lin-0033

1.2 The universal obstruction axiom

For each pair \((\mathbb S,D)\), let \(\mathcal E_{\mathbb S,D}\) be a category of typed obstruction values with a distinguished trivial object \(0_{\mathbb S,D}\). No initial, terminal, or zero-object property is assumed unless a later law states it explicitly. For \(Z\in \mathcal E_{\mathbb S,D}\), write

\[ \operatorname {Diag}_{\mathbb S}(D;Z) \]

for the set of admissible, structure-respecting diagnostics of the declared factorization problem with values in \(Z\). Changing \(Z\) changes how the failure is observed; it must not silently change the declaration whose failure is being observed. For a morphism \(u:Z\to Z'\), functoriality provides postcomposition

\[ \operatorname {Diag}_{\mathbb S}(D;u): \operatorname {Diag}_{\mathbb S}(D;Z) \longrightarrow \operatorname {Diag}_{\mathbb S}(D;Z'). \]
Axiom 1.1 Universal obstruction

For every learning sketch \(\mathbb S\) and admissible realization \(D\), the diagnostic functor

\[ \operatorname {Diag}_{\mathbb S}(D;-): \mathcal E_{\mathbb S,D}\longrightarrow \mathbf{Set} \]

is representable. Thus there is an obstruction object \(\mathcal O_{\mathbb S}(D)\), unique up to unique isomorphism, and natural bijections

\[ \operatorname {Diag}_{\mathbb S}(D;Z) \; \cong \; \operatorname {Hom}_{\mathcal E_{\mathbb S,D}} \bigl(\mathcal O_{\mathbb S}(D),Z\bigr) \]

for every admissible observer \(Z\). Moreover,

\[ \mathcal O_{\mathbb S}(D)\cong 0_{\mathbb S,D} \quad \Longleftrightarrow \quad \operatorname {Fact}_{\mathbb S}(D)\neq \varnothing . \]
Proposition 1.2 Universal diagnostic factorization

Under the representing bijection, the identity of \(\mathcal O_{\mathbb S}(D)\) determines a universal diagnostic

\[ \eta _{\mathbb S,D}\in \operatorname {Diag}_{\mathbb S} \bigl(D;\mathcal O_{\mathbb S}(D)\bigr). \]

For every diagnostic \(\delta \in \operatorname {Diag}_{\mathbb S}(D;Z)\), there is a unique \(\widehat\delta :\mathcal O_{\mathbb S}(D)\to Z\) such that

\[ \delta = \operatorname {Diag}_{\mathbb S}(D;\widehat\delta ) (\eta _{\mathbb S,D}). \]
Proof ▼

Define \(\eta _{\mathbb S,D}\) as the element corresponding to \(1_{\mathcal O_{\mathbb S}(D)}\). Naturality of the representing bijection shows that postcomposing \(\eta _{\mathbb S,D}\) by \(\widehat\delta \) corresponds to \(\widehat\delta \circ 1_{\mathcal O_{\mathbb S}(D)} =\widehat\delta \). Taking \(\widehat\delta \) to be the morphism corresponding to \(\delta \) proves existence. Injectivity of the representing bijection proves uniqueness.

The uniqueness belongs to this factorization of diagnostics. It does not imply that a repair exists, that an obstruction identifies its cause, or that two successful repairs must coincide.

Proposition 1.3 Non-scalar primacy

Whenever \(\mathbb R_{\geq 0}\) is an admissible observer object, every admissible structure-respecting scalar diagnostic for the declared failure is represented by a morphism

\[ \ell :\mathcal O_{\mathbb S}(D)\longrightarrow \mathbb R_{\geq 0}. \]

Consequently, changing the scalarization changes the view of the obstruction, not the obstruction itself.

Proof ▼

Apply Axiom 1.1 with \(Z=\mathbb R_{\geq 0}\). Representability gives the unique morphism \(\ell \) corresponding to the chosen scalar diagnostic.

This is the book’s central reversal of the ordinary optimization picture. The primitive failure is typed and potentially structured. A scalar may rank, threshold, or aggregate candidate repairs, but it can discard location, direction, interaction, provenance, and symmetry information carried by \(\mathcal O_{\mathbb S}(D)\).

11. Axiom 1.1 is a representability claim, not an observability theorem. The diagnostic functor may be poorly chosen or empirically incomplete. Observer adequacy remains a modeling commitment. ↩