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3001 bytes, as of the version from 2026-10-31 20:52 (current). Recorded here, not run.
# Argument selection for T640
Policy: the coordinated canonical skill archived in
`agents/runs/20261031T203213Z-build-skills/manifest.json`. Both archived versions
and the snapshot canonical source have SHA-256
`a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278`.
The attached `skill-evidence.json` records the source manifest and both hashes.
This is a build justification, not an independent critique acceptance.
The sole free parameter is the exact positive rational dilation `lambda`.
Its role is **documented_application** throughout the selection, including 1.
There is no additional continuously varying coupling or probability parameter.
Báez-Duarte, arXiv:math/0202141v2, §1 and Theorem 1.1, selects the integer
dilates rho_n(x) = {1/(n*x)} in L²((0,∞),dx). Their Gram matrix satisfies
G_jk = A(j/k)/j, by the change of variable t=1/(j*x). Thus the full prefix
1 ≤ j,k ≤ N requires exactly the reduced positive rationals p/q with
1 ≤ p,q ≤ N. BBLS, arXiv:math/0306251, §9, names and defines A, and
Proposition 89 supplies the rational formula. These sources were read in this
run and archived as bbls.pdf, bd.html and bc.pdf; sources.json records hashes.
We select the entire family at N=30. Thirty is a practical, noncanonical
initial bound for small dense Gram systems and the shared approximation
problem; it is not an intrinsic mathematical threshold or an entry target.
It agrees with the family’s cotangent companion T639, which supplies all
required nontrivial residue classes and their modular inverses. Increasing N
would add ratios for larger approximation spaces, rather than improve the
precision of the chosen small-space data. Those larger spaces are deferred.
The selection includes integer dilations, their reciprocals, and coprime
noninteger ratios with prime, prime-power and composite numerators and
denominators. This breadth serves complete Gram matrices, including every
scale and off-diagonal correlation, rather than extending only integers or
pruning to points with short closed forms. Both reciprocal inputs remain:
A(p/q) = (p/q) A(q/p), so the numbers differ in scale. This is one argument
selected by a documented application, not two continuous parameters sampled
on a rectangular mesh. It needs no exploratory reference_sample component.
The default generator enumerates each reduced fraction exactly once, by
increasing max(p,q), then p, then q. Integer identities are reduced to their
integer spelling. The domain in Parameters is all positive rationals; the
bound appears only here, in the generator default and in the completeness
note. No quantile, percentile, median or statistical critical value is involved.
There are 555 selected ratios, including the unit dilation. The build measures
their written lengths and total block size before filling, and records these
in numerical-evidence.json. Independent critique must reassess this full
selection and fingerprint the live document and attached source dependencies.