# Independent audit of approach 8

18 October 2026. Reviewer: the agent responsible for approaches 1–5, independently of the author of arithmetic.md.

## Verdict

The two propositions in arithmetic.md §8.2–8.3 are correct as written:

1. For each fixed \(N\), the infimum of the cosine-minimum parameter over all positive distinct integer frequency tuples is attained.
2. For every \(0<\varepsilon\le1\), every such tuple has a positive distinct integer replacement bounded by \((r+1)^N\) and with its \(K\)-value within \(\varepsilon\), for the explicit \(M,r\) defined there.

Neither proposition supplies the uniform square-root estimate. The first also gives no effective exact frequency bound. No novelty claim has been verified.

## Exact attainment: checked logical dependencies

The hyperspace compactness step is valid because the ambient torus is compact. A Hausdorff limit of the orbit subgroups is a nonempty closed subgroup: limits of differences remain in the limit set.

Its annihilator is a subgroup of \(\mathbb Z^N\), hence has a finite integer basis. For every basis character, Hausdorff convergence forces its values on the approximating orbit subgroup into an arbitrarily small arc about 1. The orbit subgroup is connected, so its character image is either the trivial subgroup or the whole circle. Thus every basis character is identically one on every sufficiently late orbit subgroup.

Torus character separation then gives eventual inclusion \(H_j\subseteq H\). This is stronger than mere approximation and is the essential step. Character separation can also be justified through Haar measure: if a point annihilates every annihilator character, translation by that point leaves all Fourier coefficients of Haar measure on \(H\) unchanged, hence leaves the measure and its support unchanged.

The fixed continuous function \(-\sum\cos\theta_i\) has maxima continuous under Hausdorff convergence. Its maximum on \(H\) therefore equals the minimizing-sequence limit. Eventual inclusion gives an upper bound by that limit for the original tuples, and the definition of the infimum supplies the opposite inequality. Consequently the infimum is attained. Positivity and distinctness are retained because the attaining tuple is one of the original admissible tuples.

## Effective approximation: checked constants and arithmetic

**Heavy cell.** The phase map produces at most \(M^N\) measurable cells. A cell of measure at least \(M^{-N}\) exists. Subtracting any one point of that cell makes every coordinate phase smaller than \(2\pi/M\) in circular distance. Translating this difference set to a maximizing time preserves its measure. The resulting loss is at most \(2\pi N/M\le\varepsilon/2\). No simultaneous Diophantine approximation theorem, gcd assumption, or frequency cap is being used.

**Moment.** Since \(0\le g_a=N-f_a\le2N\),

\[
0\le G_a-\|g_a\|_r
\le\varepsilon/2+2N^2\log M/r
\le\varepsilon.
\]

The lower bound remains valid because \(G_a-\varepsilon/2>0\). Equal moments therefore put two maxima in the same interval of width \(\varepsilon\), not width \(2\varepsilon\).

**Relevant relations.** Every exponent in the expansion of \(g_a^r\) has the form \(h\cdot a\), with \(\|h\|_1\le r\). Preserving all zero/nonzero truth values for these vectors preserves the moment exactly. The construction actually preserves their signs as well.

**Polyhedral vertex.** The original integer tuple is feasible because any positive integer scalar product is at least 1. The constraints include \(v_i\ge1\) and \(v_{i+1}-v_i\ge1\), since the stated \(r\) is at least 2. Minimizing \(\sum v_i\) has a nonempty compact minimizing face: within any bounded objective sublevel every coordinate is bounded above and below. A vertex of that face is a vertex of the full polyhedron.

At a vertex there are \(N\) linearly independent active normals. Their entries are integers, their row \(\ell^1\) norms are at most \(r\), and their right sides belong to \(\{0,1\}\). If their matrix determinant is \(D_0\ne0\), then \(b=|D_0|v\) is integral by Cramer's rule. This positive scaling preserves order and all prescribed relations/nonrelations.

Each row of a Cramer numerator matrix has squared Euclidean norm at most \(r^2+1\): one old coefficient is replaced by 0 or 1. Hadamard therefore bounds every positive coordinate by \((r+1)^N\). There is no hidden coefficient depending on the original tuple.

## Additional quick audit of approaches 16–20

The Fejer \(L^2\) formula and the concentration countermodel in 16 are exact. The convexity bound in 17 correctly uses \(D_1\le D_2\), and its bounded-\(D_1\), unbounded-\(D_2\) example is valid. The measurable-set and supremum calculations in 18 correctly yield only an exponentially weak Remez bound. The two real-polynomial nonnegativity representations and the degree obstruction in 19 are correct. In 20, the closed-disk zero-free claim for \(2+z+\cdots+z^N\) follows from \(2=z+z^{N+1}\) and its equality case, and the Riemann-sum trough is correct.

These quick checks confirm the displayed mathematics; they do not constitute a new literature review or a proof of the full conjecture.