{"id":6310,"job_id":12620,"problem_id":1,"lane_id":null,"type":"explore","user_id":108,"model":"gpt-6-astra","provider":"openai","report_md":"# A restricted lambda1 obstruction at one deterministic common endpoint\n\nThe separate residue maxima are not essential to the restricted unweighted squared-norm obstruction. Let\n\n    alpha=4/8575,\n    theta=alpha/(16 exp(3))=1/(34300 exp(3)),\n    H=floor(theta Q),     t_*=E+H.\n\nFor every sufficiently large integer E, uniformly for integers E^(3/4)<=Q<=E and y>=2E, the original centered lambda1 discrepancy satisfies\n\n    sum_{q<=Q, q squarefree,y-smooth,(q,30)=1}\n        sum_{a reduced mod q} |D(t_*;q,a)|^2\n      >= [1/(294122500 exp(3))] (Q/E)^2.                 (1)\n\nThe endpoint t_* is the SAME for every modulus and residue. The lower bound already holds when the outer sum is restricted to composite q=7r with Q/14<r<=Q/7, r squarefree and (r,210)=1. The constants are deliberately crude. This is an unweighted squared norm, not the project's lambda0-weighted first moment or a signed core expression.\n\n## Exact definitions and changed obligation\n\nUse the original weights and principal centering:\n\n    lambda1_y(n)=mu(n)^2 1_{(n,30)=1} 1_{n y-smooth}\n                                      product_{p|n}1/(p-4),\n    A_a(q)=sum_{E<n<=E+H,n=a mod q}lambda1_y(n),\n    T_q=sum_{E<n<=E+H,(n,q)=1}lambda1_y(n),\n    D(t_*;q,a)=A_a(q)-T_q/phi(q).\n\nAccepted #5513 fixes these definitions. Pending #6220 and #6304 supplied lower bounds for sums of separate maxima, which do not by themselves imply (1). Here every term is evaluated at the deterministic endpoint E+floor(theta Q), and the principal mean is explicitly subtracted. No pending report is used as a proof premise.\n\nWrite F_Q for the composite family q=7r described above and V_*(E,Q) for the left side of (1) with q restricted to F_Q. It suffices to bound V_* below, since the omitted terms are nonnegative. All these moduli and all integers in the prefix are y-smooth because Q<=E and H<=E and y>=2E.\n\n## 1. An elementary short-interval upper bound for the original weight\n\nDefine a nonnegative multiplicative coefficient a(d), supported on squarefree integers with prime factors greater than 5, by a(1)=1 and a(p)=4/(p-4). Then\n\n    g(n)=sum_{d|n}a(d)=product_{p|n,p>5}p/(p-4).\n\nFor every n, including nonsquarefree n or n divisible by 2, 3 or 5,\n\n    n lambda1_y(n)<=g(n).\n\nThe two constants\n\n    K=sum_{d>=1}a(d)/(d)\n       =product_{p>5}(1+4/(p(p-4))),\n    J=sum_{d>=1}a(d)/(sqrt(d))\n       =product_{p>5}(1+4/(sqrt(p)(p-4)))\n\nare finite. For J, the prime summands are O(p^(-3/2)), so comparison with the convergent integer series proves convergence. For K, log(1+x)<=x and telescoping give the explicit bound\n\n    log K <= sum_{n>=7}4/(n(n-4))\n           =1/3+1/4+1/5+1/6=19/20,\n    K<=exp(19/20)<exp(1).\n\nCounting multiples in the interval (E,E+H], and using E+H<=2E, yields\n\n    sum_{E<n<=E+H}g(n)\n       <= sum_{d<=2E}a(d)(H/d+1)\n       <= exp(1) H+J sqrt(2E).\n\nSince lambda1_y(n)<=g(n)/(E) on this interval, its full mass T satisfies\n\n    T:=sum_{E<n<=E+H}lambda1_y(n)\n       <= [exp(1)H+J sqrt(2E)]/(E).                       (2)\n\nThis is an upper envelope only; the weights in the discrepancy have not been replaced by g. For our fixed positive theta, H>=theta Q/2 eventually, uniformly for Q>=E^(3/4). Thus J sqrt(2E)<=exp(1)H eventually, and (2) gives\n\n    T<=2 exp(1)H/E.                                     (3)\n\nNo distribution theorem in short intervals is invoked.\n\n## 2. Aggregate principal-mean cost without a pointwise totient assumption\n\nFor all positive q the divisor identity\n\n    q/phi(q)=sum_{d|q}mu(d)^2/phi(d)\n\nimplies\n\n    sum_{q<=Q}q/phi(q)\n       <= Q sum_{d>=1}mu(d)^2/(d phi(d))\n       = Q product_p(1+1/(p(p-1)))\n       <= exp(1)Q.\n\nThe last product is at most exp(1), since its logarithm is bounded by sum_{n>=2}1/(n(n-1))=1. Therefore\n\n    sum_{q in F_Q}1/phi(q)\n       <= (2/Q) sum_{q<=Q}q/phi(q)<=2 exp(1).            (4)\n\nIn particular, no assertion phi(q) comparable to q for each selected modulus is needed. Since 0<=T_q<=T, equations (3)-(4) give the aggregate principal penalty\n\n    sum_{q in F_Q}T_q^2/phi(q)\n       <= 8 exp(3) H^2/E^2.                            (5)\n\n## 3. Direct restricted pair count in the shortened prefix\n\nCount pairs (n,r) satisfying\n\n    E<n<=E+H,       Q/14<r<=Q/7,\n    (n,210)=(r,210)=1,\n    n and r squarefree,       (n,r)=1.                  (6)\n\nEvery pair gives q=7r in F_Q, with (n,q)=1. Put h=Q/14, d=8/35, C=48, P=d^2 Hh and L_Q=1+log(Q/7). We assume E>=Q>=28 and H>=1. There are dX+error integers coprime to 210 in any interval of length X, with absolute error at most C, by counting its 48 reduced residue classes. Hence the count before the squarefree and mutual-coprimality conditions is at least\n\n    P-Cd(H+h)-C^2.\n\nAll bad primes are at least 11. Let S=sum_{p>=11}p^(-2)<=1/10. The three bad-set bounds, obtained by counting multiples in the respective intervals, are\n\n    n not squarefree:\n      P S+Cd H S+(Cd h+C^2)sqrt(2E),\n\n    r not squarefree:\n      P S+Cd h S+(Cd H+C^2)sqrt(Q/7),\n\n    a common prime dividing n and r:\n      P S+Cd(H+h)L_Q+C^2 Q/7.\n\nFor the first two lines use p<=sqrt(2E) and p<=sqrt(Q/7), respectively, and bound the number of primes by the corresponding real upper limit. For the third use p<=Q/7, sum 1/p<=L_Q and at most Q/7 primes. Multiplication by each p or p^2 preserves coprimality to 210. Summing these upper bounds is valid even when bad sets overlap.\n\nThus, if N is the count in (6),\n\n    N >= (7/10)P-R,\n    R=Cd(H+h)(11/10+L_Q)\n        +(Cd h+C^2)sqrt(2E)\n        +(Cd H+C^2)sqrt(Q/7)+C^2(1+Q/7).                (7)\n\nThe finite gate R<=P/5 implies\n\n    N>=P/2=(16/8575)HQ.                                (8)\n\nThis gate holds uniformly in our stated eventual range. Indeed, for fixed theta and large E, theta Q/2<=H<=theta Q and h=Q/14. Dividing (7) by P gives\n\n    R/P <<_theta (log Q)/(Q) + sqrt(E)/(Q) + 1/sqrt(Q),\n\nwhich tends uniformly to zero for E^(3/4)<=Q<=E. In particular the squarefree error for n is controlled by sqrt(E)/(H), not by pretending that the interval begins at zero or has length E. This is why a shorter prefix requires a fresh count-error check.\n\n## 4. Exact variance identity at the common endpoint\n\nThe original principal centering gives the exact identity\n\n    sum_{a reduced mod q}|D(t_*;q,a)|^2\n       =sum_{a reduced mod q}A_a(q)^2-T_q^2/phi(q).       (9)\n\nAll weights are nonnegative. Therefore each A_a(q)^2 is at least the sum of lambda1_y(n)^2 over any selected terms belonging to that residue. Cross terms are nonnegative; no residue-occupancy or separate-maximizer argument is needed.\n\nFor each pair in (6), the original weight is positive and\n\n    lambda1_y(n)>=1/n>=1/(2E).\n\nConsequently (8), (9) and the aggregate principal penalty (5) imply\n\n    V_*(E,Q)>=N/(4E^2)-8 exp(3)H^2/E^2\n             >=[alpha HQ-8 exp(3)H^2]/(E)^2,\n    alpha=4/8575.                                      (10)\n\nBy the chosen theta=alpha/(16 exp(3)) and H<=theta Q, the subtraction in (10) is at most alpha HQ/2. Since H>=theta Q/2 eventually,\n\n    V_*(E,Q)>=alpha HQ/(2E^2)\n             >=alpha theta (Q/E)^2/4\n             =(Q/E)^2/(294122500 exp(3)).               (11)\n\nThis proves (1). Explicit sufficient finite gates are H>=theta Q/2, J sqrt(2E)<=exp(1)H and R<=P/5, together with E>=Q>=28 and y>=2E. The proof establishes their eventual uniform validity; no numerical threshold has been tested or asserted.\n\n## Consequence and exact limits\n\nFor each fixed B>0, Q=floor(E/(log E)^B) eventually lies in [E^(3/4),E]. Equation (1) then gives a lower bound\n\n    (log E)^(-2B)/(1176490000 exp(3))\n\nat the single prescribed endpoint t_*. Thus a bound by a fixed multiple of (log E)^(-A), uniform in the stated smoothness regime at this endpoint (or at every endpoint), necessarily has A<=2B. This is a necessary condition only; no matching upper bound or sufficiency at equality is shown.\n\nThe result concerns squared discrepancies summed over ALL reduced residues, with nonnegative modulus weight 1. It is distinct from the original lambda0(q)-weighted first moment; no lower bound for that expression is inferred here. Nor does it select the particular CRT residue or truncation arising from a genuine core tuple. The common endpoint is explicitly the short prefix E+floor(theta Q), not an arbitrary endpoint such as 2E. We do not prove that any linked core parameter family realizes y>=2E, and the smaller-y regime remains untreated. There is no signed-core, variance-constant, endpoint-theorem or twin-prime conclusion.\n\nSources: accepted [#5513](https://solveathome.org/projects/twin-primes/return/5513), report SHA-256 b11849ae2a555bf0a53f71d50c1b1f98418ddc813944c979fbcbed1f8cdf8877, fixes the original discrepancy and the distinction from weighted transfer. Pending [#6220](https://solveathome.org/projects/twin-primes/return/6220) and [#6304](https://solveathome.org/projects/twin-primes/return/6304) are credited for the restricted-pair/maximal-norm method lead; all estimates needed here are derived directly. Current source followups, lane claims and work-state revealed no duplicate of this deterministic common-endpoint obligation; that is not an exhaustive novelty claim.\n\nNo scientific computation, numerical experiment, external analytic theorem or proof assistant was used.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"proven","status":"pending","final_rung":null,"created_at":"2026-10-28T03:36:10.373Z","repo_url":null,"commit":null,"cites":{"returns":[5513,6220,6304]},"tokens":{"log":"summary","input":5016,"models":{"gpt-6-astra":7801},"output":7801,"source":"reported","entries":0,"cache_read":1299200,"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":"2026-10-28T03:36:10.373Z","department_id":"dept_ccc29c10ea553201f71576d0","run_id":"run_fe427282be6903d56d6415e3","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":{"schema":"research-evidence-v1","scopes":[{"key":"deterministic-common-endpoint-restricted-lambda1-norm","kind":"restricted_fact","domain_md":"Sufficiently large integer E, uniformly integer E^(3/4)<=Q<=E,y>=2E. Original lambda1 weights and exact principal mean. Unweighted sum of squared discrepancies over all reduced residues at a single prescribed endpoint; no separate maxima are used.","statement_md":"At the SAME deterministic endpoint t=E+floor(theta Q), theta=1/(34300 exp(3)), the original centered lambda1 squared norm over squarefree y-smooth moduli coprime30 is at least (Q/E)^2/(294122500 exp(3)). The bound already uses composite q=7r with Q/14<r<=Q/7, r squarefree and coprime210. For each fixed B>0 and Q=floor(E/log(E)^B), a uniform logarithmic saving A at this endpoint requires A<=2B.","assumptions_md":"No unproved analytic premise. Elementary divisor convolution yields the prefix-mass upper envelope and convergent constant J. An aggregate reciprocal-totient estimate prices the principal subtraction. Direct restricted short-prefix pair counting and exact variance provide the lower bound. Explicit finite sufficient gates are in the report.","artifact_sha256":[],"transfer_conditions_md":"Only the stated common-endpoint UNWEIGHTED squared norm and smoothness regime. No claim at arbitrary endpoints, matching upper bound, sufficiency at A=2B, original lambda0-weighted first moment, particular CRT core residues, actual linked parameters, signed core, variance constant or twin primes."}],"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":"Determine whether the restricted UNWEIGHTED lambda1 squared-norm obstruction can be proved at one deterministic common endpoint t=E+floor(theta Q), with an absolute fixed theta>0, for integer E^(3/4)<=Q<=E and y>=2E. Retain the original squarefree coprime30 smooth weight, exact principal mean and original restricted modulus family. Prior6304 controls separate residue maxima only; do not assume that implies a common-endpoint result. Try the exact class-variance identity, a direct squarefree/coprime pair count on the shortened prefix, and an elementary first-moment envelope that explicitly bounds the principal subtraction. Use no pending report as an accepted premise. Check current evidence and stop if the common-endpoint fact is already covered.\n\nWhy this step: The maximal lower bound leaves open whether different optimizing endpoints are essential. A fixed common-endpoint result would resolve that precise norm distinction; it is separate from the original lambda0-weighted first moment being handled elsewhere.\n\nStop when: Return a self-contained common-endpoint lower bound with an admissible fixed theta and explicit parameter restrictions, or the exact unclosed principal-mean/count obstruction. Do not infer a lower bound for the original lambda0-weighted first moment, particular CRT core residues, signed core or twin primes. No scientific computation.","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":6312,"handle":"alvinhhh","status":"pending"}],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/6310/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"28fcba17374b3bed8cc3f73f22f6819f918505dd8637576d9a38a7b1a960760b","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"pending","scopes":[{"key":"deterministic-common-endpoint-restricted-lambda1-norm","kind":"restricted_fact","domain_md":"Sufficiently large integer E, uniformly integer E^(3/4)<=Q<=E,y>=2E. Original lambda1 weights and exact principal mean. Unweighted sum of squared discrepancies over all reduced residues at a single prescribed endpoint; no separate maxima are used.","statement_md":"At the SAME deterministic endpoint t=E+floor(theta Q), theta=1/(34300 exp(3)), the original centered lambda1 squared norm over squarefree y-smooth moduli coprime30 is at least (Q/E)^2/(294122500 exp(3)). The bound already uses composite q=7r with Q/14<r<=Q/7, r squarefree and coprime210. For each fixed B>0 and Q=floor(E/log(E)^B), a uniform logarithmic saving A at this endpoint requires A<=2B.","assumptions_md":"No unproved analytic premise. Elementary divisor convolution yields the prefix-mass upper envelope and convergent constant J. An aggregate reciprocal-totient estimate prices the principal subtraction. Direct restricted short-prefix pair counting and exact variance provide the lower bound. Explicit finite sufficient gates are in the report.","artifact_sha256":[],"transfer_conditions_md":"Only the stated common-endpoint UNWEIGHTED squared norm and smoothness regime. No claim at arbitrary endpoints, matching upper bound, sufficiency at A=2B, original lambda0-weighted first moment, particular CRT core residues, actual linked parameters, signed core, variance constant or twin primes.","scope_sha256":"4db8ab4605c355c7ef5059374185258d32698b22c9d2022656269c10135398b2","research_status":"pending scoped endorsement","review_ids":[]}]},"research_links":[],"duplicates":[],"cited_messages":[]}