back to table · edit · history · where entries came from · files · download
6012 bytes, as of the version from 2026-10-27 01:11 (current). Recorded here, not run.
# T510 build handoff
Draft: https://numberdb.org/T510
Title: Values of the regular Coulomb wave function $F_\ell(\eta,\rho)$.
Family: numberdb-data #210. The draft is filled, unpublished, and not offered for review. No human review action was taken. The separate critique and repair stages remain the controller's responsibility.
The live skill and local skill were read; the stricter local argument-selection guidance was followed. The family normalization and parameter order (ell, eta, rho) were adopted without mathematical changes. The domain is ell >= 0 integral, eta rational, rho > 0 rational. The finite selection is ell = 0,1,2,3; eta = 0,+/-1/2,+/-1,+/-2; rho = 1/2,1,2, with rho = 5 additionally at eta = 2. There are 88 entries. Couplings are ordered by magnitude, positive before negative. This is one quantity, F, with its signs preserved.
## Corpus and sources
`already_here('Coulomb')` found unrelated Thomson energies, critical exponents, and connective constants; `already_asked('Coulomb')` found the family itself. The three corrected API lookups, at (0,0,1), (1,1,1), and (0,2,5), returned no hits. See screen.json and probe.log. An initial lookup used an unsupported q parameter; the successful recheck uses text and computed samples.
The titles and addresses of T187 (Bessel J) and T210 (Kummer M) were read before linking. Their actual URLs were used. Existing tags are special functions, special values, physics. The proposed quantum mechanics tag remains absent. It would connect this table with T404 (quartic anharmonic oscillator eigenvalues), T406 (perturbation coefficients), T408 (pure even-power oscillator eigenvalues), and T411 (quartic double-well eigenvalues), all checked through the live API. No new tag was created. Bessel values are related by a nontrivial prefactor, so this is not the same stored computation and no repeats declaration was made.
DLMF 33.2, 33.4, 33.5, and 33.22 were read through the web tool, as were the mpmath Coulomb docs. Direct urllib GETs of DLMF and mpmath were refused with 403; the web reader succeeded. Both internal table links and the original A&S page returned 200 directly. A&S page 543 was downloaded and visually read; images and source-page HTML are preserved in the work directory. The failed page 545 download was not needed.
## Computation and evidence
The generator uses Sage's complex-ball Kummer M and gamma evaluation. Sage 10.9 has no direct ComplexBall Coulomb method and python-flint was not installed; using the family's alternative M representation avoided adding dependencies. Arguments and divisions remain exact or enclosed throughout. Working precision is 515 bits (`numberdb.bits(150)`). All output balls had at least 509 relative-accuracy bits, and the largest radius was 1.9450726e-154. Every written number has 100 significant digits and its implied +/- one-ulp interval contains the entire computed ball. There are 85 positive and 3 negative values.
The supplied dry_run.py and check.py passed. The longest final written number is 106 characters, and the measured decimal-entry block is about 12.23 KiB (the full-precision ball display measured by dry_run is about 19.6 KiB).
Local and database read-back checks passed for:
- 88 comparisons with mpmath at each of 150 and 200 decimal digits;
- 88 differential-equation residuals;
- 88 Wronskians, using G and G' from the independent U representation;
- 88 upper-order derivative identities, 66 lower-order derivative identities, and 66 order recurrences, computing order 4 as an auxiliary where needed;
- 12 free-case Bessel J identities and 3 sine identities;
- 8 printed mpmath example controls (4 F and 4 F');
- 8 original A&S example controls to absolute tolerance 1e-4.
Controls for the checking machinery include F_0(0,1)=sin(1), G_0(0,1)=cos(1), Wronskian=1, and rejection of deliberately wrong values. Residual checks require finite, narrow balls, not merely overlap. F derivatives use differentiated M with a separate product normalization. H+ uses U and Gamma/abs(Gamma), so no phase unwrapping is guessed.
The original A&S p.543 Example 1 has rounding loss in its normalization: for example 5727.97 * 1.7344e-4 = 0.9934591168 explains its printed 0.99346, while the high-precision F_1(2,5) is about 0.9935056752. The test reproduces the original arithmetic and uses an explicitly coarse tolerance for that worked example. The 100-digit result is supported by balls and independently checked implementations, not the handbook's last printed digit.
Both Programs snippets were executed successfully. `RegularCoulomb().verify(sample=None)` reported 88/88 matched, 0 differing, 0 missing, 0 extra. Evidence is in local-checks-final.log, stored-checks.log, verify.log, dry-run.log, program-checks.log, and numerical-evidence.json. Software: Sage 10.9, mpmath 1.4.1, numberdb 0.1.10.
The first audit flagged only the definition length. The equation and normalization were moved into labeled Formulas without changing numbers. The final API audit returned `findings: []`, `clean: true`. The final document and the earlier document are retained as document-final.json and document.json; the stored numbers were checked to remain identical during that prose change.
## Website artifacts and remaining work
Attached to the draft: generate.py, checks.py, the unchanged shared check.py, numerical-evidence.json, and this handoff. The build used one empty claim, one atomic numerical fill with the fully checked document, and one audit-driven prose revision. Attachments are revision-attached sources. No published table was edited, no deployment was performed, and no commit or push was attempted.
A separate critique session should independently test both mathematics and prose. The controller may offer the draft only after critique and repair stages succeed. No queue completion or issue closure was performed by this build; there is no explicit historical request `(answers #N)` on this proposal. The exact table handoff JSON names only T510.