Update README.md

README.md

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 ---
 title: Cosmology Gravity Lab
 emoji: 👀
-colorFrom: green
-colorTo: yellow
+colorFrom: indigo
+colorTo: purple
 sdk: gradio
 sdk_version: 6.1.0
 app_file: app.py
 pinned: false
 license: other
 short_description: 'Welcome to RFTs cosmology and gravity model '
+thumbnail: >-
+  https://cdn-uploads.huggingface.co/production/uploads/685edcb04796127b024b4805/TOp7INVJ6aRZAKMDObeGn.png
 ---
 
-Check out the configuration reference at https://huggingface.co/docs/hub/spaces-config-reference
+
+# Rendered Frame Theory — Cosmology & Gravity Lab
+
+This space is the primary interactive lab for **Rendered Frame Theory (RFT)** in cosmology and gravity.  
+Every module is driven by the same global coherence operator
+
+\[
+U_{\rm RFT} = \Phi\,\Gamma\,\mathcal{R}\,(1 - \Xi)\,\Psi
+\]
+
+and by the pair
+
+\[
+\Phi\Gamma = \Phi \cdot \Gamma
+\]
+
+which controls redshift mappings, lensing amplification and analytic rotation-curve scaling.
+
+The point is simple: you see exactly how changing the **same five sliders** reshapes rotation curves, κ maps, BAO/CMB structure, black hole radii, and observer-driven collapse — in one coherent framework, not a zoo of unrelated “free parameters.”
+
+---
+
+## 🔧 Global controls — what the sliders actually do
+
+These are the sliders at the top of the app. They feed every tab.
+
+- **Φ (phase / amplification)**  
+  Overall strength of coherence amplification. Higher Φ makes RFT effects stronger across all modules (rotation curves, lensing, redshift, BH radius, collapse).
+
+- **Γ (recursion scale)**  
+  Sets the microscopic recursion strength that builds BAO/CMB-like structure. Through  
+  \(\Gamma_{\rm eff} = \Gamma (1 + \Phi (1 - e^{-\Gamma}))\)  
+  it controls the effective BAO scale \(r_{\rm BAO}\) and the ℓ-peaks.
+
+- **ℛ (curvature gain)**  
+  Large-scale curvature gain. Multiplies the entire coherence operator \(U_{\rm RFT}\). Increasing ℛ globally boosts cosmological deformations (expansion, BH radius, coherence crest strength).
+
+- **Ξ (susceptibility)**  
+  How close the system is to collapse. As Ξ → 1, \(U_{\rm RFT}\) is suppressed and the system is driven towards measurement / collapse. In the observer tab we explicitly track when Ξ_total crosses 1.
+
+- **Ψ (observer phase weight)**  
+  Strength of the observer’s coupling into the field. Higher Ψ tightens the link between coherence and what is actually rendered. It appears directly in \(U_{\rm RFT}\) and in the collapse drive \(\lambda_{\rm RFT}\).
+
+Everything you see in the plots is just these five numbers being pushed through the same mathematics in different physical contexts.
+
+---
+
+## 🧩 Modules in this lab
+
+Each module is a tab in the UI.
+
+### 1. Coherence dashboard
+
+- Shows the derived quantities from your current global sliders:
+  - \(U_{\rm RFT}\), \(\Phi\Gamma\), \(\Gamma_{\rm eff}\)
+  - \(r_{\rm BAO}\) (coherence BAO scale)
+  - Approximate CMB-like peaks \(\ell_1, \ell_2, \ell_3\)
+  - Two redshift mappings: exponential and compression
+- Includes a BAO-scale vs \(\Gamma_{\rm eff}\) plot with your current point highlighted.
+
+**What to look for:**  
+Dial Φ and Γ and watch how \(\Gamma_{\rm eff}\) and \(r_{\rm BAO}\) move together. This is the “engine room” of the whole lab.
+
+---
+
+### 2. Rotation curves (analytic)
+
+- Standard baryonic rotation curve \(v_{\rm bar}(r) = \sqrt{GM(r)/r}\) for a disk + bulge.
+- RFT curve:
+  \[
+  v_{\rm RFT}(r) = v_{\rm bar}(r)\,\sqrt{\max(\Phi\Gamma, 0)}.
+  \]
+- You choose:
+  - Disk mass, bulge mass, disk scale length.
+- The plot shows:
+  - “Baryons only” vs “RFT amplified”.
+  - A 220–240 km/s band for Milky Way-like flatness.
+
+**What to look for:**  
+How much of the “dark matter” effect can be mimicked purely by \(\Phi\Gamma\) without touching the baryonic profile.
+
+---
+
+### 3. RFT gravity disk sim (N-body toy)
+
+- N-body toy galaxy disk with:
+  - Central mass + particle disk.
+  - Same initial conditions evolved with:
+    - Newtonian gravity.
+    - RFT-deformed gravity,
+      \[
+      g_{\rm RFT} = \tfrac{1}{2}\left(g_N + \sqrt{g_N^2 + 4 g_N a_0}\right),
+      \]
+      where \(a_0\) is tied to \(\Gamma_{\rm eff}\).
+- Outputs:
+  - Rotation curves for Newton vs RFT.
+  - Final spatial distribution plots for both runs.
+
+**What to look for:**  
+Whether the RFT run can sustain a flat, high-velocity outer disk without inserting a “dark halo,” purely by changing the gravitational law via \(\Gamma_{\rm eff}\).
+
+---
+
+### 4. Lensing κ maps
+
+- Computes a κ map for a Gaussian or Plummer lens using:
+  - Standard critical density:
+    \[
+    \Sigma_{\rm crit} = \frac{c^2}{4\pi G} \frac{D_s}{D_l D_{ls}}.
+    \]
+  - RFT convergence:
+    \[
+    \kappa_{\rm RFT} = \frac{\Sigma}{\Sigma_{\rm crit}} (\Phi\Gamma).
+    \]
+- Uses `astropy` distances if available; falls back to a simple mapping otherwise.
+
+**What to look for:**  
+How κ_RFT scales with \(\Phi\Gamma\) at fixed baryonic mass profile. This is the transparent alternative to “dark lens” explanations.
+
+---
+
+### 5. BAO + CMB (toy recursion spectrum)
+
+- Builds a toy recursion spectrum:
+  \[
+  P_{\rm RFT}(k) \propto k^{-1} \left[1 + \Phi\Gamma \cos\!\left(\frac{kD_A}{\sqrt{\Gamma_{\rm eff}}}\right)\right].
+  \]
+- Reports:
+  - \(\Gamma_{\rm eff}\)
+  - \(r_{\rm BAO}\)
+  - Approximate harmonic peaks \((\ell_1, \ell_2, \ell_3)\).
+
+**What to look for:**  
+How a single recursion parameter set (Φ, Γ, ℛ, Ξ, Ψ) maps to both the BAO scale and CMB-like peak spacing.
+
+---
+
+### 6. Redshift mapping
+
+- Compares three curves:
+  - FRW baseline: \(z_{\rm metric} = z_{\rm obs}\)
+  - Exponential mapping:
+    \[
+    1 + z_{\rm eff} = e^{\Phi\Gamma}
+    \]
+  - Compression mapping:
+    \[
+    1 + z_{\rm RFT} = (1 + z_{\rm obs}) \frac{1}{1 + \Phi\Gamma}.
+    \]
+- Plots \(z_{\rm rendered}\) vs \(z_{\rm obs}\) for your current ΦΓ.
+
+**What to look for:**  
+How much “redshift stretch” can be reinterpreted as coherence/observer effect instead of a hard-wired expansion history.
+
+---
+
+### 7. Black holes & LISA coherence crest (toy)
+
+- Classical Schwarzschild radius:
+  \[
+  R_S = \frac{2GM}{c^2}.
+  \]
+- RFT radius:
+  \[
+  R_{\rm RFT} = U_{\rm RFT} R_S.
+  \]
+- Adds a **coherence crest** on top of a toy LISA-band chirp by making \(\Phi\Gamma(t)\) peak around merger and perturb the frequency.
+
+**What to look for:**  
+How a temporary spike in coherence would show up as a small, structured deviation from a GR chirp in the LISA band.
+
+---
+
+### 8. Observer field & collapse
+
+- Effective susceptibility:
+  \[
+  \Xi_{\rm total} = \Xi_{\rm baseline} + \lambda_{\rm obs}\,\kappa_{\rm obs} + \Xi_{\rm slider}.
+  \]
+- Collapse drive:
+  \[
+  \lambda_{\rm RFT} = \Phi\,\Gamma\,\Xi_{\rm total}\,\Psi.
+  \]
+- The module:
+  - Plots Ξ(t) and λ_RFT(t).
+  - Flags whether Ξ_total ≥ 1 (“collapse triggered”) or not.
+
+**What to look for:**  
+How much observer coherence κ_obs you need, at your chosen Φ, Γ, Ψ, to push the system over the collapse threshold.
+
+---
+
+### 9. Math & case notes
+
+Static summary of:
+
+- The defining equations used in the lab.
+- How each module ties back to \(\Phi\Gamma\), \(U_{\rm RFT}\), and \(\Gamma_{\rm eff}\).
+- Enough detail for anyone to trace what the app is doing without guessing.
+
+---
+
+### 10. Provenance
+
+- Every run of a module logs:
+  - Module name
+  - Timestamp (UTC)
+  - Inputs (slider values, physical parameters)
+  - Outputs (key scalars)
+  - A **SHA-512 hash** over those fields
+- Records are stored in memory and (when allowed) appended to a `*.jsonl` file.
+
+You can inspect the table in the **Provenance** tab.
+
+---
+
+## ⚙️ Running the lab
+
+Locally:
+
+```bash
+pip install -r requirements.txt
+python app.py
+On Hugging Face Spaces, the app runs automatically with gradio as defined in app_file: app.py.
+
+Basic usage pattern:
+	1.	Set your global coherence field (Φ, Γ, ℛ, Ξ, Ψ) at the top.
+	2.	Pick a tab (rotation curves, lensing, disk sim, etc.).
+	3.	Adjust the physical parameters (masses, distances, timesteps).
+	4.	Click the button in that tab to compute.
+	5.	Interpret the plot using the descriptions above. All dependencies on Φ, Γ, ℛ, Ξ, Ψ are explicit.
+
+⸻
+
+🔒 Legal position & allowed use
+	•	Authorship
+Rendered Frame Theory (RFT), its coherence operators, field equations and applied models are authored by Liam Grinstead.
+	•	Protection
+This work is protected under UK copyright law and the Berne Convention. All rights are reserved unless explicitly granted in writing.
+	•	You are allowed to
+	•	Use this lab to explore and understand RFT.
+	•	Make plots, screenshots, and share results for education, open research, and discussion, with proper attribution.
+	•	Cite RFT in scientific work, referencing the relevant Zenodo DOIs.
+	•	You are not allowed to
+	•	Use RFT, its equations, or this lab to design or optimise weapons or harmful systems.
+	•	Repackage the RFT framework, operators, or code, rebrand them, and claim ownership.
+	•	Commercially exploit RFT (products, services, proprietary models, or derivative frameworks) without explicit written permission from the author.
+	•	Intent
+RFT was built to crack open the gatekeeping around cosmology and consciousness, not to hoard knowledge.
+This lab is intentionally public and mathematically transparent.
+The line is clear: no weaponisation, no quiet commercial theft, no erasing authorship.
+
+⸻
+
+📚 Citation
+
+If you use this lab or RFT concepts in your work, cite at least:
+
+Grinstead, L. (2025). Rendered Frame Theory’s Mathematical model. Zenodo.
+https://doi.org/10.5281/zenodo.17644885
+
+(Include any additional RFT DOIs that match the specific equations or predictions you use.)
+---
+Check out the configuration reference at https://huggingface.co/docs/hub/spaces-config-reference