Appendix 50 · The Information Geometry of Buddha‑Worlds

50.1 Introduction: Buddha‑Worlds as Information Manifolds

This appendix introduces a complementary viewpoint: Buddha‑worlds as information manifolds.

• What is the information distance between two Buddha‑worlds?
• What is the curvature of a Buddha‑world in information space?
• Why are Buddha‑worlds “inconceivable yet perfectly coherent”?

50.2 Probability Distributions over World‑Patterns

p(ω) for ω ∈ Ω

Each Buddha‑world corresponds to a structured distribution p(ω).

𝓜 = { p_θ(ω) ∣ θ ∈ Θ }

θ encodes vows, karmic resolutions, and geometric data.

50.3 Fisher Information Metric on Buddha‑Worlds

g_ij(θ) = 𝔼_pθ [ (∂ log p_θ(ω) / ∂θⁱ) · (∂ log p_θ(ω) / ∂θ^j) ]

• High sensitivity
• High coherence

50.4 Entropy, Curvature, and Purity

H[p] = − Σ_{ω ∈ Ω} p(ω) log p(ω)

Curvature → 0  Effective dimension → 1

All parameters align along a single “vow–wisdom” direction.

50.5 KL Divergence between Worlds and Buddha‑Worlds

D_KL(p ∥ q) = Σ_{ω ∈ Ω} p(ω) log [ p(ω) / q(ω) ]

• Practice reduces D_KL
• Approaching Buddhahood → p → q
• Buddhahood → D_KL = 0

50.6 Coherence and Mutual Information in Buddha‑Worlds

I(X;Y) = H(X) + H(Y) − H(X,Y)

I(X;Y) maximized for all relevant pairs

• perfect interpenetration
• maximal coherence
• “one in all, all in one”

50.7 Huayan Reading: Information Geometry as a Modern Mirror

• Fisher metric → sensitivity to vows and wisdom
• Low entropy → purity
• High mutual information → interpenetration
• KL divergence → distance from Buddhahood

50.8 Summary: The Information Geometry of Buddha‑Worlds

• Buddha‑worlds = structured distributions p(ω)
• Fisher metric measures sensitivity
• Entropy low due to perfect organization
• Mutual information maximized
• KL divergence quantifies approach to Buddhahood

A Buddha‑world is an information manifold of perfect coherence and minimal confusion, where every pattern of beings and events is arranged by wisdom and vow into a single, boundless adornment.