The Pfaffian’s 2^n n! Is One Block: Where the Yamagishi Euler-Characteristic Unpacking Cut (4 pi)^n n!. Third edition.
This paper showed that the normalisation constant of the Chern-Gauss-Bonnet theorem in 2n dimensions is (4 pi)^n n!, and settled that the factor of 2 in 32 pi^2 = 2 x 16 pi^2 is the n! = 2! of the Pfaffian’s antisymmetrisation. THAT SETTLEMENT IS CORRECT AND IS NOT ALTERED – neither the first edition’s “double buffering” nor Paper 32’s “dimension of the SU(2) fundamental / double cover of S^2” is reinstated. What the third edition corrects is THE WAY (4 pi)^n n! WAS CUT.
FIRST, FOLLOWING THE DERIVATION FACTOR BY FACTOR PUTS THE SEAM ELSEWHERE. The Euler class is e(TM) = (1/(2 pi)^n) Pf(Omega), and the Pfaffian is Pf(Omega) = (1/(2^n n!)) eps^{a1…a2n} Omega_{a1a2} wedge … So the normalisation constant of chi(M) = integral of e(TM) is (2 pi)^n x (2^n n!), and 2^n n! IS ONE BLOCK – the Pfaffian’s antisymmetrisation factor. Verified here for n = 1, 2, 3, 4: (2 pi) x 2 = 4 pi; 4 pi^2 x 8 = 32 pi^2; 8 pi^3 x 48 = 384 pi^3; 16 pi^4 x 384 = 6144 pi^4, each equal to (4 pi)^n n!. THE NUMBER IS THE SAME; THERE ARE TWO WAYS OF CUTTING IT. The second edition tore the 2^n out of that block and glued it to (2 pi)^n to form (4 pi)^n.
SECOND, THIS CONTRADICTS THE SECOND EDITION’S OWN FINDING. It correctly identified “n! as the factor appearing in the definition of the Pfaffian, corresponding to the number of ways of ordering the pairs when 2n indices are split into n antisymmetric pairs”. BUT 2^n COMES FROM THE SAME PLACE: 1/(2^n n!) is a single denominator in the definition of the Pfaffian, and the whole of it is the antisymmetrisation count – n! for the ordering of the pairs, 2^n for the swap within each pair. ONE CANNOT ASSIGN n! TO COMBINATORICS WHILE HANDING 2^n TO GEOMETRY. The quantity (4 pi)^n = (2 pi)^n x 2^n is made by tearing the 2^n out of the Pfaffian block and gluing it to the normalisation. This is a shape the series itself has forbidden – Papers 13 and 14 laid down that a module block must not be chopped into “pi times a number”, and what happens here is the same operation in the other direction: borrowing a factor from one block to build a new one.
THIRD, THERE ARE TWO STANDINGS. (2 pi)^n is the normalisation that makes the periods of characteristic classes integral (the Chern-Weil convention), coming from the way one enters the curvature two-form as Omega/2 pi – not the geometry itself, but the scale on which the geometry is read as integers. 2^n n!, by contrast, is combinatorics. HENCE THE 4 PI APPEARING IN (4 pi)^n IS A CONVENTION’S 2 PI TIMES A COMBINATORIAL 2, AND IS NOT A SOLID ANGLE. This is the same shape Papers 45 and 46 found inside themselves – in 45 an AREA was called a solid angle, in 46 THE SQUARE ROOT OF A LOOP FACTOR; here it is the product of a convention and a combinatorial count.
FOURTH, ONE LINE OF SECTION 4 IS THEREFORE CORRECTED. The second edition separated the origins of the factor 2 in three topological invariants: the Euler characteristic (1/32 pi^2) from COMBINATORICS (Pfaffian antisymmetrisation), the instanton number (1/16 pi^2) from COMPONENT EXPANSION (writing out F wedge F; Paper 45), and the theta term (g^2/32 pi^2) from GENERATOR NORMALISATION (Tr(T^a T^b) = delta/2). THIS ANALYSIS IS CORRECT AND IS KEPT UNCHANGED – the discipline the series wrote in Paper 86 (“a common ancestor is not a common root”) and Paper 100 (“a resonance is a signature, not a proof”) is working correctly here. What is corrected is the closing line: the second edition wrote that “what they share is not the factor of 2 but the substrate (4 pi)^n”, but as shown above (4 pi)^n is a quantity regrouped across a convention and a combinatorial count and cannot serve as a substrate. WHAT THE THREE SHARE IS THE (2 PI) OF THE CHARACTERISTIC-CLASS NORMALISATION – the 1/(2 pi)^n of the Euler class, the normalisation of the second Chern class, the way the theta term is written, each carrying a 2 pi serving the same purpose of making the periods integral, with factors of different origins sitting on top. THIS IS A CORRECTION, NOT A RETRACTION: the second edition’s intuition that there is a shared substrate was right, and the name of that substrate lay one level deeper.
ALL NUMBERS ARE UNCHANGED – 4 pi in two dimensions, 32 pi^2 in four, 384 pi^3 in six. The two-dimensional consistency was re-verified: at n = 1 the theorem gives chi = (1/4 pi) integral R sqrt(g) d^2x, and from the classical Gauss-Bonnet integral K dA = 2 pi chi together with R = 2K one gets (1/4 pi) integral R dA = (1/4 pi) x 2 x 2 pi chi = chi.
THE SCOPE, RESTATED as eleven rows. (1) Normalisation = (2 pi)^n (2^n n!): a FACT (checked for n = 1 to 4). (2) 4 pi in 2D, 32 pi^2 in 4D, 384 pi^3 in 6D: FACTS (values unchanged). (3) Two-dimensional consistency with classical Gauss-Bonnet: a FACT. (4) That the factor 2 of 32 pi^2 is n!: SETTLED (second edition; unchanged). (5) That 2^n n! is one block: THE DEFINITION OF THE PFAFFIAN. (6) (2 pi)^n: a CONVENTION. (7) The grouping (4 pi)^n: RETRACTED. (8) That its 4 pi is a solid angle: RETRACTED. (9) That the three factors of 2 have different origins: SETTLED (unchanged). (10) That the shared substrate is (4 pi)^n: CORRECTED (it is (2 pi)). (11) Double buffering, SU(2) dimension, double cover: RETRACTED and not reinstated.
IMPORTANT (scope): a structural interpretation and self-audit of established standard facts (Layer 2). The mathematics used is entirely standard (Chern’s Gauss-Bonnet of 1944 and 1945, the Euler class and the Pfaffian, the normalisation of characteristic classes). NO NEW THEOREM AND NO PREDICTION. What is not claimed: 2^n is not separated from n! (the same Pfaffian denominator; the shape of the prohibition on splitting modules in Papers 13, 14) / the 4 pi of (4 pi)^n is not called a solid angle (Papers 45, 46) / the three origins of the factor 2 are not merged (the second edition’s judgement is kept; Papers 86, 100) / the first edition’s double buffering and Paper 32’s SU(2) dimension / double cover are not reinstated / no new theorem about the Euler characteristic is asserted / nothing is read into the Gauss-Bonnet term beyond the standard fact that it does not contribute to the field equations in four dimensions / 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). The Pfaffian’s $2^n n!$ Is One Block — 3rd ed.: Where the Yamagishi Euler-Characteristic Unpacking Cut $(4\pi)^n n!$ — [Paper 48]. Zenodo.
DOI: 10.5281/zenodo.21602525
ORCID: 0009-0006-8587-3450