IMM Paper 24: The Visibility Budget — Certified Detection Depths and Instrument Economics for Hypothetical Off-Critical-Line Zeros

This paper does not prove the Riemann Hypothesis (RH) and does not claim progress toward proving it. Its contribution is instrumentation: for a hypothetical zero at s= 12+ +i _0 ( >0 ), we derive and measure the depth at which each of six classical RH-equivalent instruments --- the Mertens contrast, the Nyman--Beurling (NB) distance, the de Bruijn--Newman flow, the truncated-Weil Toeplitz family A(m) , the Li coefficients _n , and the Hankel reciprocal-moment family H_k --- would make the violation visible, against the baseline of direct critical-line scanning. Two empirical anchors are new and registered. First, a pre-registered, fully certified-arithmetic comparison (E-LI-2: exact rational plants, interval arithmetic end to end, verdicts only on strict interval separation, no stochastic components) races the Hankel family of [TSF, Theorem 4.1] against the Li sequence on twelve planted configurations: the registered verdict is hankel wins (cap-driven; two clean cells at 634 and 1079 ), robust under all three registered depth-matching conventions, with every decisive value independently replicated by a second implementation on a disjoint code path. Second, the measured Hankel row obeys a zero-count law k^* 1.83\, ( _0)+5.4 , whose frozen-ex-ante prediction test (E-HK-1) confirmed it on fresh heights across its fitted range (7/8 cells within 2 ) and measured a +4.5 bend at the extrapolation column, nearly independent of the offset --- exposing a third data channel in the instrument taxonomy: germ-side instruments, which read the zero multiset entirely through derivatives of at one point and pay in zero count rather than in offset resolution or height. Analytic budget laws, a floor law for the NB distance validated at toy saturation, kill-records for three generations of candidate positivity-propagation laws, and a worked-numbers table at the verified-height floor complete the atlas. Every claim carries an explicit evidence grade; every number is reproducible from committed, hash-manifested scripts.