[chris-ai1]

Proves the public Arora–Barak draft’s Theorem 11.4: connectivity of a labelled undirected simple graph on m vertices has exact deterministic edge-query complexity m.choose 2. The bridge adversary handles repeated queries and the empty/singleton conventions. Concrete clients distinguish a connected path from a triangle with an isolated vertex having the same edge count.

Extracts the existing clique edge encoding into Cslib.BitString.Graph, preserving its numbering and Samuel Schlesinger’s credit, and migrates clique/proof-complexity clients. Graph/word round trips make the new theorem usable with ordinary labelled graphs. The general spanning-bridges lemma lives in its own graph foundation module. The inventory distinguishes this single endpoint from the open book-wide objective #4711. Closes #4984.

Validation: strict graph/connectivity and representative clique, closure and proof-complexity clients pass; the local whole-library build also passes. Local complete advisories examine 7 definitions, 19 statements and 174 proofs without candidates or omissions. The local full run is not passing: its Python gate exposes the reproducible umask-dependent fixture #5004, and whole-library test/import/lint gates hit the confirmed host mapping limit #5012. The latter trace records mmap ENOMEM followed by a fallback read into a null buffer at65,531 mappings against vm.max_map_count=65,530; reported files are intact and import in isolation. The remaining cache re-elaboration was explicitly stopped after these failed gates before the checkout changed for authorized integration work; no local full-pass claim is made. Hosted full CI and actual advisories remain required. Independent source/math/API review and fresh both-manual provenance/rendering are clear. Queue sequencing waits for the confirmed audit conflict with #4764 to be reconciled after it lands.

Manual impact: the Operator’s Manual documents the model, exact bound, graph conversion, namespace migration and checked client command. Agent Operator’s Manual roles, procedures and handoffs are unaffected, so no edit is needed.

AI-assisted implementation and review; new work credited to Christian Reitwiessner, with original extraction credit preserved.

[chris-ai1]

Independent non-author reviewer review_bpp_reconciliations: APPROVE — source/math/API and both-manual impact only at 5616092085ae145df1adbd1ff660862220bcb068 for #4984, retaining the complete mathematical review at e79ac47d4f6df5be41d46f5f395a26f1a5a2ff1a.

Verdict: APPROVE source/math/API and both-manual impact, with no remaining source finding. The source-version metadata finding is resolved at56160920: Connectivity.lean and the registry now cite the existing AroraBarak07 entry matching the inspected 2007 draft and locator. I inspected the exact delta: only those citations, module API/reference lists and shared graph roundtrip documentation changed; all definitions, theorem statements, proofs, imports and client code retain the reviewed meaning. Final validation/readiness is still pending.

I read the complete new graph encoding, connectivity proof, bridge helper and checked client, the migrated clique source and all migration diffs, affected manual and result records, and relevant existing partial-assignment, decision-tree and Mathlib bridge APIs. I visually inspected actual primary PDF228/printed212, SHA256 da0881782a35bde6a0779a2eb1a8d480f39fbddf70bbb92824d1dc5550a033b2. The theorem matches the source’s exact deterministic edge-query bound m.choose2 on every labelled undirected simple graph. The source’s bridge-adversary argument is correctly realized using partial assignments: the upper possible graph remains connected and all confirmed edges remain bridges; a correct leaf forces lower and upper graphs equal, so no free edge remains. Fixed-coordinate branches handle repeated queries without incorrectly counting them as new information. Empty and one-vertex conventions give zero complexity; no promise, randomization or machine-runtime bound is introduced.

The reusable Preconnected.eq_of_forall_isBridge is valid even for infinite/empty vertex types, and uses existing path/cycle/bridge APIs. Placement in the Mathlib-targeted graph module is suitable. Existing bridge deletion and monotonicity lemmas are reused. The moved Edge/edgeCount/edgeEquiv/graph/edge definitions retain the old underlying types, Fintype.equivFin numbering, graph interpretation and original Samuel Schlesinger credit. Roundtrip ofGraph/graph APIs are meaningful; all moved-name clients migrate through Cslib.BitString.Graph without compatibility wrappers. Checked clients distinguish a connected four-vertex path and a triangle plus isolate with the same three-edge count, cover zero through four vertices, and exhibit a redundant repeated-query tree. These are substantive model checks, rather than an edge-count surrogate.

The Operator Manual explains the exact graph/query model, free local computation, namespace migration, graph/word conversions, noncomputable numbering, empty/one-vertex conventions and complete client command. Agent Operator roles, procedures and handoffs are unchanged; no edit is needed there. Inventory/registry scope preserves the bounded Chapter11 milestone and #4711’s broader remaining objective; the source-version key finding is now resolved. Newly created files credit Christian Reitwiessner, and extracted encoding credit preserves Samuel Schlesinger.

This is independent agent review, not human approval. No source edits, Lean builds or subagents were used by this reviewer; no queue or watch action. Targeted author validation is recorded, but final complete validation, actual advisory-report review and rendered/provenance review for both manuals remain outside this source verdict. Full hosted CI, all three complete actual advisory reports and fresh provenance/rendered review for both manuals remain required before final readiness approval or queueing.

[chris-ai1]

Independent non-author review_bpp_reconciliations: both-manual provenance/rendered QA clear at 5616092085ae145df1adbd1ff660862220bcb068, retaining the completed source/math/API/manual-impact review.

Fresh artifact14012901623 has matching API/upload/download ZIP SHA256 2db5fb51f5a073f164d6b303322536669a3d547edd24c20da3dd926439629bcd. Both PDF hashes and clean exact-head covers/manifests reconcile, as do all72+1 chapter inputs, three converters, bibliography source hash and364 ordered keys. Operator PDF278–279/printed270–271 clearly renders the exact binomial query bound, total graph model, repeated-query and empty/singleton conventions, migrated namespace/unchanged numbering, graph roundtrips, path-versus-triangle client and complete command. The bridge-adversary explanation and deterministic query/runtime distinction are readable. Agent review/coordinator guidance renders clearly; this mathematical API change alters no roles, procedures or handoffs, so its no-edit assessment carries. Generated TeX/four diagram outputs are absent from the ZIP; the successful hosted manual build validates those outputs, not local regeneration.

Final readiness remains pending. At the single snapshot, CI43371773671 and Guide are still in progress; no terminal six-gate/tested-merge or complete advisory-report clearance is claimed. The separate local Python fixture/umask issue is tracked in #5004 and is not a source finding for this PR. No polling, source edits, builds, subagents, queue or watch action. Independent agent review, not human approval.

[chris-ai1]

Integration sequencing: a read-only git merge-tree of this head with earlier queued #4764 (0c6ee9c8) reproduces a content conflict only in results/arora-barak-2007.md. Hold queueing until that audit actually lands, then merge main and reconcile its full Chapter 11 audit with this positive connectivity milestone. Preserve the other audited chapters and source/model qualifications. This is a documentation collision, not a mathematical prerequisite. Source and manual approvals remain recorded above; full current-head CI and actual advisories still need terminal review.

[edemaine]

#4764 actually landed as 4dc1eb26, clearing the actual-landing wait in the sequencing note. The retained implementation/review owner can now merge main and reconcile the full Chapter 11 audit with this bounded connectivity milestone, preserving the other audited chapters and model qualifications. Current head remains 5616092085ae145df1adbd1ff660862220bcb068; the resulting reconciliation needs bounded delta coverage and current validation before queueing.

The separate fixture repair is now published as #5016, with its complete validation and limitations recorded there. The focused Linux checks and full Python rerun pass. Its outside review and hosted checks remain pending; it does not replace this PR’s required hosted validation.