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# Checked research for T712
Definition and normalization checked in Bagchi, arXiv:math/0607733v1:
p.137 gives weight 1/[m(m+1)], constant target 1 and g_k(m)={m/k};
pp.143–145, equation (5) and proof of Theorem 1 give the step-function isometry;
p.145 Remark 8 (using Remark 4) gives squarefree support; p.146 defines D(L).
The proposed locator “Remark 3” is erroneous. The finite table uses squared
D, real coefficients (equivalent to complex), ordinary fractional parts,
unnormalized weights/basis, denominator bound N>=2. No branch choices occur.
The RH closure statements are sourced equivalences, not conclusions from data.
Bagchi's asymptotic suggestion on p.146 is conjectural and is not stated here.
BBLS arXiv:math/0306251v1 §4 defines V(p,q)=sum {rp/q}cot(pi*r/q);
Propositions 87 and 89 (pp.38–39) supply A(1) and rational A(p/q), p,q coprime.
Recomputed kernel G_jk=A(j/k)/j, anchored K_jk=G_jk-G_j1/k-G_1k/j+G_11/(jk).
Target b_k=log(k)/k follows by subtracting <chi,rho_1>/k from
<chi,rho_k>; the independent direct periodic sum checks this normalization.
Báez-Duarte arXiv:math/0011254v1, equations (1.14)/(1.17), explicitly gives
B_N and V_N. Exact coefficient sums prove zero tails; V_N uses squarefree
indices, whereas B_N may introduce its nonsquarefree endpoint.
Independent derivation: for L-periodic h, regroup 1/[m(m+1)] as
(1/L) sum_r h(r)[psi((r+1)/L)-psi(r/L)]. This follows from the convergent
partial-fraction digamma difference; there is no numerical tail truncation.
Use h=g_j*g_k at L=lcm(j,k), and h=g_k at L=k. For the squarefree systems use
principal submatrices, recomputing each optimum, never inferring it from the
all-support scalar. All g_k are linearly independent: differences at m give
sum alpha_k/k - sum_{k|m}alpha_k; m=1 forces the first sum to vanish, and
induction m=2..N forces each alpha_m=0. Thus K is positive definite. The target
is outside each finite span because every basis vector vanishes at multiples
of the common period, a set of positive sequence weight.
Discovery refreshed: cached authenticated catalogue, text search, already_here,
already_asked, tags and candidate metadata. No duplicate. T641 is unrestricted;
T707/T708 are particular coefficient choices. T640/T639 supply kernels;
T226 supplies digamma context. Read authoritative definitions, not generators
or sibling computation conclusions. Sample lookup N=2 finds T343/T433, other
contexts for 1-log2; N=3 and N=4 have no hits. These coincidences are not
family duplication. Tags number theory, functional analysis, sequence exist;
Nyman-Beurling absent (recommendation only, no unrelated tag changes).
Supporting artifacts: sources.json names PDF URLs/hashes; catalogue.json,
search-links.json, screen.json, lookup.json, T*-metadata.json/T*-refreshed.json;
argument-selection.md; offline-results.json; independent_check.py and logs.
The numerical comparisons are recorded separately when run; they are not
assumed here. Website T712 is a private claimed draft, and initial message cites
proposal #316 and contributes-to #273 (checked issue text). No other wanted
issue is a direct origin. No unresolved normalization alternatives remain.
Canonical archive: ../20261111T001542Z-build-skills/manifest.json, SHA-256
c5ee30e128246f511fb7c0e8e37f8b7d2419d5fad25227688a6111fdf9686049;
local/live bundle d6661c2f21613524889fa3a57d3c95f8330344ca62e5a856211c08a282b7dbab.