L6 - The RMS-Simplicial Quadratic Gauge

Situation. The first five papers handed forward five structures: L1's native simplicial chart and square-root sphere reading of Fisher–Rao; L2's scalar generator \(Φ=log Z\) and its cumulant tower; L3's cubic character selection rule \(a+b+c≡ 0 N\) on \( Z/N\); L4's Helmert residue ladder, with the prime radical field clock \(K_N= Q(√ p:p≤ N)\); and L5 (https://doi.org/10.5281/zenodo.20388419)'s harmonic reciprocal gauge, with \(A\) and \(H\) as polar partners around the self-dual \(G\). Gap. The root-mean-square had been hiding in plain sight since L1: the square-root sphere, the \(2\) Fisher–Rao distance, and the \(√k(k+1)\) Helmert normalizers are all \(L^2\)-phenomena. RMS is not a fifth average — it is the gauge of *squared substance*. Surprise. RMS is the common mechanism behind both L1 and L4. On the positive cone it gives Born closure; on the mean-zero Helmert ladder it gives the unit balances: \[ p_i=x_i^2/∑_j x_j^2, e_k={h_k}{√k(k+1)}. \] The amplitude sphere and the radical ladder are two restrictions of one operation: \[ L^2-normalize the ray. \] Action. We formalize RMS as the quadratic gauge \(R(x)=√1N∑_i x_i^2\) with section \(∑_i z_i^2=N\), and prove five results: 1. Quadratic Gauge Theorem. For amplitude-like ledgers (positive ray, additive quadratic mass), RMS is the unique canonical gauge, with closure \(p_i∝ x_i^2\) and projective angle \(dFR(p,q)=2· y/\|x\|\|y\|\). 2. Power-Pair Parity Theorem. The pair \((M_r,M₋ᵣ)\) splits \(K_y(r)\) into even and odd cumulant ledgers; RMS is the \(r=2\) case. 3. RMS–Conductor Mixing Theorem. Square closure is quadratic convolution in the character basis: conductor packets mix by the selection rule \(a+b-c≡ 0 N\). 4. RMS Radical Universality Theorem. For \(N≥ 4\), RMS normalization of integer amplitude ledgers realizes every quadratic field \( Q(√ d)\); for \(N=2,3\) the missing fields are exactly the classical sum-of-squares obstructions. 5. Born–Helmert Bridge Theorem. The amplitude sphere and the unit Helmert ladder are the positive and centered faces of one \(L^2\)-normalization principle. Harvest. The trilogy located the cubic as the first layer beyond Riemannian geometry; L4 hid primes inside \(√k(k+1)\); L5 turned harmonic skew into a scalar witness of odd cumulants. L6 names why the factor \(2\) was always meaningful: RMS lifts nonnegative coordinates onto amplitude rays, and Fisher–Rao distance is twice their spherical angle. The thesis: RMS is the gauge of amplitudes; amplitudes square into probability.