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Executing Causal Structure Learning with Linear-Attention Transformers

Amartya Roy, Sayar Karmakar

arXiv:2610.10395v1cs.LG

TL;DR

The paper asks whether a transformer can execute a continuous causal-discovery solver, which must enforce acyclicity while updating a candidate graph. It constructs a fixed-weight executor that carries the graph and multiplier between updates; experiments show faithful solver replay, but not reliable execution by ordinary attention models or guaranteed causal recovery.

  • Problem

    Executing a continuous causal-discovery solver requires combining an acyclicity constraint, a sparse proximal step, an allowed-edge mask, and multiplier state across updates.

    Source: §1 · p. 1

  • Method

    The paper constructs a fixed-weight transformer that encodes the current graph and multiplier and executes the complete Poly-prox update for supplied controls.

    Source: §1 · p. 2, §4 · p. 3, §3 · p. 3

  • Results

    The constructed block agrees with a reference update to floating-point precision and reproduces its outcomes, including errors, on synthetic data and seven benchmark topologies; ordinary attention models tested do not reliably execute or transfer to larger graphs.

    Source: §1 · p. 2

  • Takeaways & Limitations

    Accurate execution of the solver is distinct from accurate causal recovery because replay also reproduces the solver’s statistical and optimization errors.

    Source: §1 · p. 2

    The update-count guarantee requires contraction, a region the iterates cannot leave, and a verified error bound; it excludes the adaptive controller and does not itself ensure acyclicity or graph recovery.

    Source: §9 · p. 10

Abstract

from arXiv · show

Transformers can execute algorithms on data given in their input. We ask whether they can do the same for causal discovery. We study a standard continuous method that repeatedly updates a candidate causal graph while enforcing acyclicity. We explicitly construct a fixed-weight transformer whose forward pass exactly reproduces one update of this method, so repeated blocks reproduce its optimization trajectory. The transformer carries the current graph and the algorithm's multiplier between updates. We show that retaining the multiplier is essential for exact execution, since different multiplier values can lead to different next updates. We also give conditions under which, within a fixed stage, the number of updates needed to reach a target accuracy can be computed in advance and rounding errors stay bounded as depth grows. Experiments show that the constructed block agrees with a reference update to floating-point precision, while arithmetic replay on synthetic data and seven published benchmark network topologies inherits the reference solver's successes and failures. This separates accurate algorithm execution from accurate causal recovery. In contrast, the ordinary attention models tested under our training budgets do not reliably execute the update or transfer to larger graphs. Whether gradient training can learn an executor in the architecture class of the construction remains open.

1 Introduction

The paper asks whether a fixed-weight transformer can execute a continuous causal-discovery solver, whose updates combine an acyclicity constraint with graph and multiplier state. It constructs such an executor and tests its agreement with the solver, distinguishing faithful execution from successful causal recovery.

1 Introduction

  • A causal-discovery executor must compute a matrix-power constraint and sparse proximal step while preserving the multiplier needed by later updates.The multiplier tightens the acyclicity penalty; the allowed-edge mask also constrains the graph.Source: §1 · p. 1
  • The fixed-weight block encodes both the weighted adjacency matrix W and multiplier α, then maps that state to the next solver state.The penalty schedule and stopping rule remain external.Source: §1 · p. 2
  • One shared construction executes an update for graphs up to pmax variables in depth O(log pmax) and width O(pmax).The multiplier is carried between blocks; losing it can change the next update.Source: §1 · p. 2
  • The constructed block matches a reference update to floating-point precision, while replay reproduces the reference solver’s outcomes—including its mistakes—on synthetic data and seven benchmark topologies.Ordinary attention models tested under the stated budgets do not reliably execute the update or transfer to larger graphs; learning an executor in the construction’s architecture class remains open.Source: §1 · p. 2

2 Related Work

Prior work constructs transformers that execute specified computations, while other work predicts or learns causal structure. The paper situates its contribution as executing a complete continuous causal-discovery update and analyzing its requirements.

2 Related Work

  • Constructed transformers have implemented gradient and second-order updates, proximal methods, and looped programs and graph algorithms.The paper extends this execution line to the complete Poly-prox update and its depth, memory, and numerical-accuracy requirements.Source: §2 · p. 2
  • Learning latent causal structure and amortized graph prediction target different tasks from executing a specified numerical update.Source: §2 · p. 2
  • Continuous DAG methods include NOTEARS’ matrix-exponential constraint, polynomial constraints with the same acyclic zeros, and alternative formulations such as DAGMA, GOLEM, and SDCD.Exact execution still inherits the solver’s statistical and optimization limitations.Source: §2 · p. 3

3 The Causal-Discovery Algorithm

The section specifies a masked least-squares objective with a polynomial acyclicity constraint and a Poly-prox augmented-Lagrangian update. It also describes how feasible points affect regularity and KKT conditions.

3 The Causal-Discovery Algorithm

  • The solver minimizes masked least squares over W supported on the allowed-edge mask, with a zero diagonal and edge weights Wij representing i → j.With equal noise variances, the population least-squares loss over acyclic supports has W† as its unique minimizer.Source: §3 · p. 3
  • For N ≥ p, the polynomial constraint h(W) vanishes exactly on acyclic support graphs, and the solver uses a NOTEARS-style augmented Lagrangian with a proximal-gradient primal step.The fixed schedules for ρℓ and γℓ and the control bℓ are externally supplied; Poly-prox denotes this reference implementation.Source: §3 · p. 3
  • The update combines the least-squares gradient, the acyclicity-constraint gradient, and entrywise soft-thresholding restricted to allowed edges.The gradients of the loss and constraint are built from matrix powers; the Lasso proximal map is M ⊙ softγλ(W).Source: §3 · p. 3
  • At every feasible point, the equality-constraint regularity condition fails, and a feasible point is KKT exactly when it minimizes the convex masked Lasso problem.Source: §3 · p. 3

4 How the Transformer Executes an Update

The construction encodes the solver’s current graph and multiplier in persistent transformer state, then implements its update with linear attention and bilinear/ReLU operations. Its exact update has logarithmic depth, and repeated blocks reproduce the solver trajectory.

4 How the Transformer Executes an Update

  • The prompt encodes one row token per variable and a control token; persistent registers carry W and α across iterations, while scratch registers reset.Source: §4 · p. 3
  • One block uses 2⌈log2 pmax⌉ + 8 layers and one weight set for every p ≤ pmax to reproduce the Poly-prox update exactly in real arithmetic.The block returns the same (Wℓ+1, αℓ+1) as the reference iteration.Source: §4 · p. 4
  • Repeated squaring computes the constraint and its gradient in s shared stages, with each doubling implemented by an attention layer.The remaining gradient, soft-threshold, and second constraint evaluation use a fixed number of layers.Source: §4 · p. 4
  • Repeated blocks reproduce the first L reference iterations at depth L(2s + 8), with each block’s computed multiplier available to the next.The outside schedule supplies each iteration’s controls (ρℓ, γℓ, bℓ).Source: §4 · p. 4
  • Uniform exact execution of the specified fixed-N update requires Θ(log pmax) depth in this primitive class with affine encoding and readout.The bound is for the exponent-N recurrence, not every DAG solver.Source: §4 · p. 5

5 Memory and Numerical Accuracy

Exact execution requires retaining the solver’s multiplier alongside the graph, while numerical guarantees depend on explicit assumptions about error propagation and arithmetic. The analysis also distinguishes state error from graph error and qualifies where its bounds apply.

5 Memory and Numerical Accuracy

  • Near an acyclic state, multiplier errors can persist as state error because nothing pulls α back; whether they reach W depends on the next stage.The example has state error growing to Lε while the graph remains at the origin.Source: §5 · p. 5
  • If a trajectory approaches a feasible point that is not a masked-Lasso solution while becoming stage-stationary, its effective multiplier must diverge in magnitude.The effective multiplier is c_k = α_k + ρ_k h(W_k).Source: §5 · pp. 5–6
  • Under the stated relative-error assumptions, assembled-gradient error is at most (2ν + ν^2)C_k, even as the effective multiplier grows.The covariance is assumed exact; the relative bounds are implementation hypotheses.Source: §5 · p. 6
  • The general growth bound need not be tight, and finite-trajectory stationarity diagnostics do not establish its asymptotic premises.A two-node family has c_t ≍ t^-3.Source: §5 · p. 6
  • Propagating rounding-error bounds through both squaring passes gives a uniform block bound, with enclosures tending to zero as arithmetic and conversion errors vanish for fixed input bounds.These bounds depend on the specified arithmetic model and implementation choices.Source: §5 · p. 6
  • If an encoder merges two multiplier values, some decoded primal prediction has error at least gJ|α1 − α2|/2; discarding α is the K = 1 case.The bound concerns retained information, not a particular register layout.Source: §5 · p. 6

6 How Many Updates Are Enough?

Within a fixed penalty stage, contraction and bounded-error conditions yield a sufficient update count and corresponding execution depth. The practical tests show that these conditions do not hold across all examined regions.

6 How Many Updates Are Enough?

  • When contraction, invariant-region, and uniform rounding-error conditions hold, a sufficient update count gives execution depth DblkLtol, apart from initial encoding cost.Each update block has Dblk = 2⌈log2 pmax⌉ + 8 layers.Source: §6 · pp. 6–7
  • Uniform Hessian bounds with µ > 0 give contraction for 0 < γ < 2/LJ; choosing γ = 2/(LJ + µ) gives q = (LJ − µ)/(LJ + µ).The curvature bounds are required throughout the convex region.Source: §6 · p. 7
  • Global, local, and interval-arithmetic tests can check curvature bounds; a two-node example passes the local test even when the global bound fails.The local bounds are uniform on C when µloc > 0.Source: §6 · p. 7
  • If the primal map contracts on C and decoded transitions stay in C with uniformly bounded error, the trajectories remain in C and the exact map has a unique fixed point there.The stated theorem also requires d + ε ≤ (1 − q)r.Source: §6 · p. 7
  • None of the three tests passed on 270 balls around 90 saved states; after further objective optimization, the interval test passed at 46/90 centres, including 25 saved stage endpoints.The bound excludes earlier iterations and the additional optimization.Source: §6 · p. 7

7 When Accurate Updates Recover the Graph

Graph recovery depends on computation, optimization, and statistical error together, not on accurate execution alone. The benchmark replay matches the solver’s thresholded graphs, including its mistakes.

7 When Accurate Updates Recover the Graph

  • Across benchmark topologies, the transformer and reference solver have no thresholded-graph disagreements; black marks are empty in every row.Figure 3 depicts a median-SHD draw at m = 104 and thresholds the final graphs at 0.3.Source: Figure 3 · p. 8
  • If total error δ = ∆L + aL + ζm is below βmin/2, thresholding within [δ, βmin − δ) recovers the true edge set exactly.Here βmin is the weakest edge weight of a nonempty true graph; ζm can include regularization bias that persists with infinite data.Source: §7 · p. 8
  • Exact execution removes computation error, but optimization and statistical error can still prevent graph recovery.The solver’s output is therefore kept distinct from the true graph when assessing replay.Source: §7 · pp. 7–8

8 Experiments

The experiments test solver recovery, exact execution, numerical bounds, and whether training can learn the update. The appendices prove properties of the update and construction, while the discussion limits the guarantees to stated conditions and leaves learned execution open.

8 Experiments

  • Maximum relative discrepancy was 3.0 · 10^-15 in W and 1.1 · 10^-13 in α across 254 finite-reference cases; 220 replay datasets matched reference graphs exactly.Replay also reproduced the solver’s mistakes. At p = 5, smaller-corpus state-input models scored one-step errors of 0.94 ± 0.00 (softmax) and 0.98 ± 0.01 (linear); transfer to p = 10 failed, and predicting from scratch raised errors.Source: §8 · pp. 9–10

9 Discussion and Conclusion

  • Update-count guarantees require contraction, a region the iterates stay within, and a verified error bound; they do not ensure acyclicity or graph recovery.The adaptive controller is excluded, and learning an executor in the full constructed architecture remains open.Source: §9 · p. 10

A Proofs and Derivative Bounds

A.1 Proof of Proposition A.1

  • The least-squares gradient is Σ(W − I), and the masked Lasso proximal map applies entrywise soft-thresholding only to allowed edges.Source: §A.1 · p. 13

A.2 Proof of Lemma A.2

  • The constraint gradient vanishes exactly on DAGs, but a feasible point need not be a KKT point when masked-Lasso stationarity fails.Source: §A.2 · p. 13

A.3 Proof of Theorem 4.1

  • A single block executes the Poly-prox update exactly for every p ≤ pmax, using shared weights and depth 2⌈log2 pmax⌉ + 8.Its output encoding is a valid input to the next block.Source: §A.3 · pp. 13–14

A.4 Proof of Theorem 5.1

  • Decoded state error follows eℓ+1 ≤ εℓ + κℓeℓ, with each step’s error bounded by its transition error plus the Lipschitz-scaled prior error.Source: §A.4 · p. 14

A.5 Proof of Lemma A.3

  • When the constraint gradient is nonzero on an edge and thresholding does not erase the difference, distinct multiplier values produce distinct next updates.An exact deterministic executor must retain information distinguishing those multiplier values.Source: §A.5 · p. 14

A.6 Proof of Proposition 5.3

This proof establishes bounds and conditions for stationarity, convergence, graph recovery, and contraction, and specifies how fixed-weight transformer operations store and update the algorithm’s state. It details the register layout and routing used to implement matrix and scalar computations.

A.6 Proof of Proposition 5.3

  • A feasible limit of approximately stationary iterates that is not a masked-Lasso minimizer forces the effective multiplier magnitude to diverge at least inversely with distance to the limit.A two-node example realizes exact stationarity with c_t ≍ t^-3, showing that growth can be faster than the general inverse-distance lower bound.Source: §A.6 · p. 15

A.7 Proof of Corollary 5.4

  • The computed gradient error is bounded by (2ν + ν^2)C_k under the corollary’s hypotheses, while relative accuracy of h alone does not guarantee relative accuracy of c = α + ρh.Signed α can cancel with ρh; with exact c and absolute gradient error at most ϵ, the resulting bound is |c|ϵ, which can be amplified as |c| grows.Source: §A.7 · p. 15

A.8 Proof of Theorem 6.1

  • If P is q-Lipschitz on a closed ball and d + ε ≤ (1 − q)r, exact and approximate updates remain in the ball and P has a unique fixed point there.The theorem also bounds iteration error geometrically and gives a residual certificate; the fixed point may be cyclic, and the update bound assumes fixed controls.Source: §A.8 · p. 16

A.9 Proof of Proposition 7.1

  • Graph thresholding recovers the exact support when δ < β_min/2 and δ ≤ τ < β_min − δ, using the strict rule |cW_L,ij| > τ.Within a verified stage, the stated error and reference-distance conditions provide such a thresholding guarantee.Source: §A.9 · p. 16

A.10 Derivative Bounds and Convergence Within a Stage

  • On bounded masked domains, derivative bounds yield Lipschitz masked gradients, with constraint-related bounds of orders R^4, R^3, R^2, and R at the origin, uniformly in N ≥ 2.Strong monotonicity and gradient Lipschitzness give contraction for 0 < γ < 2μ/L_J^2; uniform Hessian bounds provide a sharper spectral-factor condition.Source: §A.10 · pp. 17–18
  • A local Hessian certificate gives q = 0.7 for a radius-0.01 neighborhood of a cyclic fixed point, but does not guarantee graph recovery.In the example, μ = 0.42, L_J = 2.38, γ = 5/7, and the center displacement is zero.Source: §A.10 · p. 20

B Transformer Construction and Computational Cost

B.1 Where the State Is Stored

  • The state uses P = p_max and dimension d = 13P + 18, with 13 wide registers for stored values.The listed registers include the mask, current graph, covariance, and intermediate quantities.Source: §B.1 · p. 20
  • Variable rows store graph-state data and separate broadcast scalars, while a control row stores the five inputs; this prevents resets from deleting input controls.The first seven scalar coordinates hold flags and controls; remaining scalar coordinates and wide scratch registers start at zero.Source: §B.1 · p. 20

B.2 Attention and Feedforward Weights

  • Fixed attention-head patterns implement matrix products and transposes, route values to multiple registers, and reduce per-variable scalars to the control row or broadcast control scalars.The construction uses identity and selector registers, with control-row identity zero and a padded variable-row identity.Source: §B.2 · p. 20
  • Feedforward weights implement linear increments, routed coordinate products, residual overwrites, and soft-thresholding using ReLU units.The linear increment construction needs at most 2d ReLU units; thresholding writes to a register that is zero on entry to that layer.Source: §B.2 · pp. 20–21

B.3 What Each Layer Computes

The indexed schedule computes each Poly-prox update through two polynomial-constraint passes, gradient and threshold operations, and multiplier writeback. Its state-preservation rules keep the next block’s input clean while bounding depth, width, and storage.

B.3 What Each Layer Computes

  • The schedule initializes constraint products, computes gradients and thresholds, then updates the primal graph and multiplier across two constraint-evaluation passes.It uses eight single layers and two s-layer passes; all heads read the same old state, and each feedforward branch uses already-formed inputs.Source: §B.3 · p. 21

B.4 State Preservation, Width and Depth

  • Masked primal registers preserve forbidden-edge and padding zeros, while the final layer clears scratch and writes the next W and α state.The allowed-edge mask is needed for persistent and masked primal registers, not for scratch matrices such as S, Q, SW, or ST.Source: §B.4 · p. 21
  • The schedule has depth O(log P), width O(P), at most four heads per layer, and O(P^3 log P) arithmetic per update for P ≥2.Streaming activations uses O(P^2) storage; separately materializing dense untied parameters takes O(P^2 log P), versus O(P log P) nonzero selector coefficients.Source: §B.4 · pp. 21–22, §B.5 · p. 22

C Why Exact Execution Needs Logarithmic Depth

For a specified class of linear-attention and bilinear/ReLU networks, polynomial degree growth limits exact execution depth. A constructed two-variable input family yields a logarithmic lower bound for full-state execution, with explicit scope exclusions.

C Why Exact Execution Needs Logarithmic Depth

  • The hard family yields a primal-output degree of 4N −1 and a full-state multiplier-output degree of 2N(4N −1).Here N = 2⌈log2 pmax⌉; exactness on an open interval forces polynomial identity with the constructed outputs.Source: C Why Exact Execution Needs… · p. 22
  • Exact full-state execution requires depth of order log pmax in the stated architecture class, matching the construction’s asymptotic depth order.The lower bound follows from a polynomial degree bound of 6D for depth D and a hard family whose multiplier output has degree 2N(4N −1).Source: C Why Exact Execution Needs… · p. 22
  • The lower bound does not cover every two-variable DAG solver, approximation, finite-test-set exactness, normalized attention, or uncounted nonlinear preprocessing or readout.The hard family uses the specified exponent N; for N = 2 and λ = 0, the stated outputs are w2 = −t7/2 and α+ = t28/32.Source: C Why Exact Execution Needs… · p. 22

D Error from Losing Multiplier Information

The next primal update depends on the multiplier, and the paper bounds the error incurred when multiplier information is compressed or lost. It also specifies an arithmetic model and input conditions for bounding rounding error through the constructed block.

D Error from Losing Multiplier Information

  • On a fixed active branch, multiplier changes produce a proportional primal-update difference; an encoder merging distinct multipliers cannot predict both updates arbitrarily accurately.For an encoder with at most K values over an interval of length A, the bound requires A ≤ 2Kε/gJ; equal-bin midpoint decoding attains the lower bound when all nonzero-gradient entries stay on fixed active branches.Source: D Error from Losing Multiplier… · p. 23

E.1 Arithmetic and inputs

  • The rounding analysis assumes scalar operations satisfy a relative-plus-absolute error model, with finite intermediates and specified handling of underflow and matrix products.The stored controls require λ, ρ ≥0, γ > 0, and a nonnegative threshold; altered kernels, casts, or reassociation need a new bound.Source: §E.1 · p. 23
  • To compare one update at its current decoded W, that W is the nominal input; deviations from an earlier exact trajectory are handled separately.Bounds assume clean scratch and include covariance or control conversion when intended values differ from stored values.Source: E Rounding Bounds for the Constructed… · p. 23, §E.1 · p. 24

E.2 Bounding the Error in Each Layer

The paper propagates operation-level error bounds through the literal transformer computation and states conditions for selecting precision and update counts within a fixed stage. Its tests show close update agreement and bounded fixed-control continuations, while recovery results remain those of the reference solver.

E.2 Bounding the Error in Each Layer

  • The operation-level enclosure propagates magnitude and error bounds through products, sums, entrywise products, ReLU, and the recorded layer graph.It also covers residual additions and the two constraint passes; uniform certificates require uniform input bounds and outward rounding or exact bound computation.Source: §E.2 · p. 24

E.3 Replacing Stored Values and Recomputing the Constraint

  • Residual writeback cannot generally be bounded by relative error on the new value alone, so the analysis carries absolute initialization error through later operations.This matters when overwriting W, initializing the second-pass S, and replacing diagonal scratch; the first-pass relative model cannot simply be restarted.Source: §E.3 · p. 25

E.4 Choosing Precision for a Target Accuracy

  • The precision test requires εW(u, ω) ≤ (1 − q)r − d and εW(u, ω) < (1 − q)tol; then the error floor is εW/(1 − q).Under the stated invariant-ball and contraction conditions, the sufficient update count is 2⌈log2 pmax⌉+ 8 transformer layers per update; no fixed dtype is guaranteed for every input or stage.Source: §E.4 · p. 25

F.1 Single Updates and Repeated Execution

  • The indexed block’s 50 one-block comparisons had maximum Frobenius W error 9.56 · 10−17 and zero multiplier error; three-block reuse also remained close.With explicit final-scratch reset, three-block maximum errors were 1.52 · 10−16 in W and 1.53 · 10−14 in α, unlike direct reuse that left contaminating scratch.Source: §F.1 · p. 26

F.2 Rounding Errors at Different Precisions

  • In 135 serial-arithmetic cases, maximum W errors were 7.03 · 10−4, 1.68 · 10−7, and 2.14 · 10−16 for float16, float32, and float64.The uniform range test passed 129/135 cases; six float16 cases exceeded the range majorants despite finite sampled intermediates, and the bounds were conservative.Source: §F.2 · p. 26

F.3 Testing the Update Bound with Repeated Blocks

  • Six fixed-control float64 continuations stayed in verified balls and reached the 10−6 fixed-point distance target within predicted update counts.The tests cover specific fixed-control cases, not all passing balls or the adaptive controller; their targets need not be acyclic or statistically correct.Source: Table 2 · p. 27, §F.3 · p. 27, Table 1 · p. 27

F.4 Additional diagnostics for the reference solver

  • On raw linear SEM data up to 20 variables, continuous solvers and sort-regress had lower SHD than PC and GES; Poly-prox had slightly higher SHD than L-BFGS solvers.Poly-prox’s median constraint value was 7·10−6 versus 10−8 for L-BFGS, while its thresholded outputs were acyclic throughout the raw grid.Source: §F.4 · pp. 27–28, Figure 6 · p. 28

F.5 Benchmark topology protocol

  • Across 220 benchmark datasets, arithmetic replay matched the reference solver with exactly zero weight gaps, multiplier gap, and thresholded graph disagreement in float64.The result establishes inheritance of these reference computations, not recovery of the underlying graph or transfer to other weightings of the same topology.Source: §F.5 · pp. 28–29

F.6 Training Models with and without State Updates

The appendix tests whether trained ordinary attention models reproduce solver updates, then checks numerical conditioning and stationarity in arithmetic replay. Results distinguish residual write-back performance from the constructed executor and do not establish convergence from the reported trajectories.

F.6 Training Models with and without State Updates

  • Ordinary models use four tied attention/ReLU repeats, train at p = 5, and are evaluated at p = 10 without retraining; they do not instantiate the guaranteed full-depth bilinear architecture.Training mixes random transitions and solver-visited states, and the models receive no explicit input identifying the two update policies.Source: §F.6 · p. 29
  • Removing residual write-back raises smaller-corpus one-step errors from 0.98 to 10.44 for linear attention and 0.94 to 10.98 for softmax; at larger-corpus p = 10, softmax errors are 14.82 versus 1.56.At larger-corpus p = 5, softmax errors are 1.70 versus 0.67. The comparison supports residual parameterization under this recipe, not the necessity of a particular state encoding.Source: §F.6 · p. 30

F.7 Checking the Relative-Error Bound

  • Adding 10^-4 to each off-diagonal gradient entry amplifies error from 0.0035 to 6.7 as c grows from 25 to 4.75 · 10^4, while relative perturbations satisfy the stated bound.The figure also reports undamped multiplier error accumulating as ℓ · 10^-3 when W = 0.Source: §F.7 · p. 30, Figure 7 · p. 31

F.8 How Stationarity Changes During Optimization

  • Across 30 saved Gaussian linear-SEM trajectories, final median ηmin/dM is 0.00338, 0.00661, and 0.0136 for p = 3, 5, and 10, respectively; none reaches ηmin ≤ 10^-6 in absolute norm.These arithmetic-replay diagnostics do not demonstrate ηk → 0, convergence to an acyclic non-Lasso limit, multiplier divergence, or certification of the adaptive outer controller.Source: §F.8 · p. 32
  • Finite controller changes can increase the augmented residual as the constraint and its gradient decrease, and Figure 8’s normalized progress axes do not align outer-stage boundaries.Its lines are medians and bands are minima and maxima over ten input-seed trajectories per dimension, not confidence intervals.Source: Figure 8 · p. 32
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