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# Argument selection for T673
Instruction evidence: `agents/runs/20261102T085118Z-build-skills/manifest.json`.
Both archived versions have SHA-256
`a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278`.
This build justification is not the independent critique's acceptance.
The quantity is the signed supertrace c_[g](n) in the Harvey–Rayhaun Thompson
module proved to exist by Griffin–Mertens. The acting class and the actual
Fourier grade are discrete indices of this one quantity. They do not select
different functions by an unrelated quantity name or continuous input grid.
* `g` (`Symbolic`, `discrete_index`): all Thompson conjugacy classes of element
order at most five: 1A, 2A, 3A, 3B, 3C, 4A, 4B, 5A. These are the first eight
columns of GM Appendix A and the first eight names in the author's
`Coeffs_all.m`. This gives the identity, every prime-order class through five,
and both composite-order-four classes. In particular all three order-three
classes and both order-four classes are retained, including their different
multipliers, sparse series and signs. Their exact eta-basis descriptions are
public and independently checkable. The order-five boundary is noncanonical;
it is a practical first low-order subfamily, not an intrinsic stopping point.
Higher orders remain future extensions of this same table, as issue #298's
coverage map specifies. Bounds are on group order, independent of notation.
* `n` (`Z`, `discrete_index`): every n=4m and n=4m+1 for 0<=m<=50, namely
0<=n<=201 with n congruent to 0 or 1 modulo 4. These are genuine Fourier
grades, not numerical samples of a continuous function. Covering complete
consecutive pairs preserves both parity sectors of the supermodule, admits
structural zeros, and supports the even/odd theta-splitting check without
uneven boundary treatment. The noncanonical cutoff m=50 supplies a moderate
initial graded reference segment across every selected class before deeper
extension of the identity alone. The measurements are in numerical-evidence.json.
The common polar term 2q^-3 is stated in the normalization and checked, rather
than repeated in eight searchable rows. Grade zero is retained because it is
the character of the 248-dimensional representation, not a common polar term.
No class is removed for repeating a small integer; no sign or zero is discarded.
The shared g,n parameter order and q=exp(2*pi*i*tau) convention agree with the
O’Nan companion T672. The grade residues differ because Thompson weight 1/2
uses 0,1 modulo 4, whereas O’Nan weight 3/2 uses 0,3. The two modules are not
merged. Thompson includes grade zero; O’Nan's selected positive grades do not.
Checked sources: Harvey–Rayhaun, arXiv:1504.08179, equations (1.4)–(1.11),
Tables 1, 2 and 6, and Conjecture 3.1; Griffin–Mertens, arXiv:1607.03078v2,
Theorem 1.2, equation (2.6), Tables A.1, A.5 and A.6, and Appendix B; Michael
Mertens, https://www.math.rwth-aachen.de/~Michael.Mertens/EtaCoeffs.html and its
linked exact eta bases and coefficients. These establish the family and
conventions; the finite bounds are our stated reference choice.
This is wholly discrete-index coverage. There are no reference_sample,
continuous, mixed-role, decimal-grid, probability or quantile parameters.
The separate critic must evaluate both indices and enumerate the attached
`generate.py`; this note does not supply a reviewer verdict or sidecar.