Relations Acting on Relations: A Layered Conjecture Statement with Definitions, Status and Refutation Conditions

Abstract

The research programme asks whether, in systems whose parts are relations, a relation can act on the update rule of another relation, and whether such second-order structure is functionally indispensable. This statement fixes one core claim, an existence claim (C1), defines its terms operationally (intervention-based “effective rule”, functional necessity) and proves it in a toy class by a closed-form comparison: with regime-dependent observation noise, the best state-dependent gain has strictly lower steady-state error than any uniform gain (14–27% in state-estimation error, 1–4% in prediction error for the parameters checked). The toy result did not transfer to the delta-rule engine of the testbed: an oracle regime-dependent scalar gain gave no measurable benefit, and an oracle relation-specific gating gave a structure-dependent but sub-percent benefit. Four further questions are explicitly not part of the core claim and are stated with their status: whether such rules emerge from learning (Q2), reflexive organisation (Q3), a conditional functional candidate for consciousness (Q4), and a heuristic question about non-propagating connection (Q5). The statement gives a status table, refutation conditions, the relation to existing work (metaplasticity, neuromodulated plasticity, fast weights, hypernetworks, adaptive networks) and a list of challenges for readers. Not a proof; not peer reviewed. Companion to doi:10.5281/zenodo.23006539, 23192515, 23373288, 23384438 and 23420727. Proposition 1 is reproduced by the attached script prop1_check.py. AI use: derivations, scripts and drafting were produced with substantial assistance from AI systems; the author is responsible for all content.