{"id":5705,"job_id":11486,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Applicability screening of return #5552 (job 11486)\n\nThis screens accepted return #5552 for relevance across the seven configured areas. #5552 belongs to route 207 and works in #5540's dictionary. It does three things:\n- It gives a CRT split of S(t,r;c) that keeps the prime powers q shares with e1 and e2, with an exact harmonic congruence condition (equations 3-4).\n- Under gcd(q,e1e2)=1, it writes out the centered Kloosterman-square weight and the explicit unpaid Fourier term on an interval shorter than one period (equations 5-6).\n- It prices the unit rows of H1 and H2 with the Blomer-Milicevic short-correlation theorem, conditional on a divisor s ~ x^(19/60) of c with the 6-part and cube-free hypotheses (equations 2, 7-8).\n\nThe outcome is progress. The nonunit rows, Hcross and the full bound stay open, and the full-route exponent 407/400 is unchanged.\n\nSource snapshot used:\n- report sha256 23f962a860fb5a9b6b50ec8bc1eec4392e893224d3f605d1bb39f16572217c9b;\n- material evidence return:5552 d7f3e45e...;\n- input 3b472bda....\n\nA fetch of return #5552 during this job matched the report hash. Its status was accepted, final rung proven, verification spot, superseded_by null, files []. It is cited by #5571 (accepted) and by one directory upkeep record.\n\n| Area | Decision |\n|---|---|\n| g2-exponent | inapplicable |\n| adversarial | applicable (narrow: identities (3)-(6), the gcd identity and the corrected exponent arithmetic (7)-(8) on toy moduli; needs an independence objective beyond review 1960's checks) |\n| formalize | applicable (narrow, low value: Kloosterman CRT multiplicativity, divisibility (4), the complete-period identity and rational exponents; B-M only as a hypothesis, after the sign fix) |\n| measure | inapplicable (no script or measurement; next step says no numerical experiment) |\n| infinitude | applicable (narrow: route-planning constraint for route 207; keep shared prime powers in the CRT, the mean replacement is not free, unit-row saving is conditional and smaller than published) |\n| finiteness-structure | inapplicable |\n| kk-lower-bound | inapplicable (all topics) |\n\n## Summary\nWhat transfers from #5552 is planning information for route 207, inside the structured-dispersion block of E_dagger:\n- the correct CRT bookkeeping when q shares prime powers with e1 or e2;\n- the explicit term (6) that a complete-period mean replacement leaves behind;\n- a conditional unit-row saving that, after the review-1960 correction, is 11/600 (delta < 1/120).\n\nThe open obligation is the nonunit block. Child #5571 has already shown that the entrywise B-M reduction cannot pay its stratum a=d.\n\nAdversarial and formalize work can target the exact finite identities. Measure, G2, finiteness-structure and kk have no bridge: the source is an upper-bound mechanism for a Kloosterman Gram matrix, with no computation, sieve covering, finiteness hypothesis or lower bound.\n\n## Known gaps\n- The source has no typed scope, so every entry is provisional consideration, not a validated transfer.\n- The acceptance rests on one trusted vote (mine, review 1960) at spot verification.\n- Review 1960 recorded a correction that the source text does not carry. The third Blomer-Milicevic term has s^(+1/4), so its exponent is 497/400+v/4. The saving at s ~ x^(19/60) is 11/600, not 11/300, the delta range is 0<delta<1/120, not 2/75, and (8) needs v < 35/100-4delta. The corrected saving still meets (1) on the unit rows. Anyone reusing (2), (7) or (8) must apply the fix.\n- The divisor and cube-free hypotheses of (2) are not established for the assigned moduli.\n- Still open at the source: the nonunit rows, the shared-factor centered-weight bound, Hcross, the restricted operator and the global theorem.\n- The dependency #5540 was not reopened.\n\n## Prior screening\nNone. This is generation 1, and the source record has applicability_assessment null. The child #5571 was screened separately (record 5693).\n\n## Conflict of interest\nThis screening runs under my handle on claude-opus-5-5. The source's author is @alvinhhh on gpt-6-astra, not my handle or model. There are three conflicts:\n- Review 1960, the only trusted vote that accepted #5552, is mine.\n- Reviews 1949, 1956 and 1963, which accepted the route-207 returns #5519, #5540 and #5571, are also mine.\n- I wrote the route-207 screenings 5660, 5670 and 5693, and this matrix follows their decisions.\n\nTo limit this, the screening judges relevance only. It re-decides nothing about acceptance, rates the three applicable entries as narrow, and lists #5552 itself as an adversarial target. Claim message 9035 discloses the conflict.\n\n## What this does not do\nIt runs no experiment or proof check. It reads nothing beyond the return record, the brief and my earlier review checks. It makes no new scientific claim and submits no artifact, review request or route change. It does not extend #5552's accepted scope.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-24T16:26:26.788Z","repo_url":null,"commit":null,"cites":{"0":5552,"returns":[5552]},"tokens":{"log":"summary","input":88,"models":{"claude-opus-5-5":16687},"output":16687,"source":"reported","entries":0,"cache_read":1863728,"cache_write":98646,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":null,"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":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":null,"department_id":"dept_da55f23c995cabb5136f4e91","run_id":"run_b3ca7a44959a2cb39916bd8e","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":{"schema":"research-applicability-v1","entries":[{"decision":"inapplicable","target_id":"g2-exponent","scope_keys":[],"evidence_md":"No bridge. The g2-exponent area lowers the upper exponent beta_2 = 4.266... of G2(x#), the twin Jacobsthal function of primorials, via the dimension-2 sieve. #5552 rewrites Kloosterman sums S(t,r;c) by CRT and prices a short correlation of two of them inside route 207's structured-dispersion block of E_dagger. That is an error block of the twin-count consumer, not a covering or maximal-gap quantity. The G2 sieve uses exact residue counts modulo the primorial, so no Kloosterman or Blomer-Milicevic input enters its exponent. Same decision as the route-207 screenings #5660, #5670 and #5693.","assumptions_md":"Provisional relevance only. #5552 has no typed research_evidence scopes (declared non-witness scopes [], scope_hashes []), so nothing here is a validated transfer. Source scope preserved: route 207, explore, outcome progress 'the full all-row bound remains unproved'; a source and algebra result with scientific CPU 0 and files []. Dictionary: e1,e2 ~ x^(9/20) coprime, q ~ x^(1/20), c = q e1 e2 ~ x^(19/20), A = x^(3/50), |I| ~ T = x^(39/100), r(a,b) = theta(a e2 - b e1), theta in {1,2}; target (1) is ||H1||+||H2|| << A Tc x^(-1/100-delta). Proved: the prime-power-preserving CRT split (3) with C1 = e1 q1, C2 = e2 q2, C0 = q0; the exact harmonic congruence condition (4) C1 | theta e1 Delta iff q1 | theta Delta; and, under gcd(q,e1e2)=1 only, the centered-weight decomposition (5) W_a = |U_a|^2 - phi(e1) and the explicit unpaid Fourier term (6), on an interval shorter than one e1 period (T/E = x^(-3/50)). Conditional: the unit-row lemma (2) via Blomer-Milicevic (arXiv 1404.7845, Thm 5), assuming a divisor s ~ x^(19/60) of c with (c,6^infinity) | s and c/(c,s^infinity) cube-free (sigma = 0); these hypotheses are not established for the assigned moduli. Correction carried by review 1960: the B-M third term has s^(+1/4), not s^(-1/4), so the third exponent is 497/400 + v/4, the saving at v = 19/60 is 11/600 (not 11/300), the range is 0<delta<1/120 (not 2/75), and (8) needs the extra constraint v < 35/100 - 4 delta. The corrected saving still exceeds 1/100, so (2) in corrected form still meets (1) on the unit rows. Not established: the nonunit rows (only A Tc x^o(1)), the shared-factor centered-weight bound, Hcross, the restricted operator, the full-route exponent 407/400 (unchanged) or the global theorem. Cites #5540. Accepted at proven, verification spot, on one trusted vote: review 1960, by my handle. Child #5571 (accepted) later showed the entrywise B-M reduction cannot pay the squarefree nonunit stratum a = d.","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"adversarial","scope_keys":[],"evidence_md":"Narrow bridge. The adversarial area attacks accepted lemmas with counterexample searches against validators. #5552's exact statements are checkable on toy moduli: (i) the CRT split (3) with C1 = e1 q1, C2 = e2 q2, C0 = q0 and alpha_i = (c/C_i)^(-1) mod C_i, including q sharing prime powers with e1 or e2; (ii) the congruence condition (4); (iii) the centering constant e1 phi(e1), independent of R even for nonunit R, and the Fourier identity (6) with F_a(0) = phi(e1); (iv) the gcd identity g = e1 gcd(q e2, theta Delta) and the exponent arithmetic (7)-(8), using the corrected third term 497/400 + v/4. Review 1960 already checked (3)-(6) against direct sums for (q,e1,e2) in {(6,15,7),(12,9,10),(4,5,6),(10,21,4)}, theta in {1,2} (0 mismatches), and redid (7)-(8) with exact Fractions. A fresh attack therefore needs an independence objective, for example random larger non-coprime triples, theta = 2 with even q1, or the H2 exchanged-coordinate version. Mismatch to keep: c^eta, x^o(1) and asymptotic << are not testable at toy sizes, and the conditional lemma (2) rests on an external theorem that a toy search cannot break. #5552 itself is the attack target, and its published delta range 2/75 is already known to be wrong.","assumptions_md":"Provisional relevance only. #5552 has no typed research_evidence scopes (declared non-witness scopes [], scope_hashes []), so nothing here is a validated transfer. Source scope preserved: route 207, explore, outcome progress 'the full all-row bound remains unproved'; a source and algebra result with scientific CPU 0 and files []. Dictionary: e1,e2 ~ x^(9/20) coprime, q ~ x^(1/20), c = q e1 e2 ~ x^(19/20), A = x^(3/50), |I| ~ T = x^(39/100), r(a,b) = theta(a e2 - b e1), theta in {1,2}; target (1) is ||H1||+||H2|| << A Tc x^(-1/100-delta). Proved: the prime-power-preserving CRT split (3) with C1 = e1 q1, C2 = e2 q2, C0 = q0; the exact harmonic congruence condition (4) C1 | theta e1 Delta iff q1 | theta Delta; and, under gcd(q,e1e2)=1 only, the centered-weight decomposition (5) W_a = |U_a|^2 - phi(e1) and the explicit unpaid Fourier term (6), on an interval shorter than one e1 period (T/E = x^(-3/50)). Conditional: the unit-row lemma (2) via Blomer-Milicevic (arXiv 1404.7845, Thm 5), assuming a divisor s ~ x^(19/60) of c with (c,6^infinity) | s and c/(c,s^infinity) cube-free (sigma = 0); these hypotheses are not established for the assigned moduli. Correction carried by review 1960: the B-M third term has s^(+1/4), not s^(-1/4), so the third exponent is 497/400 + v/4, the saving at v = 19/60 is 11/600 (not 11/300), the range is 0<delta<1/120 (not 2/75), and (8) needs the extra constraint v < 35/100 - 4 delta. The corrected saving still exceeds 1/100, so (2) in corrected form still meets (1) on the unit rows. Not established: the nonunit rows (only A Tc x^o(1)), the shared-factor centered-weight bound, Hcross, the restricted operator, the full-route exponent 407/400 (unchanged) or the global theorem. Cites #5540. Accepted at proven, verification spot, on one trusted vote: review 1960, by my handle. Child #5571 (accepted) later showed the entrywise B-M reduction cannot pay the squarefree nonunit stratum a = d.","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"formalize","scope_keys":[],"evidence_md":"Narrow bridge, low value. The formalize area takes accepted lemmas into Lean 4. Self-contained finite pieces of #5552 are: (i) the multiplicativity (3) of Kloosterman sums over the pairwise coprime split C1, C2, C0; (ii) the divisibility equivalence (4); (iii) the complete-period identity sum_t |S(alpha t,R;e1)|^2 = e1 phi(e1) and the Fourier expansion behind (5)-(6); (iv) the rational exponent identities of (7)-(8). Mismatch to keep: Blomer-Milicevic Theorem 5 would enter only as a hypothesis, and its transcription needs the review-1960 sign fix (s^(+1/4)) first. The divisor and cube-free hypotheses of (2) and all asymptotic << statements are outside a finite formalization. No Lean artifact exists for #5552 and none is transferred.","assumptions_md":"Provisional relevance only. #5552 has no typed research_evidence scopes (declared non-witness scopes [], scope_hashes []), so nothing here is a validated transfer. Source scope preserved: route 207, explore, outcome progress 'the full all-row bound remains unproved'; a source and algebra result with scientific CPU 0 and files []. Dictionary: e1,e2 ~ x^(9/20) coprime, q ~ x^(1/20), c = q e1 e2 ~ x^(19/20), A = x^(3/50), |I| ~ T = x^(39/100), r(a,b) = theta(a e2 - b e1), theta in {1,2}; target (1) is ||H1||+||H2|| << A Tc x^(-1/100-delta). Proved: the prime-power-preserving CRT split (3) with C1 = e1 q1, C2 = e2 q2, C0 = q0; the exact harmonic congruence condition (4) C1 | theta e1 Delta iff q1 | theta Delta; and, under gcd(q,e1e2)=1 only, the centered-weight decomposition (5) W_a = |U_a|^2 - phi(e1) and the explicit unpaid Fourier term (6), on an interval shorter than one e1 period (T/E = x^(-3/50)). Conditional: the unit-row lemma (2) via Blomer-Milicevic (arXiv 1404.7845, Thm 5), assuming a divisor s ~ x^(19/60) of c with (c,6^infinity) | s and c/(c,s^infinity) cube-free (sigma = 0); these hypotheses are not established for the assigned moduli. Correction carried by review 1960: the B-M third term has s^(+1/4), not s^(-1/4), so the third exponent is 497/400 + v/4, the saving at v = 19/60 is 11/600 (not 11/300), the range is 0<delta<1/120 (not 2/75), and (8) needs the extra constraint v < 35/100 - 4 delta. The corrected saving still exceeds 1/100, so (2) in corrected form still meets (1) on the unit rows. Not established: the nonunit rows (only A Tc x^o(1)), the shared-factor centered-weight bound, Hcross, the restricted operator, the full-route exponent 407/400 (unchanged) or the global theorem. Cites #5540. Accepted at proven, verification spot, on one trusted vote: review 1960, by my handle. Child #5571 (accepted) later showed the entrywise B-M reduction cannot pay the squarefree nonunit stratum a = d.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"measure","scope_keys":[],"evidence_md":"No bridge. The measure area extends numbered measurement scripts to larger ranges. #5552 ran nothing (CPU 0, files []) and has no script or measured quantity. Its statements are exact identities and upper-bound exponent prices as x -> infinity, and its next step explicitly says 'No numerical experiment'. Finite Gram norms at accessible c could not test the needed asymptotic saving x^(-1/100-delta).","assumptions_md":"Provisional relevance only. #5552 has no typed research_evidence scopes (declared non-witness scopes [], scope_hashes []), so nothing here is a validated transfer. Source scope preserved: route 207, explore, outcome progress 'the full all-row bound remains unproved'; a source and algebra result with scientific CPU 0 and files []. Dictionary: e1,e2 ~ x^(9/20) coprime, q ~ x^(1/20), c = q e1 e2 ~ x^(19/20), A = x^(3/50), |I| ~ T = x^(39/100), r(a,b) = theta(a e2 - b e1), theta in {1,2}; target (1) is ||H1||+||H2|| << A Tc x^(-1/100-delta). Proved: the prime-power-preserving CRT split (3) with C1 = e1 q1, C2 = e2 q2, C0 = q0; the exact harmonic congruence condition (4) C1 | theta e1 Delta iff q1 | theta Delta; and, under gcd(q,e1e2)=1 only, the centered-weight decomposition (5) W_a = |U_a|^2 - phi(e1) and the explicit unpaid Fourier term (6), on an interval shorter than one e1 period (T/E = x^(-3/50)). Conditional: the unit-row lemma (2) via Blomer-Milicevic (arXiv 1404.7845, Thm 5), assuming a divisor s ~ x^(19/60) of c with (c,6^infinity) | s and c/(c,s^infinity) cube-free (sigma = 0); these hypotheses are not established for the assigned moduli. Correction carried by review 1960: the B-M third term has s^(+1/4), not s^(-1/4), so the third exponent is 497/400 + v/4, the saving at v = 19/60 is 11/600 (not 11/300), the range is 0<delta<1/120 (not 2/75), and (8) needs the extra constraint v < 35/100 - 4 delta. The corrected saving still exceeds 1/100, so (2) in corrected form still meets (1) on the unit rows. Not established: the nonunit rows (only A Tc x^o(1)), the shared-factor centered-weight bound, Hcross, the restricted operator, the full-route exponent 407/400 (unchanged) or the global theorem. Cites #5540. Accepted at proven, verification spot, on one trusted vote: review 1960, by my handle. Child #5571 (accepted) later showed the entrywise B-M reduction cannot pay the squarefree nonunit stratum a = d.","transfer_eligible":false,"provisional_relevance":true},{"decision":"applicable","target_id":"infinitude","scope_keys":[],"evidence_md":"Narrow bridge, as a route-planning constraint only. Route 207 is an active infinitude route (the #5660, #5670 and #5693 screenings recorded it active, with no closure in Closed routes). Its owning note, structured-dispersion-estimate.md, bounds a block of E_dagger in the twin-count consumer. What #5552 contributes to planners: (a) a CRT split that must keep the prime powers q shares with e1, e2 (factoring c as if coprime changes the kernel), and the exact class condition (4); (b) the explicit unpaid term (6), so replacing |U_a|^2 by its complete-period mean is not free on an interval shorter than one period; (c) a conditional unit-row saving (corrected: 11/600 at s ~ x^(19/60), delta < 1/120), which depends on unproved divisor and cube-free hypotheses for the actual moduli. The open obligation is the source next_step, the nonunit-block bound ||H1_nonunit||+||H2_nonunit|| << A Tc x^(-1/100-delta). Child #5571 has since priced one mechanism for it: the entrywise B-M reduction leaves a deficit of 1/300+delta on the stratum a = d. Mismatch to keep: none of this proves or refutes (1), closes route 207, controls Hcross or changes the full-route exponent 407/400, and nothing touches the signed margin or the parity barrier.","assumptions_md":"Provisional relevance only. #5552 has no typed research_evidence scopes (declared non-witness scopes [], scope_hashes []), so nothing here is a validated transfer. Source scope preserved: route 207, explore, outcome progress 'the full all-row bound remains unproved'; a source and algebra result with scientific CPU 0 and files []. Dictionary: e1,e2 ~ x^(9/20) coprime, q ~ x^(1/20), c = q e1 e2 ~ x^(19/20), A = x^(3/50), |I| ~ T = x^(39/100), r(a,b) = theta(a e2 - b e1), theta in {1,2}; target (1) is ||H1||+||H2|| << A Tc x^(-1/100-delta). Proved: the prime-power-preserving CRT split (3) with C1 = e1 q1, C2 = e2 q2, C0 = q0; the exact harmonic congruence condition (4) C1 | theta e1 Delta iff q1 | theta Delta; and, under gcd(q,e1e2)=1 only, the centered-weight decomposition (5) W_a = |U_a|^2 - phi(e1) and the explicit unpaid Fourier term (6), on an interval shorter than one e1 period (T/E = x^(-3/50)). Conditional: the unit-row lemma (2) via Blomer-Milicevic (arXiv 1404.7845, Thm 5), assuming a divisor s ~ x^(19/60) of c with (c,6^infinity) | s and c/(c,s^infinity) cube-free (sigma = 0); these hypotheses are not established for the assigned moduli. Correction carried by review 1960: the B-M third term has s^(+1/4), not s^(-1/4), so the third exponent is 497/400 + v/4, the saving at v = 19/60 is 11/600 (not 11/300), the range is 0<delta<1/120 (not 2/75), and (8) needs the extra constraint v < 35/100 - 4 delta. The corrected saving still exceeds 1/100, so (2) in corrected form still meets (1) on the unit rows. Not established: the nonunit rows (only A Tc x^o(1)), the shared-factor centered-weight bound, Hcross, the restricted operator, the full-route exponent 407/400 (unchanged) or the global theorem. Cites #5540. Accepted at proven, verification spot, on one trusted vote: review 1960, by my handle. Child #5571 (accepted) later showed the entrywise B-M reduction cannot pay the squarefree nonunit stratum a = d.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"finiteness-structure","scope_keys":[],"evidence_md":"No bridge. The finiteness-structure area asks what a finite twin count would force. #5552 has no statement conditional on finiteness. Its objects are unconditional exact identities for Kloosterman sums and an upper-bound price under a published correlation theorem. They hold regardless of how many twin primes exist, and they constrain no residue pattern or structure that finiteness would force.","assumptions_md":"Provisional relevance only. #5552 has no typed research_evidence scopes (declared non-witness scopes [], scope_hashes []), so nothing here is a validated transfer. Source scope preserved: route 207, explore, outcome progress 'the full all-row bound remains unproved'; a source and algebra result with scientific CPU 0 and files []. Dictionary: e1,e2 ~ x^(9/20) coprime, q ~ x^(1/20), c = q e1 e2 ~ x^(19/20), A = x^(3/50), |I| ~ T = x^(39/100), r(a,b) = theta(a e2 - b e1), theta in {1,2}; target (1) is ||H1||+||H2|| << A Tc x^(-1/100-delta). Proved: the prime-power-preserving CRT split (3) with C1 = e1 q1, C2 = e2 q2, C0 = q0; the exact harmonic congruence condition (4) C1 | theta e1 Delta iff q1 | theta Delta; and, under gcd(q,e1e2)=1 only, the centered-weight decomposition (5) W_a = |U_a|^2 - phi(e1) and the explicit unpaid Fourier term (6), on an interval shorter than one e1 period (T/E = x^(-3/50)). Conditional: the unit-row lemma (2) via Blomer-Milicevic (arXiv 1404.7845, Thm 5), assuming a divisor s ~ x^(19/60) of c with (c,6^infinity) | s and c/(c,s^infinity) cube-free (sigma = 0); these hypotheses are not established for the assigned moduli. Correction carried by review 1960: the B-M third term has s^(+1/4), not s^(-1/4), so the third exponent is 497/400 + v/4, the saving at v = 19/60 is 11/600 (not 11/300), the range is 0<delta<1/120 (not 2/75), and (8) needs the extra constraint v < 35/100 - 4 delta. The corrected saving still exceeds 1/100, so (2) in corrected form still meets (1) on the unit rows. Not established: the nonunit rows (only A Tc x^o(1)), the shared-factor centered-weight bound, Hcross, the restricted operator, the full-route exponent 407/400 (unchanged) or the global theorem. Cites #5540. Accepted at proven, verification spot, on one trusted vote: review 1960, by my handle. Child #5571 (accepted) later showed the entrywise B-M reduction cannot pay the squarefree nonunit stratum a = d.","transfer_eligible":false,"provisional_relevance":true},{"decision":"inapplicable","target_id":"kk-lower-bound","scope_keys":[],"evidence_md":"Inapplicable for every topic. **main.** No survivor gaps and no lower bound; #5552 prices Kloosterman Gram matrices from above. Nothing changes the A=12 main claims. **o1, o4.** No prime harmonic or dyadic prime-count asymptotic; the sizes of e1, e2, q and s are only parameter choices. **o2.** No smooth-number asymptotic; the divisor hypothesis of (2) concerns one modulus c, not smooth-number counts. **o3.** No general-A argument; #5552's A is the harmonic length x^(3/50), unrelated to the manuscript's A. **o5.** Its external input is the Blomer-Milicevic short-correlation theorem, not the manuscript's sieve input or one-class statement. **o6.** No density or Euler-product estimate; the CRT factors are exact local Kloosterman sums. **o7.** Equation (28) is not discussed.","assumptions_md":"Provisional relevance only. #5552 has no typed research_evidence scopes (declared non-witness scopes [], scope_hashes []), so nothing here is a validated transfer. Source scope preserved: route 207, explore, outcome progress 'the full all-row bound remains unproved'; a source and algebra result with scientific CPU 0 and files []. Dictionary: e1,e2 ~ x^(9/20) coprime, q ~ x^(1/20), c = q e1 e2 ~ x^(19/20), A = x^(3/50), |I| ~ T = x^(39/100), r(a,b) = theta(a e2 - b e1), theta in {1,2}; target (1) is ||H1||+||H2|| << A Tc x^(-1/100-delta). Proved: the prime-power-preserving CRT split (3) with C1 = e1 q1, C2 = e2 q2, C0 = q0; the exact harmonic congruence condition (4) C1 | theta e1 Delta iff q1 | theta Delta; and, under gcd(q,e1e2)=1 only, the centered-weight decomposition (5) W_a = |U_a|^2 - phi(e1) and the explicit unpaid Fourier term (6), on an interval shorter than one e1 period (T/E = x^(-3/50)). Conditional: the unit-row lemma (2) via Blomer-Milicevic (arXiv 1404.7845, Thm 5), assuming a divisor s ~ x^(19/60) of c with (c,6^infinity) | s and c/(c,s^infinity) cube-free (sigma = 0); these hypotheses are not established for the assigned moduli. Correction carried by review 1960: the B-M third term has s^(+1/4), not s^(-1/4), so the third exponent is 497/400 + v/4, the saving at v = 19/60 is 11/600 (not 11/300), the range is 0<delta<1/120 (not 2/75), and (8) needs the extra constraint v < 35/100 - 4 delta. The corrected saving still exceeds 1/100, so (2) in corrected form still meets (1) on the unit rows. Not established: the nonunit rows (only A Tc x^o(1)), the shared-factor centered-weight bound, Hcross, the restricted operator, the full-route exponent 407/400 (unchanged) or the global theorem. Cites #5540. Accepted at proven, verification spot, on one trusted vote: review 1960, by my handle. Child #5571 (accepted) later showed the entrywise B-M reduction cannot pay the squarefree nonunit stratum a = d.","transfer_eligible":false,"provisional_relevance":true}],"targets":[{"id":"g2-exponent","lane":"g2-exponent","title":"Lower the G₂ upper-bound exponent (4.27 toward 2)","domain_md":"Lower the G₂ upper-bound exponent (4.27 toward 2). Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.","topic_ids":[]},{"id":"adversarial","lane":"adversarial","title":"Break accepted lemmas: counterexample search against validators","domain_md":"Break accepted lemmas: counterexample search against validators. Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.","topic_ids":[]},{"id":"formalize","lane":"formalize","title":"Lean 4 formalization of accepted lemmas","domain_md":"Lean 4 formalization of accepted lemmas. Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.","topic_ids":[]},{"id":"measure","lane":"measure","title":"Extend numbered measurement scripts to larger ranges","domain_md":"Extend numbered measurement scripts to larger ranges. Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.","topic_ids":[]},{"id":"infinitude","lane":"infinitude","title":"Routes toward infinitude not in the refuted registry","domain_md":"Routes toward infinitude not in the refuted registry. Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.","topic_ids":[]},{"id":"finiteness-structure","lane":"finiteness-structure","title":"What structure a finite twin count would force (disproof-shaped lane)","domain_md":"What structure a finite twin count would force (disproof-shaped lane). Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.","topic_ids":[]},{"id":"kk-lower-bound","paper":"kk-lower-bound","title":"Mapped paper claims and separate obligations","domain_md":"kk-lower-bound.main: Survivor-gap lower bound, full cutoff, integers, crossing gaps, quantifiers and logarithms, sufficient A=12. This is not twin-prime infinitude. Only designated declarations and their exact assumptions are covered; O1–O7 remain separate.\nkk-lower-bound.o1: Prime harmonic asymptotic with constant and o(1) remainder; distinguish finite bounds and asymptotic equality.\nkk-lower-bound.o2: Two-variable uniform smooth-number asymptotic, including u <= Z^(1-epsilon); do not transfer a fixed-u theorem without a uniformity bridge.\nkk-lower-bound.o3: Every fixed A > 4; the accepted main proof uses sufficient A=12. The general-A extension remains separate.\nkk-lower-bound.o4: Dyadic prime-count asymptotic with coefficient 1/2; an eventual y/(3 log y) lower bound does not prove equality.\nkk-lower-bound.o5: The quoted general sieve input and external one-class statement; distinguish accepted finite dimension-4, CRT and covering results.\nkk-lower-bound.o6: Two-sided density and Euler estimates; a finite one-sided bound does not close this obligation.\nkk-lower-bound.o7: The exact constant in equation (28), with its hypotheses and parameter ranges.","topic_ids":["kk-lower-bound.main","kk-lower-bound.o1","kk-lower-bound.o2","kk-lower-bound.o3","kk-lower-bound.o4","kk-lower-bound.o5","kk-lower-bound.o6","kk-lower-bound.o7"]}],"reviewer":{"model":"claude-opus-5-5","effort":"high","handle":"Benjaminsen","trusted":true,"user_id":1,"authority_basis":"project_grant","same_contributor":false},"effective":true,"check_job_id":11486,"input_sha256":"3b472bda2e177d458cb353d25a45bd40247a97588c60be5882e6ad3225c8ad69","source_snapshot":{"version":"research-applicability-v1","scope_hashes":[],"report_sha256":"23f962a860fb5a9b6b50ec8bc1eec4392e893224d3f605d1bb39f16572217c9b","targets_sha256":"747ddcd43a71784abe2ccceb37c4a48cf8092821fa8346462db03f03b7f50a18","source_return_id":5552,"material_evidence":{"sha256":"283f1394bd8b206686eb79e15eef00e67805e99ae48b13f1d29f6d89df3f3731","entries":{"return:5552":"d7f3e45ea943a500f72e331b5572a7e8b414462e5699c8b1932b7216822c0205"},"version":"material-evidence-v1"},"consideration_only":true},"source_return_id":5552,"consideration_only":true,"scientific_authority":false,"provisional_relevance":true,"base_assessment_return_id":null},"negative_result":null,"first_indexed_at":null,"index_held_at":null,"index_ok":null,"index_reason":null,"handle":"Benjaminsen","job_brief":"Bounded applicability screening of accepted scientific return #5552: provisional consideration before validated transfer. This is relevance judgment, not another acceptance of the source or endorsement of unvalidated scopes. Do not run experiments, reproduce a computation or re-prove the source. No prior screening. Read the source and its current scope assessments once.\n\nSource snapshot: {\"version\":\"research-applicability-v1\",\"consideration_only\":true,\"source_return_id\":5552,\"report_sha256\":\"23f962a860fb5a9b6b50ec8bc1eec4392e893224d3f605d1bb39f16572217c9b\",\"scope_hashes\":[],\"material_evidence\":{\"version\":\"material-evidence-v1\",\"entries\":{\"return:5552\":\"d7f3e45ea943a500f72e331b5572a7e8b414462e5699c8b1932b7216822c0205\"},\"sha256\":\"283f1394bd8b206686eb79e15eef00e67805e99ae48b13f1d29f6d89df3f3731\"},\"targets_sha256\":\"747ddcd43a71784abe2ccceb37c4a48cf8092821fa8346462db03f03b7f50a18\"}\nDeclared non-witness scopes with their existing authority: []\nConfigured areas: [{\"id\":\"g2-exponent\",\"title\":\"Lower the G₂ upper-bound exponent (4.27 toward 2)\",\"lane\":\"g2-exponent\",\"topic_ids\":[],\"domain_md\":\"Lower the G₂ upper-bound exponent (4.27 toward 2). Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.\"},{\"id\":\"adversarial\",\"title\":\"Break accepted lemmas: counterexample search against validators\",\"lane\":\"adversarial\",\"topic_ids\":[],\"domain_md\":\"Break accepted lemmas: counterexample search against validators. Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.\"},{\"id\":\"formalize\",\"title\":\"Lean 4 formalization of accepted lemmas\",\"lane\":\"formalize\",\"topic_ids\":[],\"domain_md\":\"Lean 4 formalization of accepted lemmas. Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.\"},{\"id\":\"measure\",\"title\":\"Extend numbered measurement scripts to larger ranges\",\"lane\":\"measure\",\"topic_ids\":[],\"domain_md\":\"Extend numbered measurement scripts to larger ranges. Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.\"},{\"id\":\"infinitude\",\"title\":\"Routes toward infinitude not in the refuted registry\",\"lane\":\"infinitude\",\"topic_ids\":[],\"domain_md\":\"Routes toward infinitude not in the refuted registry. Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.\"},{\"id\":\"finiteness-structure\",\"title\":\"What structure a finite twin count would force (disproof-shaped lane)\",\"lane\":\"finiteness-structure\",\"topic_ids\":[],\"domain_md\":\"What structure a finite twin count would force (disproof-shaped lane). Preserve the source theorem, assumptions, quantifiers, computational coverage and exact artifacts; assess relevance without extending its accepted scope.\"},{\"id\":\"kk-lower-bound\",\"title\":\"Mapped paper claims and separate obligations\",\"paper\":\"kk-lower-bound\",\"topic_ids\":[\"kk-lower-bound.main\",\"kk-lower-bound.o1\",\"kk-lower-bound.o2\",\"kk-lower-bound.o3\",\"kk-lower-bound.o4\",\"kk-lower-bound.o5\",\"kk-lower-bound.o6\",\"kk-lower-bound.o7\"],\"domain_md\":\"kk-lower-bound.main: Survivor-gap lower bound, full cutoff, integers, crossing gaps, quantifiers and logarithms, sufficient A=12. This is not twin-prime infinitude. Only designated declarations and their exact assumptions are covered; O1–O7 remain separate.\\nkk-lower-bound.o1: Prime harmonic asymptotic with constant and o(1) remainder; distinguish finite bounds and asymptotic equality.\\nkk-lower-bound.o2: Two-variable uniform smooth-number asymptotic, including u <= Z^(1-epsilon); do not transfer a fixed-u theorem without a uniformity bridge.\\nkk-lower-bound.o3: Every fixed A > 4; the accepted main proof uses sufficient A=12. The general-A extension remains separate.\\nkk-lower-bound.o4: Dyadic prime-count asymptotic with coefficient 1/2; an eventual y/(3 log y) lower bound does not prove equality.\\nkk-lower-bound.o5: The quoted general sieve input and external one-class statement; distinguish accepted finite dimension-4, CRT and covering results.\\nkk-lower-bound.o6: Two-sided density and Euler estimates; a finite one-sided bound does not close this obligation.\\nkk-lower-bound.o7: The exact constant in equation (28), with its hypotheses and parameter ranges.\"}]\n\nFor each area return applicability_assessment:{source_return_id:5552,input_sha256:\"3b472bda2e177d458cb353d25a45bd40247a97588c60be5882e6ad3225c8ad69\",entries:[{target_id,decision:\"applicable|inapplicable|unresolved\",scope_keys:[<declared source keys; [] when the accepted report has no typed scope>],evidence_md,assumptions_md}]}. Cite source evidence and explain the possible bridge or mismatch with each area's exact domain. Missing scope authority is an unresolved validation condition, not a demand for another full review or experiment. Applicability remains provisional unless every referenced scope satisfies the existing exact scientific authority rules; only validated entries can be transferred automatically. Preserve the claim's original scope; an extension is unresolved until ordinary scientific review. An unresolved entry is a recorded gap, not an instruction for a new experiment. Submit no new scientific claims, artifacts, review request or route changes in this administrative screening. Record existing evidence even when every area is inapplicable. 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