Two Staircases: Separating Value from Generator – The Sphere-Volume Recursion, Its Parity Split, and Where This Series’ Constants Live.
This series rests on 4pi (Papers 01, 02) and obtains 2pi^2 in four dimensions (Paper 08). Paper 109 recorded that the THREAD along which the power of pi rises with dimension is the sphere-volume recursion Vol(S^n) = (2pi/(n-1)) Vol(S^(n-2)). This paper takes ONE FURTHER STEP, as a LAYER-2 SELF-AUDIT CENSUS. FIRST: a VALUE and a GENERATOR answer DIFFERENT QUESTIONS. 4pi = Vol(S^2) is a value answering “how much on this floor”; the recursion is a generator answering “how one moves from floor to floor”. Different questions, therefore DISTINCT ROOTS. 4pi is the floor of the third storey, NOT the staircase. (This does not demote 4pi: the total solid angle as Gauss flux, the Euler-characteristic door 4pi = 2pi chi(S^2), and the constant of the n=2 uniqueness theorem all stand.) SECOND: because the recursion STEPS BY TWO, parity is preserved and the staircase SPLITS INTO TWO CHAINS THAT NEVER MIX – odd, Vol(S^(2k+1)) = 2 pi^(k+1)/k! ; even, Vol(S^(2k)) = 2^(k+1) pi^k / (2k-1)!! . Hence 4pi (S^2) lives on the EVEN staircase and 2pi^2 (S^3) on the ODD one: adjacent as dimensions, but NOT adjacent on the recursion (after 4pi comes 8pi^2/3; before 2pi^2 comes 2pi). The move from Paper 01 to Paper 08 is therefore not “one step up” but “across to the corresponding height on the other chain”. THIRD: the root of the split is the GAMMA FUNCTION, not geometry. Gamma gives factorials (rational) at integers and rational x sqrt(pi) at half-integers, so only the even spheres pick up a double factorial – an ANALYTIC root, not a topological one (86). FOURTH: a SEPARATE geometry does bear on parity – chi(S^even) = 2, chi(S^odd) = 0, and the hairy-ball theorem (a nowhere-vanishing tangent field exists only on odd spheres). But this is a DIFFERENT ROOT: the volume staircase and chi share only an INDEX and must not be joined causally. THE BRAKE (the most careful point): PARITY and 1,2,4,8 ARE DIFFERENT SELECTION PRINCIPLES AND MUST NOT BE MERGED. The odd staircase carries S^1 = U(1) and S^3 = SU(2), tempting the generalisation “odd spheres are groups” – which is FALSE. S^5 is odd yet neither a Lie group nor parallelisable. Only S^0, S^1, S^3 are Lie groups; only S^0, S^1, S^3, S^7 are parallelisable – consequences of the normed division algebras R, C, H, O having dimensions 1, 2, 4, 8 (Hurwitz; Bott-Milnor; Kervaire; Adams; Paper 114), unrelated to parity. One counterexample suffices. CONSEQUENCE: S^3 IS SELECTED TWICE, by parity (chi = 0) and by 1,2,4,8 (R^4 = H; SU(2) = Spin(3); the Hopf fibration) – but this is a DESCRIPTION, a SIGNATURE, NOT a proof that S^3 is necessary (100), and the two principles are not merged. SELF-AUDIT: the series’ two clusters were separated by parity – Papers 02, 13, 67 gather on the EVEN staircase around 4pi (topological root: chi = 2, hairy ball, Gauss flux), while Papers 08, 68, 104 gather on the ODD one around S^3 – yet the power of that odd cluster (groups, quaternions, Hopf) comes NOT from parity but from 1,2,4,8. PARITY FIXES THE ADDRESS; A DIFFERENT PRINCIPLE FIXES THE PROPERTIES. IMPORTANT (scope): a Layer-2 structural interpretation of entirely standard facts; no new theorem, no new physics, no prediction; the recursion itself was already recorded in Paper 109 and no discovery of it is claimed. NO MEANING IS READ INTO INDIVIDUAL NUMBERS such as 8pi^2/3 or 16pi^3/15 (94, 101); the volume staircase and chi are not joined causally; the double selection of S^3 is not taken as necessity (100). Layer-3 lenses are quarantined with a non-scientific label.
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). Two Staircases — Separating Value from Generator: The Sphere-Volume Recursion, Its Parity Split, and Where This Series’ Constants Live — [Paper 115]. Zenodo.
DOI: 10.5281/zenodo.21967686
ORCID: 0009-0006-8587-3450