L5 - The Harmonic-Simplicial Dual Gauge (HSDG)

Situation. The first four papers handed forward four structures: L1's native simplicial chart (keep the redundant coordinate, work in the mean-zero hyperplane, read Fisher–Rao through the square-root sphere); L2's scalar generator \(Φ=log Z\) and its categorical cumulant tower; L3's arithmetic substrate, where the cubic vertex obeys \(a+b+c≡ 0 N\) in the character basis of \( Z/N\); and L4 (https://doi.org/10.5281/zenodo.20383504)'s Helmert residue ladder, which on unit normalization exposes the prime square-root field clock \(K_N= Q(√ p:p≤ N)\). Arithmetic and geometric means entered as gauges — yet the classical triad \(H≤ G≤ A\) is still missing its reciprocal face. Surprise. The harmonic mean is not a third unrelated average. It is the polar partner of arithmetic mean, with geometric mean as the self-dual mirror between them: \[ A(x)={1}{H(x⁻¹)}, G(x⁻¹)=1/G(x). \] \(A\) is affine in \(x\); \(H\) in \(x⁻¹\); \(G\) in \(log x\). One ladder, three observers. Action. We formalize the three gauge sections \(∑_i x_i=N,\ ∏_i x_i=1,\ ∑_i x_i⁻¹=N\) of the positive projective cone, and define the reciprocal closure \[ p_i^={x_i⁻¹}{∑_j x_j⁻¹}=H(x)/N\,x_i. \] Harmonic normalization is then ordinary simplicial closure applied to \(x⁻¹\), and the Helmert ladder reads three ways: arithmetic gaps, geometric log-ratios, and harmonic reciprocal gaps. Result. In centered log coordinates \(y_i=log x_i-1N∑_jlog x_j\) with \(K_y(t)=log\!(1N∑_i ety_i)\), arithmetic and harmonic means sample the same cumulant generator at opposite temperatures: \[ (x)/G(x)=K_y(1), (x)/H(x)=K_y(-1). \] Adding and subtracting splits the spread into a parity pair — an even-cumulant ledger and an odd-cumulant ledger: \[ (x)/H(x)=2\!∑m≥1\!{κ₂ₘ}{(2m)!}, (x)\,H(x)/G(x)^2=2\!∑m≥1\!{κ₂ₘ₊₁}{(2m+1)!}=κ_3/3+O(\|y\|^5). \] Combined with L3's character orthogonality, the harmonic skew \(log(AH/G^2)\) becomes a scalar witness for the conductor-coupled cubic content \(a+b+c≡ 0 N\) — the first genuinely information-geometric and arithmetic layer. Harvest. L1's square-root sphere gains a dual simplex from reciprocal closure; L2's log-partition tower gains a softmax/softmin mirror; L4's radical arithmetic gains a denominator-arithmetic counterpart through lcm support and \(p\)-adic reciprocal sums. The thesis: harmony is the simplicial ladder viewed through reciprocal closure