{"id":64,"job_id":19,"problem_id":1,"lane_id":1,"type":"explore","user_id":13,"model":"claude-fable-5-1","provider":"anthropic","report_md":"# Job #19 (explore, g2-exponent): three inputs that are not one-point divisor counts, priced against β₂ = 4.26645\n\n## Disposition\n\nNo input priced here moves the exponent. Three candidates were taken to the point where the arithmetic or the first unmatched hypothesis stops them:\n\n1. **Moment inputs from the exact pair correlation** (the window-count variance and higher central moments over the rotation ensemble) price to exponent effect **0** for the sup-over-windows problem: the k-th moment controls the number of empty windows of length L among the e^{θ(x)} positions only once E^{k/2} ≫ e^{θ(x)}, so a polynomial L = x^β needs a moment of order k ≥ 2θ(x)/(β ln x) ≍ x/ln x, and inside the Λ²Λ⁻ sieve a moment over positions does not enter at all, since the weights are linear in the one window's |A_d|. This re-prices, with exact arithmetic, closures the record already holds (concentration family, OUTCOMES.md L2804; Lemma V mean-square, L2769; almost-all exponent 0, sift-limit-attack.md §7e). Rung for the pricing: proven arithmetic on stated hypotheses.\n2. **A bilinear (Type II) input for arbitrary windows** requires Möbius-type cancellation in an interval of length L = x^β = (ln W)^β placed anywhere among the integers up to W = e^{θ(x)}: a polylogarithmic short interval at every position. The first unmatched hypothesis is the quantifier \"every position\": short-interval results for multiplicative functions (Matomäki–Radziwiłł; Mangerel for divisor-bounded f) hold for almost all positions at length h → ∞, none for a chosen position at length (ln N)^β. Exponent effect: unpriced, because the needed theorem does not exist at any exponent; what it would give if it did is stated in §3. Rung: analysis.\n3. **The Vatwani lead** (\"Divisor-bounded multiplicative functions in arithmetic progressions\", Roy, Savalia, Vatwani, listed as submitted on the author's publication page, 2026-09-11; no preprint, arXiv id or manuscript located) cannot be read at source and stays UNREAD. What is visible from its class alone: a theorem about divisor-bounded multiplicative f in progressions concerns a different class from the sequence Λ(n)μ(n+2) that Murty–Vatwani's Theorem 1.1 needs (hypothesis EH_{μ₂}(x^{1/2+ε})), which is not multiplicative; that class mismatch is the first unmatched hypothesis and the corpus flags it (`research/RESEARCH-EXECUTION.md` §4). Exponent effect via the p² rule: a count-side input prices to 0 (the weak Zone Postulate and also the strong, anchored one, since each constrains one gap per level and G₂ is the maximum over all e^{θ(x)} positions) or to β ≤ 2 outright (the all-positions form G₂(x#) < x′² − 2); nothing in between (§3, Lemma P). Rung: source unreachable; the class statement is proven.\n\nClosed routes checked by name (OUTCOMES.md Closed routes, L2730–2826): ρ maximal law / REC(s, u₀) (L2826, truth gap at rung derived): not used; DI/Pascadi Y_N axis (L2815): not used; improving β₂ itself (L2776): not attempted; covering economy (L2778): not used; local-lemma family (L2790): not used; bounded-differences / concentration (L2804): item 1 re-prices it and agrees with the closure; Lemma V mean-square (L2769) and the repulsive/attractive point-process row (L2819): item 1 touches the same object and agrees; Murty–Vatwani divisor swap (L2733): item 3 does not use the swap. TODO item 0 is not reopened: nothing here is a defect in the recorded remainder obstruction or an estimate with changed inputs outside its scope.\n\n## 1. Moment inputs from the exact pair correlation: exponent effect 0\n\nNotation: W = x# = e^{θ(x)}; T_x the twin slots mod W; δ = (1/2)∏_{2 W·Var. Writing Var = cE this is δL > cW, i.e. L > c e^{θ(x)} ln²x/(0.41621 + o(1)). A statement about a window needs L ≤ W, so Chebyshev is non-vacuous only if c < δ, i.e. only if the Fano factor decays like 1/ln²x; measured, c rises 0.152 → 0.396. The overshoot c/δ = c ln²x/0.41621 is 5.4 to 14.3 at x = 41 and diverges like ln²x. The record's table says the same in its own currency: at @19, Var/E² = 8.38·10⁻⁶ against 1/W = 1.031·10⁻⁷ (factor 81); at @37, 4.22·10⁻¹¹ against 1.348·10⁻¹³ (factor 313). Exponent effect: 0, and not a weak β: no finite β, since the requirement exceeds the period.\n\n**(b) k-th central moment.** Hypothesis M_k = E_s|N − E|^k ≤ C_k E^{k/2}. An empty window is a deviation of exactly E, so #{s : N(s) = 0} ≤ W C_k E^{−k/2}, which is < 1 iff E > (W C_k)^{2/k}. With E = δL and L = x^β,\n\n> β ln x > (2/k)(θ(x) + ln C_k) + 2 ln ln x − ln 0.41621 + o(1),\n\nso a polynomial window needs k > 2(θ(x) + ln C_k)/(β ln x − 2 ln ln x + O(1)) ~ (2/β)·π(x). Numerically the k needed at β = 2 is 4.0, 5.5, 8.2, 9.0, 16.1 at x = 13, 19, 37, 41, 79; 139 at x = 1009; 8,659 at x = 10⁵; at β = 4.26645 it is 1.9, 2.6, 3.9, 4.2, 7.5, 65, 4,059. This reproduces `research/sift-limit-attack.md` §7d item 3(b) independently (k* ≥ (1 + o(1)) x/(B ln x) under G₂ = O(x^B); measured k* = 2, 4, 5, 8, 10, 12, 14, 16, slope 1.066 ± 0.099 over eight levels). For every fixed k the exponent effect is 0. The only object in the family that prices to anything is the uniform-in-k limit, a sub-Gaussian maximal law, which `research/phase1-T4-maximal-law.md` records as TPC-implying; `paper/anchored-note.md` Proposition 1(i) is the general statement: a bound with exceptional fraction ε admits εW members and decides a designated position only when εW < 1.\n\n**(c) Inside the sieve.** A second moment over s does not enter the Λ²Λ⁻ lower-bound sieve for one window, by definition. Theorem 9.1 of Diamond–Halberstam has hypotheses and conclusion that are functions of the one-window data D(A) = (X, ω, (r_d)_{d | P(z), d < y}) alone, and β_κ = inf{u : f_κ(u) > 0} is an infimum over the whole class satisfying Ω(κ, L) at level y = z^u. Two sequences with the same D(A) receive the same conclusion, so no theorem in the class beats X V(z) f_κ(u) while the class contains a near-extremal member with the same data; at κ = 1 it demonstrably does (Selberg's A± = {n ≤ x : λ(n) = ±1}, giving β₁ = 2), and at κ = 2 no extremal example is in print (Halberstam, Bull. AMS 40 (2003) p. 117; `research/sift-limit-attack.md` DP5), so 4.26645 is an achieved price, not a proven floor. The certificate at fixed s is T(s) = Σ_d λ_d |A_d^{(s)}|, linear in one window's counts with λ independent of s; Var is a functional of the family {A^{(s)}}, constrains the distribution of T over s, and the sieve needs min_s T(s) > 0, which no distribution bound admitting exceptions yields. This is discard point DP1 (`sift-limit-attack.md` §1: \"positions of elements relative to each other never enter\"), priced there at +2.32 of exponent on a floor of 3.3152, 71% of the loss (MEASURED, LP to x = 43).\n\nExponent effect: **0**. First unmatched hypothesis: for the moment route, ε < e^{−θ(x)}; for the sieve route, a lower-bound sieve theorem whose hypotheses include two-point data, which does not exist. Rung: proven (the counting and the indistinguishability argument); measured (the 71% share). Agrees with the closed rows OUTCOMES.md L2769 (Lemma V mean-square), L2790 (local lemma), L2804 (bounded differences), L2819 (point-process models).\n\n## 2. A bilinear (Type II) input for arbitrary windows: unpriced at exponent 2, priced through the one κ = 2 consumer at 2.579 at best\n\n**(a) What is needed.** With a_m, b_n arbitrary bounded and no Möbius there is nothing to prove (a = b = 1 gives Σ_{v ∈ (s, s+L]} τ(v) > 0), so the estimate must carry μ in one variable: in the Friedlander–Iwaniec shape, Σ_{m ≤ M} |Σ_n μ(n) 1_{mn ∈ (s, s+L]}| ≪ L (ln W)^{−A} uniformly in s ∈ [0, W) [MEMORY for the normalisation]. Elements have size ≈ W = e^{θ(x)} and the window is L = x^β = (ln W)^β (1 + o(1)). Since #{n : mn ∈ (s, s+L]} ≤ L/m + 1, for m > L the inner sum has at most two terms and the form is not bilinear; the only range is M ≤ L = (ln W)^β, N ≥ W/L, and the term count is Σ_{v ∈ (s, s+L]} τ(v) ≍ L ln W. The estimate therefore is Möbius cancellation over intervals of length L/m ≤ (ln W)^β at height about W/m, at every position: polylogarithmic short intervals relative to their position, for every position.\n\n**(b) State of the art** [MEMORY]. Matomäki–Radziwiłł (Ann. of Math. 183 (2016)): for multiplicative |f| ≤ 1 the average over [t, t + h] matches the long average for almost all t ∈ [X, 2X] for any h → ∞; for μ, Σ_{t < n ≤ t+h} μ(n) = o(h) for almost all t. For every position, nothing unconditional below h = X^{7/12+ε}, and nothing below X^{1/2+ε} on RH. Matomäki–Radziwiłł already covers the length, since (ln W)^β → ∞, and fails only on the quantifier: its exceptional set has measure o(W) while the requirement is ε < 1/W = e^{−θ(x)}. First unmatched hypothesis: \"every position\" at length (ln N)^β, the same inequality as §1. The record prices position-uniformity on the source side too: Bettin–Chandee (2015) Remark 1 admits the window phase at cost (1 + hx/MN)^{1/2}, O(1) only for x ≪ H^{1.212157}, while x reaches exp(H^{0.2344}) (`research/history/staging/attack-0829n-rml-proof.md` §4.5).\n\n**(c) Price.** A Friedlander–Iwaniec-type asymptotic sieve at κ = 2 cannot be priced: Bombieri's asymptotic sieve and Friedlander–Iwaniec (Ann. of Math. 148 (1998)) are dimension-one instruments, and no asymptotic sieve admitting Ω(2) exists in print. The one κ = 2 mechanism on record that consumes bilinear remainder input is the Brüdern–Fouvry vector sieve, threshold u > K/θ_total with K = 5.158064680330 at the Brüdern–Fouvry optimal upper/lower split (2(1 + √e) = 5.297442541400 symmetric) and θ_total = log_H(D₁D₂). Absolute-value accounting caps θ_total at 1, and the cap is sharp (Σ|r| reaches 0.978 of the trivial pair-count bound at a CRT-constructed doubly-smooth position; `sift-limit-attack.md`). Identifying a signed saving L^{−η} with θ_total = 1 + η [INFERRED]:\n\n> β(η) = 5.158065/(1 + η): η = 0 → 5.158; η = 0.2090 → 4.26645 (break-even); η = 0.25 → 4.126; η = 0.5 → 3.439; η = 1 → 2.579.\n\nη ≤ 1 is forced by D₁D₂ ≤ H², so the floor is 2.579 and this route never reaches 2. With published inputs it is negative: the needed γ ≤ 0.824975 against Deshouillers–Fouvry–Iwaniec (1997) 1.009638 and Bettin–Chandee 0.970624, i.e. θ_total = 1.030303 against the needed 1.208983, returning 5.090707 > β₂ (`attack-0829n-rml-proof.md` §4.5; `research/G2-STATE.md` parked paragraph).\n\nExponent effect: **5.158065/(1 + η) − 4.26645**; best conceivable −1.688 (to 2.579); with published estimates +0.824. First unmatched hypothesis: position-uniform signed cancellation, the sup form of Lemma V, whose mean-square form is proven (⟨R²⟩_H ≤ B(z, s) H, B ∈ [1.27, 1.68]) with almost-all exponent 0 and whose u_sup route is closed (`sift-limit-attack.md` §7e; OUTCOMES L2769, L2826). Rung: proven for the reductions in (a) and for the mean-square form; inferred for the θ_total identification; closed for the sup route as the record grades it.\n\n## 3. The p² rule: what any count-side input can do to the exponent\n\nLet β* = inf{β : ∀ε > 0 ∃C ∀x, G₂(x#) ≤ C x^{β+ε}}. The zone of p is (p, p′²); every hole of T_p in it is prime (`research/ZONE-POSTULATE.md` §1).\n\n**Lemma P.** (i) H₁ = \"for infinitely many p the zone of p contains a twin slot\" is equivalent to TPC (proven both ways, elementary; ZONE-POSTULATE §2). H₁ constrains, for infinitely many x, one of the ≍ δW gaps of T_x, the one at the origin, while G₂ is their maximum; H₁ is consistent with every value of β*. Δβ* = 0. (ii) H₂ = \"for every p the first twin slot of T_p lies below p′² − 2\" (the strong Zone Postulate; verified to 10¹¹, 4,118,054,813 primes, zero failures, per the record) implies H₁ and still constrains one gap per level; the Gap Reformulation runs G₂ < x′² − 2 ⟹ H₂ with no converse. Δβ* = 0. (iii) H₃ = \"for all large x and all s, (s, s + x′² − 2] meets T_x\", i.e. G₂(x#) < x′² − 2, is β* ≤ 2 (x′ ~ x) and implies H₂. All the way to 2. (iv) Dichotomy: a conclusion C of the form \"|{s : (s, s+L] ∩ T_x = ∅}| ≤ εW\" bounds G₂ iff the set is empty, iff ε < e^{−θ(x)}. Count-side inputs come in two kinds: averaged over modulus or position (Bombieri–Vinogradov, Elliott–Halberstam, Murty–Vatwani's twisted EH_{μ₂}, Hardy–Littlewood in progressions), for which ε ≥ (ln W)^{−A} ≫ e^{−θ(x)}; and primality-based, which by the p² rule certify slots only at s < x′² (above x′² a twin slot need not be a twin prime), an exceptional set of size W − O(x²). Either way Δβ* = 0; the only count-side route to a nonzero effect is a count uniform over all e^{θ(x)} positions at length x^{2+o(1)}, which is H₃ itself. The crossing is exactly ε = e^{−θ(x)}.\n\nExponent effect: **0** for H₁ and H₂; **all the way to 2** for H₃; nothing between. First unmatched hypothesis: ε < e^{−θ(x)}; every published count-side input has ε ≥ (ln W)^{−A}. Rung: (i), (iii), (iv) proven; (ii) proven plus verified to 10¹¹ as the record states; the absence of a type-H₃ input is the registry's.\n\n## Common finding\n\nAll three inputs fail at one inequality, εW < 1 with W = e^{θ(x)}: Chebyshev by the factor c ln²x/0.41621, fixed-order moments unless k ≍ (2/β)π(x), Matomäki–Radziwiłł by the quantifier, count-side inputs by the p² rule's confinement of primality to s < x^{2+o(1)}. The record's sentence that \"the entire distance from 4.2665 down to 2 is a positivity-method problem and zero percent a distribution problem\" (`sift-limit-attack.md` §5) survives all three prices. The only priced decrement anywhere here is 1.688, from a signed bilinear saving at its ceiling η = 1 through the vector sieve, landing at 2.579 and not at 2; with published inputs the same route prices above β₂.\n\n## 4. The Vatwani lead, read as far as it can be\n\n**Corpus position.** `research/RESEARCH-EXECUTION.md` §4: \"the listed submitted paper on divisor-bounded multiplicative functions in progressions is an UNREAD lead, with no theorem imported. In particular, f(n) = Λ(n−2)μ(n) is not made multiplicative by its name or divisor bound.\" The corpus's read Vatwani sources are different papers: Murty–Vatwani, *Twin primes and the parity problem*, J. Number Theory 180 (2017) 643–659 (`research/consumer-comparison.md` §1, PRIMARY from a Wayback capture), and Vatwani, Math. Z. 293 (2019) 285–317 (author preprint of 2018-10-02; `research/moving-cutoff-parity.md` §1).\n\n**Source, 2026-09-11.** The author's publication page (https://sites.google.com/view/akshaa/publications) lists \"Divisor-bounded multiplicative functions in arithmetic progressions\" with coauthors Arindam Roy and Aditi Savalia, status submitted, no venue and no arXiv id. Web search, the coauthors' pages and arXiv returned no manuscript. The theorem, its level Q, its divisor-bound parameter and its hypotheses cannot be stated from the text; a distinct published paper by two of the authors, Savalia–Vatwani, *Limitations to equidistribution in arithmetic progressions*, Res. Math. Sci. 10 (2023), is not this lead and was not read.\n\n**What the class alone says.** Murty–Vatwani's Theorem 1.1 consumes EH_{μ_h}(x^η): a Bombieri–Vinogradov-type bound for Λ(n)μ(n+h) in progressions with main term (1/φ(q)) times the full sum, at level η > 1/2 (at h = 2, the corpus's EH_{μ₂}(x^{1/2+ε}); `consumer-comparison.md` §1, which records \"no case of EH_{μ_h}(x^η) for any fixed η > 0 was located\"). The sequence Λ(n)μ(n+2) is not multiplicative and Λ is not divisor-bounded; a theorem whose hypothesis is \"f multiplicative with |f| ≤ d_k\" does not apply to it without a bridge, and no bridge is on record. That is the first unmatched hypothesis, and it is visible before the paper is read. If the lead did supply EH_{μ₂} at level above 1/2, the consequence would be infinitely many twin primes (Murty–Vatwani) and, by §4's Lemma, exponent effect 0 on G₂ unless the statement were uniform in every large zone.\n\n**Short windows.** For the sup-window problem itself, any progressions estimate for multiplicative functions says nothing about one window of length (ln N)^β at a chosen position: the short-interval literature for divisor-bounded multiplicative functions (Mangerel, arXiv:2108.11401) works in \"typical intervals of length h(log X)^c with h → ∞\", an almost-all statement; the \"every position\" quantifier is unmatched (item 2 above).\n\n## 5. Sources\n\n`research/G2-STATE.md` §0 and the parked paragraph (lines 233–258); `TODO.md` item 0 (lines 127–134); `research/SEARCH-CONVENTIONS.md` §4, §5; `research/OUTCOMES.md` Closed routes L2733, 2749, 2769, 2776, 2778, 2790, 2804, 2813, 2815, 2819, 2826; `research/sift-limit-attack.md` §1 (DP1–DP5), §5 L432–433, §7e L549–604, L306–309; `paper/beta2-note.md` §§1–3, §6 item 5; `research/history/staging/attack-0829n-rml-proof.md` §3, §4.5; `research/history/staging/attack-0830-rec-cheapest.md` §4; `research/06-variance-theorem.js`; `research/natal5-variance.js` reading 6; `paper/wall-note.md` §2 Face 1; `research/RESEARCH-EXECUTION.md` §4; `research/consumer-comparison.md` §1–§3; `research/moving-cutoff-parity.md` §1, §5; `paper/anchored-note.md` §2. Web (2026-09-11): Vatwani publication page; Savalia page; arXiv:2108.11401, 1012.3809, 1511.00601 abstracts. Diamond–Halberstam, *A Higher-Dimensional Sieve Method* (with Galway), Cambridge Tracts 177 (2008), p. 79, as quoted in `paper/beta2-note.md`; Booker–Browning, Discrete Analysis 2016:8, arXiv:1511.00601, for the twenty digits (not re-extracted here). Nothing local-only. Compute: none beyond arithmetic.\n\nTwo corrections to the brief's own framing, recorded rather than silently absorbed: `paper/beta2-note.md` has no §6.5 (the fallback exponent ≈ 19 + ε is §6 item 5), and the phrases \"bilinear input with a non-smooth modulus profile\" and \"exact pair correlation as a second-moment input to a weighted sieve\" do not occur in `research/RESEARCH-EXECUTION.md` §4; they are the brief's candidates, priced here as posed.\n\n## Transcript\n\nAttached, scrubbed as data (token and session id prefix-matched, UUID keys, absolute paths outside the working directory, environment values, emails other than the project contact and the attribution address); lines before the `GET /start` that received job #19 dropped; sub-agent transcripts started after it concatenated. No files uploaded: today's quota is exhausted.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"proven","status":"recorded","final_rung":"recorded","created_at":"2026-09-11T13:56:49.476Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[41],"messages":[]},"tokens":{"log":"claude-code","input":496,"models":{"claude-opus-5":688,"claude-sonnet-5":8129,"claude-fable-5-1":29970},"output":38787,"source":"claude-jsonl","entries":96,"cache_read":16152804,"cache_write":461668},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe\n\nNo computation beyond arithmetic. To check the pricing: (1) with delta = 0.41621/ln^2 x and Var = cE, verify that W Var/E^2 < 1 is L > cW/delta and evaluate c ln^2 x/0.41621 at x = 41 for c = 0.152, 0.396 (5.4, 14.3); verify the k-th moment inequality k > 2(theta(x)+ln C_k)/(beta ln x - 2 lnln x + O(1)) and its values at the listed x (theta from the primes <= x). (2) verify beta(eta) = 5.158065/(1+eta) at eta = 0, 0.2090, 0.25, 0.5, 1 and the 5.0907 figure against research/history/staging/attack-0829n-rml-proof.md s4.5. (3) read the closed-route rows research/OUTCOMES.md L2733, 2749, 2769, 2776, 2778, 2790, 2804, 2813, 2815, 2819, 2826 and research/RESEARCH-EXECUTION.md s4; fetch https://sites.google.com/view/akshaa/publications and confirm the submitted title and coauthors.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":106},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":null,"research_route_id":null,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":null,"run_id":null,"triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"zemaj","job_brief":"Read first: `CLAUDE.md`, `research/G2-STATE.md` section 0 and its \"Earlier exponent route (parked under TODO 0)\" paragraph, `TODO.md` item 0, `research/SEARCH-CONVENTIONS.md` sections 4 and 5, and the full \"Closed routes\" table of `research/OUTCOMES.md` (`research/REFUTED.md` is a pointer). The achieved bound is G2(x#) << x^(4.26645+eps) from the DHR dimension-2 sieve; a fixed exponent below 2 would prove twin primes; the band (2, 4.26645] is a proof gap, not a truth gap for the bound itself, though several specific routes across it are closed at rung derived-and-red-teamed (the rho maximal law / REC(s, u0) route, the DI/Pascadi Y_N axis, improving beta2 itself, the covering economy, the local-lemma and concentration families).\n\nTODO item 0 reopens only with a specific defect in the recorded remainder obstruction or an estimate using changed inputs outside its proved scope, and asks you to read the REC and CRT-planted-position arguments before making that distinction. The distance is recorded as a positivity problem, not a distribution problem: a perfect distribution oracle moves the exponent by nothing (`research/ZONE-POSTULATE.md` section 3).\n\nOpen-ended job: find one input that is not a one-point divisor-class count |A_d| (the sieve consumes only those) and that is not a rephrasing of the Zone Postulate, and price what it would do to the exponent. Candidates the repo names as unworked or unread: the Vatwani lead on divisor-bounded multiplicative functions in progressions (UNREAD, execution section 4), a bilinear input with a non-smooth modulus profile, the exact pair correlation as a second-moment input to a weighted sieve. Before computing, check the source in the owning convention. Every closed route must be checked against your idea by name.\n\nReturn a note posted to the lane thread, execution contract order, with the exact exponent arithmetic and the first unmatched hypothesis of any theorem you import. Reviewers assign the rung. A priced route that fails is a legitimate return if the failed step is named exactly; a route that \"looks promising\" without a priced inequality is not a return.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/64/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}