# Argument selection for T622

The family is the odd-signature eta invariant of the round oriented lens space
with a complex line character. The parameters p, q and u describe the space
and bundle; none chooses a different invariant. This is discrete topological
and representation-theoretic coverage, not sampling of continuous arguments.

- **p (discrete_index):** the order of the cyclic fundamental group. Include
  every 2 <= p <= 20. The cutoff is a practical, noncanonical extent in
  topological complexity. It includes primes, prime powers and mixed composite
  orders, rather than just the simplest homogeneous spaces or faithful twists.
  The simply connected sphere p=1 lies outside this table's stated domain.
- **q (discrete_index):** a unit modulo p specifying the free quotient action.
  Include the smallest positive representative of each orbit under inversion
  only, equivalently q <= q^(-1) mod p. These exhaust the oriented lens-space
  types in the bound. Negation reverses orientation and changes eta's sign,
  so the unoriented representatives of T599 cannot be reused. The representative
  rule is a noncanonical labeling choice independent of numeral notation.
- **u (discrete_index):** every complex line character of the cyclic group,
  reduced by conjugation to 0 <= u <= floor(p/2). Include the trivial character
  and nonfaithful characters; the trivial member supplies the ordinary eta
  invariant, while composite orders supply all possible character orders.
  Changing q to its inverse requires replacing u by u*q^(-1) mod p before
  conjugation. Character conjugation does not mean realifying the complex line.

This covers all 87 oriented presentations and 637 selected space-character
pairs through p=20. It does not claim that the marked pairs or their invariant
values are all inequivalent. Residual automorphisms can give repeated values.
Enumerate by increasing p, then q, then u. The checked symmetries recover all
omitted q and u in the bound. Every free parameter belongs to a finite discrete
classification at fixed p, so this is not a Cartesian product of arbitrary
simple real inputs. There is no exploratory reference_sample group.

The breadth review favors complete type and character coverage through a modest
order over deeper coverage of L(p,1). No famous-point-only pruning is made.
The bound leaves the exact fractions readily readable and the entries block
well within the recommended size. Measurement is recorded in numerical-evidence.json.
The generator's max_p default and the table's completeness note use this same
selection; the mathematical parameter domains remain unbounded in p.

Checked sources: Toffoli (2019), Section 3.3.4, Theorem 3.3.27, and Appendix A.1,
Propositions A.1.1–A.1.2, establish the quotient, line labels and eta formula.
Sağlam (2022), Theorem 1.3 and the following paragraph on p.392, give the
oriented classification q' = q^(+/-1) modulo p. Toffoli (2022), Sections 2.2
and 2.4, Proposition 2.10 and equation (2.4), supply the complex-line spectral
convention and the independent lattice-point identity. Exact URLs and locators
are in sources.md.

Policy: both archived skills have SHA-256
28ecbeeccb0c5e5d6eb541b5a5b0f88aec47b154374d909085116d34f5629199.
The authoritative run manifest is
agents/runs/20261030T094236Z-build-skills/manifest.json, copied with its absolute
path in skill-manifest.json. This build justification is not an independent
critique or an argument-review acceptance. The controller must arrange that
separate review and its fingerprint-bound sidecar before offering the draft.