Flow Matching and the Geometry of Transport Codex (AI) | Commissioned review v1 | 2026-10-22 Flow matching learns a velocity field from a chosen path of probability distributions. This review distinguishes the endpoint coupling, conditional interpolation, marginal regression target, and numerical flow produced at inference. Three primary sources establish the context of flow matching, rectification, and continuous-depth models. An independent Gaussian construction demonstrates that straight conditional lines can induce a curved marginal flow, while an anticorrelated coupling creates an interior singularity despite valid endpoint distributions. Reproducible variance and integration-error figures expose these distinctions. The article develops a proposed evaluation design centered on coupling, conditioning, endpoint correctness, and numerical cost, without claiming new generative-model benchmark results. 1. A path specification contains more than two distributions To transform noise into data, it is not enough to name a source and a target distribution. One must also specify how points are paired, how paired points move through intermediate times, and how a single velocity is assigned to each location visited by many such paths. Different choices can share identical endpoints and still produce very different regression problems. This review uses one-dimensional Gaussian examples to separate those choices. They are deliberately simple enough to calculate without training a network. The resulting lessons concern identification and numerical conditioning, not the relative image quality of commercial or research models. In particular, the adjective 'straight' can refer to a conditional line between two sampled endpoints or to a trajectory of the learned marginal ordinary differential equation. Those are different objects. A rigorous account states which one is measured, under which coupling, and with respect to which time parameter. 2. Contributions that should remain distinct Lipman and colleagues formulate flow matching as regression on vector fields associated with conditional probability paths; the conditional objective supplies the appropriate marginal gradient relationship. Liu, Gong, and Liu study rectified flow through straight endpoint interpolation and regression, including transport-cost properties under their stated construction. Chen and colleagues provide the continuous-depth and continuous-normalizing-flow framework in which a learned differential equation can transform states and track densities. These works connect path design, learning, and numerical integration, but they do not collapse those stages into a single guarantee. A simulation-free training objective still requires a sampling procedure. A conditional displacement interpolation does not, by that phrase alone, establish that the endpoint coupling is globally optimal for a chosen transport cost. The examples below are this article's own deductions and tests. They do not recreate the cited experiments or assert that every practical flow-matching implementation uses the same path family. References: dl-12-1, dl-12-2, dl-12-3 3. Conditional regression averages intersecting instructions Let X0 and X1 be paired endpoint variables, and define an intermediate state by linear interpolation. Each pair carries the constant conditional velocity X1 minus X0. Several pairs can pass through a neighborhood with different velocities. Squared-error regression assigns the conditional mean velocity given the intermediate state and time. This averaging is exactly where an ensemble of straight conditional instructions becomes a single marginal field. The unresolved conditional variance of velocity is an irreducible regression term. It does not vanish merely because the training lines are straight. A model trained with sufficient capacity and ideal optimization approaches the conditional mean, not a unique hidden endpoint pair for every intermediate observation. Distinguishing these targets prevents a common diagnostic mistake: treating nonzero conditional training loss as proof that the marginal transport is wrong. The correct reference is the Bayes regression risk for the chosen coupling and interpolation, where it can be calculated. 4. Independent Gaussian endpoints produce a variance valley Choose independent standard-normal endpoints. The interpolated variance is the sum of the squared interpolation weights, reaching one half at the midpoint and returning to one at the end. The marginal velocity is a time-dependent coefficient times the current state, with coefficient equal to two t minus one divided by that variance. It contracts before the midpoint and expands afterward. The exact marginal trajectory starting at x0 is x0 times the square root of the interpolated variance. It returns to its initial position at the endpoint, even though most of the sampled conditional pairs have different endpoints. Thus the deterministic flow induces a different coupling from the original independent pairing while reproducing the prescribed Gaussian marginals. Figure 1 makes the variance valley visible. This is an explicit counterexample to the assertion that straight conditional paths force every marginal trajectory to be straight, and it requires neither approximation error nor limited network capacity. 5. Coupling can create or avoid an interior singularity Compare the independent coupling with two other pairings that have the same standard-normal endpoint marginals. Under the identity pairing, every point stays fixed and variance remains one. Under the anticorrelated pairing X1 = minus X0, variance becomes the square of one minus twice time and collapses to zero at the midpoint. All points meet there before separating again. The anticorrelated example is not a valid demonstration of a globally regular invertible flow through that midpoint. Its purpose is precisely to expose the missing regularity assumption. A locally Lipschitz deterministic ODE with unique trajectories cannot map all distinct starting points to one state and later recover their distinctions. A chosen family of probability measures can therefore be algebraically well-defined without admitting the kind of smooth, invertible velocity field intended by a normalizing-flow model. Endpoint plausibility does not remove the need to inspect intermediate conditioning. 6. A transport-cost label needs a specified coupling For independent standard-normal endpoints, the expected squared displacement is two. For identity coupling it is zero, and zero is the minimum possible squared cost between two identical distributions. The anticorrelated coupling has expected squared displacement four. These numbers are exact properties of three couplings sharing the same marginals. The comparison shows why 'optimal transport' must identify the optimization problem and what is actually optimal. A pointwise straight line is shortest between its own two endpoints under Euclidean length, but the pairing of those endpoints may be costly. Moreover, minimizing a population transport objective, approximating a minibatch matching, and choosing a conditional Gaussian path are distinct operations. None should silently inherit the guarantee of another. For generative-model evaluation, report the pairing mechanism and its approximation budget. If no global endpoint optimization was performed, the description should say so instead of letting geometric terminology imply it. 7. Discrete solvers do not inherit exact marginal preservation The independent-Gaussian marginal flow returns to the original distribution at time one. A one-step explicit Euler solver does not: its initial velocity coefficient is minus one, so every starting value is sent to zero. A one-step Heun update sends each starting value to half its original value. Both miss the exact endpoint scale of one, despite using the exact field. Figure 2 refines the uniform step count and measures absolute endpoint scale error for the two methods. The reference solution is calculated independently from the known variance path. This construction separates solver error from learning error because no network is present. It also avoids interpreting function-evaluation count as wall-clock cost: Heun uses two field evaluations per step here, whereas Euler uses one. A fair implementation study would report both evaluations and measured resource use. Convergence in this smooth example does not establish stability near the singular coupling considered above. 8. Time reparameterization changes difficulty without changing endpoints A monotone change of the time coordinate can preserve a geometric trajectory while redistributing its speed. The transformed velocity includes the derivative of the new clock, so regions compressed into short time intervals can require larger local velocities and tighter solver control. Training with uniform samples in the new time variable also changes the effective weighting of the original path. Consequently, an apparently easier regression objective may reflect an altered measure over time rather than an intrinsically simpler transport. An analysis should distinguish geometric curvature, speed variation, and the derivatives controlling numerical truncation error. These are related but not identical. The supplied figures keep the original unit interval and equal time steps to make their comparisons unambiguous. For an empirical study of scheduling, one should compare several clocks with both fixed evaluation budgets and controlled integration tolerances, while retaining endpoint metrics that detect incorrect concentration or loss of support. 9. Likelihood computation requires its own numerical evidence A deterministic transport can be used for sample generation without ever evaluating a density. Computing likelihood through a continuous normalizing flow additionally requires tracking the divergence of the velocity along trajectories and handling the base density and data transformation consistently. Numerical error in state evolution and in the density calculation can interact. A sample that looks reasonable does not validate this density estimate. Conversely, a plausible likelihood on a narrow test distribution does not demonstrate adequate coverage of application-relevant modes. In the Gaussian diagnostic, density and variance are analytically available, providing an independent reference for any future density-tracking implementation. The current companion code verifies variances and endpoint scales only; it does not implement a likelihood estimator. Stating that limit matters because a normalizing-flow vocabulary can otherwise make an unimplemented capability appear to have been tested. Report generation, density estimation, and inversion as separate operations. 10. A stronger experimental contract for flow models A useful evaluation contract records four layers: endpoint distributions and coupling; conditional path and time measure; the trained marginal field; and the numerical inference procedure. The contract should include behavior near endpoints and any interior low-density or highly concentrated region. It should also record whether a straightness statistic is computed on conditional training lines or on trajectories sampled from the learned ODE. The proposed protocol begins with the Gaussian controls in this article and then expands to multimodal distributions with known density. Only afterward should one introduce a real-data benchmark where exact transport references are unavailable. Such a sequence cannot prove that a large model is correct, but it can reveal whether an apparent gain comes from path design, regression capacity, or solver budget. The central standard is not that every trajectory must be straight. It is that the claimed geometric property and the tested numerical object must refer to the same construction. Executed finite checks Conditional covariance and independently known Gaussian flow endpoints 106 assertions passed. Gaussian coupling identities and two explicitly coded scalar solvers; the singular anticorrelated path is only a counterexample, not a valid regular flow. { "independent_midpoint_variance": 0.5, "anticorrelated_midpoint_variance": 0, "Euler_N1_scale": 0.0, "Heun_N1_scale": 0.5, "Euler_N64_error": 0.019885668352616848, "Heun_N64_error": 2.5536292165284635e-06 } Code: analysis/examples.py with models.py and part_a.py through part_d.py; results: analysis/example-results.json Equations (1) X_t=(1-t)X_0+tX_1,\qquad u_t(x)=\mathbb E[X_1-X_0\mid X_t=x] Linear interpolation under a specified endpoint coupling; squared-error regression yields the marginal conditional mean where defined. (2) v(t)=(1-t)^2+t^2,\quad u_t(x)=\frac{2t-1}{v(t)}x,\quad x(t)=\sqrt{v(t)}\,x(0) Independent standard-normal endpoints. This exact marginal trajectory is generally curved in time despite straight conditional paths. (3) v_{\mathrm{id}}(t)=1,\qquad v_{\mathrm{anti}}(t)=(1-2t)^2 Identity and anticorrelated Gaussian couplings. The latter collapses at t=1/2 and violates the regular invertible-flow interpretation there. (4) x_{k+1}^{\mathrm E}=x_k^{\mathrm E}+h\,a(t_k)x_k^{\mathrm E},\quad a(t)=\frac{2t-1}{(1-t)^2+t^2} Explicit Euler for the independent-Gaussian marginal ODE, h=1/N. Heun in the code averages the initial and predicted endpoint slopes; exact endpoint scale is one. Analytical figures Figure 1. Same Gaussian endpoints, different interior variances Constructed standard-normal endpoint couplings. The anticorrelated case collapses at t=0.5 and is not a regular invertible ODE flow through that time. Independent coupling has midpoint variance 0.5. figures/flow-matching-transport-1.csv Figure 2. Exact velocity does not remove integration error Independent-Gaussian marginal ODE x'=a(t)x, x(0)=1. Exact x(1)=1. Error is computed from deterministic solver trajectories; no network is trained. Heun uses twice as many field evaluations per step. figures/flow-matching-transport-2.csv Comparison Construction | Endpoint marginals | Intermediate or coupling property Identity pairing | Both standard normal | Variance one; squared displacement cost zero Independent pairing | Both standard normal | Midpoint variance one half; cost two Anticorrelated pairing | Both standard normal | Midpoint collapse; cost four Proposed validation protocol 1. Declare endpoint marginals, coupling generation, path equation, and time-sampling measure. 2. Verify conditional means and marginal variances on Gaussian cases with independent analytical answers. 3. Include the singular anticorrelated case as a rejected regular-flow assumption rather than a valid smooth-flow benchmark. 4. Measure Euler and Heun endpoint errors against exact trajectories and count field evaluations. 5. For learned experiments, separate regression error, irreducible conditional variance, density error, and sampling error. 6. Archive pairing seeds, numerical tolerances, endpoint clipping, precision, inversion checks, and independently specified evaluation distributions. Limitations 1. The executable examples use scalar Gaussian distributions and do not predict image synthesis performance. 2. The singular anticorrelated path is a counterexample to regularity assumptions, not a recommended generative construction. 3. No learned network, likelihood estimator, or global optimal-transport solver is trained or benchmarked here. 4. Three selected foundational sources do not constitute a systematic survey of flow-matching variants. Research questions 1. Which coupling statistics predict marginal-field stiffness before a large model is trained? 2. How should time sampling balance regression variance against numerical integration difficulty? 3. Can coupling quality be assessed without conflating conditional straightness with global transport optimality? Source evidence dl-12-1: Conditional path regression provides a scalable flow-matching training construction. Boundary: Conditional path terminology alone does not certify a globally optimal endpoint coupling. Locator: Sections 3.1–3.2 and 4; arXiv v2. dl-12-2: Rectified flow studies straight interpolation, conditional regression, and induced transport properties. Boundary: The stated cost and straightness results depend on the construction and assumptions; no cited benchmark is reproduced. Locator: Abstract and the rectification formulation; arXiv v1. dl-12-3: Continuous-depth neural models and continuous normalizing flows make ODE-based state and density transformations explicit. Boundary: Numerical solvers and density tracking introduce separate implementation errors. Locator: Continuous normalizing flows section; original framework, with current arXiv record version noted. References dl-12-1. Yaron Lipman, Ricky T. Q. Chen, Heli Ben-Hamu, Maximilian Nickel, Matt Le. Flow Matching for Generative Modeling. arXiv preprint (2022). https://arxiv.org/abs/2210.02747 | DOI: 10.48550/arXiv.2210.02747 dl-12-2. Xingchao Liu, Chengyue Gong, Qiang Liu. Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow. arXiv preprint (2022). https://arxiv.org/abs/2209.03003 | DOI: 10.48550/arXiv.2209.03003 dl-12-3. Ricky T. Q. Chen, Yulia Rubanova, Jesse Bettencourt, David Duvenaud. Neural Ordinary Differential Equations. arXiv preprint (2018). https://arxiv.org/abs/1806.07366 | DOI: 10.48550/arXiv.1806.07366 dl-12-1 access: Primary metadata, abstract, and selected accessible method text; no exhaustive supplementary-material audit. Version note: Initial preprint 2022; v2 revised 8 February 2023. dl-12-2 access: Primary metadata, abstract, and selected accessible method text; no exhaustive supplementary-material audit. Version note: v1, submitted 7 September 2022. dl-12-3 access: Primary metadata, abstract, and selected accessible method text; no exhaustive supplementary-material audit. Version note: Initial preprint 2018; current v6 dated 9 October 2026 includes corrected figure counts according to the record. This review uses the framework, not those counts. Provenance A focused analytical review of three selected foundational primary sources, with original worked examples. This is not an exhaustive or systematic survey of the literature through 2026. Source access and version notes are recorded with the bibliography. Written and technically self-reviewed by Codex (AI). References identify the original researchers; the analytical constructions and proposed evaluation designs are attributed to this article. Covers are conceptual ImageGen illustrations. Technical figures are computed from declared models or finite examples, with Python and CSV supplied. They are not measurements of deployed products or reproduced performance benchmarks. This local draft has no independent peer-review or journal-acceptance record. Bounded executable examples check the stated small models; they are not formal verification of the cited systems.