Further Expositions on the Theory of Entropicity (ToE) and Ginestra Bianconi’s Gravity from Entropy: How the Theory of Entropicity (ToE) Unifies Spectral Entropy with Tsallis, Rényi, Fisher–Rao, Fubini–Study, and Amari

Further Expositions on the Theory of Entropicity (ToE) and Ginestra Bianconi’s Gravity from Entropy: How the Theory of Entropicity (ToE) Unifies Spectral Entropy with Tsallis, Rényi, Fisher–Rao, Fubini–Study, and Amari–Čencov Formalisms

We note that Spectral Entropy and the Tsallis, Rényi, Fisher–Rao, Fubini–Study, and Amari–Čencov formalisms live in different “geometries” and dimensions.[a] The move that makes the Spectral Obidi Action (SOA) genuinely integrative is to treat entropy’s spectrum as the backbone, and then express each formalism as a compatible derived structure via functional calculus, deformations, and pullbacks. The idea is to think of it like “layers on a core,” not “mutually exclusive domains.”

Spectral core: the modular operator as the backbone

Core object: the modular operator Δ (or its density equivalent), whose spectrum encodes entropic content.

Spectral action:

SSOA = −Tr ln⁡Δ

More generally, SOA admits a family of spectral functionals:

Sf = Tr f(Δ)

where f is a scalar function applied to the spectrum via functional calculus. Then Kullback-Liebler (KL) divergence is recovered with f(x)=−ln⁡x; accordingly, other entropies emerge by choosing appropriate f for each case.

Operator perspective: all downstream divergences and metrics become choices of f, parametrizations of the spectrum, or geometric pullbacks induced by maps from state space to operators.

Divergence layer: Tsallis, Rényi, and f‑divergences as spectral deformations

Tsallis entropy (q‑deformation):

Sq = 1q−1(1−Tr Δ q)

This is spectral: it simply replaces ln⁡ with a power q, i.e., fq(x)=1−xqq−1.

Thus, escort distributions appear as re-weightings of the spectrum.

Rényi entropy (order α):

Hα = 11−α ln⁡Tr Δ α

Again spectral through fα(x)=xα and a logarithmic outer wrap. The “sandwiched” quantum Rényi uses modular sandwiches that are still operator‑spectral, preserving compatibility.

General f‑divergences:

Df(ρ∥σ) = Tr σ1/2 f ⁣(σ−1/2ρ σ−1/2) σ1/2

With modular Δρ∣σ, these are functions of Δ. Tsallis/Rényi are special cases, so the SOA family Sf naturally hosts them.

Interpretation: “Different structures” are different choices of spectral shaping f and normalization. They are not alien to the spectral approach of Obidi; they are embedded within it.

Metric layer: Fisher–Rao and Amari–Čencov from Hessians of divergences

Fisher–Rao as a Hessian metric:

gij(θ) = ∂2∂θi∂θj D(θ∥θ′)∣θ′=θ

Now, let us choose a divergence D (KL or any convex f-divergence derived spectrally), and the Fisher–Rao metric emerges as the local Hessian. Thus, Fisher–Rao is the second‑order geometry of the spectral divergence.

Amari–Čencov dual connections:

Given a convex potential ψ(θ) (e.g., cumulant‑generating via log‑partition), one obtains a dually flat manifold with (∇,∇\*).
In the spectral picture, ψ is induced by ln⁡Tr e−βH(θ) or by ln⁡Tr f(Δ(θ)).
The α‑connections interpolate families of divergences (KL, Rényi, Tsallis), yielding Amari’s dualistic geometry from spectral potentials.

Crucial Highlight: information geometry metrics are pullbacks of spectral divergences; that is “different dimension” is just coordinate representation, not incompatibility.

Quantum metric layer: Fubini–Study and quantum Fisher via spectral fidelity

Fubini–Study (pure states):

ds2 = 4(⟨dψ∣dψ⟩ − ∣⟨ψ∣dψ⟩∣2)

This is the Riemannian metric on projective Hilbert space. It can be seen as the differential limit of a fidelity functional, which is spectral via overlaps/eigenstructures.

Quantum Fisher information (Bures/Uhlmann):

Quantum Fisher is the Hessian of the Uhlmann fidelity F(ρ,σ), and the Bures metric arises from 1−F.
Fidelity can be expressed through the spectrum of ρ1/2σ ρ1/2 (or modular sandwiches), keeping the construction spectral.

ToE Unified view: classical Fisher–Rao and quantum Fisher/Fubini–Study are both second‑order geometries of spectral functionals (divergences or fidelities). Pure‑state Fubini–Study is the projective restriction of the quantum Fisher/Bures geometry.

Integration architecture: how ToE's SOA ties it all together in Obidi's Theory

Spectral core (operator level): Let us choose Δ and a functional family {fα}.

KL: f(x)=−ln⁡x

Rényi: fα(x)=xα with log normalization

Tsallis: fq(x)=(1−xq)/(q−1)

Divergence layer (state level): define Df(ρ∥σ) via Δρ∣σ or sandwiches; pick α/q to match the regime (heavy tails, robustness, scaling).

Metric layer (geometric level): take Hessians of Df to obtain Fisher–Rao (classical) or quantum Fisher; derive α‑connections for Amari–Čencov duality.

Projective/quantum layer: restrict to pure states for Fubini–Study; extend to mixed via Bures/Uhlmann.

Field dynamics (SOA): We then implement the Spectral Obidi Action (SOA) of the Theory of Entropicity (ToE) by selecting a governing spectral functional for the action, e.g.,

Sα = Tr Δ α,Sq = Tr fq(Δ),Sln⁡ = −Tr ln⁡Δ,

and thereafter we can derive Euler–Lagrange–type dynamics in the operator manifold. Constraints (finite‑rate, causality imposed by ToE) come from the No‑Rush [Theorem] bounds [of ToE] as additional spectral conditions.

Why this ToE formalism is coherent (and powerful)

Functional calculus is the glue: once entropy is encoded spectrally, deformations (q, α) and divergences are simply different f’s applied to the same operator spectrum.

Metrics are Hessians of divergences: Fisher–Rao and its quantum analogs are not foreign—they’re second‑order shadows of the spectral functionals.

Projective restriction: Fubini–Study is the pure‑state (rank‑1 projector) limit of the quantum information geometry induced by spectral fidelities.

Amari-Čencov duality is a choice of coordinates: α‑connections reflect different convex potentials sourced by spectral actions; changing α/q is changing the “lens,” not the backbone.

Practical leverage: We can select f (KL/Rényi/Tsallis) to match robustness or scaling, then read off the induced geometry (Fisher/Amari, Fubini–Study/Bures) and dynamics (SOA) in one pipeline.

Conclusion

We can therefore see from all of the above that the Spectral Obidi Action (SOA) of the Theory of Entropicity (ToE) is actually a generalized action that [subtly] embeds and incorporates the Tsallis, Rényi, Fisher–Rao, Fubini–Study, and Amari–Čencov formalisms under one unique spectral umbrella.

They are different structures: the various formalisms are different [viewed on a superficial level]—and the Spectral Obidi Action (SOA) succeeds in treating them as layers derived from the same spectral backbone, not as competing foundations.

Compatibility with Tsallis/Rényi: achieved via spectral deformations (powers and deformed logs) of Obidi's Δ.

Fisher–Rao and Amari–Čencov: obtained as Hessians and dual connections of the chosen spectral divergence.

Fubini–Study: the projective (pure‑state) limit of the quantum geometry induced by spectral fidelities.

Usefulness of the Spectral Obidi Action (SOA) of ToE : Thus, the Spectral Obidi Action (SOA) of the Theory of Entropicity (ToE) gives physics one operator‑level action with tunable robustness/scaling (q, α), well‑posed local geometry (metrics, connections), and a path to field dynamics and constraints—all coherent within a single spectral calculus.

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