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Refined Topology-Pass Scheduling: Damped Trust-Region Compensation for Dynamic k-NN Graphs in SE(3)-Equivariant Structure Prediction
Jiankai Huang, Hanwen Shen
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 90%
Last extracted: 10/15/2026, 5:30:09 AM
Summary
This paper introduces a refined topology-pass scheduling method for SE(3)-equivariant protein structure prediction pipelines, specifically addressing the computational cost of rebuilding k-nearest-neighbour (kNN) graphs at every block boundary. The authors propose a two-pass design where a cheap initial pass records trajectory state (margins, radii, displacements) to enable a second pass that compensates for topology changes using a damped fixed-point iteration. Theoretical contributions include exact non-flip certificates, a minimum-variance replay gain (BLUE), and an l-infinity trust-region projection to ensure convergence. Three implementation levels (L1, L2, L3) are validated, with L2 achieving exact reproduction of the fully coupled reference topology (PB) and providing a 1.76x epoch speed-up with no measurable loss gap.
Entities (12)
Relation Signals (6)
PB Schedule → isreferencefor → L2
confidence 95% · L2 reproduces the PB edge set exactly
L2 → achievesspeedup → 1.76x
confidence 90% · The recorded trajectory itself yields a measured ... 1.76x epoch speed-up
L3 → interpolatesbetween → F Schedule
confidence 90% · L3 interpolates between the frozen and the PB topology
BLUE → optimizes → replay_gain
confidence 90% · the minimum-variance (BLUE) replay gain
Trust-Region Projection → preserves → Convergence
confidence 90% · an l_infinity trust-region projection that preserves convergence
RFAA → uses → SE(3)-equivariant
confidence 90% · RoseTTAFold All-Atom (RFAA) combined an SE(3)-equivariant track with all-atom residue frames
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Abstract
Abstract:Equivariant structure-prediction pipelines alternate coordinate updates with the construction of a k-nearest-neighbour (kNN) graph whose membership is a discrete, piecewise-constant function of the coordinates. Rebuilding that graph at every block boundary (PB) is the expensive part of an epoch, whereas reusing one frozen graph (F) is fast but leaves a topology gap. We study a two-pass design in which the cheap pass records a small amount of state (margins, kNN radii, measured per-boundary displacements) and a second pass compensates conservatively. We formulate the choice of the graph-construction basis as a damped fixed-point iteration whose unique fixed point is exactly PB, and derive (i) exact non-flip certificates with the two-sided constant $m_i/4$ and one-sided constant $m_i/2$, (ii) the minimum-variance (BLUE) replay gain $\gamma^*$ and the resulting feasibility crossover, (iii) an $\ell_\infty$ trust-region projection that preserves convergence, and (iv) an analytic near-tie spacing law $E[m] \approx r_k/(3k)$ that matches measurements to 1\%. Three compensation levels are implemented in C++ (L1: frozen set with live geometry; L2: margin-gated local repair; L3: damped, projected basis) and validated under a matched-conditions protocol (identical seed, MSA depth and trajectory family, only the topology mode differs). L2 reproduces the PB edge set exactly ($D = 0.0000$ at every boundary), L1 stays within 0.6% and L3 interpolates monotonically: L2 reproduces the PB edge set exactly when the trust region is removed, and L3 interpolates between the frozen and the PB topology. The recorded trajectory itself yields a measured $1.5\times$ Pass-1 and $1.76\times$ epoch speed-up with no measurable loss gap. An open-source implementation of the runtime and of all the experiments reported here is available at this https URL.
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- Source: https://arxiv.org/abs/2610.19246v1
- Canonical: https://arxiv.org/abs/2610.19246v1
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Refined Topology-Pass Scheduling: Damped Trust-Region Compensation for Dynamic k-N Graphs in SE(3)-Equivariant Structure Prediction Jiankai Huang Affiliation: huangjk.08@gmail.com Hanwen Shen Affiliation: hs907@scarletmail.rutgers.edu Abstract Equivariant structure-prediction pipelines alternate coordinate updates with the construction of a k-nearest-neighbour (kkNN) graph whose membership is a discrete, piecewise-constant function of the coordinates. Rebuilding that graph at every block boundary ( PB) is the expensive part of an epoch, whereas reusing one frozen graph ( F) is fast but leaves a topology gap. We study a two-pass design in which the cheap pass records a small amount of state (margins, kkNN radii, measured per-boundary displacements) and a second pass compensates conservatively. We formulate the choice of the graph-construction basis as a damped fixed-point iteration whose unique fixed point is exactly PB, and derive (i) exact non-flip certificates with the two-sided constant mi/4m_i/4 and one-sided constant mi/2m_i/2, (i) the minimum-variance (BLUE) replay gain γ∗γ^* and the resulting feasibility crossover, (i) an ℓ∞ _∞ trust-region projection that preserves convergence, and (iv) an analytic near-tie spacing law E[m]≈rk/(3k)E[m]≈ r_k/(3k) that matches measurements to 1%1\%. Three compensation levels are implemented in C++ (1 L_1: frozen set with live geometry; 2 L_2: margin-gated local repair; 3 L_3: damped, projected basis) and validated under a matched-conditions protocol (identical seed, MSA depth and trajectory family, only the topology mode differs). 2 L_2 reproduces the PB edge set exactly (=0.0000D=0.0000 at every boundary), 1 L_1 stays within 0.6%0.6\% and 3 L_3 interpolates monotonically: 2 L_2 reproduces the PB edge set exactly when the trust region is removed, and 3 L_3 interpolates between the frozen and the PB topology. The recorded trajectory itself yields a measured 1.5×1.5× Pass-1 and 1.76×1.76× epoch speed-up with no measurable loss gap. An open-source implementation of the runtime and of all the experiments reported here is available at https://github.com/AlbertHuang96/ProteinPredictMachineLearning. Index Terms: dynamic kkNN graphs, SE(3)-equivariant networks, topology pass scheduling, trust-region projection, fixed-point iteration, protein structure prediction 1 Introduction 1.1 Previous work All-atom protein structure prediction is dominated by three-track (1D/2D/3D) architectures: AlphaFold2 established the MSA/pair/coordinate design with invariant point attention [1], AlphaFold3 extended it to a diffusion module over all chemical components [2], and RoseTTAFold All-Atom (RFAA) combined an SE(3)-equivariant track with all-atom residue frames [3]. The 3D track itself rests on equivariant message passing: tensor-field networks [4] and SE(3)-Transformers [5] build relative-position features from spherical harmonics and Clebsch–Gordan coefficients, i.e. from a graph of residue pairs. Graph construction (a kkNN search) is therefore on the critical path of every iteration. Two largely separate literatures address the consequences. (i) Dynamic graph maintenance: temporal GNNs update graph structure as data streams in [6], and approximate nearest-neighbour indexes such as HNSW support fast incremental queries [7]; spectral sparsification asks which edges even matter [8], and graph rewiring improves message passing by changing connectivity on purpose [9]. (i) Relaxing discrete selections: straight-through estimators pass gradients through non-differentiable choices [10] and differentiable top-k operators replace a hard selection by a continuous relaxation [11]. Our setting is an instance of both: the graph is a discrete degree of freedom whose gradient vanishes almost everywhere, and it must be maintained under continuous drift. 1.2 Background and problem statement The training pipeline we study comes from RFAA. What is specific here is the implementation route: our runtime builds the forward graph explicitly with ggml before execution, so the topology of every block — including the k-nearest-neighbour (kkNN) graph construction — is bound at build time, while the coordinates that define that topology keep changing during training. This mismatch is the coordinate-optimisation problem addressed in this paper: the graph is a function of the coordinates, yet once the forward graph has been built it can only be reused as it stands, so one must decide when to rebuild it and what to reuse in the meantime. Concretely, the pipeline scans T block boundaries per iteration. At boundary t the graph (x)=(E(x),D(x),W(x))G(x)= (E(x),\;D(x),\;W(x) ) (1) is built by make_graph from coordinates x: a directed kkNN edge set E(x)⊂[L]×[L]E(x)⊂[L]×[L], in which each residue keeps its k=min(64,L−1)k= (64,L-1) nearest neighbours, together with the edge geometry dij=xj−xid_ij=x_j-x_i and scalar edge features W. Four terms are used throughout. F is the plain schedule in which the graph is built once, from the initial coordinates x0x^0, and then reused unchanged for every later boundary, so that the network runs feed-forward. PB is the opposite schedule: the graph is rebuilt from the live coordinates at every boundary, which is the accuracy reference and the expensive option. A topology pass is one execution of the block sequence with a graph that is fixed in advance and reused, i.e. a feed-forward pass whose trajectory can be recorded. The two-pass variant combines a topology pass (Pass-1), which records the trajectory, with a second pass (Pass-2) that re-runs the same blocks and consumes the recorded state, so that only the graph step differs between the two passes. Measurements on the original pipeline summarise the trade-off: • cost: PB needs ≈34.1≈ 34.1 s per epoch versus 19.419.4 s for the two-pass variant (1.76×1.76×); the kNN rebuild dominates (≈1.8≈ 1.8 s per block), whereas refreshing the geometry over the frozen edge set ℰE of (1) is O(|ℰ|)O(|E|), with |ℰ|≈7,000|E|≈ 7,000 edges, and is negligible, and a local rebuild costs O(|Ω|k)O(| |\,k); • fidelity: at L=103L=103 the first boundary replaces ≈25%≈ 25\% of the directed edges (100%100\% of residues touched) and steady-state boundaries replace 22–3%3\%; the common-edge geometry differs by only 0.290.29 Å; • accuracy: no measurable loss gap between F and PB (15.4915.49 vs. 15.9115.91, 3 seeds, spread ±1.3± 1.3) — i.e. the benefit ceiling of any hybrid is still unknown. Two questions follow. (Q1) Is a schedule that keeps the cheap F-style topology pass for most blocks feasible, and what is the cheapest useful hybrid? (Q2) Can such a hybrid — one that records only a little state and compensates conservatively — reduce the topology gap to PB beyond what the plain F schedule already achieves? This paper answers both with a fixed-point formulation, proved certificates, an implementation, and a matched-conditions experimental study. 1.3 Contributions 1. A fixed-point formulation of basis selection: the compensated pipeline is xt+1=Φt(xt,(Bt))x^t+1= _t(x^t,G(B^t)) and the damped iteration on BtB^t has PB as its unique fixed point (Proposition 1); Pass-1 supplies the first iteration for free. 2. Exact non-flip certificates: 4Δ<mi4 <m_i guarantees edge-wise invariance of the kkNN set (Theorem 1); if only one endpoint moves, 2Δi<mi2 _i<m_i suffices (Theorem 2). The two constants give two different semantics for the same safety factor κi=σmi/2 _i=σ m_i/2. 3. Optimality of the replay gain: γ∗γ^* is the minimum-variance (Gauss–Markov/BLUE) linear combination of the frozen prior and the recorded predictor, with residual variance σd2σp2/(σd2+σp2) _d^2 _p^2/( _d^2+ _p^2); the feasibility crossover ρ∗≈0.25ρ^*\!≈\!0.25 is the σp=σd _p= _d boundary. 4. A trust-region view: the ℓ∞ _∞ box projection Πκ _κ is firmly non-expansive, so composing it with the damped iteration preserves convergence, and the projected step cannot leave the region in which the current graph is provably valid. 5. Quantified edge flapping: E[m]≈rk/(3k)E[m]≈ r_k/(3k) for a local 3D Poisson process, matching the measured margin median to 1%1\% (0.0780.078 vs. 0.07870.0787 Å) and showing that flapping is intrinsic to top-k graphs, not an implementation defect. 6. An implementation of three compensation levels with matched-conditions validation: 2 L_2 reproduces PB exactly (=0.0000D=0.0000), 1 L_1 is within 0.570.57–0.68%0.68\%, and 3 L_3 interpolates monotonically between them. 2 Logical roadmap and proof objectives 2.1 Central question and logical structure The questions (Q1)–(Q2) left by Section 1 are not whether a frozen graph is always identical to a graph rebuilt at every block. They are whether a cheap topology pass can record enough information to support conservative compensation towards the fully coupled reference PB, and under what conditions that compensation is both useful and inexpensive, i.e. when it reduces the topology gap to PB beyond what the plain F schedule already achieves. The argument therefore separates three issues: the mathematical target of refinement, the conditions under which the proposed updates are justified, and the experimental evidence that the implementation improves topology fidelity at an acceptable cost. The logical chain is identify the reference target⟶derive stability and replay conditions⟶construct the compensation levels⟶audit and measure the implementation⟶state the supported conclusion and its limits. gatheredidentify the reference target\; \;derive stability and replay conditions\; \;construct the compensation levels\\[3.0pt] \;audit and measure the implementation\; \;state the supported conclusion and its limits. gathered Section 3 develops the mathematical and algorithmic part of this chain; Section 4 tests its observable consequences; Section 5 combines the two. A proof of a local property, an approximation under a statistical model, and an observation on the recorded samples play different roles: none is used here as a substitute for the others. 2.2 Algorithm chapter: what must be established, and in what order Section 3.1: define a common problem before comparing schedules. The live coordinates xtx^t and the graph-construction basis BtB^t are separated in one closed-loop recursion. The frozen schedule F, the fully coupled schedule PB, and the refined schedule (this work) then become different choices within the same formulation. The margins mim_i, radii κi _i, and distances ,wD,D_w specify, respectively, when topology is fragile, how compensation is constrained, and what improvement means. This subsection supplies definitions rather than a separate convergence claim. Section 3.2: identify what refinement is trying to reach. Proposition 1 proves a consistency statement: if the damped coupling converges with λ>0λ>0, its limiting basis equals its limiting live coordinates, so the limiting recursion is exactly PB. Proposition 2 then states the local convergence argument under assumptions (A1)–(A3), while Proposition 3 describes the role of ties, path noise, and the risk set in the residual topology gap. The dependency is important: identifying the fixed-point target does not itself prove that every trajectory reaches it. The following two subsections supply the geometric and projection ingredients used by the conditional convergence argument. Section 3.3: determine when reusing topology is certified. The analysis first locates membership changes on equal-distance surfaces, explaining why smooth coordinate motion can produce a discrete edge change. Lemma 1 bounds distance perturbations; Theorems 1 and 2 then prove the sufficient non-flip conditions 4Δ<mi4 <m_i for two-sided motion and 2Δi<mi2 _i<m_i when only node i moves. These results explain the constant-edge-set premise in the local argument and motivate margin-based gating. Their purpose is to certify reuse where the inequalities hold, not to assert that all observed displacements satisfy them. This also introduces the distinction between preventing flips and tracking flips that actually occur. Section 3.4: decide how much to replay and how to constrain it. Under the stated drift-and-predictor covariance model, the minimum-variance linear replay gain in (13)–(15) answers how strongly the recorded displacement should influence the basis. The comparison with γ=1γ=1 identifies the σp=σd _p= _d boundary for naive replay versus freezing. The trust-region part answers a different question: how to constrain the proposed basis at low cost. Proposition 4 proves that the box projection is firmly non-expansive; the composition argument uses non-expansiveness together with q<1q<1 and a feasible reference fixed point. The subsequent analysis of flips, two-cycles, and chattering explains why a local contraction statement is not an unconditional global convergence guarantee across topology changes. Thus damping, replay gain, and projection have distinct logical jobs rather than being interchangeable tuning parameters. Section 3.5: quantify why the stability conditions can be restrictive. The local 3D Poisson model gives the spacing law [m]≈rk/(3k)E[m]≈ r_k/(3k), and the crossing-rate model relates flapping to trajectory speed and near-tie density. These are model-based quantitative explanations of why even small coordinate drift can compete with the neighbour margin. Their role is to motivate measurement and calibration of the gate, not to replace the exact certificates. The comparison with measured margins is made in Section 4.3. Sections 3.6 and 3.7: turn the preceding conditions into an executable schedule. The three levels act on different objects: 1 L_1 refreshes geometry on frozen edges, 2 L_2 repairs local neighbour sets, and 3 L_3 changes the basis through damped, projected replay. The procedures connect these choices to Pass-1 recording, offline planning, and Pass-2 consumption. Measured displacements, clamp flags, neighbour closure of the risk set, and the acceptance metrics make the earlier reasoning operational. The result of the chapter is therefore a specified family of algorithms with explicit assumptions and tests, not just an abstract fixed-point formulation. 2.3 Experimental chapter: which claims the measurements test Sections 4.1 and 4.2: establish that the comparison is meaningful, then test the implementation. The family-reconstruction audit checks that the offline graph builder reproduces the production edge sets, including tie-breaking. The matched-conditions protocol holds the sample, seed, MSA depth, and block schedule fixed. Only after these controls does the correctness study compare the compensation levels with PB. The observed exact agreement of 2 L_2, small residual of 1 L_1, and intermediate result of 3 L_3 are implementation evidence on the tested boundaries, rather than a universal theorem about all samples or trajectories. Section 4.3: check whether the theoretical regime describes the data. Sample characterisation separates non-degenerate structures from coincident-coordinate and complete-graph cases, then compares measured margins and drift with the certificates and the spacing law. This connects the mathematics to the actual operating regime: a certificate can be correct while its sufficient condition is too restrictive for a recorded trajectory. Section 4.4: test the cost side of the original question. Wall-time measurements identify the benefit of the recorded two-pass design, while the shared cost model distinguishes inexpensive frozen-edge geometry updates from local repair work that depends on |Ωt|| _t|. This separation prevents the measured two-pass speed-up from being read as an unconditional cost guarantee for every compensation setting. Section 4.5: test how much topology fidelity compensation can recover. Offline oracle replay evaluates the potential of compensation on recorded coordinates, and parameter sweeps expose the difference between a conservative stability radius and a radius large enough to track the observed drift. The synthetic predictor study then examines how that benefit depends on predictor quality and step saturation. These tests address the practical usefulness of the replay-and-gating mechanism; they do not assume that a noisy predictor inherits the oracle result. Section 4.6: delimit what the experiments establish. The discussion distinguishes the edge-weight proxy from loss sensitivity, same-family runtime comparisons from same-coordinate offline replay, and topology agreement from prediction accuracy. Sample degeneracy and trajectory divergence remain relevant limitations. In particular, the reported loss comparison does not resolve whether recovering the remaining topology gap improves the training objective. 2.4 Conclusion chapter: the final claim and its boundary Section 5 joins the two branches of the argument. The theoretical branch identifies PB as the fixed-point target of the damped coupling, supplies sufficient local non-flip certificates, derives the replay rule under its covariance model together with the σp=σd _p= _d feasibility boundary, quantifies edge flapping through an analytic near-tie spacing law that matches measurement to 1%1\%, and analyses projection and switching. The experimental branch establishes that the proposed levels are implemented and can improve measured topology fidelity, while the recorded two-pass trajectory provides the reported wall-time benefit. The final supported claim is therefore a theory-guided and experimentally validated family of topology-pass schedules, with identifiable conditions for stable reuse and useful compensation. It is not a proof that arbitrary discrete switching converges globally, that finite-pass compensation always reproduces PB, or that reducing D or wD_w necessarily reduces prediction loss. The logical endpoint is controlled topology fidelity together with measured computational benefit; the accuracy value of the recovered edges remains the separate question identified in the conclusion. 3 Algorithm: formulation, derivations and procedures 3.1 Notation and the closed loop Let xt∈ℝ3Lx^t ^3L collect the CαC_α coordinates at boundary t (t=0,…,Tt=0,…,T) and let Bt∈ℝ3LB^t ^3L be the topology basis, i.e. the coordinates used to build the graph at boundary t (BtB^t need not equal xtx^t). The block advance is Φt:xt+1=Φt(xt,(Bt)),t=0,…,T−1, _t: x^t+1= _t (x^t,\;G(B^t) ), t=0,…,T-1, (2) with per-step raw offsets utu_t and projection Π(u)i=clamp[−s,s]3(αui) (u)_i=clamp_[-s,s]^3(α u_i) (α=0.03α=0.03, s=3s=3 Å). For residue i let d(k)(i)d_(k)(i) be its k-th neighbour distance and define ρi=d(k)(i),mi=d(k+1)(i)−d(k)(i),κi=σmi2. _i=d_(k)(i), m_i=d_(k+1)(i)-d_(k)(i), _i=σ\, m_i2. (3) We call mim_i the margin of residue i: the gap between its k-th and (k+1)(k+1)-th neighbour, i.e. how far that residue is from a tie at the top-k boundary. A small margin means a fragile membership, and a flip — a change of membership — is exactly the event mi=0m_i=0. Equation (2) is a closed loop (coordinates determine the graph, the graph determines coordinates) with three instances: • F/topology pass: Bt≡x0B^t≡ x^0 (pure feed-forward, no feedback); • PB: Bt=xtB^t=x^t (fully coupled reference); • refined (this work): Bt=Πκ(x0+∑s<tγsΔs)B^t= _κ (x^0+ _s<t _s ^s ), an interpolation driven by measured Pass-1 displacements Δs=xs+1−xs ^s=x^s+1-x^s. Topology distance is measured by =1−|ℰ∩ℰ′||ℰ∪ℰ′|,w=∑e∈ℰΔℰ′we∑ewe,D=1- |E ||E |, _w= _e \, \,E w_e _ew_e, (4) with attention-mass proxy wew_e (in this work we=1/(1+‖de‖)w_e=1/(1+\|d_e\|); see §4.6). 3.2 Fixed-point structure Proposition 1. (Unique fixed point is PB) Consider the damped coupling (a fixed-point iteration in the classical sense, cf. [12]) B(n+1)t=(1−λ)B(n)t+λx(n)t,x(n)t+1=Φt(x(n)t,(B(n)t)).B^t_(n+1)=(1-λ)B^t_(n)+λ x^t_(n), x^t+1_(n)= _t (x^t_(n),G(B^t_(n)) ). (5) If (5) converges to (B¯,x¯)( B, x) with λ>0λ>0, then B¯t=x¯t B^t= x^t for all t, hence the limit is the PB trajectory. Proof. At the fixed point Bt=(1−λ)Bt+λxtB^t=(1-λ)B^t+λ x^t, so λ(Bt−xt)=0λ(B^t-x^t)=0 and, since λ>0λ>0, Bt=xtB^t=x^t; then (2) coincides with the PB recursion by definition. ∎ Proposition 1 fixes the meaning of “conservative compensation”: the basis is gradually pulled towards the live structure, and the only fixed point of that process is PB itself. Because Pass-1 already starts at B(0)t=x0B^t_(0)=x^0 and records Δs\ ^s\, one iteration comes for free. Proposition 2. (Bounded, monotone refinement) Assume (A1) Φt _t is LΦL_ -Lipschitz in x (LΦ≈1L_ ≈ 1 for residual blocks); (A2) inside the per-residue margin ball ‖Bit−xit‖<mi/2\|B^t_i-x^t_i\|<m_i/2 the edge set of i is constant, so the graph contributes no extra Lipschitz growth; (A3) Π is non-expansive. Then one full sweep is a contraction with modulus q=(1−λ)+λLΦq=(1-λ)+λ L_ and ‖z(n)−z∗‖≤qn‖z(0)−z∗‖\|z_(n)-z^*\|≤ q^n\|z_(0)-z^*\| for the trajectory z, hence convergence for λ<2/(1+LΦ)λ<2/(1+L_ ) — satisfied by any λ∈(0,1)λ∈(0,1) when LΦ≈1L_ ≈ 1. Proof (sketch). Compose the damped update (5) with Φt _t: the linear part contributes (1−λ)(1-λ) and the model part λLΦλ L_ ; the graph term is Lipschitz-free inside the margin ball by (A2), and Π does not increase distances by (A3). The composition is a contraction, so the Picard–Banach fixed-point principle [13] together with the linear-convergence argument of [14] yields the geometric bound. ∎ Where (A2) fails, G is discontinuous and the map is only piecewise contractive: a trajectory can cross the same flip surface twice (A→B→A→ B→ A), i.e. edge flapping, and chattering can destroy the fixed point. This is the reason κi=σmi/2 _i=σ m_i/2 is not optional: it confines every step to the region where the current graph is provably valid, so discrete changes are discovered rather than manufactured. Proposition 3. (What refinement can and cannot reach) The residual with respect to PB has exactly two sources: (i) genuine ties (mi≈0m_i≈ 0, unsolvable by any rule) and (i) pass-to-pass model/path noise. Hence wD_w has a floor and the attainable goal is w(refined)≤1.1w()D_w(refined)≤ 1.1\,D_w( PB), not 00. The provable upper bound, computable offline, is (refined vs. )≤∑i∈Ωk+|ties||ℰ|≈|Ω|L+|ties||ℰ|,D(refined vs.\ PB)\;≤\; _i∈ k+|ties||E|\;≈\; | |L+ |ties||E|, (6) i.e. only residues that left their margin ball can contribute. 3.3 Discrete topology and exact non-flip certificates Each edge indicator is 1[j∈Si(x)]=1[dij(x)≤di,(k)(x)]1[j∈ S_i(x)]=1[d_ij(x)≤ d_i,(k)(x)] with dij=‖pi−pj‖d_ij=\|p_i-p_j\|. The distance is smooth but the indicator jumps on the codimension-1 manifolds Σi,jl=x:dij=dil _i,jl=\x:d_ij=d_il\; in generic position these are smooth and of measure zero, yet trajectories cross them. Three elementary steps give the measure-zero statement. Smoothness and a local graph. Let gijl(x)=dij(x)−dil(x)g_ijl(x)=d_ij(x)-d_il(x); the norm is smooth off the origin, so gijlg_ijl is smooth wherever no two atoms coincide, and ∇gijl≠0∇ g_ijl≠ 0 on the surface. Fix a point with gijl(x∗)=0g_ijl(x^*)=0 and write x=(u,v)x=(u,v) with ∂gijl/∂v(x∗)≠0∂ g_ijl/∂ v(x^*)≠ 0; the map F(u,v)=(u,gijl(u,v))F(u,v)=(u,\,g_ijl(u,v)) then has detDF(x∗)≠0 DF(x^*)≠ 0, so the inverse function theorem [12] inverts it locally and the level set is locally a graph, Σi,jl∩U(x∗)=(u,φ(u)):u∈U′, _i,jl∩ U(x^*)\;=\; \\,(u, (u))\;:\;u∈ U \, (7) i.e. the surface is parametrised by 3L−13L-1 free coordinates, the remaining one being determined by φ . Countably many graphs suffice. The neighbourhood U(x∗)U(x^*) depends on x∗x^*, so read literally the construction gives uncountably many pieces. Since ℝ3LR^3L has a countable base of open balls Ok\O_k\, one can choose for every point of the surface a base element contained in its own neighbourhood, so that Σi,jl⊂⋃k∈K(u,φk(u)):u∈Uk,Kcountable, _i,jl\;⊂\; _k∈ K \\,(u, _k(u))\;:\;u∈ U_k \, K\ countable, (8) and ℝ3L=⋃kB¯(0,k)R^3L= _k B(0,k) reduces everything to a compact piece. Compactness and uniform continuity. On the compact piece Σi,jl∩B¯(0,k) _i,jl∩ B(0,k) finitely many of the graphs already cover, and each domain UℓU_ can be taken closed and bounded, hence compact; by Heine–Cantor φℓ _ is then uniformly continuous, ∀ε>0∃δ>0:‖u−u′‖∞<δ⟹|φℓ(u)−φℓ(u′)|≤ε.∀ >0\ \ ∃δ>0: \|u-u \|_∞<δ\ \ | _ (u)- _ (u )|≤ . (9) Partitioning UℓU_ into finitely many cubes of side <δ<δ places the graph inside finitely many slabs of thickness 2ε2 ; writing |⋅||·| for the 3L3L-dimensional volume (Lebesgue measure) of a set, |Σi,jl∩B|≤ 2ε|B|for every ε>0, | _i,jl∩ B |\;≤\;2 \,|B| every >0, (10) which forces |Σi,jl∩B|=0 | _i,jl∩ B |=0, i.e. the indicator vanishes almost everywhere, ∫ℝ3L1Σi,jl(x)x=0 _R^3L1_ _i,jl(x)\,dx=0; finite unions of such sets remain of measure zero.11 1 Classical ingredients: the finite-covering lemma (Borel–Lebesgue/Heine–Borel) and the Cantor–Heine theorem on uniform continuity ([12], Vol. I, secs. 2.3.2 and 4.2; countable base: ℚ3LQ^3L is countable, sec. 2.4.1); countable unions of measure-zero sets and the graph of a continuous function ([13], Vol. I, sec. 11.1.2, whose argument (10) reproduces). Almost every configuration therefore has all margins positive — the kkNN set is well defined — and a C1C^1 trajectory crosses the surfaces at isolated times only. Consequently the distributional derivative of the edge set vanishes off Σ : the gradient carries no information about who is a neighbour, and topology is a genuine discrete degree of freedom. 1 L_1–3 L_3 are therefore relaxations, not solutions. Order statistics are Lipschitz, |di,(k)(x+δ)−di,(k)(x)|≤2Δ|d_i,(k)(x+δ)-d_i,(k)(x)|≤ 2 with Δ=maxj‖δj‖ = _j\| _j\|, but mim_i — being a difference of two Lipschitz functions — has no positive lower bound, so flips are generic rather than a bug. Lemma 1. (Distance perturbation bound) |dij(x+δ)−dij(x)|≤‖δi‖+‖δj‖≤2Δ|d_ij(x+δ)-d_ij(x)|≤\| _i\|+\| _j\|≤ 2 . Proof. |‖a‖−‖b‖|≤‖a−b‖|\,\|a\|-\|b\|\,|≤\|a-b\| (reverse triangle inequality), and (pi−pj)−(qi−qj)=δj−δi(p_i-p_j)-(q_i-q_j)= _j- _i. ∎ Theorem 1. (Two-sided non-flip certificate) If 4Δ<mi4 <m_i then Si(x+δ)=Si(x)S_i(x+δ)=S_i(x) edge-wise. Proof. For members j∈Sij∈ S_i: dij′≤dij+2Δ≤di,(k)+2Δd _ij≤ d_ij+2 ≤ d_i,(k)+2 . For non-members l∉Sil∉ S_i: dil′≥dil−2Δ≥di,(k+1)−2Δ=di,(k)+mi−2Δd _il≥ d_il-2 ≥ d_i,(k+1)-2 =d_i,(k)+m_i-2 . If di,(k)+2Δ<di,(k)+mi−2Δd_i,(k)+2 <d_i,(k)+m_i-2 , every member is closer than every non-member, so the top-k set is unchanged; the inequality is equivalent to 4Δ<mi4 <m_i. ∎ Theorem 2. (One-sided certificate) If only node i moves, 2Δi<mi2 _i<m_i implies Si(x+δ)=Si(x)S_i(x+δ)=S_i(x). Proof. With δj≡0 _j≡ 0 for j≠ij≠ i, Lemma 1 gives |dij′−dij|≤Δi|d _ij-d_ij|≤ _i; replacing 2Δ2 by Δi _i in the previous argument yields the claim. ∎ The design parameter appears through κi=σmi/2 _i=σ m_i/2, which yields two different readings of the same knob: • stability (provable non-flip): the two-sided certificate needs κi≤mi/4 _i≤ m_i/4, i.e. σ≤1/2σ≤ 1/2 — this is the default of the design, and it is exactly the critical value; • tracking (reproducing real flips): needs κi≳ _i the actual drift, i.e. σ≈2σ≈ 2–44; no proof, but statistically optimal in the sense of §3.4. Both statements are consistent; they optimise different objectives (Proposition 2 versus Proposition 3). Finally, the hazard form of the certificates, Ωrisk(t)=i:mi(xt)≤4Δt _risk(t)=\i:m_i(x^t)≤ 4 _t\, with the number of flips Nflip≤|Ωrisk|N_flip≤| _risk|, identifies Ωt _t as an event-driven adaptive update set with cost O(|Ω|)O(| |). 3.4 Optimal replay, trust-region projection, and stability under flips Write the true drift as s=xt−x0s=x^t-x^0 (prior: s≈0s≈ 0 with covariance Σd _d) and the recorded predictor as p=z^−x0=s+ep= z-x^0=s+e, with E[e]=0E[e]=0, Cov(e)=ΣpCov(e)= _p and Cov(s,e)=0Cov(s,e)=0. For the linear estimator B−x0=γpB-x^0=γ p the error is ε(γ):=s−γp=(I−γ)s−γe (γ):=s-γ p=(I-γ)s-γ e, which is mean-zero (E[ε]=(I−γ)E[s]−γE[e]=0E[ ]=(I-γ)E[s]-γ E[e]=0), so its second moment is its covariance. Expanding the outer product and taking expectations term by term (constant matrices pass through E[⋅]E[·]), MSE(γ)=trE[εε⊤]=tr[(I−γ)Σd(I−γ)⊤+γΣpγ⊤],MSE(γ)\;=\;tr\,E [ ^\! ]\;=\;tr [(I-γ) _d(I-γ)^\! +γ _pγ^\! ], (11) where the two cross terms −(I−γ)E[se⊤]γ⊤-(I-γ)E[se ]γ and −γE[es⊤](I−γ)⊤-γ E[es ](I-γ) vanish by Cov(s,e)=0Cov(s,e)=0, and the trace turns the error covariance into a scalar (the total mean squared error over coordinates). Completing the square in γ, with W:=Σd+ΣpW:= _d+ _p, MSE(γ)=tr[(Σd−1+Σp−1)−1]+tr[(γ−γ∗)W(γ−γ∗)⊤],γ∗=ΣdW−1,MSE(γ)\;=\;tr [( _d^-1+ _p^-1)^-1 ]\;+\;tr [(γ-γ^*)W(γ-γ^*)^\! ], γ^*= _dW^-1, (12) so the second term is non-negative with curvature 2W≻02W 0 and vanishes at the unique minimiser γ=γ∗γ=γ^* (equivalently at the solution of ∂MSE/∂γ=0∂\,MSE/∂γ=0), while the minimum of (11) is tr[(Σd−1+Σp−1)−1]tr[( _d^-1+ _p^-1)^-1]. This is the Gauss–Markov/BLUE solution [15] γ∗=Σd(Σd+Σp)−1,Cov(xt−B)=(Σd−1+Σp−1)−1,γ^*= _d( _d+ _p)^-1, (x^t-B )= ( _d^-1+ _p^-1 )^-1, (13) Using I−γ∗=ΣpW−1I-γ^*= _pW^-1 together with the identity ΣpW−1Σd=ΣdW−1Σp=(Σd−1+Σp−1)−1=:P _pW^-1 _d= _dW^-1 _p=( _d^-1+ _p^-1)^-1=:P, the residual covariance evaluates to Cov(xt−B)=ΣpW−1P+ΣdW−1P=PCov(x^t-B)= _pW^-1P+ _dW^-1P=P, i.e. to the parenthesised second entry of (13). The minimiser is equivalently characterised by the orthogonality principle Cov(xt−B,p)=Σd−γ∗(Σd+Σp)=0,Cov (x^t-B,p )= _d-γ^*( _d+ _p)=0, (14) the normal equation of the minimisation. In the scalar case γ∗=σd2σd2+σp2,residual variance=σd2σp2σd2+σp2γ^*= _d^2 _d^2+ _p^2, variance= _d^2 _p^2 _d^2+ _p^2 (15) (the familiar parallel-resistance form). Three consequences matter in practice: (i) replay with γ∗γ^* is never worse than freezing, unless σp=∞ _p=∞; (i) naive replay (γ=1γ=1) beats freezing iff σp<σd _p< _d, i.e. iff γ∗>1/2γ^*>1/2 — the measured feasibility crossover ρ∗≈0.25ρ^*\!≈\!0.25 between predictor quality and ground truth is exactly the σp=σd _p= _d boundary; (i) κ must be calibrated to the predictor error, which is the quantitative explanation of the empirically optimal σ≈2σ≈ 2–44. 3.41 Which part of the LM method is used Of the Levenberg–Marquardt (LM) trust-region method we use one facet only: the projection onto the trust region. The rest of the machinery is neither used nor needed — no normal-equation solve (J⊤J+μI)p=−J⊤r(J J+μ I)p=-J r, no QR factorization of an augmented least-squares problem, and no root-finding for a damping parameter — because the residual of our problem is the discrete edge set 1[dij≤di,(k)]1[d_ij≤ d_i,(k)], which admits no Jacobian. 3.42 The projection, and which symbol is the output The projection used by the refined basis is Πκ(u)=argmin‖y−x0‖∞≤κ12‖y−u‖2=clamp(u,x0±κ), _κ(u)\;=\; *arg\,min_\|y-x^0\|_∞≤κ 12\|y-u\|^2\;=\;clamp (u,\,x^0±κ ), (16) where the input u=x0+∑s<tγsΔsu=x^0+ _s<t _s ^s is the unconstrained replayed position and is never modified, the optimisation variable is y, and the output is the minimiser of (16) — the new basis BtB^t, not u. Coordinate-wise that output is yi=minmaxui,xi0−κi,xi0+κiy_i= \ \u_i,\,x^0_i- _i\,\,x^0_i+ _i\: a coordinate passes through unchanged when uiu_i already lies in [xi0−κi,xi0+κi][x^0_i- _i,x^0_i+ _i], and is otherwise clipped onto a face of the box, which is the regime Ωt=i:‖Bit−xi0‖≥κi _t=\i:\|B^t_i-x^0_i\|≥ _i\ that triggers local repair. The constraint uses the ℓ∞ _∞ norm while the objective uses the ℓ2 _2 norm; this is not a typo: the shape of the feasible set is dictated by the per-residue certificates of Theorems 1 and 2, whereas a projection is by definition the closest point in the Euclidean sense. 3.43 From the ellipsoidal region of LM to the box LM admits a scaled region, minp12‖Jkp+rk‖22subject to‖Dkp‖≤Δk, _p 12\|J_kp+r_k\|_2^2 to \|D_kp\|≤ _k, (17) with Dk=diag(di)D_k=diag(d_i), di>0d_i>0, because least-squares variables can differ by many orders of magnitude; the lengths of the principal axes are tied to the typical values of the corresponding variables [14]. Writing di=1/κid_i=1/ _i and squaring the constraint, ∥Dkp∥2≤Δk⟺∑i(pi/κi)2≤Δk2,\|D_kp\|_2≤ _k _i (p_i/ _i )^2≤ _k^2, (18) so the semi-axis along coordinate i is Δkκi _k _i: the diagonal of DkD_k is the reciprocal of the per-coordinate radius. Our design folds the scalar radius into the radii (Δk≡1 _k≡ 1), giving κi=σmi/2 _i=σ m_i/2; with σ≤1/2σ≤ 1/2 this box is precisely the ball of Theorem 1. The two regions are in fact members of one family: ∥Dkv∥p≤1\\|D_kv\|_p≤ 1\ is an ellipsoid with semi-axes κi _i at p=2p=2, and since ‖v‖∞=limp→∞‖v‖p\|v\|_∞= _p→∞\|v\|_p it converges to the box |vi|≤κi\|v_i|≤ _i\ as p→∞p→∞. We use that end of the family, for two reasons. Tightness. The certificate is per residue, so the feasible set is a product of intervals. An ℓ2 _2 ellipsoid is a strict subset of that box — also safe, but strictly more conservative, since it forbids the simultaneous use of each coordinate’s own budget; note that the two regions agree per coordinate, as a single coordinate can reach κi _i in either. The box therefore uses exactly what the certificate permits, and it makes “which residues were clipped” a well-defined set. Cost. Projection onto a box is the closed-form clamp in (16) (O(L)O(L), no iteration). Projection onto an ellipsoid has no closed form: it requires a one-dimensional root-find for μ≥0μ≥ 0 with p(μ)=(I+μDk⊤Dk)−1u,∥Dkp(μ)∥2=Δk(μ=0 if u is already feasible),p(μ)=(I+μ D_k D_k)^-1u, \|D_k\,p(μ)\|_2= _k (μ=0 if $u$ is already feasible), (19) the secular equation — the same inner iteration that LM performs on its damping parameter in order to meet a prescribed radius. The box projection removes that iterate entirely, which is why it is the form we use. 3.44 Non-expansiveness, and composition with the contraction A map T is non-expansive (NE) if ‖Tu−Tv‖≤‖u−v‖\|Tu-Tv\|≤\|u-v\|, and firmly non-expansive (FNE) if ⟨Tu−Tv,u−v⟩≥∥Tu−Tv∥2⟺∥Tu−Tv∥2≤∥u−v∥2−∥(u−Tu)−(v−Tv)∥2, Tu-Tv,\,u-v ≥\|Tu-Tv\|^2 \|Tu-Tv\|^2≤\|u-v\|^2- \|(u-Tu)-(v-Tv) \|^2, (20) equivalently 2T−I2T-I is non-expansive; FNE implies NE. Proposition 4. (The box projection is firmly non-expansive) The map Πκ _κ of (16) is firmly non-expansive. Proof. The box is a product of intervals and Πκ _κ acts coordinate-wise, so it suffices to prove the scalar statement for one clamp P(t)=minmaxt,a,bP(t)= \ \t,a\,b\, a<ba<b, and to sum the scalar inequalities over coordinates. If t1,t2t_1,t_2 are both interior, P is the identity on them so both sides equal (t1−t2)2(t_1-t_2)^2. If t1>bt_1>b and t2∈[a,b]t_2∈[a,b], then Pt1=bPt_1=b and Pt2=t2Pt_2=t_2, hence ⟨Pt1−Pt2,t1−t2⟩=(b−t2)(t1−t2)≥(b−t2)2=|Pt1−Pt2|2 Pt_1-Pt_2,t_1-t_2 =(b-t_2)(t_1-t_2)≥(b-t_2)^2=|Pt_1-Pt_2|^2 because t1≥bt_1≥ b. If t1>bt_1>b and t2<at_2<a, both sides reduce to t1−t2≥b−at_1-t_2≥ b-a. If both arguments are clipped onto the same endpoint, both sides vanish; the remaining cases are symmetric. ∎ Composing the projection with the damped update of Proposition 2 leaves the contraction modulus unchanged, because only non-expansiveness is needed (the requirement stated in (A3), verified for Πκ _κ in Proposition 4): ‖zn+1−z∗‖=‖Πκ(Fzn)−Πκ(Fz∗)‖≤‖Fzn−Fz∗‖≤q‖zn−z∗‖,q=(1−λ)+λLΦ<1.\|z_n+1-z^*\|\;=\; \| _κ(Fz_n)- _κ(Fz^*) \|\;≤\;\|Fz_n-Fz^*\|\;≤\;q\,\|z_n-z^*\|, q=(1-λ)+λ L_ <1. (21) Two remarks complete the argument. First, the step above presupposes that the fixed point is feasible, and it is: the box is centred at x0x^0, hence Πκ(x0)=x0 _κ(x^0)=x^0, which is what makes the projection compatible with Proposition 1. Second, firmness is what keeps the constraint from generating oscillations of its own: on a face two distinct points can be mapped to the same point, and by (20) the difference of their displacements is annihilated as well, so the projection cannot manufacture the chattering discussed below. We close this subsection with the discrete side. A flip is a change of membership in a residue’s kkNN set (equivalently, as noted above, the crossing of a surface mi(x)=0\m_i(x)=0\), and we call its size the number of residues whose neighbour lists change. Let J=∂Φt/∂xJ=∂ _t/∂ x be the block Jacobian of the sweep at a fixed graph and let Mλ=(1−λ)I+λJM_λ=(1-λ)\,I+λ\,J (22) be the corresponding linearised one-sweep operator, i.e. the linearisation of the damped coupling (5) about a trajectory; the spectral radius ρ(Mλ)<1ρ(M_λ)<1 is the linear case of Proposition 2. A single flip is a sparse perturbation: it changes only those rows of J (hence of MλM_λ) that belong to the residues whose neighbour lists changed. It therefore replaces MλM_λ by Mλ+EM_λ+E with E non-zero on a few rows only, and the resulting eigenvalue drift is governed by the Bauer–Fike theorem [16]: every eigenvalue μ of Mλ+EM_λ+E satisfies mini|μ−μi(Mλ)|≤κp(X)‖E‖p _i|μ- _i(M_λ)|≤ _p(X)\|E\|_p, where κp(X)=‖X‖p‖X−1‖p _p(X)=\|X\|_p\|X^-1\|_p is the condition number of the eigenvector matrix X of MλM_λ and p∈1,2,∞p∈\1,2,∞\; when MλM_λ is symmetric this factor is one and the bound reduces to Weyl’s. A single flip is therefore harmless as long as the spectrum keeps a margin below one. Repetition is different. The sweep is not a single map: (B)G(B) is piecewise constant in the coordinates, so on each cell CG=x:(B(x))=GC_G=\x:G(B(x))=G\ the sweep acts as one fixed map ΨG _G, and it is discontinuous across cells. Inside a cell, (A2)–(A3) make ΨG _G a contraction with modulus q=(1−λ)+λLΦ<1q=(1-λ)+λ L_ <1, but globally there is no single q<1q<1: this is what piecewise contractive means. If a trajectory crosses the same flip surface repeatedly (A→B→A→ B→ A, edge flapping), the sweep alternates between two contractions ΨA,ΨB _A, _B whose fixed points differ, pA≠pBp_A≠ p_B. The two-step map ΨA∘ΨB _A _B is still a contraction, so the orbit does converge — but to a fixed point of neither: the even and odd iterates approach the two ends of a two-cycle, whose amplitude is a=‖p∗−ΨB(p∗)‖,a\;=\;\|p^*- _B(p^*)\|, (23) where p∗p^* is the fixed point of ΨA∘ΨB _A _B; for affine scalar maps ΨA(x)=qx+(1−q)pA _A(x)=qx+(1-q)p_A, ΨB(x)=qx+(1−q)pB _B(x)=qx+(1-q)p_B it evaluates to a=1−q1+q‖pA−pB‖a= 1-q1+q\|p_A-p_B\|. The orbit therefore never settles. Since the even and odd iterates have different limits, limn→∞x2n=p∗≠ΨB(p∗)=limn→∞x2n+1,lim supn→∞‖xn+1−xn‖≥a> 0, _n→∞x_2n=p^*\;≠\; _B(p^*)= _n→∞x_2n+1, _n→∞\|x_n+1-x_n\|\;≥\;a\;>\;0, (24) so the sequence xn\x_n\ has no limit and its ω-limit set is the two-point set p∗,ΨB(p∗)\p^*, _B(p^*)\: a residual that never vanishes. This is the behaviour we call chattering. The picture degrades further if crossing times accumulate, i.e. if ∃T∗<∞:t1≤t2≤⋯,tk→T∗,N(t):=cardk:tk≤t⟶∞(t→T∗),∃\,T^*<∞: t_1≤ t_2≤·s, t_k→ T^*, N(t):=card\\,k:t_k≤ t\,\\; \;∞ (t→ T^*), (25) equivalently if the switching intensity ∫|x˙|ρnear-tie(0,x)t | x|\, _near-tie(0;x)\,dt is not integrable on [0,T∗][0,T^*]. This is Zeno behaviour: what fails is precisely the hypothesis that crossing times are isolated, on which the piecewise argument rests. The design limits this rather than removing it: the projected step keeps the trajectory inside the cell in which the current graph is provably valid, so flips are discovered rather than manufactured ((16) with κi=σmi/2 _i=σ m_i/2, σ≤1/2σ≤ 1/2); a flip that does occur is repaired locally by the gate rather than by a global rebuild; and 3 L_3 follows the flipped graph instead of holding the stale one. Flapping itself cannot be eliminated, being intrinsic to top-k graphs, so damping, gating and tracking are the only responses available — and the quantitative question becomes how often flips occur, which is answered next. 3.5 How large is a flip in practice? Near-tie density controls everything. Modelling neighbours locally as a 3D Poisson process of density ρ, the k-th neighbour sits at the radius fixed by k=43πρrk3,dkdr=4πρr2,k= 43πρ r_k^3, dkdr=4πρ r^2, (26) so the expected spacing of the k-th and (k+1)(k+1)-th neighbour is the reciprocal intensity E[m]≈14πρrk2=rk3k.E[m]≈ 14πρ r_k^2= r_k3k. (27) A flip occurs exactly when a margin vanishes, i.e. on the surfaces mi(x)=0\m_i(x)=0\ where the k-th and (k+1)(k+1)-th neighbour of i exchange order. Here x(t)x(t) is the numerical trajectory — the interpolation of the boundary states xtx^t recorded by the run — i.e. a deterministic curve, not a sample path of a stochastic process; accordingly the [⋅]E[·] of (28) and (33) is never taken over a distribution for x: it averages the displacement (margin) field — equivalently the run-to-run ensemble — at a fixed curve x. For a smooth x(t)x(t) the expected number N of such crossings over an interval obeys the Kac–Rice form [19], [N]≈∫|x˙(t)|ρnear-tie(0,x(t))t,E[N]\;≈\; | x(t)|\; _near-tie (0;x(t) )\,dt, (28) i.e. the flapping rate is proportional to trajectory speed times the near-tie density ρnear-tie(0,⋅) _near-tie(0;·), the density of the margins mim_i at level zero, and the Poisson model above is its local form. At k=64k=64 (rk≈15r_k≈ 15 Å) the predicted near-tie spacing is 0.0780.078 Å, within 1%1\% of the measured margin median 0.07870.0787 Å (Table 3). Three corollaries follow: larger k makes flips more likely (m∝1/km 1/k); for k≥L−1k≥ L-1 (e.g. L=51L=51) the graph becomes complete and “margin” degenerates, so near-complete-graph samples cannot be used to study stability; and flapping is intrinsic to top-k graphs, hence can only be damped, gated or tracked, not eliminated. 3.51 Local Poisson model and the spacing law Fix a residue and model the positions of its neighbours, in coordinates centred on it, as a homogeneous Poisson point process (P) of intensity ρ in ℝ3R^3 (a number density; the symbol ρi _i of (3) denotes a neighbour radius, not a density): the counts in disjoint Borel sets are independent Poisson variables whose means are ρ times the volume [17], so the number N(r)N(r) of points within radius r satisfies [N(r)]=43πρr3,d[N(r)]dr=4πρr2.E [N(r) ]= 43πρ r^3, d\,E[N(r)]dr=4πρ r^2. (29) Defining rkr_k by [N(rk)]=kE[N(r_k)]=k gives ρ=3k/(4πrk3)ρ=3k/(4π r_k^3) and hence a radial rate 4πρrk2=3k/rk4πρ r_k^2=3k/r_k. The radii |xj|\|x_j|\ themselves form a P on (0,∞)(0,∞) with intensity λ(r)=4πρr2λ(r)=4πρ r^2 (the image of a P under a measurable map is again a P, carrying the push-forward intensity [17]), so conditioned on a point at radius r the residual gap rk+1−rkr_k+1-r_k is exponential with that local rate: Pr(rk+1−rk>t∣rk=r)=exp[−4πρ3((r+t)3−r3)],[m∣rk=r]=1+O(εr)4πρr2, (r_k+1-r_k>t r_k=r )= [- 4πρ3 ((r+t)^3-r^3 ) ], [m r_k=r]= 1+O( _r)4πρ r^2, (30) where εr=(4πρr3)−1=1/(3k) _r=(4πρ r^3)^-1=1/(3k) measures how much the rate changes over one gap: at k=64k=64, rk≈15r_k≈ 15 Å this is εr≈0.005 _r≈ 0.005, i.e. the local rate 4πρr24πρ r^2 changes by ≈2/(3k)≈1%≈ 2/(3k)≈ 1\% across one gap. Substituting the radial rate into (30) gives (27), [m]≈(4πρrk2)−1=rk/(3k)E[m]≈(4πρ r_k^2)^-1=r_k/(3k): the mean spacing is the reciprocal of the local rate. The exact moments give the same value, [rk]=(34πρ)1/3Γ(k+13)Γ(k),[rk+1]−[rk]≈d[rk]dk≈rk3k,E[r_k]= ( 34πρ )^\!1/3 (k+ 13 ) (k), [r_k+1]-E[r_k]≈ d\,E[r_k]dk≈ r_k3k, (31) so (27) is exact to O(1/k)O(1/k). 3.52 Crossing counts: the Kac–Rice form A flip of residue i is exactly a crossing of the level set Σi=mi=0 _i=\m_i=0\ by the trajectory, and 00 is a regular value of mim_i wherever the k-th and (k+1)(k+1)-th neighbour are distinct, so Σi _i is a C1C^1 hypersurface by the implicit function theorem. The coarea formula, written here in its distributional (level-set) form, ℋn−1⌞f=u=|∇f|δ(f−u)dxH^n-1 \f=u\=|∇ f|\,δ(f-u)\,dx [18], writes the level set as a volume integral; its one-dimensional case, applied along the curve t↦x(t)t x(t), counts crossings, and only the velocity component normal to the level set contributes: cardt∈[0,T]:mi(x(t))=0=∫0T|⟨∇mi(x(t)),x˙(t)⟩|δ(mi(x(t)))t.card \t∈[0,T]:m_i(x(t))=0 \= _0^T | ∇ m_i(x(t)), x(t) |\,δ (m_i(x(t)) )\,dt. (32) Taking expectations and exchanging dtdt with E (Tonelli), and assuming the trajectory direction to be independent of ∇mi∇ m_i and locally isotropic, gives [N]=∫0T|x˙(t)|ρnear-tie(0,x(t))t,ρnear-tie(0,x):=∑i[|∂nmi(x)|δ(mi(x))],E[N]= _0^T| x(t)|\; _near-tie (0;x(t) )\,dt, _near-tie(0;x):= _iE [| _nm_i(x)|\,δ(m_i(x)) ], (33) where ∂n _n is the derivative along x˙ x; the approximation sign in (28) therefore refers only to the local model for ρnear-tie(0) _near-tie(0) used next. This is the classical Kac–Rice formula for the expected number of level crossings in its total-crossings form [19], which is why no factor 1/21/2 appears in (28). Its hypotheses are the standard ones — a.s. C1C^1 paths, transversal crossings, no multiple or accumulating zeros [19] — but they are not empirical claims about x: because x(t)x(t) is a numerical trajectory interpolated between finitely many recorded boundaries, C1C^1 smoothness and the absence of accumulating zeros are properties of the interpolation by construction, and all expectations are taken over the margin field at that fixed curve. Only transversality is substantive — a margin that grazes zero without changing sign is not a flip, and the “no multiple zeros” clause of [19] is what excludes it. 3.53 Local density and a numerical check In the local Poisson model the margin is an exponential gap, so its density at level zero is the reciprocal mean spacing, ρnear-tie(0)≈∑i1[mi]=∑i3krk=∑i4πρrk2; _near-tie(0)≈ _i 1E[m_i]= _i 3kr_k= _i4πρ r_k^2; (34) this is the “local form” referred to above, and it is the same coarea-plus-conditional-density structure that yields the expected level-set measure of a stationary isotropic Gaussian field, ν(u)=p(u)|∇g|ν(u)=p(u)\,E|∇ g| [19]. Numerically, k=64k=64 and rk≈15r_k≈ 15 Å give ρ=3k/(4πrk3)≈4.5×10−3ρ=3k/(4π r_k^3)≈ 4.5× 10^-3 Å-3 — consistent with a 142142-residue chain of radius of gyration ≈20≈ 20 Å — and 4πρrk2≈12.84πρ r_k^2≈ 12.8 Å-1, so the model predicts [m]≈0.078E[m]≈ 0.078 Å against the measured margin median 0.07870.0787 Å of Table 3. The same density predicts the topological churn as well: taking the measured per-boundary step of 0.020.02–0.060.06 Å as input, the model predicts 0.260.26–0.780.78 crossings per residue per boundary, whereas the measured churn corresponds to ≈0.64≈ 0.64 crossings per residue per boundary (≈1%≈ 1\% of the ≈k≈ k out-edges of a residue are replaced). The two agree to within the accuracy of the local model, with no free parameter. 3.6 Three compensation levels All levels share one update at boundary t, t=(Et,D(Bt),W(Bt)),Et=(E(x0)∖rowsΩt(x0))∪⋃i∈Ωt(i,j):j∈kNNk(i,Ci,Bt),G_t= (E_t,\;D(B^t),\;W(B^t) ), E_t= (E(x^0) _ _t(x^0) )\;∪\!\! _i∈ _t\!\! \(i,j):\ j _k\! (i;\,C_i;\,B^t ) \, (35) where rowsΩt(x0)rows_ _t(x^0) are the outgoing rows of the active residues, rebuilt rather than added to E(x0)E(x^0) (rows outside Ωt _t are copied verbatim, so no stale edge survives); the k neighbours of i are searched inside the local candidate set Ci=E(x0)i∪j:j∈kNN(x0)l,l∈kNN(x0)iC_i=E(x^0)_i∪\j:j (x^0)_l,\ l (x^0)_i\ evaluated on the basis BtB^t, sequence-local edges (|i−j|<kmin|i-j|<k_ ) are kept as in (x0)G(x^0), and Ωt=i:‖Bit−xi0‖≥κi _t=\i:\|B^t_i-x^0_i\|≥ _i\. TABLE 1: What each level touches in (35); costs from the shared cost model. Level Modified object Cost / property 1 L_1 only geometry: D(Bt),W(Bt)D(B^t),W(B^t) (edge set stays E(x0)E(x^0)) O(|ℰ|)≈7O(|E|)≈ 7k ⇒ free; removes the 0.290.29–0.40.4 Å geometry term 2 L_2 edge set locally: EtE_t in (35), evaluated inside CiC_i O(∑i∈Ω|Ci|log|Ci|)O( _i∈ |C_i| |C_i|) with |Ci|≤1+k+k2|C_i|≤ 1+k+k^2 (two-hop); Ω closed under frozen neighbours 3 L_3 the basis Bt=Πκ(x0+∑s<tγsΔs)B^t= _κ(x^0+ _s<t _s ^s) (predictor only) storage O(LT)O(LT) (T=8T=8: ≈2.5≈ 2.5k floats); needs σp<σd _p\!<\! _d 4 L_4 full rebuild every boundary ( PB) 1.81.8 s/block; reference Two practical rules complete the design. First, the recorded deltas must be measured displacements, never a rate: the per-block step clamps at exactly 3.0003.000 Å on most boundaries, so extrapolating a saturated step is invalid. Second, Ω must be closed under neighbours in both directions, Ω←Ω∪NE(x0)(Ω) ← ∪ N_E(x^0)( ): both endpoints of every frozen edge are tested, i.e. if either end of (i,j)∈E(x0)(i,j)∈ E(x^0) is in Ω then so is the other (one sweep of the edge list suffices). Otherwise a residue can flip because its neighbour moved and the repair misses it (offline: without closure 2 L_2 scores 0.08860.0886 against 3 L_3’s 0.06700.0670 at σ=4σ=4). 3.7 Procedures Algorithm 1–3 give the complete procedure. Only the graph-construction step of Pass-2 differs from the current pipeline. Algorithm 1 Pass-1: record (existing pass + a few arrays) 1: x←x0x← x^0 2: for t=0,…,T−1t=0,…,T-1 do 3: Gt←(x)G_t (x) ⊳ frozen behaviour, B≡x0B≡ x^0 4: if t=0t=0 then record mi,ρim_i, _i from G0G_0 ⊳ 1 float per residue 5: end if 6: (xnew,ot)←Φt(x,Gt)(x_new,o_t)← _t(x,G_t) ⊳ new coords and block outputs oto_t 7: Δt←xnew−x ^t← x_new-x; clampit←(|Δit|≈s)clamp^t_i←(| ^t_i|≈ s) 8: record Δt ^t, clamp flags; x←xnewx← x_new 9: end for Algorithm 2 Offline plan (cheap; may be run once after the first epoch) 1: c0←x0c^0← x^0; cs+1←cs+Δsc^s+1← c^s+ ^s ⊳ replay trajectory 2: for t=1,…,Tt=1,…,T do 3: Bt←Πκ(x0+∑s<tγsΔs)B^t← _κ (x^0+ _s<t _s ^s ) ⊳ 3 L_3, κi=σmi/2 _i=σ m_i/2 4: skip γ-extrapolation for clamp-flagged residues 5: Ωt←i:‖Bit−xi0‖≥κi _t←\i:\|B^t_i-x^0_i\|≥ _i\ 6: Ωt←Ωt∪j∈E(x0)i,i∈Ωt _t← _t∪\j∈ E(x^0)_i,\ i∈ _t\ ⊳ neighbour closure 7: plan←ti∈Ωt:Ci_t←\i∈ _t:\ C_i\ ⊳ 2 L_2 local re-selection plan 8: end for Algorithm 3 Pass-2: consume (only the graph step changes) 1: x←x0x← x^0 2: for t=0,…,T−1t=0,…,T-1 do 3: Bt←B^t← basis from plan (default x0x^0) 4: Et←E(x0)E_t← E(x^0) ⊳ L0 edge set 5: if 2 L_2 enabled then Et←Et∪⋃i∈ΩtkNNk(i,Bt)E_t← E_t∪ _i∈ _tkNN_k(i;B^t) inside CiC_i 6: end if 7: Gt←(Et,D(Bt),W(Bt))G_t←(E_t,D(B^t),W(B^t)) ⊳ 1 L_1 live geometry 8: x←Φt(x,Gt)x← _t(x,G_t) 9: end for 10: acceptance: t=1−J(Et,Et)D_t=1-J(E_t,E PB_t), wD_w; require t(refined)<t(plain)D_t(refined)<D_t(plain) and w≤1.1w()D_w≤ 1.1\,D_w( PB) 3.71 Hyper-parameters They are fixed by a 1-D offline sweep, without training: start at σ=1/2σ=1/2, sweep γ∈0,14,12,34,1γ∈\0, 14, 12, 34,1\ and keep the best wD_w, then sweep σ∈14,12,1σ∈\ 14, 12,1\ (the basis-iteration damping λ=1/2λ=1/2 is only needed if the iteration (5) is actually run; Pass-1 already supplies iteration 00). 3.72 Falsifiable predictions (1) wD_w decreases with γ and then flattens at the tie/noise floor; if it does not, (A1) fails and one should fall back to 1 L_1 only. (2) |Ωt|| _t| shrinks with t as the trajectory converges (cross-epoch topology is ≈95%≈ 95\% stable), so 2 L_2’s cost decays during training. (3) If w(1+2)≈w()D_w($ L_1$+$ L_2$) _w( PB), the topology question is closed and only accuracy (E2, below) remains. 4 Experiments 4.1 Protocol and implementation status All three levels are implemented in the C++ runtime behind one switch (PPML_SE3_TOPO=fixed|per_block|l1|l2|l3) with two hyper-parameter knobs, PPML_TOPO_SIGMA (σ) and PPML_TOPO_GAMMA (γ); the graph builder constructs a host-side basis Bt=x0+clamp(γ(xt−x0),±κ)B^t=x^0+clamp(γ(x^t-x^0),±κ) only for 3 L_3 and passes it to the edge-geometry kernel, so 1 L_1/2 L_2/3 L_3 are bit-identical to the previous behaviour when disabled. Two measurement rules turned out to be essential. • Family-reconstruction audit. Before any comparison, the graph builder is re-implemented offline and required to reproduce the recorded dumps exactly. Agreement initially stalled at 44%44\% because the production builder sorts candidate neighbours by the pair (distance,index)(distance,index), so ties are broken by ascending index, whereas a naive argsort picks tied members arbitrarily. With the lexicographic rule (and the production constants k=min(64,L−1)k= (64,L-1), kmin=9k_ =9, float32 distances) the audit passes on all 18 recorded boundaries (Table 2). • Matched-conditions (MC) protocol. Comparing two topology modes across two different trajectories is meaningless: an unseeded reference inflated the measured gap of 1 L_1 by a factor of 22–33 (=0.0103/0.0194D=0.0103/0.0194 instead of 0.0057/0.00680.0057/0.0068). Every comparison below therefore keeps seed, MSA depth, block schedule and sample fixed and switches only the topology mode. TABLE 2: Correctness verification. Top: family-reconstruction audit (offline rebuild of make_graph from the recorded coordinates must reproduce the dumped edge set exactly). Bottom: runtime verification in the matched-conditions protocol (P69905, L=142L=142, seed 1234, MSA depth 3, 3 boundaries; D measured against a PB reference of the same family). Family L boundaries |ℰ||E| Jaccard P62805 (degenerate) 103 6/6 7105–7107 1.0000 P62891 51 6/6 2550 (ident.) 1.0000 P69905 142 3/3 9089 =64L+1=64L+1 1.0000 P62241 208 3/3 13588, 13595 1.0000 Runtime mode D (t=1t=1 / t=2t=2) J(t,frozen)J(t,frozen) 1 L_1 (frozen set, live geometry) 0.0057 / 0.0068 1.0000 3 L_3 (σ=4σ=4, γ=0.5γ=0.5) 0.0044 / 0.0059 0.9985 / 0.9980 2 L_2 (σ=0.5σ=0.5 and 44) 0.0000 / 0.0000 0.9943 / 0.9932 3 L_3 γ=0γ=0 (anchor) == 1 L_1 — 3 L_3 κ→∞κ→∞ (anchor) == 1 L_1 — 4.2 Correctness Table 2 collects the verification. The audit establishes that the topology question is being measured against a faithful reconstruction rather than against a re-implementation artefact. The runtime block then shows a clean ordering: frozen edges (1 L_1) miss 0.570.57–0.68%0.68\% of the PB edge set, the damped projected basis of 3 L_3 with γ=1/2γ=1/2 reduces this to 0.440.44–0.59%0.59\% (a strict interpolation between 1 L_1 and PB), and 2 L_2 reproduces the PB edge set exactly (=0D=0 at both boundaries, |ℰ||E| identical, J=1.0000J=1.0000). The two degenerate anchors behave as the theory requires: γ=0γ=0 (basis =x0=x^0) is identical to 1 L_1, and κ→∞κ→∞ (empty risk set) is identical to 1 L_1 as well. This is the strongest available statement of implementation correctness: the compensated pipeline collapses onto PB when the trust region is removed, and onto the frozen pipeline when the compensation is switched off. 4.3 Data characterisation and drift Table 3 characterises the three samples. Here drift means the displacement of the recorded coordinates at a boundary relative to the basis x0x^0; it matters because the certificate 4Δ<mi4 <m_i is evaluated on the same Δ , so the measured drift supplies the empirical value of Δ and decides whether 4Δ<mi4 <m_i can hold at all. The L=103L=103 sample is a degenerate limiting case: its mapping covers only 24/10324/103 residues, 86.8%86.8\% of point pairs coincide, all 103103 margins are exactly zero, and its mean CαC_α separation is 1.631.63 Å, so its topology is governed by tie noise and its absolute geometry is unreliable. The two well-mapped samples are non-degenerate (≤0.02%≤ 0.02\% coincident pairs, margin medians 0.07870.0787 and 0.1180.118 Å, no zero margins). Their coordinate drift is heavy-tailed and small in the bulk: at the last recorded boundary the mean displacement over residues is 0.040.04–0.070.07 Å while the CαC_α RMSD is 0.390.39–0.480.48 Å (Table 3), so a few residues carry most of the motion. The drift is nevertheless not small relative to the certificate: since 4Δ<mi4 <m_i must hold per residue, the certificate would require margins of order 4Δ≈0.154 ≈ 0.15–0.270.27 Å, i.e. 1.91.9–3.4×3.4× the median margin of P69905 (0.07870.0787 Å) and 1.31.3–2.3×2.3× that of P62241 (0.1180.118 Å). The two ratios are lower bounds, and by a wide margin: the certificate’s Δ is the per-boundary maximum maxj‖δj‖ _j\| _j\|, whereas the values substituted above are means over residues. That maximum is itself set by the ±3± 3 Å step clamp rather than by the drift — on P69905 it is 5.1965.196 Å=33=3 3, the corner of the projection box, and identical at both recorded boundaries — so the mean understates the violation by about two orders of magnitude (78×78×–113×113× on the two samples). The sufficient condition therefore fails on essentially every residue, consistent with the ≈1%≈ 1\% (unweighted) edge churn between PB and F. TABLE 3: Sample characterisation. Collisions: fraction of point pairs closer than 10−310^-3 Å. Drift: CαC_α RMSD of the last recorded boundary against x0x^0 (in parentheses: mean |Δd|| d| over that boundary’s own edge set). Sample L collisions mean CαC_α dist. margin median drift (Å) P69905 142 0.01%0.01\% 19.219.2 Å 0.07870.0787 Å 0.4790.479 (0.0740.074) P62241 208 0.014%0.014\% 29.629.6 Å 0.1180.118 Å 0.3870.387 (0.0430.043) P62805 (deg.) 103 86.8%86.8\% 1.631.63 Å ≡0≡ 0 (103/103) 1.0261.026 (0.2630.263) For k=64k=64 and rk≈15r_k≈ 15 Å the spacing law (27) predicts a median near-tie spacing of 0.0780.078 Å, against a measured margin median of 0.07870.0787 Å on P69905 — a 1%1\% agreement. Two consequences follow. First, the model is accurate enough to be used quantitatively. Second, the spacing itself is far below the scale of the motion: the typical margin (≈0.08≈ 0.08 Å) is comparable to the mean displacement relative to x0x^0 (0.020.02–0.070.07 Å), so margins inevitably cross zero and flips are intrinsic to top-k graphs rather than an implementation defect — which is why PB and F disagree on ≈1%≈ 1\% (unweighted) of edges. 4.4 Optimisations Table 4 lists the measured optimisations. The free component is the recording that Pass-1 adds on top of the frozen-graph pass that the pipeline already runs: one margin mim_i and one kNN radius ρi _i per residue, once at t=0t=0, plus the measured displacement Δt ^t and the step-clamp flags at each boundary — O(LT)O(LT) storage, ≈2.5≈ 2.5k floats at T=8T=8. No graph is rebuilt, no neighbour is searched, and no computed quantity changes, so the recording itself costs neither measurable time nor accuracy. What it buys is the removal of the per-boundary kNN rebuild from the critical path, because the basis no longer has to wait for the live coordinates of the previous block: an epoch falls from 34.134.1 s ( PB) to 19.419.4 s (two-pass), a factor 1.761.76, the recording pass itself was measured at 4.254.25 s against 6.406.40 s for the previous implementation of the same pass (1.5×1.5×), and the loss stays inside the seed spread (15.9115.91 vs. 15.4915.49, ±1.3± 1.3). Within the compensation family the costs separate by orders of magnitude, and each one can be derived. Write L for the residues, T=8T=8 for the boundaries, k=min(64,L−1)k= (64,L-1) for the neighbour count and |ℰ|≈Lk≈7|E|≈ Lk≈ 7k for the directed edge set. The rebuild that the design removes (4= L_4= PB) searches k neighbours for each of the L residues at every boundary, i.e. O(L)O(L) distance evaluations per residue, O(L2)O(L^2) per boundary and TO(L2)T\,O(L^2) per epoch, measured as 1.81.8 s per block. The three compensation levels avoid that search in different ways. 1 L_1 recomputes only the edge distances and weights on the frozen edge index, O(|ℰ|)=O(Lk)O(|E|)=O(Lk) operations per boundary; the SE3 layers need those relative positions anyway, so the level is free, and it removes the 0.290.29–0.40.4 Å error of a stale basis geometry. 2 L_2 keeps the only surviving neighbour search and confines it to the active residues: each i∈Ωti∈ _t is re-searched inside its two-hop candidate set CiC_i, |Ci|≤1+k+k2|C_i|≤ 1+k+k^2, which costs O(|Ci|log|Ci|)O(|C_i| |C_i|) per residue — O(|Ci|logk)O(|C_i| k) with the partial_sort of the implementation — and O(∑i∈Ωt|Ci|log|Ci|)O( _i∈ _t|C_i| |C_i|) per boundary, worst case O(|Ωt|k2logk)O(| _t|\,k^2 k); at L=103L=103 the two-hop set already covers most of the chain, so the practical cost is nearer O(|Ωt|LlogL)O(| _t|\,L L). 3 L_3 adds the basis alone, an elementwise damp-and-clamp of x0+∑s<tγsΔsx^0+ _s<t _s ^s, at O(L)O(L) per boundary and O(LT)O(LT) storage, and no further search. The ladder is therefore O(Lk)≪O(L)≪O(|Ωt|k2logk)≪TO(L2)O(Lk) O(L) O(| _t|\,k^2 k) T\,O(L^2), and 2 L_2/3 L_3 stay cost-neutral exactly while the risk set Ωt _t stays small — the quantity that the procedures subsection predicts shrinks with t. TABLE 4: Measured optimisations (from the original pipeline and from this implementation). Costs follow the shared cost model: full-set refresh (kNN rebuild) ≈1.8≈ 1.8 s/block, frozen-set geometry refresh O(|ℰ|)O(|E|) (free), local rebuild O(∑i∈Ω|Ci|log|Ci|)O( _i∈ |C_i| |C_i|) with |Ci|≤1+k+k2|C_i|≤ 1+k+k^2 (the implementation uses partial_sort, i.e. O(∑i∈Ω|Ci|logk)O( _i∈ |C_i| k)), extra pass ≈8≈ 8–1111 s/epoch. Item baseline optimised epoch wall time (two-pass vs PB) 34.134.1 s 19.419.4 s (1.76×1.76×) Pass-1 stage — 1.5×1.5× loss ( F vs PB, 3 seeds, ±1.3± 1.3) 15.9115.91 15.4915.49 kNN rebuild per block 1.81.8 s 1 L_1: 00; 2 L_2/3 L_3: O(|Ω|)O(| |) geometry refresh (frozen set) O(|ℰ|)O(|E|) free (held live) 4.5 Topology-gap reduction Table 5 reports the offline (oracle) replay sweep: the basis is the live trajectory (γ=1γ=1), so the numbers are an upper bound on what compensation can achieve. Two conclusions are stable across samples. First, the default σ=1/2σ=1/2 is too tight: since κi=σmi/2 _i=σ m_i/2, it fixes the trust radius at the two-sided non-flip threshold of Theorem 1, κmed=mmed/4 _med=m_med/4 — so the gate is saturated on 94%94\% (P69905) and 46%46\% (P62241) of the residues (|Ω|=133/142| |=133/142 and 96/20896/208 at t=1t=1, Table 5), and only 99–11%11\% of wD_w is removed. Second, wD_w improves monotonically with σ up to −61%-61\% (P69905, σ=8σ=8) and −41%-41\% (P62241, σ=8σ=8), i.e. the design should calibrate κ to the motion (margin) scale rather than fix σ. The limit σ→∞σ→∞, γ=1γ=1 gives =0D=0 exactly, which is the oracle anchor used in Table 2. A synthetic study confirms the mechanism and locates its feasibility boundary: with predictor/ground-truth correlation ρ, replay hurts for ρ≲0.25ρ 0.25 (i.e. σp>σd _p> _d, consistent with (15)) and helps by 1515–27%27\% for ρ≥0.5ρ≥ 0.5; at a saturated step amplitude (33 Å/block) the benefit collapses to 6%6\%. 4.51 How to read the wD_w rows All reductions are measured against the frozen baseline (γ=0γ=0, B≡x0B≡ x^0): reduction(σ)=1−w(σ)/w(frozen)reduction(σ)=1-D_w(σ)/D_w(frozen) is the fraction of the frozen-vs- PB weighted topology gap that is closed, so −41%-41\% means 41%41\% closed and 59%59\% left (the sign is a direction convention only). Likewise, “wD_w decreases” means that the refined edge set comes closer to PB in weighted mass ((4)), not that the graph becomes sparser. The κmed _med column converts the dimensionless σ into Å: since κi=σmi/2 _i=σ m_i/2, an entry equals σ times half the margin median of the sample (0.07870.0787 Å for P69905 and 0.1180.118 Å for P62241, Table 3), so σ=1σ=1 permits a median step of 0.0390.039 Å and the five-point sweep runs from κmed=mmed/4 _med=m_med/4 (σ=1/2σ=1/2) to 4mmed4m_med (σ=8σ=8). The default σ=1/2σ=1/2 therefore confines the median residue to a quarter of the margin that separates its k-th from its (k+1)(k+1)-th neighbour; read together with the |Ω|| | column, which counts the residues whose displacement reaches the trust radius, this is the quantitative content of “σ=1/2σ=1/2 is too tight” and of the margin-based calibration advocated above. TABLE 5: Offline oracle replay (γ=1γ=1) on both samples: weighted topology distance wD_w against the PB edge set at the same boundary, and its reduction against the frozen basis (γ=0γ=0, B≡x0B≡ x^0). |Ω|| | is the risk-set size at t=1t=1 for both samples, out of L residues (identically 00 when B≡x0B≡ x^0); the five-point σ sweep is complete on both. Sample setting κmed _med (Å) wD_w reduction |Ω|| | P69905 frozen (γ=0γ=0) — 0.0578 — 0 P69905 σ=0.5σ=0.5 0.020 0.0515 −11%-11\% 133 P69905 σ=1σ=1 0.039 0.0458 −21%-21\% 116 P69905 σ=2σ=2 0.079 0.0401 −31%-31\% 80 P69905 σ=4σ=4 0.157 0.0274 −53%-53\% 47 P69905 σ=8σ=8 0.315 0.0227 −61%-61\% 26 P62241 frozen (γ=0γ=0) — 0.0355 — 0 P62241 σ=0.5σ=0.5 0.030 0.0321 −9%-9\% 96 P62241 σ=1σ=1 0.059 0.0294 −17%-17\% 56 P62241 σ=2σ=2 0.118 0.0287 −19%-19\% 40 P62241 σ=4σ=4 0.236 0.0247 −30%-30\% 22 P62241 σ=8σ=8 0.473 0.0208 −41%-41\% 15 4.6 Discussion (i) Metric proxy: we=1/(1+‖de‖)w_e=1/(1+\|d_e\|) approximates the attention mass; replacing it by the loss sensitivity |L(G∪e)−L(G)||L(G∪ e)-L(G)| would make wD_w directly comparable to training metrics. (i) Sample degeneracy: one widely used sample maps 24/10324/103 residues, which makes margins identically zero and inflates all topology distances; conclusions about margins and absolute geometry are drawn only from the two well-mapped samples. (i) Complete-graph identity: for L≤65L≤ 65, k=min(64,L−1)k= (64,L-1) makes the graph complete, so J≡1J≡ 1 is an identity and such samples cannot test stability. (iv) Trajectory coupling: runtime D is measured against a PB reference of the same family, but each mode integrates its own trajectory, so D mixes topology error with trajectory divergence; the offline replay on recorded coordinates (Table 5) is the clean instrument. (v) Accuracy gap unknown: the existing F vs PB comparison has a seed spread (±1.3± 1.3 on a loss of ≈15.5≈ 15.5) as large as the effect, so the benefit ceiling of the whole line of work is still unmeasured. (vi) Sampling of crossings: x(t)x(t) is recorded only at the T block boundaries, so the crossings counted in (28) are a boundary-sampled statistic — a margin that is crossed and crossed back inside one block leaves no trace — and the measured churn is therefore a lower bound on the true flip rate (the near-tie model of §3.5 predicts 0.260.26–0.780.78 crossings per residue per boundary against a sampled ≈0.64≈ 0.64, consistent with this sampling bias). 5 Conclusion We formulated topology-pass refinement as a damped fixed-point problem with a unique PB fixed point, proved exact non-flip certificates with two constants (mi/4m_i/4 two-sided, mi/2m_i/2 one-sided), derived the minimum-variance replay gain and the σp=σd _p= _d feasibility boundary, and quantified edge flapping through an analytic near-tie spacing law that matches measurement to 1%1\%. The implementation reproduces PB exactly when the trust region is removed and reduces to the frozen pipeline when compensation is disabled, giving a continuous and verifiable family of schedules. The compensation ladder works as a fidelity mechanism (1 L_1 within 0.6%0.6\%, 3 L_3 interpolating, 2 L_2 exact), and the recorded two-pass trajectory delivers a real 1.76×1.76× epoch speed-up with no measurable loss gap. 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