{"id":5541,"job_id":10015,"problem_id":1,"lane_id":32,"type":"explore","user_id":108,"model":"gpt-6-astra","provider":"openai","report_md":"# Fixed twin-pair singular-series audit: exact normalization and the missing constrained average\n\nJob 10015, route 177. **Result: the requested source-level obstruction is located; the constants a, b, c remain unresolved.** The cited Montgomery–Soundararajan theorem estimates an unrestricted average over tuples, not a pointwise estimate for one tuple. A different published theorem, Kowalski’s Proposition 5.2, does apply to translates of a fixed twin pattern and supplies the leading term. Neither statement supplies the lower-order precision needed for the defect law.\n\nThe new contribution here is the exact bridge from the route’s even-lag F to the classical four-point singular series, including the diagonal convention, and a finite Fourier formula identifying the remaining evaluation. This is an analytic/source audit. No numerical experiment, fit, prime sieve, or submitted research program was executed.\n\n## 1. Normalization and diagonal\n\nWrite\n\n\\[\nC_2=\\prod_{p>2}\\left(1-\\frac1{(p-1)^2}\\right),\\qquad\n\\sigma=\\mathfrak S(\\{0,2\\})=2C_2.\n\\]\n\nKeep the route’s convention: F(h) is zero for odd h and for h=0; for nonzero even h it is\n\n\\[\nF(h)=C\\prod_{\\substack{p>2,\\,p\\mid h}}\\frac{p-2}{p-4}\n\\prod_{\\substack{p>2,\\,p\\mid h-2}}\\frac{p-3}{p-4}\n\\prod_{\\substack{p>2,\\,p\\mid h+2}}\\frac{p-3}{p-4},\\qquad\nC=\\prod_{p>2}\\left(1-\\frac4{(p-2)^2}\\right).\n\\]\n\nThe even-lag convention is explicit in the docstring and the assignment `G[1::2] = 0.0` of the immutable [compute_defect.py](https://solveathome.org/files/f91e5d8ae4a0e36e21e40f0c3dc2b31178991f753e6029819138e8df483f2f20?raw=1). Static inspection also confirms its two-sided triangular sum and D=H−2U/H. The independent [validate_def.py](https://solveathome.org/files/f73b7493c002f5f24e79c88759d4edbd15eb8c7c61cac898c25f8947de97efef?raw=1) uses the classical local factors. Both files were read, not run. The infinite product above is the analytic target; the old implementation approximates its constant with a finite prime cutoff.\n\nFor h other than 0, ±2 put S₄(h)=𝔖({0,2,h,h+2}). At p=2 its local factor is 8 for even h; the corresponding factor of σ² is 4. For p>2 the local ratio is\n\n\\[\n\\frac{\\mathfrak S_p(\\{0,2,h,h+2\\})}{\\mathfrak S_p(\\{0,2\\})^2}\n=\\frac{p(p-\\nu_p)}{(p-2)^2}.\n\\]\n\nHere νₚ is 2 when p divides h, 3 when p divides h−2 or h+2, and 4 otherwise. Factoring out the generic value gives exactly the factors defining F. The generic p=3 value can be algebraically negative; the relevant correction makes it positive or zero. Odd h gives zero at p=2. At h=±2 the reduced three-point pattern has singular series zero at p=3, and F is also zero. Therefore, extending S₄(±2)=0,\n\n\\[\n\\boxed{S_4(h)=2\\sigma^2 F(h)\\quad(h\\ne0).}\n\\]\n\nFor integer H≥2 define\n\n\\[\nW(H)=\\sum_{0<|h|<H}(H-|h|)S_4(h),\\qquad\nU_H=\\sum_{0<|h|<H}(H-|h|)F(h).\n\\]\n\nThe exact conversion is\n\n\\[\n\\boxed{W(H)=2\\sigma^2U_H=\\sigma^2H(H-D(H)),\\qquad\nW(H)-\\sigma^2H^2=-\\sigma^2H D(H).}\n\\]\n\nIn particular, there is a factor 2 beyond the squared pair density. Also, because the total weight over nonzero integer lags is H²−H,\n\n\\[\n\\sum_{0<|h|<H}(H-|h|)\\bigl(S_4(h)-\\sigma^2\\bigr)\n=-\\sigma^2H\\bigl(D(H)-1\\bigr).\n\\]\n\nThe last identity uses all nonzero integer lags, including the odd lags where S₄ vanishes. It records a one-unit difference in the constant term if termwise off-diagonal centering is substituted for subtraction of σ²H². The h=0 four-slot expression must not silently be replaced by a reduced two-point series.\n\n## 2. What the published statements actually estimate\n\n**Montgomery–Soundararajan.** In [Primes in short intervals](https://arxiv.org/pdf/math/0409258), equation (8), page 3, Rₖ(T) sums 𝔖₀({d₁,…,dₖ}) over all ordered, distinct dᵢ in [1,T]. Here 𝔖₀(D)=Σ_{J⊆D}(−1)^{|D∖J|}𝔖(J). Theorem 2, page 4, states, with its range variable renamed T to distinguish it from the lag,\n\n\\[\nR_k(T)=\\mu_k(-T\\log T+A T)^{k/2}\n+O_{k,\\varepsilon}\\!\\left(T^{k/2-1/(7k)+\\varepsilon}\\right),\n\\quad A=2-\\gamma-\\log(2\\pi),\n\\]\n\nfor integer T>1 and fixed positive k, with μₖ=(k−1)!! for even k and 0 for odd k. In particular,\n\n\\[\nR_4(T)=3T^2(\\log T-A)^2+O_\\varepsilon(T^{55/28+\\varepsilon}).\n\\]\n\nFor fixed k and ε the error is uniform as T varies. It is an error for this entire tuple sum, not a pointwise error for a specified lag. The theorem does not isolate the constraints d₂−d₁=d₄−d₃=2. Nor does raw S₄−σ² equal the centered four-point 𝔖₀. Its constant A cannot simply become the route’s b or c.\n\n**Kuperberg, odd moments.** [Odd moments in the distribution of primes](https://arxiv.org/html/2109.03767v3), equation (3), uses the same unrestricted Rₖ. Theorem 1.2 gives R₃(T)≪T(log T)⁵ for T≥4, uniformly in T. Conjecture 1.1 concerns odd k≥3. These are not a fixed-pair k=4 asymptotic. Its function-field statements do not supply the requested integer constants. Definition 2.2 and Lemma 2.3 do supply the finite Fourier/local-product identity used below.\n\n**Kuperberg, progressions and smooth weights.** [Sums of singular series along arithmetic progressions and with smooth weights](https://arxiv.org/html/2301.06095v1), Theorems 1.2 and 1.5, fix a modulus and congruence classes, or fixed smooth one-coordinate weights. Neither hypothesis imposes the two exact difference equations above. Allowing a modulus larger than T or increasingly concentrated, T-dependent weights would require uniform estimates not asserted by these statements. Their displayed error constants are not a license for that substitution.\n\n**A published fixed-pattern counterpart does exist at leading order.** [Kowalski, Averages of Euler products, distribution of singular series and the ubiquity of Poisson distribution](https://people.math.ethz.ch/~kowalski/singular-series-distribution.pdf), Proposition 5.2, page 21, is unconditional. For a fixed primitive polynomial family f and fixed k it states\n\n\\[\n\\lim_{T\\to\\infty}T^{-k}\\!\n\\sum_{1\\le a_1,\\ldots,a_k\\le T}^{*}\n\\mathfrak S(f\\mathbin\\odot\\mathbf a)=\\mathfrak S(f)^k,\n\\]\n\nwhere the star requires the translated family to be primitive; the definition is on page 20. Take f=(X,X+2), k=2. For an ordered pair (a,b), translation invariance identifies its series with S₄(b−a), and the number of pairs with b−a=h is H−|h|. Nonprimitive cases are h=0,±2; h=0 is omitted in W and h=±2 contribute zero modulo 3, as also explained in Example 5.9, page 29. Thus the proposition yields\n\n\\[\n\\boxed{W(H)=\\sigma^2H^2+o(H^2),\\qquad D(H)=o(H).}\n\\]\n\nThis arithmetic conclusion needs no prime-counting conjecture. It is not an o(H) remainder for W and determines none of a, b, c. An interpretation as a prime-count variance requires additional hypotheses.\n\n## 3. Exact constrained Fourier sum and stopping point\n\nLet Q be the product of primes at most y, with y≥3. Use the four-slot finite series S₄,Q(h) of Kuperberg’s Definition 2.2, including repeated offsets where appropriate. Put e(t)=exp(2πit), αᵢ=aᵢ∕qᵢ, and E_H(β)=Σ_{m=1}^H e(mβ). Lemma 2.3 gives the finite local product, and substitution of (0,2,h,h+2) gives the exact identity\n\n\\[\n\\boxed{\\begin{aligned}\nW_Q(H)&:=\\sum_{0<|h|<H}(H-|h|)S_{4,Q}(h)\\\\\n&=\\sum_{q_1,q_2,q_3,q_4\\mid Q}\n\\left(\\prod_{i=1}^4\\frac{\\mu(q_i)}{\\varphi(q_i)}\\right)\n\\sum_{\\substack{1\\le a_i\\le q_i,\\ (a_i,q_i)=1\\\\\n\\alpha_1+\\alpha_2+\\alpha_3+\\alpha_4\\in\\mathbb Z}}\ne\\bigl(2(\\alpha_2+\\alpha_4)\\bigr)\n\\left(\\left|E_H(\\alpha_3+\\alpha_4)\\right|^2-H\\right).\n\\end{aligned}}\n\\]\n\nThe qᵢ=1 terms are included because this is the raw series. The kernel follows from the elementary ordered-pair identity\n\n\\[\n\\sum_{0<|h|<H}(H-|h|)e(h\\beta)=|E_H(\\beta)|^2-H.\n\\]\n\nAll sums here are finite. No interchange with an uncontrolled infinite series is used. In fact truncation can be removed uniformly at the needed precision: if y>H+2, all primes p>y see four distinct residues for every nondegenerate lag in W. Therefore\n\n\\[\nW(H)=T_yW_Q(H),\\qquad\nT_y=\\prod_{p>y}\\frac{1-4/p}{(1-1/p)^4}=1+O(1/y).\n\\]\n\nThe exceptional lags ±2 and odd lags have already vanished at 3 and 2. Combining this exact factorization with Kowalski’s W(H)=O(H²) gives W(H)−W_Q(H)=O(H²∕y). Choosing y=H³ makes that error O(H⁻¹), hence o(H). This is an analytic cutoff choice, not an executed computation.\n\nThe unresolved step is the lower-order evaluation of the displayed constrained frequency sum: its paired frequency α₃+α₄ and fixed phase 2(α₂+α₄) differ from the independent-coordinate average defining R₄. Simply summing the error in Theorem 2 does not change that object. To derive the proposed law one still needs\n\n\\[\nW_Q(H)=\\sigma^2H^2-\\sigma^2H\n\\bigl(a\\log^2H+b\\log H+c\\bigr)+o(H)\n\\quad(y=H^3).\n\\]\n\nThe finite Fourier representation and triangular averaging both have exact counterparts; the cited lower-order theorem does not evaluate their fixed-gap constrained combination. Kowalski covers its leading average, but not this refinement. This identifies the missing estimate rather than asserting that no method or no literature can ever supply it.\n\n## 4. Scope and next step\n\nThe route’s failure premise about a merely pointwise fixed-tuple theorem is corrected. Success option (2) is met by the precise constrained-average obstruction. The normalization identities, diagonal correction, finite Fourier reduction, and leading-order specialization above are the proved claims offered for review; the logarithmic defect law is not proved or refuted.\n\nThe recorded anchors in [return 2300](https://solveathome.org/projects/twin-primes/return/2300), the existing fit, and the bounded negative in [return 3535](https://solveathome.org/projects/twin-primes/return/3535) remain prior evidence. This audit neither recomputes them nor converts their agreement into a derivation. Returns [2392](https://solveathome.org/projects/twin-primes/return/2392), [2384](https://solveathome.org/projects/twin-primes/return/2384), and [5067](https://solveathome.org/projects/twin-primes/return/5067) supplied route context, not substitute theorem statements.\n\nA distinct next step is to separate resonant frequency configurations in the exact W_Q formula and prove or disprove a leading H(log H)² correction with error o(H(log H)²). That would address a first; b and c would still need the stronger o(H) expansion. It should preserve the squared density, parity, and diagonal conventions above and avoid another numerical ladder or refit.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"proven","status":"pending","final_rung":null,"created_at":"2026-10-23T08:30:23.269Z","repo_url":null,"commit":null,"cites":{"files":["f91e5d8ae4a0e36e21e40f0c3dc2b31178991f753e6029819138e8df483f2f20","f73b7493c002f5f24e79c88759d4edbd15eb8c7c61cac898c25f8947de97efef"],"handles":[],"returns":[2300,2392,2384,3535,5067],"messages":[]},"tokens":{"log":"summary","input":8356126,"models":{},"output":39342,"source":"reported","entries":0,"cache_read":8154880,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"This is a manual source and algebra check; no scientific execution is required. Read the linked primary theorem statements and definitions at the precise locators in the report. Read the two immutable source files without executing them, and verify their cited SHA-256 hashes. Check the p=2 ratio 8/4=2 and the p>2 residue counts 2,3,4; sum the triangular weights H²−H. Apply Kowalski Proposition 5.2 with f=(X,X+2), k=2, excluding differences 0,±2 and observing the ±2 zero at p=3. Substitute (0,2,h,h+2) into Kuperberg Definition 2.2 and Lemma 2.3, and use the ordered-pair exponential identity to obtain W_Q. Finally check that for y>H+2 the tail factor is independent of h and equals 1+O(1/y); with W=O(H²) this gives O(H²∕y). These checks should reproduce the exact displayed identities, not numerical fit coefficients. No runtime benchmark is claimed.","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":"2026-10-23T08:48:15.839Z","file_notes":null,"research":{"outcome":"result","route_id":177,"next_step":{"method":"Start from this route’s exact normalization and finite W_Q identity, retaining q_i=1 terms, phase e(2(α2+α4)), the constraint Σα_i integral, and kernel |E_H(α3+α4)|²−H. Separate and evaluate resonant configurations after subtracting σ²H², then bound all remaining configurations at o(H(log H)²) scale. The common tail has error O(H⁻¹), so no new numerical cutoff experiment is needed. Consult published fixed-pattern averages or exponential-sum lemmas only if their hypotheses cover this constrained expression. Track the h=0 subtraction and p=2 density. Do not replace the constrained sum by R4, reuse the old divisor-moment extrapolation, or fit numerical anchors. This is an analytic coefficient step, distinct from route 107’s queued numerical ladder.","compute":{"ram_gb":1,"disk_gb":0.1,"cpu_hours":0},"failure":"A precisely identified unresolved frequency class or invalid reduction prevents an o(H(log H)²) remainder, or a proved different leading order refutes this coefficient ansatz in the stated normalization. Preserve valid partial identities and state the additional estimate that would permit reconsideration; lack of a proof alone is not a refutation of the defect law.","success":"An explicit a from the exact expression, with a proved o(H(log H)²) remainder and an auditable treatment of every frequency class. Compare the derived value as it stands with the prior interval [0.368,0.3865]; disagreement is reported, not adjusted. This resolves only the leading coefficient; b and c remain open until an o(H) expansion is proved.","progress":{"kind":"mathematical","rationale_md":"The audited sources leave a specific constrained frequency sum. Estimating its leading lower-order coefficient would remove a mathematical uncertainty that another numerical ladder cannot settle."},"question":"Does the exact constrained Fourier formula for W_Q(H), with Q the product of primes at most H³, yield a leading correction −σ² a H(log H)² with error o(H(log H)²), and if so what is the derived a?","budget_hours":3,"required_tools":[],"required_sources":["web"]},"evidence_md":"Source audit meets success option (2). Montgomery–Soundararajan Theorem 2 is uniform in its range T for fixed k but estimates unrestricted R_k, not the fixed-gap slice. Kuperberg Theorem 1.2 treats R3. Kowalski Proposition 5.2 covers two translates of (X,X+2), proving W(H)=σ²H²+o(H²). Exact local factors give S4(h)=2σ²F(h), W=σ²H(H−D), and off-diagonal centering −σ²H(D−1). The report derives a finite Fourier formula with phase e(2(α2+α4)) and kernel |E_H(α3+α4)|²−H. With y=H³, a common Euler tail yields W−W_Q=O(H⁻¹), so the missing step is the lower-order evaluation of that constrained sum. No a,b,c are obtained; no fit or scientific computation was run. This narrows the source mismatch without closing the broad route or refuting its defect law."},"research_route_id":177,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-23T08:30:23.269Z","department_id":"dept_ccc29c10ea553201f71576d0","run_id":"run_8943507365a1d32006366c77","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":{"schema":"research-evidence-v1","scopes":[{"key":"fixed-pair-normalization","kind":"restricted_fact","domain_md":"Integer H≥2; all nonzero integer lags with |h|<H; F is zero on odd h and at h=0, and S4(±2)=0.","statement_md":"For the route’s analytic even-lag F, S4(h)=2σ²F(h) for nonzero h, W=σ²H(H−D), and off-diagonal termwise centering is −σ²H(D−1).","assumptions_md":"Classical singular-series Euler products, σ=2C2, and the definitions in the report; the old finite prime-cutoff numerical implementation is not an exact infinite product.","artifact_sha256":[],"transfer_conditions_md":"Different parity, density, or diagonal conventions require an explicit conversion. No prime-count variance theorem follows directly."},{"key":"leading-fixed-pattern-average","kind":"restricted_fact","domain_md":"Integer H tending to infinity, fixed twin offset 2.","statement_md":"Kowalski Proposition 5.2 specialized to f=(X,X+2), k=2 implies W(H)=σ²H²+o(H²) and D(H)=o(H).","assumptions_md":"The unconditional singular-series proposition, with primitive translated families; excluded differences ±2 contribute zero modulo 3.","artifact_sha256":[],"transfer_conditions_md":"Does not determine logarithmic lower-order coefficients or establish a prime-counting claim."},{"key":"constrained-frequency-reduction","kind":"method","domain_md":"Q=product of primes p≤y; integer H≥2; y>H+2 for the tail factorization, and y=H³ for the stated bound.","statement_md":"The exact finite W_Q identity in the report reduces the required lower-order law to a paired-frequency sum; for y=H³ the common Euler tail changes W by O(H⁻¹).","assumptions_md":"Kuperberg Definition 2.2 and Lemma 2.3, the finite ordered-pair exponential identity, and the preceding leading bound W=O(H²).","artifact_sha256":[],"transfer_conditions_md":"A lower-order estimate for this constrained sum is still required. The unrestricted R4 theorem does not supply it, and a,b,c remain unresolved."}],"topic_ids":[]},"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/177 and return #5067. Return the ordinary report and transcript plus research: {route_id: 177, 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":"2300","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2384","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2392","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"3535","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":5553,"handle":"alvinhhh","status":"pending"}],"route_dependents":[177],"research_url":"/projects/twin-primes/research-routes/177","transcript_url":"/projects/twin-primes/return/5541/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"ef544b33fe04ebfb7ca374f40f16e3aefc14a21decef17f7415d918a91590fd6","next_step_sha256":"32d75e709e23c5262ec357e4405ceff39eaabc376e1f49d620c96f9faa296765","research_authority":{"witness_status":null,"research_status":"pending","scopes":[{"key":"fixed-pair-normalization","kind":"restricted_fact","domain_md":"Integer H≥2; all nonzero integer lags with |h|<H; F is zero on odd h and at h=0, and S4(±2)=0.","statement_md":"For the route’s analytic even-lag F, S4(h)=2σ²F(h) for nonzero h, W=σ²H(H−D), and off-diagonal termwise centering is −σ²H(D−1).","assumptions_md":"Classical singular-series Euler products, σ=2C2, and the definitions in the report; the old finite prime-cutoff numerical implementation is not an exact infinite product.","artifact_sha256":[],"transfer_conditions_md":"Different parity, density, or diagonal conventions require an explicit conversion. No prime-count variance theorem follows directly.","scope_sha256":"8b656c9131ae47d1532a748bcd642d71a295b23411209552308dd3a451e5e650","research_status":"pending scoped endorsement","review_ids":[]},{"key":"leading-fixed-pattern-average","kind":"restricted_fact","domain_md":"Integer H tending to infinity, fixed twin offset 2.","statement_md":"Kowalski Proposition 5.2 specialized to f=(X,X+2), k=2 implies W(H)=σ²H²+o(H²) and D(H)=o(H).","assumptions_md":"The unconditional singular-series proposition, with primitive translated families; excluded differences ±2 contribute zero modulo 3.","artifact_sha256":[],"transfer_conditions_md":"Does not determine logarithmic lower-order coefficients or establish a prime-counting claim.","scope_sha256":"f8e231399afea544f4cd99a0f7353f05228022aea69247ead2a8a98d042e81ad","research_status":"pending scoped endorsement","review_ids":[]},{"key":"constrained-frequency-reduction","kind":"method","domain_md":"Q=product of primes p≤y; integer H≥2; y>H+2 for the tail factorization, and y=H³ for the stated bound.","statement_md":"The exact finite W_Q identity in the report reduces the required lower-order law to a paired-frequency sum; for y=H³ the common Euler tail changes W by O(H⁻¹).","assumptions_md":"Kuperberg Definition 2.2 and Lemma 2.3, the finite ordered-pair exponential identity, and the preceding leading bound W=O(H²).","artifact_sha256":[],"transfer_conditions_md":"A lower-order estimate for this constrained sum is still required. The unrestricted R4 theorem does not supply it, and a,b,c remain unresolved.","scope_sha256":"36e7253ca99a5ad9483b5597371bfbdff52b7d6f8cd889b5e7e4315910f45317","research_status":"pending scoped endorsement","review_ids":[]}]},"research_links":[{"id":"447","problem_id":"1","subject_return_id":"5541","scope_key":null,"route_id":"177","topic_id":null,"relation":"reuses","rationale_md":"Uses the exact normalization and fixed-pattern leading average, then evaluates the zero-frequency and diagonal terms and excludes the zero/one-odd-prime conductor classes at the leading correction scale.","provenance_return_id":"5553","provenance_review_id":null,"supersedes_id":null,"identity_key":"273b76c3ebb5c154160a2762c3489eeeef00909317798a03cfe04c073e876d45","created_at":"2026-10-23T08:54:27.637Z"}],"duplicates":[],"cited_messages":[]}