# T706 research (checked in build, 2026-11-10)

- Source conventions checked: https://arxiv.org/html/math/0202141v2 §1,
  equation (1.1), gives F_n as the untapered Möbius sum targeting -chi in
  L^2(0,infinity). https://arxiv.org/html/math/0011254v1 §1.2, (1.18),
  calls that sum S_n; its (1.19) F_n is dual and excluded here.
- Kernel checked: https://arxiv.org/html/math/0306251v1 §4 defines the signed
  V(p,q), including modulus one; §9 Propositions 87 and 89 give A(1) and
  the rational-argument cotangent formula. Change-of-variable and error-bound
  derivations are in derivation.md. No unresolved normalization alternatives.
- Live discovery: catalogue.json (702 records, including accessible drafts),
  screening.json, lookup.log, and T639–T642 documents/metadata are local evidence.
  No duplicate family. Lookup N=1 matches T642's N=2 (same rho_1 in a different
  context); N=2,3 return no matches. Related published tables T640 (kernel),
  T641 (minimum), T642 (taper), T639 (cotangent sum) remain unmodified.
- Existing tags checked: number theory, functional analysis. The absent
  Nyman-Beurling criterion tag is recommended for this family and T639–T642;
  no published tables were retagged. Only existing tags are used in T706.
- The 30-index consecutive selection and its noncanonical bound are justified
  in argument-selection.md. No continuous free parameter or quantile input.
- Reused code: T641 generate.py, SHA-256
  7e155af1d4a4963f874e0c536051c1cf0b7afd01e21b93d64a674c7fab8592c8;
  T641 checks.py, SHA-256
  3a6ea0c72d6588b2fdda2d91a0414232f1d64092a5e928cd4121cc4d05798678.
  Only computational components are reused, with inspected derivations and
  fresh controls/calculations. Sibling values and verdicts are not evidence.
- Provenance mapping checked against family-313/proposal.json and live wanted
  issue #273: proposal #313, contributes to #273 only. Both URLs were included
  in the first draft request; mathematical prose excludes request history.
- Numerical evidence and source hashes: numerical-evidence.json;
  reproducible generator: generate.py; separate-route check: checks.py plus
  integral_kernel.py; resumable exact-endpoint kernels: kernel-checkpoints.jsonl.
  Archived instruction bundle: skill-manifest.json, verified local/live bundle
  d6661c2f21613524889fa3a57d3c95f8330344ca62e5a856211c08a282b7dbab.
- No universal asymptotic or monotonicity assertion is inferred from this data.
  A fresh independent scientific and argument-selection verdict is outstanding.