Paper 10 — Loop, Threshold and Residue: The Algebraic Structure of the A₃–A₄ Transition

Three interconnected consequences of the Cayley–Dickson algebraic threshold established in Papers 1–9 are developed.

Objective (i) (pre-dynamical structure of the 𝒜₃↔𝒜₄ loop): The feedback loop 𝒜₃→𝒜₄→𝒜₃ is formalised as a conditional discrete dynamical system. The algebraic skeleton — the tower map T: ϱ(n)↦ϱ(n+1) — is derived unconditionally. It has no fixed point; its unique asymptote is ϱ=0; its aeon-to-aeon rate is (2ⁿ−1)/[(2ⁿ⁻³−1)·16], converging to 1/2 from above. The cosmological 5%/95% ratio is shown to be not derivable — a precise negative result. A conditional conjecture characterises exactly what a future QFT over 𝒜₄ must supply to close this gap.

Objective (ii) (Black Circle metric ansatz): The Schwarzschild radius rₛ is identified as the 𝒜₃ projection of the 𝒜₃→𝒜₄ algebraic transition. A metric ansatz is constructed: exact Schwarzschild for r≥rₛ, de Sitter interior for r < rₛ with de Sitter radius L=rₛ. The Kretschmann scalar at the origin is K(0)=6ΔN/rₛ⁴=24/rₛ⁴, finite and algebraically fixed by the norm-gap ΔN=4. The Barrabès–Israel thin shell has σ=0, p=1/(64πG³M²), T_total=1/(4G) — verified by SymPy. The Vaidya dynamic extension gives σ=0 in all three regimes; algebraic discreteness forces thin null shell formation. Objective (iii) (sedeonic residue between aeons): The residual algebra level of aeon k is defined algebraically as n_res(k)=3+k. The ZD content satisfies |ZD_dark(k)|=|ZD(3+k)|, proved from the Fano filtration theorem. The sign-parity constraint πₐπ_Bπ_C=−1 is inherited by the residue across every CCC crossing. The first-aeon problem of Penrose CCC does not arise.

All three objectives are governed by the single map T: n↦n+1, and by the algebraic chain S₄⊂PSL(2,7)⊂SU(3)⊂G₂⊂SO(7)←Cliff(0,7), verified by GAP 4.15.1. A unifying result shows that N is the unique algebraic structure satisfying simultaneously: universal persistence through all doublings, gravitational coupling at every level, and coincidence with the Born Rule probability measure exactly at n=3. At the IIB threshold these two roles split: gravity persists into 𝒜₄; the Born Rule does not.