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# T475 source and checking notes
This file accompanies the unpublished table **Scaling dimensions of the 3D O(N) CFTs**.
## Selection and conventions
The family conventions specify three Euclidean dimensions, primary
operators, the class keys `xy` and `heisenberg`, and operator keys `phi`, `s`,
`t`. The table is one quantity (scaling dimension) over operators in the two
spectra. Internal rank is not spatial spin. The singlet excludes the identity.
Only the six leading scalar dimensions for N=2,3 are included: these are the
XY and Heisenberg classes and the complete mixed-correlator dimension set in
the selected papers. No Ising dimensions, OPE coefficients, critical exponents,
central charges or one-sided rank-four bounds are entered as extra quantities.
The parameters describe the family; the completeness note states the selected
range. Higher N and higher primaries remain open. These source estimates
cannot support 100 significant digits. Every source uncertainty is preserved,
without padding, in `centre +/- radius` form at rigour `assumed-bound`.
## Primary sources
- Chester et al., *Carving out OPE space and precise O(2) model critical
exponents*, JHEP 06 (2020) 142, arXiv:1912.03324v2,
https://arxiv.org/html/1912.03324v2, equations (40)-(42), PDF page 32.
- Chester et al., *Bootstrapping Heisenberg magnets and their cubic
instability*, Phys. Rev. D 104 (2021) 105013, arXiv:2011.14647v2,
https://arxiv.org/html/2011.14647v2, equations (21)-(23), PDF page 20,
and Table 1.
Both distinguish interpolated best-fit regions from conservative enclosing
regions. This table uses the conservative results, not the smaller starred
uncertainties. In particular, O(2) uses uncertainties (22),(22),(11), and
O(3) uses (67),(81),(32). The O(3) abstract swaps the labels for phi and s;
its equations and Table 1 consistently assign the values used here. The
narrower O(3) values in its abstract and equation (20) reflect interpolation,
not a separate incompatible determination. This resolves both apparent
source discrepancies without guessing a convention.
The intervals are conditional numerical-bootstrap estimates, not statistical
confidence intervals or unconditional mathematical enclosures. The document
records the spectral assumptions, cutoff Lambda=43 and twist gap. Full
bootstrap programs and excluded/allowed point sets are supplied by the papers;
this run does not reproduce those large computations. The assumed bounds
adopt the papers' convex hulls of excluded points in boundary Delaunay
cells as outer enclosures of the physical solution. This depends on that
enclosure assumption and on the reliability of the finite-precision solves,
conformal-block approximations and finite spin sets described in appendix B
of each paper. The uncertainties are not certified by interval arithmetic.
See repair-evidence.md for the independent checks supporting this correction.
## Independent numerical checks
- Hasenbusch, arXiv:1910.05916v2, abstract and Table I: Monte Carlo nu and eta
for XY. https://arxiv.org/abs/1910.05916
- Hasenbusch, arXiv:2005.04448v2, abstract and equations (32),(37): Monte Carlo
nu and eta for Heisenberg. https://arxiv.org/abs/2005.04448
- Hasenbusch and Vicari, arXiv:1108.0491v1, Table II: Monte Carlo Y2 for
both classes. https://arxiv.org/html/1108.0491v1
Each interval is compared with an independent Monte Carlo dimension obtained
from (1+eta)/2, 3-1/nu, or 3-Y2. All six pairs overlap at the sources' quoted
uncertainties. This is a consistency check, not independent confirmation of
all the bootstrap precision. `sources.json` holds the independently transcribed
parenthetical data and downloaded-PDF SHA-256 hashes.
`checks.py` verifies rational controls and rejects deliberately disjoint and
wide intervals before using 256-bit directed interval arithmetic. It checks
source-centre and radius transcription exactly, proves the five scaling
relations as rational-function identities, and verifies them on both input
interval boxes. It also checks all twelve derived leading exponents against
T472, and checks all five relations on each T472 class. With `--stored` it
repeats the checks on fresh API data, rather than generator output.
`check.py` is the repository's reusable check helper, copied unchanged so
these attachments are self-contained. The dry run reports six entries, a
longest value of 21 characters, and an entry block under 1 KB. These numbers
are small because they faithfully transcribe a precise, useful source set,
not because a storage limit was reached.
## Reproduction
Install `numberdb` in SageMath, download the attached files to one
directory, and run `sage -python checks.py`. For a private draft, provide an
authorized `NUMBERDB_API_KEY` and run `sage -python checks.py --stored` and
`sage -python generate.py`. Only `generate.py --fill` writes entries; no
attached program offers, reviews, or publishes the draft.
The build used Python 3.12.3, numberdb 0.1.10 and 256-bit MPFI checking
arithmetic. PDF extraction used pypdf 6.19.0; the arXiv HTML equations and
Monte Carlo Table II were also read to check labels and uncertainty notation.