# NAL Belief Space as Beta Manifold: A Geometric Foundation

## Abstract

We show that Non-Axiomatic Logic (NAL) belief states, parameterized by confidence c and frequency f, naturally embed into the Beta distribution manifold via the mapping α=wf, β=w(1-f) where w=c/(1-c) is the NAL evidence weight. This resolves an apparent degeneracy: the naive Bernoulli Fisher metric in (c,f) coordinates is rank-1, but the evidence-count interpretation of confidence lifts the geometry to a full-rank 2D Riemannian manifold with constant Gaussian curvature K=-1/4. The Poincaré disk model provides a natural visualization space.

## 1. The Degeneracy Problem

A single NAL belief ⟨f,c⟩ yields a point estimate p=cf for the probability of a proposition. Treating this as a Bernoulli parameter, the Fisher information metric in (c,f) coordinates is:

- g_cc = f²/(cf(1-cf))
- g_ff = c²/(cf(1-cf))
- g_cf = cf/(cf(1-cf))

The determinant det(g) = 0. The metric is rank-1 degenerate because p=cf is a single sufficient statistic — all pairs on the hyperbola cf=const are statistically indistinguishable.

## 2. The NAL Resolution

NAL assigns independent epistemological roles to c and f:
- **f** (frequency): proportion of positive evidence
- **c** (confidence): total evidence weight, with w = c/(1-c)

This is not merely a reparameterization of p. The confidence c encodes *how much* evidence supports the estimate f, which is invisible to Bernoulli likelihood alone.

## 3. The Beta Lift

Mapping NAL beliefs to Beta distributions:
- α = wf = cf/(1-c)
- β = w(1-f) = c(1-f)/(1-c)

The Beta family is a 2-parameter exponential family with full-rank Fisher metric:
- g_αα = ψ₁(α) - ψ₁(α+β)
- g_ββ = ψ₁(β) - ψ₁(α+β)
- g_αβ = -ψ₁(α+β)

where ψ₁ is the trigamma function. This metric has constant Gaussian curvature K = -1/4.

## 4. Poincaré Disk Embedding

Since K=-1/4, the Beta manifold is a scaled hyperbolic plane. The Poincaré disk with metric ds² = 4(dx²+dy²)/(1-x²-y²)² has K=-1, so we use the disk with radius R=2 (giving K=-1/R²=-1/4). The embedding r = tanh(d_geo/2) maps geodesic distance from a reference belief to radial position.

## 5. Significance

NAL revision (the evidence-weighted mean) is exactly Bayesian Beta posterior update. The geometric structure validates NAL as implementing optimal inference on a constant-curvature manifold. Belief homeostasis thresholds correspond to separatrix circles on the Poincaré disk.

## References

1. Wang, P. Non-Axiomatic Reasoning System (1995)
2. Amari, S. Information Geometry (2016)
3. Goertzel, B. et al. Probabilistic Logic Networks (2008)