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5083 bytes, as of the version from 2026-10-29 10:07 (current). Recorded here, not run.
# T606 repair verification
The controlling skill manifest is `agents/runs/20261029T095925Z-repair-skills/manifest.json`.
The archived local and live skills both have SHA-256
`28ecbeeccb0c5e5d6eb541b5a5b0f88aec47b154374d909085116d34f5629199`.
The source snapshot canonical skill has the same digest. `repair-skill-evidence.json`
records the manifest digest and both archived versions.
## Finding 1: the limit is a norm
Authenticated live document and rendered page reads on 2026-10-29 confirmed
the original Gaussian comment. The sentence incorrectly identifies a random
sum with a scalar moment. Its preceding sentence restricts the discussion to
p > 2. The replacement preserves that restriction and identifies the limiting
quantity as the L^p norm B_p.
Let S_n = n^(-1/2) sum_(j=1)^n epsilon_j, for independent unbiased signs.
The central limit theorem gives S_n converging in distribution to a standard
real Gaussian Z. To justify convergence of absolute moments, choose an integer
m with 2m > p. Coefficientwise, cosh(t) <= exp(t^2/2), since
1/(2k)! <= 1/(2^k k!) for every k >= 0. All coefficients are nonnegative,
so raising to the n-th power preserves this coefficient comparison. Thus
E S_n^(2m) = (2m)! [t^(2m)] cosh(t/sqrt(n))^n <= (2m-1)!!,
uniformly in n. Consequently
E(|S_n|^p 1_(|S_n|>R)) <= (2m-1)!! / R^(2m-p).
This proves uniform integrability of |S_n|^p. Applying convergence in
distribution to bounded continuous truncations and then removing the
truncation gives E|S_n|^p -> E|Z|^p. Taking positive p-th roots gives
||S_n||_p -> (E|Z|^p)^(1/p) = B_p for p > 2.
Each coefficient vector (1/sqrt(n),...,1/sqrt(n)) has Euclidean norm 1.
The limit therefore supplies the lower bound for the optimal upper constant;
together with the sharp inequality it establishes sharpness over all finite
lengths. For p < 2 the Gaussian norm is not B_p, so the existing p > 2 scope
is essential and was retained.
Source independently read: Olaf Mordhorst, The optimal constants in
Khintchine's inequality for the case 2 < p < 3, Colloquium Mathematicum 147
(2017), 203-216, Theorem 1.1 (p. 203) and Section 2 (pp. 204-205),
https://www.impan.pl/shop/publication/transaction/download/product/91999 .
The paper states the optimal constants and Gaussian limiting lower bound;
the uniform-integrability argument here justifies convergence of the moments.
The downloaded PDF digest is in `repair-source-evidence.json`.
## Computations run before editing
Command:
```sh
cat "$NUMBERDB_KEY_FILE" | agents/sage.sh \
"$TMPDIR/repair-limit-check.py" "$TMPDIR/generate.py" \
> "$TMPDIR/repair-limit-check.log" 2>&1
```
The live table and unchanged attached generator were read. All 60 ordered
parameter keys equal the generator's default enumeration. Both unit values
are exact. All 58 nonconstant generator balls are finite and wholly inside
the written decimal intervals. All 30 integer values were independently
checked using Gaussian integration-by-parts moment recurrences, without Gamma
evaluation, at 700 bits with radii below 10^(-180).
The script checks the coefficientwise moment bound through order 34, a single
higher even order covering all currently selected exponents p <= 32. The proof
above applies to all fixed p > 0, by choosing a sufficiently large m.
At n=16,64,256,1024 it exactly checks E S_n^2=1 and E S_n^4=3-2/n and uses
exact binomial probabilities with Arb propagation of sqrt(n) and rational p.
For p=13/6,3,4,19/2,32 the finite-sum L^p norms lie strictly below their
Gaussian limits, with successively smaller gaps and ball radii below 10^(-180).
These twenty finite comparisons are corroboration, not a proof of convergence.
`repair-numerical-evidence.json` records every comparison and dependency versions.
An initial checker assumed unit rows were dictionaries; this table uses bare
strings. That local checker error was fixed before the successful computation;
the failed log is retained as `repair-limit-check-attempt1.log`. No table
value disagreement occurred. No values, parameter keys, definition, selection,
completeness note, program snippet, or existing generator attachment was changed.
## Finding 2: precision label
The authenticated page visibly renders `digits: 100 (Unknown key)`.
The source snapshot's data-property map in `numberdb_app/views.py` lacks
`digits`, and the fallback appends `(Unknown key)`. This is a renderer bug,
not a numerical failure. Site-source edits and deployment are forbidden in
this artifact run. The valid `digits: 100` metadata is preserved and this
finding is left for a person. `environment-findings.md` records the site work.
## Independent critique boundary
The original `agents/critiques/T606-arguments.json` is preserved byte for byte.
Changing the Gaussian comment changes the version-1 selection fingerprint,
which includes all comments even though the argument set itself is unchanged.
A fresh independent critique is required on this same draft. This repair does
not accept, replace, or refresh the critic's verdict or hashes, and performs
no offer, publication or human review action.