Thirteen exact algebraic theorems (T-A through T-N) decompose every Collatz orbit via the Kodama Observable Kₖ = Sₖ − Aₖ·log₂ (3), a logarithmic coordinate that separates the dynamics into an exact multiplicative backbone and an additive residual. The framework transforms the Collatz conjecture from a question about integer trajectories into a clearly delimited analytic inequality.

Thirteen exact algebraic theorems (T-A through T-N) decompose every Collatz orbit via the Kodama Observable Kₖ = Sₖ − Aₖ·log₂ (3), a logarithmic coordinate that separates the dynamics into an exact multiplicative backbone and an additive residual. The framework transforms the Collatz conjecture from a question about integer trajectories into a clearly delimited analytic inequality. Main Contributions 1. Complete Reduction Chain (Theorems T-A through T-N): Collatz Conjecture ⇔ Kfinal bounded ⇔ Q universally bounded on primitive orbits ⇔ T-G chain termination (proved finite via 3-adic valuation, Lemma T-G). Every orbit flows back to a primitive (non-T-G-reachable) seed. The remaining open problem—bounding Σkᵢ across all primitive orbits—is a single, clearly delimited gap logically equivalent to the conjecture itself. 2. T-M Theorem (Generalized Universal Tail): For every odd v appearing in any Collatz orbit, there exist rational coefficients (αᵥ, βᵥ) such that Qfinal = αᵥ·Qᵥ + βᵥ, obtained by composing the T-J injection formula along the odd chain from v to 1. 3. T-N Theorem (Aggregated k-Bound): Σkᵢ = Sₜotal − A for every orbit. The inequality Q < 1 is equivalent to Σkᵢ < A·log₂ (3) + log₂ (n) + 1. This transforms the problem from orbit simulation into a combinatorial optimization over partitions of S. 4. Boundary k Classification Lemma: For λ = n/3ᵇ, the 2-adic valuation v₂ (3λ−1) = j follows an alternating residue pattern: λ ≡ rⱼ (mod 2^j+1) when j is odd, λ ≡ rⱼ + 2ʲ (mod 2^j+1) when j is even, where rⱼ = 3^ (−1) mod 2ʲ. This gives the exact distribution P (k=j) = 2^ (− (j−1) ) for j ≥ 2, verified exhaustively against 200, 000+ values. Importantly, this lemma is independent of the Collatz conjecture—it is a pure 2-adic fact about 3λ−1. 5. Three Convergent Frameworks: Two independent contemporaneous works converge to the same structural boundary from different coordinate systems: • Williams (2026, arXiv: 2607. 01718): crown triangles via 3-smooth factorization a·2ᵃ·3ᵇ • Shakibaei Asli (2026, arXiv: 2601. 04289): circle rotation formulation with log₆ geometry These are convergent maps of the same territory, not partial proofs—ruling out further algebraic simplification of the boundary problem. 6. Computational Verification: Verified against 100M+ orbits (50M odd orbits, ~5. 4B steps), confirming n=993 as the global Q-maximum at Qₘax = 0. 2531421444. . . The Q-maximum is determined purely by (Sfinal, Afinal), not by n itself. The Kodama Observable Kfinal − log₂ (n) converges to ≈0. 32555 in the statistical limit. Gap Statement This work does NOT claim a proof of the Collatz conjecture. The remaining gap—bounding the Kodama Observable Q below 1 across all primitive orbits beyond 10⁸—is clearly delimited and is shown to be equivalent to the conjecture itself through the reduction chain. Three independent frameworks (Kodama, Williams, Shakibaei Asli) converge on the same boundary phenomenon, suggesting the problem is genuinely hard and further algebraic simplification is unlikely. Formalization and Reproducibility 13 theorems formalized as a Lean 4 skeleton (5 files, 558 lines) • 60 automated tests (5 new covering T-N theorem and boundary-k classification) • Full Python codebase for orbit analysis, record scanning, and neural verification • Open source: github. com/joelcanary/kodama-observable • All computational results reproducible via demonstrated scripts