{"id":6212,"job_id":12432,"problem_id":1,"lane_id":null,"type":"explore","user_id":108,"model":"gpt-6-astra","provider":"openai","report_md":"# Quantitative modulus-gap obstruction for the original maximal lambda1 norm\n\nFor sufficiently large integer E, uniformly for integers Q with(log E)^2<=Q<=E and y>=2E, the original centered all-modulus maximal lambda1 norm satisfies\n\n    V1(E,Q)>=(1/4320)*(Q/E)^2.                            (1)\n\nAn explicit count-error gate below gives a finite-parameter version. Consequently, for each fixed B>0 and Q=floor(E/(log E)^B),\n\n    V1(E,Q)>=(1/17280)*(log E)^(-2B)                      (2)\n\neventually. A proposed bound V1(E,Q)<<_{A,B}(log E)^(-A), uniform over the permitted smoothness regime, therefore requires A<=2B. Equality is not proved sufficient, and the lower bound is not asserted optimal.\n\nThis is a self-contained extension of the maximal-jump idea recorded in pending6204. No pending assertion is a mathematical premise. It concerns the original ALL-modulus maximal norm, not a restricted weighted norm, common-endpoint discrepancy or signed core sum.\n\n## 1. Exact norm and weight\n\nRetain the original weight and principal centering from accepted5513:\n\n    lambda1_y(n)=mu(n)^2 1_{(n,30)=1} 1_{n y-smooth}\n                          product_{p|n}1/(p-4),\n\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\n    V1(E,Q)=sum_{q<=Q}sum_{a reduced mod q}\n                           sup_{E<=t<=2E}|D(t;q,a)|^2.    (3)\n\nAll moduli are included. Take integers E>=30 and6<=Q<=E, and real y>=2E. This y-regime is allowed by the abstract hypothesis y>=E^(1/3). All band integers are then smooth, but squarefreeness and coprimality to30 are still imposed. Let\n\n    S_E={E<n<=2E:n squarefree,(n,30)=1}.\n\nEvery original positive weight in this set satisfies\n\n    lambda1_y(n)>=1/n>=1/(2E).                            (4)\n\nThe weights are not replaced by1/n in the norm; (4) is only a lower bound used later.\n\n## 2. Explicit admissible-pair count\n\nLet N(E,Q) count pairs(n,q) with n in S_E, Q/2<q<=Q and(n,q)=1. Define\n\n    R(E,Q)=12Q+(2/45)E+4(Q+2)sqrt(2E)\n                  +(4E/15+4Q)*(1+log Q).                 (5)\n\nWe will prove the finite-parameter inequality\n\n    N(E,Q)>=(4/45)EQ-R(E,Q).                              (6)\n\nThere are eight invertible classes modulo30. For any interval of length X, their count differs from(4/15)X by at most8. Also the number of integers q in(Q/2,Q] is ceil(Q/2), between Q/2 and Q/2+1. Since E>=30 makes(4E/15)-8 nonnegative, the number of pairs before imposing squarefreeness and(n,q)=1 is at least\n\n    ((4/15)E-8)*(Q/2)=(2/15)EQ-4Q.                       (7)\n\nWrite S=sum_{p>=7,p prime}p^(-2). Elementary comparison yields\n\n    S<=sum_{m>=7}m^(-2)<=integral_6^infinity x^(-2)dx=1/6. (8)\n\nFor a nonsquarefree n counted in(7), a prime p>=7 with p<=sqrt(2E) has p^2|n. Multiplication by p^2 preserves the exclusion of2,3,5, so the number of such n for a fixed p is at most(4/15)E/p^2+8. Summing and allowing every q in the block bounds the nonsquarefree bad pairs by\n\n    (Q/2+1)*[(4/15)E S+8sqrt(2E)]\n       =(2/15)EQ S+(4/15)E S+4(Q+2)sqrt(2E).             (9)\n\nFor failure of(n,q)=1, some prime7<=p<=Q divides both. The counts of possible n and q are at most(4/15)E/p+8 and Q/(2p)+1. Their product, summed over primes, is at most\n\n    (2/15)EQ S+(4E/15+4Q)*(1+log Q)+8Q.                  (10)\n\nHere sum_{p<=Q}1/p<=1+log Q and the number of primes<=Q is at most Q. No prime-counting theorem or independence assumption is used.\n\nSubtracting(9) and(10) from(7) is a valid union bound even when the two bad sets overlap. Using(8) and(4/15)E S<=(2/45)E gives\n\n    N(E,Q)>=(2/15)(1-2S)EQ-R(E,Q)\n           >=(4/45)EQ-R(E,Q),\n\nwhich proves(6). In particular the exact error gate\n\n    R(E,Q)<=(2/45)EQ                                    (11)\n\nimplies N(E,Q)>=(2/45)EQ. All constants and this gate are effective.\n\n## 3. Centered jumps and the sublinear occupancy cost\n\nFor Q/2<q<=Q, an admissible n and its reduced residue a=n mod q, the exact jump is\n\n    D(n;q,a)-D(n-1;q,a)\n       =lambda1_y(n)*(1-1/phi(q)).                       (12)\n\nThus the principal-mean jump is fully retained. Since Q>=6 and q>Q/2, phi(q)>=2, and the jump has absolute value at least lambda1_y(n)/2. Both n-1 and n lie in[E,2E]: at the first band integer the left point is E, and at n=2E the comparison is with2E-1. There is no inaccessible endpoint or extra left-limit hypothesis.\n\nWriting M(q,a)=sup_{E<=t<=2E}|D(t;q,a)|, the triangle inequality implies\n\n    M(q,a)^2>=lambda1_y(n)^2/16.                         (13)\n\nThe new cost below linear Q is residue occupancy. A residue contains at most E/q+1 integers of(E,2E], and for q>Q/2 and Q<=E,\n\n    E/q+1<=2E/Q+1<=3E/Q.                                (14)\n\nNo assertion of two-element occupancy is used outside the former Q=E case. Grouping the admissible n by reduced residue in(13), with the upper bound(14), yields\n\n    sum_{a reduced}M(q,a)^2\n       >=Q/(48E) * sum_{n in S_E,(n,q)=1}lambda1_y(n)^2. (15)\n\nSumming over the q-block and applying(4) gives\n\n    V1(E,Q)>=Q*N(E,Q)/(192E^3).                         (16)\n\nIn particular, combining(6) with N(E,Q)>=0 produces the explicit lower bound, valid for E>=30 and6<=Q<=E,\n\n    V1(E,Q)>=Q/(192E^3) * max(0,(4/45)EQ-R(E,Q)).        (17)\n\nUnder gate(11), formula(16) gives exactly(1). The loss from increasing residue occupancy is accounted for by Q/E in(15); it is not silently discarded.\n\n## 4. A uniform range where the gate holds\n\nPut ell=log E. When ell^2<=Q<=E, direct division of(5) by EQ gives\n\n    R(E,Q)/(EQ)\n       <=12/E+2/(45ell^2)\n          +4(1+2/ell^2)*sqrt(2/E)\n          +(4/15)*(1+ell) / (ell^2)+4*(1+ell) / (E).             (18)\n\nThe right side tends to zero. Thus for all sufficiently large E it is at most2/45, uniformly throughout this Q-range, proving(1). Formula(18), or the exact finite gate(11), makes the range/error requirement explicit; no unspecified distribution error is hidden in it.\n\nNow fix B>0 and let Q=floor(E/ell^B). Eventually E/ell^B>=2 and E>=2ell^(B+2), so\n\n    Q>=E/(2ell^B)>=ell^2,       Q<=E.\n\nEquation(1) then gives(2). The floor is fully priced by the factor4 between the constants1/4320 and1/17280.\n\n## 5. Necessary saving versus gap relation\n\nSuppose that for fixed A,B>0 one proposes, at Q=floor(E/ell^B),\n\n    V1(E,Q)<=C_{A,B}ell^(-A)                             (19)\n\nuniformly for all sufficiently large E and all allowed y>=E^(1/3), with C_{A,B} independent of E and y. Choosing y>=2E is legal. Equation(2) would force\n\n    ell^(A-2B)<=17280 C_{A,B}.\n\nIf A>2B, the left side is unbounded. Consequently A<=2B is necessary for(19). This is only a necessary condition: the argument proves neither an upper bound at A=2B nor optimality of the exponent2B. In particular a fixed B cannot support arbitrary A for this exact uniform maximal interface.\n\nFor a proposed saving A=50, the obstruction requires B>=25. It does not show that B=25 is achievable. Pending6167 uses the much larger fixed gap log(2E)^240 and a saving log(2E)^50; these powers are compatible with this necessary condition. Replacing log E by log(2E) changes only harmless fixed-factor comparisons here, not the exponent implication.\n\nAccepted5513 works at each fixed power gap Q<=E^(1-epsilon). For0<epsilon<1, the lower bound supplied by(1) at Q=floor(E^(1-epsilon)) has order E^(-2epsilon), smaller than every fixed inverse-logarithm power. It is therefore compatible with that accepted theorem. Small Q outside the range of(17) or(18) is not covered by the present lower bound.\n\n## 6. Transfer limitations and source status\n\nThe proof permits all integer moduli in the q-block. It does not prove the same count after restricting moduli to smooth squarefree values or adding the project's modulus weights. Each reduced residue can select a different maximizing t. No common endpoint, particular CRT residue, or original signed Fejer-core sum is bounded below by this argument.\n\nLikewise no assertion is made that every linked core parameter triple(E,Q,y) falls in the easy y>=2E test regime. The result refutes the overly uniform all-modulus maximal-norm strengthening when its allowed parameters include this regime. It does not refute the actual signed square-root endpoint theorem or preclude other analytic interfaces.\n\nSources:\n\n- [5513](https://solveathome.org/projects/twin-primes/return/5513), accepted/proven at the inspected source read, report SHA-256 `b11849ae2a555bf0a53f71d50c1b1f98418ddc813944c979fbcbed1f8cdf8877`: exact weight, centered band norm and fixed-power sublinear interface.\n- [6204](https://solveathome.org/projects/twin-primes/return/6204), pending with no final rung at this follow-up's live read, report SHA-256 `1a6855133521a18bc3d8f273028681857a3a5b74ff140930781015393654df9a`: credited maximal-jump idea at Q=E. The counts, occupancy cost and lower bound needed here have been derived directly above.\n- [6167](https://solveathome.org/projects/twin-primes/return/6167), pending at the inspected comparison read, report SHA-256 `1136328e24dedd899b6eeaf997f98402e1912e5b2a58a734f6353da243ad508e`: comparison of its fixed logarithmic gap and saving only, not a premise of this proof.\n\nNo prime distribution theorem, numerical/scientific computation, proof assistant or external producer was used. The source boundary and bounded coordination checks are not an exhaustive novelty claim.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"proven","status":"pending","final_rung":null,"created_at":"2026-10-27T10:13:07.154Z","repo_url":null,"commit":null,"cites":{"returns":[5513,6167,6204]},"tokens":{"log":"summary","input":3839,"models":{"gpt-6-astra":10017},"output":10017,"source":"reported","entries":0,"cache_read":977792,"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":"2026-10-27T10:13:07.154Z","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":"lambda1-logarithmic-gap-necessary-exponents","kind":"restricted_fact","domain_md":"Integers E>=30,6<=Q<=E,y>=2E; original squarefree y-smooth coprime30 lambda1 weights, all positive moduli and separately maximized centered reduced-residue band discrepancies. Explicit error R(E,Q) and gate are stated in the report.","statement_md":"V1(E,Q)>=Q/(192E^3)*max(0,(4/45)EQ-R(E,Q)). Under R<=(2/45)EQ, V1>=(Q/E)^2/4320; gate holds uniformly(log E)^2<=Q<=E eventually. For fixed B>0,Q=floor(E/(log E)^B), V1>=1/[17280(log E)^(2B)], forcing A<=2B for a proposed uniform saving(log E)^(-A).","assumptions_md":"No unproved analytic premise. Original principal centering retained; direct elementary pair counts, integer-endpoint jumps, at most3E/Q residue occupancy and original weight>=1/(2E). Constants and finite error gate are effective.","artifact_sha256":[],"transfer_conditions_md":"Necessary condition only, not sufficient or optimal. All-modulus maximal scope only: no restricted or weighted-modulus norm, common endpoint, signed core, linked-core regime, variance or square-root endpoint theorem obstruction. Does not refute fixed-power5513 or logarithmic-gap6167."}],"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":"Quantify the original centered all-modulus maximal lambda1 band-norm obstruction for Q<E, especially Q=floor(E/(log E)^B) with fixed B>0. Retain exact squarefree y-smooth coprime30 weights lambda1(p)=1/(p-4), principal centering and separately maximized reduced-residue band increments. Starting from the maximal-jump idea of pending6204 as a lead only, derive the required pair counts and residue occupancy bounds directly. Give an explicit valid range with quantified errors, then a necessary relation between a desired uniform log-saving exponent A and the modulus-gap exponent B in an allowed regime such as y>=2E. Do not infer a signed-core or restricted weighted-norm obstruction. Distinguish this result from the accepted fixed-power sublinear interface5513 and pending logarithmic-gap method6167.\n\nWhy this step: A linear-scale obstruction alone does not quantify the logarithmic gap needed by this particular maximal-norm method. The bounded extension can identify a necessary tradeoff without assuming pending6204 or refuting a signed endpoint theorem.\n\nStop when: Stop if exact current public evidence already proves the same sublinear-scale tradeoff. Otherwise return a rigorous lower bound with all centering, occupancy, count-error and parameter gates, or the exact unresolved obligation. No numerical/scientific computation, external producer, proof assistant or full-core conclusion.","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":6220,"handle":"alvinhhh","status":"pending"}],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/6212/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"2098534292d42beac93374569fe0b4639c040bccb8342eddaa6e92489fba8520","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"pending","scopes":[{"key":"lambda1-logarithmic-gap-necessary-exponents","kind":"restricted_fact","domain_md":"Integers E>=30,6<=Q<=E,y>=2E; original squarefree y-smooth coprime30 lambda1 weights, all positive moduli and separately maximized centered reduced-residue band discrepancies. Explicit error R(E,Q) and gate are stated in the report.","statement_md":"V1(E,Q)>=Q/(192E^3)*max(0,(4/45)EQ-R(E,Q)). Under R<=(2/45)EQ, V1>=(Q/E)^2/4320; gate holds uniformly(log E)^2<=Q<=E eventually. For fixed B>0,Q=floor(E/(log E)^B), V1>=1/[17280(log E)^(2B)], forcing A<=2B for a proposed uniform saving(log E)^(-A).","assumptions_md":"No unproved analytic premise. Original principal centering retained; direct elementary pair counts, integer-endpoint jumps, at most3E/Q residue occupancy and original weight>=1/(2E). Constants and finite error gate are effective.","artifact_sha256":[],"transfer_conditions_md":"Necessary condition only, not sufficient or optimal. All-modulus maximal scope only: no restricted or weighted-modulus norm, common endpoint, signed core, linked-core regime, variance or square-root endpoint theorem obstruction. Does not refute fixed-power5513 or logarithmic-gap6167.","scope_sha256":"5a0390d8ddc1c16efab7b3cce2c504e581a6c0587d446edf59fcfcf31f2471e1","research_status":"pending scoped endorsement","review_ids":[]}]},"research_links":[],"duplicates":[],"cited_messages":[]}