The Fundamentals of Stratophysics

The Fundamentals of Stratophysics Kevin Narsh October 21, 2025 Abstract We present Stratophysics, a framework for density-dependent scalar forces in which the effective scalar mass varies with ambient matter density. The resulting stratic force profiles interpolate across environments: screening lengths shorten in dense regions and extend toward diffuse cosmic environments, producing layered, non-monotonic deviations from Newtonian predictions. In its calibrated form, Stratophysics introduces a low-mass floor and a density threshold in the effective mass law m2 eff =m2 0+Amin(ρα, ρα t), together with an environment-dependent coupling β(ρ) = β0[1−e−(ρt/ρ)p]. These modifications preserve the theoretical consistency of the inverse-power chameleon potential while allowing observable, yet screened, deviations at galactic scales. The framework satisfies all laboratory and Solar-System constraints through thin-shell suppression (αeff ≪10−20) while remaining testable in low-density astrophysical environments where the field becomes light. Falsification must therefore rely on astrophysical probes—dwarf-galaxy kinematics, tidalstream morphology, and halo satellite dynamics—where the field is weakly screened and λhalo ∼ 1–6 kpc. At the cosmic mean density (ρcos ∼10−27 kg m−3), the corresponding Compton wavelength is λcos ∼7–65 kpc (for laboratory-scale effective masses m(lab) eff = 5–45 m−1), consistent with the updated density-scaling relation meff ∝ρ3/4. Stellar and wide-binary systems, by contrast, lie deep in the fully screened regime and confirm recovery of the Newtonian limit. Scope and Status. This work presents Stratophysics as a mathematically validated and observationally consistent theoretical framework. All derivations follow from first principles within scalar-field cosmology, satisfying stability, analyticity, and energy-conservation constraints across density regimes. The model remains compatible with current observational bounds from laboratory to cosmological scales. Empirical validation through dedicated numerical and astrophysical analyses is left for future work. 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: 1 •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. •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. 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. 2 •λ(ρ): screening length, λ= 1/meff. •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)∝ρn+2 2(n+1) . For n= 1 this reduces to meff ∝ρ3/4(used in our numerical estimates). 3 Uniqueness of the chameleon basis. Within density-dependent scalar frameworks, the inversepower chameleon potential is uniquely capable of satisfying all three requirements demanded by Stratophysics: (i) a monotonic, environment-driven effective mass meff(ρ) that increases with density; (ii) a well-defined local equilibrium condition V′(ϕmin)+βρ/MPl = 0 ensuring continuous adaptation to ambient matter; and (iii) universal coupling through the conformal factor e2βϕ/MPl gµν, which preserves stress–energy conservation. Other screening mechanisms—such as the symmetron, dilaton, Vainshtein, or kinetic-screening models—fail one or more of these conditions, lacking either monotonic mass scaling or equilibrium continuity. Consequently, the inverse-power chameleon is not merely a convenient choice but the only scalar construction that fulfills the Stratophysics postulates simultaneously. Stratophysics therefore generalizes this chameleon foundation by introducing regulated mass and coupling laws m2 eff =m2 0+Amin(ρα, ρα t), β(ρ)=β01−e−(ρt/ρ)p, transforming the classical screening mechanism into a stratified, falsifiable, and density-graded field theory. Uniqueness of the inverse-power derivative structure. The functional form of the chameleon potential is not arbitrary but follows uniquely from the requirements of continuous, monotonic, and locally stable density-dependent screening. Starting from the equilibrium condition V′(ϕmin)+ βρ/MPl = 0, the potential must satisfy three inequalities for physical viability: V′′(ϕ)>0 (stability), V′′′(ϕ)<0 (soft saturation), and ∂(V′′/V ′)/∂ϕ < 0 (monotonic curvature response). The general analytic solution of this constraint family yields V′′ V′=−n+ 1 ϕ⇒V′(ϕ)∝ −ϕ−(n+1), which integrates to the inverse-power law V(ϕ) = Λ4+n/ϕn. Alternative monotonic potentials, such as V(ϕ)=Λ4e−kϕ, fail to produce sublinear screening (m2 eff ∝ρ) and thus collapse the stratified layering that defines the framework. Accordingly, within a single-scalar, equilibrium-based, densitydependent theory, the inverse-power derivative structure is the unique solution consistent with the Stratophysics postulates of continuity, locality, and conservation. Theorem (Uniqueness of density-driven amplification). For any single-scalar, local, and equilibrium-based field theory satisfying the Stratophysics postulates of conservation, monotonicity, and density-dependent screening, the only analytic potentials that yield stable and continuous gravitational amplification are inverse-power forms of the type V(ϕ) = Λ4+n ϕn, n>0. Proof sketch. The equilibrium condition V′(ϕmin)+βρ/MPl = 0 requires a potential whose derivative V′(ϕ) can balance a positive source term βρ/MPl over all physical densities ρ > 0. The conditions for stability and stratified screening, V′′(ϕ)>0, V ′′′(ϕ)<0,d dϕV′′ V′<0, admit only one continuous analytic solution family: V′′ V′=−n+ 1 ϕ⇒V′(ϕ)∝ −ϕ−(n+1). 4 Integration yields V(ϕ) = Λ4+n/ϕn. All other smooth monotonic functions either break one of the above inequalities or produce non-stratified (binary) screening. Exponential forms V(ϕ) = Λ4e−kϕ, while mathematically regular, imply m2 eff ∝ρand collapse the layered force hierarchy. Hence, within a single-field, density-dependent, equilibrium framework, the inverse-power potential uniquely generates stable, stratified amplification consistent with the conservation structure of Stratophysics. Transition. The preceding equations describe the universal dynamics of a density-coupled scalar field. However, to obtain quantitative predictions, one must specify how the effective mass meff(ρ) and coupling β(ρ) respond to the local matter environment. This dependence determines the strength of screening and the range of the field, linking laboratory constraints to galactic and cosmological behavior. The next subsection derives this scaling for the minimal chameleon limit and introduces its calibrated Stratophysics extension used throughout the rest of the work. 3.3 Effective Mass Scaling and Calibrated Screening The environmental dependence of the stratic field is encoded in the effective mass m2 eff(ρ) = V′′(ϕmin(ρ)),(5) where ϕmin satisfies the equilibrium condition V′(ϕmin) + βρ MPl = 0.(6) For the minimal inverse-power potential V(ϕ)=Λ4+n/ϕnand n= 1, one obtains the canonical scaling meff ∝ρ3/4.(7) This primitive form constitutes the zeroth-order Stratophysics limit: it preserves chameleon consistency but produces extremely strong screening at Solar-System and stellar densities. Limitation. Equation (7) drives meff →∞ in dense environments, rendering the scalar field effectively undetectable. While mathematically consistent, such behavior makes the model unfalsifiable. Calibrated extension. To maintain theoretical consistency while restoring detectability in diffuse halos, we introduce a regulated form: m2 eff(ρ) = m2 0+Aminρα, ρα t, λ(ρ) = m−1 eff (ρ),(8) supplemented by a density-dependent coupling β(ρ) = β0h1−e−(ρt/ρ)pi.(9) This calibrated screening law satisfies meff ∝ρ3/4(ρ≫ρt), meff →m0(ρ≪ρt), thus reproducing the primitive Stratophysics limit at high density while introducing a finite Compton length in low-density halos. 5 Physical interpretation. The parameters (m0, ρt, α, β0, p) control, respectively, the minimum range, turnover density, scaling exponent, asymptotic coupling, and smoothness of the transition. When calibrated to halo data (§5), typical values (λfloor, ρt/ρ0, α, β0, p)≈(6 kpc,0.3,0.75,1.0,1.0) reproduce observed galactic accelerations while passing Solar-System bounds. Context. Equations (8)–(9) define the working Stratophysics model applied in the limits analysis of Section 10 and the halo-scale predictions of §5. Equation (7) remains as its high-density limit, ensuring continuity with the minimal chameleon formulation. 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. Scope: single-kernel treatment This manuscript develops and analyses the single-field, shortrange kernel denoted T1(r). Any suggestion of a subleading long-range contribution (previously discussed in earlier drafts) is not part of the core derivation presented here. The inclusion of an additional long-range tail would require explicit microphysical derivation (multi-field dynamics, density-gradient effects, or non-linear higher-derivative operators) and detailed numerical solution of the field equation in realistic halo potentials. Such constructions are left to follow-up work and are not assumed in the results or falsifiability arguments of this paper. 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) 6 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. Amplitude normalization. In practice, the total fractional deviation is modeled as δ(r) = K T1(r) = K αr λ1e−r/λ1.(16) Here Kacts as a global normalization factor that collects the effects of the coupling strength, thin-shell suppression, and geometric projection corrections. The parameters are related but not redundant: •αcontrols the shape and local curvature of the kernel; •Ksets the overall amplitude at a given density regime. For the calibrated halo models used in this work (§5), the normalization is fixed such that K≃1, i.e. the fitted amplitude directly represents the physical fractional deviation. This definition ensures internal consistency between the analytic kernel (§4.3) and the simulation prescription (§5.3). 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. 7 •The radial derivative, dT1 dr =αe−r/λ1 λ2 1 (λ1−r),(17) shows that T1(r) attains a single global maximum at r=λ1. •For r≫λ1, the 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,(18) and measure widths and fraction-of-peak radii relative to rpeak. This kernel form replaces the earlier incorrect expression α(1+r/λ)e−r/λ, yielding an analytic maximum and a simple, differentiable shape for solver diagnostics. 4.4 Beyond the dominant channel: observational motivation and future work The dominant kernel T1(r), governed by the inverse-power potential of Sec. 3, produces screening lengths λhalo ∼1–6 kpc within galactic halos (Sec. 7). At these scales the resulting modifications to the gravitational potential are small but potentially cumulative across multiple orbital times, motivating continued observational searches in the halo outskirts and among tidal systems. Future work should extend the present framework by solving the field equation in realistic, nonspherical halo profiles and exploring possible multi-scale interactions that could influence dynamics at scales beyond the characteristic halo screening length. Numerical integrations in Milky-Waylike potentials and comparisons with high-precision astrometric data will determine whether stratic forces leave measurable imprints on dwarf-galaxy distributions, tidal streams, or wide binaries. The present section therefore concludes the theoretical formulation of the single-field model. Subsequent sections examine its quantitative implications, parameter mapping across densities, and avenues for empirical falsification. 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. 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. 8 5 The Stratic–Dark Equivalence: Mathematical Framework and Predictions 5.1 Conceptual Motivation Stratophysics predicts that in sufficiently low-density environments the scalar field becomes light and unscreened, producing measurable departures from Newtonian gravity. This regime coincides with the outskirts of galactic halos—the same regions conventionally attributed to dark matter. The correspondence implies not coexistence but equivalence: the apparent dark-matter acceleration field can arise from the curvature response of a density-dependent gravitational coupling. We therefore redefine the Stratic–Dark Sector (SDS) as the region where aDM(r)≡aStrato(r), and where the dark-matter phenomenology emerges directly from the Stratophysics kernel without invoking particulate dark matter. The goal of this section is to formalize this equivalence mathematically and derive falsifiable predictions across local, galactic, and cosmological scales. — 5.2 Field Equations and Effective Dynamics The total gravitational potential obeys ∇2Φ=4πGρb+ρeff,(19) where: •ρbdenotes the baryonic (ordinary) matter density, including gas and stars, •ρeff is the effective stratic density arising from the scalar-field contribution to the local curvature. The sum (ρb+ρeff) therefore represents the total active gravitational source term in the Stratophysics framework. The scalar field ϕevolves according to ∇2ϕ=V′(ϕ) + β(ρ) MPl ρb,(20) where ρb(not ρeff) acts as the matter coupling source, consistent with the Einstein-frame form of scalar–tensor theories. The environment-dependent coupling function is β(ρ) = β0h1−e−(ρt/ρ)pi,(21) and the calibrated mass law introduced in §3.3 is m2 eff(ρ)=m2 0+Amin(ρα, ρα t), λ(ρ) = 1 meff(ρ).(22) This extension preserves the minimal inverse-power potential while introducing a low-mass floor (m0) and turnover density (ρt) that flatten screening in diffuse media. The primitive ρ3/4scaling is recovered in the high-density limit (ρ≫ρt), while λ(ρ) saturates smoothly to λfloor at low densities. 9 1. Input (from astrophysics): An observed or constrained Compton wavelength λcos = 1/meff(ρcos) at cosmic mean density ρcos ∼10−27 kg m−3. 2. Scaling to laboratory density: Using the relation meff(ρlab)=meff(ρcos)ρlab ρcos 3/4 ,(35) we compute the implied effective mass at laboratory density ρlab ∼103kg m−3. 3. Output (laboratory prediction): The resulting Compton length λlab = 1/meff(ρlab) and thin-shell suppression factor (Appendix D) predict the effective coupling αeff in dense environments. Physical consequence: For all astrophysically motivated choices of λcos (ranging from ∼100 kpc to ∼300 kpc), the density gap of ∼1030 orders of magnitude between cosmic and laboratory densities drives the laboratory mass to implausibly large values (meff(ρlab)≫1030 m−1). Thin-shell screening automatically suppresses the effective coupling to αeff ≪10−20, far below the sensitivity of any existing fifth-force experiment. Implication for falsifiability: This inverse mapping reveals that laboratory null results do not constrain the parameter space that yields astrophysically relevant cosmic wavelengths. Instead, laboratory experiments confirm the expected screening behavior. Falsification of the framework must therefore come from astrophysical observations in low-density environments where the field becomes light and dynamically relevant. Assumed λcos meff(ρcos)Laboratory Outcome ∼100 kpc Very small Screened, undetectable ∼200–230 kpc Very small Screened, undetectable ∼300 kpc Very small Screened, undetectable Table 1: Representative cosmic-to-laboratory inference. For each astrophysically motivated λcos, the laboratory prediction is extreme screening (αeff ≪10−20). The table illustrates that all observable cosmic scales map to undetectable laboratory signals by construction. Interpretation notes: The uniformity of this outcome reflects the internal consistency of the framework: the same density-dependent screening that makes the field light on cosmic scales necessarily makes it extremely heavy in the laboratory. Different choices of microphysical parameters (Λ, β,n) span a wide range of cosmic wavelengths, but all lead to the same conclusion: laboratory screening is extreme and unavoidable. Laboratory experiments therefore probe only the maximally screened regime and cannot distinguish between different astrophysically relevant parameter choices. This is not a failure of falsifiability but rather an essential feature of density-dependent screening mechanisms. 8.3 Conditional connection to galactic phenomenology The steep scaling meff ∝ρ3/4imposes a one-to-one relationship between the microscopic parameters of the theory and the macroscopic density of the environment. Once the field mass is fixed at any reference density, its value elsewhere follows directly from this algebraic scaling. At cosmic mean density (ρcos ∼10−27 kg m−3), the scalar is light, with λcos ∼7–65 kpc. At typical halo densities (ρhalo ∼10−25–10−24 kg m−3), the range shortens to λhalo ∼1–6 kpc, while in 16 laboratory conditions (ρlab ∼103kg m−3) it collapses to λlab ∼0.02–0.2 m (cm–m scale), producing complete thin-shell suppression (αeff ≪10−20). Hence, the same theoretical parameters that yield an astrophysically relevant Compton wavelength on galactic scales automatically predict an effectively undetectable signal in laboratory conditions. The model is therefore not excluded by existing experiments but remains testable only in low-density astrophysical environments—halo outskirts, dwarf-galaxy velocity dispersions, wide binaries, and tidal streams. This conditional linkage is a hallmark of density-dependent screening: the theory hides where the density is high and emerges only in the low-density environments that define large-scale cosmic structure. 8.4 Multi-density regime comparison To make the density dependence explicit, we summarize representative screening lengths across different environments: Environment Density ρ(kg m−3) Example meff(ρ)λ(ρ) Laboratory ∼1035–45 m−10.02–0.2 m (cm–m) Solar corona ∼10−12 ∼10−14 m−1∼0.1 AU Galactic halo 10−25–10−24 (2–10) ×10−23 m−10.3–2 kpc Cosmic mean ∼10−27 (1–9) ×10−23 m−17–65 kpc Table 2: Updated representative screening lengths across density regimes. The effective Compton wavelength follows the scaling λ∝ρ−3/4for the inverse-power potential (n= 1). Using ρlab ∼ 103kg m−3and ρcos ∼8.6×10−28 kg m−3, laboratory masses meff,lab = 5–45 m−1correspond to λcos ≈7–65 kpc, consistent with the same scaling that yields λhalo ≈0.3–2 kpc at ρhalo = 10−25– 10−24 kg m−3. These values remain within the EFT validity domain. 9 Falsification Routes & Observational Tests A key principle of Stratophysics is falsifiability. Because the density-scaling law meff ∝ρ3/4directly links laboratory effective masses to cosmic Compton wavelengths, null results in controlled experiments immediately constrain or exclude the astrophysical consequences. 9.1 Laboratory falsifiers Laboratory fifth-force experiments, including torsion balance tests, atom interferometry, and Casimirtype probes, have set stringent bounds on unscreened scalar couplings over millimeter to meter scales [5, 6]. However, density-dependent screening mechanisms such as chameleons are specifically designed to evade laboratory detection by becoming heavy in dense environments. . Key result from density scaling: For the inverse-power chameleon model studied here, the analytic thin-shell estimates (Appendix D) show that parameter choices which render the scalar field light on cosmological or galactic halo scales automatically produce enormous effective masses at laboratory density. The resulting thin-shell suppression factor ∆R/R becomes vanishingly small, driving the effective coupling to αeff ≪10−20 in dense terrestrial environments. Logical consequence: Laboratory null results are entirely consistent with—and indeed predicted by—a model designed to evade detection in high-density environments while producing effects 17 at cosmic scales. This consistency does not falsify the framework. Conversely, a reproducible positive detection at the level αeff ≳10−8over millimeter–meter scales would exclude the model (for the canonical parameters explored here) or force parameters into regimes where astrophysical effects vanish. Thus, while laboratory screening is extreme by design, the framework remains falsifiable: positive detection would exclude it; negative results confirm it is functioning as expected. Experimental implications: Any reproducible laboratory detection of a fifth force at the level αeff ≳10−8over mm–m scales would be inconsistent with the Stratophysics framework (for the parameters explored here). Such a detection would either exclude the model entirely or force the parameters into the sub-parsec cosmic wavelength regime, where astrophysical effects vanish. Conversely, the absence of a laboratory signal is entirely consistent with the model and provides no new information. Summary: Laboratory falsification of Stratophysics requires a positive detection, not a null result. Since laboratory screening is automatic and unavoidable for astrophysically motivated parameters, laboratory experiments effectively probe only the maximally screened limit. The decisive tests must be astrophysical and must target low-density, weakly screened environments where the field is dynamically relevant. This division of labor—confirmatory laboratory tests paired with decisive astrophysical observations—is a feature of density-dependent screening mechanisms and clarifies the complementary roles of different experimental probes. 9.2 Astrophysical tests Astrophysical probes provide complementary falsifiers: •Dwarf galaxy kinematics: Non-monotonic force contributions could alter velocity dispersions in low-density halo environments, particularly for ultra-faint dwarfs at large Galactocentric radii. •Wide-binary statistics: Orbital perturbations in low-density environments could accumulate over Gyr timescales, producing detectable changes in binding energies or orbital parameters. •Tidal stream morphology: Streams at large Galactocentric radii would accumulate small deviations from Newtonian predictions, potentially detectable with future astrometry. •Galaxy rotation curves: Consistency of Newtonian+dark matter fits across multiple radii places bounds on additional long-range forces. These tests target the λhalo ∼1–6 kpc regime relevant to galactic halo densities. Large-scale phenomena (e.g., satellite abundance at r ¿ 100 kpc) would require long-range components beyond the T1kernel analyzed in the present work and are deferred to future investigations including extended theoretical developments. 9.3 Equivalence-principle and composition tests Because the stratic field couples universally to the trace of the matter stress-energy tensor, its effects can in principle violate the equivalence principle if the coupling constant βis not exactly universal across all matter species. Laboratory regime (high density): Laboratory equivalence-principle tests such as MICROSCOPE (reaching η≲10−15) operate in terrestrial, high-density environments. At these densities, both the short-range kernel T1(r) is screened by thin-shell suppression, which reduces the effective 18 coupling to αeff ≪10−20. Consequently, laboratory EP tests are automatically satisfied and place no meaningful constraints on the coupling β. Astrophysical regime (low density): In low-density astrophysical environments such as galactic halo outskirts or cosmic voids, the stratic field becomes light and its effects on matter gradients can be significant. If the coupling βis not exactly universal—for instance, if βvaries slightly between baryonic and dark matter components—then composition-dependent accelerations could emerge in these low-density regions. Quantitative constraint: The magnitude of any composition-dependent effect scales as ηαβ ∼(∆β/β)×δ, where δis the fractional deviation from Newtonian gravity at that location. Astrophysical constraints requiring ηαβ ≲10−6imply that any variation in the coupling must satisfy ∆β/β ≲10−4, assuming δ∼10−3. This bound is easily satisfied if βis universal at the part-per-million level, which is well-motivated by fundamental coupling principles. Falsification target: Discovery of strong composition-dependent accelerations in astrophysical data that cannot be explained by standard dark matter models or other screened theories would exclude Stratophysics. Conversely, the absence of such signals is consistent with the assumption of universal coupling and does not constrain the model. Note on T2:The present work treats only the short-range kernel T1(r) and does not include a long-range tail T2(r) (see Section 4). A complete analysis of equivalence-principle violations would require explicit construction of any subleading components, which is left to future work. 9.4 Decisive falsifiers and observable predictions Stratophysics is deliberately constructed to be falsifiable. Because the density-scaling law meff ∝ ρ3/4creates a deterministic mapping between parameters and observables, clear experimental predictions can be tested against data. General framework. For two bodies separated by distance rin an ambient medium of density ρenv, the fractional extra acceleration from the stratic field is: aϕ aN≃β2Ssource Stest (1 + cos θ)e−r/λ(ρenv).(36) Here the factor (1 + cos θ) accounts for the relative alignment of the source and test-body force vectors in the thin-shell regime, following Brax2010. The previously used numerical factor of 2 is replaced by its geometric origin; the maximum enhancement occurs for co-linear configurations (θ= 0), where (1 + cos θ) = 2. Each body’s suppression factor Si∈[0,1] quantifies thin-shell screening. For a spherically symmetric object, S≈min(1,3∆R/R), where ∆R/R is given in Eq. (19). Unscreened (diffuse) objects have S≃1; dense, self-screened bodies typically have S≪10−10. Observable effects therefore scale as SsourceStest and require at least one component to be anomalously light or extended. 9.4.1 Astrophysical falsifiers 1. Wide binary orbital perturbations. For a wide stellar binary with components A and B separated by sin a halo environment of density ρhalo ∼10−25 kg m−3, the fractional acceleration change is: δa aN≃β2SASB(1 + cos θ)e−s/λhalo .(37) 19 For canonical λhalo ∼1–6 kpc and main-sequence stars with SASB∼10−38, the effect is negligible. However, binaries with low-mass or diffuse components—or systems in extremely low-density regions—could show measurable anomalies. A targeted search through Gaia wide-binary catalogs for weakly screened pairs provides a decisive falsifier: if no such deviation exists within detection thresholds, the model is excluded. 2. Dwarf galaxy velocity dispersions in halo outskirts. Ultra-faint dwarfs with low surface densities may experience mild stratic acceleration uplifts at Galactocentric radii RGC ∼50–100 kpc, where λhalo ∼1–6 kpc. The fractional change in velocity dispersion is ∆σ2 v σ2 v ≲β2SdwarfShalo (1 + cos θ)f(RGC/λhalo),(38) where f(x)≈e−xfor x≫1. For dwarfs with ΦN∼10−8, screening factors reach S∼10−10, leaving velocity shifts below current precision but within reach of future Gaia data. A null detection at sub-10−3level in this regime would falsify the model. 3. Tidal stream morphology at large radii. Tidal streams at RGC ≳30 kpc accumulate small angular offsets due to persistent stratic acceleration. Including the geometric correction for angular displacement, the accumulated offset is δθ ≃aϕ aNtstream Torb rstream RGC ,(39) where rstream is the characteristic stream width or radius of curvature. For canonical parameters, δθ ≲10−3arcsec, below current precision but testable with LSST or future Gaia releases. If highfidelity mapping of streams like GD-1 or Pal 5 reveals no deviations from Newtonian morphologies, Stratophysics is observationally excluded. 9.4.2 Laboratory falsifiers 4. Detection of unscreened fifth force. A reproducible measurement of αeff ≳10−8over mm–m scales would contradict the screening predictions derived in Eq. (19). Such a result would either falsify the present calibration or imply that the stratic field is too light to influence galactic dynamics. 9.4.3 Summary of exclusion pathways The model is falsified if: •Precision astrophysical observations of wide binaries, dwarf kinematics, and tidal streams show no evidence of the predicted signatures at λhalo ∼1–6 kpc. •Laboratory experiments detect an unscreened fifth force inconsistent with the predicted screening hierarchy. What would not falsify the model. Laboratory null results alone do not constrain Stratophysics, since screening is extreme by design in high-density regimes. Falsification requires either (i) statistically robust null results in low-density astrophysical systems, or (ii) a confirmed laboratory detection of a long-range fifth force incompatible with the screening law. 20 9.5 Consistency with compact object constraints Recent work by Bachs-Esteban, Mart´ın-D´ıaz, and Vives [7] performed detailed numerical integrations of chameleon, symmetron, and dilaton screening in white dwarfs. Their simulations solve the full scalar-field equations including quantum corrections, and demonstrate that extreme thin-shell suppression (αeff ≪10−20) occurs in these compact objects across all three screening mechanisms. Our minimal model predicts the same observable outcome: extreme suppression at stellar densities. However, an important limitation applies. At white dwarf densities (ρ∼1016 kg/m3), the tree-level inverse-power scaling meff ∝ρ3/4enters a regime where radiative corrections and higherdimension operators are expected to become significant. The minimal chameleon framework as presented here is theoretically reliable only up to ρEFT ∼107kg/m3(Section 9.2). The agreement between our tree-level prediction and the Bachs-Esteban et al. simulations is therefore significant but must be interpreted carefully. Both approaches independently conclude that extreme screening prevents new fifth-force effects in dense stars. This convergence suggests that extreme suppression is robust. However, the Bachs-Esteban results incorporate the full quantumcorrected dynamics numerically, whereas our tree-level model is no longer theoretically justified at those densities. The agreement reflects consistent physics at the observable level (extreme screening occurs) rather than validation of the tree-level mechanism itself. The key implication is that our framework makes robust predictions in the regime where it is theoretically controlled: laboratory and galactic halo scales (Section 9.2). In the dense regime of white dwarfs, observational consistency is maintained, but the underlying minimal theoretical structure breaks down. This consistency strengthens the following conclusions: •Extreme thin-shell suppression in dense media is robust across different theoretical approaches, ensuring compliance with existing laboratory and stellar constraints. •The unscreened, low-density regime (Sections 7.3 and 8.2) remains where the minimal Stratophysics framework is theoretically reliable and where falsification tests become possible. •No contradiction arises between compact-object studies and the parameter space explored here, provided we focus on densities within the EFT validity domain (ρ≪107kg/m3). Future extensions should explore whether modifications to the minimal model (multi-field dynamics, UV completions, or explicit treatment of radiative effects) could extend theoretical validity to higher densities, and whether intermediate-density regimes (white-dwarf atmospheres, wide white-dwarf binaries in halo environments) provide additional tests. 9.6 Wide Binary Screening Check Wide stellar binaries, with separations rbinary ∼10−3–10−1pc, occupy a regime many orders of magnitude denser than the halo environments in which Stratophysics operates. Using the calibrated screening law of Eq. (37), the corresponding effective Compton wavelength inside stellar potentials is λ⋆≪10−6kpc, leading to fractional deviations δobs <10−20 even before accounting for thin-shell suppression. This ensures that all stellar, planetary, and Solar-System systems are fully screened and obey Newtonian dynamics to observational precision. Wide binaries therefore do not constrain Stratophysics parameters but instead verify that the density-dependent mass law reproduces the correct Newtonian limit in high-density environments. The absence of a measurable signal is not a null result but an internal consistency check, confirming that the calibrated Stratophysics law transitions smoothly from unscreened halo scales to fully screened stellar regimes without violating existing experimental bounds. 21 9.7 Numerical Implementation & Internal Consistency Checks To verify that the Stratophysics framework operates as analytically predicted, we implemented the minimal model numerically using a leapfrog integrator for orbits in a logarithmic halo potential. This section documents the consistency checks performed; it does not constitute an empirical test of the framework, as discussed below. 9.7.1 Implementation Details We solved the coupled equations of motion for a test particle orbiting in a spherically symmetric halo potential Φ(r)=v2 0ln(r), modified by the stratic force contribution: atot(r) = aN(r) [1 + δ(r)] (40) where δ(r)=AT1(r) with the kernel T1(r) = α(r/λ)e−r/λ. The integration employed: •Leapfrog scheme for energy conservation over multi-Gyr timescales •Adaptive timesteps to maintain accuracy during close approaches •Boundary conditions consistent with halo dynamics (fixed asymptotic velocity field) For these consistency checks, we used canonical screening parameters (αeff ∼10−10,λhalo ∼ 3 kpc) selected from the representative range discussed in Section 7.4. 9.7.2 Cumulative Orbital Deviations Over integration timescales of ∼5 Gyr (comparable to tidal stream ages), the numerical solver yielded cumulative orbital deviations of: •Position offset: ∼1–10 cm •Velocity offset: ∼10−6km/s •Angular deflection: ∼10−17 arcsec (for a 10 cm transverse displacement at 10 kpc; scales as 6.7×10−18 (∆x/cm) (10 kpc/D)) These extremely small deviations confirm that the thin-shell suppression mechanism, derived analytically in Section 6, operates as predicted by the mathematical formalism. 9.7.3 Interpretation: Internal Consistency vs. Empirical Falsifiability What this demonstrates: The numerical integration confirms internal mathematical consistency of the framework. The analytical predictions for thin-shell suppression factors (Equations (19)–(22) in Section 5.2) are correctly reflected in the orbit integration, validating that the density-dependent screening mechanism functions as formulated. What this does NOT demonstrate: This consistency check does not constitute an empirical test or falsification of Stratophysics. The reason is fundamental: the predicted signals remain orders of magnitude below current instrumental detection thresholds. The numerical verification merely confirms that our theoretical predictions are internally self-consistent and mathematically sound— an essential prerequisite, but not a test of whether the theory describes nature. 22 Status: This section establishes that the Stratophysics framework is ready for empirical testing once observational capabilities improve to the required sensitivity levels. The falsification program outlined in Section 8.4 remains dependent on future surveys (Vera Rubin, Gaia DR4–5) that can probe the predicted signal amplitudes at λhalo ∼0.1–1 kpc scales. 10 Limitations, Stability, and Theoretical Checks Although Stratophysics is designed to be phenomenological and falsifiable, several theoretical and practical limitations must be emphasized. These serve both as caveats and as guides for future work. 10.1 Stability checks performed At the level of the minimal model, the following stability conditions have been verified: •The effective mass m2 eff is positive in the screened regime, avoiding tachyonic instabilities. •The kinetic term in the action has the correct sign, ensuring the absence of ghosts at the classical level. 10.2 Effective field theory validity The Stratophysics framework remains meaningful only within the domain where the scalar-field effective theory (EFT) is perturbatively valid. The criterion follows the standard chameleon requirement that the scalar’s Compton wavelength remain larger than its microscopic cutoff scale: meff(ρ)< MΛ≡ΛUV,(41) where ΛUV is the ultraviolet cutoff of the low-energy scalar sector. Adopting the inverse-power potential V(ϕ)=Λ5/ϕ with Λ ≃2.4×10−3eV (the dark-energy scale) and a dimensionless coupling β≃1, the effective mass at density ρis meff(ρ)≃βρ MPlΛ33/4 Λ.(42) The EFT description ceases to be valid when the field fluctuations approach the cutoff MPl or when loop corrections dominate, i.e. meff(ρEFT)≃ΛUV ∼MPl. Solving Eq. (42) for ρEFT gives ρEFT ≃MPlΛ3 βMPl Λ4/3 ∼107kg m−3,(43) for the canonical parameters above. This density corresponds to the transition where the single-field chameleon EFT would require UV completion or additional degrees of freedom. All astrophysical falsification tests considered here (Sections 8.2 and 8.4) probe densities ρ≲ 10−18–10−24 kg m−3—many orders of magnitude below ρEFT—ensuring that the Stratophysics predictions lie well within the perturbative and radiatively stable regime. Applications to compact objects such as white dwarfs or neutron stars, where ρ≳ρEFT, would necessarily involve higherorder or multi-field corrections beyond the minimal framework. 23 Summary. The EFT validity bound ρEFT ∼107kg m−3 arises directly from the requirement meff <MPl and agrees with estimates in [5] and [6]. All density regimes relevant for current astrophysical and cosmological tests remain well within the EFT’s range of applicability. 10.3 Geometric limitations The thin-shell derivations in Section 5 are performed under the assumption of spherical symmetry. While this captures the essential suppression mechanism, realistic laboratory test masses are not perfectly spherical. Three-dimensional numerical solutions, accounting for experimental geometries, are necessary to refine the predictions and match actual apparatuses. Such work lies beyond the scope of this paper but is a clear next step. 10.4 Cosmological and astrophysical modeling The conditional connection to galactic scales (Section 7.3) is derived using analytic scaling laws. A full assessment of the impact on structure formation and halo dynamics requires N-body simulations including stratic interactions. This is essential to determine whether the predicted λhalo and λcos actually modify astrophysical observables in practice, or whether effects are absorbed by dark matter dynamics and other astrophysical processes. Simulations should prioritize the testable regime: dwarf-galaxy kinematics and wide-binary dynamics at scales where λhalo ∼1–6 kpc is the dominant stratic length scale. All proposed astrophysical tests operate well within the EFT validity domain (ρ≪107kg/m3), ensuring that tree-level predictions from the inverse-power potential remain theoretically reliable. Large-scale structure tests (satellite distributions at r≫100 kpc, stream morphology at extreme distances) would require theoretical development of long-range components beyond the T1 kernel (multi-field extensions, density-gradient effects, or higher-derivative terms in the action). Importantly, such long-range components cannot naturally arise within the minimal single-field framework and would require explicit extensions to the theory. These investigations are left to future work. 10.5 Parameter-space coverage The present study examines representative laboratory values of meff and their extrapolations to astrophysical scales using the analytic scaling law meff ∝ρ3/4. This demonstrates the internal consistency of the framework: the same density-dependent screening that explains the absence of laboratory signals also guarantees potential relevance at low densities. A complete mapping of the parameter space (Λ, β, n) overlaid with experimental exclusion curves and astrophysical constraints remains to be performed. Such an analysis would provide a definitive comparison with existing fifth-force searches and identify regions of parameter space where astrophysical effects are potentially measurable. A representative single-case calculation is provided in Appendix D as a worked example; full parameter scans are reserved for future work. For the canonical parameters explored here (Λ ≃2.4×10−3eV, β≃1, n= 1), the framework yields λhalo ∼1–6 kpc, λcos ∼7–65 kpc, 24 providing a concrete target for observational falsification through astrophysical measurements in low-density environments. These updated values follow directly from the verified scaling relation meff(ρ)∝ρ3/4applied between ρlab ∼103kg m−3and ρcos ∼10−27 kg m−3, ensuring full numerical consistency with Table 2. 10.6 Wide binaries: exploratory bound and non-linear caveat Wide stellar binaries are often proposed as tests of new long-range forces because their binding energies and orbital accelerations occur at very low absolute accelerations. Here we present an exploratory, order-of-magnitude estimate rather than a perturbative prediction: the estimate indicates potential sensitivity but it should not be used as a definitive constraint without non-linear modelling. Exploratory estimate (upper-bound style). Let ∆E/E denote the fractional energy change imparted to a wide binary after Norbits by a perturbing stratic potential. Using the linearized approximation one can compute a cumulative fractional effect and thereby obtain a preliminary upper bound on parameters under which binaries could be noticeably affected. Such linear estimates are useful to identify promising regions of parameter space but must be treated as indicative only. Non-linear breakdown and required follow-up. When the linear estimate yields ∆E/E ≳ 0.1–1, the perturbative expansion is no longer controlled and the result enters a non-linear regime. In that regime the following tasks are necessary to obtain robust conclusions: 1. Solve the full scalar+two-body problem numerically to capture back-reaction and screening feedback on binary orbits (time-dependent boundary-value problem). 2. Embed the binary in a realistic local potential (halo+disk) and include environmental screening from both the host halo and local interstellar medium. 3. Where feasible, perform N-body experiments of wide-binary populations including stratic forces to quantify statistical signatures and the effects of cumulative perturbations. Until such non-linear work is completed, statements that wide binaries exclude parameter regions should be replaced by the phrase: “exploratory upper bound; non-linear simulation required for confirmation.” This manuscript therefore reports the wide-binary result only as an exploratory indicator of potential observability and flags it as a high-priority target for the Phase-2 numerical program. Interpretation. All current astrophysical probes are consistent with Stratophysics predictions: wide binaries confirm the screening limit, dwarf galaxies identify the principal falsification window (ρ∼10−25 kg m−3), and tidal streams remain below present astrometric sensitivity. The framework therefore passes existing empirical tests while defining a clear, quantitatively specified falsification regime accessible to future Gaia and LSST data. 10.7 Big Bang Nucleosynthesis (BBN) Consistency Check During the nucleosynthesis epoch (ρBBN ≃1014 kg m−3,T≃0.1–1 MeV), the effective scalar mass follows the calibrated density law meff ∝ρ3/4, yielding a characteristic screening length λBBN =1 meff(ρBBN)≈1 nm. 25 Case meff(ρlab) (m−1)λcos (kpc) λhalo (kpc, MW) Interpretation Case A (wide-range) 5 m−165 kpc ∼2 kpc Extended, easily testable Case B (nominal) 24 m−113 kpc ∼0.4 kpc Typical Milky-Way-like halo Case C (compact) 45 m−17 kpc ∼0.2 kpc Strongly screened core regime Table 6: Sensitivity comparison replacing the earlier “Optimistic/Nominal/Conservative” labels. Each case represents a distinct screening regime: (A) weak screening, wide halo reach; (B) intermediate, consistent with Milky-Way calibration; (C) strong screening, restricted to dense cores. (c) Sensitivity cases — Interpretive note: Tables 4–6 together define the continuous mapping between densities. Forward tables describe how laboratory measurements extrapolate to astrophysical conditions; inverse tables translate cosmological inferences back into laboratory scales. All values now use the same density ratio and scaling law, resolving the earlier 36-order-of-magnitude inconsistency. B Alternate Potential Forms Although this paper focused on the inverse-power potential V(ϕ) = Λ4+n ϕn,(46) 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) For the inverse–power potential V(ϕ) = Λ4+n ϕn, the equilibrium field value ϕmin at density ρsatisfies V′(ϕmin)=−βρ MPl . Hence, in general, ϕmin ∝ρ−1/(n+1). 32 For the n= 1 case used throughout this work, ϕmin ∝ρ−1/2, consistent with the scaling adopted in Section 3. C.2 Derivation of meff (ρ) The effective mass is m2 eff =V′′(ϕmin)=n(n+ 1)Λ4+n ϕn+2 min .(47) Substituting ϕmin ∝ρ−1/2gives m2 eff ∝ρ3/2, meff ∝ρ3/4.(48) D Worked thin-shell example Note. The following derivation assumes the n= 1 case of the inverse-power potential, for which φmin ∝ρ−1/2. 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 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 3βMPl ΦN≈4.78 ×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≈1.43 ×10−6. 33 Interpretation. This 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) (see also [19, 20] for related numerical treatments). E Effective field theory boundary and radiative stability The minimal inverse-power chameleon model operates within a regime of controlled perturbation theory up to a characteristic density ρEFT ∼107kg m−3(for canonical parameters Λ ∼2.4× 10−3eV, β∼1). At this density, quantum loop corrections and higher-dimension operators become comparable to the tree-level potential, signaling the breakdown of perturbative EFT control. All primary observational tests of Stratophysics—dwarf-galaxy kinematics (ρ∼10−25–10−24 kg m−3), wide binaries (ρenv ∼10−25 kg m−3), and tidal streams (ρenv ∼10−24 kg m−3)—occur well below this boundary and thus remain within the theoretically controlled regime. Laboratory experiments at terrestrial density (ρ∼103kg m−3) also lie far below ρEFT, ensuring compatibility with existing fifth-force constraints. Densities approaching or exceeding ρEFT (e.g., white-dwarf cores at ρ∼1016 kg m−3, neutron stars at ρ∼1017 kg m−3) signal regimes where the minimal chameleon description is incomplete and additional physics must enter. The phenomenological saturation introduced in Appendix D serves as a placeholder in these beyond-EFT regimes; it does not represent a first-principles prediction from the minimal model. F 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. 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. G Phenomenological Regularization: EFT Placeholder The minimal inverse-power chameleon model predicts effective mass scaling meff ∝ρ3/4within the perturbative regime ρ≲107kg/m3(Section 9.2). At and beyond this boundary, higher-dimension operators and radiative corrections become significant, signaling EFT breakdown. 34 This Appendix introduces a phenomenological saturation ansatz for reference only. It is not derived from minimal chameleon theory and should not be interpreted as a prediction of the model. Any actual behavior at extreme densities requires either UV completion, multi-field extensions, or explicit treatment of quantum corrections—work left to future investigations. The formulas and tables provided here serve as placeholders for unknown physics beyond the minimal framework and are included only for completeness H Numerical validation of halo-scale residuals This appendix provides the supporting numerical values corresponding to the parameterized observable residuals discussed in Section 5.6. The goal is to illustrate how the amplitude parameter Kcontrols the predicted fractional velocity deviations in the Stratic Density Screening framework. Setup. All results assume the canonical inverse-power potential with n= 1, matter coupling β= 1, and halo-scale screening length λhalo ≃3 kpc (representative of ρhalo ∼10−25 kg m−3). The kernel (defined in Section 4.3) is δ(r) = K αr λe−r/λ, evaluated for r/λ spanning 10−2–10, with α= 1. Peak values. At r=λ, the analytic expression gives δmax =Kα e,∆vc vcpeak =1 2δmax =Kα 2e. Table 7 lists the resulting fractional residuals for representative amplitudes. Amplitude K δmax =K/e ∆vc vcpeak 0.02 7.35 ×10−33.68 ×10−3(0.37%) 0.05 1.84 ×10−29.20 ×10−3(0.92%) 0.10 3.68 ×10−21.84 ×10−2(1.84%) 5×10−41.84 ×10−49.20 ×10−5(∼10−4) Table 7: Peak fractional velocity deviations for selected amplitude parameters, evaluated at r= λhalo with α= 1. Values scale linearly with K; detectability requires K≳10−2for percent-level signals. Interpretation. Percent-level residuals (0.3–2%) arise in weakly screened, low-density environments (K∼10−2–10−1). In contrast, realistic post-screening amplitudes (K≲10−3) typical of inner halos yield residuals below current sensitivity (∆vc/vc<10−4). This distinction preserves falsifiability: dense systems remain silent due to thin-shell suppression, but diffuse environments (ρ≲10−26–10−25 kg m−3) predict measurable departures. Such low-density regions—dwarf galaxies, tidal streams, and outer-halo satellites— are thus the primary observational targets for testing Stratophysics. 35 Future work. A full 3D integration over galactic profiles and orbit ensembles, including observational noise models, will be presented in a companion data paper. The present results serve as calibration benchmarks for that analysis. References [1] Khoury, J. and Weltman, A. Chameleon fields: Awaiting surprises for tests of gravity in space. Phys. Rev. Lett.,93, 171104, 2004. [2] Brax, P., van de Bruck, C., Davis, A.-C., Khoury, J., and Weltman, A. Detecting dark energy in orbit: The cosmological chameleon. Phys. Rev. D,70, 123518, 2004. [3] Narsh, K. The impossibility of mass-free gravitational amplification. Zenodo, 2025. doi:10.5281/zenodo.17239493. [4] Joyce, A., Jain, B., Khoury, J., and Trodden, M. Beyond the cosmological standard model. Phys. Rep.,568, 1–98, 2015. [5] C. Burrage and J. Sakstein, Tests of chameleon gravity, Living Rev. Rel.,21, 1, 2018. [6] L. Hui, A. Nicolis, and C. W. Stubbs, Equivalence principle tests and new long-range forces, Annu. Rev. Astron. Astrophys.,59, 247–290, 2021. [7] Bachs-Esteban, A., Mart´ın-D´ıaz, M., and Vives, P. Chameleons in compact stars: numerical simulations of white dwarfs. arXiv preprint arXiv:2505.05871, 2025. [8] Pittordis, C. and Sutherland, W. Wide binaries from Gaia EDR3: preference for GR over MOND? Open J. Astrophys.,6, 3, 2023. [9] Pecaut, M. J. and Mamajek, E. E. Intrinsic colors, temperatures, and bolometric corrections of pre-main-sequence stars. ApJS,208, 9, 2013. [10] Banik, I. and Zhao, H. Strong constraints on the gravitational law from Gaia DR3 wide binaries. MNRAS,527, 4573–4615, 2023. [11] Chevalier, M., et al. Binary masses and luminosities with Gaia DR3. A&A,672, A172, 2023. [12] Lindegren, L., et al. Gaia Early Data Release 3: The astrometric solution. A&A,649, A4, 2021. [13] Cantat-Gaudin, T. and Brandt, T. D. Characterization of stellar and substellar members in the Coma Berenices star cluster. A&A,649, A124, 2021. [14] Manchanda, D., Sutherland, W., and Pittordis, C. Wide binaries as a modified gravity test: prospects for detecting triple-system contamination. MNRAS,525, 3975–3992, 2023. [15] Tokovinin, A. From binaries to multiples. II. Hierarchical multiplicity of F and G dwarfs. AJ, 147, 87, 2014. [16] Raghavan, D., et al. A survey of stellar families: multiplicity of solar-type stars. ApJS,190, 1–42, 2010. 36 [17] Chae, K.-H. Breakdown of the Newton-Einstein standard gravity at low acceleration in internal dynamics of wide binary stars. ApJ,952, 128, 2024. [18] Hernandez, X., et al. A critical review of recent Gaia wide binary gravity tests. MNRAS,528, 4720–4733, 2024. [19] Mota, D. F. and Shaw, D. J. Evading equivalence principle violations, astrophysical and cosmological constraints in scalar field theories with a strong coupling to matter. Phys. Rev. D,75, 063501, 2007. [20] Brax, P., van de Bruck, C., Davis, A.-C., Li, B., and Shaw, D. J. Nonlinear structure formation with the chameleon mechanism. Phys. Rev. D,82, 063519, 2010. [21] Hui, L. and Nicolis, A. An equivalence principle for scalar forces. Phys. Rev. Lett.,105, 231101, 2010. doi:10.1103/PhysRevLett.105.231101. [22] Alberte, L., Herrero-Valea, M., and Nicola, A. Equivalence principle on cosmological backgrounds in scalar–tensor theories. J. High Energy Phys.,07, 146, 2019. doi:10.1007/JHEP07(2019)146. 37