# Sources and corpus evidence for T606

The controlling skill manifest is
agents/runs/20261029T093155Z-build-skills/manifest.json.
Both versions have SHA-256
28ecbeeccb0c5e5d6eb541b5a5b0f88aec47b154374d909085116d34f5629199.

## Primary mathematical source

Olaf Mordhorst, The optimal constants in Khintchine's inequality for the case
2 < p < 3, Colloquium Mathematicum 147 (2017), 203-216.
DOI: 10.4064/cm6861-7-2016. arXiv:1601.07850.
https://www.impan.pl/shop/publication/transaction/download/product/91999

Theorem 1.1 on p. 203 gives the convention and sharp formula. Section 2,
pp. 204-205, gives the Gaussian moment as the limiting lower bound using
normalized equal-coefficient signs. The source's optimal upper constant and
our B_p have the same norm and probability-average normalization. The real
positive power removes any branch ambiguity. The formula for p <= 2 follows
from Holder and equality for a single coefficient. For p > 2, the Gaussian
Lp norm is strictly greater than its L2 norm, so the max selects G(p).

PDF archived locally as mordhorst.pdf, SHA-256: 992f206cbbf8b9be5a5747f38fc1f890003b06beef93f0e459c72fc72255a536.
The publisher full text and arXiv first pages were both read. Haagerup's
1981 publisher bibliographic page was reachable, but its PDF download returned
403. A Peskir survey PDF timed out. Neither inaccessible PDF was used as checked
evidence. Mordhorst supplies the needed primary proof and conventions.

## Derived identities checked independently

For a standard real Gaussian Z, M_p = E|Z|^p is the integral of
sqrt(2/pi) x^p exp(-x^2/2) on (0,infinity). Integration by parts gives
M_(p+2)=(p+1) M_p for p > -1. M_0=1 and M_1=sqrt(2/pi) yield
M_(2m)=(2m-1)!! and M_(2m+1)=2^(m+1/2)m!/sqrt(pi).
The double factorial counts pairings of 2m labelled points: choose one point's
partner in 2m-1 ways, then recurse. checks.py compares these exact constructions
with the stored powers. Quadrature is independent of Gamma evaluation.

## Live corpus, 2026-10-29

screen.already_here and screen.already_asked were run for the full title and
Khintchine, Khinchin and Rademacher. No duplicate upper family appeared.
The requests for Khintchine and Rademacher found only family issue #241.
Search by 3^(1/4) = 1.31607401295249246 and 15^(1/6) = 1.57041780247501967
returned no matches. The exact value 1 matched unrelated objects; shared values
are not a duplication of this family. The source snapshot's screening.log and
corpus-details.json retain the actual responses. One extra approximate probe
was not verified as a family value and is not used in the duplication conclusion.

T605 was read through numberdb.table and its public page. Its long address is
Sharp_lower_constants_in_the_Khintchine_inequality, read from that page.
It is the companion lower constant and uses the same normalization and rational
parameter convention. The Gamma table was found in search and its document
read at Values_of_the_Gamma_function_at_rational_numbers. No address was derived
from a title. T605 is linked from this new draft; this run does not edit T605.

The live tag page and API search confirmed functional analysis, inequality,
and probability theory. Their tag URLs are functional+analysis, inequality,
and probability+theory. No new tag is needed.

The proposal did not settle the final extent. The chosen 60-entry mixed
selection and all cutoffs are explained in argument-selection.md. GitHub
campaign provenance is kept outside mathematical table prose.