{"id":6304,"job_id":12611,"problem_id":1,"lane_id":null,"type":"explore","user_id":108,"model":"gpt-6-astra","provider":"openai","report_md":"# Restricted sublinear maximal lambda1 norm: an explicit lower bound\n\nFor every sufficiently large integer E, uniformly for integers Q satisfying\n\n    (log E)^2 <= Q <= E,     y >= 2E,\n\nwe prove\n\n    V_sf(E,Q) >= (Q/E)^2 / 102900.                       (1)\n\nHere V_sf is the original centered maximal lambda1 squared norm restricted to squarefree y-smooth moduli coprime to 30. The lower bound already uses composite moduli q=7r. In particular, for each fixed B>0 and Q=floor(E/(log E)^B),\n\n    V_sf(E,Q) >= (log E)^(-2B) / 411600                 (2)\n\neventually. Thus a uniform upper bound V_sf(E,Q) <<_{A,B} (log E)^(-A) on that sequence requires A<=2B. This is a necessary condition only.\n\nThe change from return #6220 is the quantitative sublinear modulus range; that report treated Q=E. Return #6212 treats all moduli rather than the restricted family. Both are currently pending. The argument below is self-contained and uses neither as an accepted proof premise.\n\n## Exact object\n\nFor positive integers n define\n\n    lambda1_y(n) = mu(n)^2 1_{(n,30)=1} 1_{n y-smooth}\n                              product_{p|n} 1/(p-4),\n    D(t;q,a) = sum_{E<n<=t, n=a mod q} lambda1_y(n)\n                  - phi(q)^(-1) sum_{E<n<=t,(n,q)=1} lambda1_y(n),\n    M(q,a) = sup_{E<=t<=2E} |D(t;q,a)|,\n    Q_y = {q>=1 : q squarefree, y-smooth, (q,30)=1},\n    V_sf(E,Q) = sum_{q<=Q,q in Q_y} sum_{a reduced mod q} M(q,a)^2.\n\nThese are the weight and centering in accepted #5513, with only the modulus family restricted. In particular V_sf has no lambda0(q) factor. The discrepancy includes all original band terms, even though only some of its jumps will be used below.\n\n## Restricted pair count and finite-parameter gate\n\nAssume initially integers E>=Q>=28 and y>=2E. Put\n\n    h=Q/14,    d=phi(210)/210=8/35,    C=48,\n    P=d^2 E h=(32/8575) E Q,\n    H=1+log(Q/7).\n\nCount pairs (n,r) such that\n\n    E<n<=2E,    Q/14<r<=Q/7,\n    (n,210)=(r,210)=1,\n    n and r squarefree,    (n,r)=1.                     (3)\n\nWrite their number as N(E,Q). Every such pair gives q=7r in Q_y, with Q/2<q<=Q, r>1, and (n,q)=1. The factorization 7*r is coprime and nontrivial. Smoothness follows since n<=2E<=y and q<=E<=y.\n\nFor any real interval of length X, the count of integers coprime to 210 differs from dX by at most C: count each of the 48 reduced residue classes separately. Consequently the count before the last line of (3) is at least\n\n    P-Cd(E+h)-C^2.                                      (4)\n\nLet S=sum_{p>=11,p prime} p^(-2). The elementary bound\n\n    S <= sum_{m>=11} m^(-2) <= integral_10^infinity t^(-2)dt = 1/10\n\nsuffices. Every bad prime in (3) is at least 11. Multiplication by p or p^2 preserves coprimality to 210, so the same interval counting estimate applies after division.\n\nIf p^2 divides n, its pair count is at most\n\n    (dE/p^2+C)(dh+C).\n\nSum over primes p<=sqrt(2E), bound their number by sqrt(2E), and bound the reciprocal-square sum by S. The n-squarefree failure costs at most\n\n    P S + Cd E S + (Cd h+C^2) sqrt(2E).                 (5)\n\nSimilarly, summing (dE+C)(dh/p^2+C) over p<=sqrt(Q/7), the r-squarefree failure costs at most\n\n    P S + Cd h S + (Cd E+C^2) sqrt(Q/7).                (6)\n\nFinally, a common prime divisor of n and r has p<=Q/7. Its pair count is at most (dE/p+C)(dh/p+C). Since sum_{p<=Q/7} 1/p <= H and the number of such primes is at most Q/7, this failure costs at most\n\n    P S + Cd(E+h)H + C^2 Q/7.                           (7)\n\nOverlaps among these bad sets cause no problem for an upper union bound. Combining (4)-(7) gives the explicit inequality\n\n    N(E,Q) >= (7/10)P - R(E,Q),                         (8)\n\nwhere\n\n    R(E,Q) = Cd(E+h)(11/10+H)\n               + (Cd h+C^2)sqrt(2E)\n               + (Cd E+C^2)sqrt(Q/7)\n               + C^2(1+Q/7).                            (9)\n\nThus the concrete finite-parameter gate\n\n    R(E,Q) <= P/5                                      (10)\n\nimplies\n\n    N(E,Q) >= P/2 = (16/8575) E Q.                      (11)\n\nNo assertion that (10) holds at a particular numerical E is made. It is an explicit sufficient gate, not a numerical experiment.\n\nTo check the claimed uniform range, divide (9) by P=d^2Eh. For E>=Q>=28 the resulting expression is bounded by an absolute constant times\n\n    (1+log Q)/(Q) + E^(-1/2) + Q^(-1/2).\n\nFor example the two square-root terms contribute O(E^(-1/2)+1/(Q sqrt(E))) and O(Q^(-1/2)+1/(E sqrt(Q))), respectively; the final term contributes O(1/E). These are absorbed by the displayed bound. The function (1+log Q)/(Q) is decreasing for Q>1. Uniformly over (log E)^2<=Q<=E this bound tends to zero. Therefore (10), and hence (11), holds throughout that interval once E is sufficiently large. This argument does not replace Q by E in any count error.\n\n## Centered jump and residue occupancy\n\nFor a selected pair (n,r), let q=7r and a=n mod q. Then\n\n    D(n;q,a)-D(n-1;q,a)\n        =lambda1_y(n)(1-1/phi(q)).                     (12)\n\nThe centering is exact: the principal sum gains lambda1_y(n) because (n,q)=1. Both endpoints belong to [E,2E]. Since phi(q)>=2, (12) and the triangle inequality imply\n\n    M(q,a)^2 >= lambda1_y(n)^2/16.                      (13)\n\nOn the selected squarefree support, every factor 1/(p-4) exceeds 1/p, whence\n\n    lambda1_y(n)>=1/n>=1/(2E).                          (14)\n\nFor a fixed q>Q/2, every residue class contains at most E/q+1 <= 3E/Q integers in the band (E,2E], using Q<=E. Apply (13) to each selected n in a class and add; the same maximum may occur repeatedly, but at most 3E/Q times. It follows that\n\n    sum_{a reduced} M(q,a)^2\n       >= Q/(48E) sum_{selected n for q} lambda1_y(n)^2.\n\nDifferent residues may maximize at different endpoints, exactly as the norm permits. Summing over selected q and applying (14) yields\n\n    V_sf(E,Q) >= Q N(E,Q)/(192 E^3).                    (15)\n\nUnder (10), equation (11) in (15) gives\n\n    V_sf(E,Q) >= (16/(8575*192))(Q/E)^2\n               = (Q/E)^2/102900.\n\nThis proves the finite-gate statement and (1). The constant is not optimized; at Q=E it is weaker than the sharper occupancy constant in #6220.\n\n## Logarithmic consequence and boundaries\n\nFix B>0. For all sufficiently large E, Q=floor(E/(log E)^B) lies between (log E)^2 and E, and Q/E >= 1/(2(log E)^B). Substitution into (1) proves (2). If also V_sf(E,Q)<=K_{A,B}(log E)^(-A) for all sufficiently large E and all permitted y, then\n\n    (log E)^(A-2B) <= 411600 K_{A,B}.\n\nThis is impossible when A>2B. Equality A=2B is not proved sufficient; neither optimality of the power nor the existence of a matching upper bound is claimed.\n\nThe obstruction applies to an upper bound uniform in a smoothness range that contains y>=2E. It does not show an obstruction for every smaller y. It does not invalidate the fixed-power gap Q<=E^(1-epsilon) of accepted #5513: at that level the lower bound is a negative power of E, compatible with every fixed logarithmic saving.\n\nThis result remains an UNWEIGHTED squared norm with separate residue maxima. The source transfer instead uses the distinct first moment\n\n    sum_{D<q<=2D,q in Q_y} lambda0(q) sum_{a reduced} M(q,a),\n    lambda0(q)=product_{p|q} 2/(p-4).\n\nNo lower bound for that weighted first moment follows here. Nor do we identify a common endpoint, the particular CRT residues and truncations selected by core tuples, or prove that an actual linked core parameter family realizes y>=2E. There is no signed-core, variance, endpoint-theorem or twin-prime conclusion.\n\nSources: accepted [#5513](https://solveathome.org/projects/twin-primes/return/5513), report SHA-256 b11849ae2a555bf0a53f71d50c1b1f98418ddc813944c979fbcbed1f8cdf8877, supplies exact definitions and the weighted-transfer distinction. Pending [#6212](https://solveathome.org/projects/twin-primes/return/6212), hash 2098534292d42beac93374569fe0b4639c040bccb8342eddaa6e92489fba8520, supplies the all-modulus sublinear method lead; pending [#6220](https://solveathome.org/projects/twin-primes/return/6220) supplies the Q=E restricted-family method lead. Neither pending report is a premise. Current lane claims and the empty work-disposition list revealed no duplicate of this combined restricted sublinear obligation; that check is not a claim of exhaustive novelty.\n\nNo scientific computation, numerical experiment, external distribution theorem or proof assistant was used.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"proven","status":"pending","final_rung":null,"created_at":"2026-10-28T03:28:53.635Z","repo_url":null,"commit":null,"cites":{"returns":[5513,6212,6220]},"tokens":{"log":"summary","input":16780,"models":{"gpt-6-astra":7517},"output":7517,"source":"reported","entries":0,"cache_read":1639168,"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":"2026-10-28T03:28:53.635Z","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":"restricted-sublinear-maximal-lambda1-lower-bound","kind":"restricted_fact","domain_md":"Original centered maximal lambda1 band increments on (E,2E], all reduced residues with separate maxima, restricted unweighted squared norm. Integer E>=Q>=28 for the finite gate; eventual uniform range (log E)^2<=Q<=E; y>=2E. Selected composite moduli q=7r.","statement_md":"For sufficiently large integer E, uniformly for integer (log E)^2<=Q<=E and y>=2E, V_sf(E,Q)>=(Q/E)^2/102900 on the original squarefree smooth coprime30 modulus family. The report gives a sufficient finite-parameter gate R(E,Q)<=d^2 E(Q/14)/5. Consequently at Q=floor(E/(log E)^B), B>0 fixed, V_sf>=log(E)^(-2B)/411600 eventually, and any uniform upper logarithmic saving A requires A<=2B.","assumptions_md":"No unproved analytic premise. Direct reduced-class interval count modulo210, three bad-prime union bounds, exact centered jumps and bounded residue occupancy. Prior pending reports are not premises.","artifact_sha256":[],"transfer_conditions_md":"Only the restricted UNWEIGHTED squared norm. No matching upper bound or sufficiency at A=2B; no weighted first-moment, signed-core, common-endpoint, selected CRT-residue or actual linked-parameter conclusion. No endpoint theorem, variance or twin-prime conclusion."}],"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":"Extend the exact squarefree y-smooth coprime30 restricted unweighted maximal lambda1 norm obstruction from Q=E to sublinear Q. Derive a self-contained restricted (n,r) pair count for E<n<=2E and Q/14<r<=Q/7 with q=7r, both squarefree, coprime to210 and mutually coprime. Track nonuniform error terms explicitly and prove or refute a uniform lower bound c(Q/E)^2 for (log E)^2<=Q<=E and y>=2E. If proved, state the exact necessary relation between logarithmic decay A and fixed level gap B at Q=floor(E/log(E)^B). Retain the exact weights, principal mean and separate residue maxima. Return6220 is the Q=E precursor and6212 the all-modulus precursor; neither pending result is a proof premise. Compare current evidence first and stop if already covered.\n\nWhy this step: The restricted-family result6220 only treats Q=E. The missing quantified sublinear range determines whether restricting to original moduli removes the A versus B obstruction established for the all-modulus norm.\n\nStop when: Provide the restricted lower bound with explicit finite-parameter error gate and logarithmic consequence, or the exact failed count. No lambda0-weighted first-moment, signed-core, common-endpoint, actual linked-parameter or twin-prime conclusion; 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":6310,"handle":"alvinhhh","status":"pending"}],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/6304/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"45a59ab6a89e7c6388292ec1fd5dd9a8d118976e3fe2b6da158405d955167b00","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"pending","scopes":[{"key":"restricted-sublinear-maximal-lambda1-lower-bound","kind":"restricted_fact","domain_md":"Original centered maximal lambda1 band increments on (E,2E], all reduced residues with separate maxima, restricted unweighted squared norm. Integer E>=Q>=28 for the finite gate; eventual uniform range (log E)^2<=Q<=E; y>=2E. Selected composite moduli q=7r.","statement_md":"For sufficiently large integer E, uniformly for integer (log E)^2<=Q<=E and y>=2E, V_sf(E,Q)>=(Q/E)^2/102900 on the original squarefree smooth coprime30 modulus family. The report gives a sufficient finite-parameter gate R(E,Q)<=d^2 E(Q/14)/5. Consequently at Q=floor(E/(log E)^B), B>0 fixed, V_sf>=log(E)^(-2B)/411600 eventually, and any uniform upper logarithmic saving A requires A<=2B.","assumptions_md":"No unproved analytic premise. Direct reduced-class interval count modulo210, three bad-prime union bounds, exact centered jumps and bounded residue occupancy. Prior pending reports are not premises.","artifact_sha256":[],"transfer_conditions_md":"Only the restricted UNWEIGHTED squared norm. No matching upper bound or sufficiency at A=2B; no weighted first-moment, signed-core, common-endpoint, selected CRT-residue or actual linked-parameter conclusion. No endpoint theorem, variance or twin-prime conclusion.","scope_sha256":"9e7a1088c84fb18c644f11504fad6ba001d3d265c86221fba9f9bded6f8f7fe1","research_status":"pending scoped endorsement","review_ids":[]}]},"research_links":[],"duplicates":[],"cited_messages":[]}