Prime Numbers The Wobble Calculus: Displacement, Strain, and the Invariance Ledger of Purple Mathematics

The structures of Purple Mathematics — the wall, the sensors, the helix — were drawn on a dividing line assumed laser-sharp. This paper asks what the framework's instruments detect when the line wobbles, and what they detect when the entire structure wobbles. The boundary of the claims is set in the first paragraph: the wobble itself is the classical remainder u(x) = ψ₀(x) − x, whose deep properties belong to von Koch and Littlewood and are used here only as attributed background; everything claimed is about the framework's own objects. Four results. The Invariance Ledger: every purple observable is classified into three tiers by its behaviour under deformations of the line — arithmetic invariants (signals, κ, T, reach, domains, the ultrametric), affine covariants (strike coordinates, strain), and metric covariants (gaps, apex heights, A(p), receptions). The gauge lemma: a rigid wobble of the whole structure is undetectable by every sensor; only differential wobble — strain — is observable. The instruments, verified: the sensors read the local displacement (93.8% sign accuracy from 150 zeros over an interval where the line changes sign 126 times), and the wall's apex heights are a strain gauge whose readings satisfy the compatibility law du/dx = −strain, measured at correlation −0.9997 over a hundred windows to 10^8: the wall and the line cannot disagree. The discovery: the wobble does not couple to determination — corr(u(p), κ(p)) = +0.00004 at the top decade, sign agreement 0.4999 — while the program's earlier measurement found that strain does couple to determination (corr(A,T) ≈ −0.51/ln x). The coupling between the geometric and arithmetic layers of the framework lives entirely in the derivative: displacement is silent, strain speaks.