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 \ (= 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 \ (KN= 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 \ (L²\) -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: \ pᵢ=xᵢ²ⱼ xⱼ², eₖ=hₖk (k+1). \ The amplitude sphere and the radical ladder are two restrictions of one operation: \ L²-normalize the ray. \ Action. We formalize RMS as the quadratic gauge \ (R (x) =1Nᵢ xᵢ²\) with section \ (ᵢ zᵢ²=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 \ (pᵢ xᵢ²\) and projective angle \ (d₅ₑ (p, q) =2 y\|x\|\|y\|\). 2. Power-Pair Parity Theorem. The pair \ ( (Mᵣ, M-ₑ) \) splits \ (Kᵧ (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 \ (L²\) -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.
Leonardo Murillo Montero (Tue,) studied this question.