When does self-duality cause physics? A machine-checked criterion, tested across sixteen 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 formalization of 116 theorems across 14 Lean modules (extracted unchanged from a 257-theorem library) states the mathematical skeleton of sixteen 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: Planck's error bars sit within about 13% of the cosmic-variance floor for 5≤ℓ<30 and within a factor of two up to ℓ=50, so a perfect temperature measurement could tighten it by at most about 1.4× (this corrects the earlier version's “within about 15%”, which rested on nine sampled multipoles). Twelve of the sixteen candidate cases receive one of the criterion's four verdicts (forced; not forced; constraint, not cause; organisation, not cause) — including, new in this version, the q-state Potts model's random-cluster self-dual point, proved critical for every q≥1 (Beffara–Duminil-Copin), a third “forced” case; 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 Sector Causality Theory. Its dedicated repository, https://github.com/xaviercallens/SocrateAI-Scientific-SectorCausalityTheory, holds this paper and its two companions, the self-contained Lean formalization, reproduction notebooks, a training guide, and an MCP server for scientists and AI agents. The theory grew out of the SocrateAI-Scientific-QuantumFluids programme, whose full 257-theorem library, raw experimental results and further companion papers are archived as software at the concept DOI 10.5281/zenodo.22855581. New in this version (v3). A third “forced” case (Potts/random-cluster), making sixteen candidates and twelve verdicts; an erratum on the CMB forecast's quantitative basis (the earlier “within about 15%” rested on nine sampled multipoles; a full scan gives the ranges stated above, with the conclusion unchanged); and the theory's move to its own repository. 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.