External nilpotence of oriented projective-relation Chern coefficients

Let $S$ be a quasi-compact quasi-separated derived scheme, and let $\mathcal{E}$ be a homotopy-commutative linearly oriented motivic ring spectrum over $S$ satisfying the source projective bundle formula with powers of the actual tautological first Chern class. Let $Y$ be a smooth qcqs derived $S$-scheme and let $V$ be a vector bundle of constant rank $r\ge 1$ on $Y$. For every $1\le i\le r$, the coefficient $c_i^{\mathcal{E}}(V)$ defined by the quotient-projective relation of Definition 4.4.1 is nilpotent in ${\mathcal{E}}^{\ast }(Y)$: some positive integer power is zero. No ${\mathbb{A}}^1$-invariance or absolute smoothness over a field is assumed.

Import precisely Annala–Iwasa Lemma 4.4.3(i), using Definition 4.4.1, from the author PDF dated 14 April 2025, printed pp. 30–31. Section 2.1.1 fixes the smooth qcqs relative site; the base itself may be singular. The positive-rank statement and its nilpotence are established source input, not a locally reconstructed splitting argument. The source supplies existence of an exponent for each coefficient, rather than a dimension bound. The rank-zero, above-rank and finite-clopen extensions used by the consumer are separate explicit conventions and local deductions. Application to the actual EM model, including verification of orientation, PBF, products and coefficient comparison with base $S=X$, is made in that consumer.