Natural Ratios as the Locus of Physical Law: An Interpretive Framework for the Coherence Constraint

J. Rogers
Independent Researcher, SE Ohio

---

Abstract

The recently proven Coherence Constraint [1] establishes that the dimensional constants of physics (c, h, G) are necessarily the components of an invertible Jacobian mapping between arbitrary human unit charts and a unique, invariant natural coordinate system (the Planck scale). This paper develops the physical and ontological interpretation of that theorem. We demonstrate that the invariance of physics cannot reside in dimensional quantities, which depend on unit conventions, but must be located in the dimensionless ratios of the natural chart itself. All physical "laws" thus emerge as coordinate projections of a single, ratio-structured manifold. This reframes physical law as a generative geometry rather than a prescriptive code, aligning fundamental physics with principles of structural realism.

---

1. Introduction: From Mathematical Proof to Physical Ontology

In a preceding work [1], we established a formal proof based on the internal consistency of physical theory. We demonstrated that the fundamental constants are not independent parameters but must satisfy strict coherence relations, forming a solvable and invertible system of equations. The mathematical necessity of this structure proves that the constants function as a Jacobian transformation matrix connecting any arbitrary unit system to a unique, invariant natural scale.

The purpose of the present work is to act on the consequences of this proof. If the existence of a unique natural coordinate system and the transformative nature of constants are taken as established mathematical lemmas, what are the implications for the ontology of physical law? We argue that this architecture forces a relocation of the "locus" of physics from the dimensional equations of our arbitrary charts to the dimensionless geometric structure of the invariant core.

---

2. Formal Ontology of the Coherence Architecture

Based on the results of [1], we establish the following formal ontological structure for physical description:

Definition 1 (The Invariant Core,

        $\mathcal{C}_0$
      

): The unique, unit-free coordinate manifold in which all fundamental physical relations reduce to direct, dimensionless ratios (e.g.,

        $1:1$
      

). Its existence and uniqueness are guaranteed by the invertibility of the system of constants proved in [1].

Definition 2 (The Arbitrary Periphery,

        $\mathcal{C}_a$
      

): The infinite set of possible human-defined measurement charts (SI, cgs, etc.), each characterized by an arbitrary choice of base units that breaks the inherent symmetry of

        $\mathcal{C}_0$
      

.

Definition 3 (The Jacobian Mapping,

        $J$
      

): A smooth, invertible transformation

        $J: \mathcal{C}_a \to \mathcal{C}_0$
      

whose components are explicitly constructed from the measured values of the fundamental constants (

        $c, h, G, \dots$
      

) specific to chart

        $\mathcal{C}_a$
      

.

From these definitions, we state the foundational lemma derived from the Coherence Constraint:

Lemma 1 (Jacobian Coherence): The invariance of physical reality across all arbitrary charts

        $\mathcal{C}_a$
      

requires that the Jacobian

        $J$
      

be globally coherent. The empirical constants must satisfy constraint relations such that

        $J$
      

remains invertible and maps to the same unique

        $\mathcal{C}_0$
      

regardless of the choice of

        $\mathcal{C}_a$
      

.

---

3. Constants as Coordinate Covariants

The traditional view regards constants as fundamental, invariant scalars. Under the architecture defined above, this view is untenable. Because each arbitrary chart

        $\mathcal{C}_a$
      

defines distinct numerical bases for measurement axes, the components of the mapping

        $J$
      

must vary to preserve invariance under transformation to

        $\mathcal{C}_0$
      

.

Thus, the measured constants are not universal magnitudes but coordinate covariants—they adapt to each specific chart to maintain the structural integrity of the mapping.

Proposition 1 (Physical Equivalence via Composite Mapping):
For any two distinct arbitrary unit systems

        $\mathcal{C}_a$
      

and

        $\mathcal{C}_b$
      

, the numerical values of the constants differ (

        $J(\mathcal{C}_a) \neq J(\mathcal{C}_b)$
      

), yet they maintain physical equivalence through the composite identity mapping:

        $$J^{-1}({\mathcal{C}}_a)\circ J({\mathcal{C}}_b)=I$$      

The constancy of physics is achieved not by the invariance of the dimensional quantities themselves, but by the invariance of the transformation structure that relates them.

---

4. The Projection Principle

If the constants are adaptive components of

        $J$
      

, and the unit charts

        $\mathcal{C}_a$
      

are arbitrary, the locus of fundamental law must reside in the invariant core

        $\mathcal{C}_0$
      

. The familiar dimensional laws of physics are therefore not fundamental, but are representations generated by projection.

Definition 4 (Projection of Law):
Let

        $\mathcal{L}_0$
      

be a fundamental geometric relation holding in the invariant domain

        $\mathcal{C}_0$
      

. Its projection into an arbitrary chart

        $\mathcal{C}_a$
      

, denoted

        $\mathcal{L}_a$
      

(an apparent physical law), is defined by the transformation:

        $$\mathcal{L}_a = J^{-1}(\mathcal{C}_a) \circ \mathcal{L}_0 \circ J(\mathcal{C}_a)$$
      

Example:
Consider the equivalence of energy and mass.
In

        $\mathcal{C}_0$
      

, the relation

        $\mathcal{L}_0$
      

is the simple identity:

        $E_{natural}=M_{natural}$      

.
Under projection to the SI chart via

        $J^{-1}_{SI}$
      

, dimensional artifacts (units) are introduced, and the necessary Jacobian scaling factors appear:

        $${\mathcal{L}}_{SI}\hspace{0.25em} ⟹\hspace{0.25em} E=m{c}^2$$      

The "constant"

        $c^2$
      

arises not as a physical entity, but as a necessary component of

        $J^{-1}$
      

required to map the

        $1:1$
      

ratio of

        $\mathcal{C}_0$
      

onto the disparate scales of the SI meter and second.

---

5. The Generative Nature of Physical Law

This framework demonstrates that physical laws are generated by the act of projection, rather than discovered as independent empirical facts.

Corollary 1 (Law Generativity):
Given a ratio manifold

        $\mathcal{C}_0$
      

defined by

        $n$
      

measurement axes, the Jacobian projection into any chart

        $\mathcal{C}_a$
      

generates a complete set of

        $n(n-1)\mathrm{/}2$      

pairwise relational equations. Each projection yields a distinct apparent "law" in

        $\mathcal{C}_a$
      

, while all remain expressions of the same unified structure in

        $\mathcal{C}_0$
      

.

This corollary has been computationally demonstrated via LawForge [3], an algorithmic engine that reconstructs known equations of physics (and predicts untested ones) by projecting the ratio relations of

        $\mathcal{C}_0$
      

into arbitrary unit charts. The successful generation of these laws proves they are consequences of the geometry of

        $\mathcal{C}_0$
      

rather than independent prescriptions.

---

6. Structural Sufficiency and the End of Prescription

The traditional view of physics is prescriptive: laws are external rules governing matter. The Coherence architecture implies a shift toward ontic structural realism [4, 5].

If all apparent laws can be generated by projecting the invariant core into arbitrary coordinates, then what persists under transformation is not prescription, but structure—the invariant topology of the ratio manifold.

Theorem (Structural Sufficiency):
Because all physical behavior is invariant under the Jacobian transformation to

        $\mathcal{C}_0$
      

, the complete fundamental physics of the universe is contained within the geometric structure of

        $\mathcal{C}_0$
      .

Therefore, there exist no independent, prescriptive laws beyond this invariant structure. Apparent laws are epistemic artifacts of measurement, while the underlying geometry is ontological.

---

7. Conclusion

The mathematical proof of the Coherence Constraint [1] necessitates that physical constants form a coherent Jacobian between arbitrary and natural coordinate systems. The physical interpretation of this proof forces the following conclusions:

1. Locus of Law: The invariant locus of physics is the ratio-structured manifold

           $\mathcal{C}_0$
         

   (the Planck chart), not the dimensional equations of human unit systems.

2. Nature of Constants: Dimensional constants are coordinate covariants—adaptive components of the Jacobian transformation

           $J$
         

   .

3. Generativity: Apparent physical laws arise as projections of invariant structure into arbitrary charts (

           ${\mathcal{L}}_a=J^{-1}\circ {\mathcal{L}}_0\circ J$      

   ).

Consequently, physical law is recast from a discovery of external rules to a generative consequence of coordinate projection. The universe, in its fundamental form, is a self-consistent geometric structure from which all "laws" inevitably follow.

---

Appendix A: Physical Meaning of the Jacobian Partials

Simply stating that constants are Jacobian components can remain abstract. We can ground this by identifying them as partial derivatives relating coordinate axes across the transformation.

If we view the constants as the bridge between an arbitrary coordinate

        $x_a \in \mathcal{C}_a$
      

and a natural coordinate

        $x_0 \in \mathcal{C}_0$
      

, the constants represent the local scaling factors:

•         $c = \frac{\partial(\text{Length}_a)}{\partial(\text{Time}_a)}$
        

  constrained to map to

          $\frac{\text{Length}_0}{\text{Time}_0} = 1$
        

  .

•         $h=\frac{\mathrm{\partial }({\text{Energy}}_a)}{\mathrm{\partial }({\text{Frequency}}_a)}$      

  constrained to map to

          $\frac{\text{Energy}_0}{\text{Frequency}_0} = 1$
        

  .

•         $G_{factor} = \frac{\partial(\text{SpaceTime}_a)}{\partial(\text{MassEnergy}_a)}$
        

  constrained to map to the identity ratio in

          $\mathcal{C}_0$
        

  .

The constants are the differential rates of exchange required to maintain the unified structure of

        $\mathcal{C}_0$
      

when projected onto axes that have been arbitrarily scaled relative to one another.

---

References

[1] Rogers, J. "The Coherence Constraint: Why Physical Constants Must Form a Consistent Jacobian." (Preprint).
[2] Rogers, J. "LawForge: The Physics Law Discovery Engine." (Computational Implementation).