Paper 32 — The Fano Bifrontality: Gravity and Matter at the Cayley–Dickson Boundary

The Fano plane PG(2,F₂) appears twice in the Cayley–Dickson algebra tower with opposite operational roles. In the octonions A₃ — the last normed division algebra — its seven lines encode the multiplication rules among imaginary units. In the sedenions A₄ — the first non-alternative algebra — the same lines encode zero-divisor annihilation rules, while non-incidence identifies the kernel of left multiplication (ker(L_a) ≅ V_ni(a), Paper 29). We call this the Fano bifrontality. From this single structural observation, seven interlocked results across Papers 1–31 assemble into a coherent framework: the quadruple algebraic protection of the 14 massless Level-2 modes (mass zero, gravitational decoupling, gauge invisibility, Majorana statistics — all from a single mechanism); their Fano partition into a bosonic on-line sector irr₆ and a spinorial off-line sector irr₈; the S₄-singlet irr₁ as the unique CCC-crossing candidate; the factored structure Level-3 = 3×Level-2; the partition of Black Circle thermodynamic observables into three algebraic classes; the explicit holographic formula S_BH = (π/4)·‖L_{C₀}‖²_F·M²; and the interpretation of the Black Circle as the logical mirror of A₃. A new synthesis result is identified: the Majorana threefold isolation, unifying algebraic, gravitational, and entropic protections of the Majorana sector through a single Fano non-incidence structure.