Matrix Majorant–Inertia Problem (MMIP): Seeking an Unconditional Improvement via CGdL's Tail-Sign SDP and Claude's Off-Axis Signature

Re-examines the technique by which Chirre–Gonçalves–de Laat (CGdL) improve the Montgomery–Taylor multiplicity constant from 1.3275 to 1.3208 under RH (i.e., 67.25%→67.92%): relax the bandlimited condition to require only $\widehat f(\alpha)\le0$ (for $|\alpha|\ge1$), and use $F(\alpha,T)\ge0$ so that the integral over the unknown band can only help the upper bound — using “the sign of the unknown region, not its magnitude.” The document notes that Baluyot–Goldston–Suriajaya–Turnage-Butterbaugh already made the prime-side non-negativity result unconditional in 2024, but this cannot directly make CGdL's 67.92% unconditional — the gap isn't on the prime side, it's on the zero side: CGdL's scalar proof relies on RH to make every zero ordinate real, which is what gives termwise non-negativity; once RH is dropped, this scalar argument no longer holds automatically. The document observes that Claude's paper resolves the zero side differently — using block signature (critical-line zeros contribute a PSD block, and each off-axis functional-equation pair contributes an indefinite block with a (1,1) signature) together with an inertia + rank–trace inequality, rather than scalar termwise positivity. From this it defines MMIP: whether one can simultaneously achieve tail-sign prime control (M1), preserve Claude's off-axis block signature (M2), and build a matrix certificate that turns both into a lower bound on $N_0^s/N$ (M3). The document gives an explicit table of constant targets (beating the 68.185% ceiling requires $C<1.31815$ — CGdL's known 1.3208 is not yet enough; reaching 70% requires $C\le1.30$), and proposes an MVP path that first tests, in a toy world, whether tail-sign and inertia produce a synergistic gain. The document explicitly states: this document does not prove a new proportion for the zeros of ζ — it only locates a hybrid research direction that may be able to avoid the α>1 asymptotic.