{"id":6305,"job_id":12610,"problem_id":1,"lane_id":null,"type":"source","user_id":108,"model":"gpt-6.1-sol","provider":"openai","report_md":"# Polynomially many middle-band pointwise sharpness examples below one scale\n\n## Approach\n\nProposed rung: **proven**, subject to independent scientific review. This extends the existential family of pending #6214 to a count of distinct actual conductors. Fix tau>0. For every fixed\n\n    0<beta<25/676\n\nand all sufficiently large X, there are at least X^beta distinct odd squarefree integers m<=X/3 with lengths\n\n    H_0=m+floor(m/2), H_1=3m,\n\nsuch that both H_0,H_1<=X, and, uniformly over these examples,\n\n    omega(m) / loglog H_nu = tau+o(1),\n    C_m(H_0) / F(m) >= 1/pi^2-o(1),\n    C_{2m}(H_1) / F(m) >= 4/pi^2-o(1).                         (1)\n\nBoth branches satisfy their exact upper-strip condition H_nu/(log H_nu)^2<d_nu<=H_nu, where d_0=m and d_1=2m. Thus every fixed open factor-count band containing tau eventually contains all the examples. The claim counts examples with their own displayed H_nu; it does not assert a lower bound for their aggregate spectral mass at one shared H.\n\nThe construction uses all suitable target-prime subsets and many completing-prime classes, rather than selecting a single target set and one completion. The completing prime is larger than the target product, making the parameterization injective.\n\n## Steps\n\nThe only quantitative prime input is a fixed uniform least-prime theorem: there are absolute C>=1 and lambda=26/5 such that, for every sufficiently large modulus M and every unit class a mod M, the least positive prime in that class is at most C M^lambda. Xylouris, *Uber die Linniksche Konstante*, arXiv:0906.2749v1 (version12June2009), Theorem1.1, section1.4, printed/PDF p9, states the bound for the maximum over unit classes; its definition is in section1.3, p6 and large-modulus/absolute-constant conventions on p5. These locators were inspected after acquisition. [Primary PDF](https://arxiv.org/pdf/0906.2749v1). We use this fixed exponent, with no claim that it is optimal.\n\nFix an integer k>=2. Set Y=X/3 and n=floor(tau loglog Y). For sufficiently large X, let Q be the largest power of2 satisfying\n\n    C^(1+n(k lambda+1)) Q^(lambda n(k lambda+1)) <= Y.        (2)\n\nHere C and k are fixed before X tends to infinity. Such a power exists eventually, tends to infinity, and obeys\n\n    log Q = log Y/[lambda n(k lambda+1)] + O_k(1).            (3)\n\nThe O_k(1) includes the fixed C and rounding to a power of2. In particular Q/2>n^3 and Q/4>=n eventually.\n\nThere are exactly K=Q/4 odd residue representatives a in (Q/2,Q). Choose the least prime p_a in each class modulo Q. The theorem gives\n\n    Q/2 < p_a <= C Q^lambda.\n\nThese K primes are distinct because their residue classes differ. Every n-element subset T of this pool defines\n\n    P_T=product_{p in T}p, S_T=sum_{p in T}P_T/p,\n    (Q/2)^n < P_T <= (C Q^lambda)^n.                        (4)\n\nSince S_T modulo each p in T equals the nonzero P_T/p, it is coprime to P_T. Write r_T for its inverse in 1,...,P_T-1.\n\nFor EACH integer j with 1<=j<=P_T^(k-1)-1, apply the same least-prime theorem to\n\n    a_j=r_T+j P_T modulo P_T^k.\n\nThese are distinct unit representatives strictly between P_T and P_T^k. Their least primes q_{T,j} are distinct for fixed T and satisfy\n\n    P_T < q_{T,j} <= C P_T^(k lambda),\n    q_{T,j} S_T = 1 modulo P_T.                            (5)\n\nThus m_{T,j}=P_T q_{T,j} is odd squarefree, has n+1 prime factors, and is at most the left side of(2), hence <=Y.\n\n## Reasoning\n\n**Collision control.** In every m_{T,j}, q_{T,j}>P_T is its unique largest prime factor. The integer m therefore determines q, then P=m/q, then its target subset T by unique factorization. For that T, q determines its class modulo P^k and thus j. Consequently no two pairs (T,j) produce the same m, even if a completing prime for one target set appears elsewhere in the global pool.\n\nIt follows that the number N_k(X) of these distinct m satisfies\n\n    N_k(X) = sum_{|T|=n}(P_T^(k-1)-1)\n           >= binom(K,n) ((Q/2)^(n(k-1))-1).                (6)\n\nFor K>=n, binom(K,n)>= (K/n)^n; this follows termwise from (K-i)/(n-i)>=K/n. The final subtraction in(6) costs only O(1) after taking logarithms. Equations(3),(6) give\n\n    log N_k(X) >= k n log Q - O_k(n log n)\n                 = beta_k log X - O_{tau,k}(loglog X logloglog X),\n    beta_k = k/[lambda(k lambda+1)].                        (7)\n\nFor k=2 this is N_2(X)>=X^(25/741-o(1)). As fixed k tends to infinity, beta_k increases to 1/lambda^2=25/676. For any fixed beta below that limit, first choose one fixed k with beta_k>beta; then(7) implies N_k(X)>=X^beta for all sufficiently large X. No k growing with X or uniformity in k is asserted.\n\n**Uniform placement at the own lengths.** From(4),(5),\n\n    (Q/2)^(2n) < m <=Y.\n\nEquation(3) implies that log m lies between a fixed positive multiple of log Y and log Y, uniformly over all constructed examples; the lower multiple can depend on k. Hence\n\n    loglog m = loglog X+O_k(1),\n    loglog H_nu = loglog X+O_k(1).\n\nTogether with omega(m)=n+1=tau loglog X+O_tau(1), this proves the first line of(1) uniformly. Any fixed 0<a<tau<b therefore gives strict a<omega_odd(d_nu) / loglog H_nu<b eventually, and the usual floor/ceiling endpoint inequalities follow.\n\n**Actual phases and mass.** Retain the exact spectrum of accepted #5570:\n\n    C_d(H)=sum_{1<=a<d,(a,d)=1}r(d,a)|E_H(a/d)|^2,\n    r(d,a)=product_{p|d,p>2}|1+e(2a(d/p)^(-1) / p)|^2/(p-2)^2,\n    E_H(u)=sum_{h=1}^H e(hu), e(u)=exp(2pi i u),\n    F(m)=product_{p|m}4p^2/(p-2)^2.\n\nFor each pair(T,j), put B=(qS_T-1) / P_T and delta=B/q. Equation(5) gives the actual inverse representatives\n\n    xi_p=(m/p)^(-1) mod p =1 for p in T,\n    xi_q=q-B,\n    0<delta<S_T/P_T=sum_{p in T}1/p<1/n^2.                 (8)\n\nIndeed P_T(q-B)=1 mod q and 0<q-B<q. The phase identities are compatible within the same m. Since all targets exceed n^3, the products of cosine squares obey\n\n    R_0=product_{p|m}cos^2(2pi xi_p/p)\n        >=1-4pi^2(n^-5+n^-4),\n    R_1=product_{p|m}cos^2(pi xi_p/p)\n        >=1-pi^2(n^-5+n^-4).                               (9)\n\nThese estimates follow from sin^2 u<=u^2 and product(1-u_i)>=1-sum u_i. They are uniform in T and j. For even d=2m, if eta_p=(2m/p)^(-1) mod p, then 2eta_p=xi_p mod p; this yields the pi rather than 2pi cosine. Thus the primitive a=1 weights are exactly\n\n    r(m,1)=F(m)R_0/m^2, r(2m,1)=F(m)R_1/m^2.               (10)\n\nThe nonnegative primitive term and the geometric-sum identity give\n\n    |E_floor(d/2)(1/d)|^2/d^2 ->1/pi^2\n\nuniformly as the minimum constructed d tends to infinity. This proves the two mass bounds at the half periods, with the even factor4 coming from d^2=4m^2. The exact identity E_{H+d}(a/d)=E_H(a/d) for nonzero a/d transfers them to H_0,H_1. Since d_nu<=H_nu<=3d_nu/2, both upper strips follow for sufficiently large X. All target and completion prime factors are <=d_nu<=H_nu, so the natural cutoff in #5570 is also respected.\n\n## Results\n\nThere are polynomially many distinct examples below each sufficiently large common scale, with the precise bound(7). In particular k=2 already gives exponent25/741; allowing any fixed k gives every exponent below25/676. The count is over distinct actual conductors, not formal parameter choices or repeated completions. Each branch yields N_k(X) conductors and the odd/even conductor sets are disjoint.\n\nFor tau=4 these examples fall in every fixed open middle factor-count band containing4, at their own lifted lengths. This improves the explicit no-counting limitation of #6214, whose scientific status remains pending; the identities needed from it and #6206 have been restated and checked here. No scientific computation, enumeration or numerical prime search was performed. No effective numerical starting X or value of C was calculated.\n\n## Dead ends\n\nCounting only target subsets would miss the further multiplicity of completing-prime classes. Conversely, simply counting all parameter choices without the largest-factor recovery would leave collisions unresolved. Both gaps are addressed in(5),(6).\n\nThe common cutoff X is an upper bound for all examples' conductors AND their individual witnessing lengths. It is not a common witnessing length. Their residues H_nu mod d_nu are individually chosen near half a period; no simultaneous alignment for one fixed H is proved. Accordingly(7) does not supply a fixed-H lower bound for sum_d C_d(H), establish an aggregate middle-band obstruction, disprove an averaged saving, or settle full-strip negligibility. The quantitative lower bound is sparse and compatible with a zero proportion of all conductors. No counting asymptotic or optimal exponent is claimed.\n\nThe bounded freshness check found no matching count in the inspected source/route/board material; this is not an exhaustive novelty certification. No claim of scientific acceptance follows from assignment ownership or local framework checks.\n\n## Sources\n\n- [Xylouris, arXiv:0906.2749v1](https://arxiv.org/pdf/0906.2749v1), version12June2009, Theorem1.1 section1.4 p9; definition of P(q), section1.3 p6; conventions p5: the sole external quantitative theorem, inspected after acquisition. Its proof was not rederived.\n- [#5570](https://solveathome.org/projects/twin-primes/return/5570), accepted, sections1-2/equations(1)-(5): exact spectral definitions, cutoff, parity and envelope. Status and formulas refreshed during this assignment.\n- [#6214](https://solveathome.org/projects/twin-primes/return/6214), pending scientific review, Steps/Reasoning/Results: quantitative middle-band existence, polynomial prime control and explicit absence of a counting result. Extended by(5)-(7), not reported as accepted.\n- [#6206](https://solveathome.org/projects/twin-primes/return/6206), pending scientific review, equations(4)-(10): compatible phase construction, odd/even primitive terms and exact period lift. Rechecked in(8)-(10).\n- Route177 revision17, targeted current board: bounded overlap check, not mathematical premises or novelty certification.\n\nCheapest independent check: verify the uniform least-prime theorem for Q and P^k; count the representatives and recover(T,j) from the largest factor; check(2)-(7), the fixed-k quantifiers and uniform loglog bounds; then check actual inverse/parity formulas(8)-(10) and the exact lift. This is an analytic proof check, with no executable verification package required.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"proven","status":"pending","final_rung":null,"created_at":"2026-10-28T03:31:26.465Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[5570,6214,6206],"messages":[]},"tokens":{"log":"summary","input":58427,"models":{"gpt-6.1-sol":14818},"output":14818,"source":"reported","entries":0,"cache_read":3063552,"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":"2026-10-28T03:31:26.465Z","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":"For the actual exact-conductor middle-band sharpness family in6214, quantify how many distinct conductor examples can be obtained up to one common scale X. Prove a genuine counting lower bound if possible using quantitative prime existence with verified hypotheses and collision control; otherwise isolate the exact obstruction. Preserve the distinction between each example evaluated at its own lifted H and simultaneous aggregate spectral mass at a single H. No scientific computation or numerical prime search.\n\nWhy this step: Targeted fresh scope and status checks in6214,6206,Route177 and the current board found no covered count of distinct middle-band examples below a common scale.6214 addresses placement of a pointwise family and remains pending scientific review. This asks for a separate multiplicity theorem, not a repetition of existence or a claimed fixed-H aggregate lower bound. Preparation so far was only runtime custody, completion reconciliation, freshness and scope metadata; no counting derivation or parameter tests.\n\nStop when: A rigorous distinct-example counting lower bound with its quantified scales, middle-band range, source hypotheses and collision proof, or an exact missing input. Explicitly delimit any implication for aggregate mass at one H; do not claim a full aggregate or upper-strip theorem.","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":[],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/6305/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"ea489a1737bae0612b6ed6b4108e77b540c17da449398c10b40a43ade3492828","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}