One Trade-Off, Not Two: Self-Shielding and Mutual Shadowing for Realistically Imperfect, Extended Bodies

July 2026

Authors

Matthew Foutch, with Claude (Anthropic) and Grok (xAI) as AI collaborators

Abstract

Paper 15 established an exact result for an idealized case — a point test particle outside a single, perfectly opaque sphere — and named two open problems as the actual crux of whether push-gravity shadowing can work at all: self-shielding (does a realistically imperfect, finite-opacity absorber preserve exact mass-proportionality?) and mutual shadowing between two extended bodies, neither attempted there. This paper attempts both together, and finds a precise, quantified answer that is narrower than first hoped for. Replacing Paper 15’s perfectly-opaque sphere with real Beer-Lambert attenuation, the force law reduces, in the optically-thin single-scattering limit, to an exact closed form: $F_{thin}(d)=\kappa M/d^2$ for a point receiver outside one extended, finite-opacity sphere — exact mass-proportionality, no residual dependence on the body’s size or density individually — verified analytically and numerically. Extending to two extended, spherically symmetric bodies in the same limit, the mutual force reduces via the ordinary Newtonian shell theorem to $F\propto\kappa_A\kappa_B M_AM_B/d^2$ at any non-overlapping separation, confirmed independently by Monte Carlo volume integration. This is a real result: self-shielding is not merely reduced but exactly absent, at every order needed for observed mass-proportionality, in this regime. It is not, however, a resolution of the historical Le Sage objection. The optically-thin regime that eliminates self-shielding is precisely the regime Maxwell (1875) and Poincaré (1908) already identified as the one that maximizes the drag-and-heating problem Paper 14 found unresolved — the same physical trade-off, not two independent problems. This paper’s contribution is a precise, first-order ($\mathcal{O}(\tau_A\tau_B)$) quantification of one side of a century-old, already-settled trade-off, not new grounds to reopen it.

Keywords: push gravity, shadowing mechanism, self-shielding, Le Sage gravity, mutual shadowing, Beer-Lambert attenuation, Shell Theorem, Maxwell, Poincaré

Introduction

Paper 15 showed the geometric core of push-gravity shadowing is exact, not approximate, for a point particle outside a single perfectly opaque sphere — and was explicit that this idealization leaves the actual historically-fatal question untouched: real bodies are not perfect, mathematical blockers. If a body is opaque enough to cast a shadow at all, does the shadow it casts stay exactly proportional to its mass once its own finite absorption is taken into account? This is Poincaré and Maxwell’s specific, century-old objection to Le Sage’s 1748 mechanism [1] — not a detail alongside a secured foundation, but the reason the mechanism has never been rescued. Paper 15 also left mutual shadowing between two extended bodies as a defined but unattempted calculation. This paper attempts both.

Theory Description

Paper 15’s mechanism assumed each blocking body is perfectly opaque: every ray within the geometric shadow cone is fully stopped. Real matter isn’t like that. A real absorber transmits flux according to Beer-Lambert attenuation, $T=e^{-\kappa\rho l}$, where $\kappa$ is a material-specific mass-attenuation coefficient, $\rho$ is local density, and $l$ is the path length through the body along a given ray. This paper replaces the perfectly-opaque assumption with this real attenuation law and asks what survives.

Mathematical Formalism

Single extended, finite-opacity sphere, point receiver. For a ray from an external point at distance $d$ from the sphere’s center, at angle $\theta$ from the point-to-center axis, the impact parameter relative to the center is exactly $b=d\sin\theta$ (standard tangent-line geometry, exact for any $\theta$, confirmed independently). The chord length through a sphere of radius $R$ at that impact parameter is $l(\theta)=2\sqrt{R^2-d^2\sin^2\theta}$ for $\theta<\alpha=\arcsin(R/d)$, and zero (ray misses the sphere) otherwise. Generalizing Paper 15’s cosine-weighted force integral:

$$F(d) \propto 2\pi\int_0^{\alpha}\left[1-e^{-\kappa\rho\,l(\theta)}\right]\cos\theta\sin\theta\,d\theta$$

This recovers Paper 15’s exact $\pi(R/d)^2$ in the perfectly-opaque limit ($\kappa\rho R\to\infty$): confirmed numerically to 1 part in $10^{15}$ by $\kappa\rho=1000$.

The optically-thin limit. For $\kappa\rho R\ll1$ (realistic, non-saturating opacity), $1-e^{-x}\approx x$, and the integral solves in closed form via the substitution $u=d\sin\theta$:

$$F_{thin}(d) \propto \frac{4\pi\kappa\rho}{d^2}\int_0^R u\sqrt{R^2-u^2}\,du = \frac{4\pi\kappa\rho R^3}{3d^2} = \frac{\kappa M}{d^2}$$

exactly, with no separate dependence on $R$ or $\rho$ — only through mass $M=\rho V$. This is the same fact behind the general Cavalieri/Fubini identity that chord length integrated over a cross-section equals volume for any shape, not only spheres. Verified numerically: the ratio of the full (non-linearized) integral to this closed form converges to 1.000000 as $\kappa\rho\to0$ (checked from $\kappa\rho=10^{-2}$ to $10^{-6}$), and at fixed total mass $M$, varying $R$ from 0.5 to 3.0 (density adjusted to hold $M$ fixed) converges $F\cdot d^2/M$ to a single constant regardless of how mass is distributed between size and density.

Two extended bodies. In the same thin limit, body A’s deficit field at any external point is exactly $g(r)=\kappa_AM_A/r^2$ — a genuine $1/r^2$ field about a point, mathematically isomorphic to Newtonian gravity (the force on a point test mass from any mass distribution reduces to the same Newtonian integral form once the linear/single-scattering approximation is made). The net force on an extended, spherically symmetric body B sitting in this field therefore follows the ordinary Newtonian shell theorem: the volume average of a $1/r^2$ field sourced outside a sphere equals the field’s value at the sphere’s center, exactly, at any separation greater than the sum of the two radii — not only far-field. This gives:

$$F(d) \propto \kappa_A\kappa_BM_AM_B/d^2$$

Confirmed independently via Monte Carlo volume integration (2,000,000 sample points per configuration): the volume-averaged field over body B matched the point-at-center prediction to within 0.05% (consistent with Monte Carlo sampling noise) across $R_B/d$ from 0.05 to 0.89, and net force was confirmed independent of how B’s own mass is split between radius and density at fixed $M_B$.

Results

What is established, precisely: in the optically-thin, single-scattering regime — realistic bodies must be in this regime for gravity to stay exactly mass-proportional across the observed range of masses, since a non-thin (saturating) shadow would make force sub-linear in mass for large bodies — mass-proportionality and exact inverse-square both survive for two extended, finite-opacity, spherically symmetric bodies, at every non-overlapping separation. This is a genuine result: self-shielding, in this specific and physically-necessary regime, is not merely small — it is exactly absent at the order computed.

What this result is not: it is not a rescue of Le Sage-type gravity, and should not be read as one. The optically-thin regime this result depends on is not a free assumption; Maxwell (1875) [2] and Poincaré (1908) [3] already identified it as the only regime in which mass-proportionality survives, and already showed it is the same regime that requires an enormous ambient momentum flux to produce ordinary gravitational strength with such weak per-encounter absorption — which is exactly what produces the catastrophic heating problem Paper 14 found unresolved. Self-shielding and drag/heating are not two independent open problems in this mechanism; they are the same physical trade-off, viewed from opposite ends. Avoiding one by construction (going thin) is what maximizes the other.

Scope limits, stated plainly, not deferred silently:

1. This calculation is first order in both bodies’ optical depth, $\mathcal{O}(\tau_A\tau_B)$. Genuine mutual shadowing — depletion of flux already attenuated by one body before it reaches the other — is a higher-order effect not computed here.
2. The exact reduction to a point-mass-like force depends on spherical symmetry for the receiving body, the same restriction ordinary Newtonian gravity’s own shell theorem carries — not a unique weakness of this mechanism, but not a general-shape result either.
3. This mechanism additionally requires $\kappa$ to be a universal constant across all ordinary matter, regardless of composition — a stronger requirement than mass-proportionality alone, since a composition-dependent $\kappa$ would itself produce the composition-dependent free-fall equivalence-principle experiments already exclude to roughly 1 part in $10^{13}$. (Superseded 2026-07-31: MICROSCOPE (2022) gives $\sim10^{-15}$, 100x tighter — see revision note.)
4. No calculation here ties $\kappa$ to the measured value of $G$. Quantifying exactly how thin “thin enough” is for the Earth, a laboratory Cavendish mass, or Jupiter would require that normalization, which is blocked on the same first-principles gap Paper 14 already found in $\varepsilon_0$ — asserting a numeric bound here would mean fitting a number to an already-acknowledged reverse-engineered constant, not computing one. This is stated as an open gap, not filled with an invented figure.
5. This result does not touch, and does not claim to touch, the further objections already raised against Le Sage-type mechanisms independent of drag/heating: aberration and retarded-force effects (Laplace’s original objection — a finite corpuscle propagation speed implies an effective gravity speed far exceeding $c$ to match observed orbital stability), preferred-frame kinetic drag from any real relative motion through the flux, an Olbers-like cosmic energy-density problem (sourcing and replenishing the required flux at cosmological scale), the absence of a Lorentz-invariant completion, and the absence of any known particle candidate with the required universal $\kappa$ and flux magnitude.

Discussion

The honest reading of this result is closer to a well-defined negative result than progress toward a viable mechanism. Poincaré and Maxwell’s objection was, before this paper, argued qualitatively and via order-of-magnitude heating estimates. This paper’s contribution is a specific, closed-form, independently-verified force law confirming exactly where the boundary sits — the precise regime in which mass-proportionality and inverse-square both survive for realistically imperfect, extended bodies is exactly the regime already known to be fatal on other grounds. That is a real clarification, not a reopening.

It is worth being explicit about what changed between drafting and publishing this paper. The first draft of this work described self-shielding as “resolved.” A dedicated adversarial review — the same process that caught Paper 15’s geometry error — rejected that framing, not the mathematics: the calculations were confirmed correct on first pass, but presenting a 150-year-known trade-off as a new finding would not, in the reviewer’s words, survive contact with a historian or a well-read referee. The paper below reflects that correction.

Testable Predictions

1. The $\mathcal{O}(\tau^2)$ correction is a defined next calculation. The genuine mutual-shadowing term (depletion of already-attenuated flux, not computed here) is where any residual composition- or size-dependence would appear, and is directly bounded by existing equivalence-principle experiments (~1 part in $10^{13}$) (Superseded 2026-07-31: MICROSCOPE (2022) gives $\sim10^{-15}$, 100x tighter — see revision note.) the same way Paper 15 proposed for the single-body case.
2. A first-principles normalization tying $\kappa$ (and the required ambient flux) to $G$ remains the actual prerequisite for quantifying this paper’s regime numerically — without it, “thin enough” is a qualitative, not a checkable, claim. This is the same open normalization gap Paper 14 identified in $\varepsilon_0$, not a new one.

Conclusion

Self-shielding and mutual shadowing, attempted together for realistically imperfect extended bodies, resolve to a single clean result: mass-proportionality and exact inverse-square survive fully in the optically-thin, single-scattering limit — and that limit is not a free parameter choice but the same regime Maxwell and Poincaré already identified as fatal for independent, still-unaddressed reasons. Paper 15 asked whether shadowing preserves mass-proportionality for real, imperfect bodies. The answer is yes, precisely in the one regime where doing so costs the mechanism everything else.

References

1. Le Sage, G.-L. (1748). Essai de Chymie Méchanique.
2. Maxwell, J. C. (1875). “Attraction.” Encyclopædia Britannica, 9th ed. — corrected 2026-07-31: the Le Sage critique is in Maxwell’s article “Atom,” not “Attraction.” See revision note below.
3. Poincaré, H. (1908). Science and Method (English translation, 1914, Book II, Ch. 2); also the earlier Saint-Louis International Congress of Arts and Science lecture (1904). — corrected 2026-07-31: the chapter and lecture attributions could not be verified. See revision note below.
4. Edwards, M. R. (Ed.). (2002). Pushing Gravity: New Perspectives on Le Sage’s Theory.
5. Newton, I. (1687). Philosophiæ Naturalis Principia Mathematica, Book I, Proposition 71 (the Shell Theorem).
6. Touboul, P., et al. (2022). “MICROSCOPE Mission: Final Results of the Test of the Equivalence Principle,” Physical Review Letters 129, 121102. — added 2026-07-31.

Revision Note — 2026-07-31

Four corrections, found during the consolidation audit that produced Paper 17. The paper’s central mathematics was independently re-derived and confirmed correct (see item 5). Original text above is preserved unchanged.

1. The opaque-limit convergence figure is wrong by nine orders of magnitude. The Mathematical Formalism section states that the full attenuated integral “recovers Paper 15’s exact $\pi(R/d)^2$ in the perfectly-opaque limit: confirmed numerically to 1 part in $10^{15}$ by $\kappa\rho=1000$.” That figure is not correct. The convergence can be solved exactly rather than sampled. Substituting $w=\sqrt{R^2-u^2}$ (after $u=d\sin\theta$) reduces the integral to a closed form, giving a relative deviation from the opaque result of

$$\Delta(a,R) = \frac{2}{a^2R^2}\left[1-e^{-aR}(1+aR)\right], \qquad a=2\kappa\rho$$

which for $aR\gg1$ approaches $\Delta \to 1/[2(\kappa\rho R)^2]$ — a power law, not exponential convergence. At the stated $\kappa\rho R=1000$ the true deviation is $5.0\times10^{-7}$, not $10^{-15}$. Reaching $10^{-15}$ requires $\kappa\rho R\approx2.24\times10^{7}$. The qualitative claim — that the attenuated integral does recover Paper 15’s opaque result in the saturating limit — holds and is unaffected; only the stated precision was wrong. The likely origin of the error is a quadrature artifact: at $\kappa\rho R=1000$ the transition region at the shadow’s rim is narrower than $\sim10^{-7}$ in $u$, so any uniformly-spaced numerical grid coarser than that silently fails to resolve it and returns a spuriously small residual. (This same trap was hit and caught during the re-check that found this error — the first attempted verification used Simpson’s rule on a uniform grid and reproduced a misleading floor of $1.7\times10^{-6}$, which is why the integral was solved in closed form instead.)

2. The thin-limit convergence statement is imprecise. The same section states the ratio of the full integral to the closed form “converges to 1.000000 as $\kappa\rho\to0$ (checked from $\kappa\rho=10^{-2}$ to $10^{-6}$).” The convergence is real but linear in optical depth, with exact small-$aR$ behaviour

$$\text{ratio} = 1 - \tfrac{3}{4}\kappa\rho R + \mathcal{O}\left((\kappa\rho R)^2\right)$$

so at $\kappa\rho R=10^{-2}$ the ratio is $0.9925$, not $1.000000$. Only from about $\kappa\rho R\lesssim10^{-6}$ does it round to $1.000000$. The limit statement is correct; the parenthetical range is not.

3. Maxwell’s article is “Atom,” not “Attraction.” Reference 2 is corrected above. Maxwell’s Le Sage critique — including the passage stating that absorbed flux energy “would in a few seconds raise it, and in like manner the whole material universe, to a white heat” — appears in his article “Atom” (Encyclopædia Britannica, 9th ed., 1875), verified by direct quotation from the article text and independently corroborated.

4. The Poincaré citation’s specifics are unverified. Reference 3 asserts Science and Method, “Book II, Ch. 2,” plus a 1904 Saint-Louis lecture. The year (1908) and substance (a quantitative heating critique of Le Sage) are verified. The chapter localization and the 1904 lecture attribution were both pursued in primary text and could not be confirmed; they should be treated as unverified rather than relied upon. Stated here rather than quietly corrected, since the honest status is “unknown,” not “known to be X.”

6. Rendering fix (display only, no change of meaning). Both display equations in this paper’s Mathematical Formalism section were rendering with literal commas in place of LaTeX thin spaces, because CommonMark strips the backslash from spacing macros before KaTeX sees them. Corrected by escaping them. Same defect found and fixed in Papers 14 and 15 and the Reference Guide by site-wide sweep. Mathematics unaffected.

5. What was checked and found correct. The paper’s core mathematics was independently re-derived from scratch and confirmed. The impact-parameter relation $b=d\sin\theta$, the chord $l(\theta)=2\sqrt{R^2-d^2\sin^2\theta}$, and the substitution $u=d\sin\theta$ all check out, and the thin-limit integral does evaluate exactly to $\kappa M/d^2$ — with no small-angle approximation anywhere, since the substitution absorbs $\cos\theta$ exactly. The two-body result was reproduced by independent Monte Carlo across $R_B/d$ from 0.05 to 0.89. The result was additionally confirmed by a completely different derivation route in Paper 17: converting the angular integral to a volume integral via $ds\,d\Omega = dV/s^2$ shows the thin-limit shadow integral is formally identical to the Newtonian volume integral, so Newton’s Shell Theorem applies verbatim. Two independent derivations, same closed form. The equivalence-principle threshold cited in scope limit 3 and Testable Prediction 1 should be $\sim10^{-15}$ (MICROSCOPE, 2022, reference 6), not $\sim10^{-13}$ — a 100x tighter constraint than stated.

Update (2026-07-29): the two open items this paper left standing — whether the heating side of this trade-off can be made mathematically precise, and whether drag has any remaining fix — were both pursued directly. For a plain-language account of what was found (a real, proven heating calculation; a decisive contradiction when checked against actual published physics experiments; drag still unresolved), see “Where This Stands: A Plain-Language Progress Report (July 29, 2026)”.