Coefficient Regimes and Tail Compensation for Rademacher Sums at sqrt(2)

For a finite normalized Rademacher sum, this paper studies the conjectured strict bound P(X>sqrt(2))<=1/8 under coefficient restrictions. Conditional exact-cover compositions reach the third-coefficient threshold 41/100 and 2/5 in the positive chamber. Written results give atom-sensitive six-support classification, clipped budgets and joint loss/gain compensation with arbitrary finite actual remainders. Partitioned head layers and a diffuse bound are retained; a conditional eight-head cover reaches a8>=3/10, with its inclusive conclusion restricted to positive remainder variance. The unrestricted conjecture remains open.