Projection Obstruction Anchor: Product Projections and Exact Short-Time Defect Bounds

Title. Projection Obstruction for Factorized States: Nonlinear Dynamics, Conditional ℓ¹ Diagnostics, and Exact Short-Time Defect Bounds Abstract. Product-state projection of finite many-body dynamics defines local normal residuals. A declared family of local observation maps can turn those residuals into a 1-cochain; calling that cochain a coboundary additionally requires a specified repair differential and an exactness witness. Under explicit replica-extensive observer axioms, finite cochains admit a conditional edgewise ℓ¹ diagnostic. For finite one- and two-local Hamiltonians, a direct double-commutator argument gives the short-time lower bound Dₑ(t) ≥ νₑt − 24g²t² for the raw two-site product defect, where g is the full local interaction norm and νₑ is the native normal product-retraction rate. The bound becomes uniform in system size under explicit uniform-family hypotheses. The results concern finite carriers and finite times. Quotient and exactness statements are restricted to finite-dimensional or closed-range presentations; the quantum results assume the standard finite-dimensional Hilbert-space framework rather than deriving it. Why a researcher should care. Mean-field, Hartree-Fock, product-state ansätze, and tensor-factor approximations compress correlated states into tractable product data. This paper isolates a finite structural cost of that compression: a normal residual and its scalar rate, a declared path from local residuals to observable diagnostics, an exact local short-time defect bound, and a conditional uniform-density corollary. The value is diagnostic rather than metaphysical. What this does not claim. This paper does not claim a finite-structural ℓ¹ norm, a thermodynamic-limit theorem, infinite-time persistence, global trace-norm equivalence for the declared edge-sum diagnostic, exclusion of arbitrary nonlinear smooth conjugacies, or a new physical theory. A positive product-retraction defect certifies departure from product form and therefore non-product correlation, but does not by itself certify entanglement. The paper does not derive Hilbert space, Born-rule uniqueness, continuum physics, or governance authority. AuthorityEffect: None. Key closed forms and replayable values. Result Formula or value Bell product-retraction defect (Example 6.1) D₁₂(π/(8∥J∥)) = 1/√2 + 1/4 ≈ 0.9571067811865476, for either sign of J TFIM normal rate (Example 6.2) νₑ = 2∥J∥; for J = 1, νₑ = 2 Short-time lower bound (Theorem 5.1) Dₑ(t) ≥ νₑt − 24g²t² Tripartite Local-Green/Global-Red witness (§7.4.5) I(s) = 6 and J^α(s) = 0 for all three declared observers Version and DOI chain. Concept DOI: 10.5281/zenodo.18896776. Immediately preceding published version: 10.5281/zenodo.21752743. This release: v6.3, version DOI 10.5281/zenodo.22289548.