An Area Is Not a Solid Angle: The Hopf Factorisation Holds at Every Radius, and R = 2 Is a Choice of Unit. Third edition.
This paper treats the normalisation factor 1/(16 pi^2) of the topological charge and made three claims: (A) that it descends from the component form of the second Chern class, (B) that 16 pi^2 is the volume of the three-sphere of radius R = 2, with base and fibre both equal to 4pi in the Hopf decomposition, and (C) that the instanton action can be written S = 1/(2 alpha_eff). (A) IS CORRECT AND IS NOT ALTERED: writing c_2 = (1/8 pi^2) Tr(F wedge F) in components gives k = (1/16 pi^2) integral d^4x Tr(F_munu Ftilde^munu). What the third edition corrects is (B) and (C), and both are corrections of a kind visible from the dimensions.
FIRST, THE HOPF FACTORISATION ITSELF IS GENUINE AND DOES NOT REQUIRE R = 2. For the Hopf bundle S^1 into S^3_R onto S^2_{R/2}, the base area 4pi(R/2)^2 = pi R^2 times the fibre circumference 2 pi R equals pi R^2 x 2 pi R = 2 pi^2 R^3 = Vol(S^3_R), AND THIS HOLDS FOR EVERY R (verified here symbolically). That the volume of the total space factorises into the area of the base times the length of the fibre is the genuine geometric fact about the Hopf bundle, and it needs no specification of a radius. The second edition drew attention to the number 16 pi^2 appearing on substituting R = 2, but the factorisation does not depend on R: R = 2 is only how that fact looks in one particular scale.
SECOND, “R = 2 IS THE UNIQUE RADIUS” IS NOT GEOMETRY BUT A CHOICE OF UNIT. The second edition solved pi R^2 = 2 pi R to get R = 2. But the left side is an AREA (dimension L^2) and the right side a CIRCUMFERENCE (dimension L^1), and joining them with an equals sign has no meaning until a unit for R has been fixed. Rescaling by R = lambda R’ gives R’ = 2/lambda (verified here): lambda = 1 gives R’ = 2, lambda = 2 gives R’ = 1, lambda = 1/2 gives R’ = 4, lambda = 10 gives R’ = 1/5. WHAT CHANGES ITS ANSWER WHEN THE SCALE CHANGES IS NOT A GEOMETRIC NECESSITY (Paper 95). Corollary 2.3, “the double appearance of the Yamagishi patch is a geometric necessity”, is therefore RETRACTED. What is retracted is the necessity, not the factorisation.
THIRD, AND THE LARGEST CORRECTION HERE: THE 4PI OF THE BASE IS NOT A SOLID ANGLE. The solid angle of any sphere is 4pi steradian – independent of radius, and dimensionless. The AREA of S^2_{R/2}, by contrast, is pi R^2 with dimension L^2: pi at R = 1, 4pi at R = 2, 9pi at R = 3 (verified here). The solid-angle column does not move; the area column moves as R^2. They take the same value at the single point R = 2, and that is A COINCIDENCE OF NUMBERS, NOT AN IDENTITY OF QUANTITIES. Calling the area a “solid angle” erases that difference, and the moment it is erased a quantity independent of R stands in the same column as one that depends on R – precisely the error this series named in Paper 16 (“the equals sign has distinct roots”) and Paper 116 (“quantities of different dimension cannot be the same quantity”). WHAT THE SECOND EDITION READ AS A “GEOMETRIC NECESSITY” WAS THIS RENAMING.
FOURTH, S_inst = 1/(2 alpha_eff) IS NOT A THEOREM BUT A RESTATEMENT OF A DEFINITION. The second edition called it “the core result of this paper” and named it the perturbative-non-perturbative duality theorem. Writing alpha_eff = g^2/X in general gives 1/(2 alpha_eff) = X/(2 g^2), and this equals the BPST value S_inst = 8 pi^2/g^2 precisely when X/2 = 8 pi^2, that is X = 16 pi^2 (verified here symbolically). The equality holds BECAUSE 16 pi^2 was placed inside the definition of alpha_eff. Everything the “theorem” states is the single line 8 pi^2 = 16 pi^2 / 2: a definition was made, and then used to rewrite the expression. This is the same shape Paper 40 found inside itself (l_p/t_p = c holds because hbar G cancels), and an equation with no violable null hypothesis is not called a theorem (Papers 21, 16). The name is therefore retracted and rewritten as “a restatement of a definition”. The algebra is right; the classification changed.
FIFTH, WHETHER THE TWO 16 PI^2 SHARE A ROOT IS UNDECIDED. The second edition stated that “perturbation theory and non-perturbative effects are governed by the same geometric quantity 16 pi^2”. The loop-measure side gives 2 pi^2/(2 pi)^4 = 1/(8 pi^2) from the angular part Vol(S^3_1) = 2 pi^2 of the integral d^4k/(2 pi)^4; the topological side gives S = 8 pi^2 |k|/g^2 from the normalisation integral Tr(F Ftilde) = 16 pi^2 k and the BPS bound. Both descend from four-dimensional integration – one from the measure on momentum space, the other from the second Chern class – so they are not unrelated. But saying they are “governed by the same geometric quantity” requires grounds beyond the appearance of 16 pi^2 in each. This series wrote in Paper 86 that a common ancestor is not a common root, and in Paper 100 that a resonance is a signature, not a proof. This edition therefore records the matter as UNDECIDED, calling it neither a common root nor a rhyme.
SIXTH, THE HOPF PICTURE DOES NO WORK IN THE INSTANTON COUNT. The second edition made one correct correction: instantons are classified by pi_3(SU(2)) = pi_3(S^3) = Z, not by the Hopf map’s pi_3(S^2). Both are Z, so the conclusion that the instanton number is an integer is unchanged; what changed is the reason. And once that correction is accepted, the Hopf decomposition is doing nothing for this paper’s subject – the origin of 16 pi^2 is the second Chern class, and it does not pass through the Hopf bundle. The factorisation of Section 2 is correct as geometry, but this paper does not need it. It is the same shape Paper 41 found inside itself: two correct results welded together by a title.
THE SECTION ON THE STRONG CP PROBLEM KEEPS THE SECOND EDITION’S DELIMITATION UNCHANGED. What observation constrains is theta_eff = theta_QCD + arg det M, not theta_QCD. In the Standard Model CP is in fact violated and the CKM matrix carries an O(1) complex phase, so arg det M is generically O(1); hence imposing theta_QCD = 0 still leaves theta_eff ~ O(1) much greater than 1e-10. THE REAL DIFFICULTY IS NOT “SETTING THETA TO ZERO” BUT EXPLAINING WHY TWO QUANTITIES OF INDEPENDENT ORIGIN CANCEL TO ONE PART IN 1e10. The initialisation-condition approach addresses the former and does not answer the latter. This delimitation is accurate, and the comparison with existing approaches (Peccei-Quinn; m_u = 0, excluded by lattice QCD; Nelson-Barr) is also unchanged. Only one thing changes: the cosmic-OS lens the second edition placed in the body is removed as Layer 3. The content of the delimitation never depended on that lens.
IMPORTANT (scope): a structural interpretation and self-audit of established standard facts (Layer 2). The mathematics and physics used are entirely standard (Chern-Weil theory, the BPST instanton 1975, the Hopf bundle, pi_3(S^3) = Z, the nEDM measurement, lattice QCD giving m_u nonzero). NO NEW THEOREM AND NO PREDICTION. What is not claimed: an area and a solid angle are not treated as the same quantity (different dimensions; Papers 16, 116) / R = 2 is not called a geometric necessity (a choice of unit; Paper 95) / a restatement of a definition is not called a theorem (Papers 40, 21, 16) / the two 16 pi^2 are not declared to share a root (undecided; Papers 86, 100) / the Hopf bundle is not said to classify the instanton number (it is pi_3(SU(2))) / no claim is made to have solved the strong CP problem (arg det M is unsolved) / no alternative to the axion or to Nelson-Barr models is proposed / the cosmic-OS lens 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). An Area Is Not a Solid Angle — The Hopf Factorisation Holds at Every Radius, and $R=2$ Is a Choice of Unit — [Paper 45] 3rd ed. [Paper 45]. Zenodo.
DOI: 10.5281/zenodo.21602337
ORCID: 0009-0006-8587-3450