Finite-Sample Approximation of Hessian-Guided Perturbed Wasserstein Gradient Flows

1 Introduction

• Finite implementations approximate both the drift and Hessian-guided jump covariance empirically, while particle dependence limits direct use of standard independent-sample bounds.The paper frames this as the question of when a finite interacting-particle system accurately approximates population PWGF.Source: §1 · p. 2
• Controlled-curvature reference paths admit particle and objective-value tracking bounds under regularity assumptions and a prescribed deterministic common jump schedule.The bounds retain time dependence and weight ODE residuals and jump mismatches by separate curvature-response factors.Source: §1 · p. 2
• A population-first Gaussian coupling preserves reference particles’ conditional i.i.d. law and reduces jump comparison to covariance-factor mismatch.The coupling reproduces population-field evaluations while driving the empirical update.Source: §1 · p. 2
• In a regularized product-measure matrix-factorization model, the paper establishes attraction along an anti-symmetric branch, transverse descent, and positive second variation in two regions.These local results motivate a positive-negative-positive curvature scenario.Source: §1 · p. 2
• The tracking result is limited to reference paths satisfying the controlled-curvature condition, and deriving that condition from model primitives and synchronizing adaptive jump times remain open.The variance-plus-cosine example verifies the condition and gives a growing-horizon guarantee with one prescribed jump.Source: §1 · p. 3

2.1 Population WGF and Hessian-guided perturbations

• Between perturbation times, the population curve follows Wasserstein gradient flow; the finite-sample comparison instead uses deterministic jump times fixed in advance and common to both systems.The population PWGF described by Yamamoto et al. uses adaptive perturbation times.Source: §2.1 · p. 3

2.2 Empirical particle dynamics

• At each empirical jump, a finite-dimensional Gaussian vector with covariance determined by the empirical Hessian-guided kernel drives the particle update.The resulting jump can be simulated from an Nd-dimensional standard Gaussian vector and an Nd × Nd covariance factor.Source: §2.2 · p. 4

2.3 The auxiliary reference particles

• Reference particles start from the empirical particles’ initial positions and evolve with population drift and jump covariance, remaining conditionally i.i.d. given the population fields.The coupling uses a common Gaussian vector constructed from population-field evaluations and independent auxiliary Gaussian noise, leaving the population path unchanged.Source: §2.3 · p. 4

2.4 Tracking metric and decomposition

• The tracking comparison uses paired-particle distance and combines it with concentration of the reference cloud’s feature moments to compare objective values.The resulting analysis gives joint tracking and objective bounds under a prescribed common schedule.Source: §2.4 · p. 5

3.1 Curvature-response factors

• The curvature propagator transports errors from their creation time to the current time, discounting or amplifying them according to intervening curvature.The sampled curvature is the empirical discretization of the Hessian along the reference particles; positive values correspond to contraction and negative values to local expansion.Source: §3 · p. 5, §3.1 · p. 5

3.2 Main theorem

• Theorem 3.1 gives simultaneous particle and objective tracking bounds on controlled-curvature reference paths, under stated regularity, moment, non-spikiness, and prescribed-schedule assumptions.The theorem considers horizons T ≤ N^(1/4−υ−ζ), schedules with at most L0 jumps, and a high-probability event.Source: §3.2 · pp. 6–7
• Positive curvature damps past errors, whereas negative curvature can preserve or amplify them; later recovery can make an earlier jump mismatch fade.Continuous errors are generated throughout the flow, while a jump mismatch is introduced at a perturbation time.Source: §3.2 · p. 6
• For the variance-plus-cosine example with T = N^(1/8), the bound is sup over 0 ≤ t ≤ T of [r_t + |F(ρ_Xt) − F(μ_t)|] ≤ CεN^(−3/16+ε).The stated event has probability at least 1 − N^(−1/16), for fixed ε > 0 and sufficiently large N.Source: §3.2 · pp. 6–7
• The illustrative curvature pattern is positive on [0,1) and [2,3], negative on [1,2), with a unit jump at t = 1.5.The jump mismatch is then amplified under negative curvature and decays under positive curvature.Source: §3.2 · p. 7

4 Motivating geometry in a matrix factorization model

• The model establishes local attraction, transverse descent, and positive second variation in two regions, motivating a positive–negative–positive curvature scenario.These population results do not construct a connecting PWGF trajectory or verify its controlled-curvature path condition.Source: §4 · p. 7

4.1 Product-measure model

• The product-measure model uses two spherical marginals and a mean-field feature map that averages activation features over weights.The teacher is an antipodal two-point measure.Source: §4.1 · p. 7

4.2 Branches and standing conditions

• The standing restrictions place model and teacher measures in equal-mass antipodal caps and require the activation’s nonzero Hermite coefficients to have finitely many odd degrees.Transverse descent additionally requires a non-cancelling transverse teacher bias with a positive margin away from the equator.Source: §4.2 · pp. 7–8

4.3 Local geometry

• For 2 < pW < 3, a globally well-posed characteristic product WGF on M− that starts sufficiently close to the target within the cap class remains there and converges to the target.The cited proposition supplies the precise initialization conditions and rate.Source: §4.3 · p. 8
• With sufficiently large λreg, the spherical product pushforward second variation is positive in every nonzero admissible direction on M− where Z±(µ) ≥ ζ².The required regularization threshold depends on ζ and fixed model parameters.Source: §4.3 · p. 9
• Under the non-cancelling transverse teacher-signal condition, a same-density perturbation from the target decreases F by γesc h² plus a lower-order term, with γesc > 0.The regularizer contributes o(h²).Source: §4.3 · p. 9
• If Z±(µo+) > 0, the second variation has a positive uniform margin in a sufficiently small product-Wasserstein neighborhood of the teacher, relative to the cap class.The teacher need not be stationary for the regularized objective.Source: §4.3 · p. 9

5.1 Technical Assumptions

• The proof assumes a structured weighted-moment objective with controlled third derivatives, regular coupled particle paths, and a deterministic prescribed schedule of common Gaussian jumps.The coupling error is also assumed non-spiky to control a cubic Taylor remainder.Source: §5.1 · pp. 9–10

5.2 Proof sketch

• The population-first coupling preserves the reference particles’ conditional i.i.d. law given population fields, enabling concentration for drift, Hessian, and jump residuals.The population Gaussian fields and empirical transitions satisfy the specified coupling assumptions.Source: §5.2 · pp. 10–11, §5.1 · p. 10
• Between jumps, linearizing the drift and comparing Hessians bounds discrepancy growth using the reference configuration’s empirical curvature and localized residual terms.The proof sketch gives the differential inequality while the discrepancy remains within its localization radius.Source: §5.2 · p. 10
• At jumps, aligning Gaussian vectors reduces their discrepancy to a covariance-factor mismatch, whose bound combines smooth measure dependence with empirical covariance fluctuation.The square-root covariance dependence turns an N^-1/2 covariance-sampling bound into an N^-1/4 jump estimate.Source: §5.2 · pp. 10–11
• Integrating continuous forcing and summing jump forcing weights errors by curvature-response factors, while the recursive comparison retains nonlinear amplification and jump multipliers.Residual concentration and localization close the estimate under the theorem’s conditions, and the proof also derives an objective bound.Source: §5.2 · p. 11

6 Conclusion

• Tracking and objective-value bounds retain accumulated curvature, allowing later positive curvature to damp errors amplified earlier.The analysis covers paths satisfying the controlled-curvature condition under a prescribed deterministic common jump schedule.Source: §6 · p. 11
• The population-first coupling preserves conditional independence of reference particles and reduces Gaussian jump errors to covariance comparison.The comparison propagates continuous and jump residuals through sampled curvature.Source: §6 · p. 11
• Beyond the variance-plus-cosine example, deriving controlled-curvature and non-spikiness conditions from model primitives and synchronizing adaptive jump times remain open.The formalization covers stated results under explicit hypotheses, but not unconditional WGF well-posedness or the numerical experiments.Source: §6 · p. 11

AI use statement

• AI tools assisted with mathematical claims and proofs, drafts, experiments, figures, and Lean 4 implementation; the authors checked the proofs and retain responsibility for them.The statement says the models are not components of the optimization method.Source: AI use statement · p. 12

A A General Comparison Argument via the Auxiliary Particle System

• The appendix defines operator, Hilbert–Schmidt, and Frobenius norms, reserving the latter two labels for distinct operator and finite-matrix settings.In finite-dimensional matrix spaces, the Hilbert–Schmidt and Frobenius norms coincide.Source: A A General Comparison Argument via… · p. 16
• The appendix distinguishes tracking curvature, which retains both Hessian terms, from jump covariance, which uses the integral Hessian term.The additional spatial-Hessian bound used elsewhere is not imposed globally here.Source: A A General Comparison Argument via… · p. 17
• Reference particles have a conditional product law given population fields, while Gaussian comparison coordinates are conditioned on the pre-jump history.The comparison coordinates are standard Gaussian conditionally on that history.Source: A A General Comparison Argument via… · p. 17

A.1 Overview of the finite-particle comparison

• Curvature propagation replaces worst-case Lipschitz Gronwall growth, retaining the effect of sampled curvature along the comparison path.The comparison uses curvature-response weights for both dynamical and functional bounds.Source: §A.1 · pp. 17, 19
• The auxiliary reference particles share the empirical system’s initial sample but follow population drift and population Gaussian fields, separating pathwise stability from concentration around the population law.The reference cloud has a conditional product law given the population fields.Source: §A.1 · p. 18
• The fixed-schedule comparison requires a nonlinear stability condition; a sufficient horizon is T ≤ N^1/4−υ−ζ under the stated curvature-response and structural bounds.The theorem’s failure probability is bounded within the nonlinear stability and regularity events.Source: §A.1 · pp. 18–19
• Near-N^-1/2 residual concentration can still yield an N^-1/4 jump contribution because jump errors enter through square-root covariance perturbations.Continuous residuals enter additively along ODE intervals.Source: §A.1 · p. 19
• After particle tracking is established, the appendix bounds the functional gap using the same tracking error and residual pair, without introducing a new stochastic object.The comparison is converted into objective-value control in Section A.7.Source: §A.1 · p. 19
• In the variance-plus-cosine model, returning positive curvature makes the jump curvature-response factor decay, yielding tracking and an objective-gap bound for C log N ≤ t ≤ N^1/8.The example uses bounded mean-zero initialization and one prescribed Gaussian jump.Source: §A.1 · p. 19

A.2 Settings and Assumptions

• The comparison uses fixed deterministic jump times and amplitudes, common initial positions, and Gaussian perturbations coupled at prescribed jumps.The appendix assumes a characteristic realization of the population path and pathwise regularity conditions for the comparison.Source: §A.2 · p. 20, §A.2.1 · p. 20

A.2.1 Assumptions

• The assumptions include structured weighted moments, Gaussian perturbations, moment bounds, piecewise-Lipschitz paths, and an anti-spikiness condition on comparison paths.The anti-spikiness restriction is imposed on an event; the text does not assert that this event has probability one in general.Source: §A.2.1 · pp. 21–23

A.2.2 Local consequences used in the comparison

• A weighted-series estimate supplies a common envelope for the localized comparison and includes a bound for each fixed c > 0.The proof uses a weighted-series bound to combine the coordinate moment estimates.Source: §A.2.2 · p. 23
• Moment summability and integral formulas yield finite weighted bounds, anchor-based estimates, and formulas for changes in G and its derivative along feasible moment segments.On a population event of probability at least 1 − p0, the population path has conditional moment scale Kpop; the associated localization lemma places population measures, empirical particle measures, and interpolations in the domain where the local formulas apply.Source: §A.2.2 · pp. 24–26
• The local envelope has growth and derivative bounds with constants independent of K, D, N, T, and p.The envelope is defined using the anchor norms and localization scales; in the localized comparison, its moment scale Kpop may depend on N, T, and p.Source: §A.2.2 · p. 26
• The weighted feature map J0 is C2, with bounded derivatives through order two and derivative-difference bounds on each bounded spatial region.The bounds are obtained from coordinate-wise regularity and weighted summability.Source: §A.2.2 · pp. 27–28
• Local comparison bounds control measure-dependent gradients and Hessians, their magnitudes, and spatial differences using common moment weights.The bounds apply when the measures' moments lie in the localized region and the spatial points lie in the specified ball.Source: §A.2.2 · pp. 28–29

A.3 Bound rt =

• On each ODE interval, the particle discrepancy is analyzed through the segment joining the two configurations.The subsection then defines an interpolating particle configuration.Source: §A.3 · p. 30

Recall from lemma A.10 the population-measure Hessian

• Combining measure-slot and point-slot Hessian estimates gives directional mismatch control under population moment, sample localization, and anti-spikiness conditions.A fourth-moment bound can supply the anti-spikiness condition used in this estimate.Source: §A.3 · pp. 34–35, Recall from lemma A.10 the… · p. 36
• The continuous-time estimate packages residuals into a common moment fluctuation and bounds discrepancy by a stopped curvature-response inequality.The inequality applies on each open ODE interval and includes the zero-discrepancy case through an upper Dini derivative.Source: Recall from lemma A.10 the… · pp. 36–37

A.3 Bound rt =

• The ODE discrepancy bound separates sampled curvature, Hessian mismatch, and empirical drift residuals, yielding a stopped curvature-response inequality.The empirical vector field is linearized through its block Hessian, whose smallest eigenvalue drives contraction or expansion of the coupling error.Source: §A.3 · pp. 31–33, Recall from lemma A.10 the… · p. 36

A.4 Bound of rt including jump errors

• Conditional on population Gaussian fields and the jump schedule, reference particles remain i.i.d.; using the same Gaussian seed reduces jump error to covariance mismatch.The finite-dimensional jump vector is represented conditionally on the pre-jump history as a centered Gaussian with covariance determined by the reference particles.Source: §A.4 · pp. 38–39
• The aligned square-root construction yields symmetric matrices and a positive-semidefinite covariance, with an orthogonal alignment for comparing the finite factors.The stated factor-difference estimate uses the finite-dimensional Powers–Størmer inequality.Source: §A.4 · p. 39, If we define · p. 40

If we define

• The population-first coupling gives a conditionally standard-Gaussian finite coordinate, reproduces the population evaluation, and retains the prescribed empirical conditional covariance without changing the population path.Repeating the construction at prescribed jumps retains the conditional product law for reference particles.Source: If we define · pp. 41, 43
• The coupling preserves exchangeability of particle pairs and the reference conditional product law, but does not assert independence of empirical particles.Exchangeability yields identical one-particle moment bounds.Source: If we define · p. 43
• With square roots aligned, Gaussian perturbation error reduces to kernel-covariance mismatch, split into smooth measure dependence and reference-cloud empirical covariance fluctuation.The two terms are handled differently: localized kernel bounds apply to the sample-supported measure change, while the population-law covariance residual is controlled by weighted-feature concentration.Source: If we define · pp. 43–45
• The one-jump comparison combines Lipschitz empirical-perturbation and covariance-mismatch bounds into a jump recursion on moment and sample localization events.The comparison is stated on the event intersected with the population-moment and sample-localization events.Source: If we define · p. 46
• The aligned matrix and whitening coordinate compare the empirical transition with a finite evaluation of the population field; reference-particle conditional i.i.d. is established separately by a population pushforward argument.The independence argument does not condition on the finite Gaussian coordinate.Source: If we define · p. 48

A.5 Concentration of the stochastic residuals ΓN t and eΓN

• The section develops concentration bounds for the stochastic residuals ΓN_t and eΓN by converting empirical-functional sensitivity estimates into high-probability estimates.For ΓN, the stated approach truncates features, applies conditional McDiarmid at grid times, and removes the truncation.Source: §A.5 · p. 48, §A.5.2 · p. 52

A.5.1 Preliminary

• Conditional on population-level randomness, reference particles are treated as independent inputs for McDiarmid concentration; the jump covariance residual uses a Hilbert–Schmidt second-moment estimate and McDiarmid’s inequality.The jump-covariance control is identified as Lemma A.30.Source: §A.5.1 · pp. 48–49

A.5.2 Concentration of ΓN

• The population-first jump scheme preserves conditional i.i.d. reference particles with law µt after conditioning on common population jump randomness.This conditional structure is the basis for applying concentration bounds to the reference empirical cloud.Source: §A.5.2 · p. 49
• With probability at least 1 − p0, a population-measurable event gives uniform-in-time conditional moment bounds for X and Y, and places µt in Pα,Kpop.The result assumes deterministic branchwise Lipschitz paths and derives conditional bounds on the event rather than imposing them almost surely with the same K0.Source: §A.5.2 · pp. 49–50
• Under conditional-i.i.d. and feature assumptions, Lemma A.21 gives localized uniform high-probability control of Γ_N across time traces and removes clipping on the localization events.The argument combines concentration on a time grid with interpolation and a conditional truncation-bias bound.Source: §A.5.2 · pp. 53–55
• On the population event, the conditional truncation bias is bounded by the tail envelope; a moment-bound-only alternative is available for D ≥ eKpop.The bound compares βG_T(D) with βT(D).Source: §A.5.2 · p. 55
• A(D, K), Λ(D, K), and S(D, K) are each bounded by C exp(C{(log(e + D))^q + (log(e + K))^q}), with constants independent of D, K, N, T, and p.The bound applies for κ = κdec > max{1, κgr} and q = κ/(κ − 1).Source: §A.5.2 · p. 55

Suppose the population localization lemma gives

• With a uniformly bounded B, the BT/N term can be replaced by T/N; if Ljump is also uniformly bounded, Ξε is equivalent up to constants to {N(e + T)/p}ε.These simplifications apply to the high-probability bound under the stated population-localization premise.Source: Suppose the population localization… · p. 57

A.5.3 Weighted-factorization control of eΓN

• The jump-covariance residual is controlled by factoring the Hessian kernel through weighted features, concentrating feature covariance in Hilbert–Schmidt norm, and converting to a nuclear-norm bound.The weighted Hessian factorization represents the spatial Hessian kernel using a bounded self-adjoint operator on the feature space.Source: §A.5.3 · pp. 57–58
• The feature-bound results include continuity in Hilbert–Schmidt norm, strong measurability, and a positive trace-class covariance operator under the stated conditional-moment setup.The bounds are formulated with explicit conditional moment dependence.Source: §A.5.3 · p. 59
• A deterministic envelope at sampled particles bounds each particle’s exceedance probability by δ/(8N).The envelope threshold also satisfies τN,δ ≤ C exp{Cχ(RN,δ)}.Source: §A.5.3 · p. 62
• At each fixed time trace, the full jump residual is controlled with conditional probability at least 1−δ by combining Hilbert–Schmidt covariance concentration with a nuclear-norm bound.The covariance concentration result controls the full weighted covariance in Hilbert–Schmidt norm.Source: §A.5.3 · pp. 61–63
• For deterministic prescribed jump times, a union bound combines the fixed-time guarantees without requiring independence across times.The result uses q = p/(Ljump + 1) for Ljump jump times and confidence parameter p.Source: §A.5.3 · p. 64
• When Ξ is polynomial in N, the full jump residual is N^-1/2+o(1), and its square-root jump comparison is N^-1/4+o(1).The stated rates apply under the prescribed-jump-time growth conditions on Kpop and D.Source: §A.5.3 · p. 64
• The covariance argument needs neither a feature cutoff nor a nuclear-norm rank loss, because concentration is applied to clipped operators in S2(E).The nuclear norm is bounded using the empirical feature trace and the Hilbert–Schmidt covariance error on one event.Source: §A.5.3 · p. 64

Theorem A.1

• With probability at least 1-p, Theorem A.1 bounds tracking simultaneously at all times and left limits on the regularity and nonlinear-stability events, using residual envelopes and curvature amplification.The theorem assumes r0 = 0 and integrable curvature on [0, T].Source: §A.6 · p. 69

Probability scope

• The failure bound applies within the stated events; it does not bound conditional failure given the nonlinear-stability event or lower-bound the probability of the controlled-curvature path condition.The main-text probability statement likewise asserts neither a conditional failure probability nor a lower bound on the probability of the path condition.Source: §A.6 · p. 69

Relation to the main-text growing-horizon theorem

• The main-text result uses controlled-curvature paths and the sufficient horizon T ≤ N^(1/4−υ−ζ); the explicit stability requirement T(δN,T,p + c_D R) ≤ 1/4 can also hold on longer trajectories.The strict margin ζ > 0 absorbs subpolynomial concentration and localization factors in the sufficient regime.Source: §A.6 · p. 69

Zero-forcing case

• When the continuous residual and jump forcing vanish, starting from r0 = 0 yields exact tracking, r ≡ 0, by branchwise integration and the zero-preserving jump recursion.The argument uses no zero radius in the first-exit lemma.Source: §A.6 · p. 72

A.6 Fixed-schedule comparison with explicit high-probability rates

• For T ≤ N^(1/4−υ−ζ), where υ + ζ < 1/4, condition RN implies nonlinear stability for sufficiently large N and yields the tracking bound with failure probability at most p.The guarantee is stated for every fixed ε > 0 within the specified high-probability event.Source: §A.6 · pp. 72–73
• The exact tracking theorem continues to apply at longer horizons provided its nonlinear stability event holds.Source: §A.6 · p. 73

• A deterministic lemma bounds functional discrepancy using pathwise particle discrepancy and the same empirical-moment residual, without adding a stochastic residual.Source: §A.7 · pp. 73–74
• Theorem A.2 gives a uniform functional-value comparison on the tracking event under Theorem A.1.Source: §A.7 · p. 74
• Under corollary A.4 assumptions, the comparison gives a sufficient-horizon rate on the controlled-curvature event using the tracking theorem's curvature-response factors.Source: §A.7 · p. 74

• The example uses bounded mean-zero initialization and one prescribed Gaussian jump, then verifies the comparison assumptions.Source: §A.8 · pp. 75–76
• At T=N^1/8, particle discrepancy is at most CεN^-3/16+ε; after C log N, objective gap is at most CεN^-1/2+ε, on an event of probability at least 1−N^-1/16.Source: §A.8 · pp. 75, 81
• After recovery, the curvature-response factor is at most CN^-1/4, while population objective excess is at most CN^-1/2 for t≥C log N.Source: §A.8 · p. 78
• After the jump, the mean moves away from unstable even integers; by the recovery time, cos(πm_t)≤−1/2 and positive curvature damps the jump response.Source: §A.8 · pp. 75, 79

A.9 A fixed-time comparison of squared gradients

• At a deterministic time inside an ODE interval, Lemma A.36 compares the empirical and reference squared-gradient statistics with failure probability at most δ on the specified population and sample events.At the same prescribed time, a threshold decision agrees when the statistic discrepancy is below the threshold; this does not synchronize adaptive jump times.Source: §A.9 · pp. 81–84

A.10 Finite-feature transfer of negative curvature at a fixed time

• Under finite-feature, spatial-Hessian, near-stationarity, and sample-size conditions, the reference-sample full Hessian retains negative curvature with failure probability at most δ.The result applies to the reference sample, not the empirical dynamics, and does not bound a whole-path curvature-response factor.Source: §A.10 · pp. 84, 85, 88

B.1 Tensor notation and Hermite tensors

• The tensor conventions define order-r tensors, contractions over matched indices, and symmetric tensors, with full contraction giving the Frobenius inner product.The subsection also collects conventions for Hermite polynomials, Gaussian chaos, and the Wiener chaos product formula.Source: §B.1 · pp. 88–89
• Under the unit-sphere convention, normalized Hermite tensors contract with u^⊗r as ψ_r(u⊤x); using non-unit weights introduces a norm-dependent factor.The unit-norm convention ensures w⊤z is standard normal when z is standard Gaussian.Source: §B.1 · pp. 89–90

B.2 Wasserstein gradient flow on product measures

• For product measures, admissible motions act separately on each marginal, and the restricted Wasserstein gradient projects the full gradient onto coordinate-wise product directions.The associated dissipation identity is justified under regularity premises on the characteristic solution, for almost every time.Source: §B.2 · p. 91

B.3 Overview of the nonlinear-moment landscape

• The section motivates matrix factorization as a compact testbed for nonlinear moment landscapes, whose geometry can include saddles, symmetries, and multiple factor branches.Source: §B.3 · p. 92
• The model transfers matrix- and tensor-factorization geometry to a mean-field product-measure setting, where tensor moments are active coordinates but the dynamics remain Wasserstein flows.In the Gaussian-input spherical specialization, the fitting loss is a squared residual system for tensor blocks, and the metric regularizer controls Wasserstein distance to the antipodal anchor.Source: §B.3 · p. 92
• The appendix establishes local attraction, transverse descent, and positive second variation in two separate regions, each under its own hypotheses.The moment implications used in these results are not converses.Source: §B.3 · p. 93

B.3.1 Summary of the section

• The appendix studies product WGF, not unrestricted WGF, and defines curvature through full spherical product pushforward second variation; its geometric claims prove neither a connecting trajectory nor finite-sample escape.The second variation includes both measure and spatial-Hessian derivatives, even at stationary states.Source: §B.3.1 · p. 93
• For 2<pW<3, globally well-posed product WGF starting sufficiently near the antipodal target on the antisymmetric branch remains cap-confined and converges to that target, including for atomic initial measures.The confinement and convergence conditions use the mean-distance barrier and pointwise fitting-field estimate; the stationary-state conclusion uses a direct smallness condition.Source: §B.3.1 · p. 94
• On fixed-assignment annuli with Z± ≥ ζ2_min, an explicit domination condition gives positive spherical product pushforward second variation for every admissible product tangent field, without asserting annular geodesic convexity.The metric regularizer's positive curvature compensates for a bounded negative fitting-term contribution.Source: §B.3.1 · p. 94
• A suitable same-density perturbation from the antipodal saddle has zero fitting loss and decreases the objective quadratically, while the regularizer contributes only o(h^2).The transverse teacher signal supplies a positive escape coefficient; this establishes an available direction, not that every perturbation follows it.Source: §B.3.1 · p. 95
• Near the same-density teacher, the fitting residual vanishes and its second variation is a nonnegative square; when its metric distance is positive, the regularizer yields a locally positive curvature margin.The result is local to the fixed-assignment cap class and does not imply the teacher is stationary for the regularized objective.Source: §B.3.1 · p. 95

B.4.1 Product-measure objective and product WGF

• The model combines a matrix-valued product-network fitting loss with a metric regularizer anchoring each spherical marginal to an antipodal Dirac pair; the regularizer is Hessian-flat at the target.The dynamics evolve the two marginals under product WGF, rather than unrestricted WGF over all couplings.Source: §B.4.1 · pp. 95–96, §B.4 · p. 95

B.4.2 Hermite and moment coordinates

• Gaussian-input Hermite coordinates diagonalize the fitting term's L2 geometry, while the finite odd Hermite block supplies diagnostic active coordinates for the landscape.The active coordinates do not fully parametrize the metric-regularized objective.Source: §B.3.1 · p. 93, §B.4.2 · p. 97

B.4.3 Standing assumptions

• The geometric verification assumes controlled Hermite expansion and cap constraints, a finite odd activation block with symmetric teacher factors, and a quantitatively coherent transverse teacher imbalance.The transverse imbalance provides the signal for the same-density perturbation escaping the antipodal saddle.Source: §B.4.3 · pp. 98–99
• The transverse imbalance condition excludes global point-reflection symmetry and requires a coherent component of sliced transverse barycenters along e⋆.It supplies the direction used to construct a same-density escape perturbation from the antipodal saddle.Source: §B.4.3 · p. 99

B.4.4 Fixed-assignment metric geometry

• Cap separation fixes the optimal assignment to the two antipodal Dirac atoms, and local spherical contraction moves each point toward its assigned atom while preserving its cap.The fixed-assignment formulas apply locally to pushforward perturbations whose supports remain in the two caps.Source: §B.4.4 · pp. 100–101

B.4.5 Tensorized objective

• The fitting loss becomes a squared Hilbert–Schmidt norm of tensor residuals, while the metric term controls distance to the antipodal pair.This tensor residual system is the coordinate system used in the curvature arguments that follow.Source: §B.4.5 · p. 105

B.5 Ambient first-order formulas

• The subsection records first-order ambient formulas for analyzing branch invariance, descent directions, and spherical Wasserstein gradients.It begins by differentiating the tensorized fitting loss with respect to the active moment tensors.Source: §B.5 · p. 105

Define the active fitting coefficients

• The fitting terms’ first variations yield ambient particle-wise gradients and their spherical projections under assumptions B.1 and B.2.The active sums are finite under assumption B.2.Source: Define the active fitting coefficients · pp. 106–108
• For positive Z±, the regularizer’s second-variation contribution is nonnegative; at Z± = 0, its EXP-pushforward second derivative exists and equals zero.The nonnegativity uses pW > 2.Source: Define the active fitting coefficients · pp. 108, 110
• The full spherical product pushforward second variation combines fitting and regularizer contributions, with the regularizer contribution zero at Z± = 0.The formula applies along admissible paths of the stated product-pushforward form, including such paths that are product Wasserstein geodesics.Source: Define the active fitting coefficients · p. 111
• The proposed finite-sample curvature-transfer theorem for this spherical product model remains open; the central calculation is a second-variation identity, not an application of the general C3 theorem.The identity is the spherical product counterpart of the full Hessian, and stationarity alone does not remove its spatial term.Source: Define the active fitting coefficients · p. 112

B.6 Symmetric and Antisymmetric Sub-Manifolds

• The symmetric and antisymmetric branches imply zero symmetric and antisymmetric active-moment blocks, respectively, but neither converse follows from the finite active blocks.The coordinates are S_n = (T1,n + T2,n)/2 and A_n = (T1,n − T2,n)/2; the antisymmetric branch is the candidate toward the antipodal saddle, while the symmetric branch contains the same-density teacher configuration.Source: §B.6 · pp. 113–114

Define the active-coordinate zero fiber

• At the active-coordinate zero fiber, the antipodally symmetric, equal-factor measure with zero odd moments has vanishing Wasserstein gradients and is first-order stationary.The proof uses antipodal symmetry, oddness of the active moments, and regularizer flatness at the antipodal target.Source: Define the active-coordinate zero fiber · p. 114
• On C0, T1,n = T2,n = 0 for all n ∈ O, so the fitting first variation vanishes and the full spherical gradient is only the regularization contribution.Source: Define the active-coordinate zero fiber · p. 114

The fitting part satisfies

• The symmetric-teacher reference measure belongs to the same-density branch M+, and the distinguished zero-fit global minimum of the data term lies there.At the reference point, the fitting objective is zero and is nonnegative everywhere.Source: The fitting part satisfies · pp. 114–115

B.7 The anti-symmetric branch and descent towards the saddle

• Under a local initial condition, the anti-symmetric branch is invariant and the flow has cap confinement and convergence to the antipodal Dirac target.This summarizes the section’s stated scope; the following points give the invariance and dissipation conditions.Source: §B.7 · p. 115

B.7.1 Invariance of the anti-symmetric branch

• The anti-symmetric branch M− is invariant on every interval where the characteristic product WGF is uniquely defined and remains in the stated admissible class.Reflection–exchange symmetry of the first variation and uniqueness imply Rµt = µt throughout that interval.Source: §B.7.1 · p. 116

B.7.2 Dissipation towards the antipodal saddle point on the anti-symmetric branch

• For 2 < pW < 4, states in M−∩X±(rcap) satisfying the stated smallness condition have quantitative radial dissipation, with ν± as the only stationary state in that region.The flow conclusion applies while it remains in M−∩X±(rcap), has positive distance, and satisfies the smallness condition.Source: §B.7.2 · pp. 120–121
• For 2 < pW < 3 and λreg > 0, initial data with 0 ≤ Z0 ≤ z⋆ remain in the basin and satisfy ˙Z ≤ −λreg Z^pW/2 whenever Z > 0.The proposition assumes a globally well-posed characteristic product WGF and a prescribed small threshold z⋆.Source: §B.7.2 · pp. 122–123
• The flow remains branch-invariant and confined to the caps for all time, and Z0 = 0 gives the stationary target; no density or atom-mass lower bound is needed.For positive initial distance, the distance bound prevents cap exit; the resulting dissipation yields convergence to µ† and A(µt) → 0.Source: §B.7.2 · pp. 122, 124

B.8 Curvature away from the saddle on the antisymmetric branch

• On a fixed metric annulus away from the antipodal target, sufficiently strong regularization dominates the spherical product pushforward’s fitting curvature.This is the stated scope of the second-variation bound.Source: §B.8 · p. 125
• On bounded anti-symmetric active sets, the fitting term’s negative curvature is uniformly controlled, and the fitting-curvature estimate requires no metric annulus.The curvature bound considers arbitrary ambient product tangent perturbations, not only perturbations tangent to the anti-symmetric branch.Source: §B.8 · p. 125

Then there exists a constant

• The second-variation proof uses a finite-dimensional chain rule in active moment coordinates and retains the moment path’s acceleration term.The argument applies the chain rule to the moment map and fitting polynomial.Source: Then there exists a constant · p. 126
• The metric regularizer contributes positive spherical product-pushforward second variation on fixed-assignment caps.The squared spherical-distance Hessian has positive eigenvalues within caps of radius less than π/2.Source: Then there exists a constant · pp. 127–128
• The annulus need not be geodesically convex; a strong-convexity inequality between states requires their comparison geodesic to remain in the region.Source: Then there exists a constant · p. 130
• Positive spherical product-pushforward second variation follows on the anti-symmetric annular states under the stated fixed-assignment conditions.The result combines the metric regularizer bound with an available fitting lower bound.Source: Then there exists a constant · p. 130

B.9 Descent direction to escape the saddle point on the antisymmetric branch

• The proposition establishes existence of a negative-curvature direction, but does not show that Hessian-guided perturbations select it.The geometric verification alone does not establish selection by the perturbation dynamics.Source: §B.9 · p. 134
• Transverse imbalance yields a visible tangent direction and a same-density spherical perturbation at the antipodal Dirac pair with second derivative −2γesc.The metric regularizer contributes no quadratic descent term at the antipodal Dirac point.Source: §B.9 · pp. 130, 134, 137

B.10 Ambient positive curvature near the same-density teacher

• Positive second variation near the same-density teacher does not establish that an escaping gradient-flow trajectory reaches that neighborhood or that the teacher is stationary for the regularized objective.Source: §B.10 · p. 137

B.10.1 Invariance of the same-density branch

• The same-density branch M+ is invariant under the antipodal metric objective for product WGF solutions in the stated fixed-assignment cap class.This follows from exchange symmetry of the objective and uniqueness of the product WGF solution.Source: §B.10.1 · p. 138

B.10.2 Positive curvature at and near the same-density teacher

• At the exact-fit same-density teacher, fitting and metric curvature combine to give strict ambient positive curvature that persists locally for cap-supported states.The result requires no density assumption on the teacher measure.Source: §B.10.2 · pp. 138–139
• A strictly positive curvature bound holds within a sufficiently small teacher neighborhood when the limiting regularizer coefficient is positive and the measure remains in the fixed-assignment caps.The stated condition requires Z_o > 0; when Z_o = 0, the result does not apply. Wasserstein proximity alone does not ensure support remains in those caps.Source: §B.10.2 · p. 141

C Numerical sanity checks

• The numerical sanity checks examine Assumption 5.2(iii) and adaptive PWGF with each system making its own perturbation decisions in a smooth matrix-factorization model.The model is separate from the spherical model, and the trigger uses the gradient norm under each system’s current distribution.Source: C Numerical sanity checks · p. 142

C.1 Model and population–particle comparison

• The comparison evolves a fixed 32,768-atom population and sampled empirical particles, coupling Gaussian innovations population-first by attempt index rather than common physical time.The empirical sample sizes range from 2,048 to 32,768, and the reference particles do not enter the controller.Source: §C.1 · pp. 142–143

C.2 Adaptive controller and recorded diagnostics

• Each system tests its own L2-gradient norm, applies a jump when its threshold is met, waits 10 time units, then tests for objective decrease.A failed decrease test returns the saved pre-jump candidate, distinct from the executed state.Source: §C.2 · p. 143
• Across all 15 conditions, the largest recorded cubic ratio is 1.017416, supporting non-spiky paired errors at recorded times, not continuously or over random paths.Source: §C.2 · p. 144, Figure 4 · p. 144
• At N = 32768, recorded peaks of r_t range from 0.584 to 0.601; discrepancies can be larger at smaller N and are not monotone in N.All 15 particle systems and three population paths stopped before the cap, and empirical returned objectives ranged from 1.12 × 10−4 to 1.41 × 10−4.Source: §C.2 · p. 144
• Halving the inspection interval preserved event counts but shifted some event times by 0.05 and changed individual returned objectives by up to 4.89 × 10−6.Source: §C.2 · p. 144

C.3 Paired trajectories and objective values

• Figure 6 shows paired error and objective gaps at common physical times, while Figure 7 shows objective values until each system stops.Together, the diagnostics show particle discrepancies and optimization progress under separately triggered jumps.Source: §C.3 · p. 146, Figure 6 · p. 146, Figure 7 · p. 147