ODER: Observer-Dependent Entropy Retrieval Framework

Version 1.1 — Adds τ_char fitting, γ(τ) inversion, Δ_fail diagnostics, and envelope validation (Lemma C.5).

This update enables direct testing of ODER predictions on experimental entropy data. ODER implements an observer-dependent entropy retrieval model for black hole information recovery. It replaces global Page-curve bookkeeping with a modular retrieval law defined in observer proper time.

This repository accompanies:

Cooper, Evlondo. (2025). Modular entropy retrieval in black-hole information recovery: A proper-time saturation model.
https://doi.org/10.5281/zenodo.15654115

Quick Start

Main Framework Simulation

Use this notebook to explore how entropy retrieval evolves across different observers and modular rates.

Notebook: ODER_Black_Hole_Framework_Complete_Simulation_(V2).ipynb

• Run all cells to generate retrieval curves, entropy traces, and modular diagnostics
• View plots saved to the figures/ directory
• Reproduce Figures 1–5 from the paper

Retrieval Fitting and Validation

Use this notebook to extract parameters from retrieval data and validate against theoretical predictions.

Notebook: ODER_Retrieval_Inversion_And_Validation.ipynb

• Extract τ_char and γ(τ) from retrieval curves
• Compute retrieval horizon and failure thresholds
• Validate modular signatures and envelope structure
• Test sensitivity and self-consistency

Installation

pip install -r requirements.txt

Theoretical Framework

ODER models entropy retrieval in black hole spacetimes as an observer-dependent, Lorentzian-causal process. It replaces global Page-curve bookkeeping with a local proper-time recovery law derived from Tomita–Takesaki modular flow.

Key Distinctions

• No replica wormholes or Euclidean saddles
• No island prescriptions or ensemble averaging
• Retrieval occurs in proper time, not bulk-extremal coordinates

Modular Retrieval Law

dS_ret/dτ = γ(τ)[S_max - S_ret(τ)]tanh(τ/τ_char)

Parameters:

• τ_char: Characteristic retrieval timescale
• γ(τ): Observer-dependent retrieval rate
• S_max: Maximum retrievable entropy

Simulation Features

Model Specifications:

• 48-qubit lattice simulation
• Bond dimensions D = 4 and D = 8 for modular resolution depth
• Bounded flow toward S_max with g² correlations
• MERA used as geometric anchor (no explicit tensor contractions)

Generated Visualizations:

• Retrieval rate profiles γ(τ) across observer types
• Entropy retrieval curves S(τ) for stationary, accelerating, and free-falling observers
• Bootstrap confidence bands with 200-trace resampling
• g²(τ) correlation showing tanh-modulated oscillatory patterns
• g²(t₁,t₂) heatmaps with diagonal tanh-fringed retrieval envelopes
• Bond dimension impact comparisons

Repository Structure

File	Description
ODER_Black_Hole_Framework_Complete_Simulation_(V2).ipynb	Main framework simulation
ODER_Retrieval_Inversion_And_Validation.ipynb	Parameter fitting and validation tools
requirements.txt	Python dependencies
figures/	Output directory for plots and visualizations

Extending the Framework

This framework encourages exploration and extension:

• Modify parameters: Bond dimension, qubit count, τ intervals
• Add observer classes: New spacetime backgrounds and reference frames
• Extend retrieval law: Include back-reaction effects
• Adapt platforms: BECs, photonic lattices, analog black holes
• Apply fitting tools: Experimental or simulated entropy data

Contributions that improve, stress-test, or generalize the framework are welcome.

Citation

Cooper, Evlondo. (2025). Modular entropy retrieval in black-hole information recovery: A proper-time saturation model.
https://doi.org/10.5281/zenodo.15654115

Cooper, Evlondo. (2025). ODER modular entropy simulation (Version 1.0) [Software]. Zenodo.
https://doi.org/10.5281/zenodo.15428312

License

MIT License - see LICENSE for details.

Contact

Evlondo Cooper
Email: evlocoo@pm.me