lin-0211

17.2 Foundries and the apex

Let \(\mathcal U=\{ U_i\} \) be typed database views. Foundry \(i\) owns:

  1. a carrier \(V_i\) of database keys and provenance;

  2. a local section \(z_i:V_i\to \mathbb R^2\);

  3. a private fuzzy graph \(G_i=(E_i,\mu _i)\); and

  4. a restriction interface declaring what may be compared with other views.

For every private edge \(e=(p,q)\), the initial distance determines a band

\[ \ell _{ie} \leq \lVert z_i(p)-z_i(q)\rVert \leq u_{ie}. \]

A signed anchor triangle prevents rank collapse:

\[ \sigma _i \det [z_i(b_i)-z_i(a_i),z_i(c_i)-z_i(a_i)] \geq \rho _i. \]

Let \(y_v\in \mathbb R^2\) be a global coordinate and \(H_i=(R_i,t_i)\in SE(2)\) carry foundry \(i\) into the global frame. Incidence compatibility is

\[ g_{iv} = R_i z_i(v)+t_i-y_v = 0. \]

The join-aware geometric-transformer view initializes \(y\); the \(R\) and \(Q\) faces are aligned to it. Symmetric averaging would incorrectly declare the information-forgetting faces equivalent to their apex.

Definition 17.2 Admissible relational realization

An admissible realization satisfies every declared incidence, private fuzzy band, non-collapse anchor, chart-centering constraint, and global Euclidean gauge. When learned foundries disagree, the terminal incidence obstruction is reported rather than hidden inside a weighted embedding objective.