sec-collider-oracle-declaration

2.1.3 Running example: the collider declaration

The particle-physics example of Section 0.8 will run through the book in a fixed notation. Let \(\mathbf H_{\mathrm{phys}}\) be a declared category of admissible theory hypotheses, let \(\mathcal P\) be a category of experimental probes, and let

\[ \mathsf{Resp}_H:\mathcal P^{\mathrm{op}}\longrightarrow \mathbf{Stoch} \]

assign to each hypothesis \(H\) its predicted outcome law. A probe may encode beam species, energy, polarization, event selection, and detector context; its morphisms encode admitted changes of experimental resolution or setup. We package the identification problem as

\[ \mathfrak C_{\mathrm{coll}} =\bigl(\mathbf H_{\mathrm{phys}},\mathbb S_{\mathrm{phys}}, \mathfrak D_{\mathrm{phys}},\mathcal P,\mathsf{Resp},\mathcal Q\bigr), \]

where the sketch and doctrine delimit the permitted generators, relations, symmetries, and models, while \(\mathcal Q\) declares which comparisons of responses count as questions.

For a probe subcategory \(\mathcal P_0\hookrightarrow \mathcal P\), define

\begin{equation} H\simeq _{\mathcal P_0}H' \quad \Longleftrightarrow \quad \mathsf{Resp}_H|_{\mathcal P_0}\simeq \mathsf{Resp}_{H'}|_{\mathcal P_0} \end{equation}

2.1

in the answer equivalence declared by \(\mathcal Q\). Thus ORACLE does not initially recover “the true ontology.” It refines a quotient of hypotheses by the experiments performed and the distinctions the observer can resolve. The word Yoneda remains qualified here: collider responses form a restricted, noisy probe family, not necessarily the full representable embedding of a category.

Design principle

A scientific running example should expose all six parts of its declaration: hypothesis category, generating sketch, preservation doctrine, probe category, response semantics, and observational quotient. Without them, “learning a theory” conflates presentation, prediction, and ontology.