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5826 bytes, as of the version from 2026-10-29 09:12 (current). Recorded here, not run.
# Build handoff: T605
Built and filled https://numberdb.org/T605, **Sharp lower constants in the
Khintchine inequality**. It remains a private, unpublished draft. No offer,
publication, human review action, or independent critique acceptance was made.
The controller must run the separate critique and repair stages before any
offer. Family: numberdb-data issue 241; title unchanged from the proposal.
## Scope and decisions
The table holds A_p itself for finite real coefficient vectors and independent
uniform signs, with probability expectation. The only parameter p has type Q
and domain p > 0. The mathematical definition extends to real p > 0.
There are 36 ordered, distinct reduced-fraction identities: 2 intrinsic
first/second-moment cases, 19 documented application exponents 2m/(m+1),
2 <= m <= 20, and 15 remaining exploratory fractions with denominator at most
6 in (0,2). The finite cutoffs are noncanonical practical reference choices;
the table is incomplete. The application family reaches both nonconstant
branches. The constant region p > 2 is stated by formula, avoiding repeated
ones. The generator returns A_(2/5)=1/4, A_(2/3)=1/2 and A_2=1 exactly.
The proposal did not settle its finite extent; argument-selection.md explains
the choices and coverage review. The application paper's A_{R,p} is the
reciprocal of this table's A_p; that conversion is explicit in the table.
No quantile or critical-value selection was made.
## Numerical evidence
- agents/table-build/dry_run.py passed: 36 entries, exact values or finite
error-carrying balls, no disguised exact decimals, named Sage imports.
- Arb computation uses 429 bits, including a 96-bit guard. Every inexact
value has relative radius < 1e-125 and its full ball is contained in the
corresponding 100-significant-digit stored decimal interval.
- The independent checker uses explicit four-sign averages and 150-digit
Gaussian integration, with no Gamma evaluation in that independent path.
Controls for moments 0, 2, 4 gave errors below 1e-140. Known sign averages
passed. A deliberately shifted ten-ulp decimal was rejected.
- All 36 independent values lie inside the written claims. The largest
discrepancy from a rounded midpoint is 0.466446642241 last places.
- Every two-point row passes its exact rational power identity. All selected
exponents pass defining-inequality checks on five explicit coefficient
vectors. The selected values are strictly increasing.
- Readback verify(): **36/36 matched, 0 differing, 0 missing, 0 extra**.
The independent checks were repeated on all values read from the database.
- Final /api/table/T605/audit: **clean: true, findings: []**.
- Longest stored number: 103 characters. Dry-run ball rendering: maximum
147 characters; the measured entries block was about 6.5 KB.
The independent quadrature is a diagnostic, not a proof of the optimal-constant
theorem or the error bound. The theorem is sourced from Mordhorst, Theorem 1.1;
Arb supplies the numerical enclosures. Full precision records and dependency
versions are attached in numerical-evidence.json and stored-numerical-evidence.json.
## Sources and corpus
The live already_here/already_asked and api/lookup checks found no duplicate
family. Three requested tags exist. The Gamma table was fetched before linking
its returned slug. The sibling upper-constant table did not have an identifier
when this build searched for it; a later sibling build can add the relation
using resolved identifiers under its authorized workflow. No slug was guessed.
Full texts checked: Mordhorst's sharp-constant theorem and Bayart, Pellegrino,
Seoane-Sepulveda, section 3 / Proposition 3.1 / Corollary 3.3. Haagerup's original
publisher download was unavailable (HTTP 403 in the browser tool); the checked
Mordhorst theorem provides the same complete formula with explicit attribution.
Sources, PDF hashes, and downloaded PDFs remain in the run artifacts.
## Instruction and selection provenance
Canonical instruction manifest:
`agents/runs/20261029T085909Z-build-skills/manifest.json`.
Both archived skill-local.md and skill-live.md were read and both have SHA-256
`28ecbeeccb0c5e5d6eb541b5a5b0f88aec47b154374d909085116d34f5629199`.
The source snapshot's canonical skill has the same digest. The original
manifest is attached to T605 as skill-manifest.json. The numerical evidence
also cites that manifest and digest.
Current selection fingerprint:
`95010b7e517ab777e5b9c0c6ff8c74a772dcc0aae6d428c51a13876fc3b3838b`.
`fingerprint.json` records website-read SHA-256 hashes of generate.py,
checks.py, argument-selection.md, sources.md, skill-manifest.json and
numerical-evidence.json. This is a fingerprint only, not acceptance. The separate
critic must perform its own assessment, recomputation and fingerprint call,
then write its own required -arguments.json with its NUMBERDB_RUN_ID.
Reproduction attachments: generate.py, checks.py, argument-selection.md,
sources.md, skill-manifest.json, numerical-evidence.json. All enumeration is
in generate.py; it imports no selection-controlling source file.
## Repairs made before handoff
The first dry numerical check exposed two checker API assumptions, corrected
privately: Entries is not exported by numberdb.sage, and Sage RealBall uses
membership rather than .contains(). The first stored check needed to handle
the API's nested Numbers mapping. No numerical disagreement was found.
The site audit requested installation instructions and a shorter definition;
both were fixed, moving the sign convention into Comments. Values and selected
arguments were retained. Original failed-check logs remain as artifacts.
The builder's proposed lessons are in 20261029T085909Z-build-lessons.md in the
work directory. No pipeline source, skill, Git checkout, or controller receipt
was edited.