The Scalar (Galileon) Sector of the Unwinding Spiral: Consolidated Derivations

This note consolidates the derivations for the scalar (Galileon) sector of the unwinding spiral framework (GIT). It is an autonomous companion to the framework's main paper (DOI: 10.5281/zenodo.20709541), following from it and opening the door to the geometric branch (TAG). The scalar sector is treated as a single structure: the 2D object (modulus and phase of the spiral) and its reading in 3D/4D after the transposition. Entropy, the Born rule, the Bekenstein coefficient, the de Sitter scale and the Galilean symmetry itself are shown to be 3D names for what, in 2D, are the spiral and its generator. Consolidated results include: the Vainshtein radius r_V ≈ 0.12 kpc with the cubic form uniquely yielding the kpc scale; Cassini cleared with five orders of margin; the identification of the Galileon as the unique Wick–Born operation of the transposition, with uniqueness closed by dimensional counting of the chain (exactly two projection jumps); and the coefficient 1/4 derived as the angular complement of a_Pl in the C₃ partition of the tetrahedron's Gauss–Bonnet defect (3/4 radial | 1/4 angular). On the cosmological tensions the note is deliberately conservative: the H₀/S₈ tension is not resolved by this sector. The Galileon is structurally barred from raising H₀ (the same symmetry that shields Cassini forces a weak cosmological effect); the Weyl route to S₈ is falsified (the mode decouples, effect ~10⁻¹²⁰); and a candidate single-variance reading x(1−x)/3 ≈ 6.7% is reported as an open frontier, not a result, since the divisor was selected a posteriori and is not yet derived independently of the observed value.