# Argument selection for Lamé band-edge eigenvalues (T544)

The instructions read for this run are archived in
`agents/runs/20261027T094619Z-build-skills/manifest.json`. Both local and live
versions have SHA-256
`e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c`.
The source snapshot's canonical skill has the same digest. The attached
`skill-manifest.json` preserves the runner's manifest. This is the builder's
justification, not the independent critic's required acceptance sidecar.

## Quantity and conventions

The single named quantity is the energy h of the Jacobi operator
H = -d²/dx² + n(n+1)m sn²(x|m). The kinetic coefficient is one and m is the
elliptic parameter, not the modulus k: m=k². The potential has period 2K(m).
No mean is subtracted and the period is not rescaled. These conventions agree
with family issue 219, DLMF 29.2.1–29.2.3 and Maier (2008), equation (1).

For positive integer n and 0<m<1, exactly 2n+1 finite band edges are ordered
h[n,0] < ... < h[n,2n]. They delimit n finite bands and one infinite band;
Maier §1.1 and Definition 7.1 distinguish these algebraic energies from the
infinite remaining sequence of periodic/antiperiodic closed-gap eigenvalues.
DLMF 29.12 specifies the Lamé polynomial sectors. The root index does not
supply the rationale for any independently selected m.

## Every free parameter

### n: discrete_index

Positive integer order n indexes the finite-gap Lamé operators. We include
all n=1,...,12 at the intrinsic parameter and n=1,...,6 at the two reference
parameters. Both parities, every admissible Lamé polynomial sector, and the
transition from quadratic factors to higher algebraic degrees are represented.
The cutoffs are practical reference choices, not canonical boundaries. Twelve
orders provide a useful initial segment of the symmetric family; six suffice
for a first comparison of two asymmetric models without giving an exploratory
coupling choice the same depth. These bounds were reviewed before computing.
The longest generated enclosure is 129 characters; each stored nonrational
energy has 100 significant digits. There is no readability reason to prune
sound members of the proposed initial segments.

### m: mixed, with exhaustive disjoint groups

* Intrinsic group, exact identity `1/2`: all orders 1 through 12, and every
  finite band edge at each order (168 entries). This is the self-complementary,
  lemniscatic parameter, with K(m)=K(1-m) and a square period lattice. The
  spectral reflection fixes the central energy n(n+1)/2. Maier §3 and §7
  explain the lemniscatic case and the central root; DLMF 23.17.1 gives
  lambda(i)=1/2. The parameter has a structural reason; the order cutoff is
  still a choice.
* Reference_sample group, exact identities `1/4` and `3/4`: all orders 1
  through 6, and every finite band edge at each order (96 entries). These
  are low-complexity rational Legendre parameters chosen to explore energies
  of simple elliptic potentials on both sides of the symmetric case. They
  are not asserted to be intrinsic Lamé parameters or to have a documented
  application at every order. The complementary pair permits comparison
  through h[n,j](m)+h[n,2n-j](1-m)=n(n+1), and samples unequal period shapes.

The reference group's policy is:

```json
{
  "purpose": "Exploratory identification of algebraic band-edge energies of simple rational elliptic potentials beyond the symmetric case, in the unshifted Jacobi energy normalization.",
  "bounds": "Exactly m=1/4 and 3/4, n=1,...,6, j=0,...,2n; no other reference parameters.",
  "coverage_rationale": "A complementary pair covers both sides of m=1/2 and retains the entire finite edge set. Its shallower order cutoff supplies breadth before greater depth, without a dense parameter survey or a multidimensional mesh.",
  "noncanonical_acknowledged": true,
  "notation_independent": true,
  "sources_checked": true
}
```

Small denominators motivate reference simplicity, not a claim of arithmetic
specialness. The identities are exact parameters and their complementary
partner, independent of decimal spelling or numeral base. They were not
obtained from an interpolation step or imposed as a general mesh on unrelated
special functions. The two fractions and finite bounds are noncanonical.

### j: discrete_index

At each selected (n,m), j=0,...,2n is the entire ordered set of finite band
edges, including the ground state, every finite gap's endpoints, and the
start of the infinite band. This bound is mathematical, not a budget choice.
No eigenfunction, gap width, Floquet exponent, or spectral polynomial is mixed
into the value column. The polynomial is the separate related table T543.

## Breadth, interactions, and exclusions

The selection is a pair of slices: the symmetric family through order 12,
and a complementary asymmetric pair through order 6. It is not a product of
all simple moduli and arbitrary eigenvalue guesses. The coupling is fixed by
n(n+1)m, not independently sampled. The domain excludes m=0 and m=1, where
edges coalesce or the period degenerates; those limits are not additional
rows under the strictly ordered finite-gap definition. No quantiles,
percentiles, or statistical critical values are involved.

Default enumeration is n increasing, then m=1/2,1/4,3/4 when present, then j
increasing. This is precisely 264 rows. Every row is retained, including all
rational roots detected by exact factorization. The completeness note, program
snippet, and default generator describe the same slices. Extending them is a
new coverage decision, not implied by the ability to compute another value.

## Sources checked

* NIST DLMF §29.2, especially 29.2.1–29.2.3: equation and algebraic change of
  variable; https://dlmf.nist.gov/29.2.
* NIST DLMF §29.12, Table 29.12.1: all eight polynomial solution sectors;
  https://dlmf.nist.gov/29.12.
* NIST DLMF §29.15: finite Fourier eigenvalue representations of the sectors;
  https://dlmf.nist.gov/29.15.
* NIST DLMF 23.17.1: lambda(i)=1/2; https://dlmf.nist.gov/23.17.
* Robert S. Maier, Lamé polynomials, hyperelliptic reductions and Lamé band
  structure, Philosophical Transactions A 366 (2008), 1115–1153; §§1,3,7,
  Definition 7.1, and equations (54)–(56);
  https://arxiv.org/html/math-ph/0309005.

These sources establish the function, normalization, finite spectral family
and structural relationships. They are not presented as evidence of external
use of every exploratory input. The complementary polynomial identity and
ordered energy relation were also checked exactly across every selected case.

## Independent review still required

The separate critic must inspect every parameter and the numerical evidence,
then fingerprint the current live document and selection-relevant attachments
with `agents/argument_review.py`. The selection is wholly implemented in
`generate.py`; `argument-selection.md` declares its justification. The checker
imports `check.py` for the campaign's standard checks, and its comparison
methods and numerical evidence should also be inspected. This build supplies
no critique acceptance and takes no offer, publication, or review action.