{"id":4526,"job_id":9287,"problem_id":1,"lane_id":null,"type":"formalize","user_id":102,"model":"gpt-6-astra","provider":"openai","report_md":"# Centered finite-HL cutoff error: a certified reduction and a remaining signed-energy obstruction\n\nJob 9287. AI-generated work by gpt-6-astra at High on behalf of the verified contributor. Documents and proof work only; no data runs, prime enumeration, numerical integration or contributed-code execution. This is a prose proof, not a Lean/kernel-checked result.\n\n## Result and exact scope\n\nFor the 19 specified cells, the contribution involving products of two or three exceptional omitted-prime corrections is less than **0.0022** in absolute value. This leaves one explicitly defined signed first-order term whose bound would decide the requested 0.01 criterion. The complete error is not certified below 0.01.\n\nThe earlier uniform relative bound can be improved from 8e-6 to **4e-6** because every relevant integer d, d-2 and d+2 is smaller than Q^2. Nevertheless, the absolute-value certificate obtained from 8e-6 is provably larger than 0.01 in all four new cells; the improved 4e-6 certificate is still provably larger than 0.02 in the two h=2^24 cells. These are lower bounds on the size of an *upper-bound certificate*, not lower bounds on the actual signed error.\n\nThe exact remaining term is given in (7) and its centered Fourier representation in (12). It retains the within/between-block cancellation and the diagonal subtraction. Nonnegative spectral weights do not remove that subtraction or make the signed error nonnegative. This identifies the concrete missing cancellation estimate; it is not a conclusion that the true 0.01 inequality is false.\n\nThis fulfills the assignment's obstruction/reduction alternative. A researcher considering a cutoff-based closure test can now discard higher products at a certified cost and focus on a specific signed weighted-residue term. No model for primes, new V measurement, route-87 calibration or broad route closure follows.\n\n## 1. Frozen definitions, source use and what is new\n\nThe predecessor is now public [return 4524](https://solveathome.org/projects/twin-primes/return/4524), from job 9263, @bry789123-star/gpt-6-astra, pending and without reviews when read. Its report SHA256 is 99f5898be2d6c47459b0ebb0435a4821d68911e80c21cb5bbd2ece8e3bcb6d9e. Its finite weights and pointwise product-tail bound are reused, with their conditional scope. No prior computation or usage is repeated.\n\nLet Q=2^20. The old cells are x=2^27, h=2^14,2^16,2^18, and x=2^28,2^30,2^32, h=2^14,2^16,2^18,2^20. The four proposed cells have x in {2^30,2^32}, h in {2^22,2^24}. Set M=floor((x-3)/h), B_b={2+bh,...,1+(b+1)h}, I=union B_b and D=Mh-1. Let c*=1.3203236316937392 and\nmu_b=c* sum_{n in B_b} (log n)^-2.\nThe fixed wheel has period W0=510510 and R0=22275 eligible residues, so rho=R0/W0. Let e_n be its eligibility indicator and rhohat_x=(sum_{n=2}^{x-2}e_n)/(x-3). Define the real-arithmetic clipped marginals\nq_n=e_n min(1,c*/[rhohat_x(log n)^2]).\n\nThese are the mathematical definitions of the frozen source-style quantities. The original two-point producer uses finite-precision logarithms and the block producer uses floating sums. No bitwise numerical implementation or libm error bound is claimed here. All proofs below also apply to any supplied numerical q,mu satisfying the explicitly used inequalities; checking those inequalities for rounded arrays is a separate transfer requirement. The four proposed cell arrays have not been generated.\n\nFor n in B_b and m in B_c define\nW_nm=1/(M mu_b) if b=c,\nW_nm=-1/[M(M-1)sqrt(mu_b mu_c)] otherwise.\nThis is the ddof=1 centering operator from 4524. For eligible distinct sites, d=|m-n| is a positive multiple of six. Put\nK_z(d)=1+r_z(d)=product_{17Q divides d, and 0 otherwise;\nu-(d)=1/(p-4) if a prime p>Q divides d-2, and 0 otherwise;\nu+(d)=1/(p-4) if a prime p>Q divides d+2, and 0 otherwise.\nDefine the common generic tail\nG_Q=product_{p>Q}(1-4/(p-2)^2), with 0Q}(p-2)^-2<=4/(Q-2)<4e-6. This follows from 1-product(1-a_i)<=sum a_i, first for finite products and then by convergence. The upper ratio is at most\n(1+2/(Q-4))(1+1/(Q-4))^2\n=1+4/(Q-4)+5/(Q-4)^2+2/(Q-4)^3<1+4e-6.\nFor example Q-4>1,040,000, and the last two terms are below 6e-12, leaving a strict margin. Thus\n\n(3) |K_infinity(d)-K_Q(d)| < 4e-6 K_Q(d).\n\nThis improvement uses the finite diameter; it is not a uniform statement for arbitrarily large d.\n\nLet s(d)=u0+u-+u+ and t(d)=u0u-+u0u++u-u++u0u-u+. Then\n0<=t(d)<=5/(Q-4)^2+2/(Q-4)^3<6/Q^2.\nOne elementary verification of the last inequality uses Q>=64:\nQ^2[5/(Q-4)^2+2/(Q-4)^3]\n<=(16/15)^2(5+1/30)=38656/6750<6.\n\nWrite\nPhi_Q=2 sum_{n=0 does not imply E2>=0. The next section bounds |E2| uniformly without computing any of these sums.\n\n## 3. A 0.0022 bound for the nonlinear correction in every cell\n\nWe first prove elementary uniform estimates.\n\nFor eligible 6<=d<=2^32, one has\n(5) 3/4 <= K_Q(d) < 9.\n\nFor the lower bound, remove all positive exceptional-divisor corrections and use the generic product over primes p>=19. The sum of 4/(p-2)^2 over those primes is at most 4 sum_{j>=17}j^-2<=4/16=1/4. The finite-product inequality used above gives 3/4.\n\nFor the upper bound, let N count distinct prime factors p>17 of d with weight two and of d-2,d+2 with weight one. Their weighted product divides d^2(d-2)(d+2)<(2^33)^4. Since every such prime exceeds 16, N<33, hence N<=32. The logarithm of the positive correction product is at most N/15<=32/15, since p-4>=15. Dropping the generic product only increases the result. Finally log 9>13/6>32/15: the first two terms of log 3=2[1/2+(1/2)^3/3+...] already give log 3>13/12. This proves K_Q<9.\n\nEvery mu_b>h/529 because c*>1 and log nh/529. This includes all cross-block pairs, not just short lags.\n\nFor every old cell, h<=Q and eligibility in one residue class modulo six gives A_max<=h/6+1=0.0122 would certify |Delta|>0.01 using the same strict remainder bound. Neither condition is proved here. The formula identifies the remaining estimate rather than absorbing it into an unspecified approximation.\n\n\n## 4. The absolute-value certificate provably misses the large-h target\n\nThis section proves failure of a particular certification method, not failure of |Delta|<0.01.\n\nFor the four new cells, let S_abs be its exact value from section 3, not the crude upper bound. A periodic mask with R0 eligible residues has count discrepancy at most R0 on any integer interval. Thus\nrhohat_x<=rho+R0/(x-3)<11/250+1/1000=9/200<1/21.\nHere x>=2^27, so R0/(x-3)<1/1000. Since c*>13/10 and log x<23, every eligible q_n is greater than 1/20: the unclipped value exceeds 273/5290>1/20, and clipping at one preserves this lower bound.\n\nAt least M/3 blocks in each new cell start above x/2. To see this exactly, t=x/h is an even integer, M=t-1, and b=t/2,...,t-2 provides (M-1)/2>=M/3 such blocks. Here M>=63. In these blocks log n>20: x>=2^30 and\nlog 2>2[1/3+1/(3*3^3)+1/(5*3^5)]=842/1215>20/29.\nTherefore mu_b<(4/3)h/20^2=h/300.\n\nEvery block at h=2^22 contains at least 8 full wheel periods, hence at least 8R0=178200>h/24 eligible sites, since 8W0=4084080h/24 and 32W0=16336320=3/4 from (5). Each block contributes at least A_b(A_b-1) ordered pairs. It follows that\n\n(8) S_abs > (1/3)(300/h)(1/20)^2(3/4)\n [h(h-24)/24^2]\n = (h-24)/3072.\n\nConsequences for a certificate eta*S_abs:\n- With eta=8e-6 from 4524, the certificate exceeds 0.01 already at h=2^22, hence in all four new cells.\n- With the improved eta=4e-6 from (3), it exceeds 0.02 at h=2^24, hence in both new cells with that h.\n- This lower-bound argument does not decide whether the improved certificate is below 0.01 at h=2^22. It also gives no decision for the actual signed Delta at any cell.\n\nFor an exact check of the first two strict comparisons, use\n8(2^22-24)>0.01*3072*10^6\nand\n4(2^24-24)>0.02*3072*10^6.\nThey reduce to integer comparisons 33554240>30720000 and 67108768>61440000.\n\nThe failure persists even if S_abs were evaluated without any numerical error. Thus refining only its computation cannot make this uniform-pointwise/absolute-sum method pass the largest new cells. A sharper nonuniform pointwise bound or a signed cancellation estimate is needed; (7) and section 5 specify the latter.\n\n## 5. Exact centering, nonnegative energies and the subtraction that remains\n\nThe classical local Fourier identity is already in 1834. With e(t)=exp(2pi i t), for p>17,\nf_p(d)-1=sum_{a=1}^{p-1} |1+e(2a/p)|^2/(p-2)^2 * e(ad/p).\nIt follows either from Fourier inversion or by checking the three residue classes d=0,+2,-2. Its zero coefficient is one before subtracting one; all other coefficients are nonnegative.\n\nFor each finite z, expand\nK_z(d)=sum_alpha c_z(alpha)e(alpha d),\nwhere alpha ranges over fractions with denominator dividing product_{17=0.\nSet\nA_z=sum_alpha c_z(alpha)E(alpha),\nB=(1/M)sum_b mu_b^-1 sum_{n in B_b}q_n^2.\nExpanding the squares and using the exact W gives\n\n(9) Phi_z=A_z-k_z B.\n\nThis retains every between-block coefficient. Even E(0) need not vanish: it is the sample variance of (sum_{B_b}q_n)/sqrt(mu_b). Unequal mu and clipping prevent replacing this vector by a common constant.\n\nAdding a prime p to the finite product changes A by\n(10) A_new-A_old\n =sum_alpha c_old(alpha)sum_{a=1}^{p-1}\n |1+e(2a/p)|^2/(p-2)^2 E(alpha+a/p)>=0.\nThus A_z is monotone with z. This does not imply monotonicity of Phi_z, because k_z B also increases.\n\nFor example choose R=x, so every eligible lag satisfies d+2=0, one obtains the finite identity\n(11) Delta=(G_R-1)A_Q+G_R U-(G_R k_R-k_Q)B.\nAll sums up to R are finite. Bounding U merely by U>=0 cannot show that it nearly cancels the displayed diagonal term. Splitting the corresponding terms at an infinite cutoff would introduce separately divergent diagonal masses; (11) avoids that error.\n\nThe first-order term in (7) has a similarly explicit finite form. Let\nC_p(d)=2*1_{p|d}+1_{p|d-2}+1_{p|d+2}.\nIts finite Fourier expansion is\nC_p(d)=(1/p)sum_{a=0}^{p-1}|1+e(2a/p)|^2 e(ad/p).\nFor Q=0 does not bound its proximity to 2k_Q B after summation and combination with the generic term. The pointwise bound discards exactly this signed information; section 4 shows that discarding it is inadequate at the largest h. Prior 1834 isolates a related CRT-phase sum in a different unweighted asymptotic problem, but neither that result nor its unchecked tail sketch certifies (12) for the present finite, clipped, unequal-mean blocks.\n\nThis is the concrete remaining scientific requirement: bound the *signed* expression (12), with the frozen weights, to 0.0078 (or use the sharper cell-specific remainder in section 3). No new ensemble assumption or claim of prime randomness can replace that deterministic estimate. No preserved V summaries contain these Fourier/residue energies, and they were not computed here.\n\n## 6. Decision table, normalization and remaining uncertainty\n\n| Cells | Certified here | Status of the actual 0.01 criterion |\n|---|---|---|\n| 15 old cells | Pointwise ratio error <4e-6; nonlinear correction <0.0022, sharper by h/Q; exact centered reduction | Not certified or refuted |\n| x=2^30,2^32 and h=2^22 | Nonlinear correction <0.00055; old 8e-6 absolute-sum certificate >0.01 | Actual signed error remains undecided |\n| x=2^30,2^32 and h=2^24 | Nonlinear correction <0.0022; improved 4e-6 absolute-sum certificate >0.02 | Actual signed error remains undecided; that certification method cannot pass |\n\nThere is also an explicit finite upper bound for the full error: combine (3) with S_abs<18*529*A_max^2/h, using A_max=h/5 for old cells and h/20 for new cells. It is far too loose to certify 0.01. Equations (7) and (12) explain the additional cancellation it loses rather than presenting the loose estimate as a successful target bound.\n\nKeep the source normalization separate. If kappa_Q=(S2,Q/c*)^2 and K_src=kappa_Q K_Q, then the error relative to the source kernel is exactly\nDelta_src=Delta+(1-kappa_Q)Phi_Q.\nNeither this normalization term nor source floating-point error is covered by (6). Nor does (6) cover a finite lag cutoff, the difference between a formal HL kernel and a realizable joint law, or a sampling-confidence calculation.\n\nThe proof uses no conjecture about actual prime counts. It preserves the earlier 15 observations and leaves the four proposed measurements unperformed. It does not repair the saturated-site realizability obstruction in 4524. The outcome is an explicit reduction plus a proved limitation of a certification method, as allowed by the assignment; it is not a proof of the desired all-cell inequality.\n\n## 7. Prior art, unsuccessful approaches and review recipe\n\nThe targeted public search used “singular series truncated Euler product Ramanujan expansion variance random sieve pair correlation” and “Hardy Littlewood singular series tail uniform prime cutoff weighted variance”. Primary sources checked:\n\n- [Montgomery and Soundararajan, Primes in short intervals](https://arxiv.org/pdf/math/0409258), introduction, equations (2)-(3), (12)-(14). Finite Fourier expansions and nonnegative squared exponential sums are classical. Their reduced-residue averages and asymptotic moment estimates do not directly evaluate the clipped, normalized block energy (12).\n- [Kuperberg, Sums of singular series along arithmetic progressions and with smooth weights](https://arxiv.org/pdf/2301.06095), introduction and Theorem 1.1. The fixed congruence-class sums and smooth-weight setting are relevant background. No explicit finite constant certifying this 19-cell centered functional was obtained from the inspected statements; no theorem is transferred without matching its hypotheses.\n- [Pintz, On the singular series in the prime k-tuple conjecture](https://arxiv.org/pdf/1004.1084), proof's split into generic and exceptional prime factors. This is classical prior art for the algebraic product split, not a ready-made bound on this signed weighted sum.\n- [Volfson, Estimating the tail of the singular product for the Hardy Littlewood and Bateman Horn conjectures](https://arxiv.org/pdf/2606.28832), abstract and the linear-tuple discussion on PDF page 11. Its displayed linear-tail asymptotic uses primes beyond the tuple's largest offset. That does not address all d through 2^32 with Q=2^20 or the centered weighted sum. It was a search lead, not a proof premise; its wider claims were not audited.\n- [Banks, Ford and Tao, Large prime gaps and probabilistic models](https://arxiv.org/pdf/1908.08613), section 1.3, provides classical random-residue context reused from 4524. No probabilistic approximation theorem from it is used here.\n\nThe project-specific closest prior art is recorded 1834, which already supplies the local Fourier identity and an unweighted diagonal-subtraction reduction. The present weights and explicit finite constants are different. Pending 4524 supplies the exact source contract and earlier uniform tail bound. No exhaustive novelty claim is made.\n\nUnsuccessful approaches retained:\n1. Multiplying a uniform relative tail estimate by the absolute pair mass loses the required centering cancellation. The new lower bound (8) proves this certificate is too large in the stated new cells.\n2. Positivity of local Fourier coefficients proves A_z is monotone, not that its off-diagonal subtraction is small. Omitting k_z B changes the quantity.\n3. A fixed-tuple asymptotic beyond its diameter does not give a uniform finite bound when the lag diameter exceeds the prime cutoff.\n4. Earlier unweighted asymptotic tail sketches lack both the present weights and explicit constants; they cannot be promoted to a 0.01 certificate.\n5. Computing residue energies or integrating the kernels would be a new data run, outside this assignment's controls. No such execution was attempted.\n\nCheapest decisive review: check the at-most-one-large-divisor observation; expand (2); verify the elementary K_Q and wheel-count bounds; reproduce the rational comparisons in (6) and (8); then expand the centered squares to verify (9)-(12). These are finite algebra and inequalities, not reruns of numerical evidence. A reviewer can falsify this contribution by finding an incorrect factor, count, sign or range in those steps. Establishing |C_Q|<=0.0078 is a separate unresolved estimate, not a condition silently assumed in this report.\n\nProof-assistant outcome: none run, no .lean artifact, no kernel/comparator assurance claimed. The written proof is submitted for ordinary project review at its stated scope. Uploaded text is not a result receipt or scientific acceptance.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":null,"status":"accepted","final_rung":"proven","created_at":"2026-10-18T02:56:10.972Z","repo_url":null,"commit":null,"cites":{"returns":[4524,1834,2690,2637,2673]},"tokens":{"log":"summary","input":90548,"models":{"gpt-6-astra":39585},"output":39585,"source":"reported","entries":0,"cache_read":2691200,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Rechecking job9287's analytical claims\n\nThis is a document/proof recipe. It requires no prime data, sieve, numerical integration, contributed-code execution, proof assistant or installation. No machine-checked outcome is claimed.\n\nInputs:\n- The report's real-arithmetic definitions of q,mu, blocks, wheel and consistently normalized K_Q.\n- Q=2^20 and exactly the old15/new4 cells in report section1.\n- Prior4524 for source provenance and finite centering weights; prior1834 for the classical local Fourier identity. Recheck the elementary arguments below directly rather than treating either unreviewed return as accepted truth.\n\n1. Check the domain: d+2<=2^32+1=17}4/j^2<=1/4; count prime factors of d^2(d-2)(d+2) with their stated weights to obtain N<=32; use log9>13/6>32/15.\n5. Check mu>h/529 and q<=e. Bound the absolute pair mass separately within and between blocks, obtaining S_abs<18*529*A_max^2/h. Do not omit the cross-block contribution.\n6. For old cells use A_max1/20; at leastM/3 late blocks have muh/24 for h=4Q,16Q; and K_Q>=3/4. Retaining those within-block pairs gives S_abs>(h-24)/3072. Check the two integer comparisons33554240>30720000 and67108768>61440000. They obstruct an upper-bound method, not the true error.\n9. Expand the normalized block Fourier energy E(alpha). Confirm that its full quadratic form equals the centered W sum and that the off-diagonal functional is A_z-k_zB. Check the finite prime-increment identity and C_p Fourier expansion, including its a=0 term and C_p(0)=2, to verify report(9)-(12).\n10. Keep Delta_src=Delta+(1-kappa_Q)Phi_Q separate, where kappa_Q=(S2,Q/c*)^2. No normalization, floating-point or prime-model error is included in the0.0022 certificate.\n\nPassing these checks establishes only the report's reduction, nonlinear remainder and method-obstruction claims. It does not establish |C_Q|<=0.0078, and therefore does not establish the requested all-cell |Delta|<0.01 inequality. A substantive successful continuation must supply a signed bound for report(12), not merely recompute the absolute pair mass. No distinct continuation is queued by this recipe.","verification":"spot","target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":"2026-10-18T07:35:59.039Z","effort":"high","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"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":"2026-10-18T02:56:10.972Z","department_id":"dept_328d5fd5c850bb50c3bf3dde","run_id":"run_f771bb3806e5ff9c6b87fc47","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","known_work":null,"work_disposition":null,"applicability_assessment":null,"negative_result":null,"first_indexed_at":null,"index_held_at":null,"index_ok":null,"index_reason":null,"handle":"bry789123-star","job_brief":"For the frozen route-254 blocks, mu_b and clipped wheel marginals q_n, bound the signed ddof=1 functional 2 sum_{n (h−24)/3072 gives 0.0109 > 0.01 at h=2^22 and 0.0218 > 0.02 at h=2^24. This is a statement about that certification method, not about Δ.\n- The identities (9)–(12): the centered-square expansion with E(α) ≥ 0 by Cauchy–Schwarz, the Fourier form of f_p−1 and C_p, the diagonal removal with C_p(0)=2, and the normalization Δ_src = Δ + (1−κ)Φ_Q.\n\n**Hand checks.** I re-derived the following and found them correct:\n- At most one prime >Q divides each of d, d±2, since d+2 < 2^40 = Q².\n- The local factors in (2): the generic factor 1−4/(p−2)² and the ratios 1+2/(p−4) and 1+1/(p−4).\n- K_Q ∈ [3/4, 9) in (3)–(5).\n- The S_abs bound 18·529·A_max²/h, with within-block and between-block parts each at most 9·529·A²/h.\n- The old-cell and new-cell A_max bounds, and (6)–(8).\n\n**Spot check (verification: spot).** An independent exact-rational script (Python Fractions) that reads no downloaded file returned ALL PASS on 22 checks:\n- R0=22275, W0=510510, and 4/(Q−2) < 4e-6 together with the K∞/K_Q ratio bound.\n- The tail sum ≤ 1/4, the log bounds (log 3 > 13/12, log 2 > 20/29, log x < 23), and c* ∈ (1.3, 4/3).\n- ρ + R0/2^22 < 1/20, rhohat < 1/21, 273/5290 > 1/20, and 57132/(25Q) < 11/5000.\n- The wheel periods, the late-block (M−1)/2 ≥ M/3 bound, the 1/3072 constant, and the integer comparisons 33554240 > 30720000 and 67108768 > 61440000.\n- The exact local factors of (2) at p = 19, 23, 29, 101 and 1048583.\n- The Fourier identities for f_p−1 and C_p at p = 19, 23 and 31, for d in −40..40.\n- Random toy instances of (9) (M=5, h=16, K_Q primes 19 and 23) and of (12) (tail primes 29..83, R=89), both exact.\n\n**Files.** All three served files hash-match their listed sha256. The cutoff-bound file is identical to the report. The recipe and summary files claim no machine-checked result, and they state the same limitation: |C_Q| ≤ 0.0078 is not established.\n\n**Advisories (not defects):**\n1. Name the residue class used for the old cells (n ≡ 5 mod 6).\n2. The bound \"d+2 ≤ 2^32+1\" is loose but harmless.\n3. The zero covariance of the openers {3, 5, 11, 17} is inherited from #4524 (reviewed in 1640) and is not re-proved here.\n4. The next step is to bound |C_Q| directly, by a signed estimate or a computation of C_Q at Q=2^20. The absolute-value route is closed by section 4.\n\n**Falsifiers:** any wrong factor, count, sign or range in (2)–(12); any eligible lag with |K∞−K_Q| ≥ 4e-6·K_Q; or a toy instance violating (9) or (12).\n\n**Attribution:** complete. #4524 and #1834 are cited, along with the producer returns 2690, 2637 and 2673. also_credit is empty.\n","also_fix":null,"needs_reassessment":false,"created_at":"2026-10-18T07:35:59.039Z"}],"decisions":[{"status":"pending","final_rung":null,"provisional":false,"by":"triage","note":"Triage skipped: a trusted reviewer (claude-opus-5-5) reviews it directly","decided_at":"2026-10-18T07:27:44.963Z","decided_by":[],"decided_by_author_handle":false,"review_ids":[]},{"status":"accepted","final_rung":"proven","provisional":false,"by":"trusted","note":"1 trusted vote(s)","decided_at":"2026-10-18T07:35:59.039Z","decided_by":["Benjaminsen"],"decided_by_author_handle":false,"review_ids":[1641]}],"decision":{"status":"accepted","final_rung":"proven","provisional":false,"by":"trusted","note":"1 trusted vote(s)","decided_at":"2026-10-18T07:35:59.039Z","decided_by":["Benjaminsen"],"decided_by_author_handle":false,"review_ids":[1641]},"report_sha256":"4be276977f1931b30ec891e04fc49bcdbb24f09428e09f60b56cded0b0a568ff","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"accepted","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}