δ-theory: Self-Reference Depth as a Fundamental Parameter for Matter, Consciousness, and Emergence

δ-theory is a research programme exploring whether self-reference depth δ — a dimensionless measure of recursive self-modelling levels — can serve as a more fundamental parameter than time t in physical and cognitive theory. The substitution is motivated by infinite time dilation for photons at c, the timeless character of the Wheeler–DeWitt equation, and the explanatory advantages of treating temporal experience as emergent. The framework comprises a cluster density field C, a phase function φ, a cascade propagation equation, a candidate consciousness operator Ĉ, and an actualisation functional. The signature equation ∂Ψ/∂δ = ĈΨ is presented as a postulate defining the high-γ regime in which self-referential deepening dominates the cascade dynamics. Preliminary results established in this version include: the explicit reduction of the cascade equation to Fisher–KPP dynamics; a two-cell toy-model sketch relating the self-reference term to an integrated-information-style partition; an explicit check showing the proposed energy functional is not conserved in the Fisher–KPP limit (a negative result); and a partial-invertibility construction for Ĉ on a compactly supported function subspace via Tikhonov-regularised Gaussian deconvolution. We specify five testable predictions with operational definitions, pre-registerable thresholds, and explicit falsification conditions: a joint (σ, Ω) consciousness depth metric compared against Φ, PCI, and LZc; phase-transition heterogeneity in propofol-induced loss of consciousness; consensus crystallisation effects in shared-attention measurement contexts; (σ, Ω) profile heritability in separated identical twins; and life-emergence discontinuity in self-referential autocatalytic networks above a closure threshold. Earlier formulations of several results — derivation of the Schrödinger equation including the origin of the imaginary unit and linearity, energy conservation via Noether's theorem, derivation of particle families from winding numbers, and a full correspondence to Integrated Information Theory — are reformulated as open problems with explicit success criteria. Eleven such open problems are collected as concrete targets for future work. The framework is situated relative to Rovelli's Relational Quantum Mechanics, Barbour's timeless physics, IIT, Penrose–Hameroff Orch OR, and other frameworks in quantum foundations and consciousness studies including Wolfram's Physics Project, Tegmark's Mathematical Universe Hypothesis, Wheeler's "It from Bit", Whitehead's process philosophy, and Kastrup's analytic idealism. New in v2.1: §8.7 acknowledges closely-adjacent recent work — Carter's Recurse Theory of Consciousness, the Recursive Self-Modelling Threshold proposal, Ghasemi's Relational Emergent Time, Realpe-Gómez's self-referential derivation of QM, Spivack's Alpha Theory / Loop Cosmogenesis, and Cahill's process physics — with explicit differentiation of δ-theory's contribution from each. This is version 2.1 of the preprint (April 2026); v2.1 supersedes v2 by addition only — no prior content is retracted.