The Geometric Core: A Consolidated Statement of the Shadowing Mechanism's Newtonian Reduction, With Its Exact Domain of Validity

July 2026

Authors

Matthew Foutch, with Claude (Anthropic) as AI collaborator

Abstract

Paper 15 established that the force-relevant geometric factor for a shadowing mechanism is exactly $\pi(R/d)^2$ outside an idealized opaque sphere, and Paper 16 established that mass-proportionality and exact inverse-square both survive for two realistically imperfect, extended, spherically symmetric bodies in the optically-thin limit. Those results are correct but scattered across two papers, each stating its scope in prose caveats attached to different sections. This paper introduces no new physics. It consolidates both into a single self-contained statement: a numbered assumption set (A1–A8), three theorems derived in one continuous chain, an independent numerical verification of each, and an explicit exclusion list (E1–E10) in which every excluded phenomenon is mapped to the specific assumption it would relax. The purpose is to fix a foundation precisely enough that subsequent work can be addressed to a labeled hypothesis rather than re-deriving or re-litigating the geometry. One bridging result, implicit in Papers 15 and 16 but never stated directly, is made explicit here because it is the logical hinge of the whole chain: Theorem 1 alone does not yield Newtonian gravity. An opaque body’s shadow scales with its cross-sectional area, not its mass — three bodies of identical mass are shown to cast shadows differing by a factor of 16. It is precisely the linearization in optical depth that converts area-scaling into mass-scaling, by rendering the shadow integral formally identical to the Newtonian volume integral and thereby handing Newton’s own Shell Theorem the exact form it requires. The geometric core is exact within A1–A8 and is claimed for nothing outside them. Stated numerically: within a parameter band of $10^{-2}$ to $10^{12}$ m, masses from laboratory to solar, and slow motion ($v/c\le1.6\times10^{-4}$), PBT, Newtonian gravity and General Relativity agree to better than 99.9999975% — PBT reproducing Newton analytically exactly, and departing from General Relativity only by the post-Newtonian factor $GM/rc^2$, which does not exceed $2.55\times10^{-8}$ anywhere in that band. Outside it — notably at $v=c$, where the two differ by a factor of two — the agreement breaks rather than degrades, which is why the band is stated as a hard condition.

Keywords: push gravity, shadowing mechanism, Newtonian reduction, Shell Theorem, Beer-Lambert attenuation, optical depth, domain of validity, self-shielding

Introduction

This series has reached a point where the same geometric result is cited repeatedly as settled while its boundary conditions live in prose caveats spread across two papers. That is a fragile arrangement. A caveat attached to a Results section in one paper and a scope note attached to a Discussion section in another are easy to cite past, and this series has already had one real instance of a claim outrunning its stated scope — Paper 15’s own first draft measured blocked solid angle and treated it as force, and Paper 16’s first draft claimed to have “resolved” self-shielding when it had quantified one side of a trade-off. Both were caught in adversarial review before publication. Both were failures of scope discipline, not of arithmetic.

This paper is the response to that pattern: an anchor. It restates what is actually established, in one place, with hypotheses numbered and exclusions enumerated. It claims no new physical result. Its contributions are two, both structural:

1. A single continuous derivation chain from ambient flux to the two-body force law, with the logical hinge between Theorem 1 and Theorem 2 stated explicitly rather than left implicit (see the Area-Mass Obstruction below).
2. A formal hypothesis set with every exclusion mapped to the assumption that generates it, so that future work can state precisely which assumption it is attempting to relax.

Everything in the Exclusions section is excluded. Nothing there is claimed to be solved, and several items are already known to be closed negatively. Establishing a correct foundation is not the same as establishing a viable theory, and this paper should not be read as claiming otherwise.

Assumptions

The results below hold under the following, and are claimed nowhere else.

Mathematical Formalism

Theorem 1 — Exact inverse-square geometry (point receiver, opaque sphere)

A perfectly opaque sphere of radius $R$ occludes a cone of half-angle $\alpha$ as seen from a point receiver at distance $d > R$, where $\sin\alpha = R/d$ exactly, from the tangent-line construction — an exact relation at any $d>R$, not a far-field approximation.

The quantity determining net force is not the blocked solid angle but the cosine-weighted momentum deficit, since off-axis blocked directions contribute to the axial force only by their projection:

$$F_{geo}(R,d) \propto \int_0^{2\pi}\int_0^{\alpha}\cos\theta\sin\theta \, d\theta \, d\phi = 2\pi\cdot\frac{\sin^2\alpha}{2} = \pi\sin^2\alpha$$

Substituting $\sin\alpha = R/d$:

$$\boxed{F_{geo}(R,d) \propto \pi\left(\frac{R}{d}\right)^2 \quad \text{exactly, for every } d>R}$$

This is exact by construction — the identity $\sin^2(\arcsin x) = x^2$ — not a leading-order term with corrections. It is the same mechanism underlying Newton’s Shell Theorem: the raw solid angle overcounts near-field occlusion, and the cosine projection compensates for it exactly.

The Area-Mass Obstruction

Theorem 1 does not yield Newtonian gravity. This is the hinge of the entire chain and is stated here explicitly because it is easy to read Theorem 1’s exactness as more than it is.

The coefficient in Theorem 1 is $\pi R^2$ — the geometric cross-section. For a perfectly opaque body, the shadow is a function of the body’s silhouette area, and carries no information about what is behind the surface. Mass is a volume quantity; an opaque shadow is a surface quantity. They do not scale together.

Quantitatively, for three bodies of identical mass $M = 4.1888$ (in units where $\rho R^3$ sets the scale):

A factor of 16 spread in force at fixed mass. Real gravity is proportional to mass to roughly 1 part in $10^{15}$ (MICROSCOPE, 2022 [8]). An opaque shadowing mechanism is therefore not merely imprecise as an account of gravity — it is excluded by many orders of magnitude. This is the Poincaré/Maxwell self-shielding objection in its sharpest form, and it is fatal to the opaque case.

Theorem 2 exists to escape it.

Theorem 2 — Mass-proportionality in the thin limit (point receiver, extended body)

Replace perfect opacity with Beer-Lambert attenuation. A ray traversing column mass density $\Sigma = \int\rho \, d\ell$ is transmitted with probability $e^{-\kappa\Sigma}$, so the absorbed fraction is

$$1 - e^{-\kappa\Sigma} = \kappa\Sigma - \frac{(\kappa\Sigma)^2}{2} + \mathcal{O}(\tau^3)$$

Under A4, retain only the first term. The physical content of that truncation is that every mass element absorbs independently — no element lies in another’s shadow at this order.

Write $\Sigma(\hat{n})$ for the column mass density along the ray leaving the receiver at $\mathbf{r}$ in direction $\hat{n}$. The net force is the cosine-weighted deficit summed over all directions:

$$\mathbf{g}(\mathbf{r}) = \mathcal{K}\kappa\int \hat{n}\,\Sigma(\hat{n})\,d\Omega = \mathcal{K}\kappa\int d\Omega\,\hat{n}\int_0^{\infty}\rho(\mathbf{r}+s\hat{n})\,ds$$

Now convert the line-of-sight integral to a volume integral. The volume element in spherical coordinates centred on the receiver is $dV = s^2\,ds\,d\Omega$, hence $ds\,d\Omega = dV/s^2$. Substituting, with $s=|\mathbf{r}’-\mathbf{r}|$ and $\hat{n}=(\mathbf{r}’-\mathbf{r})/s$:

$$\mathbf{g}(\mathbf{r}) = \mathcal{K}\kappa\int_V \rho(\mathbf{r}’)\frac{(\mathbf{r}’-\mathbf{r})}{|\mathbf{r}’-\mathbf{r}|^3}\, dV’$$

The Jacobian factor $1/s^2$ is the entire origin of the inverse-square law in this mechanism. It is not assumed and not put in by hand: it appears purely from converting a column integral along lines of sight into an integral over volume. This is worth stating plainly because it is the step that makes the result feel inevitable rather than fitted.

The resulting integral is formally identical to the Newtonian volume integral. That identity is the whole content of Theorem 2: linearizing in optical depth replaces a surface quantity with a volume quantity, and in doing so hands Newton’s Shell Theorem exactly the form it requires. Therefore, for spherically symmetric $\rho$ and $d > R$:

$$\boxed{g(d) = \frac{\mathcal{K}\kappa M}{d^2}}$$

exact in $d$, with no residual dependence on $R$ or $\rho$ individually — only on the total mass $M$. The 16× spread of the Area-Mass Obstruction collapses to zero.

Theorem 3 — Two extended bodies

For a receiving body $B$ that is itself extended, spherically symmetric, and thin, its absorbed momentum is proportional to its own total mass by the same argument. A spherically symmetric body in an external inverse-square field experiences a net force equal to its mass times the field at its center. Hence for $d > R_A + R_B$:

$$\boxed{F(d) = \frac{\mathcal{K}\kappa_A\kappa_B M_A M_B}{d^2}}$$

Correspondence with Newton is then $G \equiv \mathcal{K}\kappa^2$ under A3. No calculation in this paper fixes $\mathcal{K}$ or $\kappa$ against the measured value of $G$ (see E3).

Dimensional check. With $\kappa$ a mass-absorption coefficient (m²·kg⁻¹), the correspondence $G=\mathcal{K}\kappa^2$ requires

$$[\mathcal{K}] = \frac{[G]}{[\kappa]^2} = \frac{\text{m}^3\text{kg}^{-1}\text{s}^{-2}}{\text{m}^4\text{kg}^{-2}} = \text{kg}\,\text{m}^{-1}\text{s}^{-2} = \text{Pa}$$

$\mathcal{K}$ carries units of pressure — which is exactly what it should be, since it is the ambient momentum flux of the substrate. This is a consistency check, not a derivation: it confirms the bookkeeping closes, and it makes explicit that E3 is not an arbitrary missing constant but a missing ambient pressure scale. In a theory named Pressure-Based, the one unfixed normalization is a pressure.

Two independent routes to the same result

Theorem 2 is derived here by converting the angular integral to a volume integral and invoking the Shell Theorem. Paper 16 reaches the identical closed form by a different route: evaluating the cosine-weighted angular integral directly, using the exact chord $l(\theta)=2\sqrt{R^2-d^2\sin^2\theta}$ and the substitution $u=d\sin\theta$, which absorbs $\cos\theta$ exactly and yields $4\pi\kappa\rho R^3/3d^2 = \kappa M/d^2$ with no small-angle approximation. Two structurally different derivations converging on the same expression is a stronger check than either alone, and both were re-verified independently for this paper.

How opaque is “opaque”? — an exact convergence rate

A4 is a limit, and limits invite the question of how fast they are approached. The same integral can be solved exactly rather than sampled. Substituting $w=\sqrt{R^2-u^2}$ reduces it to closed form, giving the relative deviation of the finite-opacity result from the perfectly-opaque one:

$$\Delta(a,R) = \frac{2}{a^2R^2}\left[1-e^{-aR}(1+aR)\right] \; \xrightarrow[aR\gg1]{} \; \frac{1}{2(\kappa\rho R)^2}, \qquad a = 2\kappa\rho$$

Convergence to the opaque limit is a power law, not exponential — a fact that matters, because it means “effectively opaque” is a much stronger requirement than intuition suggests. This result also corrects a stated precision figure in Paper 16 (see that paper’s 2026-07-31 revision note).

Numerical Verification

Each theorem was verified independently of its derivation, in pure Python, seed 20260731. Script and full output preserved in this project’s research log for reproducibility.

Theorem 1 — Simpson quadrature of the cosine-weighted integral against the closed form, plus the blocked solid angle shown for contrast:

Agreement at floating-point precision across the full range to contact. The final column is the quantity Paper 15’s first draft mistakenly used; its error reaches 97% at contact, which is what made that error consequential rather than cosmetic.

Theorem 2 — Monte Carlo over a uniform sphere (400,000 samples), compared against $\kappa M/d^2$:

Theorem 2b — the decisive check against the Area-Mass Obstruction: total mass held fixed at $M=1$, radius varied by more than an order of magnitude, receiver at $d=3$:

The opaque case varies by 16× across this range; the thin case is constant to Monte Carlo noise.

Theorem 3 — two extended bodies, $M_A = M_B = 1$, $d = 1$:

All residuals are consistent with Monte Carlo sampling noise ($\propto N^{-1/2}$) and show no systematic trend with separation, radius, or density.

Domain of Validity

Regarding scale: Paper 14’s 0.01% numerical agreement with Newton was computed at galactic-model radii ($\approx6\times10^{19}$–$1.2\times10^{21}$ m) and extrapolated to the everyday band ($10^{-2}$–$10^{12}$ m) by scale invariance, not independently recomputed there. The present paper’s Theorems 1–3 are scale-free — they reference only relative geometry — so they carry no separate scale restriction of their own. But the extrapolation in Paper 14’s table remains an extrapolation, and is not upgraded by anything here.

Quantified Agreement with Newton and General Relativity

The statements above are qualitative. Stated plainly and numerically: within the parameter band below, Pressure-Based Theory, Newtonian gravity, and General Relativity all agree to better than 99.9999975%.

The parameter band

Agreement is claimed only within all of the following simultaneously.

Caution: these ranges are not independently variable, and treating them as a free combination gives wrong answers. The separation and mass entries describe the terrestrial and solar-system configurations that happen to satisfy the field-strength condition. They are not a Cartesian product from which any pair may be selected. Real configurations sit inside the nominal size and mass ranges while violating the field-strength condition by orders of magnitude:

Where the field-strength condition and the size/mass ranges disagree, the field-strength condition governs. The 99.9999975% figure is the worst case among configurations satisfying $GM/rc^2 \le 2.55\times10^{-8}$, not among everything nominally fitting the separation and mass rows.

The agreement, case by case

PBT versus Newton: exact. Not “very close.” Theorems 1–3 reproduce Newton’s law analytically, with zero deviation at $\mathcal{O}(\tau)$; independent numerical verification agrees to $2.4\times10^{-14}$, which is double-precision floating-point noise rather than a physical residual. There is no separation, mass, or density within the band at which the two differ.

PBT versus General Relativity: since PBT reproduces Newton exactly, its departure from GR is precisely the Newton-GR departure — the post-Newtonian correction, of order $GM/rc^2 \simeq (v/c)^2$:

The worst case anywhere in the band is Mercury’s orbit, at $2.55\times10^{-8}$ — a difference of 0.0000025 percent. Everywhere else the agreement is tighter, by up to eighteen further orders of magnitude at laboratory scale.

Two qualifications, both essential

Small is not the same as unobservable. The per-instant difference at Mercury is $2.55\times10^{-8}$, but it accumulates secularly. Over 415.2 orbits per century the GR advance $6\pi GM/[a(1-e^2)c^2]$ totals 42.98 arcseconds per century, against the measured value of approximately 43 — the classic test GR passes and Newton fails. (That this calculation independently reproduces the known figure is a check on the table above, not a new result.) A theory reproducing Newton exactly therefore reproduces Newton’s failure here exactly as well. The agreement percentages describe instantaneous force, not century-scale accumulated observables.

Outside the slow-motion parameter, agreement fails completely. At $v=c$ — light — the naive Newtonian deflection at the solar limb is 0.8756 arcseconds while GR gives 1.7512: a factor of two, a 100% disagreement, not a part in $10^8$. This is precisely why $v/c\ll1$ appears in the parameter table as a hard condition rather than a formality. The band is not a matter of convenience; step outside it and the agreement does not degrade gracefully, it breaks.

This is Newton, not Einstein — and the difference is the whole of E10

A clarification that belongs in the main text rather than a footnote, because the title invites exactly the wrong reading. “Geometric” in this paper’s title refers to the geometry of shadowing — solid angles, chords, occlusion cones. It does not refer to General Relativity’s geometric account of gravity as spacetime curvature. The two senses are unrelated, and conflating them would overstate this result substantially.

What Theorems 1–3 deliver is Newton’s law: inverse-square in separation, linear in each mass. Newton, not Einstein. At classical scale those are not the same theory, and the entire measured difference between them is the post-Newtonian sector — Mercury’s perihelion precession of 43 arcseconds per century, light deflection at 1.75 arcseconds (exactly twice the Newtonian value), Shapiro time delay, gravitational redshift, and frame dragging.

A6 and A7 exclude precisely those effects. Post-Newtonian corrections are velocity-dependence and retardation; assuming a static configuration and instantaneous propagation removes them before any calculation begins. This is not a case of the mechanism attempting them and falling short — the assumptions delete the phenomena. The consequence is worth stating bluntly: the regime in which this mechanism reproduces gravity is exactly the regime in which Newton and General Relativity already agree. Where the two measurably differ, the results established here are silent.

A7 is worse than silent; it is in direct tension with measurement. General Relativity has gravitational influence propagate at $c$, and the coincident detection of GW170817 with GRB 170817A constrains the speed of gravity to within $-3\times10^{-15}$ and $+7\times10^{-16}$ of $c$ [9]. This series’ own stated escape from drag and from Laplace’s aberration objection (E6) runs in the opposite direction, requiring substrate speed $v\to\infty$. In a mechanism where gravitation is the substrate flux, both cannot hold.

Finally, the one place this series reports agreement with General Relativity — Paper 2’s light-bending value of 1.748 arcseconds — is explicitly labelled in that paper as a consistency check using GR’s own weak-field formula, “not an independent test distinguishing this model from GR.” It is a borrowed result correctly evaluated, not a derivation from shadowing. Papers 2 and 8 do invoke an Einstein-aether field for Lorentz invariance, but that is a published framework built atop GR’s own mathematical structure, so any GR-recovery there is imported rather than produced by the mechanism (see also E8).

None of this counts against the mechanism, and it should not be read that way. That PBT, Newton and General Relativity all agree throughout the regime where Newton and GR themselves agree is exactly what a correct mechanism ought to produce — it is the expected result, not a shortfall. E10 is a statement of scope: what these assumptions do and do not reach. It is not a record of something attempted and missed.

What E10 does mark as genuinely open is the A7 tension above. Reconciling the substrate speed this series requires with the measured propagation speed of gravity is real, unfinished work, and it is the gate through which the post-Newtonian sector would have to be earned rather than assumed.

Exclusions

Each entry names the assumption it would relax, and its current status. Nothing in this table is claimed to be solved.

A note on the relationship between E1, E4, and E5, since treating them as independent has already caused one error in this series: they are not independent. Paper 16 established that the optically-thin regime (A4) required to eliminate self-shielding is precisely the regime Maxwell (1875) and Poincaré (1908) identified as maximizing drag and heating, because weak per-encounter absorption demands an enormous ambient flux to reach ordinary gravitational strength. Avoiding E1 by construction is what maximizes E4 and E5. The assumption set A1–A8 is not a list of independently relaxable conditions; A4 in particular is load-bearing in both directions.

Discussion

What this paper establishes is narrow and should be read narrowly: within A1–A8, the shadowing mechanism reproduces the form of Newtonian gravity exactly — inverse-square in separation, linear in each mass, independent of size and density at fixed mass. Not approximately, not asymptotically, not in a far-field limit.

What it does not establish is that A1–A8 can be simultaneously satisfied by any real physical medium. That is the actual open question, and the Exclusions table is the honest inventory of it. Two entries are already closed negatively (E4, E9), one is contradicted wherever it has been tested (E5), and four are untouched (E6–E8, E1).

The strongest objection to this result, stated by us rather than left for a referee

An informed reviewer will notice that Theorem 2 comes very close to assuming what it proves, and the objection deserves to be met head-on rather than waited for.

The objection: once absorption is linearized, any isotropic, single-scattering, mass-coupled attenuation mechanism will reproduce Newtonian form. The $1/s^2$ arrives from the volume Jacobian, the mass-linearity arrives from the $\mathcal{O}(\tau)$ truncation, and the Shell Theorem does the rest. Nothing in the derivation is specific to a substrate flux, to push gravity, or to PBT.

That objection is correct, and we accept it. Theorem 2 is best understood as a statement about an entire class of mechanism, not as evidence for this one: any mechanism in which a body removes momentum from an isotropic ambient field in proportion to column mass, weakly enough not to saturate, yields exactly Newtonian form. PBT is one member of that class.

But it is not a defect, and reading it as one mistakes what kind of program this is. This work does not propose a rival to Newtonian gravity competing to out-predict it. It proposes a mechanism for a law already known to be correct. In that genre, reproducing the known law exactly is the success condition, not a null result — the same sense in which the kinetic theory of gases succeeds by reproducing the ideal gas law from molecular collisions rather than by predicting something different from it. Deriving $PV=nRT$ from moving particles was the achievement; nobody counts kinetic theory weak for agreeing with the law it explains.

Two things follow, both favourable. First, a mechanism that failed to reproduce Newton would be dead immediately — Paper 14 says exactly this — so passing is necessary, and passing exactly rather than approximately is a stronger result than the minimum required. Second, genericity is robustness: that an entire class of absorption mechanisms yields Newtonian form means the form is structurally forced rather than finely tuned. It cannot be broken by revising substrate details, and any future modification within A1–A8 inherits it automatically.

What genericity does establish is where the discriminating content lives: outside A1–A8, in E3’s normalization and E5’s flux requirement, where the specific ontology finally matters. That is a map, not a demotion. An anchor’s job is to tell you where a distinguishing test can and cannot be found, and this one does.

The value of an anchor is that it makes the next question well-posed. Before this paper, “does the geometry work?” was answerable only by reading two papers and reconciling their caveats. After it, the geometry is a fixed, verified, numbered result, and every remaining problem is addressed to a specific assumption. A future paper attempting E1 is attempting to relax A4 and must say what replaces it. A future paper attempting E6 is attempting to relax A7 and inherits Laplace’s constraint as its target.

That is a genuine gain in tractability. The theory’s empirical standing at other scales is unchanged by this paper and remains as Paper 14 reported it — but within the classical regime, what this paper establishes is a mechanism reproducing the known law exactly, which is what a mechanistic account is for.

Testable Predictions

1. A3 is directly falsifiable now. The requirement that $\kappa$ be universal across composition is strictly stronger than mass-proportionality. Any finite-opacity model predicting composition-dependent $\kappa$ is constrained by existing equivalence-principle data: the MICROSCOPE satellite mission’s final result [8] gives $\eta(\text{Ti},\text{Pt}) = [-1.5\pm2.3(\text{stat})\pm1.5(\text{syst})]\times10^{-15}$, roughly 100x tighter than the ground-based torsion-balance bound of $\sim10^{-13}$ that Papers 15 and 16 originally cited. This is an available test against already-published data, not a proposed future observation.
2. E1 is a defined, tractable calculation. Extend Theorem 3 to $\mathcal{O}(\tau^2)$ and determine whether mass-proportionality survives at second order, and at what optical depth the deviation reaches the $10^{-15}$ equivalence-principle bound. That threshold, combined with E5’s flux requirement, would convert the Maxwell/Poincaré trade-off from a qualitative argument into a closed inequality — either an explicit viable window in $(\kappa, \mathcal{K})$, or a proof that none exists.
3. E5’s short-range gap is a specific number nobody has computed. The stellar-cooling bound weakens for a force of shorter reach. Computing the correct bound in the range this mechanism requires would either close E5 negatively or identify the one surviving window.

Prediction 2 is the recommended next step: it is fully specified, requires no new physical assumption, and its outcome is decisive in either direction.

Conclusion

The geometric core of the shadowing mechanism is exact within a stated set of eight assumptions, verified numerically at floating-point precision for the pure geometry and to Monte Carlo noise for the extended-body cases. An opaque shadow scales with area and is excluded by 16× at fixed mass; linearization in optical depth converts it to a volume integral and recovers exact mass-proportionality via Newton’s own Shell Theorem. That chain is now stated once, in one place, with its hypotheses numbered.

Nine exclusions are enumerated, two of them already closed negatively and one contradicted wherever tested. The foundation is fixed; the building on it has not begun, and several of the load paths are known to be broken. Stating precisely where they break is the point of this paper.

References

1. Le Sage, G.-L. (1748). Essai de Chymie Méchanique.
2. Maxwell, J. C. (1875). “Atom,” Encyclopædia Britannica, 9th ed. — the Le Sage critique and the “white heat” passage. Article title verified against the article text; Papers 15 and 16 previously cited this incorrectly (see their 2026-07-31 revision notes).
3. Poincaré, H. (1908). Science et Méthode. — cited at book level deliberately: the year and substance are verified, but a chapter localization could not be confirmed from primary text, so none is asserted.
4. Newton, I. (1687). Philosophiæ Naturalis Principia Mathematica, Book I, Proposition 71 (the Shell Theorem).
5. Newton, I. (1713). Principia, 2nd ed., General Scholium (“hypotheses non fingo”).
6. Edwards, M. R. (Ed.). (2002). Pushing Gravity: New Perspectives on Le Sage’s Theory.
7. Bradley, J. (1728). “A Letter… Giving an Account of a New Discovered Motion of the Fix’d Stars,” Philosophical Transactions of the Royal Society.
8. Touboul, P., et al. (2022). “MICROSCOPE Mission: Final Results of the Test of the Equivalence Principle,” Physical Review Letters 129, 121102.
9. Abbott, B. P., et al. (2017). “Gravitational Waves and Gamma-Rays from a Binary Neutron Star Merger: GW170817 and GRB 170817A,” Astrophysical Journal Letters 848, L13.

Appendix: Corrections Made to Papers 15 and 16

Consolidating a result is itself a check on it. Re-deriving Papers 15 and 16 from scratch for this paper surfaced five errors in them, all now corrected in place via dated revision notes on those papers. Recorded here because an anchor that silently inherited its sources’ errors would be worse than no anchor.

Two of these deserve comment. The $10^{15}$ figure was almost certainly a quadrature artifact: at $\kappa\rho R=1000$ the shadow’s rim transitions over a width of order $10^{-7}$, so a uniform grid coarser than that silently misses it and returns a spuriously small residual. The first attempted re-verification for this paper fell into the identical trap, returning a misleading floor of $1.7\times10^{-6}$ — which is why the integral was ultimately solved in closed form rather than sampled. A numerical check that cannot resolve the feature it is checking will return a confident wrong answer rather than an error.

The equivalence-principle correction runs the opposite direction from most corrections in this series: it does not weaken a claim, it strengthens a constraint. Papers 15 and 16 both proposed the EP bound as the falsification test for any future finite-opacity model. That test is 100 times sharper than they stated.