Objectivity as What Survives Between Perspectives

I have uploaded a new version of my article Observer Equivariance as a Condition for Shared Physical Law: A Lean-Verified Categorical Model.

It is available here:

https://doi.org/10.5281/zenodo.20589336

The article uses an abstract mathematical language — category theory, fibrations, group actions, and formalization in Lean — but its starting question is fundamental: what must be shared between different perspectives for us to speak of the same physical law?

Physics is not about how the world looks from an absolute standpoint. Physics is about what can be shared between different perspectives.

This may sound philosophical, but it is already part of modern physics. Two observers in relativity may measure different times and different lengths. They may even disagree about the order of certain events. Yet they can still describe the same physics. What makes this possible is not that one description is "the real one" and the other an illusion. It is that there are definite transformations between the descriptions. What is preserved under these transformations is the objective content.

The same pattern appears in gauge theory. An electromagnetic potential can be described in different ways without changing the physical situation. The difference between such descriptions is not a difference in the observable reality, but a difference in presentation. The physics lies in what is invariant, or more generally in what can be transported coherently between different presentations.

This is the idea the article formulates mathematically.

One can think of two levels. On one level there are the concrete ways in which the world appears to an observer: a coordinate system, a choice of phase, a gauge choice, a reference frame, or some other form of presentation. On the other level there is what can be shared between such perspectives: the common structure that allows us to say that different observers are describing the same physics.

The article describes this as a projection

\[ p : O \to S \]

where \(O\) is the category of observer-dependent presentations and \(S\) is the category of common structure.

The point is not to eliminate the observer, but to understand what happens when we disregard the part of a description that belongs only to its presentation. Objectivity is not a "view from nowhere", a standpoint outside all perspectives. Objectivity is what can be shared between perspectives.

Nor is it enough that two perspectives can be compared. The deeper requirement is that an entire system of local changes of perspective can be glued together coherently. It is only there that symmetry becomes more than a translation between two descriptions: it becomes a condition for a law to be shared.

The main claim of the article is that physical law, if it is to be shared between observers, must be compatible with such changes of perspective. I call this observer equivariance: the law must not depend on an arbitrary choice of presentation, but it must be possible to lift and compare it between presentations.

In the simplest version, this leads to a short exact sequence:

\[ 1 \to G \to \operatorname{Aut}(O/p) \to \operatorname{Aut}(S) \to 1 \]

This formula may look abstract, but its meaning is quite concrete. The group \(G\) describes changes that take place only "in the fiber": changes of presentation that do not change the common structure. \(\operatorname{Aut}(S)\) describes symmetries of the common structure. The group in the middle describes symmetries that act at the level of presentations while still respecting the projection down to common structure.

In other words: in the normalized model studied in the article, a symmetry of the common structure can be lifted to the level of presentations. But the lift is not unique. It is unique only up to a fiber translation, that is, up to a change of presentation that does not change the common content.

This is the strict case of the article.

But physics also requires a more general case. An important example is the Poincaré group in relativity. It is not simply a direct product of translations and Lorentz transformations. It is a semidirect product:

\[ \mathbb{R}^{1,3} \rtimes O(1,3) \]

This means that Lorentz transformations act on translations. A boost or a rotation does not leave translations unchanged, but transforms them according to the Lorentz structure.

For this reason, the article also introduces a "twisted" version of observer equivariance. Instead of requiring

\[ F(x \cdot g) = F(x) \cdot g \]

one allows

\[ F_A(x \cdot g) = F_A(x) \cdot \theta_A(g) \]

Here \(\theta_A\) means that the base symmetry \(A\) also acts on the fiber group \(G\). It is precisely this twisting that gives semidirect products, including the Poincaré group.

This distinction matters. It shows that the first, strict theorem is not enough for all of physics. It captures the direct-product case. The twisted theorem captures the semidirect-product case. The article is therefore built so that it first treats the clean model and then shows how it must be generalized in order to fit more realistic symmetries.

Another part of the article concerns Wigner's theorem in quantum mechanics. The point is not that the article proves Wigner's theorem again. The point is rather that Wigner's theorem fits the same pattern. In quantum mechanics, the physical point is not the individual state vector but the ray in Hilbert space. Global phase is presentation, not physical content. The passage from vectors to projective Hilbert space is therefore an example of the same basic idea: objective content arises when presentation-dependent information is projected away, or simply not counted as physical content.

The article is also formally checked in Lean. This does not mean that all of physics has been formalized. The physical examples and the philosophical interpretation are still pen-and-paper arguments. But the abstract mathematical core — the strict and twisted lift theorems, the exact sequence, and the identification with a semidirect product — has been formalized in Lean without sorry.

This matters to me for two reasons.

First, Lean enforces a higher degree of mathematical discipline. It is not enough for an idea to feel right or for a diagram to look convincing. All definitions must be precise enough for the proof actually to go through.

Second, the formalization makes clearer what the article does and does not show. It does not show that all of physics has been derived from observer equivariance. Nor does it show that relativity, gauge theory, or quantum mechanics have been fully reconstructed within this framework. What it shows is more limited, but also more exact: given a certain strict category-theoretic model of observer-dependent presentations and common structure, one obtains a definite classification of how symmetries must be lifted between these levels.

The philosophical consequence is nevertheless quite strong.

Objectivity need not be understood as something wholly independent of perspective. It can instead be understood as what is stable under controlled transitions between perspectives. The objective is not what is seen from nowhere, but what can be shared between different somewheres.

This is a different picture of the basic structure of physics. The observer is not an external force that can change the common structure at will. But neither is the observer something one should ideally remove. Even if probabilities in the quantum case can be understood as perspective-bound, the structure that makes the law shareable must remain stable under transitions between perspectives. The observer is the place where a presentation is given, and physics is what survives when such presentations are compared.

Symmetry is then not merely an elegant addition to physics. Symmetry is the expression of the fact that the world can be described jointly even though every description is given from a perspective.