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Transformers with CoT Discover Heavy-Ball Acceleration Through Training

Yuxuan Cai, Yuan Cao

arXiv:2610.31089v1cs.LGstat.ML

TL;DR

The paper asks whether CoT transformers can discover algorithms beyond imitating a gradient-descent teacher. It trains first with gradient-descent supervision and then with true regression weights, finding that the model learns Heavy-Ball and accelerates over the teacher without Heavy-Ball supervision.

  • Problem

    Prior work showed that gradient-descent teacher supervision can teach CoT transformers to implement gradient descent for in-context weight prediction; this paper asks whether they can discover an algorithm beyond that teacher.

    Source: §1 · p. 2

  • Method

    A two-stage procedure first supervises CoT with a gradient-descent teacher, then trains the trajectory using true regression weights.

    Source: §1 · p. 2, §6 · p. 13

  • Results

    The learned model discovers Heavy-Ball and achieves acceleration over the gradient-descent teacher used in Stage 1, without Heavy-Ball teacher supervision.

    Source: §3 · p. 7, §6 · p. 13

  • Takeaways & Limitations

    The result indicates that transformers with CoT can discover unseen algorithms through training rather than only recover a teacher algorithm.

    Source: §3 · p. 7

    The analysis is limited to linear self-attention for in-context linear regression and gradient-flow training; more practical attention, broader tasks, and other training algorithms remain future directions.

    Source: §6 · p. 13

Abstract

from arXiv · show

Modern transformers often solve complex tasks by generating and refining intermediate reasoning steps. To develop a theoretical understanding of chain-of-thought (CoT) reasoning, a notable recent study by Huang et al. (2025) viewed transformers as algorithm executors, and showed that when transformers' CoT steps are supervised by a gradient-descent teacher, they can learn to implement gradient descent for in-context weight prediction in linear regression. In this paper, we study the same in-context weight prediction task but ask whether CoT reasoning can go beyond merely imitating the teacher algorithm. We show that, after initial supervision from a gradient-descent teacher, further trajectory-level supervision using the true regression weights leads the model to implement the Heavy-Ball method, thereby achieving acceleration over the original teacher. Notably, this accelerated algorithm emerges without supervision from any Heavy-Ball teacher. Our results provide a concrete theoretical example in which transformer reasoning can discover new algorithms through training, rather than merely imitate the algorithm used for supervision.

1 Introduction

This work asks whether CoT transformers can discover algorithms beyond a gradient-descent teacher, rather than merely imitate it. Two-stage training yields Heavy-Ball-like reasoning trajectories and theoretical guarantees for this discovery.

1 Introduction

  • Earlier work showed transformers can implement optimization algorithms for in-context learning, including gradient descent for linear regression and CoT steps matching gradient-descent updates.Huang et al. (2025) showed that gradient-descent teacher supervision leads each CoT step to implement one gradient-descent update.Source: §1 · pp. 1–2
  • The two-stage model learns CoT trajectories approximating Heavy-Ball and accelerating over the gradient-descent teacher, without Heavy-Ball supervision.Stage 1 uses a gradient-descent teacher; Stage 2 uses the true regression weights.Source: Figure 1 · p. 2, §1 · p. 2
  • The analysis shows both training stages succeed despite nonconvex losses and proves that learned CoT iterates approximate Heavy-Ball.Auxiliary parameters and recursively defined matrix polynomials support the convergence and algorithm-characterization analysis.Source: §1 · pp. 2–3
  • The study uses a Marchenko–Pastur spectral-limit setting to distinguish Heavy-Ball from gradient descent, unlike settings where their convergence rates coincide.The paper also notes that Stage 2 lacks teacher forcing and has no clean analytical loss form because outputs are autoregressive.Source: §1 · p. 2

2 Preliminaries

The preliminaries define the in-context weight-prediction task, the simplified linear self-attention model with CoT, and the two-stage training procedure. The second stage uses the model’s autoregressive trajectory and true regression weights rather than a classical algorithm’s teacher forcing.

2.1 In-Context Weight Prediction with CoT

  • The task predicts an unknown linear-regression weight vector from context training pairs, using a simplified linear self-attention model with merged projection/value and key/query matrices.The ground-truth weight vector is sampled from a standard Gaussian distribution.Source: §2.1 · p. 3
  • CoT steps represent optimization iterations, beginning from zero initialization; the model recursively appends each predicted weight vector to its input.The setup adds a one-hot positional encoding for the initialization token and specifies a maximum number of reasoning steps.Source: §2.1 · pp. 3–4

2.2 Two-Stage Training

  • Stage 1 uses teacher forcing to train predictions of gradient-descent teacher iterates, then Stage 2 trains on true regression weights using the model’s own autoregressive sequence.Stage 1 is trained until its loss falls below a threshold; Stage 2 has no Heavy-Ball teacher supervision.Source: §2.2 · pp. 4–5
  • Without teacher forcing, the Stage 2 loss has no clean analytical form because its autoregressive sequence can become complex.The paper cautions that its analogies to pretraining, supervised fine-tuning, and reinforcement-learning post-training do not establish conclusions about general pretraining or post-training.Source: §2.2 · p. 5

3 Main Result

The paper proves that two-stage training can make a transformer’s CoT approximate Heavy-Ball iterations and achieve acceleration without a Heavy-Ball teacher. The result is established in a proportional sample-size regime under stated training and initialization assumptions.

3 Main Result

  • Assumption 3.1 constrains parameter initialization and freezes one block of the W matrix during training.The authors characterize this as a technical assumption and note related theoretical studies have used similar constraints.Source: §3 · p. 6
  • With n = Θ(d), the sample covariance follows the Marchenko–Pastur regime rather than concentrating near the identity, where Heavy-Ball and gradient descent have the same rate.This proportional regime is chosen to distinguish Heavy-Ball from gradient descent.Source: §3 · p. 6
  • Under Assumptions 3.1 and 3.2, Stage 1 reaches LGD ≤ d^-1/4 and Stage 2 reaches L − min L ≤ ϵ.Theorem 3.3 gives these loss guarantees when n,d = Ω(exp(log^4(1/ϵ))) and the specified gradient-flow training durations are met.Source: §3 · p. 6
  • The learned CoT algorithm approximates Heavy-Ball and achieves an accelerated convergence rate without Heavy-Ball supervision.The authors describe the learned algorithm as essentially equivalent to Heavy-Ball and identify acceleration over gradient descent.Source: §3 · p. 7
  • The theorem requires n,d = Ω(exp(log^4(1/ϵ))), a super-polynomial but subexponential-in-1/ϵ size condition that the authors call relatively strong.They note that setting ϵ = 0.5 still yields an expected-iterate error bound relative to Heavy-Ball.Source: §3 · p. 7

4 Proof sketch

The proof tracks CoT iterates as matrix polynomials of the sample covariance, then uses auxiliary parameters to establish convergence in both training stages. It compares the limiting polynomials with Heavy-Ball to show that late CoT steps approximate its iterates.

4 Proof sketch

  • Auxiliary parameters address the distinct proof challenges created by n = Θ(d) and Stage 2’s lack of teacher forcing.The sample covariance does not reduce to the identity in this regime, and Stage 2 feeds the model’s own outputs back into its reasoning trajectory.Source: §4 · pp. 7, 9
  • Throughout both stages, the transformer parameters retain a structured manifold form, reducing analysis of the full V,W matrices to a smaller parameterization.The proof identifies this preserved structure before analyzing the polynomial iterates.Source: §4 · p. 7
  • Each CoT iterate has the form wt(τ) = pt,θ̄(τ)(S)w⋆, where the polynomial has degree at most t and zero constant term.This representation lets the analysis express both stage losses as functions of matrix polynomials in S, which matters because S cannot be approximated by the identity when n = Θ(d).Source: §4 · pp. 7–8
  • The PL inequality with respect to barθ is established only in the later phase: Lemma 4.5 holds for every τ ≥ τ0, with τ0 = d^o(1).The analyzed training dynamics remain gradient flow in the original parameters; the auxiliary parameters are only a proof tool.Source: §4 · pp. 8–9
  • In Stage 2, the model’s use of its own previous outputs makes the loss polynomials recursive, but auxiliary parameters still establish convergence to within ϵ of the minimum loss.The resulting training time is O(d^(1/α+o(1))ϵ^(-1/α)) for a constant α > 0.Source: §4 · pp. 9–10
  • For late steps T/2 ≤ t < T, the limiting learned polynomials approach the Heavy-Ball polynomials as their highest-degree coefficient decays exponentially.The proof combines polynomial comparison estimates to show that trained CoT iterates can be made close to the Heavy-Ball iterates.Source: §4 · pp. 10–11

5 Experiment

Experiments compare two-stage training and its variants with Heavy-Ball and gradient descent in two settings. The two-stage transformer closely matches Heavy-Ball, while first-stage-only and no-positional-encoding variants perform similarly to gradient descent.

5 Experiment

  • Experiments use (d, n) = (16, 32) and (12, 24), with a 24-step maximum CoT and online Adam training.Stage 1 uses a gradient-descent teacher with η0 = 0.5; the models use float64, learning rate 10^-6, batch size 6144, and independently initialized V and W.Source: §5 · p. 11
  • The experiments remove teacher forcing in Stage 1, use stochastic Adam and Gaussian initialization without frozen blocks, and test settings that do not satisfy the theory’s d/n < 0.1 condition.The authors say strictly following the theoretical setting yields similar results.Source: §5 · p. 11
  • In both tested settings, the two-stage transformer closely matches Heavy-Ball, while first-stage-only and no-positional-encoding variants perform similarly to gradient descent.The authors report this as consistent with Theorem 3.3 and as evidence for the two-stage procedure and positional encodings.Source: §5 · p. 11, Figure 2 · p. 12
  • Figure 3 pairs learned-weight heatmaps after two-stage training with second-stage-only training traces across four dimension and learning-rate combinations, marking nonfinite gradients and termination.Training retries with a newly generated independent dataset after nonfinite gradients and stops only after 20 consecutive such retries.Source: Figure 3 · p. 12

6 Conclusion

The paper studies whether CoT transformers can discover algorithms beyond a supervised teacher in in-context weight prediction. It reports that two-stage training yields Heavy-Ball and acceleration, while its analysis remains limited to linear self-attention for linear regression.

6 Conclusion

  • Two-stage training first supervises CoT with a gradient-descent teacher, then trains using only ground-truth regression weights, and the learned model discovers Heavy-Ball with acceleration over that teacher.The paper presents this as algorithm discovery through training rather than imitation of the supervised algorithm.Source: §6 · pp. 12–13
  • The analysis is limited to linear self-attention for in-context linear regression and assumes gradient-flow training.The authors identify softmax attention, more general tasks, other training algorithms, and other positional encodings as future directions.Source: §6 · p. 13

A Related Works

Related work examines transformers as algorithm executors, their training guarantees, and chain-of-thought computation. The paper builds on CoT training analyses by asking whether trajectory-level training can discover accelerated algorithms beyond the teacher.

A Related Works

  • Work on transformers as algorithm executors studies in-context learning, expressivity, and generalization, including linear regression, Bayesian-network modeling, and hierarchical vision models.Examples include Li et al. (2023), Chen and Zou (2024), Cao et al. (2025), and Shi et al. (2025).Source: A Related Works · p. 13
  • Training-guarantee studies analyze learned structures and mechanisms, while related work also examines CoT as intermediate computation and its expressive power.Examples include Jelassi et al. (2022), Nichani et al. (2024), Chen et al. (2024), and Merrill and Sabharwal (2024); Feng et al. (2023).Source: A Related Works · pp. 13–14
  • Huang et al. (2025) analyzed CoT training dynamics for in-context learning and showed that CoT-supervised transformers can implement multi-step gradient descent.Other cited studies consider tree path finding and randomized depth-first search with policy-gradient training.Source: A Related Works · p. 14
  • This work asks whether CoT trajectory-level training can let transformers discover accelerated optimization algorithms beyond the supervised teacher.Source: A Related Works · p. 14

B Proof of Stage 1 convergence

The appendix establishes Stage 1 parameter structure through gradient and symmetry arguments, then applies shared scaling assumptions in proofs of later lemmas.

B.1 Stage 1 invariant parameter structure

  • Stage 1 training preserves the parameter-matrix form specified in Lemma B.1, the Stage-1 part of Lemma 4.1.The proof uses gradients at that form and symmetry of the loss and initialization to establish invariant structure.Source: §B.1 · pp. 16–17
  • Coordinate-exchange symmetry implies the specified blocks V31, W13, and V33 are scalar multiples of the identity matrix.Source: §B.1 · p. 18

B.2 Proofs of Lemmas 4.3–4.5

  • The proofs of Lemmas 4.3–4.5 use the appendix’s shared scaling assumptions.The supplied passage does not specify their full form.Source: §B.2 · p. 18

B.3 Proof of Lemma 4.3

This section analyzes Stage-1 loss and training dynamics using spectral polynomial projections, flow invariants, and estimates with conditions on dimension and training-time horizon. It proves bounds in Lemmas 4.3–4.5 and gives the Stage-1 loss and timing result used at the handoff to Stage 2.

B.3 Proof of Lemma 4.3

  • The Stage-1 loss is expressed through spectral polynomial inner products, orthogonal projections, and a decomposition into block contributions.The proof first derives the residual and projection expansion, then substitutes them into the loss.Source: §B.3 · pp. 18–20
  • Along Stage-1 flow, I1(τ) = I1(0) = −σ2/d and I3(τ) = I3(0) = −σ; also, v33(τ) ≥ v33(0) > 0.The positivity bound follows from v33(τ)2 ≥ v33(0)2, continuity, and v33(0) > 0.Source: §B.3 · p. 22
  • For sufficiently large d, Lemma B.3 establishes a bound for the specified parameterization, using coefficient projections and finite-Wishart moment bounds.The supplied passages do not include the bound’s displayed expression.Source: §B.3 · pp. 22–24
  • Under condition (3.2), Lemma B.4 gives uniform estimates through H = d^o(1), and Lemma B.5 gives a relation on every interval within that horizon.The proof of Lemma B.5 uses projection coefficients, the Lemma B.4 bounds, and integration by parts.Source: §B.3 · pp. 25, 26, 28, 29
  • The argument shows that v31(τ) ≠ 0 for every τ ≥ τb, that event (B.59) occurs by H/2, and that (B.27) and (B.28) are proved.Source: §B.3 · pp. 39, 42, 44

B.4 Proof of Lemma 4.4

  • Combining the moment estimate for A(S) with bound (B.17) and loss decomposition (B.12) proves the claimed estimate in Lemma 4.4.Source: §B.4 · pp. 44–45

B.5 Proof of Lemma 4.5

  • The proof of Theorem 3.3 shows that L_GD ≤ ϵ_GD = d^-1/4 is reached within d^o(1) + O(T log d) time.For Lemma 4.5, the proof constructs a right inverse for the auxiliary map; the handoff passage also states a Stage-1 time of d^o(1) + O(T log d) with L_GD ≤ d^-1/4.Source: §B.5 · pp. 45–46, C Handoff from Stage 1 to Stage 2 · p. 47

C Handoff from Stage 1 to Stage 2

C.1 The Stage-1 endpoint

The Stage-1 endpoint is reached within a vanishing time bound, and the resulting parameter structure is preserved at the Stage-2 handoff. The appendix establishes scalar-block symmetry and polynomial structure for Stage-2 gradient flow.

C.1 The Stage-1 endpoint

  • d/n log(d^5/4) = do(1) bounds the first time the Stage-1 loss reaches d^-1/4.The Stage-1 raw bounds remain valid at subsequent Stage-1 times.Source: §C.1 · pp. 48, 50

C.2 The Stage-2 initial loss

  • At the Stage-1 endpoint, the same raw parameters initialize Stage 2, with W33 = 0 and scalar-identity invariance preserved.The endpoint also has v33(τ1) ≥ v33(0) > 0.Source: §C.2 · p. 50

D.1 Stage 2 part of Lemma 4.1

  • Stage-2 gradient flow preserves scalar-identity form in V31, W13, and V33, while gradients of inactive blocks vanish.The cited proof also states the relevant zero gradients for V and W blocks.Source: §D.1 · pp. 53, 55

D.2 Stage 2 part of Lemma 4.2

  • For Stage-2 iterates, wt = pt,θ̄(S)w⋆, where the rollout polynomial has zero constant term and degree at most t.The induction establishes these polynomial properties at every time step.Source: §D.2 · p. 56

D.3 Proof of Lemma 4.6

This section develops polynomial, loss, and parameter-variation estimates for the Stage-2 dynamics, supporting proof bounds and the analysis of the reduced objective. It also outlines how these estimates are used in the proof of Lemma 4.7.

D.3 Proof of Lemma 4.6

  • Conditioning on S yields the Lemma 4.6 loss formula, while the Stage-2 flow preserves v33(τ) ≥ v33^init > 0.The proof also derives a leading-term property for the polynomial recurrence when eθ1 and eθ2 are nonzero.Source: §D.3 · pp. 56–57

D.4 Polynomial coefficients and loss decomposition

  • The vectors p1,…,pt span the same space as λ,…,λ^t because their coefficient matrix in the monomial basis is triangular with nonzero diagonal.The section then changes to the monomial basis to analyze the autoregressive recurrence and decompose the loss.Source: §D.4 · pp. 57–58

D.5 Finite-Wishart comparison

  • Under (B.7), finite-Wishart polynomial expectations and degree-at-most-T minimizers admit comparison estimates against the corresponding Marchenko–Pastur quantities.The section also establishes the asymptotic estimate 8192T = O(log log d).Source: §D.5 · pp. 61–62

D.6 Parameter bounds along Stage 2

  • Under (B.7), Stage 2 starts with 1/100 < eθ1 < 2 and −eθ2 in (8/25, 23/25), and its flow exists globally.Along the flow, v31 and v33 remain bounded, with the stated uniform control in (D.37).Source: §D.6 · pp. 63–64

D.7 Study of LA and LB

  • Under (B.7), the finite-Wishart reduced objective Bνd,n(−eθ2)^2/Cνd,n(−eθ2) has a unique nondegenerate global maximizer and only one positive critical point.The section also establishes bounds on the maximizer’s location and uniform landscape estimates with constants independent of d, n, and T.Source: §D.7 · pp. 66–68
  • The LA/LB analysis establishes bounds around the unique maximizer and states Lemma D.11 under condition (B.7).The proof invokes Lemmas F.12–F.14 and D.8 to obtain the bounds in (D.42) and (D.43).Source: §D.7 · p. 69

D.8 Coefficient and coordinate estimates

  • Coefficient and coordinate estimates bound recurrence coefficients, their history inverse, and loss gradients at the specified points.The displayed bounds include factors of exp{CT log(e + T)} and exp{CT^2 log(e + T)} alongside powers of n/d.Source: §D.8 · pp. 70–73

D.9 Proof of Lemma 4.7

  • The Lemma 4.7 proof applies the LA/LB bounds and coordinate chain rules with Cauchy–Schwarz to derive its estimates.It treats both the (eθ1, eθ2, eθ5) coordinates and the (eθ1, eθ2, eΘ4) coordinates.Source: §D.9 · pp. 75–76

D.10 Raw-parameter estimates

  • Along the Stage-2 flow, every variation of active auxiliary parameters has a raw lift that is a right inverse of the auxiliary map.The construction’s final estimates establish (D.56).Source: §D.10 · pp. 77–78

D.11 Proof of Lemma 4.8

The Stage-2 convergence proof applies the flow and low-loss bounds from the Stage-1 argument, then derives the Stage-2 time bound in Theorem 3.3.

D.11 Proof of Lemma 4.8

  • For the Stage-2 flow, the proof follows the Stage-1 Jacobian-bound approach, using the right inverse from Lemma D.15 and the chain rule with Cauchy–Schwarz.Source: §D.11 · p. 78
  • Stage 2 begins with global existence, parameter bounds, and entry into the low-loss region inherited from the Stage-1 convergence argument.These conditions allow Lemmas D.12 and D.16 to apply from Stage-2 time zero.Source: §D.11 · p. 79
  • The Stage-2 time bound in Theorem 3.3 follows by solving the required-accuracy inequality.The proof combines bounds whose constants may depend on fixed comparison bounds in n = Θ(d), but not on d, n, or T.Source: §D.11 · p. 81

E.1 Optimality of the Heavy-Ball trajectory

The appendix characterizes the Heavy-Ball trajectory as the unique jointly optimal admissible trajectory and analyzes the model-optimal residual’s highest-degree coefficient. It then connects the trained trajectory to an accelerated convergence-rate result and supplies supporting uniqueness arguments.

E.1 Optimality of the Heavy-Ball trajectory

  • The jointly optimal Heavy-Ball trajectory is admissible and unique, with its coefficients uniquely determined.Its recurrence is identified with Pedregosa and Scieur (2020, Theorem 3.1).Source: §E.1 · pp. 81–82

E.2 The highest-degree coefficient

  • The finite-Wishart model-optimal residual is analyzed through its highest-degree coefficient, including indices T/2 ≤ t < T.The supplied corollary gives this index range but not the coefficient comparison itself.Source: §E.2 · p. 83

E.3 The trained trajectory

  • The trained trajectory’s comparison yields the claimed Stage-2 time, which remains O(d^o(1)ϵ^-1/α) under (B.7).The triangle-inequality argument uses a residual bound of exp[−2(log d)^1/4] ≤ ϵ/4 for sufficiently large d.Source: §E.3 · p. 84

F Supporting lemmas

  • For 0 < d/n ≤ 9/20, the infinite function B(x)^2/C(x) has exactly one critical point, a nondegenerate maximum.Its location satisfies the supplied interval bound for x∞.Source: §F.1 · pp. 84–85

F.1 Function sign and rational certificate

  • A Descartes bound of two positive intersections rules out multiple critical points, establishing the unique-maximum argument.The argument also shows each critical point is nondegenerate.Source: §F.1 · p. 87
  • The rational inequalities (F.13)–(F.14) hold on the stated domain after the four numerators are shown strictly positive.The domain includes 0 < d/n ≤ 9/20 and 0 ≤ s ≤ 1.Source: §F.1 · pp. 88, 90

F.2 Finite-degree MP coefficient functions

This part develops technical estimates for finite-degree Marchenko–Pastur coefficient functions and finite-Wishart teacher polynomials, including coefficient-ratio behavior, Wishart moments, and finite-Wishart approximations. It also establishes existence and attainability results for the training-stage gradient flows and fixed-leading-coefficient Heavy-Ball bounds.

F.2 Finite-degree MP coefficient functions

  • For 0 < d/n ≤ 1/10 and T ≥ 8, coefficient ratios increase through j = 3, then decrease, with adjacent gaps at least c(d/n)^T.The resulting coefficient-function ratio has a unique nondegenerate global maximizer on the positive half-axis, which is also its unique global maximizer over R.Source: §F.2 · pp. 92, 94, 95

F.3 MP optimal-trajectory estimates

  • Under (B.7), finite-Wishart coefficient functions are positive and close to their MP counterparts for sufficiently large d.The stated coefficient comparison has constants independent of d, n, and T; the error is small enough to preserve positivity.Source: §F.3 · pp. 97–98
  • Under (B.7), the finite MP maximizer approaches the infinite-series maximizer uniformly in d/n, with curvature, derivative, and quadratic-gap estimates transferred to the stated region.The stated region is 1/150 ≤ x ≤ 10/3 with (d/n)x^2 ≤ 9/10.Source: §F.3 · pp. 98–100

F.4 Gaussian Wishart moment estimates

  • For every k ≥ 1, Gaussian sample-covariance moments are compared with MP moments by a finite-dimensional pairing argument that also applies to moment orders growing with d.The moment difference is zero at k = 0, and the remaining pairing contributions are nonnegative.Source: §F.4 · pp. 100–101

F.5 Teacher-polynomial estimates

  • Lemma F.16 gives teacher-polynomial estimates uniformly over 0.4 ≤ η0 ≤ 0.9 on the stated MP support.The supplied passage does not include the estimate values.Source: §F.5 · p. 101
  • Under (B.7), this subsection establishes finite-Wishart moment and quadratic-gap estimates and related teacher-polynomial bounds.The lemmas include estimates for degrees up to T and sufficiently large d.Source: §F.5 · pp. 102–103

F.6 Global existence of the gradient flows

  • Under Assumption 3.1, the raw gradient flows for both training stages exist uniquely for all elapsed times.The result covers Stage 1 and Stage 2.Source: §F.6 · p. 104

F.7 Raw realization of the minimizing parameters

  • For the stated parameter conditions, the recurrence has a raw representative satisfying W33 = 0 and (D.11)–(D.13), so the auxiliary minimum is attainable.Source: §F.7 · p. 104

F.8 Fixed leading coefficients and Heavy-Ball estimates

  • Under (B.7), the subsection provides fixed-leading-coefficient and Heavy-Ball estimates, including bounds uniform over the stated iteration range.The results cover 0 ≤ t < T and, in a separate lemma, 1 ≤ t ≤ T.Source: §F.8 · pp. 105–106

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