Galaxy Formation from the Confined M4 Sector: Schechter Decomposition, Riemann Zero Spacings, and a Cross-Identity for sin2 θ13

Abstract

The rendering algebra R12 = M3(C) ⊗ M4(C) admits four dynamical sectors classified by factor (M3 or M4) and confinement status. Papers XXXIV–L cover M3 confined (QCD), Paper LV covers M3 unconfined (star formation), and Papers XXXVI–XXXIX cover M4 unconfined (electroweak/PMNS). This paper completes the quadrant by identifying galaxy formation as
the confined M4 process.
Six theorems are proven at A-tier. 

(1) Cost Duality: confinement forces the rendering cost linear in mass (S ∝ M, exponential cutoff), while non-confinement forces it logarithmic (S ∝ ln M, power law).

(2) Mass Additivity: baryon number is the unique additive conserved quantity in mergers, selecting exp(−cM) and excluding exp(−cM2/3).

(3) Confinement Energy: after QCD colour freezing reduces the active sector to (1, 15), the confinement energy is Econf = C2(adj, su(k)) = k = 4.

(4) Schechter Decomposition: the galaxy stellar mass function factorises as Φ(M) ∝ M1−b0/Nc exp(−kM/M⋆), yielding Schechter slope α = 1 − b0/Nc = −4/3 (observed −1.2 ± 0.1, 1.3σ).

(5) Galaxy = M4 Confined: baryons are colour singlets (M3 frozen) and mergers are irreversible (Z > 0).

(6) Cross-Identity: sin2θ13 = (Nc!/(Nc + Tw)) exp(−k) = (6/5)e−4, linking M4 unconfined (Paper XXXIX) with M4 confined (this paper) to 0.2%. Additionally, the critical line Re(s) = 1/2 is identified as the continuous base space of M4, and the first three Riemann zeros obey γ LO n = (2n − 1)/[2n−1(π − 3)] with NLO corrections(−1) n+1ZRn derived from the AdB4
decomposition of su(k) into even (dim = b0) and odd (dim = N2 c − 1) subspaces (≤ 0.03%); the pattern terminates at n = 3 (Montgomery–Dyson universality for n ≥ 4). The characteristic galaxy halo mass M⋆ = 8.6 × 1011 M⊙ (0.07 dex from 1012) is derived from kBT⋆ = kEion = 54.4 eV (He II ionisation threshold) with mean molecular weight µ = k2/N3c = 16/27.
Two new results close the previously open coefficient:

(7) Entropy Screening: the confined running coupling c(M) = c0/ ln(M/Mref) is derived from scale entropy (RG mode counting), with the confined/unconfined asymmetry traced to w-protection.

(8) Landauer Pole: Mref = M⋆/2 follows from βc = ln 2 (Paper XLIX), fixing c0 = 2 ln 2 via Econf = k = 4. The NLO correction (1 − Z/N) brings the 10M⋆ prediction to 0.4% of the observed excess. A(8) + A90%(6) + Pred(0) = 14 results. Free parameters: 0.