The theory of $k$-record values (Type 2 $k$-records) plays an important role in the study of partial extremes and in statistical inference based on record data. A common approach reduces the analysis of $k$-records associated with a distribution function $F$ to that of ordinary records from the transformed distribution $F_{1:k}(x)=1-(1-F(x))^k$. This representation is widely used to derive distributional and inferential results, often without an explicit construction of the underlying stochastic mechanism, and relies on a structural property of order statistics that, although classical, is typically invoked without proof. We give a direct derivation of the probabilistic structure of $k$-record processes based on the sequence of running order statistics $U_n=X_{n-k+1:n}$, the $k$-th largest among the first $n$ observations. We show that $(U_n)$ forms a Markov chain with respect to its natural filtration, with an explicit transition kernel. The key step is a conditional-independence property of upper order statistics, which we isolate and prove: conditionally on $U_n$, the $k-1$ observations exceeding $U_n$ are distributed as order statistics from the distribution truncated at $U_n$, independently of the whole past trajectory. Under continuity of $F$, the usual Type 2 $k$-record times coincide almost surely with the record times of $(U_n)$. This yields a transparent construction of the $k$-record process as the record process of a Markov chain, and classical distributional results -- including the representation through $F_{1:k}$ and the joint density of the first $m$ $k$-record values -- are recovered in a unified framework. We also treat the exponential case, in which the $k$-record values form a random walk with independent exponential increments and Gamma-distributed marginals, and record a corresponding characterisation of the exponential distribution.
The class of parametric statistical models that can be treated as Riemannian manifolds is considerably larger than the classical Fisher-Rao setting allows, once one works in the space of tempered distributions. A law is represented by a tempered distribution T in S'(R^k), while an instrument - a positive Schwartz kernel, a weak regular inference function, or a weak Stein representation - extracts information from the law without being part of it. Any instrument with full-rank sensitivity and positive-definite variability induces the Godambe information G = S^T V^{-1} S, a Riemannian metric on the parameter space; the Fisher-Rao manifold is recovered exactly when the score is an admissible instrument, and every Godambe metric is dominated by the Fisher metric in the Loewner order whenever the latter exists. Four examples lie outside the Fisher-Rao class for four different reasons: a location model built on the Cantor distribution (an undominated family - no likelihood, no score, and no Fisher information exist at all), the uniform scale model (parameter-dependent support), the shifted exponential model (transform-based inference), and a stratified finite mixture (a provably biased score in a dominated model); a lattice stochastic heat equation driven by alpha-stable noise provides a fifth, dynamical example, whose closed-form weak Godambe information stabilises at a rate governed by the spectral gap of the discrete Laplacian. Quadratic Stein discrepancies induce the same local geometry, and reproducing-kernel constructions generate a hierarchy of geometries. Because there is no canonical instrument, the model carries a family of Godambe metrics; we discuss the inferential, diagnostic, geometric, and computational roles of its members, and show that weak inferential separation (nonformation) appears geometrically as block-diagonality of the Godambe metric.
