The Branch Beyond λ = 40.0: The Far-V₀ Branch Runs to λ = 50.0 on a Ceiling Stated from the Trend Alone, the Sixth Fit Gains 0.04 — and the Limiting Slope Stands Uncomputed for a Computed Reason

The far-V0 branch of the exponential rolling field is followed past paper #611's scan end. Gated digit-for-digit on the whole body of paper #611's notebook (Gates A-I and its scan), on paper #610's headroom exhibits (Gate J), and on paper #611's scan table (Gate K), the paper states a sixth scan before its results on the trend's arithmetic alone -- lambda from 40.5 to 50.0 in steps of 0.5, the widened search carried to log V0 = 1200, the ceiling set as the fifth fit's end prediction (log V0 = 1095.76 at lambda = 50) plus a margin of about one hundred, because paper #610's exercised headroom (conditioned members at 867, single trials to log V0 = 100,000) leaves no machinery ground for a ceiling. The branch is present at all twenty sampled lambda values (log V0 878.548 rising to 1096.081, w0 +0.956 to +0.978, H0 75.12 to 75.70) and does not terminate inside the scan; the census of upward crossings at step 2 in log V0 finds exactly one crossing at every lambda. The sixth release-line fit is 22.90 lambda - 48.87, completing the gains sequence 1.18, 0.24, 0.09, 0.05, 0.04 across six fits, with half-range fits 22.89 and 22.91. The limiting slope is treated by three stated extrapolations of the gains' decay (Aitken 23.0600; geometric tail at the last ratio 23.0600; geometric tail at the all-gains fitted ratio 22.9307) and is not printed as a limit: the gain ratios are rising toward one (0.20, 0.38, 0.56, 0.80), the estimates spread by 0.13, and the bound the first three ratios would have set (22.9509) is falsified by the fourth ratio, so the limit stands uncomputed for a computed reason, with the bracket 22.93 to 23.06 printed in its place. Grade: EXTENSION. Derivation-independence count: 0. Quantum gravity NOT SOLVED; vacuum catastrophe NOT SOLVED.