Relational Cosmological Superposition Theory: From the Informational Metric to Cosmology

⚠️ RETRACTION NOTICE ⚠️

This version has been retracted. See version 5 for full explanation.

Reason: AI-generated content not properly validated.
Do not use this work as scientific reference.

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We propose a complete relational theory in which physical variables are not primitive but rather emergent from patterns of correlation. We postulate two ontological regimes: a S silent state, of zero relational entropy, timeless and ageometric, and a set of R relational states, where the growth of relational entropy SR enables a notion of time and an effective metric. The observer/observed separation is modeled as a choice of factorization of the Hilbert space, and it is precisely that choice which makes variables like space, time, and gravity exist.
Formally, we define SR as a sum of mutual information terms over families of bipartitions and reconstruct a metric gμν [I] from diffusion distances in the correlation graph. We show that informational null trajectories (those with minimal variation δSR = 0) define the causal boundary and provide an informational interpretation of c; coherent electromagnetic modes approximate this limit and act as windows from the R domain to S. The effective dynamics is implemented by means of an AQUAL-type action with an environmental acceleration scale a0(S); we demonstrate stability (absence of ghosts) in the 4D scalar–tensor extension and guarantee Lorentzian signature in the metric reconstruction.
At cosmic scale, we replace initial/final conditions with atemporal boundaries in S and derive Friedmann-type effective equations with a0(S)(z). Two paths lead R → S: extreme dispersion (homogenization of correlations) and extreme concentration (operatively dimensionless curvature). We present quantitative predictions and falsifiers: RAR band and evolution, strong lenses with the same μ and a0(S), photon coherence versus ∥∇S∥, BTFR shifts, speed/non-dispersion of gravitational waves, and a CMB/BAO/f σ8 fitting scheme to compare with ΛCDM. The theory thus becomes operational and falsifiable from the galactic scale to the cosmological scale.

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**Changes in version 1.2 (September 2025):**
This version adds a short Speculative note in Section 11 discussing a possible extension from the relational R-superposition over S (modeled as a positive informational measure W[R]) to a genuine quantum superposition with amplitudes, phases, and interference.

The note outlines a future research program that would require:
1. Endowing the space of R-configurations with a linear/Hilbert structure.
2. Defining amplitudes consistent with the informational dynamics.
3. Recovering an effective Born rule from R-admissible factorizations.
4. Preserving the informational null causal structure under such a promotion.

This extension is clearly marked as beyond the scope of the present preprint, and none of the current results depend on it. The goal is simply to indicate a possible bridge to quantum foundations for further exploration.

All main results (rotation curves, RAR, strong lensing, cosmological fits) remain unchanged with respect to v1.1.

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**Changes in version 1.3 (September 2025):**

We have updated the Overleaf projects and associated PDFs to include, at the end of Section 10 (Reproducibility and materials), an additional subsection titled “Additional reproducible results: RAR test in MaNGA”. This subsection documents the automatic generation of the new figures effect_vs_lnBF.png and the 8-panel RAR scatter plots for the MaNGA cohort, produced by running RUNME_WINDOWS.bat in the package TSCR_FINAL_PACKAGE_v1 (see MANIFEST.json). Together with the already archived Phase 1 results, these artifacts illustrate the reproducibility of the strict-null RAR test with the same $\mu(x)$ and $a_0(S)$ as in the AQUAL relational framework (Sec. 4, eq. (8)), and the falsification criteria defined in Sec. 9. In addition to the updated PDFs, we have deposited the reproducibility bundles TSCR_repro_v1.zip (code) and TSCR_FINAL_PACKAGE_v1.zip (data, figures, manifest), so that referees and readers can directly regenerate all figures and tables described in the manuscript.

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**Changes in version 1.4 (September 2025):**

This release adds reproducible artifacts for the masked RAR residuals and their independent validation via the preregistered locality-of-correlation (LoC) gate.
It includes CSVs per galaxy (w1_r2 and r2_weighted) with columns galaxy_id, r_kpc, residual, good_mask, weight, plus the script build_masked_residuals_from_manga.py to regenerate them from SPARC/MaNGA inputs.
An external reviewer reported PASS (z ≈ +0.44) under r ≥ 2 kpc, w = 1, confirming that residuals behave like colored noise with no detectable short-lag non-linear structure.

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*The conceptual framework and original ideas are my own. Computational tools, including AI assistance, were used to support the mathematical formalization and structuring of the manuscript.*