{"id":"32d0cdea-8bf0-462e-9f94-25027c519486","arxiv_id":"2505.08739","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Perplexity is order-invariant by the chain rule for a true distribution, yet GPT-2 models trained on forward, backward, and permuted text deviate systematically, with permutation training producing the largest gaps.","lead":"The paper proves that sequence perplexity is invariant under any permutation of token order when the true probability distribution is used, then shows that GPT-2 models trained on different token orders deviate from this invariance. It argues that earlier forward and backward comparisons had methodological flaws and attributes the remaining gap to positional and locality biases in self-attention.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The theorem concerns one fixed joint distribution; the experiments compare two different finite-capacity estimators, so the observed perplexity gaps do not by themselves contradict or test the invariance.","rationale":"The reader's weakest assumption identifies the same load-bearing gap: the theorem concerns a fixed true distribution, while the empirical comparison involves separately trained, imperfect models whose approximation error is never characterized. I agree that this is the core weakness. The theorem itself is correct and the methodology is careful and reproducible, but the central interpretive claim—that observed perplexity differences indicate deviations from theoretical invariance or probability inconsistency—is not established. Because the issue is addressable with additional control experiments or a reframed claim, the conditional verdict remains appropriate rather than a rejection. My stress-test pass therefore does not change the reader's verdict, but it sharpens the required condition: the paper must either measure approximation error or reposition its contribution as documenting finite-model discrepancies rather than testing the invariance theorem.","tokens_in":22657,"tokens_out":6767,"duration_ms":79115,"concrete_test":"Train forward- and backward-order GPT-2 models on synthetic sequences drawn from a known finite-order Markov chain or hidden Markov model whose true joint distribution is computable, using the same three model sizes and training schedules as the paper. Because the true P is known, compute each model's approximation error as the per-token KL divergence (or NLL gap) from the true conditional distributions in its own factorization, and the Fwd-Bwd validation perplexity gap. If the Fwd-Bwd gap shrinks toward zero as model capacity or training epochs increase while approximation error shrinks correspondingly, the paper's observed discrepancies are finite-capacity artifacts rather than evidence of probability inconsistency. If the gap persists even when both models approximate the true distribution very closely, the empirical challenge to the invariance claim would be substantiated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Equation (1) asserts that for a single fixed joint distribution P, the perplexity computed from any factorization order is exp(-1/n ln P(X0,...,Xn)). The experiments in Section 3.1 do not compute this quantity for one P. Instead, they train separate GPT-2 models on forward, backward, and permuted factorizations, obtaining three different implicit joint distributions P_hat_fwd, P_hat_bwd, and P_hat_perm, and then compare exp(-1/n ln P_hat_fwd(X)) with exp(-1/n ln P_hat_bwd(X)) on the same forward text. The chain rule, and therefore Eq. (1), places no constraint on two different estimated distributions agreeing. The theorem only implies equality of the factorized perplexities when each model's conditionals coincide with the true distribution's conditionals in its own factorization. The paper never measures or bounds approximation error, so the systematic Fwd-Bwd and Perm gaps it reports are exactly what one expects from imperfect, finite-capacity estimators with different inductive biases. They do not show that any single model's probability estimates are internally inconsistent, and they do not challenge the theoretical invariance. The abstract and Section 4 claim that the empirical results reveal 'deviations from theoretical invariance' and that such discrepancies make LLM probability distributions 'inconsistent and therefore untrustworthy'; that inference is unsupported without a demonstration that the gaps exceed what approximation error alone can explain.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper makes a formal claim that sequence perplexity is invariant under any factorization order: for a fixed joint distribution over a token sequence, PP_σ = exp(−1/n ln P(X0,...,Xn)) regardless of the permutation σ used to order the conditional probabilities. It then trains GPT-2 models at three scales (124M, 355M, 774M) on forward, backward, and arbitrarily permuted versions of a neuroscience corpus, applying protocols designed to satisfy the theorem's conditions (BOS tokens, a single forward-trained tokenizer, and token-level permutations), and reports systematic perplexity gaps, attention-pattern differences, representational divergence, and BrainBench results. The paper also criticizes prior studies for methodological deviations such as missing BOS tokens, retraining tokenizers on reversed text, and conflating logical reversal with token reversal.","tokens_in":22912,"tokens_out":18584,"duration_ms":187215,"significance":"If the empirical interpretation were supported, the paper would provide a useful benchmark and diagnostic for order consistency in autoregressive LLMs. The chain-rule identity in Eq. (1) is elementary but worth stating as a conceptual baseline, and the protocol corrections for prior work are valuable and concrete. The empirical study is substantial: 27 models, multiple sizes, multiple initializations, public code and weights, and a clear description of training and evaluation setups. The descriptive findings about attention biases and the relationship between forward and backward models are interesting. However, the central interpretive claim that the measured gaps constitute deviations from the theoretical invariance is not established, because the theorem constrains a single distribution whereas the experiments compare several separately trained estimators. The contribution should be reframed as a descriptive study of training-order-induced estimation bias, with the theoretical benchmark used as a reference point rather than as a claim of theorem violation.","major_comments":[{"comment":"The central empirical inference is not supported by the stated theorem. Equation (1) establishes an identity for a single fixed joint distribution P: for any permutation σ, the perplexity computed from P's conditionals on a given event equals exp(−1/n ln P(X0,...,Xn)). It imposes no constraint on two separately trained models, each of which realizes a different estimated distribution P̂_fwd, P̂_bwd, or P̂_perm. The reported gaps in Table 2 and Figure 1 are therefore entirely compatible with the theorem if the models' approximation errors differ, and the paper never measures or bounds the per-model approximation error relative to the true distribution or to the pushforward target distributions. Consequently, the abstract's claims of 'systematic deviations from theoretical invariance' and of probability distributions being 'inconsistent and therefore untrustworthy' go beyond what the data show; the empirical contribution should be reframed as a descriptive study of order-induced estimation bias.","section":"§3.1, Table 2, Eq. (1)"},{"comment":"The paper does not specify, for the comparisons in Table 2 and Figure 1, whether the backward- and permuted-trained models are evaluated on the reversed/permuted versions of each validation sequence or on the original forward-order text. Under the theorem, only the former comparison is even notionally covered by Eq. (1); evaluating a backward model on forward-order text measures an off-factorization conditional that Eq. (1) never addresses. In addition, Appendix D.2 states that 'the starting token's probability is masked out in the softmax operation to align closely with the proof' without saying whether the corresponding term is removed from both the numerator and the denominator; if it is excluded from both, the empirical perplexity uses a different normalization than Eq. (1), which divides by n. Both points must be resolved before the quantitative comparisons can be interpreted.","section":"§3.1, Appendix D.2"},{"comment":"The theorem as stated applies to a fixed distribution P, but the experiment with permuted orders actually concerns pushforward distributions: if the training data for order σ are generated by applying σ to each natural string, the target distribution is Q_σ with Q_σ(σ(w)) = P(w). The paper does not state or prove the lemma that a perfect model of Q_σ has perplexity exp(−1/n ln P(w)) on σ(w); without this lemma, Eq. (1) does not by itself justify comparing forward, backward, and permuted models. Adding this step would also make explicit why the theoretical benchmark concerns the pair (model, its own factorization), which is exactly the point at which the approximation-error concern in Major Comment 1 enters.","section":"§2, Implications"}],"minor_comments":[{"comment":"The phrase 'the same joint probability' should specify that the chain-rule factorizations refer to the same event, or to the same observed assignment to indexed positions, because for a non-exchangeable distribution the probability of the permuted string differs from the probability of the original string.","section":"§2, first paragraph"},{"comment":"The index ranges and the meaning of t should be stated explicitly (number of real tokens after the BOS token) and matched to the denominator in Eq. (1), so that readers can verify the empirical perplexity uses the same normalization as the theorem.","section":"Appendix E, Eq. (2)"},{"comment":"The caption contains a duplicated 'across across' that should be corrected.","section":"Figure 1 caption"},{"comment":"The t-tests treat the 9,413 validation sequences as independent; given the small effect sizes for the forward-backward comparisons (Cohen's d around 0.2-0.7), a discussion of practical significance and of possible dependence between sequences would strengthen the interpretation.","section":"Table 2"},{"comment":"The sentence 'We exclude permuted-text models, which converged during training' is unclear about whether the intended meaning is that the models converged to a poor solution or did not converge; please clarify.","section":"§3.4"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is likely to be of interest to the journal's readership, and the protocol corrections to prior work are valuable. The main gap is the inference from the theorem to the empirical claims; I recommend major revision rather than rejection because the descriptive empirical findings and the protocol contributions are solid and the interpretive issue can be fixed by reframing and by adding an approximation-error analysis. The theoretical proof is elementary, and the novelty claim should be toned down accordingly."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe paper is more useful than its headline theorem suggests, and less decisive than its authors claim. The formal result—sequence perplexity is invariant under any permutation of factorization for a fixed true distribution—is just the chain rule dressed up. It is correct, and generalizing to arbitrary permutations is a one-line step; anyone working in this area already knew forward–backward equivalence followed from the chain rule. That part is not new, and the paper overstates its novelty.\n\nWhat is genuinely valuable is the empirical work. The authors identify real methodological flaws in prior studies: missing BOS tokens, retrained tokenizers on reversed text, and confusing logical reversal with token reversal. They then run a careful, well-controlled study—27 GPT-2 models, three scales, three orders, three seeds, on a 1.3B-token corpus—with a shared tokenizer and full-context sequences. The finding that forward and backward models are nearly equivalent but with small systematic gaps, while permuted training diverges strongly, is solid descriptive evidence. The attention analysis (locality and long-range biases) is also informative and extends prior attention-sink work. And they correct their own earlier BrainBench claim, which deserves credit.\n\nThe soft spot is interpretation, not measurement. The theorem concerns one joint distribution; the experiments compare three differently trained estimators of that distribution. The observed perplexity gaps do not contradict the invariance, because nothing in Eq. (1) constrains two finite-capacity models with different inductive biases to agree. The paper never measures or bounds approximation error, so the claim that deviations make the models 'inconsistent and therefore untrustworthy' is unsupported. Reframing the results as an empirical characterization of order effects—rather than a challenge to the theory—would fix most of this. The attention-based story is also correlational; the paper doesn't intervene on attention to show causation, though that is a minor point given the paper's goals.\n\nI'd send this to review. The theorem alone won't survive scrutiny as a standalone contribution, but the empirical protocol and the attention findings are worth publishing, and the overreach is fixable in revision. It is a paper that will generate useful discussion in a reading group, and I'd cite it for the careful experimental setup even if not for the proof.","headline":"Solid empirical protocols and an interesting attention analysis attached to an elementary theorem that the paper over-sells as a contradiction of theory.","tokens_in":23408,"tokens_out":1779,"would_cite":true,"duration_ms":18310,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that sequence perplexity is theoretically invariant under any token-ordering factorization, and shows that trained LLMs nonetheless deviate from this invariance in ways traceable to positional biases in self-attention.","keywords":["perplexity invariance","chain rule of probability","token order factorization","autoregressive language models","self-attention positional bias","probability consistency","LLM evaluation protocol","attention sinks"],"falsifier":"Train sibling models with the same protocol but with position information removed or made relative only, and check whether the forward/backward perplexity gaps vanish; if they persist, the cause is not simply the positional bias isolated here. Alternatively, evaluate a single fixed-permutation model on all factorizations of the same sequences via importance sampling, since a perfectly consistent model must yield a reweighted joint-probability estimate equal to its forward estimate to numerical precision.","tokens_in":22481,"feed_emoji":"🧮","tokens_out":7696,"duration_ms":73970,"temperature":0.7,"pith_summary":"The paper establishes a theorem: for any well-defined probability distribution over token sequences, perplexity is invariant under every factorization order, including forward, backward, or any fixed permutation, because the chain rule collapses the averaged conditional log-probabilities into the joint probability of the whole sequence. It then argues that earlier comparisons of forward- and backward-trained models violated the conditions of that proof by omitting begin-of-sequence tokens, retraining tokenizers on reversed text, or confusing logical reversal with token-level reversal, and therefore were comparing different sequences. Retraining GPT-2 models at three scales with corrected protocols, the paper finds small but systematic forward-over-backward perplexity advantages and much larger deviations for permuted order, which it traces to positional and locality biases in causal self-attention. The central claim is that when a trained model's perplexity depends on factorization order, that dependence is a measurable sign that the model's probability distribution is internally inconsistent, not a refutation of the theoretical invariance. A sympathetic reading of the paper is that it turns perplexity into a principled diagnostic: order-invariance is the benchmark, and deviations from it expose where a model's learned conditionals fail to form a single coherent joint distribution.","feed_headline":"Proof: perplexity is order-invariant, but trained LLMs break it","feed_subtitle":"Forward and backward orders nearly agree; permuted orders diverge sharply, and positional attention biases explain the gap.","key_machinery":"The load-bearing object is the chain-rule telescoping identity for sequence perplexity, stated as Equation (1): $PP_\\sigma = \\exp\\left(-\\frac{1}{n}\\ln P(X_0,X_1,\\ldots,X_n)\\right)$ for every permutation $\\sigma$, proved by writing each conditional as a ratio of joint probabilities and observing that the log terms cancel in pairs. That identity turns perplexity into a pure function of the joint probability, and therefore into a concrete benchmark: any two factorizations of the same sequence must give the same number if the model is a faithful probability model. The experimental machinery is the protocol that makes the comparison legitimate, namely a BOS token with $P(X_0)=1$, one tokenizer trained only on forward text, strict token-level reversal or fixed permutation inside the context window, and full 1,024-token sequences, together with normalized attention entropy and normalized attention rank as diagnostics that localize the deviations within self-attention.","core_discovery":"The paper establishes, by a telescoping-sum argument, the identity $PP_\\sigma = \\exp\\left(-\\frac{1}{n}\\ln P(X_0,X_1,\\ldots,X_n)\\right)$ for any permutation $\\sigma$ of the token indices, where $X_0$ is a begin-of-sequence token with $P(X_0)=1$. Because every chain-rule factorization of the same joint probability yields the same product, sequence perplexity is a property of the full sequence alone and not of the order in which its conditionals are read. The empirical core is a controlled comparison of 27 GPT-2 models trained from scratch at 124M, 355M, and 774M parameters on the same 1.3-billion-token neuroscience corpus in forward, backward, and fixed-permuted token orders, using identical tokenization and a BOS token on every sequence. Forward- and backward-trained models achieve near-identical sequence-level perplexities with correlations above 0.99, yet forward models are consistently lower, and the gap widens with model size; permuted training produces substantially higher perplexities and attention patterns that favor neither local nor long-range structure. The paper attributes these gaps to causal self-attention's positional biases rather than to the data, and presents them as evidence that practical LLMs do not maintain the probability consistency the theorem guarantees.","pith_inferences":["If perplexity gaps are driven by positional priors, then a model that suppresses positional bias, for example through relative position encodings or sink-token removal, should show smaller forward/backward gaps under the same protocol; that is testable.","The same protocol could serve as a calibration check before deployment: a model whose perplexity depends strongly on factorization order is assigning meaning to token position beyond content, which may predict hallucination or out-of-distribution brittleness.","Because the theorem applies to any permutation, the fixed-permutation setting is the sharpest probe: it deliberately breaks the locality statistics of natural language, so the large deviations observed there suggest that much of an LLM's language understanding is a learned local-structure prior rather than a general joint model.","A direct extension would be to test invariance without retraining, by estimating the conditionals of one factorization from a single model through importance sampling; perfect consistency would require the reweighted joint-probability estimate to match the forward estimate, so any discrepancy would quantify the model's internal inconsistency."],"forward_implications":["Earlier forward/backward comparisons that omitted BOS tokens, retrained tokenizers, or reversed characters instead of tokens were comparing different sequences, so their ordering-effect claims cannot be interpreted against the theorem.","Because the invariance holds for any well-defined distribution, factorization-order dependence in a trained model is diagnostic: it marks where the model's conditional estimates fail to cohere into one joint distribution.","Forward and backward training are nearly interchangeable in downstream performance: on the BrainBench benchmark both directions match human experts, and backward-trained models are not worse aligned with human judgments than forward-trained ones.","Attention biases toward adjacent tokens and toward tokens at maximum distance are general across pretrained transformers and plausibly explain the lost-in-the-middle retrieval failure pattern.","The empirical deviations do not refute the invariance result; they locate the inconsistency in the learned approximation, motivating attention-based and architecture-level explanations of LLM behavior."],"supporting_citations":[{"why":"Supplies the prior forward/backward training comparison that this paper corrects and re-runs with theory-aligned protocols.","marker":"[Luo et al., 2024a]"},{"why":"Defines the GPT-2 architecture, tokenizer, and training setup used for all models in the experiments.","marker":"[Radford et al., 2019]"},{"why":"Contributes the neuroscience corpus and the BrainBench benchmark used for training and downstream evaluation.","marker":"[Luo et al., 2024b]"},{"why":"An earlier study reporting forward-order advantage whose missing-BOS design is cited as a methodological deviation.","marker":"[Kallini et al., 2024]"},{"why":"Prior forward/backward work whose tokenizer-retraining and logical-reversal designs the paper flags, and a source for the BOS-token protocol.","marker":"[Papadopoulos et al., 2024]"},{"why":"Introduces attention sinks, which the paper's attention-rank analysis extends and contrasts with its permuted-order findings.","marker":"[Xiao et al., 2024]"},{"why":"Documents the lost-in-the-middle effect that the observed local and long-range attention biases are used to explain.","marker":"[Liu et al., 2023]"},{"why":"Provides The Pile corpus used to confirm that positional attention biases generalize to pretrained models.","marker":"[Gao et al., 2020]"},{"why":"Supplies the transformer and self-attention architecture whose positional biases are analyzed as the source of empirical deviations.","marker":"[Vaswani et al., 2023]"}],"fun_headline_variants":["LLMs defy perplexity invariance proved for token orders","Theory says order doesn't matter, but GPT-2 disagrees","Perplexity invariance proven, then shattered by trained LLMs","Permuted orders break LLM probability consistency despite proof","Attention bias explains why token order still matters in LLMs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The empirical comparison assumes that each trained model approximates the true conditional distribution of its own factorization well enough that perplexity gaps between separately trained models can be read as deviations from the theorem rather than as ordinary approximation error.","fun_headline_variants_meta":{"raw":{"variants":["LLMs defy perplexity invariance proved for token orders","Theory says order doesn't matter, but GPT-2 disagrees","Perplexity invariance proven, then shattered by trained LLMs","Permuted orders break LLM probability consistency despite proof","Attention bias explains why token order still matters in LLMs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000185,"raw_usage":{"total_tokens":1349,"prompt_tokens":1002,"completion_tokens":347,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":618,"completion_tokens_details":{"reasoning_tokens":265}},"tokens_in":618,"tokens_out":347,"duration_ms":3266,"temperature":1.0,"reasoning_tokens":265,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T21:47:57.686392+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Train sibling models with the same protocol but with position information removed or made relative only, and check whether the forward/backward perplexity gaps vanish; if they persist, the cause is not simply the positional bias isolated here. Alternatively, evaluate a single fixed-permutation model on all factorizations of the same sequences via importance sampling, since a perfectly consistent model must yield a reweighted joint-probability estimate equal to its forward estimate to numerical precision.","supporting_citations":[],"review_version":1}