What a Krein String Sees in Its Weights: An Application of the Exact Derivative Calculus, with the Riemann Zeros as a Case Study

A finite positive measure on the half-line determines a discrete Krein string, and an earlier research note of this programme (An Exact Derivative Calculus for Discrete Krein Strings, https://doi.org/10.5281/zenodo.22847188) gave the gradient and the Hessian of the logarithmic string coordinates with respect to the atomic weights. This note applies that calculus to a question a measurement programme on the Riemann zeros had raised and could only answer empirically: what does the string see when its weights are rearranged within blocks? The answer is analytic. The drift of the string against block-permuted weights is a Taylor series in the block-centred fluctuation of the weight field whose coefficients, for fixed nodes and block means, are fixed linear, quadratic and cubic forms computable from Christoffel–Darboux leverages and kernels; the string condenses its weight field into these forms. The note is positioned as an application of a published theorem: the proposition assembles known ingredients (leverage and kernel-square derivatives of log-determinants, the cyclic third-order kernel of determinantal cumulants, exact block-permutation moments), and the closest precedent for a Krein string of the zero measure, Suzuki's zeta string, is acknowledged with the three differences stated: finite and unconditional, arbitrary weights, a calculus. What is proved. The third derivative of the log-determinant of a weighted Gram matrix in logarithmic weight coordinates has a closed form with a cyclic three-kernel term (Lemma 2.1). The Taylor coefficients of orders one to three of the drift against uniform block permutations are exact leverage expressions, and the permutation means are exact block moments 1/B, −1/(B(B−1)) and 2/(B(B−1)(B−2)) (Proposition 3.2); a finite sample of permutations adds an exactly computable offset (Corollary 3.3). The Hessians of the logarithmic Hankel determinants are Laplacians of orthogonal projections, which gives the supremum of the first drift coefficient at given block moments and an operator-norm bound for the second that is sharp at fixed total second moment; the Hessian increments between neighbouring degrees are controlled by the leverage increments (Corollary 3.4 and the remark after it). For equal weights the second spectrum of a window is the zero set of the derivative of its node polynomial, and for arbitrary positive weights it interlaces strictly (Lemma 4.1). Under the spectral Toda flow the velocities of both telescopes are explicit in the coefficients of the Stieltjes continued fraction, and in string coordinates the velocity of the mass telescope is the end stiffness plus the spring imbalance at the mass (Proposition 6.1, recorded as an assembly of classical facts). What is certified. Every proved statement is certified in exact rational arithmetic at N = 6 against an independent route — a symbolic derivative for all ordered index triples, a string route that never forms a Hankel determinant (exact Stieltjes procedure in a truncated power-series ring and the even contraction of the continued fraction), the mean over all block permutations, exact LDLᵀ tests of semidefiniteness, polynomial identities and Sturm counts — with the standard that the reported difference is exactly zero, not small. The negative probes — a reversed sign in the cyclic term, a wrong block moment, a perturbed projection weight, a shifted Laplacian, unequal weights, a wrong block-trace denominator, a halved cross term, a dropped factor two, swapped continued-fraction coefficients and an incorrect boundary convention in the Toda equations — fail as they must. What is numerical. The expansion is quantified on three weight fields from the zeros — reciprocal squared-derivative weights and a local log-spacing field that uses no value of ζ′, on two windows of 500 ordinates each: the second-order expansion reproduces the measured drift profiles at correlation 0.975 to 0.978, the full third-order polynomial for the sampled permutations at 0.985 to 0.989, and at small amplitude the polynomial closes to about one percent of the drift. A preregistered case study, calibrated on consumed windows and fixed and hashed before the validation window was opened, places the drift inside the class of variance-matched random-matrix control worlds on two windows (an empirical comparison that the expansion motivates but does not imply), confirms a frozen prime-sum prediction of the shape and the saturation of the local log-spacing field in the spirit of Bohigas, Leboeuf and Sánchez, and replicates a blind prediction of the drift and a bounded interlacing observable. One preregistered criterion is falsified and reported as such: the additional linear contribution of a fixed prime-sum field to the weights, beyond the local field, decreases over four windows and on the validation window falls below that of random frequencies; the preregistered notion "instrument confirmed" is therefore not achieved. The phenomena met are known physics — the prime-driven stiffness of the zero spacings — read through a new instrument. What is the scope. The statements are finite and discrete: N atoms, distinct positive nodes held fixed, block structure on the atoms. The formal core uses no input beyond a finite positive measure on (0, ∞). The ordinates of the zeros and the moduli of ζ′ at them enter only as declared, hashed numerical inputs; no property of their distribution is assumed, the definitions do not contain them beyond that declared input, and no claim to produce them is made. No statement about the location of any zero is made or implied, and no observable of this note is used to infer one. This is inverse spectral information analysis, not a Hilbert–Pólya model. What remains open. Optimal bounds for the second drift coefficient at individually fixed block moments, bounds for the third coefficient and for the remainder at amplitude of order one; how the string of the finite prime measure changes as atoms are added, where the Toda flow keeps the support fixed; the dependence of the case-study quantities on the height at the scale of the published million-zero computations; and the multiplicity convention of one preregistered clause, recorded as a design error. The verification scripts, the cached ordinate lists, the world files, the sealed preregistration and the prediction curves are available at https://github.com/utehrani/krein-strings in the folder krein/ (tag v2.0.0), with a script that reproduces every number and a manifest that lists each check with its script and log. The formal core is independent of any hypothesis about the distribution of zeros, and no claim concerning the Riemann Hypothesis is made or implied.