The Observer-Dependent Reality Framework: A Categorical Reformulation of Physics and Perception

 J. Rogers, SE Ohio

Abstract

We present a complete framework for a theory of observer-dependent reality, unifying physics, metrology, and epistemology within a single categorical structure. The theory posits a coherent, undifferentiated substrate (denoted 𝒮ᵤ) which is fractured into a phenomenal world by the perceptual and cognitive apparatus of an observer. This process is formally modeled as a Grothendieck fibration π : 𝓔 → 𝓑, where the base category 𝓑 represents the observer's conceptual axes (e.g., Mass, Length, Time), and the total category 𝓔 represents the world of measurable quantities. Physical laws are not fundamental truths but emerge as Cartesian liftings of morphisms in 𝓑, while so-called fundamental constants are revealed to be cocycle data ensuring coherence across changes of unit systems. This framework elevates the observer from a passive witness to the active constructor of reality, providing a generative principle from which all species-specific "laws of physics" must emerge.

1. Introduction: The Perceptual Fracture of Unity

Traditional physics seeks a "theory of everything" — a single, objective set of equations describing the universe from a God's-eye view. This pursuit implicitly assumes a detachment of the knower from the known. Conversely, frameworks from quantum mechanics emphasize the role of the observer but often lack a formal, mathematical structure to describe the observer's constructive role.

This paper presents a synthesis: a formal, mathematical framework where physical reality is the output of a measurement-theoretic process specific to an observer. The theory begins with a single postulate: a unified, coherent substrate 𝒮ᵤ exists. This substrate is without parts, properties, or differentiation; it is pure potentiality and relation. The universe of tables, stars, forces, and laws we experience is not this substrate, but a projection of it through the lens of a specific observer.

2. The Categorical Architecture of Perceived Reality

The framework is built upon a four-layer ontological cascade:

1. The Coherent Substrate (𝒮ᵤ): The undivided, pre-conceptual ground of reality. It is inaccessible to direct description but is defined as the limit of all possible observer perspectives.

2. The Conceptual Axes (𝓑): A category where objects are the fundamental dimensions of measurement for an observer (e.g., Mass, Length, Time). Morphisms are proportional relationships between these axes (e.g., Mass → Energy). The structure of 𝓑 is not universal; it is determined by the observer's perceptual interface.

3. The Unit Charts (𝒰): A category parameterizing the specific units of measurement (SI, Planck, CGS) an observer employs. Morphisms are scaling transformations between units.

4. The Empirical World (𝓔): The total category of lived experience. Its objects are concrete, measured quantities (e.g., (9.8, m/s²)). Its morphisms are the physical laws and relationships between these quantities.

The entire structure is bound together by a fibration functor π : 𝓔 → 𝓑, which maps a measured quantity to its dimensional type, forgetting its numerical value. The fiber π⁻¹(X) over an object X ∈ 𝓑 contains all possible measurements of that type.

3. The Observer as a Functor

An observer is formally defined as a functor:

O: 𝓟 → 𝓑

• 𝓟 (The Perceptual Interface): A category modeling the observer's raw sensory and cognitive hardware (e.g., detectable wavelength ranges, temporal resolution).

• 𝓑 (The Conceptual Axes): The observer's chosen "software" interpretation of sensory data.

• The Functor O: This is the process of perceptual quantization. It is the act of fracturing the continuous flux of sensory input into a discrete set of relational, conceptual categories. This functor defines the observer's unique perspective and is the origin of their base category 𝓑.

4. Physical Law as Cartesian Lifting

Physical laws do not govern the substrate 𝒮ᵤ; they govern the relationships within the observer's constructed reality 𝓔.

A physical law is a Cartesian lifting within the fibration π. Given a morphism in the base category, φ: X → Y (e.g., a proportionality between Mass and Energy), a law is a morphism in the total category f: (x, U_x) → (y, U_y) such that π(f) = φ and f is universal among all such morphisms.

This means that complex equations like E = mc² are simply the liftings of a simple, dimensionless proportionality E ∝ m into a specific unit chart (SI units), with the constant c² acting as the necessary scaling factor.

5. Constants as Cocycles and The Planck Hub

The so-called fundamental constants (c, h, G, k_B) are demystified within this framework. They are not properties of 𝒮ᵤ but are cocycle data—connection coefficients that ensure the consistency of the fibration under changes of unit charts.

The Planck scale is uniquely identified as the universal routing hub of physics. In the Planck unit chart, the values of these constants normalize to 1:

c = 1 l_P t_P⁻¹, h = 1 J_P t_P, G = 1 l_P³ m_P⁻¹ t_P⁻²

Their seemingly arbitrary values in other unit systems (e.g., c ≈ 3e8 m/s) are merely the Jacobian components for the coordinate transformation from the Planck chart to the observer's chosen chart (e.g., SI). They are glue for interoperability, not fundamental features.

6. The Law Compiler Λ: A Meta-Theory of Physics

The framework provides a generative principle for physical laws: the Law Compiler, a functor:

Λ : Hom(𝓑) → Sect(π)

Λ takes a dimensionless relationship in the base category (the "source code") and compiles it into a family of sections of the fibration (the "executable laws" for various unit systems). This process explains why the same physical relation (e.g., T ∝ 1/M) can manifest in wildly different forms (e.g., the Hawking temperature formula) across different unit systems; each form is a different "compilation target."

7. Examples and Implications

7.1 The "Charge" Anomaly: The perceived complexity of electromagnetic units stems from the erroneous classification of charge as a fundamental object in 𝓑_Human. Reformulating it via the natural charge density ncd = amp_Force_natural * m_P * l_P (with units [kg·m]) eliminates it as a fundamental dimension, revealing it as a derived, geometric property. This is a data model refactoring that simplifies the entire axiomatic structure.

7.2 Species-Specific Physics: A species with a different perceptual interface 𝓟_Alien would have a functor O_Alien yielding a different base category 𝓑_Alien. Their "laws of physics" would be the Cartesian liftings within their fibration π_Alien. Their constants and equations would be unrecognizable to us, yet their physics would be equally valid and self-consistent, describing the same substrate 𝒮ᵤ.

7.3 Resolving Paradoxes: The framework dissolves the mystery of fine-structure constants and dimensionless numbers. They are simply geometric ratios (like π) that appear in the lifting process for a specific observer.

8. Conclusion: Reality as a Compiled Projection

This framework proposes that what we call reality is a compiled projection of the substrate 𝒮ᵤ through the specific compiler of our own perceptual and cognitive apparatus. Physics is not the study of the substrate itself, but the reverse-engineering of our own compiler's output.

The quest for a "Theory of Everything" is redefined. It is not the search for a final equation but the discovery of the meta-theoretical framework—the architecture of the compiler itself. This work presents such a framework, one where the observer is finally and formally recognized not as a nuisance to be eliminated, but as the essential agent through which a universe of law and form becomes manifest.

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References & Influences:

1. Category Theory, Topos Theory, and Grothendieck Fibrations

2. Buckingham π Theorem (Dimensional Analysis)

3. Planck (1899), Über irreversible Strahlungsvorgänge

4. Von Neumann (1932), Mathematical Foundations of Quantum Mechanics (Observer Formalism)

5. Wheeler's "It from Bit" and Participatory Universe

6. Hoffman's Interface Theory of Perception