Recursive Curvature Sampler — Research Grade

What's New in v3

• Fading trajectory trails — Visual persistence of recursive paths
• τ-on phase compass — Real-time display of recursive substrate orientation
• Curvature metrics — ⟨κ⟩ average and Var(κ) variance tracking
• Pan & Zoom — Mouse wheel to zoom, click-drag to explore phase space
• Dark/Light mode — Toggle visual theme for presentations
• Freeze & Inspect — Click any point to examine local properties
• Enhanced analytics — Entropy sparklines, acceptance histograms

1. Purpose

This simulator extends classical Markov Chain Monte Carlo (MCMC) into the UNNS substrate, where probability is treated as recursive curvature rather than a flat energy surface. Each sampler (RWM, τRHMC, Klein) explores the information manifold by folding, reflecting, and rebalancing curvature.

2. The Three Samplers

Sampler	Role	UNNS Analogue
RWM (Random-Walk Metropolis)	Classical baseline: explores local entropy.	Flat-space diffusion; "pre-recursive" mode.
τRHMC (τ-on Hamiltonian MCMC)	Adds τ-field coupling and recursive metric.	Flow through curved recursion channels; captures harmonic motion of information.
Klein-Flip	Introduces non-orientable flips (like Möbius inversions).	Simulates topological jumps in recursion depth; resembles UNNS Collapse–Repair phases.

3. Controls & Interactions

• Mouse Wheel — Zoom in/out on phase space
• Click + Drag — Pan around to explore different regions
• Keyboard: S — Start/Pause simulation
• Keyboard: R — Reset (stops simulation, clears canvas, resets all state, generates new Run ID)
• Keyboard: F — Freeze & Inspect mode (click points to inspect)
• ↻ Reset Button — Stops simulation if running, exits freeze mode, fully clears canvas and resets all chains
• Target Distribution — switch between test surfaces
• RWM σ — proposal width
• τRHMC ε / L — integration parameters
• α (Metric weight) — τ-field coupling strength

4. Visual Layers

• Fading Trails — Motion memory showing recursive persistence
• Curvature Heatmap — color encodes Φ-field curvature
• τ-on Field Lines — arrows show recursive flow direction
• Seed & Phase Display — Shows current UNNS seed and τ-phase
• Curvature Legend — κ < 0 (concave/collapse) vs κ > 0 (convex/expansion)

5. Enhanced Diagnostics

Metric	Meaning
⟨κ⟩ (Avg Curvature)	Mean curvature across recent samples
Var(κ)	Curvature variance — measure of stability
ρ(κ,Hᵣ)	Correlation between curvature and recursive entropy
τ-phase (φ)	Current recursive substrate orientation
Run ID	Unique reproducibility stamp for this simulation

6. Scientific Features

• Deterministic RNG — Mulberry32 PRNG ensures perfect reproducibility
• Adaptive Tuning — Self-adjusts toward optimal acceptance rates
• Curvature-Entropy Tracking — Monitors correlation between geometry and information
• Run ID Stamping — Each simulation gets unique identifier for experiments

7. Understanding the UNNS Seed

The seed label "UNNS-####" combines classical randomness with recursive substrate initialization. The numeric part controls the RNG, while also defining a τ-on phase:

τ₀ = e^(i·2π·(seed mod 10000)/10000)

This phase determines initial curvature basin and topological symmetry.

8. 🧪 Example UNNS Seeds to Try

Here are experimental UNNS seed examples you can try in the calculator to explore different recursive and curvature behaviors. Each seed encodes both a numeric random source and a τ_on-phase initialization.

🧩 Canonical Seeds (Baseline)

Seed	Description	Expected Behavior
UNNS-0001	Neutral flat origin	Starts near zero curvature; behaves like standard MCMC
UNNS-0420	Low recursive curvature	τon field is shallow; slower mixing, rhythmic oscillation
UNNS-1234	Balanced harmonic mode	Smooth convergence, good metric stability (default)
UNNS-3141	π-phase seed	Alternating τon resonance; slight spiral trajectories
UNNS-2718	e-phase seed	Asymmetric growth, tends toward one curvature well

🔮 Curvature-Dominant Seeds (Deep Recursive Flow)

Seed	Description	Behavior
UNNS-8888	High τon magnitude	Deep curvature wells; strong field pull, chaotic yet rich
UNNS-9999	Collapse-rebirth cycle	Causes quasi-periodic resets (Klein-Flip dominance)
UNNS-2025	Future harmonic	Stable rhythm, small ΔHr drift — near harmonic equilibrium
UNNS-7777	Resonant attractor	Self-tuned feedback loops, rhythmic τon oscillation
UNNS-4040	Null-curvature offset	τon field cancellation; flat diffusion, almost classical

🌀 Topological / Klein-Phase Seeds

Seed	Description	Behavior
UNNS-1313	Dual-fold Möbius start	Alternating inversion every ~100 steps
UNNS-6666	Negative recursion depth	Begins in inverted τon phase (anti-harmonic drift)
UNNS-8181	Bi-curved offset	Competing τon channels; two attractor loops appear
UNNS-5050	Half-phase resonance	Balanced, but often exhibits slow Klein coupling
UNNS-9990	Collapse pre-cursor	Early drift toward recursive singularity (edge of chaos)

⚙️ Practical Tips for Experiments

• Compare trajectories by fixing all parameters except the seed.
• Plot ΔH_r drift over time — some seeds stabilize (harmonic), others diverge (chaotic).
• Turn on audio mode — each seed yields a distinct harmonic signature due to its τ_on-phase.
• Use the Freeze & Inspect mode (press F) to examine specific points in phase space.
• Export your results with the unique Run ID for perfect reproducibility.

9. Export & Reproducibility

• Export JSON — Saves complete state including Run ID
• Import JSON — Restores exact simulation state
• Run ID — Unique identifier combines seed + timestamp for perfect reproducibility

10. Applications & Use Cases

The UNNS MCMC Calculator is more than a visual demo—it's an experimental, exploratory research tool that sits between computational physics, information geometry, and recursive mathematics. Its applicability spans both didactic purposes (teaching recursion–curvature relationships) and theoretical research (probing curvature-based stochastic models).

🧮 1. Scientific Modeling & Simulation

• Recursive Markov Processes — Extends standard MCMC by embedding recursion depth n and curvature κ, providing a framework for self-referential probability distributions.
• τ-on Field Dynamics — Simulates how sampling in curved information space may behave differently from Euclidean spaces, relevant for quantum information geometry, gravitational stochastic fields, and adaptive manifolds.
• Adaptive Diagnostics — Displays acceptance rates, ESS, curvature–entropy correlations—useful for studying convergence and "field coherence" across nested sampling layers.

🎨 2. Educational and Explanatory Tool

• Visualizes concepts like recursion depth, phase coupling, and curvature propagation in an intuitive way.
• Acts as an interactive textbook for students learning about Markov processes, Hamiltonian dynamics, and recursive systems.
• The color, sound, and τ-phase representations link math with sensory feedback, ideal for cross-disciplinary education (mathematics, art, computation).

🧠 3. Algorithmic Experimentation

• Serves as a sandbox for non-linear MCMC development—researchers can modify kernels, curvature mappings, or τ-field equations.
• Offers a basis for curvature-aware optimizers, bridging Bayesian inference and UNNS recursion.
• Enables testing of τ-on–guided sampling: potentially better mixing in multimodal distributions with recursive coherence.

🌌 4. Conceptual & Theoretical Insight

• Translates Shannon-style linear information flow into curvature-recursive geometries, allowing study of how information itself curves and repairs.
• Demonstrates how collapse and reflection cycles (τ-on loops) might generalize equilibrium sampling beyond classical thermodynamic assumptions.
• Acts as a bridge model for UNNS extensions toward cosmological analogs (recursive substrate fields, dark-energy-like potential flattening).

🔧 In Practical Terms

You can use this calculator to:

• Compare classical and recursive samplers side-by-side
• Measure curvature effects on convergence or entropy in real time
• Generate reproducible τ-seeded experiments with unique Run IDs
• Educate students or colleagues about recursion, τ-fields, and stochastic geometry through immediate visual and sonic feedback
• Prototype new curvature-aware algorithms or adaptive sampling methods
• Explore connections between information geometry and recursive substrate theory

Research Note: This v3 calculator is designed as a research-grade tool. All metrics are scientifically grounded, and simulations are perfectly reproducible given the same seed and parameters.