On the Principle of Least Entropic Resistance (PoLER) in the Theory of Entropicity (ToE): Superset of the Mechanical Principle of Least Action and Obedience to the Second Law of Thermodynamics

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Principle of Least Entropic Resistance (PoLER).
Bodies, particles, and all physical systems evolve along trajectories that minimize entropic resistance, or equivalently, along paths of least entropic work.

The above principle is a ToE reformulation of the Second Law of Thermodynamics via the methodology of trajectories. It thus generalizes and hence supersedes the classical least‑action principle by embedding it within a broader entropic geometry. Where classical mechanics minimizes action, ToE minimizes entropic curvature; where classical trajectories are geodesics of a fixed metric, ToE trajectories are geodesics of an entropically induced geometry.

⭐ The Key Distinction in ToE

Entropy ≠ Entropic Curvature

In ToE:

• Entropy is the field ( S(x) ).
• Entropic curvature is the geometric structure induced by gradients, divergences, and informational thresholds of that field.

They are related, but not identical.

The Second Law governs entropy:

• ( S_{\text{total}} ) must not decrease.
• Entropy production is non‑negative.
• The universe evolves toward higher entropic states.

But PoLER (the Principle of Least Entropic Resistance) governs entropic curvature, not entropy itself.

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⭐ What ToE Minimizes

ToE does not minimize entropy.
It minimizes entropic curvature, which is the “cost” of reconfiguration.

Formally:

• Entropy increases globally.
• But the path the system takes through entropic configuration space is the one that minimizes the curvature cost of that increase.

This is exactly analogous to classical mechanics:

• A particle does not minimize distance.
• It minimizes action, which determines how it moves, not whether it moves.

Likewise:

• ToE does not minimize entropy.
• It minimizes entropic curvature, which determines how entropy increases.

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⭐ Why This Does Not Violate the Second Law

The Second Law says:

Entropy must increase (or remain constant).

ToE says:

Given that entropy must increase, the universe chooses the path of least entropic resistance — the path that minimizes entropic curvature.

These two statements are perfectly compatible.

Think of it like this:

Entropy is the destination.

Entropic curvature determines the route.

The Second Law tells you the direction of travel.
PoLER tells you the shape of the trajectory.

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⭐ A Physical Analogy

In General Relativity:

• Objects follow geodesics (paths of extremal curvature).
• But spacetime curvature itself can increase or decrease depending on mass-energy.

Similarly, in ToE:

• Systems follow entropic geodesics (paths of minimal entropic curvature).
• But entropy itself increases globally.

Minimizing curvature does not mean minimizing entropy.
It means minimizing the difficulty of entropy’s reconfiguration.

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⭐ A More Precise Statement

ToE’s variational principle can be summarized as:

Among all possible ways entropy can increase, the universe selects the one that minimizes the entropic curvature functional.

This is exactly what your emergent action encodes:

[ I_{\text{Semergent}} = \int \sqrt{-g(S)} \left[ \chi^2 e^{S/k_B} (\nabla S)^2

• V(S)

• \lambda R_{IG}[S] \right] d^4x. ]

The Euler–Lagrange equation derived from this action does not force entropy to decrease.
It forces the curvature-weighted dynamics of entropy to follow the path of least resistance.

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⭐ The Deep Insight

The Second Law is about monotonicity.
PoLER is about optimality.

They operate on different mathematical objects:

Concept	Object	Law
Entropy	( S(x) )	Must increase (Second Law)
Entropic curvature	( R_{IG}[S] ) and gradient terms	Must be minimized (PoLER)

There is no contradiction because they govern different aspects of the entropic field.

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⭐ The Resolution

Entropy increases because of the Second Law; the way it increases is determined by minimizing entropic curvature according to the Theory of Entropicity (ToE).

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Further Notes

Why ToE Minimizes Entropic Curvature Without Violating the Second Law

One of the most subtle conceptual points in the Theory of Entropicity (ToE) concerns the relationship between the Second Law of Thermodynamics and the Principle of Least Entropic Resistance (PoLER). At first glance, it may appear contradictory to assert that entropy must increase while simultaneously claiming that ToE minimizes entropic curvature. This tension dissolves once we recognize that entropy and entropic curvature are distinct mathematical and ontological objects within the theory. Their roles are related but not interchangeable.

In ToE, entropy is represented by the entropic field $S(x)$, which is the fundamental ontological substrate of reality. The Second Law governs the global behavior of this field by requiring that the total entropy of an isolated system must not decrease. This is a statement about the monotonic evolution of the entropic field as a whole. Entropic curvature, by contrast, is not the entropy itself but the geometric structure induced by the gradients, divergences, and informational thresholds of the entropic field. It is a measure of how the entropic field bends, deforms, and organizes itself across the manifold of physical configurations.

The Second Law dictates that entropy must increase, but it does not specify the manner in which this increase occurs. ToE fills this conceptual gap by introducing PoLER, which asserts that the universe evolves along trajectories that minimize entropic resistance. Entropic resistance is encoded in the curvature of the entropic field, not in the entropy itself. Thus, minimizing entropic curvature does not imply minimizing entropy; rather, it determines the optimal path through which entropy increases.

This distinction is analogous to the role of the classical action in mechanics. A particle does not minimize distance; it minimizes the action functional, which determines the form of its trajectory, not the fact of its motion. Similarly, ToE does not minimize entropy; it minimizes the entropic curvature functional, which determines the form of entropy’s evolution, not the inevitability of its increase. The Second Law provides the direction of evolution, while PoLER provides the geometric and variational structure that governs the route taken.

The emergent entropic action of ToE makes this relationship explicit. The action is given by the composite expression

$$I_{\text{Semergent}}={\int }_M{d}^{4}x\sqrt{-g(S)}[{\chi }^2{e}^{S\mathrm{/}{k}_B}({\mathrm{\nabla }}_{\mu }S)({\mathrm{\nabla }}^{\mu }S)-V(S)+\lambda R_{IG}[S]]\mathrm{.}$$

This action does not attempt to reduce the value of $S(x)$. Instead, it governs the dynamics of how the entropic field reconfigures itself. The Euler–Lagrange equation derived from this action,

$$-2{\chi }^2{\mathrm{\nabla }}_{\mu }\left(e^{S\mathrm{/}{k}_B}{\mathrm{\nabla }}^{\mu }S\right)+{\chi }^2{e}^{S\mathrm{/}{k}_B}\frac{1}{k_B}(\mathrm{\nabla }S{)}^2-V^{\mathrm{\prime }}(S)+\lambda \frac{\delta R_{IG}}{\delta S}+\frac{1}{2}\frac{\mathrm{\partial }\ln (-g(S))}{\mathrm{\partial }S}[{\chi }^2{e}^{S\mathrm{/}{k}_B}(\mathrm{\nabla }S{)}^2-V(S)+\lambda R_{IG}]=0,$$

does not impose a decrease in entropy. Instead, it determines the curvature‑weighted dynamics of the entropic field. The field evolves in such a way that the entropic curvature functional is minimized, subject to the global constraint that entropy must increase. In this sense, the Second Law and PoLER operate on different aspects of the entropic ontology: the Second Law governs monotonicity, while PoLER governs optimality.

The relationship between entropy and entropic curvature in ToE is similar to the relationship between spacetime curvature and geodesic motion in General Relativity. In GR, objects follow geodesics, which are paths of extremal curvature, but the curvature of spacetime itself may increase or decrease depending on the distribution of mass-energy. The geodesic principle does not contradict the dynamical evolution of curvature; it determines the form of motion within that evolving geometry. Likewise, PoLER does not contradict the Second Law; it determines the form of entropic evolution within a universe whose entropy must increase.

Thus, the apparent contradiction dissolves once we recognize that entropy and entropic curvature are not the same quantity. Entropy is the field; entropic curvature is the geometric cost associated with its reconfiguration. The Second Law ensures that entropy increases; PoLER ensures that it increases along the path of least entropic resistance. The universe does not choose whether entropy increases; it chooses how it increases. The Second Law provides the destination, while PoLER provides the route.

In summary, ToE does not minimize entropy. It minimizes the curvature of the entropic field, which determines the optimal trajectory through which entropy increases. The Second Law and PoLER are therefore not competing principles but complementary aspects of a unified entropic ontology. The Second Law governs the global direction of evolution, while PoLER governs the geometric and variational structure of that evolution. The two principles together form a coherent and internally consistent account of how entropy shapes the dynamics of reality.