Local Green, Global Red IV: Finite-Stage Solvability and Completion-Relative Obstruction

Title: Local Green, Global Red IV: Finite-Stage Solvability and Completion-Relative Obstruction (LGGR IV) Subtitle Finite Presentations, Projective Limits, and Original-Carrier Lifting Failures Abstract Local Green, Global Red IV is an expository technical preprint examining the distinction among solvability at every finite stage, existence of a compatible inverse-limit thread, and realization of that thread by an element of the original carrier. Its central arithmetic example considers the congruences 3x + 1 ≡ 0 (mod 2^n). Explicit finite-stage solutions form a compatible thread converging to the unique solution -1/3 in the 2-adic integers, while no ordinary integer satisfies every stage simultaneously. The paper develops finite-descent and reflection-image criteria, reviews the nonemptiness theorem for inverse limits of finite nonempty sets, and summarizes the R¹ lim← lifting obstruction for countable towers of abelian groups together with its vanishing under the Mittag–Leffler condition. Companion materials include Python tests, a Jupyter notebook, reproducible figures, and Lean 4 proofs of the arithmetic core. The work is intended as a mathematical foundation for staged evaluation and governance architectures; completion-level solvability is not treated as evidence of validity in the original operational carrier. Why This Matters Systems engineering, formal verification, and multi-stage evaluation pipelines frequently assume that passing all finite approximation stages or boundary tests guarantees the existence of a global solution in the intended execution environment. This monograph formalizes the boundary where this heuristic fails. By providing exact arithmetic counterexamples and categorical criteria, it establishes that local stage green verdicts and completion green solutions cannot be laundered into original-carrier validity without explicit reflection descent certification. Core Theorems - Theorem LGGR4-M1 (Moving-Witness Collision Criterion): Under surjective observation maps, a defect predicate D fails to have finite descent if and only if at every finite stage there exist distinct elements sharing the same stage presentation but differing under D. (Lean proves collision blocks stage factorization).- Theorem LGGR4-M2 (Finite Fiber Coherence): Cofiltered inverse limits of nonempty finite sets are nonempty (Stacks Tag 086J), precluding failure of thread existence in the completion (does not guarantee descent to original carrier).- Theorem LGGR4-M3 (Explicit Finite-Stage Solution): For x_n = −∑_{k=0}^{n−1} (−2)^k, the identity 3x_n + 1 = (−2)^n holds for all n ≥ 1, yielding local divisibility 2^n | (3x_n + 1) at every stage (Local Green).- Theorem LGGR4-M4 (Sequence Compatibility): The sequence satisfies x_{n+1} − x_n = −(−2)^n ≡ 0 (mod 2^n), forming a coherent thread in the projective limit.- Theorem LGGR4-M5 (Impossibility of Integral Global Lift): There exists no integer x ∈ ℤ satisfying 2^n | (3x + 1) for all n ≥ 1 (Global Red in original carrier).- Theorem LGGR4-M6 (Completion Solution and Metric Convergence): In the 2-adic integers ℤ_2, 3 is a unit and 3x + 1 = 0 admits the unique solution x_∞ = −1/3; the compatible thread converges 2-adically with distance |x_n − (−1/3)|_2 = 2^{−n} (Completion Green).- Theorem LGGR4-M7 (Reflection Descent Criterion): A completed solution descends to the original carrier if and only if it belongs to the image of the canonical reflection embedding η (formalized in Lean via definition unfolding `rfl`).- Theorem LGGR4-M8 (Derived Limit Vanishing): For countable inverse systems of abelian groups, the Mittag–Leffler condition implies vanishing of the first derived inverse limit R¹ lim←, eliminating coherent lifting obstructions. Formal Verification and Computational Artifacts - Lean 4 Mechanization: Formalization of core arithmetic claims M3, M4, M5, reflection descent definition D3/M7, and moving-witness collision implication M1 in `lean4/core.lean` under Lake v4.33.1. All proofs pass the kernel axiom audit (`lean4/check_axioms.lean`) restricted strictly to standard kernel axioms (`propext`, `Quot.sound`), with zero `sorry` or `admit` tokens.- Python Reference Engine: Exact rational and integer arithmetic reference implementation in `code/lggr_iv_core.py`, backed by comprehensive pytest test suites in `code/tests/` achieving 100% line and branch coverage.- Publication Figures: Vector figure generation in `code/generate_figures.py` rendering the refinement tower and 2-adic completion embedding in `public_export/figure_1_refinement_tower_and_completion.pdf`.- Jupyter Companion Notebook: Headless-verified interactive walkthrough in `jupyter_notebooks/001_finite_presentations.ipynb` demonstrating thread construction, 2-adic metric convergence, and reflection descent classification. Scope Boundaries and Non-Claims - Pure Mathematics Scope: The results establish mathematical and categorical boundaries for projective limits and completion descent. They do not claim physical spacetime discreteness or quantum mechanics emergence.- No Completion Laundering: Solvability in a completion space or passing a countable tower of finite tests does not prove original-carrier existence.- Non-Authority Invariant: Computational test passes and mechanical verification receipts document local consistency and do not mint unconditioned authority certificates. Reproduction 1. Environment Setup: Ensure Python 3.10+ and Lean 4 toolchain v4.33.1 are installed.2. Run Test Suite: `python -m pytest -n 8 code/tests --cov=lggr_iv_core --cov=generate_figures --cov-branch`3. Verify Lean Proofs: `cd lean4 && lake build && lake env lean check_axioms.lean && cd ..`4. Compile Manuscript: `python scripts/build_paper_artifacts.py`5. Verify Deposit Envelope: `python verify_deposit.py --lean` Related Identifiers & Citations - Stacks Project: Tag 086J (Limits of finite sets) and Tag 07KV (Derived inverse limits and Mittag–Leffler).- Series Predecessors: Local Green, Global Red I (Evaluated Path Kernels), II (Organizational Gluing), and III (EPK/GTLA Overlay).- Monorepo Custody: