The exact row of the Toeplitz–Cayley model: coefficient law, Binet–Laguerre remainder, and certified positivity at finite rank

A companion note (Part I) showed that the loss of positivity of a discrete-sine compression of a Toeplitz–Cayley row of the Weil form, observed at a linear threshold a=qN near q≈2.4966, could not be interpreted: the last retained term of the archimedean expansion entering the row was 2394 times the quantity being signed. This note removes the truncation. We derive the exact Taylor coefficients of the archimedean part h_arch(a(1-z)/(1+z)), h=Ξ'/Ξ, in closed form: for d≥1 the coefficient is a pole term, a reciprocal term, a logarithmic term, an explicit family Σ_k c_k t²ᵏ F₂ₖ(d) with c_k = -2²ᵏ⁻² B₂ₖ/k and F₂ₖ(d) = [z^d]((1+z)/(1-ρz))²ᵏ, and a remainder with the Binet–Laguerre representation R_d(K) = -(1/2)(-1)^d ∫_0^∞ E_K(s) e-s(a+1/2)/2 L_d⁽⁻¹⁾(as) ds whose kernel E_K has sign (-1)^K and is bounded by the first neglected Bernoulli term (Temme's identity). Writing that kernel as a Laplace average turns the remainder into a probability average of the first omitted family τK+1(d) at complex parameters, and yields the enveloping inequality |R_d(K)| ≤ |τK+1(d)| for all d≤(K+1)(a-1/2)^2/a: the truncation error of the whole Toeplitz section is bounded by an explicit order budget X(K,q,N), at every rank and with a closed-form limit at a linear threshold. The row used in Part I is identified exactly as the K=2 truncation minus a diagonal debit, and that debit is 2.8 times the correct budget. We then compute the exact row in ball arithmetic at N=5000 — 5000 Hurwitz zeta values and one power-series composition at 8325 bits, the composition losing 3^N — and certify the resulting compressions. At (q,N)=(2.4966,5000) the truncation ladder λ_min(A_80(K)), K=1,...,7, converges geometrically (successive increments shrink by factors 27.9–63.9) to the certified exact value 1.8224265897·10⁻⁴ (eleven digits common to both endpoints), while the row of Part I gives -7.66960·10⁻⁸; the diagonal debit b_6 of that row is a valid Weyl–Young budget for the omitted remainder (b_6 ≥ 2 Σ_d |R_d⁽²⁾|, certified) but exceeds the remainder's actual effect on λ_min by a factor 130. At eleven thresholds between q=1.3 and q=2.625, at N=5000, the exact compression is certified positive definite, including below q=2/log 3 where the truncated row was undefined. Under the Riemann Hypothesis this positivity is expected: by a theorem of Hinkkanen the function h(a(1-z)/(1+z)) is then a Carathéodory function, so every Toeplitz section is positive semidefinite. No claim about the Riemann Hypothesis is made; what is proved is the exact structure of the row and its truncation error, what is certified is the exact finite-rank object, and what is quantified is the price of having replaced it by an asymptotic minorant. The row is a banded transform of the Keiper–Li coefficients at a moved base point; the coefficientwise envelope, uniform in the row index, is what the pointwise error bounds of that literature do not give. This preprint record contains the article PDF, the complete LaTeX source archive, and SHA-256 checksums. This is a preprint: it has not been peer-reviewed, and its mathematics has not been independently verified. Anthropic Claude assisted with preparation; Julien Lange is the sole author and assumes responsibility.