# Judge-facing evidence scorecard

- Paper ID: `m4TAzup6Yc`
- Registered claims: 6
- Assessments: 6 verified
- Source matrix: `EVIDENCE_MATRIX.json`
- Prose-local artifact references: validated

## Claim summary

| # | Literal claim | Assessment | Decisive quantitative result |
| ---: | --- | --- | --- |
| 1 | Theorem 3.1 (i.i.d. noise) and Theorem 4.1 (Markovian noise) bound the Wasserstein distance between the centered-scaled steady state Y^(α) = (X^(α) − x*)/√α and its Gaussian limit N(0, Σ_Y) by U√α log(1/α) (Theorems 3.1 and 4.1). | VERIFIED | Exact i.i.d. W1 slope is 0.506431 (R2=0.999950); finite-state Markov normalized ratios fall from 0.075469 to 0.038420, and the independent Gaussian-AR(1) W1 slope is 0.963596. |
| 2 | Proposition 3.1 establishes an explicit non-asymptotic Wasserstein bound of order O(α^{1/2} log(1/α)) for constant-stepsize SGD on smooth, strongly convex objectives (Proposition 3.1). | VERIFIED | Across nine exact stepsizes, quadratic SGD has W1 slope 0.504422 (R2=0.999962), decreasing from 0.110903 to 0.00672218; W1/(sqrt(alpha) log(1/alpha)) decreases at every step. |
| 3 | Propositions 3.2 and 3.3 extend the same O(α^{1/2} log(1/α)) Wasserstein bound to linear stochastic approximation with a Hurwitz matrix and to contractive nonlinear stochastic approximation, respectively (Propositions 3.2 and 3.3, Section 3.2.2, Section 3.2.3). | VERIFIED | The exact d=8 linear-SA W1 slope is 0.737073; the corrected 0.75-Lipschitz nonlinear SA slope is 0.657056 (R2=0.991605), with maximum two-grid W1 difference 7.25e-6. |
| 4 | For one-dimensional projections, the paper derives a non-uniform Berry-Esseen-type tail bound \|P(⟨Y^(α),ζ⟩ > a) − P(Z_ζ > a)\| ≤ C_d α^{1/4} log^{1/2}(1/α) / a, which improves as the deviation level a grows (Section 4). | VERIFIED | Exact sup_a a\|Delta(a)\| has slope 0.506389 (R2=0.999988), faster than the alpha^1/4 factor; at alpha=1/512 the absolute gap falls 58.42x from 2 to 4 standard deviations. |
| 5 | Proposition 4.1 extends the Gaussian approximation and tail bounds for SGD, linear SA, and contractive nonlinear SA from i.i.d. noise to Markovian noise (Proposition 4.1, Section 4). | VERIFIED | Exact five-state Markov W1 decreases to 0.0133154 with all bound ratios decreasing; two-seed nonlinear Markov W1 has slope 0.864854 (R2=0.990021), and its variance reaches 3.89014 versus target 4. |
| 6 | Proposition 5.1 shows a different convergence rate of order α^{1/h} for constant-stepsize SA applied to general convex objectives with Gibbs-type limiting distributions (Proposition 5.1, Section 5). | VERIFIED | For P_2, E\|X\| fits scaling exponent 0.249787 versus theory 0.25 (R2=0.999999785), while W1 to Gibbs converges with slope 1.504865 (R2=0.999997937), faster than the alpha^(1/h) upper-bound exponent; at alpha=1/256 Gibbs W1 is 2.55003e-5 versus 0.0594585 for the best-variance Gaussian. |

## Claim 1 — VERIFIED

> Theorem 3.1 (i.i.d. noise) and Theorem 4.1 (Markovian noise) bound the Wasserstein distance between the centered-scaled steady state Y^(α) = (X^(α) − x*)/√α and its Gaussian limit N(0, Σ_Y) by U√α log(1/α) (Theorems 3.1 and 4.1).

- Decisive quantitative result: Exact i.i.d. W1 slope is 0.506431 (R2=0.999950); finite-state Markov normalized ratios fall from 0.075469 to 0.038420, and the independent Gaussian-AR(1) W1 slope is 0.963596.
- Native scale: The theorem's native object is a stationary stochastic-approximation law rather than a benchmark dataset; the package evaluates seven i.i.d. stepsizes, six exact Markov stepsizes, and the retained d=1,8,10,16 sweeps without a reduced surrogate objective.
- Source locator: arXiv 2602.13960v1, Theorem 3.1 Equation (3.3) and Theorem 4.1 Equation (4.3)
- Upstream pin: `arXiv 2602.13960v1`
- Independent evidence:
  - `outputs/claim-1.json`
  - `outputs/exact.json`
  - `outputs/markov.json`
  - `outputs/oracle_checks.json`
- Executed outputs:
  - `outputs/claim-1.json`
  - `outputs/results.json`
- Independent oracle paths:
  - `outputs/oracle_checks.json`
  - `outputs/markov.json`
- Control paths:
  - `outputs/destructive_controls.json`
- Destructive or boundary control: Replacing Sigma_Y by the i.i.d. raw-noise variance leaves W1=0.330526 at the smallest stepsize with slope 0.001095; replacing Markov long-run covariance by marginal covariance leaves W1=0.349884, so both destroyed targets fail to converge.
- Rate relation: claim-consistent; mode `empirical_scaling`; measured slope `0.5064312475534698`.
  - Horizons: 8, 16, 32, 64, 128, 256
  - Repetitions per horizon: 2
  - Measurement: Over increasing inverse-stepsize horizons 8 through 512, exact i.i.d. W1 fits exponent 0.506431 while every normalized W1/(sqrt(alpha) log(1/alpha)) ratio decreases; Markov behavior is independently checked by finite-state and Gaussian-AR(1) systems.
  - Rate artifact: `outputs/claim-1.json`
- Limitation: These direct instances stress both noise regimes but do not constitute a new proof for every drift and noise distribution allowed by the theorems.
- Scope boundary: Verified for the executed stationary recursions and stepsize grids; universal constants and worst-case dimension dependence are outside this experiment.

## Claim 2 — VERIFIED

> Proposition 3.1 establishes an explicit non-asymptotic Wasserstein bound of order O(α^{1/2} log(1/α)) for constant-stepsize SGD on smooth, strongly convex objectives (Proposition 3.1).

- Decisive quantitative result: Across nine exact stepsizes, quadratic SGD has W1 slope 0.504422 (R2=0.999962), decreasing from 0.110903 to 0.00672218; W1/(sqrt(alpha) log(1/alpha)) decreases at every step.
- Native scale: The smooth strongly-convex proposition has no external dataset or trained model scale; the complete stationary law is evaluated at nine stepsizes down to alpha=1/2048, more directly than a finite training trajectory.
- Source locator: arXiv 2602.13960v1, Proposition 3.1 and Section 3.2.1
- Upstream pin: `arXiv 2602.13960v1`
- Independent evidence:
  - `outputs/claim-2.json`
  - `outputs/exact.json`
  - `outputs/oracle_checks.json`
- Executed outputs:
  - `outputs/claim-2.json`
  - `outputs/results.json`
- Independent oracle paths:
  - `outputs/oracle_checks.json`
- Control paths:
  - `outputs/destructive_controls.json`
- Destructive or boundary control: Changing the stable Hessian direction to a positive drift eigenvalue +0.25 makes the discrete multiplier 1.0078125 and amplifies that mode by 3.3034e13 in 4000 steps, eliminating a steady state.
- Rate relation: claim-consistent; mode `empirical_scaling`; measured slope `0.5044219069051806`.
  - Horizons: 8, 16, 32, 64, 128, 256, 512, 1024, 2048
  - Repetitions per horizon: 2
  - Measurement: Two FFT-grid evaluations at each horizon bind numerical error, while the full exact sweep over inverse stepsizes 8 to 2048 gives slope 0.504422 and R2=0.999962 for the registered W1 error.
  - Rate artifact: `outputs/claim-2.json`
- Limitation: The direct experiment uses a quadratic member of the smooth strongly-convex class and therefore does not sample every allowed nonlinear Hessian geometry.
- Scope boundary: The verdict covers exact stationary behavior for the executed SGD instance; it does not estimate the proposition's worst-case explicit constant U.

## Claim 3 — VERIFIED

> Propositions 3.2 and 3.3 extend the same O(α^{1/2} log(1/α)) Wasserstein bound to linear stochastic approximation with a Hurwitz matrix and to contractive nonlinear stochastic approximation, respectively (Propositions 3.2 and 3.3, Section 3.2.2, Section 3.2.3).

- Decisive quantitative result: The exact d=8 linear-SA W1 slope is 0.737073; the corrected 0.75-Lipschitz nonlinear SA slope is 0.657056 (R2=0.991605), with maximum two-grid W1 difference 7.25e-6.
- Native scale: The linear system uses eight dimensions, six projection directions and nine stepsizes, while the nonlinear system uses the complete scalar stationary density on 2048- and 4096-point grids over six stepsizes; neither is a covariance-only proxy.
- Source locator: arXiv 2602.13960v1, Propositions 3.2 and 3.3, Sections 3.2.2-3.2.3
- Upstream pin: `arXiv 2602.13960v1`
- Independent evidence:
  - `outputs/claim-3.json`
  - `outputs/exact.json`
  - `outputs/oracle_checks.json`
- Executed outputs:
  - `outputs/claim-3.json`
  - `outputs/results.json`
- Independent oracle paths:
  - `outputs/oracle_checks.json`
  - `outputs/exact.json`
- Control paths:
  - `outputs/destructive_controls.json`
- Destructive or boundary control: A linear drift with eigenvalue +0.25 and T_bad(x)=1.05x respectively produce discrete multipliers above one; the recorded amplifications are 3.3034e13 and 6.0333e6, destroying both stability assumptions.
- Rate relation: claim-consistent; mode `empirical_scaling`; measured slope `0.657056289722349`.
  - Horizons: 8, 16, 32, 64, 128, 256
  - Repetitions per horizon: 2
  - Measurement: Direct increasing inverse-stepsize measurements give exponent 0.737073 for d=8 linear SA and 0.657056 for nonlinear contractive SA; both normalized bound ratios decrease on their executed grids.
  - Rate artifact: `outputs/claim-3.json`
- Limitation: The nonlinear result is scalar and the linear result uses six fixed projections; full multivariate Wasserstein optimization is not attempted.
- Scope boundary: The verdict covers the executed Hurwitz and globally contractive systems, not unstable matrices, noncontractive operators, or all dimensions.

## Claim 4 — VERIFIED

> For one-dimensional projections, the paper derives a non-uniform Berry-Esseen-type tail bound |P(⟨Y^(α),ζ⟩ > a) − P(Z_ζ > a)| ≤ C_d α^{1/4} log^{1/2}(1/α) / a, which improves as the deviation level a grows (Section 4).

- Decisive quantitative result: Exact sup_a a|Delta(a)| has slope 0.506389 (R2=0.999988), faster than the alpha^1/4 factor; at alpha=1/512 the absolute gap falls 58.42x from 2 to 4 standard deviations.
- Native scale: The registered object is one-dimensional by construction; evaluating its complete stationary characteristic function at thresholds from 0.5 to 6 standard deviations is the native scale, including probabilities too small for the retained Monte Carlo budgets.
- Source locator: arXiv 2602.13960v1, Corollaries 3.1 and 4.1 and Section 4 tail discussion
- Upstream pin: `arXiv 2602.13960v1`
- Independent evidence:
  - `outputs/claim-4.json`
  - `outputs/oracle_checks.json`
- Executed outputs:
  - `outputs/claim-4.json`
  - `outputs/results.json`
- Independent oracle paths:
  - `outputs/oracle_checks.json`
  - `outputs/claim-4.json`
- Control paths:
  - `outputs/destructive_controls.json`
- Destructive or boundary control: Using the raw-noise variance instead of the Lyapunov variance makes the wrong-target projected discrepancy essentially flat in stepsize (W1 slope 0.001095) and leaves W1=0.330526 at the smallest alpha.
- Rate relation: claim-consistent; mode `empirical_scaling`; measured slope `0.506389109050498`.
  - Horizons: 8, 16, 32, 64, 128, 256, 512
  - Repetitions per horizon: 2
  - Measurement: At seven increasing inverse-stepsize horizons, exact sup_a a|Delta| fits slope 0.506389 with R2=0.999988, exceeding the claimed alpha^1/4 exponent while remaining finite across all seven deviation levels.
  - Rate artifact: `outputs/claim-4.json`
- Limitation: The tested thresholds stop at six standard deviations and use one skew-noise scalar projection, so they do not estimate the theorem's dimension-dependent worst-case constant.
- Scope boundary: Verified for the exact executed projection and threshold grid; no claim is made that the observed 0.506 exponent is the universal optimal exponent.

## Claim 5 — VERIFIED

> Proposition 4.1 extends the Gaussian approximation and tail bounds for SGD, linear SA, and contractive nonlinear SA from i.i.d. noise to Markovian noise (Proposition 4.1, Section 4).

- Decisive quantitative result: Exact five-state Markov W1 decreases to 0.0133154 with all bound ratios decreasing; two-seed nonlinear Markov W1 has slope 0.864854 (R2=0.990021), and its variance reaches 3.89014 versus target 4.
- Native scale: The exact Markov calculation covers six stepsizes and the nonlinear check uses 32,768 stationary particles per seed, two seeds, five stepsizes and tail thresholds through four standard deviations; the retained full d=4 sweep provides an additional multidimensional check.
- Source locator: arXiv 2602.13960v1, Proposition 4.1 and Section 4
- Upstream pin: `arXiv 2602.13960v1`
- Independent evidence:
  - `outputs/claim-5.json`
  - `outputs/markov.json`
  - `outputs/oracle_checks.json`
- Executed outputs:
  - `outputs/claim-5.json`
  - `outputs/results.json`
- Independent oracle paths:
  - `outputs/oracle_checks.json`
  - `outputs/markov.json`
- Control paths:
  - `outputs/destructive_controls.json`
- Destructive or boundary control: Ignoring autocorrelation and using marginal covariance leaves exact W1=0.349884, 26.28x the correct-target W1; in the nonlinear check the naive target is 62.75x worse at the smallest alpha.
- Rate relation: claim-consistent; mode `empirical_scaling`; measured slope `0.8648543490756267`.
  - Horizons: 8, 16, 32, 64, 128
  - Repetitions per horizon: 2
  - Measurement: Two independent seeded executions of the genuine nonlinear Markov recursion across inverse stepsizes 8 to 128 give W1 exponent 0.864854 and R2=0.990021; the exact linear Markov normalized ratios decrease at every horizon.
  - Rate artifact: `outputs/claim-5.json`
- Limitation: The nonlinear Markov component is Monte Carlo and reports calibrated sampling floors; the exact transfer calculation is reserved for the linear/SGD component.
- Scope boundary: The verdict covers the executed finite-state and two-state chains, not every geometrically mixing Markov kernel allowed by Proposition 4.1.

## Claim 6 — VERIFIED

> Proposition 5.1 shows a different convergence rate of order α^{1/h} for constant-stepsize SA applied to general convex objectives with Gibbs-type limiting distributions (Proposition 5.1, Section 5).

- Decisive quantitative result: For P_2, E|X| fits scaling exponent 0.249787 versus theory 0.25 (R2=0.999999785), while W1 to Gibbs converges with slope 1.504865 (R2=0.999997937), faster than the alpha^(1/h) upper-bound exponent; at alpha=1/256 Gibbs W1 is 2.55003e-5 versus 0.0594585 for the best-variance Gaussian.
- Native scale: This is the exact paper-native one-dimensional objective and noise choice, evaluated at five stepsizes on both 2048- and 4096-point stationary-density grids rather than a quadratic stand-in or Gaussian approximation.
- Source locator: arXiv 2602.13960v1, Proposition 5.1 Equation (5.2), Appendix E.2.1 Definition E.1 and Equation (E.10)
- Upstream pin: `arXiv 2602.13960v1`
- Independent evidence:
  - `outputs/claim-6.json`
  - `outputs/oracle_checks.json`
- Executed outputs:
  - `outputs/claim-6.json`
  - `outputs/results.json`
- Independent oracle paths:
  - `outputs/oracle_checks.json`
  - `outputs/claim-6.json`
- Control paths:
  - `outputs/destructive_controls.json`
- Destructive or boundary control: Replacing alpha^(1/4) by the classical sqrt(alpha) scaling makes the scaled first moment spread by factor 2.00128 across the grid, while the correct scaling spreads only 1.00064; a variance-matched Gaussian remains 2331.68x farther than Gibbs.
- Rate relation: claim-consistent; mode `empirical_scaling`; measured slope `1.504864832767487`.
  - Horizons: 16, 32, 64, 128, 256
  - Repetitions per horizon: 2
  - Measurement: Two deterministic grid evaluations at each inverse stepsize from 16 through 256 yield direct W1-to-Gibbs exponent 1.504865 (R2=0.999997937), faster than the alpha^(1/h) upper-bound exponent 0.25; independently, E|X| scales with exponent 0.249787 and R2=0.999999785.
  - Rate artifact: `outputs/claim-6.json`
- Limitation: The experiment directly covers h=4 with Gaussian noise; it does not resolve the proposition's stated stability and Stein-regularity conjectures for every general convex objective.
- Scope boundary: Verified for paper-native P_2 over alpha from 1/16 to 1/256; other h values, Pareto noise and global conjecture proofs remain outside scope.