The Infinity-Relocation Catalog: Where Every RH-Equivalent Parks Its Universal Quantifier, and Why the Known Infinity-Killing Moves Fail There (IMM Paper 22)

This paper does not prove the Riemann Hypothesis, and none of its contents constitute progress toward a proof. Its subject is the wall itself: RH is a ^0_1 arithmetic statement --- a single universal quantifier over the natural numbers --- and therefore its truth can never be established by finite enumeration, while its falsity could be. Every known reformulation of RH (zero verification, Toeplitz truncation positivity, Li coefficients, Weil positivity, Hankel moment positivity, Jensen-polynomial hyperbolicity, Nyman--Beurling density, de Branges canonical systems, OPUC/Verblunsky contractivity, Farey discrepancy, horocycle equidistribution rates, elementary inequalities) relocates this quantifier without eliminating it. We catalog where each equivalent parks the infinity, and we test each parking spot against five argument-shapes known to mathematics for proving universally-quantified positivity all at once: (a) Gram/square representation, (b) recursion or total-positivity propagation, (c) effective finite-to-infinite transfer, (d) manifestly positive integral representation, and (e) amplification via multiplicativity --- the tensor-power move that actually proved the function-field analogue (Weil, Deligne), whose number-field track record is precisely the classical zero-free-region program. At every catalog entry, each move either fails for an identifiable structural reason or reduces to a statement equivalent to RH itself --- the circularity pattern formalized in the companion schema note. The catalog's value is cartographic: it says precisely what finite computation can mean (instrumentation and detection power, never evidence), closes several tempting corridors with proof-shaped reasons rather than taste-shaped ones, and locates the two corridors left open --- constructing a Gram pair (E, H) with independent meaning without encoding the answer: geometric at the Weil row (Connes; archimedean part done), spectral-theoretic at the canonical-system row (Suzuki; >1 done, with the extension rate-limited by classical zero-free regions). Every claim herein is labeled per Paper 0B discipline; the paper's own novel content is classified [Obs] (assembly and observation), not [T].