Statistical Pharmacology via Kakutani Dichotomy IX: Notch-Sparing Gamma-Secretase Modulators for Alzheimer's Disease, Schur-Complement Confinement of the Catalytic Block, and a Cone-Geodesic Model of Processive Trimming

Paper IX of the series Statistical Pharmacology via Kakutani Dichotomy. Papers I–VIII built the Kakutani indices and their estimators; this is the first application to a single pharmacological target: the presenilin-1 catalytic core of γ-secretase, whose orthosteric inhibition failed in phase III because the enzyme also cleaves Notch. We seek a quadrant-II ligand of Paper IV — silent at the active site, loud in the second-order correlated Kakutani index. MODEL A harmonic (Gaussian) elastic model of the catalytic core with D = 300 coordinates in three blocks: a catalytic block of 30 (Asp257 on TM6, Asp385 on TM7, the S3 register), a TM6a/PAL gate of 70, and a TM1–TM9 channel of 200. Block topology — and only topology — is taken from the cryo-EM structures of γ-secretase bound to APP (PDB 6IYC), to Notch (PDB 6IDF), and to an inhibitor and a modulator simultaneously (PDB 7D8X). No coordinates are used and nothing is docked. RESULTS Schur-complement confinement. The catalytic block meets the trimming machinery only through a weak coupling block, so any stiffness perturbation of the machinery changes the catalytic precision matrix by an amount exactly bilinear in that coupling. The induced index is therefore second order in the coupling and its covariance part fourth order. A 64-fold scan reproduces the predicted exponents 2, 2, 2 and 4 as 1.9906, 1.9709, 1.9755 and 3.9633. This is the mechanism that makes quadrant II reachable at all. Cone geodesic. The discrete infinity-Laplacian distance on the three-residue step lattice is solved in closed form for equal weights, reproduced to 4.44 × 10⁻¹⁶. Isotropy strictly minimises it at fixed total weight: cooperativity must be shared across all three helices to be worth anything. Dual-biomarker branching law. Treating the pathogenic product line as branching at the intermediate E·Aβ42, both clinical read-outs — Aβ42/Aβ40 and Aβ38/Aβ42 — follow from one partition, and the phenomenological exponential law is its strong-modulation limit. One fitted constant reproduces it across five compounds to within 8.91 %. The two biomarkers are one measurement seen twice, not two independent ones. Compounds. Semagacestat falls in quadrant IV (Notch retention 13.31 %, both biomarkers unmoved at 1.01×); Avagacestat also in IV despite its Notch-sparing label (5.8× the safety threshold, retention 64.96 %); R-flurbiprofen in III; E2012 and the designed lead KMS-AD-309 in II, with Aβ42/Aβ40 down 2.37× and 2.61× as Aβ38/Aβ42 rises 2.82× and 3.14×. Familial disease. The PSEN1 alleles L166P and E280A are modelled as losses of gate stiffness, not gains of activity. Both rearrange the machinery as strongly as a drug and in the opposite direction, raising Aβ42/Aβ40 to 0.333 and 0.344. Both quadrant-II modulators reverse the sign of the clamp and return it to 0.135–0.156, below the wild-type 0.182, at 99.49 % Notch retention or better; the stricter target 0.12 requires 1.11–1.24× nominal occupancy. Neither inhibitor rescues at any occupancy up to 4×, and Semagacestat on a familial background delivers the whole Notch liability and none of the benefit. Screening. The selectivity index saturates, ranking E2012 above the more potent lead on a difference far below the safety threshold. A conventional catalytic activity read-out returns 120 hits out of 240 ligands with 0 % recall on quadrant II; the paper separates what in that statement is definitional from what is empirical. LIMITS The dynamics are a harmonic surrogate on a schematic contact graph; the structures fix topology, not geometry, and the model does not place E2012 at the extracellular allosteric site that 7D8X resolves. The five compounds and the two alleles enter only through their class, not their chemistry. KMS-AD-309 is a designed point in parameter space, not a synthesised compound. Three constants are calibrated, and the allele severity was set to reach the reported Aβ42/Aβ40 range. Five limits and five open problems are stated in Section 6. CONTENTS The 22-page manuscript (LaTeX source and PDF), the experiment script and its model library, the companion script that generates every table, four CSV data files, two figures, console logs, and 32 unit tests. The experiment passes 12 acceptance criteria and the companion 23 checks. The kakutani_pharma package is inherited unchanged from Papers VII–VIII. Code under the MIT License; text, figures and data under CC BY 4.0. Funding: National Natural Science Foundation of China (No. 22464010), Guangxi Natural Science Foundation of China (No. 2025GXNSFAA069294), and the 2025 Bagui Youth Top Talent Project.