A σ-Selectivity Framework via sech²-Weighted Curvature

Jason Mullings BSc (2026) · Algebraic singularity at σ = 1/2 · Implementation of Section 3
1. The σ-Selector QN(σ) — Definition 3.4
QN(σ) = Σn=2N [n−σ − n−(1−σ)]² (ln n)²

The σ-selector measures departure from the critical line σ = 1/2 using only the Dirichlet coefficient structure. It vanishes if and only if σ = 1/2 (Proposition 3.5).

N (truncation) 100
σ range
QN(σ=0.5)
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QN(σ=0.4)
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Minimum at σ
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Q″N(1/2)
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QN(σ) — Unique minimum at σ = 1/2 (Proposition 3.5)

For every fixed N ≥ 2:

QN(σ) ≥ 0 for all σ ∈ (0,1)
QN(σ) = 0 ⟺ σ = 1/2
QN(σ) > 0 for all σ ∈ (0,1) ∖ {1/2}
2. The σ-Generator g(σ; n) — Definition 3.2
g(σ; n) = n−σ − n−(1−σ)
= −2n−1/2 sinh((σ − 1/2) ln n)

The generator encodes the functional equation symmetry σ ↔ 1−σ at the coefficient level. It vanishes uniquely at σ = 1/2 (Lemma 3.3).

Reference n 7
g(σ; n) for selected n — Antisymmetric about σ = 1/2 (Lemma 3.3)

(i) Antisymmetry: g(1−σ; n) = −g(σ; n)

(ii) Unique zero: g(σ; n) = 0 ⟺ σ = 1/2

(iii) Hyperbolic factorisation: g(σ; n) = −2n−1/2 sinh((σ−1/2) ln n)

3. Non-Degenerate Curvature — Theorem 3.6
Q′N(1/2) = 0
Q″N(1/2) = 8 Σn=2N (ln n)⁴/n > 0

QN has a strict, non-degenerate quadratic minimum at σ = 1/2.

N range
Q″N(1/2) vs N — Monotone growth (Appendix B)
N Q″N(1/2) Σ (ln n)⁴/n Ratio (8× formula)
4. Taylor Expansion Near σ = 1/2 — Corollary 3.7

Writing ε = σ − 1/2, near σ = 1/2:

QN(σ) = 4ε² Σn=2N (ln n)⁴/n + (4ε⁴/3) Σn=2N (ln n)⁶/n + O(ε⁶)
N 80
σ 0.52
QN(σ) exact
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Taylor O(ε²)
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Taylor O(ε⁴)
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Relative error
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QN(σ) vs Taylor approximations near σ = 1/2