The Kernel That Builds Its Own Laser: Five Theorems on Covariant Modified Inertia, from Specification to Closure

Completes a five-theorem chain on whether MOND can be a covariant modification of INERTIA (open since Milgrom 1994), taking the question from prohibition to specification to constructive closure -- adversarially verified, all scripts committed (exit 0). Theorems I-IV (published: 10.5281/zenodo.21016309, 21139029, 21148494) close the local and passive routes and FORCE the unique survivor: a causal nonlocal-in-time kernel fed by an INVERTED (non-KMS) anharmonic medium with net inverted spectral weight strictly above the galactic band, integrated weight tuned to cancel inertia to <=1%, saturating on |a|. WE THEN BUILT IT: m_eff = m[1 - S(|a|/a0)R(omega)], one inverted line at nu_2 in [1.1e6, 9.9e7] H0 -- collapsing exactly (1e-12) to g_obs = sqrt(g_bar^2 + g_bar a0) on circular orbits -- and it PASSED every previously fatal gate: causality (KK with sign-indefinite weight; laser-class), stability (the deep-MOND |dm|->m limit IS the marginal-stability edge and the saturation bound S<=1 IS the stability clamp), Solar System and pulsars (5-29 orders safe; the rolloff even suppresses the Cassini-quadrupole inheritance), ENERGETICS (pump demand 25-470x BELOW the Lambda-scale flux -- the anticipated kill does not exist), and universality (1e-8 dex). THEOREM V then closes it: by the same Kramers-Kronig lock that lets the line cancel inertia, the inverted line is a SUPER-THRESHOLD AMPLIFIER at its own frequency (irreducible per-radian gain >= 0.94 S after maximal inhomogeneous dilution), and the cold-ISM acoustic continuum (0.001-0.09 pc, fluid regime) spans EVERY admissible placement: in steady state either the inversion CLAMPS -- capping the delivered deep-MOND boost at ~1.4 vs the observed ~10, erasing the MOND limit precisely in the cold-HI volumes where the radial-acceleration relation is measured while gas-poor spheroidals keep full a0 (tuning, universality, and the construction's own 0.108-dex fit all destroyed) -- or the refresh required to prevent clamping exceeds the pump budget by >=350x and the energetics gate fails jointly. 25-point box scan: zero live subregions. The medium the four theorems demand builds its own laser, and the galaxy is the cavity. VERIFICATION CORRECTIONS printed beside the results: headline margins deflated x2.5 (kill survives at x3.4 hostile-corner under 10x damping stress); the binary-sweep horn NULLED x2900 by spectral hole-burning and its separation-gap observable WITHDRAWN; the construction's wide-binary gamma->1.28 prediction withdrawn with the construction; disk heating checked and does not kill. SCOPE: closes the inverted-line MI realization -- NOT 'modified inertia is impossible' (one passive-anharmonic corner remains on the prior sign-theorem adjudication) and NOT the a0 reframing or its live empirical tests, which are untouched. The chain's value: a complete map -- four theorems forcing a unique object, a construction showing it passes every prior gate, and a fifth theorem showing the universe it must live in destroys it. If a sixth idea reopens the door, this chain says exactly which wall it must breach. The author retracted all earlier 'theory of everything' claims (2026-06-23); this note claims a closure and a map, not a completion.