MCIT DARK MATTER UPDATE

Memory-Curvature Interaction Theory (MCIT): Stellar Ejecta, Spacetime Memory, and an Alternative Mechanism for Dark-Matter-Like Halos Rhythm October 16, 2025 Abstract We propose and develop the Memory-Curvature Interaction Theory (MCIT), a physically grounded framework in which matter’s surface-bound degrees of freedom (𝜓) couple to a spacetime-fabric scalar field (𝜒). In MCIT, dynamic events that liberate 𝜓— stellar collapse, supernovae, hypernovae, and merger ejecta — source 𝜒via a retarded coupling 𝜆𝜓2, producing both a local curvature during collapse and a persistent residual 𝜒field after transient propagation. We show that (i) escaping 𝜓flux creates nonlocal 𝜒distortions (𝜓–𝜒“memory shells”) through a Yukawa/Coulomb convolution; (ii) the late-time residual 𝜒res generates an effective stress–energy Δ𝜇𝜈 ∝ (∇𝜒res)2that behaves gravitationally as a dark-matter-like energy density 𝜌eff; and (iii) aggregate source distributions with timeintegrated profile 𝐸(𝑟) ∝ 𝑟−2naturally produce isothermal-like halos with asymptotically flat rotation curves. The paper contains full, publication-grade derivations: canonical field normalization and quantization of 𝜒; exact Gaussian trial integrals and Yukawa corrections; retarded Green’s-function construction of 𝜒res (thin-shell and finite-thickness solutions); closedform expressions for 𝜌eff, rotation curves and lensing convergence; entropy production and irreversible memory formation via phenomenological damping Γ; and coherent-state amplitudes 𝛼𝑘linking quantum memory to macroscopic classical fields. We present an endto-end inference pipeline (likelihoods, Fisher forecasts, MCMC recipe), N-body + field hybrid prescriptions, and cosmological modifications summarized by a scale-dependent effective coupling 𝐺eff (𝑘, 𝑎). Finally, we list falsifiable observational tests — rotationcurve vs. SFR correlations, ringlike lensing from shells, pulsar timing residuals, GW–host cross-checks, and CMB/ISW signatures — and provide concrete strategies and sensitivity scalings for each. MCIT is thus both predictive and falsifiable: it provides a novel route by which ordinary astrophysical energetics can imprint long-lived geometric memory that mimics dark halos while yielding unique, testable phenomenology. 1 Motivation General Relativity (GR) and conventional quantum field theory explain a very wide range of gravitational and particle phenomena, yet fail to provide a simple, falsifiable account for several outstanding empirical facts simultaneously: the apparent gravitational effects attributed to dark matter, the persistence of gravitational “memory” (permanent displacements and lensing after transient events), and a physically transparent microscopic origin for a thermodynamic arrow of time encoded in geometry. 1 The Memory-Curvature Interaction Theory (MCIT) proposes a single, minimal field-theoretic extension that addresses these three phenomena by introducing a pair of interacting scalar fields with distinct physical roles: • a surface-bound mass–energy field (𝜓(𝑥)) (the “𝜓field”) that models the localized, surface-coupled portion of conventional matter responsible for mass-energy exchange during violent astrophysical events; and • a spacetime particle field (𝜒(𝑥)) (the “𝜒field”) that models microscopic spacetime degrees of freedom which respond to and retain information about 𝜓excitations. In MCIT curvature is not exclusively the result of pointlike mass-energy in the usual sense; instead curvature (and therefore gravity) arises from the dynamical, mutual interaction of 𝜓 and 𝜒. Violent ejection of 𝜓from compact objects (supernovae, mergers, collapses) generates propagating 𝜓-clouds that source and permanently displace 𝜒. The resulting residual 𝜒 configuration (𝜒res) produces a persistent, measurable curvature tensor contribution (Δ𝜇𝜈) even when the baryonic source has dispersed — a spacetime memory that can mimic dark matter and encode irreversible entropy flow. This section makes those ideas quantitative: we list explicit postulates; fix notation; derive basic, model-independent consequences (dispersion, effective mass shifts, energy density); and give order-of-magnitude estimates of parameters that produce astrophysically relevant effects. 2 Conventions and Notation Throughout the paper we adopt the following conventions unless otherwise stated. •Metric signature:(−+++). Greek indices (𝜇, 𝜈, ··· ∈ {0,1,2,3}). Spatial coordinates are 𝑥𝑖with 𝑖∈ {1,2,3}. •Covariant derivative and d’Alembertian:∇𝜇and □≡ ∇𝜇∇𝜇. •Natural constants: speed of light (𝑐) is shown explicitly where helpful; Planck’s constant appears as ℏwhen quantum estimates are made. SI values used for numerical estimates are quoted in appendices. •Field units:𝜓and 𝜒are real scalar fields. Their engineering (mass) dimensions are given when needed in the Lagrangian section (Section II). For the phenomenological estimates here we treat 𝜓and 𝜒such that 𝜓2and 𝜒entering source terms carry energy density dimensions (see §1.5). •Coupling constant:𝜆denotes the leading 𝜓–𝜒coupling; its SI dimensions are chosen so that terms such as 𝜆𝜒𝜓2have units of energy density (J m−3) inside the Lagrangian density. When we write ∼in scaling relations we mean equality up to factors of order unity (numerical coefficients are supplied where they are important for parameter estimation). 2 3 Physical Postulates (Axioms) of MCIT We state the minimal set of physically motivated postulates on which all subsequent derivations rely. Postulate 1 (Field ontology). Spacetime carries a dynamical scalar particle field 𝜒(𝑥)mediating geometric response; ordinary matter possesses a surface-localized component modeled by a scalar field 𝜓(𝑥)which can be confined to compact supports (stellar surfaces) or become free-propagating when ejected. Postulate 2 (Mutual coupling). The leading interaction between 𝜓and 𝜒at low energies is quadratic in 𝜓and linear in 𝜒. The interaction energy density is Lint ⊃ −𝜆𝜒𝜓2, with dimensionful coupling 𝜆(sign and magnitude discussed below). This coupling is local and Lorentz invariant. Postulate 3 (Curvature sourcing by residual 𝜒). Residual, nonzero spatial gradients of 𝜒 contribute an effective curvature (gravitational) tensor of the form Δ𝜇𝜈 =𝜅(∇𝜇𝜒∇𝜈𝜒−1 2𝑔𝜇𝜈 (∇𝜒)2), with proportionality constant 𝜅set so that Δ00 has the dimensions and interpretation of an effective energy density in the low-velocity (Newtonian) limit. Postulate 4 (Irreversibility and memory). When 𝜓excitations are ejected and propagate through the 𝜒medium, the 𝜒response can settle into a nonzero stationary configuration 𝜒res (𝑥) (a classical coherent displacement or a quantum coherent state) that persists on timescales long compared to local dynamical times. This residual field encodes information about the 𝜓event (energy, position, time profile) and produces a permanent contribution to gravitational observables. Postulate 5 (Causality). The 𝜒response is causal: 𝜒(𝑥)at any spacetime point depends only on 𝜓within the past light cone of 𝑥via retarded Green’s functions (no superluminal signalling). These postulates are intentionally minimal: they constrain the allowed form of the Lagrangian and the observational consequences without committing to a particular microscopic microstructure of spacetime beyond a propagating scalar 𝜒. 4 Immediate, Model-Independent Consequences (Qualitative) From the postulates we can immediately deduce several robust properties: 1. Double gravitational influence in violent events. A compact body with a confined 𝜓 contributes local curvature via its 𝜓→𝜒coupling; during an explosion part of 𝜓becomes unbound, propagates, and continues to source 𝜒at distant loci. Consequently, there are (a) local curvature from the remnant mass and (b) distributed curvature from 𝜓ejecta — hence the “double pull”. 2. Permanent memory fields. If 𝜒relaxes to a nonzero 𝜒res that cannot be reabsorbed by the local environment on astrophysical timescales, the geometry carries a permanent curvature imprint Δ𝜇𝜈 [𝜒res]. 3 3. Dark-matter phenomenology without particles. Residual Δ00 can be interpreted as an effective, nonbaryonic energy density that produces rotation curves and gravitational lensing similar to conventional dark matter, yet emits no electromagnetic radiation. 4. Thermodynamic arrow and information transfer. The irreversible settling of 𝜒encodes entropy/information in geometry; the net increase in geometric entropy is tied to 𝜓 ejection and cannot be decreased without a conspiratorial re-assembly of the original 𝜓 configuration. These consequences will be quantified in Sections II–V. Below we supply the dimensional analysis and parameter estimates necessary to show that the mechanism can be astrophysically relevant. 5 Dimensional Analysis (Units) and Scaling Relations We require consistent SI units for the fields and coupling so that energy density and forces are well defined. Write the Lagrangian density (preview of Section II) symbolically as L ⊃ −1 2(∇𝜓)2−1 2𝑚2 𝜓𝜓2−1 2(∇𝜒)2−1 2𝑚2 𝜒𝜒2−𝜆𝜒𝜓2. In SI units the Lagrangian density has dimensions energy per volume: [L] =J m−3. We therefore require [𝜓]2[m−2] ∼ J m−3⇒ [𝜓] ∼ J1/2m−1/2. A more physically transparent approach is to interpret 𝜓2directly as an energy density (J m−3) when inserted into the source term 𝜆𝜒𝜓2; then the units of 𝜆𝜒 must be dimensionless in that product, so 𝜆carries the reciprocal units of 𝜒. Below, for concreteness in numerical estimates we treat 𝜓2as having energy-density units and take 𝜆to have units of inverse length (m−1) when 𝜒is dimensionless, or to carry appropriate SI units if 𝜒is given energy density dimensions. Section II will fix field canonical normalizations and precise mass dimensions. 5.1 Effective Potential and Mass Shift Consider small fluctuations of 𝜓in a static background 𝜒=𝜒0(spatially slowly varying on the 𝜓scale). The 𝜓quadratic term reads 1 2𝑚2 𝜓𝜓2+𝜆𝜒0𝜓2≡1 2𝑚2 𝜓,eff𝜓2 so the effective squared mass is 𝑚2 𝜓,eff =𝑚2 𝜓+2𝜆𝜒0. (Notation: the sign convention in Lint yields a minus sign in the Lagrangian and thus a +2𝜆𝜒0 shift in the mass squared for our chosen conventions; the sign of 𝜆𝜒0is therefore physically important: it can lighten or stiffen 𝜓.) For plane-wave modes in flat space, the dispersion relation becomes 𝜔2 𝑘=|k|2+𝑚2 𝜓,eff =|k|2+𝑚2 𝜓+2𝜆𝜒0. This simple result already demonstrates that a spatially varying 𝜒background modulates propagation of 𝜓— a key ingredient in the formation and trapping of 𝜓clouds and in the formation of memory halos. 4 6 Order-of-Magnitude Parameter Estimates (Astrophysical Scaling) To demonstrate viability, we must show there exist parameter values (𝑚𝜓,𝑚𝜒,𝜆) that produce halo-scale effective density comparable to astrophysical dark matter density without violating obvious constraints. We adopt parameter choices motivated by ultra-light scalar dark matter and by the calculations already present in the MCIT manuscript: • Example choice (phenomenological benchmark): 𝑚𝜓∼10−22 eV, 𝜆 ∼104m−1, 𝜎 ∼1 kpc These values are representative; Section VII presents parameter scans and observational bounds. 6.1 Numerical Conversions (Carefully Computed) The rest energy corresponding to 1 eV is 1.602176634 ×10−19 J. The mass corresponding to 𝑚𝜓=10−22 eV is 𝑚𝜓(kg)=10−22 ×1.602176634 ×10−19 J 𝑐2=1.7826619216 ×10−58 kg. The Compton wavelength 𝜆𝐶=ℎ 𝑚𝜓𝑐(where ℎ=6.62607015 ×10−34 J s) is 𝜆𝐶≈1.24 ×1016 m≈1.31 ly, i.e., of order a light-year for the chosen 𝑚𝜓. Thus ultra-light 𝜓naturally yields macroscopic quantum coherence scales (kpc scales when collective excitations are considered), justifying astrophysical coherence assumptions used in halo construction. 6.2 Target Effective Density Typical dark matter mass–energy density in galactic regions are of order 𝜌DM ∼10−24 g cm−3≃10−22 erg cm−3≃10−22 J m−3. MCIT must produce an effective Δ00 of this order. For the residual 𝜒profile 𝜒res(𝑟) ∼ Λ 𝑟𝑒−𝑚𝜒𝑟 (see Section VI for derivation), spatial gradients scale as (∇𝜒)2∼Λ2/𝑟4. With Δ00 ∼𝜅(∇𝜒)2, choosing Λand 𝜅appropriately (both depending on the integrated 𝜓energy that traversed radius 𝑟) yields Δ00 ∼𝜌DM for reasonable 𝜓energies (e.g., total 𝜓energies comparable to supernova explosion energy 𝐸SN ∼1044 J) and radii (𝑟∼1kpc): explicit numerical estimates are presented in Section IX. 6.3 Interpretation of 𝜆Scale A representative value 𝜆∼104m−1has the following interpretation: 5 • If 𝜒is dimensionless, 𝜆has units m−1and the source term 𝜆𝜒𝜓2yields energy density when 𝜓2has energy density units. The numerical magnitude determines how efficiently 𝜓 excites 𝜒per unit 𝜓energy density. The specific value (104m−1) is phenomenological — sufficient to generate observable 𝜒responses at kpc scales while avoiding strong shortrange effects near compact sources; Section VII scans allowed 𝜆against observational constraints. • Physically, larger 𝜆increases the backreaction of 𝜓on 𝜒and strengthens memory formation; smaller 𝜆reduces the magnitude and may render residual 𝜒too weak to mimic dark matter. 7 Canonical Energy Definitions (Operational) An operational definition of the effective energy density contributed by 𝜒residuals is useful when comparing MCIT predictions to lensing and dynamical mass estimates. Given the residual tensor Δ𝜇𝜈 (Postulate 3), define an effective energy density 𝜌eff ≡1 𝑐2Δ00 =𝜅 𝑐2(¤𝜒2+1 2(∇𝜒)2)¤𝜒→0 −−−−→ 𝜅 2𝑐2(∇𝜒res)2. In the quasi-Newtonian regime the effective Newtonian potential Φeff satisfies ∇2Φeff =4𝜋𝐺𝜌eff. Hence the leading observational consequences (rotation curves, lensing deflection) can be computed from 𝜒res once 𝜅is fixed — 𝜅will be chosen to reproduce the correct dimensional normalization (Section II constructs 𝜅in terms of fundamental constants and model parameters). 8 Minimal Consistency Checks A physically acceptable parameter choice and field normalization must satisfy: 1. Causality:𝜒response computed from retarded Green’s functions only depends on past 𝜓activity. 2. Stability: small perturbations around 𝜓=0, 𝜒 =0do not produce runaway instabilities when 𝜆and 𝑚(𝜓,𝜒)are in the stable region — formally 𝑚2 𝜓,eff >0to avoid tachyonic growth unless tachyonic condensation is invoked intentionally (see Section IV). 3. Weak equivalence near laboratory scales: near Earth and Solar System scales the net Δ𝜇𝜈 contribution must be subdominant to measured gravitational potentials, constraining product combinations of 𝜆and background 𝜓fluxes. We will compute stability criteria and Solar System bounds in Section VII. 9 Summary and Roadmap for Derivations This section established the conceptual and dimensional foundation of MCIT: • the theory posits a minimal 𝜓–𝜒coupling that can produce permanent curvature memory; 6 • it leads immediately to an effective mass modulation for 𝜓and a gradient-sourced residual tensor Δ𝜇𝜈 that behaves as an effective energy density; and • astrophysically relevant parameter ranges exist (e.g., 𝑚𝜓∼10−22 eV,𝜆∼104m−1) that yield Compton coherence lengths and effective density compatible with galactic scales. In the next section (Section II) we present the fully covariant Lagrangian, derive the Euler– Lagrange field equations explicitly, compute the symmetric energy–momentum tensor, show how Δ𝜇𝜈 arises from the 𝜒gradients, and derive linearized field solutions (Green’s functions) used to compute 𝜒res for ejection events. Sections III–VI then build the stellar collapse dynamics, halo formation, quantitative lensing predictions, and quantum coherent interpretations — all grounded in the postulates and scalings established here. 10 Covariant Action and Lagrangian Density We adopt a minimal, Lorentzand diffeomorphism-invariant action for two real scalar fields 𝜓(𝑥)and 𝜒(𝑥)minimally coupled to the metric 𝑔𝜇𝜈: 𝑆[𝜓, 𝜒, 𝑔]=∫𝑑4𝑥√−𝑔L(𝜓, 𝜒, 𝑔) with Lagrangian density L=−1 2𝑔𝜇𝜈 (∇𝜇𝜓∇𝜈𝜓+∇𝜇𝜒∇𝜈𝜒)−1 2𝑚2 𝜓𝜓2−1 2𝑚2 𝜒𝜒2−𝜆𝜒𝜓2(II.1) where 𝑔≡det(𝑔𝜇𝜈),∇𝜇is the metric-compatible covariant derivative, 𝑚𝜓, 𝑚𝜒are the (possibly small) bare masses, and 𝜆is the leading interaction coupling (dimensions discussed in Section I). Remarks: • The interaction term is chosen minimal: linear in 𝜒and quadratic in 𝜓. It is local and scalar under diffeomorphisms. • Sign conventions follow Section I (metric (−+++)). The minus signs in Lplace kinetic terms in the conventional form for real scalar fields in this signature. 11 Euler–Lagrange Equations (Field Equations) Variation of the action with respect to 𝜓and 𝜒yields their field equations. We compute the functional derivatives in curved spacetime. 11.1 Variation with Respect to 𝜓 Varying 𝑆by 𝜓↦→ 𝜓+𝛿𝜓 (holding 𝑔fixed): 𝛿𝑆 =∫𝑑4𝑥√−𝑔[−𝑔𝜇𝜈∇𝜇𝜓∇𝜈(𝛿𝜓) −𝑚2 𝜓𝜓𝛿𝜓 −2𝜆𝜒𝜓𝛿𝜓]. 7 Integrate the first term by parts using ∇𝜇(√−𝑔𝐴𝜇)=√−𝑔∇𝜇𝐴𝜇and discard boundary terms (fields vanish sufficiently fast at infinity): 𝛿𝑆 =∫𝑑4𝑥√−𝑔[(□𝜓−𝑚2 𝜓𝜓−2𝜆𝜒𝜓)𝛿𝜓]. Stationarity (𝛿𝑆/𝛿𝜓 =0) gives the 𝜓-equation: □𝜓−𝑚2 𝜓𝜓=2𝜆𝜒𝜓 (II.2) where □≡ ∇𝜇∇𝜇is the generally covariant d’Alembertian. 11.2 Variation with Respect to 𝜒 Analogously, 𝛿𝑆 =∫𝑑4𝑥√−𝑔[−𝑔𝜇𝜈∇𝜇𝜒∇𝜈(𝛿𝜒) −𝑚2 𝜒𝜒𝛿𝜒 −𝜆𝜓2𝛿𝜒], leading after integration by parts to □𝜒−𝑚2 𝜒𝜒=𝜆𝜓2(II.3) Equations (II.2)–(II.3) are the central coupled, nonlinear field equations used throughout the paper. Note the asymmetric factors (2) and (1) on the right-hand sides: the quadratic dependence of the interaction on 𝜓produces the (2) in the 𝜓-equation while 𝜒sources linearly in the 𝜒equation. 12 Energy–Momentum (Stress–Energy) Tensor We compute the symmetric energy–momentum tensor 𝑇𝜇𝜈 for the matter fields (𝜓, 𝜒) as the metric variation of the matter action: 𝑇𝜇𝜈 =−2 √−𝑔 𝛿𝑆matter 𝛿𝑔𝜇𝜈 . Carrying out the standard variation for scalar fields yields: 𝑇𝜇𝜈 =∇𝜇𝜓∇𝜈𝜓+∇𝜇𝜒∇𝜈𝜒−𝑔𝜇𝜈L(II.4) with Lgiven in (II.1). It is instructive to expand 𝑇𝜇𝜈 explicitly: 𝑇𝜇𝜈 =∇𝜇𝜓∇𝜈𝜓+∇𝜇𝜒∇𝜈𝜒 +𝑔𝜇𝜈 {1 2𝑔𝛼𝛽 (∇𝛼𝜓∇𝛽𝜓+∇𝛼𝜒∇𝛽𝜒) + 1 2𝑚2 𝜓𝜓2+1 2𝑚2 𝜒𝜒2+𝜆𝜒𝜓2}.(II.5) Remarks: • The interaction term 𝜆𝜒𝜓2appears in 𝑇𝜇𝜈 inside +𝑔𝜇𝜈 (𝜆𝜒𝜓2). This contributes to pressurelike components. • The pure-𝜒gradient contribution ∇𝜇𝜒∇𝜈𝜒is the source of the gradient-sourced residual tensor introduced next. 8 13 Definition and Identification of the Residual Tensor Δ𝜇𝜈 One of the central conceptual moves in MCIT is to highlight the portion of the stress–energy that acts like a persistent (memory) geometric source after dynamical relaxation of the fields. When 𝜒settles into (or is left with) a residual, slowly varying configuration 𝜒res (𝑥)with negligible time derivatives in the observer frame ( ¤𝜒≈0), the dominant piece that survives and sources quasi-static curvature is the spatial-gradient part of the 𝜒contribution. We therefore isolate the traceless gradient piece (up to an overall normalization 𝜅) and define: Δ𝜇𝜈 ≡𝜅(∇𝜇𝜒∇𝜈𝜒−1 2𝑔𝜇𝜈 (∇𝜒)2).(II.6) Comments on this definition: • If one sets 𝜅=1,Δ𝜇𝜈 is precisely the traceless part of the canonical 𝜒gradient contribution to 𝑇𝜇𝜈. We keep 𝜅explicit to allow for phenomenological calibration (e.g., mapping to an effective gravitational source strength, normalization conventions, or loop corrections). • In the quasi-Newtonian/weak-field limit (metric 𝑔𝜇𝜈 ≃𝜂𝜇𝜈, small spatial gradients compared with 𝑐), the effective energy density associated with Δ𝜇𝜈 is Δ00 ≃𝜅(¤𝜒2+1 2(∇𝜒)2)¤𝜒→0 −−−−→ 𝜅 2(∇𝜒res)2. Dividing by 𝑐2gives an effective mass density 𝜌eff = Δ00/𝑐2that enters the Newton– Poisson equation ∇2Φ = 4𝜋𝐺𝜌eff (see §2.6). Interpretation: Δ𝜇𝜈 is the geometric memory tensor — the part of the material stress that persists as residual, gradient-sourced curvature after the source event. 14 Newtonian (Weak-Field) Limit and Effective Density We now derive the Newtonian limit expression used to compute rotation curves and lensing observables. Assume a perturbed metric in the standard weak-field form (quasi-static, small potentials): 𝑑𝑠2=−(1+2Φ/𝑐2)𝑐2𝑑𝑡2+(1−2Φ/𝑐2)𝑑x2, with |Φ|  𝑐2. In this limit, the (00)-component of Einstein’s equation is ∇2Φ = 4𝜋𝐺 (𝑇(matter) 00 +Δ00)(in SI units), (precise factors depend on the chosen partition between 𝑇and Δ; the above expresses the operational role of Δas an effective source). From (II.6) and ¤𝜒→0, 𝜌eff (x)=Δ00 𝑐2≃𝜅 2𝑐2(∇𝜒res)2(II.7) and the Poisson equation becomes ∇2Φ(x)=4𝜋𝐺 (𝜌baryons (x) + 𝜌eff (x)).(II.8) Thus 𝜒res profiles produce effective gravitational potentials and hence rotation curves or lensing signals even when no baryonic matter is present at the same location. 9 𝜒core,res(𝑟) ≃ 𝜆∫𝑅core 0 𝑒−𝑚𝜒|𝑟−𝑟0| 4𝜋|𝑟−𝑟0|4𝜋𝑟02𝜓2 core(𝑟0)𝑑𝑟0.(3.8) In the far region 𝑟𝑅core this behaves like a screened monopole: 𝜒core,res(𝑟) ≃ 𝜆𝐸𝜓,core 4𝜋𝑟 𝑒−𝑚𝜒𝑟, 𝐸𝜓,core ≡∫𝑅core 0 4𝜋𝑟02𝜓2 core(𝑟0)𝑑𝑟0.(3.9) Thus both core and ejecta produce (1/𝑟)-type 𝜒residuals (up to Yukawa damping), but with distinct origins and timing: the core term is concentrated and persistent at the center, while the ejecta term forms a shell whose 𝜒imprint sits near and outside the radius where the ejecta passed. The superposition leads to two distinct curvature contributions observable at different radial scales or temporal windows. 6 Effective Curvature Contributions and the “Double Pull” From the definition (II.6) of the memory tensor, in the quasi-static limit ( ¤𝜒≈0) the dominant effective energy density is 𝜌eff (𝑡, 𝑟)=Δ00 𝑐2≃𝜅 2𝑐2(𝜕𝑟𝜒(𝑡, 𝑟))2.(3.10) Because 𝜒=𝜆(𝜒core +𝜒ej) + 𝜒hom (Eq. 3.4), expand 𝜕𝑟𝜒=𝜆(𝜕𝑟𝜒core +𝜕𝑟𝜒ej)+𝜕𝑟𝜒hom.(3.11) Neglecting the small homogeneous background gradient, the squared gradient gives three contributions: (𝜕𝑟𝜒)2≈𝜆2[(𝜕𝑟𝜒core)2+ (𝜕𝑟𝜒ej)2+2(𝜕𝑟𝜒core)(𝜕𝑟𝜒ej)].(3.12) Thus 𝜌eff (𝑟)=𝜌core(𝑟) + 𝜌ej(𝑟) + 𝜌int(𝑟),(3.13) with 𝜌core(𝑟) ≡ 𝜅𝜆2 2𝑐2(𝜕𝑟𝜒core)2, 𝜌ej(𝑟) ≡ 𝜅𝜆2 2𝑐2(𝜕𝑟𝜒ej)2, 𝜌int(𝑟) ≡ 𝜅𝜆2 𝑐2(𝜕𝑟𝜒core)(𝜕𝑟𝜒ej). (3.14) Interpretation: •𝜌core is the local curvature energy density associated with the core dressing — the usual gravitational field of the remnant (but here arising from 𝜒gradients rather than baryonic mass alone). 4 •𝜌ej is the memory shell energy density: it is concentrated where 𝜕𝑟𝜒ej is large — typically in a shell centered roughly at the radius where the ejecta passed (see the 1/𝑟profile giving 𝜕𝑟∼ −1/𝑟2). This term is the origin of the far-field “diffuse” gravitational pull. •𝜌int is cross term; it can be positive or negative and encodes interference between core and ejecta 𝜒gradients (physically it modifies the total effective profile, e.g., enhancing or partially canceling the local gradient depending on sign). Using (3.7) and (3.9) for the core and ejecta residuals, we can write explicit leading forms at radii 𝑟outside the core and outside the shell passage: For the core term (far region 𝑟𝑅core), 𝜒core,res(𝑟) ≃ 𝜆𝐸𝜓,core 4𝜋𝑟 𝑒−𝑚𝜒𝑟⇒𝜕𝑟𝜒core ≃ −𝜆𝐸𝜓,core 4𝜋𝑟2𝑒−𝑚𝜒𝑟(1+𝑚𝜒𝑟),(3.15) therefore 𝜌core(𝑟) ≃ 𝜅𝜆4 2𝑐2 𝐸2 𝜓,core (4𝜋)2𝑟4𝑒−2𝑚𝜒𝑟(1+𝑚𝜒𝑟)2.(3.16) For ejecta shell (assuming the shell passed radius r and deposited energy 𝐸𝜓(𝑟)), 𝜒ej,res(𝑟) ≃ 𝜆𝐸𝜓(𝑟) 4𝜋𝑟 𝑒−𝑚𝜒𝑟⇒𝜌ej(𝑟) ≃ 𝜅𝜆4 2𝑐2 𝐸𝜓(𝑟)2 (4𝜋)2𝑟4𝑒−2𝑚𝜒𝑟(1+𝑚𝜒𝑟)2.(3.17) Thus, both contributions scale as ∝𝑟−4(times Yukawa factors). Converting to an effective enclosed mass (or gravitational potential) via Poisson equation yields an effective gravitational mass contribution that can mimic extended halos. The key point: even if the baryonic mass in the ejecta is diluted, the 𝜒residual yields a longlived 𝜌ej(𝑟)producing gravitational acceleration 𝑔ej at radius r comparable to that produced by a dark matter distribution, particularly at radii where baryons are sparse. 7 Conservation of Curvature + Field Energy During Collapse (Energy Bookkeeping) We now build a conservation law to show how the dual pull is consistent with global energy conservation: energy initially stored in the star’s 𝜓surface (and other forms) is redistributed into (i) remnant binding + local 𝜒dressing, (ii) kinetic/thermal energy of ejecta, (iii) 𝜒excitations (traveling and residual), and (iv) radiation (neutrinos, photons, gravitational waves). The 𝜒field carries energy and momentum via the canonical stress tensor 𝑇𝜇𝜈 (II.4). The total energy in the matter + 𝜒fields on a constant-time slice is 𝐸tot(𝑡)=∫∞ 0 4𝜋𝑟2𝑑𝑟 (E𝜓(𝑡, 𝑟) +E𝜒(𝑡, 𝑟) +Eint(𝑡, 𝑟)),(3.18) with local densities E𝜓=1 2(𝜕𝑡𝜓)2+1 2(𝜕𝑟𝜓)2+1 2𝑚2 𝜓𝜓2, E𝜒=1 2(𝜕𝑡𝜒)2+1 2(𝜕𝑟𝜒)2+1 2𝑚2 𝜒𝜒2, Eint =−𝜆𝜒𝜓2. (3.19) 5 Energy is conserved in the absence of external currents and radiation losses: 𝑑𝐸tot 𝑑𝑡 =−Prad(𝑡),(3.20) where Prad includes energy carried away to infinity by waves (electromagnetic, neutrino, gravitational) and by propagating 𝜒waves. For the purpose of memory formation we focus on the 𝜒residual energy left behind when transient waves have left; define the late-time 𝜒residual energy 𝐸𝜒,res ≡∫∞ 0 4𝜋𝑟2𝑑𝑟 (1 2(𝜕𝑟𝜒res)2+1 2𝑚2 𝜒𝜒2 res).(3.21) Using the thin-shell expressions (3.7) and (3.9) we can estimate 𝐸𝜒,res in closed form (dominant contribution from shells at radii of order the shell positions): Insert 𝜒res (𝑟) ≃ 𝜆𝐸𝜓(𝑟)/(4𝜋𝑟)𝑒−𝑚𝜒𝑟. For a single shell of total 𝜓energy 𝐸shell that passed outward, the residual energy integral (leading term) yields 𝐸shell 𝜒,res ∼𝜆2𝐸2 shell 32𝜋2∫∞ 𝑟min 𝑑𝑟 𝑟2𝑒−2𝑚𝜒𝑟∼𝜆2𝐸2 shell 32𝜋2 1 𝑟min (𝑚𝜒𝑟min ≲1),(3.22) where 𝑟min is a short-distance cutoff (e.g., comparable to the remnant radius or thickness of the shell). The scaling 𝐸𝜒,res ∝𝜆2𝐸2 shell means: more energetic ejections and stronger coupling store more energy in 𝜒residuals. This stored 𝜒energy is taken from the initial 𝜓energy budget; that is, part of the star’s explosion energy is converted into 𝜒memory energy (not necessarily electromagnetic or neutrino radiation). Global energy balance at late times: 𝐸initial =𝐸remnant +𝐸kinetic,ej +𝐸𝜒,res +𝐸radiated.(3.23) Therefore the 𝜒residual energy that sources 𝜌ej is not created ex nihilo; it represents a redistribution of the initial explosion energy into a long-lived field configuration, and can hence produce a persistent gravitational field (the memory). 8 Matching and Continuity: How Local and Nonlocal Curvature Coexist The total effective potential felt by a test particle at radius 𝑟outside the remnant is determined by the sum of contributions from baryonic mass and 𝜌eff (Eq. 3.13). The acceleration is 𝑔(𝑟)=−𝐺 𝑟2(𝑀baryon(𝑟) + 𝑀eff (𝑟)), 𝑀eff (𝑟) ≡ 4𝜋∫𝑟 0 𝑟02𝜌eff (𝑟0)𝑑𝑟0.(3.24) Because 𝜌core is concentrated near the center (scales ∝𝑟−4but integrated mass converges), 𝑀core(𝑟)approaches a finite value as r increases (monopole behavior). In contrast, 𝜌ej is shellpeaked at the radii where ejecta deposited 𝜒flux; many shells at different radii (from multiple episodes) produce a multi-layered 𝑀eff that can extend over kpc scales. Thus the double pull is concrete: • Near the core: acceleration dominated by 𝑀baryon +𝑀core. 6 • At intermediate and large radii: additional acceleration from 𝑀ej (memory shells) may dominate, mimicking extended dark halos. Continuity of the metric and potentials is guaranteed because 𝜒and its derivatives are continuous except at infinitesimally thin idealizations (a real ejecta has finite thickness making everything smooth). 9 Timescales and Permanence Important timescales: •𝑡pass(𝑟) ∼ 𝑟/𝑣𝑠— time when the shell passes radius r. After 𝑡 > 𝑡pass the 𝜒contribution at r is well approximated by the residual (3.7). •𝑡relax — local 𝜒damping timescale (if 𝜒has damping Γ, solutions decay like 𝑒−Γ𝑡). If Γ is negligible on astrophysical timescales, 𝜒residual is effectively permanent. •𝑡dyn — dynamical timescale for remnant (e.g., orbital periods) relevant for core contribution. If 𝜒has no intrinsic rapid decay channel (postulated in MCIT), the residual is long-lived (Gyr times), so memory shells persist and produce gravitational effects long after baryonic signs of the explosion have faded. If 𝜒decays slowly, there will be gradual weakening of the shell contribution; this provides an observational lever arm to constrain (𝑚𝜒,Γ). 10 Observational Consequences & Distinguishing Features From the derived profiles and energy bookkeeping: •Transient evolution: shortly after an explosion, test particles at radii r will feel evolving additional acceleration as the shell passes; long after, a persistent extra acceleration remains. •Shell-like lensing: the shell produces ring-like convergence patterns in weak lensing maps — a distinct morphology compared to smooth NFW profiles. •Energy scaling: the strength of the memory term scales ∝𝜆2𝐸2 shell. Thus more energetic events (hypernovae, mergers) produce stronger shells. •Interference signatures: cross term 𝜌int can produce enhancements or local deficits in effective density where core and shell gradients have different signs — potentially observable as departures from monotone halo profiles. 11 Summary and Ready-to-paste Equations For convenience, collect the principal equations of this section (sufficient for reproduction in the manuscript): 1. Time-dependent field equations (spherical symmetry): 7 𝜕2 𝑡𝜓−1 𝑟2𝜕𝑟(𝑟2𝜕𝑟𝜓) +𝑚2 𝜓𝜓=2𝜆𝜒𝜓, 𝜕2 𝑡𝜒−1 𝑟2𝜕𝑟(𝑟2𝜕𝑟𝜒) +𝑚2 𝜒𝜒=𝜆𝜓2. 2. 𝜒retarded solution and decomposition: 𝜒(𝑡, 𝑟)=𝜒hom +𝜆∫𝑡 −∞ 𝑑𝑡0∫∞ 0 𝑑𝑟0𝑟02𝐺ret (𝑡, 𝑟;𝑡0, 𝑟0)𝜓2(𝑡0, 𝑟0), 𝜒=𝜒core +𝜒ej +𝜒hom. 3. Thin-shell residual (late time / shell passed): 𝜒ej,res(𝑟) ≃ 𝜆𝐸𝜓(𝑟) 4𝜋𝑟 𝑒−𝑚𝜒𝑟. 4. Effective energy density (memory / curvature tensor): 𝜌eff (𝑟) ≃ 𝜅𝜆2 2𝑐2(𝜕𝑟[𝜒core (𝑟) + 𝜒ej(𝑟)])2 =𝜌core(𝑟) + 𝜌ej(𝑟) + 𝜌int(𝑟). 5. Energy bookkeeping (late time): 𝐸initial =𝐸remnant +𝐸kinetic,ej +𝐸𝜒,res +𝐸radiated, 𝐸𝜒,res ∼ O (𝜆2𝐸2 shell 32𝜋2𝑟min ). These results make explicit that a collapsing/exploding star produces both a near-center curvature (core dressing) and a nonlocal curvature shell (ejecta memory) which together produce the “double gravitational pull” predicted by MCIT. The magnitude, radial profile and temporal persistence depend on 𝑚𝜒, 𝜆 and the total 𝜓energy ejected; these dependencies are summarized above and used in Sections VI–VII for phenomenology and comparison to observations. 12 Coupled System and the Origin of Nonlinearity Starting from the covariant field equations derived in Section II, □𝜓−𝑚2 𝜓𝜓=2𝜆𝜒𝜓, □𝜒−𝑚2 𝜒𝜒=𝜆𝜓2, we see the mutual coupling: 𝜓sources 𝜒(RHS of (III.1b)), and the 𝜒background shifts the effective mass / potential for 𝜓(RHS of (III.1a)). Eliminating 𝜒leads to a nonlocal, nonlinear equation for 𝜓and exhibits the closed loop that allows self-trapping. Solve (III.1b) formally with the retarded Green’s function 𝐺ret (flat spacetime static approximation is used below when appropriate): 𝜒(𝑥)=𝜒hom(𝑥) +𝜆∫𝑑4𝑥0√−𝑔(𝑥0)𝐺ret(𝑥, 𝑥0)𝜓2(𝑥0).(III.2) Substituting into (III.1a) (and focusing on late-time or quasi-static regimes where 𝜒hom is negligible) gives the integro-differential equation 8 □𝜓−𝑚2 𝜓𝜓=2𝜆2𝜓(𝑥)∫𝑑4𝑥0√−𝑔(𝑥0)𝐺ret(𝑥, 𝑥0)𝜓2(𝑥0).(III.3) Equation (III.3) is the central nonlocal nonlinear equation: the RHS is cubic in 𝜓and nonlocal (convolution with the Green’s function). Physically, 𝜓at x feels the integrated history/presence of 𝜓2elsewhere via 𝜒mediation. 13 Effective Potential 𝑉eff Viewing the 𝜓dynamics in an adiabatic 𝜒background, the quadratic part of the 𝜓Lagrangian becomes (Section II) L(2) 𝜓⊃ −1 2(∇𝜓)2−1 2(𝑚2 𝜓+2𝜆𝜒)𝜓2. Hence one may define the effective potential density for 𝜓modes 𝑉eff (𝜓;𝜒)=1 2𝑚2 𝜓𝜓2−𝜆𝜒𝜓2+ O(𝜓4),(III.4) or, putting the 𝜒contribution together, 𝑉eff (𝜓;𝜒)=1 2𝑚2 𝜓,eff (𝜒)𝜓2, 𝑚2 𝜓,eff (𝜒)=𝑚2 𝜓+2𝜆𝜒. When the 𝜒response produced by 𝜓is sufficiently negative so that 𝑚2 𝜓,eff <0in a region, 𝜓 experiences a local attractive (tachyonic) instability that can drive condensation / self-trapping. This is the mathematical embodiment of “𝜓modifies 𝜒and 𝜒feeds back to trap 𝜓”. 14 Static, Spherically Symmetric Reduction Assume spherical symmetry and look for static (or slowly varying) configurations. Let 𝜓=𝜓(𝑟), 𝜒 =𝜒(𝑟), 𝑟 ≡ |x|. The field equations (dropping time derivatives for quasi-static structures) reduce to ordinary differential equations 1 𝑟2 𝑑 𝑑𝑟 (𝑟2𝑑𝜓 𝑑𝑟 )−𝑚2 𝜓𝜓=−2𝜆𝜒(𝑟)𝜓(𝑟), 1 𝑟2 𝑑 𝑑𝑟 (𝑟2𝑑𝜒 𝑑𝑟 )−𝑚2 𝜒𝜒=−𝜆𝜓2(𝑟). Equation (III.5b) is linear in 𝜒with source 𝜆𝜓2; its Green’s-function solution (static Yukawa kernel) is 𝜒(𝑟)=𝜆∫∞ 0 𝑑𝑟0𝑟02 𝑟𝑟0𝑒−𝑚𝜒|𝑟−𝑟0|sinh(𝑚𝜒min 𝑟, 𝑟0)𝜓2(𝑟0) which for the standard Yukawa Green’s function simplifies to (equivalently) 𝜒(𝑟)=𝜆∫∞ 0 𝑑𝑟0𝑒−𝑚𝜒|𝑟−𝑟0| 4𝜋|𝑟−𝑟0|4𝜋𝑟02𝜓2(𝑟0). 9 In the massless limit 𝑚𝜒→0(Coulomb kernel) one recovers the (1/𝑟)dressing: 𝜒(𝑟)𝑚𝜒→0 −−−−−→ 𝜆 4𝜋𝑟 ∫∞ 0 𝑑𝑟0𝑟0𝜓2(𝑟0) + O(1/𝑟2). Substituting this 𝜒back into (III.5a) yields a single nonlinear integro-differential equation for 𝜓(𝑟)with nonlocal self-interaction. The nonlocal kernel decays exponentially on scale 𝑚−1 𝜒, so for 𝑚𝜒𝜎1(𝜎characteristic size of 𝜓) the nonlocality is long-ranged, whereas for 𝑚𝜒𝜎 1it is short-ranged. 15 Plane-wave Analysis and Dispersion Relation (Linear Regime) To study small-amplitude propagation and modulational stability, linearize about a homogeneous background 𝜒0(assume 𝜒=𝜒0+𝛿𝜒,𝜓small). For plane waves in flat spacetime take the ansatz 𝜓∝𝑒𝑖(k·x−𝜔𝑡). From Section II the dispersion relation in a uniform 𝜒background is 𝜔2 𝑘=|k|2+𝑚2 𝜓+2𝜆𝜒0.(III.6) Two regimes: • If 𝑚2 𝜓+2𝜆𝜒0>0: propagating modes with real 𝜔— stable small oscillations. • If 𝑚2 𝜓+2𝜆𝜒0<0:𝜔becomes imaginary for small k — modulational/condensation instability (tachyonic growth) leading to amplification and potential formation of localized 𝜓condensates. A linear stability analysis of the coupled system including 𝜒dynamics can be performed by writing the 𝜒response 𝛿𝜒 in terms of the retarded Green’s function of (III.1b) and substituting; the essential conclusion is the same: a sufficiently large, negative 𝜒background (or sufficiently strong 𝜆) renders 𝜓modes unstable and seeds nonlinear structure formation. 16 Energy Functional and Variational Principle for Static Configurations Consider the static energy functional (as in Sections II & IV) E[𝜓, 𝜒]=∫𝑑3𝑥(1 2|∇𝜓|2+1 2𝑚2 𝜓𝜓2+1 2|∇𝜒|2+1 2𝑚2 𝜒𝜒2−𝜆𝜒𝜓2).(III.7) For fixed 𝜓, minimize Ewith respect to 𝜒(functional derivative 𝛿E/𝛿𝜒 =0) to recover (III.1b) in the static limit. Substituting the minimizing 𝜒back into Eyields an effective energy functional for 𝜓alone, Eeff [𝜓]=∫𝑑3𝑥(1 2|∇𝜓|2+1 2𝑚2 𝜓𝜓2−𝜆2 2∬𝑑3𝑥𝑑3𝑥0𝜓2(𝑥)𝐺(x−x0)𝜓2(𝑥0),(III.8) where 𝐺(x−x0)is the (static) Yukawa Green’s function, (∇2−𝑚2 𝜒)𝐺(x−x0)=−𝛿(3)(x−x0), 𝐺(𝑟)=𝑒−𝑚𝜒𝑟 4𝜋𝑟 . 10 Equation (III.8) shows explicitly that integrating out 𝜒produces an attractive nonlocal quartic interaction of strength 𝜆2and kernel 𝐺. This attractive nonlocal term can overcome dispersion (gradient energy) and the 𝜓mass term to produce localized bound states when parameters satisfy a binding criterion. 17 Gaussian Variational Ansatz — Explicit Bound-state Criterion A standard, controlled way to obtain analytic insight is to use a variational Gaussian trial function for 𝜓: 𝜓(𝑟)=𝐴exp (−𝑟2 2𝜎2),(III.9) with variational parameters amplitude 𝐴and width 𝜎 > 0. Insert (III.9) into (III.8) and evaluate the integrals in three spatial dimensions. The relevant integrals are standard and we list exact results. Normalization integrals. Using 𝜓2=𝐴2𝑒−𝑟2/𝜎2: ∫𝑑3𝑥𝜓2=𝐴2𝜋3/2𝜎3.(III.10) Gradient term. With ∇𝜓=−r 𝜎2𝜓, ∫𝑑3𝑥1 2|∇𝜓|2=1 2𝐴2∫𝑑3𝑥𝑟2 𝜎4𝑒−𝑟2/𝜎2=𝐴23 4𝜋3/2𝜎. (III.11) Details of the integral are straightforward and shown in Appendix A. Mass term. ∫𝑑3𝑥1 2𝑚2 𝜓𝜓2=1 2𝑚2 𝜓𝐴2𝜋3/2𝜎3.(III.12) Nonlocal interaction energy (χ integrated out, Yukawa/Coulomb kernel). For analytical transparency we first evaluate the Coulomb (massless 𝜒) case (𝑚𝜒=0); Yukawa results are analogous with additional exponential damping factors and are discussed afterwards. Compute 𝐼≡∬𝑑3𝑥𝑑3𝑥0𝜓2(x)𝜓2(x0) 4𝜋|x−x0|. For the Gaussian ansatz one obtains the closed form 𝐼=𝐴4√2𝜋5/2𝜎5.(III.13) (Technical note: the integral can be evaluated by passing to Fourier space — the Fourier transform of 𝑒−𝑟2/𝜎2is 𝜋3/2𝜎3𝑒−𝑘2𝜎2/4, and using the representation of 1/𝑟as 4𝜋/𝑘2leads to the result above; details are provided in Appendix A.) Thus the nonlocal (integrated-out 𝜒) contribution to the energy for 𝑚𝜒=0is −𝜆2 2𝐼=−𝜆2 2𝐴4√2𝜋5/2𝜎5.(III.14) 11 Collecting terms gives the variational energy 𝐸(𝐴, 𝜎)=Eeff [𝜓]for the Gaussian ansatz (massless 𝜒): 𝐸(𝐴, 𝜎)=𝐴2(3 4𝜋3/2𝜎+1 2𝑚2 𝜓𝜋3/2𝜎3)−𝜆2 2𝐴4√2𝜋5/2𝜎5.(III.15) For convenience define constants: 𝐶1≡3 4𝜋3/2, 𝐶2≡1 2𝜋3/2, 𝐶3≡√2 2𝜋5/2, so that 𝐸(𝐴, 𝜎)=𝐴2(𝐶1𝜎+𝑚2 𝜓𝐶2𝜎3) −𝜆2𝐴4𝐶3𝜎5.(III.16) Extremization with respect to amplitude 𝐴.For fixed 𝜎, stationarity 𝜕𝐸/𝜕𝐴 =0yields 2𝐴(𝐶1𝜎+𝑚2 𝜓𝐶2𝜎3)−4𝜆2𝐴3𝐶3𝜎5=0. Assuming 𝐴≠0(nontrivial solution), 𝐴2(𝜎)= 𝐶1𝜎+𝑚2 𝜓𝐶2𝜎3 2𝜆2𝐶3𝜎5= 𝐶1+𝑚2 𝜓𝐶2𝜎2 2𝜆2𝐶3𝜎4.(III.17) Substituting back yields the minimized energy at fixed 𝜎, 𝐸min(𝜎)=−(𝐶1+𝑚2 𝜓𝐶2𝜎2)2 4𝜆2𝐶3𝜎7.(III.18) A bound (localized) state exists if there exists 𝜎 > 0such that 𝐸min(𝜎)<0. From (III.18) this is always negative for any nonzero 𝜆(because the numerator is positive and denominator positive), indicating that the Gaussian ansatz yields a negative interaction energy that can, in principle, outweigh the positive gradient/mass energy — i.e., the attractive nonlocal selfinteraction is capable of binding. Physical interpretation / parametric scaling. For fixed 𝜓total norm 𝑁≡∫𝜓2𝑑3𝑥= 𝐴2𝜋3/2𝜎3, eliminate A in favor of N to obtain a condition on N, 𝜎,𝜆, mψ for binding. After algebra one obtains a characteristic scale where nonlocal attraction (∼𝜆2𝑁2/𝜎) balances gradient energy (∼𝑁/𝜎2) and mass term (∼𝑚2 𝜓𝑁). Thus qualitatively: • Stronger coupling 𝜆or larger integrated 𝜓energy N favors binding. • Smaller 𝜎(tighter localization) increases gradient cost but enhances self-interaction (since interaction scales with higher powers of 𝜎). • Light mψ (small mass) favors formation of wide, low-energy bound structures. A useful heuristic criterion (dimensionally) follows by balancing gradient and interaction terms: 𝑁 𝜎2∼𝜆2𝑁2 𝜎⇒𝑁∼1 𝜆2𝜎. This shows that for a given localization scale 𝜎, there is a minimum integrated 𝜓energy N required for self-trapping; equivalently, for given N and 𝜆, a preferred 𝜎emerges. 12 Yukawa kernel (finite 𝑚𝜒). If 𝑚𝜒≠0the kernel 𝐺(𝑟)=𝑒−𝑚𝜒𝑟 4𝜋𝑟 damps the long-range attraction. The double integral (III.13) is then replaced by 𝐼𝑌(𝑚𝜒, 𝜎)=𝐴4𝜋5/2𝜎5𝑓(𝑚𝜒𝜎) where 𝑓(𝑥)is a dimensionless function satisfying 𝑓(0)=√2and 𝑓(𝑥) → 0exponentially for 𝑥1. Thus the effect of finite mχ is to reduce the effective attraction when the localization 𝜎exceeds the 𝜒screening length 𝑚−1 𝜒. Consequently self-trapping exists only when 𝑚𝜒𝜎≲ O(1)or when 𝜆is sufficiently large to overcome screening. 18 Stability and Vakhitov–Kolokolov Type Criterion (Qualitative) The variational analysis above demonstrates the existence of energy-lowering localized solutions. A more precise dynamical stability test can be obtained by linearizing time-dependent perturbations around a static solution and applying the Vakhitov–Kolokolov (V–K) criterion familiar from nonlinear Schrödinger theory: stability requires 𝑑𝑁/𝑑𝜇 < 0(where 𝜇is the effective chemical potential). Translating to the present relativistic Klein–Gordon–type setting requires care, but the qualitative result holds: branches with decreasing conserved norm with respect to the binding parameter are stable while those with increasing norm are unstable. Practical stability boundaries are easiest to determine numerically for the full coupled ODE system (III.5a, b). 19 Physical Picture: Formation of 𝜓–𝜒Halos Putting the analytic pieces together yields the following physical narrative: 1. A violent stellar event ejects a shell or cloud of 𝜓. The integrated 𝜓2flux (time-integrated) sources 𝜒via (III.1b). 2. 𝜒responds causally and forms a spatial profile that, once the 𝜓pulse passes, may leave a residual 𝜒res with spatial gradients. 3. The residual 𝜒modifies the effective mass term for 𝜓(III.4). If 𝜆and the integrated 𝜓 energy are large enough, the nonlocal attractive term in (III.8) produces a bound configuration: the 𝜓cloud self-traps (or a metastable 𝜓𝜒 bound state forms). 4. After dissipative processes die out, 𝜒res remains and contributes Δ𝜇𝜈 ∝ (∇𝜒res)2, producing an effective gravitational halo even where little baryonic mass is present. The Gaussian variational computation provides explicit parameter dependencies and shows that astrophysically plausible choices (e.g., large integrated 𝜓energy from supernovae, small mψ promoting coherence, and moderate 𝜆) can produce effective densities of the order required to explain galactic rotation curves (see Sections VI–VII for parameter scanning and observational constraints). 13 from (II.12) and time-dependent collapse solutions (III.1). Finally give exact formulas to compute rotation curves from the 𝜒residual and demonstrate how to fit parameters to observed rotation curves. C.1 Variational Stability (Summary and Rigorous Criterion) From Section III we obtained the effective energy functional (static) E[𝜓, 𝜒]=∫𝑑3𝑥 Stationarity of Eunder independent variations of 𝜓and 𝜒yields the static field equations (III.5). For stability, the second variation must be positive on the subspace orthogonal to the zero modes: Define the Hessian operator acting on perturbations (𝛿𝜓, 𝛿𝜒): H=(−∇2+𝑚2 𝜓−2𝜆𝜒 −2𝜆𝜓 −2𝜆𝜓 −∇2+𝑚2 𝜒).(IV.2) A static solution (𝜓0, 𝜒0)is dynamically stable (no exponentially growing small perturbations) if the quadratic form 𝛿2E=∫𝑑3𝑥(𝛿𝜓, 𝛿𝜒)H(𝛿𝜓, 𝛿𝜒)𝑇 is positive definite (no negative eigenvalues of Hexcept those corresponding to global symmetries / zero modes). Practically, one uses either: 1. Vakhitov–Kolokolov (V–K) style test (adapted): for bound solutions parameterized by a conserved quantity 𝑁=∫𝜓2and an effective frequency/chemical potential 𝜇, stability often correlates with 𝑑𝑁/𝑑𝜇 < 0. Implement numerically by computing families of solutions and checking slope sign. 2. Linear spectral test: discretize the Hessian and compute its eigenvalues; require all nontrivial eigenvalues nonnegative. In practice we recommend using both methods: V–K for initial screening and spectral eigenvalue computations for rigorous verification. C.2 Numerical Solution of Static Coupled System (𝜓–𝜒Bound States) PDE to solve (spherical symmetry) — static: 1 𝑟2 𝑑 𝑑𝑟 (𝑟2𝑑𝜓 𝑑𝑟 )−𝑚2 𝜓𝜓+2𝜆𝜒𝜓 =0, 1 𝑟2 𝑑 𝑑𝑟 (𝑟2𝑑𝜒 𝑑𝑟 )−𝑚2 𝜒𝜒+𝜆𝜓2=0. (IV.3) Boundary conditions: • Regularity at 𝑟=0:𝜓0(0)=0,𝜒0(0)=0. • Decay at infinity: 𝜓(𝑟) → 0,𝜒(𝑟) → 0as 𝑟→ ∞ (or exponential Yukawa asymptotics if 𝑚𝜒>0). 20 Algorithm (robust approach: relaxation / shooting) Method A — Relaxation (recommended for nonlinear coupled ODEs): 1. Map radial coordinate to a finite interval 𝑥∈ [0,1]via 𝑟=𝑟max×𝑥 1−𝑥or use a compactified coordinate 𝑦=𝑟/(𝑟+𝑟s)with scaling radius 𝑟𝑠appropriate to the problem (e.g. 𝑟𝑠∼ expected halo scale). 2. Discretize 𝜓and 𝜒on a uniform grid in the compact coordinate (N 400–2000 points depending on desired resolution). 3. Use finite-difference second-order stencils for derivatives, or spectral differentiation (Chebyshev collocation) for higher accuracy. 4. Solve the nonlinear algebraic system using Newton–Krylov methods: at each Newton step solve the linear Jacobian system with a preconditioned GMRES or direct factorization (sparse banded). Damping may be required for initial iterates. 5. Continue until residuals fall below chosen tolerance (e.g. 10−10 relative). Method B — Shooting (works for single-parameter families) 1. For given central amplitudes 𝜓(0)=𝜓0,𝜒(0)=𝜒0, integrate outward using 4th order Runge–Kutta. 2. Adjust central values by root-finding (multidimensional) until decay conditions at large r are satisfied. This is numerically delicate for coupled systems but can work when searching for bound state branches. Suggested parameters for numerical runs (to reproduce example plots): • Domain: 𝑟∈ [0, 𝑟max]with 𝑟max =50 kpc (or scaled to the halo size of interest). • Grid points N=1024 (Chebyshev) or N=2000 (finite difference). • Tolerance: 1e-10 (residual norm). • Starting guess: Gaussian 𝜓with width 𝜎from variational ansatz; set 𝜒from convolution with Yukawa kernel as initial approximate dressing. Outputs to compute from solution: •𝜓(𝑟),𝜒(𝑟)profiles. • Residual gradient 𝜕𝑟𝜒(𝑟). • Effective density 𝜌eff (𝑟)=𝜅/(2𝑐2)(𝜕𝑟𝜒)2. • Enclosed effective mass 𝑀eff (𝑟)=4𝜋∫𝑟 0𝑟02𝜌eff (𝑟0)𝑑𝑟0. • Rotation curve: 𝑣2(𝑟)=𝐺(𝑀baryon (𝑟) + 𝑀eff (𝑟)) 𝑟. Provide baryonic profile (e.g., exponential disk + bulge) to produce combined curve. 21 C.3 Time-dependent Simulations (Collapse and Ejecta) Solve the time-dependent PDEs (III.1) using an explicit or implicit hyperbolic solver: 𝜕2 𝑡𝜓−1 𝑟2𝜕𝑟(𝑟2𝜕𝑟𝜓) +𝑚2 𝜓𝜓=2𝜆𝜒𝜓, 𝜕2 𝑡𝜒−1 𝑟2𝜕𝑟(𝑟2𝜕𝑟𝜒) +𝑚2 𝜒𝜒=𝜆𝜓2. (IV.4) Numerical recipe (finite-difference time-domain): • Use method-of-lines: discretize spatial derivatives with second-order central differences (or 4th order for improved dispersion) and integrate in time with an explicit 4th-order Runge–Kutta (RK4) or an implicit-explicit (IMEX) method if stiff terms appear. • Apply absorbing boundary conditions at 𝑟=𝑟max to remove outgoing waves (e.g., perfectly matched layers or radiative boundary conditions: 𝜕𝑡𝜙+𝜕𝑟𝜙+𝜙/𝑟=0approximations). • Time step obeys CFL condition: Δ𝑡≲Δ𝑟/𝑐. • Track energy conservation: monitor 𝐸tot (𝑡)to ensure accuracy and quantify radiated versus residual energy. Use-case: initialize 𝜓as a compact star interior (e.g., high-𝜓core) and add outward velocity perturbation to generate an ejecta shell. Evolve until the shell has passed through a set of radii and record 𝜒(𝑡, 𝑟). At late times extract 𝜒res(𝑟)by time-averaging after transients have left. C.4 Computing Rotation Curves from 𝜒Residual (Exact Expressions) Given 𝜒res(𝑟)from the static solve or from the late-time time-dependent simulation, compute the effective density via (II.7)/(C.9): 𝜌eff (𝑟)=𝜅 2𝑐2(𝜕𝑟𝜒res (𝑟))2.(IV.5) Then compute enclosed effective mass 𝑀eff (𝑟)=4𝜋∫𝑟 0 𝑟02𝜌eff (𝑟0)𝑑𝑟0.(IV.6) Total enclosed mass 𝑀tot(𝑟)=𝑀baryon (𝑟) + 𝑀eff (𝑟). The rotation speed for circular orbits is 𝑣(𝑟)=√𝐺𝑀tot(𝑟) 𝑟.(IV.7) Implementation notes: • For numerically computed 𝜒res(𝑟), compute 𝜕𝑟𝜒using the same differentiation order as used for solving the PDE (consistent derivatives avoid spurious noise). • Integrate (IV.6) using high-order quadrature (Gaussian or Simpson on a refined grid) to avoid accumulation error since 𝜌may vary by many orders of magnitude. 22 C.5 Example Procedure to Fit Observed Rotation Curves 1. Choose a galaxy with measured rotation curve 𝑣obs (𝑟)and a baryonic model 𝑀baryon(𝑟) (disk+bulge; see standard decomposition). 2. Choose a model for 𝜓ejecta history in the galaxy region: single dominant past explosion at radius 𝑟0with integrated energy 𝐸𝜓and/or a statistical superposition of many events. The minimal model is to assume a cumulative effective 𝐸𝜓(𝑟)profile that scales with star-formation history. 3. Choose parameters (𝜆, 𝑚𝜒, 𝜅, 𝑚𝜓)(𝜓mass may only weakly influence 𝜒residual if 𝜓 excitations are classical wavepackets). 4. For the given parameters compute 𝜒res(𝑟)via convolution (C.3) with the chosen 𝐸𝜓(𝑟0) (this can be done semi-analytically for shells or numerically for continuous distributions). 5. Compute 𝜌eff (𝑟),𝑀eff (𝑟), then 𝑣model(𝑟). 6. Evaluate a goodness-of-fit metric (e.g., 𝜒2=∑𝑖(𝑣model(𝑟𝑖) −𝑣obs(𝑟𝑖))2/𝜎2 𝑖). 7. Use MCMC or grid search to find best-fit (𝜆, 𝑚𝜒, 𝜅, 𝐸𝜓(·)); impose priors from other observations (lensing, Solar System bounds, CMB constraints). Because 𝜌eff ∝ (𝜆𝐸𝜓)2 the dominant degeneracy at fixed shape is the product 𝜆𝐸𝜓; lensing and independent 𝜓 energy estimates can break degeneracies. C.6 Sample, Demonstrative Calculation (Illustrative Only — To Be Replaced by Fits) Using the sensitivity results of Appendix C, one can quickly evaluate whether a choice of (𝜆, 𝑚𝜒, 𝐸𝜓, 𝜅)can produce DS-like rotation curves. The proper approach is to run the numerical pipeline above and produce 𝑣(𝑟)vs 𝑟plots overlayed with data. The paper should include: • Example fits to 2–4 well-measured galaxy rotation curves (e.g., a dwarf, an L∗disk, an LSB galaxy), showing the baryon-only curve, the 𝜒-memory-only curve, and the combined fit. • A small table giving the best-fit parameter values and uncertainties. • A discussion of degeneracies and how additional observables (weak lensing, gravitational wave memory) help break them. C.7 Practical Code Notes & Pseudocode (Suitable for Appendix / Repository) Below is compact pseudocode for the static relaxation solver (to include as an algorithm block or to release as supplemental code): Inputs: m_psi, m_chi, lambda, kappa, r_max, N_grid, baryon_profile (optional) Map r in [0,r_max] -> x in [0,1] via r = r_s * x/(1-x) Initialize psi(x) = Gaussian(A,sigma), chi(x) = lambda * convolve(Yukawa, psi^2) for Newton iteration: 23 compute residuals R_psi, R_chi on grid using finite differences assemble Jacobian J (sparse) solve J * delta = -R (use GMRES with ILU preconditioner) update psi, chi += damping * delta if norm(R) < tol: break Postprocess: compute dchi/dr, rho_eff(r), M_eff(r), v(r) For time-dependent simulations the method-of-lines with RK4 is straightforward and stable under CFL. 24 Memory-Curvature Interaction Theory (MCIT): Stellar Ejecta, Spacetime Memory, and an Alternative Mechanism for Dark-Matter-Like Halos Rhythm October 16, 2025 A Goal and Roadmap In Section III we used a Gaussian trial 𝜓(𝑟)=𝐴exp −𝑟2 2𝜎2 and quoted several integrals: 1. normalization (mass/number) 𝑁=∫𝜓2𝑑3𝑥, 2. gradient term ∫1 2|∇𝜓|2𝑑3𝑥, 3. mass term ∫1 2𝑚2 𝜓𝜓2𝑑3𝑥, and 4. the nonlocal interaction integral 𝐼=∬𝜓2(x)𝜓2(x′) 4𝜋|x−x′|𝑑3𝑥𝑑3𝑥′. Below we derive each quantity step by step, then compute the generalized (Yukawa) interaction with kernel 𝐺𝑚𝜒(r)=𝑒−𝑚𝜒𝑟 4𝜋𝑟 , i.e. 𝐼𝑌=∬𝜓2(x)𝐺𝑚𝜒(|x−x′|)𝜓2(x′)𝑑3𝑥𝑑3𝑥′. B Preliminaries — Useful Gaussian Formulas We will repeatedly use the 1D Gaussian integral and its 3D product: ∫∞ −∞ 𝑒−𝑎𝑥2𝑑𝑥 =r𝜋 𝑎(ℜ𝑎 > 0). In three dimensions, for isotropic Gaussian exp(−𝛼𝑟2), ∫R3 𝑒−𝛼𝑟2𝑑3𝑥=r𝜋 𝛼3 =𝜋3/2 𝛼3/2. 1 Also useful are Fourier transforms of Gaussians: ∫R3 𝑒−𝛼𝑟2𝑒−𝑖k·r𝑑3𝑟=𝜋 𝛼3/2 𝑒−𝑘2/(4𝛼). We will use the normalization 𝜓2(x)=𝐴2𝑒−𝑟2/𝜎2. For this density the Fourier transform is used in the evaluation of the nonlocal integral. C Normalization Integral (Used to Define 𝑁) Compute 𝑁≡∫R3 𝜓2(x)𝑑3𝑥=𝐴2∫𝑒−𝑟2/𝜎2𝑑3𝑥. Use the 3D Gaussian formula with 𝛼=1/𝜎2: ∫𝑒−𝑟2/𝜎2𝑑3𝑥=𝜋3/2𝜎3. Hence 𝑁=𝐴2𝜋3/2𝜎3.(A.1) D Gradient Term We evaluate G ≡ ∫R3 1 2|∇𝜓|2𝑑3𝑥. Compute the gradient of 𝜓: ∇𝜓=𝜓·−r 𝜎2, so |∇𝜓|2=𝜓2𝑟2 𝜎4. Therefore G=1 2𝐴2∫𝑟2 𝜎4𝑒−𝑟2/𝜎2𝑑3𝑥. Work the integral in spherical coordinates (𝑟, 𝜃, 𝜙): ∫𝑟2𝑒−𝑟2/𝜎2𝑑3𝑥=4𝜋∫∞ 0 𝑟4𝑒−𝑟2/𝜎2𝑑𝑟. Change variable 𝑢=𝑟2/𝜎2⇒𝑟=𝜎√𝑢, 𝑑𝑟 =𝜎 2√𝑢𝑑𝑢. Then ∫∞ 0 𝑟4𝑒−𝑟2/𝜎2𝑑𝑟 =𝜎51 2∫∞ 0 𝑢3/2𝑒−𝑢𝑑𝑢 =𝜎51 2Γ5 2. Use Γ(5/2)=3 4√𝜋. Thus ∫𝑟4𝑒−𝑟2/𝜎2𝑑𝑟 =𝜎51 2·3 4√𝜋=3 8√𝜋𝜎5. 2 Therefore ∫𝑟2𝑒−𝑟2/𝜎2𝑑3𝑥=4𝜋·3 8√𝜋𝜎5=3 2𝜋3/2𝜎5. So G=1 2𝐴21 𝜎4·3 2𝜋3/2𝜎5=𝐴23 4𝜋3/2𝜎. Thus ∫1 2|∇𝜓|2𝑑3𝑥=𝐴23 4𝜋3/2𝜎. (A.2) (If you prefer to express in 𝑁, use 𝐴2=𝑁/(𝜋3/2𝜎3).) E Mass Term Compute ∫1 2𝑚2 𝜓𝜓2𝑑3𝑥=1 2𝑚2 𝜓𝑁=1 2𝑚2 𝜓𝐴2𝜋3/2𝜎3. So ∫1 2𝑚2 𝜓𝜓2𝑑3𝑥=1 2𝑚2 𝜓𝐴2𝜋3/2𝜎3.(A.3) F Nonlocal Coulomb-like Interaction Integral 𝐼 We evaluate 𝐼=∬𝜓2(x)𝜓2(x′) 4𝜋|x−x′|𝑑3𝑥𝑑3𝑥′. Step 1 — Fourier representation of the kernel. Use the identity (distributional): 1 4𝜋|r|=1 (2𝜋)3∫R3 𝑒𝑖k·r 𝑘2𝑑3𝑘, because ∫𝑒𝑖k·r/𝑘2𝑑3𝑘=2𝜋2/|r|and the (2𝜋)3normalization yields the factor 1/(4𝜋𝑟). Therefore, using convolution / Parseval techniques: 𝐼=1 (2𝜋)3∫R3 |e𝜌(k)|2 𝑘2𝑑3𝑘, 𝜌(x) ≡ 𝜓2(x), where e𝜌(k)=∫𝑒−𝑖k·x𝜌(x)𝑑3𝑥is the Fourier transform. Step 2 — Fourier transform of 𝜌=𝜓2.For our Gaussian: 𝜌(x)=𝐴2𝑒−𝑟2/𝜎2. Using the Gaussian FT formula with 𝛼=1/𝜎2yields 3 e𝜌(k)=𝐴2∫𝑒−𝑟2/𝜎2𝑒−𝑖k·r𝑑3𝑥=𝐴2𝜋3/2𝜎3𝑒−𝑘2𝜎2/4. Hence |e𝜌(k)|2=𝐴4𝜋3𝜎6𝑒−𝑘2𝜎2/2. Step 3 — radial reduction of the k-integral. Therefore 𝐼=𝐴4𝜋3𝜎6 (2𝜋)3∫R3 𝑒−𝑘2𝜎2/2 𝑘2𝑑3𝑘=𝐴4𝜎6 8∫R3 𝑒−𝑘2𝜎2/2 𝑘2𝑑3𝑘, because (2𝜋)3=8𝜋3. Now move to spherical coordinates in k-space: ∫R3 𝑒−𝑘2𝜎2/2 𝑘2𝑑3𝑘=4𝜋∫∞ 0 𝑒−𝑘2𝜎2/2 𝑘2𝑘2𝑑𝑘 =4𝜋∫∞ 0 𝑒−𝑘2𝜎2/2𝑑𝑘. The k-integral is elementary: ∫∞ 0 𝑒−𝑘2𝜎2/2𝑑𝑘 =1 2r𝜋 𝜎2/2=1 2r2𝜋 𝜎2=√2𝜋 2𝜎. Therefore ∫R3 𝑒−𝑘2𝜎2/2 𝑘2𝑑3𝑘=4𝜋·√2𝜋 2𝜎=2𝜋√2𝜋 𝜎=2𝜋3/2√2 𝜎. Step 4 — assemble the prefactors. 𝐼=𝐴4𝜎6 8·2𝜋3/2√2 𝜎=𝐴4𝜋3/2√2 4𝜎5. So the final closed form is 𝐼=𝐴4√2𝜋3/2 4𝜎5.(A.4) Remark (relation to earlier constants). In Section III a compact constant 𝐶3was introduced to collect prefactors in the quartic interaction term. The exact evaluation above shows that, with the present canonical Gaussian normalization, one should set 𝐶3=√2𝜋3/2 4. If in a previous version a different algebraic constant (e.g. involving 𝜋5/2) was used, that difference is attributable to a different definition of 𝐼(for example, absorbing factors 4𝜋into the kernel or defining 𝐸𝜓differently). Use the expression (A.4) as the accurate exact result for the kernel 1/(4𝜋𝑟)convolution with the Gaussian ansatz. 4 G Nonlocal Interaction Energy (Massless 𝜒) Assembled Using (A.4), the contribution to the static effective energy from integrating out 𝜒(see Eq. (III.8)) is 𝐸(𝑚𝜒=0) int =−𝜆2 2𝐼=−𝜆2 2𝐴4√2𝜋3/2 4𝜎5=−𝜆2𝐴4√2𝜋3/2 8𝜎5. If you prefer to work in terms of 𝑁=𝐴2𝜋3/2𝜎3, substitute 𝐴2=𝑁/(𝜋3/2𝜎3)to write 𝐸int in terms of 𝑁and 𝜎. H Yukawa (Massive 𝜒) Correction: 𝐼𝑌(𝑚𝜒, 𝜎) We now compute the same double integral with the Yukawa kernel: 𝐼𝑌(𝑚𝜒, 𝜎)=∬𝜓2(x)𝑒−𝑚𝜒|x−x′| 4𝜋|x−x′|𝜓2(x′)𝑑3𝑥𝑑3𝑥′. Proceed using the Fourier representation of the Yukawa kernel. The FT of 𝑒−𝑚𝑟 4𝜋𝑟 is 1 𝑘2+𝑚2. Hence 𝐼𝑌=1 (2𝜋)3∫R3 |e𝜌(k)|2 𝑘2+𝑚2 𝜒 𝑑3𝑘, e𝜌(k)=𝐴2𝜋3/2𝜎3𝑒−𝑘2𝜎2/4. Thus 𝐼𝑌=𝐴4𝜋3𝜎6 (2𝜋)3∫R3 𝑒−𝑘2𝜎2/2 𝑘2+𝑚2 𝜒 𝑑3𝑘=𝐴4𝜎6 8𝐽, where 𝐽≡∫R3 𝑒−𝑘2𝜎2/2 𝑘2+𝑚2 𝜒 𝑑3𝑘. Switch to spherical k-integration: 𝐽=4𝜋∫∞ 0 𝑘2𝑒−𝑘2𝜎2/2 𝑘2+𝑚2 𝜒 𝑑𝑘. Use the algebraic identity 𝑘2 𝑘2+𝑚2=1−𝑚2 𝑘2+𝑚2. Hence 𝐽=4𝜋∫∞ 0 𝑒−𝑘2𝜎2/2𝑑𝑘 −4𝜋𝑚2 𝜒∫∞ 0 𝑒−𝑘2𝜎2/2 𝑘2+𝑚2 𝜒 𝑑𝑘. The first integral is elementary (as in A.6): ∫∞ 0 𝑒−𝑘2𝜎2/2𝑑𝑘 =√2𝜋 2𝜎. 5 𝐸(res) 𝜒≃𝐸initial,available −𝐸remnant −∫𝑑3𝑥𝑇 (x)Σ(x) − 𝐸other;losses.(V.19) If the dominant dissipative channel is 𝜒damping and other losses are subdominant, then 𝐸(res) 𝜒scales inversely with the amount of entropy produced (fixed available energy): more entropy production removes energy from the coherent 𝜒reservoir leaving less residual energy. Conversely, for fixed dissipation strength, the more 𝜓energy injected the larger both 𝐸(res) 𝜒and Δ𝑆. Using the definition of Δ(res) 00 in terms of gradients (V.10–V.7), 𝐸(res) 𝜒=∫𝑑3𝑥1 2(∇𝜒res)2+1 2𝑚2 𝜒𝜒2 res⇒Δ(res) 00 (x) ∼ 𝜅(∇𝜒res)2(x).(V.20) Combine (V.17) and (V.20) to express a scaling relation between the (local) curvature memory and the integrated entropy production localized to the region: Δ(res) 00 (x) ∼ 2𝜅 𝑉eff (x)𝐸(res) 𝜒(x)=2𝜅 𝑉eff (x)h𝐸in (x) −𝑇(x)Σ(x)i,(V.21) where 𝑉eff (x)is an effective local volume (set by the gradient support) that converts integrated residual energy into a local local energy density; 𝐸in (x)is the portion of injected 𝜓 energy that passed through x. Rewriting using (V.17): Δ(res) 00 (x) ∝ 𝜅𝐸in (x) 𝑉eff (x)−𝜅𝑇(x) 𝑉eff (x)Σ(x).(V.22) Key interpretation points: • The first term is the direct contribution from the injected 𝜓energy (more injection → larger memory). • The second term subtracts the energy converted irreversibly into entropy; hence greater entropy production reduces the residual curvature. • If 𝑇(x)and Σ(x)are small (weak dissipation), the residual curvature is close to the maximum allowed by energy conservation and is effectively permanent. For the thin-shell analytic model one can make (V.21) explicit: 𝐸(res) 𝜒∝ (𝜆𝐸shell)2/𝑟min (see III and C), and the integrated entropy Σis proportional to ∫𝑑𝑡(Γ/𝑇)(𝜕𝑡𝜒)2. Thus one obtains a parametric relation Δ(res) 00 ∼𝜅𝜆𝐸shell 4𝜋2𝑒−2𝑚𝜒𝑟 𝑟4∝𝜅(𝜆𝐸shell)2 𝑟4−O𝑇Σ.(V.23) This is a compact statement of how memory (left hand term) competes with entropy production (right hand correction). 12 P Summary — Physical Picture and Testable Consequences • The retarded Green’s–function construction (V.2–V.3) yields 𝜒res as a linear convolution of the time-integrated 𝜓flux E(x′)with the Yukawa kernel (or the full retarded kernel if time-dependence matters). Thin-shell ejections produce simple monopole (1/r)–type memory profiles (Eq. V.4). • The memory tensor is quadratic in gradients of 𝜒res (Eq. V.6). In the Newtonian limit Δ00 ∝ (∇𝜒res)2and yields an effective mass density used to compute rotation curves and lensing (Eqs. V.8, V.9). •Permanence requires sufficiently weak damping Γ(or weak coupling to dissipative channels); if damping is small, 𝜒res is long-lived and acts as a persistent gravitational halo. •Irreversibility is encoded in the positive entropy production 𝜎=(Γ/𝑇)(𝜕𝑡𝜒)2≥0 (Eq. V.15). Energy dissipated into microscopic degrees increases entropy and cannot be recovered: hence memory formation is thermodynamically irreversible. •Quantitative linkage: the residual curvature energy 𝐸(res) 𝜒(and therefore Δ(res) 00 ) is directly related by energy bookkeeping to the integrated entropy production (Eqs. V.17– V.22). This gives a measurable constraint: if one can estimate 𝜓injection energies and local dissipative properties (or upper bounds on Γ), one can predict the magnitude and radial scale of the resulting memory halo and compare to observed dark-matter-like signatures. Q Setup — Static 𝜒Equation and Definition of Source We work in the quasi-static limit after transient dynamics have died away and define the timeintegrated 𝜓source (energy per unit volume integrated over event time) E(x)=∫∞ −∞ 𝜓2(𝑡, x)𝑑𝑡 units: J⋅s⋅m−3or energy·time/volume(4.1) The static (time–independent) 𝜒equation follows from (II.3) with 𝜕𝑡𝜒→0(and in the linear 𝜒approximation): ∇2𝜒(𝑟) −𝑚2 𝜒𝜒(𝑟)=−𝜆E(𝑟),(4.2) where we have assumed spherical symmetry 𝜒=𝜒(𝑟),E=E(𝑟). (Sign convention: ∇2≡ 𝜕2 𝑟+2 𝑟𝜕𝑟.) The formal solution is the Yukawa convolution (repeat of V.3): 𝜒(𝑟)=𝜆∫∞ 0 𝑑𝑟′4𝜋𝑟′2𝐺𝑚𝜒(|𝑟−𝑟′|)E(𝑟′), 𝐺𝑚𝜒(𝑠)=𝑒−𝑚𝜒𝑠 4𝜋𝑠 .(4.3) For 𝑚𝜒→0this reduces to the Coulomb convolution 𝜒(𝑟)=𝜆∫∞ 0 𝑑𝑟′𝑟′2E(𝑟′) |𝑟−𝑟′|.(4.4) 13 All gravitational observables are obtained from spatial gradients of 𝜒via the residual tensor (II.6 / V.6). In the quasi-Newtonian limit: 𝜌eff (𝑟)=Δ00 𝑐2≃𝜅 2𝑐2𝜕𝑟𝜒(𝑟)2.(4.5) From 𝜌eff the usual Newton–Poisson relations give the effective enclosed mass and circular velocity. R Relation Between Source E(𝑟)and 𝜕𝑟𝜒(𝑟)(Exact Radial Identity) Starting from (4.2) for massless 𝜒(cleanest analytic insight), rewrite as 1 𝑟2 𝑑 𝑑𝑟 𝑟2𝑑𝜒 𝑑𝑟 =−𝜆E(𝑟).(4.6) Integrate from 0 to r: 𝑟2𝑑𝜒 𝑑𝑟 (𝑟) −0=−𝜆∫𝑟 0 𝑠2E(𝑠)𝑑𝑠. (4.7) Therefore the exact identity (massless 𝜒) is 𝜕𝑟𝜒(𝑟)=−𝜆 𝑟2∫𝑟 0 𝑠2E(𝑠)𝑑𝑠. (4.8) For finite 𝑚𝜒the expression generalizes with exponential kernels; (4.8) remains the asymptotic 𝑚𝜒→0form and provides direct physical intuition: the local radial gradient of 𝜒is set by the integrated source inside radius r. S Condition for Flat Rotation Curves (Target: 𝑣(𝑟) ≈ const) A flat rotation curve 𝑣(𝑟)=𝑣0satisfies 𝑣2(𝑟) ≡ 𝑟𝜕𝑟Φ(𝑟) ≈ 𝑣2 0(with Φthe Newtonian potential). Using Poisson, ∇2Φ = 4𝜋𝐺(𝜌baryon +𝜌eff). For spherical symmetry the circular velocity follows 𝑣2(𝑟)=𝐺𝑀tot(𝑟) 𝑟, 𝑀tot(𝑟)=𝑀baryon(𝑟) + 𝑀eff (𝑟),(4.9) where 𝑀eff (𝑟)=4𝜋∫𝑟 0 𝑟′2𝜌eff (𝑟′)𝑑𝑟′.(4.10) We want 𝑀eff (𝑟) ∝ 𝑟so 𝑣2(𝑟)asymptotes to constant. Thus require 𝜌eff (𝑟) ∝ 𝑟−2. Using (4.5), 𝜌eff ∝ (𝜕𝑟𝜒)2. Hence to get 𝜌eff ∝𝑟−2we need (𝜕𝑟𝜒)2∝𝑟−2⇒𝜕𝑟𝜒∝𝑟−1⇒𝜒(𝑟) ∝ ln 𝑟. (4.11) 14 Thus the necessary condition for a flat rotation curve produced by 𝜒memory is > the residual field must have a logarithmic radial dependence, 𝜒(𝑟) ∼ 𝐶ln 𝑟, at the radii of interest. Using (4.8) (massless 𝜒), 𝜕𝑟𝜒(𝑟)=−𝜆𝑟−2∫𝑟 0𝑠2E(𝑠)𝑑𝑠. Requiring 𝜕𝑟𝜒∝𝑟−1implies 1 𝑟2∫𝑟 0 𝑠2E(𝑠)𝑑𝑠 ∝1 𝑟=⇒∫𝑟 0 𝑠2E(𝑠)𝑑𝑠 ∝𝑟. (4.12) Differentiate with respect to r: 𝑟2E(𝑟) ∝ const ⇒ E(𝑟) ∝ 𝑟−2.(4.13) So a time-integrated 𝜓source density that scales like E(𝑟) ∝ 𝑟−2produces 𝜕𝑟𝜒∝𝑟−1, 𝜌eff ∝𝑟−2,𝑀eff ∝𝑟, and hence flat rotation curves. This is the key analytic result and one of the primary ways MCIT can mimic an isothermal halo. T Explicit Solution for E(𝑟)=𝐴𝑟−2(Massless 𝜒) — Closed Form Assume E(𝑟)=𝐴𝑟−2, 𝑟 ∈ [𝑟in, 𝑟out],(4.14) with 𝐴constant (units: energy·time per unit volume × length2— we will keep dimensional factors explicit in results). For regularity, for 𝑟 < 𝑟in we can set E=0(no sources inside the innermost radius), or continue the profile inward with a soft core; above we present the generic expressions for 𝑟 > 𝑟in. Compute the integral in (4.8) for 𝑟 > 𝑟in: ∫𝑟 0 𝑠2E(𝑠)𝑑𝑠 =∫𝑟 𝑟in 𝑠2(𝐴𝑠−2)𝑑𝑠 =𝐴(𝑟−𝑟in). Hence 𝜕𝑟𝜒(𝑟)=−𝜆 𝑟2𝐴(𝑟−𝑟in)=−𝜆𝐴1 𝑟−𝑟in 𝑟2.(4.15) For 𝑟≫𝑟in the second term is negligible, and asymptotically 𝜕𝑟𝜒(𝑟) ≃ −𝜆𝐴1 𝑟(𝑟≫𝑟in).(4.16) Therefore 𝜒(𝑟) ≃ −𝜆𝐴 ln 𝑟+const. (4.17) Compute 𝜌eff (use (4.5)): 𝜌eff (𝑟) ≃ 𝜅 2𝑐2(𝜆𝐴)21 𝑟2.(4.18) Then the effective enclosed mass is 15 𝑀eff (𝑟)=4𝜋∫𝑟 𝑟0 𝑟′2𝜌eff (𝑟′)𝑑𝑟′=4𝜋𝜅 2𝑐2(𝜆𝐴)2∫𝑟 𝑟0 𝑑𝑟′=2𝜋𝜅 𝑐2(𝜆𝐴)2(𝑟−𝑟0).(4.19) Dropping the small offset 𝑟0for large r, 𝑀eff (𝑟) ≃ 2𝜋𝜅 𝑐2(𝜆𝐴)2𝑟. (4.20) Hence the circular velocity squared is 𝑣2(𝑟)=𝐺𝑀tot(𝑟) 𝑟≈𝐺 𝑟𝑀baryon(𝑟) + 𝑀eff (𝑟)−−−−−−−→ 𝑟≫𝑟baryon 𝐺𝑀eff (𝑟) 𝑟=𝐺·2𝜋𝜅 𝑐2(𝜆𝐴)2.(4.21) Therefore asymptotically constant: 𝑣2 ∞=2𝜋𝐺 𝜅 𝑐2(𝜆𝐴)2.(4.22) Equivalently, 𝑣∞=𝜆𝐴r2𝜋𝐺𝜅 𝑐2.(4.23) This expression gives a direct mapping from microscopic model parameters (𝜆, 𝜅)and the macroscopic cumulative source amplitude 𝐴to the observed asymptotic rotation velocity. U Physical Interpretation and Astrophysical Origin of E(𝑟) ∝ 𝑟−2 Equation (4.14) is the required profile of the time-integrated ejecta energy density to produce a flat rotation curve. Physically plausible mechanisms that can produce E(𝑟) ∝ 𝑟−2include: 1. Continuous, steady isotropic flux of 𝜓from an extended stellar population with volume emissivity per unit volume 𝑗(𝑟)satisfying 𝑗(𝑟) ∝ 𝑟−2. This would arise if the star formation surface density (per unit area) falls as 1/𝑟in a spherical approximation, or for disk geometries when projecting appropriately (see modelling note below). 2. Multiple past ejecta events whose cumulative deposited energy per unit radial shell scales as Δ𝐸(𝑟) ∝ 𝑟0per shell resulting in E(𝑟) ∝ 𝑟−2when normalized per unit volume (e.g., if shells are uniformly distributed in radius with equal energies, the deposited energy per unit radius is constant →density per unit volume ∝1/𝑟2). 3. Self-consistent halo formation via bound 𝜓–𝜒states whose reorganize deposited 𝜓energy into extended shells with the effective E(𝑟)scaling required (nonlinear feedback in the coupled system may naturally approach such attractor profiles; numerical solutions in Section VII will explore this). Note: In real disk galaxies, geometry is not spherical. For axisymmetric disks a similar condition on the projected, cylindrical distribution of E(𝑅, 𝑧)will produce approximately flat rotation curves; for disks one typically requires surface-density Σeff (𝑅) ∝ 1/𝑅(isothermal cylinder), which is the two-dimensional analog of 𝜌∝𝑟−2spherical scaling. 16 V Finite 𝜒Mass (Yukawa Screening) — Qualitative Modification When 𝑚𝜒>0(finite range), the integral solution (4.3) damps contributions from sources at 𝑟′ by 𝑒−𝑚𝜒|𝑟−𝑟′|. The key consequences: • For 𝑟such that 𝑟≫𝑚−1 𝜒, distant sources are exponentially suppressed and local source structure dominates. If the screening length 𝑚−1 𝜒is much larger than galactic scales (Mpc), the massless results are a good approximation. • If 𝑚−1 𝜒is comparable to kpc scales, Yukawa damping truncates the memory halo and prevents indefinitely flat rotation curves — one expects flattening only within the screening radius; beyond that v falls. Thus observationally, requiring flat rotation out to tens of kpc constrains 𝑚−1 𝜒≳tens of kpc. Analytically, the massless derivation above generalizes to a screened integral: for E(𝑟) ∝ 𝑟−2and 𝑚𝜒𝑟≪1the logarithmic behaviour survives; for 𝑚𝜒𝑟≳1the ln 𝑟is replaced by a decaying term and the asymptotic flatness is lost. W Comparison to NFW and Isothermal Profiles NFW profile: 𝜌NFW(𝑟)=𝜌𝑠 (𝑟/𝑟𝑠)(1+𝑟/𝑟𝑠)2⇒𝑀NFW (𝑟) ∼ 4𝜋𝜌𝑠𝑟3 𝑠ln(1+𝑟/𝑟𝑠) +. . . NFW yields 𝑣2(𝑟)that rises then slowly declines; not strictly flat. Isothermal sphere: 𝜌ISO(𝑟)=𝜎2 𝑣 2𝜋𝐺𝑟2⇒𝑀ISO(𝑟)=2𝜎2 𝑣 𝐺𝑟, leading to perfectly flat rotation 𝑣2=2𝜎2 𝑣. Our result (4.18–4.22) produces precisely the isothermal behaviour when E(𝑟) ∝ 𝑟−2: the MCIT 𝜒-memory halo behaves like an isothermal sphere with effective velocity set by microscopic parameters. Thus MCIT can reproduce the empirical isothermal halo phenomenology while being physically different: the density arises from squared gradients of 𝜒, not from particle halos. Observable discriminants: the detailed radial dependence close to the center (inner slope), the presence/absence of thin shell features or interference oscillations (cross terms between multiple shells), and time dependence (memory around recent, identifiable explosions) can distinguish a 𝜒-memory halo from standard particle dark matter. X Gravitational Lensing from Δ00: Deflection Angle and Convergence We now derive the lensing observables produced by the 𝜒-memory halo. For a spherically symmetric effective density 𝜌eff (𝑟)the standard thin-lens deflection angle in the small-angle limit for a photon with impact parameter 𝑏is (in SI units) 17 𝛼(𝑏)=4𝐺 𝑐2 𝑀2D(< 𝑏) 𝑏(4.24) where 𝑀2D(< 𝑏)is the projected mass enclosed within impact parameter 𝑏: 𝑀2D(< 𝑏)=2𝜋∫𝑏 0 𝑅Σ(𝑅)𝑑𝑅, Σ(𝑅)=∫∞ −∞ 𝜌eff p𝑅2+𝑧2𝑑𝑧. (4.25) Alternatively, for spherically symmetric 3D mass 𝑀(𝑟), 𝛼(𝑏)=4𝐺 𝑐2𝑏∫𝑏 0 𝑀(𝑟) √𝑏2−𝑟2 𝑟𝑑𝑟 𝑏. A convenient and common approximation (valid for many profiles) is 𝛼(𝑏) ≈ 4𝐺𝑀(< 𝑏)/(𝑐2𝑏). Isothermal (𝜌∝𝑟−2) case — closed form: if 𝜌eff (𝑟)=𝜌0(𝑟0/𝑟)2, then Σ(𝑅)=𝜋𝜌0𝑟2 0/𝑅(for infinite extent), so projected mass 𝑀2D(< 𝑏)=2𝜋𝜌0𝑟2 0𝑏. Hence 𝛼(𝑏)=4𝐺 𝑐2 2𝜋𝜌0𝑟2 0𝑏 𝑏=8𝜋𝐺𝜌0𝑟2 0 𝑐2.(4.26) Thus for the 𝜒-memory produced isothermal case, 𝛼is constant (independent of b), producing a characteristic strong-lensing behaviour and critical curves similar to an SIS (singular isothermal sphere). As we found in (4.18–4.22), 𝜌0=𝜅 2𝑐2(𝜆𝐴)2,so 𝛼𝜒=8𝜋𝐺 𝑐2·𝜅 2𝑐2(𝜆𝐴)2𝑟2 0=4𝜋𝐺 𝜅 𝑐4(𝜆𝐴)2𝑟2 0.(4.27) Convergence (dimensionless surface density): 𝜅lens(𝑅)=Σ(𝑅) Σcrit ,Σcrit =𝑐2 4𝜋𝐺 𝐷𝑠 𝐷𝑑𝐷𝑑𝑠 ,(4.28) with 𝐷𝑑, 𝐷𝑠, 𝐷𝑑𝑠 the usual angular diameter distances. For an isothermal 𝜒halo with Σ(𝑅) ∝ 1/𝑅,𝜅(𝑅) ∝ 1/𝑅and strong-lensing signatures (Einstein rings) are possible if 𝜅 achieves order unity. Equivalence to dark matter lensing: because lensing depends only on the projected mass distribution, any 𝜒-memory 𝜌eff (𝑟)that produces the same Σ(𝑅)as a conventional dark halo will produce identical lensing observables. In particular, the 𝜒-memory is observationally indistinguishable from particle dark matter via lensing alone if the projected surface densities match. Distinguishing MCIT would therefore rely on complementary observables (timeevolution, shell around recent explosions, correlation with past explosive events, and kinematic– lensing mismatches). Y Thin-shell Signatures and Multi-shell Superposition Recall thin-shell residual from Appendix C / Eq. (3.7): 𝜒𝑎(𝑟) ≃ 𝜆𝐸𝑎 4𝜋|𝑟−𝑟𝑎|𝑒−𝑚𝜒|𝑟−𝑟𝑎|, 18 for shell 𝑎with energy 𝐸𝑎at radius 𝑟𝑎(exact prefactor for ideal 𝛿shell). The net 𝜒is the linear superposition 𝜒=Í𝑎𝜒𝑎(for linear 𝜒regime). The gradient squares produce (𝜕𝑟𝜒)2=Õ 𝑎(𝜕𝑟𝜒𝑎)2+2Õ 𝑎<𝑏(𝜕𝑟𝜒𝑎)(𝜕𝑟𝜒𝑏).(4.29) Thus a stack or continuous distribution of shells can, in the aggregate, produce an approximate 𝜕𝑟𝜒∝1/𝑟behaviour if the shell energies and radial spacing are arranged so that the cumulative interior integral ∫𝑟 0𝑠2E(𝑠)𝑑𝑠 ∝𝑟(see Sec. 4.3). In practice, a galaxy with many past energetic events—supernovae, hypernovae, mergers—may naturally realize a near 𝑟−2timeintegrated source without fine tuning, particularly if event frequency per unit radius falls as 1/𝑟. Distinctive observational signatures of shell structure: • Localized shell lensing features: a single recent energetic shell produces a ringlike enhancement in convergence at the shell radius (projected), potentially visible as an annulus of extra lensing. • Interference cross terms could produce small oscillations or dips in radial acceleration profiles. • Temporal evolution: if shells are recent (ages less than 𝜒decay time), one might measure secular changes in outer rotation curve or lensing convergence. Z Worked Numerical Example (Order-of-magnitude) — Mapping Parameters to 𝑣∞ Use the closed form (4.23) to estimate parameters producing Milky Way-like rotation 𝑣∞≈200 km s−1. From (4.23): 𝜆𝐴 =𝑣∞r𝑐2 2𝜋𝐺𝜅 . Plugging numbers (𝐺=6.674 ×10−11 m3kg−1s−2,𝑐=3×108m/s): r𝑐2 2𝜋𝐺 ≈s9×1016 2𝜋×6.674 ×10−11 ≈s9×1016 4.192 ×10−10 ≈p2.148 ×1026 ≈4.635×1013 m1/2s−1. Thus (with 𝜅left symbolic) 𝜆𝐴 ≈𝑣∞ 4.64 ×1013 √𝜅(SI units). For 𝑣∞=2×105m/s (200 km/s), 𝜆𝐴 ≈2×105×4.64 ×1013 √𝜅=9.28 ×1018/√𝜅(SI).(4.30) Interpretation: the dimensionful product 𝜆𝐴 must be ≈1019 (in SI), modulo normalization 𝜅. If 𝜅is small (e.g., 𝜅∼10−30 after canonical normalization of fields), then smaller 𝜆𝐴 suffices 19 — this shows the degeneracy and necessity to fix canonical normalizations before mapping to microphysics. The scaling result is important: observational 𝑣∞fixes 𝜆𝐴 up to 𝜅, enabling parameter inference from rotation-curve fits. Practical Pipeline for Observational Comparison (Summary) 1. Choose a model for E(𝑟)(thin shells, continuous emission, power-law). 2. Solve static 𝜒equation (4.2) numerically (or use analytic formulae if possible) to obtain 𝜒(𝑟). For nonzero 𝑚𝜒use Yukawa kernel (Eq. 4.3). 3. Compute 𝜕𝑟𝜒(𝑟)and 𝜌eff (𝑟)=𝜅/(2𝑐2)(𝜕𝑟𝜒)2. 4. Compute 𝑀eff (𝑟)and 𝑣(𝑟)=p𝐺(𝑀baryon +𝑀eff)/𝑟. 5. Compute projected surface density Σ(𝑅)and convergence 𝜅lens(𝑅)for lensing comparison. 6. Fit model parameters (𝜆, 𝑚𝜒, 𝜅, 𝐴 or E(𝑟)shape)to rotation curves and lensing, including priors from Solar System constraints and cosmology. Closing Remarks — Physical Plausibility and Tests Analytic conclusion: MCIT naturally provides a mechanism by which exploding/collapsing events deposit time-integrated 𝜓flux E(𝑟). If E(𝑟)scales as ∝𝑟−2(or a superposition of shells produces that aggregate scaling), the 𝜒memory produces an isothermal-like halo with flat rotation curves, and identical lensing to an isothermal dark matter halo. Observational tests to distinguish MCIT from particle dark matter: • Look for ringlike lensing features correlated with past high-energy events (supernova remnants, starburst episodes). • Search for secular evolution of outer rotation curves in systems with known recent energetic activity. • Compare kinematic mass (from stars/gas) and lensing mass maps for signatures of nonmonotonic cross terms from multi-shell interference. • Use combined constraints (CMB, structure formation) to bound allowed 𝑚𝜒, 𝜆, 𝜅. 20 MCIT Quantization and Cosmological Implications Rhythm October 16, 2025 1 Canonical Quantization of the 𝜒Field (Free Theory) We quantize the 𝜒field on a fixed (background) weakly curved spacetime; for transparency we present the flat-space derivation and then indicate the minimal coupling generalization. Use metric signature (−+++). Start from the free part of the 𝜒-action 𝑆𝜒=−1 2∫𝑑4𝑥(𝜕𝜇𝜒)(𝜕𝜇𝜒) +𝑚2 𝜒𝜒2. The Euler–Lagrange equation is (□−𝑚2 𝜒)𝜒=0. Canonical quantization proceeds by promoting 𝜒and its canonical momentum 𝜋𝜒≡𝜕𝑡𝜒to operators obeying equal-time commutation relations [ˆ𝜒(𝑡, x),ˆ𝜋𝜒(𝑡, y)] =𝑖ℏ𝛿(3)(x−y),[ˆ𝜒, ˆ𝜒]=[ˆ𝜋, ˆ𝜋]=0. Expand ˆ𝜒in plane-wave modes (box normalization 𝑉then 𝑉→ ∞ limit): ˆ𝜒(𝑡, x)=1 √𝑉Õ k 1 √2𝜔𝑘ˆ𝑎k𝑒−𝑖𝜔𝑘𝑡+𝑖k·x+ˆ𝑎† k𝑒𝑖𝜔𝑘𝑡−𝑖k·x, with dispersion 𝜔𝑘=q|k|2+𝑚2 𝜒and canonical creation/annihilation operators satisfying [ˆ𝑎k,ˆ𝑎† k0]=𝛿k,k0. The free Hamiltonian is ˆ 𝐻𝜒=Õ k ℏ𝜔𝑘ˆ𝑎† kˆ𝑎k+1 2. In curved or weakly perturbed backgrounds, one uses mode functions 𝑢𝑘(𝑥)solving (□− 𝑚2 𝜒)𝑢𝑘=0with the same canonical normalization; the algebra above generalizes by replacing plane waves with 𝑢𝑘(𝑥). 2𝜓as a Classical Source: Inhomogeneous Evolution and Sourceto-Mode Coupling Recall the inhomogeneous 𝜒-equation with a classical source J(𝑥)=𝜆𝜓2(𝑥): (□−𝑚2 𝜒)𝜒(𝑥)=J(𝑥). 1 12.1 Rotation Curves 1. Compute 𝜒res(𝑟)from E(𝑟)via the Yukawa convolution (Eq. (4.3) / V.3): 𝜒res(𝑟)=𝜆∫∞ 0 𝑑𝑟04𝜋𝑟02𝑒−𝑚𝜒|𝑟−𝑟0| 4𝜋|𝑟−𝑟0|E(𝑟0).(7.1) (Implement numerically with adaptive quadrature or FFT-based convolution for efficiency; for thin shells use the analytic thin-shell forms from Appendix C.) 2. Compute radial derivative 𝜕𝑟𝜒(𝑟)(numerically differentiate with consistent scheme): 𝜕𝑟𝜒(𝑟)=𝜆∫∞ 0 𝑑𝑟04𝜋𝑟02𝜕𝑟𝑒−𝑚𝜒|𝑟−𝑟0| 4𝜋|𝑟−𝑟0|E(𝑟0).(7.2) 3. Effective density (SI units): 𝜌eff (𝑟)=𝜅 2𝑐2𝜕𝑟𝜒(𝑟)2.(7.3) 4. Enclosed effective mass and total mass: 𝑀eff (𝑟)=4𝜋∫𝑟 0 𝑟02𝜌eff (𝑟0)𝑑𝑟0, 𝑀tot(𝑟)=𝑀baryon (𝑟) + 𝑀eff (𝑟).(7.4) 5. Circular velocity: 𝑣(𝑟)=r𝐺 𝑀tot(𝑟) 𝑟.(7.5) Implementation notes: • For disk galaxies, include baryonic components (exponential disk surface density Σ𝑑(𝑅)= Σ0𝑒−𝑅/𝑅𝑑) and compute the disk contribution to 𝑀baryon (𝑟)accurately (use standard exact integrations or lookup tables). • Numerical differentiation of 𝜒must be stable: when 𝜒is computed on a grid use central differences with smoothing or spectral differentiation to avoid noisy 𝜌eff estimates. 12.2 Lensing: Deflection and Convergence 1. Compute projected surface density Σ(𝑅)from 𝜌eff: Σ(𝑅)=∫∞ −∞ 𝜌eff p𝑅2+𝑧2𝑑𝑧. (7.6) 2. Convergence and deflection: 𝜅lens(𝑅)=Σ(𝑅) Σcrit , 𝛼(𝑅)=4𝐺 𝑐2 𝑀2D(< 𝑅) 𝑅, 𝑀2D(< 𝑅)=2𝜋∫𝑅 0 𝑅0Σ(𝑅0)𝑑𝑅0.(7.7) 8 3. Time delay between images (for lens systems): geometric+Shapiro-like potential term receives contributions from baryons and Φeff. Compute Φeff by solving Poisson: ∇2Φeff =4𝜋𝐺 𝜌eff (𝑟), and integrate along light paths to get potential delay. Use Eq. (6.7) style formulae for the 𝜒contribution to time delay if preferred. Implementation notes: • For weak-lensing stacks, compute tangential shear 𝛾𝑡(𝑅)from Σvia standard relations 𝛾𝑡(𝑅)=¯ Σ(< 𝑅) −Σ(𝑅)divided by Σcrit. • For strong-lensing, compute lensing potential and critical curves; compare Einstein radius with observed arcs to constrain 𝜅𝐴. 12.3 Timing Delays & Pulsar Timing 1. For a line of sight crossing a 𝜒memory structure, compute the integrated effective index perturbation / metric perturbation along ray: Δ𝑡=1 𝑐∫path 𝛿𝑛eff (𝑥)𝑑𝑠, with 𝛿𝑛eff derived from Δ𝜇𝜈 using the dispersion shift (Sec. VI, Eq. 6.13 equivalent). A practical leading-order proxy is Δ𝑡≃𝜀 2𝑐∫Δ00(𝑥) 𝑘2𝑑𝑠 ≃𝜀𝜅 4𝑐3∫(∇𝜒)2𝑑𝑠, (7.8) with units restored; convert to seconds. 2. Compare to timing residuals: for millisecond pulsars residual precision (<100) ns in best cases — use this to set limits on 𝜀𝜅(𝜆𝐸)2. Implementation notes: • Use accurate path integration and account for background gravitational potential (Shapiro from baryons) when isolating 𝜒contribution. • Pulsar timing arrays and precise binary pulsars provide strong local constraints; include their sky positions and path geometries in likelihood. 13 Statistical Framework and Likelihood Construction We present the likelihood functions for fitting datasets and give explicit expressions for the Fisher matrix and Bayesian posterior sampling. All formulae assume Gaussian measurement errors (standard in astrophysical model fitting), but extensions for non-Gaussian noise are straightforward. 9 13.1 Rotation Curve Likelihood Given observed rotation speeds 𝑣obs (𝑟𝑖)with Gaussian errors 𝜎𝑖, the log-likelihood is ln LRC (𝜃)=−1 2Õ 𝑖𝑣model(𝑟𝑖;𝜃) −𝑣obs (𝑟𝑖)2 𝜎2 𝑖−1 2Õ 𝑖 ln(2𝜋𝜎2 𝑖),(7.9) where 𝜃denotes the full set of model parameters (e.g. 𝜃={𝜆, 𝑚𝜒, 𝜅, 𝐴, Υdisk, . . .}) and 𝑣model is computed via (7.5). Include correlated errors via covariance matrix Cif available: ln LRC =−1 2Δ𝑣>C−1Δ𝑣−1 2ln det(2𝜋C).(7.10) 13.2 Lensing Likelihood For shear/convergence profiles or strong-lensing image data, build the appropriate data vector 𝑑(e.g., binned tangential shear) and covariances Cℓ. The log-likelihood is ln Llens(𝜃)=−1 2(𝑑−𝑚(𝜃))>C−1 ℓ(𝑑−𝑚(𝜃)) +const. (7.11) For imageor pixel-based strong-lensing, use the pixel likelihood with instrument PSF and noise model. 13.3 Joint Likelihood & Priors Assuming data independence (or combining covariances appropriately), multiply likelihoods: ln Ltot(𝜃)=ln LRC +ln Llens +ln Ltiming +. . . (7.12) Bayes theorem yields posterior: 𝑝(𝜃|𝐷) ∝ Ltot (𝜃)𝑝prior(𝜃).(7.13) Recommended priors (copy-paste ready): •log10 𝜆∼ U[log10 𝜆min,log10 𝜆max]wide uniform in log space (choose wide bounds e.g. 20 decades) unless canonical normalization fixes units. •𝑚𝜒: prior flat in log space over a physically motivated range, e.g. 𝑚𝜒∈ [10−28,10−12]m−1 (user should adapt). •𝜅positive; adopt log prior if poorly known. •Υpriors from stellar population synthesis (Gaussian with mean and 𝜎). • Distance and inclination priors from observational errors (Gaussians). 10 13.4 Fisher Matrix (Forecasting) For Gaussian likelihood near maximum, the Fisher matrix elements are 𝐹𝑖 𝑗 =−𝜕2ln L 𝜕𝜃𝑖𝜕𝜃 𝑗=Õ 𝑎,𝑏 𝜕𝑚𝑎 𝜕𝜃𝑖 𝐶−1 𝑎𝑏 𝜕𝑚𝑏 𝜕𝜃 𝑗 ,(7.14) where 𝑚𝑎are model predictions for data vector components. For rotation curve fitting with uncorrelated errors, 𝐹𝑖 𝑗 =Õ 𝑘 1 𝜎2 𝑘 𝜕𝑣model (𝑟𝑘) 𝜕𝜃𝑖 𝜕𝑣model (𝑟𝑘) 𝜕𝜃 𝑗 .(7.15) Use numerical differentiation (complex-step derivative recommended for stability) to compute 𝜕𝑣/𝜕𝜃. The Cramér–Rao bound gives 𝜎(𝜃𝑖) ≥ p(𝐹−1)𝑖𝑖. Application notes: • Because observables scale roughly as 𝜌eff ∝ (𝜆𝐴)2, Fisher forecasts typically produce tight constraints on A=𝜆𝐴√𝜅but much weaker on separated 𝜆vs 𝐴unless there are independent constraints on source energetics. Use priors on 𝐴from stellar population models to break degeneracy. 14 Degeneracies, Identifiability, and Null Tests Analytic degeneracy statements (helpful for interpretation): 1. Amplitude degeneracy: For broadly distributed E(𝑟)the dominant constrained direction is A=𝜆𝐴√𝜅. Observables scale as ∼ (A)2(rotation curves, lensing). 2. Range–shape degeneracy: Finite 𝑚𝜒produces a screening scale 𝑟𝑠∼𝑚−1 𝜒. A combination of smaller Awith larger 𝑟𝑠can mimic a larger Awith smaller 𝑟𝑠over a limited radial range—multi-radius data (inner + outer) alleviate this. 3. Baryon degeneracy: Disk mass-to-light ratio Υtrades with effective mass from MCIT in inner regions. Use photometry and stellar population priors to break. 4. Temporal degeneracy: If data include systems with different histories, a statistical model for E(𝑟)(e.g., linked to SFR history) provides population-level constraints. Single-galaxy fits are limited by unknown past explosion histories. Null tests: • Compare lensing-to-dynamical mass ratios; MCIT predicts equivalence in many cases but can produce shell-related discrepancies (lensing rings without expected baryons) — a mismatch is a strong signature. • Search for ringlike lensing convergence correlated with known recent starburst or SN remnant positions. • Pulsar timing null: absence of timing residuals at predicted levels constrains local 𝜀𝜅(𝜆𝐸)2. 11 15 Practical Pipeline (Algorithmic, Ready-to-Implement) Below is a reproducible pipeline suitable for publication and public code release. It is written so you can copy–paste into a methods appendix or code README. Input: data 𝐷= {rotation curves, lensing maps, timing residuals, baryonic photometry}, observational covariances, prior information. Output: posterior 𝑝(𝜃|𝐷), best-fit model predictions, model evidence (optional), credible intervals and forecast. Algorithm (MCMC + nested sampling option): 1. Precompute baryonic mass model 𝑀baryon (𝑟;Υ, . . .)from photometry. 2. Choose parametric E(𝑟;𝜙)with parameters 𝜙(e.g., 𝐴, 𝑝, 𝑟in, 𝑟out). Set parameter vector 𝜃=(𝜆, 𝑚𝜒, 𝜅, 𝜙, Υ, . . .). 3. For given 𝜃: (a) Compute 𝜒res(𝑟)via Eq. (7.1) on radial grid 𝑟𝑗(use adaptive quadrature). (b) Compute 𝜕𝑟𝜒(𝑟𝑗)(spectral/central differences). (c) Compute 𝜌eff (𝑟𝑗)via (7.3); integrate to get 𝑀eff (𝑟)and 𝑣(𝑟). (d) Project to get Σ(𝑅),𝜅lens (𝑅), and timing predictions as needed. 4. Evaluate ln Ltot(𝜃)using Eqs. (7.9–7.12). 5. Use MCMC sampler (emcee) for posterior sampling or nested sampler (MultiNest / dynesty) for evidence. Use parallel tempering if multimodal. 6. After convergence (Gelman–Rubin ˆ 𝑅≲1.02 or effective sample size), produce posterior summaries, credible intervals, and marginalized parameter constraints. 7. For forecasts use Fisher matrix (7.15) to compute expected errors and degeneracies. Computational notes: • Use vectorized convolution (FFT) for large grids (with radial Hankel transforms for spherical symmetry if desired). • Precompute Yukawa kernel on grid to speed repeated convolutions. • Use log-space sampling for positive parameters. • For model selection between MCIT and e.g. NFW, compute Bayesian evidence with nested sampling. 16 Example Constraint Strategies (Which Datasets Constrain Which Parameters Best) •Rotation curves (large sample): constrain A=𝜆𝐴√𝜅and 𝑚𝜒(if screening affects outer radii). Best to combine dwarfs and 𝐿∗galaxies to probe different scales. 12 •Strong lensing (individual lenses): constrains central 𝜌eff and provides measurement of projected mass; best for breaking Υdegeneracy when combined with stellar kinematics. •Weak lensing stacks: constrain average 𝜌eff on tens to hundreds of kpc (sensitive to 𝑚−1 𝜒). •Pulsar timing / Solar System tests: set strict local bounds on 𝜀𝜅(𝜆𝐸)2for nearby events or in the Solar System; use these to bound small-scale behavior. •Cosmological probes (CMB, LSS): MCIT may modify structure growth if 𝜒couples at early times; this requires cosmological extension of the model (beyond scope here) but can be used to constrain 𝑚𝜒and effective coupling at early epochs. 17 Goodness of Fit, Model Comparison, and Systematics 𝜒2& reduced 𝜒2:compute 𝜒2=−2 ln Land reduced 𝜒2/𝜈(degrees of freedom) for basic checks. Posterior predictive checks: Simulate replicates from posterior predictive distribution and compare to observed residuals to identify misfit patterns (e.g., shell features not captured). Bayes factor: For model selection between MCIT and alternative models use Bayes factor via evidence ratio 𝐵12 =𝑍1/𝑍2. Interpret with standard criteria (Kass & Raftery). Systematics to include: • Photometric mass-to-light uncertainties. • Distance and inclination errors. • Non-sphericity / disk geometry corrections. Include axisymmetric solutions or full 3D convolution if necessary. • Unmodeled baryonic substructures — include nuisance hyperparameters or marginalize over flexible baryon templates. 18 Forecasts and Detectability Thresholds (Analytic Scaling) Use the scaling 𝜌eff ∝ (𝜆𝐴)2/𝑟2(for the isothermal regime) to derive detectability thresholds. Assume a measurement can detect an extra acceleration 𝑎min at radius 𝑟. The MCIT extra acceleration from memory is 𝑎𝜒∼𝐺𝑀eff (𝑟)/𝑟2∼𝐺·2𝜋(𝜅/𝑐2)(𝜆𝐴)2/𝑟(using Eq. 4.20). Solve for minimal 𝜆𝐴: (𝜆𝐴)min ∼r𝑎min 𝑟 𝑐2 2𝜋𝐺𝜅 .(7.16) Plugging numbers yields detection thresholds; use this to plan target sample sizes and precision. 19 Reproducible Example: Mock Fit Recipe (Ready to Paste) Below is a copy-paste-ready recipe to do a mock constraint on a single galaxy: 1. Data: 𝑣obs(𝑟𝑖)with 𝜎𝑖, photometry giving disk scale 𝑅𝑑, distance 𝐷. 13 2. Model E(𝑟)=𝐴𝑟−2for 𝑟∈ [𝑟in =0.1 kpc, 𝑟out =50 kpc]. 3. Parameters: 𝜃=(log10 𝜆, log10 𝑚𝜒,log10 𝜅, log10 𝐴, Υdisk). Priors: uniform in log over wide ranges; Υdisk ∼ N(𝜇, 𝜎)from stellar populations. 4. Implement forward model via Eqs. (7.1)–(7.5) on radial grid spanning [0.01,50] kpc (𝑁∼1000). Use Simpson quadrature for integrals. 5. Use emcee (MCMC) with 100 walkers, 5000 steps, discard burn-in 1000. Monitor convergence. 6. Report marginalized posterior for A=𝜆𝐴√𝜅,𝑚𝜒, and derived 𝑣∞. Produce corner plots and posterior predictive rotation curves. 20 Summary & Recommended Observational Program • Short term: fit MCIT to high-quality rotation curves with well-determined baryons (SPARClike sample), simultaneously fitting E(𝑟)shapes. • Medium term: combine rotation curve fits with strong-/weak-lensing and pulsar timing constraints to break degeneracies and pin down 𝑚𝜒and 𝜅. • Long term: extend MCIT to cosmological perturbation theory and confront with CMB + LSS if model predicts early-Universe 𝜒activity. 20.1 Closing Paragraph for Section VII This section provides a complete, implementable set of equations, likelihoods, algorithms and diagnostics for confronting the 𝜓–𝜒memory theory with data. The central observational prediction — an effective 𝜌eff ∝ (𝜆𝐴)2(∇𝐺𝑚𝜒★E)2producing potentially isothermal-like halos — is falsifiable with combined dynamical and lensing data, especially when cross-checked with timing and environmental info. The parameter degeneracies are well understood (dominant 𝜆𝐴 amplitude mode and 𝑚𝜒-driven screening), and our pipeline enables robust inference and forecasting. 21 Overview and Assumptions Goal: quantify how the 𝜒field and its sourcing by 𝜓-events alter structure formation. We adopt the following assumptions: 1. Cosmological background is described by a flat Friedmann–Lemaître–Robertson–Walker (FLRW) metric with scale factor 𝑎(𝑡). We treat 𝜒as a field on this background with small energy density compared to radiation + matter at early times (so background expansion is essentially ΛCDM to leading order). Where background 𝜒energy is non-negligible we give the generalization. 2. Work in Newtonian gauge for scalar perturbations; metric 𝑑𝑠2=−(1+2Ψ)𝑑𝑡2+𝑎2(𝑡)(1−2Φ)𝑑x2. For weak anisotropic stress we can set Φ = Ψ except where 𝜒gradients induce stress— this is treated explicitly. 14 3. Treat 𝜓sources as localized astrophysical events that deposit time-integrated E(x). On cosmological scales we treat the mean source distribution statistically (source term in 𝜒equation is a stochastic/incoherent field with some power spectrum). Two regimes: (A) early-time homogeneous pumping (e.g., if 𝜓-active era at high z), and (B) late-time, discrete sources (galaxies, SN) giving a nonuniform 𝜒residual correlated with halos. We derive linear perturbation theory for regime (A) and then describe how to treat regime (B) in halo-model / N-body approaches. Notation: comoving coordinates x, physical radius 𝑟=𝑎|x|. Conformal time 𝜂with 𝑑𝜂 = 𝑑𝑡/𝑎.𝐻≡ ¤𝑎/𝑎,H ≡ 𝑎𝐻. Fourier transforms use convention 𝛿(k)=∫𝑑3𝑥 𝑒−𝑖k·x𝛿(x). 22 Background 𝜒and Effect on Friedmann Equations Start from the 𝜒action (minimally coupled) 𝑆𝜒=−1 2∫𝑑4𝑥√−𝑔𝑔𝜇𝜈 𝜕𝜇𝜒𝜕𝜈𝜒+𝑚2 𝜒𝜒2−∫𝑑4𝑥√−𝑔 𝜒 J where J=𝜆𝜓2(classical source). The background homogeneous ¯𝜒(𝑡)obeys ¥ ¯𝜒+3𝐻¤ ¯𝜒+𝑚2 𝜒¯𝜒=¯ J(𝑡).(8.1) Energy density and pressure 𝜌𝜒=1 2¤ ¯𝜒2+1 2𝑚2 𝜒¯𝜒2, 𝑃𝜒=1 2¤ ¯𝜒2−1 2𝑚2 𝜒¯𝜒2. If 𝜌𝜒𝜌tot then Friedmann equations unchanged to leading order and 𝜒only affects perturbations via 𝛿𝜒. For completeness, include 𝜌𝜒in background numerics if needed (modify 𝐻(𝑎) accordingly). 23 Linear Perturbation Equations (Newtonian Gauge) We linearize Einstein + 𝜒equations around the FLRW background. Scalar perturbations: • Metric: Ψ(𝜂, x),Φ(𝜂, x). • Matter: baryons and cold dark matter density contrasts 𝛿𝑏, 𝛿𝑐and velocities 𝜃𝑏, 𝜃𝑐. (If one is trying to replace particle dark matter by MCIT, set 𝛿𝑐absent and compute total gravitational potential from baryons + 𝜌eff.) •𝜒perturbation: 𝜒(𝜂, x)=¯𝜒(𝜂) +𝛿𝜒(𝜂, x).𝜓sources enter via Jand its perturbation. Linearized 𝜒equation in Fourier space (conformal time) is 𝛿𝜒00 +2H𝛿𝜒0+ (𝑘2+𝑎2𝑚2 𝜒)𝛿𝜒 −4 ¯𝜒0Ψ0+2𝑎2𝑚2 𝜒¯𝜒Ψ = 𝑎2𝛿J,(8.2) primes denote 𝜕𝜂. For late-time quasi-static modes (subhorizon 𝑘 H and slowly varying background) we can drop time derivatives of 𝛿𝜒 to obtain the quasi-static algebraic relation (𝑘2+𝑎2𝑚2 𝜒)𝛿𝜒(𝜂, k) ≃ 𝑎2𝛿J(𝜂, k) +metric coupling terms.(8.3) 15 To leading order in small metric perturbations, keep RHS ≈𝑎2𝛿J. The perturbed Einstein (Poisson) equation in Newtonian gauge (Fourier, conformal time) is 𝑘2Φ = 4𝜋𝐺𝑎2¯𝜌𝑚𝛿𝑚+𝛿𝜌𝜒+3H( ¯𝜌𝜒+¯ 𝑃𝜒)𝜃𝜒 𝑘2,(8.4) where 𝛿𝜌𝜒=¯𝜒0𝛿𝜒0/𝑎2+𝑚2 𝜒¯𝜒𝛿𝜒−¯𝜒02Ψ/𝑎2+. . .. In the quasi-static limit and for nonrelativistic residual 𝜒(¯𝜒0≈0), the dominant 𝜒contribution to the gravitational potential is from spatial gradients of 𝛿𝜒, which in comoving coordinates gives an effective Poisson-like term akin to V in previous sections. 24 Effective Poisson Equation and 𝐺eff (𝑘, 𝑎) We now derive the effective modification of Poisson’s equation due to 𝜒sourcing and its memory. For subhorizon linear modes and neglecting 𝜒kinetic terms relative to spatial gradients (valid for 𝑘𝑎𝐻 and slowly varying background), take (8.3) to obtain 𝛿𝜒(k, 𝑎) ≃ 𝑎2 𝑘2+𝑎2𝑚2 𝜒 𝛿J(k, 𝑎).(8.5) Assume the 𝜓source fluctuations are proportional to matter density fluctuations on relevant scales (two possibilities): • If 𝜓is tied to baryons / star formation, 𝛿J ∝ 𝛿𝑏(bias 𝑏𝜓). • If 𝜓is produced cosmologically (early uniform pumping), treat 𝛿Jas proportional to 𝛿𝑚 with some bias 𝑏𝜓. We set 𝛿J(k, 𝑎)=𝜆J0(𝑎)𝑏𝜓(𝑎)𝛿𝑚(k, 𝑎). The constant J0(𝑎)encodes the mean amplitude of source power per matter overdensity (model-dependent). Then 𝛿𝜒(k, 𝑎) ≃ 𝑎2𝜆J0𝑏𝜓 𝑘2+𝑎2𝑚2 𝜒 𝛿𝑚(k, 𝑎).(8.6) Contribution of 𝜒to the right-hand side of Poisson is through 𝛿𝜌eff. In the quasi-static, weak-field limit, and following previous definitions, the effective density that sources gravity is proportional to (∇𝜒)2. Linearizing (∇𝜒)2about background produces a term linear in ¯𝜒0and 𝛿𝜒; however if ¯𝜒is small, the leading linear contribution to potential is via the term ∝¯𝜒∇2𝛿𝜒 or via cross-correlation with other fields. To get an explicit, useful closed-form and to connect to observables, we adopt the following controlled linearization, valid when ¯𝜒is nonzero or there is a coherent time-integrated background E: Model A (coherent background ¯𝜒nonzero): if ¯𝜒≠0and slowly varying, expand (∇𝜒)2=(∇ ¯𝜒)2+2∇¯𝜒· ∇𝛿𝜒 +O(𝛿𝜒2). The linear perturbation feeding Poisson is then proportional to ∇¯𝜒· ∇𝛿𝜒, which in Fourier space gives ∝𝑖k·c ∇¯𝜒 𝛿𝜒(k). This term depends on the geometry of ∇¯𝜒and is not isotropic, complicating cosmological averages. Realistic cosmological average of many sources tends to isotropize and reduce this term. 16 Model B (stochastic source, treat quadratic 𝜒contribution as effective linear term): pragmatically, for linear cosmological perturbation theory we can treat the 𝜒contribution as providing an effective modification of Poisson proportional to 𝛿𝜒: 𝑘2Φ = 4𝜋𝐺𝑎2¯𝜌𝑚[1+𝜇(𝑘, 𝑎)]𝛿𝑚(k, 𝑎),(8.7) with 𝜇(𝑘, 𝑎) ≡ 1 ¯𝜌𝑚·𝐹𝜒(𝑘, 𝑎), where 𝐹𝜒encodes the mapping from 𝛿𝜒 to an effective mass perturbation. Using dimensional analysis and the scaling found in static halo calculations (Sect. IV), 𝐹𝜒(𝑘, 𝑎) ∝ 𝜅(𝜆J0𝑎2)2/(𝑘2+ 𝑎2𝑚2 𝜒)2times an order-unity geometric factor. To leading parametric order we therefore obtain a scale-dependent effective gravitational coupling 𝐺eff (𝑘, 𝑎)=𝐺"1+𝛼0(𝑎)1 (𝑘2+𝑎2𝑚2 𝜒)2#,(8.8) with 𝛼0(𝑎) ≡ 𝐶𝜅(𝜆J0𝑎2)2 ¯𝜌𝑚 and 𝐶an O(1)constant that depends on the precise mapping from 𝛿𝜒 to 𝛿𝜌eff (geometry / averaging). Equation (8.8) is the practical parametrization to use in Boltzmann solvers. Limiting behaviours • Large scales (𝑘𝑎𝑚𝜒) (super-Compton): (𝑘2+𝑎2𝑚2 𝜒) ≈ 𝑎2𝑚2 𝜒⇒𝐺eff (𝑘) ≈ 𝐺[1+ 𝛼0/(𝑎4𝑚4 𝜒)] — scale independent enhancement. • Small scales (𝑘𝑎𝑚𝜒) (sub-Compton): 𝐺eff (𝑘) ≈ 𝐺[1+𝛼0/𝑘4]— rapidly decreasing correction. Thus 𝜒-mediated memory tends to produce the largest fractional modifications at large scales (low 𝑘) if 𝑚𝜒is small; if 𝑚𝜒is tiny (ultra-light), modification becomes scale independent on cosmological scales, acting like an effective rescaling of 𝐺(degenerate with other cosmological parameters unless constrained). 25 Modified Growth Equation The linear growth of matter perturbations 𝛿𝑚(𝑘, 𝑎)obeys (subhorizon, pressureless matter): 𝛿00 𝑚+H𝛿0 𝑚−4𝜋𝐺𝑎2¯𝜌𝑚[1+𝜇(𝑘, 𝑎)]𝛿𝑚=0,(8.9) where 𝜇(𝑘, 𝑎) ≡ 𝐺eff/𝐺−1is given by (8.8). Converting to scale factor 𝑎derivatives (use 𝑑/𝑑𝜂 =𝑎𝐻𝑑/𝑑𝑎), the ordinary differential equation for the growth factor 𝐷(𝑘, 𝑎)is 𝑑2𝐷 𝑑𝑎2+3 𝑎+𝑑ln 𝐻 𝑑𝑎 𝑑𝐷 𝑑𝑎 −3 2 Ω𝑚(𝑎) 𝑎2[1+𝜇(𝑘, 𝑎)]𝐷=0.(8.10) This ODE can be solved numerically for each 𝑘to obtain the scale-dependent growth 𝐷(𝑘, 𝑎) and the linear matter power spectrum 𝑃(𝑘, 𝑎)=𝑃(𝑘, 𝑎init) [𝐷(𝑘, 𝑎)/𝐷(𝑘, 𝑎init)]2. Analytic approximations 17 2.4 Gravitational-wave Event Cross-correlation Data required: catalog of GW events with ejecta energy estimates (EM counterparts), host galaxy lensing/kinematic maps. Procedure: 1. For each event compute expected 𝐸(res) 𝜒from ejecta energy using Eq. (3.22). 2. Search for excess lensing, rotation-curve anomalies, or timing residuals in host within expected radial range and timescales. 3. Stack multiple events to increase S/N. Scaling: 𝐸(res) 𝜒∝𝜆2𝐸2 ejecta — strong dependence on ejecta energy favors kilonovae/merger events with large mass shedding. 3 Null Tests and Potential Falsification Criteria These are concrete criteria that falsify MCIT as a dominant explanation for dark-matter-like effects. 1. Lack of correlation with past energetic activity: If rotation-curve anomalies (or halo masses) show no statistical correlation with cumulative star-formation / explosion histories across large samples (after controlling for baryons), then MCIT (which predicts such a correlation through E(𝑟)) is strongly disfavoured. Implement via regression of inferred Avs SFR-integrated proxies. 2. Absence of shell-like lensing signatures where expected: If high-resolution lensing maps of recent, energetic remnants show no annular convergence above predicted sensitivity (for realistic 𝜆and energy budgets), shell-memory explanation for halos fails. 3. Cosmological inconsistency: If Planck + BAO + growth data tightly limit scale-dependent 𝜇(𝑘, 𝑎)to levels inconsistent with parameters required to explain galaxy dynamics (using Eq. 8.8 mapping), then MCIT as a dominant cosmic-component fails. Use joint cosmological + astrophysical fits. 4. Solar-system / local precision bounds: If local tests require Yukawa suppression scales larger than galactic scales (i.e., 𝑚𝜒too large), MCIT cannot explain extended halos. 5. Energy bookkeeping violation: If cumulative 𝜓energy available in universe (integrated SFR and catastrophic events) is demonstrably insufficient to power observed dark-matterlike gravitational fields given energy conversion efficiencies implied by fitted 𝜆and 𝜅, the model fails. Compute global budget: 𝐸total,available ≳∑ halos 𝐸(res) 𝜒 𝜂conv , 𝐸(res) 𝜒∼𝜆2𝐸2 shell 32𝜋2𝑟min , with 𝜂conv efficiency; require plausibility. These falsification tests are quantitative and require population-level analyses (Section VII pipeline). 3 4 Potential Systematic Confounders and How to Disentangle 1. Baryonic feedback mimicking: Supernova-driven baryonic outflows can reshape rotation curves in ways that mimic an extended halo. Disentangle by comparing lensing (sensitive to total mass) vs kinematics (sensitive to forces): MCIT predicts lensing signatures tied to 𝜒memory even when baryonic tracers absent. Use combined lensing + kinematics. 2. Non-sphericity and projection: Thin-shell signatures may be smeared by non-spherical geometry. Use 3D reconstruction (kinematic tomography, multi-angle lensing) and stacking to recover signals. 3. Degeneracies with WIMP-like halos: Use small-scale structure (subhalo abundance), time evolution, and correlation with recent explosions to discriminate—particle DM predicts different substructure statistics and no correlation with recent stellar activity. 4. Measurement systematics: photometric Υ, inclination, distance. Marginalize with priors; use independent mass tracers (integrated velocity dispersion, gas kinematics) to constrain baryons robustly. 5 Concrete Experimental Proposals (Observational Programs) Below are actionable proposals formatted for observational proposals / telescope time requests. 5.1 Proposal A — Deep Annular Lensing Search Around Recent Starburst Galaxies Target sample: 50 nearby (𝑧 < 0.03) galaxies with starburst history in last 100 Myr. Observation: high-resolution HST/JWST imaging for strong-lensing features (if any) and deep ground-based weak-lensing with wide-field cameras for outer rings. Goal: detect annular convergence signatures predicted by thin-shell model at radii 1–10 kpc. Forecast S/N using Eq. (C.9) and dataset noise models. 5.2 Proposal B — Pulsar Timing Targeted Campaign Behind SNR Shells Target: millisecond pulsars with lines of sight passing within tens of pc of young SNRs. Observation: augment PTA monitoring cadence for selected pulsars for 3–5 years. Goal: detect static or evolving Δ𝑡from crossing residual; compute matched-filter template from (VI.7) and estimate detection significance. 5.3 Proposal C — GW–EM Host Stacking for 𝜒Memory Target: compact-merger hosts with EM ejecta energy estimates. Observation / analysis: stack lensing maps, rotation-curves, and pulsar timing residuals of hosts to enhance signals from small individual 𝐸𝜒. Goal: test quadratic scaling 𝐸(res) 𝜒∝𝐸2 ejecta. 4 6 Theoretical Failure Modes and Consistency Requirements These are internal consistency checks for the theory. 1. Energy non-conservation loopholes: ensure all energy flowing into 𝜒residuals is accounted for in initial 𝜓budgets and radiative losses; analytic energy bookkeeping in (3.23) must hold observationally. 2. Backreaction & strong-field regime: if fitted parameters imply Δ𝜇𝜈 large enough to alter local metric signature, the weak-field approximations fail — exclude parameter region where |𝜀Δ|≳1. 3. Quantum backreaction: if |𝛼|2for many events is enormous and coherent-state overlaps pile up, check for cumulative quantum effects (decoherence, stimulated emission of 𝜒 quanta) not included in classical model. 4. Radiative decay channels for 𝜒:if 𝜒couples to standard model fields strongly, residuals will decay — use absence/presence of decay signatures (e.g., exotic EM emission) to constrain couplings. 7 Summary Conclusions and Immediate Theoretical / Observational Next Steps Conclusions • MCIT provides a physically-motivated mechanism by which stellar and catastrophic ejecta imprint long-lived curvature “memory” onto the 𝜒field; this memory sources an effective gravitational field 𝜌eff ∝ (∇𝜒)2. • The mechanism naturally produces two distinct gravitational contributions: the local remnant dressing plus nonlocal ejecta shells (the “double pull”), and can mimic standard darkhalo phenomenology (isothermal-like profiles) when aggregate source distributions obey E(𝑟) ∝ 𝑟−2. • Observational consequences are explicit and falsifiable: rotation curves, ring-like lensing, pulsar timing delays, GW–host cross-correlations, and cosmological signatures (ISW, lensing, scale-dependent growth). Immediate next steps (recommended, copy–paste ready): 1. Data analysis: implement the Section VII pipeline and fit SPARC-like rotation curves with E(𝑟)=𝐴𝑟−𝑝templates, report posterior on Aand 𝑚𝜒and test correlation with SFR histories. 2. Lensing search: perform annular convergence searches in deep cluster and galaxy lensing datasets; stack by remnant age. 3. Timing constraints: obtain targeted PTA observations for pulsars behind selected SNRs and perform matched-filter analysis for predicted Δ𝑡. 5 4. Simulations: run particle-field hybrid N-body simulations (Section 8.7) to calibrate 𝐶in Eq. (8.8) and to validate nonlinear predictions. 5. Energy-budget audit: compile global 𝜓energy budgets from stellar populations and quantify whether required cumulative energies are plausible for fitted 𝜆. 8 Final Note to Authors / Reviewers (Boilerplate for Manuscript) To make this manuscript maximally reproducible include: • A parameter table with canonical units and recommended prior ranges for 𝜆, 𝑚𝜒, 𝜅, 𝐸. • A GitHub repository with (i) the forward-model code implementing Eqs. (7.1)–(7.5), (ii) the MCMC pipeline from §7.9, and (iii) example notebooks reproducing appendix numeric estimates (Appendix C and A). • Supplementary figures: (i) a gallery of simulated shell lensing maps, (ii) rotation-curve fits to representative galaxies, (iii) forecast curves for PTAs and lensing searches. 9 Canonical Action and Field Normalizations (Derivation) Start from the phenomenological action used in Sections II–VI (metric signature (−+++)); keep general normalization constants 𝑍𝜒, 𝑍𝜓for clarity: 𝑆=∫𝑑4𝑥√−𝑔[−1 2𝑍𝜒𝑔𝜇𝜈 𝜕𝜇𝜒𝜕𝜈𝜒−1 2𝑍𝜒𝑚2 𝜒𝜒2+1 2𝑍𝜓𝑔𝜇𝜈 𝜕𝜇𝜓𝜕𝜈𝜓−1 2𝑍𝜓𝑚2 𝜓𝜓2−𝜆𝜒𝜓2+Lother]. (D.1) Here 𝑍𝜒and 𝑍𝜓are positive real constants chosen depending on the field normalization conventions used in theoretical formulae or numerical codes. (Frequently one sets 𝑍𝜒=𝑍𝜓=1; we keep them explicit to show conversion to canonical fields.) Canonical fields. Define canonically normalized fields (so that the kinetic terms take the standard form −1 2(𝜕𝜙)2): 𝜒𝑐≡√𝑍𝜒𝜒, 𝜓𝑐≡√𝑍𝜓𝜓. (D.2) Substitute into the action (D.1). The kinetic and mass pieces become canonical: −1 2(𝜕𝜒𝑐)2−1 2𝑚2 𝜒𝜒2 𝑐,−1 2(𝜕𝜓𝑐)2−1 2𝑚2 𝜓𝜓2 𝑐. The interaction term transforms as −𝜆𝜒𝜓2=−𝜆𝜒𝑐 √𝑍𝜒 𝜓2 𝑐 𝑍𝜓 =−𝜆c𝜒𝑐𝜓2 𝑐,(D.3) with the canonically normalized coupling 𝜆c=𝜆 √𝑍𝜒𝑍𝜓 .(D.4) This identity is essential: whenever the main text writes 𝜆𝜒𝜓2but numerical code or canonical perturbation theory uses canonically normalized fields, use 𝜆cfrom (D.4). In many places of the paper we omitted explicit 𝑍-factors by setting 𝑍𝜒=𝑍𝜓=1; the above shows how to restore them. 6 10 Mass (Engineering) Dimensions in 4 Spacetime Dimensions Work in natural units (ℏ = 𝑐=1) where action is dimensionless and Lagrangian density Lhas mass dimension 4. For a scalar field in 𝐷=4, [scalar field]=1(mass dimension). Therefore: •[𝜒𝑐]=[𝜓𝑐]=1. • From the interaction 𝜆c𝜒𝑐𝜓2 𝑐: mass dimension balance implies [𝜆c] +1+2·1=4⇒ [𝜆c]=1. So in natural units the coupling 𝜆chas mass (energy) dimension +1. Equivalently, one can think of 𝜆cas an energy scale. • Mass parameters [𝑚𝜒]=[𝑚𝜓]=1(as usual). These dimension-counting results will guide conversions to SI below. 11 Units and Conversion Constants (Explicit Numeric Values) We will use the following exact physical constants (SI values) for conversions. 1. Planck’s reduced constant ℏ = 1.054571817 ×10−34 J·s. 2. Speed of light 𝑐=2.99792458 ×108m/s. 3. Energy conversion: 1 eV =1.602176634 ×10−19 J. A very useful combination is ℏ𝑐(common in particle/cosmological unit conversions). Compute it step-by-step (digit-by-digit): •ℏ𝑐=(1.054571817 ×10−34 J·s)×(2.99792458 ×108m/s). • Multiply: 1.054571817 ×2.99792458 =3.1622850 . . . (compute explicitly) –1.054571817 ×2=2.109143634 –1.054571817 ×0.99792458 ≈1.052 . . . — to avoid roundoff we use a standard numerical value: • The standard, commonly quoted value is ℏ𝑐≈197.3269804 MeV ·fm. For unit conversions we convert that to eV·m: ℏ𝑐=197.3269804 MeV ·fm =197.3269804×106eV×10−15 m=1.973269804×10−7eV ·m. (This value is exact enough for our needs and matches commonly used conversion tables.) 7 From ℏ𝑐we obtain useful conversion factors: •Energy  inverse length: an energy 𝐸(in eV) corresponds to an inverse length 𝑘(in m−1) via 𝑘=𝐸 ℏ𝑐(with 𝐸in eV, ℏ𝑐in eV·m). Numerically, 1 eV ←→ 1 ℏ𝑐≈1 1.973269804 ×10−7m =5.06773065 ×106m−1. (This is the standard identity: 1 eV ≈5.0677 ×106m−1.) •Inverse: length  energy: 1 m−1= ℏ𝑐×1 m−1≈1.973269804 ×10−7eV. •Energy  SI (Joules):1 eV =1.602176634 ×10−19 J. When reporting masses 𝑚𝜒in the paper we often quote them either as inverse length (𝑚𝜒in m−1) or energy (eV). Use the conversion 𝑚𝜒(eV)= ℏ𝑐·𝑚𝜒(m−1)(with ℏ𝑐in eV·m) to switch between them. 12 Table of Parameters: Symbol, Meaning, Canonical Units, Example Ranges Below is a copy-paste-ready table you may insert in the manuscript. The example ranges are illustrative only — use observational inference (Section VII) to replace them with fitted values. Symbol Meaning (short) Natural-unit dimension SI / natural-unit conversion Example illustrative range 𝜒/𝜒𝑐spacetime memory scalar (field) / canonical field [𝜒𝑐]=1(mass) 𝜒𝑐in eV units: 1 eV  5.07 ×106m−1amplitude depends on normalization; typically write in energy units via 𝜒∼eV 𝜓/𝜓𝑐surface-bound matter field / canonical [𝜓𝑐]=1same as 𝜒model dependent 𝜆interaction coupling in original (un-rescaled) action mass (natural units) canonical: 𝜆c=𝜆/(√𝑍𝜒𝑍𝜓). 1 eV = 1.602 ×10−19 J illustrative: 10−30–103eV (very model dependent; see text) 𝑚𝜒𝜒mass (inverse range) mass (natural units) 𝑚𝜒[eV]=(ℏ𝑐)𝑚𝜒[m−1]with ℏ𝑐=1.97327 ×10−7eV ·m. Example: 𝑚𝜒=10−22 m−1≈2.07 ×10−15 eV astrophysically relevant: 𝑚𝜒∼10−28–10−14 m−1(i.e. ultra-light → galactic range) 𝜅residual tensor normalization (Sec. V) dimensionless (in our conventions) keep symbolic; if you choose a canonical gravitational normalization you may set 𝜅∼8𝜋𝐺 (optional) ∼ O(1)nominal; often fit as free parameter 𝐸shell total 𝜓energy in a shell energy SI: Joules. astrophysical: a core-collapse SN ∼1044–1046 J(order of magnitude) use observed ejecta energies 𝐴amplitude of E(𝑟)=𝐴𝑟−𝑝time-integrated source energy×time×length𝑝/volume (see text) keep units consistent in numeric code; typically quote 𝜆𝐴 combination fit to rotation curves Notes on the table. • The key observable combinations that appear in the main text are 𝜆c(Eq. D.4) and the product 𝜆c𝐸(e.g. Eq. (C.6)). Because 𝜌eff ∝ (𝜕𝜒)2∝ (𝜆𝐸)2many observables depend on (𝜆𝐸)2rather than separate 𝜆and 𝐸. • Where we write numbers such as 𝑚𝜒=10−22 m−1, convert to electronvolts with the rule in §D.3. 8 13 Worked Conversion Examples (Step-by-step) Example 1 — convert 𝑚𝜒=10−22 m−1to eV We use 𝑚𝜒(eV)=(ℏ𝑐) ×𝑚𝜒(m−1),ℏ𝑐=1.973269804 ×10−7eV ·m. Compute numerically: •𝑚𝜒(eV)=1.973269804 ×10−7eV ·m×10−22 m−1 • Multiply powers: 1.973269804 ×10−7×10−22 =1.973269804 ×10−29. • So 𝑚𝜒=1.97327 ×10−29 eV. (This shows that 𝑚𝜒=10−22 m−1corresponds to an ultra-light mass ∼2×10−29 eV.) Example 2 — convert 1 eV to inverse meters From §D.3: 1 eV ←→ 1 ℏ𝑐=5.06773065 ×106m−1. So a mass of 1 eV corresponds to an inverse length 5.07 ×106m−1(wavenumber). Example 3 — convert a canonical coupling 𝜆cgiven in eV to SI Joules If 𝜆c=1 eV, then in SI energy units: 𝜆c=1 eV =1.602176634 ×10−19 J. Because 𝜆chas energy dimension in natural units, this translation is direct. 14 How to Map Expressions in the Paper to Canonical/SI Variables — Practical Recipes Below are ready-to-use recipes you can paste next to equations in the main text so readers know how to evaluate them numerically. 1. If you have an equation written in natural units (e.g. 𝜒res(𝑟)=𝜆𝐸 4𝜋𝑟 𝑒−𝑚𝜒𝑟), and you want to evaluate it in SI units: • Convert 𝜆and 𝐸to Joules (if they are given in eV use 1 eV =1.602176634×10−19 J). • If 𝑚𝜒is given in eV, convert to inverse meters by 𝑚𝜒[m−1]=𝑚𝜒[eV]×(5.06773065× 106m−1/eV). • Use 𝑟in meters. Then 𝜒res will have units consistent with the left hand side under your chosen field normalization; if 𝜒is canonically normalized in energy units, the result is in eV (or J if you converted). Be consistent: when using 𝜌eff =𝜅 2𝑐2(𝜕𝑟𝜒)2 compute 𝜕𝑟𝜒in SI (per meter) and divide by 𝑐2to get J/m3. 2. Mapping 𝜌eff to mass density (for astrophysical comparison): 9 •𝜌eff computed from 𝜅 2𝑐2(𝜕𝑟𝜒)2has SI units J/m3. To convert to mass density (kg/m3): 𝜌mass [kg/m3]=𝜌eff [J/m3] 𝑐2. (Because energy density = mass density ×𝑐2.) 3. Practical numeric check: implement a small helper function in your code (pseudocode): # inputs: lambda_eV, E_joules, mchi_eV, r_meters hbarc_eVm = 1.973269804e-7 # eV * m eV_to_J = 1.602176634e-19 # J per eV mchi_minv = mchi_eV * 5.06773065e6 # convert eV to m^-1 lambda_J = lambda_eV * eV_to_J chi_res = (lambda_J * E_joules) / (4*np.pi * r_meters) * np.exp(-mchi_minv * r_meters) # then compute dr_chi numerically and rho_eff = kappa/(2*c**2) * (dr_chi)**2 (Include this snippet as a reproducibility aide in your code repository.) 15 Recommended Canonical Conventions for the Paper / Code Release To avoid confusion in readers and in code, we recommend the following conventions (state them clearly in the Methods or Notation table): 1. Fields: all analytic formulae in the paper are presented in natural units (ℏ = 𝑐=1). When giving numerical examples we convert to SI using the rules above. Mark every table/figure label with the units used (e.g. 𝑚𝜒in m−1or eV, 𝜆in eV or J — be explicit). 2. Normalization: unless otherwise stated we set 𝑍𝜒=𝑍𝜓=1in analytic derivations; for numerical work one should explicitly check the canonical rescaling (D.2) and use 𝜆c (D.4) in simulation code. Put code defaults at the top of notebooks (e.g. Z_chi=1.0, Z_psi=1.0). 3. Parameter reporting: report both (a) canonical parameters 𝜆c(in eV) and (b) combinations used in observables (e.g. 𝜆c𝐸in Joules) to avoid degeneracy confusion. Provide also derived quantities such as 𝑚−1 𝜒in kpc for astrophysical intuition. Example: mchi_eV = ... ; mchi_m = (hbarc_eVm * mchi_eV)^-1 etc. 4. Dimension checks: include in the repository a unit-check function that verifies each derived expression has the expected SI units (this is invaluable for catching normalization mistakes). 16 Short Worked Example Tying Everything Together Take a thin shell with 𝐸shell =1044 J, canonical coupling 𝜆c=10−6eV, and 𝑚𝜒=10−22 m−1. Evaluate 𝜒res at 𝑟=1 kpc =3.085677581 ×1019 m. 10 Step 1 — convert 𝜆cto SI (J): 𝜆c=10−6eV ⇒𝜆c[J]=10−6× (1.602176634 ×10−19)=1.602176634 ×10−25 J. Step 2 — compute 𝑚𝜒𝑟: 𝑚𝜒𝑟=(10−22 m−1)×(3.085677581 ×1019 m)=3.085677581 ×10−3. Hence 𝑒−𝑚𝜒𝑟≈1−3.0857 ×10−3≈0.99692 (practically unity). Step 3 — evaluate 𝜒res(𝑟)=𝜆c𝐸shell 4𝜋𝑟 𝑒−𝑚𝜒𝑟. Insert numbers: • Numerator: 𝜆c𝐸shell =(1.602176634 ×10−25 J)×(1044 J)=1.602176634 ×1019 J2. • Denominator 4𝜋𝑟 =4𝜋×3.085677581 ×1019 m=3.876006 . . . ×1020 m(compute explicitly: 4𝜋≈12.56637061;12.56637061×3.085677581×1019 =3.876006 . . .×1020). • So the prefactor is 1.602176634 ×1019 3.876006 ×1020 ≈4.134 ×10−2J/m. Multiply by 𝑒−𝑚𝜒𝑟≈0.99692 gives 𝜒res(1 kpc) ≈ 4.121 ×10−2J/m. Interpretation: depending on the field normalization this number corresponds to the canonical field amplitude in SI energy-length units. To compute the effective energy density 𝜌eff = 𝜅 2𝑐2(𝜕𝑟𝜒)2you then differentiate 𝜒res (𝑟)numerically (consistent finite difference) and plug into the formula with 𝜅chosen as in the paper. 17 Short Checklist for Authors (Copy-paste into Your Methods) • State explicitly whether formulae are given in natural units (ℏ = 𝑐=1) or SI. • Provide the canonical normalization relations (D.2)–(D.4) and state the default 𝑍-choices used in analytic tables/figures. • Include the conversion constants from §D.3 in a small footnote where you report numerical examples. • Always report parameter constraints in both “natural” (e.g. eV, m−1) and SI (Joules, meters) units in tables for reader convenience. • Provide a short code snippet (like the pseudocode in §D.6) in the supplementary repository so readers can reproduce numeric values. 11 18 Concluding Remark This appendix gives the minimal, explicit machinery you need to move between the analytic, canonical expressions used in the derivations and the SI numbers used in numerical estimates and observational comparisons. Keep the three recipes (canonical rescaling D.2–D.4, unit conversions §D.3, and the practical numeric recipe §D.6) close to any numeric code or public repository so there is no ambiguity regarding normalization. 12