[{"arxiv_id": "1908.03890", "claim_anchor": "lem:hierarchy (arxiv.tex:631); main equivalence th:main (arxiv.tex:247) is claim_kind equivalence, outside the audit kinds", "bound": "Strict ambiguity hierarchy of weighted automata over Q, each separation witnessed by an explicit sequence: DetWA < ambiguity-1 WA (shuffle of (2^n) and (1)); kWA < (k+1)WA (u_n = 1^n+...+(k+1)^n); FinWA < PolyWA (v_n = n); PolyWA < WA (Fibonacci). Exact combinatorial separations, no resource-tolerance parameter.", "deployed_config": "None: pure theory paper - no experiments, no results/1908.03890.json in the corpus; parsed sections run Introduction through Conclusion with no experimental setup.", "evaluation": "Nothing to evaluate against a deployment: the separations are exact (constants fully pinned - the witnesses are explicit sequences, e.g. Fibonacci's eigenvalue (1+sqrt(5))/2 is not a root of a rational, which is the entire proof obligation), but the paper deploys no system, so no config can be inside or outside the certified regime.", "verdict": "not_evaluable"}, {"arxiv_id": "2006.04768", "claim_anchor": "thm:low-rank (linformer_combined.tex:268); proof-pinned constant at appendix 'Let k=5log(n)/(eps^2-eps^3)' (Proof of Theorem 1)", "bound": "Self-attention context matrix P admits P~ of rank Theta(log n) with Pr(||P~ w^T - P w^T|| < eps ||P w^T||) > 1 - o(1). Statement rank constant is Theta-wrapped, but the proof pins the regime exactly: k = 5 log(n)/(eps^2 - eps^3), failure probability <= 2n e^{-(eps^2-eps^3)k/4}.", "deployed_config": "Linformer projection dim k = 128 and 256 at n = 512 (downstream tables, linformer_combined.tex:416); k = 256 at n = 1024 (tables) and n in {2048, 4096} (pretraining perplexity sweep 'holding projected dimension k fixed at 256'); RoBERTa-base backbone, head dim d = 64.", "evaluation": "Feasibility gate on the PROOF'S OWN SUFFICIENT SCHEDULE k = 5 ln(n)/(eps^2-eps^3) (sufficient choice, not a necessity claim): eps^2-eps^3 <= 4/27 = 0.1481, so the schedule is satisfiable only when k >= 33.75 ln(n). Thresholds: n=512 -> 210.6; n=1024 -> 233.9; n=2048 -> 257.3; n=4096 -> 280.7. Deployed k=128: below threshold at every n; moreover the raw finite-n inequality Pr > 1 - 2n exp(-(eps^2-eps^3)k/4) is vacuous at k=128 for all deployed n even at the maximizing eps=2/3 \u2014 nothing nontrivial is certified. Deployed k=256: schedule nonempty only at n=512 (eps ~ 0.488) and n=1024 (eps ~ 0.546), and substituting the schedule gives failure probability 2n^{-1/4} ~ 0.42 at n=512 \u2014 i.e. a ~49% relative-error bound holding with only ~58% success probability. At n=2048/4096 the schedule is empty; the raw finite-n bound remains technically nonvacuous but only at eps = 2/3 (67% relative error) with success probability 0.688 (n=2048) and 0.375 (n=4096). SCOPE NOTE: thm:low-rank certifies the random JL sketch P R^T R, not the learned E,F projections deployed \u2014 thm:factorization (audited separately) is the architectural theorem. [Refined 2026-07-26 after independent cross-model verification (GPT): arithmetic confirmed; sufficiency framing and finite-n nonvacuity at k=256, n>=2048 corrected.]", "verdict": "outside"}, {"arxiv_id": "2006.04768", "claim_anchor": "thm:factorization (linformer_combined.tex:333); proof-pinned constants 'k=5log(nd)/(eps^2-eps^3)' and 'k=9log(d)/(eps^2-eps^3)' (Proof of Theorem 2)", "bound": "Exists E_i, F_i in R^{n x k} with Pr(||softmax(wE^T)FVW^V - softmax(w)VW^V|| <= eps ||softmax(w)|| ||VW^V||) > 1 - o(1) when k = min{Theta(9 d log(d)/eps^2), 5 Theta(log(n)/eps^2)}. Proof pins the branches as k = 5 log(nd)/(eps^2-eps^3) (n-branch) and k = 9 log(d)/(eps^2-eps^3) (d-branch); NB the statement's d-branch carries an extra factor d (9d log d) that the proof's choice does not - a statement/proof discrepancy of factor d.", "deployed_config": "Quality configs: (n=512, k=128), (n=512, k=256), (n=1024, k=256) downstream (linformer_combined.tex:416); (n in {2048,4096}, k=256) pretraining; d = 64 (head dim; paper: 'we will not differentiate between d_k and d_v and just use d'). Speed-only sweep: n up to 65536, k up to 2048 (inference table, random data, no quality measured).", "evaluation": "With eps^2-eps^3 <= 4/27: n-branch minimum feasible k = (27/4)*5*ln(nd); at (n=512, d=64): 6.75*5*10.397 = 350.9; larger for larger n. d-branch minimum (proof reading, 9 log d): 6.75*9*ln(64) = 252.7, n-independent. Deployed k=128 < min(252.7, 350.9): NO eps < 1 certified by either branch. Deployed k=256 >= 252.7: d-branch certifies eps^2-eps^3 = 37.43/256 = 0.1462 -> eps ~ 0.62 (62% relative error), at every n; n-branch stays infeasible at all deployed quality n. Under the statement's printed d-branch (9d log d) the minimum k is 6.75*9*64*ln(64) = 16,166 - every deployed config outside on both readings. Speed sweep at (n=65536, k=2048): n-branch 5*ln(65536*64)/2048 = 0.0372 -> eps ~ 0.22, the only deployed point with a sub-0.5 certificate, but no quality is measured there.", "verdict": "outside"}, {"arxiv_id": "2006.04768", "claim_anchor": "lm:jl_1 / lm:jl_2 (linformer_combined.tex:567)", "bound": "For R (k x n), 1 <= k <= n, i.i.d. N(0, 1/k) entries: Pr(||Rx|| <= (1+eps)||x||) > 1 - e^{-(eps^2-eps^3)k/4} and Pr(||xR^TRy^T - xy^T|| <= eps||xy||) > 1 - 2e^{-(eps^2-eps^3)k/4}. Fully pinned constants.", "deployed_config": "Deployed Linformer projections E_i, F_i are TRAINED parameters (with headwise/key-value/layerwise sharing, linformer_combined.tex:359-364), never fresh i.i.d. N(0,1/k) draws; k = 128/256, n = 512-4096.", "evaluation": "The lemma's probability is over the draw of R; deployment fixes learned matrices and draws nothing, so no deployed run instantiates the hypothesis - the certificate does not attach to any trained E, F. For reference, IF the deployed shapes were Gaussian-instantiated: at k=256, eps=0.62 the per-(x,y) failure bound is 2e^{-0.1462*256/4} = 2e^{-9.36} ~ 1.7e-4 (computable, meaningful); at k=128, eps<1 gives exponent >= -(4/27)*128/4 = -4.74, failure bound >= 2e^{-4.74} ~ 1.7e-2 per pair, before the 2n-term union bound over rows (2*512*e^{-4.74} > 1: vacuous at n=512).", "verdict": "not_evaluable"}, {"arxiv_id": "2006.06147", "claim_anchor": "remark:errorbound (ika.tex:371-378); the paper's only resource-to-tolerance guarantee (its other claims are exact factorization identities eq:remark_transformer/eq:remark_gat and rate-free limits propr:square, eq:pnorm, outside the audit kinds)", "bound": "Non-stationary kernel approximation with R sampled spectral points w_1, w_2 over a compact M of diameter D in R^d: (i) P[sup-error E >= eps] <= 2^8 (D sqrt(sigma_1^2+sigma_2^2)/eps)^2 exp(-R eps^2/(2(d+1))), sigma_i^2 = E[w_i^T w_i]; (ii) E <= eps at constant probability when R = Omega((d/eps^2) log(D sqrt(sigma_1^2+sigma_2^2)/eps)). Part (i) has explicit constants; part (ii) is Omega-only.", "deployed_config": "IKA_S / IKA_NS / IKAN / MIKAN (~28.1-28.2M params) and IKAN-direct (28.3M) on text classification (MR/CR/SST/TREC), translation, regression (results/2006.06147.json). The deployed spectral sample count R is reported NOWHERE in ika.tex ('R is the number of samples determined by a modeler', ika.tex:434, with no value given); d (head dim), D, and sigma_1^2+sigma_2^2 (moments of the LEARNED implicit spectral densities q(z|h)) are likewise unrecorded.", "evaluation": "Part (i) is a computable formula, but all four deployed inputs (R, d, D, sigma_1^2+sigma_2^2) are unpinned by the paper, so the certified failure probability cannot be evaluated at any deployed config. Non-vacuousness gate for reference: RHS < 1 requires R > 2(d+1) ln(2^8 (D sqrt(sigma_1^2+sigma_2^2)/eps)^2)/eps^2. Illustration ONLY (hypothetical values NOT licensed by the paper): at d=64, eps=0.1, D sqrt(sigma_1^2+sigma_2^2)=10, the gate is R > 130 * ln(2.56e6)/0.01 ~ 1.9e5 spectral points - far beyond any plausible per-head sample count for an O(T^2 R) attention. Additionally, two of the paper's own deployed variants step outside the hypothesis irrespective of R: IKAN-direct optimizes spectral points as parameters (delta-mixture, no i.i.d. sampling at all), and IKA_NS/MIKAN sample from input-conditioned estimators q(z|h) rather than fixed densities with i.i.d. draws.", "verdict": "not_evaluable"}, {"arxiv_id": "2009.14794", "claim_anchor": "thm:uniform (theory.tex:64); generalized version uniform_convergence (appendix.tex:1193)", "bound": "If query/key L2 norms <= R, then with m = Theta((d/delta^2) log(4 d^{3/4} R/delta)) ORTHOGONAL random features, delta = eps/(h*)^2, h* = exp(l^2/2), l = R d^{-1/4}, the TRIGONOMETRIC estimator's attention matrix satisfies ||A-hat - A||_inf <= eps with constant probability. Unwound: m proportional to (d/eps^2) exp(2 R^2/sqrt(d)) log(...). Appendix version: same shape for generalized attention with K an RBF kernel.", "deployed_config": "Default m = 256 features ('our default m = 256 number of features', appendix.tex:50; hyperparams appendix.tex:36-39), d_head = 64 ((n_heads, n_layers, d_ff, d) = (8,6,2048,512) regular, (8,36,1024,512) TrEMBL, experiments.tex:16,55). Estimator actually deployed in every quality-reported run: POSITIVE features (FAVOR+, Performer-softmax) or ReLU generalized attention - NOT the certified trig estimator. The sole trig deployment is the PG-19 TRIG ablation, which the paper reports as 'highly unstable', with 'even NaN values in training' (experiments.tex:45). R (query/key norm bound) is recorded nowhere.", "evaluation": "Estimator-scope check: the theorem's mechanism precondition ('estimators SM-trig with ORFs') is violated by every headline configuration - positive features and ReLU kernels are different estimators, and the appendix uniform_convergence theorem does not cover ReLU either (ReLU is not an RBF kernel). So the certificate attaches only to the PG-19 TRIG run, which diverged. Even for that run the m-formula cannot be checked: Theta constant unpinned and R unrecorded. Constant-1 illustration, NOT licensed by the paper: at d=64, eps=0.1, and an implausibly favorable R=1: l=0.354, h*=1.065, delta=0.088, m ~ (64/0.0078)*ln(90.5/0.088) = 8208*6.93 ~ 5.7e4 features vs 256 deployed (222x short). At spiky-attention norms R^2 = sqrt(d) ln(n) (n=4096): (h*)^2 = 4096, delta = 2.4e-5, m ~ 1.9e12.", "verdict": "outside"}, {"arxiv_id": "2009.14794", "claim_anchor": "mse-lemma (theory.tex:8)", "bound": "Exact, constant-free MSE formulas at m i.i.d. features: MSE(SMtrig_m) = (1/2m) exp(||x+y||^2) SM^{-2}(x,y) (1-exp(-||x-y||^2))^2; MSE(SM+_m) = (1/m) exp(||x+y||^2) SM^2(x,y) (1-exp(-||x+y||^2)); MSE(SMhyp+_m) = (1/2)(1-exp(-||x+y||^2)) MSE(SM+_m).", "deployed_config": "m = 256 features, d_head = 64, all quality runs (protein L=1024/8192, ImageNet64 L=12288, PG-19). The data-dependent arguments ||x+y||, ||x-y||, SM(x,y) (norms of trained per-head queries/keys) are recorded nowhere in the paper or results/2009.14794.json.", "evaluation": "The formulas are fully pinned in (m, x, y) but the deployed query/key norms are unrecorded, so no numeric MSE is computable for any real run. Pinned gates at deployed m=256: the positive estimator's relative MSE, exp(||x+y||^2)(1-exp(-||x+y||^2))/m, exceeds 1 (100% relative error) as soon as ||x+y||^2 >= ln(256) = 5.55, i.e. ||x+y|| >= 2.36 - reached by e.g. aligned q,k of norm 1.18 each. The trig estimator's MSE carries SM^{-2}(x,y), diverging when SM is small - the paper's own instability diagnosis for TRIG-PG-19 ('renormalizer D^-1 robustification' motivation). No deployed run records where its inputs sit relative to these gates.", "verdict": "not_evaluable"}, {"arxiv_id": "2009.14794", "claim_anchor": "var-theorem (theory.tex:35); ort-theorem (theory.tex:47); general-var-theorem (appendix.tex:815); general-ort-theorem (appendix.tex:803); iid-lemma (appendix.tex:791)", "bound": "Orthogonal random features strictly improve on i.i.d.: MSE(SMort+_m) <= MSE(SM+_m) - (2(m-1)/(m(d+2)))(SM(x,y) - exp(-(||x||^2+||y||^2)/2))^2, and Chernoff-type tail improvements with correction term theta^4 m(m-1)/(4(d+2)) ||x+y||^4. Every one of these theorems carries the precondition m <= d.", "deployed_config": "Deployed m = 256 features per head with d = d_head = 64 in ALL model configurations (regular (8,6,2048,512), TrEMBL (8,36,1024,512), and even the 'Small' (1,6,64,64) config has d_head = 64); ortho_features = True (appendix.tex:36).", "evaluation": "Precondition check: m <= d requires 256 <= 64 - violated by a factor of 4 in every deployed model. For m > d, m mutually orthogonal vectors in R^d do not exist; the implementation necessarily falls back to block-orthogonal ensembles (4 stacked orthogonal blocks), a mechanism none of the four theorems covers. The deployed configuration therefore provably violates the certified conditions of every orthogonality-improvement result in the paper. Secondary: iid-lemma's printed tail bound has exp(+theta m a) where the Markov derivation gives exp(-theta m a) (sign typo flagged in claims file); as printed the base bound is vacuous (RHS >= 1).", "verdict": "outside"}, {"arxiv_id": "2009.14794", "claim_anchor": "reg-theorem (theory.tex:25)", "bound": "If ||A||_inf <= C for a constant C >= 1, the regularized softmax kernel entrywise underestimates A by at most a factor 1 - 2/d^{1/3} + o(1/d^{1/3}), and never overestimates (upper bound unconditional for d >= 2).", "deployed_config": "SMREG is deployed in the PG-19 experiment ('SMREG allowing faster convergence', experiments.tex:45) at d_head = 64, m = 256; trained attention-matrix sup-norms (the C in the precondition) are recorded nowhere.", "evaluation": "Two blockers: (1) the o(1/d^{1/3}) term is unpinned, so the bound has no numeric content at any finite d; (2) the precondition ||A||_inf <= C is data/training-dependent and never measured. Illustration ONLY (dropping the unpinned o-term, NOT licensed): at deployed d = 64, 1 - 2/64^{1/3} = 1 - 2/4 = 0.5 - the pinned part of the certificate permits entrywise underestimation of the softmax kernel by 50% at deployment scale, i.e. even where it applies it is far from a parity certificate.", "verdict": "not_evaluable"}, {"arxiv_id": "2012.07436", "claim_anchor": "le.M.bound (LaTeX_arxiv.tex:303)", "bound": "Deterministic, constant-free inequality on the ProbSparse query-sparsity measure: ln L_K <= M(q_i, K) <= max_j{q_i k_j^T/sqrt(d)} - (1/L_K) sum_j q_i k_j^T/sqrt(d) + ln L_K, for any q_i in R^d and key set K of size L_K (also when q_i is in K).", "deployed_config": "Informer on ETT/Weather/ECL: encoder input L = L_Q = L_K in {24, 48, 96, 168, 336, 480, 720} (LaTeX_arxiv.tex:1019), d_model = 512, heads {8, 16}; ProbSparse keeps Top-u queries with u = c ln(L_Q), c = 5 (LaTeX_arxiv.tex:299, 616), i.e. u = 23/29/33 at L = 96/336/720.", "evaluation": "The lemma's preconditions are just q, k in R^d and a finite key set - satisfied by every deployed configuration, and the inequality is exact algebra (log-sum-exp bounds), so the certificate holds at all deployed L, d. What it certifies is only that the max-mean surrogate M-bar brackets the true measure one-sidedly within an additive ln(L_K) - at deployed L = 336 the slack term ln(336) = 5.82 nats equals the entire floor value of M. It certifies nothing about the ranking accuracy of Top-u selection under M-bar, and nothing about the attention-output or forecast error of discarding the other L - u queries (at L = 336: keeps 29 of 336 = 8.6%).", "verdict": "inside"}, {"arxiv_id": "2012.07436", "claim_anchor": "prop:max-mean-ranking (LaTeX_arxiv.tex:834); proof at sec:appendix.proof2", "bound": "Under keys k_j ~ N(mu, Sigma): there EXISTS kappa > 0 such that for queries q_1, q_2 in the top interval {q | M(q,K) in [M_m, M_m - kappa)}, M-bar(q_1,K) > M-bar(q_2,K) and Var(qk_1) > Var(qk_2) imply M(q_1,K) > M(q_2,K) 'with high probability' - the correctness claim for ranking queries by the max-mean surrogate.", "deployed_config": "Same as above: L in {24..720}, c = 5, u = 5 ln(L_Q); additionally the deployed M-bar is computed from only U = L_K ln(L_Q) randomly sampled dot products with the rest zero-filled (LaTeX_arxiv.tex:316) - at L = 336 that is 1,954 of 112,896 pairs (1.7%). Trained key vectors, not Gaussian draws.", "evaluation": "Nothing is evaluable at any deployed config: (1) kappa is pure existence - no formula, so whether ANY deployed query pair lies in the certified interval is undecidable; (2) 'high probability' carries no rate (the proof routes through lognormal-sum approximations from cited engineering literature, an n -> infinity CLT appeal, and a Weiner-process approximation - no finite-sample constant anywhere); (3) the Gaussian-key precondition is unverified and generally false for trained keys; (4) the deployed zero-filled U-sample estimator of M-bar is a further approximation the proposition does not model at all. The paper's own factor sweep (c = 3..8, Fig sensi) is the only evidence the selection works.", "verdict": "not_evaluable"}, {"arxiv_id": "2106.03546", "claim_anchor": "t:independent (main_neurips_2021.tex:697), with the logistic-link constant instantiation in the remark at line 719", "bound": "Cumulative regret of the independent-outcomes UCB algorithm <= 4z sqrt(e c_sigma/(1-c_sigma) T alpha(b,d,T,delta)) d log(1+T) w.p. 1-delta, alpha = O[bD^2 + (c_sigma/c_sigma')^2 d log(1+(1/b)(T c_sigma/(1-c_sigma) + log(T/delta))) + ...] with the big-O 'hiding constants independent of T,d,b,D,delta' - i.e. tilde-O(sqrt(bT)). Logistic instantiation (remark:719): c_sigma/(1-c_sigma) = e^D, c_sigma' >= e^{-D}/4, z = 1.", "deployed_config": "Real-data bandit runs: MovieLens-25M / Million Songs / Yelp / MNIST, SVD embeddings d = 10, 100 arms per round, budgets b_t in {1, 5, 10, 50, 100}; MovieLens horizon T ~ 490 rounds (49,047 movies in non-overlapping chunks of 100). Results file: 'hyperparameters grid-searched, best configuration reported' - the deployed confidence widths are tuned, not the theorem's schedule. D (bound on u^T x for the 'true' u) is unrecorded and, on real data, u does not exist (misspecified model).", "evaluation": "Not evaluable on the paper's own terms: alpha's O(.) constant is unpinned, and D is unknowable for real datasets. Constant-1 illustration, NOT licensed by the paper (take the O(.) constant = 1, D = 1, delta = 0.05, b = 100, d = 10, T = 490): c_sigma = 0.731, c_sigma' = 0.197, (c_sigma/c_sigma')^2 = 13.8; alpha ~ 100 + 138 ln(14.3) + 34.1 ln(9.8e5) + ln(9.8e6) ~ 954; regret bound ~ 4 sqrt(e * e * 490 * 954) * 10 * ln(491) ~ 4 * 1858 * 62 ~ 4.6e5 - versus a trivial regret ceiling of ~T = 490. Under this (already generous) reading the bound becomes non-vacuous only around T ~ 1e9 rounds, ~6 orders of magnitude beyond every deployed horizon.", "verdict": "not_evaluable"}, {"arxiv_id": "2106.03546", "claim_anchor": "l:onetimeregret (main_neurips_2021.tex:1033); l:onetimeregret-dependent (main_neurips_2021.tex:1737)", "bound": "Per-round regret <= 4z sum_{i=1}^s eps_{j_i} prod_{h<i}(1 - sigma(Delta_{j_h})) (0 for the empty sequence), conditional on the width event |Delta_j - hat_Delta_j| <= eps_j for all arms (dependent version: conditional widths over prefixes plus Delta + 2eps in [-D, D]).", "deployed_config": "Same real-data runs (d = 10, budgets 1-100, grid-searched hyperparameters). Per-round confidence widths eps_j and true gaps Delta_j = u^T x_j are recorded nowhere - and on real datasets the 'true' u of the width event is a modeling fiction.", "evaluation": "The bound is fully explicit in (z, eps, sigma(Delta)) but none of its inputs is recorded for any deployed round, and the conditioning event cannot even be defined on real data without a ground-truth u. Structure worth noting: the certified per-round regret is width-driven; with tuned (grid-searched) widths replacing the prescribed ones, the width event these lemmas condition on is not guaranteed at any deployed configuration.", "verdict": "not_evaluable"}, {"arxiv_id": "2106.03546", "claim_anchor": "l:approx (main_neurips_2021.tex:1645)", "bound": "Dependent model, greedy vs Bayes-optimal at fixed length s: E[R(J',Y)] >= gamma(s) E[R(J*,Y)], gamma(s) = max{1/s, 1-(s-1)c_sigma/2} (1-1/e) c_sigma'(r_s - l_s)/(z(r_1 - l_s)) for s >= 2 (gamma = 1 for s in {0,1}). Preconditions include: all components of u non-negative; ||u||_1 <= sqrt(d')(z - (1-1/e)c_sigma')/(6(z^2 - (1-1/e)c_sigma'^2)); a reward/loss-sequence inequality.", "deployed_config": "Dep algorithm on Million Songs / Yelp / MovieLens / MNIST with GMM topic-coverage vectors (d' latent topics), budgets b_t in {1, 5, 10, 50, 100}, vanilla (r_s = 1, l_s = 0) and exponential reward/loss scenarios; the true u (its sign pattern and L1 norm) is unrecorded and on real data undefined.", "evaluation": "The gamma(s) formula is explicit but its preconditions are stated on the unknowable true u, so no deployed run can be placed inside or outside. Illustration ONLY at D = 1 (NOT licensed; D unrecorded), logistic link (c_sigma = 0.731, c_sigma' = 0.197, z = 1), vanilla rewards: the norm gate reads ||u||_1 <= 0.150 sqrt(d'); and the certified factor at deployed budgets is gamma(5) ~ 0.2 * 0.632 * 0.197 = 0.025, gamma(10) ~ 0.012, gamma(100) ~ 0.0012 - i.e. even where all preconditions held, the theorem would certify the greedy sequence only 0.1-2.5% of the Bayes-optimal reward at budgets 5-100 (the max{1/s, 1-(s-1)c_sigma/2} factor collapses to 1/s once s >= 3). Only the b_t = 1 runs get a meaningful factor (gamma = 1).", "verdict": "not_evaluable"}, {"arxiv_id": "2503.10571", "claim_anchor": "lem:conf (sections/appendix.tex:173)", "bound": "Ordering preservation of Radar's random-feature segment scores (unscaled variant): sum_i S^(n)_{u_i,z} <= sum_i S^(n)_{v_i,z} fails with probability <= 2 exp(-(sqrt(n)(a'_v - a'_u)/(sqrt(2) exp(zeta^2)))^2), where a'_v = beta_v/(beta_v+beta_u), a'_u = beta_u/(beta_v+beta_u) are the pairwise-normalized true unnormalized-attention masses of the two segments and zeta = max{||u_i||, ||v_i||, ||z||}.", "deployed_config": "Radar deployed with projection dimension n = 2048 and top k = 64 segments, sliding window 1024 (sections/3exp.tex:9, 180: 'we use n=2048 and k=64 by default'); base models Llama-2-7b-chat-hf, Llama-7b, Mistral-7B-Instruct-v0.2 (head dim d = 128), contexts up to full 4k/32k with 1024-4096 middle-token budgets (results/2503.10571.json config_notes); segment size c = sqrt(t) by dynamic restructuring.", "evaluation": "The formula is fully explicit (rare in this corpus) but its two inputs at deployment are unrecorded. Deployed Radar applies phi to k' = k/d^{1/4} (eq:random-feature), i.e. deployment instantiates the scaled variant lem:scaled-conf; for this raw variant, inverting the printed bound at n = 2048: failure probability < 1 requires exp(zeta^2) < sqrt(n/2)*(a'_v - a'_u)/sqrt(ln 2) = 38.44*(a'_v - a'_u), i.e. zeta < 1.91 even at the unattainable maximal gap a'_v - a'_u = 1. zeta (max L2 norm over trained query/key vectors) is never measured anywhere in the paper, and neither are the deployed segment-mass gaps a'_v - a'_u, so the certified confidence at n = 2048 cannot be computed.", "verdict": "not_evaluable"}, {"arxiv_id": "2503.10571", "claim_anchor": "lem:scaled-conf (sections/appendix.tex:222); feeds thm:highprob (sections/2method.tex:95)", "bound": "Scaled (deployment-matching, 1/sqrt(d) temperature) variant: segment-score ordering preserved with confidence >= 1 - 2 exp(-(sqrt(n)(a'_v - a'_u)/(sqrt(2) exp(zeta^2/sqrt(d))))^2). Downstream thm:highprob: top segment correctly retrieved w.p. 1-delta if the attention-mass gap >= (1/c) exp(zeta^2/sqrt(d)) sqrt(8 log(2(c-1)/delta)/n).", "deployed_config": "n = 2048 random features, d = 128, k = 64 selected segments, c = sqrt(t) segments (e.g. t = 4096 -> c = 64); Llama-2-7b-chat / Llama-7b / Mistral-7B-Instruct-v0.2, LongBench + perplexity evals at 1024-4096 middle-token budgets and full context (results/2503.10571.json).", "evaluation": "All constants in the bound are printed, so it inverts cleanly at the deployed (n, d) = (2048, 128): the certificate is non-vacuous (failure prob < 1) iff exp(zeta^2/11.31) < 38.44*Delta with Delta := a'_v - a'_u, giving zeta < 6.43 at the limiting Delta = 1, zeta < 3.90 at Delta = 0.1, and NO zeta (not even 0) for Delta < sqrt(2 ln 2 / n) = 0.026. For thm:highprob at t = 4096 (c = 64, delta = 0.05): required gap = (1/64)*exp(zeta^2/11.31)*sqrt(8 ln(2520)/2048) = 0.00273*exp(zeta^2/11.31), which already exceeds the entire average segment mass 1/c = 0.0156 unless zeta <= 4.44. Whether deployed Llama/Mistral key-query norms satisfy zeta <~ 4-6 is exactly what the paper never records (zeta appears only symbolically; no measured norms, no measured gaps Delta), so the deployed configs cannot be placed inside or outside. Note the norm sits in a double exponent (exp(zeta^2/sqrt(d)) inside exp(-x^2)): the certified confidence swings from ~1 to vacuous over a zeta range of ~2.", "verdict": "not_evaluable"}, {"arxiv_id": "2503.18309", "claim_anchor": "remark:efficiency (Sections/Methodology.tex:423)", "bound": "ETGPSSM reduces GP-related computational complexity from O(d_x M^3) (independent-GP GPSSM) to O(M^3) via a single shared GP; overall complexity O(M^3 + d_x^3), the d_x^3 term from EnKF operations; added shallow-NN cost claimed negligible.", "deployed_config": "All GPSSMs use 20 inducing points by default (Sections/Experiments.tex:34); Lorenz-96 scalability study uses M = 100 with d_x up to 100 (Experiments.tex:97); system-identification tasks d_x = 4 (results/2503.18309.json config_notes); shared NN d_x -> 128 -> 64 -> 2d_x; deployed architecture is exactly one shared GP + flow network with EnKF-based inference.", "evaluation": "Structural preconditions (single GP and flow network shared across d_x dimensions; EnKF inference replacing explicit variational state parameterization) hold by construction in every deployed run, so all deployed configs are inside the certified regime. Arithmetic at deployed values: (M, d_x) = (20, 4): d_x M^3 = 32,000 vs M^3 = 8,000 (certified 4x GP-cost reduction); (100, 100): 10^8 vs 10^6 (100x), overall O(M^3 + d_x^3) = 2x10^6. The paper's measured transition-evaluation times (0.36 s ETGPSSM vs 19.56 s GPSSM at d_x = 100, Experiments.tex:99, ~54x) are consistent in trend. Caveats: big-O constants unspecified, so no wall-clock number is certified anywhere; 'negligible' NN cost is asserted, not bounded (their own caption puts NN params at 258 d_x + 8384, Experiments.tex:76).", "verdict": "inside"}, {"arxiv_id": "2503.18309", "claim_anchor": "expressivity remark (Sections/Methodology.tex:161), resting on corollary:dependent_nonstationary_GP (Methodology.tex:119)", "bound": "ETGPSSM transition functions {f_d}_{d=1..d_x} are dependent and non-stationary; under linear input-dependent flows (f = alpha_t * f_tilde + beta_t, theta_t = nn_w(x_{t-1})) with Dirac weight prior p(w) = delta(w - w_bar), the transition remains a d_x-dimensional GP with the explicit block covariance Lambda_{t,t'} built from k(.,.) and alpha outer products.", "deployed_config": "Deployed ETGPSSM uses precisely the time-varying linear flow (NN maps R^{d_x} to Theta in R^{2d_x}, i.e. per-dimension (alpha, beta); Experiments.tex:34, Methodology.tex 'we focus on the linear flow described in Example time_varying_linear_flow'); two variants deployed: ETGPSSM (DNN) with deterministic weights (= Dirac prior) and ETGPSSM (BNN) with Gaussian weight prior N(0, diag(psi)); d_x = 4 (sys-ID), 100 (Lorenz-96), M = 20/100.", "evaluation": "For the DNN variant (the paper's main configuration; used exclusively in the Lorenz-96 study, Experiments.tex:97) both hypotheses hold by construction: linear input-dependent flow deployed, deterministic weights = delta(w - w_bar). The GP characterization and the covariance formula are exact identities there - deployed inside. Split caveat: the also-deployed BNN variant (system-ID tables) has a non-Dirac Gaussian weight prior, violating the corollary's hypothesis, so for those runs the 'remains a GP' characterization is uncertified (the transition is an implicit mixture process). Nothing quantitative is certified either way: the claim is structural (dependence/non-stationarity), with no tolerance, rate, or risk bound attached at any config.", "verdict": "inside"}, {"arxiv_id": "2503.24358", "claim_anchor": "thm::att_out_bound (section.tex:173)", "bound": "For any query q_n in R^d and n cached KV pairs under any quantizer: ||Attn(q_n, K, V) - Attn(q_n, K^qtz, V^qtz)||_2 <= (sum_i ||v_i - deq(v_i^qtz)||_2 / (2 sqrt(d))) * sum_i |q_n (k_i - deq(k_i^qtz))^T| + sum_i ||v_i - deq(v_i^qtz)||_2.", "deployed_config": "SQuat / SQuat-pre at 2-bit KV quantization, group size G = 32, residual length R = 32 (recent 32 tokens full precision), subspace dim r in {5,...,60}, lambda in {0.0005, 0.001}, g in {32, 64}; Llama-2-7B, Llama-3.1-8B-Instruct, Mistral-7B-Instruct-v0.3, DeepSeek-R1-Distill-Llama-8B (head dim d = 128); LM-Eval + LongBench + math-500 (appendix.tex:409).", "evaluation": "The theorem is universally quantified (any q, K, V, any quantizer) with no numeric precondition, so every deployed configuration trivially satisfies the certified conditions - inside. But the certificate is data-dependent and quantitatively empty at deployment: its inputs (per-token value residuals ||v_i - deq(v_i^qtz)||_2 and query-projected key residuals |q_n(k_i - deq(k_i^qtz))^T| under 2-bit quantization) are never recorded, so no numeric tolerance is computable for any run. Structurally the RHS grows as O(n^2) in cached length n for non-vanishing residuals (product of two sums over n, plus an O(n) term): at LongBench-scale n ~ 10^3-10^4 even per-token residual products of ~1e-4 make the bound far exceed the O(1) scale of an attention output, so the theorem functions as motivation (drive q-projected key residuals to 0, which SQuat's subspace constraint targets), not as a deployed-regime accuracy certificate.", "verdict": "inside"}, {"arxiv_id": "2503.24358", "claim_anchor": "prop::comp_reduce (appendix.tex:319); deployed path via lem::block_quan (appendix.tex:138) + lem::rm_block (appendix.tex:287)", "bound": "Iterative coordinate-wise key quantization via prop::upd_key costs O(d^4) naively (sum_t t^3); using the rank-one downdate of prop::rem_one to update A_t^{-1} reduces it to O(d^3).", "deployed_config": "Deployed SQuat quantizes g elements per iteration with g = 64 (LM-Eval: 'resulting in just two iterations of our algorithm' at d = 128) or g = 32 (some LongBench configs); subspace rank r = 5-60 << d = 128 (appendix.tex:409).", "evaluation": "The audited proposition covers the coordinate-wise (g = 1) algorithm, which no deployed configuration runs; deployment uses the block-wise generalization, but that path is separately certified by the appendix chain: lem::block_quan gives the closed-form block update and lem::rm_block the block downdate, with the same O(d^3) accounting (initial inverse of the d x d matrix I + lambda Q^T Q is O(d^3), or O(r d^2) via Woodbury per appendix.tex:325 footnote - at deployed r = 5, d = 128 that is ~5*128^2 = 8.2e4 vs 128^3 = 2.1e6). At deployed d = 128, g = 64: T = d/g = 2 iterations, cost dominated by the single d x d inverse - comfortably within the certified O(d^3) shape. Big-O constants unspecified, so no wall-clock or FLOP count is certified; the paper's throughput/latency wins (Figure fig:mem_throughput) are measured, not derived from the proposition.", "verdict": "inside"}, {"arxiv_id": "2504.04308", "claim_anchor": "thm:gla=wpgd (sec/gla_to_gd.tex:63)", "bound": "Exact identity: a single-layer GLA whose attention weights follow the restricted construction eq. (att mtx res), parameterized by (P_k, P_q), computes f_GLA(Z) = x^T beta_1^gd(P_k, P_q, Omega) - one step of weighted preconditioned GD with weights Omega built from cumulative elementwise products of the gating rows. No tolerance parameter exists (identity or nothing).", "deployed_config": "No results/2504.04308.json exists; deployed configs recovered from sec/exp.tex: K = 2 tasks, d = 10, p = 5 (delimiter dim), context n_bar in [0, 50] per task, sigma = 0 (noiseless labels), Sigma = I, sigmoid gating phi(z) = 1/(1+e^-z), delimiters ~ N(0, I_5), trained 10^4 Adam iterations, best of 10 seeds.", "evaluation": "The paper concedes the scope gap explicitly (sec/exp.tex:24): 'although the theoretical results are based on the model constructions (c.f. (att mtx res) and (att mtx res delimiter)), we do not restrict the attention weights in our implementation.' The certified regime is a measure-zero submanifold of weight space (W_k, W_q, W_v of the specific block form), deployment trains unrestricted weights, and the trained matrices are never recorded - so whether any deployed model lands on (or near; the theorem has no perturbation term) the certified construction is unverifiable. The empirical alignment of trained-GLA risk with the WPGD-optimal dashed curves is evidence, not a certificate: the theorem certifies the equivalence only on the construction.", "verdict": "not_evaluable"}, {"arxiv_id": "2504.04308", "claim_anchor": "lemma:multi layer (sec/gla_to_gd.tex:106)", "bound": "Exact identity: an L-layer GLA with per-layer restricted weights (P_{k,l}, -P_{q,l}), gating a function of the features only (G_i = G(x_i)), and zero initialization implements L steps of data-dependent WPGD (beta_hat_l recursion with (1 - alpha_l) damping). No error term.", "deployed_config": "Multi-layer experiments (sec/exp.tex:65-68): 1-, 2-, 3-layer GLA at (r_1, r_2) = (0, 1), same setup as above (d = 10, p = 5, K = 2, sigmoid scalar gating over full tokens z = (x, y, c), unrestricted trained attention weights).", "evaluation": "Two preconditions are unverifiable at deployment: (1) the restricted per-layer weight construction is not enforced ('we do not restrict the attention weights in our implementation', sec/exp.tex:24) and trained weights are unrecorded; (2) the theorem requires gating G_i = G(x_i) (features only), while the deployed gate phi(w_g^T z) reads the full token including the label and delimiter coordinates - whether the trained w_g has support only on feature coordinates is unrecorded. The paper's Figure fig:0 1 multi shows deeper models improving, 'aligning with' the L-step WPGD prediction, but the identity itself is certified for no deployed model. Exact-identity claim: no tolerance exists to evaluate even in-regime.", "verdict": "not_evaluable"}, {"arxiv_id": "2504.04308", "claim_anchor": "corol multi (sec/gla_to_gd.tex:141)", "bound": "Exact identity: L-layer linear attention with causal mask, residual connections, and restricted weights (P_{k,l}, -P_{q,l}) implements damped multi-step preconditioned GD: o_{n+1,l} = -x^T beta_hat_l with beta_hat_l = (1 - alpha_l) beta_hat_{l-1} - P_l X^T (y_l - y).", "deployed_config": "Same experimental campaign; the multi-layer runs reported are GLA models (sec/exp.tex:68); single-layer linear attention (LinAtt) deployed with unrestricted trained weights.", "evaluation": "Same blocker as thm:gla=wpgd / lemma:multi layer: the certified regime is the restricted weight construction, deployment trains free weights and never records them (paper's own statement at sec/exp.tex:24). Additionally the multi-layer experiments train GLA rather than plain linear attention, so the corollary's exact object appears in no deployed multi-layer run at all. No tolerance parameter exists.", "verdict": "not_evaluable"}, {"arxiv_id": "2504.04308", "claim_anchor": "corol:att vs gla (sec/gla_landscape.tex:137)", "bound": "L*_att >= L*_gla (gating never hurts, given Assumption assum delimiter gate); under Assumption assum order additionally L*_att - L*_gla = d * r^T (R_+^{-1} - (1 1^T)/(1^T R_+ 1)) r >= 0 with R_+ = R + (d + sigma^2 + 1) I. These are statements about optimal risks of the model classes, pinned entirely by the data/gating configuration.", "deployed_config": "Deployed synthetic configuration pins every symbol: d = 10, sigma = 0, Sigma = I, K = 2 equal context lengths n_1 = n_2 = n_bar in [0, 50]; R = blockdiag(1 1^T, 1 1^T), r = [r_1 1; r_2 1] with (r_1, r_2) in {(0,1), (0.2,0.8), (0.5,0.5), (0.8,0.2)}; delimiters c_0..c_2 ~ N(0, I_5) (linearly independent a.s.), phi = sigmoid (continuous, phi(-inf)=0, phi(+inf)=1).", "evaluation": "Preconditions check at deployed values: assum delimiter gate holds in all four settings (3 Gaussian vectors in R^5 are a.s. independent; sigmoid qualifies). Assumption assum order (E[t_1^T t] <= E[t_2^T t], i.e. r_1 <= r_2) holds for (0,1), (0.2,0.8), (0.5,0.5) but is VIOLATED at (0.8, 0.2) (8 > 2), so the exact-gap formula is uncertified for that run - consistently, the paper's own Figure fig:0.8 0.2 shows GLA collapsing to LinAtt performance there. Worked gap at (r_1, r_2) = (0,1), n_bar = 50: R_+^{-1} block = (1/11)(I - 1 1^T/61), r^T R_+^{-1} r = 50/61 = 0.820, (1^T r)^2 / (1^T R_+ 1) = 2500/6100 = 0.410, gap = 10*(0.820 - 0.410) = 4.10 (0.41 after the paper's 1/d normalization) > 0 - matching the plotted separation between the L*_att and L*_wpgd dashed curves. Minor caveat: deployed sigma = 0 is the degenerate corner of the 'Gaussian label noise sigma^2' data model (all formulas remain finite: d + sigma^2 + 1 = 11).", "verdict": "inside"}, {"arxiv_id": "2504.04308", "claim_anchor": "unlabeled multi-layer linear-transformer lemma (supp/multi_layer.tex:34)", "bound": "Exact identity: L-layer linear transformer with layers parameterized by P_l implements masked-residual GD: yhat^(l) = x^T beta^(l), B^(l+1) = B^(l) - (M (.) 1 (D^(l) - y)^T) X, B^(0) = 0.", "deployed_config": "Same campaign; deployed models trained with unrestricted weights (sec/exp.tex:24); trained parameters unrecorded.", "evaluation": "Appendix companion of corol multi with the same scope structure: certified only under the layer parameterization eq. (multi P), which the implementation does not enforce and whose satisfaction post-training is unrecorded. Exact identity, no tolerance.", "verdict": "not_evaluable"}, {"arxiv_id": "2504.04308", "claim_anchor": "unlabeled appendix GLA variant with +P_{q,l} (supp/multi_layer.tex:162)", "bound": "Exact identity: sign-flipped variant of lemma:multi layer (W_{q,l} parameterized by +P_{q,l}): beta_hat_l = (1 + alpha_l) beta_hat_{l-1} + P_{q,l}-preconditioned gate-weighted step.", "deployed_config": "Same campaign; unrestricted trained weights, feature-only gating not enforced (deployed gate reads full tokens).", "evaluation": "Same two unverifiable preconditions as lemma:multi layer (restricted per-layer construction; G_i = G(x_i)); trained parameters unrecorded. Exact identity, no tolerance.", "verdict": "not_evaluable"}, {"arxiv_id": "2504.08964", "claim_anchor": "theorem_1 (blur.tex:505)", "bound": "BLUR universal approximation: sup_{v in V} ||F-hat(v) - F(v)|| <= epsilon for any causal seq-to-seq map F, PROVIDED the LRUs L_1, L_2 have width n >= dim(V) and the MLP has width D >= O(nN/epsilon^2).", "deployed_config": "Deployed BLUR (blur.tex:956-990, Table seq_images_full): 4 LRU layers per direction, LRU latent size n = 128, embedding dim 256 ('all linear and nonlinear projections do not change the hidden dimensions'); datasets sMNIST/psMNIST (d=1, N=784), sCIFAR (d=3, N=1024), ListOps, IMDB Text, plus ETT/weather forecasting at horizons 24-960 (results/2504.08964.json).", "evaluation": "Width precondition check: the paper's own analysis setting declares non-sparsity ('In this study, we only consider the non-sparse setting', blur.tex proof of lemma_1), under which dim(V) = dN. Then sMNIST requires n >= 784, sCIFAR n >= 3072, forecasting n >= dN of the input window - deployed n = 128 violates the hypothesis on every dataset (6.1x short on sMNIST, 24x on sCIFAR). Independently, D >= O(nN/eps^2) has an unpinned constant; constant-1 illustration NOT licensed by the paper: n=128, N=784, eps=0.1 gives D >= 128*784*100 = 1.0e7 hidden neurons vs deployed MLP width ~128-256 (a ~4e4x gap); even eps = 1 gives D >= 100,352. Deployment is also a 4-layer stack rather than the single certified composition L_3.G.H.(L_1+L_2).E. If one rejects the non-sparse reading, dim(V) is unmeasured and the verdict degrades to not_evaluable - either way no deployed config is certified.", "verdict": "outside"}, {"arxiv_id": "2504.08964", "claim_anchor": "lemma_1 (blur.tex:877)", "bound": "Lossless memory: a linear RNN whose hidden dimension scales with mN can recover all encoded input prefixes (u_i)_{i=1}^k from h_k without information loss (scalar-case mechanism pinned in the proof: 'When n >= k, we can use the Moore-Penrose inverse of V_{f,k} to reconstruct u_{1:k}', blur.tex:900); reducible to O(P) only if dim(U) = P < mN (linearly compressible).", "deployed_config": "Deployed hidden width n = 128, embedding dim m = 256; sequence lengths N = 784 (sMNIST/psMNIST), 1024 (sCIFAR), ~1000-2000 (Text/ListOps).", "evaluation": "Provable violation, constant-free: exact recovery of (u_i)_{i=1}^k in R^{mk} from h_k in R^n requires an injective linear map R^{mk} -> R^n, impossible whenever n < mk (rank/pigeonhole; the paper says it itself: 'it is impossible to store O(N) real numbers in a vector whose size is smaller than O(N) through a linear Vandermonde projection, unless the inputs are sparse' - and it then restricts to the non-sparse setting, closing the O(P) escape). Arithmetic: certified regime needs n ~ mN = 256*784 = 200,704 (sMNIST) or 262,144 (sCIFAR) vs deployed n = 128 - a 1568-2048x shortfall; even the scalar-case pinned condition n >= k fails for every prefix k > 128 out of N = 784/1024. Since theorem_1's proof routes through this recovery (f_k(h_k) = F(v_{1:k})), the deployed 128-wide LRUs provably cannot realize the certified construction for any sequence longer than 128 tokens.", "verdict": "outside"}, {"arxiv_id": "2504.08964", "claim_anchor": "theorem_2 (blur.tex:881)", "bound": "Barron-type MLP width bound: D >= 2 n r^2 C_f^2 epsilon^{-2} hidden sigmoidal neurons suffice for sup_{||h|| <= sqrt(n) r} |f(h) - G(h)| <= epsilon, for f with Barron constant C_f and hidden-state radius r.", "deployed_config": "n = 128 LRU width; deployed MLP width not explicitly tabulated (architecture note implies projections preserve hidden dims, ~128-256); tasks: image/text classification and forecasting maps whose Barron constants are unknown.", "evaluation": "The numeric constant (2) is printed, so the bound inverts symbolically: certified accuracy at deployed width D is epsilon = r C_f sqrt(2n/D) = r C_f sqrt(256/D); at D = 256 this is epsilon = r C_f, i.e. no better than the product of two quantities the paper never measures. Blockers, exactly: C_f (Barron constant of the target task map - never estimated for MNIST/CIFAR/ETT targets), r (sup over deployed hidden-state coordinates - never measured), and even D (deployed MLP width not in the hyperparameter table). No deployed config can be placed inside or outside.", "verdict": "not_evaluable"}, {"arxiv_id": "2504.08964", "claim_anchor": "lemma_2 (blur.tex:932)", "bound": "BLUR head width: D <= 4 n r^2 C_{F_k}^2 ||Theta_{f,k} + Theta_{b,k}||_2^2 s^3 epsilon^{-2}, conditional on the MLP G sufficing to approximate f_k = F_k . (Theta_{f,k} + Theta_{b,k}) at level epsilon from h-hat_k.", "deployed_config": "n = 128, N = 784-1024; Theta_{f,k}, Theta_{b,k} are the reconstruction functionals induced by Moore-Penrose inverses of the forward/backward Vandermonde matrices of the deployed (trained) eigenvalues.", "evaluation": "Four unpinned quantities: C_{F_k} (Barron constant of the target), r (hidden-state radius), ||Theta_{f,k} + Theta_{b,k}||_2 (pseudo-inverse Vandermonde norms of the trained eigenvalue configuration - unrecorded, and Vandermonde conditioning is notoriously extreme), and the output-dim factor s^3's normalization. Additionally the lemma's premise inherits lemma_1's reconstruction reading, which fails at deployed n = 128 < k for k > 128 (see lemma_1 record). No arithmetic is possible at any deployed config.", "verdict": "not_evaluable"}, {"arxiv_id": "2504.09936", "claim_anchor": "thm:attn_collapse (main.tex:297)", "bound": "Impossibility for convex-combination KV merging (k_r = w_e k_e + w_c k_c, v_r = w_e v_e + w_c v_c, w_e + w_c = 1): the merged pair's attention score satisfies A'_r < A_e + A_c strictly, hence ||o'_t - o_t|| > 0 - weighted merging is never perturbation-free.", "deployed_config": "The paper itself deploys the convex-combination baselines CaM and D2O at 20% KV-cache compression on Llama-2-7B/13B, Llama-3-8B, Mistral-7B across 13 LongBench tasks (results/2504.09936.json); its own method KeepKV deploys ZIP-Merging + Electoral Votes (p_r = p_e + p_c, log-ratio key scaling - not a convex combination).", "evaluation": "Precondition check: CaM and D2O runs satisfy the convex-combination hypothesis (D2O cosine-similarity weights, w_e + w_c = 1 per main.tex:295), so the paper's own baseline deployments sit inside the certified-failure regime and the theorem certifies nonzero perturbation for every merge they perform. KeepKV's deployments are deliberately outside the regime - that is the paper's design point, not a scope mismatch. Quantitative content: none - the theorem certifies only strict positivity (||o'_t - o_t|| > 0) with no magnitude, so the observed LongBench score gaps between KeepKV and CaM/D2O are not bounded by it in either direction.", "verdict": "inside"}, {"arxiv_id": "2504.09936", "claim_anchor": "thm:KeepKV_RE (main.tex:364; generalized as thm:app:KeepKV_RE, body/appendix.tex:318)", "bound": "Multi-step perturbation of KeepKV: if |1 - s_i^{t'}-hat / s_i^{t'}| <= epsilon < 1 for ALL i in [t'] and ||v_i - v_j|| <= gamma for all i in [t'], j in the merged set, then Theta_{t'} < 2 epsilon (1 + epsilon) gamma / (1 - epsilon)^2.", "deployed_config": "KeepKV with EMA-with-bias-correction attention-score prediction (Eq. keepkv_predict; recent-window w after prefill), merging threshold T = 0.8 (cosine key similarity; 0.6-0.95 ablated), PyramidInfer allocation, 20% compression, Llama-2-7B/13B, Llama-3-8B, Mistral-7B on LongBench.", "evaluation": "The formula is explicit (e.g. at epsilon = 0.1: Theta < 0.27 gamma; epsilon = 0.5: Theta < 6 gamma; diverges as epsilon -> 1), but neither input is ever measured: epsilon must be a UNIFORM per-token relative prediction error over all i in [t'] (Figure 2(d) plots average relative errors graphically with no recorded values, and a single token with s-hat/s >= 2 voids the epsilon < 1 precondition), and gamma (max pairwise value-vector spread) appears nowhere numerically. The paper's own appendix concedes the practical gap: 'predicting attention distributions further into the future is inherently challenging, leading to a significant increase in the estimated perturbation upper bound' and 'the inherent input differences gamma cannot be ignored' (body/appendix.tex:423). No deployed config can be certified to satisfy epsilon < 1, let alone assigned a numeric tolerance.", "verdict": "not_evaluable"}, {"arxiv_id": "2504.09936", "claim_anchor": "lemma:RE_towards_0 (main.tex:369; body/appendix.tex:325)", "bound": "Zero-perturbation limits: Theta_{t'} = 0 if either epsilon = 0 (exact prediction of future attention scores) or the merged candidates are identical, (k_e, v_e) = (k_c, v_c).", "deployed_config": "Deployed merging threshold T = 0.8 on cosine key similarity (main.tex:433; appendix ablates 0.6-0.95) - i.e. merges are performed precisely on NON-identical pairs with similarity in [0.8, 1); scores predicted by EMA (Eq. keepkv_predict).", "evaluation": "Both disjuncts of the certified regime fail at every deployed config: (1) epsilon = 0 requires the EMA predictor to exactly forecast future-step attention scores, and the paper's own Figure 2(d) shows nonzero relative prediction errors for KeepKV (its appendix concedes prediction error grows with horizon); (2) (k_e, v_e) = (k_c, v_c) would restrict merging to exact duplicates, while the deployed T = 0.8 threshold exists precisely to merge non-identical pairs (an identical-only policy would compress almost nothing). So the zero-perturbation certificate covers a corner deployment never occupies - same shape as the SMYRF/EVA findings: the lossless regime and the deployed regime are disjoint by construction, and the actual deployed error is governed by the (not_evaluable) thm:KeepKV_RE bound. The lemma's legitimate deployed role is qualitative: it motivates similarity-driven candidate selection (main.tex:418).", "verdict": "outside"}, {"arxiv_id": "2505.09022", "claim_anchor": "thm.UAT (body.tex:101)", "bound": "Two-part expressivity split with the step sizes FIXED TO CONSTANTS (Delta^(i) in eq.ZOH, Delta_k^(i) in eq.ZOHmamba): (1) S4D positive UAT - for any Lipschitz non-polynomial sigma, any continuous G on [0,1]^L, any eps > 0, there EXIST d >= 1, n >= 1 and parameters with |G-tilde(u) - G(u)| <= eps everywhere; (2) S6 impossibility - for any sigma and L >= 3 there EXIST a continuous G on [0,1]^{L-1} x {1} and SOME eps > 0 such that every single-layer S6 (all d, n, parameters) errs by > eps somewhere. Certified regime of the impossibility: single-layer S6 with constant Delta_k^(i).", "deployed_config": "Toy illustration of the theorem: single-layer S6 and S4D trained on the cosine-coefficient task 'also allowing the Delta-related parameters to be learned in both cases' (body.tex:118). Main experiments are all multi-layer: 4-layer ablation models, LRA models with more layers and h=8 (supplement.tex:464), ~30-layer ~250M-param LM models on SlimPajama-6B (supplement.tex:520); the Mamba/S6 comparison baselines are standard Mamba with input-dependent Delta_k = softplus(w^T u_k + b^(i)) (eq.ZOHmamba, body.tex:77-79).", "evaluation": "Precondition check for the S6 impossibility at every deployed S6 config: (a) fixed-Delta hypothesis - violated everywhere: deployed S6/Mamba computes Delta_k^(i) = softplus(w^T u_k + b^(i)), input-dependent by construction, and even the paper's own single-layer toy experiment lets the Delta parameters be learned (body.tex:118); (b) single-layer hypothesis - violated by all benchmark models (4 to ~30 layers). The paper concedes both exits explicitly: 'In practice, Mamba architectures typically use multiple stacked layers and do not fix Delta_k^(i), both of which enhance the expressiveness' (body.tex:116). Additionally the impossibility is existential in (G, eps): it certifies failure only on a proof-constructed G at an unquantified eps, so even a compliant deployment would receive no failure margin on sCIFAR-10/LRA/wavesum targets. S4D positive part: existence-only with no (d,n)-to-eps rate - at deployed d = 128, n fixed, no tolerance is certified.", "verdict": "outside"}, {"arxiv_id": "2505.09022", "claim_anchor": "thm.B2S6UAT (body.tex:358)", "bound": "B2S6 (block-biased Mamba) single-layer universal approximation on [0,1]^{L-1} x {1}, with step sizes ^jDelta_k^(i) fixed to constants, via EITHER mechanism alone: part 1 - all bias terms ^jB_bias^(i) = 0, existence of h, p (h*p = d >= 1), n >= 1 and parameters with sup error <= eps; part 2 - h = 1, p = d, existence of d, n and parameters (including the bias) with sup error <= eps. No rate ties (d, h, p, n) to eps.", "deployed_config": "Flagship B2S6: h = 8 blocks, bias term ON, complex parameters, input-dependent ^jDelta ('For all experiments, we set h = 8', supplement.tex:464; discretization ^jDelta = softplus(^js_Delta(x) + ^jb), Algorithm block, body.tex:340-348), multi-layer (4-layer ablation, deeper LRA models, ~30-layer LM). Ablation grid at d = 128 sweeps h in {1,...,128}, p = d/h, with-bias / without-bias / real-vs-complex (results/2505.09022.json config_notes; table body.tex:396).", "evaluation": "The deployed flagship configuration satisfies NEITHER certified corner: part 1 requires bias = 0 (deployed runs bias ON, and the ablation shows the bias is load-bearing: 87.19 with bias vs 81.54 without at h=8, d=128); part 2 requires h = 1 (deployed h = 8; the h=1-with-bias ablation row scores 81.56 vs 87.19 at the deployed h=8). All deployed variants also violate the fixed-Delta hypothesis (input-dependent softplus selection) and the single-layer architecture eq.onelayer. Beyond the corner mismatch, the guarantee is existence-only: no formula maps deployed (d, h, p, n) = (128, 8, 16, n) to any eps, so no tolerance could be computed even for the hypothesis-matching ablation rows.", "verdict": "not_evaluable"}, {"arxiv_id": "2505.12252", "claim_anchor": "rmfath (neurips_2025.tex:213) + expTheo (neurips_2025.tex:318)", "bound": "rmfath: attn_K(Q,K,V) ~ RMFA_K(Q,K,V) - stated as an approximate equality with NO error term, preconditions: K is a dot-product kernel with only non-negative Maclaurin coefficients, Phi_K a random Maclaurin feature map, inputs scaled by d^{-1/4}. expTheo: E[SchoenbAt_K(Q,K,V)] = attn_K(Q,K,V) (unbiasedness), preconditions: gamma, beta 'properly trained to restore the attention output', Phi_K an RMF map R^d -> R^D. Neither statement ties D to any tolerance.", "deployed_config": "Five deployed kernels exp, inv, logi, trigh, sqrt with printed Maclaurin coefficients a_N = 1/N!, 1, 1/min(1,N), 1/N!, max(1,2N-3)/(2^N N!) (kernel table, neurips_2025.tex:230-244); RMF implemented per Wacker with p = 2 (neurips_2025.tex:397). Synthetic validation: n=100, d in [10,200], D in [10,50], 'parameters gamma and beta are set to the ideally trained values' (neurips_2025.tex:401). LRA: embedding 64, 2 heads (per-head d = 32), 2 layers, default random projection D = 128, ppSBN eps = 1e-13 (neurips_2025.tex:448); gamma, beta trained end-to-end.", "evaluation": "Precondition check: all five deployed kernels have non-negative printed Maclaurin coefficients, and the deployed feature map is the required RMF construction, so the structural hypotheses hold at every config. The gamma/beta hypothesis is satisfied by construction only in the synthetic validation (set to ideal values); in the LRA runs gamma, beta are trained for task loss with no restoration criterion, so the unbiasedness certificate is unverifiable there. Quantitatively: rmfath's '~' carries no epsilon, and expTheo certifies expectation only - D = 128 and D = 1 receive the identical certificate; zero tolerance is certified at any deployed D.", "verdict": "inside"}, {"arxiv_id": "2505.12252", "claim_anchor": "pacTheo (neurips_2025.tex:328)", "bound": "P(|SchoenbAt_K(D) - attn_K(D)| > eps) <= 2D exp(-D eps^2 / (2 S^2 d^2)) for all eps > 0, with D = random-feature dimension, d = attention input dimension, S = max |V_ij|; preconditions: |V_ij| <= S, gamma/beta properly trained, Phi_K an RMF map. All constants printed - fully invertible.", "deployed_config": "LRA flagship: D = 128 random features, embedding 64 with 2 heads (per-head d = 32; d = 64 if the un-split embedding is used) (neurips_2025.tex:448). Synthetic error study: d in [10,200], D in [10,50], n = 100, gamma/beta ideal (neurips_2025.tex:401). Speedup study: d = 50, D in [2,120] (neurips_2025.tex:405). S is data-dependent (max |V| entry) and never recorded, but it cancels: the bound's certified eps scales linearly in S.", "evaluation": "Invert the bound: it is nontrivial (probability < 1) only for eps > S*d*sqrt(2 ln(2D)/D). At the LRA config (D=128): sqrt(2 ln 256/128) = sqrt(11.090/128) = 0.294, so eps* = 9.4*S at d=32 (18.8*S at d=64); for confidence 95%, eps = 32*sqrt(2 ln(2*128/0.05)/128) = 32*0.365 = 11.7*S. Across the synthetic sweep: best case (d=10, D=50) eps* = 10*sqrt(2 ln 100/50) = 4.3*S; worst (d=200, D=10) eps* = 200*sqrt(2 ln 20/10) = 154.8*S. For the exp kernel the true attention output is a convex combination of V entries, so |attn| <= S everywhere - the certificate only activates at deviations 4.3x-155x LARGER than the largest value the true output can take. The observed errors in Figure erroring (claimed to 'confirm Theorem pacTheo') live far below any certified tolerance.", "verdict": "inside"}, {"arxiv_id": "2505.14840", "claim_anchor": "m-thm:super-const-d-lb (intro.tex:120); m-thm:poly-d-lb (intro.tex:153)", "bound": "Conditional lower bounds for approximate attention attc(n, d, B, eps): (1) assuming SETH, attc requires n^{2 - o(1)} time when d = 2^{Omega(log* n)} and B = poly(n) - i.e. any temperature/entry scale, super-constant head dimension; (2) assuming the Generalized High-Dimensional OV Hypothesis, attc requires TMUL(n, d, n)^{1 - o(1)} time when d = poly(n). Both are asymptotic (Omega/o(1) exponents, poly(n) unnamed) and conditional on unproven hypotheses.", "deployed_config": "None. Pure fine-grained-complexity paper ('Subquadratic Algorithms and Hardness for Attention with Any Temperature'): no experiments, no benchmarks, no implementation - results/2505.14840.json is empty and no section of the source mentions any run.", "evaluation": "Not evaluable twice over: (1) nothing is deployed - there is no configuration of the paper's own to place inside or outside the certified-hardness regime; (2) the regime boundaries are asymptotic with unpinned constants: d = 2^{Omega(log* n)} has an unspecified Omega constant on an iterated-log exponent (log* n = 5 for every n up to 2^65536, so whether a real head dim d = 64 at n = 10^6 is 'super-constant' in the theorem's sense depends entirely on the hidden constant), n^{2 - o(1)} leaves the o(1) unquantified, and both statements are conditional on SETH / Generalized High-Dimensional OVH.", "verdict": "not_evaluable"}, {"arxiv_id": "2505.14840", "claim_anchor": "lem:1-d-attention-alg-faster (arxiv/warmup-1-d.tex:7); thm:backwards-pass-alg (arxiv/backwards-pass.tex:42); cor:backwards-pass-alg (arxiv/backwards-pass.tex:48)", "bound": "Upper bounds: (1) attc(n, 1, B, eps) solvable in ~O(n polylog(B/eps)) time (head dimension exactly 1); (2) attention gradient aattlgc(n, d, B, eps) reducible to O(d) calls of aattc(n, d, B, eps/Theta(n d B^3)) plus O(n d^2) time; (3) for d = O(1), aattlgc in ~O(n^{2 - 1/d} polylog(B/eps)) time. Tilde-O and Theta hide all constants and polylog factors.", "deployed_config": "None. The algorithms are not implemented; the paper reports no timing, no (n, d, B, eps) instantiation, and no experiments (results/2505.14840.json is empty).", "evaluation": "No deployed config exists to evaluate. The certified regime itself sits at head dimensions no real system uses: d = 1 (the lemma) and d = O(1) with 1/d in the runtime exponent (the corollary). Constant-1 illustration NOT licensed by the paper: at a realistic head dim d = 64 the certified exponent would be 2 - 1/64 = 1.984, a speedup factor of n^{1/64} ~ 1.24 at n = 10^6 - before the unpinned ~O constants and the eps/Theta(n d B^3) accuracy inflation in the reduction (a called oracle must be ~n d B^3 times more accurate than the gradient tolerance requested). The reduction's Theta constant is unspecified.", "verdict": "not_evaluable"}, {"arxiv_id": "2505.19105", "claim_anchor": "thm:main theorem (main.tex:371; appendix restatement thm: kernel integral, main.tex:1827)", "bound": "Structural expressivity statement only: 'a latent-SSM layer approximates the integral operator G(a)(x) = int_Omega kappa(x,y) a(y) dy' on a bounded domain Omega. No epsilon, no rate, no resource parameter appears - the word 'approximates' carries no bound, and the supporting Monte Carlo lemma (lemma:monte carlo, main.tex:1778) is likewise stated with no 1/sqrt(N) (or any) error term; the proof even sets |det(grad f^-1)| = 1 'for simplification' (main.tex:1854).", "deployed_config": "LaMO on six PDE benchmarks (Darcy, Navier-Stokes, Airfoil 221x51 mesh, Elasticity point cloud, Pipe, Plasticity 101x31 mesh), settings following FNO/Geo-FNO benchmarks; ablations sweep latent tokens M in {1,...,512} (Airfoil) and {484, 1024, 1936} (Darcy) with SSM state dimension in {16, 32, 64}, expand dim 2 (results/2505.19105.json config_notes; main.tex:794-803).", "evaluation": "Nothing to evaluate: the theorem ties no deployed knob (latent-token count M, mesh size N, state dimension, layer count) to any approximation tolerance, so no arithmetic can be performed at any config. The claims file's other two claims (prop: euler ZOH identity, lemma:monte carlo) are kind 'equivalence' - outside this audit's kind filter - and are also tolerance-free identities/interpretations.", "verdict": "not_evaluable"}, {"arxiv_id": "2505.20666", "claim_anchor": "prop:multilayer-pde (main.tex:1450; error bound eq:pde-error, main.tex:1483)", "bound": "Depth-L Transformer applying ONE explicit PDE step of size Delta t per layer, periodic/zero-flux boundaries, constant diffusion/wave speed alpha > 0: (i) geometric high-frequency damping per layer (factor 1 - alpha k^2 Delta t, main.tex:1500); (ii) L layers ~ single PDE evolution of pseudo-time L*Delta t; (iii) global error ||A_approx^(L) - A_true|| <= C * Delta t * (1 + t) 'for a constant C > 0'. The proof explicitly invokes the CFL stability condition alpha k^2 Delta t < 1 (main.tex:1511-1512).", "deployed_config": "PDE-Transformer on WikiText-103: 4 layers, 256-d embeddings, 8 heads, seq len 256-1024 (tab:lm_config, main.tex:955-978); PDE variants alpha = 0.10 (diffusion, reaction-diffusion, advection-diffusion; beta = 0.02/0.03) and alpha = 0.15 (wave) (main.tex:512-515); PDE refinement steps N_t in {1, 2, 4} per attention call, with the N_t = 8 run DIVERGING to NaN (tab:pde_analysis, main.tex:484; results/2505.20666.json). Delta t is a declared hyperparameter (main.tex:802) whose numeric value is never printed anywhere in the paper.", "evaluation": "Not evaluable at any deployed config: the bound's constant C is unpinned ('for a constant C > 0') and the deployed Delta t is unrecorded, so C*Delta t*(1+t) has no computable value and the CFL precondition alpha*k^2*Delta t < 1 cannot be checked even with alpha = 0.10 in hand (at k_max ~ pi on a unit token grid it would read 0.987*Delta t < 1, satisfied iff Delta t < 1.01 - Delta t unknown). Structural mismatch besides: the proposition couples one step to one layer (L steps total), while deployment runs N_t in {1,...,8} steps inside EVERY layer (Algorithm at main.tex:850-862), i.e. 4-32 total steps for L = 4. Behavioral evidence cuts against the certified regime: for the linear diffusion variant, any (alpha, Delta t) inside the CFL region makes the update non-expansive, so more steps cannot diverge - yet deployed N_t = 8 NaNs while N_t = 4 trains; either the shared deployed (alpha, Delta t) violates the CFL hypothesis for every run, or the divergence lives in training dynamics the proposition does not model.", "verdict": "not_evaluable"}, {"arxiv_id": "2505.20666", "claim_anchor": "prop:hybrid-error (main.tex:1526; bound eq:hybrid-bound, main.tex:1551-1558)", "bound": "Hybrid sparse/kernel + PDE attention: with initial surrogate error eps_0 = ||A^(0)_approx - A_true|| and N_t explicit diffusion steps of size Delta t (T = N_t * Delta t, diffusion rate alpha > 0), the final iterate satisfies ||A_final - A_true|| <= eps_0 + delta(T), delta(T) = O(e^{-alpha lambda_min T} + Delta t), lambda_min = smallest non-zero Laplacian eigenvalue under periodic/zero-flux boundaries. O(.) constants unspecified.", "deployed_config": "PDE-Longformer-Integrated: 2-layer Longformer, max seq 1024, attention window 256, PDE refinement step inserted in each block, WikiText-103 (tab:longformer-performance region, main.tex:425-442; results/2505.20666.json config_notes). Deployed eps_0 (the Longformer-window surrogate's deviation from full softmax attention) is never measured; Delta t never printed; N_t for this experiment not stated; lambda_min never computed.", "evaluation": "Four unpinned quantities block evaluation: the O(.) prefactors, Delta t, lambda_min (computable in principle from the boundary conditions and grid but never instantiated), and eps_0, which is a measurable matrix norm the paper never reports for the deployed window-256 surrogate. Even fully evaluated, the certificate's shape deserves note: the right side is eps_0 + delta(T) >= eps_0 with delta(T) -> O(Delta t) > 0 as T grows - the proposition can never certify that PDE refinement makes the hybrid attention CLOSER to exact softmax attention than the raw sparse surrogate was; it only bounds how much further away diffusion can push it. The paper's empirical claim (PDE refinement improves the Longformer, ppl 1.40 -> 1.35 at epoch 5) is an improvement the theorem is structurally incapable of certifying.", "verdict": "not_evaluable"}, {"arxiv_id": "2506.03085", "claim_anchor": "thm:crasp1 (formal: sections/main-results.tex:170; informal: sections/new_intro.tex:33); thm:main-weaker (formal: sections/main-results.tex:181; via thm:main, sections/proof-sketch-new.tex:6)", "bound": "Non-asymptotic length generalization of the Minimum-Complexity Interpolator (MCI): (1) C-RASP^1 with complexity C(f) = max(|a|,|b|,|d|) (parameter precision): N_MCI(T) <= O(T^2) - training on ALL labeled strings of length up to O(T^2) suffices to exactly identify any ground-truth 1-layer C-RASP program of precision T, hence length-generalize to arbitrary lengths; (2) C-RASP^2 with C(f) = T(f)^{K(f)}: inputs of length O(T^{O(K)}) suffice when the ground truth has precision <= T and heads <= K (via thm:main, N_MCI(alpha) <= O(alpha^{O(1)})). All bounds are big-O with unspecified constants, and the C-RASP^2 exponent hides a second unspecified constant inside O(K).", "deployed_config": "None. Pure theory paper: results/2506.03085.json is empty, no experiments section exists, and the certified learner is never run - the paper itself acknowledges 'the Minimum-Complexity Interpolator is computationally intractable for general function classes' (sections/foundations-setup.tex:204) and that it studies MCI to 'abstract away' SGD training dynamics.", "evaluation": "Not evaluable: (1) no deployed config - neither MCI nor any transformer is trained in the paper, so there is no (T, K, training-length) triple to test against N_MCI; (2) the O(.) constants are unpinned in both theorems, and for C-RASP^2 a constant sits in the EXPONENT (O(T^{O(K)})), so even a hypothetical deployment could not be placed: with constant 1 (illustration NOT licensed by the paper) a precision-T=10, K=2 ground truth would need all ~2^{100} binary strings up to length T^K = 100 labeled - the certified training set is exhaustive enumeration up to the length bound, not a sample. (3) The transformers whose empirical length-generalization failures motivate the work (intro discussion, sections/new_intro.tex:10-17) are trained by SGD, a learner explicitly outside the theorem's scope by the paper's own admission.", "verdict": "not_evaluable"}, {"arxiv_id": "2507.03340", "claim_anchor": "thm:kernel-approx (02_method.tex:104)", "bound": "If M >= (4 N_{q,lambda}/t) * log(64 N_{q,lambda}/(delta t)) with delta in (0,1), lambda > 0, t in (0,3], and features drawn i.i.d. from the DoF-optimal density q_lambda, then (i) ||K - K-hat||^2_{L^2(rho x rho)} <= lambda t delta^{-1} C_K^(1) + t^2 C_K^(2) w.p. 1-2delta and (ii) attention-integral error <= sqrt(2)(lambda^{1-alpha} + C_K^(3)) t ||h|| w.p. 1-delta; C_K^(1..3) are unpinned 'constants depending on K'.", "deployed_config": "Algorithm 1 (02sub_algorithm.tex) sets t^{-1} = C / mean_s(N-tilde_lambda^(s)) and deploys M_s = round(t^{-1} N-tilde_lambda^(s)), i.e. M_s = N-tilde/t with the 4*log factor dropped. GPT-2: C = 64, lambda = 2^-4, per-layer M = {130,182,35,44,42,65,92,33,28,34,46,39} (tab:dimension); Pythia-1B: C = 128, lambda = 2^-8, M = {277,...,9}. Features are PRF maps subsequently TRAINED by distillation (lr 0.02 for z_1..z_M), not i.i.d. draws from q_lambda.", "evaluation": "Precondition ratio at deployment: required/deployed = (4 N/t log(64 N/(delta t))) / (N/t) = 4 log(64 M_s / delta). Even in the vacuous limit delta -> 1: M_s = 64 (GPT-2 average) needs 4 ln(4096) = 33.3x the deployed features; smallest Pythia layer M = 9 needs 4 ln(576) = 25.4x; at a meaningful delta = 0.05, M = 64 needs 4 ln(81920) = 45.2x. So every layer's dimension-setting head (the max-DoF head Algorithm 1 sizes against, N/t = M_s by construction) sits a factor ~25-45 below the certified minimum for every delta in (0,1). Only incidental low-DoF heads (e.g. GPT-2 layer-1 head-4, N-tilde = 2.0 in tab:dof: required ~ 8 ln(128/delta) ~ 57 at delta = 0.1 < deployed 130) land inside. Second violation: deployed features are gradient-trained, breaking the i.i.d.-from-q_lambda hypothesis under which N_{q,lambda} is even defined. t in (0,3] holds (t = mean(N-tilde)/C ~ 1.01 from tab:dof). C_K^(1..3) unpinned, so no numeric tolerance would follow even inside.", "verdict": "outside"}, {"arxiv_id": "2507.16577", "claim_anchor": "prop:forget (sections/appendix.tex:152); companion props prop:class (11), prop:noise1 (48), prop:noise2 (103) are structural/comparative only", "bound": "Receptive-field separation: for vanilla gated linear attention with decay alpha_t in (0,1) and q_t = k_s = all-ones vectors, the receptive field is bounded by M_t = max_k log(P_th/d) / log(max_{s+1<=j<=t} alpha_j^k); the row-sparse update achieves M_t = t by scattering decay values 'to actually equal 1' on unselected rows (exists k with alpha_j^k = 1 for all j in (s,t]).", "deployed_config": "SSE models: 600M params / 15B tokens and 2B params / 100B tokens, expansion n=4, sparsity k in {1,2}, diagonal gating, learned key feature maps (SiLU deployed; softmax ablated), plus hybrid SSE-H and a 12B conversion (results/2507.16577.json config_notes). MQAR probe models: 2 layers, 2 heads, top-k with k = d/4 on GLA/MetaLA/GSA.", "evaluation": "Certified regime requires q_t = k_s = [1,...,1] - the proof's first line assumes it so that p_ts = sum_k prod_j alpha_j^k 'characterizes the utility of only decay'. No deployed forward pass uses all-ones queries/keys: SSE deploys learned SiLU/softmax key feature maps and learned queries, so the score formula the bound is derived from never describes a deployed score. Even granting the probe setting, the bound is not numeric at deployment: P_th is a free threshold never pinned, and the learned decay spectra alpha_j (diagonal gates) are unrecorded, so M_t = log(P_th/d)/log(max alpha) evaluates to no number for any deployed model. The row-sparse half is existence-only: SSE's top-k masking does structurally realize alpha = 1 on unselected rows, but M_t = t requires the router to keep the storing row unselected for ALL j in (s,t] - a capacity statement over router choices, not a property of the trained router.", "verdict": "outside"}, {"arxiv_id": "2508.06587", "claim_anchor": "thm:hgmn-complexity (R3_main_clean.tex:262); [thm:hgmn-gd-convergence (293) out of audit scope, see notes]", "bound": "HGMN forward pass costs O(L (n d^2 + m d^2 + n k-bar d d_m + n d_m^2) + n d c) - big-O only, no constants; preconditions are purely definitional (any graph with n nodes / m hyperedges, any L, d, d_m, c).", "deployed_config": "Node classification on E118/E123/E295/E296/E297, IIP, TerroristRel, Cora, Citeseer, Pubmed plus OGB runtime tables (results/2508.06587.json). The paper records NO deployed values for d, d_m, L, or c anywhere - the hyperparameter-sensitivity section only sweeps a learning rate it calls lambda.", "evaluation": "Not evaluable on two independent grounds. (1) Asymptotic-no-constants: the theorem certifies no operation count or wall-clock at any config; the runtime table (R3_main_clean.tex tab:runtime_results, min/epoch on OGB) cannot be compared against a constant-free O(.). Constant-1 illustration NOT licensed by the paper: at Pubmed-scale n ~ 19717, d = d_m = 64, L = 2, k-bar ~ 5, c = 3 the formula reads ~ 2*(19717*4096 + m*4096 + 19717*5*4096 + 19717*4096) ~ 1.1e9 ops, but no deployed (d, d_m, L) is printed to substitute. (2) Model mismatch inside the paper: the proof's 'mamba block' is an attention-style neighborhood aggregation sum_{j in N(i)} alpha_ij X_j W_m (line 280), while the method section defines the block as a discretized selective SSM with state matrices (A, B, C) (lines 187-192); the bound contains no SSM state dimension, so it is not established for the architecture the experiments run.", "verdict": "not_evaluable"}, {"arxiv_id": "2509.09001", "claim_anchor": "thm:ANNA solves match2 (arxiv.tex:481); thm: ANN transformers solve induction head (500) / thm:ANNA solves khop (523) with quantified appendix variants (1175, 1197); MPC-equivalence theorems (388, 417, 450, 684, 882, 927); reformer-cannot-sum (1070); supporting lsh-compute-ANN (354)", "bound": "Existence constructions: Match2 solved EXACTLY by a 1-layer, 1-head, embedding-dimension-1 ANNA-transformer; induction heads / k-hop solved by L = O(1/(eps (eps'-eps)^2)) (resp. + log k/(eps'-eps)^2) layers, H = O(N^{(eps'-eps)/4}) heads, m = O(N^{eps'}) width; two-way ANNA <-> MPC simulations with O(.)-sized resources; impossibility: constant-depth fixed-hash Reformer errs >= epsilon on SUM. Underlying mechanism guarantee: LSH with quality rho < 1/3, l = Theta(N^{3 rho} log N) tables, z = Theta(log_{1/p2} N) implements ANNA with failure eta = O(1/N^{1-3 rho}), requiring c > sqrt(3).", "deployed_config": "The corpus has no results/2509.09001.json, but the source DOES deploy (sec:experiments + appendix experimental details, arxiv.tex:537-546, 1229-1330): Match2 at N=32, M=37 with a TRAINED softmax surrogate (1 layer, 1 head, m=64, MLP 4m, beta in {0.1,1,10}) distilled to ANNA with angular LSH, l in {1..16}, z in {1..6} - zero test error at l >= 8, z = 1, beta = 0.1; induction heads at k=1, N=100, |Sigma|=4 with 2 layers, 1 head/layer, m=128, layer-1 l in {32..96}, layer-2 l in {4..32}, z in {1..4} - error 0.2 down to < 0.1.", "evaluation": "No deployed run can be placed inside any certified regime because every resource bound is Theta/O with unpinned constants over free constants (eps, eps'), and the deployed models are trained-then-distilled surrogates, not the paper's constructions: the Match2 theorem certifies a specific m = 1 construction while deployment trains m = 64 weights that the existence proof says nothing about (the paper is candid: 'our ad hoc training method was effective as a proof-of-concept', arxiv.tex:577). The experiments also never instantiate the mechanism-level regime parameters: no (r, c) approximation radii are chosen, so rho, p1, p2 - and hence the certified table count Theta(N^{3 rho} log N) and failure rate O(1/N^{1-3 rho}) - are undefined for the deployed (l, z) grids. Constant-1 illustrations NOT licensed by the paper: at N=32, rho -> 0 gives l ~ log2(32) = 5 <= deployed 8 (consistent), but z = log_{1/p2} 32 is uncomputable without p2; for induction heads, deployed m = 128 > N = 100 cannot be written as N^{eps'} with eps' < 1 (needs eps' ~ 1.054), and the certified depth at eps = 0.5, eps' = 0.9 reads 1/(0.5 * 0.16) ~ 12.5 layers vs 2 deployed. The Reformer impossibility has no deployed config anywhere (the paper runs no Reformer).", "verdict": "not_evaluable"}, {"arxiv_id": "2510.05688", "claim_anchor": "vector-sum-estimation lemma (text/method_1.tex:38); corr:num / corr:den (text/theory.tex:48, 55); ratio-combination (text/method_1.tex:53); verified-SDPA theorem eq:theoretical (text/method_1.tex:64)", "bound": "vAttention is an (epsilon, delta)-verified approximation of SDPA if the sampling budget satisfies b >= min_{eps' in (0,eps), delta' in (0,delta)} max(b_D(eps'/2, delta'), b_N((eps-eps')/2, delta-delta')), with fully explicit budget functions b_N = (Phi^-1(1-delta/2) n_s sqrt(Tr(Sigma)) / (eps ||N||_2))^2 and b_D = (Phi^-1(1-delta/2) n_s sigma / (eps D))^2 - the rare bound in this corpus with every constant printed. Two regime conditions: (1) 'for a large enough b' (CLT asymptotic normality of the sample sum); (2) Sigma, sigma, ||N||_2, D are the TRUE population statistics of {exp(<K[i],q>)V[i]} / {exp(<K[i],q>)} over the residual (non-fixed) indices. Ratio step additionally needs eps_2 < 0.5 (satisfied by the eps'/2 split at all deployed eps <= 0.3).", "deployed_config": "AIME2024: Llama/DeepSeek-R1-Distill-8B with epsilon = 0.05, delta = 0.05, f_t = 0.025, f_b = 0.025, 128 sink + 128 local tokens, 32K generations (results/2510.05688.json config_notes). RULER32K/LongBench Pareto: (eps, delta) grid-searched per sparsity target - e.g. at 10% sparsity, eps in {0.025..0.2}, delta in {0.025..0.2}, f_b in {0.025..0.075} chosen for 'lowest local attention errors while meeting the target sparsity' (tables/parameter_grid.tex). In ALL deployments Algorithm 2 computes b from PLUG-IN statistics (sigma-hat, D-hat, Tr-hat(Sigma), ||N-hat||_2) estimated on a base sample of f_b * n_s tokens - the paper concedes 'In practice, Sigma and sigma as well as exact ||N||_2 and D are not known apriori' (text/method_1.tex).", "evaluation": "Whether any deployed run satisfies the certified inequality b >= b*(TRUE Sigma, sigma, ||N||_2, D) is not decidable from the paper: the true per-(layer, head, query) statistics are never recorded, and the deployed b substitutes f_b = 2.5% plug-in estimates carrying no estimation-error certificate - knowing the true constants would require touching all residual tokens, the very cost being avoided (EVA-style circularity). The second regime condition, CLT validity, is hedged as 'for a large enough b' with no Berry-Esseen-type finite-sample correction, so no deployed b can be certified as large enough. The paper's own tightness appendix (app:tightness_analysis, eps = 0.1, delta = 0.2, 5% oracle top-k, Llama-3.1-8B on RULER 16K) quantifies the gap: conservative Hoeffding budgets (average 874 samples, i.e. CLT average ~ 874/2.8 ~ 312) achieve 'near-zero failure rate (< 2%)', while the deployed 'optimistic' CLT budgets are ~ 2.8x smaller and, per the layer-16 caption, 'have higher local errors' - i.e. the paper's own measurements show deployed CLT budgets missing the (eps, delta) coverage target on middle-layer heads. Arithmetic that IS pinned: at deployed eps = delta = 0.05, Phi^-1(0.975) = 1.96, so b* = (1.96 n_s sqrt(Tr Sigma) / (0.05 ||N||_2))^2 = 1537 (n_s sqrt(Tr Sigma)/||N||_2)^2 - evaluable the moment the population ratio is recorded, which it never is. Note also the Pareto deployments invert the guarantee's direction: (eps, delta) are tuned to hit a sparsity budget, making the nominal guarantee a knob rather than a specification.", "verdict": "not_evaluable"}, {"arxiv_id": "2106.12566", "claim_anchor": "Proposition 1 rel_vanilla (main.tex:154)", "bound": "Expressivity separation: if n > d+1, there exist W_rel^Q, W_rel^K in R^{d x d} and Toeplitz RPE biases b_{j-i} such that NO vanilla parameters W_vanilla^Q, W_vanilla^K and inputs x_1..x_n in R^d reproduce the row-softmax attention matrix. Stated for single-head with x_i in R^d; 'All the conclusions remain the same for the multi-head attention setting' (main.tex:153) with no multi-head threshold restated.", "deployed_config": "NPRF-Transformer w/ RPE deployments: GLUE pre-training seq n=512, hidden d=768, 12 heads (tab:pre-train, main.tex:485-497); WikiText-103 LM n=512, d=512, 8 heads (main.tex:339); IWSLT14 MT sentence-level n (tens of tokens), d=512, 4 heads (main.tex:344); DeiT-base ImageNet-1K n~197 tokens, d=768, 12 heads (main.tex:351); ImageNet32 generation n=3072, d=512, 8 heads (main.tex:524-528).", "evaluation": "Evaluate n > d+1 with d = embedding dim of x_i, the only reading the paper pins: GLUE 512 > 769 FALSE; LM 512 > 513 FALSE (misses by exactly 2: needs n >= 514 at d=512); IWSLT n <= ~200 > 513 FALSE; DeiT 197 > 769 FALSE; ImageNet32 3072 > 513 TRUE - the sole config inside, and it is an appendix task. So at every main-paper config the proposition certifies no RPE-vs-vanilla separation at all. Under an unstated per-head reading (rank bottleneck d_h = 64 or 128, condition n > d_h+1) all configs would flip to inside, but that threshold appears nowhere in the paper. Note also the deployed model is kernelized NPRF attention (normalized q/k, PRF features), not the softmax-RPE construction the proposition separates.", "verdict": "outside"}, {"arxiv_id": "2106.12566", "claim_anchor": "thm:sample-comlex (main.tex:251)", "bound": "PRF sample complexity: for query/key l2-norms <= R, feature dimension m = Theta(n exp(4R^2) / (eps^2 delta)) gives Pr(||A - A_hat||_1 >= eps) <= delta, where A is softmax attention and A_hat its positive-random-feature approximation.", "deployed_config": "R = 1 is pinned exactly by deployment: Algorithm 1 (main.tex) normalizes Q_i W^Q and K_i W^K to unit l2 norm before phi. Deployed (n, m): WikiText-103 LM (512, 64) (main.tex:339); IWSLT14 enc (~tens-200, 16) / dec (24) (main.tex:344); DeiT ImageNet (~197, 32) (main.tex:351); ImageNet32 (3072, 32) (main.tex:528); GLUE (512, feature dim not stated in tab:pre-train).", "evaluation": "Theta constant is unspecified, so the bound cannot be inverted at any deployed (n, m). Constant-1 illustration, NOT licensed by the paper: eps^2 delta = n e^4 / m with e^4 = 54.60 at R=1 gives LM 512*54.60/64 = 436.8 (delta=1 -> eps = 20.9); DeiT 197*54.60/32 = 336.1; ImageNet32 3072*54.60/32 = 5241; MT enc at n=100: 100*54.60/16 = 341.3. Since A and A_hat are both probability vectors (PRF features are positive), ||A - A_hat||_1 <= 2 always, so any certified eps >= 2 is vacuous; even under constant-1 the trivial eps=2, delta=0.1 certificate needs m >= n e^4/(4*0.1) = 69,885 at n=512 vs 64 deployed (~1,092x short). The theorem also does not cover the deployed RPE reweighting e^{b_{j-i}} in Eq. kernel-attn-RPE.", "verdict": "not_evaluable"}, {"arxiv_id": "2110.14416", "claim_anchor": "thm:generalization (methods.tex:62)", "bound": "Subsumption: the higher-order Transformer layer Enc_{k->l} (Eq. higher_transformer_layer) generalizes the equivariant linear layer L_{k->l} (Eq. equivariant_layer); proof sets attention coefficients alpha^{h,mu} = 1 on every equivalence class mu (all b(k+l) classes) and reduces MLP_{l->l} to the full invariant bias sum.", "deployed_config": "No deployed run uses the certified full Enc: all 'Ours' rows are reduced variants - (D) lightweight linear layers, (S) sparse, (phi) Performer-kernel attention (experiments.tex:8). Headline configs: PCQM4M-LSC 3.8M graphs and GPS/MovieLens/Drug hyperedge prediction deploy Ours (S, phi); Jets/Delaunay set-to-graph deploy Ours (D) and (D, phi) (results/2110.14416.json config_notes).", "evaluation": "The kernel variants implement the decoupled approximation Eq. kernel_querywise, introduced by the paper itself as 'We propose an approximation' with '~': the value/normalizer sums run over the decoupled union set I instead of {i | (i,j) in mu}, so the proof's alpha^mu = B^mu construction (class-restricted all-ones attention) is unavailable - constant phi (attainable at Q=K=0, phi_PRF(0) = 1_m/sqrt(m)) spreads weight over all of I, leaking outside mu. Sparse variants further restrict indices to hyperedges, excluding the non-edge entries B^mu covers. Only the dense softmax rows Ours (D) (Jets/Delaunay) can instantiate the certified construction: lightweight QK layers can output constants (uniform attention; within-class key counts are index-independent by permutation symmetry, so w_mu absorbs the softmax normalization 1/c_mu) and the lightweight MLP retains the full bias term. Split: the flagship large-scale results (PCQM4M, hyperedge) all run where the subsumption certificate is not proven.", "verdict": "outside"}, {"arxiv_id": "2110.14416", "claim_anchor": "proposition:lightweight (methods.tex:156)", "bound": "Complexity: given L_{k->l}, there always exists a nonempty subset M of equivalence classes such that the restricted layer L-bar_{k->l} (Eq. lightweight) is computable for all j in O(n^l).", "deployed_config": "Deployed lightweight layers choose exactly such an M: 'we choose M among equivalence classes that do not involve summation over input', e.g. L-bar_{1->1} keeps the elementwise basis I_n and drops sum-pooling 1_n 1_n^T (methods.tex, directly after the proposition). Used in every 'Ours' variant marked (D) and inside the (S, phi) models.", "evaluation": "Precondition check: deployed M is nonempty and contains only non-summation classes, so each output entry j costs O(1) basis contributions -> O(n^l) total, exactly the certified complexity class. Big-O constants are unspecified, so no wall-clock or memory number is certified; and the proposition bounds cost only - the capability lost by dropping classes (e.g. sum-pooling) carries no error bound anywhere in the paper.", "verdict": "inside"}, {"arxiv_id": "2110.14416", "claim_anchor": "thm:message_passing (methods.tex:298)", "bound": "Two stacked sparse Enc_{2->2} layers with kernel attention can approximate any Gilmer et al. (2017) message-passing algorithm to arbitrary precision; the converse fails (a single Enc_{2->2} needs >= d(E) message-passing steps, impossible for disconnected graphs).", "deployed_config": "Deployed model class matches the theorem's: Ours (S, phi) sparse second-order Transformer with Performer kernel on PCQM4M-LSC and hyperedge datasets; synthetic chain experiment uses 2 layers, hidden dim 16 (appendix.tex:570).", "evaluation": "The proof routes every quantitative step through the universal approximation theorem (Hornik): errors eps_1, eps_2, eps_3 are introduced with no rate, no width-to-eps schedule, and 'arbitrary precision' is a limit statement over unbounded MLP width. At any deployed width (e.g. hidden 16, 2 layers in the chain experiment) the certified tolerance is undefined - there is no formula to invert, so no constant-1 illustration is even possible. The one pinned proof step - phi(Q) = phi(K) = 1_{d_K}/sqrt(d_K) constant - IS realizable by the deployed Performer kernel at Q=K=0 (phi_PRF(0) = 1_m/sqrt(m)), so the vacuousness is purely the unquantified UA steps.", "verdict": "not_evaluable"}, {"arxiv_id": "2110.14416", "claim_anchor": "corollary-mpnn-expressivity (methods.tex:307)", "bound": "Second-order sparse Transformers with kernel attention are strictly more expressive than any MPNN in the Gilmer et al. (2017) framework.", "deployed_config": "Same deployed class as thm:message_passing: Ours (S, phi) on PCQM4M-LSC, GPS, MovieLens, Drug.", "evaluation": "Purely qualitative class separation inheriting thm:message_passing's rate-free universal-approximation argument; no resource parameter, tolerance, or failure probability appears, so no deployed (width, depth, n) can be placed inside or outside a certified inequality. The separation direction rests on graphs with diameter d(E) -> infinity (disconnected), a property of inputs, not of any deployed model configuration.", "verdict": "not_evaluable"}, {"arxiv_id": "2111.00396", "claim_anchor": "thm:s4-recurrence (src/method.tex:143)", "bound": "One step of the discretized SSM recurrence (eq. 2) costs O(N) operations for any step size Delta, given A in S4's NPLR/DPLR parameterized form (inverse of a DPLR matrix is DPLR via Woodbury; A-bar is a product of two DPLR matrices).", "deployed_config": "Every deployed S4 layer is DPLR by construction (A = Lambda - P Q* with HiPPO-NPLR init, sec:s4-architecture); recurrence mode is actually run for autoregressive generation (CIFAR-10 density, appendix-experiments.tex:226: generation 'sped up by orders of magnitude from the naive implementation'). Deployed state sizes: N = 64 on all LRA tasks (appendix-experiments.tex:72), N = 256 in the efficient-Transformer benchmark (appendix-experiments.tex:24).", "evaluation": "Precondition check: A DPLR - satisfied by construction in 100% of deployed configs (the parameterization IS the model); Delta arbitrary - satisfied. The certified O(N) step is the algorithm the released generation path uses. Big-O constant unspecified, so no wall-clock number is certified at N = 64 or 256; the certificate is realized but quantitatively silent.", "verdict": "inside"}, {"arxiv_id": "2111.00396", "claim_anchor": "thm:s4-convolution (src/method.tex:151) with prop:cauchy (src/appendix-theory.tex:596)", "bound": "SSM convolution filter K-bar computable via 4 Cauchy multiplies in O-tilde(N+L) operations and O(N+L) space, for DPLR A and any Delta. The operation count routes through the black-box Cauchy cost C(M,N) of prop:cauchy: O(MN) naively, O((M+N) log^2(M+N)) in exact arithmetic, O((M+N) log(M+N) log(1/eps)) numerically to precision eps.", "deployed_config": "All deployed S4 training runs use Algorithm 1 (the 4-Cauchy-multiply reduction) on fp32 GPUs (A100/V100, appendix-experiments.tex:13,86). Configs: LRA with N = 64 fixed, L up to 16384 (Path-X = 128x128); Speech Commands raw L = 16000; efficient-Transformer benchmark N = 256, L = 1024-4096.", "evaluation": "DPLR precondition holds everywhere, and the reduction itself (4 Cauchy multiplies, O(N+L) space) is exactly what deploys. But the certified O-tilde(N+L) count depends on WHICH prop:cauchy branch the black box instantiates, and the paper never says: the exact-arithmetic branch is unavailable on the deployed fp32 hardware, and the FMM branch's precision eps is pinned for no experiment. Constant-1 illustration (NOT licensed): at the LRA config (N=64, L=16384) even the naive branch, NL = 1,048,576 ops, sits below the (N+L) log2^2(N+L) = 16448 x 196.2 = 3.23e6 envelope, so the distinction is immaterial there; at the benchmark config (N=256, L=1024) naive NL = 262,144 exceeds (N+L) log2^2(N+L) = 1280 x 106.5 = 136,320 by 1.9x, so there the certificate holds only if a fast branch is actually used. Contrast LSSL (2110.13985), which conceded its fast algorithm was unimplemented; S4 leaves the branch unstated.", "verdict": "not_evaluable"}, {"arxiv_id": "2111.00396", "claim_anchor": "ssm-filter-from-generating-function (src/appendix-theory.tex:454)", "bound": "The truncated SSM filter K_L is recoverable from the generating function evaluated at the L-th roots of unity Omega = {exp(-2 pi i k/L)} stably in O(L log L) operations (inverse FFT step).", "deployed_config": "Algorithm 1 step 4 in every deployed S4 training run: evaluate K-hat at exactly the L-th roots of unity, then inverse FFT of length L. Deployed L: 1024-16384 (LRA incl. Path-X), 16000 (Speech Commands), 3072 (sCIFAR density).", "evaluation": "Precondition check: generating-function values available at the L-th roots of unity - satisfied identically, since Algorithm 1 chooses its evaluation nodes to be exactly Omega. The O(L log L) step is a standard iFFT, deployed as stated at every L. Big-O constant unspecified (no wall-clock certified), and the 'stably' qualifier carries no quantitative error bound in the lemma.", "verdict": "inside"}, {"arxiv_id": "2210.01989", "claim_anchor": "thm:main_theorem (tex/appendix.tex:79)", "bound": "Universal approximation: for all p in [1,inf), eps > 0, f continuous on a compact domain, there EXISTS a Performer-WavSpA network w in W^{2,1,4} (2 heads, head size 1, FFN width 4, FAVOR+ kernel, input normalization) with d_p(f, w) <= eps. Proof requires the forward-backward wavelet pair to be 'invertible and exact' and controls the FAVOR+ term to eps/4 'by controlling the number of random features m' with no formula.", "deployed_config": "LRA experiments (seq length up to 4096, tex/appendix.tex:167): fixed Daubechies-2/Coiflet/Symlet WavSpA, adaptive variants AdaWavSpA (direct filter parameterization, Daubechies-20/8 init, filter length 40/16), OrthoWavSpA (length 16), LiftWavSpA (3 decomposition levels) over Transformer/Performer/Linformer/Longformer/Linear-Attention backbones (tex/experiment.tex:28; results/2210.01989.json).", "evaluation": "Existence-style UA with no rate: no eps-to-(depth, width, m) map exists anywhere - the Yun et al. construction it invokes uses piecewise-constant discretization with unbounded depth, and the FAVOR+ feature count m is 'controlled' with no printed function, so at any deployed depth/feature count the certified tolerance is: nothing. Beyond rate-freeness, the load-bearing precondition (exact invertible transform pair) is satisfied by the fixed-wavelet, orthogonal, and lifting variants, but the paper itself concedes it can FAIL for the best-performing deployed variant: 'The direct parameterization scheme... no guarantee that the trained wavelet will form an orthogonal wavelet basis... a potentially imperfect projection and reconstruction pair' (tex/method_adaptive.tex:72), and 'this constraint is not always attached to adaptive wavelets' (:24) - while experiments report 'direct parameterization almost always provides the highest accuracy elevation' (tex/experiment.tex:44). The FNet-style forward-only ablation (no inverse transform) violates it outright.", "verdict": "not_evaluable"}, {"arxiv_id": "2211.10582", "claim_anchor": "thm ex - existence/expressivity of near-init linear RNN weights (example_paper.tex:1276)", "bound": "For any stable LDS with ||C^k D|| < c_rho rho^k (rho < 1) and RNN width m > m*, w.p. >= 1 - exp(-Omega(log^2 m)) over initialization there exist weights within Frobenius radius 2 c_rho b T_max^2/sqrt(m) of init with per-step output error O(sqrt(m) rho_0^{T_max}/(1-rho_0)^3) + O(b d^2 c_rho T_max^3 log(m)/sqrt(m)) + c_rho rho^{T_max}/(1-rho).", "deployed_config": "None: pure theory paper ('Linear RNNs Provably Learn Linear Dynamic Systems') - no experiments section, and no results/2211.10582.json exists in the corpus.", "evaluation": "Not evaluable (no deployed config). Even hypothetically, the certificate could not be checked at a concrete (m, T_max, rho_0, c_rho): every term carries an O(.)/Omega(.) with unpinned constants, m* is only given as Theta-expressions (companion thm mt: m* = Theta(c_rho^2 K^4 (1-rho_0)^8 eps^2/b^6) + Theta(1/delta)), and the first error term GROWS as sqrt(m) rho_0^{T_max} - whether it is small at a given width depends on the unstated constant trade-off between m and T_max = Theta(log-scale expression). The paper's remaining claims (SGD convergence thm mt, generalization thm gen, corollary) share the no-deployment fate and are outside this audit's kind filter.", "verdict": "not_evaluable"}, {"arxiv_id": "2212.14852", "claim_anchor": "prop:attn-cme (approx.tex:92)", "bound": "CME-attention consistency: for positive definite kernel k, i.i.d. tokens within a data point, ||v^l||_2 <= 1: w.p. >= 1-delta, ||att_CME(q,K,V) - E[V|K=q]||_2 = O( sqrt(L/lambda)(2/lambda + sqrt(Gamma(lambda/L)/lambda)) log(1/delta) + lambda/L ), Gamma the effective dimension of the covariance operator C_KK.", "deployed_config": "None: pure theory paper (exchangeability/latent-variable analysis of attention) - no experiments in any section, and no results/2212.14852.json exists in the corpus.", "evaluation": "Not evaluable (no deployed config). Additionally uncheckable at any hypothetical deployment: the O(.) constant is unspecified, lambda is a free regularization parameter never assigned a value, and Gamma depends on the unknown data covariance operator C_KK, recorded nowhere. The precondition 'tokens i.i.d. within one data point' would also fail for natural-language sequences, and att_CME is a theoretical construct - real transformers run softmax attention, connected to CME only via the separate limit-style prop:kernel_attn_story1 (uniform convergence as L -> infinity, no rate).", "verdict": "not_evaluable"}, {"arxiv_id": "2212.14852", "claim_anchor": "thm:gen-simple (paper.tex:64)", "bound": "One-layer attention generalization: under spectral/Frobenius norm constraints (assumption:constraint-simple), i.i.d. bounded data (asu::data), 1-Lipschitz aggregation: w.p. >= 1-delta, E_gen = O( (D^2/sqrt(n))[sqrt(log(1+gamma)) + sqrt(log(1+zeta R)) + sqrt(log(1+kappa/zeta))] + sqrt(log(1/delta)/n) ), D = max{d, d_p, d_y}.", "deployed_config": "None: no experiments, no results file (pure theory paper).", "evaluation": "Not evaluable (no deployed config). The O(.) constant is unpinned and gamma, zeta, kappa are functions of parameter-norm bounds (omega^q, omega^k, omega^v, alpha^nn, R^q...R^nn) that the paper never instantiates numerically; no (n, D, R) is supplied to plug in.", "verdict": "not_evaluable"}, {"arxiv_id": "2212.14852", "claim_anchor": "lem::approx_error (paper.tex:176)", "bound": "Supervised approximation error: if the RKHS l_infty projection error of g* is <= eps_g(mask) (asu::approx_err_sl) and some f_theta in F_att is uniformly within eps_att of W^T att_CME, then E_approx <= 2 eps_g^2(mask) + 2 eps_att^2.", "deployed_config": "None: no experiments, no results file.", "evaluation": "Not evaluable (no deployed config). The bound is assumption-relative: eps_g is an assumed RKHS projection error never computed for any function class, and eps_att is posited by an existence hypothesis with no witness construction or magnitude - there are no numbers to evaluate even in principle.", "verdict": "not_evaluable"}, {"arxiv_id": "2212.14852", "claim_anchor": "lem::approx_ssl (paper.tex:354)", "bound": "Downstream-after-pretraining approximation error: under asu::approx_err_ss and ||B||_2^2 <= mu with B = W_dt^T (W_ssl W_ssl^T)^{-1} W_ssl (asu::effec_ssl_param): E_approx = O( mu (E_approx^pre + eps_ssl^2(mask_pre, mask_dt)) + eps_g^2(mask_dt) + eps_agg^2(B) ).", "deployed_config": "None: no experiments, no results file - no pretraining run, so no empirical W_ssl, W_dt from which mu could even be measured.", "evaluation": "Not evaluable (no deployed config). Doubly unpinned: an explicit O(.) with unspecified constant, and every input quantity (mu, E_approx^pre, eps_ssl, eps_g, eps_agg) is an assumed bound on objects (trained weight matrices, RKHS projections) that exist in no recorded run.", "verdict": "not_evaluable"}, {"arxiv_id": "2212.14852", "claim_anchor": "thm:gen (appendix-gen.tex:100)", "bound": "T-layer h-head transformer generalization: under assumption:ker/trans/constraint and asu::data, w.p. >= 1-delta, E_gen = O( (D^2/sqrt(n))[T sqrt(log(1+gamma)) + sqrt(T) sqrt(log(1+zeta R)) + sqrt(log(1+kappa/zeta))] sqrt(hT) + sqrt(log(1/delta)/n) ), gamma/kappa/zeta maxima of layerwise constants.", "deployed_config": "None: no experiments, no results file - no (T, h, n, D, R) tuple is deployed anywhere in the paper.", "evaluation": "Not evaluable (no deployed config). Same blockers as thm:gen-simple compounded by depth: the layerwise constants gamma^(t), kappa^(t), zeta^(t) (eq:mag-xt-eq:ratio) depend on per-layer norm bounds never instantiated, and the O(.) constant is unspecified; the T sqrt(hT) scaling SHAPE is the paper's real content, but no concrete tolerance is certifiable at any configuration.", "verdict": "not_evaluable"}, {"arxiv_id": "2302.00787", "claim_anchor": "th:derf (main_0911.tex:317)", "bound": "If 8A < I_d (PSD order), (B^(1))^T (I_d-4A)^{-1} B^(2) = I_d, C^(k) = -(1/2)(B^(k))^T (I_d-4A)^{-1} B^(k), D = det(I_d-4A)^{1/4}, then T_DE is an unbiased RF decomposition of the softmax kernel K^(0) and the single-sample estimator variance equals the explicit closed form eq:devar - exact, for all x, y.", "deployed_config": "Every FAVOR# deployment instantiates DERF parameters via the th:aderf/th:sderf closed forms: variance comparison d=64, L=1024, M=1 (favor_sharp.tex sec:expvars + app:exp); UCI kernel classification M = 2^4..2^7; Conformer-Transducer LibriSpeech, 20 layers, model_dim 512, h=8 heads (per-head d=64), m in {8, 32, 128}; BERT-base GLUE, 12 layers, hidden 768, 12 heads (d=64), seq len 512, M=256 (app:exp tables).", "evaluation": "The preconditions are parameter equations, and the deployed parameter sets satisfy them by construction: Theta_ADE/Theta_SDE are defined as 'satisfying conditions from Theorem th:derf' (favor_sharp.tex sec:aderf, sec 'Symmetric DERFs'). Spot check for SDERF: A_{l,l} = (1/16)(1 - 2*Lam - sqrt((2*Lam+1)^2 + 8*Lam)) <= 0 for every eigenvalue Lam >= 0 (e.g. Lam=1 -> A=-0.320, 1-8A=3.56 > 0), so 8A < I_d holds; B = (I-4A)^{1/2}(Q^(3))^T gives B^T(I-4A)^{-1}B = Q^(3)Q^(3)^T = I_d exactly. What is certified at deployment: unbiasedness plus a DATA-DEPENDENT variance value; at M features the mean estimator's variance is Var/M (e.g. Var/8 at the smallest speech config), but Var for the trained Q/K distributions is unrecorded, so no numeric tolerance follows.", "verdict": "inside"}, {"arxiv_id": "2302.00787", "claim_anchor": "th:aderf (main_0911.tex:345; nonsingularity precondition at main_0911.tex:351)", "bound": "Closed-form global minimizer of the shifted log-variance objective L-bar over Theta_ADE (via eigendecompositions of M^(1), M^(2) and an SVD) plus the exact attained objective value eq:adevar. Precondition: M^(1) = (1/L) sum_i x^(i)(x^(i))^T and M^(2) = (1/L) sum_j y^(j)(y^(j))^T are nonsingular.", "deployed_config": "ADERF is deployed in the variance comparison (d=64, L=1024, 6 regimes: normal, sphere, heterogen, mnist, cifar10, mnist/cifar10 - images resized to 8x8, scaled by sigma, flattened; 'the shifted log-variance is optimized separately on each pair of sets for GERF, ADERF and SDERF', app:exp) and in UCI kernel classification (8 benchmarks, M = 2^4..2^7). The transformer runs use SDERF, not ADERF.", "evaluation": "Nonsingularity check at deployed values: normal/sphere/heterogen draw L=1024 > d=64 vectors from continuous distributions, so M^(k) is full-rank almost surely - provably inside. mnist/cifar10 regimes: M^(k) is built from 8x8-resized images and nonsingularity is neither guaranteed nor recorded (MNIST's near-constant border pixels can push Lambda^(k) to (near-)rank-deficiency, and the construction uses (Lambda^(1))^{-1/2}, undefined at singular M^(1)); UCI feature matrices likewise unrecorded. So the continuous synthetic regimes provably satisfy the precondition; the image/UCI regimes are unverified (not provably violated). As with th:derf, the certificate is optimality of a data-dependent surrogate objective, not an epsilon on kernel or attention error at deployed M.", "verdict": "inside"}, {"arxiv_id": "2302.00787", "claim_anchor": "th:sderf (main_0911.tex:370)", "bound": "Closed-form global minimizer of L-bar over Theta_SDE driven by the eigenvalues Lambda^(3) of M^(1) + mu^(4)(mu^(5))^T + mu^(5)(mu^(4))^T + M^(2), plus the exact attained objective value eq:sdevar. Preconditions: finite data sets X, Y; combined matrix symmetric PSD with sorted eigendecomposition.", "deployed_config": "SDERF is the deployed FAVOR# method everywhere: Conformer-Transducer LibriSpeech ('applying FAVOR# with SDERF since it worked best'), 20 layers, model_dim 512, 8 heads (per-head d=64), m in {8, 32, 128} (NST: m=8); BERT-base GLUE, M=256, 12 heads (d=64), seq 512; UCI classification M=16..128; variance comparison d=64, L=1024.", "evaluation": "Preconditions hold for EVERY finite data set: M^(1)+mu^(4)(mu^(5))^T+mu^(5)(mu^(4))^T+M^(2) = (M^(1)-mu^(4)mu^(4)^T) + (M^(2)-mu^(5)mu^(5)^T) + (mu^(4)+mu^(5))(mu^(4)+mu^(5))^T, a sum of three PSD matrices, hence PSD with a valid sorted eigendecomposition; A_{l,l} <= 0 for all Lambda^(3)_{l,l} >= 0 keeps every formula well-defined (1-4A, 1-8A >= 1). So all deployed configs are inside. Scope of the certificate: it bounds nothing - it states optimality of the mean shifted log single-sample variance over whatever (X, Y) the statistics were computed on, and gives that objective's exact (data-dependent) value; the deployed transformer statistics, hence the certified variance level, are unrecorded, and no concentration statement converts Var/M at deployed m=8..256 into an attention-output error. Optimality is also relative to the statistics-set: if the implementation reuses stale batch statistics, the parameters remain valid RFs (th:derf equations still hold) but the optimality transfers only to the batch that produced them.", "verdict": "inside"}, {"arxiv_id": "2302.02451", "claim_anchor": "thm-corrctness-runtime-wexpkde-alg (main.tex:351)", "bound": "KDEformer's WExpKDE primitive: (1 +/- eps) relative-error estimates of all n weighted exponential kernel sums in time O(n d * C), C = min_{mu>0} [ eps^{-2} mu^{-tau} + #{i : (sum_j v_j e^{<x_j,y_i>}) / (sum_j v_j e^{(||x_j||^2+||y_i||^2)/2}) < n mu } ], with tau = 0.173 + o(1). Precondition: access to the fast Gaussian KDE data structure of Charikar et al. 2020 (PreprocessKDE/QueryKDE, thm-Gauss-kde-moses, main.tex:212).", "deployed_config": "GloVe single-layer benchmark: n=8192, d=100, K=Q, feature dimension k=128, n swept 256-16384 (main.tex:607-615); BigGAN cross-attention: Q 4096x64, K 1024x64, V 1024x256, pretrained ImageNet 512x512 (main.tex:618-620); T2T-ViT: n=3136 and n=784, d=64 (main.tex ~668); LRA end-to-end: seq lens 2048/4096/1024/4096/1024, 2-layer transformer, embed 64, 2 heads, feature dim 128 (main.tex:730-731; results/2302.02451.json).", "evaluation": "Three independent blockers. (1) tau = 0.173 + o(1) carries an unpinned o(1) IN THE EXPONENT (main.tex:138, 156), so the certified runtime is not a number at any finite deployed n. (2) C is a data-dependent functional - the count of queries with normalized kernel density below n*mu - never measured or recorded for GloVe/BigGAN/T2T-ViT/LRA; in the worst case C = n and the certificate degrades to O(n^2 d), i.e. no speedup is certified without density statistics the paper does not report (its own subquadratic reading at main.tex:158 needs diameter d = o(log^2 n), an asymptotic condition with no constant). (3) The precondition is access to Charikar et al.'s PreprocessKDE/QueryKDE; the deployed pipeline is the 'practical' variant (alg-practical-adapted, main.tex:471; fixed-bucket angular LSH of appendix_lsh_design, main.tex:750) whose runtime is offered only 'by an analysis similar to' corr-simplified-runtime with 'some m' (main.tex:600), and the experiments parameterize by memory budget (feature dim k=128), reporting no eps or mu - whether the Charikar structure was implemented at all is unstated. Illustration only (constant-1 reading, NOT licensed by the paper): at n=8192, mu=1/n, eps=0.1: eps^{-2} mu^{-tau} ~ 100 * 8192^{0.173} ~ 476 per-point work vs 8192 direct - a ~17x saving IF the o(1), the constants, and the density count all cooperated; none is pinned.", "verdict": "not_evaluable"}, {"arxiv_id": "2302.10866", "claim_anchor": "prop-computational-complexity (3_hyena.tex:175)", "bound": "Order-N Hyena operator on u in R^{L x D} costs O(N D L (log_2 L + D)) - subquadratic in L vs O(L^2) self-attention. Preconditions: input u in R^{L x D}; convolutions evaluated via the Fourier domain (FFTConv).", "deployed_config": "Synthetics: order-2 Hyena, 2 layers, width D=64, L = 1024..131136 (appendix/A.tex:18,41-42); WikiText103: order-3 Hyena, 125M/153M, depth 18, width 864 (appendix/A.tex:108,167); The Pile: order-2, 125M/355M/1.3B, seq length 2024 [sic], batch 256 (appendix/A.tex:141); runtime benchmark: order-2 with fused-CUDA FFTConv, batch 64, L up to ~100K (4_experiments.tex:211).", "evaluation": "Preconditions are satisfied at every deployed config: all deployed Hyenas are finite-order (N = 2 or 3) operators on R^{L x D} inputs and the deployed implementation computes the long convolutions via FFT ('Hyena uses a fused CUDA kernel to perform FFTConv', 4_experiments.tex:211), which is exactly the proof's evaluation model. So deployment is inside the certified regime. But the certificate is big-O only - no constant, hence no wall-clock or FLOP number is certified at any (N, D, L) - and the paper's own benchmark quantifies how much the constants matter: despite the certified FLOP reduction, Hyena is SLOWER than FlashAttention below the crossover 'between 4096 and 8196' and slower than plain attention below 2048, with the 100x speedup reached only at L = 64K (4_experiments.tex:211). Illustration only (constant-1 reading, NOT licensed by the paper): N D L(log2 L + D) at N=2, D=864 gives 3.1e9 vs attention L^2 D = 3.6e9 at the deployed training length L=2048 (1.2x), 4.7x at 8192, 37x at 65536 - consistent in shape with the measured crossovers.", "verdict": "inside"}, {"arxiv_id": "2305.03796", "claim_anchor": "lem:approx-bin-prod (acl_latex.tex:346; proof at acl_latex.tex:482-)", "bound": "Existence: for binary A, B in {0,1}^{n x n} there EXISTS a two-layer ReLU network with f_mlp([Flat(A), Flat(B)]) = Flat(AB) exactly. The construction is fully explicit: input 2n^2, hidden width exactly n^3 (one unit per product A_ik B_kj via ReLU(a+b-1) = ab on binary scalars), binary W^(1) in {0,1}^{2n^2 x n^3}, W^(2) = I_{n^2} (x) 1_n, output n^2. RegularGPT's FFN is then argued (informally) to implement transition composition A(sigma_{a:b}) = A(sigma_{i:b}) o A(sigma_{a:i}) under the |Q|^2 one-hot encoding of eq:state_transition.", "deployed_config": "Regular-language RegularGPT: hidden size 256, 8 heads, log_C T layers (C=2 or 3), trained on lengths 40/50, 4.3M params (tab:regular_params); tasks: Even Pairs, Modular Arithmetic, Parity Check, Cycle Navigation, D_2/D_3/D_4/D_12, Tomita 3-6 (tab:regulars); FFN inner width not stated. Natural-language RegularGPT: hidden 768, K*log_C T layers, C=32..256, K=6/12, seq 512 (tab:natural_params).", "evaluation": "The certificate attaches to a hand-constructed network, not to any TRAINED deployed FFN: no deployed run instantiates W^(1), W^(2), and the paper's own support for 'RegularGPT's FFN models the composition' is the hedged bridge 'If o_i^(l) ~ A(...) is true, Lemma 1 implies...' (acl_latex.tex:359-361) plus PCA clustering evidence in sec:veri - the lemma certifies zero tolerance for the trained weights. What CAN be checked is embeddability of the certified construction in the deployed architecture, since this lemma (unusually) pins every constant: at PARITY, |Q| = n = 2 (the paper writes out its 2 transitions): input 2n^2 = 8 <= d_model 256, hidden n^3 = 8 - fits with huge slack; same at plausible |Q| = 4-5 (Mod Arithmetic, Cycle Navigation: n^3 <= 125 <= 256). But the closure fails for the larger deployed tasks: the FFN inner width is unrecorded, and at |Q| >= 12 (D_12's state count, unpinned by the paper) the encoding itself outgrows the deployed width - 2|Q|^2 = 288 > 256 = d_model and n^3 = 1728 > 4x256 - so whether every deployed task admits the certified construction inside the deployed dimensions is not determinable from the paper's own numbers.", "verdict": "not_evaluable"}, {"arxiv_id": "2305.17118", "claim_anchor": "theo:error_bound_main (04_method.tex:54-60)", "bound": "Budgeted-KV-cache generation error: E[||x_t - x~_t||_2] <= (2.1(1 - B/T_max)/(1-eps)^2)(k - (k-1)((1-eps)/(B/T_max - eps))^{1/(k-1)}) w.h.p., under: (i) beta_{t,j} = c v_{t,j} with v ~ power-law pdf c(x+b)^{-k}; (ii) lambda_V lambda_O (1+lambda_1 lambda_2)(1+lambda_Q lambda_K) <= 1/2 (top singular values of the value/output/MLP/query/key matrices); (iii) for EVERY t, the kept set S_t is exactly the B tokens with the largest CURRENT-step beta_{t,j} (min_{j in S_t} beta_{t,j} >= max_{j notin S_t} beta_{t,j}); single-layer transformer with ||x_t||_2 = 1.", "deployed_config": "Scissorhands on OPT-6B/13B/30B/66B, C4 LM + 5-shot Hellaswag/MathQA/PIQA/Winogrande, KV-cache compression 1x-5x, i.e. B/T_max down to 0.2 ('no accuracy drop until 5x compression on OPT-66B', 05_experiment.tex:17,31-32); 2x compression combined with 4-bit quantization (05_experiment.tex:67; results/2305.17118.json). Deployed keep-policy: Algorithm alg:sampling_pivotal_token selects by a HISTORY-window counter of low-score events (window w, recent window r, drop m; B_algorithm.tex), not by current-step scores.", "evaluation": "Precondition (iii) is one no budget-B algorithm can enforce: once a token is evicted its key is gone, so beta_{t,j} for dropped j at later steps cannot be computed - selecting the true current-step top-B over ALL j in [t] requires retaining everything (EVA/SMYRF-shaped circularity). The paper concedes the gap in its own gloss: 'Our theorem assumes that dropping the tokens depends on the attention score of the current iteration' (04_method.tex:67), while the deployed rule depends on previous iterations; its own persistence-ratio measurement is 'generally over 95%, with dips in the later transformer layers' (06_observation.tex) - i.e. the history-based set provably misses part of the current pivotal set, and the theorem carries no policy-mismatch term. Independently, the certified tolerance is not computable at deployment even under the oracle: the power-law constants (k, b, u, c) are never fitted or reported for any OPT model (power-law-ness is cited qualitatively from prior work, 02_introduction.tex:33), and the spectral condition (ii) - which forces lambda_V lambda_O <= 1/2 since the other factors are >= 1 - is never measured on OPT-6B/13B/30B/66B; the single-layer, unit-norm model also differs from the deployed multi-layer OPT.", "verdict": "outside"}, {"arxiv_id": "2305.17118", "claim_anchor": "theo:budgeted_cache (A05_Proofs.tex:308-310)", "bound": "Dropped attention mass: E[sum_{j notin hat-S_t} beta_{t,j}] <= ((1 - B/T_max)/(0.98(1-eps)^2))(k - (k-1)((1-eps)/(B/T_max - eps))^{1/(k-1)}) w.h.p., under beta_{t,j} ~ power-law f(x) = c(x+b)^{-gamma} on [0, u-b) with k > 2, u >= 5b, and S_t the top-B current-step keep-set of theo:appendix_err_bound.", "deployed_config": "Same deployments as theo:error_bound_main: OPT-6B/13B/30B/66B at 1x-5x KV compression (B/T_max >= 0.2) with the history-counter eviction policy of alg:sampling_pivotal_token.", "evaluation": "Inherits the same oracle keep-set precondition (S_t = current-step top-B over all j in [t], including already-evicted tokens) that the deployed budget-B algorithm cannot evaluate, let alone enforce - the certified regime is disjoint from any run of Algorithm alg:generation by construction, with the paper's persistence ratio (~95%, dips in later layers) quantifying the mismatch its theorem does not model. And the bound's value at the deployed B/T_max = 0.2..0.5 is uncomputable regardless: k, b, u are unpinned for every deployed model (no power-law fit is reported), eps is a free parameter, and T_min/T_max for the C4/few-shot runs are unrecorded. The paper's Figure figure:change_ratio ablation (attention-score change ratio centered at 0 with a -1 spike from dropped important tokens, 05_experiment.tex:79-80) is offered as an echo of the theorem, not an evaluation of it.", "verdict": "outside"}, {"arxiv_id": "2306.15799", "claim_anchor": "claim:one (flurka.tex:35)", "bound": "FLuRKA incurs fewer FLOPs than BOTH its kernel and low-rank constituents whenever N > d_k(H+2) > d_m > d_k > d_h (FLOP-counting argument, appendix.tex:142-173; the derivation substitutes d_m = H d_h).", "deployed_config": "Headline speed benchmark (evaluation.tex:42): d_m = 2600, d_k = 1500, d_h = 325, H = 8 ('following claim 1'), N swept 7.05k to 55.5k in 3k increments; measured culminating speedups 1.86x/1.37x (Lin.-Perf.) at the long end. Quality models (table:settings): MLM/NLU 12 heads, head dim 64 (d_m = 768), LRA 2 heads x 64; their d_k and sequence lengths are NOT recorded.", "evaluation": "Chain evaluated at the benchmark values: d_k(H+2) = 1500 x 10 = 15,000 > d_m = 2600 (ok) > d_k = 1500 (ok) > d_h = 325 (ok); H d_h = 8 x 325 = 2600 = d_m, matching the derivation's substitution. The binding condition is N > 15,000: satisfied by 14 of the 17 sweep points (N >= 16,050), where the culminating speedups are measured - inside. But the three shortest deployed points N = 7,050 / 10,050 / 13,050 provably violate N > d_k(H+2), and the main text's tightness gloss 'resulting in faster models for input sequence lengths as short as several thousand' (flurka.tex:41) is exactly the range the certificate does not cover at the deployed parameters. Quality-model deployments cannot be checked (d_k, N unrecorded); structurally, for the 12-head d_m = 768 models the chain forces N > d_k(H+2) > 768, so no d_k whatsoever satisfies it at any sequence length <= 768 - the GLUE/MLM quality wins are uncovered by the FLOP certificate unless their (unrecorded) N exceeded 768.", "verdict": "inside"}, {"arxiv_id": "2306.15799", "claim_anchor": "claim:two (appendix.tex:100)", "bound": "FLuRKA incurs fewer FLOPs than low-rank methods whenever N - 1 > d_k > d_h (FLOP difference a+b+c > d+e+f+g needs N > 1 + d_k and d_k > d_h, appendix.tex:146-149).", "deployed_config": "Downsampling-factor ablation (evaluation.tex:137): N = 20k, d_h = 128, d_k swept 8k to 20k in 3.2k increments; measured culminating speedups over Linformer 3.39x/3.38x/3.12x.", "evaluation": "d_k > d_h = 128 holds at every sweep point. N - 1 = 19,999 > d_k holds for d_k = 8,000 / 11,200 / 14,400 / 17,600 - inside, and these carry the reported speedups. The printed top of the sweep, d_k = 20k = N, violates N - 1 > d_k (19,999 < 20,000): at that single deployed point the certificate's advantage term fails its own precondition (the derivation needs N > 1 + d_k) and nothing is certified. FLOP counts are exact polynomial expressions (no hidden constants), so the inequality checks are sharp; what the certificate never covers is wall-clock - the paper's measured speedups are consistent with but not implied by the FLOP ordering.", "verdict": "inside"}, {"arxiv_id": "2306.15799", "claim_anchor": "claim:three (appendix.tex:104)", "bound": "FLuRKA incurs fewer FLOPs than kernel methods whenever N > d_k(H+2) and d_m > d_k (appendix.tex:153-173; uses N > d_k(2d_m+1+d_h)/(1+2d_h), then bounds that threshold by d_k(H+2) via d_m = H d_h).", "deployed_config": "Hidden-dimension ablation (evaluation.tex:139): N = 20k, d_k = 1k, d_k(H+2) = 6k (so H = 4), d_m swept 800 to 20k in 3.2k increments; culminating speedups over kernel methods 1.72x/1.72x/1.71x.", "evaluation": "N > d_k(H+2): 20,000 > 6,000 holds at every point. d_m > d_k = 1,000: holds for the six sweep points d_m = 4,000..20,000 - inside, where the reported speedups culminate. The FIRST deployed sweep point d_m = 800 < 1,000 violates d_m > d_k - notably the paper's own setup line asserts 'd_m > d_k (=1k)' while declaring the sweep starts at 800, an internal inconsistency: at that point the kernel-comparison term i > l (2N d_m^2 vs 2N d_k d_m) is not certified. Same caveat as claim:two: exact FLOP inequalities, no wall-clock content.", "verdict": "inside"}, {"arxiv_id": "2306.15799", "claim_anchor": "accuracy-claim (flurka.tex:43)", "bound": "||phi(Q W^Q) phi(E_1 K W^K)^T F V W^V - A V W^V||_inf < eps_4 ||F V W^V||_inf + eps_1 ||A||_2 ||V W^V||_2 with probability >= 1 - o(1) 'for large enough m', under: unbiased random features phi with m samples; k = 5 log(d)/(eps_2^2 - eps_3^2) with eps_1..eps_3 the constants of Linformer Theorem 2 (restated cite-only at appendix.tex:207); E = delta R, F = e^{-delta} R with R iid N(0, 1/k).", "deployed_config": "FLuRKA Linformer-{Performer, RNN, EVA} trained on GLUE (12 layers, 12 heads, head dim 64, ~108M), LRA (2 layers, 2 heads x 64, ~0.4M), WikiText-103, NMT, ImageNet (table:settings; results/2306.15799.json). The deployed downsampling factor d_k, the feature count m, and delta are not recorded for any quality run.", "evaluation": "Nothing is computable at deployment: (1) the kernel half of the bound is limit-style - eps_4 is granted only 'for large enough m' via the law of large numbers (proof sketch, flurka.tex), with probability 1 - o(1) carrying no rate, so at any finite deployed m the certified eps_4 is undefined (AlgoFormer-shaped vacuousness); (2) eps_1, eps_2, eps_3 are inherited from Linformer Theorem 2 without values, and delta is free; (3) the Linformer-side formula k = 5 log(d)/(eps_2^2 - eps_3^2) could in principle be inverted at a deployed k (the CONTRADICTIONS.md Linformer finding: eps ~ 0.40 at k = 256, n = 512), but the paper never reports the deployed d_k of its quality models, so even that inversion cannot be run here; (4) the theorem requires E, F to be scale-matched iid Gaussians (delta R, e^{-delta} R) - whether the deployed (trainable, Linformer-style) projections retain this is unstated. A guarantee-shaped statement with every knob unpinned at deployment.", "verdict": "not_evaluable"}, {"arxiv_id": "2309.07412", "claim_anchor": "prop:input-indep (acl_latex.tex:140)", "bound": "Impossibility: an input-independent LRNN (x_k = A x_{k-1} + B u_k, A constant) is inconsistent in representing subtraction - the proof forces a state collision on witness strings '0-1' vs '1-0' and '0-0-1' vs '0-1-0' (lengths 3-5), for ANY state dimension, any real A, B, any embeddings. No constants and no error rate: what is certified is impossibility of exact consistent representation, not any accuracy level.", "deployed_config": "Paper deploys S4 and S4D as baselines (plus Liquid-S4 and its own model): trained at L_tr = 40 (39 for ModArith), tested at L_ex = 500 (499), 5 trials x 40k updates (results/2309.07412.json; tab:extrp_settings acl_latex.tex:273). Paper's own remark (acl_latex.tex:160ff): Eq. (2) 'encapsulates the recurrence relation of S4, S5, and Linear Recurrent Unit' and the limitation 'also applies to models adopting diagonal linear recurrence relations' (S4D).", "evaluation": "Precondition check: deployed S4/S4D have input-independent transition matrices by construction (HiPPO-parameterized A and diagonal A are fixed after training, not functions of u_k) - the architectural precondition holds identically at every deployed size. The proposition is length- and dimension-free, and its witness strings (length 3-5) sit inside the deployed length range (39-500), so the certified obstruction is active at deployment. What is NOT certified: any quantitative failure rate - the observed 0.27 extrapolation accuracy of S4/S4D on Sum(5)/ModArith(5) is consistent with, but not a consequence of, the proposition (the proof's 'consistent' notion requires equal values to share a STATE, stronger than the equal-OUTPUT condition deployment needs since y = h(x) with possibly non-injective h).", "verdict": "inside"}, {"arxiv_id": "2309.07412", "claim_anchor": "prop:dlrnn (acl_latex.tex:182); proof in appendix 'Proof of Proposition 2' (acl_latex.tex:357)", "bound": "Impossibility: a diagonal input-dependent LRNN (x_k = diag(v_k) x_{k-1} + B u_k, v_k = f(u_k)) is inconsistent in representing subtraction. Same exact-representation semantics as prop:input-indep; no constants, no rate.", "deployed_config": "No accuracy-evaluated deployed model instantiates the diagonal input-dependent class Eq. (3): the paper's own model uses 8x8 blocks ((b,h)=(8,8), acl_latex.tex:295ff), not diagonal; S4D is diagonal but input-INdependent (audited under prop:input-indep); Liquid-S4 is A + diag(B u_k), which the paper explicitly says 'does not suffer from the limitation outlined in Proposition 2' (acl_latex.tex:189ff). The only literal instantiation is the b = 1 speed-benchmark run (acl_latex.tex:318ff, 'we set b=1 and reassess the speed': PScan 0.0008s vs Sequential 0.002s per instance), which reports timing only, no accuracy.", "evaluation": "Not evaluable for lack of a deployed config in the certified class: the class prop:dlrnn declares impossible is a motivational intermediate the paper never accuracy-tests. The b=1 timing config does satisfy the precondition (1x1 blocks = diagonal, input-dependent), so the impossibility would apply to it, but no quality number exists to be inside or outside of. (Reading S4D as the degenerate constant-f member of the class would make the regime non-empty at deployment, but the paper routes S4D through prop 1 instead, and 'v_k = f(u_k) ... depends on u_k' suggests genuine input dependence.)", "verdict": "not_evaluable"}, {"arxiv_id": "2310.01655", "claim_anchor": "thm:main_in_intro (intro.tex:79)", "bound": "Randomized feature map phi': R^h -> R^{r^2}, r = Theta(p eps^{-2} log(1/delta)), built from RANDOM Gaussian sketch matrices (PolySketchNonNegative, sketch.tex alg:polysketch), with (i) nonnegative inner products, (ii) sum_{i,j} |<phi'(q_i),phi'(k_j)> - <q_i,k_j>^p|^2 <= eps^2 sum_{i,j} ||q_i||^{2p} ||k_j||^{2p} w.p. 1-delta, for even p >= 2, eps in (0, 0.5).", "deployed_config": "Polysketch attention with p = 4, head size h = 64, sketch sizes r in {32, 64} (feature dim r^2 = 1024/4096), contexts 512-32k. Flagship configs (main table exp.tex:16-47, Fig. 1: all GPT-2 Small/Medium/Large rows) use LEARNED sketches: each random Gaussian G_1, G_2 replaced by a trained 3-hidden-layer gelu dense network of size [8r, r, 8r] with layernorm (app.tex:100-104; learned_sketch.tex). Random-sketch runs exist only in the appendix PG19/Wiki-40B tables: 'Polysketch (random, r = 32/64)' at contexts 512-32k (app.tex:134-139; results/2310.01655.json config_notes).", "evaluation": "Two deployment families. (a) Learned sketches (all main-table and headline speed/quality rows): provably OUTSIDE the certified construction - the theorem's phi' is a draw from a specific Gaussian-sketch distribution and its guarantee is a probability over that draw; a trained nonlinear dense network is not a draw from any sketch distribution, so properties (ii)-(iii) certify nothing about it (the paper itself frames learning the sketches as 'a heuristic', intro.tex). Property (i) nonnegativity DOES survive, since it comes from the self-tensoring x -> x^{ox 2}, not from randomness. (b) Random sketches (appendix rows): structural preconditions hold (p = 4 is even, >= 2), but the certified eps at r = 32/64 is uncomputable - the Theta constant is inherited from 'Results from Section 4 of ahle2020oblivious' (sketch.tex, proof of thm:main_in_intro) and never pinned. Constant-1 illustration (NOT licensed by the paper): eps = sqrt(p log(1/delta)/r) at delta = 0.1 gives eps ~ sqrt(4 x 2.303/64) ~ 0.38 for r = 64 and ~ 0.54 for r = 32 - the latter exceeding the theorem's own eps < 0.5 domain, i.e. under a constant-1 reading r = 32 would not be certifiable at all.", "verdict": "outside"}, {"arxiv_id": "2310.01655", "claim_anchor": "thm:tensoring-preserves-amm (sketch.tex:71)", "bound": "If S in R^{h^{p/2} x r} satisfies the (eps, delta, t)-JL and (eps, delta, 2t)-JL moment properties (t even), then for any C, D with h^{p/2} columns: ||(CS)^{ox2}((DS)^{ox2})^T - C^{ox2}(D^{ox2})^T||_F <= sqrt(5) eps ||C^{ox2}||_F ||D^{ox2}||_F w.p. >= 1 - delta. Self-tensoring costs only a sqrt(5) factor.", "deployed_config": "Same as thm:main_in_intro: deployed S is either the Ahle et al. degree-p/2 Gaussian sketch with r in {32, 64}, h = 64, p = 4 (appendix random rows, app.tex:134-139), or - in all flagship configs - a learned nonlinear dense network replacing S entirely (exp.tex main table; app.tex:100-104).", "evaluation": "Learned configs: the JL-moment property is a moment condition on a RANDOM matrix distribution; a fixed trained nonlinear network is not an object of the required type, so the precondition is violated by construction. Random configs: the paper asserts the Ahle et al. sketch satisfies the JL-moment requirements, but the (eps, delta) pair certified at r = 32 or 64 is Theta-unpinned (which eps satisfies the (eps, delta, t)-JL property at r = 32 is never computable from the paper), so the sqrt(5) eps bound evaluates to no number at any deployed config. The sqrt(5) factor is the paper's one fully explicit constant, but it multiplies an unknown.", "verdict": "not_evaluable"}, {"arxiv_id": "2310.05869", "claim_anchor": "lemm_approx_D (main_arxiv.tex:450)", "bound": "Diagonal estimation: if the sample size m = Omega(kappa^7 alpha^2 eps^{-6} log n) - with alpha = n max_i ||D^{-1} A e^{(i)}||_2^2 and kappa = ratio of max to min unmasked row mass - then ApproxD's output tilde{D} satisfies the spectral error condition eq:error-D-approx w.p. >= 1 - 1/poly(n).", "deployed_config": "Monkey-patched chatglm2-6b-32k (n = 32768) and phi-1.5 (n = 2048), replacing the final ell in {7,14,21,28} attention layers; bucket size b = 256 AND sampled columns m = 256 fixed for all sequence lengths (main_arxiv.tex:661); causal masking via recursion until n < 4096; single-layer benchmarks to n = 131k (results/2310.05869.json config_notes).", "evaluation": "Not evaluable at any deployed config: the Omega constant is unpinned, kappa is never measured anywhere in the paper, and the paper's only alpha measurements are (a) avg alpha = 8.1801 at n = 3136 on T2T-ViT ImageNet - a different model than the patched LLMs (main_arxiv.tex, 'Empirical Verification of Assumption'), and (b) chatglm alpha computed EXCLUDING the first 32 columns because 'the first columns in D^{-1}A often contain heavy entries' - an exclusion the lemma's definition of alpha does not license. Constant-1 illustration (NOT licensed: Omega constant unpinned, kappa unmeasured, alpha borrowed from the ViT model): even at kappa = 1 and eps = 1, m >= alpha^2 ln n = 8.1801^2 x 8.05 ~ 539 > 256 deployed; at eps = 0.5 the requirement is 64x larger (~34,500), and kappa enters to the 7th power (kappa = 2 alone multiplies by 128).", "verdict": "not_evaluable"}, {"arxiv_id": "2310.05869", "claim_anchor": "lem:amm (main_arxiv.tex:461)", "bound": "AMM sampling: for S built from m i.i.d. rows sampled proportionally to squared row norms of V (rescaled indicator rows), m = Omega(eps^{-2} d sr(tilde{D}^{-1}A)) gives ||tilde{D}^{-1}A S^T S V - tilde{D}^{-1}A V||_op <= eps ||tilde{D}^{-1}A||_op ||V||_op w.p. >= 0.99.", "deployed_config": "Deployed implementation: 'We implement HyperAttention based on sortLSH and uniform column sampling' with the number of sampled columns m = 256 for all sequence lengths (main_arxiv.tex:636, 661); patched models chatglm2-6b-32k (n = 32k) and phi-1.5 (n = 2048).", "evaluation": "The lemma's precondition specifies the SAMPLING DISTRIBUTION: rows drawn with probability ||V_{i,:}||_2^2 / ||V||_F^2 and rescaled by ||V||_F/(sqrt(m)||V_{l_r,:}||_2). The deployed implementation replaces this with UNIFORM column sampling by the paper's own description (main_arxiv.tex:636) - a different estimator to which the certified 0.99-probability bound does not apply (they coincide only in the measure-zero case of exactly equal row norms of V, which is unrecorded and generically false for trained LLM values). Independently, even under the certified sampler no eps is computable at deployed m = 256: the Omega constant is unpinned and neither d for the patched models nor the stable rank sr(tilde{D}^{-1}A) is recorded, so the constant-1 inversion eps = sqrt(d sr / m) evaluates to no number.", "verdict": "outside"}, {"arxiv_id": "2310.05869", "claim_anchor": "main_theorem (main_arxiv.tex:487)", "bound": "HyperAttention guarantee: if eps > 1/n^{o(1)}, max_i ||D^{-1}A e^{(i)}||_2^2 <= n^{o(1)}/n (i.e. alpha <= n^{o(1)}), and kappa <= n^{o(1)}, then w.p. >= 0.98 the outputs (S, tilde{D}) of Algorithm alg_hyper_attn satisfy the spectral condition; runtime O(d n^{1+o(1)} + d nnz(M^H)).", "deployed_config": "chatglm2-6b-32k at n = 32768 with ell in {7,14,21,28} layers replaced (ppl 5.6 -> ~6.3 at 50% speedup with ~20-24 layers; up to 12 at all layers); phi-1.5 at n = 2048; b = m = 256 throughout; uniform column sampling in place of the algorithm's row-norm sampler (main_arxiv.tex:636, 661); single-layer runs to n = 131k.", "evaluation": "Outside on two grounds. (1) Algorithm substitution: the theorem certifies the outputs of Algorithm alg_hyper_attn, whose line 3 constructs S 'based on squared row norms of V as in lem:amm'; the deployed pipeline uses uniform column sampling (paper's own words, main_arxiv.tex:636), so at every deployed config the certified algorithm is not the one run - the guarantee attaches to nothing that was executed (LSSL pattern). (2) Even for the certified algorithm the regime is uncheckable at fixed n: alpha, kappa <= n^{o(1)} and eps > 1/n^{o(1)} are asymptotic family conditions with no content at a single n = 32768, and the paper's own verification substitutes a modified alpha for chatglm - computed 'excluding the first 32 columns' after observing that the first columns of D^{-1}A 'often contain heavy entries' - i.e. the theorem's alpha as actually defined is dominated at deployment by exactly the attention-sink columns the measurement throws away; kappa is never measured at all.", "verdict": "outside"}, {"arxiv_id": "2310.05869", "claim_anchor": "cor:lsh (main_arxiv.tex:499)", "bound": "sortLSH instantiation: with mask from Algorithm alg-sort-lsh, block size b = n^{o(1)}, LSH parameter r = log2 n, and at most n^{1+o(1)} pairs (i,j) with theta(Q_i, K_j) <= (pi/2)(1-o(1)), plus all main_theorem preconditions: nnz(M^H) <= n^{1+o(1)} w.p. 1 - 1/n^{o(1)}, spectral condition w.p. >= 0.98, total runtime O(d n^{1+o(1)}).", "deployed_config": "The deployed pipeline is sortLSH-based ('Hamming sorted LSH, which our experiments are based on', main_arxiv.tex:497; implementation at 636): block size b = 256 fixed for ALL n in {1k, ..., 32k, 131k}; the LSH bit-parameter r actually used is never recorded; the angle distribution of deployed Q, K rows is never measured.", "evaluation": "Not evaluable: (a) b = n^{o(1)} and the n^{1+o(1)} angle-pair cap are asymptotic conditions with unpinned o(1) exponents - no finite n certifies or refutes them, and the corollary's own success probability 1 - 1/n^{o(1)} is itself uncalibrated (n^{o(1)} could be arbitrarily close to 1, making the mask-sparsity probability arbitrarily small); (b) the deployed r is unrecorded, so even the one pinned parameter choice (r = log2 n, i.e. r = 15 at n = 32768) cannot be checked against deployment; (c) the near-orthogonality assumption is motivated by rows 'drawn uniformly at random from a d-dimensional sphere' - trained LLM queries/keys are not that, and no measurement is offered. The corollary also inherits main_theorem's preconditions and its output statement about alg_hyper_attn, hence also the uniform-sampler deviation flagged in the main_theorem record.", "verdict": "not_evaluable"}, {"arxiv_id": "2310.05869", "claim_anchor": "cor:countsketch (main_arxiv.tex:508)", "bound": "ExpanderSketch instantiation: with threshold mask M^H_{ij} = 1 iff (QK^T)_{ij}^2 >= ||QK^T e_j||_2^2 / tau, tau = n^{o(1)}, the mask is found exactly w.p. 1 - O(1/n^2), spectral condition w.p. >= 0.98, runtime O(d n^{1+o(1)}).", "deployed_config": "None: the experiments are explicitly sortLSH-based ('We implement HyperAttention based on sortLSH and uniform column sampling', main_arxiv.tex:636); no ExpanderSketch/CountSketch mask is run anywhere in the paper's experiments.", "evaluation": "Not evaluable (no deployed config): the certified threshold-mask pipeline is never instantiated, so there is no (n, tau, d) triple to test; independently, tau = n^{o(1)} is asymptotic with no pinned exponent.", "verdict": "not_evaluable"}, {"arxiv_id": "2310.16028", "claim_anchor": "constructable-presoftmax-linear-combination (rasp_aggr.tex:73)", "bound": "Expressivity closure: if pre-softmax attention matrices M_1, M_2 are each producible by some transformer weights, then alpha M_1 + beta M_2 is producible for all real alpha, beta. Existence statement with no resource parameter, tolerance, or probability; used to compile max-aggregation via M* = Z + 2|Vocab| M_1 (rasp_aggr.tex:80ff).", "deployed_config": "Decoder-only transformers trained from scratch: 6 layers / 8 heads / 64-512 dim (2 layers / 16 heads / 1024 dim for Sort), test EM evaluated 10 tokens beyond max training length, on Count, Mode, Unique/Repeat Copy, Sort, Addition, Parity (results/2310.16028.json config_notes; table at intro2.tex:52).", "evaluation": "Nothing to evaluate: the lemma (the paper itself calls it 'trivial to prove') ties no resource to any error - it is a weight-existence closure fact holding at any width, and deployment never instantiates the compiled construction; the trained models' weights are learned, not the constructed ones. The paper's actual quantitative story (RASP-L-expressible tasks length-generalize, others do not: 96-100% vs 0% EM) rests on the RASP-Generalization CONJECTURE, which is stated as a conjecture, not a theorem (claims file 'skipped' note), so no certified regime exists for the deployed accuracy numbers at all.", "verdict": "not_evaluable"}, {"arxiv_id": "2311.14652", "claim_anchor": "lem:jl_error (preli.tex:76)", "bound": "JL sketching of V changes each entry of the polynomial-method attention output by at most eps_2/sqrt(n) (per-column l2 error <= eps_2) w.p. 1 - delta, using m_2 = O(eps_2^{-2} log(nd/delta)), under ||V|| <= 1/sqrt(n), eps_2, delta in (0,1).", "deployed_config": "None: pure theory paper - no experiments in any source file (abstract/intro/analysis/preli/conclusion all match zero occurrences of 'experiment'/'empirical'/'evaluat'), and no results/2311.14652.json exists in the corpus.", "evaluation": "Not evaluable (no deployed config): no (n, d, eps_2, delta, m_2) is ever instantiated. Even hypothetically, the O(.) constant in m_2 is unpinned and the normalization precondition ||V|| <= 1/sqrt(n) is a scaling real trained value matrices are not known to satisfy - both would block evaluation if a deployment existed.", "verdict": "not_evaluable"}, {"arxiv_id": "2311.14652", "claim_anchor": "thm:formal (analysis.tex:48)", "bound": "One-pass streaming attention: under d = O(log n), ||Q||_inf, ||K||_inf <= B = O(sqrt(log n)), ||V|| <= 1/sqrt(n), a streaming algorithm uses O(eps_1^{-1} k n^{o(1)} + eps_2^{-2} n^{o(1)}) space, outputs O(k)-sparse columns T_i with ||T_i - y_i||_2 <= (1+eps_1) min_{k-sparse y'} ||y' - y_i||_2 + eps_2, success probability 0.99, decoding time O(eps_1^{-1} k n^{o(1)}).", "deployed_config": "None: no experiments and no results file (see lem:jl_error record).", "evaluation": "Not evaluable (no deployed config). Additionally the certified regime could not be checked even for a future deployment without more pinning: d = O(log n) and B = O(sqrt(log n)) are asymptotic conditions with unpinned constants (at n = 32768, is d = 128 <= C log n for the intended C? undecidable), n^{o(1)} space factors have no pinned exponent, and ||V|| <= 1/sqrt(n) is an unverified normalization for real attention inputs.", "verdict": "not_evaluable"}, {"arxiv_id": "2311.14652", "claim_anchor": "cor:formal (analysis.tex:175)", "bound": "Same guarantees as thm:formal when the stream delivers raw inputs X_1, X_2 and weights W_Q, W_K, W_V (Q = X_1 W_Q etc., d = O(log n), B = O(sqrt(log n)), ||V|| <= 1/sqrt(n)): space O(eps_1^{-1} k n^{o(1)} + eps_2^{-2} n^{o(1)}), (1+eps_1)-relative-plus-eps_2-additive per-column error, probability 0.99.", "deployed_config": "None: no experiments and no results file (see lem:jl_error record).", "evaluation": "Not evaluable (no deployed config); inherits every unpinned quantity of thm:formal (asymptotic d and B conditions, n^{o(1)} exponents, O(.) constants, unverified ||V|| normalization).", "verdict": "not_evaluable"}, {"arxiv_id": "2401.16421", "claim_anchor": "Thm:classic_PE (sections/3_methods.tex:45; appendix/proof.tex:62-78)", "bound": "Impossibility (printed garbled): for any state-set size there EXISTS an NFA N = (Q, Sigma, delta, q_0, F) such that a Transformer with APE 'needs at least O(|Q|^2) embedding size' to represent it - an Omega(|Q|^2) lower bound written with O(.). The proof's witness NFA has alphabet = all transition functions, size 2^{|Q|^2} (proof.tex:78: 'The size of the alphabet is 2^{n^2}, so we need at least O(n^2) bits to embed each input symbol').", "deployed_config": "Deployed models: arithmetic task - 3 layers, 4 heads, hidden in {48, 64, 256} (sections/4_experiments.tex:36); language modeling - 12 layers, hidden 768, 12 heads, ~155M params, pretrain length 1024, eval to 8192, SCROLLS fine-tuning (4_experiments.tex:54; results/2401.16421.json config_notes). Baselines include Sinusoidal PE (= APE) models.", "evaluation": "Not evaluable: the lower bound is existential over adversarial NFAs - no deployed task is ever identified with a concrete NFA, so no deployed |Q| exists to plug in; the bound's constant is unpinned (and the statement itself misprints Omega as O, flagged in the claims file). The proof's witness lives nowhere near deployment by the paper's own numbers: its alphabet has 2^{|Q|^2} symbols (already 2^{2304} at |Q| = 48), while deployed vocabularies are in the tens of thousands - so even granting constant 1, no deployed APE baseline processes the certified hard instance.", "verdict": "not_evaluable"}, {"arxiv_id": "2401.16421", "claim_anchor": "Thm:BiPE (sections/3_methods.tex:50; appendix/proof.tex:82-129)", "bound": "Expressivity upper bound: for any Bi-NFA with state collection {Q_1,...,Q_k}, there EXISTS a Transformer with BiPE and embedding size O(k^2 + sum_i |Q_i|^2) representing it. The certified construction uses log-precision arithmetic, ceil(log N) layers in each of two modules (proof.tex:87: 'the log-precision transformer with the GeLU activation function and O(log N) layers ... crucial'), and T5-relative inter-segment PE.", "deployed_config": "BiPE-RoPE / BiPE-ALiBi trained models: arithmetic - 3 layers, 4 heads, hidden in {48, 64, 256} (97%/95% accuracy at hidden 48, sections/4_experiments.tex:36-38); LM - 12 layers, hidden 768, ~155M, lengths 1024-8192 (4_experiments.tex:54).", "evaluation": "Not evaluable: (a) no deployed task is given a Bi-NFA model, so (k, |Q_1|,...,|Q_k|) is undefined for every experiment and O(k^2 + sum |Q_i|^2) cannot be compared to hidden sizes 48/64/256/768; (b) the O(.) constant is unpinned; (c) the deployed models are not the certified construction: fixed 3 or 12 layers vs the proof's ceil(log N) per module (log2 N = 10 at N = 1024, 13 at N = 8192 - the 3-layer arithmetic models are below the construction's depth even before constants), standard fp precision vs log-precision, RoPE/ALiBi vs the T5-relative encoding the proof uses (the proof asserts extension to RoPE/ALiBi variants 'can be easily' done but gives no such proof). The paper's claim that the hidden-48 result 'well aligns with our theoretical results' (4_experiments.tex:38) is a shape analogy, not an evaluation of the bound.", "verdict": "not_evaluable"}, {"arxiv_id": "2401.16421", "claim_anchor": "lemma:MLP_matrix (appendix/proof.tex:29)", "bound": "Approximation: for any eps > 0 there exist two-layer GeLU MLP parameters with l_inf norm O(poly(1/eps)) such that ||f(e_A, e_B) - e_{A.B}||_inf <= eps uniformly over Boolean matrices A in {0,1}^{d1 x d2}, B in {0,1}^{d2 x d3}.", "deployed_config": "No deployed instantiation: the lemma is a gadget inside the Thm:BiPE construction (used at proof.tex:118-127 to implement transition-matrix multiplication); deployed models' MLPs are trained end-to-end, and no (eps, parameter-norm) pair is recorded for any run.", "evaluation": "Not evaluable: the rate is 'O(poly(1/eps))' with BOTH the polynomial degree and the constant unspecified - the weakest possible quantitative form, uninvertible at any deployed parameter norm even if one were recorded (which none is). The lemma certifies existence of weights, not that training finds them; deployment exercises neither direction.", "verdict": "not_evaluable"}, {"arxiv_id": "2402.07443", "claim_anchor": "thm:flash-attention (inputs/io_complexity_lower_bounds.tex:18) + matching large-cache lower bound (lemma:partition-size-lb-large-mem, io_complexity_lower_bounds.tex:~135; tight characterization thm:standard-mm-attention-mechanism-i/o, intro.tex:56)", "bound": "TIGHT large-cache I/O: Q_Att(M) = Theta(N^2 d^2 / M) for M >= d^2 (M = Omega(d^2)); the abstract certifies this matches FlashAttention 'within any constant factors for any M >= d^2'. Regime boundary at M = d^2, where N^2 d/sqrt(M) = N^2 d^2/M = N^2.", "deployed_config": "No results file exists for 2402.07443 (pure theory). FlashAttention standard deployed configs pulled from 2205.14135 / 2307.08691 per the special note: A100 192KB SRAM/SM -> M = 98,304 fp16 elements (49,152 fp32); head dim d = 64 (BERT MLPerf, GPT-2, GPT3-1.3B), d = 80 (GPT3-2.7B), up to 128 in common practice; benchmark seqlens N = 128 (BERT), 512, 1024 (GPT-2), 2k-8k (GPT3 FA-2), 2k-16k (long-range/block-sparse).", "evaluation": "The regime test here is M vs d^2 (NOT the d <= M <= Nd sub-regime that scope-sweep-A used for 2205.14135). d=64: d^2 = 4,096, M/d^2 = 98,304/4,096 = 24 >= 1 -> M >= d^2. d=80: d^2 = 6,400, M/d^2 = 15.4. d=128: d^2 = 16,384, M/d^2 = 6.0. Every standard fp16 head dim places every deployed seqlen in the LARGE-CACHE regime, where this paper proves Theta(N^2 d^2 / M) tight to constant factors. Edge case: fp32 (M = 49,152) at d = 256 gives d^2 = 65,536 > M -> small cache, but d=256 fp32 is not a standard FlashAttention deployment. Theta constants are unspecified, so no absolute HBM count is certified anywhere - only the regime placement is.", "verdict": "inside"}, {"arxiv_id": "2402.07443", "claim_anchor": "thm:small-mem-tiling-alg (inputs/io_complexity_lower_bounds.tex:111) + thm:i/o-lb-small-mem (:257) + thm:small-cache-fmm-equiv (inputs/attention_fmm.tex:463)", "bound": "TIGHT small-cache I/O: Q_Att(M) = Theta(N^2 d / sqrt(M)) in regime M = o(d^2) (upper bound by the tiling algorithm, lower bound thm:i/o-lb-small-mem); and for M <= d^2, Q_Att(M) is sandwiched between the I/O of MM(N,d,N) and MM(N,d,N)+MM(N,N,d) (thm:small-cache-fmm-equiv), even with fast matrix multiplication.", "deployed_config": "Same FlashAttention configs: A100 M = 98,304 fp16 elements; head dim d = 64-128; seqlens N = 128-16384.", "evaluation": "Precondition M = o(d^2) / M <= d^2: deployed M = 98,304 vs d^2 = 4,096 (d=64) - M exceeds d^2 by 24x, so M = o(d^2) is provably FALSE and M <= d^2 is violated at every deployed config (min M/d^2 = 6.0 at d=128, still > 1). The small-cache tight bound Theta(N^2 d/sqrt(M)) and the fast-MM equivalence characterize a regime (cache smaller than one head's Gram block) that no standard FlashAttention deployment enters. At deployment N^2 d/sqrt(M) >= N^2 d^2/M (since sqrt(M) >= d when M >= d^2), so the min() in the paper's tight characterization always selects the large-cache branch.", "verdict": "outside"}, {"arxiv_id": "2402.07443", "claim_anchor": "thm:i/o-compression-protocol (inputs/attention_fmm.tex:78); lemma:d-rank-cache-lb (:133); lemma:matrix-mult-io-lb (inputs/io_complexity_lower_bounds.tex:251)", "bound": "OneWayCC(MEC_{B*}) <= 2 M log q; OneWayCC(MEC_B) >= min(d sqrt(B/2), B/4) log q for Q,K in F_q^{N x d} with q > N; standard-MM I/O Q(M) = Omega(m k n / sqrt(M)). Together these yield the arbitrary-algorithm (fast-MM) large-cache lower bound Omega(N^2 d^2 / M).", "deployed_config": "No deployed config instantiates these: they are one-way communication-complexity reductions over a finite field F_q with q > N and a B-entry matrix-compression game; FlashAttention runs over fp16/fp32 reals, not F_q.", "evaluation": "Not evaluable (no deployed config): the finite-field model (F_q, q > N) and the matrix-compression protocol are proof devices with no deployed counterpart, and the MM lower bound is constant-free (Omega) and standard-MM-restricted. They certify the generality of the large-cache lower bound (holds even under fast matrix multiplication) but expose no resource-to-tolerance quantity checkable at any concrete FlashAttention config.", "verdict": "not_evaluable"}, {"arxiv_id": "2403.01590", "claim_anchor": "mamba-vs-attention-expressivity (main.tex:725, restated 840); mamba-count-in-row (main.tex:864); head-cannot-count-in-row (main.tex:883)", "bound": "Expressivity / impossibility with NO rate and NO tolerance: one selective-SSM (Mamba) channel expresses every function a single transformer head can, and realizes the count-in-row function exactly for all lengths; a single transformer head cannot express count-in-row for sequence length L > 2. Proved under Assumptions 1-2 (main.tex): discretization ignored and hidden dimension = 1, i.e. all parameters A_i,B_i,C_i,W^Q,W^K,W^V are scalars in R.", "deployed_config": "Vision Mamba-Small (24 layers) vs ViT-Small on ImageNet and ImageNet-Segmentation; Mamba-130M vs BERT on IMDB (results/2403.01590.json records perturbation-AUC and segmentation pixel-acc/mAP/mIoU explainability metrics). Deployed widths are hundreds-dimensional, not scalar; count-in-row is a synthetic task never trained or evaluated.", "evaluation": "Not evaluable on three independent grounds: (1) the theorems carry no resource-to-tolerance bound - they are pure function-class containment / impossibility statements; (2) the proofs fix hidden dimension = 1 with scalar weights, a regime every deployed model (multi-head, multi-channel, dim in the hundreds) violates by construction, and larger dimension does not map to any certified accuracy; (3) count-in-row is never a deployed task - the reported metrics are ImageNet segmentation / perturbation-AUC and IMDB sentiment, none certified by these lemmas. There is no (eps, size) pair to place inside or outside.", "verdict": "not_evaluable"}, {"arxiv_id": "2403.09347", "claim_anchor": "tp-runtime (example_paper.tex:631); burst-runtime-total (example_paper.tex:662)", "bound": "Exact per-Transformer-block runtime models: TP (Megatron-V3) T = 4 t_comm + T_attn/G + T_ffn/G; BurstAttention T_total = max(T_attn_f, t_comm_attn_f) + max(T_attn_b, t_comm_attn_b) + T_ffn + t_comm_weights. Explicit precondition (remarks at :635 and :668): FULL attention only - the analysis 'does not account for sparse, approximate, or CAUSAL attention.'", "deployed_config": "LLaMA-2 7B / 13B decoder-only inference, first-token latency, seqlens 4K-256K, batch size 1, on 8xA100 (single node, PCIe) and 32xA100 (4 nodes, 600 Gb/s RoCE) (results/2403.09347.json).", "evaluation": "Precondition violated: LLaMA-2 is a causal decoder-only LM, so prefill/inference uses CAUSAL attention - exactly the case the propositions' own remarks exclude. The full-attention runtime model therefore certifies none of the deployed latencies. Independently, T_attn, T_ffn and the effective bandwidth b are never pinned numerically; the reported latencies (e.g. BurstAttention-7B @32K = 3.27s, Megatron-V3 @32K = 5.67s) are measured, not derived from the formula - so the model yields no certified number even setting the causal issue aside.", "verdict": "outside"}, {"arxiv_id": "2403.09347", "claim_anchor": "tp-comm-time (example_paper.tex:627); burst-activation-comm-time (:649); burst-weight-comm-time (:657)", "bound": "Exact byte-count / bandwidth communication-time formulas: t_comm = (B N Z' d) M (G-1)/(b G); t_comm_attn_f = 2 B N' Z d M (G-1)/(b G); t_comm_weights = (4 H Z d + 2 H d_ffn) M (G-1)/(b G). Precondition: uniform cluster bandwidth b across all G devices; M bits per tensor element.", "deployed_config": "8xA100 single node (all PCIe) and 32xA100 = 4 nodes x 8 GPUs (intra-node PCIe + inter-node 600 Gb/s RoCE); LLaMA-7B/13B, seqlens 4K-256K, fp16 (M=16).", "evaluation": "Not evaluable: the effective bandwidth b is never given a number, so these exact algebraic formulas produce no certified communication time at any deployed config. Additionally, the 'uniform b across all G devices' precondition is provably violated in the flagship 32xA100 setup, where inter-node RoCE (600 Gb/s) differs from intra-node PCIe - so even these mask-independent byte counts cannot be reduced to a single certified t_comm there.", "verdict": "not_evaluable"}, {"arxiv_id": "2404.00560", "claim_anchor": "thm:full_causal_function (IAES2024.tex:429); thm:recursive_dag_solved (:458); thm:learn_gamma_is_possible (:689); thm:find_input_vertices_is_possible (:562); thm:full_dynamic_function (:316); thm:recursive_markov_dp_solved (:347)", "bound": "Realizability, no tolerance/rate: if the local causal-function domain is finite (|bold-X| < infinity) with full coverage D = bold-X, and either the maximal input-element distance R < infinity or the problem is (n,r)-consistent, then an exact interpolant f-hat exists that solves every input length (perfect length generalization).", "deployed_config": "3-6 Transformer-encoder models trained (50k batches, Adam, ~12.8M CoT steps) on CoT reasoning and tested to 5x train length. Per-problem regime (Table experiment_property): arithmetic-in-F7 |X|=13^5, R=4, (1,17)-consistent; parity-[2-line] |X|=3^6, R=1, (1,3)-consistent; addition-[2-line] |X|=17^18, R=inf, (3,3)-consistent; addition-[3-line] |X|=14^9, R=1, (1,3)-consistent; multiplication-[8-line] |X|=19^216, R=inf, (9,3)-consistent.", "evaluation": "The precondition is combinatorial and the paper pins it per problem. All five listed problems have finite local domain |bold-X| < infinity and are (n,r)-consistent (addition-[2-line] and multiplication-[8-line] via (n,r)-consistency despite R=inf; the other three additionally have R < infinity). Each deployed problem therefore provably satisfies the positive theorems' precondition -> INSIDE. Observed: all five reach 100% accuracy at every LG test length (Fig. experiments), consistent with the realizability certificate.", "verdict": "inside"}, {"arxiv_id": "2404.00560", "claim_anchor": "thm:not_full_causal_function_inf_x (IAES2024.tex:448); thm:not_full_dynamic_function_inf_x (:338)", "bound": "Impossibility, no rate: if the domain is infinite (|bold-X| = infinity) and the range is nontrivial (|f(bold-X)| > 1), then for every m > 0 there is an interpolant agreeing with f on the finite data D yet wrong on more than m points - exact recovery, hence guaranteed length generalization, cannot be certified.", "deployed_config": "arctan (|X| = infinity, R=1; 3-fully-connected-layer model) and multiplication-[1-line] (|X| = infinity, R = infinity, not (n,r)-consistent; 3-Transformer-encoder model) (Table experiment_property).", "evaluation": "Both deployed problems provably satisfy the impossibility's precondition |bold-X| = infinity (arctan: continuous annulus inputs; multiplication-[1-line]: unbounded repeat count b makes the domain infinite). So both sit INSIDE the certified impossibility regime. Observed: arctan accuracy decays monotonically as the test annulus grows, and multiplication-[1-line] shows poor LG - consistent with (though, being existence-of-adversary statements, not forced by) the corollaries.", "verdict": "inside"}, {"arxiv_id": "2404.00560", "claim_anchor": "positive LG theorems (thm:full_causal_function IAES2024.tex:429; thm:learn_gamma_is_possible :689; thm:find_input_vertices_is_possible :562) evaluated at addition-[1-line]", "bound": "Same positive-LG precondition: finite local domain AND (R < infinity OR (n,r)-consistent) => exact interpolant / perfect LG.", "deployed_config": "addition-[1-line]: |X| = 14^4 (finite), R = infinity, NOT (n,r)-consistent for any (n,r) (Table experiment_property); 3-Transformer-encoder model, trained on <=7-digit addition, tested to 20-digit.", "evaluation": "Finite local domain holds, but R = infinity AND the problem is provably not (n,r)-consistent (carry distance grows without bound with the number of digits), so both alternative sufficient conditions of the positive theorems fail -> the deployed config is OUTSIDE the certified-LG regime. The infinite-domain impossibility does not apply either, since |X| < infinity - the config falls in the theory's gap. Observed: addition-[1-line] fails to generalize beyond 7 digits.", "verdict": "outside"}, {"arxiv_id": "2405.05219", "claim_anchor": "thm:conv_formal (4_conv.tex:86) / cor:conv_exact (:109); lem:alg_error (8_app_conv.tex:438); lem:softmax_lip / lem:softmax_lip_v2 (10_app_tools.tex:98, 148)", "bound": "||Y - Y-tilde||_inf <= 2(exp(2 eps) - 1) ||V||_inf in O(k n d log n) time, PROVIDED M o (QK^T) is an eps-close (T, delta)-non-degenerate k-conv basis matrix (with eps <= delta/(5T)). Exact recovery (eps = 0) requires k = n, costing O(n^2 d log n).", "deployed_config": "Llama3-8B-Instruct with attention replaced by the conv approximation, IMDB, context n = 2048, number of conv bases k swept over {64,...,2048} (k = 2048 = exact baseline); accuracy comparable to the original model at k = 512 (6_exp.tex, 4.1_conv_exp.tex).", "evaluation": "Not evaluable: thm:non_degen guarantees the k-conv structure holds for SOME (k,T,delta,eps), but eps is co-determined with k by the actual Llama3 attention matrix's conv-basis spectrum, which the paper never measures. At the deployed k = 512 (n = 2048) no eps is pinned, so the certified tolerance 2(exp(2 eps) - 1)||V||_inf is an unknown; the paper reports empirical relative-MSE and accuracy, not a certified eps. The O(.) is also constant-free, giving no wall-clock. Exactness sits only at k = n = 2048 (no speedup), a config never used for efficiency.", "verdict": "not_evaluable"}, {"arxiv_id": "2405.05219", "claim_anchor": "thm:conv_gradient_main (4.2_conv_gradinet.tex:54); thm:conv_gradient (8.2_app_conv_gradient.tex:358)", "bound": "If u(x) is a 1/poly(n)-close (T,delta)-non-degenerate k-conv basis matrix, the training forward pass runs in O(k n d log n + Tmat(n,d,d)) and the backward gradient in O(d^2 k n log n), both to 1/poly(n) sup-norm error.", "deployed_config": "None: the only experiment (6_exp.tex) is inference-only on a frozen Llama3-8B; the conv training method is never run, and no (n, d, k, T, delta) training config is recorded.", "evaluation": "Not evaluable (no deployed config): the paper trains nothing with this method, so the k-conv precondition on u(x) is never instantiated, and the complexity is big-O with unspecified constants, certifying no runtime.", "verdict": "not_evaluable"}, {"arxiv_id": "2405.05219", "claim_anchor": "thm:low_rank (5_low_rank.tex:47) / thm:low_rank_formal (9_app_rank.tex:86); lem:low_rank_approx (9_app_rank.tex:29)", "bound": "||Y - Y-tilde||_inf <= 4 eps ||V||_inf, eps in (0,0.1), with mask-specific runtimes O(knd) / O(kd sum_j B_j) / O(knd log n) / O(rnd); rank k <= binom(2(g+d), 2g) with g = O(max{log(1/eps)/log(log(1/eps)/B^2), B^2}), precondition ||Q||_inf, ||K||_inf <= B (cited Alman-Song lemma).", "deployed_config": "None: the low-rank variant is not the deployed algorithm (experiments use the conv-basis Algorithm conv_forward); no entry-magnitude bound B or eps is recorded.", "evaluation": "Not evaluable: no deployed config exercises the low-rank theorem, and even in principle the rank bound k <= binom(2(g+d), 2g) carries an unspecified O-constant inside g and no recorded B, so the certified rank/error pair cannot be computed at any concrete setting.", "verdict": "not_evaluable"}, {"arxiv_id": "2405.19036", "claim_anchor": "thm:input_copying (02_copy_and_recall.tex:55); thm:associative_recall (:108); lem:key-lemma (:141); thm:induction_heads (03_additional_results.tex:20); thm:selective_copying (:54)", "bound": "Existence of two-block SSMs of POLYLOG size solving copying (err <= eps), associative recall (exact), induction heads (exact), and selective copying (prob 1-eps): size params D, L, W, S, log B <~ log^37(1/eps) log^42(V) log^8|W| (copying); <~ log^13|W| (recall); <~ log^5 V log^8|W| (induction); <~ log^84 V log^89(1/eps) log^8|W| (selective copying). All leading constants hidden by the <~.", "deployed_config": "Experiments (02_experiment.tex; 03_addition_experiments.tex: A100, hyenadna-tiny-1k-seqlen) use a PRETRAINED Hyena fine-tuned on the Genomic Benchmark human_enhancers_cohn dataset - a token-masking sparsity study, NOT any of the copying / recall / induction / selective-copying tasks; no instance of the constructed SSM class S(M,U,D,L,W,S,B) is trained or measured.", "evaluation": "Not evaluable (no deployed config in the certified class): the theorems are existence statements about a hand-constructed polylog-size SSM, while the deployed model is an off-the-shelf Hyena on a different task. Even as existence claims the sizes are un-pinnable: the <~ hides the leading constants and the exponents are enormous (e.g. at V = 1024, the log^42(V) factor is ~10^42-scale before any constant, and selective copying carries log^89(1/eps)), so no finite network is certified - a BigBird-style (2007.14062) vacuousness.", "verdict": "not_evaluable"}, {"arxiv_id": "2405.19036", "claim_anchor": "thm:approximation_piecewise_gamma_smooth (02_nonparametric_regression.tex:220); thm:estimation_piecewise-gamma-smooth (:248); thm:estimation_general (03_proof_estimation.tex:5); thm:ssm_covering_number (:39)", "bound": "Approximation ||F_circ - hat F||_{2,P_X} <~ 2^{-T} with M <~ T^{1/alpha}, width W <~ 2^{T/a-dagger} T^c log^3 V; ERM excess risk R(hat F, F_circ) <~ n^{-2 a-dagger/(2 a-dagger + 1)} (log n)^{c'} log^13 V, with c' <= 21 + 10/alpha + 20 beta/alpha and a-dagger the effective smoothness.", "deployed_config": "None: no regression experiment is run; the effective smoothness a-dagger and the smoothness parameters alpha, beta of any real target are never estimated, and the only deployed run is the pretrained-Hyena classification/masking study.", "evaluation": "Not evaluable (no deployed config): the rate exponent depends on a-dagger (unknown for any deployed target), the width scales as 2^{T/a-dagger}, and every bound carries an unpinned <~ constant; no (n, T, a-dagger) triple is instantiated, so neither the 2^{-T} approximation nor the n^{-2a/(2a+1)} estimation rate can be evaluated at any recorded config.", "verdict": "not_evaluable"}, {"arxiv_id": "2407.03310", "claim_anchor": "thm:rasp-simulates-tm (turing_machine.tex:56) - the paper's only theorem", "bound": "Existence: for any Turing Machine T that (1) generates no repeated n-grams and (2) operates in-memory, there EXISTS a RASP program P of O(n) lines that correctly simulates T for exp(n) steps on every input x without repeated n-grams. No constant on O(n), no base on exp(n), and the n-gram length n is a free parameter never tied to any deployed setting.", "deployed_config": "Trained 150M-parameter GPT-NeoX transformer (12 layers, D=1024, FFN 4096, H=16 heads, Hard-ALiBi with M=6 masked heads, context 500, token space 24), trained on operand/tape lengths 2-50, evaluated at length 100 digits (addition 98%, n x 1 / n x 3 multiplication 97%), 80 in-context examples (SGD 95%), and tapes L >= 50 for 10 random 5-state/15-symbol Turing machines (addition.tex:26-37, other_algorithms.tex:133-146; results/2407.03310.json).", "evaluation": "Three independent blockers. (a) Object mismatch: the certificate attaches to a hand-constructed RASP program (full code in appendix.tex:81-375); deployment runs a transformer TRAINED from data, and the paper nowhere verifies the trained weights implement P - the paper itself frames the theorem as 'a length-generalization-friendly construction' motivating, not certifying, the experiments (turing_machine.tex:44). (b) Unpinned constants: O(n) lines and exp(n) steps carry no constants, and no concrete n-gram length n is fixed for any deployed task (the appendix code takes n as a parameter; next_token_turing calls hash_n_gram_iter(x, 1) with a separator-count disambiguator, but no theorem constant is derived from it). (c) Precondition status at deployment unverifiable: 'no repeated n-grams' is guaranteed only w.h.p. for uniformly random sequences of length <= exp(n) (turing_machine.tex:54); deployed addition/multiplication scratchpads are structured, and whether a length-100 digit tape avoids repeated n-grams depends on the unpinned n (illustration ONLY, constant-1 reading NOT licensed by the paper: uniform digit tapes of length 100 collide on n-grams with probability ~ L^2/10^n, requiring n >= ~5-6 for high-probability uniqueness).", "verdict": "not_evaluable"}, {"arxiv_id": "2408.08567", "claim_anchor": "lem:0 (S3Attention_arxiv.tex:225)", "bound": "Skeleton/CUR guarantee: for rank-r, mu-incoherent X (n >= d), uniformly sampling O(mu r log n) columns AND rows of noisy X+E yields X-hat with ||X - X-hat|| <= O(||E|| sqrt(nd)/(mu r log n)) w.p. >= 1 - O(n^{-2}). Three unpinned O(.)'s: sample count, error constant, failure probability.", "deployed_config": "S3Attention samples s_1 = s_2 = 8 columns/rows in the default config on all five LRA tasks (appendix tab:config_888; paper boasts 'by only sampling 8 rows and columns... 64.11% average', S3Attention_arxiv.tex:757); best per-task configs go up to (s_1, s_2) = (128, 32) (Pathfinder), ablations sweep s_1 to 256. Sequence lengths n = 1024 (Image, Pathfinder), 2048 (ListOps), 4096 (Text, Retrieval); BERT-style pretraining n = 512. Hidden dim d for LRA is unrecorded in paper and appendix.", "evaluation": "Regime condition: samples >= C mu r log n with C unspecified and (mu, r) of the deployed token matrices never measured - real activation matrices are not exactly rank-r, and the paper never identifies the X vs E split for its data. Constant-1 illustration (NOT licensed by the paper): even at the absolute floor mu = 1, r = 1, C = 1, the requirement is ln(4096) = 8.32 (or log2 = 12) samples vs 8 deployed - the flagship '8 rows and columns' config sits at/below the most charitable reading of its own sampling theorem, and any effective rank r >= 2 doubles the requirement (r=8: 66 samples at C=1). Object mismatch on top: the deployed output is layernorm(V-hat_1) + layernorm(V-hat_2) with softmax-reweighted sampled columns/rows (eq:cur_output, S3Attention_arxiv.tex:209), not the CUR pseudo-inverse estimator C-hat U-hat R-hat that the proposition bounds; the proof (via chiu2013sublinear Thm 1.1 + hamm2021perturbations Cor 4.3) attaches to the latter only.", "verdict": "not_evaluable"}, {"arxiv_id": "2408.08567", "claim_anchor": "lem:1 (S3Attention_arxiv.tex:326)", "bound": "Smoothing: for |x_t| <= a_max, |x_t - x_{t-1}| <= b_max and i.i.d. (sigma^2/n^2)-subgaussian filter taps l_i, the causal convolution satisfies |f(t)-f(t-1)| <= b_max sigma sqrt(log(2n/delta)/(2n)) + a_max sigma sqrt(log(2/delta)/(2n^2)) w.p. >= 1-delta. Constants fully explicit GIVEN (sigma, n, delta).", "deployed_config": "Deployed smoother: learnable frequency-domain complex matrix L in C^{(floor(n/2)+1) x d} applied via rFFT/irFFT (eq:fft_1), n = 1024-4096 (LRA). The proposition is invoked for the EARLY TRAINING stage only, via 'let us assume we use Kaiming normal initialization' making time-domain taps Gaussian with variance sigma^2/(2n^2) (S3Attention_arxiv.tex:409); the actual initializer used in the experiments is never stated, and the sigma constant's d-dependence is explicitly dropped ('Here we omit the dependence in d for brevity', footnote at 409).", "evaluation": "Hypothesis check by training stage. (a) Trained models - the source of every reported number (LRA, GLUE, forecasting): the paper itself states 'the learnable matrix L will be apart from the Sub-Gaussian initialization and converge to some data-driven matrix' (S3Attention_arxiv.tex:417), so the i.i.d.-subgaussian hypothesis fails at every evaluated checkpoint; no graceful-degradation term covers trained L. (b) Step-0 models: hypothesis plausibly satisfied IF Kaiming/Xavier is used (unrecorded), but sigma is then an untracked O(1) constant with omitted d-dependence, so the certified contraction factor is not computable. Illustration ONLY (sigma = 1, NOT licensed): n = 4096, delta = 0.01 gives |f(t)-f(t-1)| <= 0.0408 b_max + 3.97e-4 a_max, a ~25x contraction of adjacent differences at initialization.", "verdict": "not_evaluable"}, {"arxiv_id": "2408.08567", "claim_anchor": "lem:2 (S3Attention_arxiv.tex:419)", "bound": "Memorization: for bounded X and the segment-average S of eq:s_DEF (d = s r), there EXIST G, L with ||(XS)_{1:t} - X^smooth_t G_{1:t}|| <= O(r^{3/2} t log(n) d^{-1/2}), G block-repeated over an orthogonal polynomial basis. Single O(.) with no constant; existence-only in both G and L.", "deployed_config": "Deployed r = 8 segment-averages (default on all LRA tasks; sweep {1, 8, 16, 32, 64} in tab:varying-r with r = 8/16 best), n = 1024-4096, t up to n; hidden dim d for the LRA models is unrecorded; L is TRAINED, never the constructed matrix of the proof.", "evaluation": "Not evaluable on three counts: (a) the O(.) constant is unspecified; (b) d is unrecorded for the deployed LRA models, and d^{-1/2} is the only decaying factor; (c) the certificate is existence-style - the trained L is never shown to coincide with (or approach) the constructed L, so no deployed forward pass instantiates the bound. Constant-1 illustration (NOT licensed): at r = 8, t = n = 4096, hypothetical d = 64: 8^{3/2} * 4096 * ln(4096) / sqrt(64) = 22.63 * 4096 * 8.32 / 8 ~ 96,400 - orders of magnitude above the trivial bound ||(XS)_{1:t}||_F <= sqrt(t d) * max|X| ~ 512 * max|X|, i.e. vacuous at full sequence length; the linear-in-t growth means the bound can only bind for t << n or d >> t^2, regimes the LRA deployment never occupies.", "verdict": "not_evaluable"}, {"arxiv_id": "2408.11656", "claim_anchor": "expTheo (aaai25.tex:341)", "bound": "Unbiasedness: for Q, K in l_2(0,1) (the unit l2 ball, 'Let l_rho(0,lambda) denote the set {||X||_rho <= lambda}', aaai25.tex:339) and Phi a Random Maclaurin Feature map, E[RMFA_K(Q,K,V)] = attn_K(Q,K,V) for every V. No D-dependence: any feature count receives the identical certificate.", "deployed_config": "Macformer on LRA Text/ListOps/Retrieval: embedding dim 64, hidden 128, 2 layers, 2 heads, random projection dimension D = 128, ppSBN eps = 1e-13, p = 2 (aaai25.tex:567, 497); ppSBN is architecturally always applied around RMFA (Algorithm ppSBN steps 1-2: batch-normalize then divide Q, K by their l2 norms). Toy RMFA experiment: random 16 x 8 x length x 64 data, lengths 200-4000, D swept in powers of 2.", "evaluation": "Precondition check: ppSBN step 2 sets Q <- Q/||Q||_2, K <- K/||K||_2, so every deployed Macformer forward pass has ||Q||_2 = ||K||_2 = 1 <= 1 - inside l_2(0,1) exactly (boundary point, inclusive as defined). Deployed p = 2 additionally makes the Maclaurin-degree distribution exact: sum_eta p^-(eta+1) = 1/(p-1) = 1, a valid probability measure (it would be deficient for any p > 2). The identity is D-free, so deployed D = 128 receives the same certificate as D = 1: expectation-correctness only, zero tolerance certified per sample. Toy-experiment inputs' norms are unrecorded (raw random data; ppSBN application there unstated).", "verdict": "inside"}, {"arxiv_id": "2408.11656", "claim_anchor": "pacTheo (aaai25.tex:364) - skipped in claims/2408.11656.json because the parsed body was truncated, but the source pins the full bound; audited here as the paper's only resource-to-tolerance guarantee", "bound": "P(|RMFA_K(Q,K,V) - attn_K(Q,K,V)| > eps) <= 2D exp(-D eps^2/(2 L^2)), under Q, K in l_2(0,1) and |V_ij| <= L. Fully pinned constants; note the prefactor GROWS with D (the proof union-bounds over the D feature draws), so the bound is nonvacuous only for D > (2L^2/eps^2) log-ish terms.", "deployed_config": "Macformer LRA models: D = 128 random projection dimension, ppSBN-normalized Q, K (so the l_2(0,1) hypothesis holds by construction), V the trained value activations with L = max|V_ij| unrecorded (aaai25.tex:567). Toy experiment sweeps D in powers of 2 at d = 64, lengths 200-4000.", "evaluation": "Preconditions hold at deployment (ppSBN enforces ||Q||_2 = ||K||_2 = 1; |V_ij| <= L is definitional with L the realized max). Evaluating the printed formula at D = 128, in units of L (the bound is scale-covariant, so unrecorded L does not block this): failure prob = 256 exp(-64 (eps/L)^2). Nonvacuity gate: eps/L > sqrt(ln(256)/64) = 0.294. For 99% confidence: eps/L = sqrt(ln(25600)/64) = 0.398; for 95%: 0.365. So at the deployed feature count the theorem certifies entrywise error <= ~0.40 L w.p. 0.99 - a ~40%-of-max-value-scale tolerance, the Linformer-magnitude regime. Doubling to D = 256 improves this only to eps/L = sqrt(ln(51200)/128) = 0.291.", "verdict": "inside"}, {"arxiv_id": "2409.03187", "claim_anchor": "min-Msumu, Theorem 4.1 (supp.tex:1211), restated at main.tex:281; companion order_Csumu_decrease_lam (supp.tex:1319) is claim_kind 'other', outside the audit kinds", "bound": "Memory-capacity floor for noisy linear recurrent networks: Msumu >= N/(1+r), r = <v^2>/<u^2> the noise-to-signal ratio, with equality iff the noise PSD is flat (white noise). Proven under tr[C_v] = K (variance-normalized autocorrelation) and convexity of 1/(1+rx); the Msumu it bounds is the LIMIT formula Msumu = sum_i 1/(1 + r lambda[C_v]_i), derived under N = K -> infinity and full rank (main.tex:268-271), with no finite-N error term.", "deployed_config": "No results/2409.03187.json exists; configs recovered from the paper text. Own numerics: N = 10^4 nodes for the (beta, r) phase diagram (main.tex:400), NSR swept to r = 100 (main.tex:421, 438, 446); noise PSDs linear, exponential 10^{-alpha f} (alpha = 1, 10), 1/f^beta (beta = 1, 2), Lorentzian (N_w = 5e3) (supp.tex:1200); EEG-noise RNN simulations (Fz/Cz/Pz areas) at r = 100 observing Msumu/Msum > 0.7 (main.tex:443-446).", "evaluation": "Hypothesis check at deployment: tr[C_v] = K holds identically (C_v is defined variance-normalized, C(0) = 1, main.tex:249); 1/(1+rx) is convex for all r, x > 0 (second derivative 2r^2/(1+rx)^3 > 0); so every deployed (N, r, PSD) satisfies the theorem's own conditions. Evaluating the floor at the headline config N = 10^4, r = 100: Msumu >= 10^4/101 = 99.0, i.e. Msumu/N >= 0.0099 - consistent with (and far below) the observed Msumu/N ~ 1 at beta > 2, and the observed EEG-noise Msumu/Msum > 0.7 sits above the flat-PSD floor as the ordering theorem predicts. Residual gap: the certified object is the N = K -> infinity closed form; the phase-diagram numerics evaluate that formula directly (supp.tex: 'we numerically calculated Eq. (MC_diff_large_func)'), so they instantiate the certified regime by construction, but the finite-N EEG/ESN simulations inherit the bound only up to an unquantified finite-size correction.", "verdict": "inside"}, {"arxiv_id": "2409.15647", "claim_anchor": "prop-parity (iclr2025_conference.tex:229); program Listing lst:parity (572-602)", "bound": "Existence: an n-RASP-L program with EXACTLY T(n) = n loop steps (P_pre o (P')^n o P_post, all components RASP-L) solves n-bit parity under the FOP format x_1..x_n > #^{n'} -> *^n y #^{n'}, n' >= 0. Step budget fully pinned, program exhibited in full; no size bound is given for compiling P' to a transformer.", "deployed_config": "Trained looped decoder-only GPT-2 block: depth 1, 64 heads, 256 embedding dims, NoPE, input injection (tab:detail at iclr2025_conference.tex:809, 828); trained on lengths [1, 20] with oracle step count T = n supervised end-to-end (no intermediate supervision, iclr2025_conference.tex:417); tested to length ~40+ near-perfectly (465); inference stops via oracle T = n or max-confidence rule. results/2409.15647.json is empty (0 results) - configs recovered from source.", "evaluation": "Checkable regime elements all pass: deployed data format matches the proposition's (padding n' >= 0 by construction), and the deployed loop schedule T = n equals the certified budget T(n) = n exactly - the deployment is step-count-faithful to the construction. What does not transfer: the proposition certifies the hand-written program of Listing lst:parity, not the TRAINED block; no theorem verifies the learned 1-layer P'-block implements parity_step, and RASP-L-to-transformer compilation carries no stated depth/width requirement, so whether a depth-1, 256-dim block can even express the certified P' is unpinned. The observed generalization (20 -> 40+) is an empirical property of training, certified by nothing.", "verdict": "not_evaluable"}, {"arxiv_id": "2409.15647", "claim_anchor": "prop-copy (iclr2025_conference.tex:241); program Listing lst:copy (604-624)", "bound": "Existence: an n-RASP-L program with exactly T(n) = n loop steps solves n-symbol (repeated-token) copy under format x_1..x_n > #^{n'} -> *^n x_1..x_n #^{n'-n+1}, requiring padding n' >= n-1. Fully pinned budget; program exhibited (n iterations of shift_right).", "deployed_config": "Trained looped GPT-2 block: depth 2, 8 heads, 256 dims, NoPE, input injection; trained lengths [1, 20), oracle T = problem length n; tested at length 35 near-perfectly (stopping-criterion figure at iclr2025_conference.tex:517; 'for tasks like addition and copy... our looped model can still perform almost perfectly', 465).", "evaluation": "Checkable regime elements pass: deployed T = n equals certified T(n) = n; the deployed data format allocates n' >= n-1 padding by construction (the output must hold x_1..x_n after n '*' tokens within the same length - the paper's example uses n = 5, n' = 6 >= 4). Non-transferable part identical to prop-parity: the certificate attaches to the exhibited shift-loop program, not the trained depth-2 block, and no compilation bound connects them.", "verdict": "not_evaluable"}, {"arxiv_id": "2409.15647", "claim_anchor": "prop-addition (iclr2025_conference.tex:265); program Listing lst:add (626-670)", "bound": "Existence: an n-RASP-L program with exactly T(n) = n + 1 loop steps solves n-digit binary addition under format x_1..x_n + y_1..y_n > #^{n'} -> *^{2n+1} z_1..z_{n+1} #^{n'-n}, requiring n' >= n. Budget pinned at n + 1 (n shifted-XOR/AND iterations plus final carry absorption).", "deployed_config": "Trained looped GPT-2 block: depth 3, 8 heads, 256 dims, NoPE, input injection; trained lengths [1, 20); deployed step supervision 'set T to be the same as the problem length' = n (iclr2025_conference.tex:419), oracle inference reuses the dataset rule; tested at length 30 (510).", "evaluation": "Off-by-one regime mismatch inside the paper itself: the proposition certifies a T(n) = n + 1 budget, but the deployed data generation and oracle stopping use T = n (text at 419) - the deployed loop count is provably one step short of the exhibited program's requirement at every n. Since the claim is existence (it says nothing about what T = n budgets can or cannot do), this does not VIOLATE the proposition - it means the deployed schedule does not instantiate the certified construction even at the step-count level, unlike parity/copy where the budgets match. Padding precondition n' >= n is met by the data format (example: n = 2, n' = 3). Object mismatch as in the other two records: trained depth-3 block never verified to implement addition_step, no compilation size bound.", "verdict": "not_evaluable"}, {"arxiv_id": "2409.16592", "claim_anchor": "pro4 (MambaJSCC-v7.tex:365); the paper's other claims (expand at 257, R^-1 = R^T remark at 278, VSSM-0 at 319, pro5 at 397 - all claim_kind equivalence - and the CSI-forgetting remark at 425, claim_kind stability) are outside the audit kinds", "bound": "Expressivity: the two-GSSM design u = U^{kappa_1} z + U^{kappa_2} z (kappa_1 identity scan, kappa_2 full reversal) 'is sufficient to capture global information' - proof shows u_t = sum_{s<=t} C^1_t A^{1x}_{t:s} B^1_s z_s + sum_{s>=t} C^2_{T+1-t} A^{2x}_{(T+1-t):(T+1-s)} B^2_{T+1-s} z_s, so every u_t combines every z_s through learnable NON-ZERO coefficients. Nonzeroness is the entire quantitative content: no lower bound on coefficient magnitude, no task, no error.", "deployed_config": "MambaJSCC deploys exactly the certified design: every VSSM-CA block processes each of its c_z vectorized sequences with two GSSM modules using kappa_1 = I and kappa_2 = reversal, independent parameters (MambaJSCC-v7.tex:359, 481). Full system: 4-stage encoder/decoder, blocks [2,2,6,2], dims [128,192,256,320], images cropped to 128x128 (CelebA), 256x256 (DIV2K), 512x512 (AFHQ), CBR = 1/48, AWGN + Rayleigh (results/2409.16592.json); ablation confirms GSSM number = 2 as the deployed default (vs 4, 'GSSM number = 2/4' config notes).", "evaluation": "Precondition check: R_{kappa_1} = I and R_{kappa_2} anti-diagonal are hard-wired into the deployed architecture, so every deployed forward pass instantiates the proposition exactly - inside, with no free constants (the claim is structural). What the certificate is worth at deployed scale, using only the paper's own numbers: the coefficient tying z_s to u_t contains the cumulative gate product A^x_{t:s}, and the paper's own CSI-forgetting remark (MambaJSCC-v7.tex:425) pins typical per-step gate values at 0.95-0.97 with A^x_{t:0} ~ 6.37e-11 by t = 500. The same decay applies to the pro4 coefficients: a 'global' interaction across Delta t = 500 positions is certified nonzero but has magnitude ~1e-10 at the paper's typical gates (0.95^500 = 7.2e-12, 0.97^500 = 2.4e-7), and deployed vectorized sequence lengths run well past 500 (stage-1 sequences for 256x256+ images are thousands of tokens). Nonzero != usable: the theorem cannot distinguish exponentially-attenuated coupling from genuine global mixing.", "verdict": "inside"}, {"arxiv_id": "2410.02140", "claim_anchor": "thm:comm-comp-limit-transformer (main_arxiv_noComments.tex:904)", "bound": "One-way communication: for a LimitTransformer T satisfying Periodic and Local, on x in Sigma^{2N} Alice can send C log N bits so Bob computes all second-half activations, 'where C depends on T but not N'.", "deployed_config": "No LimitTransformer is ever instantiated: experiments deploy from-scratch HF GPT-2 decoder-only transformers (layers {1,2,4}, heads {1,2,4}, dim {16,64,256}; 12L/12H/768 for ADDITION), APE with random offsets, fp32, trained Len <= 50 (or <= 128), tested to 150/384 (sec:ap-exp-setup, main_arxiv_noComments.tex:3398ff; results/2410.02140.json).", "evaluation": "Two blockers: (1) the theorem's object is the idealized limit object (infinite family, p-bit precision, Periodic + Local positional logits) \u2014 no deployed model is one, so there is no config at which to evaluate C log N; (2) C is existence-only, 'depends on T' with no formula, so even for a hypothetical LimitTransformer no bit count is computable at N = 150 or 384. Nothing to plug in on either side.", "verdict": "not_evaluable"}, {"arxiv_id": "2410.02140", "claim_anchor": "corr:comm-comp-no-gen (main_arxiv_noComments.tex:917); training-length-failure (1775); non-offset-invariant-failure (5350)", "bound": "Copying arbitrary strings and n-digit addition are not expressible by Periodic+Local LimitTransformers; hence (via thm:guarantee) the idealized Inference Procedure lacks guaranteed length generalization \u2014 for some training schedule, infinitely many T_n fail. Companion proposition: without offset-invariance in the hypothesis class, bounded regularizer + fitting to n/2 does not imply generalization at n.", "deployed_config": "COPY (repeat) and ADDITION experiments: SGD/AdamW-trained GPT-2-style models (ADDITION: 12 layers, 12 heads, dim 768, lr 1e-4/3e-5, 60k steps), trained Len <= 50, tested [51,100] and [101,150]; early stop at 100% accuracy on a 2000-sample in-distribution test set (sec:ap-exp-setup:3398ff).", "evaluation": "The certified-failure regime is (a) the idealized Inference Procedure \u2014 exact fit of f on ALL inputs of length <= n/2, R minimized within 1/n over Theta_n (p-bit precision, translation-invariant positional product functions) \u2014 and (b) asymptotic: 'infinitely many T_n fail' with no finite n* pinned. Deployed runs satisfy neither: AdamW on sampled batches is not the Inference Procedure (fit checked on 2000 samples, not all inputs; R never minimized or measured; fp32, invariance unenforced \u2014 the paper only checks it empirically in its appendix), and no theorem output constrains lengths 51-150. The observed ADDITION/COPY-repeat failures are consistent with, but not certified by, the corollary; the paper is explicit that the guarantee is for 'a specific idealized learning strategy' (main_arxiv_noComments.tex:319, 1006: 'incorporating SGD training dynamics... is an interesting problem for future research').", "verdict": "not_evaluable"}, {"arxiv_id": "2410.02140", "claim_anchor": "cor:expressivity (main_arxiv_noComments.tex:1829); thm:crasp-to-limit-main (802); crasp-empty-example-languages (819); lemma:regular-languages-crasp (859); prop:crasp-maj-log (2305); crasppl-inverse-morphism-closure (2829); lemma:limit-to-transformer (3660); lemma:translation-positional (4212)", "bound": "Expressivity package: C-RASP[periodic, local] is contained in Periodic+Local LimitTransformers (and in MAJ2-hat[<,+1,MOD] / linear-size uniform TC0); MAJORITY, DYCK-1, a^n b^n c^n in C-RASP[empty]; PARITY and Sigma*be* outside C-RASP[periodic, local]. Only quantitative element: cor:expressivity \u2014 there EXISTS T_N correct at all lengths <= N with p-bit precision (p independent of N), constant depth/heads, width d = O(N).", "deployed_config": "Trained models use FIXED widths across all test lengths: dim in {16, 64, 256} at Len up to 150, and the same architectures for the Len <= 128 -> 384 sweep (2 layers/1 head APE, dim {64,256,512} in the COPY UNIQUE t-c study, main_arxiv_noComments.tex:6013ff); 12L/768 for ADDITION.", "evaluation": "The class-containment and inexpressibility results carry no resource-tolerance knob \u2014 nothing to evaluate at any config. cor:expressivity is existence-only with the O(N) width constant unpinned: whether d = 256 at N = 384 is 'O(N)-sized' is undecidable without the constant, and the corollary's T_N is a construction (from the LimitTransformer emulation lemmas), not the trained network \u2014 the paper never claims the SGD solution is that T_N. Constant-1 illustration NOT licensed by the paper: d >= N would require width 384 at the longest tested length; deployed NoPE/APE widths 64-512 straddle it, but the actual constant could be anywhere.", "verdict": "not_evaluable"}, {"arxiv_id": "2410.03810", "claim_anchor": "constant_copy (acl_latex.tex:326); constant_copy_attention (acl_latex.tex:335)", "bound": "Constant-size SSM approximates COPY to ||y_i - o_i||_inf <= eps IF c_i^T b_{pos(i)} >= c(1/a_min)^{L-1} + d (c, d 'constants related to eps, delta'; appendix proof pins the threshold as (1 - eps/(2M||Delta||_inf))/alpha_{pos(i)} + (delta/2)(a_max/(1-a_max) + (1/a_min)^{L-1}/(1-a_min))); constant-size softmax attention needs only q_i^T k_j >= log(Ltilde) + c. Both under the (L,delta)-matching-set assumptions assum:copy / assum:attention-copy.", "deployed_config": "Copy-task experiments (no results file in corpus; configs from app:ex, acl_latex.tex:1034ff): Mamba-135M (20 layers, hidden 1024), Mamba-D-67M (20L/720), Mamba-L-67M (10L/1024), Mamba-LD-69M (40L/512); TF-126M (10L/1024, RoPE), TF-63M (5L); trained on lengths [N_min, N_max] up to [30,40], context 220, AdamW lr 1e-5.", "evaluation": "Every quantity in the certified score threshold is a property of the TRAINED model on actual inputs: a_min, a_max (per-position forgetting coefficients), alpha_{pos(i)} (cumulative forgetting), delta and L (matching-set parameters of assum:copy), M, ||Delta||_inf \u2014 none is recorded anywhere in the paper for any trained model, and c, d are never given as numbers in the theorem statement (only the appendix inequality). Moreover both theorems are one-directional sufficient conditions for success: when the threshold fails (the paper argues it is 'difficult to satisfy' since a_min < 1 makes it exponential in L), NO error level is certified \u2014 the observed Mamba copy failures at N_max = 40 are outside the theorems' content in both directions.", "verdict": "not_evaluable"}, {"arxiv_id": "2410.03810", "claim_anchor": "linear_copy (acl_latex.tex:343)", "bound": "Existence: for any x_1..x_N in [-M,M]^d there EXISTS an SSM module of size O(N) performing COPY exactly (y_i = o_i for all i); proof states the required size is O(Nd) with parameters bounded by O(poly(M,N)).", "deployed_config": "Deployed Mamba models have FIXED size across all tested lengths: hidden 1024 / 20 layers (Mamba-135M) etc., evaluated at N_max in {10, 20, 30, 40} (app:ex; Figure fig:ex right).", "evaluation": "The exactness certificate attaches to a hand-built construction (deactivated-SSM gadgets storing all N records), not to any trained model: no deployed network is the construction, and the O(N)/O(Nd) constant is unpinned, so no deployed size can even be tested against the requirement. Constant-1 illustration NOT licensed by the paper: at N = 40, d = 1024 the construction would need state size ~ Nd = 40,960 per module; deployed Mamba state is d_state x d \u2014 but the paper never says its trained models realize the construction, and their failure to learn copy at N_max = 40 is uncertified either way.", "verdict": "not_evaluable"}, {"arxiv_id": "2410.03810", "claim_anchor": "DP_Mamba (acl_latex.tex:399); localDP_Mamba (acl_latex.tex:416)", "bound": "Under assum:dp (constant bounds N_s, N_dp, N_A on dependency sets; polynomial bounds on state space/values; transition/aggregation functions approximable by constant-size SiLU MLPs 'with polynomial efficiency', i.e. parameter norms O(poly(M, 1/eps)) with unspecified polynomials), there EXIST Mamba layers of size O(T) solving any such DP problem with CoT answers of length <= T; for m-locality DP problems, size O(m) independent of T suffices.", "deployed_config": "CoT experiments (app:ex): LIS at task lengths L in {10, 30, 50, 70} and Arithmetic at L in {4..7}; model dims d in {32, 64, 128, 256}; Transformers 3 layers, Mamba ~2x layers to match size; AdamW, 300 epochs.", "evaluation": "Both theorems are existence constructions with unpinned constants: neither the O(T)/O(m) size constants nor the poly(M, 1/eps) parameter bounds of the MLP-approximation lemmas (lemma:multiply through lemma:indicator) are numeric, so whether a deployed d=256, ~6-layer Mamba is 'size O(T)' at T ~ L = 70 is undecidable, and the trained models are not the constructions. One precondition IS decidable from the paper's own text: the flagship CoT task LIS violates m-locality \u2014 app:m-locality states 'This does not satisfy m-locality because m is at least n and cannot be significantly smaller than the output length T' \u2014 so the size-O(m) certificate of localDP_Mamba covers the Arithmetic experiments' task class ('most common arithmetic tasks satisfy the m-locality assumption') but excludes the LIS experiments entirely; only the O(T)-size DP_Mamba certificate speaks to LIS, and no deployed model is certified to be that size.", "verdict": "not_evaluable"}, {"arxiv_id": "2410.05690", "claim_anchor": "thm:full_rank (iclr2025_conference.tex:297); thm:misspecification (350); thm:main_thm (929); coro:OLS (980)", "bound": "Constrained ERM over A(D) = {||M_A||_op <= D} recovers A_star at ||hatA - A_star||_F^2 <= C(delta) D^2 p d^2/(N(T-p)) polylog(kappa,p,d,N,T) w.p. 1-delta (misspecified variant: C(delta) D^2 (D'+1)^2 p' d r/(N(T-p))); the UNCONSTRAINED OLS inherits the rate only under the burn-in condition N(T-p') >= C(delta) eta^2 tau p'^2 d r (coro:OLS).", "deployed_config": "Synthetic experiments (app:experiments, iclr2025_conference.tex:1849ff; no results file in corpus): A_star = p orthogonal d x d matrices scaled down by alpha*p, alpha = 0.5; d in {5,10,15}, p in {5,10,15}, N in {1,5,10}, T in {1,5,10,25,50} x pdr/N (so beta/gamma = NT/(pd^2) in {1,...,50}); estimator for the full-rank/misspecification figures is plain OLS, not the constrained ERM.", "evaluation": "Preconditions on the ground truth hold by construction: each ||A_k^star||_op = 1/(alpha p) = 2/p, so sum_k ||A_k^star||_op = p(2/p) = 2, and prop:bounded_op_norm(i) gives ||M_{A_star}||_op <= 2 \u2014 hyp:bounded_op_norm holds with D = 2 (noise distribution is unstated in the text but any Gaussian choice satisfies hyp:subgaussian). But the deployed ESTIMATOR is OLS, which the theory covers only through coro:OLS's burn-in, and both C(delta) and the polylog factor tau are unpinned, so the burn-in threshold is uncomputable at every grid point. Constant-1 illustration NOT licensed by the paper (C(delta) = tau = eta = 1, well-specified p' = p, r = d): the condition becomes NT >= p'^2 d^2 = p * gamma, i.e. beta/gamma >= p in {5,10,15} \u2014 the plotted points beta/gamma in {1,5,10} (over half of each curve in fig:rate_main) would sit BELOW the burn-in under that reading, while {25,50} would sit above; where the actual threshold falls is undecidable from the paper.", "verdict": "not_evaluable"}, {"arxiv_id": "2410.05690", "claim_anchor": "thm:low_rank (iclr2025_conference.tex:320)", "bound": "Rank-constrained ERM hatA in argmin over A_r(D) = {A in A(D): rank(A_k) <= r for all k} achieves the improved rate ||hatA - A_star||_F^2 <= C(delta) D^2 p r d/(N(T-p)) polylog(...) w.p. 1-delta.", "deployed_config": "Low-rank experiments (fig:rate_rank): d = 15, r = 5 fixed, p in {5,10,15}, N and T varied as in the main grid; deployed estimator is gradient descent on the group-nuclear-norm REGULARIZED loss eq:nucl_estim with lambda grid-searched over {1e-1..1e-7} and lr in {1e-1,1e-2,1e-3}, plus an OLS (lambda = 0) baseline.", "evaluation": "The certified estimator is never run \u2014 by the paper's own admission it cannot be: 'this estimator cannot be computed in polynomial time, since the underlying optimization problem involves a non-convex constraint on the rank' (below thm:low_rank), and for the relaxation actually deployed: 'We leave the analysis of the nuclear norm estimator for future work' (below eq:nucl_estim). So every deployed low-rank run uses an algorithm carrying no theorem, and the algorithm carrying the theorem is excluded from deployment by computational hardness. The prd/N(T-p) rate observed for the regularized estimator in fig:rate_rank is uncertified.", "verdict": "outside"}, {"arxiv_id": "2410.10101", "claim_anchor": "lem:error-identifiability-noise (8-appendix.tex:450), with lem:cert-simple (2-pre.tex:112) as the exact-ERM special case", "bound": "For any two eps-approximate ERMs Theta, Theta' on a realizable dataset with bounded output noise: ||MHLA_Theta(Z) - MHLA_Theta'(Z)|| <= [(eps - (1/N) sum zeta_i^2 + (B^2/N) log(1/delta)) / lambda_min(Lambda_D)] * ||Z||_F^6, B = max_i ||Z_i||_F. Exact-ERM version: lambda_min(Lambda_D) > 0 implies all ERMs compute the same function.", "deployed_config": "Associative-memory experiments (5-experiments.tex): d = 4 (token dim 2d = 8), N = 2^14 samples, ground truth an explicit 1-head MHLA (V, Q block matrices), output noise zeta_i = 0 (the noise column sits inside Z, outputs are exact); mixed Gaussian/unitary datasets engineered so lambda_min(Lambda_D) ranges from 0.06 (identifiable) down to 0 (non-identifiable); MHLA with 1, 2, 4, 8 heads trained by SGD to ~zero loss ('the minimum of the loss, which is in our synthetic case equal to zero').", "evaluation": "Regime condition is lambda_min(Lambda_D) > 0, and the paper MEASURES it at deployment: the identifiable dataset has lambda_min = 0.06 > 0 (fig:assoc-memory-certificate caption) \u2014 provably inside; realizability holds by construction (data generated by a ground-truth MHLA) and zeta_i = 0. At the certified configs the observed behavior matches (learned and true parameters coincide in p-feature space). The companion non-identifiable dataset (lambda_min = 0) is a DESIGNED exit from the regime \u2014 the bound's denominator is 0, certifying nothing \u2014 and there SGD indeed converges to spurious solutions, i.e. the paper deploys on both sides of its own boundary as a controlled demonstration. Caveat: the numeric tolerance at the inside configs is not computed anywhere (eps and B are not reported as numbers, and the ||Z||_F^6 factor makes the certificate input-dependent), so 'inside' certifies function-equality up to an unevaluated constant.", "verdict": "inside"}, {"arxiv_id": "2410.10101", "claim_anchor": "MHLA-program expressibility lemma (8-appendix.tex:503); cor:learning-utm (8-appendix.tex:708)", "bound": "Every Autoregressive-MHLA-form program is implemented EXACTLY by some MHLA parameters; consequently, with sample complexity N = poly(d, 1/eps, log(1/delta), n, t), Algorithm alg:poly learns an MHLA whose computation history reproduces a Turing machine M on input x w.p. >= 1 - eps, within c(t) <= O((n+t)t) autoregressive steps.", "deployed_config": "None: no Turing-machine-learning or program-learning experiment is deployed anywhere in the paper (results/2410.10101.json is empty; the experiments are Gaussian linear-attention data at d in {2,4}, N in {512, 2048}, T = 100, and the associative-memory task).", "evaluation": "Not evaluable twice over: (1) no deployed config exists for the TM pipeline \u2014 nothing was trained on computation histories; (2) the sample complexity is an unspecified polynomial ('poly(d, eps^-1, log(delta^-1), n, t)') with no degrees or coefficients, and the step bound c(t) <= O((n+t)t) has an unspecified constant, so no N could be checked against the requirement even hypothetically. The expressivity lemma is exact but existence-style, with no bound on the head count/width the construction consumes at any concrete program.", "verdict": "not_evaluable"}, {"arxiv_id": "2410.11687", "claim_anchor": "prop-one-step-gd-multivariate (main_arxiv.tex:266); prop-multi-step-gd (676)", "bound": "Constructive exactness: a diagonal LRNN with sliding window of size 3 and stride 2 on interleaved (x_j, y_j) tokens admits explicit parameters (A = I, window-mixing Q = e2 e1^T, readout q = e3, beta = eta/t \u2014 all pinned in the text) whose output at every step is EXACTLY the one-step in-context GD prediction hat-y_{j+1} = -(Delta W)^T x_{j+1}; L stacked layers implement L GD steps.", "deployed_config": "Synthetic regression experiments (sec:exp; appendix:expt-1d/nd-regression): single-layer GRIL, hidden dim 20, context N = 10, interleaved tokens, window 3 / stride 2, data from a linear model W ~ N(0,I), x ~ U(-1,1)^f, n_o = f; two-layer GRIL for multi-step. Crucially the paper ALSO deploys the constructed parameters themselves ('GD instantiated with our constructed parameters', fig:lossnmatrices A) alongside SGD-trained models.", "evaluation": "Every structural precondition is satisfied by the deployed configuration by construction: diagonal LRNN (yes \u2014 GRIL block), window size 3 / stride 2 (yes \u2014 'width-3 local window' throughout), interleaved (x, y) tokens from a linear model (yes \u2014 the data generator is exactly the assumed linear model). The certified object (the explicit construction) is itself run and matches the trained model in loss, predictions, sensitivities, and parameters (fig:lossnmatrices A-G) \u2014 the certificate is exact, so no constant is left unpinned. One caveat on the multi-step variant: the deployed explicit construction 'can be sensitive to numerical scaling', so a 1000-step calibration phase is applied (appendix, multi-step details) \u2014 the two-layer certificate is realized only up to that calibration.", "verdict": "inside"}, {"arxiv_id": "2410.11687", "claim_anchor": "prop-nonlinear-gd (main_arxiv.tex:719; appendix:nonlinear)", "bound": "As stated: LRNN (window 3, stride 2) + MLP admits a construction whose output is EXACTLY hat-y_{j+1} = -(Delta W)^T x_{j+1} for one GD step on a NON-linear model h(w^T x).", "deployed_config": "Non-linear regression experiments (sine-transformed targets): GD-dagger = recurrent GRIL parameters fixed to the linear construction with the auxiliary MLP TRAINED; plus fully trained non-linear GRIL (fig:result-regression-multi-layer B).", "evaluation": "The paper's own proof text revokes the exactness the proposition asserts: the required MLP mapping rho(sum (w0^T x_i - y_i) x_i) = sum (h(w0^T x_i) - y_i) h'(w0^T x_i) x_i 'cannot in general be represented EXACTLY due to the information compression induced by the linear accumulation. Instead, a sufficiently expressive MLP can learn an approximate mapping ... over the training distribution' (appendix:nonlinear), and the main text downgrades it to 'a constructive approximation result' (sec:nonlinear) with no epsilon, no rate, and no MLP-size bound. So the exact-GD regime the proposition certifies contains no realizable configuration, and the deployed GD-dagger (trained MLP surrogate) carries zero certified tolerance.", "verdict": "outside"}, {"arxiv_id": "2410.11687", "claim_anchor": "prop-cross-entropy-gd (main_arxiv.tex:364)", "bound": "IF the local context at step t contains embedded copies of phi(x_t), the classification error vector r_t = p_t - y_t, and the next query feature phi(x_{t+1}), THEN a diagonal GRIL accumulates the cross-entropy gradient sum_i (p_i - y_i) phi(x_i)^T and reads out hat-z_{t+1} = -(eta/t) Z_t phi(x_{t+1}).", "deployed_config": "Classification experiments (sec 479ff; appendix:classificationdata, appendix:clsexp): binary and 3-class synthetic tasks, input dim 10, sequence lengths 10/100; context tokens contain raw inputs x and CLASS LABELS y only (labels from argmax(softmax(W^T x))); the feature map phi is a LEARNED MLP; single-layer model, 5 seeds.", "evaluation": "The load-bearing precondition fails at every deployed config: r_t = p_t - y_t requires the model's own predicted probabilities p_t inside the local context, but p_t is a state-dependent quantity ('In practice, the probability term p_t is produced by the classifier logits' \u2014 the paper's own concession directly after the proposition) which no embedding of the input tokens can supply; the deployed data format provides labels y_t, never residuals. Likewise phi is assumed given but is 'a learned non-linear feature map' at deployment ('in practice this requires a learned non-linear feature map', sec 479ff). The certified gradient-accumulation identity therefore holds for no deployed run; the observed GRIL-tracks-GD-dagger behavior (fig:classification-main) is empirical, with GD-dagger itself only an uncertified reference construction.", "verdict": "outside"}, {"arxiv_id": "2410.16179", "claim_anchor": "theorem: unbias (ICLR_2025_Template/rethink2.tex:90)", "bound": "ORACLE sampling \u2014 indices drawn i.i.d. from the exact normalized attention distribution w \u2014 gives an unbiased estimator of the attention output (E[o_bar] = sum_i w_i v_i) with trace of covariance monotonically decreasing in the budget B.", "deployed_config": "MagicPIG deploys LSH-based SELF-NORMALIZED importance sampling from an LSH collision proposal u != w: SimHash with (K, L) in {(10,220), (10,150), (9,120), (8,75), (11,300), ...} on Llama-2/3.1 (8B-70B), MegaBeam-Mistral, contexts 16K-96K, cost 2-5 percent of full attention (results/2410.16179.json). The oracle estimator itself is run only as a diagnostic reference on infini_igsm, where it 'requires true attention scores' and costs ~0.5 of full attention.", "evaluation": "The certified regime requires sampling from the exact w, and the paper states this is unattainable for the efficient method twice over: 'We call it oracle because it assumes that the exact attention vector w is known, which is not true for sparse attention approximations' (rethink2.tex) and 'Oracle sampling estimation cannot go beyond 2x wall clock speed up because obtaining distribution w requires full computation of all q k_i^T' (system.tex). The deployed estimator X^IS (eq: self normalized is) is self-normalized importance sampling, which is BIASED at every finite budget \u2014 the paper claims only almost-sure consistency as B -> infinity \u2014 so the unbiasedness certificate holds for zero efficiency-bearing deployed configs; its variance expression (eq: variance) is explicitly an approximation tied to an LSH proposal u that is never computed exactly (collision probability heuristically taken as 0.5^K, hyper.tex:53). The only deployment inside the regime is the oracle-reference ablation row at cost_total ~0.5 \u2014 exactly where the claimed 50x efficiency vanishes.", "verdict": "outside"}, {"arxiv_id": "2410.16179", "claim_anchor": "theorem: cost (ICLR_2025_Template/rethink2.tex:110)", "bound": "Under oracle sampling with budget B from w, the expected number of DISTINCT tokens touched satisfies E[|S|] <= 1 + B*epsilon with epsilon = 1 - max_i w_i \u2014 fully explicit given (B, max_i w_i).", "deployed_config": "All efficiency numbers (cost_total 0.02-0.05, TBT 12-18 ms, throughput up to ~54 tok/s at 96K context) come from LSH sampling with (K,L) configs, where the retrieved set is determined by hash-table collisions, not by i.i.d. draws from w; the oracle-reference rows report sampled-token ratios {0.02, 0.1, 0.3} but never report B or the measured max_i w_i.", "evaluation": "The bound's hypothesis (i.i.d. draws from the exact w) fails for every LSH deployment \u2014 deployed |S| is the union of hash-bucket collisions under proposal u, a quantity the theorem does not address (the paper sizes it heuristically via collision probability 0.5^K per table, hyper.tex:53, not via theorem: cost). For the oracle-reference rows the bound WOULD be evaluable \u2014 epsilon = 1 - max_i w_i is measurable and B is set by the experimenter \u2014 but neither number is reported, so even the in-regime deployment cannot be checked; e.g. at B = 0.3*16384 = 4915 the bound 1 + 4915*epsilon is uncomputable without the unreported epsilon.", "verdict": "outside"}, {"arxiv_id": "2410.18541", "claim_anchor": "prop sufficient (mybibfile.tex:346) and prop suff-nonident (mybibfile.tex:384)", "bound": "Non-identifiability conditions for attention weights: d_s > d_v => the map f_A from attention weights to outputs fails to be 1-1 (AW non-identifiable); d_s > d_v + 1 => A non-identifiable AND null space of [T,1] nonzero (the condition licensing the paper's efficient-attention projection). Fully explicit inequalities, no hidden constants (d_s = input sequence length, d_v = d/h = attention head dimension).", "deployed_config": "Wiegreffe & Pinter encoder-attention-decoder architecture (LSTM/RNN/MLP encoder, mybibfile.tex:525) on IMDB (4356 test), AGNews (3798), SST (1725), 20News (357) (Table 1, mybibfile.tex:507-517). Neither d_s (per-sample token length) nor d_v (encoder hidden/head dimension) is printed anywhere in the paper; results/2410.18541.json carries no accepted results and its config_notes record no dimensions.", "evaluation": "The certified condition d_s > d_v (+1) is one of the rare fully-pinned inequalities in the corpus - evaluating it needs only two deployed integers, and the paper supplies neither. No hidden size for the W&P encoder is stated, and no sequence-length statistics for the four datasets are given; quoting W&P's own hyperparameters would be inventing constants not in this paper. Indirect evidence the regime is intended to hold: the paper asserts 'in cases with relatively long inputs, non identifiability is real' (mybibfile.tex:320), and the experiments only produce nontrivial projections (A_eff != A, common efficient core for adversarial pairs, Tables 1-2) if Ker([T,1]') != {0}, which requires d_s > rank([T,1]) - so the deployed runs behave as if inside the non-identifiability regime, but no arithmetic d_s vs d_v check can be exhibited from the paper's own numbers.", "verdict": "not_evaluable"}, {"arxiv_id": "2411.01035", "claim_anchor": "thm:expressivity (tensors.tex:24)", "bound": "Tensorized spectral filtering expressivity: for any LDS (A,B,C,D) with A PSD, ||A|| <= 1, there exist M', M'', {M_ij} such that the TSF predictor with the autoregressive form y_t - 2y_{t-1} + y_{t-2} = M' u_t + M'' u_{t-1} + sum_{ij} M_ij u_{t-2:1}.(phi_i tensor phi_j) satisfies ||y_t^LDS - y_t^TSF|| ~ e^{-k/log(T)} for all t in [T]. Written with a tilde: no constant prefactor, no O(.), and the log base is unspecified.", "deployed_config": "No results/2411.01035.json exists (no accepted results extracted). From experiments.tex: (a) synthetic LDS runs with hidden dim d_A = 512, T = 2^14 = 16384, k = 24 filters (experiments.tex:12); (b) deep STU models: 2 tensordot STU layers, filters of length T = 256, tensorized variant replaces k filters of length 256 with k^2 filters formed from components of length 16 (experiments.tex:55, 76) - and 'Following prior STU architecture implementations we use no autoregressive components' (experiments.tex:57); vocab 4, induction-heads/copy tasks.", "evaluation": "Three independent blocks. (1) The bound is a tilde-asymptotic with no constant: at the LDS config (k=24, T=2^14) the certified error is c*e^{-24/log(16384)} with c unknown and even the log base unpinned - constant-1 illustration NOT licensed by the paper: natural log gives e^{-24/9.70} = 0.084, log2 gives e^{-24/14} = 0.18 - an unnormalized 'error' spanning nothing to everything depending on unstated conventions. (2) The deployed STU drops the certified predictor's structure: the theorem's TSF form requires the autoregressive terms 2y_{t-1} - y_{t-2} and the M'u_t + M''u_{t-1} inputs; the deployed networks use NO autoregressive components by the paper's own statement, so the existence certificate is for a predictor class deployment does not instantiate. (3) The STU tasks (induction heads, copy) are not LDS-generated, and the theorem only covers outputs of a PSD-A LDS; nothing is certified about them. k for the deployed STU is never printed.", "verdict": "not_evaluable"}, {"arxiv_id": "2411.12537", "claim_anchor": "thm:parity (main.tex:366)", "bound": "Impossibility: a finite-precision LRNN with finitely many layers can solve parity for arbitrary input lengths only if some layer has an input x with an eigenvalue of A(x) outside {lambda in R : lambda >= 0}. Contrapositive certified-failure regime: all transition-matrix eigenvalues real and nonnegative => parity unsolvable (for arbitrary lengths).", "deployed_config": "Chomsky-hierarchy parity: 2-layer models, embedding 128, 4 heads, trained on lengths 3-40, evaluated 40-256 (main.tex:1287-1288). Mamba [0,1]: A = Diag(s(x_t)), s in [0,1]; DeltaNet [0,1]: A = I - beta_t k_t k_t^T, beta_t in [0,1], ||k_t|| = 1 (Table tab:eigenvalue_range, main.tex:561-562). Extended variants: Mamba [-1,1]: Diag(2s-1); DeltaNet [-1,1]: I - 2 beta_t k_t k_t^T.", "evaluation": "Hypothesis check for the [0,1] deployments: Diag(s(x)) has eigenvalues s_i(x) in [0,1] subset {lambda >= 0}; I - beta k k^T (||k||=1) has eigenvalues {1 (n-1 times), 1-beta} subset [0,1] - every deployed [0,1] matrix provably satisfies the failure precondition; finite precision (bf16/fp32 training) and finitely many layers (2) hold. Certified: these models cannot solve parity at arbitrary lengths - observed scaled accuracy 0.0 (Mamba [0,1]) and 0.017 (DeltaNet [0,1], best of 3) vs 1.0 for both [-1,1] variants, whose eigenvalues 2s-1 and 1-2beta reach negative values, exiting the certified-failure regime by construction. Caveat on strength: the proof gives failure for k >= some unpinned k-bar ('for large enough k', main.tex:904), so failure at the specific tested lengths <= 256 is consistent with but not individually certified by the theorem.", "verdict": "inside"}, {"arxiv_id": "2411.12537", "claim_anchor": "thm:modcount (main.tex:374)", "bound": "Impossibility: a finite-precision L-layer LRNN can count modulo m (recognize (1^m)^*), m not a power of two, only if for some layer i and inputs x_1..x_{2^{i-1}} the product A(x_1)...A(x_{2^{i-1}}) has an eigenvalue with nonzero imaginary part. Certified-failure regime: all such products have only real eigenvalues.", "deployed_config": "Modular arithmetic tasks with m = 5 (not a power of 2; main.tex:1297-1300), 3-layer models (main.tex:1288), lengths 3-40 train / 40-256 eval; plus Z_60 addition (app:cyclic). Mamba [0,1] and [-1,1]: diagonal real transitions at any depth. DeltaNet [0,1]: symmetric PSD GH factors I - beta k k^T; DeltaNet [-1,1]: I - 2 beta k k^T.", "evaluation": "Mamba (BOTH ranges): products of diagonal real matrices are diagonal real for every layer and depth, so the failure precondition holds at every deployed Mamba config - certified unable to count mod 5 or mod 60 at arbitrary lengths; observed mod-arith scaled accuracy 0.066-0.241 across ranges, far from 1.0, consistent. DeltaNet [0,1]: for layers i = 1, 2 the products of 2^{i-1} <= 2 symmetric-PSD GH factors have real eigenvalues in [0,1] by the paper's own prop th:ghexpress(iii) (main.tex:425, part restricted to k <= 2); but the deployed depth is 3, and for layer 3 the products of 4 GH [0,1] factors are NOT certified real by anything in the paper - so 3-layer DeltaNet [0,1] membership in the certified-failure regime is unproven (though its 0.194-0.314 scores are consistent with failure). DeltaNet [-1,1]: products of >= 2 reflections realize rotations (complex eigenvalues), so it can satisfy the necessary condition - outside the failure regime, matching its 0.971 (w/o brackets) best score.", "verdict": "inside"}, {"arxiv_id": "2411.12537", "claim_anchor": "prop th:ghexpress (main.tex:425)", "bound": "GH product classes: (i) every N in M_k^n([-1,1]) has ||N|| <= 1; (ii) every contraction is in M_{3n}^n([-1,1]), orthogonal matrices in M_n^n({-1,1}), permutations in M_{n-1}^n({-1,1}); (iii) eigenvalues of N in M_k^n((-1,1]) are 1 or |lambda| < 1; if N in M_k^n([0,1]) with k <= 2 then lambda in [0,1].", "deployed_config": "DeltaNet [-1,1] deploys A(x_t) = I - 2 beta_t k_t k_t^T, beta_t in [0,1], ||k_t|| = 1 - a single GH matrix with eigenvalue parameter 1 - 2 beta_t in [-1,1], i.e. an element of M_1^n([-1,1]), at every token of every layer (340M/1.3B LM models, Chomsky and S5 models). DeltaNet [0,1] deploys M_1^n([0,1]) elements.", "evaluation": "Every deployed DeltaNet transition matrix is literally a member of the certified classes, so (i) certifies ||A(x_t)|| <= 1 (state-norm stability) and (iii) certifies the spectrum shape at every deployed config with no precondition to check beyond beta in [0,1], ||k|| = 1, which hold by construction (sigmoid/normalization). The certificate is tolerance-free, however: it ties no task error to any parameter - it is the algebraic scaffolding for thm:groups/thm:regular, not a performance bound.", "verdict": "inside"}, {"arxiv_id": "2411.12537", "claim_anchor": "thm:groups (main.tex:522); thm:regular (main.tex:539); modular-addition-reflections (main.tex:1184)", "bound": "Constructive expressivity: (a) any FSA with group transition monoid is implementable by a ONE-layer LRNN with A: Sigma -> M_{k-1}^n({-1,1}), k = max states moved by one transition, n minimal with T(A) <= S_n; cyclic Z_m needs M_2^2([-1,1]), parity scalars {-1,1}; (b) any FSA implementable with s <= 2^{|Q|} layers, n <= |Q| states, A -> M_n^n([-1,1]); (c) addition mod m with two layers of single reflections (layer 1: scalar A in {-1}; layer 2: A in M_1^2({-1}), 2m states).", "deployed_config": "DeltaNet applies exactly ONE GH matrix per token per layer. S5 experiments (main.tex:602-612, app:state-tracking): embedding 128, 4 heads, 1-5 layers, train length 32 (64 for 4-token variant), test to 500; four input regimes: swaps-only, up-to-3-cycles, 4-tokens-per-element, standard S5. Chomsky mod arithmetic: 3 layers, m=5. LM: DeltaNet 340M/1.3B.", "evaluation": "Regime membership varies by instance, and tracks the results exactly. Swaps-only S5: transitions move k = 2 states => construction needs M_1({-1,1}) - one full reflection per token, precisely the deployed DeltaNet [-1,1] class (beta saturates to 1 in finite precision, main.tex:826): deployed 1-layer model is INSIDE and fully learns the task. 4-tokens-per-element: 4 GH factors available per group element within one layer, matching the M_4 requirement for k = 5 - inside; DeltaNet [-1,1] 5L extrapolates to ~length 200. Standard S5: single tokens move up to 5 states => construction requires products of 4 GH per token, deployed architecture provides 1 - OUTSIDE the constructive regime (stacked layers do not multiply per-token GH factors), and all models sit at 0 sequence accuracy before length 30 (main.tex:611). Modular addition: the 2-layer all-reflection construction (c) is contained in deployed 3-layer DeltaNet [-1,1] - inside; mod-arith w/o brackets best 0.971. General regular languages via (b): 2^{|Q|} layers vs deployed <= 5 - the certified construction never fits deployed depth (upper-bound construction, so deployment is uncertified rather than provably incapable).", "verdict": "inside"}, {"arxiv_id": "2411.12603", "claim_anchor": "unlabeled proposition (supplement.tex:72)", "bound": "Expressivity containment: the kernel family Phi(t_k, t_i) = C_k (prod_j exp(A_j Delta_j)) B_i of eq. appendix-kernel-discrete contains convolutions with rational kernels as a special case. Proof route: restrict to the LTI corner (A, B, C constant for all t), then invoke 'LTI state-space models can represent every rational transfer function' (supplement.tex:74-85).", "deployed_config": "STREAM deploys the selective (Mamba-style) parameterization: A_i = A a learned REAL DIAGONAL matrix (main.tex:230, supplement.tex:96 'we choose A_i = A as a diagonal matrix'), B_i = Gamma_i Linear(u_i), C_i = Linear(u_i) input-dependent, Delta_i = (t_i - t_{i-1}) softplus(delta) (main.tex:240-242); state dim m = 32, model dim n = 32 (main.tex:469); ModelNet40 / ScanObjectNN / DVS128-Gestures / Spiking Speech Commands.", "evaluation": "The certified special case is doubly disjoint from deployment. (1) LTI precondition: deployed B_i, C_i are input-dependent by construction - selectivity is the method's point - so no deployed run sits in the constant-A,B,C corner where the containment is proved. (2) Even restricting deployment to its LTI shadow, A is real diagonal: the resulting kernels are sums of m = 32 real exponentials C e^{A Delta} B = sum_j c_j b_j e^{a_j Delta}, a_j in R, which cannot represent any rational transfer function with a complex-conjugate pole pair (every damped-oscillation kernel) - the 'every rational transfer function' scope needs poles the deployed parameterization excludes. The paper itself notes oscillating kernels require 'complex A in C' (main.tex:209), a parameterization it does not deploy.", "verdict": "outside"}, {"arxiv_id": "2411.12603", "claim_anchor": "unlabeled remark (supplement.tex:99)", "bound": "Complexity: the recurrent form h_k = e^{A_k Delta_k} h_{k-1} + B_k u_k (eq. appendix-recurrent-ssm) allows asynchronous streaming inference in O(1) time per coordinate, O(N) for the full stream. No constants; precondition-free.", "deployed_config": "Deployed recurrence uses diagonal A with state dim m = 32 (main.tex:230, 469); event-vision experiments are 'processed as a stream of events with our purely recurrent neural network without the usage of frames' (main.tex:90); training uses the parallel selective-scan CUDA kernel (main.tex:92; appendix:scan gives O(log N) depth).", "evaluation": "Structurally satisfied at every deployed config: with diagonal A, one recurrent step costs a fixed O(m) = 32-multiply update independent of stream length N, so O(1)/coordinate holds wherever the recurrent mode runs (the event-stream evaluations). Big-O only: no per-event latency, throughput, or wall-clock figure is certified, and the training path is the O(N log N) scan, not the O(N) recurrence.", "verdict": "inside"}, {"arxiv_id": "2412.10540", "claim_anchor": "kronecker-universality theorem (sample-sigconf.tex:276)", "bound": "Expressivity: any fourth-order attention tensor A in R^{N x N x T x T}, reshaped to A in R^{NT x NT}, equals sum_{i=1}^R (B_i Kronecker C_i) for some R <= min(N^2, T^2); the representation 'becomes exact' as R approaches min(N^2, T^2). No error term is stated for any R below the threshold (the SVD-truncation proof drops the Eckart-Young remainder rather than bounding it).", "deployed_config": "StockNet: N = 88 stocks, T = 5-day lag window (sample-sigconf.tex:362, 174); the deployed model replaces the certified rank-R summation with multi-head attention - 'In practice, we adopt the multi-head attention mechanism instead of the summation shown in Equation (eq:att_approx)' (sample-sigconf.tex:267) - with heads grid-searched over {1, 4, 8, 16}, hidden dim {32, 64, 128} (sample-sigconf.tex:373-374); kernel (Performer-type) attention is additionally applied (sample-sigconf.tex:222).", "evaluation": "Exactness threshold at deployment: min(N^2, T^2) = min(88^2, 5^2) = min(7744, 25) = 25. Deployed R (reading heads as R, the paper's own analogy) is at most 16 < 25 - every deployed config sits strictly below the certified-exactness rank, and the theorem attaches NO error bound to any R < 25, so nothing quantitative is certified at deployment. Two further exits from the certified object: (1) the deployed factors are softmax-normalized parametric attention matrices produced by multi-head layers, not the free SVD factors B_i, C_i of the proof, so the construction does not apply to the deployed parameterization even at R = 25; (2) the added kernel-attention approximation (via choromanski2022rethinking) is a second uncertified layer between theorem and deployment.", "verdict": "outside"}, {"arxiv_id": "2412.12465", "claim_anchor": "prop: reachability (example_paper.tex:635)", "bound": "Expressivity/connectivity: with causal masking, the CCA attention score 'fully satisfies the reachability condition' a_ij > 0 from every earlier token j to every later token i at each layer, plus an exact closed form for the output: o_j = sum_q A^L_{i,q} V_{q,j} for i <= g, else the pooled-core-token expression with weights Phi and window w = s + (i-s) mod g. No magnitude lower bound on a_ij and no approximation-error bound to full attention.", "deployed_config": "CCA-LLM on LLaMA2-7B-32K/80K, LLaMA3.1-8B-Instruct-128K, Qwen2.5-7B-128K (example_paper.tex:377-411); contexts to 64K/128K; deployed knobs are exactly the proposition's parameters: group size g swept 16 -> 2 and local window s swept 4096 -> 1024 at inference (example_paper.tex:547); SlimPajama training; A800 GPUs, 7.9x speedup at 128K claimed (example_paper.tex:195).", "evaluation": "The preconditions are the method's own definition (causal masking, Eqn cca_att), so every deployed (g, s, L) satisfies them by construction; in exact arithmetic softmax and pooling weights are strictly positive, so a_ij > 0 holds identically at g = 16, s = 4096, L = 128K as at any other config. But the certificate is connectivity-only: a_ij > 0 is satisfied by arbitrarily small weights, so it bounds no approximation error to full self-attention at any (g, s) - the paper's own sweep shows the certified property is config-invariant while quality is not (LongBench-E 20.54 at g=16/s=4096 vs 22.69 at g=2/s=1024, example_paper.tex:547). Footnote on precision: in deployed bf16, weights below the underflow threshold round to exactly 0, so even the positivity claim is exact-arithmetic-only at 128K-scale logit spreads.", "verdict": "inside"}, {"arxiv_id": "2412.19350", "claim_anchor": "commutativity proposition (main.tex:650)", "bound": "Impossibility: a single-layer selective SSM under the paper's FSA mapping (x0 = enc(q_init), b(u_t) = 0) whose transition matrices A(u_1),...,A(u_T) are simultaneously diagonalizable can only emulate commutative automata.", "deployed_config": "The paper's own C-diagonal selective SSM variants (main.tex:552-566): complex diagonal A(u_t) on the circle of radius beta, single layer, deployed both with B=0 and B!=0 and linear/nonlinear readouts on C_2 x C_30 (commutative) and D_30 (non-commutative dihedral), trained to length 90, eval to 600 (tab:C2xC30_vs_D30; results/2412.19350.json). SD-SSM (dense, non-diagonal) deployed on the same tasks.", "evaluation": "Precondition check for the deployed B=0 diagonal variants: (a) diagonal matrices are trivially simultaneously diagonalizable (W = I); (b) b(u_t) = 0 holds by config ('B=0' rows); (c) single layer by config. Deployed learned x0 differs from the mapping's enc(q_init), but the proof's mechanism (x_T = (prod_t Lambda(u_t)) x_0, order-invariant since diagonal matrices commute) holds for ANY fixed x0 and any readout of x_T, so the impossibility binds at deployment: on D_30, where the final automaton state is order-dependent, the certified conclusion is failure - observed max length-gen accuracy 8.35% (B=0, both readouts) vs 100% on commutative C_2 x C_30 (B=0 linear). B!=0 variants and dense SD-SSM violate the preconditions by design (b(u_t) != 0 / not simultaneously diagonalizable) and are unconstrained by the proposition: D_30 accuracy 61.0 (B!=0 nonlinear) and 100 (SD-SSM).", "verdict": "inside"}, {"arxiv_id": "2502.07861", "claim_anchor": "thm:main-theorem (arxiv_camera_ready.tex:221)", "bound": "BalanceKV streaming guarantee: with batch size t = O~(sqrt(d) e^{2r^2/sqrt(d)}/eps) and compression rate 2^{-T} with T = log(n/t), every stream step satisfies the attention-approximation objective w.p. >= 1 - 1/poly(n), using total memory O~(d^{3/2} e^{2r^2/sqrt(d)}/eps) (n-independent) and per-step runtime O~(d^2 e^{4r^2/sqrt(d)}/eps^2). Preconditions: ||q_j||, ||k_j|| <= r for all j.", "deployed_config": "End-to-end LongBench-E (Llama-3.1-8B-Instruct, Qwen2.5-14B/32B, head dim d = 128): 'We set b=256 and T=2, achieving a consistent compression rate of 0.25 across all inputs' (arxiv_camera_ready.tex:840); LongBench-E contexts bucketed 0-4k/4-8k/8k+. Single-layer experiments: compression rates {1/2,1/4,1/8,1/16}, batch sizes {64,128,256} (tex:353, 370). Low-compression addendum: rates 0.8/0.9/0.95 (tex:944).", "evaluation": "The theorem's parameter coupling T = log2(n/t) is provably violated at deployment: with t = 256, the certified T is log2(4096/256) = 4, log2(8192/256) = 5, log2(32768/256) = 7 for the 4k/8k/32k inputs, but deployed T = 2 for ALL inputs (matching the coupling only at n = 1024). Consequence: deployed retained memory is 0.25n - Theta(n), growing linearly - whereas the certified operating point retains ~t = 256 tokens with n-independent memory O~(d^{3/2} e^{2r^2/sqrt(d)}/eps); at n = 8192 deployment keeps 2048 tokens, 8x the theorem's target. The certified algorithm configuration is run on no LongBench-E input. Secondary blockers even at the certified T: eps is never chosen, r = max deployed query/key norm is not recorded in results/2502.07861.json, the O~ constant is unpinned, and 1/poly(n) is unnamed, so no numeric tolerance could be computed anyway.", "verdict": "outside"}, {"arxiv_id": "2502.07861", "claim_anchor": "thm:BALANCE-vectors (arxiv_camera_ready.tex:298)", "bound": "SoftmaxBalance halving: outputs C' with |C'| <= |C|/2 such that for any FIXED query q, w.p. >= 1 - delta, the signed softmax-weighted discrepancy is <= O(sqrt(s) log(ns/delta) e^{||q||^2/(2 sqrt(d))} e^{max_j ||k_j||^2/(2 sqrt(d))} max_j ||v_j||_2). Runtime O((d+s)n^2), memory O((d+s)n). Preconditions on (K, V, delta) are unconditional.", "deployed_config": "SoftmaxBalance is the recursion core of every deployed BalanceKV run (LongBench-E with b=256, T=2; single-layer TriviaQA experiments at rates 1/2-1/16, d = 128). Deployed norm profiles ||q||, max||k||, max||v|| of the Llama-3.1/Qwen2.5 layers are not recorded in results/2502.07861.json.", "evaluation": "The theorem is precondition-free over (K, V, delta), so deployed inputs trivially fall in its stated scope - but the certified tolerance is not computable at any deployed config: the leading O(.) constant is unpinned, and the bound depends exponentially on unrecorded quantities e^{||q||^2/(2 sqrt(d))} and e^{max||k||^2/(2 sqrt(d))} (at d = 128, sqrt(d) ~ 11.3; a norm of 10 vs 15 changes the factor by e^{(225-100)/22.6} ~ e^{5.5} ~ 245x). Additional deployment gap: the guarantee is per fixed q over the algorithm's randomness, while decoding-phase queries depend on outputs computed from the compressed cache (adaptive queries), a case the single-query statement does not cover (the stream-level union bound is delegated to thm:main-theorem, audited above).", "verdict": "not_evaluable"}, {"arxiv_id": "2502.09287", "claim_anchor": "lower bound white noise (sections/lower_bound.tex:19)", "bound": "For any positive integers S, K: every order-S diagonal linear-RNN filter c (form c_k = sum_{s=1}^S a_s^k b_s) approximating the shift-K filter under white-noise input has L_time(c,d) >= 1 - S/(K+1). Fully explicit, no hidden constants.", "deployed_config": "Experiments (sections/experiments.tex; appendix tab:xp-compare_init): diagonal linear RNN with S = 128 hidden neurons, task shift K* = N - t* = 1500 - 200 = 1300, input AR(rho) with rho swept over {0, 0.2, 0.4, 0.6, 0.8, 1} - the rho = 0 runs are white noise (u_n = eps_n ~ N(0,1); only u_1 ~ U(0,1) deviates). Robustness experiment: S = 128, K* = 2000, but at rho = 0.7 only.", "evaluation": "Precondition check at the deployed rho = 0 config: the trained model y_t = C x_t, x_{t+1} = A x_t + B u_t with diagonal A is exactly of the form Eq. (cab) with S = 128 (effective weights b_s = C_s B_s), and the input is white noise. Bound evaluates to L_time >= 1 - 128/1301 = 0.9016: no trained instance of the deployed architecture can push the (population, normalized) copy loss below 90.2% at the deployed (S, K). At the robustness config's sizes (S = 128, K = 2000) the same formula gives 0.9360, though that experiment deploys rho = 0.7, not white noise (covered by the autocorrelated bound, next record).", "verdict": "inside"}, {"arxiv_id": "2502.09287", "claim_anchor": "theorem autocorrelated lower bound (sections/lower_bound.tex:74)", "bound": "For integers S, K and AR(rho) input: L_time(c,d) >= (1 - 3S/(K(1-rho)))_+ for every filter c of the form Eq. (cab). Fully explicit.", "deployed_config": "Same deployed filters, S = 128. Left experiment: K* = 1300, rho in {0, 0.2, 0.4, 0.6, 0.8, 1}, data generated exactly as the assumed AR process u_n = rho u_{n-1} + eps_n, eps ~ N(0, 1-rho^2) (appendix tab:xp-compare_init). Right experiment: K* = 2000, rho = 0.7 (tab:xp-robustness).", "evaluation": "Preconditions match deployment exactly (the experimental generator IS the assumed AR autocorrelation). Evaluating 3S/(K(1-rho)) with 3S = 384: right config (K=2000, rho=0.7): 384/600 = 0.640 -> certified floor 0.360. Left config (K=1300): rho=0 -> 0.295 -> floor 0.705 (weaker than the white-noise bound's 0.902 at the same config); rho=0.2 -> floor 0.631; rho=0.4 -> 0.508; rho=0.6 -> 0.262; rho=0.8 -> 384/260 = 1.477 -> (1-1.477)_+ = 0, bound vacuous; rho=1 -> 1-rho = 0, vacuous. So the certificate is active at 5 of the 7 deployed (K, rho) pairs and empty at the two most-correlated ones.", "verdict": "inside"}, {"arxiv_id": "2502.09287", "claim_anchor": "thm:informal_upper (sections/introduction.tex:111); Lemma param of bs (sections/upper_bound.tex:26); thm upper bound (sections/upper_bound.tex:71); convergence to window (sections/upper_bound.tex:84)", "bound": "Constructive upper bound: the filter with poles a_s = e^{-alpha/K} e^{i pi s/K} and weights b_s = (e^{-alpha}(e^{2alpha}-e^{-2alpha})/(2K))(-1)^s achieves L_time(c,d) ~ 1 - (e^{-2alpha}(e^{2alpha}-e^{-2alpha})/2)(S/K), matching the lower bound up to a constant - stated only asymptotically as S, K -> infinity with S/K -> 0 (weights lemma: K -> infinity; transfer-function window statement: same limit). No finite-(S,K) remainder term anywhere.", "deployed_config": "The experiments initialize with exactly these (a_s, b_s) at alpha = 1, S = 128, K_init = 1300 (left), K_init in {250,...,32000} vs true K* = 2000 (right) - finite values, S/K in [0.004, 0.512].", "evaluation": "All four statements are asymptotic equivalences ('~' / limits) with no error term at finite (S, K), so nothing is certified at the deployed S = 128, K <= 2000 - the optimal-weights lemma does not even certify that the deployed b_s are optimal at finite K. Leading-order illustration only (NOT licensed by the paper at finite sizes): e^{-2}(e^2 - e^{-2})/2 = 0.4908 at alpha = 1, giving L_time ~ 1 - 0.4908 x 128/1300 = 0.9517 (left) and 1 - 0.4908 x 128/2000 = 0.9686 (right) - consistent with, and within ~5% of, the certified floors 0.9016/0.9360, but the o(S/K) remainder is uncontrolled.", "verdict": "not_evaluable"}, {"arxiv_id": "2502.10297", "claim_anchor": "th:groups config (i) (neurips_2024.tex:338) with companion th:suborth (neurips_2024.tex:937)", "bound": "Existence: one-layer DeltaProduct with n_h = n-1 solves the S_n word problem (imported Grazzi et al. Thm 3, keys of size n); th:suborth: n_h = n suffices for any group embeddable in O(n), or in SO(n+1) for n even - for SO(3) subgroups this means n_h = 2 with keys of size 3.", "deployed_config": "State-tracking (neurips_2024.tex:458-468, app at 1286): single-layer DeltaProduct_{n_h}[-1,1], n_h in {1,...,4}, 12 heads, head dim 32, trained on length-128 sequences, eval to 512, on S3, S4, A5, S5 word problems.", "evaluation": "Deployed single-layer configs vs certified thresholds: S3 (n=3) at n_h = 2 = n-1 -> exactly the th:groups(i) configuration; S5 (n=5) at n_h = 4 = n-1 -> exact match; keys of size n <= 5 fit the deployed head dim 32. S4 and A5 at deployed n_h = 2 sit BELOW th:groups(i)'s threshold (n-1 = 3 and 4) but exactly AT th:suborth's: both are isomorphic to SO(3)-subgroups (rotation groups of cube/dodecahedron), SO(n+1) with n = 2 even -> n_h = 2, keys of size 3 <= 32. So every deployed config the paper reports as succeeding is covered by one of its own existence theorems, and the empirical thresholds (S3 needs n_h=2, S5 needs n_h=4, S4/A5 work at n_h=2) coincide with the certified ones. Caveats: theorems are existence-of-weights only (finite precision, no learnability statement), and deployed sub-threshold configs (e.g. S5 at n_h in {1,2,3}) carry no certificate in either direction - the theorems are upper bounds, not impossibilities.", "verdict": "inside"}, {"arxiv_id": "2502.10297", "claim_anchor": "th:groups configs (ii)/(iii) (neurips_2024.tex:338); lm:permreset (neurips_2024.tex:981); th:rwkv7deltaprod_app (neurips_2024.tex:1010)", "bound": "Multi-layer route: 3 layers (n_h > 1) or 4 layers (n_h = 1) solve any S_n word problem / permutation-reset FSA / regular language (RWKV-7-product variant), PROVIDED the second-last layer's MLP computes a lookup table of size 2m x (n!)^{2m} (or 2m x s^{2m}), m = ceil((n-1)/n_h). No MLP width/parameter count is stated for realizing the table.", "deployed_config": "Multi-layer state-tracking runs (fig state_tracking_layers, caption at tex:452): DeltaProduct_1[-1,1]/DeltaNet with layer count varied on S3, S4, A5, S5; 12 heads, head dim 32 (app:1286), i.e. standard MLPs with ~1e5-1e6 parameters. No RWKV-7-product model is deployed at all.", "evaluation": "The lookup-table precondition is evaluable in size but not in representability: at n_h = 1, m = n-1, so the required table has 2m x (n!)^{2m} entries = 4 x 6^4 = 5,184 (S3) - modest; 6 x 24^6 = 1.15e9 (S4); 8 x 120^8 = 3.44e17 (S5). For the deployed S4/S5 multi-layer runs the certified construction routes through an MLP encoding ~1e9-3e17 table entries inside a network with ~1e5-1e6 parameters; whether a fixed-width MLP can represent these specific tables is not quantified by the paper (no width bound is stated), so no in/out verdict can be closed - but the constant-free reading puts the S5 construction 11+ orders of magnitude beyond any deployed MLP. th:rwkv7deltaprod_app additionally has no deployed config whatsoever.", "verdict": "not_evaluable"}, {"arxiv_id": "2502.10297", "claim_anchor": "th:groups config (ii) instantiated at S_2 = parity (neurips_2024.tex:338); lm:countmodm (neurips_2024.tex:914); thm:dihedral (neurips_2024.tex:1029)", "bound": "Parity/cyclic/dihedral toolkit: counting mod d with 1 layer and n_h >= 2 (or 2 layers, n_h = 1, prior work); D_m word problem with 1 layer and n_h >= 2 (or 2 layers, n_h = 1, >= 2 heads in layer 1); S_2 (= parity) via th:groups(ii): 3 layers, n_h > 1, lookup table 2m x (n!)^{2m} with n = 2, m = 1.", "deployed_config": "Chomsky-hierarchy experiments (app at tex:1301-1303): parity and modular arithmetic; DeltaProduct_{n_h}[-1,1] with n_h in {2,3,4}, 3 layers, 1 head, head dim 128; DeltaNet[-1,1] = n_h 1 baseline; [0,1]-range ablations; trained lengths 3-40, tested 40-256.", "evaluation": "Deployed parity config (3 layers, n_h in {2,3,4}, [-1,1]) exactly matches th:groups(ii)'s certified configuration (3 layers, n_h > 1) instantiated at n = 2, where the lookup-table precondition is trivial: m = ceil(1/n_h) = 1, table size 2 x (2!)^2 = 8 entries - representable by any deployed MLP. So the deployed parity models sit inside a certified existence regime; observed scaled accuracy 0.896-0.982 ([-1,1] variants). The [0,1]-range deployments exclude the eigenvalue -1 (beta <= 1) that every parity construction uses, and score 0.233-0.295 (near random) - outside the constructions' requirements, consistently failing. lm:countmodm's own 1-layer (n_h >= 2, one-head) config and thm:dihedral's configs are NOT deployed (all Chomsky runs use 3 layers; no D_m task is run), so those specific claims have no deployed instantiation.", "verdict": "inside"}, {"arxiv_id": "2502.10297", "claim_anchor": "th:regular (neurips_2024.tex:342)", "bound": "For any n_h >= 1 and any regular language L there exists a Gated DeltaProduct with a FINITE number of layers - the layer count depends on the language and is not given by any formula.", "deployed_config": "Gated DeltaProduct_2[-1,1] (393M) deployed only for language modeling on FineWeb (results/2502.10297.json); no deployed run targets a specific regular language with the gated variant (the Chomsky experiments use non-gated DeltaProduct).", "evaluation": "Two independent blockers: (1) the certified layer count is 'finite, language-dependent' with no formula, so no deployed depth can be checked against a threshold; (2) the deployed gated models are run on natural-language modeling, not on any regular-language recognition task, so there is no (language, depth) pair to evaluate even in principle.", "verdict": "not_evaluable"}, {"arxiv_id": "2502.16792", "claim_anchor": "thm:length-extrap (length-generalization.tex:269)", "bound": "For a k-sparse functional attention class H(G_pred, G_val) with L_local-local, relative G_pred, and an ensemble with k-sparse planted correlations that is delta-approximately realizable with position-coverage constants eta_ell: the population ERM over the class trained on lengths ell ~ Unif([L/2, L]) has test loss at length L_bar at most eta_L x eta_{L_bar} x L x L_bar^2 x delta, provided L_local | (L_bar - L) and L >= 4 L_local.", "deployed_config": "Sparse-parity experiments (tex:340-401, app:1062): 12-layer/16-head/1024-dim GPT-NeoX with RoPE+PoSE trained by AdamW on lengths 2l in [20,50], tested to 2l in [20,500], sparsity K_train in {4,6,8,10,12}, K_test to 16; special positions uniform over the whole sequence. Natural-language runs: 24-layer OLMo on C4, eval contexts 64/128 (results/2502.16792.json).", "evaluation": "Two provable violations at the sparse-parity deployment. (1) Locality: the deployed distribution places the 2k planted tokens uniformly over the sequence (the paper's own appendix at tex:1100 notes the assumptions hold only 'under the additional assumption' that the planted positions lie within L_local of each other - an assumption the deployed generator does not impose). Any L_local large enough to realize the deployed ensemble at test lengths must be ~500, and then the precondition L >= 4 L_local reads 50 >= ~2000 - violated by ~40x; even the train-only reading (L_local ~ 50) gives 50 >= 200, violated 4x. The paper's own translation ('our theoretical guarantee only holds for... Ktrain <= L/4', tex:399) puts deployed K_train = 12 at the boundary 12.5 - and the authors' figure comments record that exactly the K=12 models length-generalize worse. (2) Hypothesis class: the certified object is the population ERM over the abstract k-sparse functional attention class, which by the paper's own bridge (prop:model-attn-head-formal, tex:627) coincides with transformers only for a SINGLE attention head at k = 1; the deployed 12-layer multi-head SGD-trained GPT-NeoX at k up to 12 is provably not a member. The C4 runs have no planted sparse structure at all (delta, k, eta unknown).", "verdict": "outside"}, {"arxiv_id": "2502.16792", "claim_anchor": "prop:pc-length-extrap (length-generalization.tex:1013)", "bound": "Position-coupling extension: if the planted ensemble is amenable to L_local-position coupling and psi_ell(S*) satisfies coverage with constants eta_ell, the position-coupled class PC[H_pred] achieves (L, L_bar, eta_L x eta_{L_bar} x L x L_bar^2 x delta)-length generalization on PC[P], same side conditions (L_local | L_bar - L, L >= 4 L_local), removing the locality requirement on the raw planted set.", "deployed_config": "Scratchpad experiments with position coupling (tex:403-470): parity-with-scratchpad and variable-assignment-with-scratchpad, 12-layer GPT-NeoX trained with coupled position IDs; multiple seeds, 95% CIs.", "evaluation": "The deployed tasks are the intended instantiations of position coupling, but the paper never instantiates the proposition's parameters for them: no L_local, delta, or eta_ell is derived for the deployed scratchpad ensembles (the appendix verifies the assumptions only for sparse parity under an artificial locality restriction, tex:1100), so the certified loss eta_L x eta_{L_bar} x L x L_bar^2 x delta cannot be evaluated at any deployed (L, L_bar) - with delta = 0 it would read 0, but exact realizability of the deployed scratchpad distributions by PC[H_pred] is asserted for no concrete G_pred/G_val. The class mismatch of the previous record applies verbatim: the trained transformer is not the PC[H_pred] population ERM.", "verdict": "not_evaluable"}, {"arxiv_id": "2502.16792", "claim_anchor": "prop:noasm-nolg (length-generalization.tex:163)", "bound": "No-free-lunch: if the ERM h_hat disagrees with h* by more than epsilon somewhere at length L_bar, there EXISTS an ensemble extension P on which the class fails (L, L_bar, epsilon)-length generalization - i.e., without distributional assumptions, fitting lengths <= L certifies nothing at L_bar.", "deployed_config": "None: the adversarial extension P is a construction internal to the proposition; no deployed experiment instantiates it (the deployed data distributions are fixed benign ones).", "evaluation": "Not evaluable by construction: the claim quantifies existentially over ensembles chosen adversarially AFTER the hypothesis class and training lengths; no deployed configuration realizes (or could realize) the certified object, and the proposition assigns no quantity to any benign deployed distribution.", "verdict": "not_evaluable"}, {"arxiv_id": "2502.16792", "claim_anchor": "prop:necessity-asms (length-generalization.tex:807)", "bound": "Necessity: there is a universal constant c > 0 such that dropping any one of the four assumptions (locality, relativity, exact realizability, eta_ell = O(ell) coverage) admits a 2-sparse class/ensemble pair (L_local = 2) failing (L, L_bar, c)-length generalization for every L_bar >= 2L.", "deployed_config": "None: the four counterexample class/ensemble constructions are never trained on or sampled in any experiment.", "evaluation": "Two blockers: the universal constant c is never pinned ('There is a constant c > 0'), and the counterexample ensembles have no deployed instantiation, so there is no config at which to evaluate the failure level c.", "verdict": "not_evaluable"}, {"arxiv_id": "2503.03588", "claim_anchor": "receptive-field DAG theorem (070appendix.tex:26)", "bound": "For the DAG on vertices 1..n with edges (i,j) iff i - j = 2^k: (1) every out-degree is < log n; (2) for any j < i, dist(i -> j) <= log n (via the binary decomposition of i - j; log base 2).", "deployed_config": "PowerAttention deployed on 28-layer Qwen2-7B with 256-token blocks (040experiments.tex:19, 107): the mask keeps block-pairs at power-of-2 block distance plus a local window, same-block, and a sink block (Algorithm pseudo-code, 030method.tex). Contexts: RULER at 4k/16k/32k (sparsity up to 0.94), passkey to 64k, efficiency to 128k -> n = #blocks = 16/64/128/256/512.", "evaluation": "The deployed pattern instantiates the theorem's graph at block granularity, as a SUPERSET of E (window + sink + same-block edges added). Distance certificate: inherited a fortiori at every deployed n - at 32k, n = 128 blocks, any block reachable in <= log2(128) = 7 hops (worst case d = 127 = 1111111b, exactly 7), and deployed depth 28 >= 7 (also 8 at 64k, 9 at 128k), so the information-propagation precondition 'distance <= #layers' holds at every deployed context; the paper's observed full-coverage passkey retrieval at 32k within ~6-7 effective hops matches. Out-degree: two caveats. (a) As printed the strict bound fails at the deployed 32k config: vertex i = 128 has out-degree #{k : 2^k <= 127} = 7 = log2(128), so 'out-degree < log n' is false at exactly n = 128; the proof actually establishes <= log n (off-by-one in the statement). (b) The certified out-degree covers only the power-of-2 component - deployed blocks additionally attend to the window and sink, so deployed per-block out-degree exceeds log n by design (accounted in the measured 0.94 sparsity, not by the theorem).", "verdict": "inside"}]