The Barbero-Immirzi Field as a Hyperbolic Axion-Dilaton: a Fixed Kinetic Mixing λ = −2√(2/3) and the Spiral Twist Hypothesis

Promoting the Barbero–Immirzi (BI) parameter to a field in N = 1 supergravity is equivalent to adding a dilaton–axion chiral superfield with a fixed Kähler potential (Gates, Ketov & Yunes 2009). We point out that this Kähler potential has metric K = 3/(Φ+Φ̄)², a hyperbolic field space of curvature −2/3 (the α = 1 attractor geometry), so the BI axion's kinetic term is multiplied by eλχ of the canonical dilaton with λ = −2√(2/3) ≈ −1.633, fixed, with no free parameter. This is the kinetic-mixing structure shown to produce an apparent phantom crossing in DESI-type analyses while obeying the null energy condition (Toomey et al. 2025, arXiv:2511.23463), whose fit gives λ = −1.3 (+1.3/−1.5). We are not aware of earlier work deriving this geometry, and hence λ, from the BI field. We report what survives four consistency checks: (i) with matter on the supergravity frame the dilaton has a GR point at φ = −1/2 that is a cosmological attractor, but Cassini pins it within 0.0072 MPl today, switching the mechanism off; scale (Weyl) invariance offers a known route to matter blindness; (ii) the simplest supergravity potential (constant superpotential) turns exactly at φ = −1/2 but at negative energy; (iii) the same α = 1 geometry as an inflaton gives ns = 0.967 at N = 60, 2.4σ below ACT DR6; (iv) the dilaton cannot be the early-dark-energy field. A first background calculation with the decay constant fixed by the Nieh–Yan period behaves as a thawing dilaton (w0 ≈ −0.84, wa ≈ −0.24); a literature cross-check of the illusion is not yet reproduced. We state the model ("Spiral Twist") and the tests that can falsify it. Status: the algebra is exact and verified symbolically; everything after it is a hypothesis, with failed and tension checks reported. Scripts for every number: https://github.com/dannyeldridge54/the-twist-torsion-cosmology. Companion paper: doi:10.5281/zenodo.23867116.