# Argument selection for T543: Lamé spectral polynomials

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## Quantity and every free parameter

There is one table parameter, the positive integer order `n`, classified as
`discrete_index`. It determines the finite-gap operator
`-d²/dx²+n(n+1)m sn²(x|m)`. The stored quantity is the full monic polynomial
`P_n(h,m)` of degree `2n+1` in the energy `h`, expanded in the ordered
variables `(h,m)`. The kinetic coefficient is 1 and `m` is the squared modulus.
The polynomial variables `h` and `m` are not sampled table parameters. No
energy index, solution-sector selector, free coupling or modulus mesh has been
introduced. This is one named polynomial family, not a mixture of quantities.

## Selection, breadth, and practical extent

The default enumeration is exactly `n = 1,2,3,4`, in increasing order. It is a
complete initial segment of the positive-order family, not a list of famous
special cases. Integer order makes the finite invariant spaces exist; the
polynomials describe exact algebraic conditions for finite band edges of
periodic elliptic potentials. Each entry retains all admissible sectors and
the dependence on the entire elliptic parameter, so there is no pruning to a
few distinguished moduli or individual eigenvalues.

Breadth was reviewed before depth. Both even and odd orders are retained.
The odd-sector types sn, cn, dn occur at order 1 and sn cn dn at order 3;
the even-sector types 1, sn cn, sn dn, cn dn occur at order 2. Thus the
initial segment reaches all eight polynomial solution types in DLMF
Table 29.12.1. Order 4 also includes a cubic sector factor beyond the linear
and quadratic sectors of orders 1 through 3.

The proposal suggested orders 1 through 8. We computed and independently
checked that entire range privately to measure its expanded presentation:

| n | Characters in the expanded polynomial | Terms |
|---|---:|---:|
| 1 | 51 | 8 |
| 2 | 192 | 19 |
| 3 | 438 | 34 |
| 4 | 843 | 53 |
| 5 | 1462 | 76 |
| 6 | 2324 | 103 |
| 7 | 3510 | 134 |
| 8 | 5039 | 169 |

The cutoff 4 is a practical readability decision, not an intrinsic upper bound
or a claim of demand for each order. The selected entries occupy about 1.7 KiB;
block size is not the binding constraint. The expanded eighth-order entry is
several pages of coefficients, while the fourth-order entry remains readable.
The generator can compute further orders. The omitted measured values and
checks remain in the run artifacts; there were no earlier stored entries to
remove. The table is explicitly incomplete and its completeness note states
this exact range and rationale. No quantiles or statistical critical values
are involved. There is no exploratory reference_sample group in this symbolic
table, since no continuous input is sampled.

## Sources actually checked

- NIST DLMF §29.2, equations 29.2.1–29.2.3:
  https://dlmf.nist.gov/29.2 . Operator and algebraic transformation, with
  DLMF's modulus `k` translated as `m=k²`.
- NIST DLMF §29.12, Table 29.12.1 and equation 29.12.9:
  https://dlmf.nist.gov/29.12 . Integer orders and all eight solution types.
- NIST DLMF §29.15(i): https://dlmf.nist.gov/29.15 . Terminating series and
  finite matrix eigenvalue problems.
- Robert S. Maier, arXiv:math-ph/0309005v4, Definition 7.1, equations
  (8),(9),(28),(53) and Table 3: https://arxiv.org/html/math-ph/0309005 .
  Full spectral polynomial and the independently tabulated Weierstrass factors.

DLMF was readable through the browser research tool. Direct archival HTTP
requests were refused, and this limitation is recorded in source-manifest.json.
Maier's HTML and both live tag-list pages were archived in the run directory.
These sources justify the family and conventions, not the finite cutoff.
The independent critique must assess this selection; this document supplies no
acceptance sidecar and does not claim to be that critique.