Profiled stability bounds cell leverage

Finite level, positive weights, nonsingular $I$ and $I_{\lambda\lambda}$ at the current state. (Audit: merged atoms are not needed — the proof never uses row distinctness, confirmed exhaustively on unmerged-duplicate configs; the operative hypotheses are positive weights and a non-singleton source with positive co-weight, $W_a>w_i$.) At any one-point exchange-stable profiled-$D_s$ state, for every point $(s_i,w_i)$ in a non-singleton cell $a$ and every $b\ne a$:

$$\boxed{ s_{aa}-s_{bb}\le\beta_i\,q_{aa}q_{bb}\le w_i\,q_{aa}q_{bb}, \qquad \beta_i=\frac{w_iW_b}{W_b+w_i}, }$$

with $s_{xx}=u_x^\top G_su_x$, $q_{xx}=u_x^\top I^{-1}u_x$, $u_x=s_i-\mu_x$.

Proof. By the exact rank-two determinant ratios (D3 applied to $I$ and to $I_{\lambda\lambda}$), the exact profiled gain of the move satisfies $\Delta F_s\le0$ iff

$$(1+\alpha q_{aa})(1-\beta q_{bb})+\alpha\beta q_{ab}^2 \le (1+\alpha r_{aa})(1-\beta r_{bb})+\alpha\beta r_{ab}^2,$$

where $r_{xy}=u_{x\lambda}^\top I_{\lambda\lambda}^{-1}u_{y\lambda}$. Expanding and using $s_{xx}=q_{xx}-r_{xx}$:

$$\alpha s_{aa}-\beta s_{bb} \le\alpha\beta\bigl[(q_{aa}q_{bb}-q_{ab}^2)-(r_{aa}r_{bb}-r_{ab}^2)\bigr] \le\alpha\beta\,q_{aa}q_{bb},$$

dropping $q_{ab}^2\ge0$ and the Cauchy–Schwarz-nonnegative $r_{aa}r_{bb}-r_{ab}^2$. Since $\beta\le w_i\le\alpha$ and $s_{bb}\ge0$ ($G_s\succeq0$), divide by $\alpha$ and use $\beta/\alpha\le1$. Degenerate post-move states: if $I'_{\lambda\lambda}$ is singular then so is $I'$ (Fischer, $\det I'\le\det I'_{\psi\psi}\det I'_{\lambda\lambda}$); in either singular case both determinant-ratio formulas take their true values with the left one zero, the displayed inequality holds without any stability input, and the expansion proceeds unchanged — so the bound also covers moves whose destination state is infeasible. ∎

Complements DS6: the leverage form needs no balancedness or mass margin; tiny or ill-conditioned cells surface through the leverage factors $q_{aa}q_{bb}$ instead. It is the finite half of the DS14 bridge. Exact-arithmetic regressions: 2,706 admissible moves at 110 float-screened, exactly re-ranked global optima plus 32 moves on both canonical fixtures, zero violations (NUMERICAL_EVIDENCE.md row N-DS-LEVERAGE); independently, the audit's exhaustive enumeration checked 1,748 admissible moves at all 171 exchange-stable states of five adversarial datasets — vector nuisance, vector POI, unmerged duplicates, unequal weights, 230 singular-destination moves — with zero violations (row N-DS-AUDIT-LEVERAGE, tests/test_research_claims.py::test_ds13_leverage_bound_at_every_stable_state_with_vector_nuisance).