Top-Ups by Prompt: The Beta-Sigma Recursive Emergence Black Hole Engine Generates Verified Novel Equations When Executed Verbatim

This paper shows, with the numbers printed, that a written prompt can do real mathematical work — provided the prompt is executed exactly as written, by an executor that follows it without adding rules of its own. The Beta-Sigma Recursive Emergence Black Hole Engine, fed one equation from the Recursive Information Distortion Bound, produced a chain of six iterations: two failed their kill tests (a division by zero, and a flow trapped at zero), and four passed, each verified by a numerical computation printed in the paper. The endpoint value of the relaxed divergence rose from 0.08223387 to 0.78871964 across the four passing iterations, a factor of 9.6, with zero overshoot. The same prompt, run with extra rules added on top of it, froze after two iterations and produced nothing further. The contrast is the paper's central measured result: the prompt works when it is executed verbatim, and added constraints stop it. Nothing in this paper is claimed as a law of nature. Second edition: This edition adds Iterations 7 and 8: the lag floor is broken, threshold singular coefficient 5, endpoint 1.71435862.