🌀How the Obidi Action Turns Information Geometry into Physical Spacetime through Einstein Metric–Action Principle (EMAP) in the Theory of Entropicity (ToE)

🧲Most people stop at: “information geometry is deep.”

ToE goes further:

Spacetime is not assumed—it emerges from the Einstein Metric–Action Principle (EMAP) applied to information geometry through the Obidi Action.

✨ 1. Information geometry is powerful—but not yet physical

Information geometry gives us:

📐 Fisher–Rao metric

🔀 α‑connections

🔄 statistical curvature

But it has no dynamics.

No evolution, no field equations, no physical content.

It’s a geometric language without a physical story.

⚡ 2. EMAP + Obidi Action: “Dynamics creates reality”

Einstein’s insight:

A metric becomes physical only when governed by an action.

The Einstein–Hilbert action makes gμν into real spacetime.

Obidi extends EMAP to entropic information geometry:

🎛️ defines an action on the information manifold

🧭 introduces a variational principle

📉 yields Euler–Lagrange–type equations

🔁 produces conserved currents and curvature responses

This is where information geometry becomes dynamical.

🧩 3. From information manifold to emergent spacetime

🔹 Step 1 — Start with the information manifold

A manifold of probability distributions with Fisher–Rao metric, entropic gradients, and entropic curvature Rij.

This is pre‑spacetime.

🔹 Step 2 — The Obidi Action on the information manifold

The Obidi Action

O[gij,S,∇,…]

is built from:

the Fisher–Rao metric

entropic curvature

α‑connections

entropic fields and potentials

Crucially: it is defined on the information manifold, not on spacetime.

🔹 Step 3 — The Master Entropic Equation

Varying the Obidi Action gives:

field equations

curvature evolution

entropic geodesics

conservation laws

These govern the dynamics of information geometry.

🌉 4. The Obidi Metric and the disformal Obidi Transformation

Here is the real bridge.

From the entropic side, we construct the Obidi Metric g~ij on the information manifold.

Via a disformal Obidi Transformation, is mapped to a spacetime metric gμν with Lorentzian signature.

The transformation enforces:

Rij⟶Rμν

Rij: entropic curvature on information manifold

Rμν: spacetime curvature in emergent Lorentzian sector

It is disformal, dynamical mapping encoded by the Obidi Action.

🌌 5. Geodesics, matter, Einstein‑type equations

Entropic geodesics (information‑optimal paths) yield spacetime geodesics (particle trajectories).

The emergent Lorentzian metric satisfies an Einstein‑type equation:

Gμν=8πG Tμν(entropic)

Gravity appears as the curvature response of entropic information.

🌟 The deep shift

Earlier approaches tried to interpret information geometry as spacetime.

ToE does something different:

Information geometry dynamical via the Obidi Action.

Use the Obidi Metric and disformal Obidi Transformation to extract a Lorentzian sector. Spacetime emerges as a solution.

📚Ref: The Canonical Archives: https://lnkd.in/gnwMP-Py