# Argument selection for Higher Mahler measure of cyclotomic polynomials (T681)

This build follows family issue 302, read on 2026-11-02, and the coordinated skill archived at `agents/runs/20261102T162707Z-build-skills/manifest.json`. Both archived versions have SHA-256 `a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278`. This is the builder's justification; the controller requires a separate independent critique.

The quantity is the raw signed logarithmic moment

\[m_k(\Phi_n)=\int_0^1 (\log|\Phi_n(e^{2\pi i t})|)^k\,dt.\]

The measure has mass one, the logarithm is natural, and no factorial or power of pi is divided out. The permanent parameter order is `(n, k)`, both integers. The general domain is n >= 1 and k >= 2. Root angles are derived from the primitive n-th roots, never selected as free numerical inputs. This is one named invariant at its discrete moment orders, rather than a selector between different quantities.

## Cyclotomic order n: discrete_index, with arithmetic equivalence reduction

The inputs are the complete set of odd squarefree integers 1 <= n <= 105. Cyclotomic order identifies a standard irreducible integer polynomial, independently of a numeral base or computation library. The family is studied explicitly by Lalín–Sinha, Definition 1, Theorems 5–6, and Proposition 22 (arXiv:1106.1304v1). Although its ordinary Mahler measure vanishes, its second and third logarithmic moments are nontrivial arithmetic periods.

For q = rad(n), the identity Phi_n(x) = Phi_q(x^(n/q)) preserves every moment by Haar invariance. For odd r, Phi_(2r)(x) and Phi_r(-x) have the same absolute value on the circle (there is a sign at r = 1). Rotation therefore removes the factor 2 as well. Every positive order reduces to the product of its distinct odd prime factors. We enumerate these representatives directly; we do not suggest that distinct representatives are proved to have distinct moments.

The cutoff 105 is a practical, noncanonical extent. It includes every representative up to the first three-prime order, 3*5*7, so one-, two-, and three-prime structures are all represented, together with the linear base case. Keeping the full prefix avoids retaining only famous small cases while pushing prime depth. The largest polynomial degree is 102, at n = 103; 105 itself has degree 48. Repeated prime powers and even orders provide verification cases, not redundant stored rows. The selection neither covers all positive n nor claims a canonical endpoint.

## Moment order k: discrete_index

For every selected n, store k = 2 and k = 3. These are the first nonzero even and odd moments, revealing magnitude and sign behavior that the identically zero first moment cannot. This is a meaningful two-order slice of the higher-moment family, not an arbitrary multidimensional product of input grids. Lalín–Sinha Theorems 5 and 6 give finite formulas at these orders. Kurokawa–Lalín–Ochiai, Theorem 14 and section 4.2 (arXiv:0908.0171v1), supply the independent gamma-ratio control for n = 1.

Higher k belongs in this same table, but is deferred until a rigorously controlled general implementation and an independent check are available. Extending only the easy linear case to high k would add depth while leaving composite cyclotomic structure uncovered. The chosen breadth is computationally practical at 100 significant digits.

## Interactions, completeness, and scope

Enumeration is increasing n, then k = 2, 3. There are no additional couplings, scales, angles, or reference probabilities. Both parameters are actual discrete family indices, so no `reference_sample` policy or `mixed` grouping is claimed. The finite bounds are acknowledged as editorial/computational choices rather than intrinsic endpoints.

The completeness note must state the odd squarefree n <= 105 selection and k = 2,3, the rotation/substitution reason for the representatives, and the arithmetic breadth reason for the cutoff. The general parameter constraints remain n >= 1 and k >= 2. Higher moments of geometric sums are a distinct proposed companion, not extra rows under this title; no unpublished companion identifier is invented. The prime equality Phi_p = 1+x+...+x^(p-1) explains future overlap.

Sources were independently fetched and read in this build. Their exact PDF bytes and SHA-256 hashes are in `sources.json`; the instruction provenance is in `skill-evidence.json`. The numerical evidence records precision and actual measurements rather than using a target entry count to justify the selection.