docs: sync dewey doctor synthesis layer diagnostics

[yvonnedevlinrh]

What Changed

PR #107 adds a Synthesis Layer section to dewey doctor output. This is new user-facing diagnostic output that should be documented on the website.

Closes dewey#102.

Affected Pages

• content/docs/dewey/commands/doctor.md (or equivalent) — add Synthesis Layer section description
• Any page documenting dewey doctor output format

What Needs Updating

The dewey doctor command now reports a Synthesis Layer section (between Embedding Layer and MCP Server) with:

• Provider type (ollama, vertex, or unconfigured)
• Resolved endpoint (after precedence resolution)
• Model name
• Connectivity status (Ollama: HTTP health check; Vertex: config completeness)
• Model availability (Ollama only — checks if model is pulled)

Three provider states:

State	Output
Ollama configured	Endpoint, connectivity, model availability
Vertex configured	Model, project, region, config completeness
Unconfigured	"not configured (optional)" — PASS

[jflowers]

Triage: VALID — Unanimous (5/5)

Category: docs | Objectivity: OBJECTIVE | Split: Not needed

All five Divisor agents confirm this is a well-formed documentation sync issue following the Website Documentation Sync convention. The upstream PR (dewey#107) adds a Synthesis Layer section to dewey doctor output with three provider states (Ollama configured, Vertex configured, Unconfigured).

Scope: Single page — content/docs/dewey/commands/doctor.md

Notes from panel:

• Adversary: Use sanitized placeholder values in example output (e.g., http://localhost:11434, not real infrastructure endpoints)
• Testing: Acceptance criteria are objectively verifiable — three provider states either documented or not
• Guard: Clean provenance chain from synthesis-endpoint-env OpenSpec task 3.3 → dewey#71 → PR docs: add Quality Gates page — complete feedback pipeline #107 → this issue

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Divisor Review Panel

Agent	Verdict	Category	Objectivity
Adversary	VALID	docs	OBJECTIVE
Architect	VALID	docs	OBJECTIVE
Guard	VALID	docs	OBJECTIVE
SRE	VALID	docs	OBJECTIVE
Testing	VALID	docs	OBJECTIVE