back to table · edit · history · where entries came from · files · download
8218 bytes, as of the version from 2026-10-25 10:19 (current). Recorded here, not run.
# Build handoff: T482
Table: https://numberdb.org/T482
Title: Asymptotic amplitudes of the classes of trees
Family: numberdb-data issue #203
Build stamp: 20261025T094345Z
This is a build, not the separate critique or a human review. The draft must
remain unpublished and not offered for review by this session. The controller
owns the later critique, repair and offering stages.
## Scope and conventions
Fourteen entries match the fourteen class keys of T448. Parameters are
`class, size, labelling`; every size is `nodes`. A planted root is counted.
The value is the bare amplitude c, with no Gamma or sqrt(pi) absorbed; each
row gives alpha. Unlabelled counts use ordinary generating functions;
labelled counts use exponential generating functions. The labelled root/free
amplitudes coincide but have different exponents, 3/2 and 5/2.
Series-reduced rooted trees forbid degree 2, including at the root. They are
not the auxiliary rooted trees forbidding outdegree 1, although these two
classes have equal leading amplitudes. For the latter series H, the rooted
series-reduced generating function is
D=(1+x)H-x(H^2+H(x^2))/2. Its square-root coefficient equals that of H.
The free generating function is
(1+x)(H-H^2/2)+(1-x)H(x^2)/2. Its amplitude is
(1+rho)*b^3/(4sqrt(pi)). Planted amplitudes include a factor rho.
No arbitrary grid or extra tree classes were introduced. Completeness is
explicitly limited to these fourteen standard companion classes.
## Corpus screening
The family issue and both local and live skills were read. The local skill's
stricter range guidance was retained. `already_here` found T448 and unrelated
neighbours, with no existing amplitude table. `already_asked` returned no
matching request. Typed `/api/lookup` requests for 0.439924012571,
0.5349496061 and 0.3989422804 returned no entries. The T448 document and its
actual slug were read and saved. The draft title claim succeeded as T482.
The proposed tag `asymptotic enumeration` currently has zero tables. This build
uses the existing `combinatorics` and `generating function` tags, matching
T448. A person could introduce the proposed tag across T448 (growth constants
of trees), T447 (growth rates of power-free languages), T481 (growth constants
of planar graphs) and T482; the family supplies these proposed shared uses.
No published table was edited to add a tag.
## Computation and checks completed before filling
- Exact integer coefficient recurrences, with exact division remainders checked.
- Explicit geometric majorants for both coefficient and k-summation tails.
- Certified opposite signs enclosing each positive critical root; root error
propagated through all amplitude formulas. A positive-term derivative
majorant also verifies monotonicity for the alternating identity-tree equation.
- 100 significant digits requested; repeat at 130 requested digits agrees for
all fourteen entries. The narrowest 100-digit-run ball retains 379 bits.
- Independent explicit rooted-tree objects through 10 nodes, canonical
rerooting for free trees and automorphism tests, agree for all ten unordered
and planted classes. Control: four rooted and two free trees on four nodes.
- Six independent published OEIS amplitude decimal strings agree, at their
stated lengths (23 to 100 fractional digits); the identity-free amplitude
agrees with six safely printed fractional digits of the corrigendum.
- Genitrini's two leading coefficients agree to the first 17 fractional digits
after restoring its factor 1/sqrt(pi). Its final printed digits are not used.
- Exact normalized counts at n=200,400,800 approach every amplitude. The largest
relative discrepancy at 800 is 0.0034291 (series-reduced free trees), and
each class's discrepancy decreases at each successive size.
- Nine algebraic relations between amplitudes and their radii/closed forms pass.
- Repository `dry_run.py` passes all fourteen entries. Ball representations have
maximum length 166 characters; the annotated block is about 4.7 KB. Stored
decimals are capped at 100 significant digits by the numberdb formatter.
The post-fill log records verify(sample=None), checks on all stored numbers,
and the site audit. Nothing is published or offered by those checks.
## Source discrepancies resolved during the build
The original HRS 1975 Table 1 gives incorrect identity constants, explicitly
corrected by HRS 1986 (doi:10.1017/S1446788700033760). The corrigendum itself
prints only approximate decimals: for example its rooted amplitude ends in
0.362536424, while the independently confirmed value starts 0.362536423397.
Those last digits are not used as precision evidence.
The original Table 1's series-reduced amplitude for H is 0.6009988711803.
It does not satisfy that paper's derivative equation (11). Certified evaluation
of equation (11), using its own combinatorial recurrence, gives
0.4213018528699249210965028309... . This agrees with OEIS A001679. Including
the planted node gives 0.1924225474701550354144525345... (A001678), and the
free amplitude is 0.6844472720049140610231632797... (A000014). Explicit tree
counts and the normalized large coefficients support the same choice. The
mathematical definitions agree; the discrepancy is in the printed numerical
constants, not a choice between two tree conventions. No claim is made about
the historical implementation bug that produced the wrong printed numbers.
Genitrini Table 2 prints 0.6425790797442694714 and 0.7797450101873204419 for
c*sqrt(pi). Certified values are 0.64257907974426947269355... and
0.77974501018732044277110..., differences of about 1.29e-18 and 8.71e-19.
The independent OEIS decimals of c and our certified intervals support these
values. Genitrini describes numerical approximations, not certified last-place
bounds, so its first 17 fractional digits are used as a lower-precision check.
The discrepancy is preserved rather than treated as 19-digit agreement.
## Reproduction and artifacts
`generate.py`: reproducible generator, naming T482 in the first docstring line.
`checks.py`: explicit enumeration, source numbers, convergence, high-precision
repeat, amplitude identities and optional checks on stored values.
`check.py`, `dry_run.py`: unchanged copies of the repository checks.
Run `sage -python checks.py`, or with access to the draft,
`NUMBERDB_API_KEY=... sage -python checks.py --stored`.
Run `sage -python dry_run.py generate.py` for the non-writing dry run.
Run `sage -python generate.py` for generator verification.
The generator's --fill writes entries only; it has no publication action.
Research PDFs, their extracted text, corpus screening, draft claim, numerical
logs and the complete local table document remain in the run work directory.
The independent OEIS b-file downloads were blocked with HTTP errors; the
checks transcribe the visible published page data instead and do not claim
to have validated downloaded b-files.
## Final build result
T482 was filled once with fourteen 100-significant-digit values. The normal
generator submission then found all fourteen unchanged and attached the five
reproducibility files. Sage 10.9 and numberdb 0.1.10 ran the computation.
`verify(sample=None)` reports 14/14 matched, no differing, missing or extra
entries. The independent source comparisons and all nine amplitude identities
also pass on the values read from the website (`stored-checks.log`).
The audit's long-definition finding was fixed by shortening the definition.
One structural finding remains: the labelling parameter has two named values.
This is an explained false positive, not a clean audit: issue #203 explicitly
requires this parameter; it specifies the objects counted, and all rows store
the same quantity, the leading asymptotic amplitude. The two labellings do not
repeat a common class grid (the twelve unlabelled and two labelled class keys
are disjoint). No extra quantity is hidden in a parameter. The draft prose
now explicitly explains the role of the distinction. The separate critique
should assess that explanation. See `audit-final.json`.
The draft remains unpublished. No offer-for-review or human-review endpoint
was called. The controller's requested build handoff names only T482.