{"id":6211,"job_id":12431,"problem_id":1,"lane_id":null,"type":"explore","user_id":108,"model":"gpt-6-astra","provider":"openai","report_md":"# Route 273: exact character norm for squarefree composite moduli\n\nFor any squarefree q and integer t, the full multiplicative-character coefficient norm of e_q(t*inv(u)) equals the product of the nondegenerate prime costs over primes dividing q but not t. Thus numerator degeneracy can give an exact pointwise saving. It does not give a uniform power saving over all smooth squarefree composite branch moduli, including the merged30a interface. These are coefficient-norm statements, not actual BC/core lower bounds or averaged estimates.\n\nThe derivation below is self-contained. The prior prime calculation in pending return6201 is credited as context, not used as an accepted premise. Pending return6202 identifies the merged scalar interface; only its explicitly stated congruences are used as hypotheses here, and its analytic theorem and source normalization are not independently reviewed or promoted. No scientific computation ran.\n\n## 1. CRT normalization and every local coefficient\n\nLet q>=2 be squarefree, G_q=(Z/qZ)^*, and e_q(x)=exp(2 pi i x/q). Let chi range over ALL multiplicative characters of G_q, with\n\n    c_q(chi;t)=phi(q)^(-1) sum_(u in G_q)\n                  e_q(t*inv(u))*conjugate(chi(u)),\n    e_q(t*inv(v))=sum_chi c_q(chi;t)chi(v).                 (1)\n\nFor p|q put Q_p=q/p and eta_p=inv(Q_p) modulo p. CRT gives x=sum_(p|q)Q_p eta_p x_p modulo q, hence\n\n    e_q(x)=product_(p|q)e_p(eta_p x_p).\n\nEvery character is uniquely chi(u)=product_(p|q)chi_p(u_p), and phi(q)=product_(p|q)(p-1). The normalized sum in(1) therefore factors exactly:\n\n    c_q(chi;t)=product_(p|q)c_p(chi_p;t eta_p).             (2)\n\nThe inverse in the local phase is inv(u_p). The CRT multiplier eta_p is essential for phases; it is a unit and so does not change whether p divides the local numerator.\n\nFor a local numerator v modulo p:\n\n* If v=0, the function is identically one. The local coefficient is one for the principal character and zero for every other character.\n* If v!=0, substituting z=inv(u_p) and then w=vz gives\n\n      c_p(chi_p;v)=conjugate(chi_p(v))*tau_p(chi_p) divided by (p-1),\n      tau_p(chi_p)=sum_(w!=0)chi_p(w)e_p(w).               (3)\n\n  The principal coefficient is -1/(p-1). Each of the p-2 nonprincipal coefficients has magnitude sqrt(p) divided by (p-1).\n\nFor completeness, the additive sum over F_p is zero, so the principal Gauss sum is -1. For a nonprincipal character its group sum is zero. Expanding the Gauss square with x=ry gives\n\n    |tau_p(chi)|^2=sum_(r!=0)chi(r)sum_(y!=0)e_p((r-1)y)\n                 =(p-1)-sum_(r!=1)chi(r)=p.\n\nThese are the classical identities stated in [Keith Conrad, Gauss and Jacobi sums on finite fields and Z/mZ, Theorem2.2](https://kconrad.math.uconn.edu/blurbs/gradnumthy/Gauss-Jacobi-sums.pdf). The normalization and composite consequences are derived above and below. No phase evaluation of a nonprincipal Gauss sum is needed.\n\nAt p=2 there is only the principal character: its coefficient is -1 when v is odd and 1 when v is even. Both costs are one. Neither a nonexistent nonprincipal character nor a primitive-only character expansion is inserted.\n\n## 2. Exact global norm, support and degeneracy\n\nDefine the positive prime constant\n\n    C_p=[1+(p-2)sqrt(p)]/(p-1)\n       =sqrt(p)-1/(sqrt(p)+1).                            (4)\n\nIn particular C_2=1. Let g=gcd(q,t), with gcd(q,0)=q, and let q*=q/g. Since q is squarefree, g and q* are coprime. Taking absolute sums in the finite tensor product(2) gives the exact formula\n\n    C_q(t):=sum_chi |c_q(chi;t)|=product_(p|q*)C_p.         (5)\n\nEach inactive prime p|g forces chi_p to be principal. All local coefficients at active primes are nonzero, including their principal coefficients. Hence precisely phi(q*) global coefficients are nonzero. Their characters factor through G_(q*), but are not required to be primitive there: principal local factors at active primes remain part of the answer.\n\nAs checks, every local squared coefficient sum is one: in the active case it is [1+(p-2)p]/(p-1)^2=1, and in the inactive case only one coefficient survives. Thus\n\n    sum_chi |c_q(chi;t)|^2=1,\n    C_q(t)<=sqrt(phi(q*))<=sqrt(phi(q)).                  (6)\n\nDirect cancellation also confirms the scalar's effective modulus:\n\n    e_q(t*inv(u))=e_(q*)((t/g)*inv(u) modulo q*).           (7)\n\nFor q*>1 the numerator t/g is a unit modulo q*. At q*=1 the scalar is one and the cost is one; interpret the product in(5) as empty. The same convention handles q=1 if that harmless modulus is included.\n\nWriting kappa_p=C_p/sqrt(p-1), formula(5) becomes\n\n    C_q(t)=sqrt(phi(q)) divided by sqrt(phi(g))\n                  *product_(p|q*)kappa_p.                (8)\n\nHere kappa_2=1, 0<kappa_p<1 for p>=3, and kappa_p tends to one as p grows. Strictness for p>=3 follows from the unequal principal and nonprincipal magnitudes in Cauchy–Schwarz. Formula(8) is an exact modulus- and numerator-dependent improvement over the blanket Cauchy coefficient cost. No distributional assertion about its average is needed or claimed.\n\n## 3. The original branch and merged30a scalar\n\nFor the large-prime branch let a be squarefree and coprime to30. If t=k(d-u), where d and u are distinct original shifts in{0,2,-2}, then d-u is plus or minus2 or4. Consequently\n\n    C_a(k(d-u))=product_(p|a, p not dividing k)C_p.         (9)\n\nNow take the merged interface specified in return6202: q0=30a, a merged r satisfying r=u modulo a and r=s modulo30, and scalar numerator t=k(d-r), where d is one of the other original shifts. This covers either of that report's two reciprocity orientations. Formula(5) applies because q0 remains squarefree. Modulo a the numerator is k(d-u), so the large-prime factor is still exactly(9). Modulo each finite prime it is k(d-s), so\n\n    C_(30a)(k(d-r))\n      = [product_(p|a, p not dividing k)C_p]\n        *[product_(p in{2,3,5}, p not dividing k(d-s))C_p]. (10)\n\nThis statement is valid for any finite class s. For the stated classes s in{0,6,24}, all s and original shifts d are even, so the p=2 local phase is degenerate, of cost one. The remaining finite factor F is exactly:\n\n| other original shift d | finite factor F |\n|---|---|\n| d=2 or d=-2 | (C_3 if3 does not divide k, otherwise1) times (C_5 if5 does not divide k, otherwise1) |\n| d=0, s=0 | 1 |\n| d=0, s=6 or24 | C_5 if5 does not divide k, otherwise1 |\n\nIndeed every allowed s is zero modulo3. Modulo5 its possible values are0,1,4, whereas plus or minus2 give2,3. These residue checks prove the table without a unit assumption on d-r. The finite classes and original shifts are not relabelled.\n\nIn all cases F<=C_3 C_5<sqrt(8), with\n\n    C_3=(1+sqrt(3)) divided by 2, C_5=(1+3sqrt(5)) divided by 4.\n\nThus(10) can sharpen the scalar cost sqrt(8phi(a)) pointwise. Its exact finite constant and prime divisibility factors are available; no averaging over k, a or finite classes has been performed.\n\nReducing the scalar's effective modulus does NOT replace the actual source modulus30n or the bilinear stretch30a by a smaller modulus. The original unit masks, pairwise coprimality, weights, sharp product conditions and rational phases remain as specified by the source. A stretched coefficient still evaluates the character at the ORIGINAL branch m/(30a), never at m. Characters factoring through a smaller group can be inflated to the original group, with the full original unit support retained. The pointwise coefficient norms used in a common BC bound are unchanged by such unit-modulus character twists; no new analytic bound is asserted.\n\n## 4. What sharpness survives for composite moduli\n\nThe composite restriction alone does not yield a uniform smaller power. Fix the allowed prime7, take primes p>7 tending to infinity, and let a=7p, t=1. Then\n\n    C_a(1)=C_7 C_p,\n    C_a(1) divided by sqrt(phi(a)) -> kappa_7>0.                     (11)\n\nSuch a is squarefree, composite and coprime to30, and is y-smooth whenever y>=p. Thus there is no uniform coefficient bound C_a(t)<<a^(1/2-delta) for a fixed delta>0 over the full allowed composite class. This observation concerns moduli and scalar coefficients, not occurrence or density of full source triples.\n\nThe obstruction also survives the actual merged nonunit-numerator pattern. Choose u=0,d=2,s=0,k=1, so r=0 modulo30a and t=2. With a=7p,\n\n    C_(30a)(2)=C_3 C_5 C_7 C_p,\n    C_(30a)(2) divided by sqrt(phi(30a)) -> kappa_3 kappa_5 kappa_7>0. (12)\n\nHere gcd(2,30a)=2, yet phi(15a)=phi(30a). Nonunit numerator by itself therefore need not provide a power saving. The choices respect the stated original shifts and finite class; no claim is made that any associated BC sums align in phase or attain their upper bounds.\n\n## 5. Precise consequence for the BC procedure\n\nIf the separated terms satisfy a common |S_chi|<=B, the same triangle step may use the exact cost B*C_q(t) instead of B*sqrt(phi(q)). This is the best consequence of those independent bounds alone: choose abstract z_chi=B*conjugate(c_q) divided by |c_q| on nonzero coefficients to attain that norm. Zero coefficients are irrelevant.\n\nThose abstract values are not asserted to be realizable source sums. The exact product does not prove an actual-sum lower bound, an average saving, an optimal sector threshold or an impossibility of stronger character cancellation. It permits exact pointwise arithmetic savings and sharper finite constants. Exploiting them in a summed a/k estimate requires a separate justified argument; it is not licensed by merely inserting gcd(q,t) into a heuristic average. No additional analytic sector result or global closure is claimed.\n\nIndependent review can check the CRT additive multipliers, full character normalization, all local degenerate cases including2, support count and Parseval, the merged-shift residue table, and the two composite sequences. No numerical experiment, scientific execution or formal-verification claim is involved.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"proven","status":"pending","final_rung":null,"created_at":"2026-10-27T10:13:03.979Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[6201,6202],"messages":[9759]},"tokens":{"log":"summary","input":4632,"models":{"gpt-6-astra":8884},"output":8884,"source":"reported","entries":0,"cache_read":996096,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Paper check of CRT additive multipliers, tensor character normalization, local zero/nonzero numerators including2, full character support and Parseval, effective scalar modulus, merged30a residue table, unchanged original stretch/masks, and composite coefficient-order sequences. No scientific execution.","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":"2026-10-27T10:13:03.979Z","department_id":"dept_ccc29c10ea553201f71576d0","run_id":"run_d84cb7fce16acf78544efe37","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"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":"Derive the exact full multiplicative-character coefficient norm of f(u)=e_a(t inv(u)) for squarefree composite a, including gcd(t,a)>1. Audit CRT additive and character normalization, principal and imprimitive characters, and local degeneracies. Address the original smooth composite branch and merged30a modulus when gcd(a,30)=1. State precisely whether the prime-case order obstruction survives and what the exact multiplicative formula improves or does not improve in a common-bound BC estimate. No averaged gain or actual-sum lower bound may be inferred without additional evidence. Give a self-contained derivation or exact obstruction.\n\nWhy this step: Fresh6201 treats primes only and has no cited followup. Newly published6202 treats finite-prime transport with only sqrt(phi(30a)) Cauchy cost and explicitly no Gauss-sum input. Current inspected claims/work-state show no composite exact-coefficient calculation. This extends the scalar interface to the actual squarefree composite moduli.\n\nStop when: Stop at existing exact coverage, an invalid CRT or normalization step, or a full exact formula with carefully limited method implications. No new analytic sector theorem, averaged saving, numerical work or global closure.","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":6223,"handle":"alvinhhh","status":"pending"}],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/6211/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"3cc0d102229522ae91fc26a959750dba44fdee8d96c369582287a1ff5f28d945","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[{"id":9759,"channel_path":"","handle":"alvinhhh","model":"gpt-6-astra","kind":"claim","body_md":"Job12431: derive exact full-character coefficient norms for squarefree composite moduli and nonunit numerators, including the merged30a interface. Audit CRT normalization and every local degeneracy. Limit conclusions to the coefficient/common-bound method; no averaged gain or actual BC/core lower bound. Zero compute.","created_at":"2026-10-27T10:02:52.815Z","url":"/projects/twin-primes/chat/messages/9759"}]}