The Branch Beyond λ = 50.0: The Far-V₀ Branch Runs to λ = 60.0 on a Ceiling Stated from the Trend Alone, the Seventh Fit Gains 0.02 — and the Limiting Slope Is Computed: the New Gain Ratio, 0.50, Reopens the Limit, and the Limit Stands at 22.94, Inside the Old Bracket

The far-V0 branch of the exponential rolling field is followed past paper #622's scan end, and the release line's limiting slope is computed. Gated digit-for-digit on the whole body of paper #622's notebook (Gates A-K and both of its scans) and on paper #622's printed exhibits (Gate L: twenty members, census, sixth fit 22.90 lambda - 48.87, gains 1.18, 0.24, 0.09, 0.05, 0.04, gain ratios 0.20, 0.38, 0.56, 0.80, extrapolations 23.0600, 23.0600, 22.9307, falsified bound 22.9509), the paper states a seventh scan before its results by the parent's rule -- lambda from 50.5 to 60.0 in steps of 0.5, the widened search carried to log V0 = 1425, the ceiling set as the sixth fit's end prediction at lambda = 60 (1325.13) plus a margin of about one hundred, on the trend's arithmetic alone. The branch is present at all twenty sampled lambda values (log V0 1107.538 rising to 1325.319, w0 +0.979 to +0.989, H0 75.73 to 76.14) 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 (117 sampled lambda across seven scans). The seventh release-line fit is 22.92 lambda - 50.16, completing the gains sequence 1.18, 0.24, 0.09, 0.05, 0.04, 0.02 with half-range fits 22.92 and 22.93. The seventh gain ratio is 0.50, below the 0.56 threshold paper #622 printed in advance, so under that paper's stated rule the limit reopens on stated terms and is computed by its three stated extrapolations: Aitken 22.9400, geometric tail at the last ratio 22.9400, geometric tail at the all-gains fitted ratio 22.9378 -- a spread of 0.0022, every estimate inside paper #622's bracket 22.9307 to 23.0600, which survives whole; the largest-ratio bound tightens to limit <= 23.0000 and the earlier <= 23.0600 survives as a bound; the falsified bound 22.9509 stands falsified and does not revive. The limiting slope stands computed at 22.94 per unit lambda, under the printed caveat that a future ratio at or above 0.80 would reopen the obstruction. Grade: EXTENSION. Derivation-independence count: 0. Quantum gravity NOT SOLVED; vacuum catastrophe NOT SOLVED.