{"id":5539,"job_id":11173,"problem_id":1,"lane_id":2,"type":"explore","user_id":108,"model":"gpt-6-astra","provider":"openai","report_md":"# Route 273: the smooth divisor endpoint estimate and the lam0 band transfer\n\n**Author proof claim, submitted for independent review.** The missing endpoint estimate for `I_y*I_y` follows from Harper's smooth-indicator theorem, a hyperbola decomposition, and the multiplicative large sieve. The imprimitive-character correction is written explicitly; it is not an application of an indicator theorem to divisor weights. The resulting estimate also gives the lam0 band-discrepancy input at every fixed sublinear level. It does not establish the complete two-branch cell, the balanced endpoint, the large-M tail, the three-branch cell, the variance constant, or a twin-prime conclusion.\n\n## 1. Endpoint statement and centering\n\nLet `I_y(n)` indicate positive y-smooth integers, let `tau` be the ordinary divisor function, and set\n\n`a_y(n)=(I_y*I_y)(n)=tau(n) I_y(n)`.\n\nThe equality holds because every divisor of a smooth integer is smooth. For reduced residues define\n\n`Delta_a(x;q,b)=sum_{n<=x,n=b mod q} a_y(n) - phi(q)^(-1) sum_{n<=x,(n,q)=1} a_y(n)`,\n\n`V_a(x,Q)=sum_{q<=Q} sum_{b mod q,(b,q)=1} |Delta_a(x;q,b)|^2`.\n\nFor each fixed `eta>0` and `H>0`, uniformly for sufficiently large x, `y>=x^eta` and `1<=Q<=x`,\n\n**`V_a(x,Q) <<_{eta,H} x^2 (log(2x))^(-H) + x Q (log(2x))^6`.** (1)\n\nThe assigned range `y>=x^(1/3), Q<=x^(1-epsilon)` is included. In that range the second term divided by `x^2` tends to zero faster than every fixed inverse logarithm. Constants in the logarithm-saving term inherit Harper's ineffectiveness. The exponent 6 is an upper-bound cost, not a sharp variance asymptotic. If `y>x`, the sequence is simply `tau(n)` on `n<=x`.\n\nWrite `ell=log(2x)`, and for a primitive nonprincipal character chi of conductor r put\n\n`S(t,chi)=sum_{n<=t} I_y(n) chi(n)`, `A(t,chi)=sum_{n<=t} a_y(n) chi(n)`.\n\nCharacter orthogonality gives the exact equality\n\n`V_a(x,Q)=sum_{q<=Q} phi(q)^(-1) sum_{chi mod q,chi!=chi_0} |sum_{n<=x}a_y(n)chi(n)|^2`.\n\nThus the principal term is the coprime mean written above throughout. A nonprincipal character has primitive conductor `r>1`; there is no principal-conductor term in this equality.\n\n## 2. The two source inputs\n\nHarper, [arXiv:1208.5992](https://arxiv.org/pdf/1208.5992), Theorem 2, printed p.3, bounds the indicator variance by\n\n`C_J Psi(t,y)^2 [exp(-c u/log^2(u+1)) (log t)^(-J) + y^(-c)] + C Psi(t,y) R`,\n\non `log^K t<=y<=t` and `R<=Psi(t,y)`. Its arbitrary-J form is ineffective. The same paper's Multiplicative Large Sieve 1, printed p.18, gives\n\n`sum_{r<=R} (r/phi(r)) sum_{chi mod r primitive} |sum_{n<=z} v_n chi(n)|^2 <= (floor(z)+3R^2) sum_{n<=z}|v_n|^2`.\n\nOnly these two analytic estimates are imported. The convolution, removal of coprimality restrictions, cutoff sums, and maximal transfer below are derived here. The large-sieve statement was also checked against Pascadi, [Smooth numbers in arithmetic progressions to large moduli](https://www.cambridge.org/core/journals/compositio-mathematica/article/smooth-numbers-in-arithmetic-progressions-to-large-moduli/D8E4EE07BDAA6D41CCD8C5455D81CD7A), Lemma 3.4.\n\nWe use two elementary bounds. First, `tau(n)^2<=d_4(n)`, since at a prime power the inequality is `(j+1)^2<=binom(j+3,3)`. Summing the four-fold convolution gives\n\n`sum_{n<=z} a_y(n)^2 <= sum_{n<=z}tau(n)^2 <= z(1+log z)^3`. (2)\n\nSecond, `sum_{d<=D}1/phi(d)<=phi(u)phi(v)`.\n\n## 3. Removing induced-character restrictions exactly\n\nSuppose chi modulo `q=rd` is induced by primitive chi* modulo r. Then `chi(n)=chi*(n)1_{(n,d)=1}`. Define a multiplicative coefficient c supported on cube-free integers by\n\n`c(p)=-2`, `c(p^2)=1`, and `c(p^j)=0` for `j>=3`.\n\nLet `m(k)=rad(k)`. Multiplication of the local Dirichlet series `(1-chi*(p)z)^(-2)` by `(1-chi*(p)z)^2` gives the exact finite identity\n\n`sum_{n<=x,(n,d)=1}a_y(n)chi*(n) = sum_{k<=x,m(k)|d} c(k)I_y(k)chi*(k) A(x/k,chi*)`. (3)\n\nPrimes dividing r disappear because `chi*(p)=0`; primes above y already have no factor. This handles repeated prime factors of q as well as squarefree q. The coefficient `-2` and the `p^2` term are essential.\n\nUse weighted Cauchy with weights `|c(k)|/(k)`. The first factor satisfies\n\n`B_d=sum_{m(k)|d}|c(k)|/(k) = product_{p|d}(1+1/p)^2 <= (1+log d)^2`.\n\nThe other factor is `sum_{m(k)|d}|c(k)| k |A(x/k,chi*)|^2`, after harmlessly discarding smoothness and `|chi*(k)|<=1` restrictions. Furthermore,\n\n`sum_{d<=Q/r,m|d}1/phi(d) <= phi(m)^(-1) sum_{e<=Q/(rm)}1/phi(e) << ell/phi(m)`.\n\nConsequently, for any selected set C of conductors greater than 1, its part of the variance is at most\n\n`C ell^3 sum_{k<=x, k cube-free, m(k)<=Q} [|c(k)| k/phi(m(k))] sum_{r in C, r<=Q/m(k)} phi(r)^(-1) sum_{chi* mod r primitive} |A(x/k,chi*)|^2`. (4)\n\nAll terms after Cauchy are nonnegative. Crucially, the upper limit on r is `Q/rad(k)`, not Q independently of k.\n\nThe following sums converge absolutely, with constants independent of y and x:\n\n`C_0=sum_k |c(k)|/[k phi(m(k))] < infinity`,\n\n`C_1=sum_k |c(k)|/[sqrt(k) phi(m(k))] < infinity`,\n\n`C_2=sum_k |c(k)|/[m(k) phi(m(k))] < infinity`. (5)\n\nTheir Euler factors are respectively\n\n`1+2/[p(p-1)]+1/[p^2(p-1)]`,\n\n`1+2/[sqrt(p)(p-1)]+1/[p(p-1)]`,\n\n`1+3/[p(p-1)]`.\n\nIn particular `sum_{k>K}|c(k)|/[k phi(m(k))] <= C_1 K^(-1/2)`.\n\n## 4. Large conductors\n\nOn a dyadic conductor interval `RR0} << (x^2/R0+xQ) ell^6`. (6)\n\nNo density estimate for smooth numbers and no factorization theorem for a generic divisor-bounded function is being assumed in this step.\n\n## 5. Small conductors and the large-k tail\n\nTake `R0=ell^B` and a second cutoff `K0=ell^D`, with B,D fixed below. For `k<=K0`, put `z=x/k`. All arguments `t` between `sqrt(z)` and z have `log t` comparable to ell. When `y<=t`, Harper's hypotheses hold eventually: `y>=x^eta` exceeds every fixed power of `log t`, and the elementary lower bound\n\n`Psi(t,y)>=floor(min(t,y))`\n\nalready exceeds R0. Thus no fixed-density theorem is needed. Since `Psi(t,y)<=t`, `y^(-c)` is smaller than every fixed inverse power of ell, and the exponential factor is at most 1, Harper gives\n\n`sum_{1t`, the sequence is all integers. Each reduced-class discrepancy from its coprime mean is at most 1, so its total variance is `O(R0^2)<=O(tR0)`. This proves (7) in that case too.\n\nThe exact Dirichlet-hyperbola identity is\n\n`A(z,chi*)=2 sum_{m<=sqrt(z)} I_y(m)chi*(m) S(z/m,chi*) - S(sqrt(z),chi*)^2`.\n\nTake the weighted Euclidean norm over the primitive characters in (7). Multiplication by `I_y(m)chi*(m)` contracts that norm. Minkowski, `sum_{m<=sqrt(z)}1/m<K0`, use the trivial bound `|A(x/k,chi*)|<=(x/k)(1+log(x/k))`. The weighted number of primitive characters with `15`, put `G_p=2p/(p-4)`; the local correction is\n\n`(1+G_p z)(1-z)^2 = 1+(G_p-2)z+(1-2G_p)z^2+G_p z^3`.\n\nFor `p<=y,p|30` it is `(1-z)^2`; for `p>y` it is 1. Thus `b(p)=O(1/p)`, while `b(p^2),b(p^3)=O(1)` away from the finitely many small primes. Uniformly in y,\n\n`sum_k |b(k)|/(k^(3/4))=1` define\n\n`f_T(n)=sum_{k|n,k<=T} b(k)a_y(n/k)`,\n\n`A_T(n)=sum_{k|n,k>T}|b(k)|a_y(n/k)`, and `A_all(n)=sum_{k|n}|b(k)|a_y(n/k)`.\n\nUsing `a_y(n/k)<=tau(n)`, `tau(lm)<=tau(l)tau(m)`, (2), and (11), direct expansion gives\n\n`sum_{n<=2E} A_T(n)^2 << E (log(2E))^3 T^(-1/4)`,\n\n`sum_{n<=2E} A_all(n)^2 << E (log(2E))^3`. (12)\n\nThe first estimate uses `d,e>T` and `(de)^delta>=T^(2delta)`; the second uses the same convergent sum without a cutoff. The extra divisor-count weight has therefore been priced in both squared estimates.\n\n## 7. A maximal band bound with all losses priced\n\nFor reduced residues let\n\n`D_0(t;q,b)=sum_{E0`, uniformly for `y>=E^(1/3)` and `Q<=E^(1-epsilon)`, the preceding endpoint estimate implies\n\n**`V_0(E,Q) <<_{epsilon,A} ell_E^(-A)+(Q/E)ell_E^(3A+24) <<_{epsilon,A} ell_E^(-A)`,** (13)\n\nwhere `ell_E=log(2E)` and the last bound allows a change of its implicit constant. Here is the complete maximal step.\n\nFor a centered unweighted tail over `(E,t]`, nonnegativity of A_T gives a squared residue norm at most `4 sum_b (sum_{E1` contributes to neither the reduced class nor its coprime mean; otherwise it permutes the reduced residues. Apply (1) at `t/k`, followed by Minkowski and\n\n`sum_{k<=T}|b(k)|/(k)<<1`, `sum_{k<=T}|b(k)|/(sqrt(k))<=(t/k)^(1/4)`. The squared endpoint norm, and hence the norm of the band difference at any fixed endpoint, is\n\n`<<_H E^2 ell_E^(-H)+EQ ell_E^6 T^(1/2)`. (15)\n\nUse a grid of `N` intervals J of length `h=E/N` on `[E,2E]`. At grid points the squared maximum is bounded by their summed endpoint energies, costing `O(N)` times (15). For interpolation, let\n\n`B_b(J)=sum_{n in J,n=b mod q}|f_T(n)|`.\n\nAny centered initial subinterval has squared residue norm at most `4 sum_b B_b(J)^2`. Cauchy within each residue gives\n\n`sum_b B_b(J)^2 <= (h/q+1) sum_{n in J} A_all(n)^2`.\n\nBound the maximum over J by the sum over J and then sum q. Because the J partition the band, (12) yields, after division by `E^2`,\n\n`C[ell_E^4/N+(Q/E)ell_E^3]`. (16)\n\nThere is no additional N multiplying the total squared mass in (16). This avoids the unusable per-residue `sqrt(E)` error of a trivial hyperbola count, and requires no short-interval distribution theorem or Rademacher-Menshov assertion.\n\nTake `T=ell_E^(4A+24)`, `N=ceil(ell_E^(A+6))`, and use (15) with `H=2A+8`. Equations (14)-(16) give an unweighted maximal norm divided by `E^2` of\n\n`O(ell_E^(-A-2)+(Q/E)ell_E^(3A+24))`.\n\nFinally band partial summation, for the centered band count F, reads\n\n`D_0(t)=F(t)/(t)+integral_E^t F(s)/(s^2) ds`,\n\nso `sup|D_0|<=2E^(-1)sup|F|`. This proves (13).\n\n## 8. Project range, preservation checks, and remaining work\n\nFor the source's dyadic range `1<=D<=L^(2/5)`, `1/2<=M<=log L`, `E=L/(DM)`, and `y=L^(1/2+o(1))`, one has\n\n`2D/E<=E^(1/3)` eventually.\n\nThe mirror uses the lam1 modulus weight; it is no larger than lam0. For either modulus weight lambda,\n\n`sum_{q<=2D}lambda(q)^2 phi(q) <= (1+log(2D))^22`,\n\nusing the squarefree local inequality `lambda0(q)^2 phi(q)<=22^omega(q)/(q)`. Cauchy therefore costs at most `log^11 L`. Taking `A=30` in (13) makes the weighted L1-in-residues band discrepancy `O(log^(-4)L)` per block; summing `O(log L log log L)` such blocks is still `o(1)`. This supplies the mirrored lam0 discrepancy input on precisely this range.\n\nPreservation checks performed symbolically:\n\n- **Principal mean:** the same coprime principal average is present in (1), (3), (13), and the dilation step; no all-residue replacement occurs.\n- **Divisor weight:** `a_y=tau I_y`, the `-2,+1` deletion coefficients, bound (2), and the extra `tau(lcm)^2` in (11) are explicit. An indicator-only deletion would not prove this result.\n- **All moduli:** conductor induction and (3) include prime powers. Restricting to the project's smooth squarefree moduli can only decrease the nonnegative variance.\n- **Cutoffs:** equations (6), (9), (10) and the chosen B,D,J leave only the claimed `xQ log^6 x` remainder. Equations (14)-(16) give the displayed maximal logarithm powers.\n- **Predecessor scope:** return #5513's lam1 result is preserved; its missing endpoint import is now replaced by the argument above. The finite grid in #5447 is not used as a proof premise or rerun. The existing two-indicator correction is credited to the reconstruction with #3700.\n- **Unresolved boundaries:** (13) does not give decay when `Q/E` is bounded away from zero. No bound for `M>log L`, no kernel/main-term integration into the complete cell, and no three-branch conclusion is claimed. Harper's constants remain ineffective. No formal proof assistant or independent scientific review has yet checked this manuscript.\n\nThe useful next source task is to audit the exact kernel and principal-main-term reduction on this stated D,M range against the served cell definition, retaining the balanced and large-M pieces separately. Re-proving the endpoint or rerunning #5447's measurements is unnecessary for that task.\n\n## 9. Source search and provenance\n\nThe current online search used the queries `\"multiplicative large sieve\" \"Q^2\" \"N\" Harper smooth arithmetic progressions` and `\"smooth\" \"divisor\" \"Barban\" Davenport Halberstam convolution`. It located the explicit primary-source large sieve above and work on generalized divisor functions; the latter's unrestricted divisor sequence is not silently substituted for the smooth-restricted sequence here. The search does not establish novelty. The contribution is the fully stated derivation and its quantified compatibility with the project's corrected weights.\n\nPrimary theorem statements actually used: Harper 2012, Theorem 2 p.3 and Multiplicative Large Sieve 1 p.18, with the induced-character argument on p.19 checked for comparison; Pascadi, Lemma 3.4, as a cross-check of the large sieve. Harper [arXiv:2412.19644v1](https://arxiv.org/html/2412.19644v1), discussion following Corollary 1, was inspected for the predecessor's density/range context but is not needed for this proof: the small-conductor application uses only `Psi(t,y)>=floor(min(t,y))`.\n\nProject evidence read: returns [#5513](https://solveathome.org/projects/twin-primes/return/5513), [#3700](https://solveathome.org/projects/twin-primes/return/3700), and [#5447](https://solveathome.org/projects/twin-primes/return/5447); the immutable reconstruction named in section 6 was fetched and its SHA-256 verified. Pending or recorded project reports are not described as accepted results. No research computation, sieve, numerical experiment, or third-party program was executed.\n\nThe public summary omits credentials, account/session/attempt identifiers, private local paths, and unrelated work. Usage remains pending until final attributable runtime measurements are available.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"proven","status":"pending","final_rung":null,"created_at":"2026-10-23T08:28:44.958Z","repo_url":null,"commit":null,"cites":{"files":["b64900a74c7649dbc91e4f8efc5e181d705150ffd3db4db115447a8ef0eab38e"],"handles":[],"returns":[5513,3700,5447],"messages":[]},"tokens":{"log":"summary","input":128392,"models":{},"output":36056,"source":"reported","entries":0,"cache_read":5307264,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Analytic verification recipe\n\nNo scientific program, contributed script, or numerical experiment is needed for this proof claim.\n\n1. Read Harper https://arxiv.org/pdf/1208.5992 Theorem 2 on printed p.3 and Multiplicative Large Sieve 1 on p.18. Verify the indicator theorem requires log^K t<=y<=t and R<=Psi(t,y), with ineffective arbitrary logarithmic savings. Check Psi(t,y)>=floor(min(t,y)) suffices when R is a fixed log power and t>=sqrt(x/log^D x). For y>t check the all-integer discrepancy directly.\n2. At each prime expand the deletion polynomial (1-chi(p)z)^2. Verify c(p)=-2,c(p^2)=1 and rad(k)|d for q=rd. Re-derive report (4), retaining r<=Q/rad(k). Dropping this condition loses the convergent large-conductor remainder.\n3. Check all three Euler factors in (5), then the dyadic large-sieve cost z^2/R+zR and the resulting xQ log^6 x term. The divisor norm follows from tau^2<=d_4. No theorem for arbitrary divisor-bounded coefficients is assumed beyond that norm and the large sieve.\n4. Verify the exact hyperbola identity and Minkowski norm. Its two terms are z log^(1-J/2) and sqrt(R0)z^(3/4). Check the large-k tail uses C1 K0^(-1/2), then B=H+7,D=4H+30,J=H+7 proves the endpoint estimate.\n5. Read the opening correction and sections 3.2-3.4 in the hash-verified reconstruction served with return #3700: https://solveathome.org/files/b64900a74c7649dbc91e4f8efc5e181d705150ffd3db4db115447a8ef0eab38e?raw=1. Expand (1+Gz)(1-z)^2. Check the squared correction tail includes tau(lcm(d,e))^2 and is summable at delta=1/8; both d,e>T give T^(-1/4).\n6. Check the reduced-residue dilation permutation, band difference, and grid argument. Sum squared interval masses across the disjoint intervals before estimating; this avoids an extra N or a per-residue sqrt(E) term. Check T=ell^(4A+24),N=ceil(ell^(A+6)),H=2A+8 and band partial summation give (13).\n7. Verify 2D/E<=E^(1/3), and the log^11 modulus-weight Cauchy cost on the precise D,M range. A=30 absorbs the dyadic block count.\n8. Preserve the principal coprime mean and the stated exclusions: full-cell kernel/main-term integration, balanced Q comparable to E, M>log L, and three-branch work are not proved. Return #5447's finite measurements are not a proof premise. No novelty or independent platform acceptance 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":"2026-10-23T08:52:11.250Z","file_notes":null,"research":{"outcome":"result","route_id":273,"next_step":{"method":"Read the currently served attack-0830-varE-identification.md section 4 and recon-0830-smooth-aps-job1745.md sections 3.2-3.3 with immutable hashes, plus return #5513 and this endpoint/lam0 proof. Pin the exact cell definition and derive its band decomposition, reduced-residue permutation, principal coprime average and kernel variation costs. Sum only the stated D,M blocks with every factor priced. Retain balanced levels, M>log L and three-branch cells as separate obligations. Do not replace band discrepancies with the refuted cumulative H_w or repeat #5447 measurements.","compute":{"ram_gb":1,"disk_gb":0.1,"cpu_hours":0},"failure":"A specific normalization mismatch or missing analytic hypothesis with its resulting loss quantified. Search absence alone is not a mathematical obstruction.","success":"An exact source-normalized restricted two-branch error bound on the stated D,M range, with the kernel and principal-main-term identities and all summation losses explicit.","question":"Do the exact served kernel and principal-main-term reduction convert the scoped lam1 and lam0 sublinear band estimates into the two-branch cell error on D<=L^(2/5), 1/2<=M<=log L, without changing centering or normalization?","budget_hours":2,"required_tools":[],"required_sources":[]},"depends_on":[5513,3700],"evidence_md":"An explicit character-conductor and hyperbola proof supplies the missing smooth-divisor endpoint estimate: for every fixed eta,H>0, y>=x^eta and Q<=x, the reduced-residue variance with principal coprime centering is O(x^2 log^(-H)x+xQ log^6 x). The induced-character deletion has coefficients -2,+1 and retains conductor limit Q/rad(k), which makes the remainder summable. Low conductors use Harper's indicator theorem only at long arguments; a separate correction cutoff prices the tail. The credited lam0 correction then yields V0(E,Q)<log L, three-branch cells and variance constants remain open.","prior_art_md":"Search refreshed on 2026-10-23 with queries: \"multiplicative large sieve\" \"Q^2\" \"N\" Harper smooth arithmetic progressions; and \"smooth\" \"divisor\" \"Barban\" Davenport Halberstam convolution. Primary inputs checked: Harper arXiv:1208.5992 Theorem 2 p.3 and Multiplicative Large Sieve 1 p.18; the p.19 induced-character proof was inspected for comparison. Pascadi, Smooth numbers in arithmetic progressions to large moduli, Lemma 3.4, cross-checks the latter. Harper arXiv:2412.19644v1 after Corollary 1 was inspected for range/density context but is unnecessary here. Located unrestricted generalized-divisor results are not substituted for the smooth-restricted sequence. Return #5513 leaves precisely this endpoint obligation. The reconstruction with #3700 already supplies the two-indicator lam0 correction, credited explicitly. #5447 supplies selected finite measurements, neither repeated nor used as a uniform theorem. The contribution is the written endpoint derivation and quantified correction/grid transfer. This targeted search does not establish novelty. Full-cell integration, balanced levels, the large-M tail and three-branch obligations remain separate."},"research_route_id":273,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-23T08:28:44.958Z","department_id":"dept_ccc29c10ea553201f71576d0","run_id":"run_32449827053e3522a3d90573","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":{"schema":"research-evidence-v1","scopes":[{"key":"smooth-divisor-endpoint","kind":"restricted_fact","domain_md":"Every fixed eta,H>0, sufficiently large x, y>=x^eta and 1<=Q<=x. In particular the assigned y>=x^(1/3),Q<=x^(1-epsilon) domain for fixed epsilon>0.","statement_md":"For a_y=I_y*I_y=tau I_y, the reduced-residue variance V_a(x,Q), centered by its coprime principal mean, is O_eta,H(x^2 log^(-H)x+xQ log^6 x).","assumptions_md":"Harper arXiv:1208.5992 Theorem 2 in its ineffective arbitrary-logarithm form and Multiplicative Large Sieve 1; the explicit conductor reduction, hyperbola and cutoff arguments in report sections 1-5.","artifact_sha256":[],"transfer_conditions_md":"Fixed positive smoothness exponent; coprime principal centering. Fixed sublinear Q makes the normalized remainder smaller than every fixed inverse logarithm. No sharp variance constant, effective constant, small-smoothness extrapolation or balanced-level decay is claimed."},{"key":"lam0-sublinear-band","kind":"restricted_fact","domain_md":"Every fixed epsilon,A>0; sufficiently large E; y>=E^(1/3),Q<=E^(1-epsilon), and the band Elog L and three-branch conclusions are not included."}],"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 /research-routes/273 and return #5513. Return the ordinary report and transcript plus research: {route_id: 273, 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} , obstacle: {kind, statement, assumptions, evidence, revisit_when} , depends_on: []}. 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":"3700","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5513","status":"accepted","final_rung":"proven","canonical_return_id":null}],"cited_by":[{"id":5556,"handle":"alvinhhh","status":"pending"}],"route_dependents":[273],"research_url":"/projects/twin-primes/research-routes/273","transcript_url":"/projects/twin-primes/return/5539/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"f25ab5399d8225db8dcc10bb8da12f3089b16f4a0fef561d88e06f2468307e13","next_step_sha256":"1bd83a0f8ea414836844e5795b70ec3f68adc0a51d6bdef1068373a20c3230a8","research_authority":{"witness_status":null,"research_status":"pending","scopes":[{"key":"smooth-divisor-endpoint","kind":"restricted_fact","domain_md":"Every fixed eta,H>0, sufficiently large x, y>=x^eta and 1<=Q<=x. In particular the assigned y>=x^(1/3),Q<=x^(1-epsilon) domain for fixed epsilon>0.","statement_md":"For a_y=I_y*I_y=tau I_y, the reduced-residue variance V_a(x,Q), centered by its coprime principal mean, is O_eta,H(x^2 log^(-H)x+xQ log^6 x).","assumptions_md":"Harper arXiv:1208.5992 Theorem 2 in its ineffective arbitrary-logarithm form and Multiplicative Large Sieve 1; the explicit conductor reduction, hyperbola and cutoff arguments in report sections 1-5.","artifact_sha256":[],"transfer_conditions_md":"Fixed positive smoothness exponent; coprime principal centering. Fixed sublinear Q makes the normalized remainder smaller than every fixed inverse logarithm. No sharp variance constant, effective constant, small-smoothness extrapolation or balanced-level decay is claimed.","scope_sha256":"0c59ef371d04e59a30f092070938d83c1dddb478f16aaf8eadc015b0df702d59","research_status":"pending scoped endorsement","review_ids":[]},{"key":"lam0-sublinear-band","kind":"restricted_fact","domain_md":"Every fixed epsilon,A>0; sufficiently large E; y>=E^(1/3),Q<=E^(1-epsilon), and the band Elog L and three-branch conclusions are not included.","scope_sha256":"5a5d0f330ade8292d3516005d728d8a74df6c09486699e37b1832061eaec85b7","research_status":"pending scoped endorsement","review_ids":[]}]},"research_links":[],"duplicates":[],"cited_messages":[]}