[t-uda]

Closes #727

The actual right modules over the interval-pair incidence algebra now give additive, fully faithful vertex representations. The explicit Hom inverse is reconstructed from the natural transformation's components; its incidence-linearity is proved using the existing matrix units and their expansion.

The finite projector decomposition identifies the canonically restricted module with the product of all vertex fibers, including diagonal vertices. It proves finite-dimensionality iff pointwise finite-dimensionality and preserves and reflects ordinary short exactness on the original maps.

This is the vertex-fiber half of the representation bridge. Essential surjectivity and projective-dimension transport remain separate, as do the Hom-cokernel estimate, relative 2n−2 bound and final signed stability. No numerical dimension bound or compatibility of the reconstructed map is assumed.

Validation

• Evaluated head: 9b780a4eba628b3d81821d74f6db204ab510cda4; base: 6fa5cb3cd99ccb06e31a35188fa4619a598d24a9; tree: 897d0f984a0b2bf56b02d7b5d3affa75a4928963.
• LEAN_NUM_THREADS=2 scripts/lean.sh lake build TDA.Persistence.IncidenceVertexRepresentation TDA.Persistence TDA.Persistence.Examples: exit 0, 9,268 jobs after the review fixes. Examples' focused target: exit 0, 3,189 jobs. git diff --check: exit 0.
• LEAN_NUM_THREADS=2 bash scripts/check.sh: exit 0 on the evaluated head, including textual guards, 9,524-job whole build, live axiom audit (11,229 declarations within the kernel allowlist), and coverage (8,990 public theorems all pinned; 8,801 private roots, 3,915 additionally checked).
• Sequential independent final-head source/math/API, exposition-flow and deslop-history review: PASS; all three findings resolved.
• Genuine ready-head hosted CI: 40509139240, exact head 9b780a4: check and full scripts/check.sh completed successfully. Draft-skipped run 40509099409 is not evidence. Squash-integrated at 2716302cff12efc15b9ba8d326332503e37ef797; the merge tree equals the evaluated tree. Direct develop push CI 40510244399 succeeded, including check and the full gate. The merged topic was removed remotely and locally after PR/head/no-extra-commit/clean-worktree checks; absence was verified and persistent/other-owner refs were preserved.

[t-uda]

Reviewed-by: independent reviewer /root/review_703

Source/math/API, exposition-flow and deslop-history: PASS at exact head 9b780a4eba628b3d81821d74f6db204ab510cda4, tree 897d0f984a0b2bf56b02d7b5d3affa75a4928963, base 6fa5cb3cd99ccb06e31a35188fa4619a598d24a9. The read-only reviewer is distinct from the sole implementer.

The complete source review checked the actual raw incidence ring and opposite/right-action convention, canonical field restriction, all-vertex projector decomposition, covariance, actual R-linear Hom reconstruction and both inverse laws, ordinary exactness in both directions, and the finite-dimensionality criterion. No supplied compatibility, finite-dimensionality, resolution, numerical bound or assumed equivalence replaces the required proofs.

Three scoped findings were resolved: the guide defines the interval-pair carrier/order before using its alias; composable f/g and their zero composite precede the range/kernel claim; and the R-linearity proof reuses matrix-unit commutation at c•m followed by k-linearity, removing duplicated scalar-commutation proofs while preserving factor order. The final narrow review verified these changes; public types, instances and pins are unchanged.

The review inspected nonzero off-diagonal action from a diagonal source in the actual regular module, zero fibers, actual finite-dimensional regular modules, recovery of a separately proved nonidentity rational endomorphism, and a split exact row. Concrete reuse includes existing matrix units/delta expansion/raw diagonal sum, pinned ModuleCat.Algebra/Algebra.lsmul, standard FullyFaithful packaging and ModuleCat exactness criteria. Source comparison uses BOOS v3 §2.4 Lemma 2.1/Remark 2.2; the related TauCeti path-algebra construction does not directly replace this incidence bridge.

Essential surjectivity, equivalence and projective-dimension transport remain separate, as do the numerical Hom-cokernel/relative2n−2/stability conclusions. No source-error claim is made. Mechanical validation and hosted CI are recorded in the PR body; review did not repeat them or waive their gates.