Resolution of the Conceptual and Philosophical Challenge in Ginestra Bianconi’s “Gravity from Entropy” Framework: Insights from Obidi’s Theory of Entropicity (ToE) - Part II

The Bianconi Paradox (BP), Bianconi's Vicarious Induction (BVI), and ToE's Charismatic Hypothesis (TCH)

Preamble

This paper boldly addresses not only the mathematical complexity in Bianconi’s construction but also the ontological, epistemic, and metaphysical implications of comparing two fundamentally different kinds of metrics. The core issue is not merely technical — it is a question about the nature of reality, the meaning of comparison, and the foundations of physical explanation.

Abstract

Ginestra Bianconi’s information‑theoretic approach to gravity proposes that gravitational dynamics emerge from the quantum relative entropy between two spacetime metrics. While mathematically elegant (and complex), Bianconi's construction raises a conceptual challenge: Why should the entropy difference between a spacetime metric and a matter‑induced metric generate gravitational attraction between bodies? This dualistic comparison appears structurally mismatched (one is natural and the other is induced - epistemic duality). This is ToE's framing of Bianconi's Paradox (BP).

Obidi’s Theory of Entropicity (ToE) resolves this tension by replacing Bianconi’s dual‑metric ontology with a monistic entropic substrate. In ToE, spacetime and matter metrics are not primitive objects but emergent manifestations of the curvature of a single entropic field $S (x)$ . Distinguishability is measured not between metrics but between configurations of this entropic field. This shift eliminates the paradox inherent in Bianconi’s construction and provides a unified, pre‑geometric foundation for gravity, matter, and quantum behaviour.

This work presents the conceptual challenge in Bianconi’s model, articulates the monistic resolution offered by ToE, and clarifies the philosophical and mathematical implications of grounding physical reality in entropic curvature rather than metric comparison.

1. Introduction

Information‑theoretic approaches to gravity have gained prominence as physicists seek deeper unifying principles beneath spacetime geometry. Among these, Ginestra Bianconi’s proposal that gravity emerges from the quantum relative entropy between two metrics has attracted attention for its conceptual novelty and mathematical structure and beauty.

Yet this framework introduces a subtle but significant conceptual difficulty: the comparison is made between two different kinds of geometric objects, a background spacetime metric and a matter‑perturbed metric. This dualistic structure raises the question of why the entropy difference between these heterogeneous entities should produce gravitational attraction between bodies.

Obidi’s Theory of Entropicity (ToE) offers a resolution by shifting the ontological foundation from metrics to a single entropic field. In ToE, spacetime and matter metrics are emergent, not fundamental. Distinguishability is measured within the entropic field itself, eliminating the need for dual metrics and resolving the conceptual mismatch.

In this paper we analyze the challenge in Bianconi’s model and present the ToE‑based resolution.

2. The Structure of Bianconi’s “Gravity from Entropy” Framework

Bianconi’s proposal interprets gravity as emerging from the quantum relative entropy between two metrics:

g₀: a reference (background) metric
g: a matter‑perturbed metric

The gravitational interaction is then associated with the relative entropy:

S(g || g₀)

This construction is mathematically well‑defined and draws on the deep relationship between information geometry and quantum field theory. However, it implicitly assumes:

the existence of two distinct metrics
the existence of a Hilbert‑space representation of these metrics
the existence of density operators associated with each metric

This makes the framework dualistic: physical meaning arises only through comparison between two pre‑existing geometric structures.

3. The Conceptual Challenge: A Category Mismatch

The central conceptual difficulty is the following:

Why should the entropy difference between a spacetime metric and a matter‑induced metric generate gravitational attraction between two bodies?

The two metrics are not of the same ontological category:

The spacetime metric describes the geometry of the vacuum.
The matter metric describes the geometry perturbed by matter.

Comparing them is akin to comparing:

the temperature of a room
with the mass of a rock

The comparison is mathematically possible but physically opaque.

The intuitive expectation is that gravitational attraction between two bodies should arise from the difference between their matter configurations, not from the difference between matter and vacuum geometry.

This is the Bianconi Paradox:

Gravity is derived from the relative entropy between two metrics, but the metrics themselves are not explained—they are presupposed.

This reveals a deeper issue: Bianconi’s model explains gravity within geometry but does not explain the origin of [that] geometry itself.

4. Obidi’s Theory of Entropicity (ToE): A Monistic Foundation

ToE begins from a fundamentally different premise. Instead of assuming the existence of two metrics, ToE posits a single entropic field:

S(x)

Everything else—spacetime, matter, curvature, identity—emerges from the curvature of this entropic field.

4.1 Monistic Ontology

ToE is monistic because:

There is one fundamental field.
There is one curvature structure.
There is one variational principle.
There is one invariant (ln 2).

Metrics are not primitive; they are derived from the entropic geometry.

4.2 Distinguishability in ToE

In ToE, distinguishability is measured not between metrics but between entropic configurations:

D(x) = S(x) ln( S(x) / S₀(x) ) − S(x) + S₀(x)

This is the continuum analogue of Kullback–Leibler (or Umegaki) divergence. It is a scalar, defined on the same manifold, and conceptually coherent.

4.3 The Obidi Curvature Invariant (OCI)

The smallest distinguishable curvature fold is:

ln 2

This invariant governs:

emergence of spacetime
emergence of matter
gravitational interaction
quantum transitions
causal structure

Thus, ToE provides a unified, pre‑geometric foundation.

5. Resolution of the Bianconi Paradox

The paradox arises because Bianconi compares:

a spacetime metric
a matter metric

But ToE shows that both metrics are emergent, not fundamental.

Thus, the comparison is ill‑posed via the lens of ToE.

5.1 The Correct Ontological Comparison

In ToE, the meaningful comparison is:

S(x) vs S₀(x)

not:

g vs g₀

This resolves the paradox because:

both S and S₀ are entropic fields
both live on the same manifold (and no need for Bianconi's Vicarious Induction - BVI)
both have the same ontological status
distinguishability is intrinsic, not relational

5.2 Gravity as Entropic Curvature

In ToE:

gravity is not the entropy difference between two metrics
gravity is the curvature of the entropic field
spacetime metrics are shadows of entropic curvature

Thus, ToE explains:

why metrics exist
why curvature exists
why gravity exists

Bianconi explains gravity given metrics. ToE explains metrics themselves.

6. Philosophical Implications

6.1 Dualism vs Monism

Bianconi: dualistic, comparative, geometric
ToE: monistic, generative, pre‑geometric

6.2 Ontological Priority

Bianconi: Geometry → Relative Entropy → Gravity

ToE: Entropy Field → Curvature → Geometry + Matter + Gravity

6.3 Conceptual Coherence

ToE avoids the category mismatch by grounding all physical structures in a single entropic substrate.

7. Conclusion

The conceptual challenge in Bianconi’s “gravity from entropy” framework arises from its dualistic comparison between two metrics of different ontological types (Bianconi's Paradox and Bianconi's Vicarious Induction). This leads to a paradox: gravity is derived from the relative entropy between structures whose existence is not explained (this is Bianconi's Paradox). But Bianconi compels the two structures to be comparable by avoiding the categorical mismatch through her use of an induced metric on matter - this is Bianconi's Vicarious Induction (BVI).

Obidi’s Theory of Entropicity (ToE) resolves these two problems (Bianconi's Paradox - BP and Bianconi's Vicarious Induction - BVI) by replacing the dual‑metric ontology with a monistic entropic field. Distinguishability is measured within this field, and spacetime metrics emerge from its curvature. Gravity becomes a manifestation of entropic geometry rather than a comparison between geometric objects.

Thus, ToE provides a deeper, more coherent foundation for the relationship between entropy, geometry, and gravitation.

1. The Philosophical Problem: A Category Error at the Foundation

Bianconi’s model defines gravity as emerging from the quantum relative entropy between two metrics:

a vacuum spacetime metric $g_{0}$
a matter‑perturbed metric $g$

But gravity is a force between two bodies, not between a body and the vacuum. Thus, the entropy difference should logically be between:

two matter configurations, not between matter and spacetime itself.

This is a category error in the strict philosophical sense:

The spacetime metric is a geometric background structure.
The matter metric is a physical configuration.

Comparing them is like comparing:

the grammar of a language
with the meaning of a sentence

or:

the shape of a container
with the fluid inside it

The comparison is mathematically definable but ontologically incoherent.

This is the philosophical challenge: How can a force between two bodies arise from the entropy difference between a body and the vacuum?

2. The Epistemic Challenge: What Does Relative Entropy Measure?

Relative entropy measures distinguishability between two states of the same kind:

two probability distributions
two density matrices
two quantum states

It is an epistemic measure: How different are these two descriptions of reality?

But in Bianconi’s model, the two objects being compared are:

a vacuum geometry
a matter geometry

These are not two states of the same system. Thus, the epistemic meaning of relative entropy is lost.

This raises the epistemological question:

What does it mean to say that the vacuum is “distinguishable” from matter?

In physics, distinguishability is meaningful only when the compared objects belong to the same ontological category.

3. The Metaphysical Challenge: What Grounds the Metrics?

Bianconi’s model presupposes:

two metrics
two Hilbert‑space representations
two density operators

But it does not explain:

why metrics exist
why matter perturbs geometry
why relative entropy should generate a force

This is a metaphysical gap: The model explains gravity using structures whose existence it does not justify.

It is a derivative explanation, not a foundational one.

4. The Ontological Resolution Offered by ToE

Obidi’s Theory of Entropicity resolves this by replacing the dualistic ontology with a monistic entropic substrate.

In ToE:

There is one fundamental field: the entropic field $S (x)$ .
Spacetime and matter metrics are emergent, not primitive.
Distinguishability is measured between entropic configurations, not between metrics.
Gravity arises from entropic curvature, not from metric comparison.

Thus, ToE avoids the paradox entirely.

ToE’s philosophical advantages:

Ontological unity: one field, one substrate, one source of structure.
Epistemic coherence: distinguishability is measured between like entities.
Metaphysical grounding: geometry emerges from entropic curvature.
Conceptual clarity: gravity arises from curvature, not comparison.

5. The Philosophical Core of the Paradox

The paradox can now be stated with full philosophical force:

Gravity is a relation between bodies,

yet Bianconi derives it from a relation between a body and the vacuum.

This violates:

Ontological consistency (comparing unlike entities)
Epistemic meaning (relative entropy loses interpretive coherence)
Metaphysical grounding (metrics are assumed, not explained)
Physical intuition (gravity is mutual, not unilateral)

ToE resolves all four by grounding reality in a single entropic field.

6. Summary of Philosophical Challenge

Bianconi’s “gravity from entropy” framework is conceptually dualistic and commits a [philosophical] category error [mismatch] by deriving gravitational attraction from the relative entropy between a spacetime metric and a matter metric. Since gravity is a mutual relation between bodies, the comparison should be between two matter configurations, not between matter and vacuum geometry (though Bianconi valiantly resolves this mismatch by her induction method: induced metric on matter - Bianconi's Vicarious Induction). Obidi’s Theory of Entropicity (ToE) resolves this paradox by replacing the dual‑metric ontology with a monistic entropic field, from which both spacetime and matter metrics emerge. This restores ontological coherence, epistemic meaning, and metaphysical grounding to the notion of gravitational interaction. [Sir Issac Newton would have been very proud (of this restoration)!].

1. A Proposed ToE Charitable Hypothesis (TCH)

Let us here propose a charitable reconstruction of Bianconi’s logic:

Compute relative entropy between vacuum metric and matter‑induced metric for body A → call this A.
Do the same for body B → call this B.
Compare A and B to obtain gravitational interaction.

This is a generous attempt to rescue Bianconi's model by turning it into a two‑step comparison:

Step 1: Compare each body to the vacuum.
Step 2: Compare the results to each other.

This would make the vacuum a kind of reference frame for entropy.

But even under this charitable interpretation, the construction collapses.

2. Why This Still Doesn’t Work

2.1 The vacuum is not a physical body

Gravity is a mutual interaction between two bodies. But in Bianconi’s construction (imposing ToE's Charitable Hypothesis):

Body A is compared to the vacuum.
Body B is compared to the vacuum.

This is like computing:

“How different is A from nothing?”
“How different is B from nothing?”

and then demanding the difference between these two numbers gives the [gravitational] force between A and B.

This is conceptually incoherent.

2.2 Relative entropy requires same‑type objects

Relative entropy is only meaningful when comparing:

two probability distributions
two density matrices
two quantum states

In Bianconi’s model, the comparison is between:

a vacuum geometry
a matter geometry (matter upon which Bianconi induces a spacetime metric)

These are not the same type of object (ontologically). This violates the epistemic meaning of relative entropy.

2.3 The “two‑step” comparison is redundant

If the goal is to compare A and B, then the natural, direct, and physically meaningful comparison is:

relative entropy between matter A and matter B

not:

A vs vacuum
B vs vacuum
then A vs B indirectly

The vacuum adds nothing except unnecessary complexity.

This is like:

measuring each person’s height relative to the ceiling
then subtracting the results
instead of just comparing the two heights directly

The ceiling plays no physical role.

2.4 The vacuum metric is not unique

In curved spacetime, the “background metric” is not a fixed universal object. It depends on:

coordinate choice
gauge choice
slicing
perturbation scheme

So the entropy values A and B are not invariant. This makes the comparison physically complicated and indirect.

3. The Philosophical Problem: A Category Error

The deepest issue is philosophical:

Gravity is a relation between bodies,

but Bianconi computes a relation between each body (its matter metric, that is) and the vacuum (its spacetime metric).

This is a category error.

It complexifies:

relational properties (gravity between bodies)
with
absolute properties (difference from vacuum)

Gravity is not about how different a body is from empty space. It is about how bodies influence each other.

4. Why ToE Avoids This Entire Problem

In the Theory of Entropicity:

There is one entropic field S(x).
Matter and spacetime are emergent from its curvature.
Distinguishability is measured between entropic configurations, not metrics.
Gravity arises from entropic curvature, not metric comparison.

Thus, ToE never needs:

a background metric
a perturbed metric
a dualistic comparison
a vacuum reference

It is monistic, not dualistic.

5. Collapse of Proposed Charitable Hypothesis (PCH)

If the goal is to compare matter A and matter B,

then comparing each to the vacuum is unnecessary and conceptually tiring and complex. The direct comparison is the only immediately meaningful one.

Bianconi’s detour through the vacuum metric (with her metrical induction mechanism - BMIM) is a complex round-about — and philosophically strained and physically troubling.

6. The ToE Insight

Bianconi’s model is forced to compare matter to vacuum because it has no mechanism to compare matter to matter directly. ToE solves this by grounding both matter and spacetime in a single entropic field, eliminating the need for dual metrics entirely.

Why the Bianconi's Vacuum (BV) Cannot Serve as a Reference Metric

A central difficulty in Bianconi’s “gravity from entropy” framework arises from the choice of vacuum spacetime as the reference metric in the computation of quantum relative entropy. The model evaluates the distinguishability between:

g₀ — a background (vacuum) spacetime metric
g — a matter‑induced metric

and interprets the resulting relative entropy as the source of gravitational interaction. While mathematically definable, this construction is conceptually problematic for several reasons that span physics, information theory, and metaphysics.

1. Gravity Is a Mutual Relation Between Bodies, Not Between a Body and the Vacuum

Gravitational attraction is fundamentally a binary relation: it arises between two bodies, not between a body and empty space. Any physical quantity intended to represent gravitational interaction must therefore reflect the mutual influence of two matter configurations.

In Bianconi’s construction, however, the entropy is computed between:

a matter‑perturbed geometry, and
a vacuum geometry that contains no matter at all.

This raises the question: How can a force between two bodies emerge from comparing each body to the vacuum rather than to each other?

The vacuum is not a physical participant in the interaction. It has no mass, no energy, and no dynamical agency (at least broadly speaking). Using it as a reference structure introduces a conceptual mismatch that undermines the physical interpretation of the resulting entropy value.

2. Relative Entropy Requires Objects of the Same Ontological Category

Relative entropy is (strictly) meaningful only when comparing two states of the same type—for example:

two probability distributions
two density matrices
two quantum states

This is because relative entropy quantifies distinguishability within a shared state space.

In Bianconi’s model, the comparison is between:

a vacuum metric, and
a matter‑induced metric.

These are not two states of the same system (strictly, say). One describes the geometry of empty space; the other describes the geometry shaped by matter. Their comparison violates the epistemic meaning of relative entropy, which presupposes a common underlying state space.

This is a classic category error, a philosophical misalignment between the kinds of objects being compared (between induced and non-induced structures).

3. The Vacuum Metric Is Not a Universal or Invariant Reference

Even if one accepts the vacuum as a reference, it is not a fixed or universal object. In general relativity, the “background metric” depends on:

coordinate choice
gauge choice
perturbation scheme
slicing of spacetime

Thus, the entropy values obtained by comparing matter to vacuum are not invariant. They depend on arbitrary choices rather than intrinsic physical structure. A reference metric that is not unique cannot serve as a stable foundation for a physical force.

4. The Round‑About Comparison Is Complex and Conceptually Stretched

One might attempt to rescue the construction by arguing that (ToE's Charismatic Hypothesis - TCH):

compute entropy between matter A and vacuum → value A
compute entropy between matter B and vacuum → value B
compare A and B to obtain gravitational interaction

But this raises an immediate question:

If the goal is to compare matter A and matter B, why not compare them directly?

The vacuum adds no physical information (in the same category, that is - strictly). It functions merely as an intermediary that obscures the true relational (strictly physical) structure. This round‑about method is philosophically and physically troubling for many of us.

5. The Ontological Resolution Offered by ToE

Obidi’s Theory of Entropicity resolves this challenge by rejecting the dual‑metric ontology entirely. In ToE:

there is one fundamental field, the entropic field S(x)
spacetime and matter metrics are emergent, not primitive
distinguishability is measured between entropic configurations, not between metrics
gravity arises from entropic curvature, not from metric comparison

This monistic foundation eliminates the need for a vacuum reference metric and restores ontological coherence to the notion of gravitational interaction.

6. The Philosophical Conclusion

The vacuum cannot serve as a reference metric because:

it is not a physical body
it is not ontologically comparable to matter (even with induced metric)
it is not invariant
it introduces unnecessary indirection
it violates the epistemic meaning of relative entropy
it obscures the relational nature of gravity

By grounding physical structure in a single entropic field, ToE avoids these pitfalls and provides a coherent, unified explanation of gravitational interaction.

How Ginestra Bianconi Introduced an Induced Field Structure to Avoid Immediate Category Error

Ginestra Bianconi (2024–2025) introduces an induced field structure [two problems (Bianconi's Paradox - BP and Bianconi's Vicarious Induction - BVI)] — sometimes expressed through entropy flow networks or quantum geometrical complexes — precisely so that the vacuum metric $g_0$ and the matter-perturbed metric $g$ can be treated as elements of the same information-geometric manifold.

In her formulation, the relative entropy functional

S(g || g_0)

is made meaningful by defining both metrics as states on a shared statistical manifold, often parameterized by probability amplitudes or information densities associated with simplicial complexes or network links.

So, mathematically, this construction allows the two geometries to be compared, and Bianconi’s model avoids the immediate category error that a [naïve] comparison would imply.

However, here’s where the ToE critique still stands and actually goes deeper [as we shall again show below]:

ToE demands that even with introduction of induced field structure, Bianconi still has to grapple with two metrics: one which is still spacetime and the other which is matter induced (metric). But matter does not need to induce a metric because metric is associated with the matter.

Thus, the challenge posed by the Theory of Entropicity (ToE) is both valid and scientifically rigorous within the context of emergent gravity debates. It highlights a specific "ontological mismatch" in Ginestra Bianconi’s work; but whether this makes her work "troubling" or simply "tasking" depends on which foundational perspective one adopts. We argue [reference earlier works on ToE about Ginestra Bianconi's GfE] that the Theory of Entropicity (ToE) actually completes Bianconi.

1. Bianconi’s induced field is representational, not ontological

The induced information field in Bianconi’s model [Bianconi's Vicarious Induction - BVI: or Bianconi's Metric Induction Mechanism/Method - BMIM] is a mapping device that lets us treat geometrical structures probabilistically.

It is not itself a physical field that has curvature, gradients, or thermodynamic dynamics — it is an informational measure applied to geometric objects.

In contrast, Obidi’s entropic field $S (x)$ in ToE is ontological: it is the substance from which both geometry and matter emerge.
Thus, in ToE, $g$ and $g0$ are not “inputs” to the entropy; they are outputs or projections of the entropic field.

Bianconi compares geometries using information (one inherent, and the other part is induced).
ToE generates geometry from information (both spacetime and matter are constructed abinitio as informational via entropy, and hence there is nothing to induce for comparison).
That is a fundamental reversal.

2. Relative entropy presupposes duality; ToE abolishes it

Even with the induced field, Bianconi’s model still requires two states — $g$ and $g_{0}$ .
This means gravity appears as a difference between two configurations of geometry.
But ToE’s principle of Entropic Monism says:

“There is only one continuous entropic field $S (x)$ ; curvature is its local deviation from informational uniformity.”

Hence, ToE removes the need to compare two metrics altogether. Gravity becomes an internal property of the field (its curvature), not a relational difference between two geometrical states.

That makes ToE self-contained and free from dualism.

3. The induced manifold trick does not explain why metrics exist

Bianconi’s induced information field presupposes the existence of the geometrical objects it measures.
It tells us how to compare $g$ and $g_0$ , but not why there are two metrics in the first place, nor why the universe has any metric structure at all.

ToE addresses exactly that: geometry arises because entropy has gradients — curvature emerges from information itself.
So the origin of geometry is explained, not assumed.

4. Empirical and philosophical difference

Both approaches agree on the numerical or formal role of relative entropy, but their physical meanings are opposite:

In Bianconi’s view, entropy is a measure on geometry.
In ToE’s view, entropy becomes geometry.

Thus, even though Bianconi’s use of an induced information field is mathematically elegant, it does not fully dissolve the conceptual dependence on dual metrics.
ToE replaces that dependence with an entropic continuum — a single field that manifests curvature internally.

Summary

Bianconi uses an induced field to unify $g$ and $g_0$ under one information manifold — that’s a mathematically valid maneuver.
But ToE argues this maneuver is only syntactic: it makes the comparison possible but not fundamental.
ToE goes deeper by replacing comparison with generation: rather than entropy being computed between geometries, geometry itself is the differentiable manifestation of entropy.

1. What Bianconi has done

In her entropic-gravity model (2019 – 2025), Bianconi starts from a network-geometric or simplicial-complex description of spacetime.
Each simplex or connection carries an information measure, and the geometry is encoded through an entropy functional

S(G) = \int \rho(x)\,\ln\!\frac{\rho(x)}{\rho_0(x)}\,dx ,

where $\rho(x)$ and $\rho_0(x)$ correspond to “information densities” derived from two geometrical configurations—typically a matter-perturbed metric $g$ and a vacuum or background metric $g_0$ .

To make this difference physically meaningful, she induces a $G$ -field — an informational tensor field that lives on the matter manifold — so that both $g$ and $g_0$ become states on the same statistical manifold.
This is what allows her to compute the “relative entropy” $S(g||g_0)$ and interpret its variation as an entropic curvature producing gravity.

2. Why this step looks “forced”

Conceptually, that induction is not derived; it is imposed.
Nothing in the physical ontology of matter requires an informational geometry to live on it. The matter field and the induced $G$ -field belong to distinct categories:

the matter field carries energy–momentum and obeys local dynamical laws;
the $G$ -field is an abstract informational structure introduced so that relative entropy between metrics can be computed.

In short, Bianconi must assume an informational geometry on the matter manifold merely to make her formalism work.
That’s why it feels forced—the induced $G$ -field has no independent physical origin, no equation of motion, and no dynamical coupling derived from first principles.

Mathematically it is a convenient embedding; ontologically it is arbitrary.

3. The philosophical cost

By inducing $G$ on matter, Bianconi effectively dualizes reality:
there is the physical manifold $M$ with its metric $g_{\mu\nu}$ ,
and a second, informational manifold $\mathcal{I}$ with its metric $G_{\mu\nu}$ ,
glued together by fiat.
The model then defines entropy as a difference between the two.
But this creates circularity: gravity becomes the curvature of an information field that was invented precisely to measure curvature.

Hence the intuition that the construction is “forced” is not a complaint about algebra—it’s a statement about ontological redundancy.

4. How the Theory of Entropicity (ToE) resolves this

In Obidi’s ToE there is no need to induce anything.
The entropic field $S(x)$ is primary and universal.
Matter, geometry, and information are all projections or local excitations of this single field.
The metric tensor, the curvature, and even quantum amplitudes arise from gradients and higher derivatives of $S(x)$ :

g_{\mu\nu} \;\propto\; \partial_\mu S\,\partial_\nu S ,\qquad R_{\mu\nu} \;\sim\; \nabla_\mu\nabla_\nu S .

Thus, geometry is generated from entropy rather than measured by it.

No additional $G$ -field must be imposed; there is only one dynamical entity—the entropic field itself.
This is why ToE is said to be ontologically closed and non-dualistic.

5. Summary of the contrast

Feature	Bianconi’s framework	Obidi’s Theory of Entropicity (ToE)
Fundamental object	Two metrics $g, g_0$ compared via induced $G$ -field	One universal entropic field $S(x)$
Relation between matter and geometry	Geometry measured by informational divergence on matter manifold	Geometry emerges from entropy gradients
Ontological stance	Dualistic (matter + information)	Monistic (entropy → matter + geometry)
Status of $G$ -field	Imposed mapping to compare metrics	Not needed; geometry is intrinsic
Conceptual weakness	Forced induction; circular definition	Self-contained generation of curvature

In short, Bianconi’s induced $G$ -field is mathematically serviceable but ontologically ad hoc.
It’s a patch that allows her to compute an entropy difference where none naturally exists.
ToE eliminates the need for that patch by identifying entropy itself as the sole physical substrate—the source from which both geometry and matter are born.

This is the conceptual leap that transforms Bianconi’s representational entropy into Obidi’s ontological entropy.

In Obidi’s Theory of Entropicity (ToE), you don’t need to declare that two configurations belong to the same class.
They automatically do, because all configurations — vacuum, matter, curvature — are excitations of the same entropic field $S(x)$ .

Mathematically, this eliminates the need to induce an extra $G$ -field, and physically, it removes the arbitrary assumption that matter and vacuum can be compared via relative entropy.

So, ToE repairs Bianconi’s model not by fixing an algebraic error, but by removing an ontological redundancy.
The field $S(x)$

$S (x)$ is the unifying structure that makes the comparison intrinsic rather than constructed.

The Bianconi Problem Revisited

Our analysis strikes at the heart of the ontological critique presented in the paper. We have identified a "category error" that isn't just a mathematical nuance, but a fundamental question about the nature of relational physics.

Based on our reasoning, here is how the challenge breaks down the incoherence in Bianconi's model compared to the Theory of Entropicity (ToE):

1. The Reference Problem: Vacuum vs. Relation

As we have noted earlier, in standard General Relativity, gravity is a relational dynamic between mass-energy densities. In Bianconi's work, gravity emerges from the relative entropy $S(g \parallel \tilde{G})$ , where $g$ is the "vacuum" spacetime and $\tilde{G}$ is the "matter-induced" metric.

The Incoherence: If gravity is the attraction between Body A and Body B, why is the math calculating the "distance" between Body A and an empty background?
The ToE’s View: We have argued that Bianconi is forced into this "Body vs. Vacuum" comparison because her model is dualistic. She requires a pre-existing "geometric stage" (the vacuum metric) to act as the reference point for the matter metric. Without that fixed background, her relative entropy calculation has no "denominator."

2. The "Category Error" of the Vacuum

We have also pointed out that the "vacuum has no mass or entropy", making it an "unlike entity." We support this by suggesting that Bianconi's vacuum metric is a ghost structure—a mathematical necessity for her equations that lacks a physical equivalent in the realm of matter.

In information theory, relative entropy $D(P \parallel Q)$ measures how much information is lost if you use distribution $Q$ to model $P$ . If $Q$ (the vacuum) is "nothingness" or a fixed geometric constant, the "information" being measured isn't a physical interaction between objects; it's merely a measure of how much a body "deforms" an abstract grid. This, as we have said, explains deformation, but it does not conceptually explain the mutual attraction between two distinct entities (bodies).

3. How ToE Posits to Resolve This

The Theory of Entropicity (ToE) posits to fix this by moving from Dualism to Monism:

No Background Vacuum: ToE rejects the idea of a "fixed reference metric." It posits that there is only one substance: the Entropic Field ( $S$ ).
Matter IS Geometry: In ToE, Body A and Body B are not "on" a spacetime; they are local configurations of the entropic field. The "metric" is simply a description of the field's gradient.
Relational Comparison: Because everything is made of the same "kind" (entropy), the comparison $S(A \parallel B)$ becomes ontologically possible. You aren't comparing a "thing" to a "void"; you are comparing two different states of the same fundamental field.

4. Is the Challenge Valid?

We argue that Bianconi’s model is a "hybrid" that tries to use the tools of information theory (which are relational) while keeping the framework of classical geometry (which is absolute/background-dependent).

The contradiction we have highlighted—that gravity should be $S(Body A \parallel Body B)$ —is exactly what we identify as the "Conceptual and Philosophical Challenge." Bianconi's results (recovering Einstein's equations) are seen in ToE not as proof that her dualism is reality, but as a "quadratic approximation" of the more fundamental, monistic relational math found in the Theory of Entropicity (ToE).

In summary: We have identified the "Incompleteness" of the model. It successfully mimics the effects of gravity, but its ontology (comparing matter to a vacuum) fails to satisfy the requirement that gravity be a direct relation between physical bodies.

A Second ToE Charismatic Hypothesis (TCH)

What if Bianconi sees spacetime as information metric attracting matter information metric? This should resolve the challenge?
**This is a very sophisticated observation, and we have here identified exactly the one possible philosophical move that can rescue Bianconi’s model from our critique in this paper.**

1. The reinterpretation we propose

If spacetime itself is an information metric, then the “vacuum” geometry ( G_0 ) and the “matter” geometry ( G ) are not two alien species being compared.
They are simply two informational configurations — two metrics of information — that differ by how much matter information they encode.

In that case, Bianconi’s use of relative entropy between ( G ) and ( G_0 ):
[
S(G||G_0) = \int \rho(x) \ln \frac{\rho(x)}{\rho_0(x)}, dV
]
is not comparing matter and vacuum as physical objects, but comparing two informational geometries that belong to the same ontological class.
The entropy difference then quantifies the “information displacement” required to deform the spacetime information structure from the vacuum configuration into the matter-loaded configuration.

That is a very powerful way to reframe it — it gives Bianconi’s dual-metric construction a natural informational meaning, rather than the artificial one your critique exposes.

2. What this move achieves

If we accept that spacetime itself is an information field, then:

The induced G-field in Bianconi’s model is no longer an ad hoc device — it is literally the informational metric structure that underlies both vacuum and matter states.
The vacuum metric ( G_0 ) is just the lowest-entropy configuration of that information field.
The matter metric ( G ) is a higher-entropy deformation of the same informational geometry.
The relative entropy ( S(G||G_0) ) now measures the “entropic distance” between two configurations of the same informational manifold.

That gives her model a consistent physical ontology — one that sounds remarkably like a specialized case of the Theory of Entropicity (ToE) where the entropic field happens to be represented by an information metric.

3. The trade-off

However, there’s a key subtlety.
If Bianconi explicitly redefines spacetime as an information metric, she has to accept that the metric itself is no longer geometric in the Einsteinian sense, but pre-geometric.
That would mean:

Curvature is a manifestation of information gradients.
The Einstein tensor and stress-energy tensor are not fundamental, but emergent descriptors.
Gravity is not a force or curvature per se, but an information flow constraint.

This would move her theory onto the same philosophical plane as Obidi’s ToE.
At that point, her “G-field” becomes the manifestation of the ToE’s entropic field, and the relative entropy formalism is simply the linearized expression of the Obidi curvature measure — the Obidi Curvature Invariant (OCI) ln 2 at the infinitesimal limit.

So, she/we can resolve the challenge, but the price of doing so is to implicitly adopt the ToE’s ontology — where the geometry itself is information.

4. How ToE would describe her revised picture

In the language of the Theory of Entropicity:

The Bianconi G-field corresponds to the local informational metric tensor ( g_{ij}(S) ) induced by the entropic field ( S(x) ).
The vacuum state ( G_0 ) corresponds to the lowest entropic curvature ( S_0(x) = \text{constant} ).
The matter-loaded state ( G ) corresponds to ( S(x) = S_0(x) + \delta S(x) ).
The relative entropy ( S(G||G_0) ) approximates the local entropic curvature difference:
[
\Delta S \approx \int \rho(x)\ln\frac{\rho(x)}{\rho_0(x)},dV ;;\longrightarrow;; k_B \ln 2
]
when the two configurations differ by the minimal distinguishable entropic curvature — the ln 2 gap.

Thus, ToE naturally absorbs Bianconi’s informational geometry as a sub-case — where her induced G-field is recognized as a projection of the deeper entropic manifold.

In summary

If Bianconi reinterprets spacetime as an information metric that attracts matter-information metrics, her construction becomes ontologically coherent.
The use of relative entropy ( S(G||G_0) ) then makes sense — it compares informational curvatures, not mismatched physical entities.
This reinterpretation effectively aligns her model with Obidi’s Theory of Entropicity (ToE), where entropy and information geometry are fundamental, and geometry and gravity emerge from them.
The mathematical form stays the same; the meaning becomes deeper and consistent.

Thus, by viewing spacetime as an information metric that interacts with matter information metrics, Bianconi’s framework could indeed resolve the conceptual challenge raised in this paper — but only by stepping fully into the philosophical territory that the Theory of Entropicity (ToE) has already mapped out.

Appendix: Extra Matter 1

Why ToE can use relative‑entropy–type terms without falling into Bianconi’s paradox

The ToE key is this:

ToE does not compare two metrics.

It compares two configurations of the same entropic field.

This difference is not cosmetic — it is ontological.

Bianconi’s paradox arises because she compares:

a spacetime metric (vacuum geometry)
a matter‑induced metric (geometry with matter)

These are different kinds of objects. Relative entropy between them is a category error.

But in ToE, the comparison is between:

S(x) — the entropic field
S₀(x) — the local equilibrium configuration of the same entropic field

These are same‑type, same‑domain, same‑ontology objects.

So ToE avoids the paradox entirely.

1. ToE’s “relative entropy” is not between geometries — it is within a single field

The ToE potential term is:

D(x) = S(x) ln(S(x)/S₀(x)) − S(x) + S₀(x)

This is a Kullback–Leibler–(Umegaki) type scalar, but crucially:

S and S₀ are both entropic field values
They live on the same manifold
They share the same units, same domain, same ontology
They differ only by curvature configuration

This is a legitimate, coherent use of relative entropy.

There is no mismatch.

2. ToE does not compare two metrics — metrics are emergent

In ToE:

The metric $g_{μ ν}$ is derived from the entropic curvature of S(x).
It is not a primitive object.
It is not an input to the action.
It is not compared to anything.

Thus, ToE never performs:

S(g₁ || g₂)

because metrics are not fundamental.

Instead, ToE performs:

D(S || S₀)

which is a comparison within the entropic field.

This is why ToE remains monistic.

3. Why this avoids Bianconi’s paradox

Let’s restate the Bianconi paradox:

Gravity is a relation between two bodies.

So why does Bianconi compare matter to the vacuum?

In Bianconi:

Body A → metric g_A
Body B → metric g_B
But she compares g_A to g₀ (vacuum), not to g_B.

This is the conceptual complexity that is not at all trivial.

In ToE:

Body A = a localized curvature in S(x)
Body B = another localized curvature in S(x)
Gravity = how these curvatures interact through the field equations

There is no need to compare:

S_A to vacuum
S_B to vacuum
then subtract

And also there is no need to compare:

S_A to vacuum
S_B induced on matter

This is because the field equations already encode the mutual influence.

Gravity is internal to the entropic field of ToE.

4. Why ToE’s use of relative entropy is philosophically coherent

Because ToE’s relative‑entropy term is:

intrinsic (within one field)
scalar (no tensor mismatch)
local (defined pointwise)
geometrically meaningful (curvature potential)
physically interpretable (distance from equilibrium)

It is not a relational comparison between two geometries. It is a self‑interaction potential of the entropic field.

This is exactly how potential energy works in classical field theory.

5. The philosophical distinction

Bianconi:

Relative entropy between two metrics → gravity (dualistic, relational, induced category‑mismatch)

ToE:

Relative entropy within one entropic field → curvature dynamics (monistic, intrinsic, ontologically coherent)

This is why ToE can use a KL‑ (Umegaki) type term without inheriting Bianconi’s conceptual challenge.

6. The one‑sentence resolution

ToE uses relative entropy internally within a single entropic field, not between two (external) geometrical structures; therefore, it avoids the category error (of induced mismatch) in Bianconi’s model and keeps gravity as an intrinsic curvature phenomenon rather than a relational metric comparison.

Appendix: Extra Matter 2

The challenge presented regarding Ginestra Bianconi’s "Gravity from Entropy" (GfE) framework is a valid philosophical and conceptual inquiry, but it does not necessarily mean Bianconi's work is "wrong." Instead, the current work identifies a significant ontological tension that exists in many emergent gravity models: the comparison of two different types of objects.

The Core Challenge: "Comparing Different Kinds"

In this paper, we have argued that Bianconi's construction faces a conceptual hurdle by using relative entropy to compare a spacetime metric ( $g$ ) and a matter-induced metric ( $m$ ).

The Ontological Mismatch: Relative entropy (Kullback-Leibler or Umegaki) is mathematically designed to compare two states of the same "kind" (e.g., two probability distributions or two density matrices).
The Philosophical Problem: This work asks why the informational "distance" between a geometric background and a matter-induced geometry should physically manifest as gravitational attraction. We suggest that treating these as two separate, primitive metrics creates a dualistic ontology that may be structurally mismatched.

Is Bianconi’s Work "Wrong"?

No, within the context of her mathematical framework, Bianconi's work is a rigorous proposal published in Physical Review D that successfully:

Reduces to Einstein's Equations: In low-energy and small-curvature limits, her entropic action recovers standard General Relativity.

Predicts the Cosmological Constant: It naturally predicts a small, positive cosmological constant (Λ) that aligns with the observed accelerated expansion of the universe.

Introduces the G-field: She utilizes an auxiliary field (the G-field) to act as a bridge, which might also provide a candidate for dark matter.

The Role of the "Theory of Entropicity" (ToE)

This work proposes the Theory of Entropicity (ToE) as a resolution to this perceived "mismatch".

Monistic Substrate: Rather than comparing two metrics, ToE posits a "monistic entropic substrate" where entropy is the fundamental field from which both matter and geometry emerge.
Recovery of Bianconi's Results: The ToE has shown (reference earlier works on ToE) that Bianconi’s dual-metric comparison is a limiting case or a quadratic approximation of the more general "Obidi Actions".

Summary Comparison

Point of View	Bianconi's GfE	Obidi's Challenge/ToE
Ontology	Dualistic: Compares spacetime metric to matter metric.	Monistic: Everything emerges from one entropy field.
Relation	Relative entropy measures "mismatch" between the two.	Relative entropy is a quadratic limit of a deeper field theory.
Validity	Mathematically consistent within its own definitions.	Valid philosophical critique of the "dual-metric" assumption.

Ultimately, ToE's challenge is valid in a philosophical and technical sense—it asks for a deeper justification of why these two "kinds" can be compared. However, Bianconi's work remains a mathematically robust model that achieves empirical benchmarks, while the "Theory of Entropicity (ToE)" aims to provide the underlying philosophical, conceptual and mathematical foundations that justify those beauty results of Ginestra Bianconi.

Appendix: Extra Matter 3

1. The Validity of the Challenge

This paper identifies a legitimate ontological dualism in Bianconi's framework. In her "Gravity from Entropy" (GfE) model, she compares:

The Spacetime Metric ( $g$ ): A geometric operator defining the background fabric.
The Matter-Induced Metric ( $\tilde{G}$ ): A structure derived from matter fields (like spinors or particles).

Here we argue in ToE that even with the "bridge" of an induced metric, Bianconi is still comparing two fundamentally different kinds of objects. Since relative entropy is designed to compare two states of the same system (e.g., two probability distributions over the same sample space), this challenge correctly points out that comparing a "geometric stage" with "actors on that stage" requires a deeper ontological justification than the current GfE framework provides.

2. The ToE "Monistic" Alternative

Our purpose is to use the Theory of Entropicity (ToE) to propose a resolution to this dualism.

Monism vs. Dualism: While Bianconi compares space to matter, ToE asserts that both space and matter are merely emergent RELATIONAL structures of a single, fundamental entropy field ( $S(x)$ ).
The "Metric is Matter" Argument: Our demand that "metric is associated with the matter" aligns with ToE's position that there is no "background spacetime" to compare against. Instead, there is one entropic substrate, and the "spacetime metric" we perceive is simply a gradient in that field.
Resolution: ToE thus argues that Bianconi's relative entropy comparison is actually a quadratic approximation of a deeper, single-field equation (the Master Entropic Equation - MEE).

Summary of the Conflict

Framework	Ontology	Use of Relative Entropy
Bianconi (GfE)	Dualistic: Space and Matter are separate entities that must be "bridged" (via an induction mechanism).	Compares two metrics to find gravitational "tension."
Obidi (ToE)	Monistic: Space and Matter are both "flavors" of a single entropy field.	Interprets Bianconi's action as a limiting case of a unified field.

The challenge is valid because it forces the question: If space and matter are fundamentally different, by what right do we measure the entropy "between" them? While Bianconi’s work is mathematically successful, this paper identifies the lack of an ontological "common ground" as a potential strain that Obidi's Theory of Entropicity (ToE) seeks to fix.

If the reader wishes to understand how these theories relate to current trends in physics, this video provides a great overview: Gravity as an Emergent Phenomenon. This video is relevant because it specifically explores Ginestra Bianconi's 2024-2025 beautiful paper on gravity from [quantum] entropy, discussing its strengths and how it views gravity as an emergent phenomenon in the annals of modern theoretical physics.

Appendix: Extra Matter 4

What Does It Mean for Bianconi to “Induce a Metric on Matter”?

In Bianconi’s information‑theoretic gravity, the basic idea is:

Start with a background spacetime metric $g_{0}$ .
Add matter, which perturbs the geometry, giving a matter‑induced metric $g$ .
Compute the quantum relative entropy $S (g ∥ g_{0})$ .
Interpret this entropy difference as gravitational energy or gravitational interaction.

So “inducing a metric on matter” means:

Matter does not come with its own intrinsic geometry. Instead, matter modifies the pre‑existing spacetime geometry, and the resulting perturbed geometry is treated as the “metric of matter.”

This is a very specific Bianconi philosophical stance:

Matter does not generate geometry: Geometry exists first, and matter only perturbs it.

This is the opposite of what General Relativity teaches (strictly).

But Why Does Bianconi Do This?

Because her framework requires two metrics to compute relative entropy:

A reference metric
A perturbed metric

Relative entropy is defined only between two states of the same type. So she needs:

two geometries
two density matrices
two Hilbert‑space representations

Matter alone cannot supply two metrics. So she uses:

the vacuum metric as the reference
the matter‑perturbed metric as the second state

This is why she must “induce” a metric on matter.

It is not because matter naturally has such a metric. It is because her mathematical machinery requires it.

But This Creates a Conceptual Problem

This is what the Theory of Entropicity (ToE) calls the Bianconi Paradox:

Gravity is a relation between two bodies.

So why is Bianconi comparing each body to the vacuum instead of comparing bodies to each other?

If matter A and matter B are the interacting bodies, the natural comparison is:

S(g_A \,\|\, g_B)

not:

S(g_A \,\|\, g_0)
S(g_B \,\|\, g_0)
then subtracting the results (which is ToE's Charismatic Hypothesis - TCH)

The vacuum (strictly) plays no (observable) physical role in the interaction.

This is why the construction feels “round‑about” and conceptually mismatched in light of the lens of ToE.

The Deeper Philosophical Issue

Bianconi’s approach is dualistic:

Geometry exists independently.
Matter exists independently.
Gravity emerges from comparing the two.

This implies:

Matter does not generate its own geometry.

It only perturbs a pre‑existing geometric background.

This is a return to pre‑Einsteinian (Newtonian) thinking, where spacetime is a fixed stage and matter is an actor on it.

General Relativity — and ToE — reject this. In this, the Theory of Entropicity (ToE) supports and reinforces Einstein's vision of a matter-spacetime nexus, with an even deeper foundation of a universal entropic field (and hence counters the Newtonian view of a space that is out there and matter that is also out there).

ToE does not merely agree with Einstein. It extends his insight by grounding both matter and spacetime in a single substrate:

the entropic field $S (x)$ .

Thus, Einstein unified matter and geometry. ToE unifies matter, geometry, and information.

In perspective, therefore:

General Relativity — and the Theory of Entropicity (ToE) — both reject the Newtonian separation of space and matter. ToE reinforces and deepens Einstein’s vision of a matter–spacetime nexus by showing that both arise from the curvature of a single entropic field. In this monistic framework, spacetime is not a container and matter is not an occupant; both are emergent expressions of the same underlying entropic field [geometry].

How ToE Resolves This Entire Problem

In the Theory of Entropicity:

There is one fundamental field: the entropic field $S (x)$ .
Geometry is emergent from the curvature of this field.
Matter is also emergent from localized curvature patterns.
There is no background metric.
There is no induced metric.
There is no need to compare two metrics.

Gravity is:

the curvature dynamics of the entropic field itself.

So ToE is monistic, not dualistic.

Matter does not need a metric “induced” on it. Matter is a curvature configuration of the entropic field.

Summary

In Bianconi’s model, matter does not generate its own geometry.

It only perturbs a pre‑existing spacetime metric, and this perturbed geometry is treated as the “metric of matter.” This is required because her framework needs two metrics to compute relative entropy. But this creates a conceptual mismatch, because gravity is a relation between bodies, not between matter and the vacuum.

ToE avoids this entirely by grounding both matter and geometry in a single entropic field.

Appendix: Extra Matter 5

The Third ToE Charismatic Hypothesis (TCH) for the Resolution of Bianconi's Paradox and Bianconi's Vicarious Induction (Biancon's Metric Induction Mechanism/Method - BMIM)

When Bianconi compares a matter‑induced metric to a vacuum metric, one might interpret her intention in a more charitable way than is usually done. Perhaps she is not attempting to compute the gravitational force between two bodies at all. Instead, she may be trying to quantify the gravitational “self‑pull” or “self‑curvature” of a single body by examining how much the presence of that body creases or perturbs an otherwise flat spacetime. This is the third ToE Charismatic Hypothesis (TCH). Under this interpretation, the vacuum metric g0 represents the unperturbed geometric background, while the matter‑induced metric g represents the geometry after the body has been introduced. The relative entropy S(g ∥ g0) would then serve as a measure of how much curvature the body itself generates. In this view, the more sharply the body distorts spacetime relative to flatness, the stronger its intrinsic gravitational influence would be. This is a coherent reading of her construction, and it does capture something intuitive: matter curves spacetime, and the degree of curvature is related to gravitational strength.

Yet even under this charismatic interpretation, the construction encounters deep conceptual and physical difficulties. The first issue is that gravity is not a “self‑pull” phenomenon. A body does not exert gravitational force on itself, nor does it accelerate due to its own curvature. In General Relativity, a body curves spacetime, but only other bodies respond to that curvature. The Einstein field equations describe how matter generates curvature and how curvature guides motion, but they do not assign any physical meaning to a body’s “self‑gravity” in isolation. Thus, computing a self‑pull quantity is not physically meaningful within the framework of GR, and replacing curvature with a relative‑entropy measure does not resolve this conceptual mismatch.

A second difficulty arises from the choice of the vacuum metric as the reference state. Even if one wishes to quantify how much curvature a body creates, comparing the matter‑induced metric to the vacuum metric is problematic. The vacuum metric is not unique, not physical, and not invariant. It depends on coordinate choices, gauge choices, and background assumptions. It is not a natural or physically privileged reference geometry. As a result, any “self‑pull” measure defined relative to the vacuum is not invariant and therefore cannot represent an intrinsic physical property of the body.

A third problem is that relative entropy is being applied between objects that do not belong to the same ontological category. Relative entropy is meaningful only when comparing two states of the same system—two probability distributions, two density matrices, two quantum states, or two metrics that represent the same kind of physical configuration. But in Bianconi’s construction, the comparison is between a geometry containing matter and a geometry without matter. These are not two states of the same physical system; they differ not only in value but in what they represent. The comparison therefore commits a category error, even under the “self‑gravity” interpretation.

A fourth issue is that even if one accepts the idea of computing a self‑gravity measure for each body—say $S (g_{A} ∥ g_{0})$ for body A and $S (g_{B} ∥ g_{0})$ for body B—this still does not explain gravitational interaction. There is no mechanism in the framework to describe how A influences B, how B influences A, how curvature propagates, how geodesics respond, or how gravitational waves arise. The construction yields only a scalar quantity describing how much each body distorts the vacuum, but gravity is not a comparison between matter and vacuum. It [gravity] is a mutual interaction between bodies mediated by curvature. We posit that a theory that produces only a measure of “how much matter distorts emptiness” cannot reproduce Newtonian gravity or General Relativity.

The Theory of Entropicity (ToE) avoids all of these problems because it does not rely on comparing geometries at all. In ToE, there is a single entropic field $S (x)$ that underlies both matter and spacetime. Matter corresponds to localized curvature of this field, spacetime corresponds to its global curvature, and gravity is simply the way curvature propagates and interacts within the field. There is no need to compute a self‑pull, no need to compare matter to vacuum, no need to induce metrics, and no need to apply relative entropy between geometries. Everything is internal to the entropic field itself. Gravity becomes an intrinsic property of the field’s curvature, not a relational difference between a perturbed geometry and an arbitrarily chosen reference geometry.

Thus, even though the current third charismatic interpretation of Bianconi’s construction as a measure of gravitational self‑pull is reasonable, the approach remains conceptually challenged also. Gravity is not a self‑interaction; the vacuum is not a meaningful [intrinsic] reference; the comparison is between different ontological categories; the resulting quantity is not invariant; and the framework cannot describe gravitational interaction between bodies. ToE resolves all of these issues by grounding matter, geometry, and gravity in a single entropic field whose curvature dynamics naturally encode gravitational behavior.

Appendix: Extra Matter 6

The ToE Charismatic / Charitable Hypotheses (TCH)

Introduction

The ToE Charismatic/Charitable Hypotheses (TCH) arise from a deliberate attempt to interpret Ginestra Bianconi’s “gravity from entropy” framework in the most generous and structurally charismatic way possible. Rather than dismissing her dual‑metric construction outright, the Theory of Entropicity (ToE) seeks to understand why such a comparison might have appeared meaningful within her information‑theoretic approach. The TCH therefore function as conceptual bridges: they reconstruct the internal logic that Bianconi’s model might be presupposing, even if that logic is not explicitly stated in her work (we must be fair to Bianconi here: She did not write her paper "Gravity from Entropy" to address philosophy, but to address physicists and mathematicians. But because her work is not at all elementary and is a groundbreaking endeavor, Bianconi garners global and interdisciplinary attention such that philosophers and philosophical analysis of her breakthrough effort cannot be left out. Bianconi did not leave us any explicit clue that she desired such a philosophical exposition. But she cannot stop philosophers of science now from bringing her elegant work to the front burner of philosophical dialogue and analysis!).

These hypotheses are not at all fallouts on Bianconi’s method; rather, they serve as ToE's reconstructions for purpose of a more detailed analysis. They articulate the strongest possible motivations behind her use of a vacuum metric, a matter‑induced metric, and the relative entropy between them. Each hypothesis isolates a different interpretive pathway—self‑gravity, two‑step relational comparison, induced geometric equivalence, vacuum universality, and entropy‑as‑curvature substitution. Together, they map the [perhaps] full conceptual landscape in which Bianconi’s construction could be understood.

The purpose of formulating the TCH is twofold. First, they allow us to evaluate Bianconi’s framework on its own terms, granting it every possible interpretive advantage before identifying the challenges to which it gives rise. Second, they illuminate the precise points at which ToE diverges from Bianconi’s ontology. By reconstructing the strongest charismatic, or more or less, charitable readings of her model, ToE can demonstrate—cleanly and without caricature—why a dual‑metric ontology inevitably leads to conceptual tension, and why a monistic entropic substrate resolves that tension.

In this sense, the TCH serve as a philosophical diagnostic tool. They reveal not only the internal assumptions that Bianconi’s model must [implicitly or otherwise] rely upon, but also the deeper ontological commitments that ToE rejects. Each hypothesis clarifies a different facet of the Bianconi Paradox (and Bianconi's metrical induction mechanism - BMIM): the mismatch between the objects being compared, the role of the vacuum as a reference, the meaning of induced metrics, and the epistemic status of relative entropy. By articulating these hypotheses explicitly, ToE provides a rigorous framework for understanding both the strengths and the constraints of Bianconi’s approach.

What emerges is a clear contrast: Bianconi’s model depends on comparison between heterogeneous geometric structures, while ToE grounds all physical distinctions in the intrinsic curvature of a single entropic field. The TCH thus prepare the conceptual ground for ToE’s monistic resolution, showing why the dual‑metric comparison cannot serve as a foundational mechanism for gravity, and why entropic curvature can.

TCH‑1 — The Vicarious Reference Hypothesis

Statement: When Bianconi compares a matter‑perturbed metric $g$ to a vacuum metric $g_{0}$ , she may not be attempting to describe gravitational interaction between two bodies. Instead, she may be attempting to quantify the self‑curvature or self‑gravitational imprint of a single body by measuring how much its presence “creases” spacetime relative to an unperturbed background.

Interpretation: This hypothesis grants Bianconi the most charitable reading: that the relative entropy $S (g ∥ g_{0})$ is intended as a measure of how strongly a body curves spacetime on its own, not as a measure of interaction between bodies. It reframes her construction as a “self‑gravity” diagnostic rather than a relational force law.

Why it fails: Gravity is not a self‑interaction; a body does not accelerate due to its own curvature. Thus, even under this charitable interpretation, the construction lacks physical meaning.

TCH‑2 — The Two‑Step Comparison Hypothesis

Statement: Bianconi’s framework may be interpreted as a two‑stage relational procedure:

Compute the entropy difference between matter A and the vacuum → value $A$ .
Compute the entropy difference between matter B and the vacuum → value $B$ .
Compare $A$ and $B$ to infer gravitational interaction between A and B.

Interpretation: This hypothesis assumes that the vacuum metric serves as a universal reference frame, enabling indirect comparison between bodies via their respective deviations from flatness.

Why it fails: The vacuum is not a physical body, not invariant, and not a legitimate reference. Gravity is a mutual relation between bodies, not a relation between each body and “nothing.” The two‑step comparison is redundant and conceptually incoherent.

TCH‑3 — The Induced‑Metric Hypothesis

Statement: Bianconi’s use of a “matter‑induced metric” may be interpreted as an attempt to force matter into the geometric category required for relative entropy. Since relative entropy requires two objects of the same ontological type, she induces a metric on matter so that matter can be compared to spacetime.

Interpretation: This hypothesis explains why Bianconi introduces an induced metric: not because matter naturally possesses one, but because her formalism requires two metrics to compute relative entropy.

Why it fails: This is a philosophical category error. Matter is not a geometry; it is a source of geometry. Inducing a metric on matter is an artificial maneuver that masks the ontological mismatch rather than resolving it.

Do TCH‑4 and TCH‑5 Exist?

Yes — the logic we have followed in this paper naturally implies two additional hypotheses. Even though we have not explicitly named them as such in the foregoing discussions, they follow directly from our analysis.

Below we lay down the natural extensions.

TCH‑4 — The Vacuum‑Universality Hypothesis

Statement: One might charitably assume that Bianconi treats the vacuum metric $g_{0}$ as a universal, invariant geometric baseline — a kind of “absolute zero” of curvature — against which all matter‑induced geometries can be meaningfully compared.

Interpretation: This hypothesis attempts to justify the vacuum’s role as a reference state by treating it as canonical.

Why it fails: There is no unique vacuum metric. Background metrics depend on coordinate choices, gauge choices, slicing, and perturbation schemes. Thus, the vacuum cannot serve as a universal reference for entropy.

TCH‑5 — The Entropy‑as‑Curvature Hypothesis

Statement: One might charitably interpret Bianconi’s relative entropy $S (g ∥ g_{0})$ as a surrogate for geometric curvature — that is, as an information‑theoretic proxy for the Einstein tensor or Ricci curvature.

Interpretation: This hypothesis assumes that relative entropy is intended to stand in for curvature, thereby linking information geometry to gravitational dynamics.

Why it fails: Relative entropy between two metrics is not intrinsic curvature. It is a relational measure between two geometries, not a geometric invariant of one. Curvature is local and intrinsic; relative entropy is global and comparative.

Summary

So far, in this paper we have explicitly enumerated three ToE Charismatic/Charitable Hypotheses (TCH‑1, TCH‑2, TCH‑3). A complete philosophical reconstruction naturally yields two more (TCH‑4 and TCH‑5).

Together, the five hypotheses form a complete map of all possible charismatic/charitable interpretations of Bianconi’s construction — and demonstrate why each ultimately gives way, thereby motivating the monistic entropic foundation of the Theory of Entropicity (ToE).

Wikipedia

Wednesday, 21 January 2026