Of 16 pi^2, Only Omega_3 = 2 pi^2 Is Geometry: The (2 pi)^4 Is a Fourier Convention, and the 4 pi of 4 pi v Is Not a Solid Angle. Third edition (Paper 46 and 47, merged).
This paper derived that the 1/(16 pi^2) of one-loop corrections and the 1/(16 pi^2)^2 of two-loop corrections are the loop measure of four-dimensional momentum space, and stated the general rule that an L-loop amplitude carries (16 pi^2)^(-L). THEOREM 2.1 IS CORRECT AND IS NOT ALTERED: the integral d^4k_E/(2 pi)^4 of F equals (Omega_3/(2 pi)^4) times the integral of k^3 dk F, which is (1/(8 pi^2)) times that integral, and equals (1/(16 pi^2)) times the integral of dk^2 k^2 F (with Omega_3 = 2 pi^2 the surface area of the unit three-sphere, the last step using k^3 dk = (1/2) u du; re-verified here symbolically). The L-loop rule also stands, being the same applied to each integral independently. What the third edition corrects is calling that 16 pi^2 “a purely geometric consequence”.
FIRST, DECOMPOSING 16 PI^2 PRODUCES THREE FACTORS OF THREE DIFFERENT STANDINGS. 16 pi^2 = 2 x (2 pi)^4 / Omega_3 (verified here), where (2 pi)^4 is a CONVENTION (where one parks the Fourier 2 pi’s), Omega_3 = 2 pi^2 is GEOMETRY (the surface area of the unit three-sphere), and the 2 is a JACOBIAN (k^3 dk = (1/2) u du). Omega_3 is a quantity the unit three-sphere itself possesses; IT ALONE IS GEOMETRY. The (2 pi)^4 is not: it comes from the way one writes the integral d^4k/(2 pi)^4. Whether the Fourier 2 pi’s go entirely on the momentum side, are split between the two, or are omitted, the physics is the same in every case and only the written number changes. The 2 is not either: it is the Jacobian of trading the radial variable k for k^2, and attends the choice of which variable to write in.
SECOND, THE PROOF THAT IT IS A CONVENTION IS THAT CHANGING THE CONVENTION CHANGES THE NUMBER. Computing the same combination 2 (2 pi)^n / Omega_3 gives 16 pi^2 in the QFT standard convention d^4k/(2 pi)^4, but 4 in the symmetric convention d^4k/(2 pi)^2, and 1/pi^2 with no 2 pi at all (verified here). The physical amplitude is the same in every convention, since propagators and vertices pick up compensating 2 pi’s. WHAT CHANGES IS ONLY WHETHER THE NUMBER 16 PI^2 GETS WRITTEN DOWN. What changes its answer when the convention changes is a convention, not a fact (Paper 95) – the same judgement Paper 45 passed on R = 2. Hence “containing no dynamical assumption” is correct and “a purely geometric consequence” is not: CONTAINING NO DYNAMICS AND BEING GEOMETRY ARE TWO DIFFERENT THINGS. A convention, too, contains no dynamics.
THIRD, THE ISOMORPHISM OF COROLLARY 2.2 IS AN AGREEMENT OF NUMBERS WHOSE TWO SIDES ARE COMPOSED ENTIRELY DIFFERENTLY. The second edition wrote 16 pi^2 = (2 pi)^4/Omega_3 x 2 = (4 pi) x (4 pi). The left side is CONVENTION x GEOMETRY x JACOBIAN; the right side is CONVENTION x CONVENTION (the patch constant may be any positive value; Paper 116). The numbers agree and the compositions differ entirely. The second edition’s Note 2.3 was right to state that “what this paper asserts strictly is Theorem 2.1 alone, and the isomorphism with the Yamagishi geometry is presented as an interpretive correspondence”. THE THIRD EDITION SUPPLIES THE REASON FOR THAT LIMITATION: because the left side is contaminated by a Fourier convention, even finding a transformation between the two would not make it a correspondence between geometries. The isomorphism is therefore weak even as an interpretation, and this edition records it as an agreement of numbers and no more.
FOURTH, THE 4 PI OF LAMBDA LESS THAN OR ABOUT 4 PI v IS NOT A SOLID ANGLE. From Lambda^2/(16 pi^2) less than or about v^2 one gets Lambda less than or about sqrt(16 pi^2) v = 4 pi v; with v = 246 GeV this is 3091 GeV = 3.09 TeV, and including the top-Yukawa term -6 y_t^2 gives 4 pi v / sqrt(6) = 1262 GeV = 1.26 TeV (verified here). The number, the testability and the note of observational tension with the LHC are all correct and are not changed. WHAT IS CORRECTED IS ONLY THE PHRASE “a bound expressed as the solid angle 4 pi”: the 4 pi here is sqrt(16 pi^2), THE SQUARE ROOT OF THE LOOP FACTOR, which as above is convention x geometry x Jacobian. It is not a solid angle. This is the same shape Paper 45 found inside itself, where an AREA was called a solid angle. The correct name is “the 4 pi counting of naive dimensional analysis” (Manohar-Georgi 1984, already in the second edition’s bibliography). The bound itself does not change at all.
WHAT THE SECOND EDITION RIGHTLY RETRACTED IS NOT REINSTATED HERE. The first edition’s voxel-cutoff argument – introducing Lambda_voxel = (16 pi^2)^(-1/2) by hand to render the divergence finite – is circular, since the conclusion (finiteness) is already contained in the assumption (a finite cutoff); the content of Goroff-Sagnotti, perturbative non-renormalisability, is not dissolved by introducing a cutoff. The other disciplines the second edition kept are also unchanged: the explicit statement that the LHC has seen no new physics to the TeV scale and the naturalness bound is already under observational tension; the refusal to give 209/2880 a geometric interpretation (it comes from the combinatorics of cubic Weyl contractions, a different origin from 16 pi^2); and the placement of general relativity as an effective field theory. The reading “each loop consumes one double-torus boundary capacity” was removed from the body as Layer 3 – the suppression itself (16 pi^2 approximately 158 putting one-loop corrections at the 1e-2 level) is kept as standard loop counting, but the reading presupposes that 16 pi^2 is geometry.
THE SCOPE, RESTATED as twelve rows. (1) Theorem 2.1: a FACT. (2) That L loops carry (16 pi^2)^(-L): a FACT. (3) Omega_3 = 2 pi^2: GEOMETRY. (4) (2 pi)^4: a FOURIER CONVENTION. (5) The 2 of k^3 dk = (1/2) u du: a JACOBIAN. (6) That 16 pi^2 is “purely geometric”: RETRACTED. (7) The isomorphism 16 pi^2 = (4 pi)^2: AN AGREEMENT OF NUMBERS. (8) Lambda less than or about 4 pi v = 3.09 TeV: a FACT (under observational tension). (9) That its 4 pi is a solid angle: RETRACTED. (10) A cutoff rendering the divergences finite: RETRACTED. (11) A geometric interpretation of 209/2880: NOT GIVEN. (12) An ultraviolet completion of quantum gravity: NOT GIVEN. Of the twelve rows, four are settled facts.
IMPORTANT (scope): a structural interpretation and self-audit of established standard facts (Layer 2). The physics used is entirely standard (the four-dimensional Euclidean momentum integral, Omega_3 = 2 pi^2, the quadratic divergence of the Higgs self-energy, Goroff-Sagnotti 1986, naive dimensional analysis). NO NEW THEOREM AND NO PREDICTION. What is not claimed: “contains no dynamics” is not reread as “is geometry” (Paper 95) / 16 pi^2 and the patch’s (4 pi)^2 are not treated as the same geometric quantity (different compositions; Paper 116) / 4 pi is not called a solid angle (the square root of a loop factor; Paper 45) / no claim that a cutoff renders the divergences finite (circular; retracted in the second edition) / no claim to have solved the hierarchy problem (the bound is under observational tension) / no geometric interpretation of 209/2880 (different origin) / no ultraviolet completion of quantum gravity / the reading “each loop consumes a geometric capacity” is not placed in the body (Layer 3, quarantined).
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). Of $16\pi^2$, Only $\Omega_3=2\pi^2$ Is Geometry — 3rd ed.: The Three Standings inside the Yamagishi Loop Geometric Measure (Convention, Geometry, Jacobian) — [Paper 46]. Zenodo.
DOI: 10.5281/zenodo.21602422
ORCID: 0009-0006-8587-3450