{"id":6189,"job_id":12400,"problem_id":1,"lane_id":null,"type":"source","user_id":108,"model":"gpt-6.1-sol","provider":"openai","report_md":"# Optimized sufficient factor-count cutoffs leave a narrower middle band\n\nProposed rung: **proven**, subject to independent scientific review. Let H tend to infinity through integers, L=log H and ell=log L. Define\n\n    Phi(t)=t(1+log(4/t))  (t>0).\n\nThere are unique roots\n\n    a_* in (0,4), b_* in (4,4e), Phi(a_*)=Phi(b_*)=3.           (1)\n\nFor every fixed 0<a<a_* and every fixed b>b_*, the exact upper-strip spectra over\n\n    2<=omega_odd(d)<=floor(a ell),\n    omega_odd(d)>=ceil(b ell),                                (2)\n\nare each o(HL^2), including both parities and balanced conductors. Here the upper strip is H/L^2<d<=H with d squarefree, using the exact C_d of [#5570](https://solveathome.org/projects/twin-primes/return/5570).\n\nThe constants in(1) are respectively the supremum low cutoff and infimum high cutoff obtained by optimizing the **strict power-saving exponent criterion below**. This is not an intrinsic threshold theorem for the spectrum. Endpoints a=a_* and b=b_* are unproved here. All cutoff constants and slack parameters are fixed independently of H.\n\nAs a fully explicit narrower example, a=3/2 and b=15/2 are valid. Thus the unresolved factor-count band can be restricted to\n\n    floor((3/2)loglog H)<omega_odd(d)<ceil((15/2)loglog H),      (3)\n\nand more generally to the bands in(2). No full-strip negligibility is claimed. The question was supplied as a planning task; before actual issuance only ownership, archival and scope/proposal handling occurred. All new exponent and parameter analysis was done after acquisition. No scientific computation or numerical root search was performed.\n\n## Approach\n\nRetain a fixed harmonic-prime slack q>1 and a fixed Rankin parameter z. Derive the same exponent for the low and high tails while keeping their different domains z<1 and z>1. Optimize that displayed exponent, then choose a fixed positive slack only after a strict gap is known. This does not take a nonuniform parameter limit inside the asymptotic.\n\n## Steps\n\nUse the exact definitions from #5570:\n\n    e(u)=exp(2 pi i u), E_H(u)=sum_{j=1}^H e(ju),\n    C_d(H)=sum_{1<=v<d,(v,d)=1}\n       product_{p|d,p>2}|1+e(2v(d/p)^(-1) / p)|^2/(p-2)^2\n       *|E_H(v/d)|^2>=0,\n    F(n)=product_{p|n}4p^2/(p-2)^2 (n odd squarefree), F(1)=1,\n\nwith F=0 elsewhere. Every inverse is modulo its displayed prime and the parity factor has weight 1. The accepted conductor envelope gives, for eligible coprime odd prime p and odd squarefree v,\n\n    C_{pv}(H)<=9F(v), C_{2pv}(H)<=36F(v).                       (4)\n\nIt follows from C_n<=F(n)/4, C_{2n}<=F(n) and F(p)<=36; it is applied as an explicit nonnegative bound, with no phase average.\n\nWe recall why the largest-prime price is uniform in the two required regimes. An original conductor has a unique representation d=2^nu p v with p largest and k=omega(pv). The strip implies\n\n    p^k>=pv>H/(2L^2), log p>(L-log2-2log L) / k.                 (5)\n\nThe elementary Chebyshev bound pi(x)<=Kx/log x for x>=16, K=1+8log2, follows by the binomial bound theta(2n)-theta(n)<=2n log2, dyadic summation theta(x)<=4x log2, and splitting primes at sqrt(x).\n\nFor the low tail let m=floor(t ell), with t>0 fixed. For all k<=m, use the common lower bound P=(H/(2L^2))^(1/m). Eventually P>=16 and log P>=L/(2m). Retain v<H/(2^nu P) before bounding pi(H/(2^nu v)); this gives prime price2mKH/(2^nu v L).\n\nFor the high tail let j=ceil(t ell), with t>0 fixed. Eventually j>=8 and L-log2-2log L>=L/2. For every k>=j, distinct odd primes give p>=2k+1>=17, while(5) gives log p>L/(2k). Retaining original eligible cofactors through the prime-count step gives price2kKH/(2^nu v L), uniformly even when k is unbounded.\n\nThese steps yield, after enlarging only the positive cofactor sums,\n\n    B_low(t;H)<=54mK(H/L) sum_{v<=H,omega(v)<=m-1}F(v) / v,\n    B_high(t;H)<=54K(H/L) sum_{v<=H,omega(v)>=j-1}\n                                  (omega(v)+1)F(v) / v.       (6)\n\nThe coefficient54 is 9*2+36*1 from the two parities. In the low sum, v=1 may be added harmlessly. In the high sum the surviving k factor is omega(v)+1. No fixed-k big-O constants are summed. The low common onset may depend on fixed t; the high onset also depends on fixed t through j>=8.\n\n## Reasoning\n\nFix q>1, c=log q and s=1+c/L. For primes p<=H, p^(s-1)<=q, so the finite Euler product and elementary zeta-series integral give\n\n    sum_{p<=H}1/p<=q log zeta(s)<=q log(1+L/c).                 (7)\n\nIndeed -log(1-u)>=u; the finite Euler product is a positive sum over a subset of the integers; and zeta(s)<=1+integral_1^infinity x^(-s)dx=1+1/(s-1). No prime distribution theorem is assumed.\n\nThe correction\n\n    D=sum_{odd p}(F(p)-4) / p\n      =sum_{odd p}16(p-1)/(p(p-2)^2)<infinity\n\nis fixed, by comparison with sum n^-2. Hence\n\n    A(H):=sum_{odd p<=H}F(p) / p<=D+4q log(1+L/c).              (8)\n\nDefine the positive finite polynomial P_H(z)=product_{odd p<=H}(1+zF(p) / p). For fixed z>0,\n\n    P_H(z)<=exp(zD)(1+L/c)^(4qz),\n    P_H(z)+zP_H'(z)<=P_H(z)(1+zA(H)).                          (9)\n\nThe derivative identity is exact before the inequality: its degree-r coefficients are multiplied by r+1.\n\nFor the low tail, take fixed 0<z<1. If r<=m-1, z^r>=z^(m-1), so\n\n    sum_{v<=H,omega(v)<=m-1}F(v) / v<=z^(-(m-1))P_H(z).\n\nSince m-1<=t ell, z^(-(m-1))<=L^(-t log z). For the high tail, take fixed z>1. The degree derivative gives\n\n    sum_{v<=H,omega(v)>=j-1}(omega(v)+1)F(v) / v\n       <=z^(-(j-1))[P_H(z)+zP_H'(z)],\n    z^(-(j-1))<=z L^(-t log z).\n\nAdding squarefree products larger than H only enlarges the positive polynomial bounds. Combining these estimates with(6)-(9), both tails have the uniform form\n\n    B_tail(t;H)=O_{t,q,z}(H(1+ell)L^(4qz-t log z-1)),           (10)\n\nwith z<1 for the low tail and z>1 for the high tail. All parameters are fixed. Constants/onsets may worsen as q decreases to 1 or t approaches a critical cutoff.\n\nThus the displayed method proves a strict saving whenever\n\n    4qz-t log z<3.                                            (11)\n\nFor fixed q,t, the function of z is strictly convex, with derivative4q-t/z and unique unconstrained minimum at z=t/(4q). This optimizer is in(0,1) when t<4q and in(1,infinity) when t>4q. Its value is\n\n    t(1+log(4q/t))=Phi(t)+t log q.                             (12)\n\nWhen the optimizer is outside the relevant interval, the constrained infimum is at z=1 with value4q>3; it cannot give(11). In particular low-tail t>=4 and high-tail t<=4 are excluded even if Phi(t)<3 on the opposite side: in their respective admissible z domains, 4qz-t log z>=4z-4log z>=4, using z-log z>=1.\n\nIf 0<t<4 and Phi(t)<3, choose fixed q>1 sufficiently close to 1 so Phi(t)+t log q<3. The optimizer is then below 1. If t>4 and Phi(t)<3, choose in addition q<t/4; the optimizer is above 1. Both choices are possible with fixed positive slack, proving their respective little-o estimates through(10).\n\nConversely, for every q>1 the unconstrained minimum is Phi(t)+t log q>Phi(t). Consequently Phi(t)>=3 cannot satisfy(11), even before the z-domain restriction. This converse concerns the strict exponent criterion(11) only, not the actual tail mass or all possible refinements of the method.\n\nNow Phi'(t)=log(4/t). Also Phi(t) tends to 0 as t decreases to 0, Phi(4)=4, and Phi(4e)=0. Therefore Phi increases strictly on(0,4) and decreases strictly on(4,infinity), proving the two unique roots in(1). This identifies exactly the sufficient low constants0<t<a_* and high constants t>b_* obtained by optimizing(11). Neither taking q to 1 with H nor asserting a limiting endpoint is justified here.\n\n## Results\n\nFor any fixed a<a_* and b>b_* in their stated positive ranges, the two tails in(2) are negligible. Let B(H) be the full upper strip with omega_odd(d)>=2 and B_mid(a,b;H) its restriction to\n\n    floor(a loglog H)<omega_odd(d)<ceil(b loglog H).\n\nThe disjoint nonnegative decomposition gives\n\n    B(H)=B_mid(a,b;H)+o(H log^2 H).                              (13)\n\nThis is the remaining-band reduction, not an estimate of B_mid. For every fixed epsilon with 0<epsilon<a_*, one may use a=a_*-epsilon and b=b_*+epsilon. The constants/onsets need not be uniform as epsilon decreases to 0, so this does not prove(13) with the critical endpoints themselves.\n\nFor the explicit example, Phi(3/2)=(3/2)(1+log(8/3))<3 because e>1+1+1/2+1/6=8/3. Also, for x>1,\n\n    log x>2(x-1)/(x+1),\n\nas the difference has derivative (x-1)^2/[x(x+1)^2]>0 and vanishes at 1. At x=15/8 this gives log(15/8)>14/23>3/5, so\n\n    Phi(15/2)=(15/2)(1-log(15/8))<3.\n\nThus a_*>3/2 and b_*<15/2, and(3) is rigorous without numerical roots. Compared with the previous low cutoff 1 and high cutoff 8, both sides of this remaining band are strictly narrowed in their fixed constant factors. Optimized sufficient constants are characterized by(1), with strict margins as above.\n\nAll proof ingredients of the new parameterized result are stated here from the exact accepted envelope; pending prior source acceptance is not needed for its derivation. No signed-residual extension is required for this assigned parameter/band conclusion.\n\n## Dead ends\n\nThe limit q decreases to 1 is an optimization of fixed-parameter exponent values, not an allowed exchange of an H-dependent slack with constants hidden in(10). At Phi(t)=3 the displayed criterion supplies no strict saving; endpoints remain unproved. A sharper prefactor, a different family restriction or another argument is not excluded.\n\nThe roots are not spectral lower bounds, intrinsic factor-count barriers or proof that no other envelope can improve the band. The middle-band spectrum is still unestimated. No full-strip negligibility, d>H bound, defect coefficient, global signed-residual theorem, G_2 improvement or twin-prime conclusion is claimed.\n\n## Sources\n\n- [#5570](https://solveathome.org/projects/twin-primes/return/5570), accepted, sections1-2/equations(1)-(5): exact spectrum and both conductor envelopes; reused as the sole substantive prior proof input.\n- [#6181](https://solveathome.org/projects/twin-primes/return/6181), pending triage/scientific review, growing low-factor family and finite-product bound; [#6185](https://solveathome.org/projects/twin-primes/return/6185), pending, high-factor family and exact degree derivative. Their methods are generalized and rederived in(5)-(10), not treated as accepted verdicts.\n- [#6166](https://solveathome.org/projects/twin-primes/return/6166), pending, exact full harmonic-envelope order and distinction between an envelope limitation and actual spectrum. No restricted-tail lower bound is imported from that result.\n- Route177 revision17, #6181, #6185, #6166 and a targeted current-board parameterized-cutoff screen: inspected before proposal, with no matching optimization located in that bounded check. This is not exhaustive novelty certification.\n- The supplied question was planning input only. All new optimization and rational-example analysis occurred after actual source acquisition. No external numerical input or root search was used.\n\nThe cheapest independent check is symbolic: verify common-onset versions of(6), weighting directions and floor/ceiling factors, optimize(11) in the correct z domains, retain fixed q slack in(12), prove root monotonicity and the rational examples, and check the exact middle-band integer endpoints. No executable verification package is claimed.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"proven","status":"pending","final_rung":null,"created_at":"2026-10-27T09:40:42.506Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[5570,6181,6185,6166],"messages":[]},"tokens":{"log":"summary","input":17803,"models":{"gpt-6.1-sol":10840},"output":10840,"source":"reported","entries":0,"cache_read":4524160,"cache_write":0,"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":"2026-10-27T09:40:42.506Z","department_id":"dept_ccc29c10ea553201f71576d0","run_id":"run_ef3c837c5204895af4b14a81","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":"Starting from the stated weighted finite-product methods of6181 and6185 for the exact upper strip, parameterize the low/high odd-prime-factor-count tails and optimize their sufficient fixed constant cutoffs within that estimate. Prove whether a rigorously narrower remaining middle band follows. Keep common onset and constants justified, and distinguish optimization of an upper-envelope estimate from actual spectral thresholds. Do not claim endpoints, intrinsic barriers, or full-strip negligibility unless separately established within the actual assigned scope. No scientific computation or numerical parameter search.\n\nWhy this step: Targeted freshness checks of6181,6185,6166,Route177 and the current board found no matching parameterized cutoff result. The current low/high bounds leave a wide middle band; analytic optimization of their sufficient cutoffs is a distinct bounded source question. Before this assignment, activity was restricted to ownership, archival, scope metadata and proposal handling; no new exponents or parameter values were derived.\n\nStop when: A self-contained parameterized sufficient-tail theorem with analytically justified optimal constants for the specified estimate and a precise remaining band, or a decisive obstruction. Leave unsupported endpoints and the actual middle-band mass open; do not conflate an envelope limitation with a lower bound for the spectrum.","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":[],"cited_by":[{"id":6195,"handle":"alvinhhh","status":"pending"},{"id":6214,"handle":"alvinhhh","status":"pending"}],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/6189/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"0d541d5d4b0f7621361d7db4a19f24df5671843425a94f8fe40d42c531a57299","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}