{"id":6204,"job_id":12419,"problem_id":1,"lane_id":null,"type":"explore","user_id":108,"model":"gpt-6-astra","provider":"openai","report_md":"# A positive lower bound for the original all-modulus maximal lambda1 norm at Q=E\n\nFor every sufficiently large integer E and every y>=2E, the exact centered maximal band norm from accepted5513 satisfies\n\n    V1(E,E)>=1/2880.                                      (1)\n\nThe proof is elementary and uses the original squarefree, coprime-to30 weights and principal centering. It exploits jumps inside the original supremum. In particular, no uniform bound V1(E,E)<<_A(log E)^(-A) can hold for any fixed A>0 over the allowed regime y>=E^(1/3).\n\nThis is an obstruction to a proposed uniform ALL-modulus maximal-norm extension. It is not a lower bound for a restricted or modulus-weighted norm, an endpoint-only discrepancy, a signed three-branch core, or the actual square-root endpoint theorem.\n\n## 1. Exact object and admissible parameters\n\nFor positive integers n define the original weight\n\n    lambda1_y(n)=mu(n)^2 1_{(n,30)=1} 1_{n y-smooth}\n                      product_{p|n}1/(p-4).\n\nThis is exactly f_y(n) / (n) from5513. For a reduced residue a modulo q put\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.       (2)\n\nAll positive moduli occur, and the centering is the original coprime principal mean at the same t. We neither replace the weights by 1/n nor delete this mean.\n\nTake E to be an integer tending to infinity, Q=E, and y>=2E. This smoothness condition is allowed by the abstract band interface y>=E^(1/3). Every integer n<=2E is then y-smooth, but the squarefree and coprimality conditions remain in force. Write\n\n    S_E={E<n<=2E: mu(n)^2=1, (n,30)=1}.\n\nFor n in S_E the original weight is positive and obeys\n\n    lambda1_y(n)=product_{p|n}1/(p-4)>=1/n>=1/(2E),        (3)\n\nbecause all its prime factors exceed5. Equation(3) is a lower bound on the original weight, not a substitution in(2).\n\n## 2. A centered jump survives the principal mean\n\nFix an integer q in(E/2,E] and n in S_E with(n,q)=1. Let a be n modulo q. At t=n the exact jump in that residue's centered discrepancy is\n\n    D(n;q,a)-D(n-1;q,a)\n       =lambda1_y(n)*(1-1/phi(q)).                        (4)\n\nThe principal mean jumps by lambda1_y(n) / (phi(q)), and this is explicitly retained. For sufficiently large E, all these q have phi(q)>=2. Consequently the absolute jump is at least lambda1_y(n)/2.\n\nBoth comparison points are legitimate in the original supremum: E and n are integers, so n-1>=E and n<=2E. For the first possible n=E+1 the left value is D(E); for the last n=2E the left value is D(2E-1) and the right value is D(2E). Thus no inaccessible left limit or enlarged band is used. Equivalently, the left-limit value equals D(n-1) because the step function has no intervening integer.\n\nBy the triangle inequality, at least one of |D(n;q,a)| and |D(n-1;q,a)| is at least lambda1_y(n)/4. Therefore, with M(q,a)=sup_{E<=t<=2E}|D(t;q,a)|,\n\n    M(q,a)^2>=lambda1_y(n)^2/16.                          (5)\n\nFor q>E/2, any residue class contains at most two integers in(E,2E]: three would span a distance at least2q>E. Grouping the admissible n by their reduced residue in(5) consequently gives\n\n    sum_{a reduced}M(q,a)^2\n       >=(1/32)sum_{n in S_E,(n,q)=1}lambda1_y(n)^2.       (6)\n\nThis step uses the sum of individual residue maxima in(2), each allowed to choose its own t. It does not assert the same bound at a common endpoint.\n\n## 3. Enough admissible integer-modulus pairs, without prime counting\n\nLet N_E be the number of ordered pairs(n,q) satisfying\n\n    E<n<=2E,  E/2<q<=E,\n    (n,30)=1,  n squarefree,  (n,q)=1.                   (7)\n\nWe prove\n\n    N_E>=(2/45)E^2                                      (8)\n\nfor every sufficiently large integer E. The estimates below have absolute, effective constants and are uniform in y>=2E.\n\nThere are eight invertible classes modulo30, so the number of integers coprime to30 in any interval of length X is (4/15)X+O(1), with absolute error at most8. Before imposing squarefreeness or(n,q)=1, the number of pairs in(7) is therefore\n\n    ((4/15)E+O(1))*(E/2+O(1))\n       =(2/15)E^2+O(E).                                 (9)\n\nPut S=sum_{p>=7,p prime}p^(-2). No prime-counting theorem is needed: comparison with all integers and an integral gives\n\n    S<=sum_{m>=7}m^(-2)<=integral_6^infinity x^(-2)dx=1/6. (10)\n\nIf an eligible n is not squarefree, then p^2 divides n for some prime p>=7, p<=sqrt(2E). Since p is invertible modulo30, writing n=p^2 m preserves(m,30)=1. Hence the number of possible n for this p is at most\n\n    (4/15)E/p^2+8.\n\nMultiplying by at most E/2+1 choices of q and summing over such primes bounds the number of pairs removed for nonsquarefreeness by\n\n    (2/15)E^2 S+O(E^(3/2)).                              (11)\n\nFor failure of(n,q)=1, some prime p>=7 divides both n and q; necessarily p<=E. Writing n=pm again preserves the coprimality with30. The numbers of possible n and q are at most\n\n    (4/15)E/p+8,       E/(2p)+1,\n\nrespectively. Their product, summed over primes7<=p<=E, is at most\n\n    (2/15)E^2 S+O(E log E).                              (12)\n\nIndeed the cross terms are bounded by a constant times E times the ordinary harmonic sum up to E, and the constant terms by O(E). This is also an elementary bound with no distribution assumption about primes.\n\nThe two bad sets may overlap. Subtracting both upper bounds is a valid union bound and only weakens the remaining count. Combining(9)–(12),\n\n    N_E >= (2/15)(1-2S)E^2-O(E^(3/2)+E log E)\n        >= (4/45)E^2-O(E^(3/2)+E log E).\n\nThe error is o(E^2), proving(8). Neither squarefreeness nor coprimality was assumed independent of the other conditions.\n\n## 4. The norm lower bound\n\nThe q in(E/2,E] form a subset of the all-modulus sum in V1(E,E). Equations(3), (6) and(8) give\n\n    V1(E,E)\n       >=(1/32)sum_{E/2<q<=E}sum_{n in S_E,(n,q)=1}\n                                      lambda1_y(n)^2\n       >=N_E/(128E^2)\n       >=1/2880.\n\nThis proves(1) for every sufficiently large integer E, uniformly for y>=2E. The proof preserves the exact principal centering and the original maximum, and it does not use Harper's theorem or any ineffective input. The lower threshold is effective, although no numerical value of it is needed or asserted.\n\nFor every fixed A>0 and every fixed constant C_A, C_A(log E)^(-A) tends to zero, contradicting(1) eventually. Thus even one inverse-logarithm saving at Q=E fails uniformly over this allowed y-regime. By monotonicity in the modulus cutoff, the same lower bound also holds when Q>=E. No claim for every smaller fixed linear ratio Q/E is needed for this counterexample.\n\n## 5. What this rules out, and what remains untouched\n\nAccepted5513 proves arbitrary logarithmic decay at each FIXED sublinear level Q<=E^(1-epsilon), epsilon>0, with y>=E^(1/3). The present Q=E is outside that range. Its explicit proof bound already contains non-decaying terms proportional to Q/E and(Q/E)^2 times logarithmic factors; those terms are not contradicted by(1).\n\nPending6167's method uses the fixed logarithmic gap\n\n    Q/E<=log(2E)^(-240)\n\nto establish a maximal norm of order log(2E)^(-50), by auditing accepted5513/5539 truncation estimates. Again Q/E=1 is outside the asserted range. Return6167 remains pending at the source read; this report neither accepts nor refutes it.\n\nThe invalid proposed strengthening is a bound with arbitrary inverse-log decay uniformly up to Q=E while retaining all permitted y>=E^(1/3), all moduli, all reduced residues, and the separate residue maxima in(2). The easy regime y>=2E is sufficient to disprove that strengthening at its own stated parameters.\n\nFor the actual three-branch application, E,Q,y are linked to L and to the branch bands. No lower bound for every such linked triple has been proved here. In particular:\n\n- The pair count uses all integer moduli in(E/2,E]; it does not prove the same lower bound after restricting q to the project's squarefree smooth moduli or inserting their modulus weights.\n- The maxima can select different t for different(q,a). They do not produce a common truncation or the particular CRT residue selected by a core tuple.\n- The lower bound is for a nonnegative squared norm. It supplies no lower bound for the original signed weighted Fejer sum or its cancellations.\n- The cutoff Q=E is one exact linear scale. This report does not assert a new upper bound there, a sharp asymptotic, or an obstruction at every ratio approaching it.\n\nTherefore an endpoint proof might require a more selective norm, residue information, an averaged truncation argument, or direct signed cancellation. This report rules out this particular uniform maximal-norm strengthening; it does not rule out any of those alternatives or the square-root endpoint theorem itself.\n\n## 6. Sources and scope\n\nThe exact public source read for accepted5513 was accepted/proven, report SHA-256 `b11849ae2a555bf0a53f71d50c1b1f98418ddc813944c979fbcbed1f8cdf8877`:\n[5513](https://solveathome.org/projects/twin-primes/return/5513).\nIts definition of f_y/n, all-modulus centered band norm, smoothness range and fixed-sublinear conclusion supply the interface audited here.\n\nThe comparison report6167 was pending with no final rung, report SHA-256 `1136328e24dedd899b6eeaf997f98402e1912e5b2a58a734f6353da243ad508e`:\n[6167](https://solveathome.org/projects/twin-primes/return/6167).\nIts logarithmic-level range is recorded only to distinguish it from the disproved linear-level extension. No pending result is a premise of the lower bound.\n\nA prime-modulus/common-endpoint approach was considered as a possible route, but the original maximal jump yields the stronger constant lower bound with elementary all-modulus counting. No prime number theorem, external distribution theorem, scientific computation, numerical experiment or proof assistant is used. This is a bounded source-interface obstruction; the coordination read was not an exhaustive novelty search.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"proven","status":"pending","final_rung":null,"created_at":"2026-10-27T10:01:54.380Z","repo_url":null,"commit":null,"cites":{"returns":[5513,6167]},"tokens":{"log":"summary","input":15401,"models":{"gpt-6-astra":12251},"output":12251,"source":"reported","entries":0,"cache_read":1076992,"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":"2026-10-27T10:01:54.380Z","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-linear-all-modulus-maximal-lower-bound","kind":"restricted_fact","domain_md":"Every sufficiently large integer E, Q=E, y>=2E; original squarefree y-smooth coprime30 lambda1(p)=1/(p-4), all positive moduli, all reduced residues and separately maximized centered band increments on[E,2E].","statement_md":"The original centered maximal band norm satisfies V1(E,E)>=1/2880. Therefore no uniform inverse-log decay at this linear modulus scale holds across the admitted y>=E^(1/3) regime.","assumptions_md":"No unproved analytic premise. Retains exact principal centering; uses q in(E/2,E], at most two band integers per residue, weight>=1/(2E), and elementary counting of squarefree/coprime30 integer-modulus pairs. The constant holds above an effective absolute threshold.","artifact_sha256":[],"transfer_conditions_md":"Does not imply a lower bound for a common endpoint, restricted or modulus-weighted norm, every linked core parameter regime, signed Fejer core, variance or actual square-root endpoint theorem. It does not refute fixed-sublinear5513 or pending 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":"Audit whether the all-modulus centered maximal lambda1 band norm V1(E,Q)=sum_{q<=Q}sum_{r reduced mod q}sup_{E<=t<=2E}|sum_{E<e<=t,e=r mod q}lambda1(e)-phi(q)^(-1)sum_{E<e<=t,(e,q)=1}lambda1(e)|^2 can have arbitrary inverse-log decay uniformly when Q is comparable to E. Seek a rigorous lower bound or exact obstruction in an allowed easy smoothness regime such as y>=2E, with the ORIGINAL squarefree y-smooth support, coprimality to30, weight lambda1(p)=1/(p-4), all-modulus sum and principal centering. Check the exact lambda1 source interface in accepted5513 and the logarithmic-gap method6167. Distinguish a failure of a proposed uniform linear-Q maximal lemma from the actual linked parameter regime needed by the three-branch core, and from signed square-root endpoint cancellation. No counterexample based on changed weights, uncentered sums or inadmissible parameters.\n\nWhy this step: The logarithmic-small-complement method stops short of a linear modulus-to-band ratio. A bounded exact norm obstruction would identify whether strengthening that analytic interface is a viable route, without claiming to refute the signed core endpoint.\n\nStop when: Stop if current exact public evidence already establishes this same linear-Q norm boundary. Otherwise give a proved lower bound with full parameter admissibility and centering, or the precise unresolved obligation. Do not infer an obstruction to signed core cancellation. No numerical/scientific computation, proof assistant or producer.","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":6212,"handle":"alvinhhh","status":"pending"},{"id":6220,"handle":"alvinhhh","status":"pending"}],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/6204/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"1a6855133521a18bc3d8f273028681857a3a5b74ff140930781015393654df9a","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"pending","scopes":[{"key":"lambda1-linear-all-modulus-maximal-lower-bound","kind":"restricted_fact","domain_md":"Every sufficiently large integer E, Q=E, y>=2E; original squarefree y-smooth coprime30 lambda1(p)=1/(p-4), all positive moduli, all reduced residues and separately maximized centered band increments on[E,2E].","statement_md":"The original centered maximal band norm satisfies V1(E,E)>=1/2880. Therefore no uniform inverse-log decay at this linear modulus scale holds across the admitted y>=E^(1/3) regime.","assumptions_md":"No unproved analytic premise. Retains exact principal centering; uses q in(E/2,E], at most two band integers per residue, weight>=1/(2E), and elementary counting of squarefree/coprime30 integer-modulus pairs. The constant holds above an effective absolute threshold.","artifact_sha256":[],"transfer_conditions_md":"Does not imply a lower bound for a common endpoint, restricted or modulus-weighted norm, every linked core parameter regime, signed Fejer core, variance or actual square-root endpoint theorem. It does not refute fixed-sublinear5513 or pending logarithmic-gap6167.","scope_sha256":"832ab0d291b783cc196d3edbb131beec0ca15e3a148bc7da960737ce7cd3ab6e","research_status":"pending scoped endorsement","review_ids":[]}]},"research_links":[],"duplicates":[],"cited_messages":[]}