build-lectures: no `_build/.doctrees` clear (ipywidget mimetype rendering)

v0.8.0 validation study: two native pipelines clear the Sphinx doctree cache before the final HTML build to fix ipywidget mimetype rendering — lecture-python-intro (.github/workflows/publish.yml:90), lecture-python-advanced.myst (.github/workflows/publish.yml:91). Composite build-lectures has none: nothing here touches .doctrees; its pre-build HTML steps are only Stage PDF (build-lectures/action.yml:51) and Stage notebooks (build-lectures/action.yml:70).

Intra-job trigger, not a stale cache: jb writes one shared _build/.doctrees, so HTML in a three-format job reads doctrees from the earlier pdflatex and jupyter builds; reproduces cold. That pdflatex → jupyter → html sequence is the migration target (docs/MIGRATION-GUIDE.md:313-332); single-builder consumers like lecture-dp (builder: 'html') are out of scope. Unreproduced — inferred from the native BUG: comment.

No zero-build A/B — no clear-free repo emits widget output:

• lecture-python-advanced.myst/lectures/matsuyama.md: sole ipywidgets import among migrated repos, cells skip-execution (no mimetype emitted), and that repo clears.

• lecture-python-intro (also clears): executed rich output in markov_chains_I.md (HTML(anim.to_jshtml())) and inequality.md (plotly).

• lecture-python-programming.myst: clear-free in production, but no executed widget/animation/plotly output.

• QuantEcon/test-actions-lecture-intro: composite, no clear, no cache restore, but its canary TOC builds five rich-free lectures (geom_series, short_path, lln_clt, solow, linear_equations) under only_build_toc_files: true, unpublished since 2026-02-10; widening cuts against _toc.yml's no-content-drift criteria — separate decision.

• Repro: build intro's markov_chains_I and inequality through the composite pdflatex → jupyter → html with no clear; compare to the published native site.

• Matches → the clear is not load-bearing for current content; close wontfix with the negative result recorded.

• Diverges → add an optional clear-doctrees input (default off) to the build-lectures HTML path, plus a MIGRATION-GUIDE note for repos coming from native pipelines.

• Lower priority: the cache stores _build wholesale (restore-jupyter-cache/action.yml:109), so .doctrees rides along on restore — re-check once #92 Phase 1 turns on save-cache: 'true'.

Start with build-lectures/action.yml and the native workflow steps at .github/workflows/publish.yml:90-91. Reproduce the intro markov_chains_I and inequality builds through the pdflatex → jupyter → html sequence without clearing doctrees, then compare the output with the published native site. Done means recording a negative result for closure or documenting the divergence and updating the named migration and action paths.