Derivations of the Obidi Action of the Theory of Entropicity (ToE): The Actions and Action Principle of the Theory of Entropicity (ToE)—The Pre-geometric Action Construction, the Emergent Action, the Local Obidi Action (LOA) and the Emergence of Spacetime, the Spectral Obidi Action (SOA) and the Quadratic Approximation of Bianconi Gravity, and the Emergence of Entropy-Weighted Metrics via the Cauchy Functional Equation

**The Obidi Action** is the foundational variational principle in the **Theory of Entropicity (ToE)**. It governs the dynamics of the entropy field \(S(x)\) (treated as a fundamental, ontic scalar field) and gives rise to the **Master Entropic Equation (MEE)** / **Obidi Field Equations (OFE)**, from which spacetime geometry, gravity, quantum behavior, and other physical laws emerge.

Its derivation is **not postulated ad hoc** but follows rigorously from a set of minimal axioms and information-geometric principles, ensuring pre-geometric invariance, thermodynamic consistency, and unification of classical/quantum structures. There are two complementary formulations: the **Local Obidi Action (LOA)** (differential, spacetime-ready) and the **Spectral Obidi Action (SOA)** (global, operator-algebraic). They are equivalent in appropriate limits.

### 1. Conceptual Genesis (Axiomatic Foundation)

The derivation begins with the **Obidian Dictum**: entropy dictates information flow, and information flow encodes all geometry and dynamics. Entropy \(S\) is the sole primitive field on a pre-geometric configuration space \(U\) (with coordinates \(\xi^a\)), before spacetime emerges.

Key axioms (derived from minimality, invariance, and consistency):

- **Entropic primacy**: Only \(S(\xi)\) exists initially.

- **Diffeomorphism invariance**: The action must be a scalar density under reparametrizations \(\xi \to \xi'(\xi)\).

- **Locality**: Lagrangian depends only on \(S\) and its first derivatives \(\partial_a S\).

- **Compositionality** (for independent subsystems \(S_1, S_2\)): The weight factor \(W(S)\) satisfies \(W(S_1 + S_2) = W(S_1) W(S_2)\). By the Cauchy functional equation (with continuity), this forces \(W(S) = e^{\alpha S}\). Thermodynamic/dimensional consistency fixes \(\alpha = 1/k_B\), yielding the universal exponential weight \(e^{S/k_B}\).

- **Information-geometric compatibility**: Use Fisher–Rao (classical), Fubini–Study (quantum), and Amari–Čencov \(\alpha\)-connections as the base metric \(h^{ab}_{(IG)}\) on \(U\).

- **Positivity & stability**: Kinetic term bounded below; reduces to standard entropy production in weak limits.

- **Measure covariance**: Pre-geometric volume element \(p^{-\Lambda(S)} d^4\xi\) (where \(\Lambda(S)\) encodes entropic content) transforms into the spacetime measure \(\sqrt{-g(S)}\) via the emergence map \(\Phi_S: U \to M\).

These axioms uniquely determine the minimal Lagrangian density. No higher-derivative terms appear at leading order (effective-field-theory principle).

### 2. Pre-Geometric Construction

On configuration space \(U\), the Obidi Action starts as:

\[

I_S^{(\text{pre-geom})} = \int_U p^{-\Lambda(S)} \left[ \frac{\chi^2}{2} e^{S/k_B} h^{ab}_{(IG)} (\partial_a S)(\partial_b S) - V(S) + \Lambda_{IG}(S, \partial S) \right] d^4\xi,

\]

where:

- \(\chi\) is the entropic coupling constant,

- \(h^{ab}_{(IG)}\) is the information-geometric metric,

- \(V(S)\) is the self-interaction potential,

- \(p^{-\Lambda(S)}\) is the entropy measure density (ensures scalar invariance).

The exponential \(e^{S/k_B}\) arises directly from compositionality and couples entropy to the kinetic term (amplifying flow in high-entropy regions).

### 3. Emergence of Spacetime and Local Form (LOA)

As \(S\) self-organizes, the map \(\Phi_S\) induces spacetime \(M\) with metric:

\[

g_{\mu\nu}(S) = e^{S/k_B} g_{(IG)\mu\nu}

\]

(the entropy-weighted deformation of Fisher–Rao or Fubini–Study). The volume element transforms accordingly, yielding the **emergent** (then **classical**) Local Obidi Action:

\[

A_{\text{Obidi}}[S] = \int d^4x \sqrt{-g} \left[ \frac{1}{2} g^{\mu\nu} (\nabla_\mu S)(\nabla_\nu S) - V(S) + J(x) S \right],

\]

where \(J(x)\) is a source term (matter/stress-energy coupling, later identified as \(\eta T^\mu_\mu\) or similar). This is the standard form used in most derivations.

**Variational derivation of the Master Entropic Equation**:

Vary \(A_{\text{Obidi}}\) w.r.t. \(S\) (fixed \(g\)) using the Euler–Lagrange equation. The kinetic term contributes the d'Alembertian, the potential the force term, and the source the coupling:

\[

\square S - \frac{dV}{dS} + J(x) = 0 \quad (\square \equiv \nabla^\mu \nabla_\mu).

\]

This is the **MEE / OFE** — the entropic analogue of Einstein’s or Schrödinger’s equations. Varying w.r.t. \(g^{\mu\nu}\) yields the entropic stress tensor \(T^{(S)}_{\mu\nu}\), recovering dressed Einstein equations in the limit.

### 4. Spectral Obidi Action (SOA) — Global Formulation

The global, operator-algebraic version (unifying bosonic/fermionic and classical/quantum regimes) uses the modular operator from non-commutative geometry and Araki relative entropy:

\[

\Delta = G[S] \, g[S]^{-1},

\]

where \(G[S]\) is the entropy-weighted metric and \(g[S]\) the reference. The Spectral Obidi Action is:

\[

S_{\text{Obidi}} = -\operatorname{Tr} \ln(\Delta).

\]

(This is the spectral invariant of relative entropy; eigenvalues of \(\Delta\) encode probabilities/density-matrix elements, reducing to Boltzmann/Shannon/von Neumann/Tsallis/Rényi entropies in limits.)

**Variation** (via heat-kernel or modular expansion) recovers the local form and dressed Einstein equations:

\[

G_{\mu\nu} + \Lambda_{\text{ent}} g_{\mu\nu} = 8\pi G_{\text{eff}} [T_{(m)} + T_{(S)} + \cdots],

\]

with entropic cosmological term \(\Lambda_{\text{ent}} = \langle (\nabla S)^2 \rangle\).

### 5. Equilibrium Expansion & Quadratic Approximation (Link to Relative Entropy / Bianconi Gravity)

Expand around equilibrium \(S = S_{\text{eq}} + \delta S\) (where \(\square S_{\text{eq}} = 0\) and \(V'(S_{\text{eq}}) = J\)):

\[

V(S) \approx V(S_{\text{eq}}) + \frac{1}{2} M_S^2(x) (\delta S)^2, \quad M_S^2 = V''(S_{\text{eq}}).

\]

The quadratic action becomes:

\[

A^{(2)} \approx \int d^4x \sqrt{-g} \left[ \frac{1}{2} (\nabla \delta S)^2 - \frac{1}{2} M_S^2 (\delta S)^2 \right].

\]

Spectral expansion (\(\Delta = I + \epsilon\), \(|\epsilon| \ll 1\)) gives:

\[

-\operatorname{Tr} \ln(\Delta) \approx \frac{1}{2} \operatorname{Tr}(\epsilon^2),

\]

matching the local quadratic term. This recovers **relative entropy** \(D_{\text{rel}}(S \parallel S_{\text{eq}}) \simeq \frac{1}{2} k_B \int c(x) (\delta S)^2\), exactly Bianconi’s gravity-from-entropy functional (quadratic approximation of the Obidi Action).

### 6. Entropy-Weighted Metrics Emergence

Varying the action w.r.t. the metric or via the exponential weight directly deforms base information metrics:

- Fisher–Rao (classical): \(g^{(S)}_{ij} = e^{S/k_B} g^{(FR)}_{ij}\),

- Fubini–Study (quantum): analogous conformal factor,

- Amari–Čencov \(\alpha\)-connections incorporated as deformations.

This unifies Fisher–Rao / Fubini–Study / Amari–Čencov into a single entropic geometry.

The full derivation of the Obidi Action is therefore mapped like this: pre-geometric → emergent → classical/spectral limits, with all physics (GR, QM, thermodynamics) recovered as approximations. The **Obidi Curvature Invariant (OCI = \ln 2)** enters as the minimal nonzero entropic distinguishability threshold (curvature divergence) consistent with the information-geometric Hessian and distinguishability cost, though it is not needed for the action’s variational core.

This framework is fully self-consistent and derives Newton’s laws, Einstein equations, Schrödinger dynamics, Landauer’s principle, etc., as entropic consequences. Detailed appendices in the source preprints provide the complete axiomatic and variational steps.