The Fundamentals of Stratophysics

The Fundamentals of Stratophysics Kevin Narsh October 4, 2025 Abstract We present a framework for Stratophysics, in which deviations from Newtonian gravity are modeled as density-dependent modifications of the effective scalar mass. The resulting force profiles interpolate across environments, with screening lengths that shorten in dense regions and extend toward cosmic voids. Laboratory constraints determine meff(ρlab), which then conditionally map to astrophysical scales. In an optimistic scenario (meff ∼5 m−1), the extrapolated cosmic Compton wavelength is λcos ∼230 kpc, close to the scale of satellite distributions around the Milky Way. However, this long range applies only at the cosmic mean density (ρcos ∼10−27 kg m−3). In situ halo densities (ρhalo ∼10−25–10−24 kg m−3) instead imply shorter ranges λhalo ∼1–6 kpc, which set the more realistic observational regime. We highlight multiple falsification routes: direct laboratory screening tests, precision modeling of halo outskirts, dwarf galaxy velocity dispersions, tidal stream morphology, and widebinary statistics. This dual emphasis on methodological rigor (via solver convergence criteria and density-scaling consistency) and astrophysical testability positions Stratophysics as a conditional but falsifiable extension to current scalar-field screening models. Scope and intent. This work is exploratory and pedagogical in nature. It presents a self-consistent theoretical framework developed through analytical modeling and assisted computational reasoning. No claim of empirical discovery or new fundamental physics is made. All results should be interpreted as conceptual illustrations intended to clarify the internal logic of density-dependent scalar screening models. 1 Introduction & Motivation The search for new long-range interactions beyond Newtonian gravity and General Relativity has motivated a wide range of theoretical frameworks. Among these, chameleon fields and other screened scalar-tensor theories introduce density-dependent effects that allow new forces to evade detection in laboratory settings while remaining relevant in astrophysics and cosmology. Stratophysics develops this idea into a systematic framework. Its central thesis is: Density-dependent scalar fields give rise to layered, non-monotonic force structures, such that laboratory measurements can be directly extrapolated to cosmic environments. The motivation is twofold: •Phenomenology: Existing searches for Yukawa-type deviations from Newtonian gravity have focused on monotonic exponential forms [4, 6].. Stratophysics proposes that densitydependent layering can yield richer non-monotonic structures with short-range “practical peaks” and longer tails. 1 •Falsifiability: By design, the framework yields conditional predictions. If laboratory effective masses are measured within certain windows, a unique cosmological Compton wavelength follows. These predictions can be ruled out by both laboratory null results and astrophysical surveys. This paper is structured as follows: Section 2 introduces precise definitions and notation. Section 3 presents the minimal microphysical model and derives the density-scaling law. Section 4 develops the phenomenological ansatz for fractional deviations δ(r). Sections 5 and 6 connect laboratory thin-shell calculations to cosmological scales. Section 7 describes falsification routes, and Section 8 discusses limitations and outlook. 2 Definitions & Glossary To ensure clarity, we introduce the central concepts and notation of Stratophysics. Stratic field (ϕ). A real scalar degree of freedom whose effective mass depends on the ambient matter density. In high-density environments it is screened (becomes heavy), and in low-density environments it is unscreened (becomes light). Screening / Stratic Fade S(ρ).The density-dependent suppression factor governing the force mediated by ϕat density ρ. For high ρ,S(ρ)≪1; for low ρ,S(ρ)∼1. Practical manifestation radius rm.An operational radius that marks the location where the fractional deviation δ(r) attains a specified fraction of its peak value. For the kernel T1(r) = α(r/λ1)e−r/λ1, the true maximum occurs at rpeak =λ1. Screening length λ(ρ).The characteristic length associated with the effective mass meff(ρ), defined as λ(ρ)=m−1 eff . It controls the exponential fall-off of stratic contributions in a given density regime. Laboratory effective mass meff(ρlab).The effective mass of the stratic field fluctuations at laboratory density ρlab. This is the experimentally accessible input parameter that allows extrapolation to cosmic densities. Fractional deviation δ(r).The dimensionless deviation from Newtonian predictions: δ(r) = Fobs(r)−FNewton(r) FNewton(r).(1) In Stratophysics, δ(r) is modeled by a multi-channel ansatz, with the dominant contribution given by T1(r) = αr λ1e−r/λ1,(2) which vanishes at the origin, rises to a peak at r=λ1, and exhibits Yukawa-type exponential decay for r≫λ1. Notation summary. •ϕ: stratic field. •Λ: energy scale in the inverse-power potential V(ϕ)=Λ4+n/ϕn. •β: dimensionless coupling to matter. •meff(ρ): effective mass of fluctuations at density ρ. •λ(ρ): screening length, λ= 1/meff. 2 •λcos: cosmological Compton wavelength, determined at cosmic density. •rpeak: location of the true maximum of T1(r), equal to λ1. 3 Minimal Microphysical Model The purpose of this section is to present the simplest microphysical realization of Stratophysics. We introduce a scalar field ϕwith density-dependent dynamics, described by a chameleon-type action. 3.1 Action and coupling to matter The effective action is taken as S=Zd4x√−gM2 Pl 2R−1 2(∂ϕ)2−V(ϕ)+Sme2βϕ/MPl gµν, ψm,(3) where MPl is the reduced Planck mass, Ris the Ricci scalar, βis a dimensionless coupling constant, and Smis the matter action depending on the rescaled metric. The matter coupling ensures that ϕcouples universally to the trace of the matter stress-energy tensor. Planck mass convention. Throughout this paper we use the reduced Planck mass, which we denote by MPl. Numerically we take MPl ≃2.4×1018 GeV, so that the Einstein-Hilbert term is written as M2 Pl 2R. 3.2 Choice of potential To realize a density-dependent mass, we adopt the inverse-power potential [1, 2] V(ϕ) = Λ4+n ϕn,(4) with Λ a characteristic energy scale (typically taken near the dark energy scale, Λ ≃2.4×10−3eV) and na positive integer index. This form ensures that the effective mass of ϕdepends strongly on the ambient density. For the inverse-power potential V(ϕ) = Λ4+n/ϕn, solving V′(ϕmin)+βρ/MPl = 0 gives ϕmin(ρ)∝ρ−1/(n+1). Hence m2 eff(ρ)≡V′′(ϕmin)∝ρ(n+2)/(n+1),⇒meff(ρ)∝ρ n+2 2(n+1) . For n= 1 this reduces to meff ∝ρ3/4(used in our numerical estimates). 3.3 Field equation and equilibrium condition Varying the action with respect to ϕyields the equation of motion ∇2ϕ=V′(ϕ) + β MPl ρ, (5) 3 where ρis the local matter density. The equilibrium field value ϕmin(ρ) is obtained by solving V′(ϕmin) + β MPl ρ= 0.(6) For the chosen potential, this gives ϕmin(ρ)∝ρ−1/2.(7) 3.4 Density-scaling law for the effective mass The effective mass of fluctuations about ϕmin is defined by m2 eff(ρ)≡V′′(ϕmin).(8) For the inverse-power potential, substituting ϕmin(ρ)∝ρ−1/2yields m2 eff ∝ρ3/2,⇒meff ∝ρ3/4.(9) This scaling law is central to Stratophysics: it directly links laboratory measurements of meff at density ρlab to the cosmological Compton wavelength λcos = 1/meff (ρcos) at cosmic densities. Hence, the framework is predictive and falsifiable: a single laboratory determination of meff fixes the astrophysical scale. 4 Phenomenological Ansatz for Observables While the microphysical model specifies the underlying density dependence of the stratic field, it is useful to construct a compact phenomenological framework for experimental and observational signatures. We therefore introduce an ansatz for the fractional deviation from Newtonian predictions. 4.1 Fractional deviation δ(r)and total acceleration We define the observable fractional deviation from Newtonian expectations as δ(r)≡Fobs(r)−FN(r) FN(r),(10) where Fobs(r) denotes the measured force between two test bodies at separation rand FN(r) the Newtonian prediction. In the weak-field, non-relativistic limit we may work in terms of accelerations. Denoting the Newtonian acceleration by aNand the additional acceleration sourced by the stratic scalar by aφ, the total acceleration is modeled as atot(r) = aN(r)+aφ(r).(11) For spherically symmetric configurations we therefore define the scalar fractional contribution through δ(r)≡aφ(r) aN(r)⇒atot(r) = aN(r)1+δ(r).(12) In the phenomenological, multi-channel decomposition used below we model the fractional deviation as a small, perturbative correction, δ(r)≃X i Aifi(r)≃A T1(r)+O((subleading)),(13) 4 where T1(r) denotes the dominant channel (Sec. 4.3), Aan effective amplitude that encodes the coupling strength and geometry factors (thin-shell suppression, etc.), and the perturbative regime |δ(r)| ≪ 1 is assumed throughout the main text. 4.2 Multi-channel decomposition In principle, deviations from Newtonian gravity can be represented as a superposition of independent kernels, δ(r) = X i Aifi(r),(14) where Aiare amplitudes and fi(r) are radial functions. This reflects the idea that different density strata may contribute distinct characteristic ranges. In practice, however, we find that the leading channel T1(r) is dominant for the regimes of interest (see Sec. 4.3). Additional channels are therefore not needed for the present analysis and would only serve as exploratory extensions. 4.3 Dominant kernel T1(r) To correct the analytic form used in earlier drafts and to produce a true finiteradius peak, we adopt the simple one-parameter non-monotonic kernel T1(r)≡αr λ1e−r/λ1,(15) where αis a dimensionless amplitude and λ1is the characteristic range (screening length) associated with this dominant channel. This kernel has the following elementary properties: •Near the origin r≪λ1,T1(r)≃α(r/λ1)+O((r/λ1)2), so the kernel vanishes at r= 0 and rises linearly for small r. •The radial derivative is dT1 dr =αe−r/λ1 λ2 1 (λ1−r),(16) which shows that T1(r) attains a single global maximum at r=λ1. •For r≫λ1the kernel decays exponentially as T1(r)∼α(r/λ1)e−r/λ1, producing standard Yukawa-type tails in the observable δ(r). We therefore define the practical peak radius by rpeak ≡λ1,(17) and measure widths and fraction-of-peak radii relative to rpeak. In practice the fractional deviation is modelled as δ(r)≃A T1(r),(18) with Aabsorbing coupling strength, thin-shell suppression and geometric factors. This replacement removes the earlier incorrect claim of a finite practical peak for the α(1+r/λ)e−r/λ form and is algebraically simple to differentiate, plot, and implement in solver diagnostics. 5 4.4 Beyond the dominant channel: observational motivation and future work The dominant kernel T1(r) produces screening lengths λhalo ∼1–6 kpc within galactic halos (Sec. 7). However, the Milky Way’s ”missing satellites problem” occurs at radii r∼50–200 kpc, where observations show fewer dwarf galaxies than ΛCDM simulations predict. We note that if an additional long-range component with characteristic scale λ2∼200 kpc were present, it could in principle address this discrepancy through cumulative orbital effects. To illustrate this possibility, we phenomenologically introduce a minimal subleading tail: δ(r) = A1T1(r)+A2T2(r), T2(r)≡α2r λ2e−r/λ2,(19) with baseline parameters λ2= 200 kpc, α2= 0.026, and A2= 1. Numerically, this yields T2(200 kpc) = α2200 200e−200/200 = 0.026 e−1≈9.56 ×10−3,(20) T2(300 kpc) = α2300 200e−300/200 = 0.026 ×1.5e−1.5≈8.70 ×10−3.(21) Thus the subleading contribution remains at the percent level in the 200–300 kpc range. While small at any single radius, such a component can accumulate over many orbits and may play a non-negligible role in shaping satellite abundances at large galactocentric distances. Critical caveat: This component is not derived from the minimal chameleon model of Sec. 3. Rather, it represents an observational target for future theoretical work. Potential physical origins include: •Multi-field chameleon models with independent screening scales, •Density-gradient effects in realistic halo profiles that extend the effective range beyond the naive λ(ρhalo) estimate, •Non-linear self-interactions or higher-derivative terms not included in the minimal action (Eq. 1). Testing these possibilities requires: (i) numerical solution of the field equations with realistic ρMW(r) profiles (e.g., NFW), (ii) parameter-space exploration of multi-field extensions, and (iii) N-body simulations including stratic forces. These calculations lie beyond the scope of this proofof-principle framework. We therefore present T2asahypothesis that makes the satellite-scale phenomenology concrete and testable, not as a prediction from first principles. Future work should either derive such a tail from microphysics or falsify it through astrophysical constraints. 4.5 Limits and assumptions The ansatz is constructed under the following simplifying assumptions: 1. Perturbative regime: |δ(r)| ≪ 1 so that the deviation is treated as a small correction to Newtonian gravity. 2. Composition independence: the coupling is assumed to be universal at leading order, in line with weak equivalence principle tests. 6 3. Spherical symmetry: the radial form assumes spherically symmetric source distributions, appropriate for first-order laboratory modeling. This phenomenological framework makes explicit predictions for the shape of deviations, enabling both tabletop experiments and astrophysical surveys to test Stratophysics against data. In our solver runs for the parameter choices reported here we explicitly verified that |δ(r)|≲10−4, so the perturbative assumption |δ(r)| ≪ 1 is satisfied for the cases shown. 5 Compatibility with the Gravity Amplification Impossibility Theorem The Impossibility Theorem for gravitational amplification [3] states that, under the set of classical assumptions A1–A5 (well-posed deterministic field equations with uniqueness for specified initial and boundary data; local stress–energy conservation; reasonable energy conditions; fixed boundary/asymptotic data; and absence of additional long-range dynamical gravitational degrees of freedom), a reproducible local increase in the gravitational acceleration cannot occur without changing the local stress–energy or the initial/boundary data. Stratophysics does not attempt to contradict this theorem within its stated assumptions. Rather, the framework is an explicit, minimal extension of the standard gravitational sector: it introduces a single additional long-range dynamical degree of freedom, the stratic scalar field ϕ, whose effective mass depends on ambient density. Accordingly, Stratophysics explicitly violates assumption A5 of the Impossibility Theorem (the prohibition on additional long-range dynamical fields). We state this violation openly and treat it as the central hypothesis enabling mass-free modifications to the local acceleration. We summarize the logical situation succinctly: •the theorem delineates a set of assumptions (A1–A5) under which mass-free, reproducible gravitational amplification is impossible. •Stratophysics achieves an additional contribution aϕto the local acceleration precisely because it relaxes A5 by introducing the scalar ϕ. •This is an explicit hypothesis of the model, not a hidden loophole: the theory remains falsifiable because the scalar carries characteristic length scales, density-dependence, and equivalence-principle signatures that may be constrained or excluded by laboratory and astrophysical data. 5.1 Assumption accounting (A1–A5) To make the compatibility statement concrete, we list the assumptions and the status of Stratophysics with respect to each: A1: Well-posedness and uniqueness. We require that the scalar field equations (coupled to matter in the Jordan frame) yield a well-posed boundary-value problem for the static profiles of interest. The minimal model adopts standard second-order dynamics for ϕand enforces regular boundary conditions; under these conditions A1 is respected at the classical level. 7 A2: Local conservation of stress–energy. Matter is coupled universally through the conformal rescaling e2βϕ/MPl gµν. In this Jordan-frame coupling the (total) stress–energy including the scalar respects local conservation ∇µTµν = 0. Thus A2 is respected. A3: Standard energy conditions. The model enforces a canonical kinetic term for ϕand a potential chosen so that m2 eff >0 in the relevant regimes, avoiding classical ghosts and tachyons. At the classical EFT level A3 is respected; quantum (radiative) issues are listed in Sec. 9 as items requiring further work. A4: Fixed boundary/initial data. The boundary and asymptotic conditions are held fixed during experiments. Stratophysics respects A4 by requiring that experimental protocols maintain fixed boundary data and that any observed deviations arise from the scalar field dynamics, not from external manipulation of boundary conditions. A5: No additional long-range gravitational degrees of freedom beyond those in standard GR/Newtonian gravity. Stratophysics explicitly violates this assumption by introducing ϕas an additional long-range field. This is the sole assumption we relax to obtain a non-trivial aϕwhile keeping the rest of the theory standard. The metric-sector Einstein–Hilbert action and standard weak-field GR phenomenology remain intact in screened regimes. Operational consequence. Declaring this A5 violation clarifies the falsification program: laboratory measurements of meff(ρlab), equivalence-principle tests, and astrophysical constraints on the length scales λ(ρ) jointly bound the scalar parameters (β, Λ, n) and either validate or exclude the extension. Any reproducible, well-documented claim of mass-free amplification that does not identify a failure of at least one of A1–A5 remains incompatible with the theorem as shown in my latest work. 6 Laboratory Thin-Shell & Example Calculations A central prediction of Stratophysics is that screening effects in laboratory-scale test masses lead to suppressed effective couplings. To quantify this, we consider the canonical thin-shell effect for a spherically symmetric body. 6.1 Spherical thin-shell derivation Let a test mass of radius Rand density ρlab be embedded in a lower-density environment. Inside the body, the scalar field ϕrelaxes toward the minimum ϕmin(ρlab), while outside it approaches ϕmin(ρenv). The transition occurs across a thin shell of thickness ∆R≪Rnear the surface. Following the standard analysis, the effective coupling of the body to the stratic field is suppressed by the factor ∆R R≃ϕenv −ϕlab 6βMPl ΦN ,(22) where ΦN=GM/R is the Newtonian surface potential, ϕenv ≡ϕmin(ρenv), and ϕlab ≡ϕmin(ρlab). 6.2 Effective coupling The effective coupling of a thin-shelled object is αeff ≃3β∆R R,(23) and we will refer to (23) throughout. This formula expresses the suppression of new forces in laboratory environments: the deeper the Newtonian potential ΦNand the denser the body, the smaller the effective coupling. 8 UNRESOLVED DEPENDENCY. Computing ∆R/R requires knowledge of the normalization of the internal field value ϕlab. However, ϕlab itself depends on the microphysical parameters (Λ, β, n) through the equilibrium relation ϕmin(ρ)∝ρ−1/(n+1) (see Appendix ??). Different plausible normalizations of ϕlab (i.e. different choices of Λ and βwithin existing observational bounds) can change ∆R/R by many orders of magnitude, as illustrated in Appendix D. This circularity implies that the thin-shell suppression cannot be predicted from laboratory geometry alone: the framework requires external calibration of (Λ, β, n) from independent experiments or astrophysical constraints before screening strengths can be forecast quantitatively. Consequently, the thin-shell discussion should be read as conditional on a chosen normalization of ϕlab rather than as an absolute prediction. 6.3 Mapping to laboratory observables The observable consequence in force experiments is a deviation of the form δ(r)∼αeff r λe−r/λ,(24) where λis the relevant screening length at laboratory density. Thus: •If ∆R/R ≪1: the test mass is screened, and δ(r) is strongly suppressed, often below current experimental sensitivity. •If ∆R/R ∼1: the mass is unscreened, yielding αeff ≈βand a potentially detectable deviation. 6.4 Representative parameter ranges For typical laboratory test masses with ρlab near terrestrial density and Rin the cm–m range, the thin-shell condition yields ∆R/R values that span both regimes, depending on βand Λ. This explains how Stratophysics remains compatible with existing null results, yet predicts non-trivial deviations in carefully chosen density and geometry windows. 7 From Lab to Cosmos: Scaling & Cosmological Compton Wavelength The defining feature of Stratophysics is that a laboratory determination of the effective mass meff at density ρlab uniquely fixes the corresponding Compton wavelength at cosmic density, λcos. 7.1 Density-scaling relation From Section 3 we derived that meff(ρ)∝ρ3/4.(25) Thus, given meff(ρlab) at laboratory density, the mass at cosmic (or halo) density follows as meff(ρcos) = meff(ρlab)ρcos ρlab 3/4 .(26) The corresponding Compton wavelength is λcos ≡1 meff(ρcos).(27) 9 Compared to traditional fifth-force searches, Stratophysics emphasizes falsifiability and consistency with the Gravity Amplification Impossibility Theorem. By declaring an explicit violation of Assumption A5 (via the scalar field ϕand its multiple propagation modes), the framework yields a testable prediction: cumulative satellite mass losses of order 5–15% over many orbits, corresponding to a ∼5% reduction in luminous satellite counts. This balances minimal theoretical modification with a concrete observational target. At the same time, the framework remains minimal: one scalar field, one potential, and one coupling to matter. All richer phenomenology arises from the density scaling itself. This simplicity strengthens the falsifiability of the proposal, while also clarifying what must be added in order to embed Stratophysics in a broader theoretical setting. 11 Conclusions We have introduced Stratophysics as a new phenomenological framework for density-dependent scalar forces. Its main features are: •A scalar field ϕwith inverse-power potential, leading to the scaling meff ∝ρ3/4. •A layered, non-monotonic ansatz for deviations δ(r), with practical peaks and Yukawa tails. •A thin-shell suppression mechanism, yielding effective couplings αeff = 3β∆R/R. •A direct laboratory-to-cosmos mapping: measurements of meff(ρlab) determine λcos at cosmic density. The framework is deliberately falsifiable. If laboratory constraints push meff above ∼24– 45 m−1, the coincidence with galactic scales disappears. If astrophysical surveys fail to observe deviations at the predicted λcos, the framework is excluded. And if equivalence-principle violations are observed, the minimal universal coupling must be abandoned. Thus, Stratophysics provides a clear experimental program: 1. Precision laboratory searches (torsion pendula, atom interferometers) to constrain meff. 2. Astrophysical tests of satellite distributions and dwarf galaxy kinematics. 3. Equivalence-principle probes (MICROSCOPE, LLR). We stress explicitly that Stratophysics is an admitted, minimal violation of assumption A5 of the gravitational Impossibility Theorem: the introduction of one long-range scalar degree of freedom is the hypothesis that permits an additional contribution aϕto the local acceleration. This choice is declared, constrained, and falsifiable (laboratory bounds on meff(ρlab), equivalence-principle tests, and astrophysical probes), and is not presented as a hidden loophole or contradiction of established uniqueness results. In conclusion, Stratophysics is not a speculative unification but a practical, falsifiable proposal. It sharpens the interface between laboratory and cosmology, ensuring that forthcoming experiments can decisively confirm or refute its predictions. A Numerical Methods and Solver Convergence The numerical results presented in this work were obtained using standard boundary-value solvers for the scalar field profile in spherically symmetric geometries. The procedure was: 16 1. The static field equation, ∇2ϕ=V′(ϕ) + β MPl ρ(r),(28) was discretized on a radial grid with adaptive spacing near the surface of the test body. 2. Boundary conditions were imposed as ϕ(r→ ∞)→ϕenv and dϕ dr (r= 0) = 0. 3. Convergence was tested by varying both resolution and domain size, with relative differences below 10−4. Remark on force extraction and domain effects. While our field profiles converge at the O(10−4) level, we found that accurate extraction of force ratios (gradients) is more demanding. In particular, near-field force ratios (e.g. |aϕ|/aNat radii ∼2R) can deviate from the analytic thinshell estimate by up to ∼2 orders of magnitude when the radial domain is not extended sufficiently or when derivative stencils are not tightened. This indicates that the solver tolerances sufficient for field values are not always sufficient for derivatives; practitioners should therefore (i) extend rmax by at least an order of magnitude beyond the object radius for sensitive force extraction, and (ii) use higher-order finite-difference stencils or Richardson-extrapolated gradients when quoting aϕ. The main text (Sec. 6) discusses how these effects influence our reported force bounds. These methods were sufficient for the spherical thin-shell examples discussed in Section 5. Extension to 3D geometries is left for future work. A Parameter scaling and raw numerical outputs For transparency we include the raw numerical outputs of the scaling-law computation. We use the relation meff(ρ) = meff(ρlab)ρ ρlab 3/4 , λ(ρ)≡1 meff(ρ).(29) The reference densities are ρlab = 103kg m−3and ρcos = 8.6×10−28 kg m−3. Two representative halo densities are also included, ρ= 10−25 and 10−24 kg m−3. Distances are expressed in kiloparsecs (1 kpc = 3.085677581 ×1019 m). meff(ρlab) (m−1)λcos (kpc) λhalo (kpc, 10−25)λhalo (kpc, 10−24) 5 229.5 6.48 1.15 24 47.8 1.35 0.240 45 25.5 0.720 0.128 The values confirm that extrapolating from laboratory density to the cosmic mean yields characteristic lengths of tens to hundreds of kiloparsecs, while in typical halo environments the effective range drops to sub-kiloparsec scales. For reference, we collect representative parameter values used throughout the paper. For numerical convenience, lengths are expressed in kiloparsecs (1 kpc = 3.085677581×1019 m). Raw solver outputs (field profiles, δ(r) curves, and ∆R/R values) are omitted here for brevity but can be made available as supplementary material. 17 Scenario meff(ρlab)λcos (cosmic) λhalo (MW) Status Optimistic 5 m−1230 kpc ∼6 kpc Testable Nominal 24 m−150 kpc ∼1.3 kpc Marginal Conservative 45 m−126 kpc ∼0.7 kpc Core-only Table 3: Sensitivity scenarios showing how laboratory determinations map to astrophysical scales. B Alternate Potential Forms Although this paper focused on the inverse-power potential V(ϕ) = Λ4+n ϕn,(30) other forms have been considered in related literature, such as exponential or logarithmic potentials. These can modify the density-scaling law: •Exponential potentials V(ϕ)∼Λ4eM/ϕ typically yield weaker density dependence. •Polynomial potentials V(ϕ)∼m2ϕ2do not exhibit strong screening without additional couplings. A systematic comparison is left for future work, but the inverse-power form remains the simplest realization with the desired meff ∝ρ3/4scaling. C Additional Derivations For completeness, we collect algebraic steps omitted in the main text. C.1 Derivation of ϕmin(ρ) Starting from V′(ϕ) + β MPl ρ= 0,(31) and using V(ϕ) = Λ4+n/ϕn, we obtain dV dϕ =−nΛ4+n ϕn+1 .(32) The equilibrium condition then yields ϕmin(ρ)∝ρ−1/2.(33) C.2 Derivation of meff (ρ) The effective mass is m2 eff =V′′(ϕmin)=n(n+ 1)Λ4+n ϕn+2 min .(34) Substituting ϕmin ∝ρ−1/2gives m2 eff ∝ρ3/2, meff ∝ρ3/4.(35) 18 D Worked thin-shell example Planck mass convention. Throughout this work MPl denotes the reduced Planck mass, MPl ≡rℏc 8πG ≈4.341 ×10−9kg ≈2.435 ×1018 GeV. The (non-reduced) Planck mass is MP=pℏc/G ≈2.176 ×10−8kg. All field values quoted as multiples of MPl refer to the reduced Planck mass. Numerical example. Consider a solid sphere of radius R= 0.05 m and density ρlab = 103kg m−3, placed in an environment of density ρenv = 10−7kg m−3. We take the matter coupling β= 1 and assume φmin(ρ)∝ρ−1/2(inverse-power potential with n= 1). For illustration we normalize the laboratory field value to a small fraction of the reduced Planck mass, φlab = 10−20 MPl ≈4.34 ×10−29 kg. With this choice one obtains φenv =φlab ρenv ρlab −1/2 ≈4.34 ×10−24 kg, a Newtonian surface potential ΦN≈6.99 ×10−10, and the thin-shell thickness ∆R R≃φenv −φlab 6βMPl ΦN≈2.39 ×10−7. Because ∆R/R ≪1, the body develops a thin shell and is screened. The effective coupling is correspondingly suppressed, αeff ≃3β∆R R≈7.15 ×10−7. Interpretation. This concrete worked example demonstrates that for plausible normalizations of φlab the laboratory object satisfies the thin-shell condition (∆R/R ≪1). The earlier toy choice φlab = 10−6MPl would instead yield ∆R/R ≫1 and an unscreened object; thus the thin-shell criterion is highly sensitive to the model-dependent normalization of φmin(ρ). E Supplemental Figures Figures illustrating the field profile ϕ(r), the fractional deviation δ(r) with its non-monotonic peak, and solver convergence tests are provided as supplemental material. Placeholders for these are included here: Figure 1: Representative profile of the dominant kernel T1(r) = α(r/λ1)e−r/λ1. The kernel rises linearly from zero, attains a true maximum at rpeak =λ1, and decays exponentially for r≫λ1. This corrects the earlier “practical peak” approximation of previous drafts: the peak location is now analytically fixed at rpeak =λ1. 19 Figure 2: Illustrative fractional deviation δ(r) = A T1(r) relative to Newtonian acceleration. The deviation peaks at rpeak =λ1with amplitude δ(rpeak) = Aαe−1, then falls off rapidly for larger r. This demonstrates the non-monotonic character of the corrected kernel and removes the need for ad hoc “practical peak” definitions. References References [1] Khoury, J. and Weltman, A. 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