The 4 in the light-bending angle is not 4 pi but 2 times (1 + gamma) – a census of lensing.
The deflection of light by a point mass is 4GM over c squared b. This paper writes what that 4 is made of. NO NEW MATHEMATICAL THEOREM AND NO NEW PHYSICAL LAW IS CLAIMED. Einstein (1915, 1936), Zwicky (1937), the PPN formalism (Will), Brans-Dicke (1961) and the Cassini measurement of gamma (Bertotti et al. 2003) are all standard. Because the corpus rests on 4 pi (Papers 01, 02), a 4 invites being read as a relative of 4 pi, and this paper stops that.
First, the 4 is not 4 pi. It is 2 times 2. The first 2 is the value of an integral. Integrating the transverse gravitational force along a straight trajectory, symbolic computation here gives the full-line integral of GMb over (b squared plus z squared) to the three halves as exactly 2GM over b, and dividing by the square of the speed of light gives the Newtonian deflection 2GM over c squared b. This 2 is the value of an integral, neither a solid angle nor a period, and no pi occurs anywhere in it.
Second, the exact deflection integral confirms it. In the Schwarzschild metric the exact deflection is an inverse-square-root integral minus pi. This paper removes the endpoint singularity by substitution, divides out the simple root and quadrates. With the impact parameter at 500 times GM over c squared the ratio to the weak-field value is 1.00593350, at 1000 times it is 1.00295595, and at ten thousand times it is 1.00029464. That last figure agrees to seven digits with the known second-order term, one plus fifteen pi over sixteen times the ratio, namely 1.00029452. So in a weak field the general-relativistic value is exactly twice the Newtonian one.
Third, one of the two cores. The second 2 is a measured quantity. In the parametrised post-Newtonian formalism the deflection is (1 + gamma) times 2GM over c squared b, where gamma measures how much spatial curvature is produced per unit mass, and general relativity predicts gamma equal to one. This splits the 4 in two: the first half, two times one, comes from the warping of time and is nearly forced by the equivalence principle, while the second half, two times gamma, comes from spatial curvature and is a measured quantity. And gamma moves between theories. In Brans-Dicke theory gamma is (1 + omega) over (2 + omega), and the computation here gives the four as 3.33333333 at omega equal to one, 3.83333333 at ten, 3.98039216 at a hundred and 3.99995000 at forty thousand. The four moves. The Cassini radio-delay measurement puts gamma minus one consistent with zero at the level of ten to the minus five, so the four is a quantity measured to five digits and not a mathematical necessity. Within one formula the two halves are of different standing, which is an instance of Paper 95’s sorting.
Fourth, the other core. The doubling belongs not to gravity but to travelling at the speed of light. The weak-field deflection of a particle of speed v is standardly 2GM over b v squared, times one plus v squared over c squared, so the ratio to the Newtonian value is one plus v squared over c squared. The table here gives 2.00000000 at v over c equal to one, 1.25000000 at one half, 1.01000000 at one tenth, 1.00010000 at one hundredth and 1.00000001 at one ten-thousandth. So saying that light falls twice as fast is not right: the factor is one plus v squared over c squared, and it is two only because a photon travels at the speed of light, while a slow particle barely feels the second half. The reason can be written too. The Newtonian half is carried by the time component of the metric and the other half by the spatial components; a slow particle’s worldline points almost entirely along time and sweeps almost none of the spatial part, whereas light sweeps time and space equally. The two is the consequence of that.
Fifth, so where is the 4 pi? In the critical surface density. Writing the lens equation produces a critical surface density equal to the square of the speed of light times the distance to the source, divided by four pi times the gravitational constant times the distance to the lens times the distance from lens to source. That four pi comes from Poisson’s equation, the same one Paper 35 placed on the side of fact. And the address matters: the four pi appears where an observed distortion is turned back into a mass distribution, not in the bending angle itself. This is the same shape as Paper 130, where the four pi lived not in Helmholtz’s theorems but in Biot-Savart, that is, in the inverse problem. Lensing repeats it (Papers 89, 99). To point at a four pi, look not at the bending angle but at the formula that reads the mass back.
Sixth, the content of the Einstein ring is that the mean convergence inside it is exactly one. The square of the Einstein radius is four GM times the lens-to-source distance divided by the square of the speed of light times the distance to the lens times the distance to the source. Computing the mean convergence inside that radius, that is the mean surface density divided by the critical one, symbolic computation here gives exactly one. So the Einstein ring is the radius at which the projected mean density equals the critical density, and that single sentence is the condition for the ring to exist. The square root appearing in the Einstein radius is moreover a consequence of the lens equation being quadratic and did not come from the geometry of a sphere. Paper 28 said that an exponent is not a dimension, and here too the exponent announces the degree of the equation that was solved.
IMPORTANT (scope): a census of established standard material (Layer 2). Every work cited is standard. What is not claimed: any new mathematical theorem (there is none) / any new physical law (there is none) / that the general-relativistic deflection angle is in error (it is not) / that gamma might differ from one (that is not claimed; measurement is consistent with one, and what is stated is the standing of the quantity, namely that it rests on measurement). What is added: splitting the four into two times two and making explicit that the second half is measured, confirming the exact deflection integral down to the second-order term, tabulating the doubling as a property of the speed of light, placing the address of the four pi at the critical surface density, and writing that the ring’s content is a mean convergence of one. What remains: do not let a four call four pi to mind. Until a number is broken into its parts, what it is saying is not known.
On the making of this work: The ideas and content of this work stem from the author’s own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author’s. Feedback and corrections are sincerely appreciated.
Reference
Yamagishi, Y. (2026). The $4$ in the Light-Bending Angle Is Not $4\pi$ but $2(1+\gamma)$ [Paper 134]. Zenodo.
DOI: 10.5281/zenodo.22080727
ORCID: 0009-0006-8587-3450