A Correction: The Variational Object for Complex-System Emergence Is a Large-Deviation Action, Not a Mechanical Lagrangian

Abstract

Abstract Version 1 of this paper claimed a Least Action Principle for Complex Systems Emergence: that the evolution of any complex system extremizes a complexity action functional built on a Lagrangian of the form kinetic minus tension minus shock plus driving, and that the existing variational principles of complex-systems science — Friston's free energy principle, Onsager's minimum entropy production, Ziegler's maximum entropy production, self-organized criticality, Jaynes's maximum entropy and Kahneman's cognitive miser — are special cases of it under limiting conditions.The central claim of that version does not hold, and this version withdraws it. The stationarity condition of a Lagrangian of that form yields a second-order conservative equation containing no first-order damping term; the damping was introduced in V1 by a step described as adding the dissipation term, which is the conclusion inserted by hand. No amount of repair to the Lagrangian recovers the result, because the obstruction is structural: an Euler–Lagrange equation is second order and time-reversal covariant, and the macroscopic evolution of a dissipative system is neither.A correct variational object exists, and V1 already contained it, in a fourth-level subsection. Section 4.6.3 of V1 wrote the Onsager–Machlup action for the stochastic dynamics. That functional, and the Freidlin–Wentzell large-deviation theory of which it is the rate function, is a genuine variational principle for the phenomena V1 most wanted: the quasi-potential is the minimum of that action, Kramers' escape formula is its small-noise corollary, and the most probable escape path is its minimizer. The correct principle was written as a subsection of an application; the incorrect one was the title.This version therefore does three things. It states the withdrawal precisely, item by item, with the algebra that shows why each fails. It restates the surviving content on its correct foundation, with the variational status of each item graded in three tiers and with the gradient/non-gradient distinction — never made in V1 — as the organizing line. And it states the precise obstruction that any recovery of the original claim would have to overcome, so that the withdrawal is constructive rather than merely negative.Six items are withdrawn outright: the main principle in Lagrangian form; the derivation of the evolution equation from it; the claim that Onsager and Ziegler's principle are both special cases of one principle; the claim that Jaynes's maximum entropy and Kahneman's cognitive miser are special cases of it; the variational account of free will; and the criteria for artificial consciousness. Three are demoted: the derivation of self-organized criticality becomes an argument; the derivation of downward constitution becomes an assumption plus arithmetic; the emergence index becomes a proposed composite. Three survive and are strengthened by being placed correctly: the two-stage transition path, which is the large-deviation result and was right for reasons V1 did not have; the relaxation classification, as a reading of landscape structure rather than a derivation; and the correspondence with Onsager reciprocity, which is the one relation in V1's unification table that is a theorem.On the manner of the correction. V1 is deposited. A deposited claim that is wrong is not repaired by quiet substitution: a reader who cited it is entitled to know what was withdrawn and why, and a version that only presents the improved theory conceals the size of the change. This version therefore leads with the withdrawal and keeps the erroneous derivations in view long enough to show precisely where they fail.