Hopfion XII - The Profile Normalisation Conjecture: Proof via CMB Energy Identity, Icosahedral Newton Series, and Two-Route Equivalence

The Profile Normalisation Conjecture: Proof via CMB Energy Identity, Icosahedral Newton Series, and Two-Route Equivalence We prove the profile normalisation conjecture φ9√(J2aJ4) = 10 of the density-feedback Faddeev–Niemi Hopfion series, closing the last open step in the derivation of the electron mass from the CMB temperature alone. The proof is a pure algebraic equivalence requiring no numerical input: the conjecture is shown to be equivalent to the thermodynamic equilibrium condition Efb / Vtorus = ρCMB, both conditions reducing to the same geometric constraint R0C*2 = 5 / (φ23π2) on the condensate torus. Five results are established. Theorem A (CMB energy identity): The condensate scale Λcond = TCMB(π2/150)1/4 satisfies ρCMB = 10Λcond4 exactly (150/15 = 10, a pure arithmetic identity), where 10 = ord(q2I) = 2(k+2)|k=3 is the icosahedral group-order invariant. The condensate scale is the unique energy scale at which ρCMB/Λcond4 is an integer, and that integer is the topological invariant of the binary icosahedral group. Theorem B (Pentagon identities): φ = 2cos(π/5) and with μ* = 3 − φ: sin(π/5) = √μ*/2, sin(2π/5) = φ√μ*/2. These are the trigonometric expressions of the Pentagon Theorem (k+2 = 5), and they drive the Newton series of Theorem C. Theorem C (Newton–icosahedral series): The virial fixed point b* satisfying (15/8) arctan(√(8b*)) / √(8b*) = φ has the quadratically-convergent icosahedral Newton series q* = π/5 + d1 + d2 + … with b* = tan2(q*)/8, first correction d1 = (φ√μ*/12) · (8√μ* − 3π) / (5φ√μ*/4 − π) ≈ +4.19×10−3, and subsequent terms satisfying |dn+1| = O(|dn|2). The near-identity 8√μ* ≈ 3π (error 0.21%) is the same near-identity that underlies the accuracy of the four-term α−1 formula of Paper IV. Theorem D (Exact toroidal integral): For the Battye–Sutcliffe profile on a thick torus of aspect ratio x = R0/C*, the toroidal correction to J2a is Itor(x) = (2/15) 2F1(2, 2; 7/2; −x2), proved in closed form via the Euler hypergeometric integral; at equal radii x = 1 this yields the exact identity Itor(1) = π − 3, proved by elementary integration. Theorem E (Two-route equivalence — main result): For any toroidal geometry satisfying the virial self-consistency V(b*) = φ, the profile normalisation conjecture (algebraic) and the CMB thermodynamic equilibrium condition (physical) are equivalent, both asserting R0C*2 = 5/(φ23π2). Since the physical condensate satisfies equilibrium, the conjecture follows. The physical condensate torus is in the ultra-inverted regime: in natural units Λcond = 1, β* = 300/π2 ≈ 30.4, C* ≈ 21.3, and the exact aspect ratio x* = R0/C* ≈ 8.21×10−10 (tube radius ~1.22×109 times the major radius). This is derived in closed form by Corollary F from the Theorem E constraint and the pentagon identity tan(π/5) = √(3−φ)/φ: at leading order x*0 = π(3−φ)3/2/(9600√6 φ26) ≈ 7.996×10−10 (2.60% error), with the full series x* = π tan3(q*)/(9600√6 φ23) converging to 8.209×10−10. The exact value Itor(1) = π − 3 is a secondary result at the equal-radii geometry x = 1, not at the physical x*. With the conjecture proved, the one-parameter derivation chain is complete and unconditional: TCMB → ρCMB = 10 → φ9√(J2aJ4) = 10 → me = Λcond φ20 e4π−3/400−α/π. The electron mass is derivable from the CMB temperature alone, with no free parameters. This also lifts the spoke formula J4/J2a = 24/3/φ5 and the four-term fine structure constant formula α−1 = 137.035 998 486 from “exact given numerically-established λ = φ6” to unconditionally exact results.