The Fourth Horn: Local Pais-Uhlenbeck Modified Inertia Is Excluded by an Exceptional-Point Cap and a Frequency-Selection No-Go

Closes the last open route to a covariant modified-inertia MOND: a LOCAL, PREFERRED-FRAME, PT-QUANTIZED higher-derivative worldline theory (the one combination not covered by the banked no-gos, since a preferred frame voids the Galilei premise of Milgrom's 1994 nonlocality theorem and Bender-Mannheim PT quantization dissolves the free Pais-Uhlenbeck ghost). Adversarially verified, MOND-generic results: (G0) no local worldline Lagrangian has F = m mu(|a|/a0) a as its exact Euler-Lagrange equation -- acceleration dependence forces a genuine 4th-order PU-class theory (a purely variational theorem replacing the voided Galilei obstruction). (EXACT IDENTITY) for the general PU-class action, circular orbits obey a[1-(Omega/w_eff)^2] = g_N with w_eff^2 = w^2/k' -- the effective inertia is a FREQUENCY law, never an acceleration law: at fixed g_bar a single 200->300 km/s comparison shifts the prediction by 0.176 dex (more than the entire 0.108-dex observed RAR scatter) and dwarf parameters admit NO circular solution; a classical identity, independent of quantization. (SIGN CONVERSE) the MOND (softening) direction obtains IFF the acceleration-squared term carries the GHOST sign -- in local higher-derivative inertia the MOND direction and the ghost are inseparable, the local counterpart of the nonlocal anti-MOND sign theorem. (EXCEPTIONAL-POINT CAP) PT quantization cures the ghost only in the unbroken phase, and unbroken PT <=> mu_eff >= 1/2 (harmonic-exact; flat-curve anchoring plausibly mu >= 0.854 pending a named Coriolis-coupled linearization) -- deeper softening fails by ORBIT NON-EXISTENCE at a Jordan-block exceptional point; deep MOND is unreachable inside the healthy phase under any anchoring. (SCALES) no constant PU scale works: hidden modes are galaxy-inert; the MOND-tuned scale survives Solar-System bounds via its built-in external-field shielding but is excluded by wide binaries at the ~1e5 level. The trichotomy of 'Scale Without Law' (10.5281/zenodo.21016309) thus becomes a QUADRICHOTOMY, and all four horns converge on one requirement: an acceleration-selected mu needs a nonlocal-in-time response kernel that knows the trajectory's frequency content, whose only sign-viable source is the pumped anharmonic medium specified by the state clause (10.5281/zenodo.21139029) -- a single, sharply specified, presently unoccupied completion, reached from four independent directions. Constructive residue: Lagrangian PU realizations remain viable only as MOND-inert UV spectators. One corrected constant (the cap is 1/2, not 3/4) and a re-graded flat-curve statement are reported transparently; gauntlet + 22-check verifier scripts committed. Results are MOND-generic; the framework enters only via the normalization a0 = c^2 sqrt(Lambda/32pi) and its preferred frame; the author's earlier TOE claims were publicly retracted (2026-06-23) and this note claims a closure, not a completion.