[t-uda]

The actual incidence-module vertex functor supplies full faithfulness and exactness, but its representation-category dimension bound still needs transport to actual modules. This change constructs the regular vertex graded-free identification and proves a finite-free-cover induction that reflects every finite projective-dimension bound for coefficient-field-finite modules. It then discharges the representation estimate to prove ordinary projective dimension at most D(PairPoset S) for actual incidence and diagonal-quotient modules, and at most 2*n on nonempty-factor coordinate grids with n >= 1.

The coefficient field is arbitrary. With A the interval-pair incidence algebra and I its diagonal ideal, the objects are left modules over Aᵐᵒᵖ and (A/I)ᵐᵒᵖ, encoding right modules over A and A/I. Finite-dimensionality uses the actual canonical restricted field action, and projective dimension is measured in the ambient module category. The numerical results take no resolution, equivalence, projectivity or dimension-bound premise. The actual Hom-cokernel application, relative 2n−2 bound and signed stability remain separate work under #661/#501.

The regular identification has public coefficient/birth laws. Its canonical scalar equivalence also lets the existing Bool fixture reuse the generic finiteness result without changing its public type. The generic pointwise-short-exactness proof is exposed in its existing kernel API and reused by the previous finite-lattice proof.

Source correspondence: BOOS arXiv:2208.00300v3, §2.4 Lemma 2.1 / Remark 2.2 and the ordinary incidence-dimension ingredient of Theorem 5.16. This is the bounded finite-dimensional module consequence via the explicit vertex functor and the already proved representation estimate; it does not construct the entire source equivalence or assert the full global-dimension/stability conclusions.

Fixes #729. Parent #661 and #501 remain open. Ordinary develop work; release version undecided.

Validation (evaluated head fa036ac75a4a73c9b44334c0cf1c914b0af97ac1, base 2716302cff12efc15b9ba8d326332503e37ef797):

• git diff --check: pass; clean worktree.

• LEAN_NUM_THREADS=2 scripts/lean.sh lake build TDA.Persistence.IncidenceVertexProjectiveDimensionExamples TDA.Persistence TDA.Persistence.Examples: pass, 9,271 jobs. Consumers cover zero/Free(0), nonzero regular and quotient vertex modules, the actual comparison at degree zero, public iso/scalar computations and actual one/two-coordinate grid applications.

• LEAN_NUM_THREADS=2 bash scripts/check.sh: PASS: 9,527-job build; live axiom audit of 11,276 declarations within {propext, Classical.choice, Quot.sound}; exhaustive coverage of 9,027 public theorems, all pinned, with private roots audited.

• Independent read-only source/math/API review and separate exposition-flow/deslop-history review: PASS at the evaluated head; all four initial findings and the degree-zero consumer advisory are resolved. The grouped review comment records findings and dispositions. Hosted evaluated-head CI 40532861207 passed check, including the full gate, at this exact head; the draft-skipped run was not used.

• Integration: expected-head squash a3f9ceefa806c1bd1631162b15bc59fd1f8218a4 has the reviewed tree; post-merge CI 40534968981 passed check and its full gate. The completed topic's remote and local branches were deleted after verifying the merged head and absence of later commits; both deletions were confirmed. Persistent branches and the protected work branch were preserved.

[t-uda]

Reviewed-by: /root/review_703

Evaluated head: fa036ac75a4a73c9b44334c0cf1c914b0af97ac1 (tree db994a86075185b0ca4163001211dfa79e2a9f6f), base 2716302cff12efc15b9ba8d326332503e37ef797.

Independent read-only source/math/API review: PASS. Separate exposition-flow: PASS. Deslop-history: PASS. No unresolved finding remains. The reviewer was distinct from the implementation writer and ran sequentially after implementation; no build or audit was repeated by the reviewer.

Comparison mode: source-based, partial source scope. Directly read BOOS arXiv:2208.00300v3 §2.4 Lemma 2.1 / Remark 2.2, Proposition 5.15 and the ordinary incidence-dimension ingredient of Theorem 5.16. The implemented finite-free comparison and unconditional finite-dimensional module bounds are the reviewed scope; full equivalence, arbitrary-module global dimension, actual Hom-cokernel evaluation, relative 2n−2 and signed stability are not claimed.

Inspected the actual regular-column identification, graded-free lifting/projectivity, finite-biproduct universe reindexing, epi preservation/projectivity reflection, actual finite-free kernel rows and ambient Ext dimension induction. The actual quotient scalar-composition isomorphism and the existing quotient comparison discharge the quotient bound; the representation and grid counts discharge every numerical premise. The coefficient field is arbitrary and canonical field-finiteness is distinguished from finite carriers. Old public types are preserved.

Findings from the first head and dispositions in the evaluated head:

1. Missing public regular-iso computations: resolved by public regularVertexIso_hom_coefficient and regularVertexIso_hom_birth, pins and an actual coefficient/birth consumer.
2. Generic finiteness reuse across different stored actions: resolved by the actual canonical-unop regularIncidenceLinearEquiv, shared table embedding/projection, and incidence_finiteDimensional transporting the generic canonical regular result into the old Bool API. No action compatibility is assumed.
3. Guide handedness: resolved by defining R=Aᵐᵒᵖ, C=(A/I)ᵐᵒᵖ and explaining left ModuleCat R/C versus right A/(A/I) modules.
4. Process wording: removed while preserving mathematical boundaries. The accepted consumer advisory is also resolved: the regular degree-zero example invokes the actual arbitrary-d comparison.
5. Canonical coverage identified four missing named public local Prop-instance pins. Those exact public roots were added without source or documentation changes and independently confirmed at the evaluated head; final canonical results are recorded in the validation block. The earlier pin-completeness assessment was corrected.

Concrete reuse checked against pinned Mathlib: graded-free lifting, projectivity reflection, finite biproducts, scalar restriction/composition, finite-free covers, kernel short-exact rows and both Ext dimension shifts. Immutable TauCeti 6ff9009694a393365b1a828b0ed1317c1cf70fa9 FGModuleCat projectivity uses the same available cover/reflection APIs, but its finite-subcategory targets and the quiver/path-algebra equivalence are not substitutes for this task's actual ambient incidence categories. No dependency or framework port is needed.

Reviewed the actual changed source, examples, owning imports, public pins and scoped API/math/reference documentation, including the corrected-head diff. Existing focused build and diff-check evidence was reused; canonical and hosted outcomes belong to the PR's single validation block. This review does not itself assert those checks ran.