Trajectory Engineering: The Framework for Closing the Loop

Trajectory Engineering (TE) specifies the formal architecture of closed-loop intervention on populations of individuals: observation 𝒪, inference ℳ, policy π, and cross-domain coupling diagnostic KD acting on a controlled stochastic differential equation with endogenous governance Θgov. Under standard regularity hypotheses, the paper proves a Closed-Loop Specification Theorem (Theorem 3. 6) and a quantitative K-truncation trajectory-error bound (Proposition 4. 1), with a four-component policy-regret decomposition into first-order attribution plus interaction/residual. A variational reformulation is sketched as a future-development direction rather than as the load-bearing proof structure. Five case studies — bariatric surgery accreditation (MBSAQIP), the 2008 financial crisis, coupled climate forecasting, geopolitical conflict trajectories, and the SARS-CoV-2 pandemic — illustrate the architecture across disciplines. The paper's contribution is a disciplined specification architecture for closed-loop, domain-coupled intervention at population scale with individual resolution; it is not a unified-law claim across applied systems. Revision v1. 3 (May 15, 2026), reviewer-driven mathematical rebuild. Substantive changes from v1. 0: (1) Master equation restated in aggregate-state Itô SDE form, with KD as the diagnostic decomposition of the drift Jacobian rather than an additive operator in the drift. (2) Proposition 4. 1 (K-truncation bound) re-proven using Young's convolution inequality, with the horizon coefficient C_α (T) and the correct L² scaling; the previous bound's scaling was corrected. (3) Theorem 3. 6 reframed as a Closed-Loop Specification Theorem; the identifiability clause is demoted and stated separately with stronger assumptions. (4) Lemma 3. 5a rewritten using conditional-law notation πₜ = P (Xₜ ∈ · | ℱₜY). (5) Schauder fixed-point claim softened to "existence under specified compactness, continuity, and convexity conditions"; non-convex governance treated separately. (6) Persistence-of-Excitation condition formalised (Definition 3. 5′) ; D-respecting bijection defined (Definition 3. 5″). (7) Symbol collisions resolved (KD for diagnostic, X̃ for truncated trajectory, πₜX for belief state). (8) Policy-regret decomposition stated as first-order attribution plus interaction/residual rather than exact equality. (9) Empirical claims tightened: MBSAQIP case-volume language source-specific; TED-spread caption acknowledges post-LIBOR transition (January 2022) ; Iran case caveat moved earlier; COVID magnitude claims framed against the Bollyky 2022 variance partition. (10) Figure–caption pairings corrected (Figures 3, 4, 5, 11). (11) Six new case-study figures inserted (Figures 6–10 plus updated Figure 11). (12) AI-assistance disclosure added covering Claude Opus 4. 7 and OpenAI GPT-5. 5 Pro use during development and review. (13) Body font set to DejaVu Serif for cross-platform clean rendering of math, Greek, and script characters. Series: The Trajectory Engineering Papers, No. II. Companion to Doctor AI: Reimagining Healthcare, Rebuilding Trust, Delivering Health 4. 0 (Blackstone Press, 7 April 2026), and to Paper I Trajectory Engineering: A Prospectus (SSRN abstract ID 6612840; Zenodo).