Birkhäuser (2017). [22] D. Bao, S.-S. Chern, and Z. Shen, An Introduction to Riemann–Finsler Geometry, Springer, New York (2000). [23] D. B. Percival and A. T. Walden, Spectral Analysis for Physical Applications: Multitaper and Conventional Univariate Techniques, Cambridge University Press, Cambridge (1993). [24] P. Billingsley, Probability and Measure, 3rd ed., Wiley, New York (1995). [25] J. F. Donoghue, General relativity as an effective field theory: The leading quantum corrections, Physical Review D 50, 3874–3888 (1994).
51 [26] G. Amelino-Camelia, Quantum-Spacetime Phenomenology, Living Reviews in Relativity 16, 5 (2013). [27] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Amsterdam (1993). [28] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego (1999). [29] U. C. Täuber, Critical Dynamics: A Field Theory Approach to Equilibrium and Non-Equilibrium Scaling Behavior, Cambridge University Press, Cambridge (2014). [30] J. C. Baez and U. Schreiber, Higher gauge theory, in Categories in Algebra, Geometry and Mathematical Physics, Contemporary Mathematics 431, 7–30 (2007). [31] A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, and R. J. Schoelkopf, Cavity quantum electrodynamics for superconducting electrical circuits: An architecture for quantum computation, Physical Review A 69 , 062320 (2004). [32] A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R.-S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics, Nature 431, 162–167 (2004). [33] L. Ljung, System Identification: Theory for the User, 2nd ed., Prentice Hall, Upper Saddle River, NJ (1999). [34] S. M. Kay, Fundamentals of Statistical Signal Processing, Volume I: Estimation Theory, Prentice Hall, Englewood Cliffs, NJ (1993). [35] W. H. Zurek, Decoherence, einselection, and the quantum origins of the classical, Reviews of Modern Physics 75, 715–775 (2003). [36] T. Leinster, Higher Operads, Higher Categories, Cambridge University Press, Cambridge (2004). [37] R. Street, Categorical structures, in Handbook of Algebra, Vol. 1, Elsevier (2004). [38] S. Weinberg, Photons and gravitons in S -matrix theory: Derivation of charge conservation and equality of gravitational and inertial mass, Physical Review 135, B1049–B1056 (1965). [39] S. Deser, Self-interaction and gauge invariance, General Relativity and Gravitation 1, 9–18 (1970). [40] R. N. Bracewell, The Fourier Transform and Its Applications, 3rd ed., McGraw–Hill, New York (2000).