{"id":"acbdb576-921f-472d-af04-bb7286e7c3d2","arxiv_id":"2608.11909","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Periodic RoPE schedules yield exactly LTL with past operators and modular predicates, while conventional non-periodic RoPE provides only a bounded, precision-limited fixed-offset look-back mechanism.","lead":"RoPE transformers with periodically repeating rotations recognize exactly the languages definable in past temporal logic with modular predicates, while standard non-repeating RoPE only simulates fixed-offset look-back up to a precision-dependent length. The paper separates these two regimes formally and shows controlled experiments where periodic schedules length-generalize on modular languages but conventional RoPE does not.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The claimed RoPEP/RoPENP separation is not fully proven: Corollary 4.1 is only a positive bounded simulation, and the empirical support is best-run maxima without code or variance.","rationale":"The paper's main theorem is a characterization in an abstract model with formula-dependent finite precision; the proof sketches are plausible and the limitations are disclosed. I do not find an internal inconsistency in Lemma 3.2 or the upper-bound argument. The strongest real soft spot is that the dichotomy the paper draws in the abstract—periodic RoPE equals modular predicates, conventional RoPE is only a bounded offset bias—is not fully supported: Corollary 4.1 is a positive bounded-simulation result and no matching upper bound on SMAT[RoPENP] is given. The paper's own experiments are the only evidence that conventional RoPE cannot recognize modular languages, and those are reported as best-run maxima (Table 2) with a scheduler 'explicitly written out in the code' but no code released. This makes the practical separation conditional on audit and on an additional analytic impossibility argument. The reader's conditional verdict is appropriate; my concern supplements rather than overturns it.","tokens_in":35382,"tokens_out":29856,"duration_ms":321650,"concrete_test":"Run the RoPENP sweep on (ab)* and (aa)* with the full distribution: 10 seeds x 5 learning rates, report mean plus or minus standard deviation and all individual N* values, and release the exact periodic-scheduler code. In parallel, attempt an analytic separation: for any irrational g/(2pi), fixed finite-precision regime F, and fixed classifier, show that the realized attention outputs at even and odd readout positions both accumulate densely in the finite value set (density of the irrational rotation), so no fixed decision boundary in F can separate parity at all lengths; if such a proof succeeds, the RoPENP upper bound is established. If instead any RoPENP run length-generalizes to N*=500 on either language, the separation claim fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central Theorem 3.1 is credible within the stated abstract model; my concern is the paper's practical separation narrative. Corollary 4.1 shows that a non-periodic RoPE transformer can L-simulate Y^k and LTL[P,Y] up to a certified length N_max(g,C,F); it does not provide any upper bound on SMAT[RoPENP]. The abstract and Section 4 conclude that conventional RoPE is 'rather than an all-length modular characterization,' but no theorem rules out a RoPENP transformer with an irrational schedule plus fixed finite-precision thresholds recognizing (aa)* or (ab)* at all lengths. Because (aa)* is the canonical separating language in §3.4, this missing upper bound is load-bearing for the claimed dichotomy. The only support for the separation is empirical: Table 2 reports best-run maxima over the run grid (no variance), Fig. 2 reports means but with the RoPEP reference as a best-run line, and no code or data is provided; the 'exact periodic scheduler' is only in code. If some conventional base configuration with different seeds or learning rates achieved N*=500 on (ab)*, the practical conclusion would collapse. This does not cast doubt on Theorem 3.1's abstract equality, but it does mean the paper's headline account of conventional RoPE as a bounded locality bias is not yet secured by argument.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the expressivity of rotary position embeddings (RoPE) in the fully uniform, finite-precision soft-attention transformer model (SMAT). The central result, Theorem 3.1, states that component-periodic RoPE (RoPEP) transformers recognize exactly the languages definable in past temporal logic with modular predicates: SMAT[RoPEP] = LTL[P,MOD] = PFO2[<,MOD]. The upper bound (Lemma 3.1) simulates the finite-precision transformer using modular predicates, while the lower bound (Lemma 3.2) gives a constructive, BOS-anchored attention mechanism that computes each modular atom. Theorem 3.2 extends the equality to periodic sinusoidal positional encodings, and Theorem 3.3 provides additional algebraic and automata characterizations (left-deterministic modular polynomials, QR syntactic monoids, PODFA^k). For conventional, non-periodic RoPE, Corollary 4.1 proves a bounded-length L-simulation of the operators Y^k and LTL[P,Y] up to a certified length N_max(g,C,F). The experiments train transformers on (aa)*, (ab)*, aΣ*, Σ*a, and Σ*abΣ*: constructed periodic schedules reach perfect length generalization (N*=500) on the modular languages, conventional RoPE and SiPE do not at any tested base, and RoPENP improves over NoPE on an LTL[Y] task while underperforming on LTL[P] tasks. The paper is explicit that the periodic lower bound uses a formula-tailored finite-precision value set and that the locality result of Section 4 is bounded.","tokens_in":35646,"tokens_out":29087,"duration_ms":303789,"significance":"Within the stated abstract model, Theorem 3.1 is a clean, constructive, and non-circular characterization: the schedule is derived from the target formula, no data are fitted, and the proofs are checkable by hand. The BOS-anchor construction is a genuinely insightful mechanism and connects the expressivity picture to the empirical attention-sink literature. Theorem 3.3 usefully extends the Dartois-Paperman and Li-Cotterell toolkit to the modular past fragment, and Proposition 3.1's reduction of binary modular predicates to unary ones is elegant. The paper is also unusually candid about scope: Section C.3 acknowledges that the finite-precision set F is tailored to the formula, and the Limitations paragraph states that the RoPENP result is bounded and not an all-length characterization. The main shortfall is that the framing of the RoPEP/RoPENP distinction, advertised in the abstract and Section 5 as a qualitative dichotomy, is stronger than what the theorems establish: no upper bound on SMAT[RoPENP] is given, so whether a non-periodic schedule can recognize (aa)* or (ab)* at all lengths is open, and only the experiments address that question.","major_comments":[{"comment":"The paper asserts a dichotomy between RoPEP and RoPENP (abstract: 'Conventional RoPE is different ... rather than an all-length modular characterization'; §5: 'The conventional RoPENP behaves qualitatively differently') that the theorems do not establish. Corollary 4.1 is a positive bounded-simulation result only; there is no upper bound on SMAT[RoPENP], and the paper's own Limitations paragraph concedes that 'The locality result of §4 is bounded and is not an all-length characterization of RoPENP.' In particular, nothing rules out a fixed-precision RoPENP transformer with an irrational schedule and unrotated subspaces recognizing (aa)* or (ab)* at all lengths; because the realized rotation sequence takes values in a finite set, it is eventually periodic, and the paper does not analyze whether eventual periodicity of the readout forbids the modular languages. Since (aa)* and (ab)* are the paper's canonical separation examples (§3.4), this gap is load-bearing for the narrative. I ask the authors to state explicitly in the abstract and Section 5 that the RoPENP upper bound is open, to either prove an eventual-periodicity upper bound or phrase the dichotomy as a conjecture with a clear falsification criterion, and to present the experiments as the only current evidence for the separation. The stress-test concern about this point is valid; it is partially mitigated by the stated Limitations, but the abstract does not carry the caveat.","section":"Abstract; §4.1–4.3; §5 (Limitations)"},{"comment":"The headline empirical claim (RoPEP accuracy 1.00 and N*=500 on both (aa)* and (ab)*) is not reproducible from the manuscript. Section E.3 specifies only that 'The component-periodic scheduler is explicitly written out in the code implementation,' but no code, no data, and no textual description of the schedule (periods m_d, angles, which dimensions are rotated or unrotated) are provided. Since the periodic schedule is the independent variable that separates RoPEP from RoPENP in the experiments, the authors should either include the schedule in the appendix or release the code and data; without this, the experimental separation cannot be verified or compared against.","section":"§E.3; §3.5"},{"comment":"The proof of Proposition 4.1 is incomplete as written. The cited result (Walters 1982, Thm 1.8) concerns density of the ideal irrational rotation i ↦ Rot_{g,i}, but the proposition concerns the realized map i ↦ round_F(Rot_{g,i}), and the inference from density to non-periodicity of the rounded map is not immediate and is not argued. A correct proof must show that for any m ≥ 1 with gm mod 2π ≠ 0, the set of points at which the rounding outcome changes under the nonzero rotation has nonempty interior (it is an open set of positive measure, since the rounding cells are finitely many arcs), and that the dense orbit necessarily intersects it. The claim is true, but the step needs to be spelled out rather than left to the citation.","section":"Prop. 4.1"},{"comment":"The formula in the proof of Proposition 3.2 is off by one. For the word w_1⋯w_m, the conjunct MOD^j_m(i) ⇒ π_{w_j}(i) over 0 ≤ j < m references the undefined symbol w_0, and because LTL formulas are evaluated at the readout position N+1 (Eq. 11), the length condition should be MOD^1_m at the readout (equivalent to N ≡ 0 mod m), not MOD^0_m. The claim itself is true; the displayed formula should be corrected, for example by using MOD^j_m(i) ⇒ π_{w_j}(i) for 1 ≤ j < m, MOD^0_m(i) ⇒ π_{w_m}(i), and the readout condition MOD^1_m.","section":"Prop. 3.2 (proof)"}],"minor_comments":[{"comment":"Please clarify what the 'Max accuracy' and N* columns are computed over. As defined in §3.5 (N* is the largest N with 100% accuracy at every tested length through N), the row '(ab)* NoPE: Acc 0.96, N* 96' would require perfect accuracy at every length 41–96 while the maximum accuracy over test lengths is 0.96; if 'maximum accuracy' instead means the best mean accuracy over the run grid, or if N* includes training lengths, please state this explicitly in the caption and the text.","section":"Table 2; §3.5"},{"comment":"Figures 2 and 3 mix best-run reference lines with mean ± 1 std markers. This is acceptable, but since Table 4 reports best-run RoPENP N* values of at most 98 on (ab)* and 65 on (aa)*, the text in §3.5 could cite those maxima directly to support the claim that 'no conventional base generalizes perfectly,' making the argument less dependent on the asymmetric best-run versus mean presentation.","section":"Fig. 2; Table 4"},{"comment":"The abstract's closing phrase 'bringing theoretical expressivity characterizations closer to models used in practice' sits in tension with the Section C.3 caveat that Theorem 3.1 holds for a formula-tailored finite-precision set F and 'does not assert that standard floating-point hardware represents 2π/m ... exactly.' A sentence acknowledging the abstract-model caveat in the abstract or introduction would align the framing with the proof.","section":"Abstract; §C.3"},{"comment":"The realized-score notation bS^{g,C,k,F}_{i,j} appears in Eq. (10) before the hat convention for realized quantities is introduced in Appendix D; moving one sentence of explanation into the main text would help the reader.","section":"§4.2; §D"},{"comment":"Since the realized rotation map takes values in the finite set F^{D×D}, it is necessarily eventually periodic; a remark noting that 'non-periodic' (no exact period m for all i) is compatible with eventual periodicity would prevent a natural confusion when reading Section 4.1.","section":"Def. 2.1; §4.1"},{"comment":"The running title on the first page reads 'Disentangling the Expressivity ofRoPE' with a missing space before 'RoPE'; this should be fixed in the camera-ready version.","section":"Title page"}],"recommendation":"major_revision","confidential_remarks":"The reader's conditional verdict matches my own assessment: Theorem 3.1 is credible and publishable, but the manuscript's framing of the RoPENP separation needs to be brought in line with what is proved, and the experimental reproducibility gap (scheduler described only as being 'in the code,' with no release) needs to be closed. On substance, I would encourage the authors to attempt an eventual-periodicity upper bound for SMAT[RoPENP]; the finite-image argument suggests a plausible route, and even a upper bound for a restricted subclass (e.g., depth-1 or single-head transformers) would materially strengthen the paper. On the experiments, the best-run versus mean asymmetry between the RoPEP reference and RoPENP curves in Fig. 2 is not reversed by the underlying numbers in Table 4, but the asymmetry should be made explicit in the revision. The citation to Walters for Prop. 4.1 is also worth checking: the density theorem is the right tool, but the rounding step is not a standard corollary and needs to be written out."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Useful paper, with one honest caveat. The formal result — SMAT[RoPEP] = LTL[P,MOD] — is real, and the proof strategy is sound. What the paper calls the practical account of conventional RoPE is a bounded simulation result, not a separation theorem; the claim that non-periodic RoPE is only a locality bias is supported by experiments and not by a lower bound. The authors are more careful in the limitations than in the abstract, but the framing overshoots a bit.\n\nWhat is genuinely new and good: the component-periodic definition via realized maps and lookup tables; Prop 2.1; the upper bound via PFO2[<,MOD]; the lower bound with the BOS anchor and modular predicates; the algebraic characterization of LTL[P,MOD] connecting to the existing literature; and the SiPEP analogue. The proofs are constructive and mostly checkable, and the finite-precision caveats are stated explicitly. The observation that BOS anchors expose positional phase and thereby make unary modular predicates computable is a nice mechanistic touch, and it connects well to the attention-sink literature.\n\nSoft spots, in proportion. First, there is no upper bound on SMAT[RoPENP]. Cor 4.1 gives an L-simulation of fixed-offset operators, not an impossibility result. So the \"disentangling\" is partial: the periodic side is exact, the non-periodic side is a bounded simulation plus an empirical story. The paper does not formally claim RoPENP cannot recognize (aa)*, and the limitations even say the locality result is bounded. Still, the abstract and Section 4 lean on a dichotomy that is not proven. I would like an explicit sentence stating that a full characterization of RoPENP remains open. Second, the experimental component is not fully auditable: no code or data is released, Table 2 reports best-run maxima without variance, and Fig. 2 uses a best-run reference line for RoPEP. The tailored finite-precision set F is a genuine scope caveat for practical transfer, but the paper says so. These are fixable with artifacts and full run distributions.\n\nVerdict: this deserves a serious referee. The central theorem is new, correct within its stated abstract model, and builds on prior work honestly. The empirical section needs revision rather than rejection. I would cite the RoPEP characterization and bring the paper to a reading group; I would also encourage the authors to state the RoPENP open problem explicitly.","headline":"Theorem 3.1 is a real, exact characterization of periodic RoPE; the non-periodic side is a bounded simulation plus experiments, not the formal separation the framing suggests.","tokens_in":36239,"tokens_out":2389,"would_cite":true,"duration_ms":24618,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["68Q45","68Q19","03B44","68T07"],"pacs":[],"model":"deepseek-v4-flash","headline":"Component-periodic RoPE transformers recognize exactly LTL[P,MOD], the past temporal logic with modular predicates; conventional non-repeating RoPE gives only a bounded, precision-limited look-back.","keywords":["rotary position embeddings","RoPE expressivity","linear temporal logic","modular predicates","transformer expressivity","finite-precision arithmetic","length generalization","positional encodings"],"falsifier":"Evaluate, in actual floating-point arithmetic, the realized score gap of an irrational-frequency offset head at increasing lengths: the paper predicts a finite certified length $N_{\\max}$ beyond which the target position $i-k$ no longer beats every competing offset by the margin $f_{\\mathrm{large}}$, whereas a periodic lookup-table schedule keeps the gap constant. A concrete search: train or hand-construct a conventional-RoPE transformer on $(aa)^*$ and scan base values beyond the tested range $10^{-12}$ through $10^{12}$ for perfect length generalization through 500; the paper's separation claim predicts that none will be found and that only the modular construction reaches it.","tokens_in":35158,"feed_emoji":"🔄","tokens_out":18118,"duration_ms":164284,"temperature":0.7,"pith_summary":"The paper sets out to settle what rotary position embeddings (RoPE) actually give a transformer, and it does so by separating two regimes that are usually conflated. When every rotary component repeats after a fixed number of positions (a component-periodic schedule, realized by a lookup table indexed by position modulo the period), a fully uniform finite-precision soft-attention transformer recognizes exactly the languages definable in past temporal logic with modular predicates, $\\mathrm{LTL}[P,\\mathrm{MOD}]$: for inputs of any length it can decide properties such as 'the string has even length' or 'the input is a repetition of $ab$'. Conventional RoPE, whose rotation frequencies are irrational and never repeat, cannot achieve this all-length modular behavior; its relative scoring yields only a bounded, precision-dependent look-back by fixed offsets. That distinction matters because it predicts when RoPE helps (fetching nearby context) and when it subtly harms models that must aggregate information from distant, position-independent sources, and the paper's controlled experiments on small formal languages confirm both sides of the prediction.","feed_headline":"Periodic RoPE matches modular temporal logic exactly","feed_subtitle":"Plain non-repeating RoPE offers only a precision-bounded local look-back, and experiments confirm the split.","key_machinery":"Three mechanisms carry the argument. The component-periodic schedule: each two-dimensional rotary block takes finitely many realized values and cycles with period $m_d$, implemented not by forming the unbounded product $\\theta_d \\cdot i$ but by a lookup table indexed by $i \\bmod m_d$, so the realized map repeats by construction and every matrix entry becomes a finite disjunction of modular predicates. The BOS-anchor head: a rotary pair compares the query's rotated phase at position $i$ against the fixed identity phase of the beginning-of-string key, while an unrotated pair supplies a constant non-BOS baseline chosen so that the winner leads by at least the finite-precision margin $f_{\\mathrm{large}}$; stable softmax turns that comparison into the exact truth value of $\\mathrm{MOD}^r_m(i)$, and the anchor is what makes a pairwise relative score computable from unary position information — the same behavior observed as attention sinks. The certified range $N_{\\max}^{(k)}(g,C,F)$: for an irrational angle $g$, a $k$-offset head scores position $i-k$ by $C\\cos(g(i-k-j))$, and $N_{\\max}$ is the largest length up to which that target outscores every other key by at least $f_{\\mathrm{large}}$; beyond it the offset head can fail, which is why the $Y^k$ simulation is bounded. Unrotated subspaces carry the NoPE-style $\\mathrm{LTL}[P]$ computation, and the pieces compose by structural induction.","core_discovery":"The paper's central claim is a paired characterization. On the periodic side, Theorem 3.1 states that $\\mathrm{SMAT}[\\mathrm{RoPE}_P] = \\mathrm{LTL}[P,\\mathrm{MOD}] = \\mathrm{PFO}_2[<,\\mathrm{MOD}]$: component-periodic RoPE transformers know exactly the languages of past temporal logic with modular predicates, equivalently the past fragment of two-variable first-order logic with unary modular predicates. Theorem 3.3 adds algebraic and automata-theoretic faces of the same class — left-deterministic modular polynomials, syntactic monoids in $\\mathcal{QR}$, and automata whose $k$-blowup is partially ordered — and Theorem 3.2 shows periodic absolute sinusoidal encodings satisfy the same equivalence. On the non-periodic side, Proposition 4.1 records that irrational rotations never repeat, and Corollary 4.1 shows that conventional RoPE can only $L$-simulate the fixed-offset operators $Y^k$ up to a certified length $N_{\\max}(g,C,F)$; within that range it simulates $\\mathrm{LTL}[P,Y]$ formulas, but beyond it, finite-precision rounding lets competing offsets tie or beat the intended target. The experiments match this split: constructed periodic schedules reach perfect accuracy through length 500 on $(ab)^*$ and $(aa)^*$, while no tested conventional base value — seven bases spanning twenty-four orders of magnitude — generalizes on either language.","pith_inferences":["Editorial inference: the BOS-anchor construction suggests a diagnostic for attention-sink behavior — if a RoPE model's sink token partly computes modular-position predicates, ablating the sink should degrade periodic regularities such as alternation more than it degrades local copying; the paper does not run this experiment.","Editorial inference: the certified-range formula gives a quantitative precision prediction — moving from a higher- to a lower-precision regime should measurably shorten the length at which fixed-offset heads fail, a test that standard perplexity metrics would not surface.","Editorial inference: the two characterizations imply a hybrid-schedule design rule — reserve some rotary dimensions for periodic modular counting, some for unrotated global aggregation, and optionally some for non-periodic local offsets — which the paper only conjectures for partial-RoPE variants.","Editorial inference: on this account, long-context failures of RoPE models are not merely missing capability; the rotation geometry actively biases against position-invariant access to distant content, which may explain reported gains from dropping positional embeddings in long-context setups."],"forward_implications":["Practical periodic schedules — the cyclic-lookup and rounded-wavelength RoPE variants used in deployed models — inherit the exact characterization, so they can recognize modular-position regularities such as $(ab)^*$ and $(aa)^*$ uniformly over all input lengths.","Changing the base of conventional RoPE cannot buy modular generalization: with irrational rotation angles the score gap that isolates an offset is finite-precision-limited, so the look-back mechanism is certified only up to a length $N_{\\max}$ that depends on the angle, the scale, and the precision set.","Conventional RoPE is a double-edged local bias: at six of seven tested bases it improves on NoPE for the look-back language $\\Sigma^*a$, but at every base it lowers the mean longest perfect length on the position-invariant languages $a\\Sigma^*$ and $\\Sigma^*ab\\Sigma^*$.","The decisive factor is periodicity of the realized position-to-embedding map rather than the relative-versus-absolute distinction, because periodic absolute sinusoidal encodings satisfy the same equivalence as periodic RoPE.","The class $\\mathrm{LTL}[P,\\mathrm{MOD}]$ stays incomparable with full past-temporal logic: it cannot recognize PARITY, and over alphabets with more than two symbols it cannot recognize the locally testable language $\\Sigma^*ab\\Sigma^*$, which needs the adjacent-position operator $Y$."],"supporting_citations":[{"why":"Supplies the fully uniform finite-precision soft-attention model and the baseline characterization LTL[P] = PFO2[<] that both inclusions of Theorem 3.1 build on.","marker":"(Li & Cotterell, 2025)"},{"why":"Establishes the link between periodic sinusoidal position encodings and modular predicates that the paper formalizes for RoPE, and provides the modular-predicate simulation reused in Theorem 3.2.","marker":"(Yang et al., 2024)"},{"why":"Provides the enriched-alphabet technique and the QR/PODFA^k framework used to prove Theorem 3.3 and the inexpressibility results.","marker":"(Dartois & Paperman, 2013)"},{"why":"The original RoPE construction: the block-diagonal rotation, the base-parameterized angular schedule, and the relative-position identity used in Eq. (5) and in the fixed-offset scoring.","marker":"(Su et al., 2023)"},{"why":"The mechanistic fixed-offset-heads account of RoPE that Section 4 formalizes as bounded Y^k simulation, including the cos(g(i-k-j)) scoring.","marker":"(Barbero et al., 2025a)"},{"why":"Theorem 1.8, cited as the source of Proposition 4.1, gives the ergodic fact that irrational rotations i -> Rot_{g,i} never repeat.","marker":"(Walters, 1982)"},{"why":"Connects R-trivial monoids with partially ordered automata, which Theorem 3.3 and the PARITY and locally-testable inexpressibility proofs rely on.","marker":"(Brzozowski & Ellen, 1980)"},{"why":"Attention sinks, which the paper connects to its BOS-anchor construction for computing modular predicates.","marker":"(Xiao et al., 2024)"}],"fun_headline_variants":["Periodic RoPE: exact match to past modulo logic","Non-repeating RoPE: just a bounded look-back","RoPE expressivity splits: periodic vs precision-bound","Periodic RoPE captures modular-temporal languages"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the finite-precision value set can be tailored to the task, representing the rotation-table entries, the margin constants, and their arithmetic exactly; if real hardware cannot represent values such as $2\\pi/m$ exactly, the exact equivalence holds only inside the abstract model and the practical case rests on the experiments.","fun_headline_variants_meta":{"raw":{"variants":["Periodic RoPE: exact match to past modulo logic","Non-repeating RoPE: just a bounded look-back","RoPE expressivity splits: periodic vs precision-bound","Periodic RoPE captures modular-temporal languages"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000207,"raw_usage":{"total_tokens":1429,"prompt_tokens":1004,"completion_tokens":425,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":620,"completion_tokens_details":{"reasoning_tokens":362}},"tokens_in":620,"tokens_out":425,"duration_ms":4994,"temperature":1.0,"reasoning_tokens":362,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T00:23:57.450142+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate, in actual floating-point arithmetic, the realized score gap of an irrational-frequency offset head at increasing lengths: the paper predicts a finite certified length $N_{\\max}$ beyond which the target position $i-k$ no longer beats every competing offset by the margin $f_{\\mathrm{large}}$, whereas a periodic lookup-table schedule keeps the gap constant. A concrete search: train or hand-construct a conventional-RoPE transformer on $(aa)^*$ and scan base values beyond the tested range $10^{-12}$ through $10^{12}$ for perfect length generalization through 500; the paper's separation claim predicts that none will be found and that only the modular construction reaches it.","supporting_citations":[{"cited_title":"doi:10.4230/LIPIcs.STACS.2013.329 , editor =","cited_arxiv_id":null,"evidence_quote":"Provides the enriched-alphabet technique and the QR/PODFA^k framework used to prove Theorem 3.3 and the inexpressibility results."}],"review_version":1}