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# Argument selection for T520
This is a table of one named quantity: the exact Taylor coefficients of the
order-zero angular spheroidal differential eigenvalue. Its parameter order is
(m,n,k). No parameter chooses between different quantities or conventions.
## Mathematical choices
- **m (integer angular order): intrinsic.** The only selected order is m=0.
It is the axisymmetric angular problem and the continuous prolate problem
used in time and band concentration. Osipov and Rokhlin, section 3.1,
equation (38), writes this order-zero differential problem explicitly.
Higher angular orders are outside this table's declared domain; m is retained
to distinguish angular order from mode number in every entry identity.
- **n (integer mode index): discrete index.** At q=0 the bounded eigenfunctions
are Legendre polynomials and the simple eigenvalues are n(n+1). Increasing
n labels their analytic branches. All n=0,...,11 are included, in increasing
order, with no omission of parity or ground mode. This is a finite initial
segment of the intrinsic spectral sequence, chosen to cover the first twelve
modes and their small-bandwidth corrections. The bound 11 is a reference
cutoff, not a mathematically distinguished last mode. It agrees with the
family's mode indexing without adopting its bandwidth samples.
- **k (integer Taylor index): discrete index.** k is the power of q, not the
power of c and not a derivative lacking its factorial divisor. All
k=0,...,12 are included for each selected mode. This keeps the constant term
and the first twelve perturbation corrections as an uninterrupted prefix.
The cutoff is editorial: measurements gave maximum rational-string lengths
128, 222, 316, and 406 at k_max=8,12,16,20 respectively. Twelve corrections
give a compact reference beyond the four explicitly tabulated correction
orders in DLMF, without filling unused capacity. The whole selected block has
156 entries. Neither 11 nor 12 is claimed to be an intrinsic endpoint.
The expansion point **q=0** is intrinsically selected: the spheroidal equation
reduces to the Legendre problem there. The coordinate is x in [-1,1]; q=c^2
is exactly the coefficient of x^2 in the differential operator. Changing to
c, S=2c/pi, or a rescaled coupling would change the coefficients. This fixes
both the point and its normalization. No free continuous parameter is sampled.
DLMF 30.3.8 defines the local expansion; 30.2.1 fixes the shift between lambda
and chi. DLMF 30.16(i) identifies small-parameter eigenvalue computation as a
use of the power series. We do not claim that the finite Taylor polynomial
approximates the eigenvalue at arbitrary bandwidths, nor give a remainder bound.
Sources inspected:
- https://dlmf.nist.gov/30.2 , equation 30.2.1.
- https://dlmf.nist.gov/30.3 , equations 30.3.3, 30.3.8--12.
- https://dlmf.nist.gov/30.16 , subsection (i).
- A. Osipov and V. Rokhlin, https://arxiv.org/html/1301.1707v1#S3.SS1 .
## Why the computation is exact
Write H(q)=D+qV in unnormalised Legendre polynomials, with D_ll=l(l+1)
and V representing multiplication by x^2. Choose v_0=e_n and v_(r,n)=0 for
r>0. Comparing powers of q gives
a_r = (V v_(r-1))_n,
(l(l+1)-n(n+1)) v_(r,l)
= -(V v_(r-1))_l + sum(j=1,...,r) a_j v_(r-j,l), l != n.
Every divisor is a nonzero integer. V changes degree by at most two, so v_r
is supported on nonnegative degrees of n's parity through n+2r. These finite
supports are exact consequences of induction, not matrix truncation guesses.
All coefficients are therefore rational and all operations occur in QQ.
The independent checks solve the differential equation in monomials using a
different eigenvector gauge, and compare with external DLMF coefficients.
The default generator, completeness note and read-back enumeration must agree
on m=0, 0<=n<=11, 0<=k<=12. Independent argument-selection acceptance belongs
to the separate critique stage, not to this build document.
Instructions: agents/runs/20261026T155642Z-build-skills/manifest.json
(attached as skill-manifest.json). Both archived versions and the freshly read
canonical and live skills have SHA-256
b4d51ded0d52cfe3dcc0859de0845c3879b0c2363273ba7e809a7e06ded40b18.