The Parameter Space of the Infinite Transformation Principle (ITP): A Universal Framework for Irreversible Dynamics and Structural Memory

Irreversible evolution with partial retention of history appears in systems that build structure, from cosmological clustering and galaxy evolution to biological inheritance and optimization dynamics in machine learning. In each field, "memory'' is introduced in different language and with different objects, which makes it hard to compare mechanisms, to identify what is minimal, or to formulate falsifiable cross--domain predictions.This paper presents a compact axiomatic and effective--action foundation for the \emph{Infinite Transformation Principle} (ITP): a class of systems that (i) exhibit monotone structural growth before saturation, (ii) respond to a finite causal segment of their history, (iii) saturate in structural capacity, and (iv) include a stabilizing long--wavelength feedback. Under these assumptions a generic ITP system admits, up to smooth reparameterization and field redefinitions, a representation governed by seven parameters \[ (\alpha, m, S_{\max}, \Delta, \beta, \mu, \eta), \] grouped into growth, memory--horizon, and memory--response couplings. These parameters arise from a minimal localizable nonlocal action (equivalently, an auxiliary--field formulation) and make explicit the conditions under which a single--timescale exponential kernel is the unique stable one--parameter memory kernel.The paper then defines the seven--dimensional parameter manifold $\MITP$, constructs simple dimensionless invariants for cross--domain comparison, and spells out what must be reported to claim empirical constraints on memory: kernel choice, structural source, parameter combinations, null tests, and data/likelihood details. A separate simulation paper derives an exponential drag kernel from TNG300--1 by coarse--graining $\sim 50\,{\rm Mpc}/h$ domains and measuring the response of an expansion--rate deviation to a velocity--dispersion source. Here that result is used as an explicit example of how the abstract ITP parameters map onto a concrete kernel measurement, and how short, Gyr--scale domain memory can coexist with a much longer, effective memory horizon in background cosmology fits.The result is a "parameter handbook'' for memory--bearing irreversible dynamics: it states clearly what is assumed, what is minimal, what is identifiable, and what would falsify the framework in real data.