Closed Planar Potential Ends the Flat Rotation Curve Problem: Hourglass Geometry, Vc²=4GM₀/R₀, V²=2GM₀/ΔR, and Blind Predictions for 8 JWST Barred Galaxies at z=1.1-3.16

Closed Planar Potential Method (CPPM): Formal Derivation of Flat Rotation Curves from a Closed Planar Potential and Hourglass Geometry + Extended Prediction Atlas to 1842 Galaxies ### ABSTRACT - Flat rotation curves are treated here as a geometric effect of a closed planar potential with differential levels, not as evidence for a dark halo or modified dynamics. . Formal Derivation (CPPM).The galactic disk potential is geometrically closed at the last isophote R0, defined at 26.5 mag/arcsec^2, with boundary condition Φ(R0)=0. The bar radius Rb acts as the neck of an hourglass potential with vertical depth b and slant length c = sqrt(R0^2 + b^2). Vc^2 = 4 G M0 / R0 (circular velocity at closure)V_obs^2 = 2 G M0 / ΔR, where ΔR = R0 - Rb and M0 = M_* + M_gas (observed plateau velocity) The factor 4 arises from integration over the closed surface: factor 2 from the Φ(R0)=0 closure, and factor 2 from two symmetric branches (upper/lower) of the hourglass. The factor 2 in V_obs is the projection of that full potential onto a single branch ΔR. Thus V_obs depends only on global quantities M0 and ΔR, which are constant for a given galaxy, explaining flatness without dependence on local radius r. The spiral arms are not circular orbits at r, but the projected fall along c. 2. Extended Atlas Catalog to 1842 Galaxies (Without V_obs, Only V_pred). Pure a priori prediction catalog based on S4G IRSA (Sheth et al. 2010). Selection: M0 [Msun] at 3.6μm, R0 = R_25.5 [kpc] >0.3 kpc, Q=1-2, N=2352 -> 1842 after null cleaning. V_pred = sqrt(4*G*M0/R0) calculated with G=6.67430e-11, Msun=1.98847e30 kg, 1kpc=3.085677581e19 m. No free parameters, no V_obs, no halo, no inclination or bulge correction. Columns: name, M_in, R_in, V_pred, desmos verification formula. External validation on SPARC 175 galaxies with V_obs: using R_HI vs R_25.5, median k=V^2 R / GM = 4.80, mean 5.20, P10 2.57 P90 8.06. For 133 galaxies with V_flat: mean error -4.28%, |err| 20.2%, RMSE 38.7 km/s. Overlap S4G x SPARC N=59: corr V_pred(S4G) vs V_obs(SPARC) = 0.9456. Correlation logM0-logV_pred for 1842: 0.973. V_max = 384 km/s, V_mean = 125 km/s, R_max = 48.2 kpc, M range 7.7e6 - 2.8e11 Msun. The 400-1000 km/s range remains empty: flat plateau up to 50 kpc without halo. The model maps Hubble morphology to physics via Eq. 3: M0/R0 = Vc^2/(4G). Sa/SBa with large L_bar ~0.8 R0 have larger R0 at fixed Vc and higher M/R. Sb/Sc with small bar have smaller R0. Critical threshold M/R = 1e10 Msun/kpc separates E/S0 from spirals. At Vc >400 km/s, only E/S0 survive. MW+M31 merger (15 kpc, 439.3 km/s) gives M/R = 1.12e10, moving to E/S0 region (handle of tuning fork). **Falsifiability - 3 independent tests:** (i) Rigorous blind test on 8 JWST barred galaxies at high-z selected a priori from literature where bars are confirmed: CEERS-2112 (z~3, Costantin et al. 2023, oldest bar), EGS-30836, EGS-24154, EGS-12823, EGS-26831, EGS-23205, EGS-24268, COSMOS-74706 (Guo et al. 2023). Protocol: External team provides only ID, M_*, M_gas, M0, Rbar, R0 at 26.5. Author predicts ΔR=R0-Rbar, V_pred=sqrt(2GM0/ΔR), with 7.8% MAPE and ±12% P90 from Atlas 193, and M/R=V_pred^2/(4G). Success criterion: ≥70% within 10% and ≥90% within 15%, same as Atlas 193. All predictions timestamped on Zenodo before V_obs reveal. If <70% within 15%, CPPM fails.(ii) R0 vs Vc diagram of 338 clean galaxies must separate Sa/SBa vs Sb/Sc by M/R.(iii) Measured b from hourglass geometry must satisfy c^2 = R0^2 + b^2 with observed pitch angle.(iv) For Extended Atlas: if future HI measurements (VLA/MeerKAT/SKA) show systematic V_obs >500 km/s or mean bias >30%, model is discarded. This deposit contains both the formal derivation (with Figure 1: lateral and top view of hourglass) and the full prediction catalog. Work in progress since 2012. CC-BY 4.0. **Keywords:** flat rotation curves, barred galaxies, JWST CEERS-2112, hourglass potential, closed planar potential, galaxy dynamics, falsifiable prediction, S4G, SPARC -