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# Argument selection for T623
This table stores one named quantity, the complex rank-one APS rho invariant
of the odd signature operator on an oriented three-dimensional lens space.
It uses the same quotient generator and parameter order p, q, u as T622 and
T599. It is not an eta table, a spin Dirac invariant, or a reduced eta class.
The governing skill is archived in
agents/runs/20261030T101710Z-build-skills/manifest.json. Both skill-local.md
and skill-live.md have SHA-256
28ecbeeccb0c5e5d6eb541b5a5b0f88aec47b154374d909085116d34f5629199.
The matching manifest is attached as skill-manifest.json.
## Parameters and extent
* p (discrete_index): the order of the fundamental group, with mathematical
domain p >= 2. Include every order 2 through 20. This practical,
noncanonical bound gives prime, prime-power, and mixed composite examples,
including nonfaithful characters. Small lens spaces are explicit models
for spectral boundary corrections and topological signature defects.
* q (discrete_index): the quotient action exponent, 1 <= q < p and
gcd(p,q)=1. Include the least positive member of each orbit under inversion
modulo p. The classification of oriented lens spaces identifies q with its
inverse, but not generally with -q. This retains all oriented types at each
selected order. Coordinate exchange relabels the character by u -> u*q^-1
modulo p. Choosing a least positive representative is a labeling convention,
independent of decimal notation. Residual automorphisms can still give equal
invariants for distinct stored labels.
* u (discrete_index): the exponent of the character chi_u(g)=exp(2*pi*i*u/p),
with nontrivial mathematical domain 1 <= u < p. Include every u from 1
through floor(p/2), giving all nontrivial characters up to conjugation.
Do not restrict to faithful characters. The trivial character has rho=0
identically and is stated as a formula rather than repeated as a full slice.
The linked selection has 87 oriented presentations and 550 entries. Enumerate
in increasing p, q, u. Coverage of all types and characters within a modest
order bound takes priority over extending only the homogeneous L(p,1) spaces.
The bound is not claimed to be canonical or a theorem threshold. These are
classification and representation indices, not a continuous-input reference
grid or a Cartesian mesh of unrelated parameters. Each p selects its own unit
residues and character group. No exploratory continuous-input subgroup is
present. No claim of uniqueness of the numerical values is made.
## Sources and conventions checked
* E. Toffoli, The Atiyah–Patodi–Singer rho invariant and signatures of links,
Proc. Edinburgh Math. Soc. 65 (2022), 404–440,
https://doi.org/10.1017/S0013091522000153, Sections 2.2–2.4, Proposition
2.10 and equation (2.4): complex multiplicities, rho=eta_u-eta_0,
finite sawtooth formula, and a weighted lattice-point computation.
* E. Toffoli, Rho invariants for manifolds with boundary and low-dimensional
topology (2019), https://d-nb.info/1194646867/34, Section 3.3.4 and
Appendix A.1: quotient action, orientation and marked generator;
Remark 3.3.23 gives conjugation, orientation reversal and coordinate exchange.
* M. Sağlam, Holomorphic curves in the symplectizations of lens spaces:
An elementary approach, Münster J. Math. 15 (2022), 389–440,
https://www.uni-muenster.de/FB10/mjm/vol_15/mjm_vol_15_12.pdf,
Theorem 1.3 and the subsequent paragraph on p.392: smooth and oriented
lens-space classification. The separate contact-topological theorem's prime
hypothesis is not being applied to the classical smooth classification.
In Proposition 2.10, the source exponent is k*q, so the stored label u requires
k=u*q^-1 mod p. The homogeneous formula 2*u*(p-u)/p-1 is used only for
nontrivial characters. At u=0 it would give -1, whereas the defining eta
difference is zero; the source's Corollary 2.11 omits this exception.
This is a settled endpoint restriction, not a choice between conventions.
The independent critique must evaluate all three parameters and their
interactions, and fingerprint generate.py and argument-selection.md together
with any later selection-relevant dependencies. This build note is not an
independent argument-review acceptance.