PERFECTION STUDIES: CONTINUITY•SYMMETRY•HARMONY GOALS August.2025
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The Qualitative Expansion Model (QEM)
by Bruce E. Camber

Abstract
The Qualitative Expansion Model (QEM) proposes a deterministic, singularity-free universe grounded in Planck-scale discreteness and symmetry. Starting with a general framework constrained by isotropy and homogeneity, QEM explores the emergence of infinitesimal spheres and polyhedra, scaling via 202 base-2 notations to the present (~13.8 billion years). Geometric gaps at the 60th notation (2⁵⁹_lP ≈ 9.3×10^-18m) may drive physical dynamics, aligning with classical visions of physics. Testable through lattice simulations, QEM offers an alternative to Big Bang cosmology, with π bridging discrete and continuous scales.

1. Symmetry and Planck-Scale Discreteness
The QEM theory begins with a general principle: spacetime at the infinitesimal scale defined by Max Planck (l_p ≈ 1.616×10^-35m, t_p ≈ 5.391×10^-44s) is discrete and governed by fundamental symmetries—namely, isotropy (spherical symmetry) and homogeneity. These align with the symmetries of the Standard Model, such as U(1) for rotational invariance, which QEM seeks to respect, following the emphasis on symmetry groups by physicist Gerard ‘t Hooft as constraints [SciAm, April 2025].

At this scale, events occur at a rate of approximately 1.8547×10⁴³ events per second.

From this discrete, symmetric foundation, QEM posits the emergence of infinitesimal spheres (each with radius r ≈ l_p/2, and volume V = 4/3 π r³). Spheres naturally embody U(1) symmetry via π’s isotropy, providing a geometric realization of the underlying principles.

2. Emergence of Geometric Structures
Constrained by symmetry and packing efficiency, spheres stack; and, given cubic-closed packing of equal spheres, naturally render tetrahedrons and octahedrons which perfectly fill space:

• Observation: All tetrahedrons (edge l_p) contain four “half-sized” tetrahedrons (edge l_p/2) and one octahedron.
• Observation: All octahedrons necessarily contain six “half-sized” octahedrons and eight tetrahedrons.
• Hypothesis: A fundamental geometry, the concresence of the four irrational numbers (with a spin state) manifest the functions of pi, phi, e, and the square root of 2 among the first geometric structures. See Planck Polyhedral Core. See ROSS for more speculative ideas.
• Observation: All octahedrons necessarily define four hexagonal plates.

This nesting reflects a hierarchical symmetry, with π ensuring continuity (infinite digits), symmetry (spherical isotropy), and harmony (Fourier dynamics). Unlike Big Bang cosmology’s “infinitely hot, infinitely dense” singular origin (https://81018.com/infinitely-hot/), QEM’s finite, ordered state avoids such extremes.

3. Base-2 Notations and Scaling
QEM scales via base-2 notations, doubling the length scale each notation (or step):

• 1st notation: l_p
• 60th notation: (2⁵⁹_lP ≈ 9.3×10^-18m)
• 143rd notation: ~1 second (length scale~c×1s)
• 202nd notation: Present (~13.8 billion years)

This logarithmic progression, constrained by initial symmetries, encapsulates all scales without origins within a singularity.

Figure 1: A logarithmic scale from the 1st to 202nd notation, showing key transitions: 60th (gap emergence), 143rd (1 second), 202nd (present). Sphere volumes scale as V∝(2ⁿ⁻¹l_P)³.

4. Gap Dynamics and Symmetry Breaking
Up to the 60th notation, geometric structures pack perfectly, reflecting the initial isotropy and homogeneity. Beyond this scale, five-tetrahedral and five-octahedral stacks introduce a 7.356° angular deficit, potentially acting as symmetry-breaking events:

• Tetrahedron Volume (edge $l$): VT=
• Gap Volume Estimate: At the 60th notation, ∼(2⁵⁹l_P)³/k,  k ≈ 10−20.

These gaps may seed physical dynamics, such as gauge fields (U(1), SU(2)), by breaking the initial symmetry, a process testable via lattice gauge theory [Detmold et al., arXiv:2410.03602]. Gradient-based optimization could simulate their emergence, aligning with ‘t Hooft’s call for incremental steps from symmetry constraints.

Five-Tetrahedral Gap

Figure 2: Physical model of five tetrahedrons (edge 2⁵⁹_lP) around a common edge, AB, showing the 7.356° angular deficit that becomes systemic hypothesized to be about at the 60th notation.

5. Cosmological Implications
QEM reinterprets Big Bang evidence like the Cosmic Microwave Background (CMB) as emergent from sphere interactions, constrained by initial symmetries and modulated by π-driven harmonics. Lattice simulations [Detmold et al., arXiv:2410.03602] and harmonic analysis (e.g., Fourier transforms involving π) could test these predictions, alongside gap-driven gravitational wave signatures detectable by LIGO.

6. Deterministic Framework
QEM’s geometric determinism aligns with Gerard ‘t Hooft’s vision of a classical foundation for physics [SciAm, April 2025]. Like ‘t Hooft’s cellular automaton, QEM posits that Planck-scale geometries, constrained by symmetry, underpin a deterministic universe, replacing quantum randomness with ordered sphere packing. QEM shares features with other models:

• Loop Quantum Cosmology (LQC): Planck-scale discreteness and a singularity-free origin [Ashtekar, A., & Singh, P., Physical Review D, 84(12), 124021, 2011]. QEM’s base-2 scaling and geometric gaps are distinct.
• Emergent Universe: An ordered, non-singular start [Ellis, G., Classical and Quantum Gravity, 2004], mirrored by QEM’s perfect filling pre-60th notation.
• String Theory: Planck-scale geometry, where QEM’s spheres might relate to brane structures, though QEM remains in 3D+time.

7. Conclusion
QEM offers a deterministic, symmetry-driven model of the universe, inviting further exploration through simulations and observations. By grounding its geometric framework in fundamental symmetries like $U(1)$, QEM addresses ‘t Hooft’s call for generality while proposing a novel cosmological perspective. Future work will explore how these symmetries constrain the emergence of Standard Model particles and forces, potentially bridging QEM with established physics.

References

• Detmold, W., Shanahan, P. et al. (2024). Exploring gauge-fixing conditions with gradient-based optimization. arXiv:2410.03602.
• ‘t Hooft, G. (2025). Quantum mechanics is nonsense. Scientific American, April 2025. Link
• Ashtekar, A., & Singh, P. (2011). Loop quantum cosmology: A status report. Physical Review D, 84(12), 124021. DOI:10.1103/PhysRevD.84.124021
• Ellis, G. F. R., & Maartens, R. (2002). “The Emergent Universe: An Explicit Construction.” arXiv:gr-qc/0211082. PDF. Published as: Ellis, G. F. R., & Maartens, R. (2004). “The Emergent Universe: Inflationary Cosmology with No Singularity.” Classical and Quantum Gravity, 21(1), 223–238. DOI: 10.1088/0264-9381/21/1/016.
• Ellis, G. F. R., Murugan, J., & Tsagas, C. G. (2003). “The Emergent Universe: An Explicit Construction.” arXiv:gr-qc/0307112. PDF. Published in Classical and Quantum Gravity, 21(1), 255–268. DOI: 10.1088/0264-9381/21/1/017.

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Ongoing:
PASCOS: Ongoing studies, 16 August 2025: https://81018.com/pascos/
Related work: 7 August 2025: https://81018.com/mathematical-model/
About how AI might help substantiate the QEM: https://81018.com/mit-iaifi-2025/
A Grok xAI assessment: https://81018.com/qrok-comparisons/
An Open Letter, May 2025: https://81018.com/ai-2/
24 May 2025: https://81018.com/chatgpt-3/
24 May 2025: https://81018.com/hyper-rational/

And more will come….

Key pages: This file — https://81018.com/qualitative-expansion — has two related files:
https://81018.com/big-ideas/ and https://81018.com/paradigm-shift/

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