Statistical Pharmacology via Kakutani Dichotomy X: Blocking Amyloid Nucleation of Intrinsically Disordered Tau and Abeta42 Monomers via Tail-Exponent Stabilization, Asymmetric Baseline Cubic Corrections, and Finite-Mode Crossover

Paper X of the series Statistical Pharmacology via Kakutani Dichotomy. Papers VII–IX worked with folded proteins and covariance. This one works with an intrinsically disordered protein, where there is no pocket to dock into and the only structure is a distribution over contacts. MODEL The monomer ensemble is a product measure on {coil, β} over contact modes. The healthy and pathogenic laws differ by a fixed hot-spot part — the hexapeptides VQIINK and VQIVYK of tau, the central hydrophobic cluster and C-terminus of Aβ — and a long-range tail ε_i = c·i^(−α). By Paper I the tail sum diverges for α ≤ 1/2 and converges above, so the two ensembles are mutually singular below the critical exponent and equivalent above it. Hyperphosphorylation lowers α; a kinetic ensemble stabiliser must raise it. The stabiliser acts on α alone, so K_hot is identical in every row and the whole effect is attributable to the tail. Block topology is taken from the cryo-EM structures of Alzheimer tau filaments (PDB 5O3L, whose ordered core begins at residue 306, the first residue of PHF6) and of the Aβ42 fibril (PDB 5OQV). No coordinates are used and nothing is docked. RESULTS Cubic correction at an asymmetric baseline. Paper I worked at p₀ = 1/2, where the symmetry p → 1−p kills the odd terms. A disordered coil is not equally likely to be β, and away from the symmetric point the first correction is cubic, with B₁ = (1/8)(p₀⁻² − (1−p₀)⁻²), odd about 1/2 and vanishing exactly there. At the coil baseline p₀ = 0.15 the measured remainder matches B₁ε/A₁ = 2.745·ε to 1.7 × 10⁻⁴. The consequence is that corrections to Paper I's asymptotic constants are O(c), not O(c²). An identity without cancellation. The textbook Hellinger formula subtracts two numbers of order one to produce a result of order ε², losing about ten digits at ε = 10⁻³. The exact rationalised identity h = (p−q)²[(√p+√q)⁻² + (√(1−p)+√(1−q))⁻²] has no cancellation. At ε = 10⁻⁶ it is 3.57 × 10⁶ times more accurate; by then the textbook form has stopped converging and begun to grow. This matters because the modes that decide the dichotomy are the far tail, where ε is smallest. Two-sided bound on the affinity. exp(−K_N/2 · (1−h_max/2)⁻¹) ≤ Π_N ≤ exp(−K_N/2), which makes "Π_N → 0 exactly when K_N → ∞" quantitative. Measured agreement 0.35 % at α = 0.85 and 6.0 % at α = 0.25. The two-term zeta formula, and its failure at the physical amplitude. K_tail(α,∞) = A·ζ(2α) − c³B₁·ζ(3α) + O(c⁴) for α > 1/2, with relative error O(c²). Reported as a negative result where it is one: the formula improves on one term by a factor ten for c ≤ 0.10 and is worse than one term at the physical c = 0.35, where the cubic over-corrects. The series in c is not usable there, and nothing downstream relies on it. The design law. Hyperphosphorylated tau (α = 0.30) is singular, with t½ = 10.0 h; KMS-Tau-204 (α_eff = 0.72) is equivalent, with t½ = 3.99 × 10⁸ h — a factor 4.0 × 10⁷ bought by the exponent alone. Finite-mode crossover. α_c = 1/2 is an N → ∞ statement. At finite N there is a crossover, not a step: the threshold α_th(N) rises from 0.1311 at N = 10² to 0.4676 at N = 10⁷, approaching 1/2 from below and never reaching it. The sub-threshold false-positive trap. A modulator that raises the exponent without carrying it across α_c buys a finite-size reprieve, not a phase change. At α_eff = 0.42 the half-time is 1.81 × 10⁶ h at N = 10³ — a complete block, on the scale of a short-peptide assay — and 9.47 h at N = 10⁵, as fast as the untreated ensemble at the smaller scale. KMS-Tau-204 changes by 5.8 % over the same range. The exponent a stabiliser must actually reach is α* = 0.51789, strictly above α_c: crossing the critical exponent is necessary but not sufficient. This offers a mechanism for a familiar and expensive pattern — aggregation inhibitors that are convincing on hexapeptide assays and disappear in full-length cellular models — in which the failure is neither potency nor delivery, but the system size at which the assay was run. Abeta42 against Abeta40. Holding the tail exponent equal, the C-terminal hot spot alone gives ΔK_hot = 0.4274 and a half-time ratio of 2.93 — the right direction and the wrong size. Inverting the model instead of fitting a second exponent: closing the gap to a ratio of 10, 20 or 50 requires the two extra residues to raise Aβ40's contact-order exponent by only +0.0073, +0.0117 or +0.0179. A modest and falsifiable claim. LIMITS Contact modes are an abstraction; the structures fix topology only. The amplitudes are chosen, not measured. One constant in the barrier law is fitted, to a single anchor, and every half-time follows from it. KMS-Tau-204 is a point in parameter space, not a compound. Nothing here describes seeding, cross-seeding or filament strains. Most consequentially, the product measure assumes the contact modes are independent, and correlation would change the very exponent the argument turns on — this is the first open problem. CONTENTS The 18-page manuscript (LaTeX source and PDF), the experiment script and its model library, the companion script that generates every table, five CSV data files, two figures, console logs, and 29 unit tests. The experiment passes 11 acceptance criteria and the companion 16 checks. The kakutani_pharma package is inherited unchanged from Papers VII–IX. 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.