SRT Part II: PRE-REGISTRATIONSRT Complexity Correlation Test (SN Ia + Weak Lensing)

Abstract

This document serves as a formal pre-registered falsification protocol for the State-Rewrite Theory (SRT). It defines fixed observables, fixed statistical models, and fixed thresholds designed to isolate any SRT-specific signal from standard General Relativity (GR) + ΛCDM weak-lensing effects. The protocol establishes that the proposed SRT mechanism is empirically falsifiable and locks all statistical parameters prior to analysis.

1. Ground Truth: GR + ΛCDM Expectations

GR + ΛCDM weak-lensing magnification implies a measurable dependence between SN Ia Hubble residuals and line-of-sight convergence. Therefore, the null hypothesis is not r = 0, but rather that any observed dependence is fully explained by standard magnification, selection effects, and survey systematics.

This baseline has been empirically detected in multiple datasets, notably in Pantheon (~3.6σ; see e.g. Shah et al. 2024) and DES-SN5YR (high significance). This protocol explicitly separates:

• (A) The standard mean magnification channel.
• (B) Any additional complexity-driven variance channel specific to SRT.

2. Observables and Sign Conventions (Frozen)

Primary observable: standardized SN Ia distance-modulus residuals

μres,i = μobs,i − μΛCDM,i

Computed after standard light-curve standardization and selection corrections. Positive μres indicates that the SN appears dimmer than predicted.

Uncertainty term: σμ,i, the published per-supernova distance-modulus uncertainty, used to normalize variance tests.

Predictors: derived from line-of-sight weak-lensing tomography κi(z):

• Mean-Shift (κ_eff,i): weighted sum/mean of convergence along the LOS.
• Complexity (K_i): multi-plane variance across tomographic bins, Ki = Varz[κi(z)].

κ(z) estimation protocol: κ_i(z) is computed from a fixed weak-lensing mass-map product using a 10 arcmin aperture and Gaussian smoothing (σ = 5 arcmin). If the κ map product is already smoothed or binned by the survey release, the release-native smoothing/bins are used without alteration.

3. The Dual-Channel Test (Fixed Models)

Validation requires passing two distinct endpoints.

3.1 Endpoint A: Mean-Shift (Baseline Lensing)

μres,i = a + γ1 κeff,i + b1 zi + b2 HostMassi + SurveyIndicators + εi

GR expectation: γ₁ < 0. Overdense LOS magnifies, making SNe appear brighter and reducing μ_res.

3.2 Endpoint B: Complexity/Scatter (SRT Channel)

Define the normalized variance proxy:

yi = (μres,i / σμ,i)²

Then fit:

yi = a + γ2 Ki + δ κeff,i + b1 zi + b2 HostMassi + SurveyIndicators + εi

SRT requirement: γ₂ > 0. Higher LOS complexity predicts additional normalized residual variance beyond baseline magnification. HostMass is included as a systematics check even if already corrected for in the main pipeline.

4. Tiered Monotonicity

Samples are binned by K (Complexity) quantiles. The tier means of y must increase monotonically:

• Tier 1: Lowest 20% K (baseline).
• Tier 4: Top 5% K (extreme tail).
• Threshold: Tier 1 vs Tier 4 separation ≥ 3σ via bootstrap.

5. No-Escape Clause

• γ₂ is consistent with zero or the 95% CI includes 0.
• Tier monotonicity fails or the 3σ separation threshold is not met.
• The γ₂ effect collapses when controlling for κ_eff.
• The sign of γ₂ does not replicate across independent datasets.

Document Hash: SRT-PREREG-V2-K-TEST-2026-LiMiT

License: CC BY 4.0. © 2026 LiMiT.

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