# T558 source checks

The primary papers were read independently in this run. Source hashes:

- Clarkson, J. Phys. A 56 (2023), 134002: `clarkson.pdf`, SHA-256
  d2cf630d6deb4e2710ea15a7bfec0f8f59c685edfecf9c8ee1089b328f30fdab.
  URL: https://kar.kent.ac.uk/100232/11/P.%20Clarkson%20-%20Classical%20Solutions%20of%20the%20Degenerate%20Fifth%20Painleve%20Equation%20-%20PPDF.pdf
  Definition 3.2 fixes the recurrence and initial conditions; Remark 3.3
  fixes the Wronskian's increasing derivative rows, odd degrees, and positive
  product normalization. The page was also inspected as an image.
  Remark 5.4 gives S_{n+1} S_{n-1}=x S_n^2 + m S_n(x;m+1) S_n(x;m-1).
- Clarkson–Law–Lin, SIGMA 19 (2023), 080: `sigma.pdf`, SHA-256
  ffda56c67e6e0994a8759833da399a26800f5d879b396ab086156aa522b563a9.
  Retrieved from https://arxiv.org/pdf/1609.00495; the downloaded PDF is the
  20-page SIGMA typeset article. The MathNet version was also read through
  the web tool. Theorems 2.3 and 2.5 and Remarks 2.6(1)–(4) supply the rational
  Painlevé III identity, coefficient generating function, normalization and
  Bessel specialization. Table 1's five explicit polynomials were compared
  exactly. Pages 5–7 and Clarkson's page 5 are preserved as images.

Normalization derivation: with p_k=[t^k](1+t)^m exp(xt), differentiation in x
sends p_k to p_{k-1}. Therefore the Wronskian of p_1,p_3,...,p_{2n-1}, with
functions in columns and derivatives in rows, has entry p_{2j+1-i} for
zero-based row i and column j. No row reversal or parity factor is introduced.
The positive multiplier is product_{j=1}^n (2j+1)^{n-j}; the source including
j=0 gives the same product because that factor is 1. As a control,
3 det[[p_1,p_3],[p_0,p_2]]=(x+m)^3-m.

One source assertion was deliberately not copied. SIGMA Remark 2.4(2) prints
S_n(x;m)=S_n(-x;-m), which fails for S_1=x+m. The correct sign is
(-1)^{n(n+1)/2}. This follows inductively from the recurrence: all terms of its
numerator have sign -1 under simultaneous reversal when S_n has definite
parity, and division by S_{n-1} gives the required triangular-degree parity.
The parity check is included for every stored index, but no unsigned symmetry
is asserted in the table. This is a source typo, not a recurrence ambiguity.

Corpus checks: live searches for Umemura, Painlevé, Laguerre, Yablonskii,
Okamoto, Schur and Bessel found no other published Umemura family. T558 itself
was re-read as the authoritative empty draft. T116's Bessel normalization and
all matching coefficients at n=0,...,5 were checked before asserting the
specialization. T556 and T557's live definitions support the named related
families. Search result slugs were used, rather than guessed addresses.

Skill evidence: agents/runs/20261028T224818Z-repair-skills/manifest.json;
local and live SHA-256 both
28ecbeeccb0c5e5d6eb541b5a5b0f88aec47b154374d909085116d34f5629199.
This is build evidence, not the separate independent critique required by the
campaign. The earlier source and evidence attachments remain in revision history.