The Yamagishi Born-Rule Geometric Structure Theory: the exponent 2 comes from two distinct roots, and here the 4π survives. Third edition.
Against the probability interpretation P=|ψ|² that Max Born laid down as an axiom in 1926, this paper argues from two routes why the exponent must be 2. First, the only power functional conserved under unitary evolution for every initial state is the integral of |ψ|² (Theorem 2.1, verified symbolically). Second, in three dimensions the condition that the probability flux through a sphere be independent of the radius, for the outgoing wave ψ=Ae^(ikr)/r, is |ψ|² proportional to r⁻² — the same algebraic condition as the n=2 uniqueness theorem of Paper 01 (Theorem 3.1). What the third edition adds is the address of these two 2s.
First, the two routes come from different roots. The 2 of Theorem 2.1 is fixed by dynamics (the packet spreads); the 2 of Theorem 3.1 is fixed by geometry (the area of a sphere goes as r²). The decisive difference is dimensional dependence. For a d-dimensional Gaussian packet, N_n is proportional to σ^(d(1−n/2)), and the exponent vanishes at n=2 for every d. The flux through a d-dimensional sphere, by contrast, goes as r^(d−1−n) and is constant for n=d−1 (Papers 01, 114). The two therefore agree only in three dimensions, where d−1=2. In four dimensions the dynamical route still says 2 while the geometric route says 3, and they part company (verified: of d=1,2,3,4,5,6 only d=3 agrees). The agreement is a fact, not an identity.
Second, the alternatives Theorem 3.1 excludes do not exist. Substituting ψ=Ae^(ikr)r^(−n/2), the state corresponding to |ψ|² proportional to r⁻ⁿ, into the free Schrödinger equation (radial Helmholtz reduction u=rψ, u”+k²u=0) gives u”+k²u = r^(−a−1)(a²−a−2ikr(a−1))e^(ikr), which vanishes for all r only when a=1, that is n=2 (verified symbolically). This is an important difference from Paper 01: there, rho proportional to r⁻ⁿ is a family in which every n is a realisable density distribution, and choosing n=2 is a choice among genuine alternatives. Here it is not — the Schrödinger equation has already forced n=2. The algebra is the same; what the algebra acts on is not. Paper 01 selects; this paper checks.
Third, the 4π does not vanish here. Paper 51 showed that in the shell theorem the 4π of the sphere’s area and the 4π of Gauss’s normalisation are the same solid angle Ω₂ appearing on both sides, and cancel completely (no π remains in the conclusion). In this paper’s Φ = 4π(ħk/m)|A|², the 4π appears only once, from the area of the sphere, and with nothing to cancel it against it survives. And the 4π of the Hawking temperature in Paper 52 was not a solid angle at all (twice the Euclidean period 2π). Across three consecutive papers the same symbol has met three different ends — it cancels, it is not a solid angle, it survives. Whether a 4π survives or vanishes is decided by how many times it occurs in the equation. The phrase that the solid angle 4π appears in both contexts therefore becomes checkable only once the address is written (Paper 17). What is common is integrating the Ω of the area element r²dΩ over the full solid angle, not the agreement of a number 4π.
Fourth, the force of Theorem 2.1 lies in the words for every initial state. In a stationary state (an energy eigenstate) |ψ(x,t)| is time-independent, so N_n is conserved for every n; looking at a single state, the exponent is not pinned down at all. What pins it down is the dynamics of spreading.
Fifth, the phrase in Section 5 about the two-dimensional projection operator P_2D of Paper 41 was corrected. The third edition of Paper 41 states explicitly that P_2D is not a projection operator — it fails idempotency, and a delta function is not even an element of the state space, so it is a notational convenience, neither a theorem nor a derivation. The correction exists so that a withdrawn construction is not called back through a citation. Section 5 (the universe-OS reading) was also quarantined explicitly as Layer 3.
The second edition’s corrections are unchanged. The first edition’s identification of the modulus of ψ with a length is dimensionally inconsistent (from the normalisation condition in three dimensions, |ψ|² has dimension L⁻³ and hence |ψ| has L^(−3/2), which cannot be a length) and stands withdrawn. What the first edition called the probability flux, the integral of |ψ|²r²dΩ, is not a flux either (its dimension is L⁻¹); the form using the probability current j=(ħ/m)Im(ψ*∇ψ) was substituted. That Theorem 2.1 is a uniqueness statement inside the family of power functionals, and is weaker than Gleason’s theorem because it assumes the density to be a power of |ψ|, was likewise stated in the second edition.
IMPORTANT (scope): a structural interpretation and self-audit of established standard facts (Layer 2). Every tool used is standard (conservation of the integral of |ψ|² under unitarity, the probability current, the outgoing spherical wave). NO NEW PHYSICAL LAW AND NO NEW THEOREM. This paper claims no complete derivation of the Born rule — as Gleason’s theorem (1957) shows, its mathematical necessity is already strongly constrained by the Hilbert-space structure, and nothing here goes beyond that. What is not claimed: that the probability density is a power of |ψ| (the assumption of Theorem 2.1) / why a conserved quantity should be read as probability (measurement theory) / a derivation of the Born rule (Theorem 3.1 is a consistency check that presupposes it) / that the agreement of the two routes is more than a fact about our dimension (they agree only at d=3) / that Paper 01 and this paper mean uniqueness in the same sense. What remains: on seeing the same number, ask first for its address.
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 Yamagishi Born-Rule Geometric Structure Theory [Paper 53]. Zenodo.
DOI: 10.5281/zenodo.21603193
ORCID: 0009-0006-8587-3450