Star Formation, Jets, and Black Holes from the Unconfined M3 Sector of R12

Abstract

The rendering algebra R12 = M3(C) ⊗ M4(C) governs two distinct dynamical modes within its colour/spatial sector M3: a confined mode (QCD, Wilson area law, multiplicative Cauchy, yielding exp(−Nc)) and an unconfined mode (gravity, Wilson perimeter law, additive Cauchy, yielding 1/Nc). The confined mode has been fully explored in Papers XXXIV–L. This
paper presents the first systematic exploration of the unconfined M3 mode and derives star formation physics from it with zero free parameters.

Six independent theorems follow from the AC axioms.

Theorem 1 (Schur Projection): the mass-bound fraction εcore = 1/Nc = 1/3, with jets as a necessity for Nc > 1.

Theorem 2 (Confinement–Projection Duality): the same M3 algebra yields exp(−Nc) (confined) and 1/Nc (unconfined), unifying nucleon structure with star formation.

Theorem 3 (Irreversibility Triad): Z > 0 simultaneously guarantees consciousness, stars, and black hole area growth.

Theorem 4 (Salpeter Slope): the IMF power-law index α = b0/Nc = 7/3, derived from the modular Hamiltonian spectrum and KMS thermodynamics—the rendering cost E(8,15) = b0 = 7 divided by the spatial observation channel E(8,1) = Nc = 3.

Theorem 5 (Confinement Transition): the compactness C = 1/Nc marks the boundary between unconfined and confined gravitational dynamics, unifying ISCO, BP jet launching, and star formation efficiency.

Theorem 6 (Multi-Generational Schur Iteration): cumulative efficiency ε(n) = (1 − qn)/(Nc(1 − q)) with q = (Nc − 1 + fr)/Nc, explaining how Landauer waste-heat recycling allows ε > 1/Nc across stellar generations.

The JWST early massive galaxy puzzle is resolved: MIRI-corrected efficiencies ε ∼ 0.30 are consistent with εcore = 1/3. The stellar lifecycle is recast as an M3 phase transition from perimeter law (molecular cloud) to area law (black hole).

A by-product: the powerlaw form of the IMF constitutes observational evidence for the AC rendering hypothesis, since specification-information cost (S ∝ ln M) produces power laws while thermodynamic assembly cost (S ∝ M) would produce exponential cutoffs. Thirteen results at A-tier, nine at A90%, two observations, two predictions; zero free parameters