When does self-duality cause physics? A machine-checked criterion, tested across fifteen candidate physical systems, an interventional experiment, and real cosmological data

Abstract

Physics is full of self-dual points: places where a symmetry exchanges two descriptions of the same system and maps it to itself. Some of these points are physically decisive — the Kramers–Wannier point fixes the Ising critical temperature exactly — and some are not: the free boson's self-dual radius is an ordinary point on a line of conformal field theories, not a phase boundary. This paper states, proves in Lean 4 against Mathlib with two independent kernels, and tests experimentally, a single criterion that predicts which is which: a self-dual point is forced to be physically distinguished if and only if it exchanges two inequivalent sectors on a discrete fixed-point set; on a continuous moduli space with enhanced symmetry, or when self-duality only constrains a fixed point without producing one, it is not. The duality is always exact; what does causal work, when anything does, is the sector it relabels — a conserved or topological label crossing a threshold, proliferating, annihilating, or being imposed. The criterion is tested, not just stated, in three independent ways. First, a machine-checked library of 257 theorems across 30 Lean modules states the mathematical skeleton of fifteen candidate cross-domain cases — from the Ising model to Calabi–Yau mirror symmetry — and every module is re-checked by a second, independent kernel (nanoda) so that a statement's acceptance does not rest on one implementation of type theory. Second, a pre-registered, energy-matched intervention in a real classical field (a projected Gross–Pitaevskii superfluid) asks whether a sector is a cause, not merely a correlate: on three equilibrated bases, injecting the same energy as topological defects lowers the condensate fraction by 0.49–0.73 where injecting it as phonons lowers it by at most 0.035 — the same energy, as topology, is 15–60× more effective. Third, the criterion is carried into cosmology (dark matter halos, dark energy, the K3 charge lattice of a black hole's dyonic charges) and against one real, present-day dataset: an independent, exact reproduction of a published CMB bound on a discrete dark-energy phase transition, built from the published equations and Planck's own data, agrees with the original to within a factor of 3.16 across the parameter grid tested, and a companion Fisher-style forecast shows that no future CMB-temperature-only experiment can meaningfully tighten this particular bound, since Planck's own measurement is already within about 15% of the cosmic-variance floor at the relevant multipoles. Eleven of the fifteen candidate cases receive one of the criterion's four verdicts (forced; not forced; constraint, not cause; organisation, not cause); three are, on inspection, not instances of the criterion at all — a classification (topological superconductors), or a duality between different theories with no self-map (three-dimensional Ising duality, AdS/CFT); and one, Calabi–Yau mirror symmetry, is left an explicit open question rather than forced into a verdict the literature does not support. Reporting these honestly is part of the theory's claim to be checked, not merely asserted. This is the standalone foundational paper of the SocrateAI-Scientific-QuantumFluids research programme's duality/sector-causality thread; the full machine-checked Lean library, the underlying numerical experiments, and five companion papers developing individual threads in more depth are archived as software at the concept DOI 10.5281/zenodo.22855581 (repository: https://github.com/xaviercallens/SocrateAI-Scientific-QuantumFluids). No novelty is claimed in the underlying mathematics of any single duality; what is new is the criterion stated precisely enough to be checked by a kernel and tested by an experiment, and the honest record of where it does, and does not, resolve.