Intersubjectivity and the Bridge of Shared Perception

A significant challenge to any observer-centric model of physics is the problem of solipsism or radical relativism: if physical laws are projections through an observer's specific perceptual apparatus ($\mathcal{A}$), how can two different observers—for instance, a human and a hypothetical alien intelligence—ever claim to be studying the same universe? Does this framework imply that their physics would be mutually unintelligible?

The PUCS framework provides a robust answer, establishing a structured, relational objectivity. It posits that a shared reality is accessible not through identical perceptions, but through measurement overlap, where a common conceptual axis serves as a "Rosetta Stone" for translating between two otherwise different perceptual grammars.

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Formalizing the Problem

Let us consider two distinct observers, Observer 1 (Human) and Observer 2 (Alien). Within the PUCS framework, each is defined by their own measurement structure:

• Observer 1 (Human): Possesses a perceptual category $\mathcal{A}_1$ (with objects like $\text{Axis(Mass)}$, $\text{Axis(Length)}$) and a conventional unit system $\mathcal{U}_1$ (e.g., SI). Their concrete measurements are produced by the bifibration $\mathcal{M}_1 : \mathcal{A}_1^{op} \times \mathcal{U}_1 \rightarrow \text{Cat}$.

• Observer 2 (Alien): Possesses a different perceptual category $\mathcal{A}_2$ (with objects like $\text{Axis(Inertia)}$, $\text{Axis(Extension)}$) and their own unit system $\mathcal{U}_2$. Their measurements are produced by $\mathcal{M}_2 : \mathcal{A}_2^{op} \times \mathcal{U}_2 \rightarrow \text{Cat}$.

The challenge is this: If $\mathcal{A}_1 \neq \mathcal{A}_2$, how can a law derived from $\mathcal{M}_1$ ever be translated into the language of $\mathcal{M}_2$?

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The Solution: The Functorial Bridge of Shared Axes

Communication and shared understanding become possible if the two perceptual categories are not entirely disjoint. That is, if $\mathcal{A}_1 \cap \mathcal{A}_2 \neq \emptyset$.

Let us postulate that there exists at least one conceptual axis common to both observers. A prime candidate for such a universal is $\text{Axis(Frequency)}$, as it relies on the fundamental act of counting periodic events.

Shared Ground

Both observers can make measurements projected onto the $\text{Axis(Frequency)}$ because it exists in both their perceptual categories. They are both capable of observing periodicity in the one underlying Universal State, $\mathcal{S}_u$.

Internal Scaling (Jacobians)

• Observer 1 relates their other conceptual axes to $\text{Axis(Frequency)}$ via their set of Base Jacobians. For example, they have a morphism $\text{rel}_{h_1}: \text{Axis(Mass)} \rightarrow \text{Axis(Frequency)}$, whose numerical realization in $\mathcal{U}_1$ is the Jacobian $\text{Hz}_{\text{kg}}$.

• Observer 2 does the same. They might relate their $\text{Axis(Inertia)}$ to $\text{Axis(Frequency)}$ via their own morphism $\text{rel}_{h_2}: \text{Axis(Inertia)} \rightarrow \text{Axis(Frequency)}$, realized by their own Jacobian, say $\text{Wobbles}_{\text{per Glarp}}$.

Building the Translation Dictionary

Using $\text{Axis(Frequency)}$ as the common term, the two observers can construct a translation functor $T: \mathcal{A}_1 \rightarrow \mathcal{A}_2$. This establishes a rigorous mapping between their conceptual axes. For example, they can relate $\text{Axis(Mass)}$ and $\text{Axis(Inertia)}$ because both are expressible in terms of $\text{Axis(Frequency)}$. The conversion factor between Mass and Inertia would be:
$(\text{Hz}_{\text{kg}})^{-1} \otimes (\text{Wobbles}_{\text{per Glarp}})$

Translating Physical Laws

With this dictionary in place, an entire physical law can be translated. A human law involving mass can be transformed into an alien law involving inertia. The constants ($G, h, c$ for the human; $Z, X, Y$ for the alien) are not barriers to communication; they are the critical gears of the translation, the numerical Jacobians that enable inter-category mapping.

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Conclusion: A Relational Objectivity

The PUCS framework therefore does not lead to radical relativism. Instead, it defines a more nuanced and powerful relational objectivity:

• Objectivity is grounded in the fact that all observers are projecting from the same singular, underlying reality ($\mathcal{S}_u$).

• Shared knowledge is possible not because we all perceive the world identically, but because our different perceptual frameworks can be bridged through shared conceptual axes.

• Interspecies physics thus becomes a problem of category alignment. The goal is not to force an alien to adopt our laws, but to construct the translation functor between our respective perceptual categories ($\mathcal{A}_1$ and $\mathcal{A}_2$).

In this expanded view, physical constants are revealed to have a dual role:

• They are the Jacobians required to maintain consistency within a single measurement grammar.

• They are also the bridges between different grammars, enabling communication and understanding across diverse observers.

They are the universal syntax that makes inter-observer communication about a shared reality possible.