Aggregate Head Layers for the Strict sqrt(2) Rademacher Upper Tail

Let X = Σ_i a_i ε_i, where the ε_i are independent uniform signs and Σ_i a_i² = 1. The conjectured strict endpoint bound is P(X > √2) ≤ 1/8. We prove a joint shifted-tail inequality for an arbitrary symmetric random variable: if d_1 + ··· + d_k ≥ 0, then Σ_i P(Y > d_i) ≤ k − 1. Averaging this inequality over set partitions controls a complete sign layer of a conditioned Rademacher head. For an eight-coordinate head, the two-negative layer is at most 21, uniformly over the independent symmetric remainder. Combined with elementary sixth-moment bounds, this yields a_8² ≥ 23/200 ⇒ P(X > √2) < 1/8. The support of the remainder is unrestricted and finite. As a complementary computer-assisted corollary of Keller and Klein's Prawitz inequality, we also certify the stronger inclusive bound P(X ≥ √2) < 1/8 whenever max_i |a_i| ≤ 6/25. Publication priority for the aggregate formulation and its application has not been established. This is an unrefereed research note. Neither the full endpoint conjecture nor optimality of the constants is claimed.