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 \ (= 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 \ (KN= 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) =1H (x^{-1) }, G (x^-1) =1G (x). \ \ (A\) is affine in \ (x\) ; \ (H\) in \ (x^-1\) ; \ (G\) in \ (x\). One ladder, three observers. Action. We formalize the three gauge sections \ (ᵢ xᵢ=N, \ ᵢ xᵢ=1, \ ᵢ xᵢ^-1=N\) of the positive projective cone, and define the reciprocal closure \ pᵢ^=xᵢ^-1ⱼ xⱼ^{-1}=H (x) N\, xᵢ. \ 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 \ (yᵢ= xᵢ-1Nⱼ xⱼ\) with \ (Kᵧ (t) =\! (1Nᵢ e^tyᵢ) \), arithmetic and harmonic means sample the same cumulant generator at opposite temperatures: \ (x) G (x) =Kᵧ (1), (x) H (x) =Kᵧ (-1). \ Adding and subtracting splits the spread into a parity pair — an even-cumulant ledger and an odd-cumulant ledger: \ (x) H (x) =2\!₌₁\!₂₌ (2m) !, (x) \, H (x) G (x) ²=2\!₌₁\!₂₌+₁ (2m+1) !=₃3+O (\|y\|⁵). \ Combined with L3's character orthogonality, the harmonic skew \ ( (AH/G²) \) 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