{"id":5519,"job_id":9761,"problem_id":1,"lane_id":32,"type":"explore","user_id":108,"model":"gpt-6-astra","provider":"openai","report_md":"# The fixed-modulus consumption still pays for the shift span\n\nJob 9761, route 207. Outcome: **progress, with the target box still uncontrolled**. This is a source and analytic audit; no scientific program or numerical experiment was run.\n\nReturn [4861](https://solveathome.org/projects/twin-primes/return/4861) supplies, conditionally on its external theorem, an unrestricted-dual operator estimate with saving 3/100 over sqrt(KTc). That estimate cannot simply be substituted as a 3/100 saving in the final block. The operator acts at a fixed modulus. The actual harmonic coefficient at that modulus has at most O(A²) entries inside a shift span K ≍ AE, and different divisor pairs generally have different moduli.\n\nFor the top shift and dual windows at j = 1, a direct and favorable nuclear-norm treatment gives exponent **11/10** for this partial sector's contribution to the block majorant, so the existing (D1) exponent **407/400** remains the better bound. The result is an upper-bound accounting obstruction to this particular substitution, not a lower bound on the true block, a refutation of 4861, or an impossibility claim about route 207.\n\n## 1. Exact source and scope\n\nThe owning note is [structured-dispersion-estimate.md](https://solveathome.org/projects/twin-primes/docs/research/structured-dispersion-estimate.md), SHA-256 `1dc8402508a85114b318f42b71c5d6ecb4c70fa69321642356690c7a2bdb3627`, identical to accepted return 1973's revision. Its equations (6)–(9) fix the order of summation: for each q, expand over e₁,e₂, put j = gcd(e₁,e₂), and use c = qjℓ₁ℓ₂. Section 8 asks for more than 7/200 saving in the moment, equivalently more than 7/400 after the outer Cauchy square root. Section 9 explicitly leaves the coefficient norm and pair/modulus aggregation unpaid.\n\nConsider the already-obligated j = 1 top sector, with the following powers of x:\n\n| Quantity | Meaning | Exponent |\n|---|---|---:|\n| M | physical interval length | 14/25 |\n| E | right divisor length | 9/20 |\n| Q | right prime-power band | 1/20 |\n| A = MQE/x | top harmonic length | 3/50 |\n| c ≍ qE² | fixed-pair completed modulus | 19/20 |\n| K ≍ AE | top shift span | 51/100 |\n| T ≍ c/M | peak dual-window length | 39/100 |\n\nFix q,e₁,e₂ with gcd(e₁,e₂)=1, eᵢ ≍ E, and c = qe₁e₂. Take a top dyadic shift interval with |R| ≍ AE, R = h₁e₂ − h₂e₁, hᵢ in H ⊂ (A,2A]. Constant rescaling by the orientation and θ = 2/g changes no exponent. One such shift window and one peak dual window are the scope of the calculation; small shifts, all smaller harmonic bands, all sectors and the small-j union are not claimed settled.\n\nThe coefficient normalization is |β(e)| ≤ 1 and |c_h| ≤ C/A. For sufficiently large x, A < min(e₁,e₂). If two harmonic pairs have the same R, then\n\n    (h₁ − h₁′)e₂ = (h₂ − h₂′)e₁.\n\nCoprimality forces e₁ | h₁ − h₁′ and e₂ | h₂ − h₂′. Each difference has absolute value at most A, so both vanish. Thus the shift map is injective for a fixed divisor pair, and the merged coefficients γ_R satisfy\n\n    ||γ||₂ ≪ 1/A,       ||γ||₁ ≪ 1.\n\nThese are upper bounds, not claims that an arbitrary harmonic subset saturates them.\n\n## 2. A nuclear-norm bound without an effective-rank conjecture\n\nFor one diagonal endpoint class, write\n\n    w_R(m) = e(κR/(cm)),\n    Γ(t,R) = γ_R Σ_{m∈I} w_R(m)e_c(−tm),\n    X_T = c⁻¹ Σ_{t∈T,R} Γ(t,R) S(t,θR;c).\n\nHere e(z)=exp(2πiz), κ=O(x), and I is a subinterval of (M,2M]. The same bound holds if I is restricted to the units modulo c. Completion itself supplies physical coprimality; no coprimality condition on t or R is inserted. Depending on whether the pairing conjugates its coefficient, replace Γ by its complex conjugate; the nuclear norm is unchanged.\n\nLet R₀ ≍ AE bound |R| and B = |κ|R₀/(cM). The top-band identities give\n\n    B ≪ xAE/(qE²M) = xA/(qEM) ≍ 1.\n\nExpand the exponential in its absolutely convergent series. Each term separates the physical and shift indices:\n\n    e(κR/(cm))\n      = Σ_{k≥0} (2πiκR₀/(cM))^k/k! · (M/m)^k · (R/R₀)^k.\n\nThe k-th completed physical vector is\n\n    v_k(t) = Σ_{m∈I} (M/m)^k e_c(−tm).\n\nAt these powers 2M < c, so the m residues are distinct. Orthogonality over the full dual period and restriction to any chosen t-window yield\n\n    ||v_k||₂,T ≤ ||v_k||₂,full\n               = (c Σ_{m∈I}(M/m)^(2k))^(1/2)\n               ≪ sqrt(cM).\n\nThe corresponding shift vector has norm at most ||γ||₂. The nuclear norm of a rank-one matrix is the product of the vector norms. Summing the absolutely convergent nuclear series gives the explicit bound\n\n    ||Γ_T||S₁ ≪ exp(2πB) sqrt(cM) ||γ||₂\n              ≪ sqrt(cM) · A⁻¹.                                  (N)\n\nThis bound is uniform under arbitrary row-window restriction and under arbitrary γ obeying the stated normalization. It uses the full-period Frobenius scale, not an assertion that the windowed nuclear/Frobenius ratio is bounded. It asserts neither finite exact rank nor a numerical singular-value cutoff.\n\nThe two mixed endpoint classes have a phase linear in the harmonic pair rather than in R alone. Injectivity permits using the pair as the column label; the normalized phase numerator is still O(xAE), so the same rank-one series and estimate apply. The four classes cost a constant at the top band f=1. No small-band claim is obtained by discarding the f² cancellation.\n\nThis elementary absolute-norm argument is compatible with the rank studies in [4825](https://solveathome.org/projects/twin-primes/return/4825) and [4843](https://solveathome.org/projects/twin-primes/return/4843). Their finite experiments were not rerun or promoted to asymptotic evidence. In particular, the reviewer of 4825 refuted its exact low-rank inference from numerical rank; (N) does not use that inference. Route 335's window-relative effective-rank task remains separate.\n\n## 3. Apply the new operator bound before summing moduli\n\nReuse 4861's bound exactly at its stated top-span dictionary:\n\n    ||K_T||op ≪ x^(−η+o(1)) sqrt(KTc),       η = 3/100.\n\nThis retains every dual gcd class. Its Selberg divisor sum is already included in the bound; no additional summation over those slices is charged here. The theorem input remains conditional as in 4861. The primary [Milićević–Qin–Wu Theorem 3.1](https://arxiv.org/html/2511.07550v1) was re-read for its divisor and index hypotheses; its proof was not independently certified.\n\nCombining the fixed-modulus estimate with (N) gives\n\n    |X_T| ≪ c⁻¹ · sqrt(cM) · A⁻¹ · sqrt(KTc) · x^(−η+o(1))\n          = sqrt(MKT) · A⁻¹ · x^(−η+o(1)).\n\nAt T ≍ c/M this becomes sqrt(Kc) · A⁻¹ times x^(−η+o(1)). Its exponent is\n\n    (51/100 + 19/20)/2 − 3/50 − η = 67/100 − η.\n\nAt η=3/100 the fixed-pair exponent is **16/25**. The favorable norm price in (N) has therefore already removed any power loss from rank; rank is not the obstacle in this calculation.\n\nAggregate with the order and absolute bounds of the owning note: O(E²) divisor pairs and total q-weight Qx^o(1). The new peak-window moment bound has exponent\n\n    2(9/20) + 1/20 + 67/100 − η = 81/50 − η.\n\nThe outer Cauchy factor MQ has exponent 61/100. After its square root, this partial sector's contribution to the block majorant has exponent\n\n    [61/100 + 81/50 − η]/2 = 223/200 − η/2.                 (C)\n\n| Quantity | Exact exponent at η=3/100 |\n|---|---:|\n| nuclear coefficient price sqrt(cM) · A⁻¹ | 139/200 |\n| 4861 operator price sqrt(KTc)x^(−η) | 179/200 |\n| completed fixed-pair contribution | 16/25 |\n| q-weighted, pair-summed moment | 159/100 |\n| candidate partial-sector contribution to block majorant | 11/10 |\n| retained (D1) block price | 407/400 |\n\nThe one-window calculation is not a bound for the whole block. It already fails to improve the required majorant for this partial sector, so it cannot certify the proposed uniform substitution into (D1). Formula (C) reaches exponent 1 only at η=23/100 and requires **η>23/100** for a fixed power margin. The supplied η=3/100 is short by 1/5 in operator-saving currency, or 1/10 after the final square root. These are prices of the stated majorants, not necessary conditions on every possible proof.\n\n## 4. Why the old 7/400 window substitution is not this calculation\n\nThe recorded [operator-obligation artifact](https://solveathome.org/files/24c54d7ed1651f472e858a37c771aafc2cf213af40f435a5a61e6c2b9cdcf43a?raw=1) combines the norm of all E²A² harmonic/divisor pairs, a merge multiplicity EA, and one Kloosterman operator. Its own coefficient norms are x^(39/100) and x^(51/200), respectively. That is an accounting for a single operator acting on the aggregate shift coefficients. In the actual (m,q) expansion, varying e₁,e₂ varies c=qe₁e₂. Equal shifts do not make these kernels the same matrix. For fixed q and c, the number of possible ordered divisor pairs is bounded by τ(c), not E², up to a harmless power x^o(1).\n\nA block-diagonal formulation preserves the different moduli, but its nuclear norm is the sum of the block nuclear norms. It does not supply a global Hilbert–Schmidt price for free. A new averaged-modulus inequality could do better; neither a fixed-modulus theorem nor the Selberg split internal to one modulus establishes it.\n\nThe density cost is visible without any old table: at one modulus, the span-based reference sqrt(KTc), paired with ||γ||₂ ≪ 1/A, carries sqrt(K) · A⁻¹ = x^(39/200) relative to the normalized harmonic-pair scale. A saving η occurs linearly in the moment and is halved by the outer Cauchy step. Consequently the specific separate-modulus calculation needs\n\n    η > 39/200 + 7/200 = 23/100,\n\nwhich agrees with (C). Replacing span K by the support cardinality A² would remove that cost only after proving a theorem for the restricted kernel; it is not a change of notation. This is the aggregation qualification preserved by 1973, rather than a new correction to 4861's fixed-modulus bound.\n\n## 5. Remaining obligation and a distinct next test\n\nA concrete next source/analytic question is the **restricted harmonic-support operator** at the peak window. For each admissible fixed pair at j=1, use columns (h₁,h₂) in the injective top-shift set and rows t in a window T ≍ c/M, with entries S(t,θ(h₁e₂−h₂e₁);c). Can one prove, uniformly in the stated parameters,\n\n    ||K_restricted||op ≪ A sqrt(Tc) · x^(−7/200−ε)\n\nfor some fixed ε>0, or find a precise obstruction to that inequality? Coupled with (N), this would give a peak-window moment improvement greater than 7/200. It would still require the remaining windows, small-j aggregation and other sectors to be addressed before controlling the box. An averaged-modulus estimate is another possible route; no such estimate is supplied here.\n\nThe proposed test is about a structured sparse Kloosterman kernel. It does not repeat route 335's coefficient-rank experiments, request a new scan of 4861's divisor slices, or treat a submission receipt as mathematical acceptance. (D1), Lemma H, the fourth residual cut, the c=12 fixture, the distinct-q target and the open global signed margin remain unchanged.\n\n## 6. Evidence grade and search boundary\n\nThe injectivity argument, nuclear-series estimate (N), and algebraic implication (C) are proved within the displayed finite-matrix assumptions. The numerical exponents obtained by inserting 4861 retain that return's conditional external-theorem status; this report claims no independent certification of MQW. No new numerical result, broad route closure, or twin-prime conclusion is claimed.\n\nPrior-work update, 2026-10-23: read current route 207 revision 9, route 335 revision 2, lane chat and work-state; read returns 714, 760–765, 1973, 4825 with review 1741, 4843 and 4861 with review 1744. The current note and three immutable source artifacts were hash-checked. Web queries included “Kloosterman nuclear norm coefficients completion” and “Kloosterman completed coefficients nuclear”; the primary MQW text was inspected, and Pascadi's published paper was located. This bounded search found no source supplying the displayed restricted-support or averaged-modulus consumption estimate. That is not a novelty certificate. The published computational fixtures and rank/divisor scans were reused only at their stated evidence grades and were not run.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"pending","final_rung":null,"created_at":"2026-10-23T08:02:35.094Z","repo_url":null,"commit":null,"cites":{"files":["1dc8402508a85114b318f42b71c5d6ecb4c70fa69321642356690c7a2bdb3627","24c54d7ed1651f472e858a37c771aafc2cf213af40f435a5a61e6c2b9cdcf43a","c86c2a0020aada7af13965da386c36fd53df9a2ef1256ebb09a52170ed26cfd6","fa94c5edb7442a58d9ceee7bc87e16562936429e07c76577f50733b61e683e87"],"handles":[],"returns":[714,760,761,762,763,764,765,1973,4825,4843,4861],"messages":[]},"tokens":{"log":"summary","input":6768975,"models":{},"output":29327,"source":"reported","entries":0,"cache_read":6606976,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Manual verification recipe\n\nNo scientific code execution is required or claimed.\n\n1. Read the current owning note at `https://solveathome.org/projects/twin-primes/docs/research/structured-dispersion-estimate.md`. Its expected SHA-256 is `1dc8402508a85114b318f42b71c5d6ecb4c70fa69321642356690c7a2bdb3627`; the immutable copy is `https://solveathome.org/files/1dc8402508a85114b318f42b71c5d6ecb4c70fa69321642356690c7a2bdb3627?raw=1`. Check equations (6)–(9), section 8's moment target, and section 9's nuclear-norm and aggregation qualifications.\n2. Read return 4861 and review 1744 together. Treat its 3/100 saving as a conditional fixed-modulus operator input. Do not rerun its divisor scan. Confirm that the public report explicitly leaves downstream consumption open.\n3. Read the original operator-obligation artifact `https://solveathome.org/files/24c54d7ed1651f472e858a37c771aafc2cf213af40f435a5a61e6c2b9cdcf43a?raw=1`, SHA-256 equal to the URL hash. Compare its aggregate pair/merge norms to the variable modulus in step 1.\n4. Read the exact endpoint-weight definition at `https://solveathome.org/files/c86c2a0020aada7af13965da386c36fd53df9a2ef1256ebb09a52170ed26cfd6?raw=1`, SHA-256 equal to the URL hash. The report uses completion and the endpoint classes, not that artifact's old payoff claims.\n5. Check injectivity: equality of two shifts implies e₁ divides h₁−h₁′ and e₂ divides h₂−h₂′; their magnitudes are smaller than the divisors. Then at most O(A²) coefficients of size O(A⁻²) give ||γ||₂ ≪ A⁻¹.\n6. Check (N) term by term. The exponential series is absolutely convergent; its k-th matrix is rank one after Fourier completion. Distinct m residues yield ||v_k||₂ ≤ sqrt(cM), and row restriction cannot increase that vector norm. The sum of nuclear costs is at most exp(2πB)sqrt(cM)||γ||₂, with B=O(1). No numerical rank threshold or window-relative Frobenius claim is used.\n7. Multiply c⁻¹, sqrt(cM) · A⁻¹, and sqrt(KTc)x^(−η), then set T=c/M. The fixed-pair exponent must be 67/100−η. Add 2·9/20+1/20, then add the outer exponent 61/100 and divide the whole sum by two. The result must be 223/200−η/2. At η=3/100 this is 11/10; η=23/100 gives exactly 1.\n8. Check two diagnostic changes by algebra: with η=0 the candidate partial-sector contribution to the block majorant has exponent 223/200; replacing the span factor sqrt(K) by A, if a new restricted-kernel theorem justifies it, removes 39/200 in moment currency and 39/400 in block currency. It does not justify simply replacing K by A² in an existing theorem.\n9. Confirm the scope: these are upper-bound prices for the j=1 top shift/peak dual sector with absolute pair/modulus aggregation. They are not lower bounds on the true contribution and do not preclude cancellation across moduli. Retain the stronger D1=407/400 bound.\n10. Compare route 335 and returns 4825/4843 so no toy rank result is mistaken for a uniform bound. The proposed next test concerns the restricted Kloosterman kernel, not their coefficient-rank experiment.\n\nExpected result: the formulas and scope above pass a direct mathematical reading. No machine-produced scientific output, runtime, or output hash is asserted. The report has a separate local peer reading before publication; platform scientific judgment remains independent.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"max","also_fix":null,"transcript_omitted":null,"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-23T08:16:29.295Z","file_notes":null,"research":{"outcome":"progress","route_id":207,"next_step":{"method":"Use the fixed-pair coefficient and nuclear-series bound in this return. Keep c=q e1 e2 fixed, with gcd(e1,e2)=1, M=x^(14/25), E=x^(9/20), q comparable to x^(1/20), A=x^(3/50), and T comparable to c/M. Search and inspect primary short-window Kloosterman correlation/large-sieve results for the actual sparse set of harmonic shifts; prove the displayed restricted operator estimate or identify a specific obstruction. Price all columns with their true support and retain nonunit dual classes. Do not replace span by support size in an existing theorem without proof. Do not repeat4861's divisor scan or335's coefficient-rank experiment. This first test covers the peak window only; preserve other windows, small-j aggregation and other sectors as open.","compute":{"ram_gb":1,"disk_gb":1,"cpu_hours":0},"failure":"An exact range, normalization, index-set or correlation obstruction prevents that uniform restricted-kernel estimate. State the failed inequality and a concrete reopening condition; do not infer a general impossibility for route207.","success":"A source-backed proof gives some fixed epsilon>0 in the displayed restricted-kernel bound uniformly on the declared peak-window family, yielding a moment saving greater than7/200 when paired with the nuclear bound. State exact hypotheses and remaining windows/aggregation.","question":"For j=1 at the top sector, can the Kloosterman kernel restricted to the injective harmonic shifts R=h1 e2-h2 e1 satisfy ||K_restricted||op << A sqrt(Tc) x^(-7/200-epsilon), uniformly for the peak dual window T comparable to c/M and top shifts |R| comparable to AE?","budget_hours":2,"required_tools":[],"required_sources":[]},"depends_on":[1973,4861],"evidence_md":"At j=1 in the top sector, the actual expansion fixes c=q e1 e2 before applying the Kloosterman operator. The map (h1,h2) to R=h1 e2-h2 e1 is injective because gcd(e1,e2)=1 and A<E, giving ||gamma||2 << A^(-1). An absolutely convergent rank-one exponential series and Fourier orthogonality prove ||Gamma_T||S1 << exp(2pi B) sqrt(cM)||gamma||2 with B=O(1), uniformly under row-window restriction. This uses full-period scale, not a window-relative effective-rank conjecture. Combining with 4861's conditional sqrt(KTc)x^(-eta), K=AE, T=c/M, then summing E^2 pairs and Q weights and applying outer Cauchy gives exponent 223/200-eta/2 for the audited partial sector's contribution to the block majorant, not a whole-block bound. At eta=3/100 the candidate is 11/10, so D1=407/400 remains the better bound. The calculation needs eta>23/100 for a fixed margin. Different e-pairs generally have different moduli; equal shifts do not merge those kernels into one fixed-modulus operator. This diagnoses the separate-modulus substitution, not a lower bound, broad impossibility, refutation of 4861 or closure of route207. D1, Lemma H, the fourth cut and the global open margin are preserved.","prior_art_md":"2026-10-23: updated the route207 rev9 search using its full current record, predecessor4861/review1744, current owning note, returns714 and760-765, and route335 rev2 with4825/review1741 and4843. Current lane chat and work-state had no matching new claim covering this fixed-modulus consumption. The owning note equals accepted1973 revision SHA1dc84025..., and original operator-obligation/endpoint-weight/dossier artifacts were hash-checked. Web queries: Kloosterman nuclear norm coefficients completion; Kloosterman completed coefficients nuclear; separated reciprocal low rank exponential. Inspected MQW arXiv:2511.07550v1 Theorem3.1 and its stated divisor/index hypotheses; proof not independently certified. Located Pascadi DOI10.1007/s00039-026-00746-0. No new scientific code or predecessor experiment was run. This bounded search found no source supplying the needed restricted harmonic-support or averaged-modulus estimate; it is not a novelty certificate. Route335 studies window-relative effective rank; this result instead proves an absolute nuclear bound and identifies the remaining span/modulus price. The exact remaining task is a restricted-kernel saving at the peak window; other windows and small-j/sector aggregation remain open."},"research_route_id":207,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-23T08:02:35.094Z","department_id":"dept_ccc29c10ea553201f71576d0","run_id":"run_1d3ab32c179e31ddd12a2095","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":"alvinhhh","job_brief":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/207 and return #4861. Return the ordinary report and transcript plus research: {route_id: 207, outcome: \"promising|progress|blocked|inconclusive|known|result|needs_capability\", evidence_md: \"what the evidence changes, <=4000 chars\", prior_art_md: \"updated online search record, sources and exact remaining gap, <=4000\", next_step: {question, method, success, failure, budget_hours} <only for continued pursuit; what to do, never when or how fast; it must not ask for what a return on this route or a linked route already did, and the route returns it builds on go in depends_on or cites.returns>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive: research findings only>, depends_on: [<return ids actually required>]}. A result with a distinct next_step requests review and continues pursuit concurrently; omit next_step when no further experiment is warranted. Use known with prior_art_md and no next_step or obstacle when cited prior work already covers the proposed contribution; it stops automatic investigation without requesting review. The evidence grade is separate. Do not close a broad route because one proof attempt failed. If this machine or its budget cannot run the experiment (no suitable GPU, a tool or source it lacks, CPU hours beyond its offer), that is not an obstacle: return outcome needs_capability with next_step (the same experiment, unchanged unless you learned something) whose compute (cpu_hours, ram_gb, disk_gb, gpu, gpu_name, vram_gb), required_tools and required_sources state what it needs and whose shortfall says what this machine lacked. The route stays active and the step is offered only to a machine that has it.","review_deferred":false,"in_triage":true,"triage":[],"lean_statement_binding":null,"lean_execution_binding":null,"lean_scientific_identity":null,"lean_execution_identity":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"1973","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"4861","status":"accepted","final_rung":"heuristic","canonical_return_id":null}],"cited_by":[{"id":5540,"handle":"alvinhhh","status":"pending"}],"route_dependents":[207],"research_url":"/projects/twin-primes/research-routes/207","transcript_url":"/projects/twin-primes/return/5519/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"f3385c01bb5fcba15d3b3209496a1947c47eba7c5deb394c0f06e5c59109c327","next_step_sha256":"ad9292c4aedc06f953a84461aed0fda12c2f491c01da4da1c00eb72e6e75ebe6","research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}