sec-collider-active-identification

6.4.1 Running example: enlarging the collider probe family

6.4.1 Running example: enlarging the collider probe family

At time \(t\), a collider learner has interrogated only a finite subcategory \(i_t:\mathcal P_t\hookrightarrow \mathcal P\) of the declaration \(\mathfrak C_{\mathrm{coll}}\) from Section 2.1.3. Its current hypothesis object is therefore the observational quotient

\[ \mathbf H_{\mathrm{phys}}/{\simeq _{\mathcal P_t}}. \]

Choosing the next beam energy or decay channel is an active UOCL step when it separates surviving classes rather than merely adding more events from an already redundant probe.

Proposition 6.5 Probe enlargement refines collider identification

If \(\mathcal P_t\hookrightarrow \mathcal P_{t+1}\) is an inclusion of exact probe doctrines, then

\[ H\simeq _{\mathcal P_{t+1}}H' \quad \Longrightarrow \quad H\simeq _{\mathcal P_t}H'. \]

Consequently the later observational quotient refines the earlier one: exact probe enlargement can split an equivalence class but cannot merge two that were already distinguished.

Proof ▼

Restrict the equivalence of response functors along \(\mathcal P_t\hookrightarrow \mathcal P_{t+1}\). Functorial restriction preserves the declared equivalence.

This monotonicity is intentionally idealized. With finite samples, changing calibrations, or approximate tests, an empirical quotient can fluctuate. Chapter 8 replaces exact response equivalence by a risk bound, while Chapter 14 records which layer should be repaired when a new channel conflicts with the current theory.