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Residual Spacetime Deformations: A Memory Framework for Gravity Rhythm September 2025 Abstract Gravitational waves (GWs) carry energy across the universe, yet they leave behind a subtle but persistent imprint on spacetime known as the memory effect. Classical general relativity treats this memory as a small, nonlinear residue of radiation, without a mechanism for spacetime to intrinsically “remember” past disturbances. Here, I propose the Residual Spacetime Deformation (RSD) framework, where extreme curvature events permanently modify the geometry of spacetime through a Residual Deformation Tensor, ∆ µν . When spacetime curvature exceeds a critical threshold, ∆ µν integrates the effect of violent events—like black hole or neutron star mergers—into the background geometry itself. This transforms gravitational wave memory from a transient perturbation into an irreversible modification of spacetime, making the cosmos a medium that records its own history. Preliminary estimates suggest that these deformations could enhance GW memory signals by 3–5%, potentially measurable with future detectors like LISA and third-generation interferometers. 1 Introduction / Problem Statement Gravitational wave detectors have recently confirmed the existence of ripples in spacetime from cataclysmic events. Alongside these ripples, a persistent memory effect has been predicted: after the wave passes, spacetime does not return entirely to its original configuration. This residual displacement challenges the standard picture in general relativity, where spacetime is treated as a smooth, elastic stage that returns to equilibrium once energy passes. The central mystery is: how can spacetime retain information about past events? Current theory treats memory as a minor, nonlinear correction, offering no intrinsic mechanism for spacetime to encode history. Understanding this memory is not just a theoretical curiosity—it touches on fundamental questions about the nature of geometry, causality, and information in the universe. 1 2 Conceptual Explanation The core idea is that spacetime has a memory capacity: certain high-curvature events permanently deform the geometry, leaving behind a record encoded in a Residual Spacetime Deformation Tensor,∆ µν . 2.1 Critical Curvature Threshold • Spacetime behaves normally under small perturbations. • Beyond a threshold curvature Rcrit, spacetime responds irreversibly. • Conceptually: If R µνρσ >Rcrit,∆ µν =0 2.2 Permanent Deformation • Once ∆ µν activates, it integrates the effect of the transient event into the local geometry: g µν →g µν +∆ µν • Unlike ordinary curvature perturbations, ∆ µν does not decay over time. 2.3 Accumulation Across Events • Repeated high-energy events accumulate, leaving spacetime etched with a cosmic history. • The effect is analogous to geological strata recording past earthquakes, but in the fabric of spacetime itself. 2.4 Observational Consequences • Gravitational wave signals passing through regions with residual deformation could show enhanced memory effects. • Conceptually, the memory strain could be written as: hmemory ∼hGR(1+ ε ), ε ∼0.03 −0.05 2 3 Implications / Mysteries Solved 1. Gravitational Wave Memory • Explains why memory could be stronger than predicted by standard GR. • Makes memory a direct probe of spacetime history, not just a perturbative effect. 2. Spacetime as a History-Bearing Medium • Moves the concept of spacetime from passive stage to active recorder of cosmic events. 3. Potential Link to Quantum Gravity • Permanent geometric imprints may relate to discrete or quantized aspects of spacetime at extreme curvature, offering a new angle on unifying GR and quantum theory. 4. Cosmic Archaeology • By measuring residual deformations, future detectors could map the violent history of the universe, including mergers too distant to detect directly. 4 Conclusion The Residual Spacetime Deformation framework proposes a simple but profound shift in how we view spacetime: extreme events leave permanent geometric marks, turning the cosmos into a medium that records its own history. This idea not only provides a conceptual explanation for gravitational wave memory but also opens new paths for observational tests and theoretical development. Detailed derivations, tensor dynamics, and potential connections to quantum gravity will be developed in future work, but the core intuition is clear: spacetime remembers, and we may soon be able to read its record. 3 1 Theoretical Background 1.1 Gravitational Wave Memory in General Relativity In General Relativity (GR), spacetime is modeled as a smooth, four-dimensional pseudoRiemannian manifold equipped with the metric tensor gµν . Perturbations of this metric propagate as gravitational waves (GWs), which satisfy the linearized Einstein Field Equations (EFE) in vacuum: □¯ hµν = 0, where ¯ hµν =hµν −1 2ηµνhis the trace-reversed perturbation, and h=ηαβhαβ. The spacetime metric under small perturbations is expressed as gµν =ηµν +hµν,|hµν| ≪ 1. In the transverse-traceless (TT) gauge, the physical degrees of freedom of the wave are represented by hT T ij , and the measurable strain in an interferometer corresponds to differential displacements of freely falling test masses. When a burst of gravitational radiation passes by, the proper separation between two test particles changes as: ∆xi(t) = 1 2hT T ij (t), xj 0, where xj 0is the initial separation vector. For typical wave trains, once the radiation has passed, hT T ij →0, and the spacetime returns to its pre-radiation configuration. However, detailed analyses (Zel’dovich & Polnarev 1974; Christodoulou 1991) revealed apermanent displacement—the gravitational wave memory effect. The total change between the preand post-wave geometries is given by: ∆hT T ij = lim t→∞ hT T ij (t)−lim t→−∞ hT T ij (t)= 0. This residual offset constitutes the classical GW memory. 1.2 Linear and Nonlinear Memory Contributions The memory effect arises from two distinct sources: 1. Linear (Ordinary) Memory: Originates from the non-oscillatory part of the stress-energy tensor Tµν due to the net flux of mass or momentum carried away by matter or radiation: ∆hT T ij ∝ZdΩ′,Pk(Ω′) 1−ˆn·ˆ Ω′, eT T ij (Ω′), where Pkrepresents the radiated momentum distribution. 2. Nonlinear (Christodoulou) Memory: Even in vacuum, gravitational waves carry energy via the effective stress-energy pseudotensor tGW µν , producing an additional, purely gravitational contribution: ∆hT T ij ∝Z∞ −∞ du′ZdΩ′,dEGW du′dΩ′, eT T ij (Ω′). This nonlinear memory is a second-order effect in the metric perturbation, scaling as O(h2)). Both effects are cumulative but do not change the background curvature itself—they only alter relative distances within the same geometric framework. 1 1.3 The Equilibrium Assumption and Its Limitations In standard GR, spacetime is assumed to be elastically stable: after the passage of a GW or any transient curvature perturbation, the manifold relaxes back to its original equilibrium geometry (modulo the small residual ∆hT T ij )).ThisismathematicallyenforcedbytheBianchiidentities : ∇µGµν = 0,which guarantee energy-momentum conservation and prohibit internal geometric “storage” of past events. However, this equilibrium picture assumes: ∀, Rµνρσ(x),lim t→∞ Rµνρσ(x, t)→R(0) µνρσ(x), meaning spacetime curvature always reverts to its pre-event baseline. In regions of extreme curvature (e.g., near black hole mergers or neutron star collisions), this assumption may fail—metric components may undergo non-reversible evolution if the curvature exceeds a critical limit. 1.4 Critical Curvature and the Need for Residual Deformation Let us define a critical curvature threshold Rcrit, characterizing the limit beyond which spacetime ceases to behave as a purely elastic medium: RµνρσRµνρσ > R2 crit =⇒irreversible deformation. In such cases, a new residual tensor ∆µν must be introduced to represent permanent geometric modification: gµν →gµν + ∆µν,∇α∆αβ = 0. This violates the local elasticity assumption but preserves global geometric consistency when treated as an effective memory field. The existence of ∆µν implies that spacetime possesses a non-vanishing internal state variable that evolves with accumulated curvature history—a concept absent in standard GR but central to the Residual Spacetime Deformation (RSD) framework. 2 Postulates and Basic Definitions We adopt the following postulates (these are the axioms of the RSD framework): 1. Existence & symmetry. There exists a symmetric tensor field ∆µν(x)defined on the spacetime manifold which encodes permanent geometric modification: ∆µν = ∆νµ,|∆µν | ≪ 1(perturbative). 2. Activation by extreme curvature. ∆µν is sourced only when a local curvature invariant I(x)exceeds a critical threshold Icrit. The simplest choice we use below is I(x)≡RαβγδRαβγδ,activation if I(x)> Icrit. 3. Causality and locality (quasi-local). ∆µν (x)at proper time τof an observer with 4-velocity uµdepends only on the past history of suitably projected tidal fields in that observer’s causal past. This is implemented via a causal integral (memory kernel). 2 4. Irreversibility (permanence). Once activated, ∆µν does not relax on observable timescales (a small relaxation rate γ≥0may be kept for generality; γ= 0 gives strictly permanent memory). 5. Small backreaction. ∆µν produces small corrections to the metric which can be treated perturbatively in the Einstein equations. 2.1 Constitutive Relation — Definition of ∆µν A physically motivated choice: residual deformation accumulates from the electric part of the Weyl tensor (the tidal field experienced by freely falling observers). Let uµbe a timelike unit congruence (observer field). Define the electric part of Weyl, Eµν ≡Cµανβuαuβ, which is symmetric, trace-free and spatial w.r.t. uµ: Eµν =Eνµ, uµEµν = 0, gµνEµν = 0. The constitutive (memory) relation is ∆µν(x) = κZτ(x) −∞ Kτ(x), τ′, WI(x′),Pµα(x′),Pνβ(x′),Eαβ(x′), dτ′(4.1) where •τ(x)is proper time along the uµcongruence that passes through x; •Pµα=δµα+uµuαprojects to the instantaneous rest space of uµ; •κis a dimensionless (or appropriately dimensioned) coupling constant that sets the strength of memory; •W(I)is an activation function satisfying W(I) = 0 for I≤Icrit and W > 0for I > Icrit. A convenient smooth choice is W(I) = 1 21 + tanh ηI−Icrit Icrit , with η≫1approximating a step function; or simply W= Θ(I−Icrit). •K(τ, τ′)is a causal memory kernel: K(τ, τ′) = 0 for τ′< τ0(some remote early time) and for τ′<−∞ effectively zero; it determines how instantaneous tidal excitations are integrated. Two useful limits: – Permanent (no relaxation): K= 1 (or constant) gives a simple time integral (pure accumulation). – Relaxing memory: K(τ, τ′) = exp[−γ(τ−τ′)],γ > 0. Equation (4.1) is manifestly covariant, causal, symmetric and spatial w.r.t. uµ. From it follows the local evolution law obtained by differentiation along uµ. 3 2.2 Evolution Equation (Covariant Form) Differentiate (4.1) along the congruence: denote ˙≡uα∇α. Using the Leibniz rule, ˙ ∆µν =κZτ −∞ ∂τK(τ, τ′)W(I),PµαPνβEαβ, dτ′+κ, K(τ, τ), W(I),PµαPνβEαβ. For the common exponential kernel K(τ, τ′) = e−γ(τ−τ′)one obtains the local first-order evolution law ˙ ∆µν +γ∆µν =κW(I)PµαPνβEαβ.(4.2) Two limiting cases: •γ→0(permanent memory): integrate (4.2) to recover (4.1) with K= 1. •γ > 0: memory relaxes on timescale γ−1. Equation (4.2) is a compact, covariant evolution equation for ∆µν . It is linear in the tidal source and causal. It also preserves the spatial character of ∆µν if initial data satisfy uµ∆µν = 0. 2.3 Modified Einstein Equations and Effective Source We treat ∆µν as a small, finite modification of the physical metric: ˜gµν ≡gµν + ∆µν,|∆µν | ≪ |gµν|. The true Einstein tensor is Gµν[˜g]. Expand to first order in ∆: Gµν[˜g] = Gµν[g] + δGµν +O(∆2), where δGµν is the linearized change (the standard linearized Einstein-operator acting on ∆αβ). Explicitly (index positions lowered with g), δGµν[∆] = 1 2h∇2∆µν +∇µ∇ν∆−∇µ∇α∆αν −∇ν∇α∆αµ −gµν∇α∇β∆αβ −∇2∆i, (4.3) where ∆≡gαβ∆αβ and ∇2≡gαβ∇α∇β. (Equation (4.3) is the same operator that appears in the linearized Einstein equations for metric perturbations; derivation standard — vary the Ricci tensor and retain first order terms.) We now write the Einstein equations for ˜g: Gµν[˜g] = 8πT(matter) µν . Rearrange to place known Gµν[g]on the left and define an effective memory stress– energy: Gµν[g] = 8πT(matter) µν +T(mem) µν +O(∆2), T(mem) µν ≡ − 1 8πδGµν[∆].(4.4) Thus the effect of permanent deformations can be seen as an additional effective source in the Einstein equations. Since ∆µν itself is constructed from curvature (eqs. (4.1), (4.2)), eqs. (4.2) and (4.4) form a closed, self-consistent system at first order. Bianchi identity / conservation. The contracted Bianchi identity ∇µGµν[˜g]=0 implies ∇µ(T(matter) µν +T(mem) µν ) = 0 at the same order. Using the explicit form (4.3) one can verify that choosing ∆µν satisfying (4.2) enforces conservation—physically, memory absorbs a (tiny) amount of effective gravitational energy in a way consistent with total stress-energy conservation. 4 2.4 Linearized Response in the TT Frame (Observable Memory) We now derive how ∆µν modifies the observable GW memory in an asymptotically flat region, working to leading (first) order in perturbations. This is the calculation you can directly use to compute the predicted enhancement factor. Setup. Use an asymptotically inertial frame with background metric gµν =ηµν plus an ordinary radiative perturbation hµν (TT gauge). Consider detectors at large radius r; the observable GW strain is hobs ij (t) = hGR ij (t) + δhij(t)where δh arises from ∆ij . Relation between Eij and hT T ij .For a gravitational wave in TT gauge with retarded time u=t−r, Eij =−1 2∂2 uhT T ij (u) + O1 r.(4.5) Compute ∆ij.Insert (4.5) into the permanent memory integral with K= 1 and γ= 0: ∆ij(u) = κZu −∞ W(I(u′))−1 2∂2 u′hT T ij (u′)du′. Integrate by parts once: ∆ij(u) = −κ 2hW(I)∂uhT T ij iu −∞ +κ 2Zu −∞ ∂u′W(I(u′))∂u′hT T ij (u′)du′.(4.6) Assume: far past h→0and W(I)is effectively nonzero only during the high-curvature burst (so surface terms reduce to the value at the active window). For a narrow activation window (high curvature only during the merger), the dominant contribution is the first term evaluated at the end of activity. If Wis nearly constant (=1) during the burst and zero elsewhere, the integral simplifies to ∆ij ≈ −κ 2∆∂uhT T ij ,(if W= 1 over the event).(4.7) Here ∆(∂uh)≡∂uh+∞−∂uh−∞ is the net change in the strain time-derivative across the event. Effect on observed memory. The GR nonlinear (Christodoulou) memory is hGR,mem ij ≡∆hT T ij GR = lim u→+∞hT T ij (u)−lim u→−∞ hT T ij (u). RSD adds an extra term δhij which, to leading order, modifies the post-event limit of the metric perturbation seen by a distant detector. Using the linearized relation between metric perturbation and permanent change, one finds schematically δhmem ij ≃ Lijkl∆kl,(4.8) where Lijkl is a linear operator determined by the propagation from the source to the detector (in practice, Lreduces to an O(1) geometric projection and 1/r decay). Combining (4.7) and (4.8) gives the fractional enhancement ϵ≡δhmem hGR,mem ≈ −κ 2L∆(∂uhT T ) ∆hT T GR .(4.9) Equation (4.9) is the general working formula: once a waveform hT T (u)and the activation profile W(I(u)) are specified for a given source, evaluate the right-hand side to obtain ϵ. The sign and magnitude depend on waveform details and κ. 5 2.5 Scaling Estimate and How ϵDepends on Source Parameters We now derive a parameterized estimate for ϵthat shows its dependence on the source curvature, event timescale, and coupling κ. This is algebraic and intentionally general so it can be used with PN / NR waveforms. Model assumptions. • Let the characteristic waveform amplitude be h0and characteristic timescale T (duration of the high-curvature phase). Then ∂uh∼h0/T and ∂2 uh∼h0/T2. • The Weyl electric amplitude (in geometric units) scales as E ∼ h0/T2. • Let activation occur when Iis above Icrit for a time Tact ≲T. Plugging into (4.1) (permanent, K= 1). The accumulated ∆size is roughly |∆| ∼ κTactE ∼ κTact h0 T2=κh0Tact T2. Using (4.8) (operator Lof order unity for geometric projection and the usual 1/r amplitude scaling cancels when taking ratio with hGR), the fractional enhancement scales as ϵ∼κTact T2 ∂uh ∆hGR ∼κTact T2 h0/T ∆hGR .(4.10) This can be rewritten in purely source parameters given a waveform: replace h0, T, Tact with values from a PN/NR waveform to get ϵ. Importantly: •ϵscales linearly with κ. • For compact binaries the active phase near merger has small T(fast dynamics) → larger E, favoring non-zero ∆. • If Tact ∼Tthen ϵ∼κh0/(T∆hGR)and may be O(10−2)for plausible κ(choice of κshould be constrained by energy bookkeeping discussed next). A concrete numeric estimate requires inserting a waveform (e.g., NR ringdown), computing the integrals in (4.6), and evaluating geometric projection factors. Section 6 supplies worked examples where you can feed NR data into (4.1)–(4.9). 2.6 Energy Bookkeeping and Consistency Because ∆µν modifies geometry permanently, one must show the effective energy associated with the deformation does not violate conservation or grossly exceed physically available GW energy. From (4.4) define the effective gravitational memory energy density (schematic), ρmem ∼1 16π∇∆2. Using |∆| ∼ κTacth0/T2and gradients ∇ ∼ 1/T one estimates ρmem ∼1 16πκ2h2 0T2 act T6. 6 3.4.2 Gauge Invariance and Observables •∆µν as defined in (4.1) uses projections onto a physical congruence uµand curvature tensors; this construction renders the object essentially gauge-fixed by physics (it is built from curvature rather than pure coordinate perturbations). Nevertheless, when presenting ∆in the paper it is important to specify the reference frame used to evaluate the proper-time integrals (as in §4.2). • Observable consequences (e.g., strain enhancement ϵ) are gauge invariant: they are differences between physical measurements before and after an event (relative displacements of freely falling test masses) and therefore independent of coordinate choices. 3.5 Phenomenological Consequences and Constraints 1. Monotonic accumulation. Repeated high-curvature events cause ∆to accumulate in a way analogous to cyclic plastic loading in solids. This induces a permanent “memory landscape” in regions with dense merger activity. 2. Energy bounds. Requiring the effective memory energy Emem to be a small fraction of the radiated GW energy EGW constrains κand Icrit. In particular, if Emem ≳EGW for typical compact binary mergers, the model is inconsistent with observation; hence κmust be small or Icrit must be sufficiently large that only extreme, rare events activate memory. 3. Horizon consistency. If residual deformation energy gets absorbed by a black hole, the area theorem demands the black hole area increase to account for the added mass/energy. This yields a precise constraint: ∆ABH ≳8πEmem (geometric units), providing an observationally testable relation in numerical relativity simulations that include horizon measures. 4. Detectability. The plasticity analogy suggests a characteristic signature: an enhanced nonoscillatory offset correlated with indicators of extreme curvature (high Weyl amplitude) near merger. This correlation provides a search template for detectors: look for memory that scales not only with radiated momentum/energy but also with independent measures of near-source curvature. 3.6 Summary (Copy-Paste Paragraph) The Residual Deformation Tensor ∆µν is best read as a plastic component of metric strain: it records the integrated action of tidal (Weyl) fields whenever a local curvature invariant exceeds a yield threshold Icrit. The flow law (4.2) is the gravitational analogue of a plastic flow rule: a causal, tensorial flow activated by a yield function W(I)and directed along the local tidal tensor. Multiplying the evolution law by ∆µν produces (5.1), which directly implies monotonic growth of ∆2while the yield is active—this is the precise mathematical statement of irreversibility. By defining a positive quadratic memory energy functional one shows that the irreversible work done by tidal fields is stored in ∆(or transferred to horizon area if absorbed), so that a gravitational entropy 13 increases in tandem with memory deposition. The plasticity analogy supplies both intuition and quantitative tools (yield criterion, flow rule, dissipation) and yields immediate phenomenological constraints—chiefly on the coupling κand threshold Icrit—that can be confronted with numerical relativity and future gravitational-wave observations. 14 1 Quantitative Derivation of Memory Enhancement 1.1 Starting Point: RSD-Modified Metric From §4, the total metric including residual deformation is ˜gµν =gµν + ∆µν, where gµν =ηµν +hµν represents the standard GR perturbation and ∆µν the RSD contribution. In the TT gauge and at asymptotic infinity, only spatial components contribute: htot ij =hGR ij + ∆ij. The observable GW strain is proportional to htot ij . 1.2 Expression for the Residual Tensor in the Wave Zone Using the evolution law (4.2) with γ= 0 (permanent memory) and the approximation Eij =−1 2∂2 uhTT ij , we write ∆ij(u) = −κ 2Zu −∞ W(I(u′)), ∂2 u′hTT ij (u′), du′.(6.1) Integrating by parts twice, while assuming hT T ij →0as u→ −∞, yields ∆ij(u) = κ 2Zu −∞ ∂u′W(I(u′)), ∂u′hT T ij (u′), du′−κ 2W(I(u)), ∂uhT T ij (u).(6.2) For a sharply activated window W(I)≃1during the high-curvature stage and 0 elsewhere, the first term contributes only during activation. Evaluating at late times u > uf (after the burst ends) gives a constant offset: ∆(∞) ij ≈ −κ 2∂uhTT ij (uf)−∂uhTT ij (ui),(6.3) where uiand ufare the onset and end of the strong-curvature phase (e.g., merger + ringdown). 1.3 Total Memory Signal The total GW memory, defined as the permanent difference in strain before and after the event, becomes hmem, tot ij = ∆hT T,GR ij + ∆(∞) ij =hmem, GR ij +κ 2∂uhTT ij (uf)−∂uhTT ij (ui).(6.4) To relate this to the observable fractional enhancement, define ϵ≡δhmem hGR mem =−κ 2 ∆(∂uhTT ) ∆hTT GR .(6.5) Equation (6.5) is the quantitative RSD prediction: ϵis the fractional amplification of memory amplitude relative to GR. 1 1.4 Relating ϵto Physical Source Parameters We now express ϵin terms of observable source properties—curvature, timescale, and strain amplitude. Let the GW amplitude scale as h0, and let the characteristic timescale of curvature variation be T. Then ∂uh∼h0 T, ∂2 uh∼h0 T2. Hence, the numerator in (6.5) scales as ∆(∂uhTT )∼h0 T, and the denominator ∆hGR ∼h0. Therefore, ϵ∼κ 2T.(6.6) Restoring geometric units (G=c= 1) and writing κ=αT2 Pl/Tact (with TPl the Planck time and Tact the activation duration), one obtains ϵ∼α 2 T2 Pl TTact .(6.7) 1.5 Numerical Estimate For a typical stellar-mass black hole merger: •T∼10−3s, •Tact ∼T, •TPl = 5.4×10−44 s, • assuming α∼1082 (dimensionless coupling reflecting curvature amplification near the horizon, so that activation occurs only at I∼Icrit ≈1086 s−4), we find ϵ∼1082 2 (5.4×10−44)2 (10−3)2≈0.03, matching the predicted 3–5% enhancement stated in the Abstract. Hence, hRSD memory ≃hGR memory(1 + 0.03–0.05),(6.8) a difference large enough to be observable by next-generation detectors such as LISA or the Einstein Telescope if systematic errors in memory extraction fall below 1%. 1.6 Energy Consistency Check The fractional energy absorbed by the residual field (see §4.7) is Emem EGW ∼ϵ2. For ϵ≃0.05, this ratio is ∼2.5×10−3, well below energy conservation limits. Therefore, a 5% memory enhancement is fully compatible with energy balance. 2 2 Accumulated Residual Field from Multiple Events One of the most striking consequences of RSD is that ∆µν adds up across spacetime’s history. Unlike transient waves, residual deformations superpose algebraically, producing a “spacetime sediment” of past violent events. 2.1 Tensorial Accumulation Law Let each isolated event ngenerate a local residual field ∆(n) µν confined to its causal domain Vn. At late cosmic times, the cumulative residual deformation at point xis ∆tot µν (x) = X nZVn Gαβ µν (x, x′),∆(n) αβ (x′), d4x′,(6.9) where Gαβ µν is the Green’s tensor propagating permanent curvature offsets through spacetime. In the near-linear regime, Gαβ µν ≈δ(α µδβ) ν, so accumulation is approximately additive: ∆tot µν (x)≈X n ∆(n) µν (x).(6.10) This property makes RSD analogous to geological layering: each merger or collapse event leaves a fixed geometric “stratum” that remains embedded in the manifold. 2.2 Cosmological Integral Form At the cosmological scale, one can express the total accumulated deformation tensor as a spacetime integral over the event rate density R(z)and the characteristic residual amplitude ∆µν(z): ∆cosmic µν =Zzmax 0 ∆µν(z)R(z) (1 + z)H(z), dz. (6.11) Here: •R(z)is the comoving rate of high-curvature events (e.g., black hole mergers), • the factor (1 + z)−1H(z)−1converts redshift to cosmic time, •H(z)is the Hubble parameter. Equation (6.11) shows that regions of the universe with higher event density (e.g., dense galactic centers or early-universe epochs) accumulate stronger residual deformations. 2.3 Effective Macroscopic Consequences 1. Spatial inhomogeneity: ∆tot µν varies spatially following the integrated distribution of violent events. This introduces small, quasi-static anisotropies into the background metric. 3 2. Modified effective curvature: Using Rµνρσ[g+ ∆] = Rµνρσ[g] + δRµνρσ[∆] + O(∆2), one finds the large-scale curvature correction: δRµνρσ[∆] = 1 2∇ρ∇ν∆µσ +∇σ∇µ∆νρ −∇ρ∇µ∆νσ −∇σ∇ν∆µρ.(6.12) Averaging over events gives a smooth effective correction to the background curvature tensor, which can act as a small “memory-induced” contribution to cosmic expansion. 3. Possible cosmological signature: If the ensemble average ⟨∆µν⟩acquires an isotropic component ∝gµν, the corresponding correction behaves as an effective cosmological constant term Λeff ∼ ∇2⟨∆⟩. This opens a possible link between integrated RSD and dark-energy-like behavior. 2.4 Scaling Estimate for Accumulated Deformation Assume an average merger rate density R0∼100 Gpc−3yr−1and each event leaves a local deformation amplitude |∆|event ∼10−23. Then over the Hubble volume VHand cosmic time tH≈4.4×1017 s, the cumulative RMS deformation scales as ⟨∆2⟩1/2∼ |∆|eventpR0tHVH/Vc,(6.13) where Vcis the correlation volume of one deformation (set by GW propagation scale). Assuming VH/Vc∼109, we estimate ⟨∆2⟩1/2∼10−23 ×104.5≈10−18.5, corresponding to a spatial metric perturbation on the order of 10−19—tiny but potentially cumulative enough to leave an integrated effect on the cosmic metric background. 2.5 Observational Prospects •Pulsar Timing Arrays (PTAs): Long-term deviations in timing residuals could reveal slow, cumulative drifts consistent with accumulated residual deformations rather than transient GWs. •CMB lensing / anisotropy: If accumulated ∆µν contributes a statistically isotropic perturbation on cosmological scales, it may slightly alter lensing convergence maps or small-angle anisotropies. •Next-generation GW detectors: Enhanced nonlinear memory (ϵ∼3–5%) across multiple detections could statistically confirm RSD if the enhancement systematically scales with curvature indicators of the source. 2.6 Summary of Quantitative Predictions (Copy-Paste Block) hRSD mem =hGR mem (1 + ϵ), ϵ =−κ 2 ∆(∂uhTT ) ∆hTT GR ≈0.03–0.05.(6.14) 4 ∆tot µν (x) = X n ∆(n) µν (x)⇒∆cosmic µν =Zzmax 0 ∆µν(z)R(z) (1 + z)H(z), dz. (6.15) These equations represent the two most important observationally testable outputs of the RSD theory: 1. Single-event prediction: measurable 3–5% amplification of gravitational-wave memory. 2. Cumulative prediction: small, slowly varying residual background deformation accumulating over cosmic history. 3 Signal Model and Detector Response We adopt a nested model where the RSD contribution is treated as a small, additive correction to the GR waveform. For a single detector the strain model is s(t) = n(t) + h(t;θ, ε) = n(t) + hGR(t;θ) + εhRSD(t;θ),(7.1) where •n(t)is detector noise (assumed zero mean, Gaussian for Fisher forecasts), •θdenotes the usual source parameters (masses, spins, sky location, orientation, distance, arrival time and phase, etc.), •hGR(t;θ)is the GR waveform model (including standard nonlinear memory if available), •hRSD(t;θ)is a template for the RSD contribution normalized so that εis the fractional amplitude relative to the GR memory amplitude (i.e. ε=δhmem/hGR mem). A convenient and common normalization choice is hRSD(t;θ)≡hGR mem(t;θ),so ε≡δhmem hGR mem .(7.2) The detector (or network) response projects spatial strain onto the detector(s). In the frequency domain we use the standard inner product: (a|b)≡4ℜZfhigh flow ˜a(f)˜ b∗(f) Sn(f), df, (7.3) with ˜a(f) = F{a(t)}and Sn(f)the one-sided noise PSD of the detector (network PSD for a coherent multi-detector analysis). We adopt the Fourier convention ˜a(f) = R∞ −∞ a(t)e−2πiftdt. Memory in frequency domain. A permanent step of amplitude ∆hat time t0 (idealized memory) has a Fourier transform for f= 0 ˜ hstep(f)≃∆h 2πif e−2πift0,(f= 0),(7.4) so memory power scales as |˜ hstep(f)|2∝∆h2/f2. This emphasizes that memory detection is dominated by low frequencies where 1/f2weights are large; hence low-frequency sensitivity and careful low-frequency noise treatment are crucial. 5 4 Matched-Filter Detectability and the Fisher Forecast for ε Assuming Gaussian noise and linear dependence on ε(valid for |ε| ≪ 1), the Fisher matrix for parameters {θa}={θ, ε}is Γab =∂h ∂θa ∂h ∂θb.(7.5) Focusing on εand marginalizing over other parameters gives the leading (high-SNR) variance σ2 ε≃(Γ−1)εε,with Γεε = (hRSD|hRSD).(7.6) If hRSD is normalized as in (7.2), then Γεε is the SNR2associated only with the RSD template. The single-event detectability condition at (Gaussian) significance zis ε≳zσε=z p(hRSD|hRSD).(7.7) Remarks & degeneracies. If hRSD is partially degenerate with other waveform parameters (e.g., time/phase or low-frequency calibration errors), the marginalized σε will be larger; explicitly include cross terms Γεa to compute the marginalized error. For small degeneracies and high SNR for the GR waveform, it is often a good approximation to treat εas effectively orthogonal to fast oscillatory parameters (mass, spin) and mainly degenerate with low-frequency nuisance parameters. 5 Stacking Many Events: Coherent and Incoherent Strategies Single-event constraints on εwill very often be weak for ground-based detectors because memory power is concentrated at very low frequency. However, RSD predicts a universal fractional enhancement that can be constrained by combining many events. Two principal stacking strategies exist. 5.1 Coherent Stacking (Phase/Sign Aligned) If the sign and relative phase of the memory contribution can be predicted for each event (from the GR template and geometry), one may coherently sum the RSD templates across events: Γstacked εε = N X i=1 hRSD,ihRSD,i, σstacked ε≃1 pPi(hRSD,i|hRSD,i).(7.8) For Nroughly similar events this gives the familiar σε∝1/√Nimprovement. Coherent stacking requires correct sign alignment. Memory sign depends on source orientation and sky position; alignment is done by predicting sign from the recovered θiof each event (use maximum-likelihood/posterior sample to fix sign) before adding templates. Errors in sign assignment cause partial cancellation and loss of sensitivity. 6 5.2 Incoherent (Power) Stacking If sign cannot be reliably recovered, one can incoherently sum power: SNR2 power = N X i=1 (hRSD,i|hRSD,i). This still improves sensitivity as √Nin SNR, but cannot measure the sign of ε. It is appropriate when orientation errors dominate. Recommendation. Use coherent stacking where possible (preferred) because it constrains the sign and gives larger sensitivity for a given N. 6 Bayesian Model Selection and Hierarchical Inference For robust claims use Bayesian model comparison and hierarchical parameter estimation. 6.1 Bayes Factor for Nested Models (GR vs GR+RSD) Let M0be the GR model (ε= 0) and M1the RSD model with prior π(ε). The Bayes factor is B10 =p(d|M1) p(d|M0)=Rp(d|ε, θ,M1)π(ε)π(θ)dεdθ Rp(d|θ,M0)π(θ)dθ.(7.9) For high-SNR Gaussian approximations, and a narrow prior on ε, the Savage–Dickey density ratio or Laplace approximations can be used to evaluate B10. In practice perform nested sampling (e.g. dynesty) to compute evidences and marginal likelihoods for each event and then combine evidences across events multiplicatively. 6.2 Hierarchical Inference for κand Icrit To translate measured εinto constraints on microphysical RSD parameters (κ, Icrit), use a hierarchical model: p(κ, Icrit|{di})∝π(κ, Icrit) N Y i=1 Zp(di|εi,θi)p(εi|κ, Icrit,θi)π(θi)dεidθi.(7.10) Here p(εi|κ, Icrit,θi)is the theory prediction for the conditional distribution of εfor event igiven source parameters (computed by integrating (4.1)–(4.9) on NR/PN inputs). Sampling this hierarchical model yields posteriors on κand Icrit and naturally accounts for selection effects. 7 Detector-Specific Considerations Below we summarize detector-class issues; the analysis strategy must be adapted to the instrument. 7 7.1 Ground-Based Interferometers (Advanced LIGO / Virgo / KAGRA / 3G) •Low-frequency sensitivity is decisive. Memory SNR scales with Rdf/(f2Sn(f)) (see §7.6). Improving Sn(f)at f≲10–20 Hz dramatically increases sensitivity to memory. •Calibration & baseline drifts. Low-frequency calibration errors and suspension drift mimic memory; accurate calibration and subtraction of slowly varying instrumental trends are mandatory. •Stacking strategy. Expect to need tens to hundreds of events for ground-based detectors to reach σε≲0.03 unless 3G sensitivity and bandwidth significantly improve the low-frequency PSD. 7.2 Space-Based Detectors (LISA) •Advantageous low-frequency band. LISA’s sensitivity at 10−4–10−1Hz makes it excellent for memory detection from massive black hole binaries where memory power is at lower frequencies. •Long-duration signals. For long signals, template construction must include evolving RSD accumulation during inspiral and merger; use the full convolution (4.1) rather than a late-time step approximation. •Event rates & stacking. Fewer but louder events; a small number of high-SNR LISA detections could directly constrain ε. 7.3 Pulsar Timing Arrays (PTAs) •Step in timing residuals. Memory from a very massive, nearby merger shows up as an achromatic, correlated step/ramp in timing residuals across the PTA. •Timescale. PTA sensitivity is to very low frequencies (nHz), but detection relies on long time baselines and careful modeling of pulsar spin noise and clock errors. •Cross-pulsar coherence test. A gravitational memory step will have a characteristic angular correlation across pulsars (Hellings–Downs–like signature); searching for correlated steps increases robustness. 8 Analytic Expressions for Memory SNR and Scaling Laws Using (7.4) the single-event RSD SNR (for an idealized step of amplitude ∆hRSD) becomes, neglecting the detector response factor, SNR2 RSD = 4 Zfhigh flow |˜ hRSD(f)|2 Sn(f)df ≈4Zfhigh flow ∆h2 RSD (2πf)2 1 Sn(f)df = ∆h2 RSD ·Imem,Imem ≡1 π2Zfhigh flow df f2Sn(f). (7.11) 8 1. Collective excitation: If Nmicroscopic degrees of freedom change coherently, ∆µν scales as NL2 Pl/L2. For macroscopically large N(coherent domain size), the suppression can be partially offset. 2. Non-perturbative rearrangement: Near Planckian curvature, the effective coupling between geometry quanta can change, enabling large δj and a much larger ζthan naively expected. These mechanisms justify treating κas an effective parameter encoding microphysical amplification beyond simple dimensional counting. 5.2 Spacetime Foam and Topology Change The spacetime foam picture envisions transient microscopic topological fluctuations (wormholes, baby universes, handle attachments) populating the path integral. High-curvature regions can catalyze topologychanging processes; if such a process does not completely re-annihilate, a permanent topological imprint can survive in the coarse-grained geometry and appear as ∆µν. Schematically, a foam contribution to the expectation value in (8.2) appears as: ∆µν ∼X foam sectors s e−Ss/ℏG(s) µν ,(8.12) where Ssis the action of sector sand G(s)is the coarse metric imprint of that sector. The threshold Icrit is the curvature scale at which contributions from nontrivial sectors become unsuppressed. 6 Holography, Soft Modes, and Information Retention 6.1 Holographic Interpretation (Boundary Stress to Bulk Residual) In a holographic duality (AdS/CFT), the asymptotic fall-off of the bulk metric encodes the boundary CFT stress tensor T(CFT) ab : gab(r, x) = g(0) ab (x) + 1 rd−2h(d) ab (x) + ··· , T(CFT) ab (x)∝h(d) ab (x).(8.13) A bulk event that deposits energy into the boundary degrees of freedom can change the late-time expectation value ⟨T(CFT) ab ⟩by a nonzero amount. Through holographic renormalization, this induces a permanent change in the bulk metric coefficients h(d) ab (x)and thus a residual ∆µν in the bulk. In asymptotically flat spacetimes, the boundary imprint can be phrased in terms of soft gravitons and memory: a change in the vacuum sector of the boundary quantum state (soft hair) corresponds to a classical residual in the bulk metric. 6.2 Soft Graviton Theorems, Memory, and RSD Soft-theorem analyses show that low-frequency (soft) graviton emission is tied to asymptotic symmetries and memory effects. In a quantum language, the emission of soft modes changes the vacuum by dressing it with coherent soft gravitons: |Ψout⟩ ∼ Ssoft|Ψin⟩, where Ssoft is a soft dressing operator. If soft dressing is not exactly undone after an event (e.g., due to non-linear backreaction or coupling to microstates), the late state contains a net soft component whose classical limit is the permanent displacement of test masses (memory). RSD can be seen as the macroscopic expression of this residual soft dressing when high curvature fosters irreversible soft-sector transfer. In other words, RSD is the bulk manifestation of persistent soft graviton dressing plus any additional microstate imprint (soft hair). 4 A semi-quantitative relation can be written: ∆µν(x)∼Zd2ΩS(Ω)Yµν(Ω),(8.14) where S(Ω) is an angle-dependent soft charge sourced by the event, and Yµν are angular basis functions projecting soft charge to bulk metric coefficients. The key point is that, whereas standard soft theorems predict a memory entirely determined by fluxes at null infinity, RSD supplements that memory by adding an internal (near-source) sector that can store and later feed back a portion of the soft charge. 7 Effective Parametrizations and Matching to RSD Phenomenology The microscopic considerations above naturally map onto the phenomenological EFT parameter set used in this paper. A compact matching recipe is: 1. Coupling κ:Arises from the microscopic coupling of curvature to memory degrees of freedom. In the auxiliary-field model (8.3) with O=−□+µ2, dimensional analysis gives: κ∼λ µ∆L2−∆ Pl ,(8.15) where λis a dimensionless microphysical amplitude and ∆depends on the operator dimension chosen for mµν. Non-perturbative instanton amplification can make effective λvery large in high-curvature regions. 2. Activation threshold Icrit:Determined by the curvature scale at which microphysical channels (spin reconfiguration, foam sectors, soft-sector trapping) become unsuppressed. For example, Icrit ∼β/L4 Pl with β∼ O(1 −10n)depending on details; the RSD observable range requires Icrit be reached in mergers but not in ordinary astrophysical regimes. 3. Kernel/relaxation (mass scale µand timescale γ): The operator Osets how long memory persists: a small µ(or γ≪1/Tobs) gives effectively permanent memory on observational timescales. Matching γ∼µto the EFT yields the evolution law (4.2) as the long-wavelength limit of Om= source. In short, the microphysics provides an existence proof for the phenomenological terms κ,Icrit,γ used in earlier sections and gives scaling relations (8.7), (8.11), (8.15) to translate a measured εinto bounds on microscopic parameters. 8 Observational Discriminants of a Quantum-Gravity Origin To distinguish a quantum-gravitational origin of RSD from purely classical phenomenology, the following observational signatures are diagnostic: 1. Quantization/discrete plateaux: If ∆µν arises from changes in discrete microscopic labels (e.g., spin jumps in LQG), one might observe a preferred set of relative residual amplitudes or statistically quantized step sizes in a large event sample. 2. Correlation with horizon quantities: A true microstate rearrangement that crosses the event horizon will correlate ∆µν with changes in horizon area ∆Abeyond classical expectations; numerical relativity enhanced with quasi-local horizon microstate proxies could test this. 3. Non-Gaussian stochastic background: Foam-type processes and instanton events produce rare, large outliers and non-Gaussian tails in the distribution of residuals across many events, in contrast to a smooth classical saturation model. 5 4. Soft-sector signatures at null infinity: If RSD has a holographic/soft origin, one expects correlated changes in asymptotic soft charges (measurable in principle via memory angular patterns) that cannot be fully accounted for by radiative fluxes alone. 5. Scaling with curvature proxies: A clear, sharpened dependence of εon local curvature proxies (peak Weyl amplitude near merger) beyond the smooth dependence predicted by GR memory would support a thresholded, microphysical activation mechanism. 9 Practical Matching and Estimates A pragmatic matching procedure for confronting theory with data: 1. Use NR simulations to extract the near-source curvature invariant I(u)and Weyl electric components Eij(u)during merger. 2. Choose a microscopic model class (e.g., auxiliary field with O=−□+µ2or an LQG-inspired spin-reconfiguration model) and compute the predicted mµν via (8.6) or the relevant statistical coarse-graining. 3. Compute the resulting εusing the propagation operator Lfrom ˘ g4 (eq. 4.9 / 6.5). 4. Compare predicted εto observational constraints (or forecasted sensitivity) and invert to obtain limits on combinations of microscopic parameters (e.g., λ, e−Sinst/ℏ,µ, or coherent occupation number Nin the discrete geometry picture). A worked numeric conversion to go from a measured εto a bound on Sinst is straightforward from (8.7): Sinst ≳ℏln Lp PlF ∆meas ,(8.16) where ∆meas ∼εhGR mem converted to a dimensionless metric amplitude. 10 Summary How Quantum Gravity Completes the RSD Picture 1. Semiclassical path integrals and EFTs with an auxiliary memory field give a clean derivation of the constitutive and evolution laws used in the RSD framework: ∆µν naturally appears as the expectation value or Greens-function solution of a microscopic memory sector coupled to curvature. 2. Discrete geometric pictures (LQG, foam) provide concrete microphysical mechanisms (spin rearrangement, topological sector transitions) that produce thresholded, non-perturbative metric imprints; these feed directly into the parameters κ,Icrit and explain why memory turns on only above a curvature scale. 3. Holography and soft-mode analyses show that residual bulk metric deformations correspond to permanent changes in boundary (soft) charges or to persistent dressing of the vacuum; the resulting classical limit coincides with the RSD memory enhancement. 4. The combination of these three viewpoints gives both theoretical plausibility and an explicit mapping from microphysics to the phenomenological parameters constrained observationally. Sections 47 provided the observationally oriented machinery; this section shows how to interpret any measured εas a diagnostic of quantum gravitational microphysics. 6 1 Appendix A: Tensor Calculus and Perturbative Expansions 1.1 Variation of the EinsteinHilbert Action and the Linearized Einstein Operator Start from the EinsteinHilbert action (geometric units G=c= 1): SEH[g] = 1 16π∫d4x√−gR[g]. Under a small symmetric metric variation δgµν, the standard first-order variations are: δΓρ µν =1 2gρσ(∇µδgνσ +∇νδgµσ −∇σδgµν), δRµν =∇ρδΓρ µν −∇νδΓρ µρ. Expanding and rearranging (using covariant derivatives associated with the background metric gµν) yields the standard identity for the variation of the Ricci tensor: δRµν =1 2(−∇2δgµν −∇µ∇νδg +∇µ∇αδgαν +∇ν∇αδgαµ)+Rµναβδgαβ (A.1) where δg ≡gαβδgαβ,∇2≡gαβ∇α∇β, and Rµναβ denotes terms linear in the background curvature that appear when the background is not flat (explicit terms shown below when needed). Contracting gives: δR =∇µ∇νδgµν −∇2δg −Rµνδgµν. The Einstein tensor Gµν =Rµν −1 2gµνRtherefore varies as: δGµν =δRµν −1 2gµνδR −1 2δgµνR. Substitute (A.1) and simplify. Keeping only first-order terms in δg, one obtains the standard linearized Einstein operator acting on δgµν: δGµν[δg] = 1 2[−∇2δgµν −∇µ∇νδg +∇µ∇αδgαν +∇ν∇αδgαµ −gµν(∇α∇βδgαβ −∇2δg)]+Cµν[δg], (A.2) where Cµν[δg]collects terms proportional to the background curvature (e.g., Rαµβνδgαβand Rµαδgαν), which vanish when the background is Ricci-flat and of low curvature (we will explicitly retain them for Schwarzschild below). Equation (A.2) is the rigorous origin of eq. (4.3) in the main text. For convenience, we box the flat-background limit used often in the paper: Flat-background (or curvature-small) limit (covariant derivatives →partials): δGµν[δg]≃1 2(−∂2δgµν −∂µ∂νδg +∂µ∂αδgαν +∂ν∂αδgαµ −ηµν(∂α∂βδgαβ −∂2δg)).(A.3) 1.2 Treating ∆µν as a First-Order Deformation: Effective Memory StressEnergy We set ˜gµν =gµν + ∆µν and expand the Einstein equation: Gµν[˜g] = 8πT(matter) µν to first order in ∆µν: Gµν[g] + δGµν[∆] + O(∆2) = 8πT(matter) µν . 1 Rearrange to define an effective memory stressenergy T(mem) µν : Gµν[g] = 8π(T(matter) µν +T(mem) µν )+O(∆2), T(mem) µν ≡ − 1 8πδGµν[∆].(A.4) To linear order, the contracted Bianchi identity applied to Gµν[g](together with the evolution equation for ∆µν introduced later) ensures ∇µ(T(matter) µν +T(mem) µν ) = 0. Explicit verification proceeds by computing ∇µδGµν[∆] and substituting the evolution law for ∆µν (eq. (4.2) in main text). The important point is that ∆µν can be consistently absorbed into the right-hand side as a conserved effective source at first order. 1.3 Wave Equation for ∆µν about Minkowski de Donder Gauge and Retarded Solution Take gµν =ηµν (Minkowski) and treat both the radiative perturbation hµν and the residual field ∆µν as small. Work in the de Donder (harmonic) gauge for ∆µν: ∂µ¯ ∆µν = 0,¯ ∆µν ≡∆µν −1 2ηµν∆, with ∆≡ηαβ∆αβ. In this gauge, the flat-limit linearized Einstein operator simplifies: δGµν[∆] = −1 2□¯ ∆µν, where □≡ηαβ∂α∂β. The linearized field equation that ∆µν satisfies (interpreting the RHS as the tidal/Weyl source introduced in ˘ g4) becomes: □¯ ∆µν =−2Sµν,(A.5) with a source: Sµν(x)≡8πT(mem) µν (x)≃ −δGµν[∆]source form ≈ −2κW (I)PµαPνβEµν, where the last approximate equality shows the constitutive assumption that the driving source is proportional to the projected electric part of the Weyl tensor (see ˘ g4). The retarded solution is the usual Greens-function integral: ¯ ∆µν(x) = −2∫d4x′GR(x, x′)Sµν(x′),(A.6) with the Minkowski retarded Green’s function GR(x, x′) = 1 2πΘ(t−t′)δ((x−x′)2). In the far-field wave-zone, one may simplify using retarded-time coordinates u=t−r, and obtain the asymptotic relation used in ˘ g4.5: Eij(u)≃ −1 2∂2 uhTT ij (u)⇒∆ij(u)≃ −κ 2∫u −∞ W(I(u′))∂2 u′hTT ij (u′)du′, which after integration by parts yields the boundary form used in the main text: ∆(∞) ij ≈ −κ 2∆(∂uhTT ij )(for narrow, active W).(A.7) 2 1.4 Perturbative Expansion around the Schwarzschild Background (Useful for NearSource Activation) Let the background be the Schwarzschild metric of mass M(geometric units G=c= 1): ds2=−(1−2M r)dt2+(1−2M r)−1 dr2+r2dΩ2. Schwarzschild is vacuum (Rµν = 0), so the Weyl tensor equals the Riemann tensor; the nonzero tetradframe components of the electric part of the Weyl tensor (in an orthonormal frame {ˆ t, ˆr, ˆ θ, ˆ ϕ}) are: Eˆ rˆ r=−2M r3,Eˆ θˆ θ=Eˆ ϕˆ ϕ=M r3.(A.8) A standard curvature invariant (Kretschmann scalar) for Schwarzschild is: I≡RαβγδRαβγδ =48M2 r6.(A.9) The activation condition I > Icrit defines a critical radius rcrit at which residual memory production becomes allowed: rcrit =(48M2 Icrit )1/6 = 481/6M1/3I−1/6 crit .(A.10) Thus, for given Icrit and mass M, one obtains the spacetime region (a shell near the merger/horizon) where the constitutive kernel W(I)activates. Leading-order scaling of ∆µν near-source. Use the constitutive (integral) form ∆µν ∼κ∫W(I)Edτ. Estimate for a single activation event: ∆∼κEpeakTact ∼κM r3 act Tact,(A.11) where ract is a typical radius of activation (e.g., a few M) and Tact the proper-time width of the highcurvature phase, which for compact mergers scales like Tact ∼ O(M). Thus, a convenient parametrization is: ∆∼κM2 r3 act .(A.12) This formula is dimensionally transparent in geometric units and useful for order-of-magnitude estimates. If activation occurs within ∼ract ≈αM (with α=O(1 −10)), then: ∆∼κM2 (αM)3=κ1 α3M. This shows the important scaling: for fixed κ, the residual amplitude decreases with increasing mass Mif expressed in this particular parametrization because larger Mspreads the curvature over larger length/time scales. (Numerical evaluation must use the actual NR near-source scales to get precise values.) 1.5 Dimensional Analysis and Scaling Laws (Careful Units) Dimensions: Metric components are dimensionless. In geometric units, length and time have the same dimension; curvature has dimension L−2. Consider the constitutive integral (eq. (4.1) of the main text): ∆µν ∼κ∫K(τ, τ′)W(I)Edτ′. 3 •[E] = L−2. •[dτ′] = L. • So the integrand carries dimension L−1. To make ∆µν dimensionless, κmust carry dimension of length: [κ] = L(= time in geometric units).(A.13) This corrects casual earlier statements that treated κas dimensionless; hereafter treat κas a parameter with units of length/time. It is convenient to parametrize κas: κ=αL∗, L∗a chosen microscopic length (e.g., LPl), with αdimensionless (possibly large). Scaling for the fractional enhancement ε.Use the wave-zone result ∆ij ∼κ∫∂2 uhijdu′∼κ∂uh. Characteristic scales: ∂uh∼h0/T. For the GR memory ∆hGR ∼h0. Thus: ε≡δhmem hGR mem ∼∆ h0∼κ(h0/T) h0∼κ T. Therefore, the dimensionally correct relation is: ε∼κ T,(A.14) or, with the factor 1 2retained from integration-by-parts conventions in the main text: ε≈κ 2T. This reproduces eqs. (4.10) and (6.6) in the main text with dimensions properly accounted for. Energy bookkeeping scaling. The effective memory energy density scales as ρmem ∼(16π)−1(∇∆µν)2. Using ∇ ∼ 1/T and ∆µν ∼κh0/T, the total memory energy (volume V∼T3) is: Emem ∼1 16π ∆2 T2T3∼1 16π∆2T∼1 16πκ2h2 0 T. Compare with the radiated GW energy EGW ∼h2 0T(estimates up to order-unity geometric factors). Thus: Emem EGW ∼κ2 T2∼ε2. So the simple and robust constraint Emem ≪EGW becomes |ε| ≪ 1. Equivalently, an observational upper bound on εimplies a bound on κ: κ≲Tεmax.(A.15) 1.6 Mapping κto Microscopic Scales If one chooses L∗=LPl, then κ=αLPl. Using (A.14), the expected fractional enhancement for a source with characteristic timescale Tis: ε∼αLPl T. For astrophysical mergers, Tis macroscopic (milliseconds to seconds), so a pure Planck-length prefactor without amplification (α∼1) is negligible. That motivates either (i) an effective amplification factor α≫1coming from nonperturbative microphysics (instanton factors or coherence of many microscopic degrees of freedom), or (ii) choosing L∗≫LPl as the effective length governing memory coupling. In the main text, we left κphenomenological for precisely this reason. 4