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# Source and verification notes for T562
This is a transcription of exact published rational sums, not an estimate of
individual exponents. EKZ Appendix A labels the individual and numerical sum
columns experimental, and labels the separate rational sum column exact.
The genus-three and genus-four exact columns on published p.326 were checked
visually and extracted independently with PyMuPDF. `source-column.json` keeps
those 23 rows; the two genus-two rows come from EKZ Corollary 2.
The convention is the real Hodge bundle and flow diag(exp(t),exp(-t)), with
top exponent one, for each component's Masur–Veech probability measure.
It does not sum all 2g exponents (whose sum is zero), and it does not use the
exponents of one chosen orbit closure inside the component. KZ Theorems 1–2
supply the component classification. All free arguments and finite bounds
are justified in `argument-selection.md`.
## Exact independent checks
Let kappa(mu) = sum_i m_i(m_i+2)/(12(m_i+1)).
1. Six hyperelliptic values are reconstructed from EKZ Corollary 1 using
the sphere quadratic orders (2g-3,-1 repeated 2g+1 times) or
(2g-2,-1 repeated 2g+2 times). Summing 1/(4(d+2)) over odd orders d gives
the sums independently of the transcribed column. This also verifies the
two displayed hyperelliptic closed forms on all their stored rows.
2. Ten nonhyperelliptic values are reconstructed by the geometric divisor
arguments in Chen–Möller (CM), Sections 5.1–5.4, 6.1–6.3 and 6.6–6.8.
If a lambda + sum_i b_i omega_i - d delta_0 has zero intersection with
the Teichmüller curve, CM Props.4.5,4.8 give
L = (sum_i b_i/(m_i+1) + 12 d kappa)/(12d-a).
The two odd-spin cases use Proposition 5.2's slope 4(g+8)/(g+2).
These are divisor-coefficient calculations, not comparisons with CM's
tabulated values. See the sign correction from Farkas in the next section.
3. Three values are additionally reconstructed from the cylinder configurations
in EMZ Appendix B.2: H(2,2,2)^even, H(2,2,2)^odd and H(3,3)^nonhyp.
EMZ Section 13.1 defines the double arrow as a cylinder, the single arrow
as direct gluing. The notation is cyclic, so the closing arrow repeats
the opening connection. If r_C is the table's c_C*zeta(2), q_C the number
of cylinders, and D=2g+n-1, the Vorobets relation in EKZ Section 2.5.2
gives L=kappa+2*sum_C(q_C*r_C)/(D-1). The genus-two H(2) case from EMZ
Example 14.7 is the known-answer control: r=5/3 for the one cylinder gives
L=4/3. EKZ itself draws on EMZ, so these are underlying-source
reconstructions, not independent publications deriving every input.
The union of these additional checks covers 17 distinct entries. The remaining
eight are exact EKZ transcriptions, checked against the source column. There
is no claim of a new independent derivation for all 25 entries.
All 25 values also satisfy the relation to the actual stored T561 area
constants: L=kappa+(pi^2/3)c_area. Its decimal uncertainties are propagated
using 512-bit Arb arithmetic, and each resulting interval must contain the
exact rational and have radius <10^-95. T561 was built from the same EKZ
sums, so this is a companion consistency check, not independent evidence.
## Disagreement 1: H(2,2,2)^even
CM Section 6.9 (both arXiv v2 and the published paper) states 166/75.
EKZ Appendix A states 737/333. Their difference is 1/8325.
This is not a component, time or area normalization change.
EMZ Appendix B.2, preprint p.100, gives five configuration coefficients
r_C: 180/37, 45/74, 225/37, 225/296 and 5/37. Their cylinder counts are
0,1,1,0,2 respectively, so the weighted sum is
45/74 + 225/37 + 2*(5/37) = 515/74.
Here D=10 and kappa=2/3. Consequently
L = 2/3 + (2/9)*(515/74) = 737/333.
The conflicting 166/75 is incompatible with the exact configuration data.
The stored row names both sources; no value was chosen for looking closer
to the experimental digits. The historical mechanism producing CM's printed
rational is not established by this run.
## Disagreement 2: signs in CM equation (8)
CM equation (8) prints -omega_1-omega_2+8lambda-delta_0 for Lin^1_3.
Together with CM Props.4.5,4.8 that would give L=7/4 for H(3,3)^nonhyp,
contradicting its own Section 6.6 value 2 and the EKZ source.
The reference actually cited there, Gavril Farkas, *Koszul divisors on moduli
spaces of curves*, American Journal of Mathematics 131 (2009), 819–867,
Theorem 4.6, printed pp.858–859, resolves the signs. At r=1,s=2 its general
coefficients are a=12, c=3/2, b_irr=3/2 with common factor 2/3. The psi terms
have a positive sign, giving +omega_1+omega_2+8lambda-delta_0 (up to the
other boundary terms absent on these curves). Substitution gives L=2.
The checker evaluates Farkas's coefficient formulas as well as the resulting
intersection relation. This is a printed sign error in CM's specialization,
not a change made to force numerical agreement.
EMZ's independent configuration input for H(3,3)^nonhyp also gives L=2:
its cylinder-weighted sum is 3699/1120+64/35+27/80+1/32=11/2,
D=9 and kappa=5/8. Thus 5/8+(2/8)*(11/2)=2.
## Reproduction and evidence
Use SageMath and the numberdb client; `generate.py` describes its commands.
`checks.py` runs all checks from the attached `source-column.json` and
`T561.json`; `--stored` instead fetches T562 and T561 from the website.
`check.py` is the unmodified campaign checker. The controller's later critic
must independently check the reasoning and selected arguments.
The run used the matching skill versions in
`agents/runs/20261027T213908Z-build-skills/manifest.json`, each SHA-256
`e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c`.
`skill-evidence.json` records both hashes and the manifest hash.
`sources.json` records downloaded PDF URLs and hashes. The exact results,
controls, software versions and measurements are in `numerical-evidence.json`.