The oscillating circle as a Polyakov stringeEmergence of the Fano plane and the algebraic foundation of the Alpha (NO FREE PARAMETER)

The oscillating circle of radius \ (R = \) is shown to be dual to a Polyakov string compactified on a circle, with tension \ (T = 8\). Its Lagrangian₆₄₎ = 4ḋ² + 1Rd² + 14R² - d² the on-shell action \ (4³ + ² + \). The quantized system is described by a two-qubit state \ (| () = |00 + |11\) ; for \ (= /6\) the CHSH correlation gives \ (S = 7\), while the Tsirelson bound is \ (22\). The difference operator = 22\, (ᵦ I₂) - 7\, (I₂ᵦ) on \ (C²²\) and has characteristic polynomial\X (x) = x⁴ - 30x² + 1, root \ (S = 22 - 7\) is the entanglement deficit. This deficit governs the renormalization flow and the graphon parameters\ = 2425ₓ₇ S, = ⁴+12425ₓ₇ S, \ (ₓ₇ = 12 (8/ (8-7) ) \). The kernel (x, y) = \, (⁴+1) \, e^-\, d (x, y) \ (S³\) emerges as the continuum limit of a graph sequence converging in cut norm. The spectral ratio satisfies \ (₁/₀ 2\) as \ (R \), reflecting the isotropy of the emergent FLRW spacetime. The construction contains no free parameters. The numbers \ (24\), \ (45\), \ (4/3\), \ (8\), \ (10\) arise from the transverse modes of the string, the bond dimension of the MPS, the hinge factor \ (Iₕ\), the string tension, and \ ( (4, 1) \), respectively — all encoded in the Minkowski period coefficients of the Fano 2-22.