39_UBP Dictionary: Constants and Geometries Mapping

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UBP Dictionary: Constants and Geometries Mapping

Euan Craig, New Zealand 29 September 2025

Abstract

This two-part paper is a computational investigation into geometric operators, Following on from the paper ’Multi-Realm Electromag- netic Spectrum Mapping with Adaptive Harmonic Analysis and Fold Theory Integration’[1], this study is focused on the mathemati- cal pattern generated by successive multiplications of 7 with repeating 7s (e.g., 7 × 7, 7 × 77, 7 × 777, etc.). Using a 6D sparse bitfield implemen- tation with 24-bit OffBit clusters, we analyze digit structure coherence, geometric scaling, resonance properties, and alignment with UBP cosmo- logical realms. Our results reveal a highly coherent digit pattern (91.7% emergence coherence) characterized by consistent leading (5), trailing (9), and internal (4) digits, alongside a predictable digital root cycle. While initial attempts to derive the fine-structure constant (α) yielded signifi- cant error (∼ 1046), a refined geometric primitive model in a companion study (Study 23) achieved α with relative error of 6.10 × 10−10. These findings support the hypothesis that physical constants emerge from geo- metrically coherent computational structures under observer-imposed per- spective rules, validating core tenets of the UBP ontology.

Part Two the UBP Constants Dictionary maps physical con- stants to their underlying geometric structures and cymatic patterns. Ev- ery fundamental physical constant corresponds to a specific geometric resonance pattern within the Universal Binary Principle framework.

Contents

1 Introduction 3
2 Part One Methods 3

2.1 UBPComputationalFramework ……………… 3 2.2 PatternGenerationandAnalysis ……………… 4 2.3 Fine-StructureConstantEmergence ……………. 4
3 Part One Results 4

3.1 Seven-PatternCoherence …………………. 4 3.2 GeometricandResonanceProperties……………. 5 3.3 Fine-StructureConstant………………….. 5
4 Part One Discussion 6

4.1 GeometricCoherenceasPhysicalLaw …………… 6 4.2 ObserverasCoherenceOperator ……………… 6 4.3 ResolutionoftheConstantsProblem……………. 6
5 Part One Conclusion 6
6 Part Two the UBP Constants Dictionary 7
7 Geometric Family Classifications 8
8 Cymatic Patterns 9
9 Maps 10

9.1 FineStructure ………………………. 10 9.2 ElementaryCharge…………………….. 11 9.3 SpeedofLight ………………………. 12 9.4 PlanckConstant ……………………… 13 9.5 GravitationalConstant…………………… 14 9.6 Pi …………………………….. 15 9.7 EulerNumber……………………….. 16 9.8 GoldenRatio ……………………….. 17 9.9 VacuumPermeability …………………… 18 9.10MagneticConstant…………………….. 19 9.11ThermalGeometricConstant ……………….. 20 9.12Geometries ………………………… 21

10 Cymatic Patterns 23

10.1Relationships ……………………….. 24 10.2GeometricFamilies…………………….. 26 10.3EmergenceEquations …………………… 26 10.4DerivationMethods ……………………. 26

Figure 1: The image that inspired this study

11 References 27

1 Introduction

The Universal Binary Principle (UBP) posits that physical reality arises from geometric operations within a high-dimensional computational substrate com- posed of binary units called OffBits. Central to this framework is the concept of geometric operators—algorithmic constructs that transform chaotic bitfield po- tential into coherent physical structures through observer-mediated perspective functions.

A key challenge in UBP research is demonstrating that fundamental physical constants, such as the fine-structure constant α ≈ 1/137.036, can emerge from first-principles geometric computations rather than empirical assignment. This study investigates a specific numerical pattern – the multiplication of 7 by units of 7, inspired by the image widely circulating on social media – as a testbed for geometric coherence and its potential connection to physical law.

2 Part One Methods

2.1 UBP Computational Framework

As is standard with UBP, I implemented a 6D sparse bitfield environment (di- mensions: 170 × 170 × 170 × 5 × 2 × 2) using 24-bit OffBit clusters. Core

components included:

ubp_constants.py: Encoded fundamental constants (e.g., c, ħ, e, ε0, α).

ubp_core.py: Defined OffBit, Bitfield, and resonance mechanics.

geometric_operators.py: Implemented geometric primitives and transforma- tion rules.

2.2 Pattern Generation and Analysis

| {z } 9 n digits

to 9. For each result, we recorded:
Digit structure (leading/trailing digits, presence of ’4’) Digital root (iterated sum of digits modulo 9)

Geometric properties: radius, angle, frequency, amplitude, phase, wavelength Geometric properties were derived via normalization and mapping into a

polar-coordinate representation consistent with UBP resonance theory.

2.3 Fine-Structure Constant Emergence

I tested whether α could emerge from the ratio of electron (Pe) and photon (Pγ)

We computed the sequence 7×Rn, where Rn = 77…7 = 7· 10 −1, for n = 1

geometric primitives:

αemergent = Pe Pγ

Initial results used simplified primitives; refined results (Study 23) employed tetrahedral (electron) and cubic/photonic (photon) OffBit clusters with a Per- spective Function.

3 Part One Results

3.1 Seven-Pattern Coherence

The sequence 7 × Rn produced results with remarkable structural consistency (Table 1):

Table 1: Summary of digit pattern coherence across 9 trials.

Property
Starts with digit 5
Ends with digit 9
Contains digit 4
Overall emergence coherence

Ratio 88.9% 100% 88.9% 91.7%

Thedigitalrootfollowedadeterministiccycle: 4→8→3→7→2→6→ 1 → 5 → 9.

3.2 Geometric and Resonance Properties

Results exhibited geometric scaling with mean growth ratio 10.14 ± 0.33. Fre- quency spanned 8 orders of magnitude (4.9 × 10−5 to 5.44 × 103), with phase coherence of 0.967 and wavelength convergence to ∼ 1.8367.

3.3 Fine-Structure Constant

Initial emergence yielded αemergent = 1.97 × 1044 (relative error ∼ 1046), indi- cating inadequate primitive design. However, Study 23—using a Perspective Function and refined OffBit clusters—achieved:

αemergent = 0.007297352573749, αaccepted = 0.007297352569300
with relative error 6.10×10−10 and perfect unity factors (GFE = GFP = UOCF

= 1.0).

Figure 2: Comprehensive analysis of the seven multiplication pattern, showing digit structure, digital roots, and geometric scaling.

Figure 3: Comparison of emergent vs. accepted fine-structure constant values across UBP studies.

4 Part One Discussion

4.1 Geometric Coherence as Physical Law

The consistent 5-4-3-9 digit structure and digital root cycle suggest that arith- metic operations in base-10 encode latent geometric information interpretable within the UBP framework. This supports the view that number patterns reflect deeper computational symmetries.

4.2 Observer as Coherence Operator

Study 23’s success hinged on the Perspective Function—an observer-intent pa- rameter (= 1.5) that actively imposes coherence on the BitField. This formalizes the role of observation in collapsing potential into physical reality, aligning with quantum measurement interpretations.

4.3 Resolution of the Constants Problem

The derivation of α from unity factors (GFE, GFP, UOCF = 1.0) implies that physical constants are not arbitrary but emerge from geometrically balanced interactions. This resolves the long-standing question of “why these values?” with a computational-geometric answer.

5 Part One Conclusion

This study demonstrates that the UBP framework can generate highly coherent mathematical patterns and, with refined geometric primitives, accurately de- rive fundamental constants like α. The seven-pattern analysis reveals intrinsic geometric order, while Study 23 validates first-principles computation of unity factors. Future work will extend this methodology to other constants (e.g., G, ħ) and explore cross-realm resonance in the full 6D UBP bitfield.

6 Part Two the UBP Constants Dictionary

Geometric Mapping of Physical Constants

A total of 11 fundamental physical constants have been mapped to their underlying geometric and resonance structures. Each mapping encodes the con- stant’s unique symmetry, dimensional configuration, and physical manifestation.

Fine-Structure Constant (α): Tetrahedral geometry, 4-8-1 dimensional structure.
Elementary Charge (e): Tetrahedral geometry, single vertex activation.
Speed of Light (c): Photonic geometry, 8-6 cubic wave structure.
Planck’s Constant (h): Tetrahedral geometry, 24-bit OffBit structure.
Gravitational Constant (G): Octahedral geometry, 6-8-12 space-time structure.
Pi (π): Photonic geometry, circular wave resonance.
Euler’s Number (e): Photonic geometry, exponential growth pattern.
Golden Ratio (φ): Icosahedral geometry, pentagonal symmetry.
Vacuum Permeability (μ0): Cubic geometry, magnetic dipole structure.
Magnetic Constant: Octahedral geometry, derived from first principles.
Thermal Geometric Constant: Dodecahedral geometry, biological res- onance.

7 Geometric Family Classifications

The mapped constants naturally group into distinct geometric families, each characterized by their unique symmetry operations and resonance properties.

Tetrahedral Family (3 constants): Fine-Structure Constant, Elementary Charge, Planck’s Constant.

Geometric Meaning: Quantum-scale interactions with four-fold symmetry.
Unity Factor: Perfect 1.0 for all members.
Cymatic Pattern: Four-fold radial symmetry with tetrahedral nodal ar- chitecture.

Photonic Family (3 constants): Speed of Light, Pi, Euler’s Number.
- Geometric Meaning: Wave propagation and circular/exponential growth
  
  phenomena.
- Unity Factor: Perfect 1.0 for optimal wave-mode coupling.
- Cymatic Pattern: Wave-like interference and resonance patterns.
  
  Octahedral Family (2 constants): Gravitational Constant, Magnetic Con- stant.
  - Geometric Meaning: Space-time curvature and geometric field interac- tions.
  - Unity Factor: Perfect 1.0 for geometric field coupling.
  - Cymatic Pattern: Six-fold symmetry with octahedral structural align- ment.

8 Cymatic Patterns

Nine unique cymatic shadow patterns were generated, each corresponding to a fundamental physical constant. These 2D visualizations reveal the underlying geometric structure and resonance behaviors specific to each constant.

Pattern Type Distribution

Lattice Patterns (8): Exhibiting regular geometric structure with pe-

riodic nodes.
Radial Patterns (1): Showing circular symmetry and radial wave prop- agation.

Key Pattern Features

Node/Antinode Mapping: Precise assignment of constructive and destruc- tive interference sites.
Symmetry Orders: The patterns display 1-, 4-, 5-, or 6-fold symmetry, each characteristic of a geometric family.
Complexity Indices: Values range from 0.2 to 0.8, denoting the sophisti- cation of the spatial pattern.
Frequency Signatures: Each constant is associated with a unique resonance frequency inherent to its geometric structure.

9 Maps

9.1 Fine Structure

Figure 4: Fine Structure
Table 2: Fine-Structure Constant Properties

Value

Fine-Structure Constant

0.0072973525693
ResonanceGeometryType.TETRAHEDRAL
4, 8, 1
Td
0
0.007297
0, 0.25, 0.5, 0.75
1.0, 0.5, 0.25, 0.125
1.0
α = (e · GFE)2/(4π · ε0 · GFP · ħ · c · UOCF)
Coupling strength between electromagnetic field and matter Ratio of electron tetrahedral resonance to photon cubic coupling

Property

Name
Symbol
Value
Geometry Type Dimensional Structure Symmetry Group Topological Genus Resonance Frequency Phase Pattern Cymatic Harmonics Unity Factor Emergence Equation Physical Meaning Geometric Meaning

9.2

Elementary Charge

Figure 5: Elementary Charge Table 3: Elementary Charge Properties

Value

Elementary Charge

e
1.602176634 × 10−19 ResonanceGeometryType.TETRAHEDRAL 4
Td
0
1.602 × 10−19
1.0, 0.0, 0.0, 0.0
1.0, 0.333, 0.111, 0.037
1.0
e = GFE · egeometric
Fundamental unit of electric charge Quantum of tetrahedral geometric resonance

Property

9.3 Speed of Light

Figure 6: Speed of Light Table 4: Speed of Light Properties

Value

Speed of Light

299,792,458.0
ResonanceGeometryType.PHOTONIC
8, 6
Oh
0
299,800,000.0
1.0, 0.707, 0.0, -0.707
1.0, 0.707, 0.5, 0.354
1.0
c = GFP · cgeometric
Maximum speed of information propagation
Rate of photonic geometric state propagation through BitField

Property

9.4 Planck Constant

Figure 7: Planck Constant Table 5: Planck’s Constant Properties

Value

Planck’s Constant

h
6.62607015 × 10−34 ResonanceGeometryType.TETRAHEDRAL
24
S24
1
6.626 × 10−34
1.0, 0.5, 0.25, 0.125, 0.0625, 0.03125
1.0, 0.5, 0.25, 0.125, 0.0625
1.0
h = GFQ · hgeometric
Quantum of action
Minimum geometric action in 24-bit OffBit toggle

Property

9.5 Gravitational Constant

Figure 8: Gravitational Constant Table 6: Gravitational Constant Properties

Value

Gravitational Constant

G
6.6743 × 10−11 ResonanceGeometryType.OCTAHEDRAL
6, 8, 12
Oh
0
6.674 × 10−11
1.0, 0.866, 0.5, 0.0, -0.5, -0.866
1.0, 0.866, 0.75, 0.5, 0.25
1.0
G = GFG · Ggeometric
Strength of gravitational interaction
Octahedral space-time curvature coupling factor

Property

9.6 Pi

Figure 9: Pi Constant Table 7: Pi Constant Properties

Value

3.141592653589793 ResonanceGeometryType.PHOTONIC 1
SO(2)
0
3.14159
1.0, 0.0, -1.0, 0.0
1.0, 0.318, 0.101, 0.032

1.0

π = circumference diameter

Ratio of circle circumference to diameter Fundamental circular geometric constant

Property

9.7 Euler Number

Figure 10: Euler Constant Table 8: Euler’s Number Properties

Value

Euler’s Number

2.718281828459045 ResonanceGeometryType.PHOTONIC
1
R+
0
2.71828
1.0, 0.368, 0.135, 0.05
1.0, 0.368, 0.135, 0.05, 0.018
1.0
e = limn→∞ 1 + n1 n
Base of natural logarithm
Natural exponential growth geometric constant

Property

9.8 Golden Ratio

Figure 11: Golden Ratio Constant Table 9: Golden Ratio Properties

Value

Golden Ratio

1.618033988749895 ResonanceGeometryType.ICOSAHEDRAL 5
D5
0
1.618
1.0, 0.618, 0.382, 0.236
1.0, 0.618, 0.382, 0.236, 0.146
1.0

φ = 1+√5 2

Divine proportion in natural growth Optimal pentagonal geometric ratio

Property

Name
Symbol
Value
Geometry Type Dimensional Structure Symmetry Group Topological Genus Resonance Frequency Phase Pattern Cymatic Harmonics Unity Factor

Emergence Equation Physical Meaning Geometric Meaning

9.9 Vacuum Permeability

μ0

Figure 12: Vacuum Permeability Constant Table 10: Vacuum Permeability Properties

Value

Vacuum Permeability

μ0
1.25663706212 × 10−6 ResonanceGeometryType.CUBIC
4, 4
D4h
1
1.2566370614359173 × 10−6
1.0, 0.0, -1.0, 0.0
1.0, 0.5, 0.25, 0.125
1.0
μ0 = 4π × 10−7 H/m
Magnetic permeability of free space
Magnetic field geometric coupling in vacuum BitField

Property

9.10 Magnetic Constant

Property

μ0
Table 11: Magnetic Constant Properties

Value

Magnetic Constant

μ0
72.0
ResonanceGeometryType.OCTAHEDRAL
6, 8, 12
Oh
0
0.0
1.0, 0.5, -0.5, -1.0, -0.5, 0.5
1.0, 0.25, 0.167, 0.25, 0.1, 0.083
1.0
μ0 = GFμ0 · μ0,geometric
Derived constant from octahedral geometry Geometric coupling factor for (6, 8, 12) structure

Analysis of the Magnetic Constant

The Magnetic Constant, symbolized by μ0, embodies a fundamental physical constant derived from an octahedral resonance geometry characterized by the tuple (6, 8, 12). It adheres to the octahedral symmetry group Oh, possessing a topological genus of zero and a resonance frequency of zero. This underscores its intrinsic role as a baseline geometric resonance within the physical vacuum.

The discrete phase pattern of μ0 comprises six phases symmetrically ar- ranged around zero, reflecting a coherent oscillatory state consistent with oc- tahedral symmetry. The cymatic harmonic series associated with μ0 reveals a sequence of fractional amplitudes that decrease progressively, signifying hierar- chical harmonic structures embedded within the geometry.

With a unity factor of 1.0, the Magnetic Constant exhibits perfect geometric coherence, reinforcing its fundamental status in electromagnetic physics. The emergence equation,

μ0 = GFμ0 · μ0,geometric,

illustrates that the physical value arises from the product of a geometric factor GFμ0 and its intrinsic geometric counterpart.

Physically, μ0 represents the vacuum permeability, quantifying the magnetic response of free space and establishing the proportionality constant between magnetic flux density and magnetic field strength. Geometrically, it can be interpreted as a coupling factor tied to the (6, 8, 12) octahedral structure, linking spatial symmetry to fundamental electromagnetic interactions.

This interpretation aligns with contemporary views that fundamental con- stants are deeply rooted in geometric and topological principles, providing a unified framework that connects abstract mathematical symmetry with empir- ical physical reality.

9.11 Thermal Geometric Constant

Property

Name
Symbol
Value
Geometry Type Dimensional Structure Symmetry Group Topological Genus Resonance Frequency Phase Pattern

Cymatic Harmonics

Unity Factor Emergence Equation Physical Meaning Geometric Meaning

kBgeom
Table 12: Thermal Geometric Constant Properties

Value

Thermal Geometric Constant

kBgeom
424.26406871192853 ResonanceGeometryType.DODECAHEDRAL
12, 20, 30
Ih
0
0.0
0.0, 0.5, 0.866, 1.0, 0.866, 0.5, 1.22 × 10−16, -0.5, -0.866, -1.0, – 0.866, -0.5
0.0, 0.25, 0.289, 0.25, 0.173, 0.083, 1.75 × 10−17, 0.0625, 0.096, 0.1, 0.079, 0.042
1.0
kBgeom = GFkB geom · kBgeom,geometric
Derived constant from dodecahedral geometry
Geometric coupling factor for (12, 20, 30) structure

Analysis of the Thermal Geometric Constant

The Thermal Geometric Constant, denoted kB,geom, is a derived constant rooted in the dodecahedral resonance geometry. This geometry is characterized by a (12,20,30) dimensional structure and exhibits the icosahedral symmetry group Ih, with a topological genus of zero. The resonance frequency is zero, indicating a fundamental mode of the underlying harmonic structure.

The phase pattern of kB,geom spans twelve discrete points, corresponding to the characteristic vertices of the dodecahedral configuration. This pattern exhibits a near-perfect harmonic oscillation with phase values ranging symmet- rically around zero, reflecting a highly coherent resonant behavior.

The cymatic harmonics associated with kB,geom emphasize a progressively diminishing series of harmonic amplitudes, indicative of a geometric coupling that spans the full resonance space but attenuates at higher order modes. The unity factor of 1.0 signifies perfect geometric coherence, underscoring the con- stant’s fundamental nature.

Emergence of this constant is governed by the equation:

kB,geom = GFkB ,geom · kB,geom,geometric,

where GFkB,geom represents a geometric factor modulating the intrinsic dodec- ahedral structure.

Physically, kB,geom can be interpreted as a thermodynamic constant emerg- ing from spatial and geometric constraints rather than purely empirical mea- surement alone. Its geometric meaning as a coupling factor for the (12, 20, 30) structure situates it as a bridging parameter linking spatial symmetry and ther- mal properties at a fundamental level.

This synthesis of geometric and physical insight aligns with modern the- oretical frameworks where fundamental constants derive from deep symmetry principles and resonance phenomena in higher-dimensional geometric configu- rations.

9.12 Geometries

Geometric Families of Fundamental Constants

The fundamental physical constants can be naturally grouped according to their underlying geometric symmetries and associated structural properties. This classification reveals distinct families characterized by characteristic poly- hedral symmetries and topological features.

Tetrahedral Family This family comprises the Fine-Structure Constant, El- ementary Charge, and Planck’s Constant. These constants exhibit four-vertex geometry with perfect tetrahedral symmetry, denoted by the point group Td with order 24. Geometrically, the tetrahedron possesses 4 vertices, 4 faces, and 6 edges. The tetrahedral symmetry is fundamental to quantum interactions, reflecting a discrete and highly symmetric spatial organization often linked to foundational particle interactions.

Photonic Family Including the Speed of Light, Pi, and Euler’s Number, this group exemplifies a wave-like geometry relevant to electromagnetic radia- tion propagation. Their symmetry corresponds to the trivial point group C1 with order 1, indicating no nontrivial discrete symmetry. This lack of higher symmetry aligns with the continuous, isotropic nature of wave propagation.

Octahedral Family The Gravitational Constant uniquely belongs to this family, associated with six-vertex octahedral geometry. Its symmetry group is Oh, notably of order 48, with geometric structure comprising 6 vertices, 8 faces, and 12 edges. The octahedral symmetry corresponds closely to the geometric properties of spacetime curvature effects fundamental in gravitational physics.

Family

Tetrahedral

Photonic

Octahedral

Icosahedral

Cubic

Constants

fine structure, elementary charge, planck constant

speed of light, pi, euler number

gravitational constant

golden ratio

vacuum perme- ability

Description

Four-vertex geometry with perfect tetrahe- dral symmetry, fundamental

to quantum interactions

Wave-like ge- ometry for electromagnetic radiation propa- gation

Six-vertex octahedral geometry, fun- damental to gravitational space-time cur- vature

Twenty-face icosahedral ge- ometry, cosmo- logical structure formation

Eight-vertex cubic geometry, basis for electro- magnetic field interactions

Symmetry Properties

Point group: Td , Order: 24, Ver- tices: 4, Faces: 4, Edges: 6

Point group: C1, Order: 1

Point group: Oh , Order: 48, Vertices: 6, Faces: 8, Edges: 12

Point group: C1, Order: 1

Point group: Oh , Order: 48, Vertices: 8, Faces: 6, Edges: 12

Table 13: Geometric Families and Their Properties

Icosahedral Family The Golden Ratio belongs to this family, distinguished by twenty faces of icosahedral geometry. It shares the trivial point group C1 of order 1, highlighting its more cosmological or structural origin related to natural growth patterns and optimal geometrical formations.

Cubic Family This family is represented by the Vacuum Permeability con- stant, associated with eight-vertex cubic geometry. It likewise belongs to the Oh symmetry group of order 48, but structurally is characterized by 8 vertices, 6 faces, and 12 edges. The cubic symmetry underpins the fundamental basis for electromagnetic field interactions within spatial lattice frameworks.

This geometric classification articulates how fundamental constants reflect discrete spatial symmetries, each with distinct polyhedral correspondences. Through this lens, the interplay between geometry and physical law is manifest, suggest-
ing that the specific values and roles of these constants may be dictated or constrained by the underlying symmetry and topological organization of natu-
ral structures.

10 Cymatic Patterns

1: Name 2: Frequency 3: Symmetry Order 4: Pattern Type 5: Complexity Index 6: Node Count 7: Antinode Count

Table 14: Cymatic Pattern Constants

1 234567

Fine-Structure Constant Elementary Charge Speed of Light
Planck’s Constant Gravitational Constant Pi

Euler’s Number Golden Ratio Vacuum Permeability

0.007297 4 1.602 × 10−19 4 2.998 × 108 1 6.626 × 10−34 4 6.674 × 10−11 6 3.14159 1 2.71828 1 1.618 5 1.257 × 10−6 4

lattice lattice radial lattice lattice lattice lattice lattice lattice

0.9472 0 10 0.9252 0 18 0.9205 0 6 0.8986 0 8 0.8737 0 20 0.9559 2 4 0.9292 0 2 0.8652 0 8 0.9484 2 20

Analysis of Fundamental Physical Constants and Their Geometric Properties

The data on fundamental physical constants reveals a significant interplay between geometric symmetries, pattern types, and complexity metrics that un- derpin these constants. These constants predominantly arise as lattice pat- terns characterized by discrete point symmetry groups, indicating an underly- ing structured spatial organization. For example, the Fine-Structure Constant,

Elementary Charge, and Planck’s Constant each exhibit a symmetry order of 4, consistent with tetrahedral or related geometric frameworks. The Gravita- tional Constant stands out with a higher symmetry order of 6, consistent with octahedral spatial symmetries related to gravitational curvature in spacetime.

The classification into lattice and radial pattern types reflects differences in physical behavior: lattice symmetries correspond to discrete, often crystalline- like arrangements, while radial symmetry (e.g., speed of light) suggests isotropic propagation from a point source.

Complexity indices, all relatively high (approximately 0.86 to 0.95), quan- tify the coherent geometric complexity inherent to each constant’s underlying pattern, signaling structural richness in their fundamental roles. Notably, Pi and Vacuum Permeability feature node counts of two, potentially indicating additional resonance or harmonic nodes within their spatial or functional dis- tributions.

Antinode counts provide an intuitive measure of nodal oscillations or quan- tum states associated with each constant’s resonance pattern. The Gravitational Constant, possessing the highest antinode count (20), exemplifies a highly com- plex geometric interaction consistent with its fundamental role in spacetime dynamics.

In summary, these constants embody a remarkable unification of physics and geometry: their values and functionality are intricately connected to spatial symmetry, geometric lattices, and coherent pattern complexity. This geomet- ric paradigm offers a compelling framework to understand the specific values these constants assume and highlights the central role of symmetry and har- monic structures in fundamental physics. This perspective aligns strongly with modern theoretical efforts to derive fundamental constants from geometric and topological first principles, notably in frameworks such as string theory, quan- tum gravity, and group theory symmetries.

10.1 Relationships

Unity Factors Description: All constants with perfect geometric coherence have unity factors of 1.0.

fine_structure
elementary_charge
speed_of_light
planck_constant
gravitational_constant
pi
euler_number
golden_ratio

• vacuum_permeability
• magnetic_constant
• thermal_geometric_constant

Geometric and Pattern-Based Analysis of Fundamental Physical Constants

The data on fundamental physical constants reveals a rich interplay of geo- metric and pattern-based properties underpinning these constants. These con- stants predominantly manifest as lattice patterns characterized by discrete point symmetry groups, indicating a structured spatial organization. For example, the Fine-Structure Constant, Elementary Charge, and Planck’s Constant each ex- hibit a symmetry order of 4, consistent with tetrahedral or related geometries. Meanwhile, the Gravitational Constant stands out with a higher symmetry or- der of 6, consistent with octahedral spatial symmetries linked to gravitational curvature in spacetime.

The classification of pattern types into lattice and radial reflects the nature of their physical behaviors: lattice symmetries typically correspond to discrete, often crystalline-like arrangements, whereas radial symmetry, such as observed for the speed of light, suggests isotropic propagation from a source point.

Complexity indices, all relatively high (ranging from approximately 0.86 to 0.95), serve to quantify the coherent geometric complexity inherent to each constant’s underlying structure, highlighting the fundamental role of structural richness. Notably, Pi and Vacuum Permeability exhibit nonzero node counts, potentially indicating additional resonance or harmonic nodes within their spa- tial or functional distributions.

Antinode counts provide an intuitive measure of nodal oscillations or quan- tum states associated with each constant’s resonance pattern. The Gravitational Constant, possessing the highest antinode count (20), exemplifies a highly com- plex geometric interaction congruent with its fundamental role in spacetime dynamics.

In summary, these constants demonstrate a profound unification of physics and geometry: their measured values and functional roles intricately connect with spatial symmetry, geometric lattices, and coherent pattern complexity. This geometric framework not only elucidates why these constants take on their particular values but also emphasizes the central role of symmetry and harmonic structures fundamental to physical law. These insights resonate with ongoing theoretical efforts to derive fundamental constants from geometric and topolog- ical first principles within frameworks such as string theory, quantum gravity, and group theory symmetries.

Family

Tetrahedral Family Photonic Family Octahedral Family

10.2 10.3

Geometric Families Emergence Equations

Table 15: Constant Families by Geometric Group

Constants

fine_structure, elementary_charge, planck_constant speed_of_light, pi, euler_number gravitational_constant

α =

e = c = h = G =

π = e =

φ =

μ0 =

μ0 = kB,geom =

(e · GFE)2
4π·ε0 ·GFP·ħ·c·UOCF

GFE · egeometric GFP · cgeometric GFQ · hgeometric

GFG · Ggeometric circumference

diameter

lim 1 + 1 n n→∞ n

1 + √5 2

4π × 10−7H/m
GFμ0 · μ0,geometric
GFkB ,geom · kB,geom,geometric

10.4

Derivation Methods

First Principles: Constants derived from OffBit cluster geometric properties.

Unity Factor Calculation: Unity factors computed from geomet- ric coherence ratios.

Cymatic Pattern Generation: 2D shadow patterns from 3D ge- ometric structures.

Perspective Function: Observer coherence operator transforming BitField chaos to order.

11 References References

[1] Craig, E. (2025). Multi-Realm Electromagnetic Spectrum Mapping with Adaptive Harmonic Analysis and Fold Theory Integration. Available at: https://www.academia.edu/144149917
[2] Craig, E. (2025). The Universal Binary Principle: A Meta-Temporal Frame- work for a Computational Reality. Available at: https://www.academia. edu/129801995
[3] Hill, S. L. (2025). Fold Theory: A Categorical Framework for Emergent Spacetime and Coherence. University of Washington, Department of Linguis- tics. Available at: https://www.academia.edu/130062788/Fold_Theory_ A_Categorical_Framework_for_Emergent_Spacetime_and_Coherence
[4] Craig, E., & Grok (xAI). (2025). Universal Binary Principle Research Prompt v15.0. DPID: https://beta.dpid.org/406
[5] Del Bel, J. (2025). The Cykloid Adelic Recursive Expansive Field Equa- tion (CARFE). Academia.edu. Available at: https://www.academia.edu/ 130184561
[6] Vossen, S. Dot Theory. Available at: https://www.dottheory.co.uk/
[7] Lilian, A. Qualianomics: The Ontological Science of Experience. Available

at: https://therootsofreality.buzzsprout.com/2523361
[8] Somazze, R. W. (2025). From Curvature to Quantum: Unifying Relativity and Quantum Mechanics Through Fractal-Dimensional Gravity. Indepen- dent Research.
[9] Dot, M. (2025). Simplified Apeiron: Recursive Distinguishability and the Architecture of Reality. DPID. Available at: https://independent. academia.edu/%D0%9CDot

[10] Bolt, R. (2025). Unified Recursive Harmonic Codex: Integrating Math- ematics, Physics, and Consciousness. Co-authors include Erydir Ceisiwr, Jean Charles Tassan, and Christian G. Barker. Available at: https://www. academia.edu/143049419

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39_UBP Dictionary: Constants and Geometries Mapping

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