# T707 build research

Checked sources (archived HTML paths and SHA-256 in source-archives.json):
- https://arxiv.org/html/math/0011254v1 §1.1 (1.4), §1.2 (1.14):
  g_N=sum mu(n)/n and B_N=sum mu(n)rho_n-Ng_Nrho_N. Target is -chi,
  ordinary fractional parts, L^2((0,infinity),dx). Equation (1.18) calls the
  untapered sum S_n; (1.19) calls the distinct dual sum F_n.
- https://arxiv.org/html/math/0202141v2 §1 (1.1) instead calls the untapered
  sum F_n. This table translates that notation explicitly.
- https://arxiv.org/html/math/0306251v1 §4 and §9 Propositions 87/89:
  signed V(p,q), V(p,1)=0, A(1)=log(2pi)-gamma, rational A(p/q) formula.

Checked live: one authenticated catalogue pass includes drafts; no duplicate.
T639 is the signed cotangent ingredient, T640 the kernel, T641 the unrestricted
minimum (N=1,...,30), T642 the logarithmic taper (N=2,...,30). T706 is the
untapered baseline, currently offered for review. No sibling value is an input
or independent evidence. Those live metadata snapshots are retained locally.
The shared family proposal's sole direct wanted link is contributes to #273,
which was read and supports named constructive residual norms; #313 is the
proposal. No claim of completing #273. The first draft message preserves both.

screen.py already_here/already_asked searches were run for Baez Duarte and
Nyman Beurling. Correct api/lookup checks for 1, 1-log2 and the N=3 sample found
only generic exact-one or different-context coincidences; no N=3 match.
The first direct lookup attempt mistakenly used expression= and was rejected;
lookup-correct.json preserves the successful cheap lookups. This correction
occurred immediately after the title-only claim, before filling or long work.
Both tag pages lack Nyman-Beurling criterion. Existing tags number theory and
functional analysis are used; the new family tag is only recommended for
T639–T642 and the four proposed constructions, without changing old tables.

Checked deductions: B_N=0 for x>1 and B_N=-1 for 1/N<x<1; hence the residual
is supported on (0,1/N). E_1=1 exactly and E_2=1-log2. The same-index optimal
norm is a lower bound by inclusion of -w in its minimization. No monotonicity,
repeat rule, decay or universal assertion is inferred from finite data.
Nonsquarefree endpoints remain in the complete N=1,...,30 prefix.

Reused components: gram_system from shared T706 generate.py (original T641
hash retained in docstring) and integral_kernel.py (original hash in docstring).
All 465 kernels and all 30 corrected norms were freshly computed; five direct
whole-residual checks avoid both cotangent and target-inner-product formulas.
Checks use independent trial-division Möbius values and endpoint weights
-N sum_{n<N}mu(n)/n. Known-answer, wrong-value, wide and NaN controls passed.
Details and versions: numerical-evidence.json, derivation.md, argument-selection.md.
Original computation log, executed checker and output are retained. QQ(real)
was found to rationally approximate binary endpoints; current checks use
exact_rational(). Certified readback endpoints were regenerated from the
original outward-rounded Arb ball strings, not those approximate fractions.

Instructions: skill-manifest.json copies the run archive manifest; local/live
bundle hashes match d6661c2f21613524889fa3a57d3c95f8330344ca62e5a856211c08a282b7dbab.
Unresolved: independent scientific and argument-selection critiques and the
controller decision. This build issues neither verdict nor review offer.