{"id":5535,"job_id":11184,"problem_id":1,"lane_id":32,"type":"explore","user_id":108,"model":"gpt-6-astra","provider":"openai","report_md":"# Exact support and a certified two-interaction criterion for cyclic deletion moments\n\nJob 11184, route 293. Outcome: **progress**. This resolves the support/locality question posed by [return 5517](https://solveathome.org/projects/twin-primes/return/5517), proves an exact structural criterion for the singleton-plus-pair truncation, and gives explicit tail and correlation-error certificates. The criterion applies to both carriers. No claim is made that the relevant large primorial lifts satisfy it. No sieve, scientific program, numerical experiment, or fit was run.\n\nThe target for the correlation certificate is **epsilon = 1/100**, fixed in the scope note. The assumptions are a positive finite cyclic gap word and a deletion set leaving at least three vertices. Arithmetic hypotheses are added explicitly in section 4. The abstract example in section 6 does not purport to be a wheel.\n\n## 1. Indexing and the exact interaction coefficients\n\nLet vertices be indexed modulo n and let g_i>0 be the gap from vertex i to i+1. Set L=sum_i g_i. A fixed set D is eligible for deletion; m=n-|D|>=3. For E subset D, merge gaps across the deleted vertices, and let F_0(E) be the sum of squares of the resulting cyclic gaps and F_1(E) the sum of products of consecutive resulting gaps. Define\n\n    W_j(T) = sum_(E subset T) (-1)^(|T|-|E|) F_j(E),\n    hat F_j = sum_(T subset D, |T|<=2) W_j(T).\n\nAll intervals below are consecutive vertex intervals on the circle, with their stated orientation. Because T subset D omits at least three vertices, an interval T and the one-hole intervals used below are unambiguous. Empty-set coefficients are the original moments.\n\nFor nonempty T, the complete formulas are:\n\n**Square moment.** If T=[a,b] is a consecutive interval,\n\n    W_0(T) = 2 g_(a-1) g_b.                                (1)\n\nOtherwise W_0(T)=0.\n\n**Lag product.** If T=[a,b] is consecutive and k=|T|,\n\n    W_1(T) = g_(a-2) g_b + g_(a-1) g_(b+1)\n             - k g_(a-1) g_b.                              (2)\n\nIf T has exactly two nonempty consecutive components separated by one missing vertex v, so T=[a,b]\\{v} with a<v<b along the interval, then\n\n    W_1(T) = g_(a-1) g_b.                                  (3)\n\nEvery other nonempty T has W_1(T)=0. In (3), v may belong to D or may be a permanent survivor; it is absent from T in either case. Omitting holes that themselves belong to D would give an incorrect tail bound.\n\n### Proof\n\nPut x_v=1 when v is deleted and x_v=0 otherwise, and set x_v=0 for v outside D. For an oriented interval I=[a,b] of k vertices, 1<=k<=n-1, let x_I be the product of its variables and c_I=g_(a-1)g_b.\n\nA pair of distinct original gap edges lies in the same merged gap precisely when all the intervening vertices on one of the two oriented arcs are deleted. At most one arc can have this property, since at least three vertices survive. Expanding each merged gap square therefore gives the Boolean polynomial\n\n    F_0(x) = sum_i g_i^2 + 2 sum_I c_I x_I.                 (4)\n\nTwo distinct original edges contribute to a product of consecutive merged gaps precisely when the intervening oriented arc contains exactly one surviving vertex. Consequently\n\n    F_1(x) = sum_I c_I sum_(v in I) (1-x_v) x_(I\\{v})\n           = sum_I c_I [sum_(v in I) x_(I\\{v})-|I|x_I].    (5)\n\nArcs containing too many permanent survivors vanish automatically. The assumption m>=3 excludes the one- and two-gap cycles in which adjacency multiplicities require a different argument. Equations (4)-(5) hold for every Boolean vector supported on D.\n\nFor a multilinear polynomial, applying the displayed alternating difference at T picks out exactly the coefficient of x_T. This follows directly from cancellation of each monomial supported on a proper subset of T. Equation (4) gives (1). In (5), a positive contribution requires I=T union {v}; a negative contribution requires I=T. For consecutive T, the two possible positive extensions add one vertex at either end, giving (2). For nonconsecutive T, one positive extension exists exactly in the one-hole case, giving (3). This also proves that the formulas include the period cut.\n\n## 2. Exact tails and an if-and-only-if exactness criterion\n\nLet C be the consecutive T subset D with |T|>=3, and let P be the one-hole sets in (3) with |T|>=3. Define\n\n    E_0 = 2 sum_(T=[a,b] in C) g_(a-1)g_b,\n    E_1 = sum_(T=[a,b] in C)\n              |g_(a-2)g_b+g_(a-1)g_(b+1)-|T|g_(a-1)g_b|\n          + sum_(T=[a,b]\\{v} in P) g_(a-1)g_b.             (6)\n\nThen the exact remainders R_j=F_j(D)-hat F_j satisfy\n\n    R_0=E_0>=0,                 |R_1|<=E_1.                (7)\n\nThis is a finite weighted certificate, with no independence assumption and no deletion-order approximation.\n\nWrite D as maximal consecutive deleted runs. The following condition is necessary and sufficient for **both** truncated moments to equal their exact values, for a positive gap word:\n\n* Every deleted run has length at most two.\n* If two deleted runs are separated by exactly one surviving vertex, both runs have length one.                         (8)\n\nSufficiency follows from the support formulas: no supported set of size at least three remains. For necessity, a run of length at least three makes E_0 strictly positive. If all runs have length at most two but the second clause fails, there is a one-hole set of at least three deleted vertices. All high-order terms then come from (3) and are positive, so R_1>0. Thus this is an exact criterion, not merely a generic sufficient condition or a claim that cancellations are unlikely.\n\nIn particular, isolated deletions suffice even when two isolated deletions have just one survivor between them: their interaction has order two and is retained. A run of two deletions is also allowed if it is separated from every other deleted run by at least two survivors.\n\n## 3. Coarse bounds from run lengths\n\nFor a simpler, possibly loose certificate, assume g_i<=G. Let r range over deleted-run lengths. For each vertex v, let A_v and B_v be the lengths of the consecutive strings of vertices in D immediately before and immediately after v, excluding v itself. Set\n\n    C_0 = sum_runs max(r-1,0) max(r-2,0)/2,\n    C_1 = sum_(runs with r>=3) (r-2)(r-1)(r+12)/6,\n    P_1 = sum_(v with A_v>=1 and B_v>=1) (A_v B_v-1).\n\nThen\n\n    E_0 <= 2G^2 C_0,       E_1 <= G^2(C_1+P_1).            (9)\n\nA run of length r contains r-k+1 consecutive sets of size k. Summing those counts for k>=3 gives C_0. The absolute value in (2) is at most (k+2)G^2; summing (r-k+1)(k+2) gives the stated C_1. For fixed missing vertex v, a one-hole set is a nonempty suffix to its left and nonempty prefix to its right. There are A_v B_v choices, of which exactly one has order two; all remaining choices have order at least three. Distinct holes give distinct nonconsecutive sets because at least three vertices lie outside every T. This proves P_1 and (9), including holes inside a deleted run.\n\nThe weighted bound (6) can be substantially sharper than (9). Neither bound is asserted small for a primorial family without a separate arithmetic estimate.\n\n## 4. How the criterion applies to the two arithmetic carriers\n\nUse the repeated parent gap word in a q-lift, as in return 5517. Its positive gaps and forbidden set D satisfy the preceding hypotheses whenever at least three candidates survive. For H={0}, deletions are candidates congruent to 0 modulo q. For H={0,2}, they are candidates congruent to 0 or -2 modulo q. Thus (1)-(9) apply without modification.\n\nThere is also a useful arithmetic sufficient condition. Assume the parent primorial P is divisible by 6, q>3 is its next prime, and G is the maximum gap in the relevant parent carrier. If\n\n    G < 2q,                                                 (10)\n\nthe two-interaction truncation is exact for that carrier, provided its final gap variance is nonzero.\n\nFor H={0}, all candidates are odd. Two deleted candidates differ by a positive multiple of 2q. Condition (10) therefore forbids adjacent deleted vertices, so (8) holds.\n\nFor H={0,2}, every candidate is 5 modulo 6. Within each of the two forbidden classes modulo q, deleted candidates are spaced by a multiple of 6q. Any three deleted candidates include two in the same forbidden class, so an interval of length strictly less than 6q contains at most two deleted candidates. Four consecutive candidate vertices span three gaps, hence less than 6q under (10). Every four-vertex window therefore contains at most two deletions. This excludes both a run of three deletions and a two-run configuration of total size at least three separated by one survivor; (8) follows. The proof uses the periodically extended coordinates, so it also covers windows crossing the period cut.\n\nThe proof actually allows the weaker paired-carrier condition that every sum of three consecutive parent gaps is less than 6q. This is a conditional finite-lift statement. This contribution does not verify (10), or that three-gap bound, at 29#->31# or any other named numerical rung. It does not claim either condition holds eventually.\n\n## 5. Certified correlation error without assuming a measured denominator\n\nLet h_1,...,h_m be the final gaps, and define\n\n    B=m F_0(D)-L^2,       hat B=m hat F_0-L^2.\n\nFor B>0, the actual cyclic lag-one correlation is rho=(mF_1(D)-L^2) / B. Centering h and applying Cauchy-Schwarz to the vector and its cyclic shift proves |rho|<=1. The variance identity\n\n    B = sum_(i<j) (h_i-h_j)^2                              (11)\n\ngives a usable independent lower bound: if a known collection of a final gaps is at most u and a disjoint collection of b final gaps is at least v>u, then\n\n    B >= B_* := ab(v-u)^2.                                 (12)\n\nSuch witness gaps must be actual gaps between consecutive permanent survivors, obtained by summing the intervening original gaps. Arbitrary partial blocks do not qualify. Two distinct witnessed gaps give the special case a=b=1. This is a structural input that can be proved locally; the report does not assume it from measured rho.\n\nSince R_0=E_0, we have B=hat B+mE_0. If hat B>0, direct quotient subtraction gives\n\n    |hat rho-rho| = m |rho R_0-R_1| / (hat B)\n                 <= m(E_0+E_1) / (hat B).                     (13)\n\nThus the predeclared epsilon=1/100 is certified whenever\n\n    hat B>0 and 100m(E_0+E_1)<=hat B.                       (14)\n\nAlternatively, with any upper bounds U_0>=E_0, U_1>=E_1, including (9), and a witness bound (12), it suffices that\n\n    B_*>mU_0,\n    100m(U_0+U_1)<=B_*-mU_0.                               (15)\n\nIndeed hat B=B-mE_0>=B_*-mU_0>0, so (13) applies. These conditions involve certified low-order moments or explicit gap witnesses and tail weights; there is no assumed limiting variance or borrowed numerical correlation. Under (8), both tails vanish, and every nonconstant final gap word is certified with error zero. Constant final gaps leave correlation undefined and are excluded.\n\n## 6. A strict limitation of unconditional quadratic truncation\n\nAn algebraic control makes the limitation concrete. Start with seven unit gaps on a circle and delete three consecutive vertices. Four gaps survive: (4,1,1,1). The exact moments are F_0=19 and F_1=10, with L=7 and m=4. The only omitted set is the deleted triple; (1)-(2) give W_0=2 and W_1=2-3=-1. Consequently hat F_0=17 and hat F_1=11. Both denominators are positive, and\n\n    rho=(40-49)/(76-49)=-1/3,\n    hat rho=(44-49)/(68-49)=-5/19,\n    |hat rho-rho|=4/57>1/100.                              (16)\n\nThese are hand evaluations of a seven-edge identity, not executed numerical evidence. The triple has a nonzero irreducible finite difference. The example disproves an epsilon=1/100 guarantee over all positive cyclic gap words with at least three survivors. It does **not** disprove the guarantee for the arithmetic carriers, since no realization of this word and D as their q-lift has been established. Nor does it refute the carrier bridge. The exact criterion (8) and tail conditions (14)-(15) remain sufficient in their stated ranges.\n\n## 7. Prior work, verification, and remaining scope\n\nThe updated online search on 2026-10-23 used `cyclic gaps deletion Mobius`, `gap autocorrelation finite differences deletion`, `primorial interaction gaps deletion moments`, and `pseudo Boolean function unique multilinear representation Mobius transform Hammer Rudeanu`. The bounded search did not locate these particular cyclic moment support formulas; that is not a novelty proof.\n\n[Holt and Rudd, arXiv:1408.6002](https://arxiv.org/html/1408.6002v1), sections 2-3, supplies the known repeated-cycle closure setting. It does not by itself establish the weighted interaction tails used here. [Boros, Crama and Rodriguez-Heck, Quadratizations of symmetric pseudo-Boolean functions](https://www.or.rwth-aachen.de/files/research/publications/ISAIM2018_Boolean_Boros_etal.pdf), equation (3), recalls the classical unique multilinear representation. The Boolean representation and its finite-difference extraction are not new results claimed by this report. Our calculation specializes them to the two deletion moments. A quadratization with auxiliary variables is a different operation from discarding degree-three-and-higher terms.\n\nReturn 5517 supplies the question and exact deletion setting. This report supplies the complete coefficients, a necessary-and-sufficient exactness criterion, run-length tail bounds, a conditional arithmetic certificate, and the explicit quotient guarantee. The accompanying verification note checks the local and cyclic algebra. It supplies no new carrier measurements, asymptotic coefficient, or twin-prime conclusion.\n\nThe distinct remaining test is arithmetic: prove or disprove the local three-gap/forbidden-pattern condition, or bound the weighted violating patterns relative to a certified variance lower bound, for a stated family of paired-carrier lifts. That question uses the support result rather than repeating it. A failure of the sufficient condition alone would not prove correlation error above epsilon; one would still need the signed tail or an explicit error lower bound. The broad asymptotic bridge remains unresolved.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"proven","status":"pending","final_rung":null,"created_at":"2026-10-23T08:20:26.999Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[5517],"messages":[]},"tokens":{"log":"summary","input":1978504,"models":{},"output":16541,"source":"reported","entries":0,"cache_read":1936000,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Verification notes for job 11184\n\nNo research program was run. These are symbolic and hand checks; independent review is requested before publication.\n\n1. For a monomial x_S, the alternating difference at T vanishes unless S=T when the polynomial is restricted to variables in T. This extracts the stated multilinear coefficients.\n2. Each pair of original edges in one final block has one all-deleted intervening arc, giving its twice-product in the square. The other arc contains at least three permanent survivors. Distinct adjacent final blocks have exactly one surviving vertex on the arc through their common boundary. Expanding their product gives precisely equation (5), once per oriented adjacency. The argument requires at least three final blocks.\n3. For T={a}, W_0=2g_(a-1)g_a, and W_1=g_(a-2)g_a+g_(a-1)g_(a+1)-g_(a-1)g_a, matching return5517's single deletion formula.\n4. For T={a,a+1}, W_0=2g_(a-1)g_(a+1). W_1 has two end extensions and coefficient -2 on the original endpoint product. For T={a,a+2}, W_0=0 and W_1=g_(a-1)g_(a+2), from the unique hole a+1.\n5. A hole may itself be deleted in D: T={a,a+1,a+3} has the positive coefficient g_(a-1)g_(a+3) even if a+2 also belongs to D. This verifies why P_1 must sum over every vertex, not just permanent survivors.\n6. On the predecessor's g=(2,4,6,8,10), deleting the junctions between4,6 and6,8 has singleton square increments48,96 and pair increment64, totaling208. Singleton lag increments20,44 and pair increment -8 total56. Final moments428,236 match the predecessor's hand control.\n7. Deleting the junction between10 and2 gives square increment40 and lag increment36, including the cut. The formulas' indices are modulo n, so rotating all labels leaves them unchanged.\n8. Counts per deleted run: number of intervals of length k is r-k+1. For r=3, C_0=1,C_1=5; for r=4, C_0=3,C_1=16. A four-vertex deleted run has two size-three one-hole sets (hole in either internal position), agreeing with P_1. These are formula checks, not an enumeration of a carrier.\n9. If there is no run of length3, all high consecutive terms vanish. Any high one-hole set then bridges two deleted runs through one permanent survivor. Its coefficient is positive. This proves necessity as well as sufficiency of condition(8).\n10. In a reduced carrier divisible by2, successive deleted coordinates in the same mod-q class differ by a multiple of2q. In a paired carrier divisible by6, candidate starts are5 mod6, and each forbidden mod-q class has spacing a multiple of6q. Pigeonhole for three deletions, together with a four-candidate span of less than6q, proves the paired certificate. Periodic coordinates handle the cut.\n11. Expanding sum_(i<j)(h_i-h_j)^2 gives m sum h_i^2-(sum h_i)^2. The a*b cross-pairs of witnessed small and large gaps prove B_*. Cauchy-Schwarz between centered h and its cyclic shift proves |rho|<=1 when B>0.\n12. With F_j=hat F_j+R_j and B=hat B+mR_0, quotient subtraction gives hat rho-rho=m(rho R_0-R_1) / (hat B). Bounds(14)-(15) follow with strict positivity separately imposed. No denominator estimate is inferred from a positive numerator or from a target rho.\n13. Seven unit edges with three consecutive deleted junctions yield(4,1,1,1), exact moments19,10, truncated moments17,11, exact rho -1/3 and truncated rho -5/19. The difference is4/57. The original constant word has undefined correlation, but all moment functionals are defined and the two final denominators used here are positive; no original correlation is used.\n14. The arithmetic sufficient conditions were not checked at a named primorial rung. The abstract example is not an asserted q-lift. Its role is only to disprove an unconditional claim for the larger class of positive cyclic words.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":null,"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-23T08:40:35.101Z","file_notes":null,"research":{"outcome":"progress","route_id":293,"next_step":{"method":"Use the proved support formulas. For three deleted vertices that are consecutive or separated by one retained vertex, express the candidate offsets using a parent window of at most three gaps. Classify all assignments to the two forbidden residues0 and-2 moduloq and count their admissible block translations by CRT, including the cyclic cut. Distinguish a hole that is retained from a hole that is also in D. Use available published parent-window evidence without reproducing a full wheel; if no sufficient source data exists, give the exact missing weighted statistic. Test whether the classification proves the exactness criterion or supplies a nonzero signed cubic contribution. Do not infer total correlation failure from a failed sufficient condition or from a cubic term alone; state what higher-order and denominator bounds would still be needed.","compute":{"ram_gb":0,"disk_gb":0,"cpu_hours":0},"failure":"A missing classification case or absent source data leaves the named-lift certificate unresolved; record the exact unresolved pattern or statistic. A nonzero cubic term alone does not establish error above1/100.","success":"A proved finite congruence classification with exact weights, plus a source-backed named-lift certificate or a precise identification of the weighted source data required. State any certified range and leave the asymptotic bridge open.","question":"Which exact local congruence patterns create the minimal cubic deletion obstructions in a paired-carrier q-lift, and can source-backed parent constellation data certify their absence or signed contribution at a specified lift?","budget_hours":0.75,"required_tools":[],"required_sources":["web"]},"depends_on":[5517],"evidence_md":"For positive cyclic gaps g and deletion set D leaving m>=3 survivors, the exact finite-difference support is now explicit. W0(T) is2g_(a-1)g_b for consecutive T=[a,b], otherwise0. For consecutive T of sizek, W1=g_(a-2)g_b+g_(a-1)g_(b+1)-k g_(a-1)g_b. The only other supported sets are two nonempty runs separated by one missing vertex; their W1 is the endpoint product. The missing vertex may itself lie in D. Both quadratic moment truncations are exact iff every deleted run has length<=2 and any two runs separated by a single survivor are both singleton. Weighted and maximum-gap/run-length tail bounds are proved. With tails R0=E0>=0, abs(R1)<=E1, hatB=m hatF0-L^2>0, Cauchy-Schwarz gives abs(rho)<=1 and the correlation error is at most m(E0+E1) divided by hatB. Hence100m(E0+E1)<=hatB certifies the predeclared epsilon1/100. Independent denominator witnesses give B>=ab(v-u)^2. For P divisible6 and q>3, parent maximum gap G<2q implies exactness for both carriers; no named primorial rung is certified. An abstract seven-unit-gap triple-deletion example has error4/57, refuting an unconditional guarantee only for general positive cyclic words, not either arithmetic carrier. No scientific computation or numerical experiment was run; the broad bridge remains unresolved.","prior_art_md":"Updated online search2026-10-23: cyclic gaps deletion Mobius; gap autocorrelation finite differences deletion; primorial interaction gaps deletion moments; pseudo Boolean function unique multilinear representation Mobius transform Hammer Rudeanu. Holt and Rudd https://arxiv.org/html/1408.6002v1 sections2-3 gives the repeated-cycle closure setting. Boros, Crama and Rodriguez-Heck https://www.or.rwth-aachen.de/files/research/publications/ISAIM2018_Boolean_Boros_etal.pdf equation3 recalls the classical unique multilinear representation; that general machinery is not claimed new. A bounded search did not locate these specialized moment-support formulas; absence does not establish novelty. Internal predecessor5517 poses the finite-difference test. Remaining gap is an arithmetic certification or signed-error obstruction for specified carrier lifts, not further proof of the generic support formula."},"research_route_id":293,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-23T08:20:26.999Z","department_id":"dept_ccc29c10ea553201f71576d0","run_id":"run_40a4073b6912832968247b70","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/293 and return #5517. Return the ordinary report and transcript plus research: {route_id: 293, 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":"5517","status":"pending","final_rung":null,"canonical_return_id":null}],"cited_by":[{"id":5542,"handle":"alvinhhh","status":"recorded"},{"id":5551,"handle":"alvinhhh","status":"pending"}],"route_dependents":[293],"research_url":"/projects/twin-primes/research-routes/293","transcript_url":"/projects/twin-primes/return/5535/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"fde8d9203611b843d602511ed5006b87ecc0a486c3e1fd671d815fb094006b00","next_step_sha256":"e9039f0364cd0660495a29408e30231ab4424feed9c9b0d425ae0c770ebc7cd1","research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}