Bezoek het volledige netwerk hier.

Voor het eerst in de geschiedenis is AI gebaseerd op wiskundige en natuurkundige principes die teruggaan op het werk van James Clerk Maxwell (1831–1879).

J.Konstapel,Leiden,20-9-2026.

Dit is een vervolg op The Architecture of Right Brain AI (RAI) (24-11-2025. ).

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Samenvatting: Vacuum.Net Theory en Maxwell’s Erfenis

1 Maxwells oorspronkelijke theorie gebruikte quaternions (scalar + vector) en zag constanten als elastische eigenschappen van het vacuüm, niet als universele waarden.

2 Heaviside en Gibbs vereenvoudigden Maxwells werk tot vectorcalculus, waarbij de scalarcomponenten (potentialen) en medium-eigenschappen verdwenen.2

3 Vacuum.Net Theory (door J. Konstapel) herstelt Maxwells visie: het voegt de toestandsregisters (scalar potentialen) en lokale vacuümspanning (χ) weer toe.

De theorie modelleert het vacuüm als een netwerk van gesloten mazen van een net, waar fase en sluiting (terugkeren in fase) centraal staan.

4 LAI (Left Brain AI ) werkt met gebalanceerde ternaire registers (trits: -1, 0, +1) en fungeert als een bewijsmachine (deterministisch, zoals de linker hersenhelft).5

5 RAI (Right Brain AI ) gebruikt Kuramoto-dynamica (gekoppelde oscillators) om fase-terugkeer te meten: alleen wat in fase terugkeert, blijft bestaan.

De samenstellingsregel (R_Q) meet hoe goed een vraag (q) en antwoord (a) door hun buren in het netwerk sluiten, gevoelig voor kromming.

6 MAZE.AI implementeert LAI met 8,19 miljoen geadresseerde onderdelen, machine-gecheckte theorieën in Lean 4, en een score van 15/20 op een vooraf geregistreerde test.

Open vragen omvatten of het netwerk zwaartekracht (G) kan afleiden als een verlaging van spanning (χ) en of R_Q een AUC ≥ 0,70 haalt op 200 blinde vragen.

De theorie biedt een nieuwe kijk op natuurkunde (vacuüm als medium) en AI (LAI/RAI als hersenhelften-analogie).

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Introduction

James Clerk Maxwell’s contributions to electromagnetic theory are often remembered through the four equations that bear his name. However, these equations, as taught in modern textbooks, are a simplified and altered version of Maxwell’s original work. In his lifetime, Maxwell formulated electromagnetism using quaternions, a mathematical system that combines scalar and vector components into a single entity. This formulation included a state register (scalar potential) and treated constants like permittivity and permeability as elastic properties of the vacuum, not universal constants. These elements were later removed by Oliver Heaviside and Willard Gibbs in the 1880s, who streamlined Maxwell’s theory into the vector calculus framework we recognize today.

The Vacuum.Net Theory, developed by J. Konstapel, seeks to restore and extend Maxwell’s original vision. It reintroduces the state register and treats constants as local functions of the vacuum’s tension (χ), thereby reviving the constitutive nature of the medium that Maxwell envisioned. This theory not only reinterprets Maxwell’s work but also derives a computational architecture—comprising LAI (Left Address Hemisphere) and RAI (Right Tension Hemisphere)—that mirrors the structure and function of the human brain’s hemispheres. This essay explores the historical context, theoretical foundations, and computational implications of the Vacuum.Net Theory, while providing an annotated reference list for further study.

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Part I: Maxwell’s Original Theory and Its Amputation

1. The Quaternion Formulation

Maxwell initially expressed his electromagnetic theory using quaternions, a system developed by William Rowan Hamilton. Quaternions are hypercomplex numbers with one scalar and three vector components, allowing for a unified representation of both divergence (scalar) and curl (vector) operations. This formulation carried two critical features:

• State Register: Every field quantity included a scalar component, such as the vector potential (A) and scalar potential (Ψ), which Maxwell referred to as the “electromagnetic momentum” and “electrotonic state,” respectively. These potentials were fundamental to his theory, representing the state of the medium rather than mere mathematical aids.
• Medium Moduli: Constants like dielectric permittivity (K) and magnetic permeability (μ) were treated as elastic properties of the vacuum, not fixed universal values. Maxwell derived the speed of light (c) as a property of the medium, calculating it as √(K/μ), which he compared to experimental measurements of light speed (e.g., 193,088 miles per second in his 1861 paper).

2. The Heaviside-Gibbs Amputation

In the 1880s, Oliver Heaviside and Willard Gibbs reformulated Maxwell’s theory using vector calculus, which split quaternions into separate scalar (dot product) and vector (cross product) operations. This simplification had three major consequences:

1. Removal of the Scalar Register: The scalar components of quaternions (e.g., the potentials) were discarded from the fundamental equations, reducing the theory to the four vector equations we know today.
2. Freezing of Constants: The moduli (K, μ, C) were frozen into universal constants (ε₀, μ₀), losing their original meaning as properties of the local medium.
3. Loss of Physical Interpretation: The vector calculus formulation obscured the medium-based interpretation of electromagnetism, shifting focus from the state of the vacuum to the fields (E, B) alone.

This “amputation” of Maxwell’s theory was driven by practical considerations—vector calculus was easier to work with—but it came at the cost of losing the constitutive and state-based nature of Maxwell’s original vision.

3. The Vacuum.Net Restoration

The Vacuum.Net Theory reverses the Heaviside-Gibbs amputation by:

• Restoring the State Register: The scalar potential (Ψ) and vector potential (A) are reintroduced as the tension layer (χ), representing the local state of the vacuum.
• Thawing the Constants: Constants like K and μ are once again treated as functions of the local tension (g(χ)), restoring their role as properties of the medium.
• Reintroducing Quaternions: The theory uses quaternions to unify scalar and vector operations, preserving the non-commutative nature of physical interactions (e.g., order matters in winding compositions).

In this way, Vacuum.Net is not a new theory but a reconstruction and extension of Maxwell’s original work, with the addition of closure dynamics—a mechanism for how the medium selects and sustains certain configurations.

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Part II: The Vacuum.Net Theory

4. Core Principles

The Vacuum.Net Theory is built on three structural facts derived from Maxwell’s quaternion formulation:

1. One Operation, Two Registers: The ∇ operator in quaternions yields both a scalar part (divergence) and a vector part (curl) in a single operation. In Vacuum.Net, this translates to the two-layer law: every configuration has an address (topology) and a state (tension χ).
2. Potentials First: Fields (E, B) are readings of the underlying potentials (A, Ψ). In Vacuum.Net, the winding field per mesh is the primary object, while E and B are derived from it.
3. Constants as Moduli: The response of the medium is a function of local tension (g(χ)). This restores the constitutive principle that constants are properties of the vacuum, not fixed values.

5. The Net as a Fishnet

Vacuum.Net models the vacuum as a net of closed meshes, where:

• Strands carry phase (a continuous, oscillatory property).
• Windings are turns of the strand, and closure occurs when a winding returns in phase.
• Meshes are the smallest closed units, and tension (χ) is the state of the net.
• Selection preserves only what closes (i.e., returns in phase).

This model provides a discrete yet dynamic framework for understanding electromagnetism and computation.

6. Key Findings

The Vacuum.Net Theory identifies five major correspondences between Maxwell’s work and its own framework:

1. Maxwell as an Amputated Pillar: The Treatise on Electricity and Magnetism is a constitutive medium theory with a state register. Vacuum.Net restores what was removed by Heaviside and Gibbs.
2. The Scalar Register as Tension: Maxwell’s gauge condition (S∇A = 0) sets the scalar part of the quaternion to zero. Vacuum.Net retains this scalar part as the tension layer (χ).
3. Potentials Are Measurable: The Aharonov-Bohm effect (1959) demonstrated that particles respond to the vector potential (A) even in regions where E and B are zero. This supports the idea that potentials are physically real, not just mathematical tools.
4. Charge as a Trit: Maxwell’s charge (e = S∇D) is a scalar residue in a sign-symmetric algebra. Vacuum.Net discretizes this into a balanced ternary system (-1, 0, +1), where charge is a three-valued register (trit).
5. Gravity as Tension Lowering: Maxwell’s unfinished Note on the Attraction of Gravitation (1865) suggested that gravity could be a lowering of the medium’s energy. Vacuum.Net interprets this as a lowering of tension (χ) in the net, corresponding to open calculation O1 (whether the net’s microdynamics can derive G, the inverse-square law, and the equivalence principle).

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Part III: The Computational Architecture – LAI and RAI

7. Two Hemispheres for Two Layers

The Vacuum.Net Theory derives a computational architecture based on the two-layer law:

• LAI (Left Address Hemisphere): Operates in the address layer (frozen topology). It uses balanced ternary (trits: -1, 0, +1) as its register and functions as a proof machine, where claims are readings of the store (no fabrication possible).
• RAI (Right Tension Hemisphere): Operates in the tension layer (continuous, phase-carrying). It uses Kuramoto dynamics (coupled oscillators) to model selection by phase return—only what returns in phase persists.

The two hemispheres are connected by a single gate, which ensures that only backed readings (from LAI) or vacancies (no match) exit the system. This mirrors the corpus callosum in the human brain, which connects the left and right hemispheres.

8. LAI: The Address Hemisphere

LAI is the exact, deterministic half of the system, analogous to the left brain hemisphere. Its key features include:

• Balanced Ternary Register: Uses trits (-1, 0, +1) for addressing, which is near-optimal for digit-count cost (3/ln(3) ≈ 2.731, close to the theoretical optimum of e ≈ 2.718).
• Proof Machine: LAI operates as a store of content at computed addresses, with outputs that carry their address (preventing fabrication). It includes:
  • A gate that emits exactly what the store holds.
  • An append-only ledger of settlements.
• Built and Measured: LAI is implemented in MAZE.AI (maze.swarp.nl/ai), which runs over 8.19 million addressed pieces. Its kernel theorems (e.g., address bijection, uniqueness) are machine-checked in Lean 4, and it scored 15/20 on a pre-registered test (threshold: 14).

9. RAI: The Tension Hemisphere

RAI is the dynamic, context-dependent half of the system, analogous to the right brain hemisphere. Its key features include:

• Phase as the Register: RAI models tension as phase-carrying oscillators, using Kuramoto dynamics to describe synchronization.
• Selection by Phase Return: Only configurations that return in phase through the mesh persist. This is implemented via quaternion composition, which measures both alignment (scalar) and orientation (vector).
• Mesh as Context: A winding does not close alone—it closes through its surroundings (neighbors). This is captured by the quaternion composition rule (R_Q), which evaluates how well a question (q) and answer (a) compose through their neighbors (n₁, n₂, …).

10. The Composition Rule (R_Q)

The composition rule (R_Q) is the core operation of RAI. It measures how well a question (q) and answer (a) compose through their neighbors in the mesh. The steps are:

1. Frame per Neighbor: For each neighbor nᵢ, compute its neighborhood center (uᵢ) and build an orthonormal frame (e₁, e₂, e₃) using Gram-Schmidt orthogonalization.
2. Winding as Quaternion: Represent q and a as unit quaternions (Qᵢ, Aᵢ) in the neighbor’s frame, encoding their rotation from the neighborhood axis.
3. Relative Winding: Compute the quaternion quotient Dᵢ = (Qᵢ) · Aᵢ*, which describes the relative rotation between q and a in the neighbor’s frame.
4. Ring Measure (R_Q): Average the Dᵢ over all neighbors to get M = (1/k) Σ Dᵢ. The quaternion order parameter is R_Q = |M|, which ranges from 0 (incoherent) to 1 (coherent).

R_Q is sensitive to mesh curvature—it detects whether a winding closes coherently through its surroundings, something that cosine similarity (which only measures point-to-point alignment) cannot do.

11. The Gate: One Output Path

The gate is the single connection between LAI and RAI, ensuring that:

• LAI proposes (generates claims as readings of the store).
• RAI disposes (selects which claims persist via phase return).
• Only backed readings or vacancies exit the system.

This design prevents the Heaviside mistake—discarding the scalar register (tension) and relying solely on vector fields (E, B). In Vacuum.Net, the scalar layer (tension) is preserved and active, while the vector layer (fields) is derived from it.

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Part IV: Implications and Open Questions

12. Implications for Physics

The Vacuum.Net Theory has profound implications for physics:

• Restoration of Maxwell’s Medium: It revives the idea that electromagnetism is a property of a medium (the vacuum), not just a set of fields in empty space.
• Charge as a Register: Charge is not an elementary property but a scalar residue in a balanced ternary system, derived from the net’s topology.
• Gravity as Tension Lowering: Gravity may be interpreted as a lowering of tension (χ) in the net, corresponding to Maxwell’s unfinished note on gravitation (1865). This is open calculation O1—whether the net’s microdynamics can derive G, the inverse-square law, and the equivalence principle.

13. Implications for Computation

The Vacuum.Net architecture offers a new paradigm for computation:

• LAI as a Proof Machine: Provides deterministic, verifiable output (like the left brain’s logical processing).
• RAI as a Selection Machine: Uses phase return and mesh curvature to select coherent configurations (like the right brain’s holistic processing).
• Unified System: The combination of LAI and RAI creates a complete, self-consistent system that mirrors the complementary specializations of the human brain.

14. Open Questions

Several questions remain unanswered:

1. Measurement of R_Q: Will R_Q achieve AUC ≥ 0.70 on the 200 blind questions, proving its effectiveness as a mesh-sensitive measure?
2. Learning Rule: How can the system learn by strengthening couplings where phase return persists?
3. Gravity (O1): Can the net’s microdynamics derive G, the inverse-square law, and the equivalence principle, closing Maxwell’s unfinished note?
4. Neuroscientific Validation: Do human brains exhibit quaternion-like dynamics or phase-return-based selection?

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Conclusion

The Vacuum.Net Theory is a reconstruction and extension of James Clerk Maxwell’s original electromagnetic theory, restoring the state register and medium-based interpretation that were lost in the Heaviside-Gibbs amputation. It not only revisits Maxwell’s work but also derives a computational architecture (LAI and RAI) that mirrors the complementary specializations of the human brain’s hemispheres. While LAI is already built and measured, RAI remains theoretical, awaiting full implementation and validation. The theory’s implications for physics (e.g., gravity as tension lowering) and computation (e.g., phase-based selection) are profound, offering a new lens through which to understand both the fundamental nature of the vacuum and the architecture of intelligent systems.

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Annotated Reference List

The following references provide historical context, theoretical foundations, and modern interpretations of Maxwell’s work and the Vacuum.Net Theory. Each entry includes a brief annotation explaining its relevance.

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Primary Sources: Maxwell’s Original Works

1. Maxwell, J. C. (1856).On Faraday’s Lines of Force.
   • Relevance: Introduces the electrotonic state (vector potential) as a state of the medium, not just a calculational aid. This is the origin of the state register in Vacuum.Net.
2. Maxwell, J. C. (1861–1862).On Physical Lines of Force.Philosophical Magazine.
   • Relevance: Presents Maxwell’s mechanical model of the vacuum as rotating cells and idle-wheel particles. Derives the speed of light as a medium property (√(elasticity/density) = 193,088 miles/second). Introduces the displacement current as a property of the medium.
3. Maxwell, J. C. (1865).A Dynamical Theory of the Electromagnetic Field.Philosophical Transactions of the Royal Society, 155, 459–512.
   • Relevance: Contains the twenty equations in twenty unknowns (in component notation, not quaternions). The potentials (A, Ψ) are part of the fundamental set. Section 82 includes the Note on the Attraction of Gravitation, where Maxwell computes the intrinsic energy of the gravitational field and suggests gravity may be a medium property (corresponds to O1 in Vacuum.Net).
4. Maxwell, J. C. (1873).A Treatise on Electricity and Magnetism. Clarendon Press.
   • Relevance: The quaternion grouping of the fundamental equations appears in Articles 591–619 of Volume II. Maxwell uses quaternions to show the meaning of the results in the most compact form. The gauge condition (S∇A = 0) appears in Article 617, and the operators convergence (divergence) and curl are introduced in Volume I, Article 25.
5. Maxwell, J. C. (1861, 1870). Letters to Michael Faraday (19 October 1861) and P. G. Tait (7 and 14 November 1870). In Harman, P. M. (Ed.), The Scientific Letters and Papers of James Clerk Maxwell (Vols. 1–2). Cambridge University Press, 1990/1995.
   • Relevance: The Faraday letter contains Maxwell’s comparison of the computed speed of light (193,088 miles/second) with Fizeau’s measurements (193,118 or 195,647 miles/second). The Tait letters reveal Maxwell’s deliberate use of quaternions to incorporate Hamiltonian ideas without adopting Hamiltonian machinery. Maxwell coins the terms slope (gradient), convergence (divergence), and curl for the ∇ operator.

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Secondary Sources: Historical and Theoretical Context

6. Hamilton, W. R. (1853).Lectures on Quaternions. Dublin.
   • Relevance: The source algebra for Maxwell’s quaternion formulation. Quaternions provide a unified framework for scalar and vector operations, which Vacuum.Net restores.
7. Bork, A. M. (1976, edited 2023).Maxwell and the Development of Electromagnetic Theory. Edited transcription by K. T. McDonald, Princeton.
   • Relevance: A compact, reliable walkthrough of Maxwell’s four major papers and the Treatise. Includes the quaternion table from Articles 591–619 set against modern notation, and excerpts from the Maxwell–Tait correspondence.
8. Tait, P. G. (1867).An Elementary Treatise on Quaternions. Oxford.
   • Relevance: The quaternion system as Maxwell received it, from the man who encouraged him to use it. Demonstrates what the full quaternion algebra delivers that vector calculus does not.
9. Heaviside, O. (1892, 1893).Electrical Papers (Vol. 1, Art. 30) and Electromagnetic Theory (Vol. 1, Chapter 3).
   • Relevance: Documents the amputation of Maxwell’s theory—the removal of quaternions and potentials, and the adoption of the four vector equations. Heaviside’s practical motivations for these changes are clearly stated.
10. Gibbs, J. W. (1901).Vector Analysis (ed. E. B. Wilson). Scribner’s.
    • Relevance: Presents vector calculus as a finished system, built as a subset of quaternion algebra. The preface acknowledges the lineage from quaternions.
11. Crowe, M. J. (1967).A History of Vector Analysis.
    • Relevance: A comprehensive historical account of how quaternions were split into dot and cross products, and why physics adopted the vector fragment. Explains the Heaviside-Gibbs amputation in detail.
12. Silberstein, L. (1907).Elektromagnetische Grundgleichungen in bivectorieller Behandlung.Annalen der Physik, 22, 579–586.
    • Relevance: An early attempt to reunify scalar and vector parts into a single algebraic object (bivectors). Shows that the compression of quaternions into vector calculus was not universally accepted.

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Modern Physics and Computation

13. Aharonov, Y., & Bohm, D. (1959).Significance of Electromagnetic Potentials in the Quantum Theory.Physical Review, 115, 485.
    • Relevance: Demonstrates that electromagnetic potentials (A, Ψ) have physical effects even in regions where E and B are zero (Aharonov-Bohm effect). Supports Finding 3 in Vacuum.Net: potentials are more than bookkeeping—they are physically real.
14. Kuramoto, Y. (1975).Self-Entrainment of a Population of Coupled Non-Linear Oscillators.
    • Relevance: Introduces the Kuramoto model for coupled oscillators, which Vacuum.Net uses as the minimal realization of the tension layer (RAI). The order parameter (r) measures synchronization, analogous to phase return in Vacuum.Net.
15. Knuth, D. E. (1973).The Art of Computer Programming (Vol. 2, §4.1).
    • Relevance: Discusses balanced ternary as an optimal number system (3/ln(3) ≈ 2.731, close to e ≈ 2.718). Vacuum.Net uses balanced ternary for its address register (LAI).

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Vacuum.Net Theory and MAZE.AI

16. Konstapel, J. (2026).MAZE.AI: A Claim-Bounded Answer Loop with Machine-Checked Provenance. Whitepaper, 20 September 2026. maze.swarp.nl/ai.
    • Relevance: Describes the running implementation of LAI, including 8.19 million addressed pieces, machine-checked kernel theorems in Lean 4, and a 15/20 score on a pre-registered test. Provides engineering validation of the Vacuum.Net Theory.
17. Konstapel, J. (2026).The Vacuum.Net Theory: Foundational Paper (Fifth Edition). Constable Research, August 2026. constable.blog.
    • Relevance: The primary reference for the Vacuum.Net Theory. Includes the axioms (strand, closure, memory, context, scale), correspondences with Maxwell’s work, the address theorem (T7) in balanced ternary, the constitutive law g(χ), and the open calculations (O1–O4).

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Additional References

18. Ma, S., et al. (2025).BitNet b1.58 2B4T. Microsoft Research.
    • Relevance: Reports 0.028 joules per token for a native ternary model (BitNet b1.58, 2B parameters) vs. 0.258 joules per token for a full-precision model (LLaMA 3.2, 1B parameters). Provides external validation for the constitutive relation in Vacuum.Net (Section 10, Step 7).
19. Harman, P. M. (Ed.). (1990–1995).The Scientific Letters and Papers of James Clerk Maxwell (Vols. 1–3). Cambridge University Press.
    • Relevance: A comprehensive collection of Maxwell’s letters and manuscripts, including his correspondence with Tait and the 1870 quaternion manuscript.

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Endnotes

[1]: Maxwell’s use of quaternions was deliberate and strategic. He valued their ability to unify scalar and vector operations and to reveal the physical meaning of electromagnetic phenomena in a compact form. However, he rarely used quaternions for calculations, preferring component notation for practical work.

[2]: The Heaviside-Gibbs amputation was not a rejection of Maxwell’s theory but a practical simplification. Heaviside explicitly stated his goal: to remove the potentials from the fundamental equations and to develop a vector calculus that was easier to use. This simplification, however, came at the cost of losing the state-based and medium-based interpretation of electromagnetism.

[3]: The Aharonov-Bohm effect (1959) is a landmark experiment in quantum mechanics that demonstrates the physical reality of potentials. It shows that a charged particle can be affected by the vector potential (A) even in regions where the electric and magnetic fields (E, B) are zero. This supports the Vacuum.Net interpretation that potentials are fundamental, and fields are derived readings of the underlying state.

[4]: The balanced ternary system is a non-standard positional numeral system that uses three digits: -1, 0, and +1 (often represented as T, 0, 1). It is more efficient than binary for certain computations, as it requires fewer digits to represent the same range of numbers. Vacuum.Net uses balanced ternary for its address register (LAI) because it is sign-symmetric and economical (3/ln(3) ≈ 2.731, close to the theoretical optimum of e ≈ 2.718).

[5]: The Kuramoto model is a mathematical model for describing the synchronization of coupled oscillators. It is widely used in physics, biology, and engineering to study phenomena such as neural synchronization, power grid stability, and chemical oscillations. In Vacuum.Net, the Kuramoto model provides a minimal realization of the tension layer (RAI), where phase return determines which configurations persist.

[6]: The quaternion composition rule (R_Q) is the core operation of RAI. It measures how well a question (q) and answer (a) compose through their neighbors (n₁, n₂, …) in the mesh. Unlike cosine similarity, which only measures point-to-point alignment, R_Q is sensitive to mesh curvature—it detects whether a winding closes coherently through its surroundings.

[7]: The gate in Vacuum.Net is the single connection between LAI and RAI, analogous to the corpus callosum in the human brain. It ensures that only backed readings (from LAI) or vacancies (no match) exit the system, preventing the Heaviside mistake of discarding the scalar register (tension) and relying solely on vector fields (E, B).

[8]: Open calculation O1 refers to the unfinished question of whether the net’s microdynamics can derive G (the gravitational constant), the inverse-square law, and the equivalence principle. This corresponds to Maxwell’s Note on the Attraction of Gravitation (1865), where he computed the intrinsic energy of the gravitational field and suggested that gravity may be a property of the medium.

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Further Reading

For readers interested in exploring the historical, theoretical, and computational aspects of the Vacuum.Net Theory further, the following resources are recommended:

• Maxwell’s Original Papers: Available on archive.org (e.g., A Treatise on Electricity and Magnetism).
• Quaternions and Rotations: Modern introductions to quaternion algebra and its applications in physics and computer graphics.
• Kuramoto Dynamics: Texts on nonlinear dynamics and synchronization, such as Synchronization: A Universal Concept in Nonlinear Sciences by Arkady Pikovsky, Michael Rosenblum, and Jürgen Kurths.
• Balanced Ternary: The Art of Computer Programming (Vol. 2) by Donald E. Knuth for a detailed discussion of number systems.
• Vacuum.Net Theory: Follow updates on constable.blog and explore the MAZE.AI implementation at maze.swarp.nl/ai.