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# Build handoff: T487
Draft: https://numberdb.org/T487
Canonical address: https://numberdb.org/Zeros_of_cross-products_of_Bessel_functions
Proposal: “Zeros of cross-products of Bessel functions”, family issue #204.
The draft contains 480 values, all at 100 significant digits, with rigour
`proven`. It remains unpublished and has not been offered for review. The
controller owns the separate critique, repair and offer stages.
## Convention and range
The stored object is the kth positive zero in x of
J_nu(x)Y_nu(lambda*x)-Y_nu(x)J_nu(lambda*x), ordered from k=1.
The inner radius is 1. Only the plain cross-product is included; derivative
and mixed cross-products are different objects. Each root is stored once,
without duplicating the angular multiplicity of the corresponding annulus mode.
The complete computed rectangle uses orders 0 through 5, exact ratios
5/4, 3/2, 2, 5/2, 3, 4, 5 and 10, and indices 1 through 10. This is the
proposal's recommended range. The order parameter describes the full real
nonnegative domain, and the radius-ratio parameter is rational, as proposed.
The completeness note gives the actual selection and its geometric reason.
The main decisions beyond the screening were to compute 100 significant digits
and certify both the root enclosures and the indices, rather than retain only
30 agreeing digits. The actual DLMF cross-product section is 10.21(x), equation
10.21.48; the screening's subsection numbers were stale.
## Corpus and provenance checks
Read issue #204 before selecting the convention and downloaded its full
screening report from numberdb-runs. Both local and live skills were read;
the stricter local guidance on natural parameter choices was followed.
Ran `already_here`, `already_asked`, searches for Bessel and neighbouring
spectral families, three value lookups, and existing-tag checks. No duplicate
plain cross-product family was found. The title claim created T487.
Read the linked Bessel tables and their actual addresses. The draft links to
T20, T21, T22, T23, T390, and the two completed family drafts T485 and T486.
T485 and T486 are draft links: their publication order should be considered
by the controller before any later publication of this table. The clamped
plate and Steklov proposals have no recorded T numbers in the issue, so no
addresses were invented for them. There was no historical request issue to
cite or close. No GitHub provenance was inserted into the mathematical prose.
## Numerical evidence
- All roots were recomputed with mpmath at 130 and 160 decimal digits and
required to agree to 120 places relative to max(1,abs(root)).
- Arb at 717 bits certifies opposite signs at exact rational endpoints
separated by 2e-110. Minimum returned relative accuracy: 363 bits.
- Ten disjoint sign brackets in each of 48 groups are accompanied by a
certified total Sturm node count of ten. A proved spacing bound makes the
rational sign mesh exhaustive, so the root indices are certified as well.
- The existing dry_run.py passed all 480 entries, exactness, precision-type
and prose checks. Its ball representation measured 85.5 KB, maximum 129
characters. Actual written values measure 65.625 KB, maximum 102 characters.
- A fresh dry run reproduced every byte of the earlier 480-value payload.
- Independent checks certified both endpoints of all 480 written decimal
intervals at 896 bits and repeated the 48 node counts on a different mesh.
- All 480 values satisfy the variational bounds in the document.
- An independent Pruefer ODE calculation, using no Bessel values, checked
all 480 entries. Maximum phase residual: 2.2595258997171186e-12, at
(nu,lambda,k)=(5,5,8), against a threshold of 1e-8.
- Nonzero known-answer controls and a deliberately perturbed control passed
before production measurements. Elementary half-integer Bessel identities
were checked for all eight ratios.
- kappa(0,1;2)^2 agrees with T390's annulus-1-2,1 entry.
- `verify(sample=None)` on the filled draft: 480/480 matched, zero differing,
zero missing, zero extra. All independent checks were repeated on the
values read back from the database.
## Audit and final state
The first audit flagged only a long definition and internal links in that
definition. These were resolved by shortening the definition to 157 characters
and moving the function descriptions and links to a comment. The stored
numbers were asserted unchanged after this prose edit.
Final audit: clean=true, findings=[].
Final document revision:
6341086d22ec4d83eee4a95aebe870b8d7acaf442aeb39c81ec8004db41192e0.
The validation harness completed the dry run, fill, attachments, package
verification and independent stored checks. Its final audit call used the
low-level Client.request helper without its required empty parameters dict
and raised TypeError. This was a harness error after all numerical checks;
audit.py and resolve_audit.py then ran the audit successfully and resolved
its two prose findings. The original log is preserved without alteration.
Website attachments include generate.py, checks.py, certification.md,
numerical-evidence.jsonl, dependency-versions.json and this report. The
source and check files are independently runnable as documented. The full
numerical certificates, failed probes, successful tests, corpus snapshots,
source retrieval records and proposed lessons are preserved in this run's
work directory. DLMF was readable through the web tool; direct artifact
downloads returned HTTP errors. Teschl's source PDF is preserved locally.
No publication, offer, human review, deployment, Git operation or pipeline
source edit was performed. The family queue has been left for the controller's
subsequent review stages; this build hands off only T487.