{"id":1757,"job_id":4023,"problem_id":1,"lane_id":null,"type":"explore","user_id":22,"model":"gpt-6-astra","provider":"openai","report_md":"# Job #4023, route 162: the symmetric truncation reverses the consumer inequality\n\n**Stop the proposed positivity/count extension for this family.** Return\n#1754's collapse identity is algebraically correct, but the resulting\nproduct is not the required lower certificate. There are explicit\ncounterexamples for both proposed truncations, including R=4 at z=29,\ninside the proposed extension range. The obstruction is pointwise\nadmissibility, before uniform mean positivity or a mean-square estimate.\n\nThis is a proof/source result. No scientific computation was run and no\npublished table was reproduced.\n\n## 1. The direction error\n\nWrite `theta(n)=1_(gcd(n,P(z))=1)` and use R for the truncation order,\nleaving r for the integer position. The required component brackets are\n\n```\nL(n) <= theta(n) <= U(n).\n```\n\nSetting `L=U=F` satisfies these inequalities only when `F=theta`\npointwise. Being a normalized divisor sum with coefficients bounded by\none does not supply the missing lower inequality.\n\nFor R=2 the size cap never bites, since products of two primes below z\nare below z^2=theta>=0`, hence\n\n```\nF_2(r)F_2(r+2) >= theta(r)theta(r+2),\n```\n\nthe reverse of the needed `cc<=theta(r)theta(r+2)`.\n#1754 writes the greater-than direction and then calls its mean a\nlower bound. A positive majorant mean does not establish the existence\nof a survivor. C4 cannot repair a failure of C6.\n\n## 2. Two exact counterexamples, with the cap checked\n\nFor R=2, take z=11, P=210, D=1331, and r=103.\nThe two gcds are 1 and 105=3*5*7. Every divisor of either gcd is below\nD. Thus\n\n```\nF_2(103)=1, F_2(105)=1-3+3=1,\ncc(103)=1, theta(103)theta(105)=0.\n```\n\nFor the proposed R=4 repair, take z=29, D=24389, and r=15013.\nHere\n\n```\nr+2=15015=3*5*7*11*13 < D.\n```\n\nThe first member is coprime to P(29): it is odd, is -2 modulo\n3,5,7,11,13, and has remainders 2,3,17 modulo 17,19,23.\nTherefore its gcd is 1 and the second gcd is exactly 15015.\nAgain the size cap is inactive on all relevant divisors. Consequently\n\n```\nF_4(15013)=1,\nF_4(15015)=1-5+10-10+5=1,\ncc(15013)=1, theta(15013)theta(15015)=0.\n```\n\nThe single-position window containing either displayed r has proposed\ncertificate 1 and true rough-pair count 0. These are direct failures\nof the pointwise certificate, not adverse mean-square measurements.\n\n## 3. The failure persists for every fixed truncation order\n\nThe elementary alternating-binomial identity gives, at k=R+1,\n\n```\nsum_(j=0..R) (-1)^j binom(R+1,j) = (-1)^R.\n```\n\nFor fixed even R, choose a fixed odd squarefree n with R+1 prime\nfactors. For all sufficiently large z, its primes are below z and\nn<=z^3. CRT supplies a position with gcds `(1,n)`: take r=-2 at primes\ndividing n, avoid both 0 and -2 at every other odd prime below z,\nand take r odd. Then the collapsed product is 1 but the true\nrough-pair indicator is zero.\n\nFor fixed odd R, choose two coprime fixed odd squarefree n1,n2,\neach with R+1 prime factors. Once both are below z^3, CRT similarly\nsupplies gcds `(n1,n2)`. Each truncated divisor sum is -1, so their\nproduct is again 1 while the true pair indicator is zero.\n\nEvery remaining odd prime has at least one allowed residue, and 2 is\nin neither gcd, so both CRT constructions are legitimate. The size\ncap is inactive on the selected gcds. This proves failure for every\nfixed R, not just a finite exception to an eventual positivity trend.\n\nSeparately, exact equal component brackets cannot be recovered merely\nby increasing R under an insufficient level cap. If\n`sum_(d|n) a_d=1_(n=1)` for every n dividing P, Mobius inversion forces\n`a_d=mu(d)` for every divisor of P, including d=P. Whenever D