All-Level Local Identifiability of Riemann Xi Toeplitz--Hankel Jets from Cumulative Second-Compound Data

Let F(x)=ξ'/ξ\!(1/1-x)=∑n≥0fₙxⁿ be the logarithmic-derivative germ of the completed Riemann xi-function, and form the symmetric Toeplitz--Hankel jet gᵢⱼ=f|i-j|-fᵢ₊ⱼ₊₁+δᵢⱼf₀. For every odd cutoff L, we restrict the second compound of the leading matrix jet to row and column pairs of total degree L, contract that block against a fixed three-term-recurrence polynomial vector, and retain the resulting diagonal polynomial QL. The cumulative forward map ΦN=(Q₁,Q₃,…,QN) is quadratic and is invariant under the exact coefficient gauge (f₀,f₁,f₂,…) (f₀+c,f₁+2c,f₂+2c,…). We prove that the sharp generic threshold for local recovery modulo this gauge is $N=7$: through $N=5$ there are exact local deformations preserving all cumulative data while changing an adjacent second minor, whereas for every odd N≥7 the quotient Jacobian has the maximal rank $2N+1$ on a Zariski-open set. The induction is governed by a cutoff-independent four-column gate Σ=V(g₀₁)∪ V(g₀₀)∪ V(g₀₃,g₁₂)∪ V(g₀₂,g₁₁). A regular level-seven jet outside Σ remains regular at every higher odd cutoff, regardless of all later coefficients. We also show that the quotient map is generically finite but not globally finite, that its generic degree is even because of the unavoidable sign involution, and that local identifiability alone does not control the signs of adjacent minors. For the actual xi-derived coefficient sequence, a zero-pair moment formula implies unconditionally that the leading 4×4 matrix block is entrywise positive, hence avoids Σ and fixes the arithmetic sign orientation. Finally, a directed-rounding MPFR computation certifies a nonzero 15×15 Jacobian minor at $N=7$. Consequently, the actual xi-derived jet is locally identifiable modulo the exact gauge at every odd cutoff N≥7. These results concern a structured inverse problem; they do not prove the Riemann hypothesis or the nonnegativity of the adjacent minors.