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3291 bytes, as of the version from 2026-10-26 18:05 (current). Recorded here, not run.
# Argument selection for T522
The table stores one quantity, the raw radial moment W_n(s) of the endpoint distance after independent, uniformly directed, unit-length planar steps. The parameters are ordered (n,s). The angular variables are integrated out, not sampled or stored. There is no adjustable length, dimension, probability, coupling, or elliptic modulus.
## n: number of steps
This is a genuine discrete index. We select n=3 and n=4, the two short walks studied together in Borwein, Straub and Wan, *Three-step and four-step random walk integrals* (2013), especially equations (1), (5), (7), (40) and (58) in the October 14, 2010 author preprint. They are the first non-elementary cases after the one-step constant and the elementary two-step gamma quotient. The selection is a short-walk reference, not a claim that every step count is covered. The model itself fixes the normalization: each step has length exactly one; no division by sqrt(n) or factorial is applied.
## s: radial moment order
The family includes integer raw radial moments with s>=-1 for n>=3. We store s=1,-1,3,5,7, in that display order for each n. The mean distance s=1 is first, and the mean reciprocal distance s=-1 is next. Both seeds have explicit evaluations in the cited paper. The positive odd orders are the initial nontrivial odd-moment sequence, closed under the shift-by-two recurrences of Borwein, Nuyens, Straub and Wan (2011), Theorem 4 and Example 1. These are arithmetic moments of a probability distribution, not integer samples of an otherwise unmotivated continuous input. They also have the Bessel-moment representations proved in Borwein, Straub, Wan and Zudilin (2012), Section 7, Lemmas 1 and 3. Even moments have elementary squared-multinomial sums; they are used as exact controls in this run rather than stored as extra small-integer rows.
The upper cutoff 7 is an editorial prefix of a meaningful discrete family. It includes the two seeds and three further odd moments for each walk; it is not an intrinsic boundary or an interpolation grid. Ten entries suffice for this initial reference. The default enumeration, completeness note and stored rows must all match this explicit list.
## Evidence and provenance
- https://arminstraub.com/downloads/pub/walks2.pdf : equations (1), (5), (7), (19), (40), (58).
- https://arminstraub.com/downloads/pub/walks.pdf : Theorem 4 and Example 1.
- https://arminstraub.com/downloads/pub/walks-densities.pdf : Section 7, equations (75), (76), Lemmas 1 and 3.
- Family issue 213 was read before work. Its linked compressed screening report returned HTTP 404. The family fixes conventions and a count of ten but omits the exact list; this build therefore independently selects and explicitly records the ten rows justified here.
- Instructions: ../20261026T172724Z-build-skills/manifest.json. Both archived skill versions and the freshly fetched live skill have SHA-256 b4d51ded0d52cfe3dcc0859de0845c3879b0c2363273ba7e809a7e06ded40b18. See instructions-read.json for the absolute manifest path and its digest.
This is the builder's justification, not the required independent critique acceptance. The controller must obtain a separate critique, numerical checks and a fingerprint-bound argument-review sidecar before offering the draft.