The Last Application · Part V — Technical Appendices

Chapter 22 — Pillars and Projections

3 min read

This chapter assigns mathematics to each pillar and to the projections that make $G$ usable without becoming a second model.

Pillars as subgraphs

Recall the pillar cover $\mathsf{Pillars}$ from Definition 21.3. For each pillar $P$ , write $G_P$ for the subgraph (objects + induced links) assigned to $P$ .

Experiences

Definition 22.1 (Experience). An experience root $e$ determines a finite tree (or DAG) of pages and components:

T_e = (N_e,\, \mathsf{parent}: N_e \rightharpoonup N_e,\, \mathsf{route},\, \mathsf{props})

with $e \in N_e$ the root. The renderer $\pi_{\mathrm{UI}}(G)$ is a function of $T_e$ and the bound design system.

Composition patterns and layout shells are data on pages/experiences — elements of $d(o)$ — not forks of renderer source.

Workflows

Workflow objects and their graphs $W$ (Chapter 20) constitute $G_{\mathsf{Wf}}$ . Runs $\rho$ are either objects in $G$ or rows in a run store keyed by workflow id; in either case they are addressable memory of process, not ephemeral chat.

Knowledge

Definition 22.2 (Knowledge). Knowledge is a triple

K = (S_{\mathrm{src}},\, C_{\mathrm{chunk}},\, \Sigma_{\mathrm{skill}})

sources, chunks, and skills. Retrieval is a map

\mathsf{Retr}: \mathsf{Query} \times K \to \mathsf{List}(\mathsf{Passage})

used by Assistant and craft. Skills are doctrine objects: versioned instructions that IDE agents pull so rails live in $G$ , not only in editor config.

Integrations

Definition 22.3 (Connector). A connector $c$ is an object with an adapter identity and sealed secret handles $h$ such that plaintext secrets never appear in $D$ , tool arguments, or chat. Effects on external systems of record are

\mathsf{Adapt}(c, h, \mathsf{cmd}) \mapsto \mathsf{result}

with audit via events. Integrations extend $R(S)$ ; they do not replace $G$ as SoT.

Events

Definition 22.4 (Organizational memory). The event stream is a time-ordered set $\mathcal{E} \subset V$ (or a store isomorphic to it). Evolution recommendations and healing are functions of $(\mathcal{E}, G)$ , not of log archaeology outside $R(S)$ .

Projections, systematically

Definition 22.5 (Projection family). A finite family $\{\pi_j\}_{j \in J}$ with

\pi_j: G \to \mathsf{View}_j

includes at least:

$j$	$\mathsf{View}_j$
UI	Rendered Experience
Q	Query API results
IDE	Page trees, related walkers, taxonomy dumps
Form	Collect / seal microforms
Assist	Retrieved passages + tool results
Brand	CSS variable map (below)

Axiom 22.1. For all $j$ , authorship that matters updates $G$ , not $\mathsf{View}_j$ alone.

Brand as structured projection

Definition 22.6 (Brand stack). Brand is a chain of objects

\mathsf{Tokens} \to \mathsf{Themes} \to \mathsf{DesignSystem} \to \mathsf{DesignSpec} \to e.\mathsf{theme}

culminating in a CSS variable map

\pi_{\mathrm{Brand}}(G) = \{ \texttt{--cp-*} \mapsto \mathsf{value} \}.

Semantic roles (surface, on-surface, border, accent, …) are the only colorimetric vocabulary components consume. Hard-coded product hex in component payloads is a SoT leak: it makes $\pi_{\mathrm{UI}}$ author color outside the brand stack.

Identity style. $\mathsf{type}$ (mark + wordmark) or $\mathsf{image}$ (logo URLs) is data on the design system — still in $G$ .

Packs

Definition 22.7 (Pack). A pack is a finite set of objects $P \subset V$ with install rules. Installing a pack is a mutation $\Delta_P$ (creates/updates under $I$ ). Packs are how product slices move between organizations without forking platform source for IA and copy.

The point

Pillars partition meaning; projections deliver it; brand is one more projection with unusual leverage. The final chapter restricts who may run which operators on which environment’s graph.