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6674 bytes, as of the version from 2026-10-27 21:27 (current). Recorded here, not run.
# Source and verification evidence for T561
Skill policy: `agents/runs/20261027T205535Z-build-skills/manifest.json`.
Both archived local and live skill files were read and compared byte for byte.
Each SHA-256 is
`e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c`.
The manifest's two recorded digests were checked against the archived bytes.
## Definition and data
* Eskin, Kontsevich and Zorich, *Sum of Lyapunov exponents of the Hodge bundle
with respect to the Teichmüller geodesic flow*, arXiv:1112.5872v6,
<https://arxiv.org/pdf/1112.5872>. Section 1.6, equations (1.3)–(1.4), defines
the area-weighted constant with divisor pi L^2 on unit-area surfaces.
Theorem 1, equation (2.1), relates it to the sum of nonnegative Lyapunov
exponents, with leading exponent 1. Corollary 2 gives the genus-two values.
Appendix A, pp. 100–101, gives exact rational sums for all genus-three and
genus-four components. The heading “Conjectural approximate values” refers
to the experimental individual exponents, not the distinct rightmost column
explicitly labelled “Exact.” Screenshots of both source tables were checked.
* Kontsevich and Zorich, *Connected components of the moduli spaces of Abelian
differentials with prescribed singularities*, arXiv:math/0201292v2,
<https://arxiv.org/pdf/math/0201292>, Theorems 1–2. These establish the
partition and component enumeration, including the genus-two and genus-three
exceptions. The same classification is set out in EMZ §5.2.
* Eskin, Masur and Zorich, *Moduli spaces of Abelian differentials: the
principal boundary, counting problems, and the Siegel–Veech constants*,
arXiv:math/0202134v2, <https://arxiv.org/pdf/math/0202134>, §§2.1–2.2,
fixes labelled zeros, retained phase, lattice covolume one and radial
disintegration. Normalizing each component's measure to probability removes
the overall volume multiplier. It does not mix components.
The data are exact rational multiples of 1/pi^2, not the scaled rational
constants pi^2*c_area/3. All rational coefficients are positive. Therefore
none of these area constants is rational; no exactly rational row has been
written as an uncertain decimal. The generator uses Sage QQ and Arb at 397
working bits for 100 requested digits (64 guard bits). The minimum measured
relative accuracy, using exact bounds on interval width divided by its positive
lower endpoint, is 393 bits.
## Checks and their limits
`checks.py` runs controls before comparisons and checks all written decimals
as exact center-plus-or-minus-one-last-place-unit intervals. An independent
512-bit MPFI calculation lies wholly inside every such interval. All 25 Arb
computations also agree with 160-digit Arb computations, with finiteness and
width checked before agreement. The exact EKZ identity is checked for all
rows, and again after reading the stored table.
The six hyperelliptic entries are checked using EKZ Theorem 3(a) and Lemma 1.1:
compute the genus-zero quadratic-stratum constant, then double it for the
orienting cover. The quadratic orders are (2g-3,-1^(2g+1)) or
(2g-2,-1^(2g+2)). This uses no tabulated Lyapunov sum. The two genus-two
values 10/(3*pi^2) and 15/(4*pi^2) are exact control cases.
Eleven entries are checked from a separate transcription of EMZ configuration
constants, multiplied by the number of cylinders in each configuration, then
divided by 2g+n-2. Conversion from the source's zeta(2) scale multiplies the
rational by 6. The Vorobets relation c_area=c_cyl/(2g+n-2) is stated in
Athreya–Eskin–Zorich, arXiv:1212.1660v4, §4.13 (published version p. 1350),
<https://arxiv.org/pdf/1212.1660>.
The checked EMZ rows are Examples 14.12–14.13 (genus three, odd (4) and
odd (2,2)), and Appendix B.2 (all genus-four components of (6), (3,3),
(4,2), and (2,2,2)). Only configurations with cylinders enter the sum;
zero-cylinder terms are explicitly omitted. For cyclic degeneration patterns,
the duplicated first/last arrow represents the same gluing. The number of
cylinder gluings counts that arrow only once.
The EMZ calculation is an independent arithmetic path and catches factors and
transcription errors, but EKZ's tabulated sums themselves use EMZ data. It is
not claimed to be an independent mathematical derivation of those sums.
Three entries additionally use the geometric arguments in Chen–Möller,
*Non-varying sums of Lyapunov exponents of Abelian differentials in low genus*,
arXiv:1104.3932v2, <https://arxiv.org/pdf/1104.3932>, §§5.1–5.2 and §6.1.
The disjoint hyperelliptic divisor gives slope 9 for (4)odd and (3,1);
the theta-null divisor gives slope 60/7 for (6)even. Proposition 4.5 converts
these to sums and hence area constants. The latter slope is checked directly
in their divisor equation. These geometric arguments do not use EMZ cylinder
counts. Their symbol c is scaled: c_CM=pi^2*c_area/3, as their equation (13)
shows. This normalization is explicit in checks.py.
Together the additional mathematical checks cover 16 distinct entries.
The remaining nine entries rely on the exact published EKZ sums and Theorem 1;
they have transcription, arithmetic, precision, and stored-interval checks,
but no independent derivation in this build. This limitation is retained for
the separate critic.
## Corpus and provenance
The family issue 224 was read first. The live `already_here` and
`already_asked` checks found no existing area-constant family. Search terms
included Siegel, Veech, Abelian, Masur, and Lyapunov. Numerical API lookups
near 10/(3*pi^2), 15/(4*pi^2), and 4/pi^2 returned no matches; the last is a
normalization diagnostic rather than a selected entry. The first two are
actual family values. Public T560 lookup fails because the companion volume
table is a draft; its entire document was read with the authorized API key.
The stable HREF target T560 is used directly, as supported by the site's
resolver, rather than inventing its private slug. Two drafts may link, but a
person should coordinate their publication order.
The paginated tag list contains 73 names. `dynamical systems` exists; `flat
surfaces` does not. The proposed new subject tag would serve T560 (volumes),
T561 (area constants), and the queued Kontsevich–Zorich Lyapunov-sum table in
family 224. This build uses the existing tag and searchable flat-surface
keywords; it does not create a tag or write a GitHub request into the prose.
The generator and source hashes are recorded in the run's numerical evidence.
Only this draft is filled. It is neither published, offered nor human-reviewed
by this build. The controller performs the subsequent independent critique.