Schrödinger Bridges on Lie Group Manifolds for Probabilistic Intrinsic Generation

• The Lie-group formulation avoids Euclidean projection and location-dependent tangent-space transport by keeping configurations on the group and velocities in a common Lie-algebra coordinate space.This addresses geometric overhead and compounded numerical errors while preserving the manifold support of periodic, rotational, and rigid-configuration data.Source: §1 · pp. 1–2
• The observed-endpoint Schrödinger problem uses two-sided entropy scaling to determine endpoint coupling, latent endpoint conditionals, and the stochastic law of intermediate trajectories.Under the endpoint-kernel condition, the entropy projection is unique and yields a forward Doob transform together with an entropy-optimal initial conditional law for latent variables.Source: §1 · pp. 2–3
• The framework models kinetic states as (g_t, ξ_t), evolving configurations intrinsically on a Lie group while representing stochastic forcing and controls in the fixed Lie algebra.When only group coordinates are observed at endpoints, the Lie algebra velocities remain latent and their conditional law is determined by the same entropy projection.Source: §1 · p. 2
• WKBC calibrates bridges from explicit periodized kinetic kernels on compact Abelian groups, whereas RCCBM learns Doob fields on compact non-Abelian groups through endpoint calibration and conditional-control matching.For rigid-frame data, compact gauge reduction removes the global rotational component before RCCBM learning; both methods target the same bridge.Source: §1 · pp. 2–3, Figure 1 · p. 3
• Experiments validate the approach on protein and RNA torsions, SO(3), U(n), and mdCATH-based protein conformational transition pathways, with ablations and consistency checks supporting the design.The reported validation spans the evaluated settings and includes numerical checks of the predicted modular error behavior.Source: §1 · p. 3

Schrödinger bridges and reciprocal matching

• Schrödinger bridge and reciprocal-matching research provides the probabilistic basis for observed-endpoint kinetic formulations and learning Markov controls from endpoint-conditioned bridge laws.The literature moves from entropy projection and potential iteration toward conditional or controlled-dynamics learning, including kinetic settings with latent velocities or momentum.Source: §2 · p. 4

Manifold and Lie group generative modeling

• Intrinsic manifold and Lie group generative models avoid Euclidean representations and ambient projection, while this work adds two-endpoint entropy projection on kinetic Lie group state spaces.Prior methods span reparameterizable Lie-group distributions, manifold flows and ODEs, Riemannian diffusion, SO(3) diffusion, and Riemannian flow matching.Source: §2 · p. 4

Schrödinger bridges on manifolds and position of this work

• Compared with prior manifold Schrödinger bridges, this work uses the kinetic state space G × g, intrinsic group reconstruction, Lie-algebra control, and general endpoint maps that leave unobserved velocities latent.Its computational realizations use wrapped kinetic kernels for compact Abelian groups and reciprocal endpoint calibration with conditional-control matching for compact non-Abelian groups.Source: §2 · p. 5

3.1 KINETIC GENERATOR AND OBSERVED-ENDPOINT SCHR ¨ODINGER SYSTEM

• Under an observed-endpoint kernel condition, the kinetic Schrödinger problem becomes a well-posed bounded two-sided scaling system, with compact-group verification available for positive continuous densities.The compact-group result supplies the endpoint regularity needed for projective contraction and uniform factor control.Source: §3.1 · pp. 5–6

3.2 BRIDGE FACTORIZATION

• The bridge admits marginal and dual-potential factorizations that identify the forward Doob dynamics used by both computational regimes.These identities connect the bridge marginals, propagated endpoint factors, and forward control representation.Source: §3.2 · p. 6

3.3 TWO COMPUTATIONAL REGIMES

• WKBC uses a periodized Gaussian-lift kinetic kernel on compact Abelian groups, providing an explicit observation kernel and numerical oracle for the observed-endpoint bridge.The wrapped kernel makes propagated terminal factors and the full kinetic marginal score computable, while generation uses the forward Doob control.Source: §3.3 · pp. 6–7
• RCCBM calibrates the two-sided endpoint reciprocal law, constructs mollified conditional-control teachers, regresses one Markov forward control, and applies constrained terminal refinement.It targets the same bridge without requiring a closed-form transition kernel, thereby addressing compact connected non-Abelian groups.Source: §3.3 · p. 7
• Endpoint calibration is exact at population level: the centered dual has a unique maximizer, and its dual gap equals both reciprocal-endpoint and path-law KL errors.This establishes calibration as an intrinsic path-law objective rather than an auxiliary heuristic.Source: §3.3 · p. 8
• Mollified conditional teachers form a Markov reciprocal law whose pooled regression targets the Markov Doob control, while Theorem 3.8 separates endpoint, optimization, generation, teaching, weighting, clipping, and reference errors.The resulting RCCBM path-law bound propagates identifiable module-level errors under the stated consistency assumptions.Source: §3.3 · pp. 8–10

3.4 REDUCTION OF SE(3)N DATA

• Reducing SE(3)^N data to a compact group transfers the stochastic guarantee to the reduced path space, while approximate Cartesian reconstruction requires separate evaluation.A C2-diffeomorphic reconstruction preserves relative-entropy comparisons; without injectivity, only data processing is guaranteed.Source: §3.4 · p. 11

4.1 EXPERIMENTAL PROTOCOL

• The protocol fixes evaluation maps and parameters before checkpoint selection, recomputes final diagnostics after refinement, and reports likelihood, geometric, calibration, control, group-constraint, and pathway-quality metrics.Compact Abelian comparisons use RDM (Huang et al., 2022), RFM (Chen & Lipman, 2024), and TDM (Zhu et al., 2025), while non-Abelian likelihoods follow the common TDM convention.Source: §4 · p. 11, §4.1 · pp. 11–12

4.2 COMPACT ABELIAN TORSION BRIDGES

• WKBC attains the lowest NLL on every compact Abelian torsion dataset, reducing absolute NLL relative to TDM by 0.10, 0.20, 0.42, 0.35, and 2.00.The reductions correspond to General, Glycine, Proline, Pre-Pro, and RNA; WKBC also improves Sinkhorn distance, Ramachandran JSD, ValidRate, and path energy on all five tasks, while GeoRMSE is lower on four protein subsets but slightly higher on RNA, 0.854 versus 0.843.Source: §4.2 · pp. 12–13, Table 2 · p. 12, Table 3 · p. 12

4.3 VERIFICATION AGAINST AN EXPLICIT WRAPPED BRIDGE

• RCCBM remains within the same numerical scale as the high-accuracy WKBC reference across source, target, intermediate-marginal, and forward-control discrepancies.WKBC has lower discrepancies across all four metrics, while RCCBM provides a close approximation to the optimal Doob control induced by WKBC.Source: §4.3 · p. 13, Table 4 · p. 14

4.4 COMPACT NON-ABELIAN SO(3) AND U(n) BRIDGES

• RCCBM attains the highest log-likelihood on all four SO(3) benchmarks, with margins of 0.036, 0.193, 0.088, and 0.406 over the second-best method.The margins correspond to SO(3)-GMM32, SO(3)-GMM64, SO(3)-GMM128, and SO(3)-RingBand, respectively; generated samples also recover multimodal and narrow annular support while preserving group constraints.Source: §4.4 · pp. 14–15, Table 5 · p. 14
• RCCBM matches the prescribed SO(3) terminal law while continuously transporting complex source and target distributions, including SO(3)-GMM64 to SO(3)-RingBand.Intermediate marginals progressively reorganize from broad multimodal support toward the target’s narrow annular support, demonstrating distribution-to-distribution modeling beyond fixed-prior-to-data generation.Source: §4.4 · pp. 15–16, Figure 4 · p. 15

4.5 PROTEIN CONFORMATIONAL TRANSITION PATHWAY GENERATION

• RCCBM achieves a 95.06% overall directness gain and a 99.21% roughness reduction over reference MD segments across 15 held-out mdCATH transitions.The gains use 32 stochastic trajectories per transition and retain prescribed endpoint constraints in the reduced representation.Source: §4.5 · pp. 16–17, Table 7 · p. 17

4.6.1 WKBC MODELING ABLATION

• The full WKBC configuration is most favorable on every General-task ablation metric, while removing endpoint smoothing or uniformizing time sampling increases endpoint and path discrepancies.Numerical sensitivity tests also find production settings stable: N = 600 gives NLL 0.593 versus 0.591 at N = 800, only 0.338% higher, with further lattice or tolerance refinement producing limited, metric-dependent changes.Source: §4.6.1 · p. 18, Table 8 · p. 18, §4.6.3 · p. 19

4.6.2 RCCBM COMPONENT ABLATION

• RCCBM component ablations show that endpoint calibration, stochastic resampling, replay, refinement, spectral conditioning, and calibrated initialization have complementary effects on control and distributional accuracy.Fixing random numbers increases held-out teacher error; removing endpoint calibration or terminal replay increases post-refinement relative MSE and validation MMD, while spectral conditioning and calibrated initialization improve U(4)-TFIM diagnostics.Source: §4.6.2 · p. 18, Table 9 · p. 18

Current limitations

• The current evaluation focuses on moderate-dimensional groups and selected molecular and geometric tasks, leaving performance on higher-dimensional groups, larger systems, and more heterogeneous datasets uncharacterized.The framework currently covers compact Abelian groups, compact non-Abelian groups, and rigid-frame problems after compact reduction.Source: §5 · p. 19

Future work

• Future experiments should examine higher-dimensional groups, larger molecular systems, and more diverse scientific datasets to clarify scalability and robustness.Source: §5 · p. 19
• Extending the analysis to genuinely noncompact Lie groups, quotient manifolds, and homogeneous spaces would broaden the framework’s geometric applicability while retaining its observed-endpoint kinetic Schrödinger bridge formulation.Source: §5 · p. 19

6 CONCLUSIONS

• The framework constrains observed endpoint variables, determines latent velocity conditionals by entropy projection, and yields Schrödinger and kinetic Doob representations on G × g.Source: §6 · p. 20
• WKBC calibrates observed-endpoint bridges with explicit wrapped kernels on compact Abelian groups, while RCCBM uses reciprocal calibration, mollified teachers, corrected regression, and terminal refinement for compact non-Abelian groups.The teacher mixture remains a reciprocal Markov law, enabling forward generation with one learned controller and a calibrated initial law.Source: §6 · p. 20
• The analysis separates endpoint, mollification, teacher, regression, initialization, and discretization errors into a bounded-Lipschitz path-law convergence statement.Source: §6 · p. 20
• Experiments across protein and RNA torsions, SO(3), U(n), and protein conformational pathways support effectiveness, component roles, and numerical stability in Abelian and non-Abelian settings.Source: §6 · p. 20

APPENDIX

• The appendix supplies implementation details, theoretical proofs, visual diagnostics, experimental settings, and reproducibility information for the main algorithms and results.Source: APPENDIX · p. 23

A.1 OBSERVED-ENDPOINT WKBC

• Observed-endpoint WKBC uses positive quadrature or Monte Carlo rules, log-domain Sinkhorn scaling, bridge-weighted control regression, and calibrated initial-state sampling.Its approximation conditions ensure convergence of normalized scaling factors and logarithmic gradients on preterminal strips.Source: §A.1 · p. 24

A.2 RCCBM AND THE REDUCTION WRAPPER

• The method samples teacher tuples from two time windows, applies exact proposal-to-target importance weights, and updates a learned forward controller using independently generated teacher tuples.The terminal-window fraction rterm selects t ∼Unif[atermT, T], while probability 1−rterm selects t ∼Unif[0, atermT).Source: §A.2 · pp. 24–25

A.3 EXACT-OU SPLITTING

• Exact-OU splitting preserves SO(3) and U(n) geometry, is exact for the frozen-control velocity substep, and satisfies a bounded-Lipschitz path error of O(∆t^1/2).The constant is uniform over control sieves sharing Lipschitz and linear-growth bounds.Source: §A.3 · p. 26

A.4 TDM-ALIGNED LIKELIHOOD EVALUATION

• The TDM-aligned benchmark uses a Haar-Gaussian prior, full-state Gaussian velocity augmentation, and intrinsic probability-flow likelihood evaluation for the prescribed-endpoint bridge protocol [Zhu et al., 2025].With an exact velocity score, the probability-flow ODE has the same one-time marginals as the stochastic benchmark generator.Source: §A.4 · pp. 26–27
• Intrinsic change-of-variables with Haar-divergence-free group fields yields the held-out LL/NLL evaluator, using an independently augmented terminal Gaussian velocity and a fixed Gaussian trace probe per trajectory [Zhu et al., 2025].The trace estimate is unbiased conditioned on the trajectory state, and the auxiliary Gaussian factor is removed under the full-state likelihood target.Source: §A.4 · pp. 27–28
• WKBC obtains its full velocity score by two-sided wrapped-kernel propagation, whereas RCCBM fits an auxiliary implicit score network after freezing the final controller and checkpoint.The RCCBM score is trained on fresh trajectories from the prior under the frozen controller and then used for held-out likelihood evaluation.Source: §A.4 · pp. 27–28

B.1 ENDPOINT-KERNEL VERIFICATION AND LATENT-STATE LIFT

• For compact connected groups with positive continuous source, target, and reference densities, group-coordinate endpoint observations satisfy the endpoint-kernel assumption after integrating out velocity.The uniform bounds support Birkhoff–Hilbert contraction and bounded scaling factors, although they are stronger than minimal entropy conditions.Source: §B.1 · p. 29

B.2 AUXILIARY DOOB AND PATH-STABILITY IDENTITIES

• The auxiliary Doob analysis relates control-law relative entropy to Girsanov changes and bounds path-law KL error through the backward-factor residual and endpoint log-factor mismatch.These identities require shared initial laws and diffusion coefficients, with the relevant Girsanov exponentials being true martingales.Source: §B.2 · p. 29

B.3 WRAPPED KERNEL AND WKBC REGRESSION DETAILS

• WKBC’s bridge-weighted regression has population minimizer s* = ∇ξ log hβ*, and its regression-to-path-law conversion has explicit constant γT without an auxiliary occupation-density-ratio constant.Endpoint reweighting of reference paths makes the regression design measure equal to the bridge occupation law, so reverse-KL control error is evaluated under the correct law.Source: §B.3 · p. 30
• Under refined lattice, quadrature, Sinkhorn, interpolation, initial-law, and bridge-weighted regression approximations, the numerical-bridge and regression terms in WKBC’s path-space decomposition vanish.The implementation uses a fixed uniform time density and self-normalized endpoint weights computed from factors already produced by WKBC.Source: §B.3 · p. 30

B.4 DETAILED RCCBM CALIBRATION, TEACHERS, AND LEARNING ASSUMPTIONS

• Alternating KL projections converge to the calibrated endpoint factors, including the source corrector omitted by terminal-only constructions.Endpoint calibration assumes bounded doubly centered classes, empirical-dual convergence, sieve approximation, and optimization consistency, with errors vanishing in probability.Source: §B.4 · p. 31
• Once the initial law and forward control are identified, exact forward sampling requires no propagated factor networks, provided L2 convergence and true-martingale conditions hold.The conditional-kernel assumptions require positivity, ξ-differentiability, valid endpoint-integral differentiation, well-posed martingale problems, and preterminal square-integrability.Source: §B.4 · p. 31
• Teacher approximation errors, mollification, replay correction, direct-control matching, clipping, and refinement are required to vanish asymptotically, while injective MMD features preserve control equivalence.Uniform mollification yields local and uniformly integrable L2 control convergence; full-rank target-whitened embeddings satisfy the identification condition for SO(3), U(n), and the compact reduced product group.Source: §B.4 · p. 32
• The self-normalized empirical initialization and Lie–Trotter discretization errors vanish in probability, with the controlled path map Lipschitz in bounded-Lipschitz distance.The path error is uniformly controlled over the learned control sieve, and dBL(Su(µ), Su(ν)) ≤ Cinit dBL(µ, ν).Source: §B.4 · pp. 32–33

B.5 REDUCED-SPACE ASSUMPTIONS AND RECONSTRUCTION

• Reduced-space claims are exact when the reconstruction is a C2-diffeomorphism onto the embedded conformation manifold; without injectivity, only data processing is asserted.Under this assumption, the induced state-path map is a Borel isomorphism onto its image.Source: §B.5 · p. 33

B.6 PATH-SPACE TOPOLOGY AND ADJOINT CONVENTIONS

• The appendix equips path space with a bounded uniform metric compatible with its Polish topology, so bounded-Lipschitz distance metrizes weak convergence and is bounded by 2 TV.It also fixes a complete metric on G and specifies the formal adjoint of the kinetic generator.Source: §B.6 · p. 33

B.7 RESIDUAL-TO-PATH IDENTITY

• The residual-to-path identity bounds the discrepancy between controlled laws induced by an approximate backward factor and the exact factor through the integrated residual along trajectories.The identity assumes equal initial-state laws and the required integrability conditions, with the residual term involving |rK(t, Xt)|.Source: §B.7 · p. 33

B.8 WKBC NUMERICAL ERROR DECOMPOSITION

• WKBC numerical errors split into numerical bridge approximation and bridge-weighted control regression, both vanishing under their respective consistency conditions without an occupation-density-ratio constant.The numerical approximation term aggregates lattice, quadrature, Sinkhorn, factor-interpolation, and initial-law errors.Source: §B.8 · p. 34

B.9 RCCBM LEARNING ASSUMPTIONS

• RCCBM assumptions require dense teacher and empirical-risk classes, controlled importance ratios, sieve clipping, uniform laws, and shared refinement sieves, yielding separated theorem error terms.The resulting terms are εapp,N, εopt,N, εgen,N, εteach,N, εiw,N, εclip,N, and δref,N.Source: §B.9 · p. 34

B.10 VERIFICATION OF THE RCCBM ASSUMPTIONS ON Tm

• On Tm, wrapped Gaussian kernels are smooth and strictly positive, while compactness and positive densities bound the endpoint likelihood ratio above and below.The lattice series and all derivatives converge absolutely and locally uniformly on every positive time interval.Source: §1 · p. 34

1. Endpoint calibration

• Endpoint calibration has unique positive factors, vanishing approximation and optimization gaps, and εend,N →0 in probability under dN log N/N → 0.Centered log-potentials are C2, and wrapped-kernel derivative envelopes verify the required endpoint regularity.Source: §1 · p. 35, §2 · p. 35

2. Conditional kernels

• Periodic heat kernels provide uniformly controlled conditional kernels, and a slowly chosen εN schedule ensures finite moments and εmol,N →0.The endpoint sieve yields uniformly bounded, equicontinuous exponentials, while preterminal derivative bounds verify the required envelope condition.Source: §2 · p. 35

3. Teacher and direct-control approximation

• In the torus verification regime, direct teacher sampling makes εiw,N, εteach,N, and auxiliary discretization errors zero, while dense control sieves and growing truncation levels remove approximation and clipping errors.Taking the refinement map as the identity also gives δref,N = 0, and empirical minimization gives εopt,N = 0.Source: §3 · p. 35

4. Calibrated initial law and dynamical stability

• Bounded initial weights, Lipschitz/linear-growth control envelopes, synchronous coupling, Grönwall estimates, and Lie–Trotter convergence imply εinit,N and εdisc,N vanish, yielding bounded-Lipschitz path-law convergence.All terms in Eq. (3.22) therefore vanish in probability.Source: §4 · pp. 35–36

B.11 UNIFORM MOLLIFICATION CONDITIONS

• Uniform mollification requires equicontinuity, integrable derivative envelopes on compact state-time regions, and uniform integrability of squared mollified controls when L2 convergence is used.These conditions are imposed relative to the corresponding occupation measures.Source: §B.11 · p. 36

B.12 FEATURE IDENTIFICATION USED IN REFINEMENT

• The MMD identification condition requires only that the identification feature Φ_id^G be continuous and injective, independently of auxiliary controller features.The spectral descriptors used in the U(n) experiments belong to Ψ_aux rather than the MMD-identifying feature.Source: §B.12 · p. 36
• Whitening applies a regularized full-dimensional linear transform whose matrix W_G is positive definite and invertible because λ_w > 0.Translation by μ_G and multiplication by W_G preserve continuity and injectivity.Source: §B.12 · p. 36
• For production groups, Φ_id^G is built from matrix-entry embeddings and periodic sine-cosine embeddings of torus coordinates.The periodic embedding is independent of the chosen angle representative.Source: §B.12 · p. 36
• The resulting Φ_id^G is continuous and injective for every production group considered.This follows from the corresponding matrix and torus embeddings together with the invertible whitening transform.Source: §B.12 · p. 36
• Compactness makes Φ_id^G a homeomorphism onto its Euclidean image, where a positive finite Gaussian-kernel mixture is characteristic and metrizes weak convergence.Thus the MMD term satisfies the required feature-identification condition.Source: §B.12 · p. 37
• At the exact population minimizer with δ_ref = 0, the refinement objective preserves the same control equivalence class.This conclusion follows from the established identification condition for Φ_id^G.Source: §B.12 · p. 37

C.1 PROOF OF THEOREM 3.2

• The composite scaling map S is a projective contraction with coefficient at most τ^2, so Banach’s fixed-point theorem yields a unique projective fixed point.The projective iterates are Cauchy after normalization, and the resulting scaling functions are constructed from this fixed point.Source: §C.1 · p. 37
• The fixed-point scaling pair defines a coupling with the prescribed marginals, and reciprocal gauge is the only remaining ambiguity.Continuity and boundedness follow from the integral operators on compact spaces, while a second scaling pair yields the same projective fixed point up to reciprocal gauge.Source: §C.1 · p. 37
• The constructed coupling is the unique entropy minimizer, and endpoint disintegration makes the associated path law unique under the observed endpoint constraint.The entropy decomposition is uniquely minimized by the constructed endpoint coupling and by matching the conditional path laws to the reference conditionals.Source: §C.1 · p. 37

C.2 DERIVATION FOR LEMMA B.2

• The kinetic Ornstein–Uhlenbeck reference process satisfies global Hörmander hypoellipticity, fixed-time controllability, and strictly positive smooth transition densities, yielding uniform endpoint-kernel bounds on compact G.The bracket condition spans the full tangent space, every state is reachable at every t > 0, and the endpoint map is a submersion; compactness upgrades positivity to uniform bounds.Source: §C.2 · pp. 38–39

C.3 PROOF OF LEMMA B.4

• The entropy chain rule shows that, for a fixed observed marginal, the conditional law of unobserved velocities is uniquely matched to the reference conditional law.The conditional relative-entropy term is nonnegative and vanishes uniquely when π(dξ | g) = Q_i(dξ | g).Source: §C.3 · p. 40

C.4–C.6

• The Doob transform preserves the group drift while adding the normalized velocity control α∇ξ log φt, and Girsanov–Pinsker arguments convert path relative entropy into marginal total-variation bounds.The terminal factor solves the backward equation, the transformed forward density satisfies the corresponding evolution, and Lemma B.5 follows by evaluation-map contraction.Source: §C.4 · p. 40, §C.5 · p. 40, §C.6 · p. 41

C.7 DERIVATION FOR LEMMA B.7

• On a torus, the kinetic Ornstein–Uhlenbeck transition density is the lift-independent periodization of a Euclidean Gaussian, with velocity integration giving the group marginal.Conversion from Lebesgue to normalized Haar measure contributes the fundamental-cell volume factor.Source: §C.7 · p. 42

C.8 PROOF OF PROPOSITION B.9

• The bridge score equals the conditional expectation of the terminal score, while bounded-Lipschitz errors decompose into control, initialization, and numerical bridge errors that vanish under uniform kernel approximation.Uniform contraction of normalized Sinkhorn maps yields convergence of scaling factors, initial laws, Doob controls, and numerical bridge path laws.Source: §C.8 · p. 42

C.9–C.10

• The entropy projection onto a prescribed endpoint law uniquely retains the reference conditional path law, establishing the path-space characterization of the calibrated bridge.The endpoint-map entropy chain rule identifies the unique minimizer for almost every endpoint.Source: §C.9 · p. 43, §C.10 · p. 43

C.11 DERIVATION FOR PROPOSITION B.12

• Alternating information projections converge geometrically to the unique two-sided Schrödinger scaling pair, whereas terminal-only correction cannot generally attain the optimum because the first projection changes the source factor.Uniform positive kernel bounds convert projective convergence into uniform convergence and convergence of the endpoint KL gap.Source: §C.11 · p. 43

C.12 PROOF OF PROPOSITION B.14

• The endpoint approximation error is bounded by the sum of sieve approximation, estimation, and optimization errors, implying convergence in probability under Assumption B.13.The proof bounds the four decomposed terms by εend,app,N, δend,N, and εend,opt,N as specified in Eqs. (B.19)–(B.20).Source: §C.12 · p. 44

C.13 PROOF OF THEOREM 3.6

• The endpoint-tilted law has conditional path law equal to the reference bridge on each endpoint fiber, with a Doob representation and an L2-optimal teacher projection valid on preterminal strips.The preterminal restriction is essential because point-pinned controls may have infinite energy at the terminal time.Source: §C.13 · p. 44

C.14 PROOF OF PROPOSITION B.17

• For the controlled-drift variational problem, the mollified Doob law Qz,ε is the unique law-level entropy minimizer, and the excess objective equals the corresponding control discrepancy.This follows by Girsanov’s theorem and nonnegativity of relative entropy.Source: §C.14 · pp. 44–45

C.15 PROOF OF PROPOSITION 3.7

• The endpoint-factorized bridge is Markov, while its conditional center law admits the stated variance identity; fixed ε > 0 removes terminal point-pinning singularity.The conditional center density is proportional to eβT(y)hy,ε.Source: §C.15 · p. 45

C.16 DERIVATION FOR PROPOSITION B.20

• Uniform endpoint-factor bounds yield the bounded-Lipschitz path-law approximation via dBL ≤ 2TV, while dominated differentiation and uniform integrability give local uniform and L2 convergence of the controls.The convergence argument uses local lower bounds and Vitali’s theorem, or dominated convergence under stronger domination.Source: §C.16 · pp. 45–46

C.17 PROOF OF THEOREM 3.8

• Theorem 3.8 bounds the final excess control risk by approximation, optimization, generalization, teacher, importance-weighting, clipping, and refinement errors, with probability tending to one.Every component converges to zero in probability under the stated assumptions.Source: §C.17 · p. 47
• Theorem 3.8 converts vanishing endpoint, mollification, control, initialization, and discretization errors into bounded-Lipschitz path-law convergence in probability.The proof combines control-risk conversion, Girsanov comparison, initial-law stability, and numerical integration error.Source: §C.17 · p. 47

C.18 PROOF OF PROPOSITION A.1

• When gk ∈ G, reconstruction by gk ExpG(∆t ξk+1) remains in G because exp(H) ∈ U(n), so the reconstruction step preserves the matrix Lie group exactly.Source: §C.18 · p. 48
• Freezing the control yields an exactly integrated affine Ornstein–Uhlenbeck velocity update with centered Gaussian noise and the coefficients specified in Eq. (A.6).The velocity interpolation follows the exact frozen-control OU flow on each interval.Source: §C.18 · p. 48
• The splitting and noncommutative reconstruction defect contributes O(h) in mean square over the full time horizon.Uniform moment and increment estimates support the global strong path estimate and constants independent of 0 < h ≤ h0 ≤ 1.Source: §C.18 · pp. 49–50

C.19 PROOF OF PROPOSITION 3.10

• Under Assumption B.23, the reduced state lift and its path-space extension are Borel isomorphisms onto their images, enabling change-of-variables for reduced path laws.The construction uses the C2-diffeomorphism, inverse differential, and Lie-algebra coordinates induced by left translation.Source: §C.19 · p. 51
• If reconstruction is noninjective, relative entropy instead satisfies the data-processing inequality; the reduced problem is therefore in the compact non-Abelian RCCBM regime when endpoint and reference assumptions hold.The domain SO(3)^(N−1) × T^K is compact and non-Abelian whenever at least one SO(3) factor is present.Source: §C.19 · p. 51

D ADDITIONAL EXPERIMENTAL RESULTS

• Projected RCCBM samples closely match dominant support geometry across oscillator- and TFIM-induced U(n) tasks, including annular, spiral, crescentlike, and anisotropic structures.The comparison covers U(4), U(6), and U(8) oscillator tasks and U(4) and U(8) TFIM tasks.Source: D ADDITIONAL EXPERIMENTAL RESULTS · p. 51, Figure 8 · p. 53
• WKBC reproduces heterogeneous RNA torsion marginals, spanning sharply concentrated modes to broader asymmetric profiles across seven periodic coordinates.These coordinate-wise marginals complement quantitative endpoint and path results in Table 3.Source: D ADDITIONAL EXPERIMENTAL RESULTS · p. 51, Figure 7 · p. 52

E.1 ENDPOINT, REFERENCE, AND OPTIMIZATION SPECIFICATION

• Experiments fix endpoint and reference specifications, bridge parameters, endpoint mollification, integration grids, and seed 1209, with WKBC and RCCBM using their respective implementations.RCCBM uses endpoint-corrector networks and DirectControlNet; the reproducibility manifest records configuration, sampling, refinement, validation, and evaluator settings.Source: §E.1 · pp. 54–55, Table 11 · p. 54
• RCCBM ablations remove endpoint calibration, replay, local-scale terms, refinement, spectral features, or calibrated initial-state sampling to probe their contributions and initial-law consistency.A fixed-common-random-number CondSOC variant and General-task WKBC variations in endpoint smoothing and time sampling are also evaluated.Source: §E.1 · p. 54

E.2.1 RCCBM ENDPOINT, TEACHER, CONTROL, AND STRUCTURE DIAGNOSTICS

• RCCBM evaluates endpoint calibration, CondSOC, direct-control matching, terminal refinement, Lie-group structure preservation, and independent-noise validation through distinct diagnostic stages.Reported control metrics include MSE, relative MSE, and cosine, while structure errors measure preservation of the Lie-group constraint under exponential updates.Source: §E.1 · p. 54, §E.2.1 · p. 55

E.2.2 MMD, SLICED-WASSERSTEIN, LOCAL PRECISION/COVERAGE, AND INTRINSIC COSTS

• Distributional evaluation combines target-whitened feature MMD, sliced-Wasserstein, local precision and coverage, debiased Sinkhorn divergence, GeoRMSE, Ramachandran JSD, and ValidRate.The procedures use fixed embeddings, direction banks, pairings, periodic partitions, validity regions, and common entropic-transport settings where specified.Source: §E.2.2 · pp. 55–56

E.2.3 NORMALIZED-CONTROL ENERGY

• Normalized control energy is comparable across methods only when the reference process, diffusion coefficient, time horizon, initial-state law, and integration grid are identical.The reported energy is defined for velocity drift correction αu_t.Source: §E.2.3 · p. 56

Evaluation setup

• Evaluation uses mdCATH domain 1jvmB00 trajectories in a reduced representation with 63 rotation factors and 192 torus coordinates.The evaluator operates in the same reduced state space as the generator.Source: §E.2.4 · p. 56

Evaluation protocol

• Both molecular-dynamics and RCCBM trajectories are evaluated on the same 101-point normalized-time grid after intrinsic interpolation.Each MD pair has 101 stored states, while RCCBM uses 128 stochastic integration steps before interpolation.Source: §E.2.4 · p. 56

Path diagnostics

• Target-aware tortuosity measures geometric detour while enforcing target attainment, with values at least one and closer to one indicating less detour.The terminal-residual term prevents paths that miss the prescribed target from receiving artificially favorable scores.Source: §E.2.4 · p. 57
• Normalized Lie algebra roughness increases with reversals, abrupt directional changes, and local oscillations, and is not bounded above by one.Its numerator measures changes in successive local displacement vectors, normalized by total local motion.Source: §E.2.4 · p. 57

Aggregation and evaluation protocol

• For the prespecified held-out pair list, target-aware tortuosity is computed from the prescribed target according to the rule defined above.Source: §E.2.4 · p. 57