The Coherence Thread Theory of Fundamental Constants: A Unified Categorical Framework for Physical Law

 J. Rogers, SE Ohio, 25 July 2025, 1426

Abstract

We present a unified categorical framework that reveals the true nature of fundamental constants as coherence threads maintaining consistency across the measurement fibration of physical law. Building on previous work demonstrating the dynamic compositional structure of the gravitational constant G and the categorical framework of physical law as Grothendieck fibration, we show that all fundamental constants—G, h, c, k_B—function as coherence morphisms ensuring that different projections of the same underlying substrate relationships remain numerically consistent. This framework elevates constants from mysterious "correction factors" to structural invariants that guarantee the coherence of physics itself. We provide computational verification demonstrating that G's different compositional faces across gravitational contexts all resolve to the same numerical value, proving the fibration's coherence. The theory reveals that physical complexity emerges not from fundamental intricacy, but from the coordinate artifacts of projecting simple substrate relationships through misaligned measurement geometries.

Note, all these planck units are the full planck units with h, not hbar.  Unit scaling has nothing to do with geometry of pi.  Confusing the two is a category error. 

1. Introduction: From Correction Factors to Coherence Threads

Traditional physics education presents fundamental constants as mysterious properties of nature—fixed numerical values that somehow emerge from the deep structure of reality. Recent work has challenged this view, first by demonstrating that the gravitational constant G exhibits dynamic compositional structure across different physical contexts, and subsequently by developing a categorical framework modeling physical law as a Grothendieck fibration over measurement space.

This paper synthesizes these insights into a unified theory revealing that fundamental constants are neither mysterious properties nor mere correction factors, but coherence threads—structural invariants that maintain consistency across the measurement fibration as we project the same underlying substrate relationships through different physical contexts.

2. The Fibration Structure of Physical Law

2.1 The Four-Layer Architecture

Physical reality emerges through a four-layer ontological cascade:

Layer 1 – The Coherent Substrate (𝒮ᵤ): Pure dimensionless relationships like T ~ 1/M, existing as morphisms in a pre-conceptual substrate.

Layer 2 – Conceptual Axes (𝒜): Decomposition into measurement categories (Mass, Time, Length, Temperature) forming a symmetric monoidal category.

Layer 3 – Coordinate Charts (𝒰): Human-imposed unit systems (SI, Planck, CGS) with morphisms as scaling transformations.

Layer 4 – Measurement World (𝓔): Concrete numerical quantities where physical laws appear as morphisms relating measured values.

2.2 The Fibration π : 𝓔 → 𝓑

We model this as a Grothendieck fibration where:

• 𝓑 represents dimensionless measurement types
• 𝓔 contains measured quantities as (value, unit) pairs
• π maps each measurement to its conceptual base
• Physical laws emerge as Cartesian liftings of morphisms in 𝓑

3. Coherence Threads: The True Nature of Constants

3.1 Beyond Correction Factors

The traditional view treats constants as correction factors compensating for unit system misalignment. While useful, this human-centric perspective misses the deeper structural role: constants are coherence threads that maintain consistency across different projections of the same underlying substrate relationships.

3.2 G as Exemplar: Multiple Faces, Single Thread

The gravitational constant G demonstrates this principle perfectly. Across different physical contexts, G exhibits distinct compositional structures:

• Force context: G = (l_P²/m_P²) × F_P
• Energy context: G = (l_P/m_P²) × E_P
• Velocity context: G = (l_P × (l_P/t_P)²)/m_P
• Length context: G = (l_P × (l_P/t_P)²)/m_P
• Acceleration context: G = (l_P²/m_P) × a_P

These are not different constants sharing a symbol, but different projective views of the same coherence thread as it maintains consistency across the gravitational sector of the measurement fibration.

3.3 Computational Verification of Coherence

Context                Formula                           Calculated G
----------------------------------------------------------------------
Force (F)             G = (l_P^2 / m_P^2) * F_P        6.67430e-11
Energy (E)            G = (l_P / m_P^2) * E_P          6.67430e-11
Velocity (v_e)        G = (l_P * (l_P/t_P)**2) / (m_P) 6.67430e-11
Length (r_s)          G = (l_P * (l_P/t_P)**2) / (m_P) 6.67430e-11
Acceleration (g)      G = (l_P^2 / m_P) * a_P          6.67430e-11

That all compositions resolve to identical numerical values proves the fibration's coherence. G serves as the structural guarantee that Newton's law of gravitation, gravitational potential energy, escape velocity, and Schwarzschild radius represent consistent projections of the same underlying substrate relationships.

4. The Cocycle Condition: Mathematical Foundation

4.1 Coherence Morphisms

In the categorical framework, constants function as coherence morphisms satisfying the cocycle condition. When composing paths through different measurement contexts:

Mass → Force → Energy
Mass → Energy (direct)

The constants must combine such that both paths yield consistent results. This is not an empirical accident but a structural necessity for the fibration to remain coherent.

4.2 G as Structural Invariant

G is the structural invariant proving that:

• Gravitational force (Newton's law)
• Gravitational potential energy
• Escape velocity
• Schwarzschild radius

...are not independent phenomena but coherent projections of identical substrate relationships through different measurement contexts.

5. Universal Principle: All Constants as Coherence Threads

5.1 The Complete Spectrum

This principle extends to all fundamental constants:

c (Speed of Light): Coherence thread for the kinematic/relativistic sector, ensuring consistency across velocity, energy, momentum, and spacetime interval contexts.

ℏ (Planck's Constant): Coherence thread for the quantum sector, maintaining coherence across energy-frequency, momentum-wavelength, and uncertainty relation contexts.

k_B (Boltzmann Constant): Coherence thread for the thermodynamic sector, ensuring consistency across energy-temperature, entropy, and statistical mechanics contexts.

5.2 Sectoral Coherence

Each constant governs coherence within its respective sector:

• G: Gravitational sector coherence
• c: Kinematic sector coherence
• h: Quantum sector coherence
• k_B: Thermodynamic sector coherence

6. The Basis Rotation Interpretation

6.1 Laws as Coordinate Artifacts

All physical laws emerge as coordinate-dependent expressions of coordinate-free proportionalities. What appears as complex relationships between different quantities are actually simple rotations between measurement axes, obscured by Jacobian factors required for expression in non-orthogonal coordinate systems.

6.2 Constants as Jacobian Elements

The constants encode the Jacobian matrix elements for rotating between measurement axes. A formula like T = c³ h/(G M k_B) appears complex, but the actual substrate relation T ~ 1/M is trivial. The constants are pure geometric bookkeeping for basis misalignment.

7. Computational Derivation: The Law Compiler

7.1 Three-Stage Process

Every physical law emerges through a canonical process:

1. Substrate Conception: Pure proportionality T ~ 1/M as morphism in 𝒜
2. Planck Normalization: Scaling to natural units T/T_P ~ m_P/M
3. Coordinate Projection: Lifting through fibration with Jacobian transformation

7.2 Buckingham Pi as Fibered Projection

The Buckingham Pi theorem finds its natural home in this framework. Pi groups represent coordinate-free substrate relationships, while their dimensional realizations are the coordinate projections requiring constants for coherence.

8. Implications for Physical Understanding

8.1 The Illusion of Fundamental Complexity

Traditional physics mistakes coordinate artifacts for deep laws. Complex formulas reflect measurement geometry, not reality's intrinsic structure. The substrate contains only simple proportionalities between measurement axes.

8.2 Unification Through Coherence

This framework provides unification not by adding theoretical epicycles, but by revealing the coherence that was always present. Different physical phenomena are unified as consistent projections of the same substrate relationships, with constants serving as the coherence infrastructure.

8.3 The Observer as Section Selector

Each observer selects:

• A conceptual decomposition (choice of axes)
• A unit scheme (choice of coordinate chart)
• A coordinate basis (point within fibers)

Physical laws become observer-indexed liftings, with constants encoding the observer's scaling imprint upon the coherent substrate.

9. Verification Through Unit Scaling

9.1 Jacobian Coordinates

The constants are revealed as Jacobian coordinates by rotating the basis from SI to Planck units. The scaling factors:

rescale_factors = [
    {"symbol": "s",  "factor": t_P,             "swap_with": "t_Ph"},
    {"symbol": "m",  "factor": t_P * c,         "swap_with": "l_Ph"},
    {"symbol": "kg", "factor": Hz_kg/t_P,       "swap_with": "m_Ph"},
    {"symbol": "K",  "factor": 1/(t_P * K_Hz),  "swap_with": "T_Ph"},
]

demonstrate that setting constants to unity is equivalent to performing a specific basis rotation to natural coordinates.  Remember, that t_P is non reduced. 

9.2 The Hz_kg and K_Hz Jacobians

Standard physics hand-waves the operation of "setting c = h = 1" without understanding that this formally requires the Jacobian coordinates Hz_kg = h/c² and K_Hz = k_B/h, which encode the fundamental scaling relationships between measurement axes.

10. Conclusions: A Meta-Theory of Lawful Perception

10.1 Constants as Infrastructure

Fundamental constants are neither mysterious properties of nature nor arbitrary human constructs. They are the infrastructure of coherence—the structural elements that allow us to build consistent mathematical models of reality from multiple observational perspectives.

10.2 The Deep Unity of Physics

This framework reveals that the apparent complexity of physics emerges from projecting simple substrate relationships through misaligned measurement coordinates. The deep unity of physics exists not as a theoretical aspiration but as a demonstrated mathematical reality, maintained by the coherence thread structure of fundamental constants.

10.3 Metrology as Foundation

Measurement emerges not as peripheral to physics but as the fundamental act that generates physical law through structured projection from the coherent substrate. Constants encode the geometry of this projection, making them central to understanding the nature of physical reality itself.

References

Rogers, J. (2025). The Dynamic Composition of G: Unifying Gravitational Physics at the Planck Scale. Internal Research.

Rogers, J. (2025). The Structure of Physical Law as a Grothendieck Fibration. Internal Research.