The first hypothesis is the one part I didn't prove myself but licensed from the literature: Platt and Trudgian's record verification (2020) that every nontrivial zeta zero up to height
lies exactly on the critical line. The criterion at our row consumes height , which sits inside the verified range with an exact margin of . The zero-height endpoint is closed classically: on the real segment a two-line alternating-series argument shows (strictly negative, so no zero hides at the bottom), and at zeta has a pole, not a zero. Nothing conjectural enters: RH is used only as a finite, machine-verified fact up to a specific height, plus the functional equation to map -zeros to zeta zeros by the exact change of variables .
A detail the audit surfaced: the margin lives entirely inside Platt–Trudgian's precise Theorem-1 height 3,000,175,332,800. With the rounded “3×10¹²” of their abstract, hypothesis (i) would fail by 92,913.5. The height the barrier consumes, X/2, sits less than one part in 10⁴ below the exact record; it is the single thinnest external dependency in the whole chain, and it is cited to the digit.
Analogy. The verified height is a foundation poured by others — a twelve-trillion-zero computation, published and checkable — that this proof builds on rather than repeats. The house goes up at the property line: a margin of 175 million against a depth of three trillion puts the fence 0.006% from the edge. Chapter 11 explains why building there is the right choice: at this height, that is where the value lies.
Check (i) secured.