Parameter fibers: finding a hidden change of variable by differentiating with respect to a parameter

Typeset copy of the article "Parameter fibers" (https://math.osp.no/parameter-fibers), version of 2026-11-07, including the corrections of 2026-11-06/07. A hidden change of variable u = u(x) can be found with a parameter. If a problem has a parameter p and the hidden map does not depend on it, then suitable quantities built from the derivative ∂/∂p are functions of u alone: they are constant on the fibers of u. Their inner expressions are candidates for u. Reading them off is a heuristic, and each candidate is tested exactly before it is used. Integrals. If f dx = g(u; p) du, then ∂p ln f is a function of u. The premise can be weakened to maps whose fibers do not move with p, and a constant of the integrand can be promoted to a parameter. Three worked examples are given: a multiplied-out cube in an exponent; a constant used as the parameter; and an integral on a Picard curve of genus 3 whose substitution appears nowhere in the integrand, with an elementary answer. Each answer is checked by differentiation, in the computer algebra system Samara and with mpmath at 40 digits, at real points in every region and at complex points (Examples 2 and 3 also exactly). In Samara the finder feeds the integrator's substitution step. Together with other substitution routes, it raised the share of Sam Blake's 2541 test integrals answered by 90 (181 failures became 91; in 80 of the 90 the candidates from this finder answered), and answered 51 more of the 8658 integrals of Rubi's test suite (15 of them through these candidates). Linear ODEs of the second order. In the normal form v'' = r v, a known family W'' + J(t; p) W = 0 pulled back by t = u(x) gives r = −u'² J(u; p) − ½{u, x}. So ∂p r is u'² times a function of u, and whatever cancels u'² gives candidates: the logarithmic derivative by one parameter (F1), the ratio of the derivatives by two parameters (F2), and ∫ sqrt(∂p r) dx (F3, the known Liouville transformation). On ten synthetic pullbacks of the Bessel, Kummer, Gauss, Airy and Hermite equations the finders find the map for 9. On recall they solve all 29 equations that Samara already solved by a change of variable. In Samara's test suites they open one more equation (Mathieu's equation after ln x). Prior art. The parts are classical: derivative-divides (Moses), functional decomposition (Ritt; Barton–Zippel), Goursat's and Blake's symmetry-based substitutions, Ovsiannikov's equivalence transformations, the Liouville transformation (Milson), and pullback algorithms (Bronstein–Lafaille, van Hoeij–Weil). The finder for integrals, and F1 and F2 for ODEs, were not found in the literature read. The article lists what was read and how each work relates. The idea was proposed by an AI agent reading a compact catalog of Samara's integration methods alongside data from automated searches, then tested on Samara's failing problems. The ODE analogue was proposed by an AI agent in Samara's discussion session on 2026-11-06, and tested by hand by another AI agent. An independent AI referee checked the article; its findings are corrected and listed in it. The author directed the work and is responsible for the content.