{"id":6223,"job_id":12455,"problem_id":1,"lane_id":null,"type":"explore","user_id":108,"model":"gpt-6-astra","provider":"openai","report_md":"# Route 273: exact character second moment and the missing correlation estimate\n\nThe full character second moment is an exact residue-product collision energy. For a degenerate scalar numerator, the collision modulus is the quotient supporting the nonzero character coefficients. The resulting Cauchy estimate improves the exact coefficient-triangle estimate only if a quantitatively specified collision bound holds. This report proves the identity and that sufficient condition; it does NOT prove the missing bound or a new BC/core estimate.\n\n## 1. Preserve the actual bilinear sums\n\nFix the scalar modulus q=a or30a, frequency numerator t, orientation, bin and any Perron or partial-summation parameters. Here q is the scalar modulus; the physical source modulus remains abc or30abc. Let D be the finite set of ORIGINAL branch pairs(b,c) retained at that stage. Every pair has gcd(bc,q)=1. All original branch labels, weights, smoothness and coprimality masks remain in D or the coefficients.\n\nWrite the character-twisted sum as\n\n S_chi=sum_((b,c) in D) W_(b,c) chi(b)chi(c)\n =sum_D W_(b,c)chi(bc). (1)\n\nThe complex W is independent of chi and retains the actual inverse phase and every other non-character factor. For example, in the orientation with stretch m=q*c, the inverse phase involves inv(q*c) modulo b, while the character factor is chi(b)chi(c)=chi(b)chi(m/q), NOT chi(m). The other orientation is analogous. No conjugation is inserted on the second branch's character.\n\nThe identities below hold for arbitrary finite W, so a sharp joint product cutoff can also remain in W. This observation DOES NOT make an arbitrary joint-cutoff coefficient eligible for the separated BC theorem. When a common BC bound U is invoked below, it must already be justified for the particular S_chi—for example at the original separated Perron/partial-summation stage. No new analytic permission follows from the finite identities.\n\n## 2. Full character second moment\n\nLet h=phi(q), and for r in G_q=(Z/qZ)^* define\n\n A_q(r)=sum_(D, bc=r mod q) W_(b,c).\n\nThen S_chi=sum_r A_q(r)chi(r). Full character orthogonality is\n\n sum_chi chi(r)conjugate(chi(s))=h*1_(r=s).\n\nFor unequal r,s the sum is zero because multiplication of the character index by a character nontrivial on r/s preserves the sum and multiplies it by a value different from one. Equality r=s gives h. Expansion of the square therefore proves\n\n (1/h)sum_chi|S_chi|^2\n =sum_r|A_q(r)|^2\n =sum_(i,j in D) W_i conjugate(W_j)\n 1_(b_i*c_i=b_j*c_j mod q). (2)\n\nThe diagonal i=j contributes D_2=sum_i|W_i|^2. The ordered off-diagonal collision sum O_q contributes the rest. It is real, since reversed pairs are conjugates, but need not be nonnegative. The total energy E_q=D_2+O_q is nonnegative. This is a weighted correlation identity, not an unweighted count and not a cancellation estimate.\n\n## 3. The contributing character quotient\n\nSuppose q is squarefree, as in the original a and merged30a interfaces. Put g=gcd(q,t), Q=q/g, and h*=phi(Q). The scalar\n\n f(u)=e_q(t*inv(u))=e_Q((t/g)*inv(u))\n\nfactors through G_Q. If Q=1 it is the constant one. In its normalized full expansion f=sum_chi c_chi chi, only characters trivial on primes dividing g contribute. They are precisely the characters of G_Q inflated to G_q, a set H of size h*. One can check this directly by factoring the normalized coefficient sum over the CRT prime factors: an inactive prime has local function one and only its principal character survives; at each active prime all local character coefficients are nonzero by the elementary Gauss-square identity. Principal and imprimitive characters are retained.\n\nNormalized Parseval gives sum_(chi in H)|c_chi|^2=1 because |f|=1. The exact coefficient norm is denoted C=sum_H|c_chi|. Equivalently, by the local calculation,\n\n C=product_(p|Q) [sqrt(p)-1/(sqrt(p)+1)], (3)\n\nwith empty product one. At an active prime the principal coefficient has magnitude1/(p-1); the p-2 nonprincipal magnitudes are sqrt(p) divided by (p-1). The latter follows by expanding the Gauss square as (p-1)-sum_(r!=1)chi(r)=p. These local facts give(3), including cost one at2. The previously submitted return6211 is context for this interface; the quotient and normalized square norm used here follow from the finite-group argument, without assuming that report has been accepted.\n\nOrthogonality over H is orthogonality modulo Q, not modulo q. With A_Q(r)=sum_(D,bc=r mod Q)W_(b,c), the exact relevant identity is\n\n (1/h*)sum_(chi in H)|S_chi|^2\n =E_Q:=sum_(r in G_Q)|A_Q(r)|^2\n =D_2+O_Q. (4)\n\nHere O_Q uses distinct ordered pairs with b_i*c_i=b_j*c_j modulo Q. Original pairs still satisfy the full q unit masks and original source conditions. Q replaces only the character collision modulus; it does not replace the actual stretch q, physical modulus, inverse phase or coefficients. Coarsening the collision buckets can merge complex sums constructively or destructively; no monotonic comparison of E_Q with E_q is assumed.\n\n## 4. The exact improvement contract\n\nLet T=sum_H c_chi S_chi be the scalar-expanded expression, and suppose a common nonnegative U has been established for all |S_chi|. The coefficient-triangle estimate is\n\n |T|<=C*U. (5)\n\nThe normalized coefficient square norm and(4) instead give\n\n |T|^2<=sum_H|S_chi|^2=h*E_Q. (6)\n\nTherefore for any desired 0