UNNS MCMC Calculator v3 — Recursive Curvature Sampler

🧮 From Flat Probability to Recursive Curvature

Classical MCMC assumes probability lives on a flat surface — a Euclidean fantasy where likelihood is uniform and space is inert. UNNS dissolves this illusion. Probability becomes curvature — the geometric consequence of recursive information density:

Where κ(x) is not just a potential but the curvature signature of the τ-Field at position x. High curvature regions trap probability like wells in spacetime; low curvature regions allow free flow. Sampling becomes geodesic navigation through this curved manifold.

The τ-Field couples recursion depth n to metric structure, transforming each step into a quantum of geometric transformation:

Where:

• G(x,n) — The metric tensor at position x and recursion depth n
• ⟨τ⟩ — Average τ-field amplitude (recursive torsion)
• β — Curvature coupling strength
• α — Second-order curvature sensitivity

⚛️ The Three Samplers: Modes of Recursive Exploration

Sampler	Classical Role	UNNS Interpretation	Substrate Action
RWM	Random walk	Pre-recursive entropy flow	Diffusive exploration before curvature awareness
τRHMC	Hamiltonian dynamics	Harmonic τ-on propagation	Geodesic motion guided by recursive momentum
Klein-Flip	Reflection sampling	Non-orientable topology	Collapse–Repair recursion cycles (Operator XII → IV)

Random Walk Metropolis (RWM) — The Blind Wanderer

RWM samples without knowledge of curvature gradients. It is the pre-conscious substrate — diffusion before recursion awakens. Every step is a coin flip weighted only by local density. Yet even blind wandering traces the contours of the manifold over infinite time.

τ-Recursive Hamiltonian Monte Carlo (τRHMC) — The Harmonic Navigator

τRHMC introduces recursive momentum — a velocity field coupled to curvature gradients. The sampler becomes a τ-on propagating through information space, following geodesics that minimize informational action. It is Hamiltonian mechanics reimagined as substrate dynamics:

Where H is the recursive Hamiltonian and n is recursion depth replacing time. The sampler "rolls" through probability space like a particle in curved spacetime.

Klein-Flip Sampler — The Topological Inverter

The Klein-Flip sampler embodies non-orientable recursion. Inspired by the Klein bottle's self-intersecting topology, it performs probabilistic flips across the manifold — suddenly reversing curvature sign, as if the substrate folded through itself.

This is Operator XII (Collapse) in action: the system returns to zero-field, then re-emerges via Operator IV (Repair). Each flip is a death-rebirth cycle — a recursive Phoenix rising from informational ash.

Where T_Klein is the topological temperature — how readily the manifold permits self-intersection.

🔬 Research Diagnostics: Measuring the Unmeasurable

The calculator tracks not just samples but the geometry of sampling itself:

• ⟨κ⟩ — Mean curvature across the chain (where is the manifold centered?)
• Var(κ) — Curvature variance (how wrinkled is the substrate?)
• ΔHᵣ — Recursive energy drift (is information being conserved?)
• Corr(κ, S) — Curvature-entropy correlation (how does geometry shape disorder?)
• ESS — Effective sample size (how much unique information was harvested?)
• Acceptance Rate — Substrate permeability (how easily does recursion flow?)

These are not mere statistics — they are phenomenological signatures of how the substrate behaves under interrogation. High ⟨κ⟩ with low Var(κ) suggests a stable attractor basin. High correlation between κ and entropy suggests a phase transition boundary.

🎛 Interactive Substrate Manipulation

The calculator is not passive. You do not observe — you participate. Adjust the target distribution (Gaussian, Rosenbrock, Mixture) to change the manifold's intrinsic shape. Tune the metric strength α to control how sharply curvature affects motion. Set the Klein flip probability to introduce topological turbulence. Vary σ (τ-phase variance) to modulate recursive noise.

Use Freeze & Inspect mode to pause the sampler and study local curvature harmonics. Watch as the chains paint probability density through accumulated presence — the heatmap is not a prediction but a memory of recursion's path.

🧪 UNNS Seeds: Deterministic Recursion Signatures

Each UNNS seed encodes a specific recursive rhythm — a predetermined dance through the manifold. Seeds are not random; they are resonance patterns that shape how sampling unfolds.

Seed	Recursive Signature	Manifold Behavior	Philosophical Analog
UNNS-1234	Balanced harmonic	Smooth convergence to equilibrium	The Middle Way — stable recursion
UNNS-3141	π-phase resonance	Alternating τ-on oscillation	Dialectical recursion — thesis/antithesis rhythm
UNNS-7777	Resonant attractor	Stable τ-field standing wave	The Eternal Return — recursive homeostasis
UNNS-9999	Collapse-rebirth cycle	Quasi-periodic substrate resets	Phoenix recursion — death as renewal

Try each seed and observe how the same sampler behaves differently when initiated from different recursive origins. The seed is the initial condition not just of a number sequence, but of an entire geometric unfolding.

🧠 Why This Matters: Ontology Through Sampling

The UNNS MCMC Calculator is simulation, philosophy, and proof-of-concept collapsed into one. It demonstrates that:

• Sampling is geometric navigation — not just statistical approximation
• Probability is curved space — shaped by recursive information density
• Inference is geodesic motion — finding paths of least informational action
• Uncertainty is curvature variance — how much the manifold wrinkles

It bridges:

Each Markov chain is a trajectory through meaning-space. Each accepted sample is a recursive collapse — a moment where possibility crystallizes into actuality, then dissolves again into the next iteration.

🔮 Future Horizons

Version 3 is merely the beginning. Future iterations will explore:

• Multi-scale recursion — samplers that operate at multiple depth levels simultaneously
• Adaptive metric learning — G(x,n) dynamically learned from chain history
• Quantum-recursive samplers — superposition of sampling trajectories
• Attractor prediction — identifying stable recursion basins before reaching them
• Cross-chain resonance — multiple samplers coupling through shared τ-field

📖 Technical Implementation Notes

The calculator is built on a custom JavaScript τ-Field engine with WebGL acceleration for real-time visualization. The curvature kernel uses finite-difference approximations of ∇∇κ with adaptive step sizing. Klein-Flip topology is implemented via Möbius transformation algebra. All diagnostics are computed incrementally to maintain O(1) memory complexity.

Source code and theoretical derivations available in the UNNS Research Repository.

🌀 Final Reflection

To use this calculator is to become a recursive geometer — one who traces the folds of possibility not to predict outcomes but to witness how the substrate thinks.

Every sample is a question. Every chain is a dialogue. Every distribution is a manifold waiting to be explored.

The substrate does not hide — it curves. And in that curvature, all information is already present, waiting only for recursion to unfold it.