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# Argument selection for T534
This table holds one named quantity: the positive interior zeros of the order-zero continuous prolate angular spheroidal wave function. The conventions follow the differential problem in Osipov and Rokhlin, §3.1, and DLMF §§30.2 and 30.4. It is not a table of radial zeros, eigenvalues, quadrature weights, or statistical quantiles.
The policy read for this build is archived at `agents/runs/20261027T020220Z-build-skills/manifest.json`. Its local and live skill copies both have SHA-256 `e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c`. The attached `skill-manifest.json` preserves that manifest. This is a builder's justification; the separate campaign critique must independently assess it and the generator enumeration.
## Convention
Put c = pi S/2. The equation is
((1-x^2) psi'(x))' + (chi_n(c) - c^2 x^2) psi(x) = 0,
-1 < x < 1,
with bounded solutions at both endpoints. Differential eigenvalues increase from n=0. The angular order is m=0, distinct from n. Choose the real eigenfunction with integral psi^2 = 1 on [-1,1] and psi(1)>0. Its positive zeros are x_(n,1) < ... < x_(n,floor(n/2)) in (0,1). Multiplication by a nonzero normalization factor does not move the zeros.
DLMF 30.4.3 gives parity, and §30.4(ii) states the number of interior zeros. Thus negative zeros are the reflections of the stored zeros; odd n also has the exact zero 0. These forced values are stated analytically. The n=0 and n=1 modes have no positive zeros. The value-column header is x_(n,j)(c), and parameters are ordered (m,S,n,j).
## Each parameter and its extent
| Parameter | Role | Selection and reason |
| --- | --- | --- |
| m | Intrinsic slice | Only m=0. This is the order in the continuous time-frequency concentration problem; it is explicitly recorded so it cannot be confused with the mode. Other angular orders are outside this family build. |
| S | Bounded exploratory reference sample | S in {1,2,4,8}, with c=pi S/2 formed at working precision. The sinc concentration operator has trace S, so the normalization measures effective dimension. Four successively doubled, small integral effective dimensions provide comparable reference problems from a single-mode scale to a modest multimode scale. Neither the four integers nor the upper bound 8 are canonical. |
| n | Discrete spectral index | All 2 <= n <= 11. The cutoff is practical, retaining low modes of both parities and modes on both sides of the concentration transition for the larger S. It provides spectral breadth without extending just one bandwidth to high mode numbers. n=0,1 contribute no positive zeros. |
| j | Discrete zero index | Every 1 <= j <= floor(n/2), increasing from the origin toward 1. There is no arbitrary root cutoff or selection of only famous roots within a chosen mode. |
For S, the reference-policy fields are as follows:
- **Purpose:** identify roots encountered in small normalized PSWF reference calculations and in quadrature nodes derived from these modes. No specific publication using each selected bandwidth is claimed.
- **Bounds:** precisely S=1,2,4,8 at m=0 and 2<=n<=11, including all their positive zeros. This is 120 entries across 40 spectral modes.
- **Coverage rationale:** doubling the effective dimension covers distinct bandwidth scales with four reference problems, rather than filling an interpolation mesh. All positive roots and both parities are retained at each scale. An integer grid in c, fractional Shannon numbers, more angular orders, or more bandwidths would be different reference selections; no claim is made to cover them. The cutoff keeps independently checked 100-digit computation practical.
- **Noncanonical choices acknowledged:** yes. The trace selects the meaning of S, not this finite set. Matching T518 and T519 is useful interoperability, not the mathematical justification by itself.
- **Notation independence:** yes. Successive doubling of effective dimensions does not depend on decimal spelling or the numeral base used to write the four integers.
- **Supporting sources checked:** yes, for the function, normalization, parity, zero count and quadrature relationship. Sources are not evidence of an application for every exploratory S.
The full domain is positive rational S, n>=2, and 1<=j<=floor(n/2), with m=0. Bounds belong in the completeness note rather than in a declaration pretending the selected four bandwidths are the domain.
## Breadth and exclusions
The c=0 limit produces Legendre zeros, already represented by the corpus's Gauss–Legendre table. It is a control and a relation, not a hidden fifth bandwidth. Negative c does not change this differential equation. Nonzero angular orders and zeros outside (-1,1) are outside the selected continuous order-zero problem. The positive half avoids replicating reflection symmetry and the exact odd central zeros while retaining every nontrivial positive node. This is a family-specific selection, not a rule to apply the same bandwidths or meshes to unrelated functions.
## Checked primary sources
- Osipov and Rokhlin, *On the evaluation of prolate spheroidal wave functions and associated quadrature rules*, [§3.1](https://arxiv.org/html/1301.1707v1#S3.SS1), for the order-zero differential and integral problems; §3.2 for the Legendre representation; §2.2 and §5.3 for the positive/negative roots as quadrature nodes.
- [DLMF §30.2](https://dlmf.nist.gov/30.2), equation 30.2.1, for the separation-constant convention chi=lambda+c^2 at m=0.
- [DLMF §30.4](https://dlmf.nist.gov/30.4), equations 30.4.1–30.4.3 and §30.4(ii), for normalization, the Legendre limit, parity and exactly n interior zeros when m=0.
- [DLMF §30.15](https://dlmf.nist.gov/30.15), for the signal-concentration problem and its order-zero angular eigenfunctions. The trace S=2c/pi follows directly by integrating the diagonal value c/pi of the sinc kernel over [-1,1].
- [DLMF §30.16](https://dlmf.nist.gov/30.16), for finite Legendre eigenproblems. Its asymptotic truncation estimate is not used as an explicit error bound.
## Computation and limits
`spectral.py` recomputes differential eigenvalues by indexed Sturm bisection. `generate.py` constructs parity eigenvectors by backward recurrence and evaluates their Legendre expansions. Four settings compare arithmetic precision (160,220 decimal digits) and basis length (80,120,180 terms in each parity block). `checks.py` independently solves the endpoint Taylor equation, at (220 digits,600 terms) and (260 digits,800 terms), with its eigenvalue fixed by the parity boundary condition at the origin. SciPy gives only double-precision initial guesses and a low-precision external check.
At least 110 relative digits must agree before 100 significant digits are retained. Each written decimal denotes plus or minus one last-place unit. Both independent endpoint computations and the largest Legendre computation must take opposite signs at those decimal endpoints. Known nonzero Legendre values, derivatives, and the root 1/sqrt(3) control the independent checker. Parity is checked on actual nonzero function values, and every mode has its expected number of disjoint positive zero intervals. These are numerical agreement checks; there is no certified infinite-series truncation bound. The stated rigour is `heuristic (agreement-checked)`.