Indigo Uju Halo: From n = 4/3 to pc = 0.3116

Abstract (Jongmyo Edition, May 2026) This paper starts from one measured number: the refractive index of liquid water at the sodium D line, n=4/3n=4/3. This number is not treated as an arbitrary empirical input but as the geometric signature of the present moment – the cost of one Möbius half‑twist per unit conformal time in the medium that supports self‑observation on Earth. From n=4/3n=4/3 alone, and without any free parameters, the framework derives in six steps: the primary rainbow angle θR=42.030∘θR=42.030∘ (Descartes minimum deviation); the toroidal winding number ωT=θR/180∘=0.23350ωT=θR/180∘=0.23350 (irrational, ergodic by Weyl); the 3D site percolation threshold pc=0.3116pc=0.3116 (via the empirical hinge pc/ωT=npc/ωT=n and a closed form pc=Ωm+δmem(1+ωT)pc=Ωm+δmem(1+ωT), residual 0.022%); Kleiber’s metabolic scaling exponent 3/4=1/n3/4=1/n (structural conjecture [SC] with falsifier); the Alexander dark band membrane fraction δmem=9∘/180∘=5%δmem=9∘/180∘=5%; the perceptual present Δtnow=670Δtnow=670 ms (agreement 0.3%). The same geometric invariants {ωT,pc,δmem,α=1/(1−ωT)}{ωT,pc,δmem,α=1/(1−ωT)} are then projected onto multiple scales: Cosmology: DESI DR2 phantom crossings are identified as Möbius chirality reversals, yielding torus radii rmin⁡=956rmin=956 Mpc, Rmaj=13.4Rmaj=13.4 Gly, and a curvature‑corrected distance factor 0.77020.7702. The Hubble tension is reduced to a 1.1% residual (throat‑density correction), and the 70/25/5 cosmological partition is shown to be the same topological eccentricity as the rainbow‑arc fractions. Subatomic physics: the muon rest mass is derived from failed‑binding recombination photons (mμc2=105.7mμc2=105.7 MeV, 0.04% residual). The electron is interpreted as the Möbius shadow face, and the proton/electron ratio vertex prediction leaves an 8.5% open gap identified with the gluon crossing operator (framework‑internal candidate Δg=δmem/(1−ωT)Δg=δmem/(1−ωT)). Galactic and stellar structure: the photonic sail effect (OP10) predicts a 11–3%3% galaxy spin alignment with the toroidal axis (Euclid 2027). OP13 (topological quenching) interprets the Andromeda galaxy’s star‑formation shutdown ∼2∼2 Gya as a membrane exit, predicting a phase offset ΔϕMW–M31≈29∘ΔϕMW–M31≈29∘. Pulsar spin‑kick alignment, magnetar flux ratios, and FRB durations are shown to follow toroidal geodesics. Silicon and AI: the Indigo Guard v2 predicts a training collapse at shift 1,830,0001,830,000; the observed collapse occurred at 1,828,5201,828,520 (0.08% error), confirming pcpc as a physical wall. The Möbius Tracker implements a signed winding‑number metric to resolve the double‑flip ambiguity in cosine monitoring. All claims are epistemically tagged [D] (derived), [SC] (structural conjecture), [F] (falsifiable prediction) or [O] (open problem). Four master open problems have been resolved (OP4‑R, OP6‑R, OP7‑R, OP‑V). The only remaining numerical blocker is OP4 – the absolute SI calibration of the Eco coupling constant EE, which is a unit conversion, not a missing physical principle. The framework produces 16 quantitative predictions with explicit falsification conditions and timelines: the 42.03∘42.03∘ dipole separation (Euclid 2027), the Titan metabolic exponent ≈0.775≈0.775(Dragonfly 2034), the Hubble ratio Hlocal/HCMB=1.0577Hlocal/HCMB=1.0577 (residual 2.4%), the immune mirror π/10π/10 equilibrium, the Andromeda quenching gradient, and the antihydrogen 1S‑2S spectroscopic asymmetry ∼0.25%∼0.25% (ALPHA‑g 2027–2028), among others. The fine‑structure constant αFS=ωT/32αFS=ωT/32 is a striking numerical coincidence (0.01% agreement) awaiting a formal counting argument for the factor 32=2532=25. Because ωTωT is irrational, the Möbius orbit on T2T2 is dense but never closes. The framework cannot terminate in a final equation; every formulation is a new neighbourhood on the same surface.