Physics:Potency of Sean Collins' Path Integral Formulation of QM Dynamics

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On the Potency of Sean Collins' Mjolnir (A137) Lattice Dynamics Path Integral Formulation of Quantum Mechanics and Its Incorporation Into the Conceptual and Mathematical Foundations of the Theory of Entropicity(ToE)

Overview

The Mjolnir (A137) Lattice Dynamics is a physics theoretical framework developed by Sean Collins.[1][2] It introduces a discrete lattice ontology in which aggregated objects, built from base masses, move under the influence of drag forces and synchronization effects. A distinctive feature of this framework is its path‑integral formulation, which modifies the standard Feynman weighting by introducing an effective Planck constant [math]\displaystyle{ \hbar_{\text{eff}} = \hbar/\gamma }[/math].

This paper examines the potency of the Sean Collins’ formulation, highlighting its mathematical structure, physical interpretation, and potential applications. It also situates the method in relation to conventional quantum mechanics and alternative approaches such as the Theory of Entropicity(ToE), as first formulated and developed by John Onimisi Obidi. [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23] [24][25][26][27][28][29][30][31][32][33][34][35][36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77][78]

Foundations of the Theory of Mjolnir (A137) Lattice Dynamics

Ontological Assumptions

  • Base units: Aggregates are composed of fundamental base masses [math]\displaystyle{ m_b }[/math].
  • Cycle speed: Each base particle cycles at a characteristic speed [math]\displaystyle{ v_{mb} }[/math].
  • Wave speed: The lattice supports a superluminal propagation mode [math]\displaystyle{ v_{\text{wave}} = \kappa c }[/math].
  • Drag force: Motion through the lattice induces a drag term

[math]\displaystyle{ F_{\text{drag}} = k \,\rho(x)\, v_{\text{wave}}\, m\, v }[/math].

  • Effective potential:

[math]\displaystyle{ V_{\text{drag}}(x) = k \rho(x) |x| \left(\tfrac{v_{\text{wave}}}{c}\right)^2 }[/math].

Discrete and Continuum Action

Discrete Action

For a discretized path [math]\displaystyle{ x(t_i) }[/math] with timestep [math]\displaystyle{ \Delta t }[/math], the local action increment is

[math]\displaystyle{ \Delta S_i = m \frac{(x_{i+1} - x_i)^2}{\Delta t} + V_{\text{drag}}(x_i)\,\Delta t . }[/math]

Summing over slices:

[math]\displaystyle{ S_{[\text{path}]} = \sum_{i=1}^{N} \Delta Si . }[/math]

Continuum Limit

As [math]\displaystyle{ \Delta t \to 0 }[/math]:

[math]\displaystyle{ S[x(t)] = \int \left[ m \dot{x}(t)^2 + V_{\text{drag}}(x(t)) \right] dt . }[/math]

Path‑Integral Formulation

Modified Weighting

Sean Collins introduces a phase synchronization parameter [math]\displaystyle{ \gamma }[/math] into the Feynman Path Integral Formulation of Quantum Mechanics, thus we can write:

[math]\displaystyle{ Z = \int \mathcal{D}[x] \; \exp\!\left(\tfrac{i\gamma}{\hbar} S[x]\right). }[/math]

This defines an effective Planck constant:

[math]\displaystyle{ \hbar_{\text{eff}} = \frac{\hbar}{\gamma}, }[/math]

hence we obtain:

[math]\displaystyle{ Z = \int \mathcal{D}[x] \; \exp\!\left(\tfrac{i}{\hbar_{\text{eff}}} S[x]\right). }[/math]

The above seemingly innocuous integral is highly ingenious and potent; making a small tweak and inserting a probing parameter in the Feynman Path Integral turns the latter into a powerful new tool for investigating the foundations of physical reality. This forms the foundation of Sean Collins' Path Integral Formulation. Hence, the above expression is the Sean Collins Path Integral.

Identification of Path‑Integral Mapping in Sean Collins' Path Integral Formalism

  • Discrete action:

[math]\displaystyle{ \Delta S_i = m \frac{(x_{i+1} - x_i)^2}{\Delta t} + V_{\text{drag}}(x_i)\,\Delta t }[/math]

  • Path action:

[math]\displaystyle{ S_{[\text{path}]} = \sum_{i=1}^{N} \Delta S_i }[/math]

  • Partition functional with phase synchronization parameter \(\gamma\):

[math]\displaystyle{ Z = \int \mathcal{D}[x] \exp\!\left(\tfrac{i\gamma}{\hbar} S[x]\right) }[/math]

  • Effective Planck constant:

[math]\displaystyle{ \hbar_{\text{eff}} = \hbar/\gamma }[/math]

Interpretations of Sean Collins' Method

  • Small [math]\displaystyle{ \gamma }[/math] → [math]\displaystyle{ \hbar_{\text{eff}} \gg \hbar }[/math]: strong interference, quantum‑like regime.
  • Large [math]\displaystyle{ \gamma }[/math] → [math]\displaystyle{ \hbar_{\text{eff}} \ll \hbar }[/math]: stationary‑phase dominance, classical regime.
  • Euclidean action governs tunneling:

[math]\displaystyle{ S_E[x(\tau)] = \int \left[ m \left(\tfrac{dx}{d\tau}\right)^2 + V_{\text{drag}}(x(\tau)) \right] d\tau }[/math]

Stationary Phase and Classical Limit

The stationary‑phase condition

[math]\displaystyle{ \delta S[x] = 0 }[/math]

recovers deterministic lattice dynamics.

In the limit:

[math]\displaystyle{ \hbar_{\text{eff}} \to 0 }[/math],

the path integral is dominated by the extremal classical path.

Short‑Time Kernel and Schrödinger‑like Equation from Sean Collins' Mjolnir (A137) Lattice Dynamics

The short‑time propagator from the above Collins' Path Integral Formulation therefore becomes:

[math]\displaystyle{ K(x+\Delta x, \Delta t; x, 0) \approx \sqrt{\tfrac{m}{2\pi i \hbar_{\text{eff}} \Delta t}} \; \exp\!\left[ \tfrac{i m (\Delta x)^2}{2 \hbar_{\text{eff}} \Delta t} - \tfrac{i \Delta t}{\hbar_{\text{eff}}} V_{\text{drag}}(x) \right]. }[/math]

The composition of kernels thus yields the following generalized Schrödinger‑like equation:

[math]\displaystyle{ i \hbar_{\text{eff}} \frac{\partial \Psi}{\partial t} = -\frac{\hbar_{\text{eff}}^2}{4m} \nabla^2 \Psi + V_{\text{drag}}(x)\,\Psi . }[/math]

Euclidean Action and Tunneling

Under Wick rotation of requisite parameters in the Sean Collins' Path Integral, that is [math]\displaystyle{ t \to -i\tau }[/math], we therefore obtain:

[math]\displaystyle{ S_E[x(\tau)] = \int \left[ m \left(\tfrac{dx}{d\tau}\right)^2 + V_{\text{drag}}(x(\tau)) \right] d\tau . }[/math]

And the following Tunneling amplitude emerges naturally:

[math]\displaystyle{ T \propto \exp\!\left( -\tfrac{SE[x_{\text{bounce}}]}{\hbar_{\text{eff}}} \right). }[/math]

Potency of the Sean Collins Path Integral Formulation

Given the above mathematical developments of the Sean Collins Path Integral Formulation, we can now enumerate the power which is inherent in it.

Conceptual Strengths

  • Economy of assumptions: Requires only a lattice, drag potential, and synchronization parameter.
  • Interpolating control: [math]\displaystyle{ \gamma }[/math] provides a tunable bridge between classical and quantum regimes.
  • Operational falsifiability: Predicts measurable deviations in line widths, tunneling rates, and interference patterns.
  • Compatibility: Reduces to standard quantum mechanics when [math]\displaystyle{ \gamma = 1 }[/math] and [math]\displaystyle{ V_{\text{drag}} \to V }[/math].

Further Expository Notes on Sean Collins' Mjolnir (A137) Lattice Dynamics

We must empathize here that the “Mjolnir lattice framework” is not the same as the MJOLNIR software package used in neutron scattering, nor the Mjolnir programming library found on GitHub. Those are unrelated to our work in this paper.

Here, Mjolnir lattice is a theoretical construct introduced by Sean Collins as the ontological substrate for his path‑integral mapping. It’s essentially a hypothetical discrete lattice of base‑particles (units of mass \(m_b\)) through which aggregated objects (atoms, nuclei, galactic structures) move. The framework is defined by:

  • Base units: Aggregates are built from fundamental base masses \(m_b\).
  • Clocking speed: Each base particle cycles at a characteristic speed \(v{mb}\), which sets the lattice’s intrinsic “tick.”
  • Wave speed: The lattice supports a superluminal propagation mode \(v_{\text{wave}} = \kappa c\).
  • Drag force: Motion through the lattice induces a drag term

[math]\displaystyle{ F_{\text{drag}} = k \,\rho(x)\, v_{\text{wave}}\, m\, v }[/math], where \(k\) is a drag coefficient and \(\rho(x)\) is the local lattice density.

  • Effective potential: This drag manifests as

[math]\displaystyle{ V_{\text{drag}}(x) = k \rho(x) |x| \left(\tfrac{v_{\text{wave}}}{c}\right)^2 }[/math].

The potent novelty of the Mjolnir (A137) Lattice framework formulated and developed by Sean Collins is that it treats this lattice as the ontological ground of physics, and then shows how a path‑integral weighting emerges naturally from it. Instead of assuming quantum mechanics as axiomatic, the Mjolnir lattice + drag potential + phase synchronization parameter \(\gamma\) introduced by Sean Collins is what actually generates Quantum Mechanics, which it does via the following Collins' algorithm:

  • A discrete action that reduces to a continuum action in the limit.
  • A modified path integral with effective Planck constant [math]\displaystyle{ \hbar_{\text{eff}} = \hbar/\gamma }[/math].
  • A stationary‑phase limit that recovers deterministic lattice dynamics.
  • A Euclidean action that governs tunneling probabilities.
  • A short‑time kernel that then yields a Schrödinger‑like equation with \(\hbar\) replaced by \(\hbar_{\text{eff}}\).

Methodological Novelty in Sean Collins’ Path‑Integral Mapping Compared to Richard Feynman's Path Integral Formulation of Quantum Mechanics (QM)

We note that Sean Collins’ formulation introduces a modified path‑integral approach distinct from the standard Feynman construction. While Sean Collins retains the functional‑integral structure, he alters both the weighting scheme and the ontological interpretation in a most potent way.

The Standard Feynman Path Integral

In conventional quantum mechanics, the propagator is expressed as

[math]\displaystyle{ Z = \int \mathcal{D}[x] \; \exp\!\left(\tfrac{i}{\hbar} S[x]\right), }[/math]

with a fixed Planck constant [math]\displaystyle{ \hbar }[/math] and an action [math]\displaystyle{ S[x] }[/math] derived from the Lagrangian.

Sean Collins’ Lattice Path Integral

In the Mjolnir (A137) lattice framework invented by Sean Collins, the discrete action increment is defined as

[math]\displaystyle{ \Delta S_i = m \frac{(x_{i+1} - x_i)^2}{\Delta t} + k \,\rho(r_i)\, |\Delta x_i|\, \Delta t \left(\frac{c}{v_{\text{wave}}}\right)^2 , }[/math]

leading to the path action

[math]\displaystyle{ S_{[\text{path}]} = \sum_{i=1}^{N} \Delta S_i . }[/math]

The weighting is modified to include a phase‑synchronization parameter [math]\displaystyle{ \gamma }[/math]:

[math]\displaystyle{ Z = \int \mathcal{D}[x] \; \exp\!\left(\tfrac{i\gamma}{\hbar} S[x]\right). }[/math]

This is equivalently written with an effective Planck constant:

[math]\displaystyle{ \hbar_{\text{eff}} = \frac{\hbar}{\gamma}, \qquad \text{hence} \qquad Z = \int \mathcal{D}[x] \; \exp\!\left(\tfrac{i}{\hbar_{\text{eff}}} S[x]\right). }[/math]

Distinctive Features in the Sean Collins' Lattice Path Integral

  • Control parameter: [math]\displaystyle{ \gamma }[/math] is physically measurable from lattice velocities, not a free constant.
  • Interpolation:
    • Small [math]\displaystyle{ \gamma \Rightarrow \hbar_{\text{eff}} \gg \hbar }[/math] (quantum‑like interference);
    • large [math]\displaystyle{ \gamma \Rightarrow \hbar_{\text{eff}} \ll \hbar }[/math] (classical stationary‑phase dominance).
  • Modified potentials: The drag‑derived potential [math]\displaystyle{ V_{\text{drag}}(x) }[/math] enters directly into the [quantum] action, thus giving macroscopic origins for effective quantum potentials.
  • Emergent dynamics: The short‑time kernel which Sean Collins introduces therefore yields a Schrödinger‑like equation with [math]\displaystyle{ \hbar }[/math] replaced by [math]\displaystyle{ \hbar_{\text{eff}} }[/math].
  • Euclidean sector: Tunneling amplitudes scale as [math]\displaystyle{ \exp(-S_E/\hbar_{\text{eff}}) }[/math], making tunneling probabilities explicitly dependent on Collins' lattice synchronization.

Interpretation of Differences Between Sean Collins and Richard Feynman Path Integral Formalisms

This approach does not replace the Feynman path integral but reinterprets it as an emergent statistical‑geometric tool. Quantum interference arises from lattice phase synchronization, while classical dynamics emerge in the stationary‑phase limit. The method is therefore falsifiable, predicting measurable deviations in line shapes, tunneling rates, and residuals compared to Standard Model expectations.

Why Sean Collins' Lattice Path Integral is Different and Important in a Major Sense

  • It reinterprets quantum interference as emerging from lattice phase synchronization, rather than being fundamental.
  • It provides a control knob (\(\gamma\)) that interpolates between classical and quantum behavior. This is Sean Collins' fine tuning parameter.
  • It embeds drag potentials and lattice density directly into the action, giving a macroscopic origin for effective potentials.
  • It yields a modified Schrödinger‑like equation with \(\hbar_{\text{eff}}\) instead of \(\hbar\), and with drag‑derived potentials.
  • Sean Collins has therefore made the \(\hbar_{\text{eff}}\) a tunable, physically interpretable parameter tied to lattice velocities, rather than a universal constant, hence removing some of the intrinsic constraints imposed by Planckian Quantum Mechanics and Dynamics.
  • The stationary‑phase limit (\(\gamma \to \infty\)) recovers deterministic lattice dynamics, while small \(\gamma\) yields strong interference.

Comparison of the Sean Collins' Lattice Path Integral Formulation to Other Path‑Integral Approaches

  • In condensed matter (e.g. polaron physics), path integrals are often used with effective actions after integrating out phonons. Collins’ method is structurally similar in spirit — but here the “environment” is the Mjolnir lattice itself, not phonons, which therefore makes Collins' methodology a more generalized approach.
  • In Euclidean path integrals, tunneling amplitudes scale as \(\exp(-S_E/\hbar)\). Collins’ version modifies this to \(\exp(-S_E/\hbar_{\text{eff}})\), thereby enforcing tunneling rates to depend explicitly on lattice phase synchronization.

In closing, we note that Sean Collins is indeed introducing a different path‑integral method: a formalism with a new weighting and interpretation that ties the effective Planck constant to lattice dynamics. This makes the Sean Collins path integral a bridge between deterministic lattice physics and emergent quantum‑like behavior, with falsifiable predictions (line shapes, tunneling rates, residuals) that differ from the Standard Model.

Relation Between Entropicity Weighting in Obidi's ToE's Vuli-Ndlela Integral and Collins’ Path Integral

Entropicity Weighting

In the Theory of Entropicity (ToE), the Feynman path integral is modified by introducing entropy‑motivated weights, resulting in the Vuli-Ndlela Integral, which forms the cornerstone of John Onimisi Obidi Theory of Entropicity(ToE). The partition functional in the Theory of Entropicity (ToE) is thus (neglecting other weighting parameters, such as configuration, vacuum, EM, Gravitation, etc.) expressed in its simplest form as

[math]\displaystyle{ Z_{\text{ToE}} = \int \mathcal{D}[x] \; \exp\!\left( \tfrac{i}{\hbar} S[x] \;+\; W_{\text{ent}}[x] \right), }[/math]

where [math]\displaystyle{ W_{\text{ent}}[x] }[/math] is an entropy‑derived functional, typically depending on information‑geometric curvature, entropy production, or coarse‑graining rates, etc. This weighting reflects the entropic structure of the underlying statistical manifold. Thus the weighting in the Vuli-Ndlela Integral of ToE is not a single parameter, but a product of a variety of functions modelled in line with the problem statement and structure to be solved.

Collins’ Weighting

In contrast, Sean Collins’ Mjolnir (A137) Lattice Dynamics introduces a weighting based on a phase synchronization parameter [math]\displaystyle{ \gamma }[/math]:

[math]\displaystyle{ Z_{\text{Collins}} = \int \mathcal{D}[x] \; \exp\!\left( \tfrac{i}{\hbar_{\text{eff}}} S[x] \right), \qquad \hbar_{\text{eff}} = \frac{\hbar}{\gamma}. }[/math]

Here, the modification is not entropy‑motivated but arises from lattice synchronization properties.

Parametric Reproduction of Sean Collins' Lattice Path Integral

By appropriate parametrization, the Vuli-Ndlela Integral of the Theory of Entropicity(ToE) can readily reproduce Collins’ formula. Specifically, if the entropy‑motivated weight is chosen such that(the reader should note that we are here still using the simplest structure of the Vuli-Ndlela Integral for ease of parametrization) we can write:

[math]\displaystyle{ W_{\text{ent}}[x] \;\equiv\; i \left( \tfrac{\gamma - 1}{\hbar} \right) S[x], }[/math]

then the ToE partition functional reduces to

[math]\displaystyle{ Z_{\text{ToE}} = \int \mathcal{D}[x] \; \exp\!\left( \tfrac{i}{\hbar_{\text{eff}}} S[x] \right), }[/math]

with [math]\displaystyle{ \hbar_{\text{eff}} = \hbar/\gamma }[/math],

which is precisely Collins’ weighting scheme imposed on the set-up of the Vuli-Ndlela Integral of the Theory of Entropicity(ToE).

We must point out here, however, that the algebra of this apparently simple parameterization between Obidi's ToE's Vuli-Ndlela Integral and Sean Collins' Path Integral becomes much more complex and complicated when we invoke more and more components of the function [math]\displaystyle{ W_{\text{ent}}[x] }[/math], in which case our parameterization becomes no longer this straightforward.

Interpretational Notes

  • Entropicity origin: In ToE, the modification is rooted in entropy and information geometry. We remember that even information too is entropy, as noted in earlier works of Claude Shannon and John von Neumann. In the seminal works of Fisher, Rao, (Fisher-Rao Information metric), Fubini, (Fubini-Study Quantum Distance metric), Amari, Čencov (Amari-Čencov alpha connections to the affine geometry of Levi-Civita), they have taught us that geometry can be associated with information. The Theory of Entropicity(ToE) has thus gone further to teach that entropy itself is what creates the information and the geometry. Hence with this premise, the Theory of Entropicity(ToE) derives the field equations of the entropic field, and there upon goes on to equally derive both the Schrödinger equations of Quantum Mechanics and the Einstein field equations of the General Theory of Relativity as quantum and classical limits on the Entropic Field. [79][80][81][82][83][84][85][86][87][88][89][90][91][16][17][18][19][92][21][93][23][94][95][96][97][98][99][100][101][102][103][104][105][106][107]

[108][109][110][111] [112] [113] [114] [115] [116] [117] [118][119] [120][121] [122] [123] [124] [125][126] [127] [128] [129] [130][131] [132][133][134][135][136] [137] [138] [139][140][141][142][143][144][145][146] [147][148]

  • Collins’ Weighting Origin: In Mjolnir lattice dynamics, the modification executed by Sean Collins is rooted in lattice synchronization and drag.
  • Equivalence of ToE's Vuli-Ndlela Integral and Collins' Path Integral Weighting Parameters: Through parametrization, the ToE can subsume Collins’ method as a special case, while retaining its broader information-geometric entropic interpretation of reality.

Selectivity, Filtering and Constraints Between Obidi's Vuli-Ndlela Integral and Collins' Lattice Path Integral

Here, we wish to bring to the reader's attention another crucial difference between Obidi's ToE's Vuli-Ndlela Integral and Collins' Lattice Path Integral.

Within the Theory of Entropicity (ToE), the Vuli‑Ndlela Integral introduces a distinctive weighting scheme for the path integral. Unlike the traditional Feynman formulation, or Collins’ lattice‑based weighting, the Vuli‑Ndlela Integral employs entropy‑motivated coefficients that act as selective filters on the space of possible histories:

[math]\displaystyle{ Z_{\text{(ToE)Vuli-Ndlela}} \;=\; \int{\mathcal{C}_{\text{ent}}} \mathcal{D}[g,\phi] \; \exp\!\left( \tfrac{i}{\hbar} S[g,\phi] + W_{\text{ent}}[g,\phi] \right), }[/math]

where:

  • [math]\displaystyle{ \mathcal{C}_{\text{ent}} }[/math]

denotes the subset of spacetimes and field configurations that satisfy entropic constraints,

  • [math]\displaystyle{ W_{\text{ent}}[g,\phi] }[/math] are entropy‑derived coefficients that suppress all other configurations, ensuring they do not contribute to the final evaluation of the action.

Thus, the Vuli‑Ndlela Integral does not sum over all possible spacetimes indiscriminately, but only over those consistent with entropic admissibility. This makes the ToE formulation both selective and constraint‑driven, with entropy acting as the governing principle of admissibility.

By contrast, Sean Collins’ Mjolnir (A137) Lattice Path Integral does not impose such an entropic selection. His formulation sums over all lattice paths:

[math]\displaystyle{ Z_{\text{Collins}} = \int \mathcal{D}[x] \; \exp\!\left( \tfrac{i}{\hbar_{\text{eff}}} S[x] \right), }[/math]

with weighting determined solely by the effective Planck constant [math]\displaystyle{ \hbar{\text{eff}} = \hbar/\gamma }[/math] and the drag‑derived potential [math]\displaystyle{ V{\text{drag}}(x) }[/math].

There is no entropic filtering mechanism: all paths are included, though their contributions are modulated by synchronization and drag.

Key Difference Between Vuli-Ndlela Integral and Collins' Path Integral

  • Vuli‑Ndlela Integral (ToE): Selective — only entropy‑admissible spacetimes contribute; others are suppressed and excluded.
  • Collins’ Lattice Path Integral: Inclusive — all lattice paths contribute, with weights determined by [math]\displaystyle{ \gamma }[/math] and drag, but without entropic suppression.

Absorption of Sean Collins’ Mjolnir (A137) Lattice Path Integral into Obidi's Theory of Entropicity(ToE)

With all the above differences and parametrization identified and noted, the Mjolnir (A137) Lattice Dynamics framework of Sean Collins can nonetheless be absorbed into the Theory of Entropicity (ToE), even though the Vuli‑Ndlela Integral constrains and filters paths by entropy. The key logic is to treat Collins’ lattice as a special‑case environment or effective sector within the broader entropic manifold. The reconciliation should therefore work as follows:

1. Path Space

  • Collins' Lattice Path Integral: All lattice paths are included, weighted by [math]\displaystyle{ \exp\!\left(\tfrac{i}{\hbar_{\text{eff}}} S[x]\right) }[/math].
  • Obidi's ToE (Vuli‑Ndlela): Only entropy‑admissible paths (spacetimes/configurations) are included; others are suppressed.

The Collins-Obidi Bridge (1):

Collins’ lattice paths can be interpreted as a subset of the ToE’s admissible paths. The entropy filter then decides which lattice trajectories survive. In other words, Collins’ formulation is the “raw” lattice sum, while ToE applies an entropic sieve on below to filter out discounted lattice blocks, vertices, and/or points.

2. Weighting

  • Collins Lattice Path Integral:

Weighting depends on [math]\displaystyle{ \gamma }[/math] (phase synchronization) and the drag potential.

  • Obidi's ToE's Vuli-Ndlela Integral: Weighting depends on entropy‑motivated coefficients [math]\displaystyle{ W_{\text{ent}}[x] }[/math].

The Collins-Obidi Bridge (2):

By proper parametrization, ToE can reproduce Collins’ weighting as follows:

[math]\displaystyle{ W_{\text{ent}}[x] \;\equiv\; i \left(\tfrac{\gamma - 1}{\hbar}\right) S[x], }[/math]

so that ToE’s entropy‑based integral reduces to Collins’ formula when entropy constraints are relaxed.

3. Interpretation of the Resolution

  • Collins' Lattice Path Integral: Synchronization and drag are physical lattice properties.
  • Obidi's ToE's Vuli-Ndlela Integral: Synchronization and drag can be reinterpreted as emergent entropic coefficients — i.e., entropy production or information‑flow constraints that mimic lattice drag.

Thus, Collins’ ontology can be absorbed as a phenomenological model inside ToE, while ToE provides the deeper entropic justification.

4. Key Difference Retained Between Obidi's Vuli-Ndlela Integral and Collins' Lattice Path Integral

  • Obidi's ToE: Entropy constraints mean not all lattice paths are admissible.
  • Collins Lattice Dynamics: All lattice paths are admissible, but their contributions are modulated by [math]\displaystyle{ \gamma }[/math] and [math]\displaystyle{ V_{\text{drag}} }[/math].

Conclusion on Reconciliation Between Obidi's ToE's Vuli-Ndlela Integral and Collins' Lattice Path Integral

Here is the conclusion of our efforts, at least so far: The Theory of Entropicity(ToE) generalizes Collins' Lattice Path Integral — ToE can reproduce Collins' formula when entropy constraints are relaxed, but it also goes further by filtering the path space.

The lattice framework can indeed be absorbed into ToE. The way to think of it is the following:

  • Collins’ path integral = “inclusive lattice sum with synchronization weighting.”
  • ToE’s Vuli‑Ndlela Integral = “entropy‑filtered sum, which can reduce to Collins’ formula under parametrization.”

This makes Collins’ method a special case of ToE, while ToE remains the more general, entropy‑driven theory.

The Theory of Entropicity(ToE) thus imposes entropic constraints on Collins' Lattice Path Integral and demands or enforces that not all the lattices and lattice points must be summed over or integrated, hence permitting some to contribute to the action integral while suppressing and/or discounting other lattice points or blocks. When such a selection and filtering mechanism is relaxed in the Theory of Entropicity (ToE), then it directly reflects and evolves the same ontology and orchestration as Collins' Lattice Path Integral Formalism. Then, entropy can infact ensure the lattice contributions are modulated by [math]\displaystyle{ \gamma }[/math] and [math]\displaystyle{ V_{\text{drag}} }[/math], since entropy can actually be constructed to mimick the [math]\displaystyle{ V_{\text{drag}} }[/math] itself.

Concluding Remarks on the Collins-Obidi Equivalence

The Theory of Entropicity and Collins’ Mjolnir lattice dynamics introduce different motivations for modifying the Feynman path integral. ToE’s Vuli‑Ndlela Integral is constraint‑driven and entropy‑filtered, while Collins’ method is synchronization‑driven and lattice‑inclusive.

However, the mathematical structures are sufficiently flexible in the simple cases such that Entropicity can reproduce Collins’ formula by parametrization. This demonstrates both the generality of the ToE weighting scheme and the compatibility of Collins’ approach as a special case within the entropic framework:

Entropy weighting (in ToE's Vuli-Ndlela Integral) → after Parametrization → yields Collins’ Path Integral formula.

Summary Note

The Mjolnir lattice framework is a conceptual lattice‑based ontology that underlies Sean Collins’ physics Theory of Mjolnir (A137) Lattice Dynamics. It is not a software tool or coding library, but a potent physics model: a powerful discrete substrate with drag and synchronization properties, designed to show how quantum‑like interference and tunneling emerge from background deterministic dynamics.

Work in Progress to Address Challenges

  • Deriving full spacetime dynamics from lattice ontology.
  • Establishing consistency across scales.
  • Identifying robust experimental signatures.

Collaboration and Attribution Endnotes

  • The Theory of Mjolnir (A137) Lattice Dynamics is the independent work of Sean Collins.
  • The Theory of Entropicity (ToE) is the independent work of John Onimisi Obidi.
  • While the above formalisms are distinct, we must recognize that Sean Collins’ potent path‑integral method, after some algebraic transformations, can be integrated and interpreted as a special case within the broader information‑geometric framework of the Theory of Entropicity(ToE).

Footnote on the Meaning of "Mjolnir (A137)" in Sean Collins' Theory of Lattice Dynamics

The term “Mjolnir” in Mjolnir (A137) Lattice Dynamics of Sean Collins is not an abbreviation or acronym. Rather, it is a chosen name adopted by Sean Collins for his lattice framework. The name is drawn from Mjölnir, the hammer of Thor in Norse mythology, symbolizing strength and foundational impact.

The appended designation “A137” serves as a technical identifier for the specific lattice construction, distinguishing this formulation from other possible lattice models.

Thus, the full title Mjolnir (A137) Lattice Dynamics should be understood as a symbolic name plus version tag, not as an acronym. This clarification prevents misinterpretation and situates the theory within its intended conceptual and metaphorical context.

See also

References

  1. ↑ Sean Collins (February, 2025). On the Theory of Mjolnir (A137) Lattice Dynamics by Sean Collins: Collected Unpublished Works and Communications with John Onimisi Obidi (Unpublished). Self-archived manuscript and materials. 
  2. ↑ Physics:Sean Collins' Path-Integral Mapping of the Mjolnir(A137)Lattice. (2025, October 7). HandWiki, . Retrieved 20:13, October 7, 2025 from https://handwiki.org/wiki/index.php?title=Physics:Sean_Collins%27_Path-Integral_Mapping_of_the_Mjolnir(A137)Lattice&oldid=3743664
  3. ↑ Physics:Einstein's Relativity from Obidi's Theory of Entropicity(ToE). (2025, August 30). HandWiki, . Retrieved 12:19, August 30, 2025 from https://handwiki.org/wiki/index.php?title=Physics:Einstein%27s_Relativity_from_Obidi%27s_Theory_of_Entropicity(ToE)&oldid=3742784
  4. ↑ Physics:Time Dilation, Length Contraction in the Theory of Entropicity (ToE). (2025, August 30). HandWiki, . Retrieved 10:01, August 30, 2025 from https://handwiki.org/wiki/index.php?title=Physics:Time_Dilation,_Length_Contraction_in_the_Theory_of_Entropicity_(ToE)&oldid=3742771
  5. ↑ Physics:Insights from the No-Rush Theorem in the Theory of Entropicity(ToE). (2025, August 1). HandWiki, . Retrieved 09:43, August 30, 2025 from https://handwiki.org/wiki/index.php?title=Physics:Insights_from_the_No-Rush_Theorem_in_the_Theory_of_Entropicity(ToE)&oldid=3741840
  6. ↑ Physics:The Cumulative Delay Principle(CDP) of the Theory of Entropicity(ToE). (2025, August 11). HandWiki, . Retrieved 09:40, August 30, 2025 from https://handwiki.org/wiki/index.php?title=Physics:The_Cumulative_Delay_Principle(CDP)_of_the_Theory_of_Entropicity(ToE)&oldid=3742101
  7. ↑ Physics:Theory of Entropicity(ToE), Time Quantization and the Laws of Nature. (2025, August 1). HandWiki, . Retrieved 09:34, August 30, 2025 from https://handwiki.org/wiki/index.php?title=Physics:Theory_of_Entropicity(ToE),_Time_Quantization_and_the_Laws_of_Nature&oldid=3741802
  8. ↑ Book:Conceptual and Mathematical Treatise on Theory of Entropicity(ToE). (2025, August 30). HandWiki, . Retrieved 09:31, August 30, 2025 from https://handwiki.org/wiki/index.php?title=Book:Conceptual_and_Mathematical_Treatise_on_Theory_of_Entropicity(ToE)&oldid=3742769
  9. ↑ Physics:Gravity from Newton and Einstein in the Theory of Entropicity(ToE). (2025, August 7). HandWiki, . Retrieved 09:19, August 30, 2025 from https://handwiki.org/wiki/index.php?title=Physics:Gravity_from_Newton_and_Einstein_in_the_Theory_of_Entropicity(ToE)&oldid=3742006
  10. ↑ Physics:Randomness and Determinism Unified in the Theory of Entropicity(ToE). (2025, August 13). HandWiki, . Retrieved 09:17, August 30, 2025 from https://handwiki.org/wiki/index.php?title=Physics:Randomness_and_Determinism_Unified_in_the_Theory_of_Entropicity(ToE)&oldid=3742233
  11. ↑ Physics:Relativity from Fundamental Postulate of Theory of Entropicity(ToE). (2025, August 30). HandWiki, . Retrieved 09:13, August 30, 2025 from https://handwiki.org/wiki/index.php?title=Physics:Relativity_from_Fundamental_Postulate_of_Theory_of_Entropicity(ToE)&oldid=3742766
  12. ↑ Physics:Artificial Intelligence Formulated by the Theory of Entropicity(ToE). (2025, August 27). HandWiki, . Retrieved 03:59, August 27, 2025 from https://handwiki.org/wiki/index.php?title=Physics:Artificial_Intelligence_Formulated_by_the_Theory_of_Entropicity(ToE)&oldid=3742591
  13. ↑ Physics:Curved Spacetime Derived from Obidi's Theory of Entropicity(ToE). (2025, August 29). HandWiki, . Retrieved 09:01, August 30, 2025 from https://handwiki.org/wiki/index.php?title=Physics:Curved_Spacetime_Derived_from_Obidi%27s_Theory_of_Entropicity(ToE)&oldid=3742730
  14. ↑ Physics:Information and Energy Redistribution in Theory of Entropicity(ToE). (2025, August 30). HandWiki, . Retrieved 09:05, August 30, 2025 from https://handwiki.org/wiki/index.php?title=Physics:Information_and_Energy_Redistribution_in_Theory_of_Entropicity(ToE)&oldid=3742765
  15. ↑ Obidi, John Onimisi (2025). Master Equation of the Theory of Entropicity (ToE). Encyclopedia. https://encyclopedia.pub/entry/58596
  16. ↑ 16.0 16.1 Obidi, John Onimisi. Corrections to the Classical Shapiro Time Delay in General Relativity (GR) from the Entropic Force-Field Hypothesis (EFFH). Cambridge University. (11 March 2025). https://doi.org/10.33774/coe-2025-v7m6c
  17. ↑ 17.0 17.1 Obidi, John Onimisi. How the Generalized Entropic Expansion Equation (GEEE) Describes the Deceleration and Acceleration of the Universe in the Absence of Dark Energy. Cambridge University. (12 March 2025). https://doi.org/10.33774/coe-2025-6d843
  18. ↑ 18.0 18.1 Obidi, John Onimisi. The Theory of Entropicity (ToE): An Entropy-Driven Derivation of Mercury’s Perihelion Precession Beyond Einstein’s Curved Spacetime in General Relativity (GR). Cambridge University. (16 March 2025). https://doi.org/10.33774/coe-2025-g55m9
  19. ↑ 19.0 19.1 Obidi, John Onimisi. The Theory of Entropicity (ToE) Validates Einstein’s General Relativity (GR) Prediction for Solar Starlight Deflection via an Entropic Coupling Constant η. Cambridge University. (23 March 2025). https://doi.org/10.33774/coe-2025-1cs81
  20. ↑ Obidi, John Onimisi (25 March 2025). "Attosecond Constraints on Quantum Entanglement Formation as Empirical Evidence for the Theory of Entropicity (ToE)". Cambridge University. https://doi.org/10.33774/coe-2025-30swc
  21. ↑ 21.0 21.1 Obidi, John Onimisi. Einstein and Bohr Finally Reconciled on Quantum Theory: The Theory of Entropicity (ToE) as the Unifying Resolution to the Problem of Quantum Measurement and Wave Function Collapse. Cambridge University. (14 April 2025). https://doi.org/10.33774/coe-2025-vrfrx
  22. ↑ Obidi, John Onimisi . "On the Discovery of New Laws of Conservation and Uncertainty, Probability and CPT-Theorem Symmetry-Breaking in the Standard Model of Particle Physics: More Revolutionary Insights from the Theory of Entropicity (ToE)". Cambridge University. (14 June 2025). https://doi.org/10.33774/coe-2025-n4n45
  23. ↑ 23.0 23.1 Obidi, John Onimisi. A Critical Review of the Theory of Entropicity (ToE) on Original Contributions, Conceptual Innovations, and Pathways towards Enhanced Mathematical Rigor: An Addendum to the Discovery of New Laws of Conservation and Uncertainty. Cambridge University.(2025-06-30). https://doi.org/10.33774/coe-2025-hmk6n
  24. ↑ Physics:HandWiki Master Index of Source Papers on Theory of Entropicity(ToE). (2025, September 9). HandWiki, . Retrieved 17:33, September 9, 2025 from https://handwiki.org/wiki/index.php?title=Physics:HandWiki_Master_Index_of_Source_Papers_on_Theory_of_Entropicity(ToE)&oldid=3743060
  25. ↑ Philosophy:Obidi's Agile Manifesto in Publishing of Revolutionary Ideas. (2025, September 9). HandWiki, . Retrieved 17:37, September 9, 2025 from https://handwiki.org/wiki/index.php?title=Philosophy:Obidi%27s_Agile_Manifesto_in_Publishing_of_Revolutionary_Ideas&oldid=3743065
  26. ↑ Physics:Entrodynamic Bellman Equation of AI RL in Theory of Entropicity(ToE). (2025, September 10). HandWiki, . Retrieved 14:46, September 10, 2025 from https://handwiki.org/wiki/index.php?title=Physics:Entrodynamic_Bellman_Equation_of_AI_RL_in_Theory_of_Entropicity(ToE)&oldid=3743125
  27. ↑ Why language models hallucinate | OpenAI. (September 2025). https://openai.com/index/why-language-models-hallucinate/
  28. ↑ Physics:On the Phenomenological Foundations of the Theory of Entropicity(ToE). (2025, September 15). HandWiki, . Retrieved 05:06, September 15, 2025 from https://handwiki.org/wiki/index.php?title=Physics:On_the_Phenomenological_Foundations_of_the_Theory_of_Entropicity(ToE)&oldid=3743204
  29. ↑ Wikipedia contributors. (2025, May 7). Entropic uncertainty. In Wikipedia, The Free Encyclopedia. Retrieved 00:34, September 20, 2025, from https://en.wikipedia.org/w/index.php?title=Entropic_uncertainty&oldid=1289229404
  30. ↑ Pawłowski, M. (2020). Entropy in Foundations of Quantum Physics. Entropy, 22(3), 371 https://doi.org/10.3390/e22030371
  31. ↑ Physics:Entropy as Foundation of Quantum Mechanics in Theory of Entropicity. (2025, September 20). HandWiki, . Retrieved 01:29, September 20, 2025 from https://handwiki.org/wiki/index.php?title=Physics:Entropy_as_Foundation_of_Quantum_Mechanics_in_Theory_of_Entropicity&oldid=3743315
  32. ↑ Physics:The Theory of Entropicity (ToE) as a New Foundation for Cybersecurity. (2025, September 20). HandWiki, . Retrieved 02:29, September 20, 2025 from https://handwiki.org/wiki/index.php?title=Physics:The_Theory_of_Entropicity_(ToE)_as_a_New_Foundation_for_Cybersecurity&oldid=3743326
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  37. ↑ A Brief Note On the Theory of Entropicity(ToE) and its General Implications - Substack https://open.substack.com/pub/johnobidi/p/a-brief-note-on-the-theory-of-entropicitytoe?r=1yk33z&utm_campaign=post&utm_medium=web&showWelcomeOnShare=true
  38. ↑ The Theory of Entropicity(ToE) Explains Mass, Time Dilation, Length, Speed of Light, etc.(Part 1) - Substack https://open.substack.com/pub/johnobidi/p/the-theory-of-entropicitytoe-explains?utm_source=share&utm_medium=android&r=1yk33z
  39. ↑ The Theory of Entropicity(ToE) Explains Mass, Time Dilation, Length, Speed of Light, etc.(Part 2) - Substack https://open.substack.com/pub/johnobidi/p/the-theory-of-entropicitytoe-explains-51c?utm_source=share&utm_medium=android&r=1yk33z
  40. ↑ Correspondence Between the Theory of Entropicity(ToE) and the Idea of the Aether and the Higgs Field Mechanism in the Standard Model of Particle Physics https://open.substack.com/pub/johnobidi/p/correspondence-between-the-theory?utm_source=share&utm_medium=android&r=1yk33z
  41. ↑ Physics:Theory of Entropicity(ToE), Idea of Aether and Higgs Field Mechanism. (2025, September 28). HandWiki, . Retrieved 23:39, September 28, 2025 from https://handwiki.org/wiki/index.php?title=Physics:Theory_of_Entropicity(ToE),_Idea_of_Aether_and_Higgs_Field_Mechanism&oldid=3743493
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  44. ↑ The Limit of the Speed of Light (c) is a Consequence of [Thermodynamic] Entropy Rather Than the Geometry of Einstein's Theory of Relativity. By John Onimisi Obidi on Medium: https://medium.com/@jonimisiobidi/the-limit-of-the-speed-of-light-c-is-a-consequence-of-thermodynamic-entropy-rather-than-the-6ee6c04aae60
  45. ↑ The Limit of the Speed of Light (c) is a Consequence of [Thermodynamic] Entropy Rather Than the Geometry of Einstein’s Theory of Relativity. https://open.substack.com/pub/johnobidi/p/the-limit-of-the-speed-of-light-c?utm_source=share&utm_medium=android&r=1yk33z
  46. ↑ Physics:The Emergence of the Speed of Light in the Theory of Entropicity(ToE). (2025, September 30). HandWiki, . Retrieved 06:37, September 30, 2025 from https://handwiki.org/wiki/index.php?title=Physics:The_Emergence_of_the_Speed_of_Light_in_the_Theory_of_Entropicity(ToE)&oldid=3743551
  47. ↑ How Does the Theory of Entropicity(ToE) Describe Nature in Modern Physics? https://medium.com/@jonimisiobidi/how-does-the-theory-of-entropicity-toe-describe-nature-in-modern-physics-ae5739a15e7c
  48. ↑ How Does the Theory of Entropicity(ToE) Describe Nature in Modern Physics? https://open.substack.com/pub/johnobidi/p/how-does-the-theory-of-entropicitytoe?utm_source=share&utm_medium=android&r=1yk33z
  49. ↑ Physics:How the Theory of Entropicity(ToE) Describes Nature in Physics. (2025, October 2). HandWiki, . Retrieved 00:10, October 2, 2025 from https://handwiki.org/wiki/index.php?title=Physics:How_the_Theory_of_Entropicity(ToE)_Describes_Nature_in_Physics&oldid=3743579
  50. ↑ So, Why Do Objects or Bodies or Particles Move At All? Explained By The Theory of Entropicity(ToE) https://open.substack.com/pub/johnobidi/p/so-why-do-objects-or-bodies-or-particles?utm_source=share&utm_medium=android&r=1yk33z
  51. ↑ So, Why Do Objects or Bodies or Particles Move At All? Explained By The Theory of Entropicity(ToE) https://medium.com/@jonimisiobidi/so-why-do-objects-or-bodies-or-particles-move-at-all-2749d400a026
  52. ↑ The Theory of Entropicity and the Laws of Newton https://en.everybodywiki.com/The_Theory_of_Entropicity_and_the_Laws_of_Newton
  53. ↑ The Theory of Entropicity(ToE) Explains Motion and the Laws of Newton https://en.everybodywiki.com/The_Theory_of_Entropicity(ToE)_Explains_Motion_and_the_Laws_of_Newton
  54. ↑ How The Theory Of Entropicity(ToE) Explains Newton's Laws Of Motion And Einstein's Theory Of Spacetime Curvature https://open.substack.com/pub/johnobidi/p/how-the-theory-of-entropicitytoe?r=1yk33z&utm_campaign=post&utm_medium=web&showWelcomeOnShare=true
  55. ↑ How The Theory Of Entropicity(ToE) Explains Newton's Laws Of Motion And Einstein's Theory Of Spacetime Curvature https://medium.com/@jonimisiobidi/how-the-theory-of-entropicity-toe-explains-newtons-laws-of-motion-and-einstein-s-theory-of-2dbd601786a0
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  57. ↑ The Theory of Entropicity(ToE) Re-interprets Newton's Gravitation and Einstein's Relativity Under One Unifying Principle https://open.substack.com/pub/johnobidi/p/the-theory-of-entropicitytoe-re-interprets?utm_source=share&utm_medium=android&r=1yk33z
  58. ↑ The Theory of Entropicity(ToE) Re-interprets Newton's Gravitation and Einstein's Relativity Under One Unifying Principle https://medium.com/@jonimisiobidi/the-theory-of-entropicity-toe-re-interprets-newtons-gravitation-and-einstein-s-relativity-under-aafa8bb6ce95
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  68. ↑ Čencov, Nikolai N. *Statistical Decision Rules and Optimal Inference*. Translations of Mathematical Monographs, vol. 53. Providence, RI: American Mathematical Society, 1982. ISBN 978-0821830979. https://bookstore.ams.org/mmono-53 (accessed October 3, 2025).
  69. ↑ Amari, Shun-ichi; Hiroshi Nagaoka (2000). Methods of Information Geometry. Translations of Mathematical Monographs. 191. American Mathematical Society and Oxford University Press. ISBN 978-0821805312. https://bookstore.ams.org/translations-191. Retrieved 2025-10-03. 
  70. ↑ Amari, Shun-ichi, and Hiroshi Nagaoka. *Methods of Information Geometry*. Translations of Mathematical Monographs, vol. 191. Providence, RI: American Mathematical Society and Oxford University Press, 2000. ISBN 978-0821805312. https://bookstore.ams.org/translations-191 (accessed October 3, 2025)
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  73. ↑ On the Mathematical Foundations of the Theory of Entropicity(ToE): A Qualitative Odyssey and Roadmap https://medium.com/@jonimisiobidi/on-the-mathematical-foundations-of-the-theory-of-entropicity-toe-a-qualitative-odyssey-and-aeb2f754addc
  74. ↑ On The Uniqueness of the Theory of Entropicity (ToE) With Respect to Other Competing Theories in Modern Theoretical Physics https://medium.com/@jonimisiobidi/on-the-uniqueness-of-the-theory-of-entropicity-toe-with-respect-to-other-competing-theories-in-29e409de7177
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  78. ↑ The Theory of Entropicity ToE: On a New Path Toward Quantum Gravity https://open.substack.com/pub/johnobidi/p/the-theory-of-entropicity-toe-onutm_source=share&utm_medium=android&r=1yk33z
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  85. ↑ Physics:Gravity from Newton and Einstein in the Theory of Entropicity(ToE). (2025, August 7). HandWiki, . Retrieved 09:19, August 30, 2025 from https://handwiki.org/wiki/index.php?title=Physics:Gravity_from_Newton_and_Einstein_in_the_Theory_of_Entropicity(ToE)&oldid=3742006
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  91. ↑ Obidi, John Onimisi (2025). Master Equation of the Theory of Entropicity (ToE). Encyclopedia. https://encyclopedia.pub/entry/58596
  92. ↑ Obidi, John Onimisi (25 March 2025). "Attosecond Constraints on Quantum Entanglement Formation as Empirical Evidence for the Theory of Entropicity (ToE)". Cambridge University. https://doi.org/10.33774/coe-2025-30swc
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  102. ↑ Physics:The Theory of Entropicity (ToE) as a New Foundation for Cybersecurity. (2025, September 20). HandWiki, . Retrieved 02:29, September 20, 2025 from https://handwiki.org/wiki/index.php?title=Physics:The_Theory_of_Entropicity_(ToE)_as_a_New_Foundation_for_Cybersecurity&oldid=3743326
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  107. ↑ A Brief Note On the Theory of Entropicity(ToE) and its General Implications - Substack https://open.substack.com/pub/johnobidi/p/a-brief-note-on-the-theory-of-entropicitytoe?r=1yk33z&utm_campaign=post&utm_medium=web&showWelcomeOnShare=true
  108. ↑ The Theory of Entropicity(ToE) Explains Mass, Time Dilation, Length, Speed of Light, etc.(Part 1) - Substack https://open.substack.com/pub/johnobidi/p/the-theory-of-entropicitytoe-explains?utm_source=share&utm_medium=android&r=1yk33z
  109. ↑ The Theory of Entropicity(ToE) Explains Mass, Time Dilation, Length, Speed of Light, etc.(Part 2) - Substack https://open.substack.com/pub/johnobidi/p/the-theory-of-entropicitytoe-explains-51c?utm_source=share&utm_medium=android&r=1yk33z
  110. ↑ Correspondence Between the Theory of Entropicity(ToE) and the Idea of the Aether and the Higgs Field Mechanism in the Standard Model of Particle Physics https://open.substack.com/pub/johnobidi/p/correspondence-between-the-theory?utm_source=share&utm_medium=android&r=1yk33z
  111. ↑ Physics:Theory of Entropicity(ToE), Idea of Aether and Higgs Field Mechanism. (2025, September 28). HandWiki, . Retrieved 23:39, September 28, 2025 from https://handwiki.org/wiki/index.php?title=Physics:Theory_of_Entropicity(ToE),_Idea_of_Aether_and_Higgs_Field_Mechanism&oldid=3743493
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  114. ↑ The Limit of the Speed of Light (c) is a Consequence of [Thermodynamic] Entropy Rather Than the Geometry of Einstein's Theory of Relativity. By John Onimisi Obidi on Medium: https://medium.com/@jonimisiobidi/the-limit-of-the-speed-of-light-c-is-a-consequence-of-thermodynamic-entropy-rather-than-the-6ee6c04aae60
  115. ↑ The Limit of the Speed of Light (c) is a Consequence of [Thermodynamic] Entropy Rather Than the Geometry of Einstein’s Theory of Relativity. https://open.substack.com/pub/johnobidi/p/the-limit-of-the-speed-of-light-c?utm_source=share&utm_medium=android&r=1yk33z
  116. ↑ Physics:The Emergence of the Speed of Light in the Theory of Entropicity(ToE). (2025, September 30). HandWiki, . Retrieved 06:37, September 30, 2025 from https://handwiki.org/wiki/index.php?title=Physics:The_Emergence_of_the_Speed_of_Light_in_the_Theory_of_Entropicity(ToE)&oldid=3743551
  117. ↑ How Does the Theory of Entropicity(ToE) Describe Nature in Modern Physics? https://medium.com/@jonimisiobidi/how-does-the-theory-of-entropicity-toe-describe-nature-in-modern-physics-ae5739a15e7c
  118. ↑ How Does the Theory of Entropicity(ToE) Describe Nature in Modern Physics? https://open.substack.com/pub/johnobidi/p/how-does-the-theory-of-entropicitytoe?utm_source=share&utm_medium=android&r=1yk33z
  119. ↑ Physics:How the Theory of Entropicity(ToE) Describes Nature in Physics. (2025, October 2). HandWiki, . Retrieved 00:10, October 2, 2025 from https://handwiki.org/wiki/index.php?title=Physics:How_the_Theory_of_Entropicity(ToE)_Describes_Nature_in_Physics&oldid=3743579
  120. ↑ So, Why Do Objects or Bodies or Particles Move At All? Explained By The Theory of Entropicity(ToE) https://open.substack.com/pub/johnobidi/p/so-why-do-objects-or-bodies-or-particles?utm_source=share&utm_medium=android&r=1yk33z
  121. ↑ So, Why Do Objects or Bodies or Particles Move At All? Explained By The Theory of Entropicity(ToE) https://medium.com/@jonimisiobidi/so-why-do-objects-or-bodies-or-particles-move-at-all-2749d400a026
  122. ↑ The Theory of Entropicity and the Laws of Newton https://en.everybodywiki.com/The_Theory_of_Entropicity_and_the_Laws_of_Newton
  123. ↑ The Theory of Entropicity(ToE) Explains Motion and the Laws of Newton https://en.everybodywiki.com/The_Theory_of_Entropicity(ToE)_Explains_Motion_and_the_Laws_of_Newton
  124. ↑ How The Theory Of Entropicity(ToE) Explains Newton's Laws Of Motion And Einstein's Theory Of Spacetime Curvature https://open.substack.com/pub/johnobidi/p/how-the-theory-of-entropicitytoe?r=1yk33z&utm_campaign=post&utm_medium=web&showWelcomeOnShare=true
  125. ↑ How The Theory Of Entropicity(ToE) Explains Newton's Laws Of Motion And Einstein's Theory Of Spacetime Curvature https://medium.com/@jonimisiobidi/how-the-theory-of-entropicity-toe-explains-newtons-laws-of-motion-and-einstein-s-theory-of-2dbd601786a0
  126. ↑ Obidi, John Onimisi. "Theory of Entropicity(ToE) Explains Newton and Einstein" Encyclopedia, https://encyclopedia.pub/entry/59079 (accessed October 03, 2025)
  127. ↑ The Theory of Entropicity(ToE) Re-interprets Newton's Gravitation and Einstein's Relativity Under One Unifying Principle https://open.substack.com/pub/johnobidi/p/the-theory-of-entropicitytoe-re-interprets?utm_source=share&utm_medium=android&r=1yk33z
  128. ↑ The Theory of Entropicity(ToE) Re-interprets Newton's Gravitation and Einstein's Relativity Under One Unifying Principle https://medium.com/@jonimisiobidi/the-theory-of-entropicity-toe-re-interprets-newtons-gravitation-and-einstein-s-relativity-under-aafa8bb6ce95
  129. ↑ Obidi, John Onimisi. "Theory of Entropicity(ToE) Reinterprets Newton and Einstein" Encyclopedia, https://encyclopedia.pub/entry/59081 (accessed October 03, 2025)
  130. ↑ Physics:The Obidi Action in the Theory of Entropicity(ToE). (2025, October 3). HandWiki, . Retrieved 17:12, October 3, 2025 from https://handwiki.org/wiki/index.php?title=Physics:The_Obidi_Action_in_the_Theory_of_Entropicity(ToE)&oldid=3743585
  131. ↑ Obidi, John Onimisi. "The Obidi Action in the Theory of Entropicity(ToE)" Encyclopedia, https://encyclopedia.pub/entry/59083 (accessed October 03, 2025)
  132. ↑ Physics:Pre-geometric Origin of Obidi Action in Theory of Entropicity(ToE). (2025, October 3). HandWiki, . Retrieved 21:34, October 3, 2025 from https://handwiki.org/wiki/index.php?title=Physics:Pre-geometric_Origin_of_Obidi_Action_in_Theory_of_Entropicity(ToE)&oldid=3743593
  133. ↑ Amari, Shun-ichi (1985). Differential-Geometrical Methods in Statistics. Lecture Notes in Statistics. 28. Springer-Verlag. ISBN 978-0387909683. https://link.springer.com/book/10.1007/978-1-4612-5056-2. Retrieved 2025-10-03. 
  134. ↑ Čencov, Nikolai N. (1982). Statistical Decision Rules and Optimal Inference. Translations of Mathematical Monographs. 53. American Mathematical Society. ISBN 978-0821830979. https://bookstore.ams.org/mmono-53. Retrieved 2025-10-03. 
  135. ↑ Amari, Shun-ichi. *Differential-Geometrical Methods in Statistics*. Lecture Notes in Statistics, vol. 28. New York: Springer-Verlag, 1985. ISBN 978-0387909683. https://link.springer.com/book/10.1007/978-1-4612-5056-2 (accessed October 3, 2025).
  136. ↑ Čencov, Nikolai N. *Statistical Decision Rules and Optimal Inference*. Translations of Mathematical Monographs, vol. 53. Providence, RI: American Mathematical Society, 1982. ISBN 978-0821830979. https://bookstore.ams.org/mmono-53 (accessed October 3, 2025).
  137. ↑ Amari, Shun-ichi. *Differential-Geometrical Methods in Statistics*. Lecture Notes in Statistics, vol. 28. New York: Springer-Verlag, 1985. ISBN 978-0387909683. https://link.springer.com/book/10.1007/978-1-4612-5056-2 (accessed October 3, 2025).
  138. ↑ Čencov, Nikolai N. *Statistical Decision Rules and Optimal Inference*. Translations of Mathematical Monographs, vol. 53. Providence, RI: American Mathematical Society, 1982. ISBN 978-0821830979. https://bookstore.ams.org/mmono-53 (accessed October 3, 2025).
  139. ↑ Amari, Shun-ichi; Hiroshi Nagaoka (2000). Methods of Information Geometry. Translations of Mathematical Monographs. 191. American Mathematical Society and Oxford University Press. ISBN 978-0821805312. https://bookstore.ams.org/translations-191. Retrieved 2025-10-03. 
  140. ↑ Amari, Shun-ichi, and Hiroshi Nagaoka. *Methods of Information Geometry*. Translations of Mathematical Monographs, vol. 191. Providence, RI: American Mathematical Society and Oxford University Press, 2000. ISBN 978-0821805312. https://bookstore.ams.org/translations-191 (accessed October 3, 2025)
  141. ↑ Physics:Pre-Geometric Origin of Obidi Action in Theory of Entropicity(ToE). (2025, October 4). HandWiki, . Retrieved 01:47, October 4, 2025 from https://handwiki.org/wiki/index.php?title=Physics:Pre-Geometric_Origin_of_Obidi_Action_in_Theory_of_Entropicity(ToE)&oldid=3743611
  142. ↑ On the Mathematical Foundations of the Theory of Entropicity(ToE): A Qualitative Odyssey and Roadmap https://open.substack.com/pub/johnobidi/p/on-the-mathematical-foundations-of?utm_source=share&utm_medium=android&r=1yk33z
  143. ↑ On the Mathematical Foundations of the Theory of Entropicity(ToE): A Qualitative Odyssey and Roadmap https://medium.com/@jonimisiobidi/on-the-mathematical-foundations-of-the-theory-of-entropicity-toe-a-qualitative-odyssey-and-aeb2f754addc
  144. ↑ On The Uniqueness of the Theory of Entropicity (ToE) With Respect to Other Competing Theories in Modern Theoretical Physics https://medium.com/@jonimisiobidi/on-the-uniqueness-of-the-theory-of-entropicity-toe-with-respect-to-other-competing-theories-in-29e409de7177
  145. ↑ On The Uniqueness of the Theory of Entropicity (ToE) With Respect to Other Competing Theories in Modern Theoretical Physics https://open.substack.com/pub/johnobidi/p/on-the-uniqueness-of-the-theory-of?utm_source=share&utm_medium=android&r=1yk33z
  146. ↑ Physics:Rényi Entropy Derived from Obidi's Theory of Entropicity(ToE). (2025, October 6). HandWiki, . Retrieved 05:56, October 6, 2025 from https://handwiki.org/wiki/index.php?title=Physics:R%C3%A9nyi_Entropy_Derived_from_Obidi%27s_Theory_of_Entropicity(ToE)&oldid=3743623
  147. ↑ Physics:The Theory of Entropicity(ToE) On a New Path to Quantum Gravity. (2025, October 6). HandWiki, . Retrieved 16:39, October 6, 2025 from https://handwiki.org/wiki/index.php?title=Physics:The_Theory_of_Entropicity(ToE)_On_a_New_Path_to_Quantum_Gravity&oldid=3743629
  148. ↑ The Theory of Entropicity ToE: On a New Path Toward Quantum Gravity https://open.substack.com/pub/johnobidi/p/the-theory-of-entropicity-toe-on?utm_source=share&utm_medium=android&r=1yk33z




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