# T709 build research, 20261110T211830Z

Checked facts, not an independent critic's verdict:
- Báez-Duarte math/0011254v1 §1.2 (1.19) names the dual approximation
  F_n=sum [M(n/k)-M(n/(k+1))] rho(k/(nx))-rho(1/(nx)). The table calls it D_N.
  §1.1 fixes ordinary fractional parts, M and the Lp(0,infinity) convention;
  (1.14) defines the distinct B_N. math/0202141v2 §1 (1.1) instead uses F_n
  for the untapered sum. All three source passages were inspected directly.
- BBLS math/0306251v1 §4 defines the signed V(p,q), including V(p,1)=0;
  §9 Proposition 89 gives the finite rational autocorrelation formula.
  Source downloads and hashes: source-archives.json (baez.html, bbls.html,
  bd2002.html in this run directory). Exact source URLs are recorded there.
- Live catalogue fetched once including drafts: no existing dual table;
  accessible siblings T706/T707/T708 are distinct offered drafts. Live metadata
  T639/T640/T641/T642/T707/T708 checked. T640 is the kernel cross-link, T707
  the natural B_N companion; no same-N T641 lower bound is claimed here.
- screen.py already_here and already_asked: only family #313, originating
  wanted #273 and older #281 are relevant. #273 inspected through GitHub;
  constructive residual norms are explicitly in scope. Initial edit message
  preserves proposal #313 and contributes-to #273, with no wider-origin claim.
- /api/lookup checked 1 and high-precision E_2,E_3,E_4. E_2 matches distinct
  contexts T343 and T433; E_3,E_4 had no matches. Small integers are generic.
  lookups.json and screen.json retain results without treating coincidences
  as duplicate families. Existing tags number theory and functional analysis
  confirmed; Nyman-Beurling criterion absent from both tag pages. Recommend
  that tag for T639–T642 and T706–T709, without changing published tables.
- Exact cancellation sum k*w_k=0 follows from Möbius divisor inversion.
  This makes a period-one step residual in t=1/(Nx), allowing whole-residual
  integration. derivation.md gives the proof and four exact small formulas.
- Fresh computation checked all 30 squared norms with independent digamma
  residual integrals, plus explicit-series tails at 1,2,3,4,30. Five negative
  controls were rejected. numerical-evidence.json contains exact binary
  endpoints, code hashes and versions. No sibling decimals were inputs.
- Reused gram_system and integral_kernel.py components from the shared family
  dossier, inspected and rederived in this run. The latter's use here is the
  rational integrator and bounded tail constants, not a copied sibling verdict.
- Archived skill: skill-manifest.json, bundle SHA-256
  d6661c2f21613524889fa3a57d3c95f8330344ca62e5a856211c08a282b7dbab.
  The build's argument-selection.md justifies all N=1,...,30 as one discrete
  prefix; no free continuous input exists. No source convention is unresolved.

Independent scientific and argument-selection judgments remain for the
controller's fresh critic. No asymptotic conjecture is promoted to a theorem.