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

Abstract

This is version 3.4. It records the outcome of an adversarial re-run of every numerical claim in v3.3 against the deposit's own definitions, and it corrects the claims that were found to be stated above what the computation supported. The corrections are recorded in CORRIGENDUM-2-v3_4.txt and CORRIGENDUM-3-v3_4.txt and carried in a new working note, WN4. The only version previously published on Zenodo is v2.1; v3.0 through v3.3 were manuscript development iterations that were not separately deposited, so as a deposit this version supersedes v2.1 directly and the published chain reads v2.1 to v3.4. δ-theory is a research programme in which the organising parameter of physical and cognitive reality is taken to be self-reference depth delta rather than coordinate time t. Ordinary time is treated as emergent, matter and measurement are described by a single equation for a self-referential cluster field, and consciousness is proposed to be the regime in which self-reference runs deep. The aim is to test the picture and to derive its claims where they can be derived, marking clearly where they cannot. It rests on the two structural results introduced in version 3, a Lorentz-covariant parent for the cascade equation and a selected functional form for the consciousness index, and it records honest limits on both that the v3.4 re-run established. First, the cascade propagation equation is reinterpreted not as fundamental but as the rest-frame, on-shell, momentum-marginal slice of a Lorentz-covariant parent dynamics for a real scalar Wigner function on relativistic phase space. The law is then Lorentz invariant by construction, the preferred cosmic frame arises by spontaneous symmetry breaking with the status the cosmic-microwave-background rest frame already has, and time is emergent. This reformulation is [Structural]. The free real diffusion is the Wick rotation of free Schroedinger evolution at the level of the identification, which fixes the diffusion coefficient at D = hbar/2m in the reaction-free sector, verified exact at leading order. Version 3.4 records a limit on how far the parent reaches: an attempt to reduce the cascade's reaction sector from the parent (open item R2a) is not supported. The parent's source is never specified, and the logistic reaction term's density-level target is non-analytic in the density, so no local Lorentz-scalar source marginalises to it; the only closed form that hits the target is non-local and non-analytic, which is the reaction term relabelled rather than derived, and the amplitude-to-density map that is exact for the free sector does not close for a source. The covariant parent therefore covers the free kinetic sector only, and every reaction-sector reading (objective collapse, the consciousness source, the dark-energy sector) inherits [Structural] at best from an unspecified source. Closing R2a requires a new commitment, an explicitly specified covariant source with the reduction and error estimate carried out. Second, the consciousness index delta_c = f(sigma, Omega) is selected as the multiplicative form delta_c proportional to sigma*Omega by a discriminability requirement together with a stated minimality assumption [Structural]. Since sigma is a length and Omega a frequency, sigma*Omega is a velocity. O(4) invariance of the self-modelling kernel fixes sigma*Omega = c as a dimensional identity, which fails at neural scales by nine to ten orders of magnitude, a result reported openly [Verified]. Version 3.3 identifies the relevant velocity as a Fisher-KPP propagation speed of the coarse-grained cascade, v_eff = alpha*sqrt(D*mu), with the correlation-speed value alpha = 1 primary and the front-speed value alpha = 2 carried as the sensitivity bound. That c cannot be recovered from the coarse-grained (D, mu) is [Verified] on dimensional grounds. With alpha fixed under the identification of the self-modelling reach with the correlation length (sigma_e = xi), which is [Structural], delta_c = sigma*Omega/v_eff carries no further free constant once sigma, Omega, D_eff and mu_eff are measured, and equals 1 at the natural operating point as a consistency identity rather than a derived number. The empirical content is the measured departure of sigma*Omega from v_eff across states. A boundedness analysis establishes that the self-reference source renders the cascade equation ill-posed in exactly the high-coupling regime the consciousness sector requires [Verified]. Version 3.3 proposed a bounded non-gradient operator with a difference-of-Gaussians kernel to resolve this. Version 3.4 corrects that claim: the operator preserves the upper bound of the field's domain, but the non-local gain turns negative under the inhibitory surround and drives the field below zero, so the field stays in its domain [0, 1] only under an explicit projection, which also lowers the deepening. The excursion is unchanged under timestep refinement, so it is not an artefact of the time integrator; it is a property of the spatially discretised operator, and because the reported minimum differs between grids we do not claim it as a property of the continuum operator. Under projection the operator stays in range and deepens in the variance sense, but the projected dynamics is a different operator from the bare one and carries the same re-derivation cost. Whether the operator forms structure at the kernel scale rather than a single large domain is [Open]. Boundedness on the upper side is [Verified, numerical]; domain preservation on the lower side is [Verified, numerical, under projection]; the variance-sense deepening is [Verified, numerical]. Propagating the projected operator through the index and collapse constructions remains [Open]. A conjecture from working note WN1, that the Feshbach-Villars negative branch is the two-sector reservoir, is tested and reported as not supported. The programme retains five pre-registerable predictions with falsification conditions, sharpens the central anaesthetic test into a differential test that distinguishes delta-theory from spatial-coherence-alone criticality, adds a further differential test against temporal complexity, and lists its open problems explicitly. A frozen dataset addendum fixes, before analysis, the datasets for those tests (Chennu et al. 2016 at the University of Cambridge repository; OpenNeuro ds005620; Purdon et al. 2013 for the phase-amplitude-coupling definition), the channel sets, the source model, and the values of D_eff, mu_eff and v_eff, and records honestly that phase-amplitude coupling under propofol reorganises rather than simply collapsing, so the premise is tested and not assumed. Amendment 1 to that addendum, also frozen before analysis, records two limits of the frozen data: neither dataset is a TMS-EEG recording, so the PCI-referenced threshold test is deferred until a PCI source is added; and both cohorts fall below the pre-registered minimum sample size, so the registered effect sizes will be reported at the power the available data supports. Neither limit relaxes a kill condition. The programme also engages a closely adjacent published result: Barontini (Phys. Rev. Research 8, L022047, 2026) realises a cold-atom platform that constructs an entropic internal time from two-sector entropy exchange and recovers an effective Schroedinger equation on a slice. This coincides in method with the emergent-time construction, the two-sector budget model, and the Feshbach-Villars route used here, and it provides an experimentally grounded precedent that such reductions can be realised and tested. It is read as convergence of method, not as evidence for the delta-theoretic claims, which stand or fall on their own reductions and predictions. The programme does not claim a grand unification, a finished derivation, a derived Born rule, a derived single-valuedness condition, or a confirmed consciousness application. Status discipline. Four labels are used and applied strictly. Verified marks a result established by explicit computation or proof that has been checked. Structural marks a construction whose ingredients are individually justified but whose controlled derivation with error estimates is not complete. Open marks something not established, with the task named. Empirical marks a neuroscientific or observational premise requiring independent literature verification. No claim is stated above its label. Where the v3.4 re-run found a claim stated above its label, the claim was lowered. This deposit contains the version 3.4 preprint, four companion working notes (WN1 on the Feshbach-Villars route and entropic time, WN2 on the two-sector entropy-export test, WN3 on the effective velocity and the derived operator, WN4 on the boundedness correction and the reaction-sector reduction attempt), a pre-registration of predictions P2 and P2' with a frozen dataset addendum and Amendment 1, a standalone reaction-diffusion boundedness note, the reproducibility scripts for all numerical results together with the figures they generate, and three corrigenda (the v3.3 framing corrigendum, the v3.4 numerical corrigendum, and the v3.4 consistency corrigendum) with the record of the reproduction run. Every result labelled [Verified, numerical] is reproduced by a deposited script; all ten were re-executed before deposit and reproduced every published number. The PDFs are the record. No result outside the corrected operator claim and the R2a open item is retracted.